m>»w CORNELL • UNIVERSITY LIBRARY BOUGHT W^ITH THE INCOME OF THE SAGE ENDOAVMENT FUND GIVEN ?IW iS^I ^BJ ' HENRY WILLIAMS SAGE Cornell University Library The original of tliis book is in tlie Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924012339689 A TEXT-BOOK ON SOUND MACMILLAN AND CO., Limited LONDON • BOMBAY • CALCDTTA MELBOURNE ' THE MACMILLAN COMPANY NEW YORK ■ BOSTON ■ CHICAGO ATLANTA • SAN FRANCISCO THE MACMILLAN CO. OF CANADA, Ltd. TORONTO A T;EXT-BOOK f '^ ON SOUND EDWII^ H, BAETON, D.Sc. (Lond.), F.RS.E. A.M.I.E.E., F.Ph.S.L. PROFESSOR OF EXPERIMENTAL PHYSICS UNIVERSITY COLLEGE NOTTINGHAM MACMILLAN AND CO., LIMITED ST. MARTIN'S STREET, LONDON 1908 Kg Uf^lV! ft&f'i Y 1- 1.# R A kY,:- PREFACE In writing the following pages the aim has been to provide students with a text-book on Sound, embracing both its experimental and theoretical aspects. To make it more widely useful, the mathematical portions are restricted to the elements of the calculus. Thus, when Fourier's Theorem or Differential Equations are introduced, the treatment is simple and the physical bearings of the problem are carefully discussed, so as to meet the needs of those readers having no previous acquaint- ance with such methods. All higher analysis is entirely excluded. Indeed, much of the work is intelligible to those not familiar with the calculus at all. For, in the parts where it is used, simpler alternative methods are often provided, chiefly for the sake of emphasising certain important physical aspects of the subject. Experiments suitable either for laboratory exercise or lecture illustration are described in some detail, and distinguished by smaller type and numbered headings. The various typical musical instruments are discussed more fully than usual, though always from the view-point of the physicist rather than that of the musician. At the close of the book a number of original examples on the various chapters and sections are given to afford a V vi SOUND gauge of the student's progress and grasp of the text. These include enunciations, proofs, numerical and descrip- tive examples, essay-writing, and actual manipulation in the laboratory. They are for the most part fairly straight- forward, it being assumed that Pass or Honours candidates for Degrees or other Diplomas will further test themselves by working special problems from papers previously set at the particular examination they are taking. Like every other modern work on Sound, the present text-book obviously owes much to the classical treatises of Helmholtz, Lord Eayleigh, and Tyndall. The excellent Continental courses on physics by MuUer-Pouillet, Willlner, and Jamin and Bouty also deserve mention. On special subjects a number of other works have been consulted with advantage, and references given in the text where necessary. The endeavour has been made to bring the treatment up to date by insertion of the more important recent researches at home and abroad, the respective authorities being in each case quoted. Thanks are hereby tendered to Messrs. Newton and Co. for permission to reproduce one of Prof. C. V. Boys' photographs of a bullet in flight; to Messrs. Taylor and Francis for a like favour with respect to any illustrations in the Philosophical Magazine; and to Prof. J. G. M'Kendrick, P.E.S., who kindly allowed some of the highly-interesting curves obtained from his Phonograph Eecorder to appear in these pages. To Mr. Ambrose Wilkinson, B.Sc, Lecturer in Physics at University College, Nottingham, special thanks are given for his valuable help in reading the proofs ; but, as it can scarcely be expected that any single reader, however PEEFACE vii careful, would succeed in noting every inaccuracy, it is feared that some errors or obscurities still remain. Should such be found by any users of the book, their kindness in forwarding either corrections or suggestions would be heartily appreciated. UNiVEKSiTy College, Nottingham. February 1908. CONTENTS PAET I. PEELIMINARY SUKVEY CHAPTER I Inteoductoey ARTS. PAGE.S 1-2. Sound and its Production 1-3 3. Noises and Musical Sounds . 3-4 4-5. Propagation of Sound .... 4-8 6-7. Vibrations — Pitch 8-10 8-9. Musical Notation and Intervals . 10-13 10. Logarithmic Cents .... 13-14 11. Intensity and Loudness 14-15 12. Quality, Order of Treatment . 15-17 PAET II. MATHEMATICAL CHAPTER II Kinematics 13-19. Simple Harmonic Motion . ... 18-25 20-25. Progressive Waves ...... 25-39 26-31. Composition of CoUinear Vibrations . . . 39-48 32-37. Composition of Eectangular Vibrations . . . 48-62 38-42. Stationary Waves ... . . 63-G9 43. Diverging Waves . . . . 69-71 44. Huyghens' Principle . . • 71-74 45-46. Plane Reflection and Refraction . 74-77 ix SOUND ARTS. 47. Reflection and Refraction at Curved Surfaces 48-50. Diffraction and Rectilinear Propagation 51-55. Fourier's Theorem and Series 56-59. Damped Harmonic Motion 60. Doppler's Principle 61-66. Refraction, etc., of Sound by Wind 6 7-6 7a. Strains and Strain Ellipsoid . 68-69. Typical Strains and Compositions 70. Different View of Simple Shear PAGES 78 78-83 83-91 91-96 96-99 99-106 106-108 108-111 111-114 CHAPTER III Elasticity 71-72. General Conceptions and Mea.5Ures 115-117 73-74. Volume Elasticity : Gases . . 117-120 75-77. Rigidity, Young's Modulus, etc. . 120-123 78. Relations between Elastic Constants . . 123-125 79-83. Experimental Determination of Elasticities 126-131 CHAPTER IV Dynamical Basis 84-86. Examples of Simple Harmonic Motions 132-135 87-88. Small Oscillations by General Method 135-137 89-90. Resisted Oscillations . . 137-140 91-111. Forced Vibrations : Theory and Experiments . 140-167 112-114. Large Vibrations, Asymmetrical and Symmetrical 167-171 115. Asymmetrical System under Double Forcing 171-172 116. Superposition and its Limitations . 172-174 117-118. Speed of Rope Waves . 174-177 119-120. Speed of Sound in a Gas 177-179 121. Newton's and Laplace's Formulae 179-181 122. Distortion of Large Waves . .181-182 123. Speed of Longitudinal Waves in Solid Prism 182-1-84 124-129. Differential Equations for Waves . . 184-190 130-134. Solution and Initial Conditions . 190-196 135-138. Reflections at Fixed and Free Ends 196-201 CONTENTS xi PAOES 139-140. Various Initial Disturbances 201-204 141-144. Plucked and Struck Strings . . 204-210 145-148. Energy and Activity ofWaves 210-214 149. Spherical Radiation in Air . . 214-215 150-155. Differential Equation and Solution . 215-223 156-162. Partial Reflections of Waves . 223-231 163. Characteristics ofWave Motion . 231-232 PART III. PHYSICAL CHAPTER V ViBEATiNG Systems 164. Types of Vibrating Systems 233-234 165-168. Strings; Stiffness: Bridges 234-237 169-172. Rods Vibrating Longitudinally . 237-241 173-174. Torsional Vibrations . 241-244 175-178. Parallel Pipes . . 244-249 179-181. End Corrections . 249-254 182-188. Conical Pipes . . 254-262 189-190. Stretched Membranes . 262-264 191-197. Rectangular, Square and Circular 264-274 198-217. Bars Vibrating Transversely . 274-297 218-220. Bent Bars; Tuning-Forks . 297-299 221-222. Vibrations of a Ring . . _ 299-301 223-226. Vibrations of Plates . 301-303 227-230. Chladni's Figures 303-309 231-234. Cylindrical Shell : Bells . 309-313 CHAPTER VI Resonance and Response 235. Order of Treatment . . . 314-316 236-244. Resonators, Theory and E.xperiments 316-326 245-249. Melde's Experiment and Theory . 326-333 SOUND ARTS. 250-251. Erskine-Murray's Phonoscope 252-258. The Human Ear and its Action 259-260. Goold's Generators . _ . 261-262. Action of Violin Bow . 263-264. Trevelyan's Rocker 2G5-277. Singing FJames 278. Electrically-driven Tuning-Fork 279-282. Sensitive Flames and Jets 283. Setting of Suspended Disc in Sound 284-286. Kundt's Dust Figures . 287. Maintenance of Compound Vibration PAGES 333-335 335-343 343-346 346-348 348-349 349-361 361-362 362-365 365-367 367-371 371-373 CHAPTER VII Interference and Combinational Tones 288-292. Interference 293-296. Beats ... 297. Combinational Tones 298-302. Differential and Summational Tones 303-306. Have they Objective Reality ? 307-314. Upper Partials of Loud Tone 315-316. Koenig's Beat Notes, and Work of . 317-320. Riicker and Edser : Rayleigh, Waetzmann 321. Everett's Theory of Resultant Tones 374-378 378-383 383-384 384-388 389-393 393-399 399-401 401-404 404-405 PAET IV. MUSICAL CHAPTEE VIII Musical Instruments 322-327. General Survey 328-330. Guitar : Harp : Mandolin 331-332. Quality of Tone from Plucked Strings 406-410 410-413 413-414 CONTENTS Xlll ARTS. PAGES 333-335. The Pianoforte . . . 414-416 336-346. The Violin Family . 416-422 347-360. Vibrations of Bowed Strings . . . 422-434 361-363. Responsive Vibrations in Monochord, etc. 434-436 364-365. Metal Reeds without Pipes . . . 436-438 366-373. Organ Pipes with and without Reeds 439-443 374-376. The Flute Family ... . 444-44,'5 377-379. The Oboe and Bassoon 446-447 380. The Clarinet Family . 447-448 381-385. Brass Instruments . 448-450 386-389. The Bugle, French Horn, etc. . 450-452 390-392. The Trombone . . . 452-454 393-398. Valved Instruments . . . 454-458 399-404. The Human Voice : Vowel Quality . 458-462 405-408. Air Pressures for Wind Instruments . 462-466 409. Temperature Variations of Pitch . . . 466-467 410-415. Partials of Typical Instruments . . . 467-471 CHAPTER IX Consonance and Temperament 416-422. Discord due to Beats 423-428. Various "Ways of Producing Beats 429-433. Consonance of Chief Intervals 434-436. Chords and their Various Positions 437—450. Reasons for Temperament 451. The Pythagorean Tuning 452-455. Mean-Tone Temperament 456-460. Eqiial Temperament 461-464. Bosanquet's Cycle of Fifty-Three 465. Ellis's Harmonical 466-467. Character of Keys 468-470. Orchestration 471-472. Chief Intervals : Historical Pitches 472-476 476-480 480-484 484-486 486-494 494-495 495-499 499-503 503-505 505-506 506-507 507-509 509-511 xiv SOUND PAET V. EXPERIMENTAL CHAPTEE X AconsTic Determikatioks ARTS. PAGES 473-479. Velocity of Sound in Free Air . . 512-518 480-481. Velocity of Sound in Water • 518-519 482. Method of Coincidences 519 483-486. Velocity of Sound in Pipes . . 519-523 487-488. Change of Speed with Intensity and Pitch 523 489-490. Indirect Methods : Velocity in Iron 523-524 491-493. Dulong's "Work with Organ Pipes . . 524-526 494-498. Wertheim's Work with Pipes and Liquids . 526-530 499-507. Velocities, etc., by Kundt's Tube 530-535 508. Mechanical Equivalent of Heat . 535-536 509-511. Hebb's Telephone Method for Air . 536-538 512-515. Speeds in Metals, Wax, etc. . . 538-542 516-520. Further Researches on Pipes . . . 542-545 521-522. Wtillner's Work on Hot and Cold Gases 545-546 523-526. Blaikley's Experiments with Brass Tubes 546-549 527-530. Later AVork : Tubes : Hot and Cold Gases 549-552 531-532. Correction for Open End . 552-553 533-534. Interval and Pitch 553-555 535-536. The Vibration Microscope 555-557 537-541. Lissajous' Figures . . 557-560 542-549. Absolute Values of Frequency : Koenig ; Ray- leigh ; Mayer ; Scheibler . . 560-564 550-552. Rayleigh's Harmonium Method . . 564-567 553-568. Simple Methods for Frequency : Wheel; Siren; Fall Plate ; Interference ; Monochord, etc. 567-576 569-570. Variation of Pitch and Decrement . . 5-76-578 571. Reaction of Resonator on Pitch of Fork . 578-579 572. Subjective Lowering of Pitch . 579-580 573. Number of Vibrations for Sensation of Pitch 580-581 574-576. Lowest and Highest Pitches Audible . 581-583 577-578j Harmonic Echoes . . 584-585 579-580. Musical Echo from Palisade, etc. 586-587 CONTENTS XV ABTS. PAGES 581. ^olian Tones ... . 587-588 582. Minimum Amplitude Audible . . 588 583-586. Rayleigh's Whistle and Fork Methods . 588-592 587-588. Dr. Shaw's Method for Minimum Audible 592-593 589-593. Long Range Transit of Sound . . . 593-596 594-597. Pressure of Vibrations : Larmor's Theorem 596-600 598-599. Direct View for Sound Pressure . . 600-601 600-601. Work of Altberg and of Rayleigh 601-602 602-606. Momentum of Radiation 602-605 607-611. Quality and Phase . . 605-608 612-616. Perception of Sound Direction 608-611 617-619. Architectural Acoustics . 611-613 CHAPTER XI Recorders and Reproducers ■620. The Phonautograph 614-615 621-625. The Phonograph . . . 615-618 626-629. M'Kendrick's Phonograph Recorder 618-621 630-632. Bevier's Phonograph Analysis . 621-624 633-637. The Telephone .... 624-627 638-639. Its Sensitiveness : Permanent Field 627-628 640-642. Edison's Carbon Transmitters . 628-630 643-644. Hughes' Microphone . . 630-631 645. Trunks and Transformers . . 631-632 646. Vibrations of Telephone-Membrane 632 647. Rhythm Electrically Perceived 632-633 648-650. The Speaking Arc . . 633-635 651-653. The Musical Arc . . 635-637 654-656. Oscillatory Discharge of a Condenser 637-C39 657-658. Forced Electrical Vibrations 639-640 659-660. Impedance of a Circuit . 640-642 661-662. Resistance and Inductance modified by Alterna- tions and Damping . . 642-643 663-666. Alternating Currents in Parallel 643-647 667-668. Electric Waves along Parallel Leads . .. 647-648 669-672. Heaviside's Distortionless Circuit . 648-650 SOUND 673-675, Reflections at Terminal Bridges 676. Heaviside's Generalised Bridge 677-679. Intermediate Bridges PAGES 650-652 652-653 653-655 EXAMPLES Examples on Chapter I II III IV V . VI VII VIII IX X . XI 657 657-661 661-662 663-667 667-669 669-671 671-672 672-673 674-675 675-678 678-680 Index 681-687 CHAPTEE I INTEODUCTORY 1. The word sound is commonly used in two different senses : (1) to denote the sensation perceived by means of the ear when the auditory nerves are excited ; and (2) to denote the external physical disturbance which, under ordinary con- ditions, suitably excites the auditory nerves. This usage will generally be followed here. It rarely leads to any ambiguity, the context generally showing in which of the two senses the word is employed. When necessary, for clearness' sake, the first or subjective sense will be represented by " the sensation of sound," and the second, or objective sense, by " sound waves," or other similar expressions. It is a matter of common knowledge that the source of sound is always a body in a state of vibration, or rapid to- and-fro motion. This body may be a solid, as the string of a harp ; or fluid, as the column of air in a wind instrument. We may, therefore, define Acoustics, or the study of sound, as that branch of physics which deals with vibratory motion as perceived by the sense of hearing. It is usual, however, to include with this a few other closely allied phenomena. 2. Production of Sound. — But in order to produce souud it is not sufficient to have some body in a state of vibration as its source. We need also (1) some medium to m 1 B 2 SOUND CHAP. I receive and transmit this vibratory motion, otherwise neither the sensation of sound nor the external disturbance would be present. (2) It is imperative that the parts of the body in vibratory motion should have such shape, size, and motion as to cause a disturbance to advance through the air, and not such as to produce a local flow and reflow of the air simply. (3) Our ears enable us to perceive the sensation of sound only when affected by to-and-fro move- ments whose number per second lies between certain limits. Therefore, to produce sound sensations, it is necessary that our vibrating body should conform to this requirement also. These points are respectively illustrated by the following experiments : — ExPT. 1. A Medium Essential. — To illustrate the necessity of a medium to convey sound from its source to the ear, hang a bell by india-rubber cords within a glass bulb fitted with a tap. Sound the bell ia the bulb full of air with the tap open, and then with the tap closed, so as to indicate its loudness in each case. Next, exhaust the bulb as com- pletely as possible by an air-pump. Detach the bulb when exhausted and shake so as to attempt to make the bell sound. No sound, or only an extremely feeble one is heard. While the bell is inaudibly shaking, open the tap so as to admit the air ; the sound is very quickly restored. Contrast with this the case of an ordinary electric glow-lamp, whose incandescent filament sends light to us across the space within its bulb although it is practically devoid of air. Light, like, sound, needs a medium for its propagation. But the medium essential to the propagation of light is the ether which we are at present unable to remove from any space. ExPT. 2. Importance of Sound Board. — Fit up two steel wires each about 1 mm. diameter — one on a monochord, the other on the bars of two 56 lb. weights. The latter should have the same length between the bars that the former has between the bridges, and they should be tuned to the same pitch. It is desirable that the 56 lb. weights should not rest upon wood. They may both be on a level tiled floor, or slab of stone or slate. Or, the wire might be 3 INTRODUCTOEY 3 vertical, the upper weight being on an iron or stone bracket from a wall, and the lower one hanging freely. On pluck- ing or bowing the wire on the 56 lb. weights, Only a very • feeble sound will be heard. But on plucking or bowing the wire on the mouochord the sound is easily heard by an audience of a thousand persons. It is thus seen that the wire whose ends are on the massive iron weights, although moving to and fro like the other, is yet unable to produce vigorous waves of sound in the air. But the precisely similar wire on the monochord passes over bridges which are moved by its vibrations. And the bridges in turn set the belly or upper board of the sound box in motion. It is true the motions of the bridges and belly are very small, but the shape of the latter forces the air near it to take up a vigorous vibratory motion which can advance to distant parts of a large room. Any mere local flow or reflow of the air, which is almost all that the wire on the weights could produce, is impossible in the case of the monochord sound box. Hence the distinction between the effects produced in the two cases. ExPT. 3. Eight Frequency needed. — Take a piece of steel about 1 m. long, 2 cm. wide, and 1 mm. thick. Clamp it firmly in a vice, allowing at first, say, 80 cm. to project. Pull this projecting end aside and let it go. It is seen to execute vibrations, but no sound is heard. Decrease the length of the projecting part, and again test the vibrations for audibility. Repeat the shortening and the test until the vibrations are audible as well as visible. This simple ex- periment shows that very slow vibrations are inadequate to produce an audible effect, although absolutely similar ones executed a greater number of times per second produce a distinct sound. We shall see later that vibrations executed too quickly fall beyond the limits of audition. 3. Noises and Musical Sounds. — All sounds may be divided roughly into two classes, noises and musical sounds. Noises are characterised by irregularity or suddenness, musical sounds by their comparatively smooth and even flow. But the line of demarcation between them cannot be sharply drawn. Throughout the irregularity of some noises may be perceived, now more and now less plainly, a persistent 4 SOUND CHAP. I musical sound. On the other hand, very few musical sounds are entirely free from accompanying noises which disturb to some slight extent their regularity and smoothness. These differences can be illustrated better than defined. The following experiments afford typical examples : — ExPT. 4. Extreme Gases. — (1) Drop from a height of 6 ft. a 2 lb. ball of iron upon a tin plate a foot square resting upon sawdust in a box on the floor. (2) Take a tuning- fork mounted upon its resonance box, and excite it gently with a carefully rosined bow. The first sound is unquestion- ably a noise, and the second just as surely a musical sound. ExPT. 5. Intermediate Cases. — Pour water from a jug into a water bottle or tall jar until the bottle or jar is full. The sound produced by the falling water seems at first to be merely a confused noise. But, after a few seconds, it will be noticed that throughout the noise a sustained musical sound is also present. Moreover, the musical sound rises as the jar becomes nearer full, and this continuous rise makes it easier to detect. Bars of steel, bells, and gongs, when struck with another hard body, furnish examples of sounds in which noises and musical elements are both clearly present. The harder the hammer with which they are struck, the harsher and less musical will the sound usually be. Since sounding bodies are in a state of vibration, and musical sounds are characterised by their regularity and smoothness, it seems natural to infer that they are produced by regular or periodic vibrations. We shall afterwards see distinct proof that this is the case. Noises, on the other hand, are produced by irregular or non-periodic vibrations. We shall, henceforth, be concerned almost solely with musical sounds, noises being practically dismissed with this brief notice. 4. Propagation of Sound.— Let us now examine the general nature of the process by which sound advances from its source to the ear or other recipient. -The medium of this advance is usually the air, and this will serve our present purpose, as the essentials with which we are now 4 INTEODUCTOEY 5 concerned are the same whatever the medium. The features in question may be illustrated as follows : — ExPT. 6. Air Waves. — Arrange a pipe AB, about 6 feet long and 4 inches diameter, with a glass funnel F and a lighted candle C in a hne with it at one end. At the end remote from the funnel introduce into the pipe two watch-glasses, one containing strong hydrochloric acid and the other ammonium hydrate, so as to produce dense white fumes. Or, instead of producing fumes in this way, the smoke from smouldering brown paper will serve. Near the end containing the fumes or smoke make a sharp report by clapping together smartly two blocks of wood or two books. When everything is rightly adjusted the candle Fig. 1.— Aie "Waves. flame is seen to duck at each clap. But, although abundant fumes may be evolved within the tube at the end next the source of sound, no trace of fume is seen to issue at the end next the candle. Hence we conclude that when sound passes from one point to another it is not the medium of its propagation which advances. On the contrary, the parts of this medium make only quite small excursions to and fro. And it is this state of minute to -and -fro motion that advances. Thus the medium as a whole, after the disturbance has passed, is practically where it was to begin with. Illustra- tions of this advance of a state of motion through a medium which does not itself appreciably advance may easily be multiplied. The following will repay consideration : — ExPT. 7. Water Waves. — Obtain a trough about 6 feet SOUND CHAP. I long, 8 inches wide, and 6 inches deep. Let this be placed level and about half-filled with water. Next float upon the water about six pieces of cork, each about 7 inches square and a quarter of an inch thick, and each carrying a straw mast and a paper flag. Agitate the water at one end by alternately depressing and raising a block of wood. The disturbed state of the water thus produced at one end is shown by the floats to advance to the other, the flags, one by one, nodding as the waves pass them. On carefully watching this phenomenon it will be observed that the corks move not only up and down, but also ex- hibit a little movement lengthwise of the trough. But if the agitation is slight they do not leave their place Fio. 2.— Water Waves. entirely to advance with the wave. The motion of the water at any point is really round and round in a closed loop in a vertical plane. ExPT. 8. Rope JFaves. — Obtain a solid india-rubber cord, about half an inch in diameter, and about 20 or 30 feet long. Fix one end on a hook or staple in a wall, and take the other end in the hand. On shaking sidewise the end held in the hand a series of corresponding displace- ments will be seen to pass along the cord to the farther end. Here the motion of the separate parts of the cord is clearly not one of advance, but only a to-and-fro motion sideways or transversely. But this state of motion advances, each part of the rope executing in turn the motion that a neighbouring part executed a little before. ExPT. 9. Spring Jfuwes.—The apparatus shown in Fig. 3 affords a valuable illustration of the phenomena now under discussion, and although more elaborate than those INTEODUCTOEY previously referred to, is well worth the expense and trouble required for its preparation. A helical coil should be wound in a lathe, the wire being of soft copper about 1 "5 mm. diameter, its turns being 1 cm. diameter, their pitch, or distance apart longitudinally, may be about 1 cm., the whole coil being about 2 m. long, thus containing about 200 turns. Each turn of the coil should be supported by a fine silk thread in the form of a V as shown, each limb of the V being about 1 m. long, its two upper ends being half a metre apart where the threads are fixed to the wood framework. The coil may require to be wound of a rather smaller diameter, and with its turns somewhat closer together, Fig. 3. — Sprinq-Wave Apparatus. in order that, when liberated, it should assume the dimensions given above, which are those it should have when finished and in use. These sizes are chosen to insure a slow advance of the disturbance from one end to the other, and cannot without disadvantage be departed from at random. A very slight increase in the diameter of the wire, for example, would greatly increase the speed at which any disturbance would advance along the coil. It would thus render it difficult for the eye to follow its movements and so detract from the illustrative value of the apparatus. To use the coil, it is sufficient for our present purpose to take one end in the hand and move it to and fro endwise. A corresponding endwise or longitudinal state of motion is then seen to pass slowly along to the other end. If this end is provided with a disc of cardboard with a pendulum bob resting against it, as shown in the figure, the bob will be 8 SOUND CHAP. I agitated on tlie reception of the disturbance. We should thus have an imitation, in a very crude fashion, of the reception of sound by the human ear. 5. In these four experiments we have illustrations of progressive wave motion, or the advance at a finite speed of a disturbance among parts which dg not themselves advance. In the case of the india-rubber cord the waves are called transverse, because the small excursions of the separate parts were performed transversely to the line of advance of the disturbance. In the case of the coil of wire the waves are called longitudinal, because the small ex- cursions of the separate coils were executed longitudinally, or along the line of advance of the disturbance. In the case of the water waves the motion of the separate parts is more complicated, being a combination of Che two forms just referred to. The question arises, which of these three, if any, represents the waves of sound in air which were produced in the first of the present set of four experiments ? We shall see later considerations which show us that waves in the air must be of the longitudinal type like those were which passed along the coil of wire. This apparatus, therefore, gives us the best illustration of the passage of sound waves through the air. 6. Vibrations. — We shall conclude thi« introdiictory chapter by defining or explaining various terii^l*iand phrases of frequent occurrence and fundamental importEwice : — Definitions (1). The period of a vibration is the time from the instant when the vibrating point passes through any position to the instant when it next passes through the same position moving in the same direction (symbol for period t — unit for its measurement, the second). (2) The frequency of a vibration is the number of vibrations performed per unit time. Thus frequency is the reciprocal of period. (Symbol for frequency N=1/t — unit for its measurement, "per second.") 5, 7 INTRODUCTORY 9 (3) The amplitude of a vibration is the maximum dis- placement assumed by the vibrating point in the course of its motion.. (Symbol, a — unit the centi- metre, foot, etc.) (4) The phase of a vibrating point at any instant is the state of its displacement and motion at the instant in question. The various methods of measuring phase will be dealt with later. (See arts. 22 and 24.) All musical sounds are characterised during their steady continuance by three features : pitch, intensity, and quality. Each of these needs a little notice here. 7. Pitch. — The pitch of a musical sound is a feature recognised by every one. It depends upon the period or frequency of the vibrations constituting the sound, and upon that alone. The greater the frequency, the higher the pitch. This may be illustrated by rubbing the finger- nail, first slowly and then quickly, across a finely-grooved surface, such as a book-cover of ribbed cloth. Pitch is specified in two distinct ways, namely : (1) scientifically, by the statement of the period or frequency of the vibra- tion, or by the logarithm of its frequency ; (2) musically, by assigning to the sound in question its position in a certain accepted series of sounds constituting a musical scale. Perhaps the simplest proof that pitch depends only upon the frequency, is that afforded by the following experiment with the siren : — ExPT. 10. Pitch fixed by Frequency. — For the present purpose it will suffice to use the disc siren due to Seebeck. This is shown in Fig. 4. It consists essentially of a disc capable of rotation about its centre, and pierced with one or more sets of holes arranged equidistantly in circles ■concentric with the axis. Opposite to each circle is a jet through which air may be blown perpendicularly to the disc. Thus as the disc revolves the air will alter- nately pass through the holes and strike tlie blank spaces between them, thus giving rise to a series of puffs which produce a musical sound. The disc may be driven by hand, 10 SOUND CHAP. I or preferably by an electric motor. Experiments with this apparatus show that the pitch of the sound obtained de- pends in no way upon the size or shape of the holes or the pressure of the blast, but only upon the number of holes which pass the jet per second. The air blast may even be replaced by a quill or cord touching the edges of the holes, but still the pitch will be found to depend simply on the frequency. Fia. 4. — Seebeok's Siren. 8. Musical Notation. — The ordinary or stnff notation of notes for the representation of musical sounds is given in Fig. 5. Underneath the several notes are placed the letters used as names for them. The various notes of the same name are distinguished by capitals, small letters, subscripts, and accents, on the plan introduced by Helmholtz. The relative frequencies are shown next, and, lastly, the actual frequencies for the pitch generally used in physical INTEODUCTOEY 11 §01 u "o i; -CJ Plrf 8 21" M 5 !>0 « a" s .|« g ^ ). U SOUND CHAP. I But by addition of (1) and (2) /^ + 7.^ = 7^(logi;-log.V) (4), so by (3) and (4) /=/, + /, (5). For k any convenient number could be chosen, as (5) shows that the relation desired is independent of it. But the late Mr. A. J. Ellis (the translator of Helmholtz's Sensations of Tone) has adopted as the unit for this logarithmic measure the cent, 1200 of which make the octave. The name cent is used because 100 cents make the semitone of those instruments in which twelve equal semitones are the intervals occurring in an octave. Hence the clue to reduction of any intervals to these logarithmic cents would be found in the following equations, where / is the interval in cents between notes of frequencies M and N: — I == k log iM I iV (6). 1200= A log 2 (7). Whence by (6)-4-(7). log M- log iV 7=1200 '- " (8). log 2 ^ ' 11. Intensity and Loudness. — The intensity of sound waves is a purely physical quantity independent of the ear. It is proportional to the wave energy passing per unit time through unit area. It will be shown later that for a given medium of propagation and given frequency of vibration the energy of a wave motion is proportional to the square of the amplitu P be V cm. per second. Then a is obviously the amplitude of the vibration. Also the period t is given by T^2iralv=2Trl(o (1), where m denotes the angular velocity of P round in radians per second, and therefore w = v/a. The frequency N is, of course, given by iV = 1/t = (B/27r, or 0) = 27rN' (2). The phase of the vibration at the instant depicted in the figure depends upon the angle POX, which may accordingly be termed the phase angle. If the point describing the circle started from X at the com- mencement of the time t, the x[ angle POX would equal cot, P being the position of the point at the time t. Hence XOX' and YOY' being perpendicular dia- meters, the displacement of the point executing the simple har- monic motion is given by OM = OP sin POX, or y = a sin cot where y denotes OM. 15. Velocity in Simple Harmonic Motion. — In figure 7, let PT represent the velocity v of the point P. Draw PK parallel to OY, and let TK be drawn parallel to OX. Then PK represents in magnitude and direction the velocity of M. But the angle TPK is equal to the angle POX, which equals the angle 0PM. Hence, we have velocity of M = PK = PT cos TPK = I'T cos 0PM = v ■ MP/a, or tj = o) MP (4), where y represents the velocity of M whose ordinate is y, the dot converting a displacement into a velocity in the Fig. 7. — Simple Harmonic Motion. in 20 SOUND oHAr. 11 same direction. In the interpretation of (4) we must take distances to the right or upwards as positive, and those to the left . or downwards as negative. Note also that the line MP must be taken in the order of the letters, and not in the reverse order. Then, on tracing the path of P throughout the four quadrants of the circle, we see that the general features of equation (4) are justified. The angular velocity to of the point P is constant. Hence, we see by (4) that the velocity of M varies directly as MP. Thus when M is at 0, and P at X, MP is positive and has its greatest value, hence the velocity of M has also its greatest value. As P moves through the first quadrant, namely, from X to Y, MP is positive, but continually decreases, hence the velocity of M is always positive, that is, directed upwards, but is diminishing, which is obviously the case. When M arrives at Y, P is at Y also, that is, MP has vanished, consequently the velocity of M has vanished. And we easily see this to be the case, for M is in the act of reversing its motion. As P moves through the second quadrant, MP is negative, and the velocity of M is negative also. But MP is increasing numerically from zero to its full value, and M's velocity is increasing also as to numerical value. While P describes the third quadrant, MP is negative, but decreasing numerically, or M's motion is still downwards, but slower and slower. Finally, as P describes the fourth quadrant, MP is positive and increasing, hence M's velocity is upward and increasing also. 16. Acceleration in Simple Harmonic Motion. — Referring again to Fig. 7, let PN represent the accelera- tion of P. This is shown in elementary text-books (1) to be along the normal PO, (2) to be directed towards the centre of the circle, and (3) to have the magnitude 2 / 2 V ja = a> a. Thus, drawing PL parallel to OY, and letting NL fall perpendicularly upon it, we have the acceleration of M represented in magnitude and direction by PL. By 16, 17 KINEMATICS 21 construction, the angle PNL equals the angles I'OX and 0PM = cot. Hence, we may write — The acceleration of M = PL = PN sin PNL = ( - w^a) ■ OM/a, or y= - (0^1/ (5), where y denotes the acceleration of M. The minus sign is needed because the acceleration PN and its component parallel to OY, PL, are negative when the displacement of M is positive. The result expressed in equation (5) is of great importance, and may be put in words as follows. The acceleration of a point executing a simple harmonic motion is proportional to its displacement, but is oppositely directed. Further, equation (5) shows us that equations (1) and (2) for period and frequency are susceptible of the follow- ing useful forms — T =2',r J -yjy, (6), and 27rN = J -yjy (7). The values of the acceleration of M may be traced throughout the whole vibration, just as was done for the velocity. It may thus be seen that when the displace- ment is zero and the velocity has its maximum numerical value, positive or negative, the acceleration is zero. Again, when the displacement has its greatest numerical value, positive or negative, and the velocity is zero, the accelera- tion has its greatest numerical value, but of sign opposite to that of the displacement. 17. Analjrtical Treatment. — By aid of the differential calculus the velocity and acceleration of a point executing simple harmonic motion are immediately obtained. Thus, we have for the displacement of the point, y = a sin cot. Differentiating with respect to the time, we get y = (oa cos cot = CO . MP. 22 SOUND A second differentiation gives y = — (i?a sin mt ■■ o (uV V These equations correspond to (3), (4), and (5) previously given. 18. Graphic Representation of Simple Harmonic Motion. — The char- acteristics of any rectilinear motion may be very conveniently exhibited by its displacement curve. This is a line whose abscissse represent the times, and whose ordinates represent the corresponding dis- placements. Such a curve for simple harmonic motion is shown in Fig. 8, the auxiliary circle to the left being used to obtain the displacements correspond- ing to equal intervals of time. This curve may be easily plotted on squared paper. The circle to the left may be dis- pensed with if a table of sines is used. This curve is identical with that often plotted in studying trigonometry, and called a sine-graph. Evidently it is also a curve described by a point which simultaneously executes a simple har- monic motion and a rectangular uniform translation, that is to say, a simple harmonic motion along YOY', and a translation with uniform speed along OX. In order to grasp the full meaning and utility of a displacement curve it is necessary to note (1) that the slope of the curve represents the velocity of the mov- ing point, and (2) that the rate of change of slope represents the acceleration of that point. These facts follow im- mediately from the method of plotting 18 KINEMATICS 23 the curve and the definitions of the terms involved. Thus, if the displacement curve is anywhere horizontal, as at A and B, we have, at the instants in question, the velocity equal to zero. This we know to be the case, for here the vibrating point is at its greatest elongation and is Just about to return. Again, at 0, C and D we have parts of the curve where the slope is greatest, and at the corresponding instants the vibrating point must accordingly have its greatest velocity. This also agrees with what we have seen as the result of equation (4), namely, that when the displacement is zero the velocity is a maximum. Eegarding now the rate of change of slope, we see that this is greatest at A and B, for there the curva- ture is seen in the figure to be greatest. Accordingly at the instants corresponding to A and B, the acceleration of the vibrating point should have its greatest value. At 0, C and D the curvatures are seen to be small and are indeed zero. These zero curvatures denote that the ac- celeration of the vibrating point at the corresponding instants is zero also. Both these results agree with equation (5). We may thus draw up the following scheme (see Table I.) as the meanings of the various features of a displacement curve : — Table I.— Displacement Curve Features of Curve. Significance. Abscissa . Ordinate Slope .... Rate of change of slope . Horizontal part Steepest part . Straight part . Sharpest curvature , Time. Displacement of yibratiug point. Velocity of vibrating point. Acceleration of vibrating point. Zero velocity, or rest of vibrat- ing point. Greatest velocity of vibrating point. Zero acceleration, or uniform velocity of vibrating point. Greatest acceleration of vibrat- ing point. 24 SOUND CHAP, n 19. Quantitative Use of Displacement Curve. — Consider now points P and P' on the curve (see Fig. 8), and draw PN parallel to OX meeting in IST, P'JST parallel to OY. Let the ordinates at P and P' be tj and ij, and the corresponding times from be < and t' respectively. Then the mean speed of the vibrating point during the short time t' — t is given by But this ratio is represented on the diagram by NP'/PN", hence we have y = tan P'PN (2). Draw PK the geometrical tangent to the curve at P, and making the angle (5), 19, 20 KINEMATICS 25 where QQ' refers to the length of the arc. But, from (3), tan yjr represents the velocity at Q, and tan yjr' represents the velocity at Q'. Also the horizontal distance between Q and Q' represents the time in which the velocity changes from one value to the other. Hence the mean accelera- tion between Q and Q' is practically given by tan ilf ' — tan -vlr •^= — Qcy — ' ^'^' where QQ' is strictly the horizontal distance between these points, but may be assimilated to the value in (5). But, since the angles are everywhere supposed small, tan i/r = -v^ nearly, and the same holds for -^fr'. Thus, (6) may be written And from (5) and (7) we obtain the approximate equation, acceleration = ij = curvature (8). By the calculus these steps may be put more briefly as follows, (lif rf(tan and x < s, hence by (9) and (10) we have strictly acceleration > curvature (H)- 20. Transverse Progressive Waves. — The simpler wave motions may now be illustrated and discussed. ExPT. 12. — The lantern-slide model represented in Fig. 9 will be found useful for demonstrating the various features 26 SOUND CHAP. II of progressive waves. It consists essentially of a sliding bar cut to a sine-graph, upon which ride a number of uprights carrying balls at their tops. These uprights slide through vertical holes in the two horizontal bars. The uprights may be suitably made from darning-needles whose eyes, when broken, will form the fork for resting on the sliding bar. The balls may be made by beads cemented on the needle- points, or by small pellets of sealing-wax. The illustration is from a photograph of the original model used by the Avriter. This shows the glass cover buttoned on, which holds all in place, but can easily be removed for any adjustments. Fia. 9. — Wave Model for Lantekn. The experiment with this model consists simply in moving the bar uniformly along, and watching the consequent motion of the beads as seen projected on the lantern-screen. From the construction of the apparatus, the points noted in, this paragraph immediately follow for the state of things when the guide-plate is sliding uniformly along at speed v, viz. — (1) Each bead executes a simple harmonic vibration vertically, the amplitude being the same for all the beads, but the phase varying continuously from bead to bead according to distance along the row. 20 KINEMATICS 27 (2) The arrangement of the beads at any instant is in a series of crests and troughs, something like gentle water waves. We call this a wave form. In the case under consideration it should be noted that a crest is like an inverted trough. (3) This wave form, without changing its type, advances horizontally at speed v. This form of wave motion is called transverse, because the vibration of the beads is at right angles to the advance of the waves. (4) The instantaneous form or arrangement of the beads exactly repeats itself at intervals called wave lengths (A,). (5) The velocity and acceleration of any one bead, at any instant, are precisely like those, at the same instant, of certain other beads distant longitudinally an integral number of wave lengths (X, 2X, 3X, etc.). (6) If X is the wave length of the pattern cut in the guide-plate, v the speed at which it is moved, and T the period of the vibration of the beads thus pro- duced, we have v = X/t = JVX (1), JV being, as usual, the frequency. Hence, a sine-graph of wave length X and amplitude a moving along its axis at speed v, represents a transverse progressive wave of simple harmonic type of wave length X and amplitude a, and propagated at speed v, the frequency of vibration of the particles being given by JS/'=v/X. The amplitude a of the waves is, of course, represented on the same scale by the extreme ordinates of the graph, and the velocity and acceleration follow in the usual manner for simple harmonic motion. Suppose now a guide plate were cut to any other exactly repeating pattern of a type that the rods could work on. Then, on moving it at constant speed, we should still have a transverse progressive wave motion executed by the beads, but now of a more complicated type. For, just 28 SOUND CHAP, n as a simple harmonic motion is the simplest type of vibra- tion, so the corresponding or simple harmonic wave is the simplest type of wave motion. We previously mentioned that any periodic motion of a particle along a straight line could be made by compounding two or more suitable simple harmonic motions. It accordingly follows that any wave motion, whose type does not change as it advances, can be formed by compounding two or more simple harmonic wave motions. 21. Longitudinal Progressive Waves. — Let us now consider the kind of wave motion executed by the spring model shown in Fig. 3. This kind is called longitudinal because the line of vibration of the particles is along the line of advance of the waves instead of transverse as in the case just considered. Suppose then we start with a straight row of particles, wliich when undisplaced are equidistant. It is evident that, as the longitudinal waves pass along them, they will remain in a straight row, but will no longer be equidistant. Thus the ari-angement is never along a wavy line as in transverse waves. Nevertheless, it is a matter of great convenience to represent the instantaneous state of all the particles by a single curved line, instead of having to show by a number of dots the position of each particle. And this convenience can readily be attained for longitudinal waves if we agree to plot a curve whose ordinates mean displacements parallel to the axis of abscissa: , the abscissae denoting, of course, the undisplaced positions of the various particles ; that is to say, the displacements, though actually occurring to right and left, are represented on the curve by ordinates up and down respectively. A longitudinal wave passing along a row of particles is exhibited in various ways by the different lines of the diagram in Fig. 10. The first line gives an actual picture of the particles, their equilibrium or undisplaced positions being indicated by small crosses, and their actual or displaced positions by 21 KINEMATICS 29 small circles. The second line exhibibs the same wave by the displacement curve drawn on the convention described above, namely, ordinates up and down denote displacements to the right and left respectively. The third and fourth lines representing the velocities and accelerations may be deduced from the displacement curve by supposing it to move to the right, while the fifth line showing the linear densities may be inferred from either the same curve or the f Conventional (ii)-( Displacement (^{ 't^ii:^^ \^^ ^ /• \ t Ace Accelerations \ Particles i l„\i Linear Densities t^'l of Par i{, N Tho Yetoclties are all reversed if the Waue travels to the left. Fig. 10. — Longitudinal Pkogressive Waves. first line which gives actual positions. The E's denote rarefactions, the C's condensations, the 'N's places of normal linear density. We may therefore state the elements of a longitudinal progressive wave of simple harmonic type as follows : — (1) Each particle executes a simple harmonic motion alony the line of advance of the wave, the amplitude being the same, but the phase varying continuously for every particle according to distance along the line. (2) The arrangement of all the particles at any instant 30 SOUND CHAP. II is along a straight line, but such as may be repre- sented by a sine-graph on the conventions that dis- placements to the right are denoted by positive ordinates, and displacements to the left by negative ordinates. (3) This arrangement, without changing its type, advances with uniform velocity, v say, and so may be represented on a working diagram or model by a uniform advance of the displacement curve already referred to. (See (2) of this article and line (ii) in Fig. 10.) ' (4) This instantaneous arrangement repeats itself at regular distances called wave lengths (and denoted by X say). (5) The velocity and acceleration of any one particle at any instant are precisely like those, at the same iustant, of certain other particles whose distances longitudinally are \, 2\, 3X, etc. (6) With the same notation as before, we have v = x/t The above statement corresponds to the like-numbered sections in art. 20 for transverse waves. We may now add here another section special to the case of longitudinal waves and not before needed. (7) It follows from sections (.3) and (4) that the linear density of the particles, or their state of condensation or rarefaction, is, at any instant, arranged in a pattern which repeats itself at intervals of a wave length. Further, this state of alternate condensation and rare- faction, without changing its type, advances with the uniform velocity v. The significance of the various features of the conven- tional displacement curve for longitudinal waves may be seen from Fig. 1 and Table II. The relation between any feature of the curve and the algebraic sign of the quantity denoted by it is best seen from the figure. It should be 21 KINEMATICS 31 noted that if the wave travels to the left the velocities of the particles are thereby reversed, all else remaining the same. Table II. — Displacement Curve foe Longitudinal Waves Features of the Curve. Their Significance. Abscissae Ordiaates Slope proportional to Kate of change of sluye proportional to Slope also indicates Lengths along line of advance of waves. Displacements along axis of abscissae. Velocity of particles longitudin- ally. Acoele'ratioli of particles longi- tudinally. Linear density differs from its normal value. As to the various quantitative relations here involved, it is easily seen that what was stated for vibrations and trans- verse progressive waves (arts.. 19 and 20) will guide us here also. To deal with the new feature of linear density, let its value at a; be o- and its normal value ctq. Then, since these densities are inversely as lengths occupied by given material, we have on reference to line (ii) of Fig. 10, o-o(PN) = o-(P]sr + NP'), or o-o/o- = 1 + NP'/PN = 1 -f tan ^, where is the angle between the axis of absciss£e and the tangent 1 1+s whence s= — tan <^ nearly (1). We may also derive from Fig. 10 the following relations :- - NP' , Velocity of particle at P = ^^-^ = - ■w tan ^ = — » X slope (2). Also acceleration at P ^ -^tan.^ -(-^taufj ^ ^^, ^ curvature (3). PN/v at P. Or, if o- = o-o(l+s), we have ^^-^— = 1 + tan j>, 32 SOUND CHAP. II 22. Phase and its Measurement. — During its motion the displacement, velocity, and acceleration of a vibrating particle wax, wane, and change sign, passing in the course of a complete vibration through the whole sequence of values possible to them. The place in this sequence which is occupied by the displacement and velocity, at any instant, of a vibrating point, is called its phase at the instant in question. Or, we may define as follows : — The phase of a vibrating point, at any instant, is its state of displacement and motion at that instant, judged with respect to those which it successively assumes in the course of its vibration. Thus, if two vibrating points pass simultaneously through their undisplaced positions in the same direction, they are at the instant of such passage in the same phase. And this is true whether their velocities were numerically equal or not. If, however, the direction of one were reversed, all else remaining the same, they would at the instant of such passage be in opposite phases. Again, if two vibrating points simultaneously reach their maximum displacements in the same direction they are then in the same phase ; if in opposite directions, in opposite phases. And this is the case whether the amplitudes are equal or unequal. The consideration which decides the phase is always the relation borne by the displacement and velocity to those possible in the given vibration. The absolute value of the displacement and velocity of two vibrating points in the same phase may be quite different. When wishing to specify phase to a greater nicety than simply equal or opposite, we must choose some standard state to reckon from, and some method or methods of measuring from that state to the state in question whose phase is to be specified. The state usually chosen as standard is that possessed by the vibrating point when passing through its undisplaced position in the positi^'e direction. As to measuring from this state, evidently 22 KINEMATICS 33 every state of motion possible to a given vibrating particle is deiined by the corresponding position of the point describing the auxiliary circle. But the position of this point in the auxiliary circle may be specified in three ways as follows : — (1) By thQ fraction of a period which has elapsed since last the point passed through its standard position (which of course corresponds to the standard state of the vibrating point). (2) By the angle which it has described since last it passed through its standard position. (3) By the fraction of a ivave length traversed by the waves since the point last passed through its standard position. This third method is, of course, applicable only when waves are proceeding from a vibrating source. It is then often very con- venient. The relation of these three methods of measuring phase difference is easily seen and remembered by taking as an example that of opposite phases. Then, clearly, this may be expressed (1) by half the period, (2) by the angle it radians, or 180°, and (3) by half a wave length of path. It is often necessary to pass from the fraction of a period to the actual time in order to insert the corresponding value in our equation, or from the fraction of a wave length to the actual length, but in each case it is the fraction that measures the phase or the phase difference in question. If the fractions are alike in two cases, then the phase differences so measured are alike, no matter what the actual times or lengths may amount to. This distinction may be illustrated by the following equations representing vibrations, in which t = 27r/(B : — y —a&m (ct (1), y^ = asm a)(t + ~\=a sin {wt + it) (2), 34 SOUND OHAP. 2/3 = a sin -< (3), 2/^ = asm^(i + ^) (4), = «sm-/ i! + -- j = asmf-!; + ;-J (5). Thus (1) and (3), at the instant t = 0, are in like phases. Vibrations (1) and (2) are at every instant in opposite phases. Equations (3) and (4) represent vibrations, which at first sight might seem to be in opposite phases also, but in reality, as shown by (5), y^ differs in phase from y^ by a quarter of a period only ; or the angle vr '2 and not tt. Here, since the a of equations (1) and (2) is replaced by -, the period is changed from t to 2t, thus the extra time T in equation (4) — is only a quarter of a period, and the phase TT angle in consequence ^ only. The measurement of phase by fraction. of wave length will be exemplified in a subsequent article (24). Epoch is the term used to denote the phase at the commencement of the time. It may be expressed in any of the ways by which phase is measurable. Thus in equation (2) the epoch is half a period, or tt radians ; in equations (4) and (5) a quarter of a period, or — radians ; while in (1) and (3) "the epoch is zero. 23. Analytical Representation of Progressive Waves. — Let a progressive wave proceed in the positive direction along the axis of x. Denote by y the displace- ment at time t of a point whose equilibrium position is x. Then, if the waves are of the simple harmonic type, they may be represented by the following equations : — 23 KINEMATICS 35 2/ = asin 27r/^--^j (1), or, ij = a sin 27r(iW - xjx) (2), where t and N care respectively the period and frequency of the vibrations, a their amplitude, and X is the wave length. The speed of advance of the waves does not appear in the above, but is evidently given by v = --^ NX (3). Instead of equation (1) or (2) we may write the equivalent form y = "^ sin ai(t — xjv) (4). This has the advantage of brevity and also introduces V explicitly, but it omits both iV" and X, which are some- times disadvantages. The three forms are easily seen to be identical. To verify their representation of progressive waves it is sufficient to compare either of them with the sections (l)-(6) of articles 20 and 21. We will take them in order, pointing out in each case how the verifica- tion is to be made. (1) Give to X any constant value, and let t grow con- tinuously from zero to t or beyond. The dis- placement is then seen to pass through the changes characteristic of simple harmonic motion, the phase depending upon x. (2) Give to t any constant value, and give to x all values possible from zero to X or beyond. The displacements simultaneously occurring at all the different points along the line are then seen to be those of a sine-graph. (3) Fix upon any point the co-ordinate of whose equilibrium position is of: say, and let its dis- placement at time t' be ij. Then, taking equation (2), we have y' = a sin 27r {Nt' — x'jX) (5). 36 SOUND CHAF. II Now, in order to find whether the wave travels without change of type, we must examine whether for any change in x a corresponding change in t can be fouQd and such as will leave y unaltered. If so, the wave evidently advances, without change of type, and at speed equal to change in a; divided by corresponding change in t to leave y unaltered. Suppose, therefore, that after the lapse of time t from the instant represented by equation (5), the point whose equilibrium position is x' + x has the dis- placement y'. Then we shall have . a/ + .«■ y' = awa.2ir{Nt' + t-——^] (6), and we must find the relation between x and t to satisfy both (5) and (6). Obviously the condition needed is that the value in the brackets is the same in each equation. A value differing by any integer would indeed make y' the same, but it would correspond to the passage from one wave to another and must accordingly be rejected. We therefore derive Nt-xl\ = Q,o\:- = N\ = v&&Y (7). Hence, no matter what the displacement y' was originally, it remains unchanged if we pass in time t to another part of the wave at a distance vt farther on. And this is true for all values of t and vt, provided v is given by equation (7). But this constitutes the verification sought that the wave, without change of type, advances at speed v. (4) Keeping t constant add to the x co-ordinate any multiple of \, and it is then seen that the displace- ment is unaltered. (5) Again, keep t constant and increase x by any multiple of X, and it is evident that we have the vibi-ation 24, 25 KINEMATICS 37 in the same phase as before, or in other words, the velocity and acceleration of the vibrating point are unchanged. (6) The statement of this section has already been verified with section (3). (7) The statement of the additional section of art. 21, occurring only in connection with longitudinal waves, follows from what lias already been given in sections (3) and (4) above. 24. Phase measured by Path of Waves. — We may now recur to the third method of measuring phase differ- ence, namely, by the fraction of a wave length passed over by the waves while the vibration changes from one phase to the other. Thus, consider the following two wave.j : — (t x\ t/j = fflj sin ^7? and Vi = «2 sin 27r( " ' 'J ' -- ) = a„ sin 27r( ^ - "' ^ ) (9). yj = ajSin 27rf-— -j (8), t + rl4: x\_ . „ A «-X,/4'^ On comparing these two equations we see that the phase difference of the vibrations at a given point for the two waves is a quarter of a period, or, by the second form of (9) we may say the second wave is a quarter wave length ahead of the other at any given time. Negatively-travelling Waves. — It should be noted that the equations just dealt with represent positively- travelling waves. In order to represent those travelling in the negative direction along the axis of x we must change the sign of X or v, whichever occurs in the form of equation used. Thus, the following equations represent negative waves : — y = asm 2'jri-+-\ = a?,\\\ai{t + xlv) (10). 25. Condensation in Longitudinal Waves. — Consider now the linear density o- of a slice of material whose zero 38 SOUND CHAP. II positions lie between x and x-\-ckc, the corresponding dis- placements being ^ and ^ + d^. Then if the normal linear density is a^, we have a^dx = a(ch: + d^) (11). Now let us write "- = 0-5(1+8) (12), s being called the " condensation." Then from (11) and (12) we have 1=(1+,) (1 + '^] so s= -'^ nearly (13). \ d.r/ dx Thus in the displacement curve where ^ is represented by y we have .= -f^ (14). dx Velocity of Particles in Progressive Waves. — Write the equation in the form (3) y = asm co{t — xjv). Then we have for the velocity of the particles — = &)« cos a){t — xjv) (15). But this may be expressed in terms of the slope of the displacement curve. For -f = a cos ai{t-x r) (16). (/,*; V TT dy dy Hence "- = —v~?- n 7\ dt dx ^ >■ Or: — velocity of particle = —v times slope of curve nearly, as obtained in equation (2) near the end of article 21. This result applies to either transverse or longitudinal waves. Acceleration of Particles in Progressive Waves. Differentiating again, we obtain for the acceleration d"y ^7, = - co^a sin co{t - xjv) (18). 26 KINEMATICS 39 Also Hence , = ^« sm (o{t — xjv) dx^ df ^ dx^ (19). (20). Or : — acceleration of particle = +v times curvature nearly, as obtained in equation (3) at end of article 21. 26. Composition of Simple Harmonic Motions along the same Line. — Por the solution of this problem several courses are open to us. It may be treated graphically by the triangle or polygon method and by the method of dis- placement curves. We shall take these in the above order, concluding with the analytical method, and point out the special cases for which each method is most convenient. Polygon Method for Vibrations of same Period. — This graphical method is suitable only for equal periods, but can deal quite well with any number of vibrations of different amplitudes and phases. When two vibrations only have to be dealt with the polygon reduces to a triangle. It is this simple case which is illustrated in Fig. 11. Let the vibrations be each of period t and occur along YOY'. Let the amplitudes and epoch angles of one be h and ^, and of the other c and 7 respectively. It is required to find the resultant motion of a point simultaneously obeying both these vibrations. In other words, we need an expression for a displacement which is at every instant the algebraic sum of the displacements at that instant of the component vibrations. Construction. — Take XOX' at right angles to YOY'. Lay off from 0, OP of length I and at the angle /3 with OX. Composition of S.H.M.'s by the Polygon Method. 40 SOUND CHAP. II Lay off from P, PE of length c and at the angle 7 with OX. Draw PM, EN parallel to OX, and meeting OY in M and N, also join OE. Then as OE rotates about with angular velocity a = ^tt/t, N shall execute along YOY' the motion which is the resultant of the two given component vibrations. In other words, the resultant sought is itself a simple harmonic motion of period t of amplitude F = OE and of epoch angle (f> = EOX. Froof. — By construction OM is the displacement at < = due to the first component vibration, while MN is that due to the other at the same instant. Hence at t = the resultant displacement is OM + MN = ON. But in order that ON shall always represent the resultant displacement, its components OM and MN must always represent their respective component displacements, i.e. the lines OP and PE must rotate each at angular speed to. Thus they remain at the same angle to each other, hence OE is of constant length and rotates at angular speed m also. ON accordingly continues to represent the resultant of the two \ibrations to be compounded, and so solves the problem. Analytical Kvprcssioii for Ilcsidtant. — In Fig. 11 draw PK and EL parallel to YO, and produce MP to meet EL in Q. Then we have for the resultant amplitude 0E2 = 0P2 + VW - 20P.PE cos OPE, or, i?'2 = &2 + c2+2&ccos(y8~7) (1), since the angle (/8 ~ 7) is the supplement of OPE. Again, for the epoch of the resultant, we have ^ ^_^ LE K P + QE tan EOX = -^^ = — ^^ — r--' OL OK + PQ & sin /3 + c sin 7 or, tan d) = ~ ~ '- (2). ^ cos yS + c cos 7 ' Finally, since the resultant of the two vibrations is "iven by the motion of N as OE rotates at angular velocity a>, we see that it may be expressed by 27, 28 KINEMATICS 41 y = F sin {(ot + (j)) (3), i^and

=^ (13)- Equation (11) illustrates the case in which two equal vibrations in opposite phases nullify one another, and is usually referred to under the term " interference." The 29 KINEMATICS 43 term is objected to by some writers on the ground that each vibration has its full natural effect. The subject is of great interest and importance both in acoustics and optics, and will be dealt with experimentally in the seventh chapter. 29. Frequencies nearly Equal: Beats. — Let the two vibrations we have to sum be represented by the right side of the following equation : — 2/ = ft sin {vi + n)t + i am {in — n)t ( 1 ), in which ii is supposed small compared to m. Hence we have to determine y to solve the problem. Assume y=fBin{mt+^) (2); then expanding the right sides of (1) and (2), and comparing, we have f cos would be zero for Vr TT TT IT OTT nt = to -, (j) = — for nt = -

pt+ . + \ cos, ]pt + \ cos 2pt + h^ cos 2>'pt+ . (2). It might naturally be supposed that the problem now before us was tl;e reduction of the right sides of these 46 SOUND CHAr. n equations just as was accomplished previously in cases apparently similar. But this is not so. We shall see later that with special values of the constants these equations can be made to represent any periodic motion of frequency pJ'Itt. It is not, however, easy, or indeed possible, in general to simplify the above expressions for the resultant displacement. They stand, therefore, as the analytical representation of what may be termed a compound harmonic motion of frequency ^/27r. If the periods of the components are not commensurate we then have a vibration which is not periodic. It never repeats itself. If, however, the periods of the components are very nearly commensurate we shall have a type of vibration which, for a time, closely approximates to that represented by equations (1) and (2). The deviation of the periods from strict commensurability would be sus- ceptible of approximate representation in equation (1) by a continuous change in the values of the q's, and in equation (2) by corresponding changes in the a's and Vb. 31. Typical Examples of Composition. — The following examples shown by displacement curves are fairly repre- sentative of the most important types of composition of simple harmonic motions. The data are given at the left and the curves at the right of Fig. 13, the components being in thin lines and the resultants in thick ones. In cases I. to III. the resultant is simple harmonic, the first being simply doubled in amplitude, the third changed in amplitude and intermediate in phase, while in the second the amplitude is obliterated, the two components being equal and opposite. Case VII., illustrating the pheno- menon of beats, approximates to a combination of cases I. and II. alternately, the components being at the outset in like phases, and after 12 periods of one and 12^ of the other in opposite phases. In case IV. we have a tone and its octave, to use the musical expression. This introduces the appearance of a compound harmonic motion 31 KINEMATICS 47 ■Si's II j";?-5 111 Q. 11. 1-5 a [J. "I (5. 00 £ Oi ^ 3 -s: 3 2? 2" 2 :^ N 48 SOUND CHAP. II already referred to when first discussing the method of composition by displacement curves. Cases V. and VI. correspond to the tone and its twelfth to use the musical term again. The diagrams illustrate the great difference in form of curve depending on the phase relation of components of given frequencies and amplitudes. Case VIII. illustrates the composition of a number of vibrations of commensurate frequencies with amplitudes diminishing regularly according to a certain law as their frequencies increase. The resultant is an example of the kind of vibration sought from the essential parts of a musical instrument. The term note has been used by Lord Eayleigh for this kind of compound musical sound whose vibrations are composed of commensurate simple harmonic motions, and of which case VIII. is the type. Both Tyndall and Eayleigh used the word tone for those sounds whose vibrations are simple harmonic. The same practice will be followed here. It should be noticed that displacement curves are most readily drawn on squared paper, the ordiuates being taken from tables of sines. 32. Composition of Rectangular Vibrations. — We shall now consider the composition of two simple harmonic motions along lines at right angles to each other. It will be convenient to start with equal frequencies, and to pass afterwards to examples of commensurate frequencies expressed by simple ratios. Each case will be dealt with graphically and analytically ; the effect of a slight departure of one frequency from the strict value assigned to it will also be discussed. Equal Frequencies : Graphical Method. — The graphical method of attacking this problem is illustrated in Fig. 14. In this figure the projection upon OX of the point P describing the lower circle uniformly gives the point M, executing simple harmonic motion along XOX'. Similarly the projection upon OY of the point Q describing the right- 32 KINEMATICS 49 hand circle, gives the point N executing simple harmonic motion along YOY.' By compounding these two displace- ments OM and ON, we obtain the resultant displacement OR. In the case drawn, the point E is found to execute along the inclined line AOA' a simple harmonic motion which is the resultant of the two given rectangular vibra- tions. This statement is verified by the graphical method, which will be readily understood from the corresponding Q Y A 2 ---Hy&- ^/^^_ -^o y 1 \\ X' ix °-. Ml ^ i... 8 O yf =^ y-' A ^x^ Y' 4 i B' f i 6 0-^ p 2 Fig. 14.— Composition of Rectangular Vibrations of Equal Periods, numerals round the circles and the dotted lines indicating the projections. The above is for the case of no phase difference, for when M is at 0, moving in the positive direction OX, N is at moving in its positive direction OY, To obtain the resultant for any specified phase difference, we have simply to shift the numerals round one of the circles by the requisite amount, allowing those on the other circle to remain where they were, then compound as before. Thus, if the case of opposite phases be desired, we have only to shift the numerals on either circle through 180°, E 50 SOUND OHAP. II the others remaining undisturbed, and we obtain the path BOB'. Again, let the phase difference be a quarter of a period, that is, its angular measure is 7r/2. It is then easily seen that the resultant is an ellipse whose axes are XOX' and YOY', and which is inscribed within the rectangle AB'A'B. If the vibrations have equal amplitudes, the auxiliary circles in the figure are equal, and the ellipse just referred to becomes Fig. Ifi. — Moue General Case op Composition. a circle also, and is described by E with constant speed. The fact that two rectangular simple harmonic motions of equal period and amplitude and phase difference 7r/2 com- pound into a uniform circular motion is also obvious from the definition of simple harmonic motion. If the phase difference is neither zero, tt nor 7r/2, we have as the resultant an ellipse with oblique axes, but still inscribed in the same rectangle as before. Such a case is illustrated in Fig. 1 5, in which the phase difference is 7r/4, or one-eighth of a period. In this case the numerals in the lower circle are shifted 32 KINEMATICS 51 forwards by the angle 7r/4, those in the other circle remaining in the standard position. The resultant inclined ellipse shown then follows from the ordinary method of projection. Analytical Method. — To treat the problem analytically it is convenient to take a general symbol for the phase difference, and afterwards assign to it any required values. We accordingly write for tlie two rectangular components : x = a sin (cot + B) (1), ■If = h sin at (2). Now to obtain the equation of the path we need to eliminate t between these two equations, we shall then have a single relation between j: and y instead of having each expressed in terms of t and other constants. The result may be written 6V - 2ahxy cos 8 + ahf = aV sin^ S (3). This is, in general, the equation of an ellipse with axes inclined to the co-ordinate axes. To test the effect of various phase differences, consider the following cases : — (i) For 8 = 0, (3) becomes (fe - ayf = (4), which represents a pair of coincident straight lines through the origin, and lying in the first and third quadrant as AOA', Pig. 14. The analytical representation of a p'^ir of coincident lines corresponds to the fact that the line AOA' is described twice (in opposite directions) in each period. (ii) For S = 7r, (3) becomes (bx+ayf = (5), representing the other diagonal BOB', (iii) For 8 = 7r/2, (3) becoijies &V+«y = a%' (6), which is the equation of an ellipse with semi-axes a and b 52 SOUND CHAP. II ■along OX and OY respectively. Further, if l = a, this becomes a circle. (iv) For 8 = 7r/4, (3) becomes &V - 2ahxyl J 2 + a\f = a^&V2 (7), which is the oblique ellipse in Fig. 15. 33. Frequencies nearly Equal. — We have just seen the results of compounding two simple harmonic motions of equal period and various phase-differences. Now, let the periods be very nearly but not quite equal, and let the initial phase- difference be zero. Then it is evident that initially the resultant will be practically that correspond- ing to vibrations of equal period and like phases. When, however, one vibration has gained an eighth of a period on the other, we shall have as resultant a figure closely resembling the oblique ellipse just dealt with above. When the phase-difference has become a quarter of a period we shall have for resultant something closely like the ellipse with axes parallel to the component vibrations, and so forth, passing through all possible phase-differences while one vibration gains a period on the other. Thus, when the difference between the frequencies is very small, the phase- difference, though gradually changing, can be regarded as almost constant through any one period. Hence, for any such period, the resultant may be taken as practically represented by that closed loop obtained for two vibrations of equal frequencies and of the phase-difference which ruled at the time in question. If, on the other hand, the differ- ence of frequencies is considerable, the resultant at any time differs materially from a closed loop, because an appreciable change in the phase-difierence has occurred in the course of a single period. 34. ExPT. 13. Com2}osition illustrated by Spherical Pendulum. — Take a thread about a ilietre long, fasten it rigidly at its upper end, and attach a metal bob to the free end below. There should be space enough for the bob to mo-^'e, say 50 cm., in any direction from its resting position. With 33, 34 KINEMATICS 53 this simple form of spherical pendulum we may illustrate the composition of two simple harmonic motions at right angles to each other with periods sensibly equal, or distinctly unequal at pleasure. These facts are seen from the following sketch of the theory of the pendulum : — It is shown in elementary text-books on mechanics that the period of a simple pendulum is given by T = 2ir -Jlfg, where I denotes the length of the pendulum and g the acceleration due to gravity. This expression is obtained for vibration through angles small enough to make their sines and circular measures practically equal. For somewhat larger amplitudes of a radians, it is shown in more advanced treatises that the period is approximately expressed by r.2 r=2^^l/g(l+y^ Thus, if our pendulum of 100 cm. long vibrates through an arc of 40 cm. we have a = 0"4, and a^/16 = 0"01. In other words, such vibrations have a period longer by one per cent than infinitely small vibrations of the same pendulum. And this would make an appreciable difference in the behaviour of our pendulum as we shall notice presently. If, however, the arcs used are 20 cm., the period is lengthened by only one-fourth per cent, for halving the amplitude reduces the correction to one-quarter. Again, for arcs of 10 cm. (or about 4 inches) we have the correction term reduced to -^ per cent, which is negligible for our present purpose. Hence for vibrations within 10 cm. amplitude, we have practically the same period as for infinitely small arcs ; whereas with the extreme of 40 cm. arcs, or 0-4 radian amplitude, the addition to the period reaches one per cent. To compound vibrations with equal periods, start the bob by a slight tap to the east say, and when it next passes through its equilibrium position give it another similar tap to the north or south. Then we shall have illustrated the composition of vibrations in like or opposite phases. Then, whether the taps were equal or not, provided they were both feeble enough to keep the amplitudes under 10 cm., we see, from the results of theory just quoted, that the periods will be practically equal in the two directions at right anffles. 54 SOUND OHA.P. n Hence the resultant will be an oblique straight line since the phase-difference is zero or tt. To obtain any other desired phase difference it is only necessary to time the two taps accordingly. Thus if the second tap is given to the bob when it is at either end of the swing due to the first, we have the phase-difference one- quarter or three-quarters of a period, and the resultant figure an ellipse with axes along the directions of the com- ponent vibrations. If, however, the second tap is given when the bob is neither at the zero position nor the end of its swing, we have a phase-diiference between zero and a quarter of a period, and, accordingly, an oblique ellipse is described as the resultant motion. Now let the taps be such as to give amplitudes of 40 cm. and 5 cm. respectively in two directions at right angles, the phase-difference being one-quarter. Then the figure initially described is approximately a long, narrow ellipse of semi- axes 40 cm. and 5 cm. But the period lengthwise is about one per cent greater than the cross period. And this slight inequality produces a striking effect which is at once noticed when performing the experiment. Before seeing this result it might be expected that the various figures described would be those treated in the theory of composition with various phase-differences, namely, figures all of which are inscribed in the same rectangle unchanged in size and position. This, however, is not the case. On the contrary, the quasi-ellipse shifts round, or " precesses " as it is termed, the direction of precession being the same as that of descrip- tion. The explanation is easy. Suppose the large ampli- tude to have been originally east and west, and the small one north and south. Then the slightly longer period corresponds initially . to the east and west vibration. Let the original phase-difference be 7r/2, so that the initial figure described is an ellipse with major axis east and west. Consider the bob as it passes round from the end of the major axis, in, say, the counter-clockwise direction. It goes north for a quarter of the shorter period, south for half that period, north for another quarter, and again crosses the major axis of the original ellipse. But its vibrations east and west have been slower, corresponding to the longer period due to large amplitude. Hence at the instant of crossing the major axis to the northward it has not quite 35 KINEMATICS 55 finished its motion eastwards. It thus crosses the axis northerly and easterly, and at a point short of the end of the axis. Thus the second description will give a quasi-ellipse whose major axis is not exactly east and west, but has turned slightly from that position in a counter clockwise direction. But the longest period still corresponds to the major axis of this displaced ellipse, hence the cause for its further rotation holds good as before, and results in its continued rotation in the sense of description. Of course, any of these pendulum curves are subject to a slight shrinkage of amplitude as time goes on, which causes them to fail of being quite re-entrant, apart fr.om the special phenomenon now under notice. 35. ExPT. 1 4. Composition illustrated hj Stbhrer's Projection Apparatus. — In the apparatus due to Stohrer and illustrated in Fig. 15a, it is seen that two opaque discs are mechanically made to execute simple harmonic motions along directions at right angles. Each disc has a slit at right angles to the direction of its own motion. Hence these slits are mutually perpendicular, and leave only a small square open for light to pass through. An optical lantern is used, and so we obtain focused on the screen an image of this square hole. When the handle of the apparatus is turned to put the discs in motion, the spot of light on the screen executes a motion which is obviously the resultant of the two rectangular simple harmonic components. By means of a nuihber of wheels supplied with the apparatus, and which may be changed at will, the periods may be made precisely equal, nearly but not quite equal, or of various simiile mtios. Moreover the phase-difference can be arranged to have any desired value by springing the wheel of one disc out of gear, and setting it by any specified angle ahead of the other. If the handle is turned slowly so that-the resultant motion is performed say only once or twice per second, the spot is seen in motion, but the exact shape of its path can scarcely be ascertained. If, however, the handle be turned quick enough to cause the whole cycle to be passed through not less than ten times per second, then the complete path of the resultant motion will be exhibited as a bright line. This is owing to the phenomenon of persistence of vision, in virtue of which the impression on the retina persists for about a tenth of a second after the removal of the stimulus which 56 SOUND oHAP. n produced it. Hence if the periods of the components are exactly equal, we have a steady or stationary figure whose form depends upon the phase-difference chosen. If, however, the periods are slightly different (say 59 : 60, by using wheels of 60 and 59 teeth driven by the same driver), then the phase-difference passes through all its possible values in 60 Fig. 15a. — Stohkbe's Apparatus. of the smaller periods, and hence the figure appears to slowly melt or dissolve from one characteristic form to another, passing in turn through all the forms possible for practically equal periods. Unlike the spherical pendulum just described, the com- ponent vibrations in this apparatus preserve their original direction, amplitude, and period. Hence the rectangle which circumscribes the initial figure contains also the whole series 36 KINEMATICS 57 of figures subsequently described. This apparatus may, of course, be used for periods as 1 : 2, 2 : 3, etc. But in such cases where several turns of each wheel are needed before the cycle of the resultant is completed, it is difficult to attain such speed of working as to make the whole figure apparent. Thus if the cycle occupied two-tentlis of a second in descrip- tion, only about half of it would be visible at any one time. The result is then far from satisfactory. Hence such cases dealt with in the next article are illustrated by a special pendulum. Fig. 16. — Composition of RECTANcnLAB Vibkations of Pkrious 2 : 1. 36. Periods Commensurate or nearly so. — Periods 2 : 1. — We shall treat this first by the graphical method previously adopted for equal periods. In Figures 16 and 1 7 this is illustrated for like phases and for an initial phase- difference corresponding to a quarter of the smaller period. It should be noted that here; since the periods are quite different, the phase is not constant, hence the term initial phase- difference. The previous description of Figs. 14 and 15, and the numerals on these two figures, will make the method of description clear enough. 58 SOUND CHAP. II Analytical Treatment. — The general case for rectangular vibrations of periods as 2 : 1 is analytically represented by a; = « sin {^.tot + h) and y = i sin tot. On eliminating t between these two. we obtain Fig. 17. — Composition with PEMODi 2 : 1, but In Different Phases. which is the general equation for any phase-difference and amplitudes. Hence, for 8=0, this becomes + - = a- (2), which is the equation of the curve in Fig. 16. For B = 7r/2 as in Figure 17, we obtain 2?/2 X -f + -_1) =0 0^ a (3), 36 KINEMATICS 59 which represents the two coincident parabolas. The equation to the single parabola may be written t = - ^(a-- - a) (4), which gives the curve shown in the figure. Equation (3) gives two coincident curves as it should do since the single curve of (4) is described twice (viz. in opposite directions) in each cycle of the resultant motion. These two results may, of course, be obtained by giving to h the appropriate value before eliminating t, but' that method lacks generality. Periods 3 : 1. — This need not be treated in detail by the graphical method. For it is clear that no new difficulty will arise beyond the more mechanical labour. On the other hand, in the analytical method, to avoid undue com- plication, it becomes advisable to sink full generality and put in at the outset the particular value of the phase- difference chosen. Thus for 8 = we have as the components x = a sin oiot and 3/ = & sin at. Whence we obtain as the equation of the resultant h^ h ^ a 3 --^+1=0 (5). This is a cubic or an equation of the third degree. In reality the path of the point is two coincident cubics described opposite ways. This would be seen from the graphical method. It could also be obtained analytically by elimination of t with S unspecified, and then writing S = in the equation so obtained. Take now the following case : — a; = a sin {icot -|- 7r/2) = a cos iat, and y ~^ sin (ot. 60 SOUND CHAP. II We then find on elimination of t, .-^l-fe i-ii-) » This is a curve of the sixth degree. Other Ratios. — For periods of any ratios whatever, com- mensurate or incommensurate, the graphical method is applicable, and gives, as readily as can be expected, the result sought. The analytical method, on the other hand, soon becomes exceedingly cumbrous, and is perhaps hardly v7orth using for ratios beyond those just given. Where the periods are nearly but not quite of a given simple ratio, the resultant may be approximately expressed as that for the simple ratio in question, but with a continuously changing phase - difference. A special case of this has already been dwelt upon at some length, namely, that in which the frequencies are nearly equal. 37. Typical Examples of Composition of Rectangular Vibrations. — Let the two vibrations be respectively X = a sin (rat + S) and y = h sin at. The type of the resultant then depends upon the value of the ratio, r, of the periods and the phase-difference as expressed by 8. The ratio of length and width of the circumscribing rectangle is of course dependent on the value of ajh, the ratio of the amplitudes, but this affects the dimensions only, and not the type of the resultant curve. Fig. 18 shows a few typical cases, the value of the ratios being giveu in the first column and those of S vary from to TT along the lines. ExPT. 15. Composition illustrated hy Blackburn's Pendulum. —All the cases just given in Fig. 18 and any others can be illustrated in a very clear and interesting manner by a simple apparatus known as Blackburn's pendulum. Fig. 19 shows an outline of the instrument, which is seen to consist essentially of a bob and funnel with a Y-shaped suspension. The stem of the Y has a length I, while the whole length to 37 KINEMATICS 61 1:3 2:3 Pig. 18. — Typical Cases of Rectangular Vibrations. 03 SOUND CHAP. II the level of the top points is L. Thus for vibrations normal to the plane of the Y the period is given by T = 27r v i//g', while for vibrations in the plane of the Y executed by the stem of the Y only, the period int = 2-K ^lljg. So the ratio of the periods is given by r = •JLjl, and by altering the L and I, r may be made what we please. To form a trace of the path the funnel may contain fine silver sand which is received on a blackened surface. Some ready means of changing the lengths is required ; one simple method consists in fixing the position A, of the top of the stem of the Y, by a mrtal clip which can be slid up and down, and remains by Fig. 19. — Blackbokn's Pendulum. friction where set. It is also desirable to have packing blocks or wedges under the board to bring it near to the funnel, as if too far away the sand spreads too much.' The bob may be a simple ring of lead round the funnel, or a piece of cast metal turned and bored true. Of course the greater the mass of the bob for a given size, the less quickly will the vibrations die away. It is obvious that with this pendulum the ratio of periods and amplitudes and the values of the phase-difference can all be varied at pleasure within very wide limits. When, as usually is the case, the adjustment of periods aimed at is not quite attained, then the figure passes successively through all the forms characteristic of the various phases for the ratio in question. Thus the forms appear successively like those in a given line of Fig. 18, but those corresponding to different lines are not mixed. 38 KINEMATICS 63 38. Composition of Linear Progressive Waves. — The problem of compounding wave motions progressing along •the same line at the same speed naturally falls under two heads, according as their directions of motion are the same or opposite. When they are the same the matter is very simple and can readily be extended to embrace more than two components. It is sufficient to consider the form of each wave at a given instant, and then to compound them by addition of their respective ordinates. The resultant instantaneous Wave-form so obtained must then be imagined to move with the speed of its components. It then repre- sents the resultant wave motion sought. Thus, let the components be y^ = a^ sin a>^{t — x/v) and y^ = «2 ^^'^ ^'2^^ ~ ^A) ^ ■*■)' and for compounding them choose the instant t = 0. The above equations then reduce to 2/j = — ttj sin {(Ujx/v) and y^= —a^ sin ((o^xjv) (2). Hence the resultant at < = is given by 3/^= —a^sm(co^x/v) — a,^sm((o^x/v) (3). Thus the initial displacement is expressed in terms of x, and the curve so obtained must be shifted along in the positive direction of x at speed v to represent the resultant wave motion. This final result can be obtained and repre- sented analytically by the simple addition of the original components as expressed by the equation (1). Thus, we have y = a^ sin (o^(t - x/v) + a.-^ sin a^^t — x/v) (4), which is equivalent to the statement made in words. It is obvious that the resultant wave-forms at any instant, in the case now before us, are identical both in shape and process of derivation with the corresponding dis- placement curves shown in Fig. 13, art. 31. But in that case the abscissa represented time, whereas now abscissae represent actual lengths just as the ordinates do in both cases. 64 SOUND If the progressive waves to be compounded are longi- tudinal instead of transverse, they may still be represented by the above curves on the usual conventions for longi- tudinal waves. 39. Formation of Stationary Waves. — Let us now turn to the case of compounding waves progressing in opposite directions. In doing so we shall restrict our examination to the simple but important case of components of equal periods and amplitudes. And this is precisely the type which oftenest occurs. For either wave train may be Instant of Time Components Resultants A !n A n: a A_ ^;n A Nl & 4-, ^!n A n:.^ A ^'iN A n!__a A !n A N' A Fig. 20.— Derivation op Stationary Waves. produced from the other by reflection, as will be seen in Expt. 16 and treated further in Chap. IV. The resultant of the components in question is a system of waves alternately waxing and waning, but without progression in either direction. To this the .term stationary waves is applied. The process of graphical composition by dis- placement curves and the result obtained are sufficiently illustrated in Fig. 20. Five sets of curves are given showing the components and resultants after equal intervals of t/8. Fig. 21 illustrates in more detail the state of things in stationary waves at two typical instants. In this figure the first three lines apply equally well to transverse or 39, 40 KINEMATICS 65 longitudinal waves; the last line has reference to longitudinal waves only. The points marked JST and A in Figs. 20 and 21 denote respectively the places of no motion and maximum motion. They are called nodes and antinodes. The last line of Kg. 21 shows linear densities of longitudinal waves, in which the U's mean undisturbed or normal densities, and the C's and E's compressions and rarefactions respectively. t = o. Displacement 1 Curues ' Velocities Accelerations Linear Densities \ of ) Longitudinal Wanes) (i iiilllii* = r/4. A H Ttd UUUIIUUUUII c; c u ?• ■li inn A N o 00000009 000000(» tnntm"4iiiiiiii'' R ?• U c C IIIIIHIIIIIIillWI FiQ. 21. — Details op Stationary Waves. 40. Characteristics of Stationary Waves. — ^We are now prepared for an explicit statement of the characteristics of stationary waves of simple harmonic form, and in giving it may fitly follow the order already adopted for progressive waves in articles 20 and 21. (1) With certain important exceptions each particle executes a simple harmonic motion. The amplitude of these vibrations varies continuously from point to point along the line of waves, passing through positive, zero, and negative values ; whereas the phase, at any instant, is the same all along the line. The points where the amplitude is zero, and where, therefore, the particles never move at all, are called 66 SOUND CHAP. II nodes. Their distance apart is called half a wave length. The intervening portions are called ventral segments. The points midway between the nodes and where the amplitudes have their maximum values are called antinodes. The term loop is often used loosely, referring sometimes to a ventral segment and sometimes to an antinode. (2) The arrangement of the particles at any instant is in general that of a sine-graph (or, if the waves are longitudinal, may be represented by one). But at certain instants, occurring twice in each period, this curve shrinks to a straight line. (3) This wave-form neither changes its type nor advances, but only shrinks to a straight line by proportional diminution of all its ordinates. It then expands proportionally ; all the ordinates being now reversed in sign, again shrinks, and so forth. (4) The instantaneous form repeats itself at certain constant intervals called wave lengths. (5) The velocity and acceleration of any one particle at any instant are precisely like those at the same instant of certain other particles whose distances longitudinally are any multiple of the wave length. (6) If X and t denote respectively the wave length and period of vibration of a system of stationary waves, we have X/t = v, where v represents the velocity of propagation of progressive waves of the same type which, by travelling in opposite directions, can produce the stationary waves in question. The above sections apply to both transverse and longi- tudinal waves, the following section to longitudinal waves only. (7) In longitudinal stationary waves the linear density of the particles, or their state of compression or dilatation is, at any instant, arranged in a pattern which repeats itself every wave length. Further, " KINEMATICS 67 this state ot linear density neither changes its type nor advances. It oiily changes, proportionally at each point, to a state of uniform and normal density. This occurs twice in each period. The deviations from normal density then reverse sign at each point, and experience proportional exaggeration until a maximum is reached, when they again suffer a shrinkage, and so forth. The nodes, or places of no change of position, are places of maximum change of linear density. Whereas the antinodes, or places of maximum change of position, are places of no change of linear density. 41. ExPT. 1 6. Fm-mation of Stationary Waves. — The actual experimental derivation of stationary waves may be demon- strated very simply by a solid india-rubber cord or thick- walled tube of rubber, about half an inch in diameter, and (say) 20 feet long. One end should.be fixed, and the other end held in the hand and moved to and fro at a suitable rate. A train of waves is thus sent along the cord from the hand to the fixed end Avhere they are reflected. Thus the incident and reflected wave trains moving in opposite directions give us stationary waves. It is easy to arrange that the whole length of the cord shall be X/2, A, 3A/2, etc., thus containing 1, 2, 3, etc., ventral segments. To form and sustain clearly marked and regular stationary waves it is necessary that the motions of the hand should be regular and sustained. ExPT. 16a. Progressive and Stationary Waves Contrasted. — If a wire is wound uniformly round a cylinder so as to form a helix, and then taken off' and viewed from a distance, it is easily seen that it must present the appearance of a sine- graph. Moreover, if it is rotated uniformly, this sine-graph will appear to move endwise, and thus represent a progressive wave. Again, if a piece of the same wire be taken and bent in a plane to the form of a sine-graph, with a piece left at each end to form the axis of the graph, it is obvious that this wire will, on rotation, represent a set of stationary waves. For the amplitudes will all change proportionally and vanish simultaneously when the wire is viewed edge- 68 SOUND CHAP. II wise. To exhibit these effects to an audience it would be inconvenient to view the wires direct. It is therefore desirable to mount the pair of wires as a small working model, and throw their image on the screen by the pro- jection lantern. Fig. 22 shows the model used by the writer for this purpose. Fio. 22. — Lantern Model for Stationary and Progeessive Waves. 42. Analytical Treatment of Stationary Waves. — Let the two systems of oppositely travelling waves be represented by y^ = a sin m{t — xjv) and y^ = a sin a){t + xjv) (1). Then the equation of the resultant system of stationary waves is obtained by addition, and may be written — 2/ = 2 cos (a>xlv)a sin wf (2). The comparison of the part played by x in (1) and (2) shows at once the leading distinction between the two kinds of waves. The contrasting features have already been set 42, 43 KINEMATICS 69 forth in sections (1) of articles 20, 21, and 40, and may well be again referred to here as derivable from equations (l)and (2) above. In the equation for progressive waves, X enters into the argument of which the sine is taken and there only. It thus, by its change, can change the phase of the vibration, but is powerless to alter its amplitude. In the equation for stationary waves, on the other hand, x enters into the coefficient and there only. It is accordingly able, by its change, to change the amplitude, but is unable to affect the phase. All the featui'es shown in Figs. 20 and 21, and detailed in article 40 may be derived from equation (2) of this article. For the statement as to longitudinal density the y in equation (2) must be interpreted as a displacement along x. To derive the characteristics from (2) it is as well to write it in the form — y =2 cos {27rx/X)a sin cot (3). Then for t=0, y = for all values of x. Also, for x = A,/4, 3\/4, 5\/4, etc., y=0 for all values of t. These points correspond to the nodes and are seen to occur at intervals of X/2. Again, for a; = 0, X/2, \ etc., y=±2a sin at (4). These points X/2 apart correspond, therefore, to the anti- nodes, since here the amplitude is a maximum, namely + 2a. By differentiating (3) with respect to t, we obtain for the velocity of any point x at time t — y = — 2q) cos (2'7rx/X)a cos cot (4). Differentiating again, we have for the acceleration — 43. Diverging Waves. — Hitherto, for the sake of simplicity, we, have considered waves advancing along a line merely. In actual fact, however, such waves are not usually met with. We must, accordingly, extend our ideas to waves starting from a given source and diverging thence in various directions, either (1) in a plane, or (2) in space of 70 SOUND CHAP, n three dimensions. Waves diverging in all directions in a plane are familiarly illustrated by the ripples formed on the surface of calm v^ater, in which a pebble is dropped. The waves diverging in solid space are illustrated by the case witli which we are here most concerned, namely, sound waves in air. To give an example in which the divergence is of the most regular character, namely, spherical shells of alternately compressed and rarefied air, we may explode a gaseous mixture in a soap bubble. For this purpose the products of the electrolysis of dilute acid may be used. Composition of Diverging Wave Systems. — Suppose we have two sources of exactly similar waves spreading in a plane. Then it is obvious that at some places the crests of one system will coincide with the crests of the other. We shall, accordingly, have unusually high crests at such places. Again, where the troughs of one system coincide with the troughs of the other, we shall have specially deep troughs. But where the crests of one system coincide with the troughs of the other, we may have a maintenance of the normal level. This phenomenon of the increased disturb- ance at some places, and the obliteration of disturbance at others, is often called the interference of the wave systems in question. The term is introduced under protest as one which has acquired such vogue that it should be known. Its reference is obviously to the annulment produced by opposite phases. But it has been pertinently asked, if that is interference in which tlie resultant is the algebraic sum of the components, what can non-interfereuce be ? The positions of the places of crest on crest, and crest on trough, can be easily seen from the circles in Fig. 23. In this the sources are the centres, the full lines show crests, and the dotted ones troughs. Hence where lines of the same kind intersect we have increased effects, where lines of opposite kinds cross we have no change from the normal level. u KINEMATICS 71 Figs. 84 and 85 are from photographs of such divergent wave systems obtained by Dr. J. H. Vincent on the surface of mercury by points from vibrating tuning-forks. To extend either of these cases to that of the composition of wave systems radiating in three dimensions we must, in imagination, rotate the whole about an axis passing through the two point sources. Thus the ripples in the form of circles on a plane give place to spherical shells, and the Pig. 23. — Composition of Divergent Waves. points where crest falls on crest become circles whose centres lie upon the axis of rotation containing the two point sources. The same applies to the points where crest falls on trough, or trough on trough. 44. Huyghens' Principle of Wavelets and Envelopes. — Let us now consider, in more detail, the advance of waves in three dimensions in space from a point source. Let the velocity of propagation be the same in every directioi^. We can then regard the advance of the waves in two ways. First, we may think of them simply as emanating from the given source without any intermediate 72 SOUND CHAP. II consideration. Second, we may think of the state of things at a certain instant as derivable from that at some previous instant when the disturbance was on its way from the source. This second view, which is often of great utility, was put forward by Huyghens in 1678 with reference to light, but may be applied equally to sound. lu the original treatise by Huyghens the explanation and discussion of this principle occupies about seven pages, and is nowhere con- cisely enunciated. It accordingly appears in very different forms in the various text-books on optics and sound. This seems likely to lead to haziness of conception on the part of the student. Consequently, in the hope of making Huygliens' principle as clear as possible we shall here attempt to condense the pith of it for our purpose, in the following definition and enunciation : — Definition. — The locus of all points just reached by a wave disturbance at any instant is called the wave front at the instant in question. Huyghens' Principle. — The wave front at any instant may be derived as the envelope of wavelets whose origins are all the points constituting the wave front which existed t seconds previously. In an isotropic medium at rest these wavelets are spherical and of radius vt, where v is the velocity of propagation of the waves in all directions in the given medium. The above statements, being higlily condensed, will be understood better on reference to a concrete example, namely, that given in Fig. 24, which is a reproduction of one of Huyghens' original diagrams. In this figure A represents the point source of a wave proceeding towards CE. At a certain instant the wave front is the arc CE whose centre is A. This, on the ordinary view, would be regarded simply as having eman- ated from A without any inquiry as to the intermediate stages in the process of propagation. But on Huyghens' view we may regard the wave front CE as derived from a KINEMATICS 73 the concentric arc BG, which was the wave front at some previous instant. To construct CE on this principle, a number of points, BbbbbG, are taken on the wave front HBGI. From these points as centres circular arcs are described with the radius EC = GE. Then CE is easily seen to be the envelope of these secondary waves or wavelets. In this extremely simple case, that of the emanation of a wave from a point source, it is evident that no simplification of construction follows from the adoption of Huyghens' principle. In fact, it is much easier to describe CE as the circular arc whose centre is A, than to describe it as the envelope of half a dozen similar arcs. But the gain lies in the in- sight thus afforded into the process of propagation. The conception thus gained may be applied easily to other cases which cannot well be resolved by any other method. [5^ The grounds for accepting TT 1 , ■ /. ,, .. Fig. 24. — Hutghens' Pkinciple. Huyghens view of the matter lie in its reasonableness, and in the agreement between its consequences and experiment. This agreement will be unfolded as we proceed, the reasonableness we may note now. Thus, since at some instant the wave starts from A, and at some later instant CE is the wave front, it may be assumed that all parts of space between A and CE have been traversed by the wave disturbance. Hence BG, concentric with CE, was at some instant the wave front. At this instant each point along BG was in a state of disturbance, and miglit therefore be regarded as the origin of an elementary disturbance ; that is, each such point may be looked upon as the centre of a circular wave. Again, looking at the matter quantitatively, let the time for A to B be /!,, and from B to C be t^, then the time from A to C 74 SOUND CHAF. n would be t^ + 1^. Thus, if the velocity of propagation be denoted by v, we have AC = v{t^ + 1^, taking the ordinary view of propagation without any reference to the inter- mediate wave front BG-. Secondly, on taking Huyghens' view, we have AU = vt^, and BC = vt^, so that AC = v{t^ + 1^ as before. Hence the ordinary view, and that obtained on Huyghens' principle, are in agreement in the case con- sidered. Let us also notice here that the direction of advance of the wave is normal to the wave front. Further, if we take a very distant point source, our wave front is practically plane and advances at right angles to it. It will often be convenient to consider waves of this nature which we may briefly designate plane waves. 45. Reflection at a Plane Surface. — Suppose now that a plane wave falls obliquely upon a plane reflecting surface. The consideration of what happens is somewhat obscure on the ordinary view, but is very simple when examined in the light of Huyghens' principle, which we accordingly now apply. Fig. 25 is again from Huyghens' classical treatise of 1678. In this figure AHHHC is the plane wave front at the instant of incidence of A upon the plane surface AKKKB of the obstacle. The angle of incidence may be measured by BAG, the angle between the wave front and the reflect- ing surface. It may be noted, in passing, that these planes have each an important physical meaning and objective reality ; whereas the angle of the same value, used in the ordinary method of geometrical optics, is taken between the normals to these surfaces, the normals being geometrical conceptions which are not in evidence in any experiment on reflection. Consider now the passage of the wave front AC ; if no obstacle were present it would pass to GB, the opposite side of the rectangle ACBG. Owing, however, to the presence of the obstacle, the wavelet from any one of the 45 KINEMATICS 75 points H can only reach in the original direction the corre- sponding point K, hence the equivalent of the remaining portion, KM, of the path must be executed, after reflection, on the original side of the surface AB. Thus, A, H, H, H, C, being the centres of the wavelets, we note first that the distance AG must be traversed wholly after reflection from A. We therefore describe from A the arc AN with radius AG. The path CB, on the other hand, is wholly described in the original direction before reflection. The paths from the intermediate points H, H, li, need treating in two parts as mentioned before. The first part HK is undisturbed in direction. The second part, equal in length to KM, is ~^^c fec^ Fio. 25. — Plane G Reflection on Huyghens' Principle. then to be treated as the radius of a wavelet whose centre is K. We thus derive the plane wave front BN, reflected from AB, as the envelope of arcs whose centres are A, K, K, K, and radii AG, KM, KM, KM respectively. The three pairs of lines, each pair lettered LK and KO, trace the disappearance of the incident wave front and the growth of the reflected wave front. It is seen that only when the incident wave has wholly disappeared is the reflected wave front completely formed, and on the point of quitting the surface which has produced it. The angle of reflection, NBA, is easily seen to be equal to that of incidence, CAB ; for AN is equal to CB, and AB is the common hypothenuse of the two right-angled triangles ACB and BNA. We accordingly have an agreement 76 SOUND between the result of Huyghens' principle and the experi- mental facts to be dealt with later. 46. Refraction at a Plane Surface. — Huyghens' method of treating this problem is illustrated by Fig. 26, taken also from his original work. Here AB represents the first face of a second medium in whicli the waves can advance, but with a speed bearing a constant ratio to that of their speed in the first medium. AC is the wave front at the instant when A is incident upon the face AB. If the first medium extended without Fig. 26. — Plane Refraction on Huyghens' Principle. interruption, this wave front would advance to GB. Owing, however, to the presence of the second medium, in which the advance of the wave is retarded, some other line, NB, is reached instead of GB. To obtain NB on Huyghens' principle, we take a number of points, H, H, H, on the wave front AC, and draw from them lines HKM, parallel to the direction of advance of the wave, cutting the refracting face at K, K, K, and reaching at M, M, M, the line BG. Thus, at the instant when the wave front reaches any point K, K becomes the origin of a spherical wavelet spreading in the second medium. The radius of this wavelet after time t will be v't, where v' is the speed of the waves in the second medium. Let v be the cor- 46 KINEMATICS 77 responding speed for the first medium. Then, to obtain the refracted wave front NB, we must (1) describe from A the arc ENS of radius AN, where AN : AG = i/ -.v; (2) describe from each point K an arc with radius ('(/'/*') times the cor- responding KM. Thus, we obtain as the new wave after refraction tlie line NB, which is the envelope of tlie wavelets just described. The three pairs of lines, LK, KO, trace the diminution of the incident wave front, and the growth of the refracted wave front, as the process of refrac- tion proceeds. Let us now seek the relation between the angles of incidence and refraction to which this principle leads. Obviously the angles in question are respectively CAB and NBA, in the two right-angled triangles whose common hypothenuse is AB. Thus, taking sines of the angles, we have by construction, sin CAB _CB/AB_'y sin NBA ~ NA/AB " v" or, sini v , , - — = ^ = ;^say, (1), sm r V where i and r denote the angle of incidence and refraction, and /x, the ratio of the velocities, is called the index of refraction for the pair of media in question. This con- sequence of Huyghens' principle, that the ratio of the sines is constant, is seen to be in accord with the well-known optical laws of refraction. The same laws are valid for acoustics also. But the above treatment of Huyghens gives further a physical significance to the index of refraction, showing it to be the ratio of the velocities in the two media. This, too, is in accord with the experimental facts both of light and sound. In the domain of optics the knowledge of this relation assisted in the rejection of Newton's emission theory and the establishment of the wave theory in its place, for Newton's emission theory required the false relation /u, = v'jv. 78 SOUND CHAP. 11 47. Reflection and Refraction at Curved Surfaces. — We have just seen that the ordinary laws of reflection and refraction may be derived from Huyghens' principle. But all the formulae of geometrical optics are based upon these laws, hence they may be derived from Huyghens' principle. This could be done directly or by the use of the laws of reflection and refraction just established. Thus all the relations in use in optics between the distances of object and image with spherical mirrors and lenses are valid for the analogous cases of the reflection and refraction of sound. The index of refraction from the first to the second of any two media is always the ratio of the speed of sound in the first to that in the second as already shown. As phenomena of this kind occupy a subordinate place in acoustics we must dismiss the subject now with this brief reference. 48. From a Small Opening Waves Spread: Dif- fraction. — We must now pass to the consideration of phenomena lying outside the domain of geometrical optics of which some are not explainable even by Huyghens' principle in its original form. First, consider the incidence of a wave motion upon a small round opening, i.e. one whose diameter is very small compared with the length of the waves falling on it. We can then regard this opening as practically a single point on the incident wave front. Hence, by Huyghens' principle, it is the origin of a spherical wavelet. Further, no other point on the wave front is free to send wavelets, for the opening in question is considered as a mere point. Hence, the new wave front, which is the envelope of wavelets originating on the old front, is, in this case, reduced to the spherical wavelet whose centre is tlie opening under discussion. And this is true whatever the form of the original wave front which encountered the small opening. Thus if it were a plane wave front, the waves would be advancing along parallel lines, but after passing the opening the wave front is spherical, and the directions of advance radial, and therefore 47-49 KINEMATICS 79 diverging. This is illustrated in Fig. 27, in which the incident waves are represented as plane. Partially opened doors and windows in a house or other building afford apertures which may be regarded as small in comparison with the wave length of many common sounds. Thus, the openings may be a few inches wide, and the wave length of the speaking voice of a man may be eight feet or more. Hence such sounds spread in all directions beyond those openings, as is well known, instead of proceeding in straight lines and giving sharp sound shadows as in the case of light through the same openings. Two such openings close together would be comparable ^iq 27 —Waves to two small point sources of waves, They Spbead from would, consequently, form an interfering ^^^^^ "^^' system like that discussed in article 43 and illustrated in Figure 23. In physical optics various phenomena due to the spreading of light waves when passing through narrow openings or round 'small obstacles are known and studied under the general term diffraction. In all such cases the openings in question bear about the same relation to the wave length of light as the openings for the analogous acoustical phenomena bear to the wave length of sound. 49. A Large Opening allows Rectilinear Propagation. — Now consider the passage of waves through an opening not small in comparison with the wave length, say of width equalling or exceeding it. We can here derive the main portion of the wave front beyond the opening, on Huyghens' principle, as the envelope of a number of wavelets. Thus, as shown in Fig. 28, we obtain from a plane wave incident upon the opening AB, a transmitted wave whose main central part, equal to that width, is plane also. But, in addition, we have feebler curved portions which appear as continuations of the main central part. These curved parts project beyond the dotted lines AC and BD, which limit the central plane part, or transmitted wave 80 SOUND CHAP. II proper, and are due to wavelets whose origins lie at A and B. In his discussion of the optical case, these curved portions were dismissed by Huyghens with the remark that the vibrations there were too feeble to produce light. This, however, falls short of a complete account of the phenomena. The fact is that, when we have rectilinear propagation, the wavelets interfere and mutually weaken each other outside the limits of the transmitted ■ beam. This application of Young's principle of interference in FiQ. 28. — Rectilinear Propagation oi' Waves. completion of Huyghens' principle was made by Fresnel. Hence, by what we may term the Huyghens -Fresnel principle, we are furnished with a satisfactory explanation of the rectilinear propagation of wave motion under certain circumstances. We thus see that the spreading, so usual in the case of sound, and the rectilinear propagation, so usual in the case of light, are simply matters of scale, and are both explicable on the wave theory. We are thus led to infer that under exceptional conditions we may have the rectilinear propagation of sound and the spreading of light after passing through a narrow slit. And both these phenomena have been observed, the phenomena due to the 49a KINEMATICS 81 spreading of light, as already noticed, being termed diffraction effects. Let us now examine a little more closely the conditions needed for the rectilinear propagation of a wave disturbance. Take a point P (Fig. 28) outside the track followed by the beam when rectilinear propagation occurs. Tlien, assuming that no resultant disturbance is felt at P, we see that, on the Fresnel extension of Huyghens' principle, this must be due to the destructive interference at P of all the wavelets whose origins lie along AB. In other words, although P is simultaneously affected by wavelets whose origins occupy the opening AB, yet the disturbances arriving at P have at every instant a zero resultant. Obviously, if AB is small compared to the wave length, the resultant could not be zero, for all the wavelets would reach P in practically the same phase. If, however, the opening AB is a number of wave lengths wide, theory and experiment alike show that a train of waves incident on the opening give a transmitted beam with rectilinear pro- pagation, the intensity off' the track of the beam being almost inappreciable. 49a. Failure of Rectilinear Propagation from Im- pulse. — Suppose now, instead of a succession of waves of constant amplitude falling on AB where the width is several wave lengths, we had only a single wave or a still shorter disturbance of an impulsive character. In that case we cannot have the simultaneotis arrival at P of wavelets proceeding from all points of AB, for that implies their starting from the various parts of AB at instants differiiig by several periods, which is now impossible, since, by the impulsive character of the source, AB is only disturbed for an instant or fraction of a period. Thus we should have, arriving successively at P, the wavelets that had started at the same, instant from various parts of AB. But, in consequence of this successive arrival, no destructive interference is possible. G 82 SOUND CHAP. 11 This interesting theoretical deduction receives a striking experimental confirmation in the instantaneous photographs by Professor C. V. Boys of bullets in flight. By kind permission of Newton and Co., the holders of the copyright, one of these photographs is reproduced in Fig. 29. The bullet passes through a dark box in front of a sensitive plate, and is photographed at the right instant by an Fig. 29. — Huyghens' Wavelets fkom Flying Bullet. electric spark let off by the completion of a circuit by the bullet itself. In the photograph here reproduced reflectors were in use to test the reflection of the air waves from the point and rear of the projectile. The Huyghens' wavelets, shown so beautifully in this plate, came as a surprise, being neither sought nor expected by the experimenters. 50. ExPT. 17. Huyghens' Principle illustrated hy Ripple Tank — The wave effects just dealt with can be illustrated very well by ripples on the surface of water in a flat- bottomed vessel. It is desirable to have, for this purpose, a tank with a bottom of plate-glass painted white and carefully levelled, and then covered with inky water to 50, 51 KINEMATICS 83 a depth of about half a centimetre. About 3 feet by 2 feet will be found a convenient size for the tank if it is desired to exhibit the phenomena to a class. To start a train of plane waves, a lath, about a foot long, may be put in and suddenly moved to and fro broadside. If, in the path of the advancing ripples, two obstacles are placed so as to leave only a small opening, then the waves passing through are seen to spread. If, on the other hand, the opening is wide enough, a nearly plane wave may be obtained, thus showing propagation which is approximately rectilinear. Again, if only an impulsive wave be sent, the Huyghens' wavelets may be seen, as shown by the bullet in flight. The use of a tank for these purposes was pointed out by Professor S. P. Thompson. 51. Fourier's Theorem illustrated. — We have already seen in articles 27 and 31 that the composition of simple harmonic vibrations of commensurate periods may result in periodic motions of various characters. We now pass to the examination of a very important theorem which may be applied to this subject. It was discovered by I'ourier, and is given in his renowned Analytical Theory of Heat (Paris, 1822). This theorem shows that the composition of commensurate simple harmonic motions of suitable amplitudes and phases is competent to produce a finite periodic motion of any form whatever. The theorem also shows how to determine the amplitudes and phases of the components requisite to produce any given resultant. In other words, it shows how to analyse any given periodic motion, however complicated, into the simple harmonic com- ponents of which it may be conceived to be built up. Thus, in Fig. 30, line (i) is the displacement diagram of a given' vibration of period t and amplitude le. The diagram consists of two straight lines at equal but opposite slopes. It accordingly denotes a uniform motion in one direction followed, without any break, by a precisely similar motion in the opposite direction. Let it be required to analyse this motion into its simple harmonic components. 84 SOUND Fourier's theorem, applied to this case, gives as the com- ponents an infinite series of simple harmonic motions whose (") (iii) (iv) O (v O' Original Axis of Abscissae X Fig. 30. — Illustration or Fourier's Theorem. periods are t, t/3, t/'5, t/7, . and whose amplitudes are respectively Hlf 111 I The phases are best seen by reference to the figure, the first three components being shown in lines (ii.), (iii.), 52 KINEMATICS 85 and (iv.). Line (v.) shows the result of compounding these three components. It will be seen that the resultant of the first three of the infinite series comes very close to the required vibration. It is, therefore, easy to believe, what at first seemed almost impossible, that an infinite number of simple harmonic motions may yield as their resultant a broken line, or, indeed, any vibration whatever. Many other illustrations might be given, but considerations of space force us to conclude with a brief outline of the mathematical treatment and its application to a few typical cases. 52. Analytical Treatment of Fourier's Theorem. — Fourier's theorem is susceptible of many forms of expression. For our purpose the following is convenient : — 2/=,^g + aj cos wi + ftjcos 2&)^+ . . ." ~^ + rt ,j cos 7ioot + . . ' , ^ , + &i sin mt + bo sin 2a}t + . . ^ ' + &,j sin na>t + . . . A, = ~[ ydt (2), 2 f^ a^=-\ y cos {n(ot)dt (3), "Jo Of'' h^=Z I y sin (ncot)dt (4) ; Jo where, as usual, cot = 27r. Here y denotes the displacements at time t, of a point executing a vibration of any type whatever of period t. The fundamental fact of Fourier's theorem, i.e. the possibility of expressing such a vibration in terms of simple harmonic components, is stated in equation (1). The values of the constants in the right-hand side of (1) are given in the three following equations. The rigid proof of this theorem is beyond the scope of this text-book. It is well, 86 SOUND cHAi-. II however, to notice that, granting the possibility of the ex- pansion given in equation (1), we may verify equations (2), (3), and (4) as follows : — Multiply (1) by dt and integrate from to t, and we obtain (2). Next, multiply (1) by cos(nat)dt and integrate from to T; and we obtain (3). Finally, multiply (1) by sm{7iQ)t)dt and integrate from to T, and we obtain (4). Equation (1) may be thrown into the form — 2/ = A(, 4- Aj cos (at — 6^) + A., cos {2Q)t — e.,) + + A„ cos (ncot — 6„) + . . or, still more compactly thus, 2/ = --^0 + ^ ^^« ^°^ 'y'^"^^ ~~ «») r(5). These forms correspond with (1) when the new constants fulfil the conditions — A,: = a,^ + b,^ (6), and tan e,j = &„/«„ (7). Thus, the constants of (5) are expressible in terms of those given in (2), (3), and (4). 53. Analytical Illustration of Fourier's Theorem. — Let us now illustrate the working of this theorem by its application to the case shown in Fig. 30. The values of ^ as a function of t are given by the ordinates in line (i.) of the figure, the middle point having co-ordinates ( — , 2k From this we have to determine the right side of equation (1) of art. 52, which expresses x/ analytically as the sum of sines or cosines, or both. It will now be convenient to take as our axes of abscissre O'X' in lines (i.) and (v.) instead of OX throughout as in the former references to this figure. Then, on turning to equations (2), (3), and (4), we see that it is necessary to express y in terms of t ready for insertion 53 KINEMATICS 87 under the sign of integration. It is obvious from line (i.) in the figure that it is desirable to take two equations for y, one for the first, or ascending half of the line, and another for the second or descending half. These are respectively y = AlaJT, from t=Q to it = t/2 n and y = 4k(T - O/t, from i! = ^ to ^ = t J ^^^• Hence each of the integrals in (2), (3), and (4) splits into two. We accordi^igly obtain , , 4/rf , ( U(t - 1) «re = Z I I — cos ncotdt + ^ cos ncotdt ■^ Jr/2 ■^ =J^.[(-ir-i] (10). 'il Cam . ^ Cu(T-t) . ) ?{fl sin nmtdt + sin nwtdt f = (11). Putting now for n in equation (1 0) the values 1, 2, 3, 4, etc., we see that for all even values of n the corresponding «'s become zero. The odd values of n give to the corre- sponding as the following values : — 1 — , ~, — o, etch (12). ttHI' 32' 52' J ^ ^ Hence inserting these values in (1) we have finally 8k( cos cot cos 3&)< cos Scot ] ^_,^ ^=^-^|-L^+~3^^+^^+"j ^''^- And from the first few terms of this expansion shown in (13) the curves of figure 30 were plotted. To test this 88 SOUND cHAi-. II equation, give to t iu succession the values 0, t/4, t/2, and T, and we find the corresponding vahies of y to be 0, Ic, 2k, and 0, as should be the case. When working out these values of y it should be remembered that the infinite series involved sums up as follows : — l+^, + i,+ ,Vad.inf.= -^ (14). This summation is given in Eiemann's Partielle Differential- gleichungen and elsewhere. 54. Partial Fourier Series. — When it is known that the function to be expanded consists of sine terms only, or of cosine terms only, and extends over only half a wave length, other forms of the Fourier expansion will be found convenient. Thus, if w^ and «'., are functions of x extend- ing over a half wave length, and involving respectively cosines only and sines only, we may write the expansions as follows : — Wi = — + «! cos fe + ... + ff„ COS inkx + . . . (15). where 2 r'" fl^ = — p iv-^ COS mkxdx (16): and W2 = bi sin Jcx + . . . + &,n sin m/.'sj + ..(17), where 2 r"'' 5,„ = —r- Wo sin mkxdx V2J0 ' (18). In these equations X and k are connected by the relation kX=2iT (19). It must also be noted that X/2 > « > (20). These equations are susceptible of verification in the same manner as equations (l)-(4). Thus, multiply each side of (15) by cos mkxdx, and integrate between the limits and x/2. Then all the terms on the right side dis- 54, 55 KINEMATICS 89 appear except that involving a„„ and the sole reraainiug term becomes a^^^XJA, thus reproducing equation (16). Similarly if we multiply each side of (17) by smmkxdx and integrate between the limits and X/2, we verify (18), all the terms on the right side vanishing except the m*'\ It should be noted that the vanishing of eill the terms but one on the right side is due to the fact that, except in the case reserved, we have after multiplying either a cosine, or products of cosines or of sines of different angles. These products will transform respectively into the sum and differ- ence of cosines. But on integrating these cosines we obtain sines, which are zero for each of the limits of integration. It is also to be carefully noticed that the values of ff,^ and &„j given in equations (16) and (18) apply only to those partial series, and ^ust not be interchanged with those in equations (3) and (4) for «„ and h^ which apply to the full series. For, in the case of the full series, we have, on multiplying, products of sines and cosines. These transform to sums and differences of sines which accordingly integrate to cosines. But these cosines assume the values + 1 for the limits and ir respectively, and so would not vanish for the integration over the half period which is employed for the partial series. It will be seen that both for the full and the partial series the value of the coefficients of the integrals which express the a's and J's are : (2-=- the extent over which the integration is to be effected). The application of one of these partial series will be found in the treatment of vibrating membranes (see next art. and Chap. V.) and struck strings (Chapters IV. and VIII.). 55. Extension of Fourier's Theorem to Two Vari- ables. — Fourier's theorem may be extended to represent an arbitrary function of two variables. For example, it may be used to represent the displacement of a surface at a given instant. For the sake of simplicity and its immediate 90 SOUND CHAP. II application to the vibrations of membranes (see Chap. V.) let us take the following concrete case : — Let the surface be a rectangle in the plane of xy and extending from the co-ordinate axes to x-=a and y = 'b. Let its displacements be represented by a sine series simply. Then, beginning first with the displacement as a function of X only, we may write ^ J{x) = S A^ sin m=l « '4=". m-n-x 2 r" , . mita-, ,,-... = i sm .- /(a) sm da (-l). In the integral a new variable, a, is written for the sake of distinction. It is evident that ssj^ variable whatever may be written in this expression for the constant A^, since in the evaluation of the definite integral the variable finally disappears. Let us now introduce the co-ordinate y into the function, but suppose it, at first, to be constant, as denoted by the subscript y outside the bracket. We thus have „ , Jl'"" . mirx 2 r ., . . mtra A^>y))dz ^L-. k^ + ~"^ k' + z" where k must not be zero. That is, a damped vibration may be conceived as a superposition of an infinite series of undamped vibrations, whose frequencies vary continuously on both sides of that principal frequency defined by n, the amplitudes of these component vibrations, however, diminishing on each side as the central frequency is departed from. 60. Doppler's Principle. — In 1842 Doppler, considering the coloured light from the double stars, showed that a motion of approach between the source of waves and tlie recipient would cause an apparent increase of their fre- quency, and that a decrease of apparent frequency would occur with a motion of separation. In the corresponding acoustical phenomena, it is perhaps as well to consider at once the effect of a possible motion of the medium in which the waves occur as well as motions of both source and recipient. Thus, let v be the velocity of the waves through the medium, u^ that of the source, w that of the recipient, and w that of the medium itself; all being along the line 1 AnnaUn der Physilc, p. 356, June 1906, aiul 2,5, p. 650, 1885. 59, 60 KINEMATICS 97 joining source and recipient, and reckoned positive when in the same direction, namely, from the source towards the recipient. These are shown in Fig. 32. Source of frequency N v = velocity of sound through air Receiver 0« ^ ^ ^ , X > > > Hj 11} = velocity of wind u^ Apparent frequency K' Fig. 32. — Doppleb's Pkinciple. Now in the ordinary case with all at rest save the sound waves themselves, we have — v = NX whence we may write X = v/N (1), and ]V= ^jx (2). Hence, with source, medium, and recipient all in motion as above described, we have, by modification of (1) — Disturbed wave length along OX = velocity of separa- tion of wave fronts and source -r- frequency of their emission from source, or, \' = {v + w — u-^)lN (3). Again, by modification of (2), we obtain — Frequency of receipt of waves = velocity of approach of waves and recipient -^ disturbed wave length, or, N' = {v + w- u^)/\' (4). Thus (3) and (4) give finally — ^Y^=^t«^. (5). ' v + w — u^ It should be noted that the above apply only to motions of source, recipient, and winds along the line joining source and recipient. An oblique wind would introduce a change, even if its component in the above direction remained H 98 SOUND Mg = w; = 0, but a cross unaltered. For example, if m wind blew at right angles to OX, and of speed exceeding v, no waves from the source could ever reach the recipient however near. This may be easily seen by the method of the next article. When, however, the motions of medium, source, and recipient are all along OX, we have to note that the apparent rise of pitch due to a motion in one direction is different from the apparent fall due to a reversed motion of the same value. Of course, for u^ = ii^, N' = N for any value of w. A few typical illustrations are given in Table III. Table III. — Doppler's Principle Velocities of Apparent Change in Pitch. Source 1(1. Recipient. 1(2. Wind Ratio N'/N. Musical Interval. vjn - vjn -v/n ) n n-1 For « = 2, note raised an octave -vjn {n+\)ln For n = 2, note raised a fiftli -vjn v/n vjn j «/(» + !) For « = 2, note lowered a fifth v/n («-!)/» For « = 2, note lowered an octave vjn -vjn -vjn' vjn' n{n' + l) m'(m- 1) n(n' -1) n'{n + l) If «= 62 = h', the change from upper to lower line involves a fall of a diatonic semitone The last case dealt with in the table corresponds to the change from a mutual approach of source and recipient to 61 KINEMATICS 99 a mutual recession, the speed of each being about 12 miles per hour. Thus, if one cyclist riding at that speed meets another riding in the opposite direction at the same speed ringing his bell, at the instant of passing the pitch of the bell would appear to fall a diatonic semitone. Of course, at much lower speeds the effect of Doppler's principle is noticeable, especially in this case of double approach followed by double recession, a fourfold effect being involved. "Doppler's principle has been experimentally verified by Buij's Ballot and Scott Eussell, who 'examined the alterations of pitch of musical instruments carried on locomotives." 61. Oblique Propagation of Sound in a Wind. — Let a region be imagined, in all parts of which the wind is horizontal and of speed m. Let a plane wave front be inclined 6 to the horizontal, and let the direction of pro- pagation of these waves of sound be inclined to the vertical and in the same vertical plane as the wind. Then, on examination of this case, we shall find that (f> is usually different from 0. Let AB in Fig. 3 3 represent the wave front at a certain instant, and let CD represent it after the lapse of a short time denoted by t. Then the Huyghens' wavelet, whose origin is A, may be conceived as radiating from A in every direction with the speed of sound v, compounded with a horizontal velocity u. Hence, the wavelet from A after any time is a circle whose radius is vt, but whose centre is transferred a distance ut horizontally in the direction of the wind. Thus, lay off horizontally AA' = ut, then from A' describe with radius vt the arc ECF, and we have the wavelet required. Similarly, we obtain the wavelet originating at B and at any other intermediate points along AB. The new wave front is the envelope CD of these wavelets, and is obviously parallel to AB. Also the direc- tion of propagation of the wave is AC, making with the vertical the angle VAC = cp ; whereas A'C makes the 100 SOUND angle with the vertical, since it is perpendicular to the wave front which is inclined to the horizontal. From C let fall ON perpendicular to the horizontal line AN. Then, for the relation between ^ and 0, we have by construction — NA tan NCA = —- ■■ NA' - + A'A ON ^ CA' cos NCA" tan = tan + -s&c0 (1). Hence the ray, instead of naaking the angle with the vertical, as it would do if normal to the wave front, makes Fig. 33. — Drift of Sound Rats in a Wind. the angle j>, which generally differs from whenever there is a wind in the region in question. The exception obviously occurs when the wave front ceases to be oblique. Thus for horizontal wind and vertical wave fronts, we have 6'= 90° and j> = 0. 62. Refraction of Sound by Wind.^ — Consider now two wind zones divided by a horizontal plane. Let the wind in the lower zone be everywhere in the same hori- zontal direction and of speed u^, and in the upper zone in the same direction but of speed w^. Let a plane wave front inclined ^^ to the horizontal while in the lower zone assume the inclination 0^ after passing into the upper zone, the directions of propagation being throughout in the same ' See Phil. Mag'., January 1901. 62 KINEMATICS 101 vertical plane as the wind. Let us determine the relation between 0^ and d^^. In Fig. 34 let AC represent the wave front incident at A upon AB, the plane of separation of the two zones. Draw CB' at right angles to AC and lay off B'B, making B'B/CB' = ujv, where v is the speed of sound. Then by art. 61, CB is the direction of propagation in the lower zone (see lines BD', D'D, and BD in Fig. 33). If t be the time occupied from C to B, we have CB' = 'i;^ and B'B = uJ. To construct the new wave front in the upper zone, we may consider A as the origin of a wavelet as in the Y Fia. 34.— Refraction of Sound by Abrupt Change of Wind Speed. first case. This wavelet is obviously a spherical one of radius A'Q = vt, described about a centre A', distant hori- zontally u t from A. Then from B draw BQ tangential to the arc representing this sphere, then BQ denotes the refracted wave front required. To obtain the law of refraction, we have by the figure, , BA' B'A-AA' + B'B_B^ u^t-u,t cosec QBA = ^ = - ^^^ ^^ , or. cosec 0j = cosec 6^ — (2). This law of refraction should be contrasted with that obtained for different conditions in art. 46. We there found : ratio of sines = constant, lilce the ordinary optical 102 SOUND CHAP, n law. Whereas here we find : ditference of cosecants for wave fronts = constant. ISTow, by equation (1), or from the figure of the present article in which QN is drawn perpen- dicular to AB, thus making the angle NQA' = 0^, 71 tan W), = tan 6,+— sec 6^ (3). This completes the solution, by giving the direction of propagation in terms of the inclination of the wave front. 63. Refraction through any Number of Wind Zones. — Let us now consider any number (?i + 1) of horizontal zones, let the wind be horizontal and in the same direction in all the zones, but differ in speed in the various zones, being v^ in the lowest, u^, u . . . and u^ in the others. Further, let the angles which the wave fronts make with the horizontal, and the angles which the rays make with the vertical, be respectively denoted by 6 and <^ with corre- sponding subscripts for the different zones. Finally, let the rays be in the same vertical plane as the winds. Then from equation (2) we have cosec 6^ = cosec ^^ — (u^ — u^^jv, cosec 6^ = cosec 0^ — (?(„ — u^jv, cosec e„ = cosec 6'„.i - {n „ - u.,^.i)/v. Hence, on addition, we obtain cosec ^„ = cosec 0^ - ^''" ~ "° (4). Also by (1) or (3) we have, for the final direction of propa- gation, u tan <^,^ = tan ^„ H sec ^„ (5). We thus see that the final inclination of wave front and direction of propagation are each independent of the constants characterising the intermediate zones, though, of course, the path of the ray will depend on these constants, and also upon the thickness of each zone. It should be 63, 64 KINEMATICS 103 noted, further, that since a cosecant cannot have a value between + 1 and — 1, if any of the zones required such a value the series must cease there, equations (4) and (5) not holding for the higher zones. But this leads us to inquire what has become of the waves which could not penetrate the higher zone. If they cannot be refracted, are they reflected ? 64. Total Reflection of Sound by Wind.— Although the cosecant law for the wave front obtaining here differs from the ordinary optical law of refraction, we still have, as in optics, the possibility of total reflection. And in this connection the distinction between wave front and direction of propagation is very striking. Thus, for the wave front, if the angle of refraction 6,, is put 7r/2, we have from equation (4) the critical case expressed by cosec^„-'^^^^°=l (6). Hence, any pair of values of 6^ and (n„ — u^), which makes the left side of (6) numerically less than unity, affords an example of total reflection, the last zone not being entered by the beam, since no real angle 6^ can be found whose cosecant is less than unity. Suppose now that we have a series of wind zones in which the wind speed increases as we ascend. Then, provided the initial inclination of the wave front were finite, it is clear that, at some point, we must have total reflection. But if, on the other hand, the wave front were initially horizontal, we should then have 0g = 0. And, by equation (4), all the ^'s would be zero also. That is, we should have no refraction of the wave front, and consequently no total reflection anywhere. Whereas the ray, initially vertical, deviates as we ascend, without limit from the vertical. This is shown by the fact that equation (5) now reduces to tan (/)„ = ujv 0)- 1 Oi SOUND CHAP, n So that, in this case, we have zero refraction of the wave front associated with unlimited refraction of the rays, total reflection being impossible. On consideration of the case by Huyghens' principle, it is seen that, when a zone cannot be penetrated and total reflection occurs, then such reflection follows the ordinary optical law — " angle of reflection equals angle of incidence." 65. Path of Sound Rays where Wind varies as Height. — Let the wind be everywhere horizontal and in the same vertical plane, but let its speed u vary continuously from one level to another according to the equation iijv = c + ay (8), where v is the speed of sound, c and « constants, and y is measured vertically upwards. The sound rays are supposed to be in the same vertical plane as the wind, and the x co-ordinate of the ray will be taken horizontally to leeward. Let us now determine the inclinations of the wave front and the rays at any point, also the path of the rays. Then we need to express 6, - 0/2) and ( - 0/3, - of 3, /g) the resultant being («, b, c). Then we have by addition /i - of 2 - 0/3 = « ) -<^A+f2-[ (6). and /3(1 + aXl - 2a) = c{l - a) + (a + by) 70 KINEMATICS 111 To produce the simple elongation («, 0, 0) by three such strains, we have merely to put & = c = in (6) and we obtain /i(l+cr)(l-25 s=o strain in which the £ prefixed denotes " limit of," and the subscript '' s = " means the strain is vanishingly small. We shall now proceed to discuss the various simpler kinds of elasticity and the relations between them. We shall confine the treatment throughout to homogeneous isotropic substances, and, except in the case of gases, it will always be assumed that the substance is maintained at the same temperature throughout its mass and during the con- tinuance of the strain. A homogeneous substance is one whose properties are the same at every point. An isotropic substance is one whose properties are the same in all directions at any point. We saw in the last chapter (art. 69) that any small strain whatever could be resolved into a uniform dilatation and two shears. Hence the corresponding elasticities are specially important. For if known they give the key to the behaviour of the substance under any stresses that may be applied. They will be accordingly taken first. 73. Volume Elasticity. — For the elasticity of volume the stress is a uniform normal pressure over the whole exterior surface, and is measured by force per unit area. The strain is the fractional diminution of volume, that is the diminution of volume per unit original volume. Hence the stress may be measured in lbs. weight per square inch or dynes per square centimetre. Whereas the strain is of no dimensions, for being the ratio of like physical quantities, its value is a pure number. Hence the dimensions of volume elasticity (or bulk modulus as it is sometimes called) 118 SOUND CHAP. Ill are those of force per unit area simply. It is thus ex- pressible in the same units as the stress. Let k denote the volume elasticity of a solid substance, then we may express its value by the following equation : — where V is the original volume of the body, i^ is the pressure to which it is subjected, and v the consequent diminution of volume. The second expression on the right, in which 3rf is written for vjV, is a reversion to the notation used in Chap. II. for fractional change of volume in the case of a uniform dilatation of amount d per unit length. It will serve equally well for a diminution when, as in the present case, the stress is a compressive one. The reciprocal of an elasticity is sometimes referred to, a name and a symbol being adopted for it. Thus the reciprocal of the volume elasticity is called the com- pressibility, and is often denoted by ^. We thus have — ,3=i=!A:=i^ (2). k p p In dealing with liquids the compressibility is often spoken of rather than the volume elasticity. 74. Elasticity of Gases. — Turning now to the case of gases we have to distinguish between the two chief methods of estimating the volume diminution. We may keep the temperature of the gas constant during compression, or, we may prevent the escape of heat from it and thus cause its temperature to rise during compression. These two methods of compression are called isothermal and adiahatic respectively. We have consequently two values for the volume elasticity of a gas. These are called the isothermal and adiahatic elasticities, and will be here denoted by Uf and E^^ respectively. The subscripts signify that in the one case the temperature denoted by t is constant, and in the 7i ELASTICITY 119 other that the qucantity of heat denoted by h is constant. Now for a gas which obeys Boyle's law we may obtain a convenient expression for Ef as follows. Let P and V be its original pressure and volume, and P+^ and V—v its pressure and volume at the same temperature after the infinitesimal compression. Then, by definition of elasticity, we have Et=pVlo (3), and by Boyle's law {P + p){V-v) = PV. But, remembering that p and v are indefinitely small quantities whose product is accordingly negligible, this second equation becomes pV-Pv=0 or, p=pVlv. Hence on comparison with (3) we have Et = P (4), or, in words, the isothermal elasticity of a gas obeying Boyle's law is equal to its pressure. The adiabatic elasticity is greater than this. In text- books on heat it is shown that the ratio of these two elasticities is equal to that of the two specific heats. Thus, we may write !<:,„% = SjS,. = y, say (5), where Sp and S„ denote the specific heats at constant pressure and at constant volume respectively. Hence the adiabatic elasticity is given by A = yP (6), 7 being the ratio of the two specific heats for the gas in question. The determination of 7 by acoustical methods will be dealt with later. We may note here that its values are approximately 1§ for a monatomic gas, 1"41 for a 120 SOUND CHAP, in diatomic gas, and 1'26 for a triatomic gas. The three cases are illustrated by argOD, hydrogen, and carbon dioxide respectively. It should be noted that gases, though limited to volume elasticity, exhibit this single kind without any imperfection. The narrowness of the limits assigned to the stresses and strains used for the value of an elasticity are here necessary simply to give definiteness to the result (see equations (4) and (6)), and not to prevent its becoming imperfect as in the case of solids. 75. Rigidity. — Eigidity or stiffness is a kind of elasticity which obviously applies only to solids. The strain which we are here concerned with is that known as a sim_ple shear, and has been already discussed (arts. 68-70). The corresponding stress we may call a shearing stress, and it may be described as follows : — Take a cube in the substance in its unstrained state and subject it to uniform normal tension over one pair of opposite faces, to an equal normal pressure over another pair of opposite faces, leaving the third pair of faces unacted upon. The shearing stress is measured by force per unit area. We may now define rigidity thus. The rigidity of a substance is the quotient shearing stress divided by the amount of the shear thereby produced, the value of the quotient being taken in the limit when the shear is indefinitely small. Hence, writing n for rigidity, p for the force per unit area in the tensions and pressures, and ■)(^ for the amount of the shear, we have The above view of the shearing stress and rigidity corresponds to the view we first took of a simple shear, namely, an elongation one way accompanied by an equal contraction at right angles to it. But we saw in the last chapter (art. 70) that another view of a shear could be taken, namely, the relative sliding, without distortion, of parallel planes, their direction for a small shear being at 75 ELASTICITY 121 angles of 45" to those of the elongation and contraction. And, corresponding to this second aspect of the strain, we have a second aspect of the shearing stress, namely, two equal but opposite couples in the same plane, which is the plane of the tensions and compressions on the other view of the stress. Fui^ther, the forces composing these couples will act along the directions of the sliding of the undistorted planes, and are ^\ accordingly at angles of 45° with the tensions and compressions which constituted the first aspect of the stress. They are, therefore, tangential forces applied parallel to the surfaces over which they C act instead of normally as in the ^- f-^^l^^r " first view of the matter. This aspect of the stress is illustrated in Fig. 38. The tangential forces are represented by the arrows, and are each of the magnitude p per unit area. Those along AB and CD obviously tend to produce the relative motion which carries CD to C'D'. But they would also tend to rotate the body clockwise since they constitute a couple. This tendency is counteracted by the forces along AD and CB which form an equal and opposite couple. And this second couple is seen to be able to produce the sliding parallel to AD and CB. We have thus shown how the second view of shearing stress fits the corresponding view of shear. We have now to trace the quantitative relation between the two" views of the stress. To change this set of four tangential forces to the pair of normal tensions and contractions, combine them as follows. Suppose all the edges to be of unit length and take together the forces along AD and CD, these yield a resultant along BD of value ps/2, but spread over the diagonal plane AC of area n/2. Similarly the forces along 122 SOUND CHAP, iir AB and CB give an equal and opposite force along BD, but acting on the plane through AC of area \/2. Thus the forces give a tension along BD of value p per unit area. Again, combining the forces along AB and AD to one resultant and the others to another resultant, we find a pressure exerted along AC, in which the force of magnitude 2}\/'2 is spread over the area v2 of the plane through BD. Thus the forces give a pressure of value j5 per unit area along AC. And, when the shear is inhnitesimal, these directions are obviously at angles of 45° with those of the tangential forces. It should be particularly noted that whereas in the two views of the strain the amount of the shear is double the elongation, in the two views of the stress the normal tensions and pressures (per unit area) have precisely the same values as the tangential ones by which they may be replaced. Thus in equation (7) we may supj)ose the ^> to refer to either aspect of the stress at pleasure, and yet the validity of the equation remains. 76. Young's Modulus. — Here the stress is a uniform tension parallel to a given axis. This third kind of elasticity, though apparently simpler than either of the foregoing, is in reality more complicated. In the first place, it involves change of both shape and volume instead of a change of only one of these. Secondly, although like the other elasticities, it is represented by a fraction whose numerator is a stress, unlike them, the denominator of the fraction takes cognisance of only one aspect of the corre- sponding strain, namely, the fractional elongation produced. The lateral contraction, which is also caused by the stress in question, is ignored in estimating the value of Young's modulus, although this contraction bears no constant relation to the elongation. "Writing q for Young's modulus, p for the tension in force per unit area of cross section, and / for the fractional elongation, we have S=p/f (8). 76-78 ELASTICITY 1-23 77. Elasticity of Simple Elongation. — Suppose now lateral forces are applied so that no change in lateral dimensions occur in spite of the longitudinal tension. Let the tension be p as before, and let the fractional elongation be a. Then denoting the corresponding elasticity by /, we have j=p/a (9). It is seen that in the case of this elongational elasticity the denominator of the fraction fully specifies the strain, while the numerator of the fraction denotes only one part of the stress, its longitudinal tension, the lateral forces being unspecified and ignored. It is therefore the converse of Young's modulus. 78. Relation between the Elastic Constants. — It is now desirable to deduce certain useful relations between the stresses, strains, and the various elastic constants already dealt with and denoted by p, f, a, k, n, q, and /. These are respectively tension or pressure, fractional elonga- tion, Poisson's ratio, volume elasticity, rigidity. Young's modulus, and elongational elasticity. By the definitions and equations of this chapter, and the composition and resolution of strains in the preceding one, we have at once a number of important relations. Thus, by article 73 and article 69, equation (8), we have j,^P=^-P (10). ;;,/ y/(l-2o-) ^ ^ Again by article 75 and article 69, equation (9), we obtain P 'P P n = (11). X 2. 2/(1+ cr) Let us next rewrite here for convenience equation (8) of article 76. q=plf (12). Finally, article 77 and equation (7) of article 69 yield 124 SOUND CHAP. Ill ■' a /(l+cr)(l-2<7) ^ '■ By cross multiplication of any pair of these equations p and / are eliminated and a relation obtained between the two elasticities concerned and Poisson's ratio. By combining the equations thus obtained it is obviously possible to express any one of the five quantities k, n, q, a, and / in terms of any one of the six possible pairs of the remaining four quantities. Some of the more important of these relations are given in Table V., and will be found useful for reference. The first column is derived immediately from equations (10) to (13), and the other columns by further elimination as indicated above. The last two lines of the Table give expressions for the elongation and lateral con- traction under a' tension p without any forces acting laterally. The results in the column headed h and n are the most important. 78 ELASTICITY 125 b •^ 'b" 1 + i-H 1 CO o "b" 1 :?- CO b + rH b 1 CO b b 1 i-H 1^1 in is< «!- &1 CH Cm -^ CO JS ^ 1 % + 1 IH 1 &. 1 4^ r« oa CO CO_ Oi CO CO 1 S Cm O b 4- 1 g 1- -1- b 1 1 i-H + b b + e S i—t CN I-H I-H i-H 0) . c5' C<1 CM CM ICO T) i-j 13 S" CO S g «> ten tM &1 H C3 55 «^ Cm 1 ^ 1 1 -* :o "1 ai 1 tM '^ ^ 1 + 1 + + + 1 s Ol " CO H^ Ci^ __^ b tr. 1 ,_^ CM CH 1 *= b 1 13 J+ Cm b 1 T-H r-4 1 tM b| &< s 1- + b IPS' '.CM Cm "-^ 1 ri; g 5m b ■■^ -& ^ £3 -i m .5- • E» r3 . OJ . -32 s^ J §1 o rt 03 O o 3 ^o , Sift 1 <6 S a> >> o "be +3 o =« o a % 3 60 » o cj'S ■lit I rt O 'I ^-- ^0-3 126 SOUND CHAP. Ill 79. Experimental Determination of Elastic Constants. — We shall now describe briefly a few simple experimental methods of determining the elastic constants. The easiest to determine directly in the laboratory is Young's modulus. Eigidity is also fairly easy to determine experimentally though more complicated theory is needed. The direct determination of volume elasticity for a solid presents such difBculties that most values for it have been deduced from the relation between it, the rigidity, and Young's modulus as given in Table V. We shall accordingly describe the experimental determination of the other two elasticities, and suppose the volume elasticity to be deduced from them. The value of the elongational elasticity may then be inferred from the values of k and n or q and m. The same applies also to Poisson's ratio. The above remarks refer, of course, to solids ; for gases, the isothermal elasticity, being equal to the pressure, is found by reading the barometer and a pressure gauge. The adiabatic elasticity could then be calculated if the value of 7, the ratio of the specific heats, were known. It is more usual, however, to find the adiabatic elasticity directly by an acoustical method, and so obtain the value of 7 indirectly. These determinations will be dealt with later (Chap. X.). ExPT. 18. Young's Modulus by Tension of a TFire. — This experiment may bo performed with any one of various methods for the measurement of the elongation. The essentials, however are the same, namely, that a wire is loaded and the consequent lengthening observed. But the method used for noting the elongation governs the fineness with which it is possible to work and so dictates the total length of wire advisable. Suppose the simplest method of measurenrent to be adopted, the scale and vernier reading to say 1/20 of a miUinietre. Then it is desirable to have a vertical wire about 10 metres long and about a millimetre diameter using loads of 1, 2, 3, and 4 kilograms successively. Further, to guard against a possible error due to the yielding of the support of the upper end of the wire when loaded, 70, 80 ELASTICITY 127 the scale should be carried by another wire hanging from the same support, the vernier being on the wire under examina- tion. Again, lest the temperature should change during the experiment it is advisable to have the two wires of the same material and diameter to obviate any unequal expansion or contraction. These arrangements being made, the experi- ment is performed by suitably loading the wire, reading carefully the scale and vernier before and after each loading. To guard against a violation of Hooke's law by exceeding the elastic limits, the readings should also be taken after removal of the weights one by one. The loads and elonga- tions may be plotted on squared paper and a curve drawn showing the relation between them. This should be a straight line through the origin. Suppose it makes an angle 9 with the axis on which the loads are plotted, then cot denotes the quotient of load divided by elongation. But Young's modulus is tension per unit area divided by fractional elongation. Thus writing r for the radius of the wire and I for its length, we have load elongation I q = — r -■ 7 = —9 cot (9. The diameter of the wire should be measured carefully in several places by a micrometer gauge, the length may easily be obtained with abundant accuracy. To obtain the value of q in c. g. s. units (dynes per square cm.), the load must be of course expressed in dynes by multiplying the weight in grammes by 9S1, the value of the acceleration due to gravity, also all linear measures must be expressed in centimetres. 80. ExPT. 19. Finer MHhods for Young's Modulus. — Instead of using the scale and vernier we may use various finer methods to measure the elongatior. in the determination of Young's modulus. Thus the spirit level and micrometer gauge are used in an apparatus devised by Mr. G. F. C. Searle. The method of optical interference was adopted by Mr. Gr. A. Shakespear (1899). The electrical micrometer has been used for this purpose by Dr. P. E. Shaw. In this arrangement the position in which the tip of the micrometer screw touches a plate carried by the wire is known by its completing an electric circuit containing a cell and a telephone. Finally, the familiar apparatus called the optical lever may 128 SOUND CHAP. Ill be used for the purpose in question. With these finer methods the lengths of the wires operated upon may be proportionately reduced without loss of accuracy. 81. Theory of Torsion of a Cylinder. — As a pre- liminary to the determination of the rigidity of a substance by experiments on a cylinder of the material, we need the following theoretical investigations : — Let a right cylinder of length I and radius a be held at one end while the other end is twisted through an angle 6 radians, and there held by the application of a couple G dyne- centimetres. It is required to obtain the relation between these quantities and the rigidity n of the substance which is assumed to be homogeneous and isotropic. Let a circular slice of thickness dl be bounded by planes at right angles to the axis of the cylinder. In this slice take a ring of radii r and r + dr where dr = dl. Lastly in this ring consider a cube of sides dr = dl. Then, when the relative twist of the ends of the cylinder is d, that of the planes bounding the slice will be 6dljl. Accordingly a face of the cube parallel to the base of the cylinder will be sheared relatively to the opposite face through the distance rOdljl. To obtain the amount of the shear we must divide this quantity by the distance dl between the faces in question. Thus the amount of the shear is given by X = rell (1). But the rigidity is defined by n =plx' where p is the tangential force per unit area. Hence we have Thus the actual force distributed over a face of the cube parallel to the bases of the cylinder will be {drY times the above expression for p. Further, the moment of this force about the axis of the cylinder will be r times the force itself. Or, for the whole ring, the moment of the force will be {2'n-r)dr, r times p. 81, 82 ELASTICITY" 129 Thus dG=-^-^rHr (3). T Hence the moment for the slice of thickness dl and radius a is obtained from (3) by integrating from to a. Turther, this moment for the slice is the moment for the entire cylinder, for each slice into which we may divide the cylinder experiences the same twisting moment. Hence ^ 217116 3 , IT e . G = --y- r^dr = -n-a"^ (4). Thus Gozd, so writing C for GjO we have C=G/e = 7rna*/2l (5) as the expression for couple per radian of twist. Accord- ingly, to determine w we have simply to find C by some experimental method for a cylinder and measure its length and radius. Some dif&culty is occasionally felt by students as to the twisting of a cylinder making no change in the volume. Some imagine that a twist involves a shortening. But this is contrary to Hooka's law. For if stress and strain are proportional, and a twist one way shortened the cylinder, a twist in the opposite direction would lengthen it. This, however, is obviously contrary to our supposition that the substance is isotropic. Probably any expectation that a twist would involve a shortening is traceable to our ex- perience of ropes and other fibrous twisted cords. 82. ExPT. 20. Static Determination of Rigidity. — To de- termine the rigidity of a substance by holding a cylinder of it twisted, the apparatus shown in Fig. 39 designed by the author is suitable. The wire in a vertical position is held at the top and twisted at the bottom by the application of a couple due to the silk threads which are fastened to the wheel. These threads pass over smaller wheels at the side, and terminate at the ends of the horizontal cross bar which is loaded by weights placed in the scale pan at the centre. The angle of twist is read in degrees by observing each end K 130 SOUND CHAP, in of the needle over the graduated circle which is provided with an anti-parallax mirror. Thus, suppose a mass of m grams in the pan produces a twist of cP, and that the radius of the tread of the pulley on the wire is r cm., the length and radius of the wire being I cm. and a cm. respectively. Fig. 39. — Eiqiditt Apparatus. Then the tension on each thread is — ^ dynes and the couple 2r times this, i.e. mgr-Gt dyne -centimetres. Further, fF = (7rrf/180) radians, or 6* = Tri^^/lSO. Hence from equation (4) or (5) we obtain for the rigidity of the material under test, 360(7;' ' »' d^ (6). 83 ELASTICITY 131 The value of the last factor mjd!^ may be conveniently- obtained from a curve plotted on squared paper from a number of loads and corresponding twists. In the apparatus shown in Fig. 39 the upper end of the wire is held by a crosshead which may be adjusted anywhere on the upright bars on which it slides. Thus the lengths of wire under test may be varied at will. When loads are applied at the scale pan, and so a twisting of the wire produced at the lower end, it is evident that an equal couple is experienced at the upper end of the wire also. But, if the arrangements to withstand this couple are insufficient the crosshead will yield appreciably, and the twist of the wire will be less than that of its lower end. Hence to test this point the two plummets shown are provided, and their readings on the circle should be observed before and after the loads are in the pan, corrections being made if necessary. 83. Other Methods for Elastic Constants. — We have only noticed so far very simple methods for the determina- tion of Young's modules and rigidity. Many others are available and practicable. Thus Young's modules may be determined by the static or dynamic flexure of a bar, since the bending of a bar stretches its convex side and compresses the other, and thus involves the elasticity in question. Again, rigidity may be determined by the torsional oscilla- tions of a mass suspended by a wire. The theories of bending and torsional oscillations will be developed in Chap. IV., but the application of them to the determinations under notice may be left to the student. An ingenious application of the method of oscillations has been made by Mr. G. V. C. Searle. He thus obtains with the same apparatus the rigidity of a wire about a foot long by torsional oscillations, and the Young's modulus by flexural oscillations. (See Phil. Mag., Feb. 1900.) A still simpler method introduced by the writer furnishes the Young's modulus for a wire or glass fibre by static bending and the rigidity by torsional oscillations. (See examples in Chap. II. No. 10.) i^ ^ \j^ ^ .^ % ^ CHAPTEE IV \ DYNAMICAL BASIS 84. Examples of Simple Harmonic Motion. — hi Chapter II. we saw that a simple harmonic motion of period T involves the relation y = — oj^y where (ot = 2-jr and y and y denote respectively the displacement and acceleration of the vibrating point. Now, by Newton's law of motion, y =flm, where / is the force impressed upon a body of mass m to produce in it an acceleration y. Hence to cause a body of mass m to execute a S.H.M. of period t we need that these two expressions for the acceleration should be equal. That is, we must have the body so conditioned that it is acted upon by a force expressed by /= -co-my . (1). This force may be due to elasticity, gravity, magnetic action, etc. Simple Pendulum. — Take first an example in which the restoring force is due to gravity. Consider a simple pendulum of length I with bob of mass m. Then when the angular displacement is 6 radians the restoring force is — mg sin 6, or, for very small values of 6, it is — mgd nearly. But, the displacement y, being measured along the very slightly curved arc, we have 6 = yjl. Hence, on substitution of this value of 6, the restoring force, being the component along the arc of the weight of the bob, is given by f=-vigyll (2). 132 84, 85 _ DYNAMICAL BASIS 133 Thus, on reference to (1) we see that w-=g\l Conse- quently, since t= 27r/(u, we obtam T=2-wJ'lfg (3), the well-known expression for the period. Elastically-suspendcd Boh. — Take now the case of a body of mass m hanging by an elastic cord, say of india-rubber, whose mass is negligible. Let the length of the cord be I, the area of its cross section c, and its Young's modulus §. Then, if the restoring force / corresponds to an extension y, we have 2=--^,or/=_^V (4). cy I Further, when the displacement is y, i.e. when the cord is stretched y more than when at rest with the mass m hang- ing on it, the total tension on the cord iaf+mg, so that the force / is free to cause upward acceleration of the bob. Similarly, when the displacement is —y, or the bob is at a distance y above its position of hanging at rest, the total tension on the cord will be mg —f, so that the force — / is free to produce a downward acceleration of the bob. Hence, by equations (1) and (4) we obtain o)^ = qcjhn, whence T = 2'7rs/ Im I c[c (5). Now, we see from (4) that the restoring force per unit dis- placement is fjy = — qc/l = —s say. Adopting this abbre- viation, equation (5) then becomes T= 2-71 \fm/s (6). 85. Generalised Expression for Period. — -This important result may be expressed in words as follows : — The period of oscillation of a particle is 27r times the square root of the quotient inertia factor divided by spring factor. On further examination this is found to be true for a rigid body executing rotary vibrations. In the linear case considered above the inertia factor is the simple mass of the bob, and 134 SOUND CHAP. IV the spring factor is the force per unit displacement. But if angular vibrations were under consideration, the inertia factor would be the moment of inertia of the rotating body taken about its axis of oscillation, ,'&nd the spring factor would be the couple per unit of angular displacement. Ilass at Middle of String vibrating transversely. — Let a body of mass m be fixed at the middle of an elastic cord of negligible mass of length I and stretched by a force F. And let the transverse vibration executed be so small that the alterations of this stretching force are negligible in comparison with the force itself. Also, let / denote the restoring force called into play by the transverse displace- ment y of the bob or mass at the middle of the cord. Then, since / is the resultant of the two forces F,F due to the two parts of the string when drawn aside, we have by the parallelogram of forces /: F= 2y : 1/2, or, 4:F f= -^y= -^y say (')• Thus, comparing with equation (1) we have m^ = AFjlm = sjm say, and therefore T = 2TrJTm^ 4:F=27r s./mjs (8). 86. Energy of vibrating Particle. — "When a mass m is executing a S.H.il. represented hj y = a sin at, it has been shown that its velocity is given by y = &>« cos (9). Thus for 2/, = 0, y^ =^^^ =f^ ^^J (10)- Thus, the variable factor in the kinetic energy, v?hose dependence on n and k we are to determine, is expressed by Eliminating 8 between this equation and (6) in which it was defined, we obtain ,h=„ 4.''-^=/''-^-?V (13). n Wj n If, therefore, values of B be taken as ordinates, and a curve plotted having for abscissa3 ^tlie corresponding values of A, we see that for k constant we have curves symmetrical about A = 0, and with a maximum value of B at this place, i.e. for n=2>. This maximum value is given by 5„,,= l/4«;2 (14). 95. Thus, for any given value of Kj the kinetic energy of the forced vibration, when the particle is passing through the zero position, is a maximum when the period of the impressed force coincides with that of the vibration natural to the system if friction were absent. Moreover, the diminution of energy consequent upon the lack of this coincidence is the same for a given ratio of frequencies whether that of the impressed force is too great or too 95, 96 DYNAMICAL BASIS 145 small. This will be seen from the second form for A in (13). For its value is there shown to depend upon that of njp minus its reciprocal pjn. Hence a change of this ratio to its reciprocal simply changes the algebraic sign of A, leaving its numerical value and therefore its square unaltered. But this ratio njp measures the musical interval between the impressed force and that of the system without damping. Thus an equal diminution of kinetic energy follows from the impressed force being a given interval above or below that which gives the maximum energy. But this symmetrical relation of ratios does not apply (except approximately, within narrow limits) to differences of frequencies. In other words, though the curves for B are symmetrical when plotted with A as abscissiE, they would not be symmetrical if plotted with n as abscissas. Thus if p= 100, the maximum B follows' when n= 100, and the same diminution in B follows when m = 200 or 50, but not when ?i= 150 or 50. 96. Referring to equation (10), we see that for the Icinetie energy to be a maximum for a given value of k we must have S = 7r/2. ■ But by (2) and (4) that involves a lag of the displacement of the forced vibration of a quarter of a period behind the impressed force. The velocity of the vibrating point is, however, in phase with the impressed force. These facts are expressed by the following equations which are easily derived from (4), (6), and (9), putting S = 7r/2, f cos pt fs'mpt In these equations p is written instead of n, as they only hold for this particular value of n. We see from (14) that if K be allowed to vary, then B^ varies inversely as the square of k. Thus, the less the damJDing, the greater is the absolute value of the maximum kinetic energy. Further, from (12) we see that for any given value of L 146 SOUND A, the less k, is the greater is the absolute value of the kinetic energy corresponding to this particular lack of tuning. The variation of B and B^^ with A and « are illustrated by the ordinates of the curves in Fig. 39a, whose abscissas are the values of A. Each curve is for a specific value of k as shown. Fig. 39a. — Kinetic Energy op Forced Vibrations. 97. Sharpness of Resonance. — We may now consider, as a function of n and k, not the absolute value of the kinetic energy, but the ratio which it bears to its possible maximum with the given «, which value is only attained for A = 0. Denoting this ratio by h, we have from equa- tions (12) and (14) ^ = ^/^'« = A^TI? (16). Thus, as we should anticipate from (12), 6 reaches its maximum value, unity, for any value of k, when A = 0. But we see also from (16) that, for a given value of A, 6 increases with increase of k, whereas with B it was just the reverse. These points are illustrated by the curves of Kg. 39b, in which the ordinates represent the values of h and the abscissae those of A, separate curves being plotted for specific values of k. 97, 98 DYNAMICAL BASIS 147 If the impressed harmonic force proceeds from one sound, and the forced vibration constitutes another, the latter is often called a resonance. This term has, however, been .extended to cases where neither driver nor driven are sounds at all, but are very different primary forces and responsive motions. Further, the greater the dependence of the energy of the forced vibrations on exactitude of , frequencies between driver and driven, the sharper is the^^"^ resonance said to be. Let us now find a mathematical expression for the sharpness of resonance. Fig. 39b. — Sharpness op Resonance. We find from equation (16) that 1-h 1 A? -A" = ^;--^ = S sa.j (in where S denotes what we may call the sharpness of resonance, since it expresses the ratio of the fractional diminution of b to the square of A which causes it. Hence the sharpness of resonance thus estimated varies inversely as the square of the damping coefficient. In other words, the smaller the frictional damping, the more important does exact tuning become in order to obtain the greatest energy in the forced vibration. 98. This dependence of sharpness of resonance on the damping has been exhibited in a different and very in- structive way by Helmholtz. 148 SOUND CHAP. IV Eliminating the damping coefEici6nt, he obtains a relation between the resonance effect and the number of complete periods required for the energy of the vibration, when unforced, to diminish to a certain fraction of its initial amount. Thus from (1*7) and (13) we may write .1) (18). But, if the amplitude of the vibration when left to itself is affected by the factor 6 after x complete vibrations, we have = e-''(^') = e-2™^/j'^ where p is written for q, to which it is practically equal for ordinary values of k. Thus logu 0= — 2irKxlp, or *=-/-l0ge^ (19). Thus (18) in (19) gives ■-Jn_ASj I (20). \p n) that algebraic sign being taken for the root which will make x positive. By this method Helmholtz obtained the numerical results given in Table VI. To deduce these we must write in (20) h = -^-q, 6"' = j-ij, and njp equal to the ratio of the frequencies defining the musical interval re- quired, thus 9/8 for a .tone, 5/4 for a major third, and so forth. These being inserted the values of x may be calculated. DYNAMICAL BASIS 149 Table VI. — Sharpness of Eesonance after Helmholtz Musical interval (of mistuning of impressed force) corresponding to the reduction of the forced vibration to ■^, i.e. interval to make b = 1/10. Number of complete vibrations in whicli the intensity of tlie vibra- tions wlien unforced sinks to -^, i.e. the valup of a: to make 02=x/lO. 1. An eiglith of a tone 2. A quarter of a tone 3. A semitone .... 4. Three-quarters of a tone 5. A whole tone .... 6. 5/4 of a tone 7. A minor third (| tone) . 8. 7/4 of a tone 9. A major third (two whole tones) . 38-00 19-00 9-50 6-33 4-75 3-80 3-17 2-71 2-37 99. Amplitude of Forced Vibrations. — We have seen how the kinetic energy of a forced vibration varies -with the frequency of the impressed forces, and that it is a maximum for n=p. Let us nov^ inquire what value of n gives the forced vibration a maximum amplitude. We shall still sup- pose the impressed force to be given by /sin nt, and keeping / constant shall examine the change in the amplitude y^ consequent upon a change in n. Thus, referring to equa- tions (4) and (8) of article 93, we have / sin (7it — S) 7/ = ^ sin (nt - 8) = / -^ ^ , ,-rT~2 (^ !)• Thus, .) f/A' = (ri" -pj + iK-n'K-'^^"^ ' (22) expresses the variable factor of the amplitude depending on n. Differentiating this to n and equating to zero, we find that (Z''^'"" .2 ■A If is a maximum for v? =p\— 2k^ the maximum value itself being given by A„, in /M.= 2«? where cf =p' — «", as in (7«) of article 92. ^..-^ (23), (24), 150 SOUND CHAP. IV Another way of arriving at the same result is to throw equation (22) into the form f'jA' = (71^ - ir + 2>cy + 4«V (25). From this we may also derive the following conclusions. If a curve were plotted co-ordinating A/f and n^, this curve would be symmetrical about the ordinate n^=p'^— 2k^, and further, for a given value of n, Ajf decreases with increase of K. If, on the other hand, we consider the ratio of A to its maximum value A^ for the same k, then this ratio may increase with increases of k. For, from (22) and (24) we have \A„J {n^ - irf + -\:ic^n' {n? - -p-f + -ijcV ■' ^ " since k is always small compared with ]>. Thus, when the relation of w^ and f? is such that 4/cV in the denominator is negligible, we see that ^Mm = ,j^2_^2 (27), or the ratio iu question varies as k. Vp'-^i.if^ It should be noted that the frequency of the impressed _ force to make the amplitude a maximum is lower than that'^ natural to the system with friction ; while the frequency f of the impressed force to make the kinetic energy a maximum is above that natural to the system with friction, f/ and equals that if friction were absent. Moreover, the squares of these three frequencies form an arithmetical ^ progression whose common difference is proportional \mK the square of the damping coefficient. Thus, the vibrations of the system with friction being proportional to e""' sin qt, the force for maximum kinetic energy varies as sin ft, ■and that for maximum amplitude varies as sin lit, where /_22 = ^2^^2_;j2 (28). 100 DYNAMICAL BASIS 151 100. Nature of Complete Solution when n = q. — Let us now return to the complete solution of vibrations executed under the action of an impressed force. We see from equation (7) in art. 92 that it consists of the sum of two sine functions with different amplitudes and different phases.. In the general case the frequencies also are different and the composition is more complicated. So, for simplicity, we take first the case in which the frequencies are alike. This is obtained by putting n = q. In other words, this is the limiting case in which the " forcing " disappears. We may, therefore, compound these two vibrations by the methods of Chapter IT., articles 26 and 28. We accordingly write the solution in the form y=C'sm(qt-y) (1), where 6" = .4- + 2^ae-''' cos (8 + e) + «'<-'"'-^"' (2), , ^ y4 sin 8 — ae""' sin e ,„n and tan7=~ — —^ (o), A cos d + ae " cos e A being written for (/sin B)/2Kq (4). Hence, if k is very small, we may regard the motion, at any instant, as approximately simple harmonic motion, but the amplitude and phase angle are continually changing. Thus for t=0, C and 7 are each the same as if no damping were present. Next, when t is great enough to make the square of e""' negligible, we may write the approximation C' = A'^+ 2 J«e-"* cos (8 + e), or, C = A l + ^e-^'cosCS + e) A nearly (5), and y=[^ + fle-"'cos(S + e)]sin(2(;-7) (6). Finally, when t is so great that e'"* is negligible, becomes A and 7 becomes S. The motion has then subsided to that of the forced vibration simply, and is, therefore, strictly simple 152 SOUND harmonic. Or, in other words, y,^ has disappeared and y has reduced to y^ This rather complicated motion may be simply illustrated graphically as follows : — Let a radius of fixed length, A, turn about a given point with constant angular velocity q. Let another radius of continually diminisliing length, ae"*', turn about the same point and with the same angular velocity, the angle between the two radii being always (S + e). Complete the parallelogram on these two lines. Then it may easily be seen that the projections upon a Fio. 40. —Complete Solution ok Forced Vibrations. certain fixed line of the moving ends of the two radii give the motions we wish to compound. Similarly, the projec- tion upon the same fixed line of the moving end of the diagonal of the parallelogram gives the motion sought which is the result of compounding the other two motions. This construction is illustrated in Fig. 40, in which OP is the radius of length A, OQ that of length nc""', the angle POX = - 8 and QOX = + e, the lines PE, QE complete the parallelogram, M being the projection of E upon the axis of y. Thus OE = C and the angle EOX = — 7. It should be noted that although one pair of opposite sides and all the angles of the parallelogram remain constant, 101 DYNAMICAL BASIS 153 yet the shrinkage of the other pair of sides causes a shrinkage in the diagonal Oli and a change in the angle it makes with the sides of the parallelogram. 101. Initial Conditions. — Let us now obtain expressions for the displacement and velocity at the commencement of the time. On equating these to the values which specify any initial condition of the vibrating system, we can then determine the arbitrary constants a and e, and thus express our solution entirely in terms of known quantities. Thus, on writing < = in equation (7) of art. 92 we have for the initial value of the displacement ?/q = — A sin h + a sin e, where A is written for (/sin S)J2Kg^, q^ being now equal to n. Hence a sin e = ^ sin 8 + yg (7). Again, differentiating to t, equation (7) of art. 92, we obtain y = qA cos (qt — B) + ae "'{q cos qt + e — k sin qt + e), so, for < = we have j^ = qA cos S + a{q cos e — « sin e), whence, applying (7) of this article, we transform to «cose=^-(«sin8-^cosS) + ^+^» (8). Here 3/0 and ^q denote respectively the initial value of displacement and velocity. Thus k, q, A and S being known from art. 91 and 92 in terms of the circumstances of the vibration system, if we now have given in addition the values of 3/0 and y,^, equations (7) and (8) above determine a cos e and a sin e, and thus give a and e. If these quantities are being determined graphically, equations (7) and (8) are in the form most convenient for use. If, however, the quantities are being determined algebraically, we may deduce from them the separate values of a and e in 154 SOUND CHAP. IV any special case. For division of the two equations gives cot 6, squaring and adding them gives c^. Let us now consider several special cases with different initial conditions in which, however, n = g throughout. 102. Case, I. — Let the initial displacement and velocity be given by 2/o = — -4 sin 8 and ^„ = Aq^ cos S (9). Then, by substitution in (7) and (8), we obtain fi sin 6 = a cos e= 0. Thusa=0 (10). That is, the motion is of the permanent type from the commencement, and accordingly suffers no change subset quently. In other words, = A and 7 = 8 from the beginning, and remain so while ever the impressed force acts. See equations (2) and (3) of art. 100. This is, of course, a very exceptional case. Case, II. — Let the initial conditions now be the very ordinary ones of no displacement and no velocity when the impressed force begins to act, i.e. let them be given by the equations 2/0 = and yo=0 (H). Then (7) and (8) become a sin ^ = A sin 8 \ A '- (12). and a cos e = — (« sin 8 — g' cos 8) j Whence cot e = cot 8. But by equation (7 a) of art. 92, /c/y = 2 cot 8. Thus cote = cot 8, or e=8 (13 j, and therefore a = A (l-i)- These values of e and a put in equations (2) and (3) of art. 100, give, together with (1), the solution required. 102-104 DYNAMICAL BASIS 155 103. Case III. — Now suppose the initial disiDlacement to be zero, but the velocity finite, the equations being 2/0=0 and 2/o = -^!Z (15). Then, on reference to (7) and (8), we have a sin e = A sin S and a cos e = '-(« sin S — ^ cos h + q), i , a: sin 8 — (7 cos h + q whence cot e = - : i q sin S But, since n = q, cot S = {p- - q^)l2Kq = A:/2y, and cosec h = ( n/4j^ + ■K^)l2q. Thus cot e = 'i"- - ' +y g +>' ^ 1 + ^ nearly (16), 2q ^ 2q ^ ^ J' the square of k being regarded as negligible in comparison with Aq^. And, to the same approximation sin S = 1 and cos S = «/22', thus a sia e- A and « cos e = ^ [ 1 + ~^~ Whence «- = A\2 + KJq) = A'2{1 + «/2,/), or, a = As/2{l+Kl4:q) (17). If, however, « is negligible in comparison with q, we have cot e= 1, or e = 7r/4, and a = ^->/2 (18). The present case gives rise to a curious phenomenon which is readily made clear by Fig. 41. 104. Since the forced vibrations start from the undis- placed position it is obvious that 7=0, and from (16) and (17) or (18) in (2) we find that G is practically equal to A. Thus in the figure OP and OQ represent the initial positions of the revolving lines whose projections on OY give the forced and free vibrations respectively. Therefore Y / / Q R \ / P / R X 156 SOUND CHAP. IV the resultant motion is represented by the projection on OY of E the moving corner of the parallelogram on OP and OQ. Hence OP has the length A, 0^=0= A nearly, and OQ = « = approximately A J 2. Further, the angle POX = 8 = 7r/2 nearly, and QOX = e = 7r/4 nearly. The line OPi, is seen to lie on OX, thus making the angle 7 initially zero. Then as time pro- ceeds OP and OQ revolve at the same angular speed q, thus keeping the angle POQ constant, but while OP is of constant value, OQ continu- ally shrinks, since its value at time t is ae'^K Consider the instant when OP is again in its initial posi- Fic. 41.— Forced Vibrations tion, but OQ has shrunk to OQ', STARTING FROM ZERO Posi- g^^^^^^ j^alf Its Initial valuc, then OR TKIN. . occupies the position OR . Hence at this instant G will be denoted by OE' and 7 by the angle E'OX. We thus see that as time increases 7 changes gradually from zero to about 90°.' But C on the other hand at first diminishes from^ to about Aj v 2 while 7 grows to about 45", and then iiicreases from this minimum to its final value A, while 7 grows from about 45" to nearly 90°. The minimum value of C occurs at the instant given approximately by e""' = 1/2 or t = -logj2. 105. — Let us now regard the matter analytically. Take the approximate values S = 7r/2, e = 7r/4, a = A v 2 and put them in equations (2) and (3), also rewriting (1). We thus obtain y=G sin (qt — 7) (18a), where G=A Vl - 2e-'''4^2e^^«^ (19), and tan 7 = c"' - 1 (20). 105, 106 DYNAMICAL BASIS 157 (iii.) when (iv.) when e' (21). An examination of these equations shows an agreement with the results already illustrated graphically, and gives in addition more detailed information. Thus at various times (18a) assumes successively the following forms : — (i.) At < = 0, « ""' = 1 and ij = A sin qt; (ii.) when e"''' = -^-, "^''' is negligibly small, 2/ = ^(1 -«""') sin (g-if-ry); '""* is negligibly small, y = Asva.{qt — K)= —A cos qt] By differentiation to t of (19) or writing it in the form A . ^'= -^^Vl+(l-26-«'f, it is easily seen that C assumes a minimum value Aj i>j2 for e-'''=l/2. 106. General Case of Forced Vibrations: Beats. — Turning lastly to the case where n is not equal to q, i.e. to cases in which there is actual " forcing '' of the vibrations out of their natural frequency, we see that equation (7) of article 92 no longer admits of the same simplification. It now represents the sum of two vibrations : one simple harmonic, the other damped harmonic, but of different periods as well as different amplitudes and phases. The cases of most interest are those in which the periods though different are but slightly so. We can treat such cases as follows. Let q = n+g where gi is very small compared with n. We may then write equation (7) in the form y = Asin (nt — 8) + ae""' sin {nt + e-\-gt) (1), and this by the composition of harmonic motions becomes 2/ =C sin (71^-7) (2), 158 SOUND CHAP. IV where C = A'' + ah--'' +2 Aae"" co&{i + e+gt) (3), A sin h — ae' '* s\i\ (e + at) , ,. and tan 7 = -3, 1- —f , , ,, (4). A cos 6 + ac "' cos (e + gt) Thus the motion may be regarded as approximately simple harmonic of frequency equal to that of the impressed force. But, in the initial stages, the amplitude and phase are each complicated by slight fluctuations whose frequencies equal the difference between the frequencies of the impressed force and that of the damped vibrations natural to the system. In the course of time both these fluctuations subside, leaving the motion strictly simple harmonic. The ampli- tude and phase thus established are those which characterise the forced vibrations of the system under the given im- pressed force. Tliey accordingly remain unchanged so long as the force in question acts. The fluctuations of amplitude here referred to, afford another example of the phenomenon of " beats." In operat- ing with any actual system they afford a valuable criterion of the closeness of the approximation between n and q. Moreover, if a change in a certain direction in one of these quantities makes the beats slower, it is known that the exact tuning is being still more closely approached. Of course, here, as in the preceding articles, a and e may be determined to suit any given initial conditions. And, if in any case a = 0, the beats will not appear, the permanent form of the vibration being immediately established. If, on the other hand, C is zero for < = 0, then the growth of amplitude from zero to A will occur gradually and be accom- panied by the fluctuations as time goes on, tlie value of 7 meanwhile changing from its initial amount to its final value h. These circumstances are clearly deducible from equations (2), (3), and (4), when the appropriate values of a and e for any given initial conditions are substituted. 107. Algebraical Method for Forced Vibrations. — The following algebraical method will serve as a check upon 107 DYJSTAMICAL BASIS 159 the results obtained by the calcuhis for forced vibrations and may be preferred by some readers. Assume that under the action of a periodic impressed force, / sin (nt + S), per unit mass of the particle, a vibration of the same frequency is maintained in it when subject to elastic restoring forces and frictional resistances. Let the resulting vibration be 1/1 = A sin nt (1). Then for the velocity and acceleration, we have by Chap. II. arts. 15 and 16, 2/1 = nA cos nt (2), and 2/1 = — n^A sin nt (3). Let the elastic force be — ^^-times the displacement, and the frictional force — 2«-times the velocity, each per unit mass. Now, applying Newton's second law of motion, equate the algebraic sum of the forces per unit mass to the acceleration of the vibrating particle. This gives /sin {nt + B) —p^A sin nt — 2^71^ cos nt= — n^A sin nt (4). Then, expanding sin (nt + B) and equating to zero the coefficients of sin nt and cos nt, since the equation holds for every instant of time, we obtain /cosS = ^(p=-«') (5), and /sinS = ^2/c« (6). Thus from (6) A J-p^^ 01 ZKn and by (6)^(5) tanS = ^^~2 ' W- These results show the legitimacy of (1) and interpret it. They also agree with equations (6) of article 92. If the vibrating system be left to itself, we see from the hypotheses that the acceleration of the moving particle is given by i/=-p'y-2Kij (9). 160 SOUND CHAP. IV Hence, by equations 9 and 10 of article 58 the vibration may be expressed by y2 = ae~'Hinqt (10), where q^ =p^ — k". Now, although the action of the impressed force causes the system to execute vibrations of the period of the force and of definite amplitude A and phase lag S given by (V) and (8), yet this action by no means precludes the co- existence of the vibrations expressed by (10). Thus, any initial state of things may be regarded as the resultant of these two vibrations given by y^ and y^ with the appropriate values for amplitude and phase of y.^. 108. Graphical Treatment of Forced Vibrations. — The subject of forced vibrations may be instructively illustrated, and parts of the problem easily solved, graphically. Thus, taking the notation of the previous article let it be required to find graphically the values of A and 8 in terms of the known constants. In Fig. 42 let the forced vibration in question be repre- sented by the projection upon OY of the point P moving with uniform angular velocity n round 0. Now, when left to itself, the point executes damped vibrations represented by (10), the restoring force called into play by elasticity and the retarding force due to friction or other resistance being respectively — ^^-tiines the displacement, and — 2«:-tiraes the velocity, each per unit mass. But the displacement at time t is OP sin nt and the velocity n . OP cos nt. To represent the corresponding forces, take on the axis of x, OS = — p^.OP; and on the axis of y, OF = — 2«:w.0P. Then, the pro- jections upon the axis of y, of S and F revolving about with angular velocity n will correctly denote the restoring and retarding forces during the forced vibration. The change from the axis of x to that of y in the case of the frictional force corresponds to the change from sin nt to cos nt. But, for the particle to vibrate as represented 108 DYNAMICAL BASIS 161 by equation (1) the resultant force upon it must be - n^OF sin nt. So take OE along the axis of x and equal to - ?i,^OP. Then, if E rotate about with angular velocity n, its projection upon the axis of y correctly represents the resultant force on the vibrating particle. It is thus evident th9,t the impressed force to be applied to the vibrating particle can be represented by the projection upon OY of some point I rotating about with angular velocity «, and such that OE must be the resultant of OS, OF, and 01. Hence the following construction for I. Take OK on the (A) Fig. 42. — Graphical Treatment op Forced Vibrations. axis of X and equal to SE and OH on the axis of y and equal to FO, complete the rectangle on OH, OK, and the new corner so obtained opposite to is I, the point required. For, resolving 01, OS, and OF parallel to OX, the resultant is OE as it should be, and resolving the same three radii parallel to OY the resultant is, as required, zero. Which shows that 01 is the radius required so as to give with OS and OF the resultant OE. Further, since 01 represents/, the amplitude of the force in equation (4) and the angle lOP is the angle S of the same equation, the figure forms the graphical solution sought, giving / if A is known, or A, i.e. OP, if/ is known. To see that this result agrees with the analytical solution previously obtained we may proceed thus : — 162 SOUND We have from the figure /sm g = OH = rO = 2KnA, and /cos 8 = OK = SR = SO - RO = (j>^ - n^)k. /smS , ^ 2kw — — and tan 6 = -s , 2«w J) — w I Oil! U iun,lb Thus ^ = -7p^- and tan 8=-^ ^2 (H)- And these agree with the results obtained in articles 92 and 107. 109. Initial Conditions treated Graphically. — We can also exhibit graphically the completed solution under any given initial conditions. In the preceding article, for convenience of drawing, the phase angle 8 was put in the expression for the force. Reverting now to the original notation of articles 91 and 92, let us write for the force / sin nt, the forced vibration then becoming A sin {nt — B). This is shown in Fig. 43, in which as before 01 and OP correspond respectively to the force and the displacement, and represent them on some convenient scale. The expressions for a sin e and a cos e in equations (7) and (8) of article 101 are now immediately applicable, and enable us to complete the solution for the given initial conditions. Thus, we have only to insert the values of y^, y^, and the other constants in the right sides of these equations to obtain a sin e and a cos e. We must then set off OK along OY to represent to scale a sin e, OL along OX to represent to the same scale a cos e. The completion of the rectangle gives the point Q, and its diagonal OQ represents to scale the initial amplitude a of the damped vibration 1/2 natural to the system, the angle QOX being its phase angle e. This complete solution may then be expressed as follows. The impressed force and the forced vibration are represented by the projections upon OY of the points I and P respectively, each moving in circles about as centre and with angular velocity n. The free but damped vibration natural to the 109 DYNAMICAL BASIS 163 system is represented by the projection upon OY of the point Q, the radius OQ rotating about with angular velocity q and at the same time shrinking in length accord- ing to the factor e'"'. Thus, if K be the projection of P, the motion of the vibrating point will be given by a point M on OY whose ordinate OM is the algebraic sum of OK and ON. The figure represents the state of things for ^ = 0. By drawing P, Q, and M for a number of positions, say, 8 Fig. 43. — Completed Solution for Forced Vibrations. or 12. times in each revolution, and continuing for several revolutions, and then combining with this actual vibratory motion of M a uniform translation parallel to OX, we could exhibit the graph showing the type of motion which the vibrating particle executes- under these conditions. When n = q no beats are possible, but we have usually a continuous growth of the vibration until the permanent amplitude and phase are attained. Though, under the conditions noted in Case III. of article 1 3, we have an exceptional case, in which a single diminution of amplitude may occur followed by a rise to the final steady state. When n is not equal to q, as 164 SOUND CHAP. IV in article- 106 (except when the initial conditions make a = 0), beats must occur. Under these circumstances we have at first repeated waxings and wanings of amplitude. These, however, die away, leaving finally the permanent form of vibration established, namely, that expressed by i/i, as due to the impressed force. 110. Experimental Illustrations of Forced Vibra- tions. — Many illustrations of the phenomena of forced Fig. 43a. vibrations may easily be called to mind or readily devised. Of these two will be noticed here, others will occur later. ExPT. 21. Rough Illustrations of Forced Vibrations. — From a slightly stretched horizontal cord about a metre long, preferably of india-rubber about half a centimetre diameter, suspend, side by side, two pendulums. These may be arranged with helical springs and heavy bobs, one of which is adjustable in mass, and each bob oscillating vertically. Or, they may be simple pendulums with metal bobs and silk thread suspensions, one being adjustable in length. Both arrangements are shown in Figs. 43a and 43b. Ori starting one pendulum in oscillation the other is soon 110 DYNAMICAL BASIS 165 seen to be moving also, and may, by its absorption of energy, almost reduce the first one to rest. In this case the "driven" becomes in turn the driver, and starts again the vibrations in the pendulum from which its own impulses were received. Indeed several such exchanges of energy between the two are possible and may be observed. The more correct the " tuning," the greater is the amplitude of the forced vibration, as theory shows. The frames shown in Figs. 43a and 4.3b are of wood about one inch square, and are stayed at the \ ^^^^^^ \ V /^ \ FiQ. 43b. bottom by strings only, in order that they may fold up for more compact storage in the apparatus cupboard. •This arrangement of forced vibrations may be modified so as to be more striking if space is available for a tight rope about 10 metres long. . In that case a single simple pendulum 2 metres long may be hung about 2 metres from one end, and at 2 metres from the other end may be hung, say, three pendulums of lengths 1'8 m., 2 metres, and 2-2 m. re- spectively. Then, on setting the single pendulum in oscilla- tion, the one of corresponding length, though so far away, is soon seen to be in vigorous oscillation, its fellows too long and too short only starting slightly and then stopping again, thus exhibiting " beats." KiG SOUND CHAP. IV 111. ExpT. 22. Lord JRayleigh's Model for Fm-ced Vibra- tions. — This arrangement, shown in Fig. 44, is more elaborate than those just described, but well repays the trouble of making. In this case the simple pendulum PQ is forced to oscillate in time with the very massive pendulum from which it is suspended. This main pendulum is seen from the figure to be attached at the fixed points A and B by four wires supporting the points C and D which are connected by the strong liar CPD, carrying the large bobs exactly beneath Fig. 44. — Rayleigh's Foeoed-Vibeations Model. the points C and D. It should be observed that the four wires are of equal lengths, thus causing the bar CD to set horizontally at right angles to the line joining the fixed points A and B. Hence, when this massive pendulum oscillates in small arcs, P moves almost horizontally. Further, this main pendulum moves with practical independence from the reaction of the forced pendulum PQ, since this latter has so small a mass in comparison with the former. By means of this apparatus the leading features of forced vibrations already enlarged upon may be readily illustrated. To facilitate accurate adjustment, the main suspensions AC, BC, AD, BD must be strong, say of steel wire. String quickly stretches and vitiates the tuning. The phenomenon of beats Ill, 112 DYNAMICAL BASIS 167 is very well shown and needs no special endeavour to attain. Setting the apparatus at random will almost certainly produce them. On the other hand, to so adjust the model that beats are absent is extremely difficult. To show the effects of damping in reducing the actual amplitude and the sharpness of resonance several methods are open to us. Thus we may let the bob Q swing in water. Or we may replace the small metal bob Q by a large woollen ball. It is important in all these gravity pendulum arrange- ments that the amplitudes be kept fairly small, otherwise the period is appreciably changed and the accuracy of the tuning accordingly impaired. From the theory previously developed it may be seen that the equation of motion of the bob Q is y + ^Ki) +p\y - j/') = where y is the displacement of P and y that of Q. But since P's motion is nearly simple harmonic we may write p^y =/ sin nt, thus obtaining the form already used, viz. — y + 2/cy + phj = / sin nt. 112. Large Vibrations. — Let us now consider' the character of vibrations performed under the influence of restoring forces not simply proportional to the displacement. In strictness this is usually the case, but when the dis- placements are so small as to make the terms involving their cubes and squares negligible, the lack of propor- tionality in the restoring force has no appreciable effect. When, however, the vibrations are large, these squares and cubes instead of being negligible may become paramount. Both the square and cube of the displacement may be involved in some actual case, but, for simplicity's sake, the effect of each will be studied separately, and, further, the frictional term will be omitted. Asymmetrical Case. — Take first the case in which, beyond the first power, only the square of the displacement is involved. The equation of motion may then be written y+fy + af=0 (1). It is thus seen that the restoring force changes its numerical value as well as sign when the displacement changes sign only. 168 SOUND CHAP. IV Obviously, the asymmetry is represented by the term involving y^, and its coefficient is supposed small. The straightforward, exact solution of this equation gives t as an elliptical integral of y, and is therefore not suitable for our purpose. We accordingly follow the method of successive approximations as used by Lord Eayleigh, and which gives y in a series of cosines involving t. Thus, at first, put a=0 in (1). This yields the approximate solution y = A cos ipt, and this value, substituted in the term ay^ of (1), gives the approxiftiate differential equation 2/ + pV= -»^(l+cos2^;0 (2). The solution of this may be written y = ^ cos in -\-B-VC cos 2ipt (3), where 5 = - — i and C = ^p (4), A being arbitrary. But, suppose this second approximate value of y were inserted in the term ay''' in (1). Then, its first term cos^<, would, in the solution, necessitate a term of the form t sin^^ This term would imply that the displacement would grow indefinitely with the time, which is inadmissible. We thus see that the period involved, namely, 27r/^, must be wrong, and so in time leads us quite astray. Let us therefore, in equation (3), replace jo by another symbol q, whose value is to.be determined. Then, on substituting in equation (1) we obtain the differential equation y +fV = ~ «(^ cos qt,^B-\-C cos 2: (10), as the condition to be fulfilled by the pressure and volume, in order that waves may be propagated unchanged in character as they advance. But this is not the law for any known gas. For obviously (10) would give a straight line on the pressure-volume diagram, whereas both isothermals and adiabatics are always curves. Suppose, however, that this straight line is taken tangentially to the actual adiabatic curve. Then, if the 182 SOUXD CHAP. IV changes of pressure and volume are very small, the corresponding straight line will practically coincide with the curve for the length in question. Hence for such small changes the condition for steady motion is practically fulfilled, and the results obtained on the supposition of unchanged propagation remain valid. If, however, the sound is very loud, then the pressures and volumes will change very considerably. The corre- sponding curve on the diagram of pressures and volumes, accordingly, departs sensibly from any straight line. Consider now the condensed portion of the wave. Here the slope of the adiabatic is appreciably steeper, and the volume less than at the normal or uncondensed parts of the wave. But the slope of the line given by (10) is expressed by the constant B which denotes v'jU'^. Thus for the condensed part of a wave of large amplitude we have v^jV^ appreciably larger than normal, and U distinctly less. Hence for both reasons v is increased. We have thus obtained the important conclusion that the highly condensed portions of intense sound waves advance faster than those parts of the same waves at which the pressure is normal. Similarly, the highly rarefied portions of the wave travel rather slower. Accordingly, the condensed parts will out- strip the rarefied parts, and the type of the wave is modified or distorted as it advances. A similar change of type with advance is noticeable in water waves on a sloping sea beach. Here the crests gain on the troughs, and the faces of the waves become steeper and steeper until they curl over and break. This effect is clearly brought out in Maxwell's Theory of Heat, in the section dealing with the velocity of sound. 123. Speed of Longitudinal Waves along a Solid Prism. — "VYe may here follow the method used for a gas, changing what needs to be changed to meet the present case. Thus let the prism be moved backwards at the mean speed V, that at which the sound waves move forward 1^3 DYNAMICAL BASIS 183 relatively to the prism. Also, let I denote the normal length of unit mass of the rod and I' that where the speed is i/. Then the equation of continuity may be written v'll' = vll (1). Let F and F\ be the stretching forces experienced by the prism over its entire cross section, where the speeds are respectively v and /. Then the equation of motion becomes ^'-^-(7)''-(l> (^>- The substitution of (2) in (1) gives i'-F=jll'-l) (3), , „ (F'-F)l whence v^ = ~ — -^ • Is (i), {I — l)s where s is the cross-sectional area of the prism. 'But the fraction on the right side of (4) is Young's modulus for the material of the prism, which we will denote by q. Again, Is is the specific volume or volume per unit mass, and is thus the reciprocal of the density p. Equation (4) may accord- ingly be transformed to V = Jqjp (5). In this case the value of Young's modulus used should, in strictness, be the adiabatical value and not that obtained by any of the ordinary statical methods which give the isothermal Young's modulus. However, the difference between the two is small and is usually neglected. If we examine the criterion for the permanence of the type of these longitudinal waves in an elastic solid, we see that it is fulfilled within the limits to which Hooke's law holds. For equation (3) shows that the increase of force is v^jl''' times that of the corresponding elongation. And by 184 SOUND CHAP. IV equation (1) v^jf is a constant. Hence (3) may be written (F' — F) on (I' — I), which is the symbolic expression of Hooke's law. Let us now inquire what stretching force is needed to make, if possible, the speed of transverse waves along a wire equal to that for longitudinal waves as just proved above. Eeferring to article 117 we have Speed of Transverse waves = \l Fja = sj Fjps (6), where o- is the linear density, p is the volume density, and s the cross-sectional area of the wire. Again, from (5) we have •Speed of Longitudinal waves = v ^//o = s/qsjps (7). Hence the stretching force required is qs, or a force of q units per unit area of cross section. But this is the force that, if Hooke's law held, would stretch the wire to double its length ! Hence longitudinal waves in wires travel considerably faster than any transverse ones which we ever obtain. 124. Partial Differential Equations for Progressive Waves. — Let us now treat progressive wave-motion by the methods of the calculus. We have, first, to derive the differential equation ; second, to obtain the solution of it ; and third, to fit this solution to the various possible initial and terminal conditions. The derivation of the differential equation is, of course, different for strings, gases, solid prisms, and extended solids, dealing with the mechanics of each special case. They will accordingly be taken separately in the above order. But we shall find that, under the restric- tion of small amplitudes, each of these cases leads to the same approximate differential equation. The solution of it is accordingly deferred, until it has been established for each case. The common form arrived at will then be considered, the general results so obtained being applicable with slight modifications to any of the cases. Transverse Waves along a Cord. — Let the mass of the 124, 125 DYNAMICAL BASIS 185 cord per unit length be o- and the stretching force F absolute units. Take the axis of x along the cord and let the transverse displacement at time t for a point whosp abscissa is x be denoted by y. Then we have seen in article 117 that Fjr expresses the pressure per unit length due to a curvature of radius r. Nov?, vv^e shall suppose_JhiB_curva^--£^^} ture to be always very small, hence itrinaybe approximately represented by d^yjclsi? instead of the fuller expression which is the above quantity divided by { 1 + {dyjdxfY^'^- Further, we may equate the restoring force to the product of mass per unit length and transverse acceleration. Thus, we obtain the differential equation of motion required, viz. (/;:- dy? where «' = Fja (2). 125. Flanc Waves in a Gas. — Let the plane of the wave front be at right angles to the axis of o:. Also let x define the equilibrium position of particles and y their actual positions at time t. But y is to be measured from the same origin and along the same liiie as x. For the waves being in a gas must be longitudinal. Then, at a neighbouring layer at the same instant t let the equilibrium and actual positions of other particles be represented by x + dx and y+ ~dx respectively. Hence, if po and p are respectively the normal and altered densities of the layei', we have Pndx = p^dx, dx or p^Ip = dyjdx. This and the adiabatic relation Fx p^ or Fp-y = PoPo"',&^Q 186 SOUND CHAP. IV Also, the mass of unit area of the displaced and deformed slice is pdy = pfflx, and the corresponding moving force is dP , dP , --~dy= -(/*•. • dy •' dx Thus the equation of motion may be written dP d?y dx dt' 126. Eliminating P between equations (3) and (4) we obtain \dx/ df pQ dx' This is the cxad equation given by Lord Eayleigh- defining the actual abscissa y in terms of the time t and the equi- librium abscissa x. We can easily transform this so as to exhibit the longitudinal displacement ^ in terms of x and t. Thus, by definition, y = x+^ and on substitution (5) becomes dxj dt Pq dx But on writing the approximate value unity for each of the quantities in brackets in equations (5) and (6), they reduce to the ordinary approximate forms, namely : — dt dx and ^=r'^^ (8), in each of which v^ = jPq/po (9). 127. Sometimes it is found convenient to use another dependent variable called the " condensation " instead of either the y or ^ just given. This is usually denoted by s and is defined by the relation p = po{l+s). Thus, since Po = p{l + d^jdx), we have s = — d^jdx. Differentiating (8) to X and then using this value of s we obtain successively 126, 127 DYNAMICAL BASIS 187 d'^ ,d'^ dL'dx d,F J d's .,d's an*! 7. = ^"?., (10). dt' dx' ^ ' We thus see that the same form of equation is obtained whether y, f, or s is the dependent variable. It forms a further useful exercise and check to derive directly the equation in s. At x let the density be p and the speed w. Then the equation of continuity for the slice between x and x + dx is z^pA,uJUx. dx dt In this equation we may write d , . du dp du E^P''^ = PTx + "'dx-P^d-x'''^''^- For, in deriving the approximate equation for small motions, u. is a small quantity and s negligible in comparison to unity. Thus u— is omitted, and for p = p^{l+s) is written dx Pq simply. Again, dpjdi on the right side of the same equation is p^ds/dt. Hence the equation of continuity in this approximate form becomes — da ds / 1 1 \ ^aT ^di ''' The corresponding moving force is dx. Thus the The mass of unit area of the slice is pdo; or p^dx nearly, -dl- dj. equation of motion may be written -dP du ~dJr=P'>It ^''^- Now the adiabatic relation may be written Pp-y = P,p,-y or p = P,{plp,r = P„(i + s)y. 18.S SOUND CHAP. IV Hence ^ = yPo(l + sy-^l" = 7P0 J nearly (13). dx cU: dx Thus, by substitution from (lo), equation (12) becomes -yP^ ds ^du. ^^^^_ Pq dx dt Then, on differentiating (11) to t and (14) to a;, and equat- ing the values thus found for the identical quantities d\/dxd/, and dhi/dtdx, we obtain d^s ryPr. d^s „d'-s ,. _. — 2 = ^^ T-2 = ^' V, (15), which agrees with (10). It should be noticed here that according to the kinetic theory of gases the speed of sound is determined by, and is directly proportional to, the velocity of mean square of its molecules. For on this theory we have F = lpV'' (16), where V is the mean square of the velocities of the various molecules and P the pressure. Hence, for the speed of sound in a gas of density pp at pressure Pg we have 128. Longitudinal Waves along a Solid Rod. — Take the axis of x along the rod and let f denote the dis- placement at time i of a plane whose equilibrium position is x. Consider a neighbouring plane whose equilibrium position is X + dx. Then its displacement at time t is expressed by f -)--?(/,«. Thus the actual elongation of this layer is -^4x dx dx dP and its fractional elongation is — . Hence, if the value of dx Young's modulus for the material is denoted by q and the restoring force per unit area due to the stretching by /, we 128 DYNAMICAL BASIS 189 have/= q d^jdx. But the moving force per unit area on a slice of thickness clx will be the increase of this force in the thickness of the slice. It is accordingly represented by '^-^dx dx = q{d^^/dx^)dx. And pdx is the mass per uuit area of the slice to be moved. Hence the equation of motion is °^' • dl-'-d ^''^' where i>^ = q/p (19). It might be supposed that the foregoing reasoning and result were rigorous ; they are, however, in reality approxi- mate, as may easily be seen thus. Since the elasticity used is Young's modulus, the supposition is that the sides of the rod are not prevented from bulging where longitudinal com- pression occurs, and shrinking laterally where elongation occurs. But this bulging and shrinking each involve a lateral motion which has not been taken into account in the equation of motion. Thus the above result is approximate and not rigidly true. It is, however, quite near enough for many purposes, provided that the wave length is great com- pared with the thickness or width of the rod or bar. This may be shown by examining the extent of the displacements longitudinally and laterally. Thus if the wave length is \, we may regard a length X/4 as being distinctly under com- pression or elongation at any one time. If the fractional elongation of this part of the rod is e, the actual elongation is eXJ-i, which measures the extent of longitudinal motion. But if Poisson's ratio for the rod is o-, the lateral contraction would be ae per unit width ; so for a width h, say, the actual contraction and possible motion would be acb. Thus for this lateral motion to be negligible we need ah to be small compared with x/4 ; or, since a is about 1/4, we may say I 190 SOUND CHAP. IV must be small compared with X. It should be noted that only the outer portions of the rod would have the full value of the lateral motion just written, but in like manner only the end portions of a part elongated or compressed would have the full value of the longitudinal motion just assigned as due to this cause. 129. Longitudinal Waves in an Extended Solid. — If, instead of a long thin rod, we have an unlimited solid, and produce infinite plane longitudinal waves in it, then the lateral bulging and shrinking which occurs in tlie rod is no longer possible. Accordingly, Young's modulus no longer applies as the elasticity appropriate to the case, its place being taken by the simple longitudinal elasticity referred to in articles 77 and 78 and Table V. Thus the differential equation may be written '^. = A (20). df fix" ^ ' where „■ = (. 4,)/, = _Ar^ (21), k denoting the volume elasticity, n the rigidity, and o- the value of Poisson's ratio. Examples of waves approximately of this type occur in the earthquake disturbances travelling through the earth's crust, but we are not deeply concerned with them here. 130. Discussion of the Differential Equation for Wave Motions. — In the four cases just considered — viz. transverse disturbances in a stretched cord, longitudinal disturbances in a gas, a rod, or an extended solid — we have found one common form of differential equation which applies strictly or approximately to each case. The constant denoted by v in these equations has, however, different values for the different cases. But this fact does not affect our discussion of the equation itself Further, the dependent variable used in the various cases is sometimes a transverse displacement, sometimes a longitudinal one, and 129, 130 DYNAMICAL BASIS 191 at others the condensation. This again does not affect our treatment of the equation itself. "We accordingly write, as the general type for solution and discussion, 5=4' (1). at' dx' This may be put into words, thus : — The second differential co-efticient of the dependent variable with respect to time is proportional to its second coefficient with respect • to the co- ordinate along the axis of wave propagation. We shall afterwards see that this constant ratio v^ is the square of the velocity of wave propagation. The solution of this equation may be written y=f,{x-vt)+f^{x + vt) (2), where each of the/'s denotes an arbitrary function. It may be seen from the respective arguments {x — vt) and {x + vt) that f-^ denotes a disturbance travelling in the positive direction, and /a one travelling in the negative direction. That this sum of functions satisfies the differential equation may be easily verified as follows : — Differentiate (2) twice with respect to t, thus giving the left side of (1), viz. 5 = rf;'(x - vt) + v'f:'{x + vt) (3), dt- where the dashes to the /'s denote differentiations of the functions with respect to the arguments in brackets. Now differentiate (2) twice with respect to x, and multiply the result by v^. We thus obtain the right side of (1), viz. 'fi = v'f^ix - vt) + ^y;'(* + vt) (4). dx' But the right sides of (3) and (4) are identical, hence we see that (2) satisfies (1) whatever the functions may be. Equation (2) gives the displacement y, for completeness' sake we may add to this the value of the velocity y. 192 SOUND CHAP. IV Thus differentiating (2) once with respect to t, we obtain ij= - ^fA''= - ^0 + '.'/2%'- + rt) (5). 131. Now this solution for the dependent variable y applies equally to the same form of differential equation in which the dependent variable was ^ or the condensation s. Thus, applying this result to equation (15) of article 127 we see that s=A{x-rt)+M.: + vt) (G). But by equation (11) of article 127, we have tir Thus differentiating (6) to t, substituting in (7) and inte- grating, we find >,= + r|//(,/; - liyir - vj/./i.': + vt)dx = ;.f^c-vi:)-vf^r+v€) (8), or, for either the positive or negative part taken separately we have II = ± vs (9), the upper sign referring to the wave disturbance denoted by /i which travels positively, and the negative sign to the wave travelling negatively. Tlrus ii, the velocity of the particles, has the same sign as the condensation in a positively-travelling disturbance. Now in a condensed part of the wave s is positive ; thus in a positively-travelling wave II is positive also, i.e. the velocity is there with the wave. Again, in the condensation of a negative wave, u would be negative, i.e. the velocity of the particle is still with the wave. On the otlier hand, where there is a rare- faction, s is negative, and the velocity ti, of the particles is against the direction of propagation of the wave. 132. Initial Conditions. — We have now to consider the various other conditions that may be imposed, and the 131, 132 DYNAMICAL BASIS 193 forms that the arbitrary functions must assume to satisfy them. Let us at present suppose both the medium and the original disturbance to be unlimited. Suppose also that the initial displacements and velocities are given as functions of x. Then the problem is to satisfy the follow- ing three equations : — cU- d.i The differential equation, -^ = v^ -^^^ (1). The initial displacement, y = (j)(jv), ior t = (2). The initial velocity, y = ■\lr(x), for t=,0 (3). We have already seen that (1) is satisfied by y =/i{^ - «0 +/2(* + «0 (4)- It remains now to define /j and/j so as to fulfil (2) and (3) also. By putting ^=0 in (4) and using (2), we have ix)=Mx)+Mx) (5). By differentiating (4) to t, then putting t = and using (8), we obtain On integrating (6) with respect to x, we find ^1 fi^^dx= -A(x)+Mx) (7). By addition and subtraction of (5) and (V) we then obtain the expressions sought for /^ and /g, viz. Ai^)=^-llf(^y^ (8). and /2(«=) = ^ + ^ft(^-y«^ (n Jo Then (8) and (9) substituted in (4) give the complete solution which fits the imposed initial conditions, viz. 194 SOUND cHAr. IV 1 f' y = ^(l){x—vt) — -:-\ ■^{j-- — vt)dx + h4>{^' + vt) + j^( ylr(x+rt)d.r (10). When the functions (j> and ^jr are of any ordinary well- known type, the integrations in (10) can, of course, be easily effected and the value of y fully expressed. Several cases of special simplicity or importance will now be considered. 133. Illustrative Examples of Initial Conditions. — Case I. — As the first illustration let the original disturbance be a displacement without velocity. Thus for t = let y = (p(x) and // = (1 1). Then, in the previous notation, -^(a') = 0. Thus we find by the foregoing that fi(x) =fi{x) = ^ then we have 2/ = a sin \^o}{t —xjv)^ — a sin {&>((; + a;/'f)} (14). Case II. — Now let the original disturbance be an impressed velocity without displacement. Thus, for < = 0, let i/ = and // = i/rOr) (15). Then, proceeding as before, we have -/i(.^)=+/2(^0 = ^f#'-K'^ (16), Jo 133, 134 DYNAMICAL BASIS 195 which by substitution in equation (4) or (10), gives I r If" y= - — I f(x- vt)di.: + — ■y\r(x + vt)dx (17). ^ h "^ Jo It is seen that in this case we have two disturbances, each half the initial one, and travelling in opposite directions. The disturbances in this case are initially equal but opposite in their displacements, but have velocities which are alike in both magnitude and sign ; this conamon velocity .being ^• Case III. — ISTow let the initial disturbance involve both displacement and velocity. Thus a,t t = let y = ^{x) and y = f(x) = - v4,'(-r) (20). Then Mx) = i(*) ~ 2" f 'p'i^)^^ = Jo Thus y = 4>(x-vt) (22). That is, we have only a single wave, it is travelling in the positive direction, and has amplitude and velocity equal to those of the initial disturbance. Hence, if for t= 0, y= — a sin (tDx/v) and y — coa cos (cox/v) (23), we find as the solution ^ = a sin [co(t — x/v)\ (2"i). 134. Original Disturbance Limited. — Now, while the medium remains unlimited as before, let the original disturbance be confined between the limits a; = and x = l. (21). 196 SOUND CHAP. IV Then the solution undergoes a slight corresponding modifica- tion, the disturbance produced having limitations in time and space. Thus, the original disturbance having length I, we have (in the general case) two disturbances, each of length I, travelling one in the positive and the other in the negative direction. If only one of these elements is present in the solution, say the positive wave, then we have only some one length I disturbed at any instant. Let us follow the single positive wave in its course. This disturbance of lengtli I moves forward at speed r. Hence, at time t, the portion of medium disturbed has abscissfe vt and I + vt. Consider now the period during which any point experiences the disturbance. Let its abscissa be I'. Then obviously it is undisturbed until t = Q,' — l)lv, when the head of the advancing wave train reaches it, it is theu disturbed until t = I'/v when the tail of the wave train passes it, after which it remains without further disturbance. Or, we may say that the point I' is disturbed only while t is not outside the limits I'fv and (/' — l)lv. Precisely similar considerations apply to points on the negative side of the origin when negatively-travelling waves are present in the solution. 135. Reflection at a Fixed End : Stopped Pipe. — In a medium of unlimited length we have seen that the solution of the differential equation may give a single positive or negati^'e wave, or both superposed, the one case or the other being dependent upon the initial conditions. Suppose now the medium to be limited in length at one end, and let the initial conditions be such as to give only a single wave travelling towards that limit to the length of tlie medium. Then we shall show that the single initial disturbance gives rise to another travelling in the opposite direction. This result is produced by reflection at the end which limits the medium. But the nature of the reflection will depend upon the nature of this end of the medium. It may be what is termed a " fixed " end where no dis- 135, 136 DYNAMICAL BASIS 197 placement is possible. Or, it may be what is termed a " free " end, where the displacement may be as large as we please. Let us consider first the fixed end. This condition is represented by the fixed end of a cord vibrating trans- versely, or of a rod vibrating longitudinally, or by the " stopped " end of an organ pipe, or, for sounds in the free air, by any large obstacle which produces the familiar phenomenon of the echo. Let this end be situated at the origin of co-ordinates, and let the medium extend only on the negative side of the origin. Suppose, further, that the initial conditions give rise to a disturbance represented by y=A(:vt-x) (1). And, in order to represent this original or incident disturb- ance together with that produced by reflection at x=Q, let us write y=Mvt-x)+fivt + x) (2). We have then to determine f.^ subject to the conditions at the fixed end, namely 2/ = at a; = for all values of t (3). Substituting (3) in (2), we have =f,{vt) +flvt) or flz) = -J\(z) (4), where s is any variable. Thus (2) becomes ]/=f,ivt-x)-f^ivi + x) (5). We see by inspection that for a- = 0, this gives ;// = for all values of t and for any form of the function /^. Thus the reflected wave is like the incident wave, but with amplitude of reversed sign. 136. Equations (1) and (5), of course, apply under certain restrictions of time and space which differ according to circumstances. Precisely what the restrictions are for any individual case can easily be ascertained by the method of the previous article. It will suffice here to examine a single case of special interest. Thus let the original 198 SOUND CHAP. IV disturbance be confined to a portion of the medium of length I. Then equation (1) applies alone until the head of the disturbance reaches the origin. Let this occur at time t=Q. Then (5) begins to apply instead of (1). But its application is restricted in space, and the restriction is different for the different terms. Thus at time t, where Kljv, the first term applies only between the origin and x= — 1-\- vt, and the second term applies only between the origin and a- = — vt. When t = Ijv, the first term ceases to apply, and the second term alone represents the value of y. That is to say, the act of reflection, which began at f=0, is now completed, and the value of y is henceforth represented by the negatively-travelling wave — y=-f,(:d + x) (6). These various stages are illustrated in Fig. 46, in which the original function consists of two straight lines. It is thus seen that as the various portions of the incident disturbance reach the reflecting point the corresponding portions of the reflected disturbance emerge from it. And to show this fact the more clearly in the figure, dotted lines are used for those parts of the reflected wave which are represented by the general equation (5), but are excluded from actual existence by the restrictions which apply to that equation. Thus, when the act of reflection is just beginning, the dotted lines on the positive side of the origin show a negatively-travelling wave as expressed by the second term of equation (5). But the dotted lines indicate that as yet this wave has no physical existence. At the next stage the incident wave has moved to the right, the part which has passed the origin being now dotted to show its fictitious character. On the other hand, the negatively-travelling wave has passed a like distance to the left, that part which has passed the origin being in full lines to show that it is now part of the actual reflected wave, the remaining portion being dotted to show that it 137, 138 DYNAJVIICAL BASIS 199 has not as yet any actual physical reality. Hence the components of the wave disturbances to tlie left of the origin may be obtained by the following construction :— 137. Draw the displacements of the incident disturb- ance, and make on the other side of the origin a copy of them derived by rotating the incident wave through 180° about the axis of z (at right angles to the axes of both X and y). Let the original disturbance proceed in the positive direction at speed v, and the copy of it move in the negative direction at speed v\ but at any instant account only those portions of these waves real which are then on the negative side of the origin along the axis of x. Components ffeSuHitfit Fio. 46. — Reflection at Fixed End. 138. Reflection at a Free End : Open Pipe. — Consider now the reflection of a positive wave at a " free " end situated at the origin of co-ordinates. A free end is repre- sented by a free or loose end of a cord hanging vertically, a free end of a rod vibrating longitudinally, or the open end of an organ pipe. The conditions imposed by such an end are repre- sented by dx - = at a; = for all values of t (n It may be seen that this is equivalent to writing for the string the end must be always parallel -to the axis of x, and for the open end of a pipe or the free end of a rod the density at the end must remain normal. 200 SOUND CHAP. IV We have therefore now to combine equations (2) and (7), and determine the form of the function/. Differentiating (2) with respect to x, we find where /' denotes the function obtained by differentiating the original function, or f'(z) denotes - . Thus equation az (7) gives f^ivt) =f^'(vt) for all values of t (8). Thus, integrating with respect to t from to t, we have jy,;{vt)dt = j'fl{rf)dt, or fpt)=f,{rf) (9). Hence, for the free end, by substituting (9) in (2) we obtain the solution 2/ =/;(''^-.'0 +/:('■< +.'•) (10). This shows that the reflected wave is like the incident one without reversal of sign of amplitude. In other words, by rotating the incident wave 180° about the axis of y, we obtain the form of a wave which, if made to move at speed V in the negative direction along the axis of x, will, on passing the origin, become the actual reflected wave. It will afford good practice for the reader to draw for himself a diagram for this case similar to that illustrating the previous article. It should be noted that various aspects of a wave motion may be singled out for consideration, and that if one is reversed by reflection another may be left unchanged. Thus, in the case of the stopped end, we saw that the dis- placenient is reversed. The condensation, however, is left unchanged in sign by the reflection. For the condensation arriving at the end with the incident wave is associated with a motion of the particles towards the end (see Fig. 10, art. 2], and equation (9), art. 131). Now by the presence 139 DYNAMICAL BASIS 201 of the stopped end this motion is exactly annulled by the equal and opposite motion of the particles in the reflected wave. But this opposite motion in the oppositely-travelling wave is also associated with a condensation as in the incident wave. Thus a condensation is reflected as a condensation from a stopped end. Similarly at a stopped end a rarefac- tion is reflected as a rarefaction. Thus, though the dis- placement is reversed by reflection from a stopped end, the state of condensation is unchanged. It is easy to see that the exact opposite of the above holds for reflection at an open end. That is, at an open end the displacement and motion of the particles are not reversed by reflection, but the condensation is thereby reversed in sign. Hence, at an open end an incident condensation is reflected as a rarefaction, and a rarefaction is reflected as a condensation. 139. Medium itself limited, Original Disturbance shorter still. — Suppose now that the medium is of finite length, and that the original or initiating disturbance occupies only a portion of this length. And let it be first assumed of such a nature as to give rise to only a single wave travelling in the positive direction. Then we see from the foregoing articles on reflection that this disturbance will course to and fro along the given length of medium suS'ering reflection at each end, the circumstances of those reflections depending on the natures of the ends. Suppose first both ends are fixed or stopped, then the amplitude or displacement is reversed in sign by such reflection. We are thus able to predict the whole future course of events. Second, suppose that both ends of the medium are free or open. Then such reflection occurs without change in magnitude or sign of the displacement. Here again we are able to follow out the whole phenomena as the wave disturbances pass to and fro along the given length of medium. Thirdly, suppose the medium in question to be terminated by one fixed and one free end. In this case 202 SOUND cHAv. IV the reflections occur alternately with and without change of algebraic sign of the amplitude. It should be noticed that where the ends of the medium are (dike in nature the original state of things is restored after the time required for the disturbance to traverse tioice the length of the medium. Whereas, when the ends of the medium are unlike, the original state of things is not restored until after the time requisite for the waves to traverse a distance equal to four times the length of the medium. We have hitherto supposed the original disturbance to give a positive wave only. Obviously, all the remarks just made would apply equally if the original wave were a negatively-travelling one. Finally, suppose the initial disturbance were quite general and, accordingly, gave rise to two waves travelling in opposite directions, and let it be required to find the state of things after time t. For this purpose it is necessary to take each wave separately, and find the state of things after time t due to each of them, and then superpose these states for the solution sought. 140. Displaced String. — Let us now consider the case of a string, extending from x =0 to x = l, displaced and then let go. We have here then a finite length of elastic medium with both its ends fixed. Further, the initial disturbance is to consist of a displacement the whole length of the string, but without velocity. The solution of the differential equation (see equation (2), art. 130) may be written ]/=/\i^^-rO+Mx + vt) (1). Let the initial conditions be expressed by y=2/(.0 and j/ = 0] from .,• = to .c = I, when t=Oj The terminal conditions may be written — For all values oi t, 7j = Q both at (and outside)] the limits ,-; = and jj = I f ^'')' 140 DYNAMICAL BASIS -203 We must now determine /^ and/^ in (1) so as to satisfy (2) and (3). We have previously seen that to satisfy (2) we must have oppositely-travelling waves of like amplitude and sign, thus y=J{^-vt)+f{x + vt) (4). To make y=Q at the limits, in accordance with equation (3), we have, on substitution in (4), o=/(-^^0+/( + ^-0. and =/(/ - vt) +f(l + vt). Putting +vt= —z in the first of these equations, and -\-l-\-vt= —ziw the second, we obtain .m= -/(-«) (5), and f(2l + z)^-f{-z) (6), whence f{2l + z)=f(z) (7). Thus, equation (5) shows that the amplitude changes sign with change of sign of x, and (7) shows that the function is periodic in length 21. Hence, for a stretched string originally displaced in any manner, but without velocity, we may construct the subsequent motion as follows : — Describe a curve of displacements for the length of the string, the abscissae denoting lengths along the string, bixt the ordinates being only half the actual displacements at the corresponding points. Beyond the string's length describe a curve obtained from the foregoing by revolving a half turn first about the axis of x, and then about that of y. We have thus described curves for the length 21, i.e. for twice the length of the string. This forms one complete pattern of the curve required, and needs repeating inde- finitely right and left. This curve must now be taken to represent, at the instant ^ = 0, two sets of component wave trains which have, at that instant, identical displacements. One of these trains is to move to the right, the other to the 204 SOUND CHAP. IV left, each with speed v. The actual displacement and motion of the string at time t are to be found as the resultants of those corresponding to the parts of the two component wave trains which are, at that instant, between the limits a; = and x = I. To avoid obtaining any resultant outside the limits occupied by the string, the component wave trains must be regarded as having no real existence outside these limits, but as being, while there, convenient but wholly abstract mathematical conceptions which give birth to the corre- sponding physical realities only when passing inside the limits x= and x = I. This construction should be compared with that for reflection at a fixed end (art. 137 and Pig. 46), which was a rather simpler example of the same essential principles. 141. Plucked String. — Suppose now the initial dis- placement to be produced by " plucking," that is, by pulling a point of the string aside and letting it go. Let the co- ordinates of the plucked point when let go be (a, 2,h). Then, for a > a; > 0, f(.>i) = hxja (8), and for />«>«,/(»■) = — ^ (/-..;) (9). I — a Further, in constructing the component wave trains for t=Q, we shall have y=0 for a;= +nl (10), where /( is any integer ; 7/= +J for ,r = «±2-»/ (11), and finally // = — 6 for ,'■ = — a ± 2iil (1 2). The points determined Ly (11) and (12) joined by straight lines give the required component wave trains at ^ = 0, equation (10) serving as a check. Fig. 47 shows these components and the resultant for a number of positions. It will be seen that the resultant form of the 141, 142 DYNAMICAL BASIS 205 vibrating string consists always of two or three straight lines. The component wave trains outside the limits of the string are shown in broken lines to indicate that they must not be taken as actually existing. The components inside the limits are shown by fine continuous lines, the resultant by a bold line. ExPT. 24. — These phenomena can easily be illustrated by an india-rubber cord about a centimetre in diameter and ten metres long, fastened at each end and plucked by the hand. x = -l Fig. 47. — Plucked Stking. 142. Alternative Method for Vibrating String. — Let the string as before extend from » = to x = l, the ends being fixed. And let its differential equation of motion be iVu It'll „dy dt" dof (!)• Then the terminal conditions are represented by At a; = 0, and at a; = /, y = for all values of t (2). The most general integral of (1) which fulfils (2), and corresponds to a periodic motion of the string, may be written 206 SOUND CHAP. IV y = sin — («! cos ft + \ sin ff) + sin - - (rt., cos 2^« + 6, sin 2p!!) + . . + . . . mrx, 1 ■ .. - /o- + sin -j-(«„ cos n'pt + 6,^ sm nj)t) + . . . + (•>J. Now suppose that the initial conditions are given by At ;; = 0, // = Fit) and y = from »■ = to a; = Z (4). Then, we have by (4) in (3), F(x) = rt, sm -J- + a, sin -^ + . + a„ sin — + . . . (5). Again, by differentiating (3) with respect to time to obtain //, and equating to zero according to (4), we obtain = b^p sin \-02 2,p sin !-■•■+ b^np sm — + ... (6), which is satisfied if the h's are all zero. Further, the values of the coefficients in (5) may be determined as shown when treating Fourier's theorem (see equation (18) of article 54). Thus, multiplying both sides of (5) by sin dx and integrating from x = to x = 1, we ' obtain "„=-\ F{x) sin ^^dx (7). •'o Hence equation (3) may be rewritten as a final solution in the form • ''TX , , . Itrx „ , y = a^ sin — Go^pt + a.^ sin cos 2pt + . . 4- «-,i sin — -' cos npt 4- . . . (8), the values of the ft's being given by (7). 143 DYNAMICAL BASIS 207 The treatment of the vibrations of stretched membranes is reserved for Chapter V. 143. Plucked String by Second Method. — Let us now treat the case of the plucked string by this second method. Let the co-ordinates of the point when plucked be (7i, h), the string as before extending from x=0 to a; = I. Then equations (4) and (5) of the previous article break into two parts each, since F{x) - kx/h for h > x > ' and F{x) = ^(L_^ for I >x > h l — h Equation (7) accordingly becomes (9). 2 I kx . WTTCC , 2lk(l-^x) . nvx , ,^ .. This, when evaluated, gives ikV' . nirh ' nVX^-A) ^"' T Hence, equation (8), with the values of the a's substituted from (11), gives the solution for the plucked string under consideration, viz. 2kl^ f . irh . 1TX y = — sm — siu — cos pt + 1 . mrh . uttx ^ , \ /-, n\ + sm sin cos?!;^<-|- ... J (1^)- n I I We see from this equation that the nth tone will disappear when sin mrh/l = 0, i.e. when nhjl is any integer, m say, or, we may say that the wth tone disappears wheu h = ml/n. Thus, if we divide the string into n equal parts and pluck it exactly at any one of the points of this division the nth tone is absent, and this is the tone whose nodes fall upon the points in question. But a node for the wth tone is 208 SOUND f'HAP. IV also a node for the 2nth, 'dnth, etc., tones ; hence all these higher tones are absent also in the case under discussion. We also see from equatipn (11), that the expression for «„ has n^ in the denominator. Hence the series in (12), though infinite, is usually quickly convergent. But its convergence does not depend simply upon the values of the ff's. For, as shown in (12), the amplitude of each com- ponent is affected by the fluctuating factor sin mrh/l, whose values depend upon the co-ordinate h of the point plucked. The amplitudes of the partials in any given case may easily be exhibited graphically as follows. From in a horizontal base line OX, draw a series of radii making with it the angles "jrh/l, iirhjl, . . mrh/l, . . . etc. Mark off on them lengths proportional to 1, l/2'-^, 1/3^ . . . l/n', . . Call these points so found on the radii Pj, P.^, . . . P,„ . etc. Then the projections on the axis of y (at right angles to the a}!;is OX) of these radii OP^, OP.,, . . . OP, etc-., show the series of amplitudes required. The relative intensities of the partials is, of course, found by taking for each the square of the amplitude multiplied by the square of the frequency. The above investigation is strictly valid only for a very thin flexible string whose possible tones form the harmonic series. If, owing to stiffness, the tones natural to the string form an inharmonic series, there is nothing in the act of plucking to coerce them. Hence the motion cannot be expressed by a Fourier series whose components are always of commensurate periods, for the higher partial tones produced will be all slightly sharper than the corresponding tones of the harmonic or commensurate series. Also, it must be remembered that, owing to various losses of energy in friction and sound radiation, all the tones of a plucked string die away, and probably each one of the various com- ponents at its own rate differing from those of the others. The approximate application of this theory to the harp and kindred instruments will be found in Chapter VIII. Ui DYNAMICAL BASIS 209 144. Struck Strings. — Consider now a string extending from x=0 tox = l, struck at the infinitesimally short region from x = h tox = h + dx, and let the velocity instantaneously imparted there be u. Thus for < = 0, we have everywhere 2/ = 0, and also y = 0, except for the length dx at U where y = u. We may thus write as the general equation for the vibration of the stiing — y = sin —Oi sm.pt + sm &2 sin 2pt+ . . . i I . nirx^ + sm ~j~o,, sin npt + . (13). The cosine terms are all absent in order to secure y =0 for i = 0. From (13) we find for the initial velocity (^)o - V^i sill {-n-xjl) + 2p\ sin (27r«//) + . . . + K/)5„ sin (wTra'//) + ... (14). Then, applying to this Fourier expression for (y)Q, equations (1*7) and (18) of article 54, and remembering that x/2 is here represented by I, we find '^'^-^ = 7 C2/)o sin -y- dx = -T M sm --- (fa;. From this, since p = 27riV, where iV is the frequency of the prime tone of the string, we obtain where c is written for udx. Substituting these values of the h's in (13) we find as the solution sought — c ttA. . -irx . 1 . mrh . n-TTX . + — sm ^;— sm — r— sm ?i«< + . . .) ( L o). Wit Thus, here again if the point excited by striking be exactly p 210 SOUND CHAP. IV at one of the nodes of a certain possible tone, that tone is not directly generated by the blow. It should be further noted that the series of tones is again infinite and convergent, yet it is more slowly convergent than that for the plucked string, since we now have in the expression for the amplitude of each component tone n itself in the denominator instead of 71/ as in the previous case when the string was plucked. The sine term is still present, producing, as before, a iiuctua- tion of amplitude dependent on the place of excitation. In Chapter VIII. a reference is made to an extension of this theory to the pianoforte, in which the blow is somewhat different. In the case of the struck string, as with the plucked one, it must be noted that stiffness may make the series of tones natural to the string inharmonic, and so cause a slight departure from the above theory. Further, the component tones will all die away, and each at its own special rate. Hence the above theory strictly applies only to the com- mencement. If terms expressing the resistance were intro- duced at the outset, the corresponding damping factors would have appeared in the solution. 145. Energy of Progressive Waves. — We shall now consider the energy of wave motion, and its distribution in place and time. Let us begin with the progressive wave represented by y = a sin (a{t — .I'jv) (1), and deal with unit area of cross section of the parallel beam of radiation. Let the total energy of the waves per unit volume at place x and time t be denoted by E, and its kinetic and potential portions by K and P respectively. Then, 'since K=2 mass X velocity squared, we find b}' differentiation of (1) and substitution, K= ^pa-Q}^ cos^ o)(t — x/v) (2), where p is the density of the medium, which we may suppose to be a gas. But the sum of the kinetic and 145-147 DYNAMICAL BASIS 211 potential energies of a vibrating point is a constant, and when one form of energy is at its maximum value the other form is zero. Tlius the maximum value of K furnishes us with the value of the constant sum of the variables K and P. Hence we have ^=^max.=i/0«''"' (3). Accordingly P=E-K= ya'w^ 1 - cos^ a{t - xjv)] (4). Further, on multiplying (3) by X, we have Total energy of the progressive wave motion per wave length = ^pKa^ar' (5). 146. Energy Current. — But this is the length passed over in time t, the period of the vibration. Hence the rate of flow of energy per unit time per unit cross section, passing along the line of advance of the wave, or the energy current per unit area, is given by C = \p\c^(i?lr = \pvc^(i? ( 6 ). It should be noted that though K and P vary with x and t, yet E and C are each independent of both x and t. If we wish to compare the energy currents in two gases of different densities, we must remember that v changes also, and in a manner related to the change in p. For it has been shown that in a gas v= ij'-i'pjp. Hence vp = <^pjv= s/^pp. Thus equation (6) may be written C = ^a?a)\plv = ^aW -Jjpp ( 7 ). Thus, if y, the ratio of the specific heats, is the same for each of the two gases, and &> and p are constant, we have Ccr.ayv OT Ccr^a^Jp (8). 147. Let us now ascertain how the energy throughout the wave length is distributed in the two forms of kinetic and potential. To do this we may slightly transform 212 SOUND CHAP. IV equation (2), and then integrate it with respect to x from :ii to .': + X. We thus obtain Kinetic energy of the motion per wave length = ya?- = IpWcc-" (9). Dealing in the same way with equation (4), we find Potential energy of the motion per wave length = \p\aW (10). Hence the energy of a progressive wave motion over any exact number of wave lengths is, at every instant, half kinetic and half potential. To apply the above to waves along a cord or rope of linear density a, it is obvious that a then replaces the volume density p of the preceding equations, and that other modifications are needed. Let us, for example, find the energy current in rope waves. Then v = s/Flcr where F is the tensile force and o- the linear density. Accordingly, va = Fjv = >J Fa, and we have C = ^aW-F/v = la^Q,^ ^IW (11). Thus, for CO and F constant, we see that Cxa^jvox Co-^a^Ja (12). 148. Energy of Stationary Waves. — Talcing now the energy of stationary waves, we shall represent the motion in question by y = 2a sin (27r,t;/X) sin at (1), and use F', K' , P', and C' to correspond with the respective unaccented letters of the last article. By differentiating (1) to t we have 2/ = 2ao) sin(27ri'X.) cos a)<, hence the kinetic energy per unit volume is given by K' = 2paW sin' (27ra;/x) cos' cot (2). Also ii^' = 7v'„„. = 2paVsin'(27r,i;;X) (:i), 148 DYNAMICAL BASIS 213 where K\„^^, denotes the maximum value of K' at place x. Equation (3) shows that E' varies with x, and fluctuates between the values zero and 4:E. (See equation (3) of preceding article.) Again, we have for the potential energy P' =E'-K'= 2pa?(o\l - cos^ wt) sin^ (27ra;/X) (4). From equation (3), by integration with respect to x from X to « + X, we obtain Total energy of the stationary waves per wave length = pkci^aP' (5). On comparing this with equation (5) of the last article, we see that this value is double that for the progressive waves. This should be the case, seeing that the waves now under consideration are compounded of two sets, each like that in the last article, except that one is reversed in direction of propagation. But in the stationary waves, as the name implies, there is no advance of this wave energy in either direction. Hence the flux of energy is zero, or C" = (6). In other words, the value of C here is the algebraic sum of the values of the (7s, the energy currents, in the two oppositely-travelling progressive waves, of which the stationary waves may be built up. But these two energy currents are equal and opposite, so the resultant C' is zero. It may be noted here that while K' and P' each vary with X and t, E' varies with x only, while C is invariably zero. This statement may be compared with the corre- sponding one on progressive waves just after equation (6) of the preceding article. Let us now integrate equations (2) and (4) with respect to X from x to x-\-\. We thus find how the kinetic and potential energies are distributed throughout the wave length. The results are — Kinetic energy of the stationary waves throughout a wave length = p\a^(o^ COS' at (7), 214 SOUND cHAi'. IV and Potential energy of same per wave length = /)XaV(l- cos^eot) (8). It is thus seen that the energy of stationary waves over any number of wave lengths is divided between the forms kinetic and potential in a way which varies from instant to instant, but is independent of place. Thus the energy is all kinetic everywhere when cos (ot=± 1, and all potential everywhere for cos wi = 0. Contrast this with the corre- sponding statement for progressive waves at the end of the previous article. 149. Spherical Radiation in Air. — We now pass to the treatment of wave motions in air, or other gases, with- out restriction to the case of plane waves. We shall simplify the matter, as far as possible, by making at the outset the following assumptions ^ ; — (1) That the action of gravity upon the medium in question is negligible. (2) That the effect of viscosity in the medium is negligible. (o) That the motion is devoid of rotations, and is a vibratory one. (4) That the vibrations are small, so that writing the density p = p^{l+s),s is a small quantity often negligible in comparison with unity. (5) That the velocities and accelerations of the medium are small quantities whose squares and products are negligible. This last assumption introduces an important simplifica- tion in the method of estimating the accelerations. In thinking of the accelerations of a fluid, two views may be taken. Thus, we may follow in thought an individual particle, and note how much its velocity is increased per second. Or, secondly, we may note what is the increase ^ See tile author's article on "Spherical Radiation," etc., Phil. Mui/., January 1908. 149, 150 DYN^AMICAL BASIS 215 per secoiid of the velocity of those particles whichever they are that, at the time in question; occupy a given fixed position. In other words, the first method notes the increase of speed of an individual in a procession, the second method notes the increase of speed of each part of tlie procession in turn at the instant when it passes a given point fixed on the route. In our use of acceleration the first of these methods should, in strictness, be taken. But with the assumption (5) the distinction drops, as the difference between the two is of the second order of quantities which it has been agreed to consider small. Under these restrictions the problem, though sufficiently general for our purpose, is comparatively simple. Especially so when, following Eiemann, we use the condensation s as the final single dependent variable, and avoid using velocity potential, the conception of which might prove an un- necessary difficulty to some readers. The plan of procedure is to derive the differential equation of the wave motion of the medium, to solve it, and apply the solution to the various cases of interest. The differential equation is itself based upon the so-called equation of continuity, and the equation of motion. These we take in order. 150. Equation of Continuity. — The so-called equation of continuity is simply the mathematical form of the state- ment that no matter is created or annihilated in the interior of the fluid. Talie a rectangular parallelepiped, whose edges dx, dy, dz are parallel to the respective co-ordinate axes, and one corner is at the origin. Then the excess of matter escaping over that entering through its faces in the time dt must equal the diminution of stuff inside the parallelepiped during this time. Let the velocity com- ponents of the fluid parallel to the axes of x, y, and z be u, V, and w respectively. Then the matter entering the face in the y-z plane is given by iidt . dy dz. p; for udi is the length passed over in time d(, multiplying by the area dy dz of the face we get the volume which- is passed •2U SOUND CHAP. IV in, finally the density being introduced gives the mass required. Similarly that passing out during time dt at the opposite face, parallel to the y-z plane, but distant dx from it, is given by d(pu) \ pu + -^^~dx\du <^~ (It- Hence the excess passing out of the parallelepiped by the pair of faces in question is ^^rf.; dy dz dt. dx ^ By considering the other velocity components v and w, and the faces to which they are respectively normal, we obtain like expressions. Thus the total mass lost by the parallelepiped in time dt is But obviously the mass lost may also be expressed as — i-dx dii dz dt. dt •' Thus, equating these two forms we obtain dp d{pu) d{pv) djpio) ^ dl'^ dx dy ^ dz ^ '' This is the equation of continuity in its general form. ^^ dp ds , „ ,,..., isow -J- =poj' and lor our purpose, by the initial assumptions, ^— = PoC 1 + *) , ^ + " y = Po r nearly, dx dx dx '^ dx the product of the two small quantities u and dpjd^j being negligible. And the like simplifications apply to the other terms. Hence we may replace equation (1) by 151 DYNAMICAL BASIS 217 ds dii dr die dt dx dy a:: And this is the form of the equation of continuity suitable for small oscillations of an elastic fluid whose weight is negligible in comparison with the elastic forces. 151. Equations of Motion. — We have now to express, for the fluid in our parallelepiped, the condition that the product of its mass into the acceleration equals the moving force to which it is subjected. The product, mass into acceleration parallel to the x axis, is {dx dy dz)p dii/dt. The moving force, parallel to the axis of x, is the excess of that due to normal pressure p on the y-z face behind over that due to the pressure p + dp on the parallel face in front. Or, in symbols, the moving force in question is p dy dz— (p + —-dx )dy dz = — -- dx dy dz. \ dx I dx Hence, equating these two, we find dM du Idp dii, — rr = p^ or - — = — ;-- (3). dx '^dt pdx dt ^ ' But p is some function of p, and whatever the form of the curve co-ordinating p and p, the small portion with which we are concerned may be regarded as straight. Hence, for our small vibrations we may write ^ = «-, a constant (4). dp It should be noted that we are not entitled to integrate equation (4), and draw from the result of that process any conclusions about the general relation between p and p. On the other hand, the relation between p and p must be determined independently, and the general value of dpjdp derived from it. Then the value of a' applicable to our case can easily be chosen. This point will be referred to again later. 218 SOUND From equation (4) we have dp .,dp c'Pod" P ' P ~/>o(l+«) d' ds nearly (5). Then the substitution of equation (5) in (•■i) gives the following equation (6), from which (7) and (8) are written by symmetry. „ ds dii ""dr-dt (B). .3 ds dr dy dt (^), , ds dm "'d^-M (8). These are the required equations of motion. 152. General Differential Equation. — We have to deal with equations (2), (6), (7), and (8). These involve not only the dependent variable s whicli we wish to retain, but also II, V, and ir. These last three we must eliminate. This is easily performed as follows. First differentiate (2) with respect to f. Tlie result of this may be written — (/".s d'u d'^r d/ir ~ dJ'^ dxdt'^ chfdt'^ d-~dt ^ ^' >>'ext, differentiate equation (6) with respect to jj, (7) to y, and (8) to .:, and add the three equations so obtained. Their sum may be written in the following form : — (/-.■-■ ij'-s d's\ d'll d'-v d~ii' d? "*" dy' "*" d?) "" dtlle "*" ilTdy "*" df^/z ^^^^' But, since the order of partial differentiation is in- different, the riglit sides of (9) and (10) are equal. "We may accordingly equate their left sides, tlius obtaining dh , , ,,. = -V- (11), 152, 153 DYNAMICAL BASIS 219 d^s dh d^s where y^s is written for , 2 + Tl + Ji- (V is pronouuced CtU/ OAj (Jj% "nabla"). Equation (11) is the general form applicable to small vibratory disturbances of a light compressible medium in space of three dimensions. As a check, let us reduce it to the form for representing plane waves advancing along the axis of x. Under these circumstances s will be a function of i and x only, being independent of y and z. For the wave fronts will always be parallel to the yz plane, and at every point in a wave front the condensation s has the same value. In this case V^s reduces to — 2,aud equation (11) is replaced by (^"s „ (T'S Now this equation is of the form already obtained (see article 127). And the two can be identified if the a^ here has the value of ir in the previous case. We therefore now proceed to examine this point. And first let us obtain another expression for dp/dp. Denote by U the specific volume. Then p= l/U and dp= — jj^dU, hence >2-«-:^. (12). 2^#^ _ dp U=I!U=B/p = y2j/p (13). dp ,dV'/ir_ ' We thus see that a" has the value, elasticity divided by density, and is accordingly identical with the value of v^ in article 127. For to our order of approximation subscripts to p and p may be retained or dropped at our option. 153. Solution for Spherical Waves. — Let us now pass to the case of spherical wave motion. For this we must transform our cartesian co-ordinates to polar co-ordinates in space of three dimensions. The relation of the two sets of co-ordinates is illustrated in Fig. 48 and expressed in the following equations, the 220 SOUND co-ordinates of P in cartesians being {x, y, z), and in polars x = r sin 6 cos <^,ij = r sin 6 sin . With this simplifica- tion (15) reduces to „ d^s 2ds 1 rf' , , V--^ = X.,+r-.= -:.77i ('•*-•) (16). dr- dr • d>- And, with this substitution, the general equation (11) is replaced by ^d!^(rs) d-(;rs) dr" df (17). 154 DYNAMICAL BASIS 221 The general solution of this equation may be written — rs^f^{r-at)+f^{r + at) (18), where /j and/^ denote arbitrary functions. 154. The solution denotes both diverging and converging spherical waves of any type whatever, but travelling with speed a. If we now restrict ourselves to diverging waves of simple harmonic type, we may write the solution as follows : — c s = - cosk(i' — at) (1-9)) where c and k are arbitrary constants. To obtain the speed of the gas along the radius r, denote this speed by u, and we then derive from equations (6) and (19) -~=a''-= — { - k sin /.■ (r - at) -'^ cos h (r - aol (2 0) dt dr r { ' r J Thus M = —i?\—dt = —\co?,k(jr — at) — — sin k (r — at) \ (2 1). J dr r [ kr ) And if the displacement along r be denoted by f, we have, by another integration ^ = I udt, or, P= - — ] sin k (r - at) + y- cos k (r - at) \ (2 2). kr{ kr ) Let us now compare equations (19), (21), and (22). We thus see that the condensation s has only the ordinary phase change inseparably associated with the advance of a progressive wave. Its amplitude, however, suffers diminu- tion by varying inversely as r. But, owing to this diminu- tion or attenuation with advance, we have in the other equations the factors - and also —^, one applying to a sine and the other to a cosine function. Thus the speed n and the displacement | each exhibit, during advance, an 222 SOUND oHAr. IV additional slight change in phase beyond that always present in a progressive wave. But at such distances as make — a negligibly small quantity, equations (21) and hr {'12) reduce to the simple approximate equations ca II = — cos /.' (?' — at) nearly (23), and ^ = — ^7 sin A' (r — at) nearly (24). Now the activity of the wave motion per unit area being proportional to the square of the amplitude is thus seen to vary inversely as r^. But the area affected by the spherical radiation varies directly as 7-^. Thus the activity of the whole wave front remains constant, as would be anticipated. It is pointed out by Lord Eayleigh {Theory of Sound, vol. ii. pp. 127-128) that for cylindrical radiation between two parallel reflecting planes, the intensity falls off in- versely as the first power of the distance. It should be noted that the attenuations here referred to are those due simply to the spreading of the waves during radiation. Anything of the nature of frictional resistance to be over- come in the passage of the waves through the medium would produce a further enfeeblement. 155. Reflection at Pole. — Let us now regard the two spherical waves represented by equation (18) as a con- verging one and a diverging one, to which the other gives rise by reflection at the pole or centre of the system. And let it be required to determine the relation between /j and /, thereby resulting. The total current across the surface of a sphere of radius r is ■iirr'u, and for r= this current must vanish, since all is symmetrical round the origin or pole. That is, u cannot be infinite and make rht finite for r = 0. But, if ■i'jrr'^ii is to vanish for all values of t, so also will -i-n-r'^du/dt. And this condition 155, 156 DYNAMICAL BASIS 223 is easier to obtain analytically. Thus we have from (6) and (18) 47rr2 ^ = - 4=7r«V'y^ = 4:7ra''{f^(r - at) +/., (r + a;!)} - 47r«.^'{//(r - at) +f,Xr + at)]. Hence, putting dii/dt = for r = 0, we have 0=f^i-at)+/,(af) (25) as the relation between /, and /. ■'1 '^2 But we see from (18) that the right side of (25) is the value of rs for r = 0. Hence we may write as tlie con- dition at the pole rs=0 for r=0 (26); or, rs must vanish with r. Thus, at the pole, a condensation is reflected as a rarefaction and vice versa, somewhat as in the case of reflection at the open end of a parallel pipe ! Hence if an initial disturbance occurs at the surface of a sphere of radius r^, we may have two spherical waves originating thence, one divergent and the other convergent. But the latter, by reflection at the centre, gives rise to a second divergent wave, in which the condensation suffers reversal of algebraic sign on reflection. 156. Partial Reflection of Waves- — We shall now consider the partial reflection and transmission of waves on encountering an abrupt change of density of the medium in which they are travelling. The waves in question may be of the transverse type, along a cord or rope which suddenly changes to another more or less massive rope, the tensile force being, of course, the same throughout both its sections. Or, the waives may be sound waves in a gas which suddenly gives place to a second gas of a different density, but having the same value for 7, the ratio of the two specific heats. A thin membrane of negligible thick- ness, mass, and elasticity may be supposed to divide the two 224 SOUATD CHAP. IV gases. The case of longitudinal waves in a solid rod will be dealt with separately. For the sake of simplicity we shall first adopt a treatment which, where it fails in rigour, is founded upon an assumption that is almost axiomatic. We shall afterwards pass to the more complete and rigorous discussion of the problem. The assumption just referred to is that no change of phase occurs in the act of reflection and transmission other than can be represented by a change in the algebraic sign of the amplitude. In other words, we assume that the phase of a derived wave is the same or opposite to that of the parent wave, hence the ratio of displacement is constant and equals that of the amplitudes. The justification or ground of this assumption is that any other phase relation involves a varying ratio of displace- ments which in turn entails a temporary accumulation of energy at the junction. But for this there is no provision, as the junction is a point or a plane, and has no volume. It now suffices to use the energy equations and the equation of continuity of displacements at the junction. This method is similar to that adopted by Fresnel for the analogous problem in optics. Let the amplitudes of the incident, reflected, and transmitted waves be respectively a, b, and c, and let v^^ be the velocity of propagation in the first medium, v.2 that in the second. 157. Transverse Rope Waves or Sound in Gases. — By the conservation of energy, the energy current in the incident waves must equal the sum of those for the reflected and transmitted waves. In the case of gases, we shall suppose the incidence to be normal, and may take the energy currents per unit cross section. In the case of the waves along a rope or string, the cross section may vary suddenly at the junction, but the energy currents must, of course, apply to the total energy throughout the cross sec- tion for the wave in question. These energy currents are given in equations (8) and (12) of articles 146-147, and the 157, 158 DYNAMICAL EASTS 225 result for our problem when 7 for each gas has the same value, is the same for both cases, viz. 2 72 ^ « -^ c- , — rr- = -1 or a -h- = fjj-' (1), ^1 ^2 where /i = Vi/'yj- The equation of continuity of displacements at the junction gives a + l = c- (2), since we have assumed the phases to be like or opposite. These are our two equations from which to determine hla and cja. Equation (1) divided by (2) gives a — h = jj,c (3). Also (3)^(2) gives ^ = ^, whence b/a = — (4) 2 Again, (2) + (3) yields cla = (5). /j,+ 1 These results are identical with those obtained by Young for light incident normally and analogous .to those by the author for electrical waves along a pair of wires. ^ 158. Longitudinal Waves in Bars. — For the partial reflection of longitudinal waves in solid rods or bars, the conditions and results are somewhat different. Thus the energy currents per unit cross section of the incident wave of amplitude a is ^"^a^p^v^ or ^a^ay^qylv^, the vibrations being expressed by sin at. Hence, if the cross sections of the rods before and after the junction are Sj and Sj, our previous energy equation (1) is replaced by (a' - hy,q^lvT_ = c\q^lv^ (6), the g''s denoting the values of Young's modulus for each part of the rod. This may be written a^-6= = /.V (7) 1 See Proc. Roy. Soc, vol. liv., 1893; Ann. der Phi/.ia-, Bd. 53, 1894. Q 226 SOUND CHAP. IV where ^, ^ ^ . M. ^ s_2P2^2 (g). But we still have, as before, a + l = c. Thus using this and (7) instead of (2) and (1) we obtain a result of the same form, but in which fj, is replaced by im' defined by (8). Hence, for our present case V«=-4^i (9). 2 and cla = —. — - (10). /i + 1 These results are in agreement with those found by Lord Rayleigh, who used the equality of tensions at the junction instead of that of the energies given and received as here done. 159. Alternative Method for Partial Reflections. — Let us now attack the problem more fully, providing for possible changes of phase, but restricting ourselves to rope waves or sound in gases. Let the waves be Incident train, 3/ = «■ sin wit — xjv-^. Reflected train, y = l sin {&)(< + xjv^ + /3}. Transmitted train, y = c sin .{a>{t — xjv^ + ly}. Then the continuity of displacement at the junction, a; = 0, gives a sin wt + h sin {ayt + ^) = c sin (cot + y). But this holds for all the values of t. Hence, on expanding and equating to zero the coefficients of sin at and of cos at respectively, we obtain « + 6 cos /3 — c cos 7 = (1) and . J sin yS — c sin 7 = (2). We have thus two equations out of the four necessary to determine the two amplitudes and the two phases. We accordingly require two more. Now the relation derived from consideration of energy would furnish us with only a single equation, for the energy current is independent of 159, 160 DYNAMICAL BASIS 227 X and t, and therefore of /3 and 7 also. Let us therefore make use of the condition that at the junction there must be continuity of dyjdx. For a sudden change anywhere in the value of dyjdx would involve a finite force acting upon an indefinitely small mass, and accordingly produce an infinite acceleration, which is not present in the waves under consideration. Hence, we first differentiate with respect to x the expressions for the three waves, next put a; = in the results, and, finally, equate the sum of the expressions so found for the incident and reflected waves to that for the transmitted wave. These operations give — a cos mt + h cos (ait + jS) = — jjlc cos (wt + 7), where, as before, /x denotes vjv^. Now this equation holds for all values of t, and therefore, on expansion, gives us the two further relations required, viz. a — h cos ^ — /J-c cos 7=0 (3), and b sin ^ + fic sin 7=0 (4). By substituting (2) in (4) we obtain (fi+ l)c sin 7=0, which is satisfied by 7 = 0. Then both (2) and (4) are satisfied by /3 = 0. Thus cos /3 and cos 7 are each equal to unity, and equations (1) and (3) reduce respectively to equations (2) and (3) of article 157. Thus the results are /3 = 7=0 (5). They are accordingly in agreement with the results found by simpler means. 160. Partial Reflections by Imaginary Analysis. — By introducing imaginary quantities the above working may be shortened a little. Thus, let the incident, reflected 228 SOUND OHAP. IV and transmitted waves be represented respectively by the real parts of in which i denotes v — 1. If the three coefficients a, I, and c are all real this will correspond to like or opposite phases ; but if, while a is real, I or c is partly imaginary, that would indicate a change of phase in the corresponding wave train. Thus if - = — 7= we should have a change of « V2 phase of tt/-! in the reflected waves. The condition of continuity of displacement at the junction x= Q gives a + l = c (6). Similarly the condition of continuity of di/'dx at «=0, ^'^^^^ a -1 = ^.0 " (7). These two equations give the entirely real ratios, ^=_/i:Lland-^ = ^J- (8), a /i+l a, /i+1 as found before, and thereby show that there is no change of phase calling for trigonometrical expression. The above, of course, applies to either rope waves or sound in gases. But the boundary conditions used are not the only ones which could be chosen. 161. Alternative Method for Gases. — To illustrate this point let us now find the reflected and transmitted waves for gases. We shall take, as before, the continuity of displacement for one boundary condition and the equality of pressures at every instant for the other. Let the incident, reflected and transmitted waves be represented respectively by ?i = ae ii4!-xlvi) and I =fc''"(' ->/'•-). 161 DYNAMICAL BASIS 229 Then the continuity of displacement, at x = 0, gives a + h = c (1). The adiabatic relation may be written p = kp' where k is a constant and 7 the ratio of the specific beats. Then, since the increase of density dp is represented by — pd^/dx, the corresponding change of pressure is given by dp = dQcpy) = kypy-\ - pd^jdx) = - r^pd^jdx. Hence, for equality of pressures at the junction at every instant, the sum of pressure increments for the incident and reflected waves must equal that for the transmitted wave. Thus, we find , ia^ id) .^ , iw . ^ And for x = this becomes a — h = ixc (2), where /x = v-^^jv^ as before. But equations (1) and (2) here are respectively like equations (1) and (3) previously obtained for the same problem, and so lead to the same results, viz. hi a = and cla = -. In applying these formulae to a numerical case it must be remembered that the activity or energy currents of the waves reflected or transmitted are proportional to pv X amplitude^, or, if the y's are the same, to V p x amplitude". (See equation (6) of article 146.) Thus for waves incident normally from hydrogen to oxygen, we have P,Ip.^ = 1/16, fi = v,lv^ = 4:. Hence h/a = — 3/5 and c/a = + 2/5. Thus the energy reflected is 9/25 of that incident, and the energy transmitted is ^16 X (c/a)^= 16/25 of the 230 SOUND cHAi'. IV incident. If the waves had originated in the oxygen, then still hja has the same numerical value but of opposite sign which disappears on squaring, so the energy reflected is, as before, 9/2 5 of that incident. The transmitted amplitude is now 8/5 of the incident ! But the energy transmitted is measured by one-quarter of the square of this amplitude, owing to the density being one -sixteenth. Thus the fraction of incident energy in the transmitted wave is, as before, 16/25 ; being now 1/4 of 64/25. Facts of this kind can be best illustrated by rope waves as in the follow- ing experiment : — 162. ExPT. 25. Partial Reflection of Rope Warns. — The phenomena of partial reflections may be suitably illustrated by the use of an india-rubber cord about 8 or 10 metres long, fixed at each end. One part, say about half the total length, may be 5 or 6 mm. diameter, the other part just double that diameter. Then the linear densities of the two parts would be in the ratio 1 : 4, corresponding to wave speeds as 2 : 1. In other words, waves from the thin to the thick part have an index ^ = 2. Hence hja= - 1/3 and c/a= + 2/3. Let the thin cord, near its fixed end, be struck by the hand so as to produce a displacement of about 30 cm. Then a wave may be seen travelling along to the junction, from which point a reflected wave of reversed displacement and amplitude about 10 cm. will be seen returning. At the same instant a transmitted wave of direct amplitude of about 20 cm. will pass on from the junction along the thick cord. Suppose now that waves are originated in the thick part of the cord by a blow of the hand as before. In this case jx is clearly of the value 1/2, hence J/rt= + 1/3 and cja= + 4/3! Thus we obtain visibly the somewhat surprising result, that the partial wave transmitted in the thin cord not only travels twice as fast, but has an amplitude exceeding that of the original wave, part of whose energy is spent in producing a reflected wave. But though the amplitude and speed are 162, 163 DYNAMICAL BASIS 231 greater, the energy current, owing to the smaller linear density, is less than that of the original wave. Indeed, the energies associated with the reflected and transmitted waves are easily seen to be respectively 1/9 and 8/9 of that of the incident wave. 163. Physical Characteristics of Wave Motion. — We may now fitly summarise some of the chief physical charaeteristics of wave motion, and with this we close the present chapter. 1. An elastic medium is required for the propagation of wave motion. 2. The speed of propagation is finite and differs for different media. 3. Eeflection usually occurs when waves reach an abrupt termination of the medium. 4. Refraction, accompanied by partial reflection, occurs at a junction of two difl'erent elastic media. 5. Diffraction occurs when the waves reach an opening or obstacle very small compared with the wave length. C. Eectilinear propagation occurs, forming sharp geo- metrical shadows when openings or obstacles are encountered which are very large compared with the wave length. 7. Interference phenomena may occur under special conditions, say, when reflection gives rise to a virtual second source. 8. Stationary waves may be formed by reflection of a train of progressive waves. 9. Energy is flowing along the line of advance of waves, hence for their maintenance we need a source of energy at the source of the waves. 10. Attenuation of waves is involved by their divergence apart altogether from any dissipation of energy which may occur owing to frictional resistances on the way. 232 SOUND OHAP. IV 11. Pressure is exerted upon an absorbing or reflecting surface by wave motion incident upon it (see articles 594-601). 12. Momentum also is sometimes associated with wave motion (see articles 602-606). CHAPTEE V VIBKATING SYSTEMS 164. Tjrpes of Vibrating Systems. — Having dwelt suflS- ciently ou the mathematical side of acoustics, we pass now to its physical aspect. In the present chapter we shall deal with the more important vibrating bodies or systems, concerning ourselves chiefly with the frequencies of the various natural tones proper to each system, and its division into "the appropriate segments while executing them. "When two or more of these free vibrations or proper simple tones are simultaneously elicited, they are called the partials of the compound tone then produced. The lowest of the partials is called the prime tone or fundamental, and the others the upper partials or overtones. The nominal pitch of the compound tone is usually that of its fundamental, an exception occurring in the case of bells. We defer to Chapter VIII. the consideration of the various qualities of compound tone due to the relative intensities of these partials in any particular case. And we shall there also examine the dependence of these relative intensities on the details of the system and the means by which its vibrations are excited. The various types of vibrating systems may be divided into the following three classes : — The systems whose possible tones (1) form the full harmonic series ; (2) form the odd harmonic series ; (3) are inharmonic. In other 233 234 SOUND CHAP. V words, the relative frequencies of the partials in the first and second classes are represented respectively by 1, 2, 3, 4, 5, 6, etc., and 1, 3, 5, 7, etc.; while those in the third class cannot be expressed by any small whole numbers. The compound tones which may be elicited from these systems require, as we shall see later, a more extended classification than the systems themselves. We now take the different systems beginning with strings. 165. Strings vibrating transversely. — Let us consider the transverse vibrations of a string of linear density o- gms. per cm., stretched by a force F dynes, and fixed at two points I cm. apart. We have to find the pitch of the prime and the relation borne to it by the other possible tones. We have as the general relation, N=vlX (1), where N is the frequency of any wave, X its length, and V its speed of propagation. In order, then, to determine N we require v and X.. But for the present case (see article 117) V = sJ¥f(T (2). Lastly, X must be found from its relation to I when both ends are fixed. This is obtained from article 39, Figs. 20 and 21. Whence we see that the longest waves admis- sible in this case, namely, those giving stationary waves with nodes I apart, satisfy the relation X^^2l (3). We see, by (1), that the greatest wave length Xj corresponds to the smallest value of N. Thus, denoting this by N-^, we have from (1), (2), and (3), iV^ = >V^ (4), giving the frequency of the prime tone or fundamental. For the other tones, we may write in place of equation (3), a generalised one for all possible X's, namely 165, 166 VIBEATING SYSTEMS 235 2Z = X^ = 2\,= 3X3= . . . =,a„ (5), where n is any integer. This is obtained by noting the condition that we must have nodes at the ends of the string, no other conditions being imposed on the remaining parts of it. By equations (1), (2), (4), and (5), we see that • jj K=2l^F/a = nlY, (6). Or, in other words, the possible tones form the full harmonic series, thus putting the string in type 1 of article 164. This second result is apparent in equation (3) of article 142. The first result as to frequency of prime may also be deduced from it, affording perhaps a better way for those familiar with the calculus. 166. ExPT. 26. Vibrations of Strings. — The above results may be roughly verified by simple experiments on the mono- chord. Let the length between the bridges be one metre, and use for the "string" a steel piano wire of say 0'6 mm. diameter. We could find the density of the wire by weighing a given length of it ; suppose it is 8 gm. per cc. Next, tune the wire to iVj= 128 per second, using a standard fork to compare it with, and adjusting the tension by weights until, on exciting the fork and plucking the string, the beats finally disappear. Then to test the result with the theoretical formula, let us write equation (4) in the form 1 N, = ^^Mgl^-^P (7), where M is the mass in grams of the weights used to produce the tension, g is the acceleration (in cm. per sec^) due to gravity, r is the radius of the wire in cm., and p is its density in gm. per cc. For our present case equation (7) gives If = 15 kilograms nearly. To test the theoretical relation of the pitches of the other tones possible to the string, leave the tension the same, but touch the middle of the string lightly with the finger, bow it at about 5 or 6 cm. from one end, and compare the tone thus elicited with a 256 fork. It will be found that the two 236 SOUND CHAP, v are practically in unison. Similarly, touch the string at one of its points of trisection, bow it at about 3 cm. from one end, and compare the tone with a 384 fork. Proceeding in this way, it will be found that equation (6) is practically correct. It may further be verified that simultaneously reducing the tension to one-quarter, and the length to one- half, has no effect on the pitch, which again is in accordance with equations (4), (G), and (7). 167. Stiffness of Strings. — If, instead of the rough lecture experiments just described, the vibrations of strings are examined with great care, deviations from the equations (4) and (6) of article 165 may be observed. This is partly owing to the stiffness of the strings. An investiga- tion of this disturbance VFas carried out by Savart, who found a result that may be expressed as follows : — N'^ = N- + N^^ (1), where N' is the true observed frequency, N that given by equation (4), and iV„ is the frequency with which the string would vibrate, in virtue of its own stiffness, without any tension whatever. Duhamel sought to give a simple ex- planation of this result by likening the effect of the string's stiffness to a constant increase, F^ say, of the tension. Thus, if instead of the actual tension F we wrote in equation (4) of article 165 the value F+F^, we should obtain Savart's equation. It was shown by August Seebeck (1846-47) that this was only an approximation, and that to be more correct we must write N' = N{1+,) (2), where, the N's, having the previous meanings, c is a cor- recting factor. Now this factor increases with increasing values of the radius of the string or the Younsf's modulus of its material. But it decreases with increasing values of the length of the string or the tension to which it is sub- jected. Lord Eayleigh also finds similar results. Thus, if 167-169] VIBRATING SYSTEMS 237 we keep to the same string we are concerned with the second remark only. Hence, if we raise the pitch by- increasing the tension, we approach closer and closer to the case where c=0, and therefore JV'-^N=1. If, how- ever, we raise the pitch by taking smaller lengths of the string at the given tension, we increase c and make (iV'-r-iV) become greater and greater. But this latter is the case when, beginning at the prime, we elicit in succession the various possible tones of a given string. Hence, when the stiffness of the string is taken into account, we see that not only is the prime tone sharpened, but that the others are sharpened still more, and accordingly no longer form with it an harmonic series. With the strings employed in music these facts, however, are practically inappreciable. 168. Yielding of Bridges.- — The modification in the vibrations of strings, due to the yielding of the bridges at their ends, has been mathematically investigated by Lord Eayleigh. The result may be stated thus : — (1) Let the end supports have a negligible mass, but a very large spring or restoring force fj,. Then, to the order of approximation used, the possible tones do not cease to form an harmonic series, but the pitches of all are slightly lowered. The effect of the yielding is the same as that due to an increase in the length of the string by an amount inversely propor- tional to fi. (2) Let the end supports have a negligible spring, but very large mass 3f. The effect is then equivalent to shortening the string to an extent depending on the pitch of the tone. There is consequently a rise in pitch, and this rise is greater the lower the component tone. Thus, the harmonic series is here violated. 169. Longitudinal Vibrations in Rods. Case I. Both Unds fixed. — In dealing with the longitudinal vibrations of bars, we shall follow the plan adopted for the transverse vibrations of strings. Ou"r two general equations are accordingly as follows : — N=vlx (1), 238 SOUND CHAP. V and v= y/q/p {'2). Our third equation, for both ends fixed, may be written 2l = x^=2\ = . . . = n\, = . . . (3), since, as in the case of vibrating strings, we must here have a node at each end. We thus obtain ^, = ~l^l/p (4), 21 and iV„, = wiVj (5), in which iV„ is the frequency of the nth tone, I is the length of the bar, q and p the Young's modulus and density of the material. Thus, the pitches of the possible tones is obtained, and they are seen to constitute the full harmonic series. It should be noted that this result will apply equally to a wire or string, provided it be made to vibrate longitudinally and not transversely, as is also possible to it. It is also noteworthy that the pitch for longitudinal vibra- tions is, to our present approximation, independent of the cross-section of the rod or wire and of the tension, if any is applied. Further, since the value of v for longitudinal waves exceeds that possible for transverse ones in a thin wire (equations (6) and (7) of article 123), the pitch for longitudinal vibrations is higher than any possible for transverse ones from the same wire. 170. ExPT. 27. Longitudinal Vibrations. — The various points embodied in equations (4) and (5), and just enlarged upon, admit of easy experimental illustration as follows : — Take a steel wire about 5 or 10 metres long, and say a milhmetre in diameter. Let one end be coiled round a wrest pin firmly fixed in a heavy bench. The other end of the wire is to be firmly attached to the centre of a wooden tray or board, say 70 cm. by 50, which is itself fastened to a bench or table. The attachments of the board to the wire and to the bench must be so related as to leave the board capable of vibration, and yet enable it to withstand the pull of the wire. Strain the wire by turning the wrest pin until 170 VIBEATING SYSTEMS 239 all appearance of looseness or kinks is removed, and then rub it longitudinally with a rosined wash-leather. It will then be found to emit a musical tone. If the wire be plucked or bowed so as to excite transverse vibrations, the pitch of the tone thus produced will be found much lower, no matter how much it is sharpened by increasing the tension. The tone given by the longitudinal vibrations, on the other hand, will be found practically the same in pitch, whatever the tension, provided both ends of the wire are really fixed. To illustrate equation (5), nip the wire at its middle between the finger and thumb, and rub one of its halves so as to make it vibrate longitudinally in two equal segments. The tone so produced is found to be the octave of the prime. Similarly, if nipped at one of the points of trisection and rubbed elsewhere, it gives the twelfth of its fundamental. Again, making a node at the quarter length, rubbing excites the double-octave of the prime, and so forth in accordance with equation (5). Now, let the length available for vibra- tions be altered by clipping the wire at some intermediate point very firmly on to a heavy block, say a 56 lb, weight. Then the dependence of frequency on length may be shown to follow the law contained in equation (4). By changing to a thicker or thinner wire of the same length and material, we find there is no change in the pitch of the longitudinal vibrations. This is in accordance with (4), since diameter does not occur in the equation. Now let us change from a steel wire to a brass one of the same length. We thus leave the density practically unaltered, while reducing the value of Young's modulus to about a half. As seen from equation (4), this should cause a lowering of the frequency to — ~, or V 2 0'707 of its former value. The musical intei-val in question is thus of the order half an octave (or three whole tones on the tempered scale), for its ratio is the square root of that defining the octave, and hence its logarithm is one-half. Again, we may note from (4) that the alteration in material may be compensated by changing the length to 0"707 of its former value. Both these views of the matter will be found in agreement with the results of the experiment. It is much better, of course, for the sake of comparison of different materials, to have the two wires simultaneously in use, each 240 SOUND OHAP. V with its own board and means of adjusting the length. It may then be shown that about 7 metres of brass wire gives longitudinal vibrations in unison with those from 10 metres of steel wire, the tones being of the same quality facilitates comparison and tuning. The longitudinal vibrations of a rubbed string have been examined in an elaborate series of experiments by H. N. Davis (Amer. Academy, May 1906). He finds a general accord between these vibrations and the transverse ones of which the string is capable. 171. Case II. Rods fixed at one End. — -For the case of a bar vibrating longitudinally with one end fixed and one free, it is clear that the general equations, (1) and (2) of article 169, still hold good. Equation (3), however, is now replaced by 4; = Xi = 3Xo = 5X3 = . . . = (2% - 1)\^ (6). For the fixed end must be a node, and the free end must be an antinode of the stationary waves set up in the rod. Thus, as may be seen by reference to Fig. 21, the length of the rod must contain an odd number of quarter wave- lengths. And this fact is mathematically stated in (.6). On substituting from equations (6) and (2) of article 169 in (1) we obtain N,= s/qJ^IU (7), and N,, = {2n-1)N^ (8). We have in this case, therefore, the odd harmonic series illustrated. 172. Case III. Rods fixed at the Middle. — It is obvious that this case is equivalent to two rods each of half the length, placed end to end, fixed where they meet and free elsewhere. Thus the possible tones will again compose the odd harmonic series as in the case of rods fixed at one end just considered. The pitch of the prime tone, however, will be raised an octave by changing the fixed point from one end to the middle. In other words, the pitch of the prime 171-173 VIBEATING SYSTEMS 241 returns to that of the first case with both ends fixed. Using as before the two general equations (1) and (2) from article 1 6 9 we have as our third equation for this case .2/ = Xi=:3X2= . , . =(2n-l)X,, (9). Thus, by substitution, we obtain N.^Jqfpl^l (10), and iV; = (2«.-l)iVi (11). This case presents an important practical advantage over the other two methods of fixing. For when the rod is vibrating longitudinally with its middle fixed, any two points symmetrically placed along it are always moving at equal speeds, but in opposite directions. Thus the whole vibra- tion is balanced and a very slight constraint is sufficient to start and maintain this fixedness of the middle point. For example, when using a wood rod a metre long and a centi- metre in diameter, holding at the middle with thumb and finger is sufficient to fix it so that rubbing it with a rosined leather near the end excites the prime tone. In the other cases previously considered very massive or rigid supports would be necessary to impose the fixedness theoretically Contemplated. In connection with the present case it should be noted that it is the fixture of the middle and not the freedom of the ends which suppresses the evenly-numbered tones of the full harmonic series. Thus if, while the ends of the bar were free, the middle were alternately fixed and free, the bar would then yield all the tones, odd and even, of the full harmonic series. 173. Torsional Vibrations of Rods. — A cylindrical rod of circular section, whether solid or hollow, may be twisted by couples in such a manner that each transverse section remains in its own plane and unchanged in shape. The forces by which the twist is resisted depend upon the elastic constant termed rigidity. This has been dealt with in 242 SOUND CHAP. V Chapter III., and was there denoted by the symbol n. To form our differential equation let us consider a cylindrical tube of radius r and thickness dr, with axis coinciding with the axis of x. Let it have a free end at the origin, extend along the positive direction of the x axis, being fixed at some distant place. Now suppose the tube to be in equilibrium under the following conditions. At the origin let a couple K act about the axis, and along the tube imagine couples to be continuously distributed of amount k per unit length. Then at the point x the couple will be K->rkx. Let the value of the angular displacement at x be 6, then at radius r this involves a tangential displacement r6. But 6 decreases with increasing x, hence the amount of the shear at « is — d(;r6)i'dx or — rdO/dx. In Chapter III. this quantity was denoted by p^ ; it may be regarded as the tangent or the circular measure of the change of angle between the sides of an elementary cube. Now, by defini- tion, rigidity is the tangential force per unit area divided by the amount of shear. Hence, in the present case, the force per unit area is — nrdO/dx. And, for the whole ring, the force is ^irrdr times this. Lastly, the moment of the force is r times the product of these two quantities. Hence, equating this expression to the couples producing the state of twist in equation, we have - 2-Trnr^drdejdx = K+ kx (1). But this involves K and x, and we wish to obtain an equation which takes cognisance only of the state of things at a point. Hence, differentiate (1) to x, and we have - 2'wnrHrd''eidx" = k (2), an equation confined to the point under consideration, and therefore independent of any special conditions which were adopted to obtain it, except the vital one that at x the total couple increases at the rate k per unit length. Now the slice of the tube of length dx situated at x is in equilibrium under the action of the external couple kdx and the reactions 174 VIBRATING SYSTEMS 243 of the neighbouring portions of the tube. Hence, if the external couple be removed the slice would be acted upon by the couple —Mx. Accordingly this quantity may be equated to the product of the moment of inertia of the slice into its angular acceleration. Now the radius of gyration of the slice is r, and its mass is 2'n-rdrdxp, where p is the density. We thus obtain •27rrMrdxp(P0/dt^ = -kdx (3). Comparing (2) and (3) we have d^e _ ^d^e df~'"dJ' .2-^^37; (4)> where v'' = njp (5). These equations being independent of r apply equally well to a tube of finite thickness or a solid rod. 174. Torsional Vibrations compared with Longi- tudinal. — We thus see that the differential equation for the propagation of torsional vibrations is identical in form with those obtained in Chapter IV. for the transverse disturb- ances of stretched strings, bars vibrating longitudinally, and the propagation of sound in gases. Hence, torsional waves may be propagated in either direction with speed v = sjnjp. For a fixed end the condition is obviously ^ = 0. For a free end, as seen by putting K and a; = in equation (1), we have ddjdx = 0. Or, in other words, we have a node at the fixed end and an antinode at the free end. Hence the whole theory of rods vibrating torsionally is like that for rods vibrating longitudinally. The speed of propagation and the frequency of any tone in any case being obtained by substituting n for q in the appropriate formulae. In article 78, Table V. shows that §/»= 2(1 4-0-), hence NIN' = vjv' = J^ = ^2(1+0-) (6), where the dashes denote the quantities applying to torsional vibrations and the others those for longitudinal ones. 244 SOUND CHAP. V Moreover the values of a for all solids lie between and 1/2. Hence ^/qjn lies between a/2 and V 3, i.e. between 1''414 . . . and 1-7.32. . . . For an ordinary value of a, say 1/3, we have s/qjn= iJ^jZ = l-&2> or 5/3 nearly. Thus, we may say roughly that torsional vibrations under any given conditions are of the order a major sixth lower in pitch than the corresponding longitudinal vibrations. ExPT. 28. Torsional Vibrations. — In the production of torsional vibrations the rods should be well polished, and then, if round, excited by a rosined leather turned round right or left. If the rod is not round, two bows may be used simultaneously crosswise and in opposite directions ; for example, one up and one down at the ends of a horizontal diameter. If only one bow is available care must be taken to avoid producing the transverse vibrations. By comparing the frequencies of the longitudinal and torsional vibrations, and using equation (6), the value of Poisson's ratio might be determined for the material of the rod. 175. Parallel Pipes. — We pass now to the elementary theory of columns of air or other gases vibrating in parallel pipes, open or stopped. Owing to the parallelism of the pipes we may assume the motion at any instant to be the same at each point in any one cross-section. Hence the methods of the previous articles apply here also. We may therefore write as follows for the frequency and speed of waves : — N=vlX (1), and V = s/'^iPU= Vo s/l+at (2), where 7 is the ratio of the elasticities and specific heats, P is the pressure in absolute units, U the specific volume, Vq is the speed of propagation at 0° C, a is the coefficient of expansion, and t the temperature in degrees cent. Case I. Ojxn Pipes. — For pipes open at both ends, since motion is possible there, and change of pressure a vanishing quantity, each end must be an antinode of the 173 VIBEATING SYSTEMS 2-45 stationary vibration. Again, the middle of the pipe may be either a node or an antinode. Hence we have, where I is the length of the pipe and X the wave length, 2l = X^ = 2\.,= -i\= . . . =n\^ (3). We thus obtain for the pitch of the fundamental, iVj = Vo s/l + at/21 = J^^j2l (4), and for the series of over-tones natural to the pipe, N^ = nN, (5). It is thus seen that equations (1) and (3) are independent of the gas used for the pipe, but that (2) (and hence (4) also) depends both on the gas and its temperature. Obviously the pitch is raised by rise of temperature. It should also be noted, as shown by (4), that the pitch of a given pipe does not depend simply on the density of the gas used, as so often implied in text-books, but upon the value of 7 also. Thus, if we change from oxygen to hydrogen, since both have two atoms in the molecule, the values of 7 are practically identical, and so for this case we have velocities and frequencies inversely proportional to the square roots of the densities. But if the comparison is made between hydrogen and carbon-dioxide, then the value of 7 changes from about 1'4 for the hydrogen to about 1'25 for the tri-atomic carbon-dioxide. And this change must be taken into account as well as that of the densities. An alternative method of deriving equation (4) is as follows : — Let a condensation start from the mouth of the pipe and arrive at the open end. It is there reflected as a rarefaction, and again traverses the length of the pipe to the mouth. It is here again changed by reflection, and so becomes a condensation as at first. Consequently, any state of density and motion completes its full cycle of possible changes by twice traversing the length of the pipe. Hence, the frequency of the tone in question will be the number of times that the double-length of the pipe can be traversed 246 SOUND CHAP. V by the wave motion per second. In other words, the frequency is the quotient 'y/2/, and this is what (4) expresses. 176. Case II. Stopped Piins. — For stopped pipes equations (1) and (2) still hold, while (3) is replaced by 4/ = Xi = 3X2 = 5X3= . . . =(2h-1)X„= . .. =(6), since we must always have a node at the stopped end and an antinode at the mouth. We accordingly obtain the following solution for this case : — N, = v^Jl+atlU (7), and N^ = C2n-l)X, (8). Here again it is instructive to derive (7) by the alternative method as used for (4). Thus, a condensation starting from the mouth traverses the pipe to the stopped end, where it is reflected as a condensation and again reaches the mouth as such. It is here changed by reflection into a rarefaction, and again passes to the stopped end and back without further change of its nature. Then, by the second reflection at the mouth it is changed into a condensation as at the outset. This completes its cycle of possible changes and involves traversing the length of the pipe four times. Accordingly the frequency of the prime is vjAl as stated by equation (7). 177. ExPT. 29. Vibrations in Organ Pipes. — The actual state of things inside a pipe when sounding may be experi- mentally illustrated by the following two methods. First, a little fine dry sand may be scattered upon a small imitation tambourine about 2 or 3 cm. diameter. This is then sus- pended by a thread at various positions within the pipe when sounding as shown in Fig. 49. If it is hung at the open top of the pipe, the sand will always be agitated, no matter what tone the pipe is made to produce. For, as shown by the letter A, the top is always an antinode what- ever tone is being produced. The same result would follow if the tambourine were lowered to the mouth of the pipe. If, however, it be suspended at the middle of the pipe 176, 177 VIBRATING SYSTEMS 247 the effect will vary with the tone produced. Thus, as stated in the first column headed prime tone, we have a node at the middle for this tone, consequently the sand will remain at rest there while the pipe speaks its prime. The same remark applies to the third tone possible to the pipe as shown in the third column. "Whereas for the second tone we have an antinode at the middle, thus the sand would be agitated Prime 2nd. Partiat $rct. Partial Tone Tone Tone -A-- -A- --A h|^ / Coal gas ': . M 1 r --N- A- in---A- A --N A N --A FiQ. 49.— Open Pipe there when the pipe speaks the octave of the prime, as it may easily be made to do by overblowing. If, now, the tam- bourine be again supplied with sand and placed one-quarter or three-quarters down the pipe while the octave is sounding, it will remain at rest, indicating nodes at these places as marked in the second column on the figure. The second method of illustration is that by Koenig's manometric flames. This is of great importance, and we shall have occasion to make further reference to it. The manometric capsules F, G, and H have coal gas supplied to 248 SOUND CHAP. V them by the pipe shown, and the jets are lit. They are each divided from the interior of the pipe by their membranes. Hence, when the pipe is sounding, a capsule at the place of maximum change of pressure will have its membrane agitated. And this effect is rendered visible by the flickering of the corresponding jet. On the other hand, a capsule at a place of no change of pressure for a given tone will remain steady while that tone is sounded, provided the pipe is not simultaneously producing one of its other tones. Now the places of maximum change of pressure are those of minimum displacement, i.e. nodes. Similarly, the places of no changei of pressure are those of greatest change of dis- placement, or antinodes. Hence, for any tone producible by the pipe the flame at F might be expected to be quiet. As a matter of fact, since it cannot be quite at the end, and this method is extremely sensitive, it will always flicker a little. The jet at H, on the other hand, will flicker very much and be extinguished when the prime tone is sounded. The jet Gr will behave in an intermediate manner for the prime tone. But when, by overblowing, the second tone, or octave of the prime, is produced, the jet G may be extinguished while H is only slightly affected. In conjunction with the mano- metric flames a rotating mirror is often used, but for the present purpose it reveals a distinct flickering in the flames under all circumstances. The special flickering of the jet when at a node can be seen quite well enough without the mirror. It is therefore better to dispense with it for this experiment. It was shown by K. Marbe^ that on passing a paper rapidly through a smoky Konig's flame a record could be obtained indicating the frequency and quality of the sound thus examined. 178. ExPT. 30. Pipe Vibrations illustrated by Projection Model. — The stationary vibrations in an open or closed pipe may also be illustrated by the optical lantern and the working model devised by the writer and shown in Fig. 50. In this model, which is used as a lantern slide, three wires are provided, each capable of rotating in concert, being coupled by wheels as illustrated in the photograph. One of these wires is a right-hand screw whose projection is a sine graph ; 1 Phys. Zeitschr., Aug. 1, 1906. 178, 179 VIBEATING SYSTEMS 249 on turning this wire by the handle provided, its projection represents a progressive vrave moving, say, in the positive direction. At the same time, a second vrire in the form of a left-hand screw gives a projection representing a negatively progressing wave. The third wire being itself in one plane and in the form of a sine graph yields, on turning, a projec- tion representing a stationary wave. And the model is so arranged, that at any position the stationary wave repre- Fio. 50.— Pkojection Model for Stationary Vibrations. sented by this third wire is that which is formed by com- pounding the progressive waves represented by the other two. Further, the brass plates at the ends are movable, and may be so adjusted that each end represents an " open end," or so that one end is open and the other a " stopped end." It thus illustrates graphically on the screen the vibrations of an open or a stopped pipe. 179. Corrections for Open Ends. — By the elementcary theory of parallel pipes advanced in article 1*75, the various proper or natural tones were determined. And in accord- :^50 SOUND CHAP. V aiice with that theory their relative frequencies for open and stopped pipes of a given length may be represented as follows : — Open pipe— 2, 4, 6, 8, 10, 12, etc. Stopped pipe 1 1 ., <5 7 q 11 etc or same length J That is to say, we established — (1) The pitches of the prime tones. (2) The simple octave relation between the open and stopped pipes of same length. (3) The frequencies inversely as the lengths of the pipes of either class, open or stopped, and (•4) The strictness of the harmonic series for each class of pipe. A closer examination shows that all these four points are subject to modification, and traces these modifi.cations to the phenomena occurring at the open ends of the pipes. The correct theory of the open organ pipe was discovered by Helmholtz, and given in his classical memoir of 1859 (Theory of Air Vibrations in Open-ended Tules). Lord Rayleigh has also given a theory, and his results are in agreement with those of Helmholtz. These theories are, however, restricted by two assumptions : first, that the diameter of the end of the pipe is small compared with the wave length; and second, that it is fitted into an infinite plane flange to which the axis of the pipe is perpendicular. No theoretical solution of the problem of an unflanged open pipe seems yet to have been given. Under the above restrictions, the correction for an open end to be added to the length of the pipe was determined by Helmholtz in 7r certam cases to be -R, while Lord Eayleigh obtained the value 0-8 2i?, where B is the radius of the end. As the unflanged end was not amenable to theory, Eayleigh sought to determine the effect of the flange experimentally. The 179 VIBKATING SYSTEMS 251 result was that 0-2B was that part of the correction due to the flange, or, when repeated by Mr. Bosanquet, 0-2 5i^. Thus, subtracting the mean of these two values from the full theoretical correction, we obtain 0-6i2 as the cor- rection for an unflanged open end, small in comparison with the wave length. Direct experiment by D. J. Blaikley on the correction for an unflanged open end, the tube being of thin brass about two inches diameter, gave values varying between 0-564^ and 0-595ii, the mean correction being 0-576i? (for details see article 531 in Chap. X). Helmholtz has shown that the correction for the open end is a function of \ and that for very short X the correction tends to vanish. Eayleigh has also shown that for certain cases the correction for an open end is a function of the length L of the neck, and that the correction is —R when ^ 4 Q L vanishes ; and approaches, but cannot reach ^ — B for L infinite. The mathematical theories of the phenomena occurring at the open end are too abstruse for introduction here either wholly or in part. A simple consideration will, however, suffice to show that the elementary theory previously developed needs supplementing. Thus, in deal- ing with the reflection at an open end (articles 138 and 17 5), it was assumed that the particles at the end were quite free, and that the change of pressure and density were negligible. But obviously these assumptions are not strictly correct. The air outside the end is not devoid of inertia. Hence, the vibrations extend beyond the actual end of the pipe. Let us note the consequences of this extension. (1) There is communication of sound to the external air. Hence the sound of the pipe may be heard. Thus, near the open end, we have insich the pipe stationary 'plaim waves, and outside, it progressive spherical waves. 252 SOUND CHA1-. V (2) The intensity of the waves reflected from the open end is less than that of the incident waves, since some energy escapes. Thus the sound of an organ pipe very quickly dies away unless the blast is maintained to supply the energy thus dissipated. (3) The virtual length of the pipe is increased, and thus the pitch is lowered. 180. It is with the third of these consequences that we are now concerned. The theories and experiments hitherto mentioned give 0'6B as the correction for the upper open end of an organ pipe. But if this end needs correction on account of lack of openness, the mouth of the pipe, being far less open, must need a still greater correction. According to Cavaille-Coll, the whole correction required for both ends is 3^E. This leaves about 2-7-S as the correction for the mouth alone. Observations by Eayleigh confirm the necessity for a very large correction for the mouth. Let us now apply these results to a stopped and an open pipe of equal lengths and diameters. Then we see that the stopped pipe needs the correction for the mouth only, while the open one needs the additional correction for the 'upper open end. Thus, not only are both lowered in pitcli by this correction of the elementary theory, but the simple octave relation between them is destroyed. In other words, to produce a given tone the stopped pipe must be shorter than the elementary theory gives, and to produce the exact octave of this the open pipe of same diameter must be shortened still more from the length given by the simple theory. Or, to put the matter in symbols, equations (4) and (7) of articles 175 and 176 will now need replacing by the following : — , For open pipes, N = (1), ■^{l + c + c) V, »Jl+at For stopped pipes, iVj = (2), 180 VIBRATING- SYSTEMS 253 where c is the correction for the mouth of the order 2-^ R, and c'=0'6-B nearly is the correction for the open end. These end corrections, while considered as fractions of the radius of the pipe and independent of wave length, have no disturbing influence on the harmonic series of tones for each pipe. These are accordingly represented to this approxima- tion by equations (5) and (8) of articles 175 and 176, the values of N^, from this article being now used in them. But this view of the matter is valid only while the assiimp- tion of the theory is fulfilled, viz. that the wave length is large in comparison with the diameter of the pipe. Thus for narrow pipes and the lower tones we have ■ practically the harmonic series. But for wide pipes the proper or natural tones distinctly depart from the harmonic series. Thus Helmholtz says, "for wide open pipes the adjacent proper tones of the pipe are all somewhat sharper than the corresponding harmonic tones of the prime." Thus to summarise : — (1) The elementary theory gives the first rough approxi- mation to the pitches of the various tones natural to open and stopped pipes. (2) The introduction of an end correction as a fraction of the radius of the pipe gives the second approxi- mation. This lowers the pitches of all tones, and destroys the simple octave relation between the open and stopped pipes of given length and diameter, but leaves the ■ harmonic series un- disturbed. (3) While, lastly, the recognition that the end correction depends slightly upon the wave length also disturbs the harmonic series for each pipe, specially so if they are wide. It should be carefully noted by the student that throughout this chapter we are discussing the natural tones, or tones proper to each system vibrating freely. That is, in the case of pipes, we are speaking of the tones which 254 SOUND CHAP. V could be most easily and powerfully excited ia them. Other tones slightly different in pitch may he elicited under the influence of forcing. When the tones of an organ pipe are produced and maintained by the blast in the usual way, that is an example of forcing for any tones lying off the harmonic series, because the whole motion must then be strictly periodic. Hence all the components that speak must constitute an exact harmonic series. The question as to what are the frequencies of the free vibrations is still, however, of the utmost importance. For evidently upon this depends the presence and relative intensity of the various tones simultaneously elicited under the influence of forcing. 181. Helmholtz has shown that a cylindrical pipe may be constructed with an end so formed as to require no end correction. The end in question is slightly widened and of trumpet form. The curvature of its longitudinal section is everywhere convex to the inside. The radius of curvature of this section decreases continually from the cylindrical part till at the mouth it is infinitesimal. The form of the section and that of the stream lines inside are confocal hyperbohe. The area of the mouth is twice that of the cross section of the cylindrical part. Or, in other words, the radius of the mouth is x/2 times that of the cylindrica,l part of the pipe. 182. Conical Pipes. — In the discussion of parallel pipes we had only plane waves to deal with whose treatment was possible by elementary methods. In the present case of conical pipes we have to consider spherical waves,^ and must therefore use the more general theory developed in articles 149 to 155. We commence with equation (17) in article 153, viz. where a is the speed of propagation of the wave motion, ' See tlie anttim-'s article oil " Vilirations in Conical Pipes," etc., Phil. Mag., Jaunary 1908. 181-183 VIBEATING SYSTEMS 255 s the condensation, and r the radius vector from the centre of disturbance. To apply this to conical pipes we must use the form of solution corresponding to stationary waves. Thus, let 7^s be everywhere proportional to cos kai. Then d\rs)/dt^ = - h\\rs), and equation (1) transforms into '^^ + /cVs)=0 (2). The general solution of this may be written rs = {A cos kr + B sin kr) cos hat (3), where A and B are arbitrary constants. These are to be determined for each case by the position and nature of the ends of the pipe. There will, accordingly, be a number of separate cases to consider. 183. Case I. Open Unds. — Pirst, let both ends of the conical pipe be open. Then obviously the condition at each end is approximately s = 0. For at the ideal open end there can be neither condensation nor rarefaction. Let the co-ordinates of the ends of the pipe be r^ and r^, measured, of course, from the vertex of the cone produced. Then we have from (.3) for the terminal conditions, A cos kr^ + -S sin kr^ = 0, and A cos kr^ + B sin kr^ = 0. Whence, by the elimination of AjB, we obtain sin k(r2 — r^ — 0, or k(r2 — r^) = mr. This may be written fid, r, - r = n\p, or iV„ = (4), where n is an integer ; for, since s is proportional to cos kat, k = 1'KNja = 27r/A., N being the frequency and X the wave length of the motion. Thus, for a conical pipe with open ends the pitch of the prime tone and the form of the series of the other natural tones are like those for an open- 256 SOUND CHAP, t ended parallel pipe. This might have been anticipated from the similarity of the differential equations, and the conditions for the open ends in each case. There is, however, this slight difference, that r — 1\ is the slant length of the conical pipe and not its length parallel to the axis. But if the conicality is slight, this involves only the second order of small quantities. Lord Eayleigh has also shown that the conicality of the pipe affects the wave length only to the second order of small quantities. Theory of Sound, vol. ii. p. 115. As to the segments into which the pipe is divided when emitting its higher natural tones, it follows from equation (4) that the antinodes are equi- distant. It will be seen from Case IT. that this simplicity does not extend to the nodes. 184. Case II. Closed Ends. — The condition at closed ends is obviously m = 0. Consequently dujdt = there also. But by equation (6) of article 151 di(,jdt= —crds/dr if u denotes the velocity along r. We may thus write as our condition for a closed end ds/dr = 0. Applying this to equation (3) of article 182, we obtain ^(cos k)\ + ki\ sin kr-^ = BQcr^ cos kr-^ — sin kr-^, and ^(cos h\ + kr., sin kr^ = B{ki\ cos kr.2 — sin h:^. On dividing out by the cosines and writing tan ^j and tan O^, for fo'i and kr^^ we may eliminate AjB between these equations, the results being written in the forms A tan ^1 — tan h\ . ,a i \ i- fa i \ B 1 + tan 6i tan lci\ or h\ — ta.\i~^kr.2 = h\ — td,i\'^ki\ (5). Here again, it 'may be shown (Rayleigh's Sound, vol. ii. p. 115) that a slight conicality has only a second-order influence on the pitch. The complicated form of equation (5) shows that in a conical pipe the nodes will not be equidistant. We shall presently determine where they are 184-186 VIBEATING SYSTEMS 25? situated in the important case of a compleite cone with open base. 185. Case III. Closed Cone. — To treat the case of a cone continued to the vertex and with the base closed, we have simply to write ^i = in equation (5), and B the slant length of the cone for r.^. This gives tan kR = kB (6). To solve this equation, which we may regard as tan x = x, we may proceed graphically. Thus plot the two graphs y = £B and y = tan x. Then their intersections will give the roots required. See Fig. 51 as an illustration of this. The equation may also be solved by successive approxima- tions. Proceeding thus, Rayleigh finds (Theory of Sound, vol. i. p. 334) — — = - = 0, 1-4303, 2-4590, 3-4709, 4-4747, 5-4818, IT IT = di, 6^, Ob, di, Or,, Of,, 6-4844, etc. Ov say (7). Thus these quantities, each multiplied by tt, give the first seven values of kB in equation (6). Now, since Oi = 0, we may write (k^B) = irOn+i- But we also have as the general relation ^,j = 2irN^la. Hence, we may write for the frequency of the nth. tone natural to the closed cone iV. = ^^„+, (8). Thus the frequencies are directly proportional to the speed of sound, inversely proportional to the slant length of the cone, and the relation of the various possible tones in the series follows that of the roots given in equation (7). 186. Case IV. Open Gone. — We now consider the case of a complete cone with base open. At the open base, as in article 183, we have the condition rs=0. And it was shown in article 155, equation (26), that at the pole s 258 SOUND cHAr. V of spherical waves rs = ; this, accordingly, is our condition for the vertex also. So that although one end is open and the other end closed, we have the apparent anomaly that the same condition applies to each. Hence if li is the co- ordinate of the base, i.e. is the slant length of the cone, we have from (4) B = m\j2, or 7V„ = ma/2B (9), where M^ is the frequency of the 7)ith natural tone, X^, its wave length, and a the speed of sound. Thus, we have the strange phenomenon of a tube open at one end and stopped at the other, in the form of a complete cone, giving practi- cally the same fundamental and the same full harmonic series of other natural tones as are obtainable from a parallel pipe of the same length open at both ends. ExPT. 30(1. Conical Resonators. — That this is the case may- be easily verified experimentally by a cone of zinc and a set of tuning-forks of relative frequencies 1, 2, 3, «tc. Apart from end corrections, the cone must be of the same length as a parallel pipe open at both ends, whose prime is in resonance with the lowest fork. This cone will then respond to each of these forks, thus verifying the surprising relations in question. 187. We see from the first form of (9) that the wave length is inversely as the order of the tone produced, hence the antinodes are all equidistant. This, however, does not apply to the nodes. To determine the positions of the nodes we must refer to equations (6), (V), and (8). Now, equation (8) gives in terms of 6 the various values of N for a closed cone of fixed slant length E. Let us, however, substitute the variable r for the constant B, and, dropping the subscript of JV, rewrite this equation as follows : — 2A>/« = 611, 02, 0s, 9i, 0i, 0^ or 0^, etc. (10). We may now regard both iV and r as variables which must satisfy equation (10), r being the slant length of a closed cone. Again, equation (9) gives the frequencies of the 187, 188 VIBRATING SYSTEMS 259 various tones natural to the open cone. Let us rewrite it, dropping from iV its subscript, and writing for m on the right side the series of natural numbers which it represents. We thus obtain 2iVB/a = 1, 2, 3, 4, 5, 6 or 7, etc. (11). Here we consider the slant length B of our open cone as a constant, and iV to vary in accordance with the numbers on the right side of the equation. Now by dividing equation (10) by (11) we eliminate the variable iV, and obtain the required relations between r and H, viz. r _ (9i ^1, or Oo^ 6-,, 6^, or 6^ 6^, 6^, 6^, or 6I4 R 1 2 3 4 ^1. ^2) ^3) ^4' 01' ^5 ^1> ^2. ^8> ^4. ^5. or ^6 5 6 01, 02, 03, 04, 05, 0e, or 0, or etc. (12). 188. It is necessary to cross combine the right sides of (10) and (11) in this way to obtain all the values sought. The denominator of any one of the fractions on the right of (12) shows the order of the tone being emitted by the pipe, while the various values of rjB obtained by taking the various 0's in the numerator of that fraction locate the nodes for the tones in question. The series of 0's in each numerator is finite, being limited by the obvious fact that rjB cannot exceed unity. The first few nodal positions are given in Table VIII. They are also exhibited graphically together with the positions of the antinodes in Fig. 51. In this diagram the graph y = x is shown by a full line, and the branches oi y = tan x by broken lines. The abscissse of their intersections show the values of the roots of the equation tan x = x. To show the segments of one pipe of fixed lengths, when emitting its various tones as expressed by equation (12) with B constant, a number of such diagrams would be needed. To 260 SOUND Table VIII. — Nodal Positions of an Open Cone Order m. of Natural Tone, i.e. Denominator of Fractions in Equation (12). Nodal Positions, i.e. Values of r/B = 9/111 in Equation (12). 1 2 0-7152 3 0-4768 0-8197 4 0-3576 0-6148 0-8677 5 0-2861 0-4918 0-6942 0-8949 i 6 0-2384 0-4098 0-5785 0-7458 0-9136 1 7 0-2043 0-3513 0-4958 0-6392 0-7831 0-9263 Values of 9'a in "j Numerator of «i 92 «3 «4 9, 9e ^7 Fractions in Equation (12). . 1-4303 2-4590 3-4709 4-4747 5-4818 6-4844 avoid this repetition, a series of pipes is shown in the upper part of the diagram. These are of various lengths, so that the longest at the top is sho-wn -with the segments corre- sponding to its seventh natural tone. Passing down the series we reach the last pipe, the shortest of all, with one segment only as when emitting its fundamental. The antinodes are indicated hy small circles, and the nodes hy crosses. The equidistant positions midway between the antinodes are shown by dots. Experiments by D. J. Blaikley have shown the existence of these nodes in positions slightly displaced from equidistances, each node being moved towards the vertex from the corresponding equidistant position. Further, it was found by Blaikley that the displacement was the greater for those nodes which are nearer the vertex. Or, in other words, the nodes in tlie cone are at increasing distances apart, reckoning from the open end, and at the apex of the cone is a node common to all the notes. These nodal positions were established by Blaikley with an experimental bugle made in sections. On 188 VIBEATING SYSTEMS 261 taking this to pieces, thin metal diaphragms were inserted at the positions of the nodes for a certain note, and their 7th. [• ^^ i =^ 1 1 ,71 J ^- J i ..1 -i / / 200 6th. { 17-5 -" /> f __^ /: K -X J X- )— X- / ' ■ — I > 1 A 1 1 1 5th. { 150 _^ — T^ / j / / / >1 -s — •S II -s: A 5 4th.{ 1 C12-5 ft -10-0, / ^ A 1 t i " / 7-5 J / -s: /'■■ Co 50 / / / - 7st.{ 2-5 "■=^ /; ' / 1 / 1 1 1 1 1 1 ( / 1 / / 1 1 1 * X 3 6. /i e 3 5?r 4 JT 7 Radi IT ins TVie tf's s/ioH/ f/ie ualuee of the roots of the equation tan x^x Fig. 51. — Vibeations in Open Cones. ' presence was found not to prevent the production of the note in question. The diaphragms had each a few small 262 SOUND oHAi'. V holes to admit the passage of the player's breath, but prevented all free vibration at the place. 189. Stretched Membranes. — We pass now to the treatment of the vibration of stretched membranes. The theoretical membrane of. acoustics is an infinitely thin and perfectly flexible solid sheet of uniform material and thick- ness. It is stretched in all directions in its plane by a tension great enough to be not sensibly increased by the small displacements supposed to occur. The problem is therefore very similar to that of the transverse vibrations of strings, but the vibrations now extend over a surface instead of, as before, only along a line. To form the differential equation we need a relation between the curvature of an element of surface and the restoring force thereby called into play. This may be obtained simply as follows : — Imagine each point of the element in question to experience an infinitesimal normal displacement. Then the work done is susceptible of two expressions. First, it is the product of the excess of pressure on the concave side into the increment of volume described by the element. Second, it is the product of the tension of the surface into the increment of area acquired by the element. Then on equating the two equivalent expressions we can derive the relation sought. Thus, let the plane of xy be tangential to the surface. And, at the point of contact, take as the element an infinitely small rectangle of sides a and /3. Let these sides be parallel to the axes of x and y respectively, and let the corresponding radii of curvature of the surfaces be r^ and r . Then, as the curvature is everywhere very small, the normal dis- placement of each point may be denoted by dz. Hence our first expression for the work of a small normal displacement may be written dW = pa^dz (1), where p is the excess of pressure on the concave side of the surface. To obtain the increment of the surface in 189, 190 VIBEATING SYSTEMS 263 consequence of the displacement, we need an expression for the increase of each side of the rectangle. Thus a a-\- da da r r^-\-dz dz' since each of these expressions is the circular measure of the angle subtended by the side of length a at the centre of its curvature. Hence da = —dz, similarly dl3 = —dz. 190. Thus, if T is the tension of the membrane, our second expression for the work of the small displacement may be written dW= Td{a^) = T{^da + ad^) = Ta/3dz(- +-) (2). Equating these two expressions for the work, we obtain The restoring forces called into play by the curvature is, of course, equal and opposite to p, and may be equated .. to the product mass of the element into its acceleration. Let the equilibrium position of the membrane be taken as the plane of xy, and let displacement parallel to the axis of z be denoted by w. Then, as all the displacements and curvatures are small, — may be replaced, to the required approximation, by j-^' *^^ divisor \'^ + \~r j \ ^^ the ex- pression for curvature being practically equal to unity. A similar remark applies to —, which is accordingly replaced '2 by d^wjdy'^. Thus the restoring force on the element of surface is (d^w dSv\ 264 SOUND CHAP. V Again, the product mass of elemeut into its acceleration is a^a'u), a denoting the surface density of the membrane. Hence the differential equation for the vibrations of the membrane may be written where c = s/ Tjcr and is of the nature of a velocity. 191. Rectangular Membrane. — Let now a particular case be taken, in which the membrane is rectangular, ex- tending in the plane of xy between the co-ordinate axes and the lines x = a and y = i. For every point in the area equation (4) must be satisfied, and for every point along the boundary we must have in addition w = for all values of t. Both conditions are satisfied by the equations — Tmrx . niry w = sm sm - , cos pt a 2 2 "Z™^ , "^^ where p — cV'l -^ + -p (5), and TO and n are any integers. These may easily be verified by examination and differentiation. This, therefore, constitutes a particular integral. The substitution of Biapt for coBpt gives another particular integral, the addition of both would give another. Following this method the general solution is found, and may be written as follows : — m=00 7t = Q0 ^ sm-— sm— |^^„cosp!;-f5^„sinj?i!| (6), m=l 71=1 where the ^'s and ^s are a series of arbitrary constants for each value of m and n, which must be chosen to suit any given initial conditions. 192. Initial Conditions. — Now let the initial displacement and velocity of the membrane be specified as follows : — 191,192 VIBEATING- SYSTEMS 265 I'or t =Q, let w^=f^(,,,y) ^7), ^^^ % =fl«', y) (8). Then we must so choose the constants in equation (6) as to satisfy equations (7) and (8). By putting ^ = in (6) and equating to (7), we have «l = CO 1l~ a a and '^"""2^^^ <^^^)' where the subscripts of N denote the values of m and n. The lines which are always at rest are readily found by pvitting w = for all values of t. Hence, for the present case, we obtain from (17) . irx . iry , ^, sm — sin~-=0 (19). a a This splits into sin irxja =0 or sin Try/a = 0. Hence equation (19) gives the lines whose equations are a;=0, x — a, y=0, y = ct', i-e. the sides of the square in question. Thus, as must be the case, there are no other nodal lines for the prime tone. Second Tone. — The next higher tone is obtained by putting m= 1 and n='2, or vice versa. In either case it 193, 194 VIBRATING SYSTEMS 267 is evident from (16) that the frequency is the same, and given by Thus two distinct vibrations are possible whose periods are the same. If these two coexist, nodal lines will be obtained, provided the vibrations are of the same phase. For we may now represent the Miiole motion by the equation f . 2i7X .try ^ . irx . 27r«"l w= Wava. sm h-Osm — sm — - \ cospt (21). I Ct CI (X CL \ To find the nodal lines put w = 0. We thus obtain . TTX . Try f ^ irx irii] sni — sm— iCeos — +Z)cos — h = (22). a a { a a ] This expression vanishes for sin — = or sin — = 0, a a which give the edges of the membrane which must be nodal. On equating to zero the factor in the brackets, we obtain, in general, a curved line, which, however, always passes through the centre of the square. In special cases this reduces to a diameter or diagonal of the square. These are easily found by writing successively in (22), C = 0, D=Q, C-\-D=Q, C=D. For representations of the nodal lines see Table IX., which illustrates all the cases dealt with. 194. Third Tone. — The next case in order of pitch occurs when we write m = n=2. The motion may be represented by . 27ra; . 27ry .-,.,, iu= sm sin— — cospt (2o). a a And, in addition to the edges, the nodal lines are deter- mined by TTX TTw a , a ^ „ , . cos — cos — ^ = ov v = — and x = — (^4). a a * 2 2 ^ ^ 268 SOUND CHAI-. V They are accordingly the two diameters. The frequency is expressed by iV',,= |^x/8 (25). Fourth Tone. — Let us now write 7ft =3 and n=l, or vice ve?-sa. The frequency is then given by ^.3 = ^3i = i-/l0 (26). When the two vibrations coexist in the same pliase, the whole motion may be written f^ . Sttx . ttw ^ . ttx . Bttv] ,„^^ w= -^ Csin sin h-^sm — sm [ cosvt (27). [ a a a a ] From this, rejecting the factors sin — sin ^, which corre- spond to the edges, we have for the internal nodal lines c/4cos^^-l')+i)(^4cos^^-l\ = (28). From this equation we see that for any ratio of CjD, the curve passes through the four points given by « = a/3 or 2a/3 and y = aj'i or 2a/3. For C=0 we have the two lines y = a/3 and y = 2a/3 as the nodal lines, for D = we have instead x = a/3 and a; = 2a/3.. For G+D—O we obtain the two diagonals of the square y = x and y = a — x. Lastly, for G= D the equation of the nodal curve is cos^ — + cos^'^ = ^ (29). a a '2 This is a closed curve resembling a circle. Its diameters parallel to the sides of the square are a/ 2, while those along the diagonals are only av2/3, i.e. 0'4714 instead of 0'5 of a. By giving to Cjl) different values any number 194 VIBKATING SYSTEMS 269 of intermediate forms could be obtained. Some of these I p o « ■ o together with those noted above, are shown in Fig. 52, 270 SOUND CHAP. V whicli should be followed round by taking the lower line from right to left. 195. Fifth Tons. — The next tone is obtained by putting TO = 2 and w = 3 or vice versa. The frequency in either case is ^.3 = ^3. = i-/l3 (30). When both possible vibrations of this frequency coexist, we may follow the plan adopted for the previous tones. Thus, write the expression for w, and equate it to zero for the nodes. Then eliminate the factors which represent the edges and transform. We thus obtain as representing the internal nodes, (7f4cos=--l')cos^ + i?cos'^Y4cos^^-lUo(31). \ a J a a \ a / For C = or D=0 we have three lines parallel to the edges, namely, two lines one way and the third across it. For C=D, equation (31) may be written cos'^ + cos'^V4cos!^cos'^-lUo (32). a * / \ « a J The two factors of this equation represent respectively the diagonal y = a — x and parts of each branch of a hyperbolic curve. For C = — Z) we have the same pattern about the diagonal y = x. By giving to the ratio Cjl) other values, we could obtain any number of curves intermediate between the simple ones already noticed. Sixth Tone, — This is obtained by putting m = 1 and TO = 4, or vice versa. The frequency is ^- = ^"=£^^ (33). The whole motion, when both systems coexist in the same phase, is f . AirX . TTW , -r, . TTX . 4:TnA , ,, ., iy = J c sm sm -^ + X* sin — sm — - [-cos pt (34). [ a a a a j 195 VIBEATING SYSTEMS 271 The nodal system is given by equating to zero the quantity in the brackets. Omitting the factors sin — . Try sm -— , which represent the edges, this reduces to Ccos — cos 1- Z* cos— cos — ^=0 (35). a a a a Por C = 0, this gives three equidistant lines parallel to the axis of x. For D=Q, the corresponding system parallel to the axis of y is obtained. Take, thirdly, the case of C~D. It is then convenient to transform cos — cos a a to 2 cos^ — — cos — , and the same with the factors in- . a a volving y. Collecting like powers of the cosines, equation (35) then becomes 2 cos^ — + cos^ -^ — cos — + cos -^ = 0, a a I \ a a I irx , iry\ or, cos — + cos — ^ X \ a a I 1 + 2 cos — COS ^-— 2 cos'' — — 2 cos'' _!^ = (36). a a a a / The first factor, equated to zero, gives the diagonal y = a — x. The second factor gives a closed curve passing through the points _,__,,«, «\ (t, «Y f «, h\ I'ia, h\ and (la, « 2 47 V4 4/ V4 2/ V2 4 / \4 4 / \4 2 — i — Tor C= —D, the same system is obtained about the other diagonal y = x. Seventh Tone. — The last tone we shall notice is that obtained by writing m = %=?>. The frequency is given by ^'33 = f.x/T8 (:^7). 272 SOUND CHAP. V As m and n are equal no variety can be effected by their interchange. It is evident also, without writing the equation, that the single pattern of internal nodal lines is a set of four straight lines, namely, two equidistant with the edges parallel to the axis of x, and two others similarly situated with respect to the axis of y. 196. Summary of the Various Oases. — The relative frequencies, values of m and n, and typical diagrams of the nodal lines for a square membrane, are collected together iu Table IX. It thus serves as a .compact summary of all the chief cases hitherto discussed. The relative frequencies in the first column are the values of fj-nv^ + n^-^ V 1^+ 1^ and so give the ratio of frequency of each tone to that of the prime for the same membrane. 197. Circular Membranes. — The mathematical treat- ment of circular membranes is far more diificult than that of rectangular ones, and cannot be given here. It is fully discussed by Kayleigh. The nodal lines are shown to be either concentric circles or diameters, the ends of the latter being equidistant. The prime tone has, of course, no nodal line except the boundary itself In the order of ascending frequencies the nodal patterns of the other tones are as follows : — One diameter, two diameters, one circle, radius 0'436 that of the membrane, three diameters, one diameter and one circle, of radius 0'546, four diameters, two diameters and one circle, of radius 0'601, two circles, of radii 0'278 and 0'638, five diameters, three diameters and one circle, of radius 0'654, six diameters, etc. The relative frequencies of the corresponding tones, including the prime, are 1, 1-594, 2-136, 2-296, 2-653, 2-918, 3-156, 3-501, 3-600, 3-652, 4-060, and 4-154. Experimental researches upon the vibrations of mem- branes have been made by a number of physicists, and have given results which, in their main features, are in harmony with theory. Many discrepancies have, however, been detected owing to various disturbing causes not allowed 196, 197 VIBEATING SYSTEMS 273 Table IX. — Vibrations of Square Membranes Tones and their Relative Frequencies Prime, 1 Second, 7-68 Values of Constants 2 I Diagrams of Typical Nodal Lines D -B m D=o C+D=o C=n D----2C Third, 2 Fourth, 2-24 Fifth, 2-55 Sixth, 2-98 C^o D=o t) C^D=o o_ C=D 3 2 3 :c=o C+D=o ■C=o - D^o C=D D=o i) m m C+P=o C=D Seventh, S 274 SOUND cHAP.v for in the theory. Membranes have been tried made of paper wetted and glued to wood rings. Drum-skins have been used and e^'en soap films also. In the case of soap films, the nodal lines have been detected by interference of light. With the solid membranes sand is used to indicate the nodes. The performance of any of these experiments, however, appears to be a matter of consider- able difficulty. 198. Transverse Vibrations of Bars. — Anything like a full treatment of bars vibrating transversely is beyond the scope of this work. For that the reader is referred to Lord Eayleigh's Theory of Sound, to which the writer is so much indebted for what follows. We shall restrict our- selves to the case of a bar of uniform section, not subject to tension, and straight when left at rest free from external forces. We shall further suppose that the curvatures are so small as to be represented by d'yulr'', and to leave the' lengths along the bar and along the axis of x practically equal. We shall neglect gravity in comparison with the elastic forces, and, finally, shall take the rotatory inertia of the bar as negligible in comparison with the inertia due to translation. This is practically the case for thin bars, since the inertia of translation varies as the thickness simply, while the inertia of rotation varies as the cube of the thickness. In other words, the two inertias in question vary as the section and as its moment of inertia respectively. Before proceeding to derive the differential equation of motion, some preliminary steps must be taken. These are dealt with in order. Bending Moment. — In attacking the problem we need, at the outset, an expression for the bending moment or couple required to bend a bar to a given curvature of radius R say. Let the bar be bent in the plane of xy, then there is in it, perpendicular to this plane, a neutral surface which is neither extended nor contracted. Outside this neutral surface extension occurs, inside it contraction. Consider 198, 199 VIBEATING SYSTEMS 275 an element of the bar of length x, distant v\ radially from the neutral surface. Then, by the bending of the bar to the radius B,, the length of this element becomes x + dx x + dx X dx ,,.■■,■, where — = — = — . ihus, the fractional elongation of B + r) B 7) " the element is given by dx/x = tj/B. Hence, by definition of Young's modulus, here denoted by q, the tensile force on this element of cross section dS is qrjdS/B. The bending moment is, of course, found by multiplying this expression by 1] the distance from the neutral surface, and integrating over the whole cross section of area co say. Then, writing K for the radius of gyration of the cross section about its intersection with the neutral surface, we obtain for the bending moment ^ ~ M = jqrfdSjB = qwi^jB = qLH'^yldx^ (1). 199. Bending Moments and Forces on a Bar at Rest. — Laying aside all attempt at complete generality, let us take the next step in the problem by finding a relation between the bending moment, the shearing forces, and the applied forces on a bar by the following method. Consider the bar bent in the plane of xy, and in equilibrium under given forces. Let it have a free end at the origin of co-ordinates and lie almost along the axis of aj.the further end being maintained at rest by clamps or any other convenient arrangements (see Fig. 53). Then sup- pose, first, an isolated force /j to act at the origin along the axis of y. shearing force in the bar is accordingly /^ at every point up to where it is clamped, but the bending moment at x is f^x. That is, the bending moment is zero at the Bak Bbnt by Foeoe at End. The 276 SOUND origin, and increases with x at the rate /^ per unit length along X. Thus, if M denotes the bending moment, we have i)/=/ja;and dMjAx^f^. Or, if M. be plotted as a graph, we obtain a straight line through the origin at an angle ^^ with the axis of x, where tan ^^ =/^ (see Fig. 54). Next, at x^, between the origin and x, let a second force /^ act together with, and parallel to, the force /j at the origin. Then the graph for M consists of two straight lines, their junction occurring over the point a;^ where the second force is applied. The bending moment at x, where x > «„ is now given by M=f^-\-f^ — x^, and is thus dependent on ajg, the place of application of the second force. But, as seen from the equation or the graph shown in Fig. 55, the vaU of increase of M is independent of x^ and depends only on the swm of the forces. Thus, for values of x exceeding a;,, rlMldx =f^ +/„. Or, Y Fig. 54. — Bending Moment of Bab WITH FOKCE AT ObIGIN. O x^ X X Fig. 55. — Bending Moments due to Two Foeoes. if the angle of the final part of the graph is S'(sin x' - sinh «') (2 0). 284 SOUND CHAP. V This is identical with (11) provided that 2P=A + G+D, 2Q = A-C-B, ■2E = B+C-I> and 2S=£-C+I). The advantage of the form of expression used in (20), as pointed out by Lord Eayleigh, is that the four quantities in the brackets repeat themselves on differentiation and vanish with x' except cos x' + cosh x' which equals 2 for x' = 0. It must be borne in mind that cosh x = ^(e" + e'") and sinh x = ^(e'' — e'^), and that accordingly — cosh ;r = sinh x and d . ^* -y- sinh X = cosh x. ax Now, for the bar free at both ends and extending from x=Q to x = l, we have to apply at each end the two condi- tions given in equation (7). Thus, for the end at the origin, we must differentiate equation (20) twice with respect to x, put a; = 0, and equate the result to zero. And similarly for the second condition in (7), differentiating this time thrice with respect to x. These operations give e=0, and>S'=0 (21). We have now to follow the same method for the other end, but can omit to begin with the terms involving the constants Q and ^S", already shown to be each zero. Hence we have u = P(cos x' + cosh x') -f i2(sin x' + sinh x') (22). Differentiating this twice and thrice respectively, and putting X = I, i.e. x' = m, and equating to zero, we obtain P( — cos m + cosh m) + B(^ — sin m + sinh m) = 'j and P(siu m + sinh m) + R{ — cos m + cosh m) = j These two equations give two expressions for the ratio P : B, and enable us by equating them to iind in. Thus, omitting 208 VIBEATING SYSTEMS 285 a constant multiplier, we may write from the second form of (23), u = (cos m — cosh m)(cos x' + cosh x') + (sin m + sinh m)(sin a;' + sinh x') (24), or a corresponding and equivalent expression from the first form of the same equation. 60 50 40 30 20 10 O o -10 -20 -30 -40 -50 -60 s m s Upper part of figure applies to 1 to II " 1 II Free-Free Bar and to ) Fixed- Fixed Bar _J / TT L _J ™ 37r _J 27r L 1 Values \in radians 0~~~ H ^ aw 2 TT Y "^ ? ^ It f — -k ™ s ■^ \ Lower part of figure II applies to ^ ^St Fixed-Free Bar ■ FiQ. 56. — Graphical Solution of sec m=+ cosh m. The simple harmonic component of this type may be denoted by y = Kucos(~m't-\-e\ (25), in which, when m is known, the K and e are easily expressible in terms of the ^'s and ^'s of equation (15). 208. Series of Tones. — We further derive from the two equations (23) the following relation : — cos m cosh m = 1,1 /26^ or, sec m = cosh m J 286 SOUND This may be solved by the aid of tables of hyperbolic cosines, or by a method of expansions and successive approximations. It may also be solved graphically as follows: — Take the second form of (26) and plot the graphs y — cosh m and y = sec m. Then the values of m at their intersections will give the roots of m sought. It will be seen from the upper part of Fig. 56 that, apart from the first value zero, the series approximates to 3, 5, 7, etc., times 7r/2, the approximation becoming closer as we proceed to the higher values. Now the frequency, as seen from equation (25), is proportional to vi^, hence for the higher values the frequencies approximate to the squares of odd numbers. The lower part of the diagram refers to another equation for the fixed-free bar to be dealt with later. 209. Table X. gives the values of m, the relation of the corresponding frequencies proportional to m^, and also the intervals between the prime and each higher tone for the free-free bar. Table X. — Tones for Fuee-Free Bar (or Fixed-Fixed Bar) Values of m. Relative Frequencies, Intervals from Prime to Approxijnate (NINi) a. m2. each Higher Tone. Notes. «!o = • Octaves, Equal Semitones «!i = 4-73 1 C m2=7'85 2 -756 1 and 5-52 F'Jt 1)13 = 10-996 5-404 2 and 5-23 F" i«,j = 14-187 . 8-933 3 and 1 -91 D'" m-, = 17-279 . 13-345 3 and 8-86 A'" The first column of Table X. is from Eayleigh's Sound, the values being obtained by computation. The second column is derived from the first by squaring and reducing the prime tone to unity. To reduce relative frequencies to equal semitones, in order to obtain the third column, we may write s = k log (N'/N), 209, 210 VIBRATING SYSTEMS 287 where s is the number of mean semitones in the interval between the tones whose frequencies are JV' and JV and Jc, some constant. Then, obviously, we have 12= A log 2. Whence _ 12 log (JSf'/N) log 2 (27). This gives the third column of the table from which the fourth easily follows, the accents representing successive higher octaves, though the letters must not be taken as denoting any absolute pitch, but are all relative only. It is thus seen that the series of tones rises much more rapidly than those of either an open organ pipe or a stretched string fixed at the ends. Further, they depart utterly from the simple relation constituting the harmonic series. From equations (14) and (25) and the table above for the values of the m's, we can obtain the actual fre- quency of any tone for a given bar. Thus, let the prime tone be required for a free-free bar of steel of rectangular section of thickness a cm., and length I cm., then k = a/ v 12, Jqlp= 523,100 cm./sec = &, whence i\r= 538,400 (a/P) per second. Multiplying this value by the other numbers in column 2 of Table X., we obtain at once the other tones for the same bar. Fixed-Fixed Bar. — On applying the proper conditions, equation (6), article 203, for a bar fixed at each end, it will be found that the same series of tones is obtained as for one free at each end. 210. Nodes for Free-Free Bar. — By inserting any one value of 7n in equations (24) and (25), the displacement curve at any instant may be obtained for that type of vibration. Hence, also, the nodes or points of no dis- placement can be found. The work is, however, somewhat long, and will not be reproduced here. It must suffice to quote the results as to position of nodes found by Eayleigh 288 SOUND for the free-free bar. The theoretical distances of the nodes from one end in fractions of the bar's length are — Table XI. — Theoretical Nodal Positions First tone Second tone Third tone The nth tone 0-2242 0-1321 0-0944 J 1-3222 (4)1 + 2 0-5 0-3558 4j9820 in + 2 0-6442 9-0007 451 + 2 0-7758 0-8679 0-9056 4i-3 in + 2 The last term in the last line indicates the position of the y-th node from the end when the bar is producing the )i-th tone. This does not apply to the few nodes near the end. ExPT. 31. Fibrations of Free-Free Bar. — These phenomena can be suitably illustrated by means of a steel bar supported at one pair of nodes by little pads of india-rubber. The one used at Nottingham was supplied by Mr. Joseph Goold, and is 29 inches long, 1:^ inch wide, and J inch thick. It may be excited by an ordinary bow, or by the special generators designed by Mr. Goold and described in the next chapter. The presence and positions of the nodes are best indicated by chalk grated upon the bar by a sharp-edged stick or by carbonate of magnesia. The bar should be kept vaselined when out of use. Before use it must be scrupulously cleaned, the last trace of grease being removed by india-rubber. The closeness of the agreement between experimental and theo- retical results for this bar is shown in the accompanying Table XII. [Table 211 VIBEATING SYSTEMS 289 Table XII. — Actual Free-Free Bar compared with Theory No. of Tone, Actual Freqwencies. Relative Theoretical Frequencies^ Distance of Nodes from nearer End according to Theory (Actual Values in Brackets). Per Second. Relative. 1 1 1261 1 0-2242 (0-224 and 0-223) 2 350 27777 2-756 0-1321 0-5 (-604 (0-134 and 0-13) and -496) 3 686 5-4444 5-404 0-0944 0-3558 (0-095) (0-359) (0-091) (0-349) i 1134 9-0000 8-933 0-0734 0-277 0-5 (0.-0759) (0-2812) (-.507) (0-718) J (0-293) 5 1696 13-4603 13-345 0-0601 0-2265 0-4091 (0-063) (0-232) (0-4146) (0-0596) (0-222) (0-4037) 6 2366 18-7777 18-63 0-0509 0-192 0-3462 OS (0-053) (0-1951) (0-351) (0-506) (0-0504) (0-1885) (0-3416) 7 3150 25-0000 .24-24 0-0441 0-1661 0-3002 0-4333 (0-046) (0-1684) (0-306) (0-438) (0-0438) (0-163) (0-294) (0-426) 211, Fixed-Free Bar. — Let the bar be fixed at a; = and free at x = l. Then -we have to determine the constants in equations (20) consistently with equation (6) for x=0, and equation (7) for x = l. Thus for y = at x=0 we 1 The theoretical frequency of the prime as determined by equations (25) and (26) is 126-03 per second. U 290 SOUND cHAr.y find P= 0. Again for dyjdx =0 at « = we find R = Q. Thus, equation (20) reduces to v,= Q (cos x' — cosh x') + S (sin x' — sinh x') (28). This has now to be made to satisfy d^yjdx" = 0, and d^y/dx^ = for x = l, i.e. for x' = m. These conditions give Q (cos m + cosh m) + *S' (sin m + sinh m) = 0, 1 , and $( — sin m + sinh ???,) + *? (cos m + cosh m) = J From these we have two expressions for the ratio Q/S. Hence, omitting a constant multiplier, we may write u = (cos m + cosh 7n) (cos x' — cosh x') + (sin m — sinli m) (sin sc' — sinh x') (30), from the second equation of (29), or a similar equation from the first. Further, by equating the two expressions for Q^S from (29), we find an equation to determine m, viz. cos m cosh m + 1 = 0, ;. I (31)- or, sec m= — cosh m J ^ ' The second form of (31) is coiivenient for graphical solution and is illustrated in the lower half of Fig. 56. From this it is seen that the values of m are approximately as the odd values of 7r/2, the approximation at first being somewhat rough, but rapidly becoming very close with higher values. The frequencies of the possible tones are as the squares of these numbers. Table XIII. gives the values of m obtained by Eayleigh, together with the intervals, also the odd values of ir/'I for comparison with the m's. [Table 212 VIBEATING SYSTEMS Table XIII. — Tones for Fixed-Free Bar 291 Odd Values ofTr/'i. Values of m. Relative Frequencies. Intervals from Prime to each Higher Tone. Approximate Notes. 1-571 4-712 7-854 10-996 14-137 17-279 m-^= 1-875 OT2= 4 '694 m3= 7-855 7714=10-996 m5 = 14-137 rrt8= 17-279 1 6-267 17-55 34-39 56-85 84-93 Octaves, Equal Semitones 2 and 7-77 4 and 1-60 5 and 1-24 5 and 9-95 6 and 4-90 c Div D'b Fvi 212. Nodes for Fixed-Free Bar. — The distances of the nodes from the free end of a bar fixed at tlje other end, as found by Seebeek and by Donkin, are as follows : — Second tone, 0-2261 Third tone, 0-1321, 0-4999 Fourth tone, 0-0944, 0-3558, 0-6439 «th tone, 1-322 4-9820 9-000'7 4i-3 471-2' 4?i-2 4m-2 4w-2 4w- 10-9993 4M-7-0175 4m.— 2 4w-2 For the nth tone,- the first three nodes and the last two are given numerically, that involving j being the general term for a yth node intermediate between the other two sets. ExPT. 32. Nodes on Large Tuning-Fork. — It is interesting to compare the theoretical results with the positions found for the nodes of a tuning-fork when higher tones are elicited. For this purpose the fork should he taken out of its resonance box which encourages the fundamental tone and that only. It may be laid with its stalk on a massive table and there held by the hands of an assistant, the prongs projecting horizontally from the edge of the table, one being exactly over the other. The fork should now be bowed at the end while a finger is lightly applied to the prong at various distances from the end. In this way higher tones may be ehcited, and the positions of the nodes shown by chalk dust scattered on the prong. An UT^ fork of 128 per second was 292 SOUND cHAP.v examined by this method. Its prongs were 10^ inches long to the inside of the hollow between them, or say 11 inches to the point which would remain at rest. Then, with the second tone, a node was found 2f inches from the end, i.e. 0-216 of its length, which compares well with the theoretical value 0'2261. On placing two fingers in the right positions as found by trial and bowing suitably, the third tone was elicited. The nodes were at 1|5 and 5 '5 inches respectively from the end. This gives 0'136 and 0-5 of the length of the prong, which again are close to the theoretical values of 0-1321 and 0-4999. If it is only desired to elicit and hear the higher partials of a tuning-fork, this may be effected as follows without removing from its resonance box : — Touch one prong lightly with the edge of a piece of cork at a node and tap the same prong at an antinode with a padded hammer. 213. Simpler Methods for Fixed-Free Bar. — In his Theory of Sound, Eayleigli has shown that the dependence of the frequency of a fixed-free bar upon its material and dimensions may be established by a simpler method, and even the absolute value of the frequency closely approxi- mated. The method consists in supposing that the vibration curve of the bar is that in which the bar would dispose itself if statically deflected by a force applied at its free extremity or some other point along it. The expressions for the potential and kinetic energies of the bar are then quoted from a former article where they were obtained by the calculus of variations. It is thus found that the frequency varies as KbjP, the symbols having their previous meanings. Further, if the bar be supposed pulled aside at the free end and at one-quarter its length from the free end respectively, the corresponding periods thus calculated have the values 0-98556 and 0-9977 of the true value calculated by the fuller theory. In other words, the pitches found by the approximate methods are too sharp. Eayleigh points out that the bar when vibrating cannot really assume the curve of one deflected by a force at the end, since this curve 213-215 VIBEATING SYSTEMS 293 violates one of the conditions for a free end, viz. — dSjIdx^ = 0. 214. In 1904, C. A. B. Garrett gave to this theory a still more elementary form, and also found experimentally for a particular bar the best place for the deflecting force. The experimental part consisted of two portions — one photo- graphical, the other microscopical. The bar was set vibrating and then instantaneously photographed. It was next deflected with the same end displacement and photo- graphed at rest, the cameta occupying the same position as at first. This was done with the deflecting force at the end, one-tenth, one-fifth, and three-tenths from the end respectively. The photographs showed the best agreement for the force at one-fifth. The second method of examina- tion was based on the following consideration : — If the bar when deflected by a force at a certain place assumed the form, it had when vibrating so as to give its prime tone, then as the vibrations died away the amplitudes at any place, say the middle of the bar, would die away continu- ously. If, however, the bar, when held pulled aside, assumed a curve distinctly different from that when vibrating for its fundamental, then the vibrations from that time will consist of the prime plus certain upper partials or overtones. And these higher forms of vibration would be of periods incommensurate with the prime, and would, accordingly, alternately increase and diminish the apparent amplitude of the middle of the bar. Thus if curves are plotted with the displacements of the middle as ordinates, and time as abscissae, we should have a wavy curve when the bar is pulled at a wrong place, and a curve showing simple subsidence, or nearly so, when the bar is pulled aside at the best place. By observing the amplitudes microscopi- cally, the best place of pulling was determined as one-fifth the bar's length from the free end, thus agreeing with the result found from the photographs. 215. Garrett's theory may be given in our notation as 294 SOUND CHAP. V follows : — For a bar of length I fixed at the origin and pulled aside by a force F at the end the equation may be written Thus the displacement y-^ at the free end is FP dqwK where the dots denote differentiations with respect to time. Thus the moment about the origin of the forces producing the motion of the bar is given by [' 11 M= lxi/po)dx= — pa>Pi/i- But the bending moment at 0, if the shape remains as when pulled, is Fl for the initial displacement, and remains pro- portional to that displacement. Or it may be obtained from the value of cl^i/ldj? at the origin. In either case, we have jl/= oqaii^y-ylP. Equating these two expressions for M we obtain '120 q k'\ P Thus the motion is simple harmonic of frequency Garrett next finds that if the bar be pulled aside at a point 23l from the fixed origin, its frequency would be on this method of computation ^^--^n-p-fF- N., = N,/ ^^ p{f-lQp+2Q) 216. A still simpler way than Garrett's is available to suggest the dependence of frequency on material and size 216, 217 VIBRATING- SYSTEMS 295 of the' bar. Thus, suppose the inertia of the bar when vibrating may be replaced by that of a smaller mass at its free; end, the rotatmy inertia of this mass being neglected. And let the deflecting force i^.be applied at the free end and produce the deflection y-^ there. Then the force per unit displacement at the frde end is And let the hypothetical mass replacing that of the bar be j(opl, where j is some proper fraction. Then for the equation of motion at the end we have j(oply = - \—jy ]y- Hence, the motion is simple harmonic, of frequency given by That is, the frequency is directly proportional to the thick- ness of the rod and to the speed & x)f .longitudinal waves in it, and inversely proportional to the square of the length. It is obvious that Garrett's theory may be regarded as a niethod of evaluating the quantity/. 217. Remaining Oases of Vibrating Bars. — The bar vibrating with one end free and the other supported k like half of a free-free bar when vibrating with a node in the middle. For, at the central node y=0, and it can be seen from Symmetry that cPyjda? = also. But these are the con- ditions for a supported end. Again, the vibrations of a rod fixed at one end and supported at the other are like those of one-half of a rod with both ends fixed and vibrating with a central node. We have now to treat very briefly the sixth and only remaining case, namely, that with both ends supported. Eeferring to equations (20) and the conditions (8) we find that ktx =0.,y~0 gives P = 0, and cPyjdx^ = gives ^ = 0. Again, at x = I, i.e. x' = m, we find y=0 gives Il — S=0 and 296 SOUND CHAP. V (fy/dx^ = gives sin m = 0, i.e. m = mr, where n is any integer. Thus, in equation (20), P and Q having vanished, and the sinh x' having disappeared also in virtue of the equality of Fig. 57. — Approach of Nodes in a Bent Bab. E and S, the expression reduces to a single term, viz. that involving sin x'. We may accordingly write „ . nirx f Kbn-ir y = K sin ~- cos ( — -^ — t + e (32), where K and e are arbitrary constants to be chosen to satisfy the initial conditions. It is thus seen that the V_> VJ. \jj \J \J Fig. 58. — Nodes for U-shaped Bar. segments of the rod in this case are like those of a string fixed at the ends. The frequencies of the possible tones, however, are quite different. For, in this case of the rod, the frequency of any tone is proportional to «^ instead of 218 VIBEATING SYSTEMS 297 to n simply as for a string. The frequency of the mth tone for the rod is evidently given by i^. = ^ = -i^. (33). 218. Bent Bar vibrating Transversely. — Chladni experimented on the transverse vibrations of a bent bar free at each end, and showed that as the bar deviated more and more from the straight form the two nodes of the fundamental vibration approached each other more and more closely. . This gradual approach of the nodes from the case of a straight bar to that of a bar bent with its two limbs parallel is shown in Tig. 57, taken from Chladni's diagram. As the bar was bent more and more, Chladni found it more and more difficult to elicit the vibration with three nodes. Until in the final U -shape with parallel limbs, this mode of vibration was quite impossible. The other modes with 4, 5, 6, 7, and 8 nodes respectively were, however, all obtained. The nodes are shown in Fig. 58, and the approximate pitches in Table XIV., both taken from Chladni's celebrated work. The bars were excited by a violin bow, and each node rendered visible by arranging that part horizontal and scattering a little sand on it. Table XIV. — Vibrations of U-shaped Bar No. of Kodes 2 3 4 5 6 7 8 Approximate Pitches c Missing '/$ /"« d" ?"« d' Numbers whose squares give approximate relative frequencies (2) (5) 3 4 .5 6 7 298 SOUND cHAP.v 219. Tuning-Forks. — The behaviour of the U-shaped bars just dealt with approximates to that of tuning-forks. But the vibration of tuning-forks is usually further com- plicated by the presence of an additional block at the centre of the bend and the stem attached thereto. Indeed, it may be a nearer approximation to regard each prong as a straight bar fixed at the end near the stem and free at the other end. On this supposition, let us calculate the pitch of the prime tone of a prong of steel of thickness ffi cm. and length I cm., the section being rectangular. From equation (14) of article 204 we have N ''■^,-J^ w- '.ttI Here K = aj ijvi, and for steel we may write s/.qlp = 523,700 cm./sec. Further, we see from Table XIIL, article 211, that ??i.j=1875. Whence, for the fundamental vibration, we have ^=84,590 «//' nearly (2). 220. Temperature Variation of Pitch. — To show the effect on a tuning-fork of change of temperature we may proceed as follows : — By a rise of temperature t° C. let the value of Young's modulus be affected by the factor (1 —yt), and the linear dimensions by the factor (1 +zt). In other words, let i?_^be the coefficient of linear expansion and y the co-efficient of decrease of Young's modulus with rise of temperature. Then, since yt and zt are small, we may use the approximate binomial expansion (1 + A)" = 1 -|- nh, where h is small compared with unity. Then, writing N^ for the frequency when the temperature is raised f C. from that where the frequency was iV, we have rt.(l-f~0 Jq{l-yt) Nt = -. 9 m J VI l\l + 2zt)Jp{l-'M) rl'' J 12 219-221 VIBEATING SYSTEMS 299 N,==N(l-tl^t) (3). or. But, in the absence of any better determination of y, we had perhaps better take the vahie of (y — s) fiom the direct experiments on a fork. Thus, Koenig found that the temperature coefficient of a fork was 0000112. This, then, is the value to be ascribed to (y — z)/2 in equation (3). From this, since 2=0'000012 nearly, we obtain 2/= 0-000236. 221. Vibrations of a Ring. — A complete ring of elastic material may be made to exhibit either longitudinal or flexural vibrations. The latter were experimentally studied by Chladni as early as 1787. Chladni recommends that the ring be placed horizontally, resting on three supports of cork or other soft material at the nodes. A vibrating segment should project beyond the table and be excited by a violin bow, the ring being meanwhile held in place by the tips of two fingers over the nodal supports. Chladni states that the vibrations are most easily excited if the bow is used vertically. It would appear, therefore, that the vibrations chiefly contemplated by Chladni were flexural ones perpendicular to the plane of the ring. It is obvious that flexural vibrations in the plane of the ring are also possible. Chladni obtained the following vibrations : — Number of nodes— 4, 6, 8, 10, 12, 14. Numbers whose squares are propor-"! tional to the frequencies of the|-3, 5, 7, 9, 11, 13. corresponding tones — / The subject has been treated mathematically by Hoppe, by Michell, and by Lord Eayleigh. The results are briefly as follows : — Let the ring be circular of radius a and of circular cross section of radius c, small compared with a. Suppose its material to be of density p and Young's modulus q. Lastly, let it vibrate so as to have s periods (or com- N^-^. "-' -'^ .1 1 a-) 300 SOUND cHAP.v plete wave lengths of the stationary vibration) in its circumference. Then, for flexural vibrations in the plane of the ring Hoppe (1871) and Lord Eayleigh found that the frequency may be expressed by 1 s(s^ - 1) ^ Vl+s' ci\ p For flexural vibrations perpendicular to the plane of the ring, Michell (1889) found 1 s($^—l) c la " 4,r Jl+a + s^ cr \J p ^"^' in which a- denotes the value of Poisson's ratio for the material of the ring. In each of the above vibrations the circular axis of the ring is supposed to remain unextended. For the other class of vibrations, in which the ring remains in its own plane, but suffers alternate extension and contraction, Hoppe found "•'^^■l-Jl P> x/l + s- J- jq 222. In equation (3), if s= 0, we have the solution for vibrations which are purely radial. In other words, the ring contracts and expands alternately, always remaining a circle and in the same plane. For the flexural vibrations a cannot be less than two, in which case we have the gravest tone with four nodes. For s = 1 we should have a translation of the ring as a whole in its own plane but Avithout deformation. The sequences of tones found from either (1) or (2) agree fairly well with those experimentally observed by Chladni. The vibrations of a ring are mentioned here, not so much for their own sake, but rather because they form the first step towards the very complicated phenomena presented 222-224 VIBRATING SYSTEMS 301 by the vibrations of bells. It should be carefully noted that the motions in the general flexural vibrations here con- templated are not purely radial. For that would be inconsistent with the supposition that no extension occurs. Thus, when s = 2, and the number of nodes is 4, the deformed ring is elliptical, cutting the equilibrium circle in 4 points which are nodes for the radial motion, but are places of maximum tangential motion. Conversely, the places of maximum radial motion are nodes with respect to the tangential motion. 223. Vibrations of Plates. — We have already seen that, in passing from the theory of the ideal o?ie-dimensional stretched string of acoustics to the ^wo-dimacsional stretched membrane, the complication is much increased. The same remark holds good if we pass from the flexible string to the elastic bar. If, therefore, we attempt to deal with the vibrations of a plate, a system at once ^wo-dimensional and elastic, we may expect to encounter still greater difSculties. And such is the case. Indeed, the problem in its generality seems not to have been solved. Fortunately, the experi- mental methods of treatment are as easy as the theory is difficult, The vibration of plates with free edges was approached from the experimental side by Chladni, and his first results published in 1787. The systems of nodal lines so obtained are still called Chladni's figures. The mathematical treatment of this subject lies beyond the scope of this work ; it may, however, be desirable to quote a few of the chief results which have been obtained. 224. Fundamental Equations. — Let the plate, when at rest, be in the plane of xy, have thickness 2h, volume density p, Young's modulus q, and Poisson's ratio a. Let w be a small displacement perpendicular to the plane of the plate at the point (x, y), and at time t. Suppose the vibrations to be flexural only, i.e. of the type which involves no extension or contraction of the middle sheet of the plate. 302 SOUND CHAP. V Then it may be shown (Eayleigh's SouTid, §| 215-217) that the differential equation is d^io/df + c*v\o = (1), where c" = ^—^ — ^ (2), If we assume that w oc cos {pt — e), equation (1) becomes (V'-/O"'=0 (4), where U = -p'^jc^ (5). 225. Boundetry Conditions. — For the boundary con- ditions of the edges parallel to the axis of y, we liave the following scheme of equations applying as shown at right and left : — - — +(- -0-) s 1 = (6) ,l:c\jl.^'^^ Vl [Free .if '^'''' '?'«' r. ,-s (edges. Supported ,, + „■ = (,) edges. ''■"' ^y } (lw=0 (8) 1 ^ ^ ^ ^ I Fixed - Equation (10) shows that the vibrational segments of a " supported " rectangular plate are like those of a stretched membrane of the same shape. Equation (11), however, shows that the analogy does not extend to the sequence of tones. The same kind of partial analogy was seen to occur between a bar with supported ends and a string, the vibration segments of the bar being like those of the string, but the tones being inharmonic for the bar and harmonic for the string. 227. Plates with Free Edges. Ohiadni's Figures. — The problem of rectangular plates with free edges is one of great theoretical difficulty which is only partly removed by making the plate square. We shall therefore entirely omit its mathematical treatment, and pass to the experimental methods introduced by Chladni. To produce each type of vibration of a plate and render visible the corresponding nodal lines, we must proceed generally as follows : — Hold it at one or more points on nodal lines, sprinkle a little sand on it, and excite with a violin bow a part of the edge 304 SOUND CHAP. V between two nodal lines. The sand is then thrown off by the vibrating portions and accumulates on the nodal lines. To this general instruction we may now add various details which will facilitate the success of the experiment. The plates may be of glass, or of any sufficiently sonorous metal, for example steel or copper. Wood yields irregular figures on account of the varying elasticity in different directions. Chladni preferred glass plates as they are easily obtained to order, and their transparency permits of the fingers being used underneath at points which are shown to be nodal by the sand above. Small plates up to six inches diameter will serve quite well for simple figures, larger ones are better for the more complicated figures. The edges of the plate must be filed or ground smooth so as not to injure the hairs of the bow. It is desirable to hold the plate where two nodal lines intersect. The plate may be held between the thumb and finger, or be clamped between the rounded ends of a stud and an opposing screw. Or, the plate may have a hole through it and be fitted on the screwed stalk of a stand and fastened down with a nut. In this case leather washers may be introduced above and below the plate -to leave it fis free aS possible to take any direction in the neighbour- hood of the part held. In addition to the place where the plate is held, it is desirable and usually imperative to touch it at another place with a finger of the left hand while bowing with the right. This place may be at the edge of the plate, or away from the edge either above or below. It is usually best to bow at an antinode rather near to the nodal line thus touched. If the desired figure appears but imperfectly, shift the place of touch slightly. To reproduce any particular figure, mark the places of touching and bowing that were found satisfactory. To exclude undesired figures touch with a finger at a second place which is also nodal for the desired figure, but not nodal for the figure which it is sought to exclude. In general simple figures correspond to grave tones, and are more easily produced by 228 VIBEATING SYSTEMS 305 the hard pressure of a slack bow moved slowly. The compli- cated figures, on the other hand, usually correspond to high tones, and are more easily produced by the light pressure of a tight bow moved quickly. It is also here desirable to remove the bow from the plate at each end of its stroke. If several tones seem to be struggling together and the sound fluctuates, it may usually be found which tone corresponds to the figure desired, and further what sort of bowing encourages it to the exclusion of the other. When E / 6 F "G Fig. 59. — Typical Chladni's Figures. this is ascertained, persevere in the right bowing till the desired figure is clearly produced and the corresponding tone heard alone. 228. ExPT,' 33. Chladni's Figures on a Square Plate. — ■ Observing the precautions just enumerated, produce with a square metal plate of about 30 cm. side and 2 mm. thick the following seven typical figures of Fig. 59. The plate is to be held throughout in the centre, touched with the fingers where marked /, and bowed where marked b. The figures A, B, E, and F are examples of exact agree- ment with forms fairly easy to predict theoretically. The seven shown in Fig. 59 are but typical of a large number possible. Chladni's original work gives illustrations of 52 figures obtained with a square plate. X 306 SOUND cHAp.v 229. Circular Plates. — To apply the theory to a circular plate, it is necessary to transform to polar co- ordinates the fundamental equation and boundary conditions. The complete solution may then be expressed in a Fourier series. But, as the analysis is rather long and involves Bessel functions, it must suffice here to quote some of the results. The nodal system for a circular plate with free edges may have n diameters symmetrically distributed round the centre but otherwise arbitrary. It may also have n' circular nodes. Lord Eayleigh points out that theory con- firms the discovery by Chladni " that the frequencies corresponding to figures with a given number of nodal diameters are, with the exception of the lowest, approxi- mately proportional to the squares of consecutive even or uneven numbers, according as the number of diameters is itself even or odd." Further, within certain limits, " the pitch is approximately unaltered, when any number is subtracted from n', provided twice that number be added to n." lu other words, each nodal circle has about twice the effect in raising the pitch possessed by a nodal diameter. For the radii of the circular nodes Table XV. presents a comparison between calculation and observation. The observations are throughout by Strehlke, while the calculations are due to Poisson or based upon the theory of Kirchhoff. The radii are given in fractions of that of the disc. 229, 230 VIBEATING SYSTEMS 307 Table XV. Eadii of Circular Nodes on Disc with Free Edge Number of Nodal Diameters. n. Number of Nodal Circles. Radii of Nodal Circles. Observation. Calculation. 1 0-678 0-681 2 /0-391 \0-841 0-392\ 0-842J 3 ' (0-256 -^0-591 io-894 0-2571 0-591 I 0-894 J 1 1 0-781 to 0-783 0-781 2 1 0-79 to 0-82 0-823 3 1 0-838 to 0-842 0-847 1 2 /0-488 to 0-497 tO-809 0-4971 0-870/ It should be noted that the circular disc in a telephone is clamped round its edge. The theory of this case has been developed by Lord Eayleigh (Theory of Sound, vol. i. pp. 366-367). Nu7riber and Varieties of Chladni's Figures. — The number of Chladni's iigures obtainable is practically unlimited. For example, beside the 52 figures for a square plate already mentioned, Chladni gives illustrations of 43 figures with a circular plate, 30 with an hexagonal plate, 52 with a rectangular plate, 26 with elliptical plates, 15 with semicircular plates, and 25 with triangular plates. 230. Simple Derivation of Chladni's Figures. — We 308 SOUND CHAP. V may often obtain a clue to the formation of a Chladni figure by noting that it is due to the coexistence of two simple vibrations of the same period. This coexistence was shown to occur in the case of membranes when treating them analytically (see Fig. 52, art. 194), and can now be applied here even in the absence of the full theory of which it is a natural result. For example, take the case of a square plate vibrating with three nodal lines parallel to one edge and the axis of x. It will have the same period if ■'///■'' ■■ -'/ '/, ' WiMlk v^- \ ■ > ,. /;•,- ; < '' '''o '^ 1 ; \ 1 1 Fig. 60. — Dbkivation op Chladni's Piquees. vibrating with three nodal lines parallel to the axis of y. Hence the two vibrations may, and usually will, coexist. The phases may be the same or opposite, and their composi- tion will produce different figures in the two cases. In Fig. 60 let the shaded portions represent the parts of the plate which are down at a given instant, the unshaded portions those which are up at the same time. Then, the first and tliird figures in the upper line represent the com- ponent vibrations just referred to. The middle figure in the upper line shows the two sets of shading superposed, and therefore represents the coexistence of the corresponding kinds of vibration. In this case it is clear that the nodal 231 VIBEATING SYSTEMS 309 lines must lie in the once shaded portions, those portions doubly shaded being displaced down by each vibration, and therefore doubly down, the unshaded portions in like manner being doubly displaced up. The nodal lines are accordingly as shown in the middle figure of the upper line. Both analysis and Chladni's experiments confirm the result thus simply obtained. The three figures in the lower line show the composition of the same vibrations, but with the right hand one reversed in phase. It is seen, as might be expected, that the resultant figure is now formed about the other diagonal, but is in other respects like the first. This simple method of derivation may be applied to obtain the figures of the nodal lines in vibrating membranes also, or as a check on results analytically derived. 231. Vibrations of a Cylindrical Shell. — The problem of the flexural vibrations of a thin cylindrical shell has been attacked by several mathematicians. For the case of a long cylinder which suffers no extension or contraction of the middle surface. Lord Eayleigh's theory furnishes the following expression for the frequency : — irjs^+l a-M {3k + 4:n)p ^ ' Here s is the number of periods, or wave lengths, into which the circumference is divided, so 2s will be the number of nodal lines, 2A is the thickness of the shell, a its radius, li is the volume elasticity of the material of the shell, n its rigidity, and p its volume density. Thus, for a given material and number of nodes, we have from equation (1) that N — 2 means that this tone on the bell was two vibrations per second flatter than the e'b on the harmonium, /" -f- 4 means that tone of the bell was four vibrations per second sharper than the /" on 312 SOUND cHAP.v the harmonium. The next tone was about midway between 6"b and h". The single numbers in brackets refer to the number of nodal meridians, and where two numbers occur the second refers to the number of nodal circles. The highest tone of the iive seems to be taken by the English bell-founders as the nominal, because just after striking it is the most prominent of the five. Hence for tuning and naming the bells of a set to be used in quick succession this must be taken as defining the pitch. 233. The endeavour of bell -founders seems to be to make the " hum - note," " fundamental," and " nominal " successive octaves ; the " fifth " and " tierce " being re- spectively the major fifth and a major or minor third above the so-called " fundamental." This, however, seems never to be attained. The great aim is to make the nominal and fundamental a true octave apart, the others being of less importance. In the example given, above, accepting the nominal as the standard, we see (1) that the fundamental is 6 per second flat, (2) that the hum-note is nearly a semitone sharp, (3) that the tierce is 4 per second too sharp for a minor third, (4) that the fifth is about ^ of a tone sharp. To any one at all familiar with the rules of harmony it might appear incredible that such tones as these five could be heard together with pleasure. But it must be remem- bered that each tone of the bell is itself simple, and not an ordinary musical tone, which is usually compound. Thus, for example, sounding the bell referred to, and striking simultaneously five keys on a piano tuned to the above pitches, the effects in the two cases would be widely different. The bell, with its simple though inharmonic tones, gives a subtle delight. The discordant effect on the piano from the mistuned compound tones would be distress- ingly painful. The reasons for this will be seen in the chapter on concord and discord. 234. Cause of Beats in Bells. — Very often, on listening 233,234 VIBEATING SYSTEMS 313 to a bell, whether a large church bell or a hand bell, beats may be observed, and indeed often several sets are heard simultaneously. Their production is explained by Lord Eayleigh as follows : — " If there be a small load at any point of the circumference a slight augmentation of period ensues, which is different according as the loaded point coincides with a node of the normal or of the tangential motion, being greater in the latter case than in the former. The sound produced depends, therefore, on the place of excitation ; in general both tones are heard, and by inter- ference give rise to beats, whose frequency is equal to the difference between the frequencies of the two tones." CHAPTEE VI EESONANCE AND RESPONSE 235. Order of Treatment. — lu this chapter we shall be concerned, chiefly with experiments which illustrate either (1) the phenomena of resonance, (2) the maintenance of vibrations, or (3) some other action on the part of a sensi- tive apparatus in response to an external periodic impulse. All the cases treated here will be of the acoustic order of frequency. The term resonance, or resounding, refers, of course, to the production of a second sound by the stimulus of a first. We shall not restrict the use of the term to the ideal case in which both sounds are of precisely the same pitch. Maintenance, on the other hand, refers to the continuance of a vibration with undiminished vigour, owing to the application of some external periodic force. In both cases we may call the first sound or otlier active agency the driver, and the second sound or vibration evoked or maintained the driven. This leads to another distinction between resonance and maintenance. In the former both driver and driven are sounds, while the term maintenance we restrict to cases in which one or other is not a sound. In all the cases of resonance with which we shall deal, and in some of the cases of maintenance, the driver and the driven have each their own definite natural periods, not necessarily alike. Here, therefore, the theory of forced vibrations developed in Chap. III. strictly applies. And, S14 235 RESONANCE AND RESPONSE 315 consequeatly, the closer the tuning between driver and driven, the fuller is the resonance, or the more efficient is the maintenance. In other cases of maintenance, the driver has no perfectly definite period of its own, or it may be no period whatever, yet, by the reaction of the driven, assisted perhaps by the care of the experimenter, the driver's action becomes periodic and of a suitable frequency.. Hence, in this case also, the theory of forced vibrations applies, or, at any rate, affords a valuable clue. The period to be ascribed to the driver is, of course, that which the reaction of the driven awakens in it, and is nearly, though not of necessity precisely, the natural period of the driven. The starting of these actions is somewhat uncertain, but once started they are stable. Lastly, we have a class of cases in which the stimulus is periodic and the response to it on the part of some sensitive apparatus continues while the stimulus lasts. The theory of such will be indicated in connection with each. The various cases to be dealt with are classified on the foregoing plan in Table XVII. For completeness' sake this table includes a few important examples whose consideration is wholly or partially deferred to the next chapter. In each such case a reference to the explanatory article is given in the table. The remaining contents of the four columns will be taken downwards in order. It will be noticed that the side headings on the left indicate a subdivision of^each column into several rows or sections, according to whether the driver and driven are respectively solid and fluid, both solid, etc. [Table 316 SOUND Table XVII. Examples of Eehonance and Allied Phenomena Nature of Driver and Driven re- spectively. Resonance Maintenance. Other Responsive Actions. Period of Driver Definite. Period of Driver wholly or partially induced by Driven. Solid Fluid 1. Tuning Fork near Air lie- son ator ■2. Adjustable Resonator J. Helmholtz Resonators (See also ch. viii.) Solid Solid 4. Hclmholtz's Experiment 7. Melde's E.Kpt. 10. Goold's Gene- rators and Bars 11. Violin Bow and String, Fork or Plate 12. Trevelyan's Rocker Fluid .'i. Voice and Air Resonator 13. Singing Flames 16. Sensitive Flames of Fluid 14. Organ Pipes (chap, viii.) various kinds Fluid Solid 6. One Fork re- sponding to another at a distance 8. Erskine- Murray's Phonoscope 9. Human Ear 17. Setting of , Disc 18. Kundt'sDust i Figures 19. Striations in same Magnetic Field 15. Electrically- driven Tuning-Fork Solid 236. ExPT. 34. Timing-Fork mid Simple Air liesonatms. — Perhaps the simplest experiment on resonance is that with a 236 EESONANCE AND EESPONSE 317 tuning-fork and a tumbler or bottle. If the natural tone of the cavity of the vessel is too flat to correspond with the fork, it may be raised by pouring in water. If, on the other hand, the pitch of the cavity is too sharp, it may be flattened by shading the mouth. The necessary adjustment can usually be effected in a few minutes. Some clue to dimensions will be afforded by the following particulars of an actual case. Here the necessary adjustments were made in a few minutes, although only the homeliest materials were available. The fork used was an ordinary musician's philharmonic a' (say 454 per second). The tumbler used was almost cylindrical in shape, 2| inches diameter at the top, and i\ inches high. Its natural tone when empty, as ascertained by gently blowing across the top, was higher than the fork. Hence it needing shading. A closely-fitting glass plate is perhaps best for this purpose ; but a small pamphlet answered well enough. The best resonance was obtained when the pamphlet covered about three-quarters of the diameter of the tumbler's mouth perpendicular to the edge of the pamphlet. Even the hand put over the tumbler's mouth answered almost as well, the necessary adjustment being quite possible during a single sounding of the fork. As an example of a second resonator initially too low, the case of a bottle may be noticed, the same fork being still used. The body of the bottle was nearly cylindrical, about 3f inches high, and If inches diameter inside. Its nearly cylindrical neck, 3 inches high, was f of an inch diameter at the bottom, but had a short constriction to | of an inch diameter near the top. To tune this bottle as a resonator to the fork, it was neces- sary to pour in water to the height of 3:^ inches, thus leaving only I inch of its body occupied by air. A sensitive method of detecting the fault when too much water is in, consists in placing a finger so as to partially shade the mouth of the neck while the fork is sounding. When too much water is in and the natural tone of the bottle is in consequence too sharp, the shading flattens its pitch and thus improves it as a resonator, which is shown by an increased volume of sound. The fact that the first two articles which came to hand were readily tuned to the only fork then available, shows how simple this process is. If it is desired to tune a vessel more permanently to a given sound, instead of water, oil or melted wax should be poured into a vessel naturally too flat. Or 318 SOUND CHAP. VI the glass plate used to shade the mouth of a vessel naturally too sharp may be fastened on with wax. The mounting of tuning-forks on resonance boxes is another application of the principle just dealt with. 237. ExPT. 35. Use of Adjustable Resonator. — The principle of partially occupying the air cavity with water to Fig. 61. — Abjustable Resonaiob. raise the pitch to correspondence with any given sound receives a more elaborate application in the apparatus now to be described. It consists essentially of a tall cylindrical glass jar connected at the base by an india-rubber tube to a reservoir of water which is adjustable to various heights on a wooden upright. The reservoir may be clamped at any desired height by means of a screw. The jar may have a graduated scale attached to it for reading the depths below its mouth corresponding to best resonance with any given 237,238 EESONANCE AND EESPONSE 319 forks or other sources of sound. One form of this simple but very useful piece of apparatus is shown in Fig. 61. This apparatus may be very rapidly and silently adjusted by raising or lowering the reservoir. Thus, the position for maximum resonance with a fork may be obtained while it is sounding. If a simple jar is used, it is usual to tune by successive trials of the water level. But this makes it hard to decide precisely which level is best. An improvement with the simple jar is the use of a stick thrust into the water and adjusted with one hand while the other holds the fork. But the apparatus shown above is much more convenient and yields better results ; about 20 inches is a convenient height for the resonating column. As an example of its use let us determine the relation between the prime tone of a fork and that with four nodes, that is, one near the end of each prong in addition to the usual two near the stem. As already seen in the theory of the lateral vibration of bars (art. 211), the ratio of the frequencies of the second tone and prime for a fixed-free bar is about 6' 3. For a series of tuning- forks examined by Helmholtz this ratio was found to vary between 5 "8 and 6 "6. Let a c' fork be used, of frequency 256 per second. And suppose 12 '6 7 inches to be the measure of the column of air above the water for the best resonance, with its prime tone obtained by bowing or by striking on a soft pad. Then by striking oii a hard surface find the resonance position for the first upper partial. Let this distance be r73 inch. Suppose the internal diameter of the cylinder to be 1'5 inch. Then, by article 179, each of the above lengths needs increasing by the end correction 0'6 a which is here '6 x -75 = 0'45 inch. The two corrected lengths are accordingly 13"12 and 2' 18 inches respectively. Hence, the ratio of the frequencies is the ratio of these lengths, that is 13-12/2-18. And this equals 6-02. Other uses can be made of this apparatus, as we shall see later. 238. Theory of Resonators. — Having noticed some simple preliminary experiments, we now pass to another and very important class of resonators, namely, those whose openings to the external air are small in comparison with the enclosed cavity. These are used to respond to particular tones as fully and precisely as possible. They thus indicate 320 SOUND CHAP. VI the presence of those tones even when very weak, and also serve to ascertain their pitch with great nicety. These qualities are due to the fact that the small opening allows of only a small dissipation, and hence a vibration once started in them persists for a considerable number of periods. Now, it was shown in articles 94-96, on forced vibrations, that the smaller the damping possessed by a vibrating system, the greater is the vibration forced upon it and the more sensitive is it to exact tuning. Hence the advantage of the form under notice and the necessity for a theory concerning it. They are included liere because they were used so extensively by Helmholtz to respond to tuning-forks close to their mouths. They thus afford an example of the driver being a solid body, and the driven or responsive substance a gas, viz. the air. It is obvious, however, that they can be used also to respond to the voice or other sounds originating in the air and not pro- duced by the vibration of a solid body. The full theory, as given by Helmholtz and others, is beyond the scope of this work. We can, however, follow the simple method given by Lord Eayleigh for the pitch of the prime tone of such resonators. 239. Imagine first the case of a stopped cylinder in which a piston moves without friction. Let the piston move to and fro very slowly. Then, as an approximation, we may suppose the pressure of the air inside to be every- where the same, namely, that due to the momentary position of the piston. Now, if the mass of the piston were con- siderable in comparison with that of the air in the cylinder, the natural vibrations of the system would occur slowly and almost as if the air had no inertia. Thus, in calcu- lating the period, it would be possible (1) to consider the kinetic energy as resident wholly in the piston, and (2) to treat the potential energy of the enclosed air as if its rare- factions and condensations were unifoi'm. Thus, the air inside acts as a spring and the form of the containing 239,240 EESONANCE AND EESPONSE 321 vessel is immaterial, the period for a given piston remaining the same provided the capacity is unchanged. When we pass from the case just described to an actual resonator of the type to be dealt with, the approximation becomes less close. For here the piston is represented simply by the air in and near the opening, and its mass cannot be great. But, since its velocity may be very great in comparison with that of the air well within the chamber, we may still regard the kinetic energy as resident chiefly in this air "piston," and write our equations accordingly. The results obtained by this elementary method afford an instructive view of the subject, and often give a fair value for the pitch of the prime tone or fundamental. They must not, however, be supposed to correspond to a rigorous treatment of the subject. 240. We have already seen (arts. 84-85) that, if a mass is urged by a restoring force proportional to its displacement, it executes simple harmonic motion of period expressed by T = 2 TT V mass -r- spring factor ( 1 ). We have, accordingly, to express for our present problem the values of mass and spring factor and substitute in (1). Let the cross sectional area of the neck, its length and the density of the gas in it, be denoted respectively by A, L, and p'. Then, for the " piston " with which we are con- cerned, we have obviously mass = ALp (2). Let S be the volume of the chamber of the resonator and p the density of the gas within it (not necessarily the same gas as that in the neck). Then the spring factor or force per unit displacement may be denoted by Adp. But it was seen (in art. 152) that -y' = ^ = TP//' (3). V being the average speed of sound in the gas occupying the chamber, and p the pressure. Y 322 SOUND CHAP. VI Thus dp = yjjdp/p (4)- Also, by the geometry of the case for a displacement of unit length in the neck, S{p + dp):=(S+A)p, whence dp/p = AjS (5). Thus, from equations (4) and (5) we obtain for the spring factor Adp = Aypdp/p = A^yp/S (6). Hence, substituting from (2) and (6) in (1), we have T=27r^LSp'/Ayp CI), or, T=2-7ris/Sp'/cyp (8), where c is written in place of A/L, and by electrical analogy may be called the " conductivity of the neck," The frequency is given by ^^s/cypJSl .^g^_ 27r 241. We thus see that the pitch is dependent on the density of the gas in the neck, but is independent of the nature of the gas in the chamber except for the small possible variation due to a change in the value of 7. If, now, we suppose the gas to be the same throughout the chamber and neck, we have from equations (3) and (8), \ = VT=2'7r\/slc (10). Hence, in this case, the wave length is independent of the gas used, and depends only on the size and shape of the resonator itself. Again, if the gas is the same throughout, (9) may be rewritten ^'=^v/c7^^ (11)- It may be noted here that the introduction of the con- ductivity c of the neck not only affords a more compact expression of the results of the simple theory, but may be 241, 242 RESONANCE AND RESPONSE 323 used to express also the frequency obtained by the more complete treatment. In the latter case, however, c is no longer the quotient AjL merely, but the value of the conductivity determined by higher analysis. 242. Comparison with Experiments. — Prom experi- ments on flasks with long necks, Sondhauss found N==Afi10-5s/AJLS (12), which corresponds in form with (11), but gives the pitch about a tone flat'ter than that obtained by the simple theory putting «= 33,200 cm./sec. For the case of resonators without necks, Sondhauss gave the formula N= 5,240^Y;S'* (13). The theory of such a resonator has been developed by Helmholtz, who found N= ^tL=^YS1 = v{AlAm-'Sy (14). Putting «= 33,200 cm./sec, this gives i\^= 5,617^*/^ nearly. The theoretically determined pitch is again sharper than the observed, but this time the discrepancy is only of the order a semitone. Helmholtz's theory is developed on the assumption that all dimensions of the chamber are vanishingly small compared to the wave length of the air, and the air opening vanishingly small in comparison with that of the chamber walls. But, in actual practice, the necks are not usually so long that end correction is unnecessary as assumed in the simple theory developed in the last article. Neither are they usually so short as to be themselves negligible as assumed in the theory leading to equation (14). We have already seen, in connection with organ pipes (articles 179, 180), that we may represent the end correction by an addition to the length. Further, this quantity is to a first approximation independent of the length of the neck L and of the wave length X. As previously noticed in connection with pipes, 324 SOUND CHAP. VI the value of the end correction depends on whether at the open end there is an infinite flange. But for our present purpose the value —B, where R is the radius of the neck, will be near enough. For this is the value theoretically found when the infinite flange is present, and when the neck is short this condition almost represents the fact, and when the neck is long the correction is itself of sraaU account. Hence, in place of equation (11) we may write The second form being more convenient if the opening is nearly but not quite circular. It must be clearly understood that the methods spoken of refer only to the pitch of the prime or gravest tone of the resonators in question. The overtones are relatively very high, and to determine them it would not be legitimate to neglect the inertia of the air in the chamber. The experimental use of resonators of this class will be sufliciently exemplified in the next chapter. 243. ExPT. 36. Helmholtz's Resonance Experiment. — Tune the wire of a monochord exactly in unison with a good fork ha\'ing a frequency of 128 per second. Prepare a number of riders of white paper each about 2 cm. long and 2 mm. broad, and bent to an L-shape. Place several riders anywhere upon the wire. Then bow the fork and immediately place it with the stem on the wire just where it passes over one of the bridges. If all has been properly performed, the wire should be set into vibration in its gravest mode as shown by its almost instantaneous rejection of all the riders. Next, place only three riders on the wire, and at the points 1/4, 1/2, and 3/4 of its length from one end. Bow a fork of frequency 2.56 per second, and place its stem on the bridge as before. This time the first and third riders should be kicked off the wire while the second remains, thus showing 243, 244 EESONANCE AND EESPONSE 325 that the vibration has two segments appropriate to the first overtone, as might be anticipated. The experiment may be repeated with a fork of 384 per second, riders placed at 1/3 and 2/3 along the string remaining, while riders at 1/6, 1/2, and 5/6 should be jerked off by the vibration of the string in three segments. Again, with a fork of 512 per second, riders placed at 1/4, 1/2, and 3/4 along the wire should remain, others at 1/8, 3/8, 6/8, and 7/8 along being thrown off. The higher the pitch of the fork, the harder it becomes to elicit from the wire vibrations of sufficient amplitude to dislodge the riders. To be successful the tuning must be very exact, and the bowing of the fork as vigorous as possible, It may also be found desirable to roughen or notch the end of the tuning-fork stem, and it sometimes acts better if placed not quite over the bridge, but a little (say 1 or 2 mm.) from the bridge towards the middle of the wire. If, however, the stem needs moving any considerable distance from the bridge before the response occurs, it is proof that the string is too flat. In all these cases the riders are for indicating to the distant members of a class that the string has responded ; those very near may hear the resonance if the experiment is well executed. 244. ExPT. 37. Air Resonators responding to the Voice. — The response to the voice of .a simple air resonator such as a bottle, flask, or vase may be demonstrated as follows : — Blow across the mouth of the vessel to ascertain its pitch, then sing powerfully into the vessel a vowel at that pitch. Immediately on ceasing to sing, apply the mouth of the resonator to the ear, whefl the response will be heard, and with some vessels lasts a second or two. If a Helmholtz resonator is available, an observer may have its nozzle applied to his ear while a tone of the right pitch is sung at some distance by another person. The reinforcement due to the resonator will then be clearly perceived. ExPT. 38. One Fork resiwnding to Another. — For this experi- ment two very good tuning-forks are required of precisely the same pitch, and each mounted on a suitable resonance box. One fork is vigorously bowed, and the two held for a few seconds with the mouths of their resonance boxes near together. Then, on stopping 'the flrst fork with the finger, the other will be heard distinctly sounding. This is classed 326 SOUND "HAP. VI in the table as a case of a fluid vibration driving a solid body, because it is the air in the resonance box of the first fork which starts that in the box of the second fork, and this in turn appears to be chiefly instrumental in generating the vibrations of the fork itself. This concludes our notice of resonance proper. 245. Melde's Experiments. — In this illustration of the maintenance of vibrations, a thread is set in regular periodic motion by a tuning-fork. The thread extends from a fixed point to one prong of the fork, and when matters are properly adjusted responds to the fork's motion. There are two chief modes of the experiment to consider, which may be called the transverse and the longitudinal positions or modes respectively. In the transverse position the length of the thread is placed at right angles to the path described by the tip of the prong to which it is attached. In the longitudinal position the thread and the path described by the tip of the prong are in the same straight line. In the former position the nature of the action is quite simple, the, latter, however, requires a special examination. Tratisverse Arrangements. — Here we may regard the stationary waves in the thread as formed by the composition of the direct waves sent from the fork, and those reflected from the fixed point at the far end. Hence, the ordinary equation for vibrating strings will apply. Thus, for the vibration of the thread in one segment we have (see equation (4), article 165) N=^'jFp (1), where N is the frequency of vibration of the thread, L is its length, F the stretching force, and a the mass of the thread per unit length. If this vibration is to be caused by a fork of frequency N, then L, F, and o- must be chosen so as to satisfy (1). If, however, we wish the same length of thread to vibrate in two segments, keeping to the same fork, we must tune the thread down an octave by reducing 245, 246 RESONANCE AND EESPONSE 32*7 the stretching force to one-quarter of its previous value. Then the fork, though still exciting a vibration synchronous with its own, elicits, not the prime, but the second vibration of which the thread is capable. So generally, if we require n segments excited in the thread by the same fork, we must tune the string down so that its prime has a frequency Njn. In other words, keeping N, L, and o- constant, we must so choose F^ the stretching force for n segments, as to satisfy the equation Fn = Fln^ (2), F being defined by (1). For values of F see Table XVIII. 246. Longitudinal Arrangements. — For the longitudinal form of Melde's experiment, suppose the thread to be horizontal, and the fork with its prongs upwards vibrating horizontally in the vertical plane through the thread. And, in considering the nature of the action involved, begin with the prongs apart and the thread consequently slack as shown in the upper line A of Fig. 62. We suppose here that the fork and the thread are each at the full extent of their excursions and have for the instant no velocity. This fact is indicated by the small circles near the prongs and the thread. Now let the prongs approach each other till their nearest position is reached, and let the thread at the same time be just straight and horizontal. Then the prongs, as before, are at the pause for an instant. The thread, however, is now moving upwards. This state of things is represented by the second line B of Fig. 62, in which the circles indicate rest and the arrows motion. Now, let the prongs separate agaiu to their fullest extent. The thread having liberty will naturally move to its position of maximum displacement. But, owing to its upward momentum when horizontal, its displacement this time will be upwards. This position is represented in the lowest line C of Fig. 62, the circles again indicating rest. Now from the position shown in line A to that shown by C, the whole period of the fork elapses, but only half a period 328 SOUND CHAP. VI of the thread's vibration. Hence for the vibration of the thread in one segment as shown we must have it tuned an octave below the fork. In other words, the action of the driver depends upon the position of the driven, and by the momentum of the driven that position and action are reversed in each succeeding period of the driver. Hence, in its effect upon the driven, two periods of the driver are Fig. 62. — Longitudinal Arrangement of Melde's Expekiment. required to complete the cycle of operations. Thus, if the frequency of the fork be N as before, and the stretching force of the thread be F', we have ^ 1 /WT (3). 2Z So F' = F/4: (4), where F is the force in the transverse form of the experi- ment for one segment. Suppose now we require n vibrating segments in the 247 EESONANCE AND KESPONSE 329 Then we have the thread when excited longitudinally, tension expressed by F\=F'jn' (5). For lecture illustration a large fork of 64 per second and a thread about 2 '5 metres long have been found convenient. The thread may be of silk and about 16 metres to the gram. The corresponding stretching forces in a set of lecture experiments were then as shown in Table XVIII. Table XVIII. — Melee's Experiment with a 64 Fork. No. of Segments in Thread, n. Transverse Mode. Longitudinal Mode. Frequency of Thread. stretching Force, Fn. Frequency of Tliread. Stre'tching Force F'n. 1 2 4 8 64 32 16 8 640 160 40 10 32 16 8 160 40 10 247. ExPT. 39. Melde's Experiment. — This experiment may be carried out on a variety of scales and in different ways according to the object in view. For a large audience the thread may be horizontal and illuminated by the beam from an arc lantern. Another method is to steep the thread in sulphate of quinine and expose it to the violet rays of the arc light as suggested by Tyndall, the string then ex- hibiting a brilliant greenish-blue fluorescence. In this case it is better if the fork is electrically driven (see article 278). But where these elaborations are not needed or cannot be attempted, the experiment may be performed in the following simple way in a laboratory, or for audiences up to a hundred : — Clamp the fork so as to project horizontally from the top of a stool, and let the thread, about a metre or more long, hang vertically from it with its weight at the bottom. The right adjustment of length and stretch- ing force may be found either by calculation or by a pre- liminary experiment with the thread horizontal and passing over a pulley. This being done for one segment with the 330 SOUND CHAP. VI transverse mode of the experiment, the key is obtained to the whole set of forces required for any other number of segments of that mode or of the longitudinal mode. Then the proper weights being hung on, a single stroke of the bow sets the fork and the thread into the desired vibration. Thus, for the thread previously mentioned, but now 1^ metre Fig. 63. — Simple Form of Melde's Experiment. long and used on a fork of 128 per second, the series of weights given in Table XVIII. will apply. The simple turning of the fork in the clamp through 90' about its own axis serves to effect the change from the transverse to the longitudinal mode of the experiment. Both modes are shown in Fig. 63. 248. Theory of Longitudinal Form of Melde's Experiment. — The somewhat unusual character of the 248 EESONANGE AND EESPONSE 331 maintenance by a driver of double frequency met with in the longitudinal form of Melde's experiment has been treated mathematically by Lord Rayleigh, whose method we follow here. The motion of the fork produces a variation of the tension of the string, and hence, if the adjustments are right and if by any means a vibration is started, then it is maintained. Let us therefore suppose the steady motion to be in existence, and examine the conditions necessary to continue it. Now, a variation in the tension produces a corresponding variation in the restoring force acting on any displaced portion of the string. Thus, limiting our con- sideration to a single vibrating point we may represent the effect of the driver by a periodic variation of the spring factor. We may thus, with our previous notation, write as follows a sufficiently representative equation of the motion: — my + ry + (s — 2ma sin 2pt)y = 0. Or, if rjm = k and sjm = n?, we obtain as our working expression, y + Ky + {n^ - 2a sm 2]pt)y = () (1). Try as a solution y = A^ sin pt + B^ cos pt + A^ sin ipt + £^ cos 2>'pt + A^ sin 5pt + B^ cos bpt-\- . . . (2), terms involving an even multiple of pt being unnecessary. Substituting in equation (1) and equating to zero the co- efficients of the sines and cosines of pt, ipt, 5pf, etc., we find A(.n^ - P^) - a-B, - KpB, - a^3 = 1 B^ln^ -/) - a-^i + x^pA -aA^=Q\ As(v^ - 9^') - aB, - 3kpB, - aB, = O] Bjy - 9/) H- a^i -I- 'iKpA^ -aA,= 0\ A,(n' - 2 5/) -aB,-5KpB, . . = 1 B,(n^-25p^) + aA^+5KpA, . . = OJ Equations (4) show that A^ and Bg are of the order a times (3), (4), (5). 332 SOUND CHAP. VI ^1 and B^. Similarly equations (5) show that A^, and, B^ are of the order a times A^ and B^. Thus, if from (3) we omit A^ and B^, we have as a first approximation A(n' -f) = B,{a + K2J)] B,(n'-p') = A,(a-Kp)j -^1 n^-2r a-K}} Ja-Kp ,^, \¥hence - - = — = — — ^^7, = . 1 1 )> A^ a + Kp n —p- s/a + Kp and {n^ —p^T = «" — i^'p"'" (S)- 249. Thus, if a be given, the value of p necessary for a steady motion is definite. And if p has this value, we have to our approximation by omission from (2) of A^, B^, etc., y = A-y sin pt + B^ sin^^, or, 2/ = P sin (^t + e) (9), where P^ = A^- + P/, and tan e = B-^jA-^ (10), in which, therefore, e is definite also. Hence the period and phase are fixed. But there is nothing as yet to deter- mine P, the absolute amplitude, whose value, therefore, depends upon other circumstances. Equation (8) shows that, for regular motion, a must be not less than Kp. If a. = Kp, we then have p = ii by equation (8), and from equations (7) and (10) we see that e = 0. Hence (9) becomes y = Psmnt (H). when a = Kp, or in words, this is the case of ideal adjust- ment considered in the elementary view of the matter, and in which the frequency of the driver is just double that of the driven, supposing friction to be absent. If a were less than Kp, the motion would not be main- tained. But if a were greater than Kp, then the energy available for supply would be greater than needed. But the balance between energy actually supplied and that dissipated could be attained by an alteration of the phase relation. Equations (10) and (7) show that e may have 249, 250 EESONANCE AND EESPONSE 333 either algebraic sign. If the sign is taken positive, then by (8), ^ is given by ■f = 71^ - >J a- - K'f (1 2). It thus appears from the equations that the adjustment of frequencies by the experimenter must be perfect in order to secure the maintenance of the motion. This is, however, subject to modification in the actual experiment. For, with the thread used, the value of w is to a slight extent dependent on amplitude. Hence the steady state may be automatically attained by the assumption of a large and determinate amplitude which secures the necessary adjust- ment of frequencies. 250. Dr. Erskine - Murray's Phonoscope. — For the graphical exhibition to an audience of the vibrations con- stituting sounds produced in their hearing, very similar arrangements seem to have been independently devised at about the same time by Professor J. G. M'Kendrick and by Dr. J. Erskine-Murray. The one here described was first publicly exhibited at Nottingham in 1901. Its action will be easily understood from Fig. 64, which shows the essentials of the device. The exciting sound is produced close to the large open end AD of the conical tube ABCD. The small end BC of this tube is covered with a thin membrane left rather slack. Near the middle of this end a small mirror E is hinged upon a wire F. The mirror is also connected near its axle by a light support (of aluminium foil, say) to a point G near the middle of the membrane. Thus every flutter of the membrane rocks the mirror about its horizontal axis. A beam of light HE from an arc lantern is passed through a small hole in a diaphragm and arranged to fall upon the mirror E. It is thus reflected to a set of mirrors J capable of rotation about a vertical axis. From the mirror J the light passes to a point K, say, on a large distant screen KL, upon which the light is focused by a lens not shown in the diagram. If the mirror J is at 334 SOUND CHAP. VI rest and the sound directed into the open mouth AD, the membrane BC flutters, the mirror E rocks about its horizontal axis, and the vertical line MN is described by the spot of light on the screen. If, on the other hand, without any sound being produced the mirror J is rotated about its vertical axis, we should have the horizontal line KL described by the spot of light on the screen. When, how- ever, the sound is directed into the instrument, and the set of mirrors at J is simultaneously rotated, we have, from the composition of the two motions, a periodic curve such as KPQRL described on the screen. In making this phono- FiG. 64. — Dr. EESKiNE-MnRRAy's Phonoscope. scope, it is of the utmost importance that the mirror E should be very light and quite free to move, but without shake, and that the membrane should be somewhat slack. The diameter of the membrane must also be chosen with reference to the pitch of the sounds for which it is to be used. 251. ExPT. 40. Phonoscope responding to Organ Pipe Tones. — Place an organ pipe with its mouth close to AD, switch on the lantern, adjust and focus. One operator should now rotate the mirror J while another sounds the prime tone of the organ pipe. The sine graph is immediately described on the screen by the spot of light. Next, suddenly place on the open end of the organ pipe, a padded board. This converts it into a stopped pipe, and therefore lowers the pitch 251, 252 EESONANCE AND EESPONSE 335 about an octave. Immediately the pitch is heard to fall the sine graph on the screen is observed to change its wave length to about double its former value. A third experiment with the same pipe is as follows : — Elicit the prime tone of the open pipe as at first, and then suddenly " over-blow " it so as to make it speak its first overtone. This is an octave higher than the prime, and is recorded on the screen by a sine graph of half the previous wave length. A fourth variation may be made by blowing so as to elicit simultane- ously the prime and the first overtone. Both sounds may be heard with care, and the beautiful curve characteristic of this compound tone at the same time appears on the screen. It is difficult to maintain this state of things for long, and indeed requires some little address to obtain at all. A fifth experiment may be made by obtaining the first overtone of the same pipe when stopped. This, as shown theoretically, will be heard and seen to be the twelfth of the corresponding prime. Sixthly, the prime and first overtone of the stopped pipe may be obtained together, and the curve showing the composition of frequencies 3 : 1 exhibited on the screen. Other uses of the phonoscope of still greater interest and importance may be made in the case of vowels and con- sonants uttered by the voice. 252. The Human Ear. — For anything approaching full details of the human ear, the various anatomical and physiological woriis should be consulted. Nothing can be attempted here beyond what is needed for understanding the physics of the phenomena with which we are concerned. And, for this purpose, it must suffice to give a bare outline of the disposition and action of the more essential parts. Starting from the outside, we have first the external ear {pinna), from which extends the ear passage {external auditory meatus). This ends at the drum-skin {memhrana tympani), ■ usually, though improperly, spoken of as the drum. Beyond this lies the cavity which is properly called the drum {tympanum). From the drum the Eustachian tube proceeds to the pharynx. This tube is usually closed, but opens each time swallowing occurs. This occasional opening of •336 SOUND CHAP. VI the tube for an instant effects, when necessary, the eqiialisa- tion of the pressure of the air inside the drum with that of the external atmosphere. In other words, it relieves the drum-skin of any tension due to a difference of air pressures on its inner and outer sides. The drum is bridged across by a train of three little bones or ossicles. These are called the hammer, the anvil, and the stirrup (or malleus, incus, and stapes respectively). One part of the hammer is in contact with the drum-skin, while another part articulates with the anvil. A process of the anvil is attached to the apex of the stirrup. The base of the stirrup is applied to a membrane which closes an oval opening {fenestra ovalis or fenestra vestibuli) in the bony wall which forms the inner or mesial limit of the drum. In this bony wall there is in addition a round opening {fenestra rotunda or fenestra cochlea) also closed by a membrane. Beyond this mesial side of the drum with its two windows lies the innermost part of the organ of hearing called the labyrinth. This may be subdivided according to the natures of its materials or according to the shapes of its parts. Following the first method of subdivision, we recognise, first, the bony labyrinth or hollow excavated in the bony substance itself; and, secondly, the membranous labyrinth, a similar structure enclosed in the other and of similar shape. The mem- branous labyrinth contains a liquid called the endolymph. It is also surrounded, wholly or partially, by another liquid called the perilymph. This fills up the space between the membranous labyrinth and the bony labyrinth. Taking now the second method of subdivision, according to shape, the labyrinth comprises the vestibule, the three semi-circular canals, and the cochlea. The vestibule is the middle portion of the labyrinth, and contains the oval window that receives the foot of the stirrup. Upward and backward from the vestibule proceed the three semi-circular canals, each of which debouches with both its ends into the vestibule. With these we are not much concerned. Forward and 253, 254 EESONANCE AND EESPONSE 537 downward from the vestibule we have that very important structure, the cochlea. This is a spiral canal in form like a Three Semicircutar Canals Cochlea Fig. 65. — General View op HnMAN Ear, Hammer] snail shell. At its entrance lies the round window previ- ously referred to. 253. Let us now try to form a working conception of these parts and their mutual relations. A reference to Fig. 65 will assist in this endeavour. But it must be borne in mind that in a single general diagram on a moderate scale it is impossible to obtain intelligibility without some sacrifice of strict accuracy. In this case, for example, the labyrinth is, for clearness' sake, drawn on a larger scale than the rest. Fig. 66 gives a larger view of the ossicles in position, and should be examined next. 254, Let us now pass to a closer study of the cochlea. z Fig. 66. — Auditory Ossicles in Position. 338 SOUND CHAP. VI Suppose the spiral canal which forms the cochlea to be straightened out. Then let a longitudinal section of it be taken without cutting through its inner or membranous portion. The result is diagrammatical! y represented, in Fig. 67. We thus see that the perilymph space is, for the greater part of the length, divided into two by a partition (the lamina spiralis) which reaches almost to the tip of the cochlea. The upper part in the diagram is called the scala vestibuli. It begins at the oval window on which rests the foot of the stirrup. The lower part is called the scala tymiMni and begins at the round window. The two are in communi- cation at the tip of the cochlea beyond the termination of Stinup, FiQ. 67. — LoNGirnDiNAL Section of Cochlea as if Straight. the lamina spiralis. It should be noted that each of these scaloe, though here shown Straightened out, is in the actual cochlea comparable to the winding stairway of a turret, hence their names. 255. We now pass to the consideration of a transverse section of the spiral canal of the cochlea. This is shown in Fig. 68. In addition to the two perilymph spaces (scala vestibuli and scala tympani) seen in the longitudinal section, this shows the membranous portion of the cochlea {canalis coch- learis). It is thus seen that in this region the membranous labyrinth is almost triangular in section. Its curved base is closely applied to the corresponding portion of the bony labyrinth instead of being separated from it by any space filled with perilymph. The apex of the triangle meets the 255 EESONANCE AND EESPONSE 339 partition (lamina spiralis) which completes the separation between the scala vestibuli and the scala tympani. The upper side of the triangle^ as seen in the diagram, is known as the membrane of Eeissner. It divides the canalis coch- learis from the scala vestibuli. It is thin and simple in character. The lower side of the triangle, on the other Pig. 68. — Diagham of a Teansvbesb Section of a Whorl of the Cochlea. hand, is very complex. Its outer part, to the right in the figure, is formed by the basilar membrane. Its inner portion (to the left) is composed of a projecting part of the lamina spiralis, which here terminates in two lips (the labium vestibulare and the labium tympani). Just beyond these lips we have the organ of Corti. This comprises a number of inner and outer rods of Corti (somewhat like rafters of a roof), and various other minute structures. 340 SOUND CHAP. VI This organ rests upon the centre part of the basilar mem- brane, which is thin and not highly specialised. The outer portion of the basilar membrane, which is free from other structures, is itself thicker and more highly specialised. It is found to be composed of radial fibres lying side by side and embedded in the homogeneous ground substance of different nature. The auditory, cochlear nerve passes up the axis of the cochlea. As it ascends it gives off fibres passing outwards in a spiral manner into the lamina spiralis. The fibres pass into and become connected with the auditory epithelium of the organ of Corti. 256. The Act of Hearingf. — Having sketched the chief parts of the human ear, we are now prepared to trace the contributions made by each to that complicated train of phenomena which constitutes the act of hearing. Suppose we have a simple tone produced at any external source in the neighbourhood, then waves of sound pass from that source to the ear. The atmosphere at the entrance to the ear is consequently vibrating in a simple harmonic manner. As the air waves fall upon the drum-skin, they force it into approximately corresponding vibrations. For the drum- skin is of such size and tension as to readily respond to any vibrations between certain wide limits. These vibra- tions then pass through the train of three ossicles. For the motion of the drum-skin is imparted to that process of the hammer in contact with it, by means of its articulation the hammer then passes the motion on to the anvil, and this in turn moves the stirrup bone. The base of the stirrup is applied to the membranous covering of the oval window, and so hands on its motion to the perilymph beyond, which occupies the scala vestibuli. This vibratory motion of the perilymph appears to affect the endolymph in the canalis coc^learis, the two liquids being separated only by the thin , membrane of Eeissner. Thus, the organ of Corti and the basilar membrane are exposed throughout the 256, 257 EESONANCE AND EESPONSE 341 length of the cochlea to vibrations synchronous with those originating at the external source in the atmosphere. From the base of the cochlea to its tip these structures are graduated in size. They seem, therefore, designed to respond sympathetically to vibrations of all different fre- quencies throughout the range of audition. Formerly it was supposed that the rods of Corti played this part. But that view was abandoned as untenable. The rdle of sympathetic vibrations has since been ascribed to the basilar membrane. For it is loose longitudinally, that- is, from base to tip of the spiral of the cochlea, but tense radially. Hence, it may be considered as consisting of a number of parallel strings somewhat like those of a harp. The radial dimensions of the basilar membrane are found to increase from the base to the tip of the cochlea. According to this view, then, as any simple vibration sweeps along the cochlea, it throws into sympathetic vibration just_ that part of the basilar membrane which is tuned to respond to motions of that particular frequency. The vibration thus excited in the membrane in turn so affects the overlying structures that auditory impulses are generated in a particular group of fibrils of the auditory nerve. These auditory impulses reaching the brain give rise to the corresponding sensation of a sound of that particular pitch, loudness, and duration. 257. Compound Tones. — Suppose now that two or more simple tones proceed from the same or different sources. "We then have the corresponding number of simple 'harmonic vibrations produced at the external ear. Or we may. regard the motion there as one compound vibration. This compound motion travels like the simple one through the drum-skin and chain of ossicles, arriving at the perilymph and endolymph in the cochlea. Then, as the composite vibration sweeps along the basilar mem- brane, we suppose it to excite sympathetic vibrations in certain parts of it, and in those only. Further, the parts 342 SOUND CHAP. VI ia question are those which are tuned to respond to the particular frequencies which characterise the simple tones of which the compound sound may be regarded as composed. But this view of the case as to the function of the basilar membrane must be taken only as a hypothesis. The dimensions of the membrane seem scarcely adequate to confirm it. In man the whole width of the membrane has been found to vary between 0-21 mm. at the base of the cochlea, to 0-36 mm. at the tip. The dimensions of the outer specially-modified part show a similar range, being 0-075 mm. at the base to 0-126 mm. at the tip. The estimated number of the radial fibres is 24,000. Thus, even the physics of hearing is very complicated, and part of it not clearly established. It is thus impossible strictly to classify the ear along with the comparatively simple physical experiments. It has been treated in this place because the action of the drum-skin, which is perhaps best understood, is certainly an example of the maintenance of vibrations. According, then, to the hypothesis just developed, we may regard as follows the perception of those features which characterise any musical sound, namely, pitch, loud- ness, and quality. The perception of pitch of a simple tone is fixed by the part of the basilar membrane excited, its loiidness by the degree to which that part is excited, while the perception of quality of a compound tone depends upon the relation of pitch and loudness which subsists between the different simple tones into which, by the ear, we suppose the compound tone to have been analysed. Por we have already seen (see articles 12, 31, 51, and 164) that the quality of a tone depends upon the number of partials of which it is composed, their relative frequencies and intensities. (See also Chap. VIII.) 258. Vibrations modified by Aural Mechanism. — We are not, however, to suppose that, in the act of hearing, each part repeats and hands on with absolute fidelity the 258, 259 RESONANCE AND RESPONSE 'Ua type of vibration received by it. Indeed, that this is highly improbable a moment's reflection will show. Eor the vibration originally in the air has to be copied in turn by an elastic membrane, a chain of three rigid solids, a second membrane, a liquid, a third membrane, a second liquid, and, finally, those minute elastic structures in connection with the auditory nerve. Further, that first elastic membrane, namely, the drum-skin, is not flat, but has a point near its middle drawn inwards. From this point radial fibres proceed which are themselves convex out- wards, see Fig. 69. Hence the drum-skin is under asymmetrical conditions as regards motions inwards and motions outwards. It ^'''- 69--EDaE View op deum-skin OF THE Eak and Ossicles. may thus be expected to display all the complication which we have previously found to characterise the behaviour of such systems when under the action of periodic forces (see articles 115 and 1 1 6). Next, in the chain of ossicles it is found that an asymmetry results from their manner of articulation. Hence all aural sensations need careful interpretation, or, in various cases, we may be led astray. This subject will be referred to again later (see article 304). 259. Goold's Generators and Bars. — Having suffici- ently considered cases of maintenance in which the period of the driver is quite definite, we now pass to those cases in which the periodic action of the driver is wholly or partially induced by the reaction of the driven whose period is definite. The first example we take is that 344 SOUND CHAP. VI furnished by bars vibrating transversely under the stimulus of a cane or other elastic rod passed in an oblique position over their surface. This method of eliciting the tones of bars, plates, etc., is due to Mr. Joseph Goold of Nottingham. It is illustrated in Fig. 70. The motion of the generator seems like that of a walking-stick (say of Malacca or other cane) vehen grasped firmly by the handle and pushed along with its point on a smooth pavement. The stick then executes a -number of vibrations which cause its point to move along by a series of short leaps or bounds. A smart blow is thus struck on the ground after each leap. Similarly a Goold's generator, when grasped in the hand and moved along the bar, seems Fig. 70. — Action of Goold's Gbheratok. to give the bar a number of taps. Now if these are roughly in tune with the bar (say within a major third), then by slightly altering the inclination of the generator or the pressure of the fingers, or both, the tuning may be brought sufficiently near to elicit the tone of the bar by transverse vibrations. Thus in this case the driver acquires the correct period not solely by the direct automatic reaction of the driven, much of the adjustment being due to the intelligence and dexterity of the operator directed by suggestions derived from the driven. The simplest form of generator adopted by Mr. Goold is that shown in Fig. 70. It consists of a piece of round cane fitted into a metal tube provided with a set screw for fixing the cane in the position found best for the purpose in view. The tip of the cane is simply rounded off. Some generators for rather low tones are tipped with leather. 260 EESONANCE AND EESPONSE 345 Others, designed for eliciting low tones of 100 to 200 per second, have the elastic part of wire fitted into a massive iron handle. These are provided with a tip of india-rubber for contact with the bar. 260. ExPT. 41. To elicit a given Partial from a Bar by a Goold's Generator: — First obtain a rectangular bar of tool steel, say 70 to 80 cm. long, and about 3 cm. by 1 cm. cross section. Then, find from the theory of transverse vibrations of bars (article 210) the positions of the nodes for the particular vibration desired, and mark these positions on the bar. At two of these nodes the bar should be supported on india-rubber. Mr. Goold makes these supports by putting three or four short pieces of round rubber, about 5 mm. diameter, into a base of wood so that the ends of the rubber just stand up in a millimetre or two, and are in a straight row. This row of rubbers forms the support for one node ; a precisely similar arrangement supports the other node. If the bar has been prepared for this experiment some-time previously, and greased to preserve it from rust, it must be cleaned as perfectly as possible before use. It is desirable to remove the last trace of grease with india-rubber. Place the bar so that it is supported on the rubber at two of the nodes for the required vibration. Now it is important to have evidence when the bar is beginning to vibrate in the mode desired, so dust of some kind must be scattered on the bar. Sand, however, should be avoided, as a single grain of it interferes with the action of the generator. Use, in preference, flakes of chalk. But even this should not be present where the generator is to be used, say a region half the width of the bar and extending from near a node to the next antinode. So cover that portion of the bar with a card or slip of wood, and then scatter over the whole the dust or flakes obtained by grating a piece of chalk with the sharpened edge of a wood stick or dry gum brush (not a knife blade). Carbonate of magnesia scattered from a pepper-box is still better than chalk. Eemove the covering from the part where the generator is to be used. Then stroke this part gently with a suitable generator. Listen for the sound, and watch whether the chalk flakes move towards the nodes. The pitch of the sound could be obtained by theory, or more simply by giving 346 SOUND CHAP. VI the bar, when properly supported, a "tap with a padded mallet or even with the knuckle. If the desired sound is not at first obtained, lengthen or shorten the projecting part of the cane or other elastic substance of the generator. Also change its inclination to the bar when in use. But do not press hard or in any way use great force. That is useless. But when the right length of cane is found, and the proper inclination, these must be strictly preserved in use. Indeed, the manipulation is much like that used with a glazier's diamond, when the adjustments are perfect a gentle touch suffices. Much time may be occupied in finding the right length of cane for a given pitch on the first occasion. The cane should then be marked and preserved in that position for future use for the given pitch. On repetition with all duly prepared, the experiment is very striking and beautiful, and well repays the time originally spent upon it. 261. Action of Violin Bow. — In the case of Goold's generators just considered, if the driver is sufficiently near being in tune with the driven, the reaction of the latter seems to complete the tuning and make the experiment succeed. But some approximate tuning is first needed. The striking peculiarity in the action of the violin bow is that' practically no tuning of the driver is needed. The driver in this case has no period whatever proper to itself Indeed, it has no periodic motion of its own at all. Still it readily excites and maintains vibrations in any string, fork, bow, or plate capable of vibrating in the direction of the bow's motion. How, then, does it act ? Consider first its construction. A good violin bow when newly haired contains about one hundred hairs from the tail of a white horse, about half their number being directed one way, and the rest the opposite way. The hairs are then rosined to enable them to attack the strings well. Let a string, initially at rest, have a bow drawn across it. The string begins to vibrate, and that vibration may be maintained till the end of the bow is reached. Thus while the bow moves continuously in one direction, the part of the strino- under 261, 262 RESONANCE AND RESPONSE 347 it moves to and fro. Hence the motion of the string is alternately with and against that of the bow. Thus the speed of the bow relative to the string is alternately less and more than its own actual speed. But the tangential frictional force called into play between unlubricated solids • is greater with a low relative speed than with a high one. Hence the urging effect of the bow when the string is moving with it is greater than its retarding effect when the string is moving against it. Thus there is, in the action of the bow, an outstanding excess of energy supplied to the string over that withdrawn. And it is this excess which is able to increase the vibrations of the string or maintain them in spite of dissipative forces. Moreover, the vibrations of the string, although thus maintained, are practically free as regards the naturalness of their frequencies. For, as we have just seen, it is the motion of the string itself which imparts periodicity to the action of the bow. The bow, as previously mentioned, has no periodicity in its own motion. Suppose a horizontal string is bowed at a point nearer the right end than the left, and that the bow moves downward. Then, it appears from the experimental investigations of Helmholtz that the place on the string bowed, for the greater part of the period, moves down with a constant but comparatively low speed. Then it suddenly flies up, this high speed being also constant. It is then caught by another part of the bow and again moves slowly down. Its speed of descent is probably about that of the bow. 262. ExPT. 42. Vibrations started and maintained hy con- tinuous Motion. — An interesting illustration of the mainten- ance of vibrations by alternate bite and release is afforded by the following arrangement : — Set up in bearings a horizontal shaft of rough wood about 2 cm. diameter provided with a small handle. From this shaft hang a Httle bob of lead by a thread about a metre long. The loop of the thread round 348 SOUND CHAP. VI the shaft should be quite slack. Now rotate the shaft slowly and steadily in one direction. The pendulum will soon be seen to be vibrating, and the vibrations may be maintained indefinitely by the continuous rotation of the shaft. This illustrates on a cruder and more visible scale the type of action involved in the use of the bow. 263. Trevelyan's Rocker. — We now pass to the first example of a vibration maintained by the periodic com- munication of heat and the consequent expansion. The experiment is due to Trevelyan, and is usually referred to as Trevelyan's rocker. The rocker consists of a prism of brass almost triangular in section, but one edge is taken Fig. 71. Fig. 72. Section of Tkeveltan's Tbeveltan's Rookee. Rocker. away and a groove made in the small flat thus produced, see Pig. 71. When working this prism rests with its groove downwards on a block of lead with a rounded top. The end of the rocker is carried by a rod terminating in a ball which rests on the counter, see Eig. "72. 264. ExPT. 43. To produce Various Tones hy Trevelyan's Rocker. — Clean the rounded top of a lead block with a steel scraper, or burnish it with a smooth steel rod. Avoid using sand-paper or emery-paper on the lead, as the gritty particles so introduced would stick in the soft lead and interfere with the good metallic contact between the lead and brass, and thus prevent the rapid heat conduction so essential for the desired action. Next, place the rocker with groove uppermost over a Bunsen burner, and leave it till the flame is tinged with green. Eemove the rocker from the flame, and quickly polish with fine emery the two narrow faces 263-265 EESONANOE AND EESPONSE 349 separated by the groove. Then immediately place it groove downwards on the lead block. The brass of the rocker is hard enough to bear the use of emery without detriment though the lead block is too soft for such handling. The rocker may spontaneously emit a musical sound immediately it is placed on the lead, or it may require a slight tap to start it rocking. Or, at first attempt, it may be too hot or too cold to work properly. But when, after a little trial, matters are rightly adjusted, it will rock rapidly and emit a humming sound. The pitch of this sound may be sharpened at will by pressing on the rod of the rocker near the ball with a round-ended rod, say a pen-holder. Also, by alternately increasing and decreasing the pressure so applied the sound emitted will rise and fall like the soughing of the wind. The explanation of the maintenance of vibrations in this case is simple. The metallic surfaces being quite clean, very rapid communication of heat is possible. Thus, when the rocker is placed on the lead with one of its narrow faces in contact, the heat is communicated to the lead, and by the consequent expansion a slight hump is raised on the lead. This tilts the rocker over on to its other face, where the operations are repeated, and the rocker thereby returned to its original position. In order to account for the maintenance, we must introduce the idea of a slight lag between contact and expansion, or the approach of the rocker would be checked, and the motion discouraged to an extent equal to its encouragement in its return motion. The slight lag decreases the discouraging effect and increases the encouraging effect of the expansion, and thus there is an outstanding excess of encouragement which accounts for the maintenance of the vibration. 265. Singing Flames. — We now deal with a still more interesting though more complicated class of examples, in which vibrations are maintained by heat, namely, those in which the vibrating body is gaseous, and the heat is supplied by a small flame which vibrates in the process and so emits 350 SOUND CHAP. VI a musical sound. Hence the term singing flames. The vibrating body is the mixture of air and other gases con- tained in the tube or chamber into which the flame is introduced. To prevent deterioration of the air in this chamber it must evidently be open both above and below, but its form may be cylindrical or bulbous, like a lamp chimney. The gas used for the flame may be the ordinary coal gas, or be hydrogen evolved in the usual way from granulated zinc and hydrochloric acid in a flask with the jet tube mounted at the top. Let us first consider the matter from the experimental side, and notice the conditions requisite for success. We shall then be in a better position to examine in detail the hypotheses which have been advanced to account for these somewhat obscure phenomena. 266. ExPT. 44. Simple Form of Singing Flames. — In a horizontal gas-pipe, about half an inch diameter, fasten an upright supply pipe about 10 cm. high with a pinhole burner at the top. Cut off from glass tubing about 1 '5 cm. diameter a piece 30 cm. long. Light the jet and place the glass tube over it supported by a chp stand so that the jet is about 7 or 8 cm. from the lower end of the tube. The flame may perhaps spontaneously "sing." If not, it must be coaxed by adjusting its position or its size, or both ; or it may be that it will start if one sounds in the vicinity the note proper to the tube, found previously by blowing across its end. Usually the " singing " of the flame may be most easily started as follows : — Begin with the jet fairly large, and very slowly turn the gas lower until the flame begins to sing. It may then be turned much larger again without danger of losing the musical sound thus started. Next try, with the same supply pipe and jet, a sound tube 15 cm. long and the same bore as before. Probably no coaxing will make the flame sing in this tube. The sound tube may be raised and lowered, moved sideways so as to bring the jet near its wall or back to the middle, and the gas turned lower, but probably all in vain. The reason for this we shall discuss later. It may easily be ascertained whether the luminous part of the flame is intermittent by moving the head rapidly from side to side. The same fact 266,267 EESONANCE AND EESPONSE 351 may be more conveniently demonstrated to an audience by the use of a rotating mirror. If several singing flames are desired to sing simultaneously, a number of equal supply tubes and jets may be arranged on the same horizontal gas-pipe, and a singing tube placed in the right position over each, the vchole being mounted on a simple vi^ooden frame. If the lengths of the singing tubes vary they will emit sounds of correspondingly diiferent pitches. Thus, if the sounding tubes are 60, 48, 40, and 30 cm. long, their internal diameters being graduated from 3 cm. to 1'5 cm., we should obtain the major chord and octave. If the lengths or diameters are not exact, and therefore the tones a little out of tune, the needed tuning may be effected by rings of paper sliding on the tops of the tubes. Eaising this paper slider lengthens the tube and lowers its tone. 267. Phase of Heat-Supply. — In the case of the forced vibration of a point, we have already seen that the best effect in encouraging the kinetic energy of the vibration occurs if the impressed force is at a maximum when the displacement is zero (see arts. 91-96). Thus if the equation' of motion is y+-2Ky-\-p^y=fs.mnt (1), the forced vibration is expressible by /sin 8 . , . where tan h = —5 5- (3). p —n And, for the kinetic energy to be a maximum, we must have sin S a maximum, i.e. B — 7r/2. Hence by (3) we see that for this case n=2^ (4). Accordingly (2) becomes, / 2Ki where the impressed force is / sin pt. Vi=-J~^ospt (5), 352 SOUND CHAP. VI But, for vibrations maintained by heat, Lord Eayleigh has shown that the vibrations are best encouraged when the heat is given at the phase of greatest condensation and abstracted at the instant of greatest rarefaction. And, in stationary waves, the states of greatest condensation and rarefaction (at peaces between nodes and antinodes) occur when the particles are at their maximum displacement. So, at first sight, this may seem contradictory to what we have said above as to forced vibrations. However, it is not so, as we shall find presently. 268. The communication of heat may be regarded either (1) as replacing the impressed force of a forced vibration, or (2) as altering the zero position of the vibrating point, and consequently the value of the restoring forces throughout the time until the next abstraction or com- munication of heat. Now, if the heat were supplied and withdrawn according to a sine function of the time, the first view of the matter and an analytical treatment might be most convenient. But if, as we suppose, the heat is given somewhat suddenly, the second view of the matter and a graphical treatment seem more convenient. The vibrations of these gaseous particles and their maintenance by heat are comparable to the vibrations of a simple pendulum, and their maintenance by a periodic shift of its point of suspension. It will therefore probably facilitate a thorough understanding of the essential features of the case if we consider the two sets of conditions side by side. 269. Thus, suppose we have a simple pendulum of length /, the bob having unit mass and the point of sus- pension moving horizontally in the plane of vibration. Then, if the vibrations are small, so that the tension of the thread may be equated to the weight of the bob, we have the restoring force represented by t (?/ — y„), where y is the horizontal displacement of the bob, and y^ that of the point 268-270 EESONANCE AND EESPONSE 353 of suspension. Hence, the equation of motion may be written 2/+2«2/+f(2/-2/o)=0 (1). Now, if 2/g oc sin nt, we may easily retain the analytical method and obtain the ordinary equation of forced vibration, namely, y+2Ky+phj=fs,mnt (2), y here denoting gjl. If, however, y^ is susceptible to sudden changes only, instead of varying harmonically, it seems simpler to adopt a graphical representation corre- sponding to equation (1). Thus let the amplitude of the vibration be a and let y^ be alternately zero and k. Thus, let 2/j be zero till the displacement is at its full negative value — a, when y^ suddenly becomes k. It is evident that the amplitude is thereby virtually changed to a + k, and that all the restoring forces are altered accordingly so long as 2/o retains its value k. The bob therefore swings in the next half-period, from the position — a, about the displaced centre of suspension k, to the other end of its swing, viz. a + 2k. If, at this instant, the point of suspension returns suddenly to the zero position, i.e. y^^ = 0, it is evident that the amplitude is thereby made equal to the displacement a + 2k. Thus, each time the displacement of the centre of suspension is suddenly changed at the right instant and in the right direction, the amplitude is increased by k, the same amount as the shift. If, in the meantime, due to the frictional resistances, the amplitude had diminished by that amount, then the dissipative forces would be just balanced and the vibrations maintained. 270. Turn now to the gaseous vibrations, and let us consider first a pipe stopped at one end E, and with an antinode at or near 0, see Fig. 73. Let the amplitude of the vibrations at be a. Then, when the particles are displaced their full amplitude from to A, let heat be 2a 354 SOUND CHAP. VI given to the portion EO, and increase its pressure so that, on the pressure falling to its normal value, is shifted to 0', where 00' = h Then, by the first of the two views of the matter previously mentioned, we could say that the impressed force is acting to urge the particles from A towards in addition to the ordinary restoring force. But, taking the second view of the matter, which is here more convenient, we can say that the zero position of the particles is suddenly shifted from to 0', and that the amplitude is accordingly a-^k instead of a only. Half a period later the particles will be at B, where O'B = O'A = a + k. If, at this instant, heat be suddenly abstracted so that 0' is brought back to 0, then the amplitude is OB = a + 2h. 271. To determine whether the period is Fig. 73. Thkoet Singing unaltered, we must examine if the restoring forces bear to the displacement the same relation as before. Now, the heat communicated causes expansion from to 0' under constant pressure, and the elasticity of a gas is proportional to the pressure. Hence the elasticities with zeros at and 0' must be equal. Let the increments of pressure for displacement y be respectively f and p' with zeros at and 0'. Then, since the volumes concerned are proportional to the lengths of the columns, we have P P V or - : P ■■ EO' : EO (3), 2//E0 {y^m^O"'- y-y + h that is, forces per unit displacement with zeros at and O' are in the ratio EO' to EO. Thus, if 00' = A; be negligible in comparison with EO, we have p y + k nearly (4). And to this approximation the periods are equal. 271,272 RESONANCE AND RESPONSE 355 272. Let us now examine the slight effect on the period when 00' is not neglected. The period of a simple harmonic motion is given by T=27r/o, where «= / fo^ce per nnit displaceme nt V mass Thus, the ratio t// of the periods with zeros at and 0' is given by t/t' = <»7« = x/EO/EO' ( 5 ). Or, in words, the period is slightly increased by the shift of zero due to heating. Another way of obtaining the same result is to note that, for the stationary waves in question, where X and v denote wave length and speed of propagation at temperature t ; X^ and v^ those at 0° C, and a the co- efficient of expansion. Thus, if, on the supply of heat and consequent expansion from to 0' at constant pressure, the temperature rises from t to t', we have t/t' = — "' • VI + at' or, since the volume EO is proportional to the factor 1 + at, t/t' = VEO/EO' (7), which agrees with (5). If, in contrast to the above cases, heat were given when the particles were passing from 0' to 0, it is obvious that the period would be slightly shortened, for the zero position would come out to meet them. On the other hand, if heat were given half a period later, when the particles were passing from to 0', the period would be slightly lengthened. In either of these cases the amplitude would not be much changed, because near the equilibrium position the restoring forces are very small, hence the speed is about the same at 356 SOUND CHAP. VI or 0', and therefore the amplitude about the same with respect to one zero position or the other. 273. Graphical Treatment of the Problem. — Let us now illustrate graphically a forced vibration, the impressed force being harmonic, in contrast with vibrations maintained by heat suddenly supplied and withdrawn. The first state of things is shown in Kg. 74, and the second in the two following figures. Harmonic and Impulsive Forces. Fig. Fig. 75. g P Forced Vibration with q Harmonic Impressed Force In most favourable phase Vibrations encouraged by sudden supply and withdrau/al of Heat Heat suddenly supplied Fig 76 and withdrawn so as to shorten the Period but not affect Amplitude nj splttcement r. Here -^^ A mplitude = a Here amplitude = a+h Period is changes ^ ^/jere amplitude 1+2/; from T to (T-2S) Thus, we may summarise as follows: — (1) If heat is supplied at the instant of greatest condensation (the particles tlien having their maximum displacement) or abstracted at 273, 274 RESONANCE AND EESPONSE 357 the instant of greatest rarefaction, or both, the vibration is encouraged and the period not much changed. (2) If the heat is supplied and abstracted when the vibrating particles are not displaced, and consequently the gas at the normal density, the amplitude is not much changed, but the period is. (3) Thus, the period is shortened if the heat is supplied a quarter of a period before the phase of greatest condensa- tion, and the period is lengthened if the heat is supplied a quarter of a period after the phase of greatest condensation. (4) If, in the general case, heat is supplied at intermediate times, then it will produce effects both on the amplitude and the period. These statements should be verified by the reader from the figures and the general theory applicable to them and already adduced. The comparison of Figs. 74 and 75 will also solve the paradox that for the best encouragement of the vibrations a harmonic force must be at its maximum when the displacement is zero, but the sudden supply or withdrawal of heat should occur when the displacement has its maximum value. 274. Place of Heat-Supply. — In considering the phase of heat-supply and its effects upon the vibrations, we have so far supposed the heat to be diffused along the quarter wave length from a node to an antinode. Let us now examine the consequences of restricting the region, and of changing its position with respect to the stationary waves in question. And, first, let the heat be supplied near a node. Then it may be seen that the effects above described will be produced however near to the node the place of heat- supply may be. In other words, heat supplied near a node alters the restoring force or spring, and is like shifting the point of suspension of the pendulum. It thus changes the zero position of the vibrations, and either their amplitude or period according to the phase of supply. If, however, the heat be supplied near an antinode, the spring is practically unaltered. In Fig. 77 let E be a node, an antinode, A 358 SOUND CHAt. VI the particles at their maximum displacement downwards from 0, and let heat be supplied in the thin layer between C and A. Then the portion CA may be expanded to CA' and the zero shifted from to 0', where AA' = 00'. But the amplitude is not thereby altered, although the heat was supplied in the -^"*''""'« most favourable phase, for A'0' = AO. Or reverting again to the pendulum, the introduction of heat in GA may be compaied to shifting simultaneously and by equal amounts both the point of suspension and the bob itself. We have thus ascertained not only the phase, but also the place of heat- supply to best encourage the vibrations. In what follows we can, therefore, dispense with the restriction of a stopped end, replacing it by any node of the stationary waves. We are accordingly now in a position to con- sider the maintenance by heat of vibrations in a tube open at both ends like those used in the experiment No. 44. 275. Length of Supply Tube. — We have now finally to consider under what conditions the heat may be supplied at the right place and times to encourage the vibrations, and thus explain the experimental facts of singing flames. It might at first sight seem probable that the greatest issue of gas would occur at the phase of least pressure in the sound- ing tube, and that the greatest heating effect would occur at the instant of greatest issue. If these things were so the flame would be unable to maintain the vibrations, but would discourage them, the circumstances being the reverse of what we have seen (in articles 2*70-274) to be requisite for encouragement. But both the above suppositions are incorrect. Lord Eayleigh has pointed out that there are Node Fig. 77. — Place of Heat-Sdpply. 275, 276 EESONANCE AND RESPONSE 359 really stationary or approximately stationary waves, not only in the larger or singing tube, but also in the smaller or supply tube. And at the jet or tip of the supply-tube we must have the condition that the pressures in large and small tubes are equal. Hence for equality of pressures it is evident that the pressures in both tubes must be increasing or decreasing simultaneously. Now in stationary waves the amplitudes and velocities increase with distance from a node. Thus, when there is motion from a node the pressures are decreasing, because more gas is being lost by the higher velocities at places remote from the node than is being received by the lower velocities at places nearer the node. Hence, both in the singing and supply tubes, the motions at any instant must be either both to or both from their respective nodes. 276. Consider now a supply tube, whose length, from the jet to the longer pipe below, into which it opens, is a little less than a quarter of a wave length for the gas being burnt and the frequency produced in the sounding tube. Let the jet be well inside the sounding tube, as in our Expt. 44. Then in the supply tube we have an antinode at the bottom where it opens into the larger pipe, and a node a little above its top. In the sounding tube we have an antinode at the bottom and a node at the middle. Thus, at any given instant we may have the motions in the two tubes near the jet either both up or both down. For, with motions up in each tube, they will be towards a node in each, and consequently the pressure will be increasing in each, and this makes it possible for the pressures to remain equal at the jet in the small tube and the large one. If, on the other hand, the motions are both down, the pressures at the jet are decreasing in each tube, which again allows them to remain equal as required. In Fig. 78 the motions in the two cases are indicated by arrows drawn in full lines and dotted lines respectively. The node and antinode for the supply tube are denoted by 360 SOUND CHAP. "VI Nj and A^, and those for the singing tube by N^ and A^ respectively. Thus we shall have gas coming out of the jet best when the air in the singing tube is condensing at IST^. Hence we might obtain the greatest heating effect at the instant of greatest condensation if we could account for a lag in that effect behind the instant of freest emission of gas from the jet. That such a lag must exist is evident from the fact that time is required for combustion and heat production, and also for sending out the air previously sucked in at the jet. 277. Thus we have the conditions required by theory for the maintenance of the vibrations, and the phenomena of the singing flames are, in the main, explained. If the supply tube is made more than X/2, but less than 3/4X, it will again encourage the vibrations and give a singing flame. If, however, the length of the supply tube is made more than X/4 and less than X/2, we should have the vibrations discouraged, and no coax- ing would make the flame sing. (This is the explanation of the failure of the 15 cm. tube in Expt. 44.) It M must be remembered that the X here used is that corresponding A, Fra. 78.— Simultaneous Motions in f„ (.],„ „„„ qnm-iliprl to thp iet SiNOiNQ AND Supply Tubes. ''" ''"^ S^^ suppliea to tne jet, and to the frequency of the note emitted by the singing tube. This frequency in turn depends on the length of the singing tube and on the nature and temperature of the gases in it. There is thus no simple geometrical relation between the X above mentioned and the length of the sounding tube. The 277, 278 EESONANCE AND EESPONSE 361 lengths of the supply tube and the effects of the corre- sponding jets in the vibrations may be thus exhibited — Lengths of Supply Tube o A/^ A/^ 3A/^ Vibrations of Singing Tube | encouraged | discouraged | encouraged | If the length of the supply tube were made exactly \/4 the vibration woiild be neither, encouraged nor discouraged, for the jet would be at a node, and no variation in the flow could occur there. What we have discussed simply concerns the mainten- ance of a vibration already started. Such vibrations may or may not start spontane- ously. Hence, in the actual experiment it is sometimes necessary to start the sound by making a sound of the right pitch near the tube. 278. Electrically-Driven Tuning - Fork. — Another good example of the main- tenance of vibrations is that afforded by a tuning-fork fitted with an electro-magnet between its prongs. The current may be derived from Pig. 79.— Diagkam of electmcally- 1, Driven Tuning-Fobk. one or two storage cells, and is interrupted by the vibrations of the fork itself. The interruption may be accomplished by a platinum wire dipping into a cup of mercury, as advised by Helmholtz. Or it may be due simply to a platinum wire on the prong and an adjustable screw on the framework, which alternately make and break contact as the fork vibrates. This arrangement is shown diagrammatically in Fig. 79. To set the fork going it is only necessary to put on the current, tap the fork with 362 SOUND CHAP. VI the finger to obtain a slight initial vibration, and then adjust the screw contact until the vibrations are increased and vigorously maintained. It is interesting to examine why the current is competent to maintain this vibration. If the end of the prong and contact wire were quite rigid, and the magnetic field and current followed their motions instantaneously, then no encouragement of the vibrations would be possible. Por in that case the magnet's attraction on the prongs assisting their closing motion would be exactly counterbalanced by its equal and equally-long exerted attraction opposing their opening motion. There would thus be no balance carried over to compensate for dissipation. But owing to self- induction there is a lag in the establishment of the magnetic field and current, and also in the annulling of the field again. And thus, for both reasons, the prongs of the fork will be more assisted in closing than resisted in opening. Hence the encouragement of the vibrations is explained. Here, of course (as shown by the theory of forced vibration, see art. 96), the maximum effect on energy of vibrations and the minimum disturbance of period will be secured only if the maximum of impressed force occurs at the instant of no displacement of the prongs. 279. Sensitive Jets and Flames. — Having sufficiently treated the maintenance of vibrations, we now pass to the consideration of various pieces of sensitive apparatus which undergo some change in response to a sound, and thus serve as indicators of its presence and reception. We begin with what are called sensitive flames. The term sensitive flame is usually applied to specially arranged flames which burn steadily of a given size and shape when no sound reaches them, but shorten and roar when excited by the receipt of appropriate sounds. But Tyndall has shown experimentally that the seat of the sensitiveness lies, not in the flame itself, but in the jet of gas issuing from the burner. Thus the term " sensitive jet " is really more 279, 280 EESONANCE AND EESPONSE 363 correct. The flame is a mere indicator of the behaviour of the really sensitive portion of the unignited gas. The problem of sensitive jets has been dealt with mathematically by Lord Eayleigh. But the analysis is imsuited for introduction here. It must suffice to quote the results as follows: — When the fluid jet is launched into still air a steady motion may satisfy the dynamical conditions, and so be in a sense possible. But under certain conditions the smallest departure from such ideal steady motion tends to increase, and often very rapidly. Thus any impressed force, however feeble, provided it is of suitable frequency and reaches the jet at the right place, produces a departure from the normal motion, which departure is quickly exaggerated and made evident by the consequent behaviour of the flame. Many types of sensitive flame have been devised. We shall deal here with three illustrative types, namely, Tyndall's vowel flame. Lord Eayleigh's enclosed jet, and an ordinary bunsen burner made sensitive. 280. ExPT. 45. Tyndall's Vowel Flame. — This is a high- pressure flame sensitive to sounds of very high pitch. The gas issues from the single orifice of a Sugg's steatite pinhole burner, and the flame reaches a height of about 24 inches. The gas may be derived from a weighted gas bag or a steel cylinder of compressed gas. In either case the pressure must be adjusted till the flame is as tall as possible without flaring. Tyndall, in his Lectures on Sound (p. 260), describes its sensitiveness as follows : — " The slightest tap on a distant anvil reduces its height to 7 inches. When a bunch of keys is shaken the flame is violently agitated, and emits a loud roar. The dropping of a sixpence into a hand already con- taining coin, at a distance of 20 yards, knocks the flame down. It is not possible to walk across the floor without agitating the flame, the creaking of boots sets it in violent motion. The crumpling or tearing of paper, or the rustle of a silk dress, does the same. It is startled by the patter of a rain-drop. I hold a watch near the flame : nobody hears its ticks ; but you all see their effect upon the flame. At 364 SOUND CHAP. VI every tick it falls and roars. The winding up of the watch also produces tumult. The twitter of a distant sparrow shakes the flame down ; the note of a cricket would do the same. A chirrup from a distance of 30 yards causes it to fall and roar." ... "To distinguish it from the others I have called this the 'vowel flame,' because the different vowel sounds affect it differently. A loud and sonorous XJ does not move the flame ; on changing the sound to the flame quivers ; when E is sounded the flame is strongly aff'ected. I utter the words hoot, boat, and heat in succession. To the first there is no response ; to the second the flame starts; by the third it is thrown into greater commotion. The sound Ah J is still more powerful." We shall see in Chapter VIII. that the different vowels have components of certain definite frequencies specially favoured. Hence the flame responds best to those vowels which contain prominent components near the pitch to which it is most sensitive. 281. ExPT. 46. Bayleigh's Sensitive Flame. — Lord Eayleigh's enclosed jet is sensitive to sounds of ordinary pitches, say those throughout the compass of the pianoforte, and possesses the great advantage of working off the ordinary gas-supply. A jet of coal gas rises from a steatite pinhole burner placed in the interior of a chamber. It then passes through a vertical tube ^ inch inside diameter and 6^ inches long, mounted on the chamber, and, on reaching the top, burns in the open air. The front of the chamber is formed of tissue paper ; on this flexible membrane the sounds are allowed to impinge, and so affect the jet just after leaving the burner. A stout rod of copper, covered with asbestos, may with advantage pass from the top of the tube, where the combustion commences, down to the supply tube, as near as possible to the burner. This rod, by conduction of heat, prevents the formation of dew on the steatite pinhole burner. The dew is to be avoided, because it would inter- fere with the proper working of the apparatus. To adjust the flame to sensitiveness, begin with the full pressure of the ordinary gas-supply. The flame then presents the appearance of a Bunsen flame when air is excluded. Next lower the gas pressure very slowly and watch the flame. When a certain pressure is reached the flame suddenly assumes a fluttering and lop-sided appearance, being at one 281-283 EESONANCE AND EESPONSE 365 side drawn down into the tube a little. Lower the pressure a little more until, though still lop-sided, the flame is steady. The flame is then right for use, and will be found sensitive to the crumpling of paper and to the sound of walkipg across a boarded floor, to the clapping of hands, and to the notes of the pianoforte. Its recovery after excitation, is rather slow. 282. ExPT. 47. Sensitive Flame from a Btmsen Burner. — Select a Bunsen burner whose upright tube is of brass, 5 inches high and 3/8 inch in bore, and of the pattern which possesses only one side hole for the admission of air, and which is perfectly closed by a half-t\im of the sleeve provided for that purpose. To obtain the sensitive state, exclude the air completely by the half-turn of the sleeve, and then reduce the pressure of the gas-supply until the flame becomes lop-sided but quiet. The maximum pressure subject to these conditions seems to give the best results. The flame is then about 4 inches high, that side of its base next the supply tube being detached from the lip of the upright tube and extending downwards into it about a third of an inch. The burner is not, however, " lit back." When responding to suitable sounds, the flame falls from its usual height of 4 inches to a height of about 1|- inch. This form of flame is insensitive to the crumpling of paper and the jingling of keys, but responds promptly to clapping the hands, shuffling the feet on a boarded floor, coughing, speaking, whistling, or singing. The flame, after responding to any sound, quickly resumes its sensitive state ready to receive and respond to another sound. Hence it is possible to whistle a slow staccato passage, each note being acknow- ledged by a ''duck" or curtsy of the flame, and each rest by a recovery to its usual height and form. 283. Suspended Disc. — When a light disc is suspended so as to be capable of rotating freely about a vertical axis, it shows a tendency to set normally to the direction of flow of the medium in which it is placed. The rigid proof of this and the evaluation of the couple experienced are beyond the scope of this work, but Lord Eayleigh gives a simple qualitative explanation as follows : — 366 SOUND CHAP. VI " That a flat obstacle tends to turn its flat side to the stream may be inferred from the general character of the lines of flow round it. The pressures at the various points of the surface BO (Fig. 80) depend upon the velocities of y-.„ the fluid there obtain- ^y^ iug. The full pressure due to the complete stoppage of the stream is to be found at two points where the cur- rent divides. It is pretty evident that upon the up-stream side this lies (P) on AB, and upon the down -stream side upon AC at the corresponding point Q. The resultant of the pressures thus tends to turn AB so as to face the stream." For the quantitative solution of the problem, the reader is referred to an article by Konig.^ In the case of a thin circular disc, whose normal makes the angle 6 with the direction of the undisturbed stream, the moment M of the couple tending to diminish 6 is given by Fig. 80.— Explanation of Couple on Disc. M=-pa^W''sm2e, p being the density of the medium streaming with velocity W, and a being the radius of the disc. If the stream be alternating instead of steady, the mean value of W^ must be employed. It is seen that the equation supports the view that might have been inferred from the elementary consideration of stream lines, viz. that the moment is a maximum for ^ = 45°. ExPT. 48. Setting of Suspended Disc. — In front of the mouth of the resonance box of a tuning-fork of (say) 128 1 Wied. Ann., xliii. p. 51, 1891. 284, 285 EESONANCE AND EESPONSE 367 vibrations per second, suspend by a cocoon fibre a disc of thin card about 1 cm. in radius. Adjust the fibre so that the equilibrium position of the disc is at an angle of about 45° with the mouth of the resonator. Then, on bowing the fork, the disc will promptly set itself approximately across the mouth. In order to demonstrate the effect to an audience, replace the disc by a small mirror, which reflects a beam of light on to a distant screen. Then, on bowing the fork, the spot of light will sharply move across the screen. The same response may also be elicited by powerfully singing the same note to the vowel oo in front of the resonator. Prof. C. V. Boys, by an arrangement of double resonators, has exalted in a remarkable degree the sensitiveness of an instrument constructed on this principle.-' 284. Kundt's Dust Figures. — To make evident the stationary waves in a horizontal glass tube and mark their length, Kundt devised the method of strewing its interior with lycopodium seed or other dust. This dust is promptly thrown into a recurring pattern when the waves are excited with sufficient vigour. The experiment was performed in several ways by Kundt himself, and further applications of the principle have been made by others. Some of these uses of the experiment will be noticed in the chapter on acoustical determinations, a brief description of one simple form sufficing here as an introduction of the method. In this form the stationary waves in the tube are excited by the longitudinal vibrations of a rod. 285. ExPT. 49. Simple Fm-rn of Kundt's Dust Figures. — Select a glass tube about 60 to 100 cm. long and 5 cm. internal diameter. This is to form what is called the wave tube in which the stationary vibrations are to be set up. They in turn, under proper conditions, produce the dust figures by which the vibrations are recognised and measured. To start and maintain the stationary waves, choose a brass rod about a metre long and a centimetre diameter. This may be called the sounding rod. One end of this rod must ' Phil. Mag., vol. xiv. p, 186, 1882. 368 SOUND CHAP. VI be provided with a disc, preferably of ebonite. This disc must be small enough to enter without touching the wave tube, but so large as to only just clear it. The sounding rod needs firmly clamping exactly in the middle of its length, and the wave tube must be supported so that the axis of rod and tube are in the same line, the disc of the rod being within the tube. The far end of the tube must also be closed by a cork or rubber bung. A simple and convenient way of effecting these arrangements is shown in Fig. 81, in which it will be noticed that the bases of the supports for rod and tube are separate and fit together so as to allow of a sliding motion endwise. When these arrangements are made, the 7nod^cs operandi is briefly as follows : — A very little lycopodium having been introduced into the wave tube and spread about it, stroke the rod with a rosined leather so as to elicit its longitudinal vibrations. FiQ. 81. — Appabatus fob Kundt's Dust Figures. This may set up stationary waves in the tube and so lead to the formation of the dust figures which mark their presence and length. Thus the actions of the rosined leather on the rod, and of the rod on the air in the tube, are examples of maintenance. But the formation of the dust figures, with which we are here principally concerned, is an example of a sensitive arrangement which passes from one state to another in response to a wave motion. Hence the inclusion of the experiment in this subdivision of the chapter. But in order to make the experiment succeed, several practical details need careful attention. For the rod will not " speak " freely if clamped anywhere. Neither will the stationary waves be set up in the wave tube unless the length between its stopped end and the disc on the rod is a multiple of the half-wave in air (or other gas in the wave tube) of the vibration executed by the rod. And, finally, the stationary waves, when set up, will not give the desired dust figures if either the tube or dust is wet or too much dust is present. So we summarise as follows : — (1) Wipe the wave 285- EESONANCE AND- EESPONSE 369 tube out with a " mop " of cotton wool on the end of a stick, and then dry it thoroughly over a Bunsen flame (use no liquid in the tube). (2) Dry the lycopodium in a desiccating chamber or in a porcelain bowl over a small flame. (3) Use only as much lycopodium as will cover a strip the length of the wave tube and about 2 mm. wide. (4) "When the wave tube is in place, tap it smartly with a metal rod or pencil to bring the lycopodium into a line along the bottom of the tube, then rotate the tube about its axis through 45° or oven 60° till the lycopodium is just on the point of slipping down. (5) Take care that the sounding rod- is^ clamped exactly at the middle so that it can " speak " freely. (6) Adjust the length in the wave tube between its stopped end and the disc on the rod, by steps of about a ,,,UljlUUIIIIIUMi,,,,,,,^,,,llllM""lMlllai,,,,_,,,,lllllllUllllli,|,,| _^^,,,lUlllllll||,lli t FiQ. 82. — Plan op Kundt^s Dust Fiqckes, First Staqe. J- FiG. 83.— Plan op Kundt's Dust Figures, Final Stage. centimetre. Stroke the rod with the rosined leather after each such adjustment until the pattern of the dust figures appears. Their first appearance will probably be due to the lycopodium remaining stationary at the nodes, and falling at the ventral segments or spaces between the nodes as shown in Fig. 82. This state is all that is required for purposes of lecture demonstration to show that the waves are present and can move the dust. It is not, however, very suitable for purposes of exact measurement. So, if this is required, (7) make the disc fit the tube as perfectly as possible by putting, if necessary, a thin paper round it till it is as large as may be without touching the tube. By a continued sounding of the rod's note, by stroking it, the dust figures should then change slowly by a motion in the ventral segments, from the disc towards the stopped end of the wave tube, until the dust is in little heaps at the nodes. This state, which is somewhat difficult to obtain, is shown in Fig. 83; The length of the sounding rod is half the wave length in 2 B 370 SOUND CHAP. VI that material of the tone it emits, also the distance between the nodes in the wave tube is half the wave length of that same tone in the air or other gas occupying the wave tube. Further, the wave lengths for a given tone are proportional to speed of propagation. Hence the experiment gives directly a comparison between the speed of sound in the material of the sounding rod and in the gas occupying the wave tube. Thus by using rods of different material and filling the tube with gases a variety of experiments is possible. These will be dealt with in the chapter on acoustical determinations. 286. Striations in Dust Figures. — As shown in Fig. 82, Kundt's dust figures not only exhibit the pattern with which we are mainly concerned, and whose length is half the wave length of the stationary vibrations in the tube, but show also a much smaller pattern or striation. And this too is of interest, being closely connected with the phenomenon of rotation of a suspended disc when exposed to vibrations or a steady current. We saw in article 283 that a suspended disc was subject to a couple, and that the couple would be experienced by any flattened, or elongated body, But the lycopodium seeds are approximately spheres, and a sphere exposed to a stream could have no tendency to turn. Further, an isolated sphere exposed to a stream would have the flow symmetrical on the up-stream and down-stream sides, and consequently the main pressures would balance one another. Thus a sphere would experience neither force nor couple in an ideally perfect fluid. But a close examination of the dust figures in a Kundt's tiibe show that, while the vibrations are in progress, a rib-like structure obtains, the lycopodium grains being at places heaped up, and the tube in the intervening spaces being nearly bare. When the vibrations cease, much of this striation is lost, but evidence of it still remains. The presence of these striations has been ex- plained by A. Konig.^ Although a single spherical obstacle ' Wied. Ann., vol. xlii. pp. 353 aud 549, 1891. 286, 287 EESONANCE AND EESPONSE 371 experiences no force nor couple from an alternating current, each of a pair of spheres at a moderate distance apart may do. First, let us note that where the velocity of the stream is greater the pressure must be less, for it is accelerated in passing from a space of greater pressure to one of less pressure. Now consider two spheres, one ahead of the other in the alternating stream. Each screens the other, so the stream speed between them will be less than outside, hence the pressure between them is greater than outside, and the spheres repel each other. Again, consider two spheres side by side in the alternating stream. It is easy to see that the stream speed between them is greater than normally, and accordingly the pressure is less than normally, hence attraction follows. The result of these forces is a tendency to aggregation in laminae across the stream. Kouig has also calculated the direction and magnitude of the forces operative on two spheres which are not too close together. From this it appears that when the spheres are oblique the direction of the forces does not coincide with the line of centres. If the spheres are rigidly connected, the forces on them reduce to a couple, tending to increase 0, whose moment is given by G = — f-^^ sm 26, where p is the density and W the speed of the fluid, a^ a^ the radii of the spheres and r their distance asunder, and 6 is the angle between the line of centres and the direction of the current. When the current is alternating the mean value of W^ is to be taken. 287. Maintenance of Compound Vibration. — Suppose we have a system capable of simultaneous vibrations of periods nearly or quite commensurate. And let it be acted upon by a periodic impressed force almost in tune with the primary free vibrations. Then it may be shown that each 372 SOUND CHAP. VI harmonic component of- the force produces in the system a harmonic motion of the same period as that of the force, provided that the squares of the displacements and velocities may be neglected.^ But, if the whole force impressed is strictly periodic, it may be regarded as consisting of a number of comviensurate harmonic forces as shown by Fourier's theorem (see article 51). Hence, the compound vibrations maintained in the system will consist wholly of commensurate harmonic motions, although the periods of the free vibrations natural to that system may differ slightly from the harmonic series. But, if these periods natural to the system do not form a harmonic series, this fact is not without effect on the character of the vibrations maintained. Tor although the periods are forced to be commensurate, the amplitudes are greater the closer the tuning between the vibrations natural to the system and those forced upon it. Hence the closer the period of any of the free vibrations lies to that of the corresponding vibrations in the strict harmonic series, the more fully will that component vibration be elicited and maintained by the impressed force. Those lying farther out of tune with the corresponding tones of the harmonic series being less powerfully brought forth (see also arts. 97 and 98). This fact has an important bearing in connection with the maintenance of vibration in strings and organ pipes. For both these systems are capable, as we have already seen, of a number of vibrations whose periods are almost but not quite a harmonic series. When exciting free vibrations in such systems (say by plucking a string), tliis departure from the harmonic series would be preserved. When, however, the string is bowed or the pipe blown, the vibrations are maintained by a periodic force, and the periods are forced into the strict harmonic series. Accord- work > Eayleigh's Sound, vol. i. p. 147 ; compare also article 116 of tlie present 287 EESONANCE AND EESPONSE 373 ingly, in view of what we have just seen, those partials are most favoured, which lie closest to the strict harmonic series, and those are feeblest heard which depart more widely from that strict series. The effect of this will be discussed further in Chapter VIII. CHAPTEE VII INTEEFEKENCE AND COMBINATIONAL TONES 288. Interference. — In dealing with the kinematics of the subject (articles 28 and 43) we saw that the composition of two vibrations of equal period and amplitude but opposite phases yielded a zero resultant. Or, in other words, two such vibrations destroy one another. This is the typical case which is often referred to as an example of interference. The term interference is used in acoustics and optics to denote the phenomena of alternate additive and subtractive character which occur, under suitable conditions, when radiations from two sources mingle. Thus, as shown in any treatise of physical optics, two sources of light may produce on a given screen equidistant bright and dark bands, the effects of the two sources being additive in the bright and subtractive in the dark bands. These phenomena are not only of importance as showing that the radiation concerned is a wave motion, but they also enable us to measure the wave length. For evidently in passing from a bright to the next dark band we pass from a place where the vibrations from the two sources are in the same phase to a place where they are in opposite phases. Suppose the two sources to be in the same phase» and that the bright band in question were equidistant from these sources. Then the neighbouring dark bands would be at distances from the two sources differing by a half-wave 374 288, 289 INTERFERENCE 375 length. Or, generaHy, if the difference of distances from the two sources of any bright band is 2n half-wave lengths, the difference of distances of the contiguous dark bands from the two sources must be {2n ± l)X/2. 289. Reserving quantitative applications of this principle to the tenth chapter, we shall now deal with the experi- mental illustrations suitable to demonstrate its reality in the domain of acoustics. It should be observed at the outset that, in dealing with the actual vibrations of material bodies, it does not follow, as a matter of course, that the simple kinematical rules developed in Chap. II. will always hold. They certainly will when the displacements are indefinitely small. And throughout the first part of this chapter, dealing with interference, we shall suppose that condition to hold good. Later, under the heading of com- binational" 'tones, we shall study what happens when the above condition of very small displacements is violated. The phenomena of interference may sometimes be noticed when a fairly regular swell is running on the sea near a pier, embankment, or sea-wall. The waves rolling towards the wall are then met by those reflected from it. At certain places crest meets crest, and a specially high crest is there formed in consequence. At other places crest meets trough, and the usual undisturbed level is scarcely departed from. This is, in fact, the extension to space of two dimensions of stationary waves which we first studied along space of one dimension, viz. a string or rope. These effects may be imitated and systematised on the small scale in a " ripple tank" (see art. 50), but are somewhat difficult to produce and observe. They have been, however, very successfully produced and photographed by Dr. J. H. Vincent.^ Tuning- forks were used to periodically disturb the surface of mercury, and the effects obtained were rendered permanent by instantaneous photography. Two of the many beautiful plates so obtained are reproduced in Figs. 84 and 85. 1 Phil. Mag., 46, 1898, pp. 290-296. 37B 290, 291 INTERFERENCE 377 290. ExPT. 50. Interference from Tuning-Fork — Perhaps the simplest illustration of interference accessible to every- one is that afforded by a tuning-fork. When the prongs are separating it is evident that a compression starts from the outer surface of each and proceeds in the direction of separation. But, at the same time, it is equally clear that a rarefaction starts from the place between the prongs and proceeds in the direction perpendicular to that before mentioned. Hence if the fork stands with its prongs up- right, while compressions are starting from it and going say north and south, rarefactions will be starting from it and travelling east and west. Hence, in directions about in- termediate between the cardinal points of the compass we shall have destructive interference, for the vibrations being in opposite phases and of equal amplitudes destroy each other. Hence if the ear can pass quickly across these regions we shall lose and regain the sound repeatedly. This is most easily effected by moving the fork and its associated alternating regions of fullest sound and zero effects. Thus, strike or bow a small fork, hold it by the stem to the ear and twirl it round, and the sound will be intermittent, being quite lost four times in each revolution. If it be wished to demonstrate this effect to an audience, bow and then turn a large fork in front of its own resonance box. Better still, if a whirling table be available, mount the fork upon it, place the resonance box with its mouth near the -prongs and facing the audience. Then bow the fork and rotate it at any suitable speed. ' It should be noticed that in this experiment the alterna- tions of sound; and silence occur primarily in regions fixed relatively to the fork. They are only made to succeed one another at a given place, viz. the ear of an obsei'vel-, by tl^e device of turning the fork. Thus, as the fork may be turned at various speeds, there is no one constant frequency of the alternating effects characteristic of the givfen fork. In the case of beats, with which we deal shortly, it is just the reverse. 291. ExPT. 51. Interference from Chladni's Plate. — The same effect may be very strikingly shown by means of a Chladni's plate. Use, , say, a square plate mounted at its centre on a stand, touch and bow it so as to elicit one of the ' sirhpler figures with large segments, say that with the 378 SOUND cHAP.vii diagonals as nodal lines. While the plate is vibrating pass the ear rapidly over the nodal lines, shown by sand, in the usual way. The sound will be lost and regained at each passage of the ear over a nodal line. To render the effect appreciable to an audience, have a wood or metal plate made to cover alternate vibrating segments. Strew the plate with sand, set it vibrating in the manner to which the cover corresponds, then bring the cover close down upon the plate and raise it again promptly. At each approach of the cover, although half the plate is thereby screened off from the audience, the sound will be noticed to be much louder, and at each removal of the cover the sound is again enfeebled. This shows conclusively that, under the ordinary conditions, the sound heard is much enfeebled, though not destroyed, by interference between those waves or radiations which proceed from the alternate segments. 292. ExPT. 52. Interfereme from an Organ Pipe. — Avery interesting illustration of interference may be sometimes obtained from a bass organ pipe in a church or other large building. The interference is obtained between the direct waves proceeding from the pipe and those reflected from a distant wall which must be large and fairly unbroken. For this purpose a single pipe must be sounded, say a 16 foot wide open pipe giving C, about 34 per second. Or the effect may perhaps be obtained from an 8 foot stopped pipe giving the same note. In either case the waves are of the Order 32 feet long; hence, if the direct and reflected waves are along the same straight line, the quarter -wave length separating a node from an antinode is about 8 feet. It is not in every church, or other building containing an organ, that a sufficiently large and unbroken surface is available for reflection and in the right position relative to the organ. But the test for this may usually be made and the experi- ment tried when an organ is being tuned or under repair. When all the circumstances are favourable the effect is very striking, the silences of the nodes being almost perfect when a wide pipe is used whose overtones are scarcely audible. 293. Beats. — When the two interfering sources are of precisely the same frequency the opposite effects, additive 292,293 INTERFERENCE 379 and subtractive, are exhibited at certain places, but remain the same at these while the sounds are maintained. If, however, we have two sounds of nearly, but not quite, the same frequency, a different state of things is produced. For evidently we shall now have, at every place reached by the two sounds, additive and subtractive effects succeeding each other as one sound gains half a vibration on the other. This periodic waxing and waning of intensity constitutes the phenomenon of heats. In other words, with sounds of the same pitch we have througlwut time, places of maximum and minimum effects ; whereas with sounds of slightly different pitch we have throughout space, times of maximum and minimum effects. The former case is usually intended by the term interference, the latter being always meant by the term beats. The kinematics of beats has already been given in articles 29 and 31, and graphically illustrated in Fig. 13. We there saw that two vibrations, a sin (m + n)t and a sin(m — n)t give a resultant which may be represented by y=2a cos nt sin mt (1). Hence, if the difference of frequencies is very small, we may regard the sound as an approximately simple harmonic motion expressed by sin mt, whose amplitude, however, changes slowly from 2 a to zero as expressed by 2 a cos nt. Or, if the frequencies of the two component tones are N-^ and N^, we have approximately a tone of frequency —^—- — ^ A whose amplitude waxes and wanes in cycles of frequency given by B = N^~N^ (2), B denoting the number of beats per second. Thus the difference of frequencies is known when the beats can be observed and timed. The following selection of experiments affords instructive examples of beats : — 380 SOUND CHAP. VII • 294. ExpT. 53. Beats from Sonometer. — Fit two strings or wires on the sonometer and tune one to a low note, say 100 per second. Tlien tune the other till beats of about four per second are heard between them. In order to bring out the beats plainly the strings should be plucked and let go. Or, they may be both very vigorously bowed simultaneously and then let go. The beats cannot be heard to advantage while the strings are being bowed. This is because bowing on a crude instrument like a sonometer usually introduces from time to time sudden changes in the phase of the vibrations. Hence the even flow of each tone and the consequent regular succession of beats is, in that case, prevented. On the violin, with a good bow well handled, the beats may be heard while the bow is still on the strings. In any case it will probably be noticed that absolute silence is not attained at the minimum. Two reasons for this may easily be seen as follows : — First, the primes may not be equal in. intensity. Hence to make the beats as perfect as possible we must endeavour to produce primes of equal amplitude both at first and all through. Plucking carefully nearly secures this. Second, suppose the prime tones in question are 100 and 104 per second. Then each string may yield also the octave of its prime of frequencies 200 and 208 respectively. Hence we should have not orjly 4 beats per second between the primes, but also 8 beats ■ per second between the first over-tones. Further, suppose a maximum of these quicker beats to be coincident with a maximum of the slower. Then the quicker beats would also have a maximum coincident with a minimum of the slower beats. And this prevents absolute silence being attained even when the primes are equal. To reduce this disturbance to a minimum choose rather thick strings and pluck near the middle. 295. ExPT. 54. Beats from Two Siiigincj Flames. — Mount two singing flames side by side, the tubes being the same length, but one fitted with a stout paper slider to tune it. Set the two flames going, but then put them out of unison by raising the slider on the tulie fitted with it. They will then produce beats. The frequency of the beats can be adjusted to any requisite nicety by corresponding movement of the slider. ExPT. 55. Beats ^ludible and Visible. — If two open organ 294-296 INTERFERENCE 381 pipes and a Helmholtz resonator, all of the same pitch, are available, the phenomena of beats may be sti-ikingly exhibited to an audience as follows : — Mount the two organ pipes on a wind chest with their mouths side by side, and both facing the mouth of the resonator. Fit the nipple of the resonator into a manometric capsule and light its jet. Now sound the pipes and, by shading the upper end of one of them with the hand, put it out of unison with the other. We then have the beats not only audible, but also made visible by the manometric fiame throbbing to the pulsations of pressure in the capsule. The audible and visible eflfects, of course, keep exact time. For a large audience it is well to use acetylene gas for the flame, and to project it with a lens on to a distant screen. It should be specially noted that a rotating mirror is both useless and detrimental for this experiment, for it analyses too much, and by showing us the separate vibrations, with which we are not concerned, masks the very beats we are wishing to observe. If, instead of the hand placed over one pipe, a lead shade is fixed, the tuning may be made whatever is desired, and so left pro- ducing permanently, say 4 beats per second. 296. ExPT. 56. Freqiiency of Beats by a Helmlwltz Siren. — By means of a Helmholtz siren we can not only produce beats, but demonstrate that their frequency is the difference of frequencies of the generating tones. The double siren in question, shown in Fig. 86, was evolved by Helmholtz from the single siren due to Dove. Dove's siren had a single chest with several circles of holes and a corresponding disc rotating over it. Any one or more of these circles can be thrown into use by means of separate " hit and miss " rings of holes. Helmholtz arranged two chests on the same vertical axis with one spindle carrying the two discs required for the two chests. The circles in the lower , disc contain respectively 8, 10, 12, and 18 holes, while those in the upper disc contain respectively 9, 12, 15, and 16 holes. Or, calling the tone obtained from the circle of 8 holes c, the lower disc gives the tones c, e, g, and d', and the upper gives d, ffj.h, and c. Hence we cannot get the characteristic effect of beats by using any of these tones in their ordinary way, that is, with the chests stationary and the discs rotating. For, when the disc was spinning fast enough to produce audible tones, the 382 SOUND CHAP. VII beats between them would be too quick for easy counting. But, to produce beats, Helmholtz has added a special device by which the upper chest, usually stationary while the disc ■ 1 ^■^^H^JJL Fig. I -Helmholtz Siren. revolves, is itself capable of a slow rotation by means of a handle and pair of wheels seen in the figure at the right hand top corner. If we use the circles of 12 holes in each chest, and do not turn the handle, we obtain tones of the 297 COMBINATIOJSrAL TONES 383 same pitch for each chest. But if, while the siren is sounding, the handle is turned, then the tone from the upper chest is sharpened or flattened and beats are heard. Further, for every complete turn of the handle the upper chest is turned through one-third of a revolution, and this corresponds to ■4 holes out of the circle of 12. Thus, according to the direction of turning the handle, the tone from the upper chest, has 12n + 4 or 12m - 4 vibrations while that from the lower has 12?i simply, where n is the number of turns of the spindle to one of the handle. And it is found that one turn of the handle always produces 4 beats, whether the discs are turning slowly or quickly. That is, the number of beats per second is the difference of the frequencies of the generating tones, and is constant for a constant difference no matter how great or small those frequencies may ,be. 297. Combinational Tones. — We now turn from the case in which the amplitudes of the component vibrations are very small, and the resultant is the simple sum of the components to the case in which the amplitudes are large enough to render invalid the process of simple addition which has just been followed in the preceding articles. That is, we pass from the cases of comparatively feeble sounds, in which the law of simple superposition of small vibrations is valid and the phenomena of interference are produced, to examples of louder sounds, well sustained, which generate new tones differing in pitch from their generators. We have already dealt mathematically with one case in which new tones are thus produced (see article 115). Let us now regard the matter from the historical and experimental standpoints. One example of combinational tones, as they are now called, was first discovered in 1745 by Sorge, a German organist. They were afterwards generally known through the Italian violinist Tartini, and from him they were called Tartini's tones. Helmholtz further examined the matter very fully. He gave the name differential tones to those known to Sorge and Tartini, because their frequency is the difference 384 SOUND CHAP. VII of the .frequencies of the generating tones. A second class of combinational tones was discovered by Helmholtz himself, and by him called summational tones, because their frequency is the sum of the frequencies of their generators. To render these combinational tones audible their generators must be loud and should be sustained. Further, it is desirable to choose the pitches of the generators so as to bring the combinational tone fairly near the middle of the compass, say somewhere on the bass staff. Thus, to hear a differential tone, the generators should be high in pitch and preferably near together, so that the differential is quite distinct in pitch from either of the generating tones. To obtain the summational, which is more difficult to hear, low notes should be used, and the interval between them may be as much as a major fifth. For either differential or summational, it is well to sound first that generator which lies nearer in pitch to the combinational tone to be produced, and then let the more distant generator join the' other. This order of procedure makes the third tone stand out more distinctly. The fallowing experiments will sufficiently illustrate these facts : — 298. ExPT. 57. Differential Tone from a Double JfTmtle.— A type of whistle consisting of two "stopped pipes" side by side has had a considerable vogue in various forms. It has been used by policemen, tram-conductors, and cyclists, also by referees at football and water polo. The type is therefore probably well known to many who have never examined the principles of its action. Whistles of this class depend for their efficiency upon the differential tone which they generate with great intensity when the two pipes are simultaneously blown in the usual manner. In some forms of the whistle the two pipes give notes whose interval is a minor third, in others the interval between the generators is a semitone only. In either case, piercing as the generators are, they sink into insignificance in comparison with the differential tone generated by them. And it is this differential tone which gives the whistle its value. This can be illustrated by covering one of the two 298, 299 COMBmATIONAL TONES 385 separate pipes successively with a finger so as to sound the other pipe alone, and then, while one is sounding, remove the finger and let it be accompanied by the other. The effect is very striking, and can be appreciated by an audience of five hundred or more. Either generator alone is a feeble, high-pitched note of whistle quality, whereas the two together generate the deep boom so characteristic of this type of whistle. 299. ExPT. 58. Differential Tone from Two Flageolet Fifes. — A yet more striking method of producing a powerful differential tone is that by the use of two flageolet fifes as mentioned by the late Mr. A. J. Ellis, the translator of Helmholtz. For this purpose, two precisely similar instru- ments in G are to be preferred. On these, one performer should produce «'^, /'^, /'^| as loudly as possible, and then the other player should join in with the note g'^"', or, the «'"> /'^ f"% and g" may be sounded successively by one player while the other holds gr'^. The fingerings for these notes and a few others on the G fife, are shown in Table XIX., in which O denotes an open hole, and % a hole closed by a finger. To produce the higher notes somewhat forcible blowing is needed. The differential tones produced by any of the above-mentioned combinations may be calculated from the law of their formation or seen by reference to Table XX. Table XIX. — ^Fingerings on G Fife ^ 1^ ;E: — ^ •f Fingers ^-^'-- r eiv /" /-# V Left 1st. • o e o o o 2nd. . • o • • • Hand, j 3rd. • o o • • , 1st. • o • • o Right 2nd. • o o o o Hand. I 3rd. • o o o o 2c 386 SOUND CHAP. VII 300. ExPT. 59. Differential Toms from Two Organ Pipes.-— For this purpose two open pipes should be selected to give two .high notes about a semitone apart, say /'"# and g^" re.spectively. It is better for these pipes to be upon a wind chest and fed with air of the right pressure, namely, that for which they were "voiced" by the organ-builder supply- ing them. By shading the /'^# pipe with the finger its tone may be flattened to f" or «'\ which causes the diflferential generated by it and the g""" pipe to rise ; or by shading the g", so as to flatten it to f% the diflferential tone may be flattened below the range of audition. This experiment when properly performed is very eflfective, and may be appreciated by a large audience. As in the case of the fifes it is well to sound the two notes singly, then sound the lower one alone, and after its tone is well asserted let it be joined by the higher one, when the deep boom of the differential is immediately heard. 301. ExPT. 60. Combinational Tones from a Harmonium or Pianoforte. — To produce and detect the diflferential tone, sound as loudly as possible on a harmonium the a' followed almost immediatel}' by c" (both in the treble staflE). These produce quite plainly the F (just below the bass staff). It is perhaps advisable to sound this F faintly before the a' and c", so as to be prepared to recognise it when present. Further, if the a' and c" are not strictly of frequencies as 5 : 6, the diff'erential produced by them may be slightly out of unison with the F on the instrument. If so, it will give beats with the F if sounded along with the a' and c". If, on the other hand, the diflferential should chance to be quite in tune with the F on the instrument, then slightly flatten the F by only partly depressing the digital. Beats will then be heard between it and the diflferential. This diflferential may also be heard, by a careful observer, on sounding the same two notes xery loudly on a pianoforte. The pairs of notes in Table XX. at the intervals of a fifth, a fourth, and a major third respectively, may also be used to obtain the diflferentials on a harmonium, but, being less striking in their eflfects, it may be diflftcult to hear the differentials from them on the pianoforte. To obtain a summational, use by preference the harmonium, and commence by sounding the c (in the bass staflf), followed by the accompaniment of the F (just below the bass staff). 300,301 COMBINATIONAL TONES 387 As soon as the F is sounded along with the c, the a (at the top of the bass staff) is heard completing the chord. Here again the a produced as a summational can be the easier recognised if it has been just faintly sounded previous to the c and F. To obtain proof of this summational on the piano, use the same generators, but proceed as follows : — Without sounding the a, hold down the digital with a finger of the right hand, so as to lift the damper off the string. Then, with the left hand, strike simultaneously the c and F as vigorously as possible, sustain the notes a little, and then raise the left hand so as to stop the generators. The a will then be heard singing, its note having been started in response to the summational in unison with it. This affords another interesting example of sympathetic resonance. It should be noted that this a is not an upper partial of either the F or the c which were used as its generators, so the effect cannot be spurious and due to an overtone of either generator. The above method must not be applied to detect a differential on the piano. For, suppose the two notes named above a' and c" were used. Their frequencies are as 5:6. Then although they would evoke the vibrations of the F whose relative frequency is represented by unity, they would also evoke, and far more strongly, those upper partials of that string which are in unison with the generating tones a and c". And whatever simply related generators are used, they or their upper partials are themselves upper partials of the differential which they can produce. Thus, if the generators are as 4:5, they are both upper partials of the differential tone represented by 1. If, again, the generators are as 3:5, their upper partials 6, 12, etc., from the 3, and 10, 20, etc., from the 5, are all upper partials of the differential represented by 2. It should be noted that these experiments with the harmonium, and still more those on differentials with the piano, can only be appreciated by a small group gathered closely round the instrument. They are quite unsuitable for demonstration to a larger audience. The resonance experiment on the piano for demonstrating the summational may, however, be heard by an audience of a hundred or more. The summational may be produced on the harmonium 388 SOUND CHAP. VII also from generators at the interval of an octave, a fourth, etc., as given in Table XXI. 302. Pitches of Combinational Tones. — Tables XX. and XXI. give respectively the differential and summational tones produced by different pairs of generators. Their pitches are in all cases chosen so as to be fairly suitable for demonstrating the effects sought. They may, howrever, be changed an octave or in other ways modified with advantage, when using any special instrument or apparatus. The table for the differentials begins with the case of two generators an octave apart, although, if the interval were perfect the differential would be lost in the lower generator. But, if the interval is not perfect, the differential gives beats with the lower generator, hence the importance of this case and its inclusion in Table XX. Table XX. — Differential Tones Intervals of Generators— Octave. Fifth. Fourth. ^Wr MinorTMrd. Whole s^^^„^^_ Relative Prr.,,u(-ii-\., ,,0,1,0 ,. i , ' " ciesofalITuncs)2-l = 13-2 = H-3=l 5-4 = 1 -5=1 9-8 = 1 16-15=1 Generators. J m Differentials. m lE^ -I- S^EESE^Esi: Table XXI. — Summational Tones Suminatiouals Generators Intervals of Generators Octave. Fifth. Fourth. Major Third. Minor Third. Relative Frequenciesof all Tone.s— 1+2 = 3 2+3=5 3+4 = 7 4+5=9. 5+6 = 11 • Really flatter than sir. f Really between /■ and /'t 302, 303 COMBINATIONAL TONES 389 303. Have Combinational Tones Objective Reality? — The foregoing experiments have been chosen as seeming most generally suitable to demonstrate combinational tones to others, but by no means exhaust the list of possibilities in this direction. For example, combinational tones may be obtained from the \doliii or the concertina, while Helmholtz himself for this purpose used his double siren. He also asserted that combinational tones sometimes have an objective existence, whereas formerly it had been believed that they were purely subjective. At that time only the differential tones were known, and they were connected with the phenomena of beats. It was thought that when these beats occurred with sufRcient rapidity, the Avaxings and wanings of intensity would produce the sensation of a new tone of frequency equal to that of the beats. But this, in the first place, leaves the summational tones entirely un- explained ; and, secondly, this supposition cannot be reconciled with the experimental law that the only tones which the ear hears correspond to pendular vibrations of the air. Lastly, Helmholtz showed " that under certain conditions the combinational tones existed objectively, independently of the ear, which would have had to gather the beats into a new tone." He showed that the condition for the generation of combinational tones is that the same mass of air should be violently agitated by two simple tones simultaneously. This occurs in his double siren if the tones are produced by two series of holes blown upon simultaneously from the same wind chest. In this case Helmholtz asserts that the combinational tones are almost as powerful as the generators. Of the objective existence of the combinational tones, Helmholtz assured himself by the sympathetic resonance of membranes tuned in unison with them. He also used his air resonators, which were more sensitive than the membranes. Helmholtz experimented also upon the harmonium as to its power of producing combinational tones. When two notes were sounded from the same 390 SOUND CHAP, vn source of air pressure, the combinational tones were clearly reinforced by resonators tuned in unison with them. Yet, in this case, it was found that the greater part of the intensity of the combinational tone was generated in the ear itself. This was shown by the following method : — One of the generatiug notes was sounded by air from the bellows moved below by the foot, and the other was blown by the reserve bellows, which was first pumped full, and then cut off by drawing out the so-called expression stop. It was then found that the combinational tones were not much weaker than before, but the objective portion which resonators reinforce was much weaker. Again, Helmholtz found that " when the places in which the two tones are struck are entirely separate and have no mechanical con- nection, as, for example, if they come from two singers, two separate wind instruments, or two violins, the reinforcement of the combinational tones by resonators is small and dubious. Here, then, there does not exist in the air any clearly sensible pendular vibration corresponding to the combinational tone, and we must conclude that such tones, which are often powerfully audible, are really produced in the ear itself" 304. Mathematical Examination of Combinational Tones. — In article 115 we treated an asymmetrical system subject to double forcing. It was there shown that, given such circumstances, differential and summational tones are produced in addition to those in unison with the impressed forces. Let us now regard the drum-skin of the ear as the system under forcing. Now we have already seen (art. 258) that the drum-skin is differently related to displace- ments inward and outward, and is therefore an asym- metrical system of the type contemplated. Hence the subjective existence of differential and summational tones is accounted for. Let us also examine, in a little more detail, the quantitative results of the theory. Looking at equation (5), article 115, we see, by the numerators of the 304,305 COMBINATIONAL TONES 391 coefficients of the terms in question, that the amplitudes of the combinational tones are proportional to the products of those of the impressed forces, which are due to the generat- ing tones. This shows why, in the experiments, it was essential to have loud generating tones. Again, tlie frequency of the natural vibration of the druiu-skin and its attached ossicles is small. Everett states it to be about ten per second.^ Accordingly, it may be almost neglected in comparison with the frequencies of the generating tones. Hence, from the denominators of the coefficients in equation (5) of article 115, we find the following result: — The amplitude of the differential is to that of the summational almost as the square of the fraction : sum of frequencies of generators divided by their difierence. Or, in symbols, D (N, + N, S \ iV.-N^ nearly, where D and *S' are the respective amplitudes of the differential and summational tones produced by generators of frequencies JV^ and JV^- This explains, what any experi- menter soon finds, namely, that the summational tones are harder to hear and demonstrate than the diiferentials. Lastly, note in the equation (.5), article 115, that all the combinational tones have as a factor the coefficient a, which expresses that the displacement is not simply proportional to the force producing it. Hence, if a vanishes absolutely, then all the combinational tones vanish also. 305. But the above theory is not limited in its application to the human ear. We may have, outside the ear altogether, systems in which the restoring force is not simply proportional to the displacement, but requires for its exact expression higher powers of that displacement. Where, e.ff., the cube of the displacement comes into account, it might be shown that results apply similar to those deduced in article 115, where the square of the ' Phys. Soc. of London, vol. xiv. p. 94, 1896. 392 SOUND CHAP. VII displacement occurred. Instead, however, of developing the matter further along these lines, take the following simple illustration. Suppose we have a system in which a force/ producing a displacement y is represented by /=ay + &/ + c/+ . . (1), and let two forces / and /g, when acting alone, produce the displacements y^ and y^ respectively. Then we have and li=^ay., + ly^^rcyi-\- . Now let these two forces act simultaneously and produce the displacement Y. Then by (1) we have f,^-f, = aY^lY''-^cY'^■ . . (3). But, from (2) by addition we obtain /i +/2 = «(yi + 2/2) + Kyi + yif + r f- a'b-h'b<"-d" Cjt G d g—b rf' ./'—(,' C G 1— c g— b\,-c'-d'-e: a c'b V\j r h^ ^' ^-'U rl" eeds. A cw 1,\, c'b (/' 6'b 1 - B> ; Bb / ib— (?' f— h'\) Eb -Bb cb g—h\f c'b - - -B,b F Bb d -f -61,.-.. man ("Soprano , Vocal J Contralto ■ds as j Tenor d' 3 d!' eds. I Bass ..V ,. -^ - - ■ istrn- "j f I Kettledrum toh. J 'If f ' Among the " plncked " str.ngs must be included instruments of the violin family wh These form, with tombones, u " Brass Band." or Tvit1, \Wr.„^ -ht: j _,. / „ INSTRUMENTS. Eemahks. -'i h"h -V\ -/' -f b J- The " Strings ' of the Orchestra. Keyboard Instru- ments, with separate Reed or Pipe for each note. The "Wood Wind " of the Orchestra and Military Bands. The " Brass " of the Orchestra. Complete Family of Valved Brass Instruments.^ ' Soli " and Chorus. i a pla I'lilita •ed "pizzicato.' y Band. " Facepage 411. 3^7-329 MUSICAL INSTEUMENTS 411 plays the three lower strings, and the fingers the upper strings, of which the highest is played by the third finger. To obtain the different notes of the scale, the strings, at the appropriate lengths, are pressed by the fingers of the left hand against the fret or little pieces of wood which cross the finger-board. Notes called "harmonics" may also be obtained by touching the strings instead of pressing them down on to the frets. These notes will be understood from what we have seen as to the vibrations of strings. Thus, touching at the middle of a string allows it to produce its octave as though it were pressed against the fret at tlie same place. Touching the string at a third of its length (where hard pressure on the fret would yield the musical fifth) gives the twelfth, touching at a fourth of its length gives the double octave, and so forth. We thus see that the thumb and three fingers of the right hand are the exciters, the six strings the vibrators, the sound-box the resonator, while the fingers of the left hand and the frets form the manipulative mechanism for produc- tion of the scale. The guitar is of feeble tone, and does not blend well with other instruments, and may be over- powered by a strong voice. 329. The Harp. — -This is the important representative of the family of stringed instruments played by the hand. The double-action harp is tuned in Cb, and is provided with seven pedals. Each pedal acts upon all the strings of a given name throughout the compass of the instrument. Moreover, each pedal may be used to raise its strings a tone or a semitone at the option of the player. Thus, by the use of the pedals, the instrument may be tuned so as to provide the major scales in each of the fifteen keys from Cb to C|l both inclusive. The minor scales cannot be " set " by the pedals if that form is required which differs in ascending and descending passages. For this instrument that form of the minor scale is to be preferred which has the interval of an augmented second between the sixth and 412 SOUND CHAP, vm seventh notes both in ascending and descending passages. This form of the minor scale can be " set " in the twelve keys from Ab to C| inclusive, which are all that are necessary. The octave "harmonics" are produced on the longer strings of the harp by touching, with the fleshy part of the palm of the hand, the centre of the string, while pluck- ing with the thumb and two first fingers of the same hand. Thus, in the harp, the thumbs and fingers of both hands are the exciters, the strings the vibrators of definite pitch, and the sound-box the resonator. Manipulative mechanism is not needed for playing the diatonic scale in a given key for which the instrument is set by the pedals. The pedals, however, supply this need in the case of accidentals, also for setting the instrument in another key. The harp's quality of tone mingles extremely well with the horns, trombones, and brass instruments generally. For the relative intensity of the various partials present in the vibrations of a plucked string see arts. 331-332. It must be remembered, however, that this refers to the string itself From a musical instrument whose vibrators are strings (whether plucked, struck, or bowed), the actual effect on the ear depends both on the nature of the vibrations executed by the strings and upon the modificatiou of the vibrations introduced by the resonator. It is, perhaps, at this point that the technical skill and experience of the instrument-maker play their most important part. Experiments have been initiated by the writer to investigate the modifications in the vibrations from a string which are introduced by the various other parts of the instrument on which that string is mounted (see arts. 361-363). 330. The Mandolin has four double strings, each pair being tuned in unison. The e" strings are of catgut, "the a strings of steel, the d' strings of copper, and the g strings of catgut covered with silver wire. The strings are plucked with a plectrum (of tortoise-shell or horn), and " stopped " upon the finger-board to produce the notes of the scale. 330,331 • MUSICAL INSTRUMENTS 413 Berlioz states that the quality of tone of the mandolin has a keen delicacy not possible on other instruments sometimes substituted for it. Violin played "pizzicato." — When instruments of the violin family are played " pizzicato," that is, by plucking with the fingers of the right hand instead of with the bow, they fall strictly under the present category, but call for no further comment in this place, being treated, as to their normal mode of playing, under the heading of bowed strings. 331. Quality of Tone from Plucked Strings. — We have already seen (art. 143) that a string of length I plucked at a point distant h from one end has the full harmonic series of overtones, the amplitude of the nth partial being proportional to (l/w^) sin (mrh/l). Thus, as we ascend the series of partials, the factor l/w^ in the amplitude quickly decreases according to a fixed law independent of where the string is plucked. But the other factor sin (nirh/l) obviously depends for its effect upon the value h/l, i.e. upon where the string is plucked. This, then, is at the option of the player in the instruments we have now under consideration. And, by exercising choice in the matter, the quality of the tone may be somewhat varied. Thus, if a string be plucked in the middle, the evenly-numbered partials are all suppressed. The tone in this case is found to be somewhat nasal. If, on the other hand, the string be plucked at one-seventh of its length from the end, the seventh partial is the first to be suppressed, and the tone is, in consequence, much pleasanter. For any specified place of plucking the relative amplitudes may be found from the theory just quoted. Further, the physical intensities of the various partials may be calculated since they are proportional to the product n^a^', or the product of square of frequency into square of amplitude. Finally, we may take the intensity of the prime tone or first partial as 100, and express the others in terms of that arbitrary value. This has been done by Helmholtz for a string plucked at one- 414 SOUND CHAP. VIII seventh of its length from an end, and the results are given in the upper line of Table XXX. If, instead of a sharp point to pluck the string as supposed in the theory, the finger or other rounded object be used, then the angle is rounded off where the string is plucked, and the series of partials is more convergent. Now, the quicker the series of upper partials dies away, the mellower is the quality of the tone produced. Hence, in the case of the mandolin, in which the strings are plucked by a pointed plectrum, we have more prominence given to the high partials, and thus a shriller or more tinkling tone results than with the guitar or harp plucked by the fingers. Further, the high partials are formed by thin strings ; hence the thicker strings used wholly or in part on any given instrument yield a softer, sweeter tone than the others which are thinner. 332. In. all the instruments in which the strings are excited by plucking, the tone obviously dies away from the instant of plucking. And, further, since the different partials usually die away at different rates, the higher ones more rapidly than the lower ones, the tone, as it continues, becomes sweeter and duller. There is also a characteristic beginning of the note due to the plucking. No doubt all these effects in various degrees help us to distinguish the family of plucked stringed instruments as a whole, and also to discriminate between the individuals of the family itself We might also note, in conclusion, that if the strings are very thick and of rigid material, the higher partials will be slightly disturbed from the harmonic relation. This again is a distinction from the instrument played with a bow which preserves strict periodicity, and therefore (by Fourier's theorem) commensurate periods of all the partials. 333. The Pianoforte. — In this instrument the only question that need detain us is the manner of exciting the vibration by the blow of the hammer and the partials thus produced. In the theory of strings (art. 144), we saw that 332,333 MUSICAL mSTEUMENTS 415 a string of length I, excited by an instantaneous blow at a point distant Ti from the end, has the full series of harmonic partials. Further, the amplitude of the wth partial is pro- portional to - sin {mrhjl). This series, therefore, having only the first power of n in the denominator, converges more slowly than that for strings plucked at a point, in which case the square of n occurred in the denominator. But it has, like the plucked strings, the sine factor which here expresses the dependence of quality on place struck. Hence the necessity, on the part of the instrument-maker, for rightly choosing where the hammers shall strike the strings. Further, to prevent the formation of the long and too obtrusive series of partials which would result from a blow with a hard hammer striking at a single point, the hammer has a rounded end and is covered with an elastic pad. Thus the motion is imparted to a considerable length of the string instead of being confined to a point ; also the contact between hammer and string is not instantaneous, but extends over a time which, although very short absolutely, is appreciable in comparison with the period of vibration of the string itself. These modifications of the phenomena Helmholtz has endeavoured to introduce in his theory. As an approximation, he writes • for the pressure of the hammer A sin rnt, and the hammer is supposed in contact with the string for half the period defined by m. He thus finds that the amplitudes of the partials are affected by factors, one of which is a sine function, and involves the suppression of the partials which have a node ut the point of impact. There is also an algebraic factor, according to which the amplitudes of the partials decrease nearly as the inverse cuhe oi n when ni is finite. When, however, m is infinite, we have returned to the perfectly hard hammer, and the amplitudes decrease inversely as the first power of n. The values of the intensities of the partials for the pianoforte calculated by Helmholtz from 416 SOUND CHAP, viit his formula when the striking point is ^, and the hammer is in contact for |- the period of the prime, are given in Table XXX., and were found to agree with the notes a little above the centre of the compass of his grand pianoforte, i.e.. near c". 334. It must, however, be remembered that this theory is but approximate. Ellis states that, with the ordinary pianoforte hammer, the partial tone corresponding to the node struck, though materially weakened, is not absolutely extinguished. As to the choice of place struck, Helmholtz writes, " In pianofortes, the -point struck is about \ to -|- the length of the string from its extremity, for the middle part of the instrument. We must therefore assume that this place has been chosen because experience has shown it to give the finest musical tone, which is most suitable for harmonies. The selection is not due to theory. It results from attempts to meet the requirements of artistically trained ears, and from the technical experience of two centuries." 335. The effect of the soft pedal in producing the muffled or veiled quality of tone by interposing an elastic pad between the hammer and the strings is a further in- teresting illustration of theory afforded by the piano. In the pianoforte, as previously in the case of the harp, we must distinguish between the quality of tone built up of the partials of the string itself, and the quality perceived by the hearer from the whole musical instrument. For, by means of the resonator, the partials of the string are rein- forced and usually in a selective or preferential manner. It is perhaps here more especially that theory fails, and the experience of musicians and instrument-makers through a long period count for so much. 336. The Violin Family. — The stringed instruments played with a bow now in ordinary use are only four in number, namely, the violin, the viola, the violoncello, and the double-bass. Of the latter, two varieties exist, those 334-338 MUSICAL INSTRUMENTS 417 with three and those with four strings. The former is more usual in England. It is generally tuned in fourths, but occasionally in fifths, the less common tuning being shown in Table XXIV. by the letters in brackets. There is another instrument of this family, the viola d'amore, which though nearly obsolete, is of scientific interest on account of its double stringing. Berlioz describes it as follows : " It has seven catgut strings, the three lowest of which are covered with silver wire. Below the neck of the instrument, and passing beneath the bridge, are seven more strings of metal, tuned in unison with the others, so as to vibrate sympathetically with them, thereby giving to the instrument a second resonance full of sweetness and mystery." Beyond the points just mentioned as to the double-bass and the viola d'amore, the only distinctions between the various members of the violin family which are of import- ance to us, namely, the tuning of the strings and the compass of each, are shown in the Table XXIV. 337. The remarks which follow apply primarily to the violin as the representative of the family, but, with certain modifications which may be easily inferred, they apply also to the other kindred instruments. The bow with its rosined hairs is the exciter, the strings of catgut, bare or covered with wire, are the vibrators of definite pitch, the sound-box is the resonator, while the finger-board, free from frets, allows the scale to be played by the fingers of the left hand. Note that the absence of frets on the finger-board makes it possible to " stop " the notes anywhere. Thus, as regards intonation, it gives full freedom to the highest refinements of the finished performer, but affords no aid to the novice. Further, the smooth finger-board permits the use of the grace called portamento, that is, the gliding of a note con- tinuously from one pitch to another. 338, The use of the fingers of the left hand to press 2 E 418 SOUND CHAP, yiii the string against the finger-board always, of course, sharpens the note ; for it shortens the portion of the string left free to vibrate, while not materially changing its tension. Hence, the inferior limit of the compass is quite definite, being that of the lowest open string. The superior limit, on the other hand, depends upon the skill of the performer. Very often, for the sake of special effects, notes called " harmonics '' are used. That is, as already mentioned in connection with the guitar, the string is touched by a finger of the left hand at the middle, one-third the length, one-fourth, and so on, thus producing respectively the octave, the twelfth, the double octave, etc., of the note given by the open string itself. These are called natural harmonics. For all beyond the simple octave there is obviously a choice as to which node shall be touched for the same note. Thus, the twelfth of the open string may be produced by tou.ching at the lower or at the higher of the two points of trisection. The former is sometimes more convenient, the latter is preferred by some as giving the nobler quality of tone. In addition to the natural harmonics on the lower and upper halves of the string, we have also what are called artificial harmonics. These are produced by pressing the string firmly against the finger-board with one finger (firm stop), and touching it lightly with another finger, (loose stop) at a point between the former place and the bridge. By this means very high notes can be obtained without moving the hand far up the string. Thus, let the firm stop be made one-ninth the length of the string from the lower end, and the loose stop at one-fourth up the part of the string left free to vibrate. Then the firm stop would give a note one tone above that of the open string, while the loose stop further raises the pitch a double octave. On the violin these stops can be made with the first and fourth fingers respectively without moving the hand from its place at the lower end of the string, called the first position. These artificial harmonics are, however, entirely tabooed by some eminent players of 339-341 MUSICAL INSTEUMENTS - 419 the violin, since their introduction encourages the use of thin strings which are held incompatible with the pro- duction of the noblest quality of tone. 339. The points above mentioned for the stops are the strictly theoretical ones which apply to a perfectly uniform string. On an actual violin the strings, when well selected, are usually a little taper, hence the theoretically assigned positions are slightly departed from. It is important in stringing a violin that the strings all taper the same way, so that the same stops on the various strings lie truly side by side. Another displacement of the stops from their ideal positions would occur if the bridge and iinger-board are so related that the string is very high above the finger- board, for then the pressing the string to the finger-board would appreciably increase its tension. This should be avoided by cutting the bridge, so that all the strings lie near enough to the finger-board, but still have room to vibrate. 340. The resonator of the violin demands special notice. It is this which gives to the instrument its valued quality of tone, and upon which the makers have bestowed their greatest pains. The wood, the shape, and the varnish are all of the highest importance. The bridge and ribs are of maple, the back of maple or sycamore, and the belly of pine. Most violins, as tested by tuning-forks held over the "/" holes, show one or two specially marked resonances ; but some of the finest violins appear to yield an almost continuous resonance between wide limits. Such instru- ments present in a high degree the justly valued quality of tone throughout their entire compass. 341. All the instruments of the violin family may be played with a mute, for the sake of the specially muffled or veiled quality of tone thus obtained, and the extreme pianissimo effects then possible. The mute is a small apparatus of wood or metal which fits on the bridge, and 420 SOUND CHAP, viu thus deadens the sound considerably. Its use where desired is indicated in the music. 342. With ordinary players of bowed instruments there are often perceptible hissing or rustling noises at the beginning and finishing of the notes. With the finest performers this feature almost disappears. For example, if the bow is not moved perpendicularly to the strings, but allowed a small lengthwise component also, this would generate in the string longitudinal vibrations of excessively high pitch. This is one possible explanation of the whistling sounds from a violin in the hands of a careless player. These objectionable noises may also be produced by a wrong adjustment between the pressure and speed of the bow. Yet other possible causes are a roughness of the hairs of the bow or the rosin on them, and imperfections in the wood of the violin, which cause sudden changes in the phase of the vibration, thus interrupting its uniform and even flow. Some of these causes may operate at any stage of the duration of a note, but their effect is likely to be most noticeable at the beginning and ending of a note, owing to the lack of perfect technique in the player's manner of putting on and taking off the bow. 343. The use of the bow as an exciter, unlike the plucking or striking action, enables the player to sustain the notes at will, and either with the same, a diminished, or an increased intensity. As to the handling of the bow to produce these various effects, the following points may be noticed : — The part of the string bowed, the pressure of the bow on the string, the speed of the bow over the string, are all at the disposal of the player and may be varied, not only from note to note, but during the continuance of any given note. An increased pressure and speed are required for a crescendo, a decreased pressure and speed for a decrescendo. But, by bowing nearer to the bridge, as usual in forte passages, there may be also a more brilliant quality of tone produced. While when bowing 342-345 MUSICAL INSTRUMENTS 421 nearer the finger-board, as is usual in piano passages, a rather softer, duller quality of tone is probably obtained. But these effects may be modified by the following circum- stances : — The bow is normally held obliquely, so that with a light pressure only a few of the hairs_ at the edge touch the string, but with a greater pressure more hairs touch, and so facilitate the production of a louder note. But this more extended place of contact of bow and string would discourage the higher partials and prevent them from being unduly obtrusive. The place on the string at which the bow is applied varies, of course, with the length of the string in vibration. Thus the bow should be applied nearer the bridge for a high stop on a given string than for a low stop or the same string open, i.e. with the whole string in vibration. Further, the bow may approach nearer the bridge for a thin string, as the e" or a', than for a thick one as the d' string. The position of bowing may also be chosen with a special view to the quality of tone desired. The extreme range of positions at which bowing is admis- sible lie between say ^ and ■^'-g of the length of the string, the normal distances being about ^^ to ^^, or roughly 1 inch from the bridge for a string whose length is 12|- or 13 inches. 344. It should also be noticed that certain passages may be taken entirely upon particular strings for the sake of the special quality of tone thus obtained. For, as previously mentioned, some of the strings are bare, and one, the g string, in the violin is covered with wire ; further, the strings differ in thickness. Hence, each string has its own character difiering slightly from each of the others, and is utilised by composers accordingly. 345. Another special power possessed by instruments of the violin family is their ability to produce the embellish- ment called the tremolo. In Spohr's Violin School this is described as follows : — The tremolo " consists in the waver- ing of a stopped note, which sounds alternately a little 422 SOUND CHAP. VIII above and a little below its just pitch, and is produced by a trembling motion of the left hand in the direction from nut to bridge. This movement, however, should be very- slight, so that the deviation from the true note may not offend the ear." . It is used " rapid, for intensifying passionate expression, and slow, for imparting tenderness to sustained and pathetic melody." 346. The " strings," besides being valuable in themselves, constitute the very foundation of the orchestra. Various reasons conduce to this. For example, the players of the " strings " have not to pause to take breath, they command the most delicate shades of expression, and their quality of tone is such as not to pall upon the bearer so soon as that of the wind instruments. One writer, comparing the qualities of tone in the orchestra to the colours in a picture, has likened the strings to the various shades of grey, the wood-wind and horns to colours moderately bright and rich, the trumpets and trombones to colours approaching to purple and scarlet. 347. Vibrations of Bowed Strings. — No complete mechanical theory of the bowed string seems yet to have been given. Helmholtz, however, by a skilful combination of exiperiment and analysis has given a solution which serves for the most important musical applications, namely, the cases in which the bow is used so as to produce a tone of fine musical quality. Something has already been said in the previous chapter as to the action of the bow. The vibration of the string is maintained by it, but is not to any appreciable extent " forced " in the sense of having its pitch thereby modified. The maintenance is here one of those examples of the action in which the " driven " selects for itself, as it were, the periodicity of the influence under which it shall come. To discover something as to the nature of the motion executed by a bowed string, Helmholtz proceeded as follows : — The a! string of a violin was placed in a vertical position and tuned up to 1%, the part of it to 346, 347 MUSICAL INSTRUMENTS 423 be observed was blackened with ink, when dry rubbed over with wax, and then powdered with starch so that a few grains remained sticking on the string. A starch grain, when illuminated, formed the point to be observed through a microscope whose objective was carried by one prong of a tuning-forlv placed horizontally, but so that its vibrations occurred vertically. The fork was Bb, exactly two octaves below the string, and was electrically driven. The string was excited by drawing the bow across it parallel to the prongs of the fork. The figure presented in the field of view by a luminous point, when both fork and string were vibrating, was evidently that resulting from the two vibrations, that of the fork vertically and that of the string horizontally. But since that of the fork is known to be simple harmonic of given period, it provides an exact measure of the time, and thus enables the experimenter to determine all the details of the other component, viz. the vibration of the string. This apparatus for observing a rapid vibration Helmholtz called a " vibration microscope." In the simplest case when the bow bites well, and the prime tone is powerfully produced, Helmholtz found that the motion of any point of a horizontal string consists of an ascent with uniform, speed followed immediately by a descent at a uniform speed. For the middle point of the string these two speeds are equal, for other points they are unequal. At the place of bowing the speed in the direction of bowing appears to be equal to that of the violin bow. " During the greater part of each vibration the string here clings to the bow, and is carried on by it ; then it suddenly detaches itself and rebounds, whereupon it is seized by other points in the bow and again carried forward." Thus, if the motion of a point on the string is represented in a diagram with time as abscissae and displacements as ordinates, we shall have for the required " curve " a two-step straight line zigzag, the slopes being in general different, see Fig. 87. Eeasoning upon these experimental results as a basis, 424 SOUND CHAP. VIII Helmholtz has given an analysis of the motion of a bowed string which, with slightly different notation, is followed here. 348. Theory of Bowed Strings by Helmholtz. — Let the motion of a point on the string be as represented in Fig. 87, and be also defined by the equations y =ft — h from t=(i io t = a\ ,^, and y ^'J {t — t)—h from i = a to t = T^ where t is the complete period of the compound vibration. Hence for < = a we have fa=g (r-a). Fig. 87. — Motion of Well-bowed String. Now let y be developed in the Fourier series, y = a^ cos tot + a.,Q.os,2a}t + . . . + a,^ cos n(x>t + . . . + &1 sin wi; +&.2 sin 2«B< 4-. • . + 6„ sin «&)< + ■ • • where wt = 2-jr, and the values of the coefficients are to be obtained from the theorem by integration (see art. 52). Thus 2 2 a„_ = — \ y cos nwtdt = — (Jt — h) cos nafdt 2^ + - {ffi"^ — ~ ^} COS ncotdt. whence (,/+^> (1 — cosnma). 348-350 MUSICAL INSTEUMENTS 425 Similarly &„ = ^"^ J^\ sin na>a. Then, by substituting these in the Fourier expansion, we may write the results as follows : — 349. But this equation only refers to the point whose motion is delineated in the iigure and expressed by equation (1). Now let X be the co-ordinate of a point on the string and I its length. Then, since in the general expression for the vibrating string the cosines of multiples of xjl are all absent, we obtain By comparing (2) and (3), we have . n-nx (f+(i)T . ntra ... I n TT T Here (f + g) and a are independent of n, but not necessarily of X. On putting n = 1 and » = 2 in (4) and dividing, we find c, "TTX 1 Tra — cos — = — cos , Ci / 4 T from which it follows that for x = 1/2 and a finite ratio of c's, a = t/2. And since, by observation, a decreases with x, and both vanish together, we obtain C2 = -' and - = a/T (5). 350. Thus, since x does not appear in the ratio of the c's, we find that (/+5') is independent of a;, but a is not. We 'shall see later that / and g are each functions of x, although their sum is constant. f+rj=2u[- + 426 SOUND CHAP, viii Now let %i be the amplitude of the vibration of the strings at the point x, then fa=g{T — a) = 2m '\ 1 \ 2mt _ 2w;^ ■ (6), ^ a. T — aj air — a) rx(l — x) . the last expression being found by (5). And, since Z+^r is independent of x, we have, putting U for the amplitude at the middle of the string, 2wZ' _^. TX{1 —X) T or u = 4.Ux{l-x)IP (7). Also (7) in (6) gives /+ ^7 = 8 UJt (8). And from (5) it follows that a and (t — a) are proportional to the corresponding parts of the string on each side of the observed point. Put the value of f-\-g from (8) in (4) and then the value so obtained of c,^ in (3). We thus find, for the expression of the string's motion under the circumstances considered, the following equation : — 8f7^"="fl . nwx . n^nrf aX} ,„, Of course the ordinary relations for strings hold here, namely, ^^l^ = N=^-=^^ (10), 27r T 2Z 2/ where N is the frequency of the prime tone, v the speed of propagation of a disturbance along the string, F the force by which it is stretched, and a- its linear density. 351. It should be carefully noted that the analysis leading to equation (9) is based and built up on the assumed motion of the observed points being as specified at the outset, namely, the two-step straight line zigzag. That 351,352 MUSICAL mSTEUMENTS 427 this is sometimes the case is the result of Helmholtz's experiments, and forms the foundation of the theory. But, as Helmholtz himself found, it is not always the ease and then all this falls to the ground. Thus, when the motion of the observed point exhibits, as it sometimes does, little crumples superposed on the main zigzag, we should then have to use corresponding values of y instead of those in (1) to obtain by integration the coefficients which appear in (2). Hence equation (9) is not to be taken as a complete solution for a bowed string corresponding to that for a string plucked at a point (art. 143). Another way of showing that this must be so, is as follows : — The above expression contains no factor depending on the place at which the string is bowed, and all the partials are included and their relative amplitudes vary as 1/m^ simply. "Whereas it is found that if a string is bowed exactly at any node of any partial, then that partial will not be elicited. Thus, if bowed very near to the point in question, the whole motion of the string would, in all probability, be slightly modified. Thus, in the full expression there should be a factor showing how the various partials fade away as the bow approaches any one of their nodes, and that they vanish when the bow precisely reaches any node. 352. Let us now proceed with the theory as developed by Helmholtz. If we put ( i( — - j = in equation (9), then y=^ for all values of x, and hence all parts of the string pass through their equilibrium position simultaneously. At that instant the velocity / at the point x is found by equations (6), (7), and (5) to be / = 2w/a =^U{1- x)llr (11). But this velocity only lasts for aj2=XTJ2l, after the instant when y = everywhere. So, within this limit, we have the general expression for the displacement, y=ft=^%U{l-x)tjlT (12), 428 SOUND CHAP. VIII and the extreme value of y reached, or the amplitude at x, will be ,(=/a/2 = 4?7(/-a;>y/!^ (13). From this point the value of y diminishes at speed given by equations (6), (7), and (5), namely, g = A'_ = ^U<1- ^ ^ 8 UxjlT (14). r — a P{r — a) Hence, after time t from when y = everywhere, y after increasing to its maximum value u will have been diminished for the time (t — a/2), so will be less than u by the amount g(t — a/2). Thus we obtain by (13), (14), and (5), _4:Uxll-x 2/ a\\ _4:UxJ T-a 2t a ^~~r\~r i\ 2J| ~rt^,^ 7 t or, y=8lTx[^-tyiT (15). 353. Hence, at time t, the configuration of one part of the string through x = l is given by equation (12), and, at the same instant, that of the other part passing through «= is expressed by equation (15). Each equation shows that the form of the part to which it refers is a straight line. The point where these straight lines intersect is given by the condition that the two values of y are equal. Thus equating the right sides of (12) and (15), we obtain (l — x)t=l- — t ]x, X I r/2 the V being identical with that occurring in equation (10). Hence the abscissa x of this point of intersection increases in proportion to the time and at the speed which enables it to traverse the length of the string in t/2. That the speed of any disturbance in running along the string must have 353-355 MUSICAL INSTRUMENTS 429 this value might have been foreseen at the beginning, and so equation (16) serves as a check on the accuracy of the previous working. 354. Further, the point of intersection itself, which is at the same time the point of the string most removed from its position of rest, passes from one end of the string to the other, and during its passage describes the parabolic arcs, for which the equations, as seen from (13), are 2/= ±^Vq,-x)xlf (17). In these equations the y must not be confused with the y previously used for the displacement of a point x at time t. Of the two algebraic signs on the right side, the upper refers to the upper curve with positive ordinates, and the lower to the lower curve whose ordinates are all negative. 355. Hence the motion and the form of the whole string at any one instant may be thus briefly described. In Figure 88 the foot D of the ordinate of the highest point moves forwards (and D' say, corresponding to the lowest point, then moves backwards) with the constant speed v along the horizontal line AB, while the highest (or lowest) point of the string describes in succession the two parabolic arcs ACjB and BCjA, and the string itself is always _^=Sj!= stretched in the two straight lines ACj, CjB or BC2, CjA. Further, the motion of the string fiq. 88.— Configuration of Bowed String. at one point x is iirst upwards with constant speed / for the time a, and then downwards with constant speed g for the time {t-o); where r is the complete period of the compound vibration ; a = xt/1, /= 8 U{1 - x)IIt and 5- = 8 Uxjh, see equations (5), (11), and (14). Hence, although the numerical sum of the upward and downward speeds at any poiut of the string is the same as that at any other point (7+^7= 8 ?7/t = const.. 430 SOUND CHAP, vin see equation (8)), yet, the separate speeds and the fractions of the period during which each is maintained vary from point to point, and in such wise as to make the amplitudes at the different places differ from each other, and become indeed the ordinates of the two parabolic curves defined by equation (17). 356. As previously mentioned, Helmholtz also found that, in certain cases, the motion of the string at the bowed place was more complicated, consisting of little crumples superposed upon the main two-step zigzag. These little crumples he failed to keep steady enough to count until using an old Italian violin by Guadanini. This uniformity of vibrational form is evidently connected with the pure smooth flow of the tone heard, and shows the superiority of fine old instruments that have also been long played upon. Examples of motions of both characters are given in the next article. "We thus see that the motion of the string itself may be modified by the resonator with which it is associated. Now the foregoing investigation relates to the actual motion of the string when associated with a resonator. Yet still, as in the case of plucked and struck strings, we have to note that the effect on the ear from any instrument with bowed strings may be further modified by the action of the resonator, certain of the partial tones being preferen- tially reinforced (see arts. 361-363). The physical intensities of the partials from the string itself when well bowed are shown in Table XXX. 357. Researches of Krigar-Menzel and Raps.— In 1891, 0. Krigar-Menzel and A. Eaps published (in the Sitzungsber. d. Berl. Akad. d. Wiss.) an account of a different method of observing vibrating strings. By this they obtained photographically a record of the motion at any point in the form of a displacement-time diagram. The essentials of the method are as follows : — Imagine a string stretched horizontally some distance in front of a vertical slit which is illuminated from behind by an arc lamp. By 356-358 MUSICAL mSTEUMENTS 431 means of a convex lens a real image of the slit is focused on the string. This image crossed by the opaque string now serves as object to a second lens, which focuses upon a sensitive film the bright slit crossed by the string's shadow. The photographic film is wound upon a drum rotated uniformly by clockwork about its axis, which is vertical. Then the up-and-down motion of the shadow due to the string's vibration will be compounded with a uniform horizontal motion due to the drum's rotation. But this is what is required to give a displacemeat-time diagram. On the original negatives these diagrams are white on a black ground, and they are so reproduced in the paper. This method obviously allows any point to be chosen as the- bowed point, and the same or other point to be chosen as the observed point. These points were variously chosen and combined by the authors, and sixty- four photographs accompany their paper. We can only reproduce a few of them here (Fig. 89). For the sake of sharp photographs, fine metal strings were usually employed, but they found that the effect was practi- cally the same with strings of other materials. 358. In the first section of Fig. 89 we have cases of special musical interest and also illustrative of Helmholtz's theory. For here the bowing is at ^gth the length of the string, which approaches to the practice of violinists. And we see by the seveii diagrams that at all the observed points the motion is a pure straight line two-step zigzag ; and, further, that the times occupied on each step are pro- portional to the lengths of the two parts into which the string is divided by the point of observation. In Section II., on the other hand, the point of observation is the same in all seven diagrams, and being the middle point, all evenly numbered partials elude notice, as they have a node there. But as the bowing passes from ^jj to ^ the type of motion changes considerably. In this section the symbol e denotes a small undetermined quantity of the order ^jj. The last Fractions of length at which Bowed Observed 1 T5 ■ -\ 1 TO 5 1-e "5^ 2-e 7 2 + e 1-e 3 1 Ttr 1 Types of Complicated Motion at Bowed Place. WVVA/A. AAA/VW XAAAAAAvAA^ v-VN/N/VN/N/N/" ^^^A^^A^v^A^/^Av^Av*Av^Av^ AvAvvVV^V^VAVV^V'VvV FiQ. 89. — Motion of Bowed Strings observed bt Krioar-Menzel and Eaps. 432 359, 360 MUSICAL INSTEUMENTS 433 diagram of this section is almost, a pure sine graph. The reason for this can easily be seen. By bowing at ^, the partials 3, 6, 9, 12, etc. are all absent from the motion of the string ; and by observing at |, the partials 2, 4, 6, 8, etc. are all absent from this particular diagram of its motion, since they have a node at the centre. Thus the partials remaining in the diagram, up to the 16th inclusive, are those numbered 1, 5, 7, 11, and 13. And their re- spective amplitudes are of the order 1, ■^^, J^, ^-^j-, and ik^- Hence we are practically reduced to the first and a very small fifth partial, the effect of the others being almost negligible. Section III. consists of two diagrams only, showing types of complicated motion at the place of bowing. It is clear that a theory of the whole motion of the string founded upon these motions at the bowed place would be more complicated than that of Helmholtz already given. For, in obtaining the coefficients for the Fourier series, we should be obliged to take y from these diagrams instead of from a simple two-step zigzag. 359. It should be especially observed in connection with bowed strings that it is no approximation to express their motion as a Fourier series. For, when the motion is properly maintained by the bow, it is strictly periodic, and therefore all the partials are compelled to be of com- mensurate periods, although the free natural periods of the same partials might depart slightly from the harmonic series. This slight departure would, of course, be allowed to assert itself when the violin is played " pizzicato." Professor W. B. Morton and T. B. Vinycomb have also followed up the subject, and published results^ illustrating plucking and resonance. 360. ExPT. 62. Projection of String Vibration Cwves. — The displacement-time curves shown in the diagrams of Fig. 89 can easily be projected on a screen visible to an audience. I Phil. Mag., Nov. 1904. 2f 434 SOUND CHAP. VIII This is attained by the following slight modification of the arrangement of Krigar-Menzel and Eaps. The drum is re- placed by a rotating mirror which is turned by one operator, while another bows the string. The ordinary metal " string " of a sonometer will do, but needs careful bowing. The light after reflection from the rotating mirror falls upon a suitably placed screen, and thus exhibits the curve while the sound is audible. Yet another method of projection is due to S. Mikola (1906), and consists in replacing the drum-film of Krigar- Menzel and Eaps by alternate black and white strips on the drum and parallel to its axis ; this then forms the screen on which the curves are seen. 361. Vibrations of the Various Parts of a Stringed Instrument. — As we have already seen, the vibrations of strings — whether plucked, struck, or bowed — are, in their main features, known to us through the work of Helmholtz and others. But, although the interest of the pure mathe- matician may cease at this point when the motion of the string is solved, and the musician's interest may only begin with the main body of sound leaving the instrument, yet the physicist may rightly inquire what happens between these two stages. For " the sound received by the ear from a stringed instrument does not come chiefly from the string direct. Indeed, if a string is mounted on very rigid and massive supports, scarcely any sound can be obtained from it, although the amplitude of its vibrations may be consider- able (see Expt. 2, art. 2). Under the usual conditions the string moves the bridges over which it is stretched, they in turn move the belly, sides, and back of the sound-box, and the air within the box pulsates in response. Probably, therefore, the chief part of the sound received by the ear comes from the belly and other parts of the sound-box, as it has a larger surface, and is therefore better able to set the external air in vibratory motion. " Now the question naturally arises, Are these vibrations of the same quality as those executed by the string itself? 36], 362 MUSICAL INSTEUMENTS 435 that is, may they be compounded of the same Fourier terms ? Probably not, for the worth of a violin does not lie in the strings, but in the sound-box. " It appears, therefore, to be a matter of some importance to attempt to trace the changes in the character of the vibrations which occur as we pass through the series.: string, bridges, sound-box, air within the sound-box, air outside the sound-box." This aspect of the matter occurred to the writer some time back, and the above passage is quoted from the first instalment of work upon the subject.-'- 362. In these experiments vibration curves were simultaneously obtained from the sound-box and string of a monochord. This new feature had the advantage of exhibiting the relations of amplitudes and phases of the two motions, and also showed whether variations in the belly's motion were traceable to faulty bowing. For the string's motion a modification of the method of Krigar-Menzel and Eaps was adopted — their film on a drum being replaced by an ordinary glass negative shot along rails. The motion of the belly, or upper face of the sound- box, was detected and recorded by means of a tiny three- legged optical lever. This lever received upon its mirror the light from an arc lantern after passing through a pin- hole and a focusing lens. The light is then reflected on to the photographic plate, where it makes a bright spot just below the image of the slit crossed by the string's shadow. When the string is excited its shadow moves up and down the slit, and the belly moving in response the bright spot also oscillates vertically. If, however, in addition the plate is moving horizontally, we have both oscillations drawn out into displacement-time curves, the upper one for the string is, in the positive print, a black line on a white ' Barton and Garrett, Phil. Mag., July 1905. 436 SOUND CHA.P. vm ground ; the lower one for the belly being a white one on a dark ground (see Fig. 89 a). 363. Further instalments of this work were carried out in 1906 by the writer and J. Penzer, in which the vibra- tions of the air^ and those of the bridge vertically and lengthwise of the string^ were dealt with. The methods for the vibration of the string were essentially the same throughout, its motion being, accordingly, always shown by a black line on a light ground. For the motions of the air the centre hole of the three on one side of the monochord was covered with a thin animal membrane, whose centre was connected by a light stalk of aluminium to a mirror capable of rocking on a horizontal axle, consisting of a needle working in vees on brackets fixed to the monochord itself. Thus, as the string was sounded, the air pulsated, and so by means of the membrane and stalk rocked the mirror. Consequently the spot of light reflected by the mirror and focused on the photographic plate rose and fell, and thus, as the plate moved horizontally, traced upon it a wavy line, showing the amplitude, phase, and character of the motions of the air. These proved to be of greater com- plication than those of the sound-box, the air apparently following more closely the highly complicated motions of the string itself (see Fig. 89 b). " The motions of the bridge were obtained by special arrangements of the rocking mirror or optical lever, and were found to be intermediate in general character and complication between those of the sound-box and those of the air (see Fig. 89c). In Figs. 89b and 89c the string is shown in the lower part of the photograph. In all the three figures the frequency of the string is 130 per sec, and the bowing is at ^^ throughout. 364. Metal Reeds without Pipes. — Examples of this class of vibrator are presented by the harmonium, the American organ, and the concertina. In these cases the 1 PhU. Mag., Dec. 1906. 2 p;„7 j^^^^ ^pj.^ i907_ 363, 364 MUSICAL INSTEUMENTS 437 reed or tongue is a thin oblong metal plate or strip fastened at one end to a block in which there is a hole behind the tongue and of the same shape. In some examples of these reeds the tongue, when in its position of rest, closes the hole in the block with the exception of a very fine chink Fig. 89a. — Bellt's Motion. Magnified 260 times that of string. Fig. 89b.^Aib's Motion. Magnified 1400 times that of string. Fio. 89o.— Bridge's Motion Longitudinally. Magnified 67 times that of string. all round its margin. "When in motion the tongue oscillates so as to alternately open and (nearly) close the hole in the block. This arrangement is termed a " free reed." Thus the mode of producing a musical note with a free reed resembles that in the siren. For in each, the passage for the air being alternately opened and closed, its stream is 438 SOUND CHAP. VIII reduced to a series of separate puffs or rushes. On the siren this is effected by the rotation of a disc pierced at regular intervals, but in the case in question by the vibration of the elastic metal tongue or reed. Now, it is a result of mathematical analysis that, the more sudden the discon- tinuity of any periodic motion, the greater the relative importance of the high upper partials into which that motion may be resolved. Thus, the more sudden the action of the reed, the more obtrusive are the high upper partials, and the more cutting or grating is the quality of tone. Moreover, since, in the class of instruments under discussion, there are no pipes whose resonance can modify this quality, it remains of that cutting character which so soon palls upon the hearer. 365. As to the distinction between the harmonium and the American organ, the former has the wind forced from the bellows through the reeds, while the latter has it sucked through the reeds into the bellows. The chief claim of the concertina to notice is the fact that it has fourteen notes to the octave, the two unusual notes occurring between D and E, and between G and A. There are thus separate notes for Dff and Eb, and again separate notes for GJf and A\>. We shall see in the next chapter to what this is due. In the various instruments now under review the wind is the exciter and maintainer, the reed and the associated puffs of air form the vibrator of definite pitch, while the keys and their connections are the manipulative mechanism for producing the scale. The reeds are tuned as follows : — To sharpen one, a little is scraped off the tip. This, while not materially weakening the spring, diminishes the mass where its effect is greatest, the frequency is accordingly increased. To flatten the pitch of any note a little is scraped off the root of the correspond- ing reed. This weakens the spring while scarcely altering the effective mass, and so the frequency is diminished. 365-367 MUSICAL INSTKUMENTS 439 366. The Organ. — For our purpose we must divide the stops of the organ into two classes, those with reeds and those without. We shall treat them in this order, Metal Reeds with Pipes. — In passing from the reeds of the harmonium to those of the organ, we have to note, first, .that the quality of the tone is now modified by the presence of a pipe ; and, second, that the tuning is effected differently. Further, the reeds themselves may be of two kinds : fi^rst, free reeds like those in harmoniums ; and, second, heating reeds which are too large to pass into the opening with which they are associated. They accordingly bend down upon the opening like a flap, and thus nearly close it. A beating or striking reed is " voiced " so as to come down with a rolling motion, and thus gradually cover the aperture. The harshness of quality consequent upon a sudden discontinuity is thus avoided. The free reed of an organ pipe is tuned by a wire clip which grasps it upon both sides near its root. The shorter that part of the reed left free to vibrate, the sharper is ■ the pitch. The beating reed, on the other hand, has simply a wire pressing upon it near the root, and by an adjustment of this wire it is tuned. By varying the make of the reeds and the shape of the pipes with which they are associated, various typical tone qualities can be obtained, imitating, more or less perfectly, the various instruments of the orchestra. In these reed pipes or reed stops of the organ we have, therefore, the following classification of parts and functions : The blast of wind is the exciter, the reed of definite pitch is the vibrator, and the pipe with which it is associated is the resonator modifying the quality of the tone produced. The manipulative mechanism for the production of the notes of the scale, expression, and quality of tone, is repre- sented by the keys, stops, bellows, etc. 367. Pipes without Reeds. — We now pass to the flue or flute stops of the organ. These consist of pipes, usually of parallel bore, and with a typical side-opening or " mouth " 440 SOUND CHAP. VIII familiar to every one. They may be of wood, and are then usually of rectangular cross-section ; or of metal, when they are usually cylindrical. Further, they may be either open at the end distant from the mouth or stopped. We have already seen (art. 175) that the open pipes have a full series of harmonic partials almost exactly in tune, especially if narrow, and that the stopped pipes, on the other hand, have only the odd series of partials. " It remains, further, to be noticed how the tone is excited and maintained, and the other details upon which the quality depends. Let us consider the latter first. The investigation of the tones natural to pipes (art. 180) referred to the tones of strongest resonance, as Helmholtz terms them. But, when the compound tone is maintained by blowing, the partials, though naturally forming a series nearly but not quite harmonic, are forced into the strict harmonic series, since according to Fourier's theorem any periodic motion has components of strictly commensurate periods. Now, in the case of narrow pipes the natural tones are very nearly the strict harmonic series. Thus, the components of any periodic motion whose frequency is that of the pipes' fundamental, will be almost exactly in tune throughout with the tones natural to the pipe. They will thus be all freely elicited, and the retinue of upper partials will be relatively full, the quality of tone from the pipe being accordingly fairly bright. In wide pipes, on the contrary, the upper partials natural to the pipe deviate considerably from the harmonic series founded on its prime. Hence, when a periodic motion of the frequency of its prime is forced upon it by blowing, the strictly com- mensurate components of this motion being considerably out of tune with the higher tones natural to the pipe, they are but slightly elicited. Hence, in this case, the retinue of upper partials is feeble, and the quality of tone is, in consequence, soft and mellow. 368. We have now to examine the action of blowing 368 MUSICAL INSTEUMENTS 441 itself. On this subject Helmholtz ^ wrote, " The means usually adopted for keeping pipes continually sounding is blowing. In order to understand the action of the process, we must remember that when air is blown out of such a slit as that which lies below the lip of the pipe, it breaks through the air which lies at rest in front of the slit in a thin sheet like a blade or lamina, and hence at first does not draw any sensible part of that air into its own motion. It is not until it reaches a distance of some centimetres that the outpouring sheet splits up into eddies or vortices, which effect a mixture of the air at rest and the air in motion. This blade-shaped sheet of air in motion can be rendered visible by sending a stream of air impregnated with smoke or clouds of sal ammoniac through the mouth of a pipe from which the pipe itself is removed, such as is commonly found among physical apparatus. "Any blade-shaped gas flame which comes from a split burner is also an example of a similar process. Burning renders visible the limits between the outpouring sheet of gas and the atmosphere. But the flame does not render the continuance of the stream visible. Now the blade- shaped sheet of air at the mouth of the organ pipe is wafted to one side or the other by every stream of air which touches its surface, exactly as this gas flame is. The consequence is that when the oscillation of the mass of air in the pipe caiTses the air to enter through the ends of the pipe, the blade-shaped stream of air arising from the mouth is also inclined inwards, and draws its whole mass of air into the pipe. During the opposite phase of vibration, on the other hand, when the air leaves the ends of the pipe the whole mass of this blade of air is driven outwards. Hence it happens that exactly at the times when the air in the pipe is most condensed, more air still is driven in from the bellows, whence the condensation, and consequently also the equivalent of work of the vibration of the air is ^ Ellis's Translation, p. 91. 442 SOUND CHAP, viu increased, while at the periods of rarefaction in the pipe the wind of the bellows pours its mass of air into the open space in front of the pipe." 369. The above accounts for the maintenance of the vibrations when once started, but scarcely explains their initiation. Experiments have been made with a blade- shaped sheet of air directed (1) wholly without the pipe, and (2) wholly within. And it has been shown that in either case the pipe will not spontaneously speak. If, however, by a puff of wind the direction of the sheet of air were changed into its usual path, then the pipe commenced and continued to speak. The phenomena of initiation and the theory of the exact adjustment of the mouth that ensures ready speech seem to be still obscure. 370. As to the dependence of quality of tone upon the special arrangements at the mouth, it must be remembered that the more sudden or discontinuous is the action of the blade-shaped stream of air, the more important are the higher components into which its periodic motion may be analysed by Fourier's theorem. And, obviously, this increase of the higher components of the impressed forces produces a corresponding increase in the upper partials elicited from the pipe. Thus Helmholtz says,^ " Wooden pipes do not produce such a cutting windrush as metal pipes. Wooden sides also do not resist the agitation of the waves of sound so well as metal ones, and hence the vibrations of higher pitch seem to be destroyed by friction. For these reasons wood gives a softer, but duller, less penetrating quaUty of tone than metal." This extract also brings into notice yet another factor to be considered as to the quality of tone obtainable from a given pipe. 371. We may now sum up as follows : — The relative intensities of the various partials present in the compound tone of an organ pipe depend upon — ^ Ellis's Translation, p. 94. 369-373 MUSICAL INSTEUMENTS 443 1. The relative intensities of the corresponding com- ponents into which, by Fourier's theorem, the periodic impressed force may be analysed. 2. The closeness of tuning between these strictly commensurate components of the force and the tones natural to the pipe ; and 3. The possible more rapid diminution of intensity of the higher partials due to friction on the interior of the pipe. The partials actually present in various pipes as determined by Helmholtz are shown in Table XXX. 372. It is also noteworthy that D. J. Blaikley finds that the pitch of the note emitted by a pipe when blown at practical pressures is a few vibrations per second sharper than its natural note of strongest resonance as determined by a tuning-fork. By over-blowing organ pipes the pitch jumps somewhat suddenly to the next higher partial. This is, of course, avoided in the organ itself, each pipe being adjusted at the mouth, or " voiced " as it is termed, so as to speak freely its proper tone at the specified pressure at which the wind is to be supplied to it in the instrument. 373. Applying our usual classification to a flue pipe, we see that the blast of wind is the exciter as before, but the blade-shaped sheet of air and the column of air in the pipe had better be regarded together as the vibrating system. The blade-shaped sheet of air might be called the vibrator, but it is not of definite pitch, and is not by deliberate intention moved at the outset in any predetermined way. The settlement of the pitch lies almost solely with the pipe, which therefore can scarcely now be looked upon as a mere resonator to intensify the sound or modify its quality as in the case of the reed pipe. A stopped organ pipe is tuned by adjusting the piece which stops the upper end. An open pipe is tuned by moving a sliding ring or by adjusting a lead piece which shades the upper end. 444 SOUND CHAP, vm 374. The Flute Family comprises those well-known instruments, fifes and flutes of various pitches, and that smallest, highest-pitched kind termed a piccolo. In all these instruments we have a cylindrical pipe open at one end and pierced with a special side mouth hole near the other end. They are thus seen to be comparable to the open flue or flute pipes of the organ in the manner of exciting and maintaining the sound; though, in the case of the flutes and piccolo, the lips and chest of the performer replace the corresponding mechanism in the organ. The air blast may be regarded as the exciter, while the vibrating system comprises (1) the blade-shaped stream of air which passes from the player's lips and strikes the sharp edge of the mouth-hole, and (2) the column of air within the cylindrical pipe. The pitch is probably governed and kept steady chiefly by the air column which, by its vibration, causes the stream of air to enter or pass over the mouth hole. The manipulative mechanism for the production of the various notes of the scale consists of holes along the side of the tube. In the very simplest forms these may be reduced to six to be covered by the fingers. But, in all instruments intended to give the full chromatic scale, these six finger holes are supplemented by others provided with keys which remain closed except when the keys are pressed. 375. We have already seen from theory that an open cylindrical pipe gives the full retinue of harmonic partials. But the limit to which this retinue extends upwards and the prominence of the higher tones depends, as we have also seen, upon the width of the tube and other circum- stances. ]SJ"ow, in the case of the flute, Helmholtz has shown that the overtones are very few and feeble. In fact, the lower notes of the flute when gently Bounded are almost devoid of overtones. Thus, a very good imitation of such a flute tone is obtained from two tuning-forks, the higher being an octave above the other and very feebly sounded. It is well known that the quality of tone of the flute is 374-376 MUSICAL mSTEUMENTS 445 sweet and pure, and in the lower register rather dull. This accords with Helmholtz's analysis of the flute's quality, and the fact that when few and weak overtones are present the quality is the opposite of strident and penetrating. The highest notes of the flute are, however, very bright and pleasing. The piccolo is valuable for its power to emit such high notes with considerable power. Its highest notes are, however, harsh and tearing, and cannot well be endured except with a powerful accompaniment. 376. The compass of flute and piccolo are shown in Table XXIV., and the overtones of the flute in Table XXX. In the former the three notes, d', d", and d'" are indicated as open notes. These are the prime, its octave, and double octave respectively, and are obtained with but slight modifications of fingering. We may, accordingly, regard the full chromatic scale as being built upon three of the tones natural to the whole tube or certain parts of it. Thus, beginning with d', we have the prime of the tube. Ascend- ing note by note we have the primes of portions of the tube shorter and shorter as more and more of the side holes are opened. But, for producing d", we have the same length of tube in use as for d', but the octave or second natural tone is now produced instead of the prime. The utterance of this higher note is favoured by opening the first hole of the six (that nearest the mouth hole), whereas for the prime all six holes were closed. We cannot profess to give any exact theory of the use of the holes and the pitches of the resulting notes. Indeed, the flute affords an excellent example of a case in which practice has outstripped theory. It is stated that the holes were originally made in the positions convenient for the fingers. Afterwards they were made in the positions and of the sizes found desirable for the notes to be produced. And the exact position and size appears still to be settled by trial and error. 446 SOUND CHAP, vm 377. The Oboe and Bassoon. — The family now to be dealt with consists of instruments characterised by double cane reeds and conical tubes terminated by bells. It com- prises the oboe, the English horn (often called by its French name, Cor Anglais), the bassoon and the double bassoon. Of these the oboe and bassoon are most used. The relation of the four as regards compass is exhibited in Table XXIV. The double cane reed vibrates transversely as well as longitudinally. In the course of this vibration the aperture at the end is alternately opened and closed. When open, the sides of the reed are curved, concave inwards ; when closed, the two sides are straight and in contact. Thus, the player's breath, alternately passing and checked by the reed, forms the exciter. The vibrating reed and the column of air in the conical tube form the vibrating system. The pitch is settled by the tube, and is much lower than the tones ■ proper to the reed itself. The manipulative mechanism con- sists of side holes and keys. 378. The strict theory of these openings, like that for those of the flute, is a matter for physicists in the futm'e. Their general effect in raising the pitch by either shortening the tube in use or favouring the elicitation of a higher partial is obvious. As to the pitch of the prime and the series of overtones which theory predicts for such an instrument as the oboe, we have seen in art. 186 that the tones proper to a conical tube closed at one end is the full harmonic series, as for a parallel tube of the same length open at each end, the lengths being the "corrected lengths" in each case. Helmholtz has shown that, when a reed is placed at the vertex of a cone, as in this case, then that end must be regarded as a closed end ; for when it is not closed it opens to admit air into the tube. Hence, as regards the power of the end to resist pressure, it is either closed and does not yield, or when open, air enters instead of being permitted to escape. Thus, the oboe by overblowing gives the octave, twelfth, 377-380 MUSICAL INSTEUMENTS 447 and double octave of its prime. Some of these natural or open tones are utilised on the instrument. The other notes required to fill the gaps and complete the full chromatic scale are obtained by the use of the holes aud keys. 379. The quality of tone obtained from the oboe is somewhat penetrating. It would appear, therefore, to con- sist of an extended retinue of partials. And this Helmholtz has found to be the case, see Table XXX. The bassoon, though differing greatly in appearance from the oboe, is, from the scientific point of view, simply a bass oboe. Because of its much greater length the tube of the bassoon is, for convenience, doubled on itself Hence the very different appearance of the instrument. For con- venience, also, the reed is fitted on a tube curved sideways. It is thus seen that the changes are in form only in order to adapt the bassoon to the performer, all the essentials being as in the oboe. 380. The Clarinet Family. — There are many clarinets m use similar to each other, but of different pitches. Thus, besides the clarinet proper in a variety of pitches, we have also the alto and bass clarinets. The basset-horn is like the alto clarinet, but has a small brass bell mouth. The compasses of the chief clarinets are shown in Table XXIV. All these instruments are characterised by their single cane reed and cylindrical pipe with a small bell mouth. Thus, as with the oboe and bassoon, we may regard the air blast produced by the performer as the exciter, the reed and the air column forming the vibrating system giving a note whose pitch depends upon the length of the column, and considerably lower than those proper to the reed. The manipulative mechanism consists of the side holes and keys. But since we have here a cylindrical pipe with a reed alternately opening and closing one end, the tones possible to the instrument form the odd harmonic series. The effect of this is twofold. Thus first, by overblowing, the first overtone obtained is a twelfth above the prime, and not 448 SOUND CHAP. Tni an octave as iu the case of the oboe and flute. Hence a sufficient number of side holes and keys must be provided to bridge this very large gap. Secondly, the quality of the tone produced is that characterised by the enfeeblement or extinction of the evenly numbered partials. It has been shown by D. J. Blaikley that some of the even partials are feebly present. The results of his experiments are shown in Table XXX. This is, however, no refutation of theory; for the clarinet, as Mr. Blaikley points out, is not cylindrical throughout ; there is the bell mouth at the lower end, and a slight constriction near the mouthpiece or upper end. The characteristic quality of tone is no doubt also de- pendent to some extent on the form of reed used. This is a beating reed, applied to the sloping end of the mouth- piece which forms a table for it. 381. Brass Instruments. — In treating of brass instru- ments it will be convenient to take first a general survey and note the features common to all, leaving to later articles the more detailed examination of several important types. Thus, in the first place, all have cupped mouth- pieces to which the player's lips are applied. Secondly, the tube is quasi-conical, or, more strictly, hyperbolical in shape with a bell mouth. If strictly conical with a closed vertex instead of a mouthpiece, the natural notes would constitute the full harmonic series of the same pitches as for a parallel tube of the same (corrected) length, and open at both ends (see art. 186). And although the vertex is not closed, and the tapering is, in some cases, interfered with by a parallel portion to allow of slides or valves, still the makers so proportion the tapering tube in- other parts as to restore, as nearly as may be, the tuning of the natural or open notes to the harmonic series. The various kinds of manipulative mechanism to bridge over these gaps and provide the chromatic scale differs in different instruments, and will be treated in subsequent articles. 381-384 MUSICAL INSTEUMENTS 449 ' 382. The intermittent stream of air issuing from the player's lips may be regarded as the exciter, the lips themselves constitute a double membranous reed, and are alternately open and closed, thus constituting along with the air column the vibrating system. If we ask what parts are played by the air column and the lips in fixing the pitch, the answer requires a little care. It is the experience of players that the adjustment of the lips to the mouthpiece, the tension of the lips, and the pressure of the air used decide which of the various notes then natural to the instrument shall be elicited. But the exact pitch of that note depends chiefly upon the tube in use, and the temperature of the air within it at the moment. If the tube is small, the note can be forced a little above or below its natural pitch. Thus, on a cornet, it is said that some players can force the note a quarter of a tone flatter or sharper than its normal pitch. With the larger instru- ments this is not so easy. 383. Helmholtz has pointed out that membranous reeds circumstanced as the human lips are in playing brass instruments produce tones which are always sharper than their pitch if isolated. He classifies the lips in this case as membranous reeds striking outwards. It should be noticed that the opposite change of pitch occurs in the ca.se of reeds which Helmholtz classifies as striking inwards. We have already seen illustrations of these in the case of the oboe and bassoon and the clarinet family. 384. We have seen that the tones natural to all brass instruments approximate closely to the full harmonic series. But the closeness of that approximation differs in different types of instruments. And this difference affects, not only the intonation of the instrument when the different notes are sounded separately, but also the quality of each such note. For, when any note is excited and maintained by the player, its various simple tones must be of commensurate periods, i.e. form a harmonic series. Now, if some of the 2 a 450 SOUND CHAP, viii tones natural to the instrument are very nearly in tune with the corresponding tones of this harmonic series, they will be strongly elicited. If, however, some of the other tones natural to the instrument are considerably out of tune with the corresponding tones of the harmonic series, these will be but feebly elicited. In a badly made instru- ment, the tones strongly and feebly elicited might occur in any erratic fashion. But, in a carefully formed instru- ment, the tones more and more removed from the prime will be less and less strongly elicited. And this falling off should follow a regular law for all, though differing in each type. Hence the distinction between good instruments of various types probably lies chiefly in the extent upwards of the retinue of overtones sounding along with their prime. Or, in other words, the distinction may be said to depend upon the degree of prominence or subordination of the upper partials. 385. As to the cause of the greater or less prominence of these partials, it is perhaps to be sought mainly in the relation of the diameter of the tubing to its length, but also in part in the proportions of the bell. Much depends, too, on the shape of the mouthpiece itself. For an un- suitable mouthpiece spoils the tone of a good instrument. If the diameters of the tubing and bell are small in com- parison with the length, as in the trumpet and trombone, the higher tones natural to the instrument are brought well into tune, and so are more strongly elicited. We thus have, from such instruments, an extended retinue of partials and a brilliant tone. This is also assisted by the somewhat shallow cup-shaped mouthpieces used. With deeper mouth- pieces and either much wider tubing or larger bell, as in the euphonium and French horn respectively, we have the higher overtones less prominent and a softer quality results. 386. Mechanism for the Scale. — On some brass instru- ments no mechanism is provided for completing the scale, which is accordingly limited to the natural or open notes 385-388 MUSICAL INSTEUMENTS 451 of the harmonic series. Examples of this class are afforded by the coach-horn, the bugle, and some trumpets. Thus, calls or fanfares performed on these instruments have a special character of their own owing to this limitation of scale, apart from the tone quality characteristic of each instrument. 387. The French Horn without valves comes next in order of simplicity. The tube of this instrument is coiled in a circular form, and is played with the left hand near the mouthpiece, the right hand being inserted in the bell, which is very large. When the natural or open notes of the instrument are required, the right hand is laid with the tips of the fingers so applied to the interior of the bell as not to materially obstruct the passage of the air. When, however, it is desired to obtain notes somewhat flatter than the open notes, the hand is placed so as to partially close the bell. This partial closing of the bell has a double effect. In addition to flattening the pitch it also deadens or muffles the tone. This latter effect may be at times of great artistic value. Sometimes, however, it is undesir- able. This flattening of the pitch by the fist in the bell may be produced at the option of the player to the extent of a quarter of a tone, a semitone, three-quarters of a tone, a whole tone, or even more. 388. It might at first sight be supposed that even then only a poor approach to a complete scale could be obtained. It must, however, be noted that the French horn has a great length of tubing, but is played with a small mouthpiece. Hence the notes chiefly used are those in the higher part of the harmonic series; they consequently lie close together. Thus the slight flattening possible by closing the bell suffices to completely bridge some of these small gaps. This may be seen on reference to Table XXIV., which shows the first sixteen open notes for the French horn in F. The fundamental (Fj) is shown in brackets, as it is scarcely obtainable with the ordinary mouthpiece, 45 2 SOUND CHAP. VIII and is rarely if ever used. It should be noted that the 7th, 11th, 13th, and 14th open notes are slightly different from any notes of the staff. They are accordingly only approximately represented by the notes given in the Table XXIV. As indicated by the full line in the table, the complete chromatic scale may be obtained for about an octave and a half, but certain gaps must occur lower down in the compass when only the fist is used in the attempt to bridge them. 389. But, in order to obtain notes in these gaps, and also to bring the more important notes of a given com- position on the open notes of the horn, the instrument is put into different keys. This is done by means of a piece of detachable tubing of the right length, and called a crooh. Thus, by taking off the F-crook and replacing it by another slightly longer one, we can put the horn into the key of E'l-i. Then all the sixteen notes shown in the table would be lowered by a whole tone. It may be urged that all the gaps in the scale are as large as before, which is true. Bu.t still the device is of great value, and very often a composer directs that some horns shall be in one key, and some in another, thus increasing greatly the open notes at his disposal. For the action of the French horn with valves, now so often used, the reader is referred to articles 393-4, where all the valved instruments are treated together. 390. The Trombone is the most familiar example of the next device for bridging the gaps between the open notes, and thus completing the scale. This device consists of the slide, a U-tube which fits upon corresponding straight parallel tubes of very slightly smaller bore. When the slide is closed up it is said to be in the' first position. The instrument is then at its normal pitch, and yields the open notes shown in Table XXIV. "When the slide is drawn out a few inches to what is called the second position, sufficient extra tubing is brought into use to put the instrument a semitone lower. We have thus at command 389-392 MUSICAL mSTEUMENTS 453 another full series of open notes like the first, but all a semitone flatter. Hence, what was accomplished with the French horn by changing a crook during a rest in the music, is instantaneously accomplished on the trombone by means of the slide without any cessation of playing. The third position of the slide lowers the pitch a whole tone, the fourth a tone and a half, and so on, to the seventh and last position, which lowers the pitch three whole tones. Now the prime tone (called the pedal) is scarcely ever used. Thus the largest gap to be bridged is a musical fifth, occurring between the second and third open notes. Hence, the flattening extending to three whole tones, obtained by the seven positions of the slide, provides the full chromatic scale in this the largest gap to be dealt with. 391. But the slide in addition to this extends the compass by three whole tones below the lowest open note used (the octave of the pedal), and also gives useful alternative ways of playing some of the higher notes where the open notes are closer, and the gaps between them are consequently more than bridged by the three whole tones of the slide. These points are shown in Table XXV., which gives the scale for slide and valved instruments. Further, the slide has the great advantage of allowing an exact adjustment for each position so as to obtain the desired intonation. It also confers, from its continuous action, the power of gliding from one note to another. 392. But the acrobatic feats required to take in rapid succession distant positions form a drawback to this mechanism which unfits the slide trombone for the execu- tion of rapid passages. It is noteworthy that these two advantages of the slide are possessed also by the violin, which, however, is free from the drawback inherent in the trombone. Indeed, to introduce the drawback of the slide trombone into a member of the violin family we ought to choose from the latter a violoncello, but restrict the instru- 454 SOUND 'iiAP. VIII nient to one string and the player to the use of one finger, but allow him the power of eliciting the harmonics at will without additional fingering. Thus in the violin family the use of several strings and several fingers obviates the difaculties of execution inherent in the slide trombone. Turning again to the advantages of the slide, it is obvious that, as in the case of the violin family, this mechanism exacts from the player higher skill and greater care as a compensation for its higher possibilities. 393. Valved Instruments. — We pass now to the con- sideration of the third and commonest kind of mechanism for producing the scale in brass instruments, namely, valves. Among the brass of the orchestra which usually have valves, we may notice the trumpet with three valves, the French horn with two or three valves, and the bass tuba with three or four valves. The French horn is occasionally fitted with only two valves, because it is chiefly used to sound its higher notes which lie near together, hence the gaps between them are easily bridged. Bass instruments, on the other hand, sometimes have four valves in order to bridge the largest gap occurring between the open notes, namely, that between the prime and its octave. 394. But we may now confine attention chiefly to the normal arrangement of three valves, which applies to the instruments mentioned above, and in addition to nearly all the valved instruments used in a purely brass band. On depressing the piston of any valve an extra piece of tubing is put into use. The valves are referred to as the first, second, and third, and when depressed they flatten the pitch by about a whole tone, a semitone, and a tone and a half respectively. Thus they enable the player to bridge the gap between the second and third open notes of the instrument which are at the interval of a musical fifth. The prime of the instrument, called the pedal, is not usually employed. The mouthpiece is not suitable for producing it, and its quality and intonation would probably 393, 394 MUSICAL mSTEUMENTS 45J be faulty on most instruments. It is thus seen that the use of these three valves (1) bridges the largest gap between the open notes in use, (2) extends the compass downward, and (3) supplies alternative fingerings for some of the higher notes. In these respects the action of the valves is closely analogous to that of the slide of the trombone. This fact is clearly exhibited in Table XXV., which shows the fingering for the scale on instruments with a slide or with three valves. The notes are shown of the pitch of the tenor trombone and euphonium ; by altering them up or down, but preserving the same relation, the table would represent the scales for other slide or valved instruments. Table XXV. — The Scale on Tenor Trombone and Euphonium Order of Open Note in Use. Positions of "1 Slide on V 1st Trombone. J 2nd 3r(l 4th 6th Oth rth Valves de- ^ pressed on V None. Euphonium. J 2 1 1} ^1 3/ iu') (/') (0 («) c F ^1 3' 9 8 6 5 4 3 2 /' d' bb f a e' » ^3 •^ .2 s i*-, pi >> P '-0 ^ s n !-< ►rt s m & &H w Ti s Ti r, P* t3^ ^ d 00 X i X X ,- ' (N ': [ X > X y. 00 o : y^ X "S; X X X o s Tji : X : X X >X X X - CO CO : : X >i Sx x x: « to 00 i-H « y. X >^ X ■-^x] x >< s k ^ '■ '■ >'^ X "" X --xxx s i IXXXXXX-.XXX w ;* iz; paddo:}g ^ ^ aodg •UMJO JO Er O a o 411,412 MUSICAL INSTRUMENTS 469 411. ExPT. 62a. Synthesis of Musical Tones by Forks. — The .facts of Table XXX. receive very valuable illustration if the respective instruments and performers are available, and also a set of forks tuned to the relative frequencies of the first eight partials. It is convenient to have the prime fork tuned to c (128), the eighth will then be c'" (1024). The seventh must be specially made the true trumpet-seventh, which is flatter than the ordinary 5"b. Its frequency is, of course, 896 if the prime tone c is exactly 128. These forks should be each mounted upon suitable resonance - boxes arranged in order before the operator. Then the modus operandi is as follows : — Having decided the instrument to be imitated and the pitch of the note, sound gently the fork to be used as prime, and let the performer tune his instrument to correspond. Next, beginning with the prime, sound loudly the combination of forks to build up the desired tone quality and with correct relative intensities according to Table XXX. Finally, let the performer sound the same note on his instrument. With eight such forks carefully bowed one obtains ex- tremely near imitations of the quality of tone of the flute, clarinet, euphonium, and trombone, which make a striking ap- peal to an audience. Eight forks are, however, insufficient to imitate the specially penetrating quality of the oboe. The tone of the clarinet can also be well imitated by the longitudinal vibrations of a wire about ten metres long stroked with a rosined cloth at or near the middle, so as to practically eliminate the evenly numbered partials. It is obvious that the experiment with the forks is open to the objection that the sounds of the higher forks die away quicker than those of the lower. Hence the relation of the intensities changes, and the tone consequently changes in quality. To test his theory and illustrate it experimentally in a manner free from this objection, Helmholtz performed the synthesis of the vowels by a set of electrically driven forks whose intensities were adjusted to and maintained at the desired values. 412. ExPT. 63. Analysis by Helmholtz Resonators. — The presence of the various partials in the tones of the voice or 470 SOUND CHAP, vm brass instruments can be well shown to an audience by the use of a set of Helmholtz resonators. There should be at least ten in number and mounted above one another. A monometric capsule is needed to each, and a tall rotating mirror, fixed on a vertical axis, near the jets. The whole arrangement is illustrated in catalogues of acoustic apparatus. Suppose the prime tone to be c (128), then a bass voice should be tuned precisely to this pitch as ascertained by blowing across the mouth of the largest resonator. The vowel ah {a as in father, but of pure Italian quality) at this pitch should then be sung forcibly and near the mouth of each resonator in turn while the mirror is rotated. It will be found that all the ten resonators pick out their respective partials for the tone produced by the voice. For lecture illustration of the presence of the partials in the tone of a brass instrument the trombone is most suitable. For, by means of the slide it can be instantly tuned to the right pitch as ascertained by another operator blowing across the mouth of the lowest resonator. 413. ExPT. 64. Analysis of Vowels ly Flames. — The fact that vowel quality is a thing apart from the qualities of tone of ordinary musical instruments may be well shown by means of the manometric flame and rotating mirror (see art. 177). For this purpose it is well, for the sake of con- trast, to have also some bell-mouthed instrument, say a cornet. On rotating the mirror and playing the scale or other passage on the cornet with its bell towards the manometric capsule, the flame images are seen to remain of the same pattern throughout, changing only in fineness as the frequency changes. When, however, any vowel is sung at different pitches in front of the capsule, it is noticed that the flame images usually change very much in type or pattern as well as fineness as the frequency changes. All the chief vowels, as shown in the vowel triangle (art. 403), should be used. The contrast between the effects of some of them and those of the instrument is in certain cases very striking. For this purpose the flame may be produced by acetylene instead of ordinary gas. By this means Mr. Merritt suc- ceeded in obtaining an effect bright enough to photograph. Another method of analysing vowels or detecting differences in different vowels is that of the specially 413-415 MUSICAL INSTEUMENTS 471 long sensitive flame used by the late Prof. Tyndal, and called by him the vowel flame (see art. 280). 414. ExPT. 65. Speech analysed hy the Flwnoscope. — By means of the phonoscope already described (arts. 250 and 251) we may illustrate to an audience the different characters of the vibrations composing various vowels and even con- sonants. For this purpose the diaphragm must be very thin and rather slack, and the whole arrangement of mirror and connections light and free. When these details have been sufficiently attended to we obtain a very instructive illustration of the vibrations in question. All the chief vowels are well responded to by this arrangement, and such forcible consonants as p and r (if well trilled) are distinctly rendered on the screen. The words papa and father, uttered with a declamatory vigour, will be found particularly suitable. 415. ExPT. 66. Analysis and Synthesis by Piano. — A valuable experiment on the analysis and synthesis of sounds may be performed with a piano (preferably a grand piano). The front or top must be removed so as to lay bare the wires. Also all the dampers must be raised by depressing the loud pedal. Next choose a note on the piano of a suitable pitch for vigorous singing. Finally, sing to this pitch a full and well -sustained tone of a distinct vowel quality. When the sound of the voice ceases, the piano is then found to be giving forth a tone of the same pitch and of approximately the same vowel quality. Here, then, we have a case of analysis, resonance, and synthesis. For, first, the strings in tune with the various components of the tone sung, perform, as it were, the analysis, and responding to those vibrational components incident upon them give us the phenomenon of resonance. Then, on the cessation of the voice, all these tones, of approximately the right pitches and relative intensities, are blended, and produce upon the ear a very good imitation of the original vowel quality sung. All the chief vowels should be tried. It will be found that some succeed better than others. The better the piano, the better the result. For some years Prof. M'Kendrick of Glasgow has worked upon the subject of speech. His work will be introduced in Chap. XI. CHAPTEE IX CONSONANCE AND TEMPERAMENT 416. Discord due to Beats. — The title of this chapter is to be understood broadly. Thus, we shall treat here of the various degrees of harmoniousness and dissonance present in different combinations of two or more tones, from the purest concords to the harshest discords. We shall afterwards inquire into the reasons which have led to a very general departure from just intonation, and shall then notice several important systems of approximate intonation or temperament, two of which have been widely adopted. Before the Christian era the Greeks had studied musical concord and discord. They knew that the notes which produced consonance could be obtained from various lengths of a given string, and that between these lengths a very simple relation existed. This relation could be expressed by small whole numbers, as 1 to 2, 2 to 3, and so forth. Lengths in the relation of 64 to 81, on the other hand, produced dissonance. 417. Then Pythagoras propounded his celebrated riddle, "Why is consonance determined by the ratios of small whole numbers ? " For over two thousand years that enigma remained unsolved. But in 1862, Helmholtz published the result of eight years' work on acoustics. This gave a masterly insight into the whole question, and 472 416-419 CONSONANCE AND TEMPERAMENT 473 resulted in the enunciation of Helmholtz's theory of con- cord and discord. Helmholtz showed that all dissonances or discords are due to unpleasant beats generated by the component notes. Consonances or concords, on the other hand, are formed by notes which fail to produce such beats. That is the theory in its bare outline. Many details must be supplied to define what constitutes the unpleasantness of the beats and to show how such beats may be produced. 418. First, let us ask why should beats be unpleasant in any case. Helmholtz's answer to this question is some- what as follows : — During the loudest phase of the beats the ear is fatigued somewhat, but during the feeblest phase it is rested, and its sensibility restored. Hence, in this specially sensitive state the recurrence of the loud phase may be distressing. This effect on the ear Helmholtz likens to the effect upon the eye if one walks near a tall palisade when the sun is shining through from the other side, the flickering light being at times very irritating. Another factor required to make the phenomenon of beats unpleasing, is that they should succeed each other at a frequency between certain limits. This holds good for the analogous case of flickering light also. The phenomena of fatigue of the eye and the irritation due to flickering at a certain rate may be illustrated to an audience by the following experiments : — 419. ExPT. 67. Fatigue of the Ejje. — Let a sharply defined object in black or vivid colours be projected by the arc-light lantern upon a white screen in a dark room. A grotesque image in vermilion is particularly suitable. Let the eyes of the audience be intently fixed upon some point image for about twenty seconds, the time being counted audibly by the demonstrator. Then let the slide be removed so that the screen becomes pure white, the eyes being still fixed upon the same point of the screen as before (secured by the presence of a small cross at the point in question). Then an image, in shape like the first, but complementary in illumination and colour, soon appears on the screen. This 474 SOUND CHAP. IX shows that the parts of the eyes concerned in receiving the impressions of the original image and screen respectively become sufficiently fatigued by the process as to be unable to give equally vivid impressions of the various parts of the screen when entirely white. Thus, if the original image were black and the screen white, the parts of the eyes concerned with the white screen would become fatigued, and afterwards show a duller screen than that shown by those parts of the eye corresponding to the black image and which were accordingly unfatigued. In other words, the after image is in this case white upon a duller ground. If the original image were red, the after image would be green. 420. ExPT. 68. Irritation due to Flickering. — Introduce in front of the projection lantern an opaque disc having a number of radial openings, so that by rotating it the screen is alternately illuminated and darkened. Start with the screen illuminated, and then rotate the disc slowly ; the effect is not unpleasant. Rotate the disc quicker and quicker until the effect of a steady, uniform illumination is obtained ; this again is not unpleasant. But before this very high speed was reached there was an intermediate rate of alternate illumination and darkening of the screen which was par- ticularly irritating. This rate may be again reached and maintained on slowing down. If properly done, the effect is so unpleasant as to cause many to avoid gazing at the screen for more than a few seconds at a time. 421. In Helmholfcz'is view the harshest effect of beats in the neighbourhood of c' is obtained when the beats are about 33 per second. But the frequency of the beats for the harshest dissonance varies in different parts of the compass. Another way of putting this is to say that it depends upon the interval between the two notes, for 33 per second corresponds to a semitone at cue part of the compass, but to an octave at another part. Thus, we may say that the roughness arising from the sounding of two notes together depends in a compound manner on the magnitude of the interval between the two notes, and on the frequency of the beats produced by them. 420-422 CONSONANCE AND TEMPEEAMENT 475 ExPT. 69. Beats by Helmholtz's Siren. — We may illustrate the various degrees of roughness due to beats by the double siren of Helmholtz (see art. 296). Thus, opening circles of twelve holes in each chest and setting the siren rotating, the air blast being also turned on, we have unison. Then, on rotating the handle which turns the upper chest, we have a "mistuned unison" and beats produced. Now, just as in the case of the experiment with flickering light, it will be found that the beats are not unpleasant either when very slow or very quick, but that between certain limits they are distinctly disagreeable. Moreover, by having the siren driven at different speeds, it may be shown that the frequency of the beats must be correspondingly modified to give the roughest effect. 422. ExPT. 70. Beats on Ellis's Harmonical. — The eflFect of beats of the same frequency, but produced by notes at different intervals, can be very well illustrated on a specially tuned harmonium designed by Mr. A. J. Ellis, and by him termed the "harmonical." It has five octaves, of which the highest has very special tuning (see art. 465). The other four octaves have the white keys tuned to just intonation, the middle C (or c') being 264 per second. Of the black keys in these octaves, Eb, Ab, and Bb are such as make a\>, c, e'b, and eb, g, Vo perfectly just major chords {i.e. with frequencies as 4:5:6). To the Db digital is assigned the note which Ellis calls grave D, i.e. a note a comma (81/80) flatter than D. Thus, the grave D is only the interval 10/9 above C, whereas the D on the white digital is 9/8 above C. The F| digital is also specially tuned, being. ''Bb as Ellis calls it. Thus, the relative frequencies of the notes c, e, g and that on the /| digital are 4, 5, 6, and 7. Having explained so much of the special tuning of the harmonical, it will be seen that the following pairs of notes all differ by 33 vibrations per second. Hence, each pair gives beats of 33 per second, but the effects of the various pairs are very different, since their intervals range from a semitone to the interval of a fifth. [Table 476 SOUND CHAP. IX Table XXXI.— Notes giving same Beats ( ({' e' d' g G c G Pairs of Notes on EUis'h I ,, ^' ' r C C Hiirmonical, all givingJ ^ a c e C yj yj beats of 33 per second. 1 (264) Intervals and Batio of/semitone. ^^'i™^ '^'X ThM. Si ^°^'^^- ^''^^■ Ftequencies. | ^^^^^ ^^^^ ^^^ g^^ 5^^ ^^3 g^^ With the tuning usual in an ordinary harmonium, piano, or organ, the above effects may be approximately obtained by using the same notes. But the examination of the point at issue is in that case disturbed, since the frequencies of the beats are only approximately constant. 423. Various Ways of producing Beats. — In the preceding illustrations on the harmonical the frequency of the beats mentioned was that of the beats between the primes of the notes in question, as they would be the beats most noticeable. But it is easy to see that beats may occur in other ways. For a note of musical quality is a compound tone or a retinue of partials, and between any simple tone of one note and any one of the other, beats may occur. Further, as we have seen (in Chap. VII.), tones may, in certain cases, give rise to other tones called differential and summational tones. And between these and any of the components of the parent tones, beats may arise. But into all the conceivable intricacies we must not enter. "We shall notice only the following more important cases : viz. beats arising between (1) two primes, (2) a prime and a differential, (3) a prime and an upper partial, and (4) two upper partials. We shall first illustrate these various ways by experiments, and then pass to the conclusions Helmholtz has drawn from a consideration of these various causes of dissonance. 424. ExPT. 71. Beats between Primes. — To illustrate the loudness of beats between two primes, use two large forks precisely similar and mounted on resonance-boxes. Make one 423-426 CONSONANCE AND TEMPERAMENT 477 a little flatter by a piece of wax near the tip of one prong (or of both). Then on bowing the two forks the beats will be very plainly heard. It is well to have the forks of pitch c' (256). In this experiment we may safely conclude that the beats heard are between the primes, for, unless the amplitude is very large the octave is not appreciably pro- duced, and if it were it would be overmasked by the resonance-box responding to the prime. 425. ExPT. 72. Beats Mween Prime and Differential. — For this purpose use forks an octave apart, say c' (256) and c" (512), taken off their resonance-boxes. Plug the hole up on the c' resonance-box, and set it mouth up on the table, or use a water resonator set to respond to c' (256). Flatten one of the forks with wax, bow both and hold over the resonator, when beats will be distinctly audible throughout an ordinary lecture-room. We may explain how these beats are produced as follows : — First, Let the higher fork have the wax on, and suppose its frequency is thereby reduced by two vibrations per second from 512 to 510. (Verify this by another unaltered c" fork, if available, both being bowed and held over a c" resonator.) Then by the tones of the flattened c" fork (510) and the c' fork (256) we have a differential tone generated of frequency 510-256 = 254; and this differential tone gives two beats per second with the c' fork (256 - 254 = 2). Secondly, Let the lower fork be flattened with wax by two vibrations per second ; it has therefore a frequency of 254 instead of 256. (Verify again if possible by an unaltered c' fork.) We thus have tones from the two forks of frequencies 512 and 254. These give a differential tone of 258, which, with the 254 fork, yields foiir beats per second. This doubled rate of beats is a very striking confirmation of theoretical expectation. 426. ExPT. 73. Absence of Beats between Fork and Mistuned Major Third. — Taking two simple tones nearly at the interval 4 : 5, let us examine what might be expected as to the formation of beats. Suppose the tones are originally c' (256) and e (320), and that the c' be flattened to 254, then between the 254 and the 320 we have a differential of 66, but nothing for it to beat with. If we pursue the formation of differentials of higher orders between the differentials themselves and the primes, we may, however, obtain beats as shown by Helmholtz. 478 SOUND CHAP. IX Thus 320 - 254= 66, Differential Tone of the first order. 254- 66 = 188 „ „ „ second,, 320-188 = 132 „ „ „ third „ 254-132 = 122 „ „ „ fourth „ And between the differentials of the third and fourth orders we have beats of ten per second (132-122 = 10). Scheibler claims to have heard beats between forks at intervals slightly differing from a major third, and Helmholtz believes he has heard them, but says that they are not of any importance in distinguishing consonances from dissonances. This statement can easily be confirmed by experimenting with the forks in the way mentioned above. It is specially striking to alternate this experiment with the previous one. Suppose the d fork to be flattened two vibrations per second, it gives so plainly two beats per second with another c' fork, and four beats per second with the c" fork. Whereas with the e fork, the beats if audible at all in this mistuned major third, are extremely feeble in comparison with those of the mistuned octave. In the case of the mistuned major third, the writer, using good forks, could not detect any beats, whereas an organ builder present thought he could. Lord Kelvin has made experiments on the "Beats of Imperfect Harmonies," * and claims that to hear beats of harmonies, other than the octave and fifth, is not so difficult as Helmholtz supposed. Thus it appears that the whole matter is involved in much difficulty, and at the present stage it seems unsafe to dogmatise. 427. ExPT. 74. Beats behoeen Prime and Upper Partial. — The beats in this case are perhaps best shown by an imperfect octave on the harmonical. Use, for example, the grave D in one octave, and the ordinary D in the next octave above or below. Another pair of notes that also shows these beats is the ordinary and special forms of Bb taken an octave apart. In either case the first upper partial of the lower note beats with the prime of the upper note, which is not exactly the octave of the lower. Of course it might be urged that these beats are due to the differential tones, as in the experiment No. 72, intended to illustrate that effect. And it must be admitted that the beats are of the same frequency, so cannot be distinguished. Probably the beats ' I'ruc, Roy. Soc. Edin., vol. ix. p. 602, 1878. 427, 428 CONSOlSrANCE AND TEMPEEAMEJSTT 479 heard are due to both causes. But the cause now under notice is fairly certain to be paramount. The effect may be illustrated, of course, upon any piano, harmonium, or organ which is out of tune, or very effectively by sounding a mistuned octave on the violin. It may also be noted here that F. Lindig ^ has shown that on sounding two tuning-forks at a mistuned octave beats occur between the higher fork and the asymmetry octave of the lower. 428. ExPT. 75. Beats between Two Upper Pariials. — For this purpose it is best to choose the interval of a mistuned fifth. For with a perfect fifth the primes are in the ratio 2 : 3, hence the third partial of the lower note is exactly in unison with the second of the upper. But, if the fifth is a little out of tune, it follows that the two upper partials in question are not exactly in unison, but produce beats. And these partials being so near their respective primes, are usually fairly strong, and consequently give the desired effect in a mai'ked degree. Now on the harmonical we have a mistuned fifth ready to hand between the D and the A, and between the d and the a. For the following major chords are in exact tune : c, e, g; /, a, c' ; and g, I, d ; their relative frequencies being 72 :90 . 108; 96 : 120 : 144; and 108 : 135 : 162. Also ci and d' make a perfect octave (81 : 162). Thus d and a (81 : 120) are a comma (81/80) less than a fifth apart, and give in consequence gritty beats. It requires the grave d (80) to make a perfect fifth with the a (120), and these notes should be found in perfect tune. The difference made in the harmony by the interval of a comma is very striking, yet, melodically considered, this interval is very small. The beats of a mistuned fifth may also be heard with any fifth on the piano, harmonium, or organ, as ordinarily tuned. They may also, of course, be obtained by mistuned fifths on the violin. The beats between two upper partials may also be 1 Ann. d. Pliysil; 11. 1, pp. 81-53, April 1903 ; Krience Abstracts, p. 168, 1904. 480 SOUND CHAP. IX illustrated on the harmonical by a mistuned fourth. For, although on this instrument the ordinary Vf> makes a perfect minor third with g, it is too sharp by a comma to make a perfect fourth with/. Thus, the relative frequencies of /, g, and h\> are respectively 120, 135, and 162 ; hence the minor third, 162/135 = 6/5, is true, but the fourth, which is 162/120 = (81/80) x (4/3), is a comma too great. On the other hand, the special ib provided on the /f digital is too flat to make a fourth with/. Thus, either of these notes for h\> will give beats with /. The beats heard are between the fourth partial of the lower note (J) and the third partial of the higher note (h\> or '^Jb). Again, we can obtain on the harmonical, beats between the upper partials of a mistuned major third. Take the notes e and ab, these are respectively perfect major thirds above c and below c. But the interval between these two notes exceeds a major third by about a fifth of a tone. For the relative frequencies of c, e, «b, and c' may be denoted by 20, 25, 32, and 40, whence the first and third intervals are 25/20 and 40/32, and each equals 5/4, a true major third ; but the second interval is 32/25 = 128/100 = (128/125) X (5/4), and thus exceeds the just major third by the interval 128/125, which equals 42 of Ellis's logarithmic cents. Thus, the notes in question on the harmonical yield very distinct beats, namely, between the fifth partial of the e and the fourth partial of the a\>. This case is important in contrast with the absence (or faintness) of beats between forks at the interval of a mistuned thii'd (see article 426). 429. Consonance or Dissonance of the chief Intervals. — We have seen that discord is due to beats, and also that beats arise in various ways. Again, these vs^ays depend partly on the tones being pov^erful enough to produce differential tones, or rich enough in upper partials to give appreciable beats between the partials of one and the other note. Hence the degree of roughness due to the presence of beats will depend, not solely on the interval between the notes, but also and to a large extent on the quality of the compound tones in question. But, to simplify matters and keep to what is of chief importance, 429, 430 CONSONANCE AND TEMPEEAMENT 481 we shall usually confine attention to notes (or compound tones) of good musical quality with harmonic partials of gradually decreasing intensity extending up to say the sixth. We shall suppose also that the notes are not powerful enough to make the differentials of great importance, and that the summationals are practically inappreciable. In Table XXXII. is given a view of the chief intervals with the upper partials of each note and the beats arising between them. The differ- entials and summationals are ajso shown, but their power to produce appreciable beats is ignored. The numbers of the beats between upper partials are those for the pitch of the harmonical, for which middle c' is 264 per second. Table XXXII. — Scheme of Various Consonances Numbers of Beats perv second produced by in- I tervals of a Tone — -, and > a Semitone ^, between I Partials ^ Upper Partials * and \ Summationals * / 66 88 88 33 20 -4 66 44 44 44 83 62-8 62-8 26-4 26-4 Prime Tones Differentials * Intervals and tlieir) relative Frequencies j" Minor Minor third sixtli 5:6 6:8 430. Helmholtz has calculated the degree of roughness of all the intervals comprised within two octaves. For this purpose he selected the quality of tone obtained from a good violin. To make the comparison quantitative, he was under the necessity of assuming a law as to change of roughness with number of beats, since the full details of the relation are not known. He chose, however, the simplest expression, which makes the roughness vanish when the frequency of the beats is either zero or infinity, and makes the roughness a maximum for thirty-three beats per second. The results of this calculation for a single 2i 482 SOUND octave are represented in Fig. 93, in which the intervals are plotted as ahscissse and the varying degrees of rough- ness as ordinates. It is seen from this curve that a very slight mistuning of the unison or octave produces a very harsh dissonance, vehereas the consonances of some other intervals (notably the thirds and sixths) are far less impaired by slight mistunings. 431. ExPT. 76. Consonmues and Dissonances of all Intervals within, an Octave. — The facts represented by the curve of Fig. 93 may be very effectively illustrated on a violin if the first or e" string be replaced by an a' string and tuned in unison with the second or usual a string. Then, by d eb e f g ab a lb c Fig. 93. — Degree of Roughness of Intervals within an Octave. bowing the two a! strings together while sliding the finger along the lower half of one of them, the other being mean- while open, we reproduce the effects predicted by Helmholtz and shown by his curve. The great nicety of tuning required to make either the unison or octave tolerable is plainly noticed. In fact, this characteristic of both these intervals is well known to violinists and others. To obtain as strikingly as possible the effects shown in the diagram, it is necessary at first to move the finger very slowly, and take a small range of one or two semitones at a time, the hearers being notified beforehand of the particular range of intervals to be dealt with. When the exact nature of the effect to be noticed is better represented, it is possible to sweep the whole octave with the finger at a single stroke, and follow with the ear almost all the details shown in the curve. 432. ExPT. 77. Consonance of gieen Interval varies with Pitch. — But the consonance or dissonance of two notes 431-433 CONSONANCE AND TEMPEEAMENT 483 depends not only upon their quality and the interval between them, but also upon their position in the range of audible sounds, i.e. on their absolute pitch. This is conveniently shown by sounding notes at the interval of a major third in the bass octaves, and then in the other octaves in succession. It is at once noticed that the given interval sounds much rougher in the bass part gf the compass than in the middle or higher octaves. Indeed, if we go high enough, the interval of a tone is but slightly dissonant ; whereas if we go low enough in the compass, an octave is the only interval that can be tolerated. These effects are preferably shown on the harmonical, but an ordinary harmonium or even piano will do. 433. But when an interval is chosen and the absolute pitch specified, the degree of roughness in some cases depends upon yet another circumstance. This case arises when the two notes composing the interval are assigned to different instruments whose tone qualities are distinctly different. It may then become a matter of moment as to which instrument takes the lower note. This has been pointed out by Mr. T. E. Harris,^ who takes as examples the oboe, with the full harmonic series of partials, and the clarinet with (practically) the odd series only. Perhaps the most striking intervals for these instruments are the fourth, which is better with the oboe below ; and the major third, which is better with the clarinet below. These cases are shown in Table XXXIII. Table XXXIII. — Consonances with Oboe and Clarinet Beats between upper Partials occur at Tones ^, and Semitones ss. *-g -m->ff-m- *^ff: Upper partials $ -,&— Instrunients Oboe Clarinet Clarinet Oboe Oboe Clarinet Clarinet Oboe Intervals Fourth Fourth Major third Major third ^ Handbook of Acoustics. 484 SOUND CHAP. IX Among the upper partials the beats occurring at an interval of a whole tone are indicated by a single dash or bar, those occurring at the interval of a semitone by a double mark. 434. Chords and their Various Positions. — More than two separate compound tones, simultaneously produced, compose what is called a chord. If, with a given note, two others are sounded, each of which is consonant with the first, the consonance or dissonance of the chord will obviously depend chiefly upon the nature of the interval between the two added notes. If these two form a dissonant interval, the chord cannot be free from dissonance. If they form a consonant interval, then the chord will be to a correspond- ing degree consonant. Thus the triads (or chords of three notes) which lie within the compass of an octave and con- sonant are as follows : — 1. C E G. 2. C Eb G. 3. C F A. 4. C F AK 5. C Eb Ab. 6. C E A. Of these, C E G, the major chord of C, and C Eb G, the minor chord of C, serve as the representatives of all other major and minor chords. It is seen that each consists of two thirds, one a major and the other minor. But in the major chord the major third is below, while in the minor chord the minor third is below. The two chords can be simply represented by numbers giving the relative frequencies of their notes: thus 20, 25, 30 represent a major chord, while 20, 24, 30 represent a minor chord. 435. The various chords have been examined by Helmholtz, and also the various positions of the major and minor chords. It is thus found that, on scientific theory, the various positions or inversions of a chord may be arranged in an order of diminishing consonance. Thus Helmholtz gives twelve positions > for tlie major triads, divided into two groups of six each, called respectively the 434-436 CONSONANCE AND TEMPERAMENT 485 most perfect positions and the less perfect positions. These are given in Tables XXXIV. and XXXV. Table XXXIV. The Most Perfect Positions of Major Triads rf ^ r—^. r— d 1 4 5 6 r— S 1 -i r -J W-^-^ S — 1 ^^S— fS rc( (=1 1 za — ^ a iJ ~sl- -m- -J- -si- i 1 & L J b— : ^ — .^lU — p — L C] ■ i — ' 1 — ' — F — ^ _!_-> — : M ' 1 ^ 1 ' =f — ° Table XXXV. The Less Perfect Positions of Major Triads In the first group the combinational tones (represented by crotchets) do not disturb the harmony. In the second group some of the combinational tones are unsuitable, and, while not making the chords dissonant, put them in about the same category as minor chords. The disturbing effect of these combinational tones is indicated in the table by dashes. 436. Of minor triads Helmholtz says that no position can be obtained perfectly free from false combinational tones. The three positions shown in Table XXXVI. he gives as the best positions, the other nine are classed as less perfect. As before, the minims denote the notes of the chord, the crotchets representing the combinational tones. 486 SOUND Table XXXVI. The Most Perfect Positions of Minor Triads Helmholtz views with favour that period of musical history when the simpler and smoother chords were used almost to the exclusion of the more dramatic effects of later times. Writing of that earlier period he has the following fine passage : — " Thus that expression which modern music endeavours to attain by various discords and an abundant introduction of dominant sevenths, was obtained in the school of Palestrina by the much more delicate shading of various inversions and positions of consonant chords. This explains the harmonic usness of these compositions, which are nevertheless full of deep and tender expression, and sound like the songs of angels with hearts affected but undarkened by human grief in their heavenly joy. Of course such pieces of music require fine ears both in singer and hearer, to let the delicate gradation of expression re- ceive its due, now that modern music has accustomed us to modes of expression so much more violent and drastic." We must omit notice of the various dissonant triads and of all chords of more than three notes, as lying rather beyond the scope of the present work. They are dealt with at great length by Helmholtz. 437. Reasons for Temperament. — We are now in a position to examine the number of notes to the octave which should be possessed by an instrument, and the exact tuning of these notes. In making this examination we must bear in mind the legitimate claims of the composer and the audience on the various musical instruments employed, and consider also the convenience of those who 437-439 CONSONANCE AND TEMPERAMENT 487 play the instruments. Each such claim or desideratum will involve certain consequences, and some of these we shall find to be conflicting. Hence, every practical solution of the problem before us is a compromise. These com- promises consist in adopting a limited number of notes to the octave, and, by an approximate tuning called tempera- ment, making them do duty for all the notes needed. 438. The claims may be grouped under three heads : — (1) purity of concords, (2) power to modulate, and (3) practical convenience. Let us deal with them in this order. First, To satisfy the cultivated ear in the case of the simultaneous sounding of two or more notes, the chords of each instrument and combination of instruments should be pure. For the major diatonic scale, it will readily be admitted that this claim involves seven notes to the octave in what is called just intonation. This has already been referred to in the Introduction (art. 8), and is shown also in the present chapter (art. 430). But we may easily show that an eighth note is often needed, namely, the grave D, mentioned in art. 428. Let us now regard the matter from another point of view. In just intonation the whole tones are of two sizes, viz. 204 and 182 cents respectively; and the diatonic semitone, 112 cents, is larger than half of either of these tones. Now c and g are at the interval of a perfect fifth, but d is, & large tone above c, while a is only a small tone above g. Hence the interval d io a in. a justly intoned scale is not a perfect fifth. Thus, if it be required simultaneously to sound the pair d and a, or the minor chord d f a, we need an eighth tone, the grave d, which is a comma (22 cents) flatter than the ordinary d. 439. Second, In order to suit another instrument or a voice, each instrument must have the power to pitch a composition in any one of a number of keys differing by semitones only. Also, for the sake of artistic effect, the instrument must have the power to modulate into all such keys in the course of a composition. Further, in all these 488 SOUND CHAP. IX keys the intonation should be the same, that is, the intervals of the various degrees of the scale in any key into which we pass must be preserved the same as the corresponding intervals of the scale in the natural key of C. And to preserve the purity of the chords as claimed under the first heading, all these scales must be in just intonation. Now, to preserve just intonation during modulation involves the provision of hvo new notes in the octave for each sharp or flat remove. Thus, for the first sharp key, that of G, we need not only F| instead of F, but also the A must be made a comma sharper than for the key of C. For in the new scale beginning with CI the first step from G- to A must be a large tone (like C to D, 204 cents), whereas for the key of C the interval from G to A is only the small tone (182 cents). When these two new notes are pro- vided we have a scale exactly like that in C, except that • every note is a perfect fifth sharper than the corresponding notes of the original scale, i.e. the key-note G is a perfect fifth above C, the second note of the scale ; the new A is a perfect fifth aljove D, the second note of the old scale, etc. Hence, the next sharp remove from G to D will again require tvo new notes for just intonation : these are C| instead of C, and in place of the ordinary E a new E a comma sharper is required. 440. Similarly, if we take the first flat remove from the natural key of C, we require two new notes beyond the ordinary seven in order to preserve just intonation. These are the Bb in place of B, and the grave D a comma flatter than the usual D. These changes give us the key of F with notes in precisely the same relation to each other as those in the key of C. Hence, for the next flat remove, to the scale of Bb, we shall again need two new notes if just intonation is to be preserved. Thus, for the natural key of C, seven sharp keys and seven flat keys, it appears that we require V + (2 X Tj + (2 X 7) = 35 notes to the octave. 440-442 CONSONANCE AND TEMPEKAMENT 489 441. But "this is on the supposition that none of the notes asked for coincide. Hence the question arises, Do any of them coincide ? To answer this query, consider for a moment the keyboard of the piano or organ with twelve digitals to the octave. It is easily seen that on ascending by fifths through seven octaves, twelve steps are taken, and every note in the scale is used in some octave or another. Thus twelve fifths appear to equal seven octaves. Further, on ascending by major thirds, it is found that three major thirds appear to equal one octave. Now, do these relations hold in just intonation ? Certainly not. Thus, take for simplicity's sake the frequency of the starting note, or 100, and ascend by octaves, by fifths, and also by major thirds, and see if any notes arrived at in these various ways ever tally. The intervals are shown in Table XXXVII. Table XXXVII. — Various Intervals in Just Intonation Relative Fkequenoies of Notes. '^Octoves l^"" ^"^ **"' *"'* ^'""' ^^"^ "**"' ^^'*'"' '*'''™Fifths l-^™ ^^° ^^^ '^'^^ ^"'^^ ''^"•^ • ■ 13,218-79 M^-ofThldJlOCl^S 1561 195A • .... 442. It is seen by the even hundreds along the first line, and the odd numbers and fractions in the second, that no coincidence can ever be reached between them. Still less can any correspondence be ever attained between the first and third lines. The fact is the numbers 2 and 3 being prime, progress by octaves and by fifths from a given note can never lead to correspondence. Thus, twelve perfect fifths up gives the interval (f/^ =531,411^4096; whereas seven octaves up gives the interval 2"^, which may be written 524,288-^4096. Hence the discrepancy be- tween them is the small interval 531,4114-524,288 or about 24 cents. Accordingly, if this interval is ignored and the seven octaves divided into twelve equal fifths, all those fifths are mistuned. And, further, since in ascending 490 SOUND CHAP. IX by fifths every white and black note of the acale was used in some octave or other, all those notes differ from just intonation, which must have its fifths perfect. Similarly, since the numbers 5 and 4 are prime, progression by octaves and major thirds can never lead to correspondence if the intervals are maintained exact. 443. Third, For the sake of convenience to all con- cerned a reasonable simplicity is needed in the construction of each instrument, its tuning and keeping in tune, and in writing, reading, and playing the music. This claim is usually interpreted as involving a restriction to about twelve notes in the octave in the case of instruments whose notes are limited by separate mechanism. This includes such instruments as the piano and organ, the stringed instruments with fretted keyboards, and the various keyed and valved wind instruments. Obviously many more notes can be produced by instruments with a continuous adjust- ment of pitch, like the human voice, and instruments of the violin and trombone families. But the higher ideal possible to these instruments cannot be realised when they are used in conjunction with the others whose notes are limited to twelve in the octave. For, obviously, all must play the same scale, and thus the limitations of part become, for the time being, the limitations of the whole. 444. And, even when voices or violins are not fettered by association with instruments of a limited scale and fixed intonation, the difficulties of just intonation are probably greater than scientific writers have usually allowed. Take, for example, the case of a violin with strings tuned to g, d', a', and e". Now suppose a composition begins in C natural and the strings are tuned to a justly-intoned scale of C. Then, if the piece modulate to the key of D, the a' string needs sharpening a comma to make a perfect fifth with d'. Of course this true, slightly-sharpened al could be obtained from the d' string. But, if the use of the open strings were sacrificed, much of the sonority of 443-446 CONSONANCE AND TEMPERAMENT 491 the violin and its power of playing chords would be lost. In fact, the technique of violin -playing would then be greatly altered, and, probably, most players would feel a difficulty in maintaining the pitch true if they were driven to abandon the use of the open strings altogether. For, obviously, they serve as a basis of reckoning after extended passages in which the hand is shifted far up the finger- board into the higher positions. Thus, without the liberty to use the open strings, probably the intonation would be worse than now and not better. 445. Somewhat similar remarks would apply to an instrument like the trombone. By the use of the slide it is true that the pitch can be varied continuously. But take the case of the open notes ; let the instrument make two of these at the interval of a perfect fifth. If the piece to be played were in such a key as to require these two open notes for the key-note and fifth respectively, then they are in perfect tune. But suppose a piece were played in such a key as to bring these open notes on the second and sixth notes of the scale. Then the perfect fifth pro- vided might not be required. Here again, as on the violin, the required interval can be obtained, and with, perhaps, a smaller departure from the ordinary technique than in the analogous case of the violin. But it would still entail a slight sacrifice, for although the note could be forced into tune by the lips, it would suffer, in consequence, some loss of brightness of tone quality. Perhaps the nearest approach to just intonation that we ever realise in musical performances is in glee - singing without instrumental accompaniment. 446. But if the difficulties of just intonation are great in the case of violins and trombones, how much greater are they in instruments with digitals and fixed notes ! Suppose the number of notes to the octave to be increased con- siderably beyond the usual twelve, and the written music correspondingly complicated, then no doubt the difficulties 492 SOUND CHAP. IX of reading and playing could be overcome to some extent. Probably, however, the power of execution would be generally limited to simpler music than that now under- taken. Further, the number of notes to the octave need not be very great if only limited power of modulation were demanded. Whether music of a less iiorid nature, or con- fined within narrower limits of modulation, played in just intonation, would be preferable to very rapid passages and unlimited modvilation, but in tempered intonation, is open to argument. Helmholtz greatly desired to see just intona- tion brought into general use. But did even he fully realise the difficulties in the way of its adoption ? Many plans have been put forward by different authors and at various times with the view of obtaining an intonation practically just. Some of these plans involved over fifty notes to the octave. But, although devised with great ingenuity, they have never come into general use. 447. Hence, desirable as just intonation may be in the abstract, we seem as far as ever from its practical realisation for general musical purposes. Perhaps musicians usually underrate the desirability of just intonation. Scientists, on the other hand, may perhaps often lose sight of the enormous difficulties that lie in the way of its attainment. We must also remember that the right of science to dictate ceases where aesthetics begins. Thus, it may be safe to affirm that discord is due to beats between certain limits of frequency and concord to an entire or comparative absence of such beats. Further, it may be agreed that the intonation must be just for certain pairs of notes simul- taneously sounded to be as free as possible from beats. But in florid music there is scarcely time for the beats to be heard. Further, in some passages it is desirable to have beats occurring only two or three times per second. Indeed, these are purposely introduced on certain stops of the organ. 448. Again, take the case of the flats and sharps. 447-449 CONSONANCE AND TEMPEEAMENT 493 For just intonation it is incontestable that the sharp, occurring between two notes a tone apart, is below the Hat occurring in the same interval. Thus, for G| to make a just major third with E', which is itself a major .third above C, we have an interval ratio from C to G| equal to (5/4) X (5/4) = 25/16 = 125/80. Whereasjor Ab a just major third below the upper C we have for the correspond- ing interval, i.e. from to Ab, the ratio 2 x 4/5 = 8/5 = 128/80. Thus, Ab is sharper than G| by the interval 128/125, which equals 42 cents. Or, to reckon by cents direct, a major third in 386 cents, so from C to G| is double that, or 772 cents, whereas from C toAb=1200-386 = 814 cents, or Ab is 42 cents sharper than G|. But suppose now these notes, G| or Ab, form a chromatic passage written for a single instrument. If descending, for the sake of avoiding needless accidentals, it would be written A Ab G-, and if ascending G- G| A. But in either case, if the instrument has the power of distinguishing ■ between G^ and Ab, who knows what the composer wants ? And if, in such cases, musicians sometimes demand that the G| should be sharper than Ab (or that B should be nearer to c than a diatonic semitone), has the physicist any right to complain ? Surely the requirements here are matters of eesthetics, dependent upon the passage in question and the musical history and cultivation of the writer or hearer. Also the notation employed is, in many cases, purely a matter of convenience ; and into this question the physicist, as such, has no title to enter. 449. As to the difficulty of tuning and keeping in tune, if just intonation is aimed at, the change of pitch with temperature should be borne in mind. This temperature variation for a number of instruments was given in the last chapter. Erom this it will be seen that, apart from the changes in a few weeks or months that cause instruments to need retuning, there is the more serious obstacle of this continuous variation which occurs as the 494 SOUND CHAP. IX room warms in the course of a single evening and affects different instruments to different degrees. 450. Granting, then, that we are usually almost driven to adopt comparatively few notes in an octave to represent the infinite number really needed, we shall presently con- sider in some^ detail two of the practical solutions that have attained a considerable vogue, namely, the mean -tone temperament and the equal temperament. The mean-tone temperament obtains a just major third, but at the expense of power to modulate (though this power may be increased somewhat by additional notes in each octave). The equal temperament has perfect power of modulation, but obtains it at the expense of fiat thirds and sixths. Before, however, proceeding to these, we must say a little about the Pytha- gorean tuning or temperament, because it paves the way for the other, and our present staff notation still retains symbols to express its notes. And afterwards, as a specimen of a more ambitious system, we shall notice Bosanquet's cycle of fifty-three notes to the octave. 451. The Pythagorean Tuning. — In this ancient theoretical system all the notes used may be expressed by ascending in a series of twenty-six perfect fifths. These are represented in Table XXXVIII. Table XXXVIII Pythagorean Tuning by Perfect Fifths Abl), Ebb, Bbb, Fb, Cb, Gb, Db, Afc, Eb, Bb, F, C, G, D, A, E, B. Fjt, CJ, Gt, Df, AJ, Ett, Bjt, Fx, Cx, Gx. It will easily be seen that, on this system, the major thirds must be a comma too sharp. Further, it is clear, from what has been said, that the notes will never correspond to those obtained by ascending in octaves. Now, as shown in Fig. 93, a great increase of roughness is introduced by making a major third a comma sharp. Hence the above tuning is unfit for harmony. But violinists may be accused of still using this temperament, inasmucli as they 450-452 CONSONANCE AND TEMPEEAMENT 495 tune their open strings to perfect fifths. It must, however, be observed that this does not commit them to sharpened major thirds, for the notes made by stopping the strings with the fingers can be varied at pleasure. Mr. Ellis gives the pitches of the above notes in cents as shown in Table XXXIX. Table XXXIX. Pitches or Notes in Pythagorean Tuning Notes. Cents. Notes. Cents. c ./■« 611-7 bn 23-5 abb 678-5 'lb 90-2 9 702-0 ci 113-7 /x 725-4 '-■bb 180-5 ab 792-2 d 203-9 Vt 815-6 C X 227-4 hbb 882-4 eb 294-1 a 905-9 dS 317-6 gx 929-3 /b 384-4 ib 996-1 e 407-8 at 1019-6 f 498-0 db 1086-3 eH 521-5 h 1109-8 ffb 588-3 c' 1200-0 452. Mean -Tone Temperament. — This temperament owes its name to the fact that it sinks the distinction between the large and small tone, and adoj)ts a tone which is the mean of the two. Thus it makes C to D and D to E, each 193 cents, instead of 204 and 182 respectively. It thus obtains the major third in just intonation. The fifths are, however, considerably out of tune, and its power to modulate is very small unless some additional notes are used beyond the usual twelve to the octave. This tempera- ment may be regarded as derived thus. Ascend by four perfect fifths from C. Then the G and d are in just in- tonation, but the succeeding a and e are each a comma sharp. Now, let the e be made in just iiitonation, the 496 SOUND necessary flattening being equally distributed over the four fifths used in ascending from C to e. Thus each interval of a fifth is made a quarter of a comma too small. Thus, whereas the Pythagorean tuning kept the fifths perfect at the expense of greatly damaging the major thirds, the mean-tone temperament keeps the major thirds perfect at the cost of appreciably spoiling the fifths. The derivation of mean-tone temperament is exhibited in Table XL. Table XL. — -Derr'atign of Mean-Tone Temperament Just Intouation C G d a e' Four ju.st fiftlis up \ from C / C C G G' d d '■■ 1 a t Mean -Tone T&mA perament, four 1 fifths lip but each J comma small . Errors of Mean- '\ Tone Tempera- [■ ment J '.a each a comma sharp. T comma flat. T comma flat. J comma sharp. 453. It is easy to see the power of modulation in mean-tone temperament is very restricted. ¥or, having begun with a just major third in the scale of C, we naturally wish to keep it in all other keys. Thus, in the key of E major we need the justly intoned G|. But in the key of Ab, in order to use the C's already provided, the Ab must be a just major third below c. It is, accordingly, forty-two cents sharper than the G|; as already referred to in art. 448. Hence the necessity for either more than twelve notes to the octave, or the restriction of modulation to the very few keys in which the original type of intervals is preserved. The limitations of mean -tone temperament may be shown in another way as follows : — Let notes be written proceeding upwards l:iy mean-tone fifths from C. Then from a C, seven octaves higher, descend by mean-tone 453-455 CONSONANCE AND TEMPEEAMENT 497 fifths, and also ascend by fifths from the high C for a few notes. Now since tlie mean-tone fifths malce the major thirds true, we never obtain again the notes from wliich we started, no matter how many intervals are taken up or down (see art. 442, where it was noted that octaves and major thirds involve ratios which are prime to each other). The scheme of notes just referred to is shown in Table XLT., in which the cents up from the lower C are also given. The notes in the bottom line are thus seen to be forty- two cents sharper than the corresponding notes in the top Line. Table XLI. — Scheme of Notes in Mean-Tone Temperament a^S G n Ot GJ m A# EJ Bt Fx Ox Gx "j .J Otl(V,5, 1393, 2089-5, 2780, 3482-6, 4179, 4876-5, 5572, C2CS-5, 0965, 7601-5, 8358, 9054-5, 9751, 10,44 42, 7SS-5, 1485, 2131-5, 2828, 3524-6, 4221, 4917-6, 5014, 0310-6, 7007, 7703-6, 8400, 9096-5, 9793, 10,489-5^ ■dVo Abb Ebb Bbb Fb Ct" Gb Db Ab Eb ]3i> F 454. This table clearly shows that for modulation into the seven sharp keys and the seven flat keys, we should need twenty-one notes to the octave, i.e. to B# inclusive in the top line and backwards to Ffci inclusive in the bottom line. With only twelve notes to the octave we should have power to play in six keys only, the key of C natural and five others, say two sharp keys and three flat ones or vice versa. And even then there would be no possibility of sounding accidental sharps when in the extreme sharp key, nor accidental flats when in the extreme flat key. 455. By reducing the above scheme to a single octave we obtain the notes of the mean-tone temperament in the compact form shown in Table XLII. 2k 498 SOUND » 'tt Table XLII. — Single Octave in Mean-Tone Temperament Diatonic Scale. Cents up from C. 193 380 303 -.5 ■096-5 889-5 1082-5 1200 C Notes. C D E P G A B Sharps and Flats. Notes. Cents up from C. CJ 76-5 llY-5 DJ 208-6 SlO-5 n 679 621 GJ 772 Ab S14 At 965 13b 1007 It may be noted that the chief intervals in the mean- tone temperament are the diatonic tone and semitone of 193 and 117'o cents respectively, the chromatic semitone of (193 — ll'Z'S =) Y5'5 cents and that between a sharp and the adjacent flat of (2 x 117-5 - 193 = ) 42 cents. To us now the mean-tone temperament derives its im- portance from the vogue which it once had, as shown by the following quotation from the late Mr. A. J. Ellis : — " This was the temperament which prevailed all over the Continent and in England for centuries, and for this, and the Pythagorean, our musical staff notation was invented, with a distinct difference of meaning between sharps and flats, although that difference was different in the two cases. This temperament disappeared from pianofortes in England between 1840-1846. But at the Great Exhibition of 1851 all English organs were thus tuned. Handel in his Foundling Hospital organ had sixteen notes, tuned from d\> to aj^, ' ascending by mean-tone fifths.' Father Smith on Durham Cathedral and the Temple organ had fourteen notes from a\> to dj^, and the modern English concertina uses the same temperament and the same number of notes. The only objection to this temperament was that the organ-builders, with rare excep- tions, such as those just mentioned, used only twelve notes to the octave, eb h\> f c g d a e b fj^ c!^ g^." Ellis thought that if it had been carried out with twenty-seven notes to 456, 457 COFSOlSrANCE AND TEMPERAMENT 499 the octave from a\)\> to gx. (as in Table XLT., omitting the Dbb), it would probably have still remained in use. In his examination of fifty temperaments, Ellis found that this was decidedly the best for harmonic purposes. He con- sidered that for simple melodic purposes the Pythagorean was preferred by violinists, but that was always absolutely impossible for harmony. 456. Equal Temperament. — The temperament now so widely in use is called the equal temperament, since it aims at dividing the octave into twelve equal intervals. It is clear that this temperament at once satisfies the re- quirements of practical convenience and, if attained with theoretical accuracy, secures perfectly the advantage of freedom in modulation. For, in the first place, it has the smallest number of tones that can be taken to represent the chromatic scale ; and, secondly, whatever note is taken as the key-note, it is possible to proceed by precisely the same sequence of the standard intervals of tones or semi- tones to the octave above. Thus, on this temperament, twelve fifths make seven octaves, and three major thirds make one octave precisely. We must now examine how far this temperament falls short in respect of supplying pure concords. The first step to this will be- to find how far the various notes deviate from the corresponding ones in just intonation. We may do this in several ways, the following is perhaps the most suitable for our purpose. 457. Since only twelve notes are used in the octave and the steps made equal, we evidently fuse C^ with Db, and B| with C, and so with the others. Hence, if we pass up by twelve fifths from C to B|, the temperament re- quired may be expressed by making the fifths all equal, and such as to bring B|; into coincidence with C. Now on passing up twelve perfect fifths, we have the interval ( - = 531,441-^-4096; whereas, on passing up seven octaves, 500 SOUND CHAP. IX we have the interval 2'^= 524,288-1-4096. Hence, to bring B| down to C, we must lower it by the interval 531,441 : 524,288. This interval is known as the Pytha- gorean comma. To express it in cents, ascend the 12 perfect fifths, and descend seven octaves all measured in cents. Thus, we have the Pythagorean comma =12 X 701-955 — 7 X 1200 = 23-46 cents, or say twenty-four cents nearly. Hence, the fifth used in equal temperament is smaller than the just or perfect fifth by one-twelfth of the Pythagorean comma, that is, by practically two cents or about one-eleventh of the ordinary comma. 458. Examine next the major third. This consists of two whole tones or four semitones of the standard equal- temperament size, so these major thirds must make an octave which contains twelve such semitones. Thus, if we ascend by the series of major thirds, Ab C E G|, the equal temperament fuses G| and Ab- Hence, if the deviation of tliis interval from just intonation be denoted by the ratio r or by c cents, we have the two equations - r j = 2 and 3 (386-314-f c)= 1200, 3/128 126 whence r= / = — — ■ nearly, and c = 13-686 cents. •V 125 125 ■" These figures express the amount by which the equal- tempered major third is larger than the same interval in just intonation. 459. It is obviously convenient to measure intervals in cents when we wish to judge of the freedom for modula- tion which any temperament admits. On the other hand, when we wish to judge whether a consonance will be pure, it is often preferable to measure the intervals concerned by their frequency ratios. Further, when wishing to know how far a deviation from just intonation will damage the purity of any concord, we must refer to Helml^oltz's diagram of degrees of roughness. To facilitate the comparison in various ways of the mean-tone and equal temperaments 458-460 CONSONANCE AND TEMPEEAMENT 501 with just intonation, Table XLIII. and Fig. 94 are given. These show clearly that the mean-tone is superior to the equal temperament in its major thirds and sixths, but is worse in its fifths. Also, when restricted to twelve notes to the octave, the mean- tone is hopelessly behind the equal temperament in scope for modulation. Table XLIII. — Just and Tempered Intonations Notes. Intervals above C and Errors of Tempered Notes. | 1 Just Intonation. Mean-Tone, Equal Temperament. Frequency Ratios. Cents. Intervals. Errors. Errors. Intervals. c D E F G A B a 1 9i8 5:4 4:3 3:2 5:3 15 :8 .2:1 204 386 498 702 884 1088 1200 Cents. 193 ■ 386 503 697 890 1083 1200 Cents. 111? 5Jt 5b 5b Cents. ib 148. 2« 2b 16S 12( Cents. 200 400 500 700 900 1100 1200 460. ExPT. 78. Effect of the Comma on the Violin. — As an illustration of the effect which the comma produces in harmony, Helmholtz gives the following simple method open to any violinist. Tune the first, second, and third strings to /', a', and d', as usual making the fifths perfect. Then stop the note b' with the first finger on the a' string. Obviously this will make a major sixth with the d' string and a major fourth with the e" string. And for melody, a single position of stop with the finger would make both the intervals passable. Try now the effect of harmony by sounding the b' along with the d' and with the e" respectively. It is at once perceived that no single position of the stop gives the best result both for the major sixth d' b' and for the major ' All cent.s given are to the nearest whole number ; this line should really read 884-359, 889-7, 5-341, etc. 50! SOUND fourth V e". On the contrary, the position of the finger to stop the V for a perfect major sixth with d' must be about 3/20 of an inch farther from the bridge than when stopping the V to make a perfect major fourth with e". On reference to the diagram Fig. 94 the reason for this is easily seen. For a major sixth (there represented by the A a sixth above C) we must have the sixth only a small tone (10/9) above the perfect fifth (G). Whereas for the perfect fourth (there represented by F a fourth above C) we must have the note a large tone (9/8) below the perfect fifth (G). Thus, to raise the pitch of the a' string from the perfect fifth to a major sixth with the cV we need to stop so as to sharpen it Mean-Tone \c Temperament ' tlust Intonation Equal Temperament en d d>t e f f» g gn a an 6 Fig. 94. — Illustbatihg Discoed due to Temperament. by only a small tone (10/9). Whereas, to diminish the perfect fifth between the a' string and the e" to a perfect fourth, we need to raise the a' string by a large tone (9/8). And the difference between the two stops for these tones, expressed as a fraction of the string's length, is obviously J^-A = TfV- So if the string is 13 '5 inches long, this difference of stops is 3/20 of an inch, a nicety of stopping quite well within the range of a player's accuracy. It should also be observed from the diagram Fig. 94 that all the intervals on which this experiment depends, the perfect fifth, the major sixth, and the fourth, are intervals which are well defined or delimited. That is to say, a slight mistuning causes a large degree of roughness. Hence it is easy to get all these intervals tuned with sufficient acciu:acy by ear alone. 461, 462 CONSONANCE AND TEMPEEAMENT 503 Bub although it is easy by this experiment to show that the best harmonies in the two cases referred to are- obtained by the diiferent stoppings, it will be found that an inter- mediate position gives a result which is tolerable, even for harmony, especially in a rapid passage and with a mellow-toned instrument. 461. Bosanquet's Cycle of Fifty-Three. — Although no temperaments but the equal and mean-tone have obtained much vogue, the cycle cif fifty-three notes to the octave, invented- and carried out by Mr. E. H. M. Bosanquet, deserves mention as the simplest of the many attempts to approach more nearly to just intonation. Indeed, Helmholtz says of it : " The ear cannot distinguish this scale from the just, and in its practical applications it admits of unlimited modulation in what is equal to exact intonation." But Mr. Ellis, the translator of Helmholtz, adds to the above passage a footnote. He allows that the intervals taken melodically might be indistinguishable from just intonation. But of the effect when used harmonically he says, "At least, as the intervals were tuned on Mr. Bosanquet's instrument, there was a decidedly perceptible difference to an ear accustomed as mine was to listen to just intonation." 462. This temperament divides the octave into fifty- three precisely equal degrees or steps, of which nine represent the large tone, eight the small tone, and five the diatonic semitone. All the fifty-three notes are provided, hence beginning at any one of them the diatonic scale can be played with the same sequence of the given standard intervals. Thus unlimited power of modulation is provided. Let us next examine the degree of approximation to just intonation which this scheme offers. It is evident that each of the fifty-three degrees or steps will be equal to' 1200-f-53 = 22'64151 cents, whence the notes and intervals are easily calculated. Table XLIV. shows the comparison of this cycle with just intonation to the first place of decimals in cents. 504 SOUND Table XLIV. — Bosanquet's Cycle of Fifty-Three Nok-s. Just Intona- tion Cents up from C. Bosanquet's Cycle. Cents up Errors in from C. Cfuts. " Steps up from C. Intervals in "Steps." D E F G A B C 203-9 386-3 498-0 702-0 884-4 1088-3 1200 203-8 384-9 498-1 701-9 883-0 1086-8 1200 0-li> 1-41, O-IJ 0-1? l-4i, 1-5? 9 17 22 31 39 48 63 } 9 } 8 } 5 }- 9 f 8 } ^ } 5 463. The table shows that throughout the diatonic scale the departure of Bosanquet's cycle from just intonation never reaches two cents. Accordingly, it has no interval so bad as the best in equal temperament, namely, the fourth and the fifth. Hence the scheme now under notice secures the advantage of practically just intonation together with that of freedom to modulate. But is it a practical temperament ? We fear not. A harmonium on this plan was made and exhibited at South Kensington in 1876. In this instrument, for convenience of fingering, the fifty- three notes in the octave were distributed over eighty-four digitals. Passing from left to right the keys were arranged in the ordinary way, seven white keys and five black ones to the octave. But, passing upward from front to back, there were tier after tier of keys somewhat like seven manuals condensed into one. This may seem a formidable array of digitals. It is, however, noteworthy that with this keyboard the inventor secured the signal advantage of making the fingering of any given scale or chord precisely the same, no matter what the key or the names of the notes. Thus, given scale passages or chords have the same form to the hand in any key instead of twelve 463-466 CONSONANCE AND TEMPEEAMENT 505 different forms according to the key as on the ordinary keyboard. 464. The notation suggested by Bosanquet for his instrument was the ordinary staff notation with the following slight modifications. The prefix of a back- ward-sloping line, thus \, indicated the flattening of a note by one of the " steps," of which fifty-three make the octave. Two such lines denoted the flattening by two steps, and so forth. The sharpening by the same amount was indicated by lines sloping the opposite way, thus /. The scale of c major would be c cZ \^ f 9 \^ \^ "'> but the notes represented by the staff notation instead of these letters. For the key of G- the / is of course replaced ^ by /# while the \a is replaced by «. It may be noted also that j j jh is the same note as \c' , since five of Bosanquet's " steps " make his diatonic semitone. 465. Ellis's Harmonical. — This instrument is a har- monium specially tuned according to the design of the late Mr. A. T. Ellis, the translator of Helmholtz's Sensations of Tone. Though already referred to in article 422, it requires a little additional notice in this place. One of these instruments was exhibited in London, and others have since been supplied by Moore and Moore to a number of universities and colleges. Its name is derived from the fact that the instrument furnishes a large number of harmonics. Thus, by special tuning, especially in the top octave, it has the first 16 partials of c, 132 per second. Whereas for C (66 per second) it provides 26 partials up to the 32nd inclusive, the 6 lacking being the 11th, 13th, 21st, 23rd, 27th, and 31st. It has what the inventor calls a " harmonical bar," by which the first 1 6 partials beginning at c can be played simultaneously ; or, if desired, the Vth and 14 th may be omitted. It is remarkable that the 7th and 14th, though not in the ordinary scales (either tempered or just), blend perfectly with the others, as theory predicts. Whereas harsh 506 SOUND CHAP. IX dissonances are caused by the use of the ordinary Bl,'s/ which are minor thirds above the G's. The instrument affords valuable illustrations of major and minor chords in just intonation (in the keys of C major and minor), but is not designed as a practical musical instrument, as it is incapable of modulation. 466. Character of Keys. — It is a matter of scientific interest to account for the special character of each key so often alleged by musicians. "What is here referred to is, of course, the absolute character of the keys C, G, D b, etc. and not the relative character of C and its dominant G into which the piece temporarily modulates. This matter is referred to briefly by Helmholtz and at considerable length by Berlioz. It appears that little if any difference of character can be attributed to the various keys for the organ or for voices with organ accompaniment or un- accompanied. But on pianos, violins, and brass instruments the difference is marked. Consider these classes of instruments in order. In the case of pianos it may be that the short, narrow, raised, black digitals are usually struck differently from the long, broad level, white ones. Hence, in the different keys these diflerences of touch are differently distributed throughout the degrees of the scale. Thus a different character may be imparted to the different keys. Again, in instruments of the violin family, the different keys bring into more or less prominence the open strings which sound fullest. Also the fingering varies considerably for different keys. So that here again we see how one key may be bright and ringing, another veiled and obscure. Similar remarks apply to the brass instru- ments. In one key the tonic and dominant may fall on open notes of the instrument, and in another key no open notes may be available for the diatonic scale. And the difference between open notes and valve notes is probably quite sufficient to give a character to the various keys which give or deny prominence to the open notes. 466-468 CONSONANCE AND TEMPERAMENT 507 467. It may also be suspected that, in the case of the various instruments, by the ordinary method of tuning current in tlie musical profession, the various keys are unequally favoured although equal temperament is supposed to be aimed at. Thus, it may be that on a piano fresh from the hands of some able tuner the keys in most frequent use deviate slightly from equal temperament in the direction of just intonation, while the least used keys may deviate slightly in the opposite direction. If so, we should usually have something rather better than equal temperament, and only occasionally something rather worse. And this difference would be available for artistic effect by giving a different character to the various keys. 468. Orchestration. — Although many of the problems of orchestration are chiefly musical and beyond the scope of this work, there are a few points of scientific interest which demand notice here. In the first place, we may naturally inquire why in an orchestra there aie several instruments of any given class but of different jjitches and taking different parts. Thus, we have first and second violins, violas, violoncellos, and basses. Again, there are usually several flutes, several clarinets, an oboe, and a bassoon. So that each class of instruments furnishes several parts of the composition with its own special quality of tone. The same holds with respect to the brass. Thus we may have two or three trumpets, two to four French horns, and usually three trombones. Instruments of very distinct character may be used singly, as the triangle and others of like nature. Perhaps two reasons may be given for the grouping of instruments of one class but of different pitches into small bands in this manner. First, it enables the composer to give a harmonic rendering of a passage in the quality of tone of which these instru- ments are capable. Second, the employment of instruments of the same quality of tone is essential to the characterisa- tion of the various consonances and dissonances. For we 508 SOUND CHAP. IX have seen that discord is due to beats, and usually for musical purposes these beats are between upper partials. Further, to make beats distinct, the sound at the minimum should be as weak as possible, that is, preferably of zero intensity. But, to secure this, the two beating tones must be of equal intensity. Hence, to secure this equality between upper partials, the two compound tones should have the same law of diminution in the retinue of upper partials, i.e. they should be of the same musical quality. Hence the necessity for having the various parts of a harmony executed in tones of the same or similar quality if the various shades of consonance and dissonance are to be clearly rendered. 469. Probably at one time only instruments of one class could be tolerated in harmony, because each class had its own style of tuning. Hence flutes might deviate from just intonation in one way, and be passable with each other, but intolerable along with some other family of instruments with different intonation. This may be the reason why at one period pieces were written for several instruments of the same class, say flutes only, or strings only, but no orchestration attempted. 470. Obviously orchestration implies standardisation both in pitch and temperament. We may thus regard an orchestra as an aggregation of small bands or families of instruments, each family being capable of harmonic utterance, but each having its own special character, dis- advantages, and limitations. Thus, we may divide the orchestra and chorus broadly into the strings, the wood- Wind, the brass, and the voices. Hence, between these various parts, whether they are required for combined or successive utterance, there must exist a suitable balance. What this should be depends, of course, upon the character of the piece to be performed, and also upon the progress made in the art of providing instruments capable of rendering artistic music. Thus, the position of the 469-471 CONSONANCE AND TEMPEEAMENT 509 orchestra at the present time is very different from that prevailing in Handel's time. The composition of an ordinary orchestra and chorus of the present day illustrates some of the preceding remarks. Of course the total number of performers differs very v^idely, and the relation of the constituents also varies greatly with the music to be rendered. Table XLV. gives a fair illustration of present practice for an orchestra and chorus of about three hundred for the purpose of oratorios and similar compositions. Table XLV. Example of Modern Orchestra and Chorus Wind Instruments, etc. Strings and Voices. 2 Flutes 12 First Violins 2 Oboes 8 Second Violins 2 Bassoons 6 Violas 2 Clarinets 6 Violoncellos 4 Horns 6 Double Basses 2 Trumpets — 1 Alto Trombone 38 1 Tenor Trombone ^^ 1 Bass Trombone 70 Sopranos 1 Bass Tuba 50 Contraltos 1 Pair Kettledrums 50 Tenors 50 Basses 4 Soloists 19 224 Band of 57 Performers, Chorus and Soli of 224, grand to tal 281, 471. Chief Intervals. — The chief intervals already re- ferred to within the range of an octave are collected together for reference in Table XLVI. The values of the intervals are given in Ellis's logarithmic cents and by their frequency ratios. The references to the notes of the scale supposes that the key of C is being used. 510 SOUND CHAP. IX Table XLVI. — Chief Intervals within an Octave Values )f Intervals. Name. and Examples of Intervals. Frequency Just Mean -Tone Equal a S Ratios. Intonation. 1 Temperament. Temperament '221 81 : 80 The ComiTia 42 128 : 125 Gjt to AlJ, etc. 70 25 : 24 D to i)i, etc. 76 70 : 67 C to Of, etc. 92 135 : 128 C to CJ, etc. 100 89 : 84 Every Semitone 112 16 : 15 Diatonic Semitone 117-0 107 : 100 Diatonic Semitone 182 10 : 9 Small wiiole Tone 193 180 : 161 Every Whole Tone 200 41:1 : 400 ■ Every Whole Tone 204 9 : 8 Large Whole Tone 300 44 : 37 Minor Third 316 6 : 5 Minor Third 386 5 : 4 Major Tliird 400 63 : 50 Major Third 498 4 :3 Fourth 500 303 : 227 Fourth 503 107 : 80 Fourth 697 163 : 109 Fifth 700 433 : 289 Fifth 702 3 : 2 Fifth 800 100 : 63 Minor Sixth 814 8 -. 5 Minor Si.xth 884 5 : 3 Major Sixtli 900 37 ■ 22 Major Sixth 969 7 : 4 Trumpet; Seventh 2 996 16 : 9 Minor Seventh, a fourth above F 1000 98 : 55 Minor Seventh 1018 9 :5 Acute Minor Seventh, a Minor Third above G 1088 15 : 8 Major Seventh 1100 168 : 89 Major Seventh 1200 2 : 1 Octave Octave Octave 1 The comma of the just intonation is strictly 21-506 cents ; each of the 53 intervals or "steiw" of Bosanquet's cycle is 22-64151 cents, and the Pythagorean comma is 23-46 cents. '-^ Tliis is the note called 7Bb in Ellis's Harmonical. 472. Historical Pitches. — We shall conclude tlie present chapter with Table XLVII., giving a few of the leading pitches that have been used at various times and 472 CONSONANCE AND TEMPEEAMENT 511 places. These are extracted from the mass of details obtained and compiled by Ellis, and given by him in an appendix to his translation of Helmholtz's Sensations of Tone. Table XLVII. — Chief Pitches from Lowest to Highest Is ID Cm h ll 1 Place. Instrument or Authority identifled with each Pitch. t 6 1 ■i 3 17 33 370 373 7 377 1648 1511 Paris Heidelberg Imaginary lowest pitch to reckon from. 66 129 384-3 398-7 1700 1854 Lille Lille Old Fork. Old Organ. Chamber Pitch (Low). 148 196 402-9 414-4 1648 1776 Paris Breslaii Spinet. Clavichord. S.2 "S g n 199 230 243 261 415 422-5 4-25-8 427-8 1754 (■1751 1820 1877 1824 1788 Dresden England Westminster England Paris Windsor Organ. Handel's Fork. Abbey Organ. /Curwen's "Tonic Sol-Fa," V Standard c" =507. Opera Pitch. St. George's Cljapel Organ (measured by Ellis, Feb. 1880, while still in " Mean- Tone"). 1 260 273 285 430 433-2 436-1 1810 1828 1878 Paris London London Fork. Sir G. Smart's own Philhar- monic Fork. Fork to which Organ tuned at H.M. Theatre. 1 O E O i t Si 288 289 305 318 323 346 350 339 380 437 437-1 441-2 444-6 445-8 452 452-9 455-3 460-8 1859 1666 1878 1877 1856 1885 1878 1879 1880 Toulouse Worcester Loudon London Paris London London London U.S. America Conservatoire. Cathedral Organ. Covent Garden Opci-a. St. Panl's Organ. Opera. International Exhibition of Inventions and Music. Kueller Hall Military School. Erard's Concert Pitch. Highest New York Pitch. Church antX Chamber Pitches (High). 368 541 726 740 457-6 506-8 563-1 567-3 1640 1361 1636 1619 Vienna Halberstadt Paris North German Organ. Organ. Mersenne's Chamber Pitch. Church Pitch. CHAPTEE X ACOUSTIC DETEEMINATIONS 473. Velocity of Sound in Free Air. — Most of the methods of acoustic measurements hitherto developed aim at the determination of a velocity of propagation, absolute or relative, or a pitch. These will be taken in the order named. "We shall then consider other problems, some of subordinate importance, whose treatment is less advanced, and in some cases has scarcely yet passed beyond the qualitative stage. Though much of the present chapter will be necessarily concerned with descriptions of classical determinations or researches, the simpler methods suitable for lecture illustration or laboratory exercises will not be overlooked. We commence, then, with some of the classical determina- tions of the absolute velocity of the propagation of sound in free dry air at 0° C. Paris Academy in 1738. — Cassini, Maraldi, and La Caille, three members of the Academy, made what appears to be the first exact determination of the velocity of sound in the open air. Their stations were the Observatory at Paris, Montmartre, Fontenay-aux-Eoses, and Monthlery, the total distance involved being 1*7 or 18 miles. The experiments were made at night, and commenced on a signal being given from the Observatory. Alternately from each of the two end stations a cannon was fired at constant 512 473,474 ACOUSTIC DETEEMINATIONS 513 intervals. At the other stations the times were observed which elapsed between seeing the flash and hearing the jeport. The distances in question were accurately measured and thus the speeds found. These observations were con- tinued for some time and under different atmospheric conditions. The conclusions arrived at are as follows : — (1) The velocity of propagation is independent of the pressure of the air. (2) It increases with the temperature of the air. (3) It is the same at each distance from the source of sound; that is, sound is propagated at uniform speed. (4) With the wind sound is propagated quicker than against the wind, the speeds being in the first case the sum, and in the second the difference of those of sound and the wind. (5) The velocity of propagation of sound in still dry air is 337 metres per second. However, by the calculations of Le Koux applied to these same experiments, the velocity of sound at 0° C. was determined as 332 metres per second. 474. Bureau des Longitudes, 1822. — These experiments were also conducted at Paris, namely, between Monthlery ^nd Villejuif. Cannons were fired from opposite ends of the line at intervals of five minutes. The observers were Humboldt, Gay-Lussac, and Bouvard at Monthlery, those at Villejuif being Arago, Mathieu, and Prony. At Villejuif were heard all the cannons fired at the other end, the mean time elapsing between flash and repoTt being 54'84 seconds. The observers at Monthlery made the mean interval to be 54'43 seconds, but of the twelve firings at Villejuif only seven were heard, hence the correction for wind was not made so perfectly as desired. The mean time was, how- ever, taken as the arithmetic mean of the above two values, namely, 54'63. The distance between the stations was determined by Arago as 18,622'27 metres. Thus, the speed 2l 514 SOUND OHA.P. X of sound at the temperature, when observed, was determined as ■y= 18,622-27-:- 54-63 = 340-8 metres per second. As the temperature was approximately 16° C, the speed at 0° C. was calculated to be 340-8 = 331-2 metres. per second. '" n/i + 0003665 X 16 Dutch Physicists, lS'£o. — A very careful determination made by Moll, van Beek, and Kuytenbrouwer, gave as the value of the velocity of sound in still dry air at 0° C, i;o = 332-26 metres per second. And from these observations a recalculation by Schroder van der Kolk afforded the value, ?Jo = 332-77 metres per second. 475. Speed along a Slope. — In 1823 Stamfer and Myrbach made determinations of the speed of sound between stations in the Tyrol differing in level by 1364 metres. In 1844 Bravais and Martins experimented between stations whose levels differed by 2079 metres. The upper station was on the Faulhorn and the lower one at the Lake of Brienz. The slant distance between the stations was of the order 9560 metres, and the slope of the straight line connecting them over 12° Eeciprocal firing from each end was practised. On the mountain eighteen shots were fired during three days by A. Bravais and Martins. At the lake C. Bravais fired fourteen shots. The directly observed speeds of the sounds were upwards 337-92 m./sec, and downwards 338-10 m./sec, the mean value being 338-01 m./sec. Eeduced to 0° C. and dry air this research gave the value, ■Wo =332-37 m./sec. The experiments in the Tyrol gave practically the same value. These are interesting as confirming the theoretical 475-477 ACOUSTIC DETEEMiNATIONS 515 prediction that the speed of sound is independent of pressure. For the Lake of Brienz is 1857 feet above the sea-level, the top of the Faulhorn 8803 feet, and obviously at these altitudes the mean pressures would be very low compared with that generally observed at or near the sea-level. 476. Experiments at Low Temperatures. — Observations on the speed of sound at low temperatures have been made in Arctic expeditions. Parry determined the velocity at temperatures from — 38°-5 F. to — 7° F., and again at 33°-5 and 35°. From the actual results the speed of sound at 0° F. would be about 1050 feet per second, and the temperature correction about 1 ft./sec. per 1° F. Thus, at 0° C. we should have a value between 1080 and 1090 ft./sec. A number of observations made by Greely at temperatures between — 10° and — 45" C. resulted in the expression, v = (33S + 0'6t) metres per second, where v is the velocity of sound at t° C. 477. Stone's Experiments, 1871. — This determination was made at Cape Town. Two observers were stationed, one 641 feet from the one o'clock time cannon at Port Elizabeth, the other at the Observatory 15,449 feet distant. The instants of hearing these reports by the observers were electrically recorded on the chronograph situated at the Observatory. There was no reciprocal firing, hence the wind velocity was riieasured and allowed for. The time taken by the electrical signals is negligible in comparison with the time taken by sound to travel. But the chrono- graph records were corrected for " personal equation." This is the term used by astronomers and others to denote the time lag or interval peculiar to any observer between his perceiving and recording an event. If, in the experi- ments under notice, the two observers possessed equal personal equations, however large or small, the results would have been unaffected by them. But, probably, each had a different personal equation, and especially a different 516 SOUND CHAP. X one for the given conditions in which one observer hears a loud sound and the other a feeble one. To eliminate this difference of personal equations, a smaller gun was fired at distances from the two observers, chosen, so as to make the loudness for each observer the same as in the actual experiment with the time gun. The time for sound to travel over this distance, about one-tenth those in the main experiment, was calculated provisionally from the experimental value found by the neglect of the personal equation. The recorded interval was greater than that calculated by O'OQ second. Consequently 0-09 was sub- tracted from the intervals recorded in the main experiments, as representing the excess of the personal equation of one observer over that of the other. The value thus obtained by Stone was i;o = 1 9 ■ 6 f t./sec. = 332 -4 m./sec. 478. Variation of Velocity with Intensity. — In 1864 Kegnault experimented near Versailles on the velocity of sound in the open air, two distances respectively, 2445 and 1280 metres being used. Eeciprocal firing of guns was employed. The instant of firing was recorded by the rupture of a wire forming part of an electrical circuit, and passing across the gun's muzzle. The arrival of the sound at the distant station was also electrically recorded. The sound-wave was received by a wood cone fixed to a cylinder and closed at the far end by a thin india-rubber membrane. The motion of the membrane due to the sound broke a second electrical circuit. The two records were made upon the same chronograph. But this apparatus had a time lag just as truly as a human observer has his personal equation. Kegnault endeavoured to evaluate the error thus introduced, and to correct for it. The mean values from a number of experiments are as follows : — For the distance of 1280 metres, Vq= 331-37 m./sec. For the distance of 2445 metres, Vg= 330-7 m./sec. 478,479 ACOUSTIC DETERMINATIONS 517 This decrease of speed with diminished intensity of the sound is in accord with other experiments, and with theory. The value for the velocity of sound according to the ordinary theory professes to be valid only for infinitely small motions and variations of pressure. For finite amplitudes a larger value can be seen to hold ; because with finite increases of pressure we have a finite rise of temperature, and hence an increased speed of propagation of the parts compressed. This fact of increased speed for great compressions is also well exemplified in the photo- graphs of flying bullets due to Prof. C. V. Boys, see Fig. 29. In these photographs it is clear that the projectile, passing at a higher speed than that of ordinary sounds, compresses the air in front of it until a point is reached at which that shell of compressed air or wave can keep pace with the bullet. In Eegnault's experiment let us suppose that this lower limit or theoretical velocity was valid throughout the distance of 1165 metres between the first receiving station and the second. Then we have, for this lower limiting velocity, difference of distances 2445 — 1280 '"" ^ difference of times " 2445 1280 330-7 331-37 = 329-9 m./sec. 7-3934-3-8628 479. Jacques experimented with a cannon at Water- town, Mass., and showed (in 1879) that the velocity of sound is different in different directions round such a source. Membranes were used as receivers, and they re- corded on a chronograph. Immediately in the rear of the cannon the velocity was less than the usual amount. A little farther back the velocity reached a maximum value distinctly above the ordinary velocity. Still farther away the velocity fell to about the usual value. Thus, with a 518 SOUND CHAP. X charge of 1^ lb. of powder, tlie velocity at the rear changed from about 1076 ft./sec. at about 20 ft. distance to a maximum of 1267 ft./sec. at about 80 ft. distance, and then decreased again. Other examples of variation of velocity with intensity will occur in connection with the propagation of sound in water and through pipes. 480. Velocity of Sound in Water. — In 1826,Colladon and Sturm measured the velocity of sound in water by experiments in the Lake of Geneva. Two boats were moored at a definite measured distance apart. From one a bell hung immersed in the lake. This was sounded by the stroke of a hammer fixed to a lever whose upper end by the same motion fired some gunpowder. Thus the instant of striking was known at the other boat by the flash. The sound travelling through the water was received at the distant boat by a tube whose lower end was immersed, the upper end being applied to the ear of the observer. Hence the interval between the starting and arri^xl of the sound was found. It was registered by a quarter-second stop-watch and the velocity calculated. This was determined to be 1435 metres per second, and the mean temperature of the water concerned was estimated at 8°"1 C. Now in water, as in air, the velocity of sound is theoretically given by the expression /3j2 = 14:37 m./sec., which agrees practically with the experiments of Colladon and Sturm. 496. Thus the velocity of sound in liquids may be found (1) directly by large scale observations, (2) theoreti- cally as the square root of (elasticity -^density), or (3) according to Wertheim, from experiments on organ pipes, a factor being then introduced to transform from velocity in a cylinder to velocity in bulk. Further, the theoretical method, where the elasticity is known, may be compared with Wertheim's organ pipe method. Or, what amounts to the same thing, the velocity being found by Wertheim's 496-498 ACOUSTIC DETERMINATIONS 529 method, the elasticity may be deduced and compared with that found by direct experiment. If the results are concordant, our confidence in Wertheim's hypothesis is naturally strengthened. , Following this plan, Wertheim made the >determination8 and comparisons set forth in Table LII., which show a very satisfactory agreement, considering the many difficulties of the work. 497. It may be noted the factor which in the case of solids transforms the square of velocity in a thin rod to that in the same substance in bulk is (1 — °~t-i- ... J ■! „, B„H i i ' ■ — .-^a — ts^ Fig. 96. — Kundt's Double Apparatus for Dust Figures. 500. Kundt found that the velocity of sound in tubes depended not only upon the diameter of the tube, but also upon the frequency of the tone employed. The comparison in these two respects is shown in the values obtained by him, and set forth in Table LIII. Table LIII. Eelative Velocities of Sounds by Kundt's Tubes Diameters of Tubes. Wave Lengths of Sounds used. 18 cm. 9 era. 6 cm. • 5-5 cm. 2-6 cm. 1-3 cm. 0-65 cm. 0-35 era. 1-01010 1-00908 1 -00000 0-98031 0-92628 1-00885 1-00842 1 -00000 0-99170 0-96666 1-00584 1-00781 1-00000 0-99176 532 SOUND CHAP. X 501. Kundt could not detect any influence of intensity upon the velocity. He found, however, tliat the powder in a narrow tube, and especially if too much were present, decreased the velocity slightly. It was practically without effect in a large tube. Also a roughening of the interior of the tube was found to diminish the velocity. Indeed, Kundt considered that all the observed changes of velocity were due to friction and exchange of heat between the gas and the sides of the tube. It is obvious that when the gas is compressed, and in consequence warmed, any con- duction of heat to the walls of the tube will diminish its temperature, and therefore diminish also the elasticity and velocity. Again, in the rarefied parts of the gas there is cooling. But here also conduction of heat, although this time from the walls of the tube, will, as before, produce a diminution of elasticity and velocity. Now, the quantity of the gas present varies as the square of the diameter of the tube, whereas the surface of wall presented for heat conduction varies only as the first power of the diameter. Hence the diminution of velocity from this cause must be greater with a smaller tube. 502. Ratio of Specific Heats by Kundt's Tubes. — Using the double form of wave tube apparatus, Kundt and Warburg (1876) determined the ratio of the specific heats of mercury vapour. The two wave tubes contained air and mercury vapour respectively, the latter being at a high temperature. Quartz sand was used to exhibit the dust figures, from which the wave length and velocity v for mercury vapour were found. But theoretical considera- tions give for the velocity v= VTp/p (see art. 121). The tempefature and the molecular weight of mercury give the value of j^Ip for the vapour, and thus 7, the ratio of the specific heats, was calculated. Its value was determined to be 1'66. Soon after the discovery of argon, the ratio of its two specific heats was determined by Lord Eayleigh using the 501-504 ACOUSTIC DETERMINATIONS 533 method just described. The values 1-65 and 1-61 were found.^ 503. Kundt's Tube for Liquids. — Using the tube method, Kundt and Lehmann succeeded in obtaining dust figures in liquids as in gases, after taking care that the liquids were absolutely free from air. As powder to form the figures, very fine iron filings were used. The results obtained agreed exactly with the criticism by Helmholtz on the hypothesis put forward by Wertheim, as to the latter's experiments on liquids in organ pipes (see arts. 495-498). In other words, as Helmholtz predicted, the speed of sound increased if the diameter of the liquid column decreased, and the strength of the walls of the glass tube used in the experiments increased. This is shown in Table LIV. Table LIV. Speeds of Sound in Water, by Kundt's Tube Diameter of Tliickness of Temperature. Speeds of Tube in mm. Wall in mm. Sound m./sec. 28-7 2-2 l°8-4 1040-4 34-0 3 17-0 1227-7 23-5 3-0 18-0 1262-2 21-0 3-5 18-6 1357-6 16-5 5-0 18-5 1360-2 14-0 5-0 22-2 1383-2 504. ExPT. 80. Velocities of Sound in Solids by Kundt's Tube. — To determine in the laboratory the velocity of sound in a rod of metal or wood the single form of Kundt's tube will suffice. The description of this and the manner of using it are given in art. 285. A thermometer is needed to observe the temperature of the air in the wave tube. From this the speed of sound in the air may be calculated from the expression D = Vy V 1 + at (1), where v„ may be taken as 331 m./sec, a the coefficient of See Nature, 7tli February 1895. 534 SOUND CHAP, x expansion as 0-003665 per 1° C, and t is the observed temperature in degrees centigrade. Let the distances be- tween the nodes as shown by the dust heaps or figures be I, and the length of the rod be I'. Then I = A/2 and Z' = k'j-l, where X and X' are the respective wave lengths in air and in the rod of the sound of frequency N say given by the longitudinal vibrations of the rod. Hence for the speed v of sound in the rod, we have v' = N\' = N-ll' (2). And for the speed of sound in the air, we have similarly ,. = NX = K-21 (3). Thus, on division, we obtain v'jv = I'jl, or V = I'vjl (4). And by ( 1 ) this may be written / = -?)gs/i + at (5), It which gives the speed sought in terms of the observed quantities and the constant v^. 505. ExPT. 81. Young's Modulus for a Solid hy Kundfs Tube. — The manipulation for this determination is just as in the foregoing experiment, except that we need, in addition, the density of the solid rod. For this purpose weighing and measuring will be most convenient if the rod is of wood. If it is of any well-known standard metal its density may be taken from tables of such constants. Thus, for east-iron, wrought-iron, steel, or brass, the densities may be taken as 7-25, 7'8, 8, and 8 gm./cc. respectively. We then use the ordinary approximate expression for v (as proved in art. 123), v'= s^q/p (6), where q is Young's modulus and p is the density of the rod. Thus v and p being known, q is calculable. 506. ExPT. 82. Speed of Sound and Ratio of Specific Heiits in Gases hy Kundfs Tube. — For this determination the double form of apparatus described in article 499 is desirable. Then using air in one tube and the other gas in the other, 505-508 ACOUSTIC DETEEMINATIONS 535 and denoting by v and *' the respective speeds of sound in them, and by I and I' the observed nodal distances, we have v'Jv = I'll, , v' 1 + at (!)■ But, by theoretical considerations, we have v' = ^'ypjp (2). Hence, the pressure and density of the gas being known, and v determined from (1), y is calculable from ('J). This application of the method has already been referred to as used for mercury vapour and for argon. 507. ExPT. 83. Velocity of Sound in Liquid's by Kundt's Dust Figures. — For this purpose a modified form of Kundt's tube used by Dvofdk is desirable, as it obviates the necessity for making sure that the liquid is absolutely air-free. Dvorak used a horizontal tube about 2 m. long, one end being turned up a little and closed, the other being turned up about 10 cm. and left open. The tube was then filled with water, except that at the closed end a large air bubble was left, the open end containing only water. This end is now to be blown as an organ pipe by blowing smartly across the pipe. The liquid is thus set in vibration, the nodal positions being conveniently shown by gunpowder free from saltpetre. By this method Dvorak obtained for water the results shown in Table LV., which are in good agreement with those due to Kundt and Lehmann and previously quoted. Table LV.^Speed of Sound in Water by Dvorak Diameter of Tube. Thickness of Wall. Speed. mm. mm. m./sec. 17-9 0-82 998 117 0-63 1046 8-46 0'52 1164 15 2 1213 11 2 1281 508. Calculation of the Mechanical Equivalent of Heat. — It is noteworthy that the value of the mechanical equivalent of heat may be obtained from acoustical 536 SOUND CHAP, x determinations and theoretical relations. Thus, let the characteristic equation of a gas be written PV=RT (1), where P, V, and T denote respectively the pressure, volume, and absolute temperature of the gas. Then B = P,alpo (2), in which P^ is the standard pressure, a the coefficient of expansion, and p^, the density at 0° C. and standard pressure. Thus, the value of B in (1) is given by. (2) and a knowledge of the tabular density. But the theory of thermodynamics furnishes the relation C,-C, = P/J (3), where C and C„ are the specific heats in heat units at constant pressure and constant volume respectively, and J is the mechanical equivalent of heat. Now, suppose that by any method the velocity v of sound has been determined. Then, from its theoretical expression V = V^ (4), knowing P and p, we can calculate y and use the relation C,/C. = 7 (5)- Hence, if Cj, is known by Eegnault's method of determination, 6\ may be calculated, and on substitution in equation (3) J" is thence determined. If, on the other hand, neither specific heat for a certain gas is known, but only their ratio, then on assuming the value of J, equations (3) and (5) serve to determine both Cy and C„. 509. Hebb's Telephone Method for Speed of Sound in Air.— At the suggestion of Michelson, T. C. Hebb in 1904 made an elaborate determination of the velocity of sound by means of telephones and parabolic reiiectors. All 509, 510 ACOUSTIC DETEEMINATIONS 537 methods involving the use of a tube were rejected on account of its attendant complications. Further, to all long-distance methods the following objections were made : — (1) Very intense sounds must- be used, thus involving a possible difference of speed near the source. (2) It is almost impossible to correct accurately for wind, temperature, and humidity over such long ranges. (3) The " personal equation " of an observer or of some recording device is involved. 510. The attempt was accordingly made to free the research from all these objections. The experiments were conducted in a room 120 feet long. The source of sound was a whistle blown so steadily as to maintain its frequency to 1 in 5000. It was placed at the focus of a parabolic mirror made of plaster of Paris, 5 feet in aperture and 15 inches in focal length. From this first mirror plane sound waves proceeded along the length of the room. These were received by a second precisely similar mirror and so converged to its focus. Near the whistle was a telephone transmitter connected to a battery and one of tico primaries of a special induction coil. At the focus of the second mirror was placed a second similar telephone transmitter, also connected to a battery and the other primary of the special induction coil. * Finally, to the secondary of this induction coil was connected a telephone receiver. When the whistle was sounding, the two transmitters were set in vibration with a definite phase relation depending on the distance between the mirrors. Further, the telephone receiver gave the resultant or vector sum of these two effects. Hence, by changing the distance between the mirrors the phase relation of these two effects was changed, and they accordingly gave alternate maxima and minima at the receiver. To make the minima as sharp as possible, the effects from the two telephone transmitters were adjusted to equality by resistances in 538 SOUND cHAi'. x their respective circuits. Thus, on obtaining the successive minima, the wave length of the sound issuing from the whistle was determined, and its pitch being found the velocity of sound easily followed. 511. In some of the experiments, instead of the whistle tone, waves of about 10 inches long were used from an e'" fork (1280 per second). Distances iip to a hundred wave lengths were taken, and the minima could be located to an inch or one-tenth of a wave. Thus an accuracy of one in a thousand was reached, and this order of accuracy was aimed at throughout the experiment. The temperature was taken by six thermometers distributed along the space involved, and there was no disturbance from wind. It was feared that some little diffraction effect occurred with the ten-inch waves, so waves of six inches long were afterwards used. From these, which were considered quite satisfactory, the final value obtained for the velocity of sound in dry air at 0° G. was 331" 2 9 metres per second with a probable error of 0"04. The determination of the pitch of the whistle was made by tuning it to unison witli a fork which was itself compared with a 512 fork by traces on a smoked glass disc. The 512 fork was then compared in the same way with a pendulum, and, finally, the pendulum with a clock. 512. Wertheim's Determination of Speeds of Sound in Solids. — By exciting longitudinal vibrations in a rod of measured length L, clamped in the middle, and observing the frequency N of its tone, we have for the speed of sound in it V = N\ = N2L. Using this method, Wertheim determined the speeds of sound for the whole series of metals. -And the comparison of the values thus experimentally obtained with those theoretically calculated showed a good agreement. The results in question are given in Table LYI. 511-514 ACOUSTIC DETEEMINATIONS 539 Table LVI. — Wertheim's Speeds of Sound in Metals Metal. Relative Speeds of Souiifl, that in Air being Unity. Experimentally determined. Theoretically calculated. Lead (drawn) Gold „ Silver „ Zinc ,, . . Copper ,, Platinum Wire Iron (drawn) Cast-Steel (drawn) . Steel Wire ,, . 4-257 6-424 8-057 11-007 11-167 8-467 15-108 15-108 14-961 3 -7.' 7 6-247 7-940 10-524 11-128 8-487 15-472 15-003 14-716 513. Velocity of Sound in Wax, etc. — The velocities of sound in -wax and other soft solids which cannot be excited by stroking were determined by Stefan as follows. A rod of another material whose longitudinal vibrations could be excited was used to originate the tone, and to one end of this was fixed a piece of the softer material under examination. From the data thus afforded the velocity of sound in the soft body was determined by a somewhat complicated calculation. In wax at 17°, Stefan found the speed of sound to be 880 m./sec. with a decrease of 40 m./sec. per 1° C. rise of temperature. 514. Warburg for the same determination used two rods, one of the soft substance under examination, and the other, say of glass or other ordinary solid. These were mounted so as to be capable of transverse vibrations, and an antinode of the hard rod was linked to' an antinode of the soft one by a light wood connector fastened with wax. The nodal lines on each rod were shown by sand, and the distances between them measured. Then, if these similar nodal distances were I and I' in the hard and soft rods respectively, and their thicknesses /( and h' in the plane 540 SOUND of vibration, we have from theory (see art. 205) for the ratio of the velocities of sound in the two rods, v~ l-h' To test this method Warburg first compared the velocities of sound in brass and glass. He found for the ratio the values 0-676 and 0-6-45, giving a mean of 0-660, whereas by Kundt's method he made the same ratio to be 0-668, which agrees with the other value within about one per cent. Warburg's results for the other bodies tested by this method are shown in Table LYII., which applies to temperatures from 15° to 17° C. Taking the velocity of sound in air at 16° C. as 340 m./sec, and that in glass as 15-65 times this, we have from Warburg's determination the velocity of sound in wax as 883 m./sec, which agrees well with Stefan's result. Table LVIl. Warburg's Speeds of Sound in Soft Bodies Speed relative to Young's Modulus Material. that in Glass as Density. relative to tliat of Unity. Glass as Unity. Glass 1 2-390 1 Stearine . 0-265 0-974 1/35 Paraffin 0-251 0-908 1/42 Wax 0-166 0-971 1/88 Tallow 0-075 0-917 1/461 515. ExPT. 84. Velocity of So I md in JFires. — The method described by Tyndall for determining the velocity of sound in a wire is effective as a lecture illustration, and forms also a useful laboratory experiment where sufficient length is available. The wires under examination are fixed in any convenient manner at one end, the other end being attached perpendicularly near the centre of a wooden tray or board. When a rosined leather is passed gently along the middle of the wire it excites its fundamental longitudinal vibrations. 515 ACOUSTIC DETERMINATIONS 541 These move the sounding-board perpendicularly to its own plane, and thus produce powerful sound waves in the air. As we have seen before (art. 169), the relation between the frequency and the other constants is expressed by where N is the frequency, A the wave length, and v the speed of sound along the substance of the wire whose length is L, Young's modulus q, and density p. Thus, if N is known by comparison witb a fork or siren, and L measured, V may be determined. Further, if p is known or determined, the Young's modulus q may be calculated. Or, if another wire of different material be used also for the sake of com- parison, we may write for it Hence for a comparison of the velocities of sound in the two wires we may vary the length of one until the N's, are the same. In this case we obviously have v'lv = L'IL (.S), from which the required ratio is at once obtained when the adjustment of lengths for unison has been effected. Further, writing (3) in the form x/YTp'/^Wp^L'/L (4), we see that the Young's modulus q' of the second wire may be obtained if that of the other and both densities are known. The sounding-board may be screwed along its middle line to the edge of a firm counter or table, or a window-board. The wire should be attached near, but not quite at this middle line. It should be noted, also, that to shorten the wire it is not sufficient to place a bridge under it as in the case of strings vibrating transversely. The shortening may be arranged by clipping the wire with a heavy pair of thumb vice firmly attached to a large mass (say 56 lb.). In order to avoid unpleasantly high notes, it is also desirable to make the wires from 10 to 30 ft. long. Fiuther, when ex- perimenting with the wires there should be just sufficient 542 SOUND CHAP. X longitudinal tension on them to take out any slight kinks and permit the rosined cloth rubber to pass smoothly along. 516. Further Researches on Sounds in Pipes. — We have already dealt with direct and indirect methods for finding the velocity of sound in pipes, but many other researches have been made in the endeavour to obtain a full insight into the phenomena concerned. The problem was theoretically attacked by both Helmboltz and Kirchhoff. The former took into consideration the friction alone, while the latter considered also the exchange of heat between the pipe walls and the contained gas. For the speed of sound in pipes both obtained like expressions, the difference between them consisting only in the significance to be attached to one of the constants. Thus both forms may be represented by in which, according to Helmholtz, the constant c is the viscosity of the gas, while according to Kirchhoff it depends upon the heat conduction between the gas and the wall of the pipe. In either case v' is the speed of " sounds of frequency N in a pipe of radius r, v being the speed in the open. Thus according to either physicist we have 517. Since neither the researches of Eegnault nor those of Kundt were extensive enough to test this relation, both Schneebeli and Seebeck undertook new experiments with this object. Both experimenters, like Kundt, set up stationary waves in the air of a tube closed at one end, but unlike Kundt, they determined directly hy the ear the distances of the successive antinodes from the closed end. In the form of apparatus adopted hy Seebeck the closed end of the tube was a movable position, which was shifted 516-518 ACOUSTIC DETERMINATIONS 543 along a graduated bar so as to bring the antinodes one by one to a fixed point in the pipe. From this point a branch side pipe started which was connected to one ear by an india-rubber tube, the other ear being meanwhile closed. When the antinode is formed at the commencement of the branch pipe we have maximum changes of place, but minimum changes of pressure there, consequently no sound passes along the tube to the ear. Hence the adjustment in question is attested by the absence of sound. By this method Seebeck found that with wave lengths of 200 to 300 mm., the deviations from the mean were usually scarcely 1 mm. 518. Taking v= 332-77, Seebeck found for the decrease of speed (v — v') due to pipes the value shown in Table LVIII. In this table the calculated values are on the supposition that the Helmholtz and Kirch hoff law holds, the constant being chosen so as to give the speed ex- perimentally found for the narrowest pipe. Table LVIII. — Seebeck's Eesearches on Pipes Diameter of Pipe in Milli- metres. c", A'=512 per sec. g', iV=3S4 per .sec. t', AT =320 per sec. (v-V). Observed. Calculated. Observed. Calculated. Observed. Calculated. 3-4 9-0 17-6 9-79 4-33 1-86 9-79 3-70 1-90 13-91 5-09 2-91 13-91 5-25 2-70 15-51 4-75 3-53 15-51 5-06 3-01 If we take the observations with the same tone in the various pipes and combine them, a value of the velocity v of sound in free air may be derived. Thus from equation (1) we may write for the same tone in two different pipes c \ I c \ r= — and v<, = v\\ — , —r — - , v-^ = v[ 1 544 SOUND where v^ denotes the speed of sound for the frequency iV" in the pipe of radius rj, and v^ the corresponding speed in pipe of radius r^. Whence, on eliminating between these two equations the constant c/2v7riV, we obtain ''l — ■''2 519. Schneebeli combined in pairs in this way all his experimental results with pipes of diameters from 14 to 90 mm. and found the mean value ■y= 332-06 m./sec. But although the experiments supported the theoretical expressions of Helmholtz and Kirchhoff as to the variation of speed with diameter of pipe, both these experimenters found results in disagreement with the theoretical ex- pression as to the dependence of the speed on pitch of the note. Thus, according to theory, the change in speed should be inversely proportional to the square root of the frequency, while according to the experiments of Seebeck the decrease of speed is inversely proportional to the square root of the cube of the frequency. Indeed, if in the table of his results (Table LYIII.) we multiply each value of v — v' for a given diameter of pipe by iV"^'^, the products for each line are approximately constant. But, from direct observa- tions, the values of v — v' for a given diameter of pipe seemed to vary inversely as the frequency itself. Hence from Seebeck's researches no definite conclusion could be drawn upon this point. 520. After this Kaiser ixsed the method of dust figures to subject the Helmholtz-Kirchhoff theory to a further test. He experimented with three tones of 2357, 3895, and 5232 vibrations per second, in five tubes whose diameters were 25-8, 33-3, 44, 51-7, and 82 mm. From his experi- ments he concluded that the dependence of the velocity change both on the diameter of the pipe and the frequency 519-522 ACOUSTIC DETEEMINATIONS 545 of the tone agreed with the theory, provided that to the constant c of equation (1), instead of the theoretical value 0'00588 a value of about four times that, namely 0-0235, be assigned. The value for the speed of sound in free air derived by Kaiser from his experiments is 332-5 m./sec. 521. Wiillner's Experiments on Various Gases and at Different Temperatures. — Although the indirect methods above described have not cleared up the problems of pipes, and although it may be doubted whether they can afford a value of the speed so trustworthy as those carried on in the open air, still they certainly offer special advantages for dealing with different gases and at various temperatures. For different gases Dulong used organ pipes, as we have already seen. And, changing both gases and temperature, WlUlner used the method of dust figures. The wave tube used had a diameter of about 30 mm. The sounding tube was of glass one metre long and gave a tone of 2539 vibrations per second. This sounding tube was clasped at its middle by a rubber bung which fitted air-tight into one end of the wave tube. The other end of the wave tube was also closed air-tight by an adjustable stopper which was set so that between it and the fixed bung an exact number of half-wave lengths extended. Both stoppers were fitted with glass cocks so that the wave tube could be exhausted and filled again with any desired gas. The middle part of the wave tube for a length of about I'l metre was immersed in melting ice or in steam. 522. According to the Helmholtz-Kirchhoff theories, the decrease in the speed of sound due to enclosing the gas in a pipe is not independent of the nature of the gas. Yet for the diameter of tube and pitch of tone chosen by Wiillner, the correction on this account was so insignificant as to be usually negligible. For air at 0° C, Wiillner found as the mean of six experiments, 'yQ= 331-898 m./sec. 2n 546 SOUND Now the theoretical expression for the speed of sound may be written v= \/'Yp(l +af)lpQ. But, if 7 the ratio of the specific heats is supposed to change with temperature so that 7 = 7o(l + /^O) we may write V = s/'y,p(l+l3t)(l+at)/p,. Finally, since (1 + /SQ (1 +at)= 1 +a + ^t nearly for moderate ranges of temperature, we obtain the compact expression v = VQ\'l + St, where 8 = a + /S. Wiillner found that S was usually less than a, so that j8 seems generally to have a negative value. The results of Wiillner's experiments are shown in Table LIX., which also includes for the sake of comparison some of the determinations of Dulong and Eegnault. Table LIX. WiJLLNER's Experiments on Gases by Kundt's Tubes Value of fi— Relative Speeds of Sound Speed of Sum of GaRPK Relative' according to Sound at Tempera- Densities. 0" 0. by ture Wiillner. Co-efflcients Dulong. Regnault. Wiillner. a and p. Air . 1 1 1 1 331-898 0-003646 Oxygen . 1-1056 0-9624 Hydrogen 0-06926 3-812-3 3 -SOI Carbonic Oxide 0-9678 1-0132 1-0158 337-129 0-003588 Carbon Dioxide 1-5290 0-7856 0-8009 0-7812 259-383 0-003401 Nitrogen' Oxide 1-527 0-7865 0-8007 0-7823 -259-636 003307 Ammonia 0-5967 1-2279 1-2534 415-990 0-003436 Ethylene 0-9784 0-9518 315-90 0-003060 523. Blaikley's Experiments with Brass Tubes. — In 1883-4 D. J. Blaikley published^ an account of his researches on the velocity of sound in air in smooth brass tubes. He felt that in the large scale experiments ' Phil ^fnq. 523, 524 ACOUSTIC DETEEMINATIONS 547 the results were usually vitiated by uncertainties as to the mean temperature and hygrometric state of the air over the range in question. The Kundt's tube methods were also dismissed by him as being beautifully adapted for comparative results, but less suitable for absolute determina- tions. Blaikley remarks that in Eegnault's experiments with pipes the diminution of velocity there found would, if extended to the diameters used in brass instruments, lead to smaller values than those actually experienced. This discrepancy is attributed to the roughness of Eegnault's tubes. Further, it is pointed out that the membrane used by Eegnault might introduce an error which would in every case lead to an under-estimation of the velocity. In illustration of this, Blaikley took a cylindrical tube with a membrane of gold-beater's skin at one end against which rests a bead hung by silk. Now, to respond to a vibration of 512 per second, this resonator needed to be 5 inches long only, whereas a pipe with a rigid stopped end must be Q^ inches long to respond to the same note. Again, Dulong's experiments are objected to because in them care was taken to obtain a pipe producing a tone of good musical quality. This, Blaikley asserts, is the worst possible state of things for the purpose in view, and that for the following reasons: — In an organ pipe speaking a good musical tone the usual retinue of overtones is present. These upper partials when elicited separately are slightly inharmonic with the prime. Thus, on blowing the pipe, these naturally inharmonic overtones, on being forced into the strict harmonic series, constrain to some extept the pitch of the prime. The prime tone is in consequence slightly different in pitch from that which without such constraint corresponds to the true velocity of sound and the wave length under observation. Hence for the wave length and constrained pitch a vitiated velocity of sound would be inferred fi-om such a pipe. 524. For these reasons Blaikley chose to experiment 548 SOUND CHAP. X with a special form of organ pipe. This had a bulb or pear-shaped portion introduced in the first quarter wave length near the mouth. Beyond this bulb the pipe continued cylindrical for a considerable distance, and in this part a sliding plug worked. "With this plug adjustments could be made corresponding to the one- quarter wave length and three - quarters wave length with an ordinary cylindrical pipe. The bulb caused the natural overtones of the pipe to be quite inharmonic, and consequently they were not elicited by blowing. "By this means a pure tone was obtained. The blast was obtained from a fan, the wind from which passed through a regulating bellows with automatic-valve action, and it was found that great care was necessary on this point. The pressure in the bellows was 2"5 inches of water, and in the speaking mouth in every case very small. The temperature was observed by means of a thermometer entering the tube, so that the actual temperature during vibration might be recorded. The wet-bulb temperature and barometric pressure were also taken for moisture correction. The pitch was taken from a carefully tested Koenig fork of 256 vibrations, and the tubes were set to give a beating rate of about four per second, the lengths being read by a micrometer and standard rods. All the notes were exceedingly feeble, the pressure in the mouth being less than 1/10 inch of water, much under the lowest which Eegnault found to influence the velocity." 525. Experiments were made with smooth brass tubes of five sizes, the frequencies of the tones ranging from about 131 to 32.3 per second. The results for the velocity are shown in Table LX. [Table 525-527 ACOUSTIC DETEEMINATIONS 549, Table LX. — Blaikley's Speeds of Sound for Dey Air AT 0"C. IN Brass Tubes Diams. of Tubes.->- 11'43 mm. 19 '05 mm. Sl-Tl mm. 62-91 mm. 88-19 mm. m./sec. m./sec. m./sec. m./sec. m./sec. SA^ 324-533 327-09 328-72 3-29 -90 330-29 s^ 324-234 327-14 328-74 329-84 330-46 :sb 326-98 328-78 329-84 330-02 1'§ 326-70 328-72 329-70 329-72 327-09 328-72 329-95 329-99 326-69 328-89 329-80 330-41 ° , so that it sounded exactly two octaves higher than the tuning-fork of his microscope which sounded B\). In Helmholtz's experiment the fork of the vibration microscope was electrically driven. Lord Eayleigh points out that " the vibration microscope may be used to test the rigour and universality of the law connecting pitch andi period. Thus it will be found that any point of a vibrating body which gives a pure musical 536, 537 ACOUSTIC DETERMINATIOlN^S 557 note will appear to describe a re-entrant curve, when examined with a vibration microscope whose note is in strict unison with its own." 537. Lissajous' Figures. — This is another optical method for the comparison of vibration frequencies and also due to Lissajous. It is susceptible of high accuracy, and forms in addition a striking lecture illustration, as the figures produced by the simultaneous perpendicular motions of two forks may be projected on a screen, and thus made visible to a large audience. It may be conveniently carried out as follows : — FiQ. 97. — Projection op Lissajous' Figures. ExPT. 86. Projection of Lissajous' Figures. — For this purpose we require two tuning-forks tuned either to the same pitch or to some simple interval like an octave, a twelfth, or a fifth. These are provided with small plane mirrors, and are mounted on adjacent faces of a cubical box or framework CO, as shown in Fig. 97. From an electric- arc lantern a small beam of light I is derived, which passes through a focusing lens L, and is then refjected in turn at P on the fork A, and Q on the fork B, arriving finally in focus on the screen, where it makes a small bright spot at 0. It is obvious that the vibration of the fork A alone will cause the spot on the screen to describe a horizontal line along XOX'. In like manner the vibration of the fork B alone will give to the spot a vertical motion along YOY'. Thus the simultaneous vibrations of both forks will give on the screen a curve executed by the bright spot which represents the resultant of the two rectangular simple harmonic motions ; in other words, the spot describes one of Lissajous' figures. 558 SOUND CHAP, x As seen from the previous discussion in the kinematical chapter, if the periods are precisely equal the figure is constant in type, its exact form depending on the relation of the amplitudes and phases of the component vibrations. If the forks are excited by bowing, both amplitudes diminish continuously after the cessation of bowing. If the forks are electrically maintained, the amplitudes can be adjusted to a desired value and kept constant. If the periods are nearly, but not quite equal, the phase relation passes slowly through all its possible values. Thus, if at any instant the figure is an oblique line through the origin, it will change to an ellipse with axes along OX and OY, while the phase difference changes from zero to one- quarter of a period. This ellipse reduces to a circle if the amplitudes are equal. 538. Suppose now the method is used to adjust two forks to perfect unison, and let us note the accuracy obtain- able. Let the forks have a frequency of the order 100 per second, and if bowed only suppose the vibrations are visible for a minute. We should then have 6000 complete vibra- tions executed, and in this number a gain of a quarter of a period of one fork on the other would be distinctly visible, since it involves the change of the figure from an oblique line to an ellipse with horizontal and vertical axes. We should, accordingly, in this simple case detect a lack of perfect unison expressed by one vibration in 24,000 ! The interval ratio would be 24,001 : 24,000, or, measured in cents of which 1200 go to the octave, the interval between the forks would be only 0'072 cent, or less than a thousandth of a semitone. With forks electrically driven this limit of accuracy could be much exceeded. If the forks have freqiiencies in the ratio of 2:1, 2:3, or other ratios expressed by small whole numbers, the corresponding figure is produced and easily recognised. Further, it remains with or without change of form according as the tuning is approximate or precise. 539. ExPT. 87. Comparison of Forh, hij Gh-aphie Records. — This is a rougher, but still very instructive method of com- paring the frequencies of two forks in which each carries a light style of aluminium foil, and thus traces a sine graph on a moving smoked surface. The surface may be smoked by exposure to the fumes from burning camphor. In the 538-540 ACOUSTIC DETERMINATIONS 559 better form of the experiments this smoked surface is pro- duced on glazed paper, whicli is then wrapped round the convex surface of a drum which may be kept uniformly- rotating by clockwork. The vibration of the style is arranged so as to be parallel to the axis of the drum. The two forks are firmly mounted side by side near the drum so as to secure this relation. Then the drum being in rotation and the forks bowed, the two sine graphs are simultaneously described on the same paper, which is then removed and the records carefully examined. It may thus be found how many vibrations of one fork correspond to a given number of vibrations of the other. A rougher method of obtaining this comparison is as follows : — The forks are mounted side by side, each fitted with a suitable style, and a piece of glass is smoked. Then while one operator bows the forks, another moves gently past them the smoked glass so as to receive the double trace. The direction of motion of the glass must be at right angles to that of vibration of the forks. If the motion of the glass is not uniform the accuracy of the result is not impaired, as only a comparison of the two frequencies is aimed at, and not any absolute determination of either. 540. ExPT. 88. Comparison of Forks hj Monochord. — Another simple but instructive method of comparing two forks is afforded by the monochord. In this we either assume the relation that the frequency of a given stretched string is inversely as its length, or experimentally confirm it by comparison with forks of known frequencies. We must then carefully determine the different lengths of the string, which are exactly in unison with the forks to be compared. Thus, if the lengths are L^ and L^, and the frequencies of the corresponding forks are denoted by iVj and TVj, we have NJN^ = L^/L^, or TV^ij = N^L^ = a constant. The adjustments to unison between string and fork in each case may be made by listening for the extinction of beats, as the bridge limiting the length of the vibrating portion of the string is moved along by steps of about one millimetre each. In this method of tuning the string should be plucked or struck and not bowed. The reason and 560 SOUND CHAP. X importance of this when listening for beats were pointed out in article 294. 541. An alternative method of tuning is that of placing the stem of the vibrating fork on the bridge, and feeling lightly with the tip of a finger at the middle of that part of the string which it is desired to tune to the fork. If the two are in unison the string immediately responds. Indeed, while the fork's vibrations continue, the string, if well adjusted, may be repeatedly started after each check by the touch of the finger. This is on the principle of Helmholtz's resonance experiment described in article 243. 542. Absolute Value of Frequency. — -Determinations ly Koewig and by Bayleigh. — We shall now proceed to notice a number of methods for the absolute determination of frequency, taking first the classical methods which have been used by eminent investigators, and afterwards the simpler methods suitable for routine work in the laboratory. To Koenig we are indebted for one of the most direct determinations of frequency which involves the use of a special instrument. This consists of a fork of sixty-four complete vibrations per second whose motion is maintained by a clock movement and escapement. This clockwork showed ordinary time and also the number of vibrations exe- cuted. The behaviour of the fork was tested by comparison between the instrument and any other clock known to be trustworthy. A standard fork of 256 complete vibrations per second was compared with this instrument by observation of the Lissajous' figure characteristic of the double octave. 543. Another celebrated determination of frequency is due to Lord Eayleigh, and is described as follows by the experimenter in his Theory of Sound : — "An electrically -maintained interrupter fork, whose frequency may, for example, be 32, was employed to drive a dependent fork of pitch 128. When the apparatus is in good order there is a fixed relation between the frequencies, the one being precisely four times the other. The higher is, of course, readily compared by beats, or by optical 541-544 ACOUSTIC DETEEMINATIONS 561 methods, with a standard of 128, whose accuracy is to be tested. It remains to determine the frequency of the interrupter fork itself. " For this purpose the interrupter is compared with the pendulum of a standard clock whose rate is known. The comparison may be direct, or the intervention of a phonic wheel ^ may be invoked. In either case the pendulum of the clock is provided with a silver bead, upon which is con- centrated the light from a lamp. Immediately in front of the pendulum is placed a screen perforated by a somewhat narrow vertical slit. The bright point of light reflected by the bead is seen intermittently, either by looking over the prong of the interrupter or through a hole in a disc of the phonic wheel. In the first case there are thirty-two views per second, but in the latter this number is reduced by the intervention of the wheel. In the experiment referred to the wheel was so arranged that one revolution corresponded to four complete vibrations of the interrupter, and there were thus eight views of the pendulum per second, instead of thirty-two. Any deviation of the period of the pendulum from a precise multiple of the period of intermittence shows itself as a cycle of changes in the appearance of the flash of light, and an observation of the duration of this cycle gives the data for a precise comparison of frequencies. 544. "The calculation of the results is very simple. Supposing in the first instance that the clock is correct, let a be the number of cycles per second (perhaps -^-q) between the wheel and the clock. Since the period of a cycle is the time required for the wheel to . gain or lose one revolution upon the clock, the frequency of revolution is 8 ± a. The frequency of the auxiliary fork is precisely sixteen times as great, i.e. 128 ± 16a. If 5 be the number of beats per second between the auxiliary fork and the 1 The phonic wheel, invented independently by M. La Cour and Lord Eayleigh, may be briefly described as a wheel whose rotation is electrically governed so that its speed remains practically constant. 2 562 SOUND CHAP. X standard, the frequency of the latter is 128 ± 16a ± ft. The error in the mean rate of the clock is readily allowed for ; but care is required to ascertain that the actual rate at the time of observation does not differ appreciably from the mean rate. To be quite safe it would be necessary to repeat the determinations at intervals over the whole time required to rate the clock by observation of the stars. In this case it would probably be convenient to attach a counting apparatus to the phonic wheel." 545. Chronographic Method of A. M. Mayer. — In this method adopted by Professor Mayer the fork under examination carries a style and marks a sine graph upon a camphor- smoked paper on a revolving drum. A novel feature of the arrangement lies in the fact that the same style automatically records time upon the trace. This is effected as follows : — The pendulum of a clock at each swing makes electrical contact with a bead of mercury, and so makes and breaks the primary of an induction coil. Now the smoked paper and the style carried by the fork are included in the secondary of this induction coil. Hence at each instantaneous current in the secondary, a small spark occurs at the smoked paper which records the instant in question. If, by the comparison of the wavy and spark traces on the smoked paper it is found that n vibrations occur in t seconds, it is clear that the frequency sought is n/t per second. 546. Thus, in tlais method, although an absolute determination of the frequency is made, uniformity in the rotation of the drum is not essential. It must, however, be noticed that the true period of the fork is slightly interfered with by the friction of the style on the smoked paper. Por, as we saw in the dynamical portion of this work, resistance to a vibration has not only a iirst order effect on the amplitude, but also a second order effect on the period. Accordingly this method must rank as inferior to those in which observations are conducted by optical or acoustical 545-548 ACOUSTIC DETEEMINATIONS 563 means, leaving the fork or other vibrator in its usual un- trammelled state. It is obvious that a slight modification of this ex- perimental arrangement could be utilised for the inverse purpose of measuring small intervals of time, the sparks giving the seconds, and the sine graph tenths, hundredths, or thousandths of a second. 547. Scheibler's Tonometer. — Scheibler of Crefeld pre- pared a set of standard forks which constituted what is termed his tonometer ; and by careful tuning and counting of beats the absolute pitch of each fork was determined. The principle of the method is as follows : — Suppose two forks are tuned exactly to an octave ; let the frequency of the lower fork be denoted by JV, then that of the higher is 2JV. Now let a number of intermediate forks be prepared, each making about four beats per second with the one below and the one above it in the series. Let the exact numbers of the beats per second between adjacent forks in the series be denoted by &j, i^, b^, etc., then clearly we have 2N=If+h, + l, + h, + . . .1 where the summation extends over the octave. Thus knowing N we easily obtain the frequency of any other fork in the series. For obviously if the frequency of the nth fork in the series beginning at the lowest is iV^,. we have N, = Il+h, + h, + . . .\., (2). 548. Thus, the principle involved is to ascertain by beats the difference of frequencies whose ratio is known to be two. Lord Eayleigh points out that if a smaller interval like a fifth, fourth, or major third were obtained precisely by Lissajous' figures, the labour of bridging the larger interval of the octave would be much reduced. Scheibler's tonometer has not only the advantage of great accuracy, but also of portability. It is thus suitable for 564 SOUND OHAP. x bell founders to take with them into belfries to ascertain the pitches of the various tones of any bell which they may have to replace. To determine the pitch of any tone by means of this tonometer it is only necessary to time the numbers of beats per second between that tone and each of the two forks in the tonometer to which the tone is nearest in pitch. Thus, if the forks of the tonometer have frequencies 64, 68-1, 71-9, 76 per second, etc., and the tone to be determined gives 5 '6 beats per second with the 64 fork, 1-5 beats per second with the 68'1 fork, 2-3 with the 71-9, and 6'4 with the 76, then its frequency is obviously 69'6 per second. 549. In order that this method may be trustworthy it is essential that the pitch of each standard fork should be known to not vary when sounded with the one below and with the one above respectively. It is the lack of fulfil- ment of this condition which destroyed the value of the tonometer which was designed by Appun, and consisted of a number of harmonium reeds. And even with the tono- meter of forks, it is necessary for extreme accuracy to know the pitch of each fork at the temperature of use, and its rate of variation of frequency with temperature. According to the observations of M'Leod and Clarke, the frequency of a fork falls by •00011 of its value for each degree centigrade of rise in the temperature. The value found by Koenig for the temperature coefiicient" of forks is -0-000112 per 1° C. Hence the frequency of a 256 fork falls 0-028672 of a vibration per second per 1° C. rise of the temperature. 550. Rayleigh's Harmonium Method for Pitch. — In spite of the fact just noticed that harmonium reeds vary a little in pitch according to whether they are sounded with the note above or below. Lord Rayleigh has utilised them for an absolute determination of pitch. Perhaps it is as well to consider first the method which must be rejected on account of this lack of permanence of pitch. Let the 549-651 ACOUSTIC DETERMINATIONS 565 frequencies of the notes forming the interval of a major third on the harmonium be x and y. Then beats may be heard and counted between the fourth partial of the higher note and the fifth of the lower. Hence if the frequency of these beats is a per second, the fourth partial being the higher of the beating tones, we have 4:i/ — 5x = a (1). Now if the value of «//* were exactly that prescribed in the system of equal temperament, or were something else but accurately known, we should then be able to determine both X and y. But since we cannot be quite sure that the interval is correctly tuned, take the next major third, calling the frequency of this third note z, and the beats reckoned as before b per second. Then we have Az-5ij = h (2). Finally, let the beats between the fifth partial of this third note and the eighth of the first note be c, the latter partial being the higher. Then 8* — 5z = c (3), whence from (1), (2), and (3) we obtain the values of x, y, and z in terms of a, h, and c. Thus x = ^{25a+20l+16c), y = ^(32a+25h + 20c), and s = l(40a + 32&+25c). 551. Now, as already mentioned, this plan would fail of high accuracy, because the frequency of the second note, say, would be slightly different when sounded with the first from that when sounded with the third note of the series, and similarly for the other combinations. In other words, the values of x, y, and a in the equations (1), (2), and (3) are not strictly identical, but have slightly different values in each equation. To remove this source of error we must be able to check 566 SOUND CHAP. X the interval at the same time that the beats are being counted. That is to say, it is necessary to obtain two relations between the frequencies at a single simultaneous sounding of the two notes. Thus, let the equal tempered whole tone be selected as the interval, and call the frequencies v and w. Then it is slightly smaller than the large tone of just intonation 9/8. Hence we may obtain between the eighth partial of the higher and ninth partial of the lower notes slow beats of frequency, say d per second. Then we shall have the relation 9 y _ 8w = rf (4). Again, this tempered interval is considerably larger than the small tone 10/9. Hence there will be quick beats between the ninth partial of the higher and the tenth partial of the lower note. Thus, if these beats are e per second, we have 9w-10v = e (5). From (4) and (5) we obtain « = 9(^ + 8e (6), and w=lQd+'de (7). 552. This was the plan followed by Lord Eayleigh in determining the pitch of the note C on a harmonium. The notes used were C and D, the interval being purposely reduced from the usual equal temperament. This made the quick beats slower, and so facilitated counting. Indeed, in the case noted below, what would naturally be the quick beats were so altered as to be slower than the others. Two observers were necessary, one to count the beats shown in equation (4), and the other to count those shown in equa- tion (5). In one experiment, the time being ten minutes, the beats thus recorded were respectively 2392 and 2341. Hence, c?= 2392/600 and «= 2341/600, 652-564 ACOUSTIC DETEEMINATIONS 5 6 7 whence the pitch of C was given from (6) by v= 67'09 per second (8). But this result is true for the C only when sounding with the D. 553. Simple Determinations of Frequency. — We now pass to those methods for the determination of frequency which are of the simple class suitable for lecture illustration and ordinary experimental work in the laboratory. We shall commence with a few which are mechanical in nature, i.e. methods which assume no acoustical laws or constants. We shall afterwards notice several methods which may be styled acoustical in that they do assume such laws and constants. ExPT. 88ffi. Smart's TVluel. — This method, as its name im- plies, is due to Savart, who first reduced it to a fair degree of accuracy. The apparatus consists essentially of a toothed wheel capable of rotation, a card or plate held so as to be caught by the teeth, a driving gear (or an electric motor could now be substituted) for setting the wheel in rapid rotation, and, finally, a counting arrangement to record the number of turns executed by it. In one apparatus used by Savart the wheel had 600 teeth, and could be driven at various speeds up to 40 revolutions per second. In this extreme case 24,000 taps per second would be given to the card by the teeth of the wheel, and hence a note of frequency 24,000 would be produced. On maintaining the speed of the wheel so as to exactly match the pitch of any note under examination, it is evident that from the counter and the time ascertained by a watch we can calculate the frequency sought. Thus, let the frequency of the note be N, and suppose that the wheel contains n teeth, and makes r complete turns in t seconds when matching this note. Then obviously N = nrlt (1). This method is somewhat rough and may be regarded as inferior to the siren. 554. ExPT. 89. The Siren. — For the purpose of determin- ing the pitch of a pipe or tuning-fork, the form of siren due 568 SOUND oHAP. X to Cagniard de la Tour will be found convenient. It consists essentially of a vertical spindle carrying a disc pierced with a ring of holes which alternately open and close a similar ring of holes on the top of the cylindrical wind chest. At the upper part of the apparatus is added a counting mechanism with fingers and dials which record the number of rotations of the disc. The two rings of holes have opposite obliquities so that the pressure of the air as it escapes through them spins the disc. Thus, by suitable adjustment of pressure, the siren may be tuned to any desired note within certain limits. To determine a pitch by the siren we have obviously to tune the siren to the note in question, observe the indications of the dials at the beginning and end of a timed period, during which the siren is maintained at the right pitch, and then calculate the result as in the case of Savart's wheel, the number of holes in the disc replacing the number of teeth in the wheel. 555. But a little care is necessary both in the tuning and in the counting. For example, at some pressures the siren is distinctly flattened by putting the counting mechanism into gear. Hence, though the siren is usually provided with a sliding motion to put the counters in and out of gear, it is safer to keep them in gear throughout the experiments. If available, a stop watch may be used in conjunction with the siren to facilitate the timing when the counting mechanism is running the whole time. Again, in order to have the air blast sufficiently under control, either of the following arrangements will suffice : — First, Derive the air blast from a large reservoir in which the air is maintained at a constant pressure by an automatic pump and safety valve, air pipes being laid round the laboratory as water pipes are. It is then desirable to connect the siren to the air tap by a rubber pipe fitted with a screw clip. This is the method in use at Nottingham, and acts very well. The screw of the clip can easily be moved an eighth of a turn at a time, and so the air pressure and consequently the pitch of the siren slowly changed. It should be noticed, however, that after even a very slight adjustment of the screw the pitch of the siren continues to creep up or down for, say, a minute or more. The fact that it is so changing is almost imperceptible, but it is better to time a minute by the watch before supposing the pitch to 555-557 ACOUSTIC DETERMINATIONS 569 have reached its final value aue to the change in the position of the screw. Second, If the above arrangement is not available, a foot bellows and wind chest with suitably weighted valve may be substituted. The final adjustment of pitch should still be made by a screw clip on the connecting tube manipulated as in the first installation. In a preliminary adjustment of the wind pressure by the screw, when the pitch is rapidly altering, the experimenter may often feel uncertain whether the siren is sharp or flat, or by how much. It is then convenient to touch the disc or the spindle very lightly with the finger, thus decreasing its speed and so lowering the pitch. By this means the siren, if too sharp, can be brought down to the pitch required, or, if already too flat, the fact immediately ascertained. 556. When wishing to make a very exact determination of pitch, it is not safe to trust either that the siren is exactly tuned, or that it will remain constant in pitch during the counted interval, the better plan being to maintain the sound under examination during the whole time of counting, and to keep the siren the whole time slowly beating with it, the beats being counted also. This delicate adjustment may be preserved by lightly touching the axle with a feather. Thus, suppose the siren to be certainly sharper than the other sound the whole time, and let the total number of beats be h in the counted time of t seconds. Also, let the total number of revolutions of the disc in this time be r, and the number of holes in the disc be n. Then the frequency N of the sound under test is obviously given by N='^ (2). 557. In order to be sure that the siren is throughout sharper than the other note, proceed as follows : — First, Make sure it is sharper to start with by ascertaining that increased pressure of the feather slows the beats. Second, Adjust the beats to about four per second. Third, Keep the beats at about this rate, taking special care that they never get slower and vanish. A typical example would be — ,-. 16x326-76 „_„ , N= -rrzr =257 per second. * 20 570 SOUND Of course, if 6= 0, equation (2) reduces to equivalence with (1) which was written for Savart's wheel. 558. ExPT. 90. Tuning -Fork and Fall Plate. — In this experiment, instead of assuming the accuracy of some watch or clock, we assume the value of the acceleration due to gravity, i.e. the quantity usually denoted by g. The apparatus consists essentially of a smoked glass plate arranged to fall past the tuning-fork whose pitch is to be determined. The fork is mounted vertically in a block A, Fig. 98,Vhich slides in a groove BB in the base board CO, which also carries a Fig. 98. — Fork and Fall Plate. vertical board D to support the smoked plate E ready for its fall. The fork F carries on one prong a light style Gr of thin aluminium foil attached by soft wax. The plate may be conveniently smoked by the fumes of burning camphor. It is hung by a fine thread passing over two nails HH, and under screws at the hack of the plate, which may be the side pinch screws of terminals used for the plates of Daniell's cells. This method of suspension causes the plate to tilt forward very slightly like pictures on a wall. Then, on adjusting matters so that to start with the style just touches the plate, on releasing the plate by burning the thread the style will. be in gentle contact during the fall. To prevent breakage 568-660 ACOUSTIC DETERMINATIONS 571 of the plate, a second loop, J, of stronger thread, may also be used, passing round a lower pair of nails on the board, and attached to the terminals on the plate so as to arrest it at the end of its fall. The arrangement will be understood from Fig. 98. 559. These details being arranged, the fork is set vibrating by a bow (or electrically driven if possible), and the thread burnt between the nails HH. Then as the plate falls, a wavy trace is marked upon it which is clearly the resultant of a simple harmonic motion horizontally, and a uniformly accelerated motion vertically. The acceleration g being assumed, we have thus the data for calculating the frequency N of the fork. But, owing to the very slow motion of the plate near the commencement of its fall, the corresponding wavy trace on the stnoked surface will be very crowded, thus rendering a counting of the waves practically impossible. It is accordingly advisable to avoid this region and proceed as follows : — Select two portions PQ and QE where the waves can be readily counted, and each containing the same number n of waves, their lengths being \ and l^ respectively. Further, let njN the time of fall from P to Q or Q to R = i Then, denoting by u the speed of the plate when P passes the style, we have by elementary kinematics l^ = ut + hgfi (1). But the speed as the point Q passes the style is v, + gt ; hence for the portion QR we have ?3 = (m + gt)t + yf (2). Thus (2) - (1) gives l2-K = 9t^ (3). "Whence, on writing for t its value n/N, we have N=ns/JL^ (4). 560. It may be noted here that the frequency of the fork is somewhat disturbed both by the mass of the style and its friction on the plate, and both these sources of error flatten the fork. These could be allowed for by tuning a second fork precisely to the one under test when without the style, and then counting the beats between the two when the first 572 SOUND CHAP. X was again provided with the style, and it was rubbing against the smoked plate. Thus if the beats between the two were h per second, we should have for the frequency iVg of the undisturbed fork, N„ = N+b = b + n\/.^ (5). If desired a correction to g may be made for the effect of friction which retards the plate in its fall. This could be approximately obtained from a very good fork whose frequency was accurately known, and finding g from equation (4). This value, slightly less than the true g, would then be inserted in (5). 561 ExpT. 91. Pitch from Stationary Waves. — The pro- cedure is as follows : — The source whose pitch is to be determined is excited and sustained, the progressive waves proceeding from it are reflected normally from a plane surface so as to set up stationary waves. The positions of the nodes or antinodes are located, and their distance apart measured along the line normal to the reflector. Then we have Z = A/2 (1), where I is the distance between successive nodes and A the length of the progressive waves. But, if N is the frequency of the tone and v the speed of sound in air, we have V = NX (2). Hence by (1), v=2Nl, \ V 1 + «' or. N=vl2l= '^^':"-' (3), where iiq is the speed of sound at 0° C, t the temperature in degrees C, and a is the coefficient of gaseous expansion (0'003665 per 1°C.). Thus, measuring I, observing i, and assuming v^, we may calculate N, the frequency sought. In the practice of this method several precautions should be noted. In the first place, the reflecting surface must be such as to insure adequate reflection. Thus a light board might only form a resonator, and a cloth would stifle the sound. A smooth plaster wall or a sheet of glass will 561-563 ACOUSTIC DETEEMINATIONS 573 do. Secondly, the surface must be large enough in comparison with the length of the waves used to obviate diffraction. Again, the distance between successive nodal or antinodal positions must be measured along the normal to the surface through the source of sound. 562. Then to find the nodal positions the sensitive flame used must be chosen so as to have sufficient but not excessive sensibility. For a small organ pipe or tuning-fork of pitch in the treble staff, the sensitive flame obtained from an ordinary small Bunsen burner has been found suitable (see article 282). Again, in using the sensitive flame it is well to locate by it the places of least disturbance, as these are usually more sharply defined than the places of maximum disturbance. Sensitive flames are usually most afi"ected by the places of maximum motion, i.e. by the antinodes. Thus, the places of least disturbance would be the nodes, and these occur at the reflector and at distances from it of A/2, 2A,/2, 3A/2, etc. If, however, the particular sensitive flame in use behaves in the opposite manner so that its places of least disturbance are the places of no variation of pressure, i.e. the antinodes, these would occur at distances from the reflector of A/4, 3A/4, 5A/4, etc. Hence in this case the first distance from the reflector would be only A/4, whereas all the intermediate distances in the first case and the others in the present case would be A/ 2. It is also well to note that the position of a node or an antinode is fixed simply by that of the reflector and the length of the waves in use, and is quite independent of the exact distance of the source from the reflector. 563. ExPT. 92. Pitch by Monochord from Difference and Ratio of Frequencies. — For this method, in addition to the monochord and source of sound whose pitch is to be determined, we require a second source of sound to beat with the first. Hence by counting beats we obtain the difference of frequencies of the source under test and the auxiliary source. Also by finding the lengths of the string which at a given tension are in unison with the two sounds, we infer the ratio of their frequencies, which are inversely as the lengths observed. Thus, let the source under test (a fork or organ pipe say) and the auxiliary have frequencies N and N' and be in unison with lengths of the string L and L' respectively, L being greater than L'. Let the 574 SOUND CHAP. X beats between the two sources be h per second. Then we have L'N' = LN (1), and N'^N+h (2), whence N = y — p (3) gives the frequency sought. 564. In the practice of this method care must be taken that the tension of the string remains precisely the same when the lengths in unison with each source are determined otherwise equation (1) would not hold. Thus, in determin- ing the lengths, the movable bridge used should be as accurately as possible of the same height as the fixed terminal bridges, and the requisite pressure of the string upon it must be supplied externally, say by the edge of a small coin pressed upon it at the place. A little consideration will show that this is not a refined method, but rather interesting from its simplicity. Thus, if the number of beats per second were five (and quicker beats are difiicult to count), we should need ten seconds uninterrupted flow of both sounds to give fifty beats, and enable the observer to determine h with an accuracy of one per cent. And equation (3) shows that the accuracy of the value obtained for N is not greater than that with which h is found. Again, if the pitches are high the method fails in accurac}^, since then the five beats per second would correspond to a very slight difference of lengths L and L'. But for pitches of about 100, or say 128 (the c of the physical apparatus makers), the method succeeds fairly well. Further, the steel wire which usually forms the " string " of the monochord is more satisfactory at these pitches than when tuned much higher. 565. ExPT. 93. Pitch by Vertical Monochord. — In this experimental determination we assume the absolute relation between the pitch and all the circumstances of the case instead of the law of dependence of pitch on length simply as in the previous experiment. And in order to know precisely what the tension is, a vertical monochord is used, having a fixed bridge at the upper end of the string and a movable one to define the lower end of its vibrating portion. 564-567 ACOUSTIC DETEEMINATIONS 5Y5 Tension is then applied directly by a weight at the bottom of the string, and the vertical position of the monochord obviates the uncertainty of tension which in other positions would arise from the friction of the string over the bridge. It is as well also to pull the string away from the lower bridge before each determination, and then press it to the bridge by the rounded edge of a metal plate carefully applied, so as not to change the tension, but simply the length as desired. Further, to avoid complications due to the stiffness of the string raising the pitch, a thin wire should be chosen. 566. With these precautions the string is brought into unison with the source of sound under test, the first rough adjustments being made by changing the weights used and the final adjustment by sliding the bridge. The length is then measured, and the mass per unit length ascertained say by weighing a measured length of the wire or other cord in use. The frequency sought is then calculated from the equation developed in art. 165, viz, where I is the length in cm. of the vibrating portion of the string, F is the tension in dynes, and o- is the linear density of the string in gm./cm. For good work the value of a- must be known at the temperature at which the string is used in vibration. It may accordingly be determined at this temperature by direct weighing and measuring, or if a well-known standard substance, it may be calculated from its tabular density /d^, at 0° C, its radius ?•„ at 0° C, and its coefficient of linear expansion s ior 1° C. Thus, at f C. we have 567. ExPT. 94. Fitch by adjustable Resonator. — An indirect method of finding the speed of sound in air from a fork of known frequency was detailed in article 5.32. This can obviously be used to find the frequency of a fork when the speed of sound is assumed. It is accordingly named here for completeness' sake, but need not be again described 576 SOUND CHAP. X as the manipulation is just the same. The experimenter has merely to treat N as the unknown instead of v. 568. ExPT. 95. Fitch from Transverse Vibrations. — A strip of steel or other metal is used fixed at one end, free at the other, and set in vibration. It is then assumed from theory (see art. 205) that the frequencies of such strips are inversely as their lengths squared. A length L is found such that the pitch of this portion of the strip is then in \mison with the fork or other source of sound whose frequency N is to be determined. The length of the vibrating portion is then increased to say L', so as to make its frequency N' so small that it can be counted and timed by a good watch, preferably a stop watch ; thus N' is known. But from the theoretical relation, we have NL^ = N'L''^, or N=N'L'^jL\ which determines N. The strip may be about 2 cm. wide, and say 1 mm. or less in thickness. It is conveniently clamped by an ordinary engineer's vice if such is available, in which case the bar may be horizontal but edge up, and the vibrations horizontal. Failing the vice, the strip may be laid on a strong table with an end projecting over the edge, and weights placed over that part on the table. In this case though the strip is horizontal the vibrations will be vertical, and hence gravity will enter as a disturbance. 569. Variation of Pitch and Decrement. — From the consideration of the determination of pitch simply we now turn to other matters intimately connected with it, taking first the variation of pitch and decrement with amplitude as investigated by K. Hartmann-Kempf. In this research the method of letting the vibrating body mark a sine graph upon a smoked surface was necessarily rejected, as the friction of the style upon the surface would change the damping and period, thus altering the very quantities which it was desired to measure. A photographic method was accordingly adopted in which the tuning-fork or other vibrator carried a mirror, thus reflecting light upon a sensitive film passing over a drum, and so yielding a wavy trace which shows the damping per wave. But it was 568-570 ACOUSTIC DETEEMIlSrATIONS 577 also desired to obtain information as to the very slight change of period, as the vibrations died away, and for this very delicate purpose the uniformity of rotation of the drum could not be trusted. Hence, in the path of the beam was interposed a disc of yellow glass with a slit in its middle. This was carried by an auxiliary tuning-fork whose vibrations were electrically maintained. White light passed through the slit for about one ten-thousandth of a second each time this auxiliary fork was at its equilibrium position. And since this occurs twice in the period of the auxiliary fork, a series of bright dashes thus obtained on the photograph correctly indicated these intervals of time. During the remainder of the auxiliary fork's motion the actinic quality of the light was much weakened by its passage through the yellow glass disc, and its power of giving an intense photograph reduced accordingly. But in spite of this reduction the light was well able to give a good photograph at the extremities of the wavy curve where the full amplitude was reached, and the bright reflected spot paused in the film before making its return motion. And it is at this very place and there only that a good print was needed in order to register the amplitude of the fork's motion. 570. Thus, the white light through the slit gave a series of bright dashes occurring at instants separated by rigidly equal intervals of time, and so by their position on the film forming an accurate time record. While the yellow light passing through the disc not only showed quite distinctly those portions of the trace required of it, namely, the turning-points of the waves, thus recording the amplitude of each vibration, but also in the other parts of the wavy trace was faint enough to avoid obscuring the bright dashes. Hartmann-Kempf found that the relation between logarithmic decrement' and amplitude for three makes of tuning-fork is almost linear. The dependence of frequency 2 p 578 SOUND on amplitude was, however, found to be more complicated. Both are shown in the curves of Fig. 99. In all cases it is seen that the decrement increases and the frequency decreases with increasing amplitudes. 571. Reaction of Resonator on Pitch of Fork. — This subject was investigated by Koenig, who found that a certain fork of 256 per second sounded for about 90 seconds OS 10 vs Amplitudes Fig. 99. — Variation op Pitch and Decrement. without a resonator. An adjustable resonator was then brought near it, but its pitch set much lower than that of the fork. The resonator was then gradually raised in pitch, and when it was still a minor third below the fork the latter failed to sound quite so long as before, and at the same time received a slight increase of frequency. As the resonator was tuned more nearly to the fork this decrease of time and increase in frequency became each more pronounced. But, at the instant when unison was 571, 572 ACOUSTIC DETEEMINATIONS 579 established between the natural tones of fork and resonator, the change in the fork's pitch suddenly disappeared. At the same time the sound was powerfully reinforced, but lasted only about 10 seconds. When the pitch of the resonator was still further raised, so that its natural tone was now sharper than that of the fork, the sound of the fork changed in the opposite direction, being now a little flatter than its proper pitch. The maximum disturbance of frequency found by Koenig was 0"035 complete vibration per second. Lord Eayleigh has shown that the above phenomena are in accordance with theory, and that " on whichever side a slight departure from precision of adjustment may occur the influence of the dependent .vibration is always to increase the error." 572. Subjective Lowering of Pitch. — Dr. C. V. Burton in 1895 called attention to the subjective lowering of pitch in the case of forks when flrst bowed. This was stated to occur to the extent of a semitone, a whole tone, or even a minor third in the case of large forks strongly bowed. The pitch recovers its normal value as the amplitude dies away. The phenomena were confirmed by other observers. To account for them the following hypothesis was put forward : — It is known that large amplitudes of a vibrator may entail a slight rise of pitch if the elastic forces are not strictly proportional to the displacements (see e.g. Chap. IV., art. 114). Thus, for large amphtudes the microscopic structures in the internal ear might be as it were tuned up above their normal responsive pitch. Hence, in response to large external vibrations of say 100 per second, we might have appreciable motions in those structures whose natural frequency for small vibrations is less than 100, but whose frequency for these larger vibrations is 100 or thereabouts. There would, accordingly, be not one, but a number of these' structures throughout a certain 580 SOUND CHAP. X range affected, and the region of maximum disturbance would be among parts whose natural frequency for small vibrations is below 100. But on HeUnholtz's theory our perception of pitch is fixed for each such structure and the mechanism by which it communicates with the brain. Thus, the pitch assigned to such a loud sound would be somewhat lower than the true pitch, being some kind of mean among the different pitches proper to the various structures stimulated. In the discussion at the Physical Society on this subject, Prof. C. V. Boys said " he found that by careful attention he could apparently persuade himself that the note in Dr. Burton's experiment was lowered or raised in pitch, or that it remained unaltered. A similar effect in the case of the eye could be obtained with stereoscopic pictures." 573. Total Number of Vibrations needed for Sensa- tion of Pitch. — Experiments to determine this point have been made by a number of physicists, perhaps the fullest being by W. Kohlrausch. He used an arc of a circle carrying a limited number of teeth and attached to a pendulum. When the pendulum was let go the teeth struck a card suitably held. The sound thus produced was compared with that of a monochord. By varying the length the string was tuned till it was just perceptibly higher or lower than the sound from the card. The ratio of these lengths defined the characteristic interval which expresses the precision with which the pitch could be estimated from the given total number of vibrations. Some of the results of these experiments are shown in Table LXIII. [Table 573,574 ACOUSTIC DETEEMINATIONS 581 Table LXIII. — Total Vibrations and Estimate of Pitch Total Number of Vibrations in Sound from Card. Characteristic Interval expressing Pre- cision in Estimation of Pitch. Ratio of Frequencies. Value in Cents. 16 9 3 2 0-9922 0-9903 0-9790 0-9714 14 17 37 50 This shows that even with so few as two vibrations a fair estimate of the pitch can be formed. 574, Lowest Pitch Audible. — To determine the lowest sound audible, Savart used a bar about two feet long re- volving on an axis between two thin wooden plates distant about a tenth of an inch from it. He maintained that a grave, continuous, very deafening sound was thus produced with 7 or 8 vibrations per second. Helmholtz, however, declined to accept this result, and attributed the sound heard to upper partials. In order to know exactly which tone was being heard, Helmholtz used his double siren. In this siren (as explained in article 296), when the rings of 12 holes are open in both wind-boxes, and the handle which moves the upper wind-box is rotated once, there are 4 beats for the primes, but 8 for the second partials, and 12 for the third partials. Now, when the pitch of the primes was 12 to 13 per second a tone could be heard, but the test by beats showed it to be the third partial. For, corresponding to each turn of the handle, 12 beats were heard. Thus, the tones actually heard were from 36 to 39 per second. Again, with primes of frequencies 20 to 40 per second, 8 beats occurred to each turn of the handle ; thus showing that it was the second partial of frequency 40 to 80 that was audible. And it was not until 80 puffs of air occurred per second that the 4 beats of the primes themselves were heard. Helmholtz further tried 582 SOUND CHAP, x experiments with wide-stopped organ pipes, as particularly- suited for giving powerful primes with but little admixture of near partials. He found that the lower tones of the 16 -foot octave, E^ to C^, began to pass over into a droning noise so that the pitch was uncertain. He considered that in the case of the 16 -foot C^ of the organ with 33 vibra- tions per second, although there was a tolerably continuous sensation of tone, it was almost possible to observe also the separate pulses of the air. Helmholtz also obtained deep tones fred from all near partials by using a thin brass string weighted with a coin. In this case the partials were inharmonic, and those nearest to the prime were distant from it several octaves. For B^, 37^ per second, there was only a very weak sensation of tone, and for B^^\>, 29 J per second, there was scarcely anything audible left. Helmholtz afterwards investigated the matter with two large tuning-forks by Koenig fitted with sliding weights. One gave from 6 1 down to 3 5 vibrations per second ; the other from 35 down to 24. For 30 vibrations per second Helmholtz could still hear a weak drone, for 28 scarcely a trace, although the amplitude was 9 mm. 575. Preyer, using loaded - tongues or reeds, considered that he heard down to 15 vibrations per second. But Helmholtz expressed doubts about such results unless checked by the counting of beats. This check, how- ever, was supplied by Mr. Ellis, the translator of Helmholtz, in experiments in South Kensington Museum, on a copy of Preyer's instruments. Ellis heard the beats at 4 per second quite distinctly from the reeds of frequencies 1 5 and 19, and thus concluded that the lowest partial of the reed at 15 per second was effective. The, lowest pair of reeds from which Ellis was able to hear the bell -like beat of the lowest partials distinct from the general crash, had frequencies 30 and 34. Probably the limit of low pitch for audibility varies from one person to another, and even with the same person may be subject to iluctuations from 575, 576 ACOUSTIC DETEEMINATIONS 583 time to time. It would appear that not much continuous tone is usually heard from vibrations slower than about 30 per second. 576, Highest Pitch Audible. — The highest pitches in musical use are, on the pianoforte a'^, say 3520 per second, and in the .orchestra, on the piccolo d", say 4752 per second. Appun and Preyer with small tuning-forks bowed reached e^'" of 40,960 per second. Ellis could hear these 100 feet away. Lord Eayleigh has constructed bird calls giving tones up to 50,000 vibrations per second. But these high frequencies were detected by sensitive flames, nothing was heard from them above 10,000 per second. Thus, the extreme limits audible by one means or another may be assigned as about 15 to 40,000 per second, the musical limits being about 40 to 4000 per second. Thus, whereas the eye sees a range of barely one octave (from the extreme red of the spectrum of frequency say 400 billions per second to about 760 billions per second at the extreme violet), the ear has an extreme range of say eleven octaves, about seven octaves being musically available. To facilitate the determination of the difference of the upper limit of audibility in different persons Gallon's whistle is very useful. It is a miniature adjustable stopped organ pipe whose setting can be read off from linear and circular scales in the manner of a micrometer gauge. The air is supplied by an india-rubber bulb compressed in the hand. Sometimes the difference between two observers is accidentally shown in a striking manner. Thus, on one occasion, a whistle was alternately sounded by air and by ordinary coal gas. "With the coal gas put on one observer exclaimed, " The pitch now rises about a minor third," but the other replied, " N"ay, the sound has ceased altogether." The explanation, of course, being that to the second observer the lower sound lay beneath his upper limit, but the higher sound lay just above that limit, and so in consequence entirely escaped his perception. 584 SOUND CHAP. X ' 577. Harmonic Echoes. — It has sometimes been noticed that the echoes returned from groups of trees are apparently raised an octave. This phenomenon has been mathe- matically treated and explained by Lord Eayleigh. Let T denote the small volume throughout which the constant of compressibility m and the density of the medium o- experience changes whose mean values are respectively Am and Ao-. Let the incident or primary waves be expressed by where Sq is the condensation, e, the base of the Naperian logarithms, i= J —\, /^ = 27r/X, and w is the velocity of sound. Then Eayleigh has shown that the effect due to the abnormal space T can be expressed by the equation s 7rr('-*'"-rAm Ao- ] -= — T-^—\ — -^—A (2)- In this s denotes the condensation of the secondary waves at the time t and at the distance r from the place of dis- turbance, /i being the cosine of the angle between x and r. "Since the difference of phase represented by the factor e"'*^'' corresponds simply to the distance r, we may consider that a simple reversal of phase occurs at the place of disturbance." It should be noticed that the amplitude of the reflected or secondary waves is inversely proportional to the first power of the distance r but to the square, of the wave length A.. Further, of the two terms on the right in equation (2), the first is symmetrical round the place of disturbance, but the second varies as the cosine of the angle between the primary and secondary rays. Thus, a place at which the compressibility m varies behaves as a point source or simple source. But a place at which the density a varies behaves like what Eayleigh calls a double source. In other words, the first may be compared with radial oscillations of a single small sphere, but the second 577, 578 ACOUSTIC DETERMINATIONS 585 with equal and opposite oscillations of two neighbouring spheres with centres on the axis of x. 578. Lord Eayleigh derives equation (2) by analysis too advanced for introduction here, but he afterwards shows that the results may be inferred from the method of dimensions as follows : — Thus Am and Act being given, the amplitude of the secondary disturbance is necessarily pro- portional to T, and in accordance with the principle of energy must also vary inversely as r. Then the only quantities dependent on space, time, and mass, of which the ratio of amplitudes can be a function, are T, r, X, u, and cr. But the ratio of amplitudes is a pure number, and hence cannot involve cr, as it is the only one of the five containing mass, nor u, as it is the only one of the five involving time. Thus, if X be involved to the power a, Tr'^X^ must be independent of the unit of length. That is (L^)L-'^L'' = i", or 3 - 1+ a = 0. Hence a = — 2, or the amplitude of the secondary waves varies inversely as the square of the wave length X.. But the intensity of a wave varies as the square of the amplitude, accordingly the intensity of the secondary waves must vary inversely as the fourth power of the wave length. Thus, " the octave, for example, is sixteen times stronger relatively to the fundamental tone in the secondary than it was in the primary sound. There is thus no difficulty in under- standing how it may happen that echoes returned from such reflecting bodies as groups of trees may be raised an octave." ^ For, in the case just mentioned, the primary sound incident upon the obstacles would be estimated as of the pitch of its fundamental tone, whereas in the secondary waves sent out from the obstacle that funda- mental would be practically masked by the exaggerated prominence of its octave, which would accordingly be taken as fixing the pitch. 1 Theory of Sound, p. 1 53 of vol. ii. 586 SOUND CHAP. X 579. Musical Echo from Palisading or Overlapping Fence. — We have just seen how the pitch may be apparently changed by a diffuse reflection or scattering. Another phenomenon, perhaps more remarkable though simpler in theory, is that in which the pitch may be said to be originated by reflection. To realise this effect a very sudden sound is needed near a palisading or overlapping fence presenting a number of equidistant and parallel re- flecting surfaces. Then regarding the original sound incident upon the fence as a single impulse or small part of a wave, it is clear that we shall have a multiple reflection from the several bars or overlapping edges of the fence. Moreover, as these successive bars are at distances from the observer which increase by nearly constant amounts, the arrival of the successive reflections build up a tone of a definite frequency. On walking along a hard pavement in nailed boots past an overlapping wood fence this phenomenon may be readily observed, especially if the heels are purposely struck down at each step more forcibly than usual. In a case under the writer's notice it was clear that the distinct musical ring often noticed when walking near a certain fence was due to this cause. For, .while the pavement consisted of asphalt along the whole of one side of a park, the over- lapping wood fence forming the first section of its boundary was followed for some distance by a rough, irregular stone wall, after which came a third section consisting of wood fence as at first. On walking past this boundary the musical echo was well heard from the first and third portions where the wood fence formed the boundary, but refused utterly to come from the wall forming the second or intermediate portion. 580. The succession of surfaces offered to sound by a palisading and an overlapping fence may be compared with the optical arrangements known as the diffraction grating and the echelon respectively. But although the palisade and fence might conceivably be used acoustically 579-581 ACOUSTIC DETEKMINATIONS 587 as the grating and echelon are used optically, they are not so used in the case referred to. For in the optical case with monochromatic light the incident waves form a con- tinuous train of definite frequency and wave length, and spectra are in consequence formed in certain definite positions depending on that wave length. In the acoustical case under discussion the original distm-bance incident on Fig. 100. — Wood's Photoqkaph of Mbsical Echo. the successive bars is almost confined to a single pulse or part of a wave. Thus the frequency of the succession of impulses received at any point depends upon the difference in length of the paths traversed from the source to the observer by the waves incident upon the various successive bars. The actual formation of these waves by reflection from such a stepwise surface is shown in Fig. 100, which is from an instantaneous photograph by Prof. E. W. Wood of the waves in air from an electric spark.' 581. iEoliau Tones. — Strouhal investigated the pheno- 1 See Phil. Mag., pp. 218-227, 48, 1899. 588 SOUND CHAP. X mena involved in the generation of the tones of the ^olian harp, which sounds when the wind plays upon the stretched wires. He found that the pitch of the Jlolian tone was independent of the length and tension of the wire, but depended on its diameter and the speed of the wind. Thus denoting these quantities by d cm. and w cm./sec. respectively, he obtained for the frequency N the relation i\^= 0-185 wjd. When, however, the ^olian tone coincided with one of the proper tones of the wire, the sound was greatly rein- forced. Under the more extreme conditions the observed frequency deviated from the value of N given above. Strouhal showed also that with a given diameter and a given speed of wind, a rise of temperature was attended by a fall of pitch. 582. Minimum Amplitude Audible. — So far back as 1870, Toepler and Boltzmann made an estimate on this subject. Their method involved an application of Helmholtz's theory of the open organ pipe. They found that plane waves of frequency about 180 are just audible if the maximum value of the condensation s is 6'5 x 10"^ Now 'ws^jx. = 2/max, = tt>ymax.) whcrc 2/ and i/ are the displace- ment and velocity of the vibrating particles, v is the speed of sound, and to = 27r times the frequency N. Thus the amplitude is given by i-s^ax 34,000 X 6-5 <, ,„ fi « = 2/.„ax. = TT^: = ., ' ,„^ — ^8 = 2 X 1 -« cm. nearly. 2-itN 2ir X 180 X 10*" 583. Baylcigh's Whistle Method. — Lord Eayleigh carried out two methods for the determination of the minimum amplitude. In the first the source of sound was a whistle mounted upon a Wolfe's battle. This was blown from the lungs so as to maintain a steady pressure measured by a water column 9-5 cm. high. This gave a sound heard without effort in both directions to a distance of 82,000 582-584 ACOUSTIC DETEEMINATIONS 589 cm. A laboratory experiment showed that the air passing through the whistle at the above pressure was 196 c.c. per second. Hence the energy expended in the whistle was E=196x9|-x981 ergs per second (1). So this is the rate at which energy passed through the surface of a hemisphere whose centre is the whistle near the ground and whose radius is 82,000 cm. If the amplitude at this surface is a, we have as the expression for the activity per unit cross-sectional area (Equation 6, art. 146), A = IpaWv (2), where /) is the density of the air and equals about O'OOIS, o) = 2'irN and v is the speed of sound which was in this case 34,100 cm. per sec, the frequency being about 2730 per second. Thus, for the hemisphere in question we have as the total activity 27rX 82,000^ times the above ex- pression (2). Hence equating this to (1) we have 27r X 82,000' X ^ X 0'0013 x a\4.ir^ x 2730') x 341,000 = 196x9^x981 (3), whence a= S'l x 10"^ cm. In this case the maximum velocity of the vibrating particles would be 0'0014 cm. /sec, and the maximum con- densation s = max. vel.-^ speed of sound = 4'1 x 10 "^ It should be noticed that the energy expended in blow- ing the whistle is not all converted into the energy of sound. Hence equation (3) is not strictly true, but gives an upper limit to the amplitude on the supposition that all the energy went into sound. 584. Rayleigh's Fork Method for Minimum Ampli- tude. — In this method the activity of the source is estimated from the decrement of the stock of energy possessed by it at a given standard amplitude. This activity is then equated to the usual expression for the activity in the wave 590 SOUND CHAP. X front at the position of minimum amplitude for audibility. The details are as follows : — Let the amplitude of the ends of- the fork prongs at time t be V = V,e-''" (1)- Then the law of the energy of the vibrating fork may be written E=E^e-'^' (2). For the energy varies as the square of the amplitude, hence its decrement involves, a double index. To ascertain the value of k, let the time t,^ be observed in which the amplitude sinks to l/wth of its initial value. Then, we have whence 21og,w But since the activity A of the fork at any instant is its rate of loss of energy at that instant, we have A= - dEjdt = kE^e " *' = kE (4). Thus, if Eq, k and t are known, A may be calculated. 585. But it is only the activity expended on sound that we require. So let k express the decrement of energy with the resonator in use and k-^ the corresponding decrement without the resonator. Then k can be divided into two parts as expressed by A ^ /Sj + Aj KpJ' in which k^ will represent the extra decrement of energy due to the presence of the resonator, and so covers that spent in producing sound, together with that which is dissipated in internal losses in the resonator. Thus the value k^ is still only a superior limit for the decrement corresponding to sound-production, but must be taken as being the nearest 585, 586 ACOUSTIC DETERMINATIONS 591 approximation attainable. Hence, for the activity produc- ing sound we may write A.^^'k^E^e-^* = li^E (6). Thus, by experiments with the resonator and without, Ic and \ are found by use of equation (3) ; /cj is then inferred from (5). The values of k^ and Ic are then used in (6). A little reflection will show that it is the h without a subscript that occurs in the index of e in equation (6). In conducting the main experiment, one operator gives the fork a large vibration, and observes with a microscope till the standard amplitude is reached corresponding with Eq in (6). He then gives a signal and withdraws. On receipt of this signal the other experimenter, who is stationed at a distance r from the fork, observes the time t for which the fork is still audible. Thus, from (6) the minimum value of A^ is obtained in terms of E^ and the /c's. Now the value of E^ for both prongs of the fork when the amplitude at their ends is t/q is calculated by Eayleigh to be where a; h, and I are the density, cross-section, and length of the prongs, and N their vibration frequency. Then, equating the expression for A^ from (6) and (7), with the usual one for the wave front taken as hemispherical of radius r, we have l^alWN^ Vo'e - "* = 2-Kr\yv{2-i7Nfa? (8), where p is the density of the air, v the speed of sound, and a the minimum amplitude for audibility. 586. In an experiment on a fork of frequency 256, the vibrations fell from 0-020 cm. to 0-010 cm. in sixteen seconds without, and in nine seconds with, the resonator, whence ^= 0-154, /&! = 0-0866, and ^-5= 0-0674 nearly. And for this fork J!o = about 4-06x10^ ergs. Thus hoK- Thus, from (1) is obtained for this case — r 7-1-1 f fV'-dxdt (pi -Po^t = rr'P" I — I — "^ -*■ 605. Hence the mean additional pressure upon the piston is now (7-I- l)/2 of the volume-density of the total energy. We fall back on Boyle's law by taking 7=1. It appears, therefore, that the result is altered when Boyle's law is departed from. Still more striking is the alteration when we take the case treated in Theory of Sound, S 250, of the law of pressure — p = constant — a^pi^jp. According to this, / {p^ = a? and /"(/jo) = — 2aypo. Then (1) gives for this case — fip,-Po)dt = (4). The law of pressure here used is that under which waves of finite condensation can be propagated without change of type. In (4) the mean additional pressure vanishes, and the question arises whether it can be negative. It would appear so. If, for example, p = constant — a^p^jip^ ; then fipo) = *^/"(/^o) = - ^»'/po- and Pn f f U^dxdt (i^i-Fo)*=-9Jj— ^ (5). 606. The question of the momentum of wave trains is then considered, and it is shown that momentum = \ P°LApI ^ I x total energy (6), I 4a' 2a I ^■' ^ ' 605-607 ACOUSTIC DETERMINATIONS 605 the a now denoting the velocity of infinitely small waves. This may be compared with (1). If we suppose the long cylinder of length I to be occupied by a train of progressive waves moving towards the piston, the integrated pressure upon the piston during a time Ija should be equal to twice the momentum of the whole initial motion. The two formulae are thus in accordance, and (6) needs no detailed discussion. It may suffice to call attention, j^rs<, to Boyle's law, where f"{po) = 0, and, second, to the law of pressure leading to (4), under which second case progressive waves have no momentum. It would seem that pressure and momentum are here associated with the tendency of waves to alter their form as they proceed on their course. 607. Quality and Phase. — In the present chapter on acoustical determinations we have dealt with the velocity of propagation of sound, also, the pitch and amplitude of musical tones. There is but little to be said here about the third feature of a musical tone, namely, its quality. The two questions respecting quality that have not been already sufficiently discussed, and may be dealt with here, are the following : — (1) Does quality depend on phase? (2) Are vowels characterised by a fixed or a variable pitch of resonance ? To answer the first question Helmholtz experimented with a set of electrically maintained tuning-forks forming a harmonic series. In this set of forks he had two ways of weakening any individual tone. One way, that of placing the resonator a little farther away, had no effect on the phase. Whereas the other way, namely, that of shading the mouth of the resonator, and thus putting it a little out of tune with the fork, both weakened the tone in question and changed its phase also. By this means a difference of phase of one-quarter of a period could be obtained, and by reversal of the electric current a difference of half a period could be made. Thus full power over the phase was 606 SOUND CHAP. X attained. Speaking of the results of experiments con- ducted thus, Helmholtz said : " So far as quality of tone was concerned, I found that it was entirely indifferent whether I weakened the separate partial tones by shading the mouths of their resonance chambers, or by moving the chamber itself to a sufficient distance from the fork. Hence the answer to the proposed question is : The quality of the musical portion of a compound tone depends solely on the number and relative strength of its partial simple tones, and in no respect on their difference of phases." 608. Helmholtz afterwards pointed out what he called an apparent exception to the above rule. Koenig, however, considered it a real exception. It is the case of two forks sounding a slightly mistuned octave. Koenig attacked the question as to the influence of phase on quality by means of his wave siren. In this instrument the wind issues from a slit-like aperture, and encounters the margin of a disc whose departures from the form of a circle consist of a set of regularly recurring waves made of any desired pattern. The slit from which the wind issues is in the position of a radius of the disc. Thus, as the wave-formed margin passes round the air blast is alternately increased or diminished according to a simple or compound harmonic law corresponding to the shape of the particular disc in use. 609. Speaking of these experiments by Koenig, Mr. Ellis, the translator of Helmholtz,^ says Koenig "compounded harmonic curves of various pitches, and with various assumptions of amplitude, under four varieties of phase: (1) the beginning of all the waves coinciding; (2) the first quarter ; (3) the halves ; and, (4) the third quarters of each wave coinciding — briefly said to have a difference of phase of 0, ^, ^, p These were reduced by photography, inverted, and placed on the rim of the disc of a wave siren, and then made to speak. He gives the remarkable curves which ' Sensations of Tone, p. 537. 608-611 ACOUSTIC DETERMINATIONS 607 resulted in a few cases, and instructions for repeating the experiments. The following are his conclusions : " ' The composition of a number of harmonic tones, in- cluding both the evenly and unevenly numbered partials, generates in all cases, quite independently of the relative intensity of these tones, the strongest and acutest quality of tone for the ^ difference of phase, and the weakest and softest for |- difference of phase, while the differences and ^ lie between the others, both as regards intensity and acuteness. " ' When unevenly numbered partials only are com- pounded, the differences of phase |- and ^ give the same quality of tone, as do also the differences and J ; but the former is stronger and acuter than the latter. 610. " ' Hence, although the quality of tone principally depends on the number and relative intensity of the harmonic tones compounded, the influence of difference of (phase) is not by any means so insignificant as to be entirely negligible. We may say, in general terms, that the differences in the number and relative intensity of the harmonic tones compounded produces those differences in the quality of tone which are remarked in musical instru- ments of different families, or in the human voice uttering different vowels. But the alteration of phase between these harmonic tones can excite at least such differences of quality of tone as are observed in musical instruments of the same family, or in different voices singing the same vowel.' " 611. Vowel Pitches. — We now come to the second question, namely, as to the resonance pitch for a given vowel. Does it vary with the pitch of the note to which that vowel is sung or is it fixed ? Or, in the words of Lord Eayleigh, is " a given vowel characterised by the permanence of partials of given order (the relative pitch theory), or by the permanence of partials of given pitch (the fixed pitch theory) ? " Willis decided the question in favour of the fixed 608 SOUND CHAP. X pitch theory, and Helmholtz seemed to hold the same opinion. " If indeed, as has usually been assumed by writers on phonetics, a particular vowel quality is associated with a given oral configuration, the question is scarcely an open one. Subsequently, under Helmholtz's superintendence, the matter was further examined by Auerbach, who along with other methods employed a direct analysis of the various vowels by means of resonators associated with the ear. His conclusion on the question under discussion was the inter- mediate one that both characteristics were concerned. The analysis showed also that in all cases the first, or funda- mental tone, was the strongest element in the sound." " Hermann pronounces unequivocally in favour of the fixed pitch characteristic as at any rate by far the more important, and his experiments apparently justify this con- clusion. He finds that the vowels sounded by the phono- grq,ph are markedly altered if the speed is varied." The theory and action of the phonograph is given in the next chapter in connection with which the subject of vowel pitches is again referred to. 612. Perception of Sound Direction. — Some years ago Lord Eayleigh ^ " executed a rather extensive series of experiments in order to ascertain more precisely what are the capabilities of the ears in estimating the direction of sounds. It appeared from these that, when the alternative was between right and left, the discrimination could be made with certainty and without moving the head, even although the sounds were pure tones. On the other hand, if the question was whether a sound were situated in front of or behind the observer, no pronouncement could be made in the case of pure tones. But with sounds of other character, and notably with the speaking voice, front and back could often be distinguished. The discrimination between the 1 Phil. Mag., xiii. pp. 214-232, February 1907, mi Science Ahstracts, p. 146, March 1907. 612.6U ACOUSTIC DETEEMINATIONS 609 right and left situations of high sounds is easily explained upon the intensity theory, the head forming a fairly effective screen which places the averted ear in a sound shadow. But this theory becomes less and less adequate as the pitch falls. At a frequency of 256 the difference of intensity at the two ears is far from conspicuous. At 128 it is barely perceptible. But although the difference of intensities is so small, the discrimination of right and left is as easy as before. 613. "There is nothing surprising in the observation that sounds of low pitch are nearly as well heard with the further as with the nearer ear. When the wave length amounts to several feet it is not to be expected that a sound originating at a distance could be limited to one side of the head. This subject has been quantitatively examined in Eayleigh's Theory of Sound, § 328, whence it appears that for a frequency of 256 the difference of intensities is only about 10 per cent of the whole intensity. A fall -in pitch of an octave reduces the difference of intensities 16 times. Thus at frequency 128 the difference would be decidedly less than 1 per cent of the whole ; and from this point on it is difficult to see how this difference could play any important part '' in the lateral discrimination. In 1906-1907 Lord Kayleigh returned to the subject and carried out experiments " which have solved the above outstanding difficulty. By theory and experiments with forks it is concluded that above c" = 5 1 2 the discrimination of right and left is made chiefly, if not solely, upon the difference of intensities at the two ears, but that at low pitch, at any rate below c= 128, phase-differences must be appealed to. 614. " To confirm this the obvious method was to conduct to the two ears separately two pure tones, nearly but not quite in unison. During the cycle, or beat, the phase- differences assume all possible values. This was reahsed by two forks of frequency 128, independently electrically 2r 610 SOUND CHAP. X maintained, placed in different rooms and isolated. The observer in a third room listened with each ear to the sounds along gas-pipes led through holes in a thick wall fro^n the resonators associated with each fork. The beat could be slowed down until it occupied 40 or even 70 sec, thus giving opportunity for more leisurely observation. The results were quite decisive ; it was found that if the vibra- tion on the right were quicker, the- sensation of right followed agreement of phase, and the sensation of left followed opposition of phase — that is, the sensation of receiving a sound from the right was experienced when the sound received at the right ear had a lead in phase over that reaching the left ear. 615. " The conclusion, no longer to be resisted, that when a sound of low pitch reaches the two ears with approxi- mately equal intensities, but with a phase-difference of one quarter-period, we are able so easily to distinguish at which ear the phase is in advance, must have far-reaching con- sequences in the theory of audition. It seems no longer possible to hold that the vibratory character of sound terminates at the outer ends of the nerves along which the communication with the brain is established. On the contrary, the processes in the nerve must themselves be vibratory, not of course in the gross mechanical sense, but with preservation of the period and retaining the characteristic of phase — a view advocated by Eutherford, in opposition to Helmholtz, as long ago as 1886. And when we admit that phase-differences at the two ears of tones in unison are easily recognised, we may be inclined to go farther, and find less difficulty in supposing that phase relations between a tone and its harmonics presented to the sa7ne ear are also recognisable. The discrimination of right and left in the case of sounds of frequency 128 and lower, so difficult to understand on the intensity theory, is now satisfactorily attributed to the phase-differences at the two ears. 615-617 ACOUSTIC DETEEMINATIONS 611 616. " Observations with sounds of the frequency 256 are next dealt with, and it is concluded that when, by passing up in pitch, difference of phase fails, difference of intensity comes to our aid in the discrimination of right and left. In conclusion, it is pointed out that, in wishing to locate a fog-signal heard on board a ship, a combination of three or four observers facing different ways offers advantages. A comparison of their judgments, attending only to what they think as to right and left, and disre- garding impressions as to front and back, should lead to a safe and fairly close estimate of direction." The above conclusions as . to lateral discrimination by phase-difference was further confirmed by Lord Eayleigh^ by experiments with a revolving magnet and two telephones. 617. Architectural Acoustics. — According to W. C. Sabine,^ " The problem of architectural acoustics requires for its complete solution two distinct lines of investigation : one to determine quantitatively the physical conditions on which loudness, reverberation, resonance, and the allied phenomena depend ; the other to determine the intensity which each of these should have, what conditions are best for the distinct audition of speech, and what effects are best for music in its various forms. Sabine's article contains contributions to each aspect of the subject. The question as to what conditions are best was attacked by investigating for a number of rooms what number of cushions and other absorbent articles were judged by musicians to give the best effect when hearing piano music. It was thus found that musical taste in such matters is very concordant and sensitive, a change of a few per cent making a room too resonant or too ' dead.' The other aspect of the question as to the physical dependence of reverberation on the presence of certain articles was elaborately investigated. 1 Phil. Mag., xiii. pp. 316-319, March 1907. 2 Aoner. Acad.Proc, xlii. No. 2, pp. 51-84, June 1906, and Saence A p. 524, October 1906. 612 SOUND CHAP. X Tests were made of the absorbing powers of different sub- stances, such as felt, curtains, an audience, and the usual furniture of an auditorium. Further, the variation of these absorbing powers with pitch was determined. Usually the absorption increases with rise of pitch ; or, in other words, the duration of reverberation is diminished by a rise of pitch." 618. In 1906 a valuable paper appeared by Marage^ which " deals with the acoustic properties of six halls in Paris, ranging from the Trocadero of volvime 63,000 cub. m. with an audience of 4500, down to the Physiological Theatre of the Sorbonne, with an audience of 150, the volume being 890 cub. m. In a hall," according to this author, "the audience hears three sets of vibrations : (1) those received direct from the source ; (2) those diffused or scattered by the walls, ceiling, etc., thus producing the sound of resonance ; and (3) those regularly reflected by the walls which give the distinct echoes. For a hall to be good from the acoustical point of view there should be no echo, and the resonance should be short enough to reinforce the original sound to which it is due, instead of trespassing upon the sound following. W. Sabine found from his experiments that the duration of the resonance can be given by <= 0'171'w/(a + *), where v is the volume of the room, a is its absorbing power when empty, and x the absorbing power of the spectators. Marage used as the sources of sound the five vowels OU, 0, A, '&, I, each synthetically produced by a siren in the place usually occupied by the speaker, the listener being successively at different points in the hall. In the largest hall, the Trocadero, holding 4500, the mean time of resonance was 2 sec. when empty and 1'4 sec. when full. To make himself distinctly heard in this hall a speaker must use a slow utterance, pausing at each, phrase. But it is not necessary to use more energy 1 Comptes Rendus, 142. pp. 878-880, 9tli April 1906, and Science Abstracts, p. 313, June 1906. 618, 619 ACOUSTIC DETEEMIHATIONS 613 than in addressing 250 in the Physical Theatre of the Sorbonne. In the large theatre of the Sorbonne, holding 3000, the resonance extended almost to 3 sec. empty, but was only 1 sec. or less when full. The acoustic properties of this hall are considered very good. Four other halls are studied and details given. 619. " The author concludes (1) in agreement with Sabine, that the resonance serves to characterise the acoustic properties of a hall (2) The duration of the sound varies with the quality, the pitch, and the intensity of the primary sound, hence a hall good for a speaker may be bad for an orchestra. (3) With the formula before given we can determine the duration of the resonance as a function of the number of auditors. (4) For a hall to be good acoustically, the duration of the resonance should be practically constant at all parts of the hall, and for all vowels, and fall between ^ sec. and 1 sec. (5) If the duration of the resonance much exceeds 1 sec, the speaker can make himself understood only by speaking very slowly, articulating distinctly, and avoiding giving to the voice too much energy." CHAPTEE XL KECOEDEES AND EEPEODUCEES 620. The Phonautograph. — In this chapter we shall treat 6f various instruments which have been devised to record or reproduce sounds. Some of these involve applications of electricity to the subject of acoustics. Hence, for the full appreciation of these parts of the chapter it is necessary that the reader should possess some conversance with electrical theory and practice. But as it is clearly outside the scope of this work to teach elementary electricity, it will be assumed that the student, has acquired from other sources a sufficient electrical knowledge to enable him to follow intelligently the descriptions and qualitative actions of the instruments dealt with. For convenience' sake, however, a few notes will be given at the end of the chapter on those parts of more advanced electrical theory which are applicable to the subject in hand and are needed for its quantitative study. We commence with the phonautograph of Scott and Koenig. This instrument was introduced in 1864, and yields a record of sounds in the form of a wavy trace on smoked paper. The paper is lixed on a rotating drum, and the curves are made by a style attached to a membrane. This membrane is set in vibration by the sounds which it receives after concentration by passage through a conical or paraboloidal funnel. By means of this instrument many 614 620-623 EECOEDEES AND REPEODUCEKS 615 curves have been obtained due to single sounds or to a combination of sounds. Its interest is now chiefly historical. For examples of curves originally due to the phonautograph, see Eig. 13, Art. 31. 621. The Phonograph. — The phonograph was invented by Edison in 1877. It not only records sounds as the phonautograph did, but also reproduces them. Thus it may be held to solve in the domain of acoustics a problem closely analogous to that achieved by photography in the domain of optics. For in each case a record is made, and then an imitation of the original facts may be in a sense reproduced from that record at will and after the lapse of time. But in the case of an ordinary photograph the form and light and shade are imitated, but the colour is lost. With the phonograph, however, not only may the pitch and relative intensities be correctly repeated, but even the quality also is reproducible and with astonishing though not perfect fidelity. 622. The essentials of the first instrument are briefly as follows : — The original sound falls upon a diaphragm whose consequent vibrations move a style which appropriately indents a sheet of tinfoil wound on the surface of a rotat- ing drum. This indented tinfoil constitutes the record of the sound in question. To reproduce it the drum is adjusted to the starting-point, and again rotated at the same speed as before. The style and diaphragm are thus caused to repeat approximately their original motions, and thereby reproduce what is recognisable as an imitation of the original sound. The details of the apparatus may be understood by reference to Figs. 101 and 102. 623. Supported upon a suitable base and bearings we have the horizontal axle aa. Fig. 101, carrying the drum or cylinder cc. Both axle and cylinder have a screw thread of the same " hand " and the same pitch. Thus, on turning the handle h, a point in one of the threads of the drum will 616 SOUND remain in the same thread although the drum is advancing. The thin diaphragm d, made of mica, animal membrane, or metal, is placed just below the mouthpiece in, see Fig. 102. Both are carried on the bar I pivoted at pp, and adjustable by the screw s, so as to make right contact with the tinfoil round the cylinder. At the centre of the diaphragm is cemented a small plate I, Fig. 102, which carries a little style t. This presses on the steel spring r, which bears a little rounded steel point n, which indents the tinfoil. After taking a record the screw s is unfastened, and the riG. 101.— HlSTOEIO Phonoskaph. bar h swung up so that the point n clears the tinfoil. The handle is then turned backwards until the cylinder is brought to its original position. The bar h is then brought- down again, the screw s adjusted, and the funnel / fitted on to the mouthpiece. The handle is now turned at the same speed as at first, and the original sound is thereby approximately reproduced. The original design of the instrument just described, and in which the record was made upon tinfoil, was followed by a modified form in which a wax cylinder replaces the tinfoil. Further, in the improved forms of the instrument uniformity of rotation is secured by clockwork, or an electro- 624 RECOEDEES AND EEPEODUCEES 617 motor, both in taking the record and in reproducing the sounds from it. 624. We have already referred to the use of the phonograph by Hermann to settle the question as to whether vowels are characterised by a fixed or a- variable pitch. We may now notice in detail how the phonograph lends itself to this determination. Suppose, for example, the vowel oo to be sung iij^o the phonograph at the pitch F, Pig. 102. — Sectional Detail of Phonogeaph. the chief resonance being / an octave higher. Then the second partial / would be specially reinforced, and this special reinforcement would be registered in the phono- graph's record. Now, if the vowel oo is characterised by a special reinforcement -of its second partial, the phonograph when reproducing the sound at any speed would retain and reproduce correctly this reinforcement of the second partial, the whole compound tone being raised in pitch if the handle was turned quicker than in taking the record, and the whole lowered in pitch if the handle were turned slower. 618 SOUND CHAP. XI 625. But if the vowel 56 is characterised by the special reinforcements of the partials near the fixed pitch /, then, on turning the phonograph either quicker or slower in reproducing the sound than in taking the record, we should have the special reinforcements of a pitch sharper or Hatter than/, and accordingly a different vowel produced. And this was the result found by Hermann, hence his adherence to the fixed-pitch theory. , " Other forms of phonographs, some termed gramophones, have been invented, in which the records are taken on a flattened disc rotating horizontally, and so arranged that the recorder describes a series of spirals diminishing from the circumference to the centre of the disc ; but they are all constructed on the general principle of the phonograph." ' 626. M'Kendrick's Phonograph Recorder. — Dr. J. G. M'Kendrick, professor of Physiology at Glasgow, has for years past carried out extensive researches with the phonograph. In addition to its powers for amusement or practical purposes, he regards the phonograph in its present state as a " scientific instrument worthy of a place iu physical and physiological laboratories beside other instru- ments of scientific research, and those employed for demonstration in teaching." Professor M'Kendrick has studied " the marks on the wax cylinder in three different ways — by casts, by photographs, and by mechanical devices." The method of casts was not satisfactory ; it had the dis- advantage of flattening out the marks. Numerous micro- photographs were taken of portions of the wax cylinder on which were records of the voice and other instruments. But from these not very much could be inferred. Finally, M'Kendrick overcame the many difficulties met with in designing and using a mechanical device for tracing out a curve showing the depths of the indentations in the wax cylinder. This perfected instrument he calls a phonograph recorder. ^ Encydopasdia Britannica, vol. xxxi, p. 679. 625-628 EECORDEKS AHD REPEODUCEES 619 627. The chief difficulties encountered in the construc- tion of such an instrument were due to (1) the inertia of the moving parts, (2) the extreme shallowness of the in- dentations to be copied, and (3) the disturbance caused by friction of the recording pen or pencil The difficulty as to inertia was overcome by driving the wax cylinder about one thousand times slower when operating the phonograph recorder than when it was used to give out tones. To represent the indentations on a scale large enough to be appreciable, the depths of the minute indentations in the wax cylinder were in the curves magnified nearly a thousand times by a series of levers, the lengths of the indentations being magnified about thirty-five times only. The difiiculty as to friction of the pen was overcome by using a fine glass syphon like that in Kelvin's syphon recorder. The strip of paper which is uniformly passing ■ lengthwise past this pen is made alternately to approach and recede. Thus, on each approach it takes a minute drop of ink at the end of the fine glass tube. Hence, without appreciable friction, a series of dots is made on the paper sufficiently represent- ing the curve sought. In the curves so obtained one foot of paper represents the fortieth of a second. With this apparatus Professor M'Kendrick was able to record the vibrations of the tones of several instruments, and also the tones of the human voice, both in speech and in song. Some of the results are shown in Fig. 103, taken from the Science Lecture for 1896 before the Philosophical Society of Glasgow. M'Kendrick has also published his record of the spoken word " Constantinople," but owing to its extreme length and complexity it is not reproduced here. 628. Eeferring to the records of spoken words Professor M'Kendrick writes as follows : — " There is not for each word a definite wave form, but a vast series of waves, and, even although the greatest care be taken, it is impossible to obtain two records for the same word precisely the same 620 SOUND CHAP. XI in character. A word is built up of a succession of sounds, all usually of a musical character. Each of these sounds, if taken individually, is represented on the phonograph- record by a greater or less number of waves or vibra- tions, according to the pitch of the sound and its duration. .... The speech sounds of a man vary in pitch from 100 to 150 vibrations per second, and the song sounds Cornet ~-./%A/'V Quick Firing Guns ' Boiler Makers Shop Soprano Bugle Bass Trumpet Piccnlo Euphonium"^' Military B'arld " ' ■'■ Fig. 103.— M'Kendbiok's Phonogkaph Records. of a man from 80 to 400 vibrations per second. The sounds that build up a word are chieily those of the vowels. These give a series of waves representing a varia- tion in pitch according to the character of the vowel sound. In the record of a spoken word the pitch is constantly moving up and down, so the waves are seen in the record to change in length. It is also very difficult to notice where one series of waves ends and where another begins. .... In ' Constantinople ' there may be 500, or 600, or 800 629, 630 EECOEDEES AND EEPRODUCEES 621 vibrations. The record of the words 'Eoyal Society of Edinburgh,' spoken with the slowness of ordinary speech, showed over 3000 vibrations, and I am not sure if they were all counted. 629. " This brief illustration gives one an insight into nature's method of producing speech sounds, and it shows clearly that we can never hope to read such records in the sense of identifying the curves by an inspection of the vibrations. The details are too minute to be of service to us, and we must again fall back on the power the ear possesses of identifying the sounds, and on the use of con- ventional signs or symbols, such as letters of the alphabet, vowel symbols, consonant symbols, or the symbols of the Chinese, which are monosyllabic roots often meaning very different things according to the inflection of tone, the variations in pitch being used in that language to convey shades of meaning. " When human voice sounds are produced in singing, especially when' an open vowel sound is sung on a note of definite pitch, the record is much more easily understood. Then we have the waves following each other with great regularity, and the pitch can easily be made out. Still, as has been well pointed out by Dr. E. J. Lloyd, of Liverpool, a gentleman who has devoted much time and learning to this subject, it is impossible by a visual inspection of the vowel curves to recognise its elements. Thus two curves, however similar, possibly identical to the eye, may give different sounds to the ear, that is to say, the ear, or ear and brain together, have analytical powers of the finest delicacy." 630. Bevier's Phonograph Analysis. — In 1900-1902 L. Bevier,^ junr., carried out in America a series of experi- ments with the phonograph designed to throw light on the composition of vowel sounds. The analysis was carried 1 Physical Mevieiv, also Sdmice Abstracts, p. 541, 1900; p. 778, 1902; pp. 113 and 301, 1903. 622 SOUND CHAP, xi out by the optical enlargement of phonograph records, ordinates and abscissae being then microscopically examined, and Fourier components to the tenth harmonic looked for. Bevier " concludes that the vowels, as produced by the human organs of speech, are composed in the first place of two elements : that due to the vibration of the vocal chords, and that due to the resonance of the mouth and nose cavities. It is not always possible to separate clearly these two elements, but the problem is quite simple for the vowel a (as in father). The fundamental is due to the vocal chords, and the overtones that are strongly reinforced are due to the mouth and throat resonance. This vowel a, at any pitch, and pronounced by any clear voice, contains the following partial tones: (1) The fundamental to which it is sung, with the first two or three overtones. (2) The overtone or overtones whose frequencies of vibration chance to fall between 1000 and 1.300 vibrations per second. This is the main characteristic of a, which serves to identify it to the ear, and remains remarkably constant, no matter what the fundamental may be. (3) The over- tone or overtones whose frequencies of vibration chance to fall between 575 and 800 per second for men's voices, with a maximum at about 675 ; or between 675 and 900 with a maximum at about 800, for the voices of women and children. This is presumably the resonance of mouth and 'throat cavities resounding as one vessel, and is not as constant as the main resonance described above." 631. Tor the vowel a (as in hat), three resonance regions were found near frequencies 1550, 1050, and 650, the first being by far the most important one. For the vowel e (as in jiet), the strongest resonance was found for frequency 1800, there being also two centres of weaker resonance at 1050 and 620. The vowel i (as in ^jii) showed a strong resonance characteristic of the vowel at about 1850, with another 631, 632 EECOEDEES AND EEPEODUCEES 623 resonance at about 575, there being comparatively little distributed resonance between the above pitches. The vowel i (as in pique) was found to be characterised by a powerfully-reinforced upper partial at about 2050, a chord-tone present with large amplitude, there being very little intermediate resonance. See Fig. 103a. This I. Eesonance curve was plotted by Bevier from all the records computed for this vowel, and was added to show the matter in convenient shape to the eye. He states, however, that this average result gives a too gradual rise and decline for the region of strong resoiiance, as had previously been noted by him for the corresponding curve ^ i. ^ J ? ^ •.^ ^ — - -■ .. 3 r^ ■- > -r = T. ' - ~- 8=- -- - ■«T- 1- lo- •,r I?* - ^^ 51T -..- 10 11 12 13 14 IS 16 17 IB 1 I 21 22 23 24 25 26 27 26 29 30 31 32 33 34 36 36 37 Fig. 103a. — I. Resonance CnEVES. for the vowel i as in pit. For the strong resonance at frequency 2048 see the last two curves of Fig. 104, which show so clearly the eighth partial of 256 and the fourth of 512. 632. Bevier considers that both physiologically and acoustically these vowel sounds, as in father, hat, p«t, pit, pique, form a true series. And in concluding the study of this series a sheet of curves was exhibited giving examples of all these palatal vowels side by side. The pitches were chosen to make the comparison most instructive, viz.: one of each at 512 per second, and one of each at or near 256, the octave below. No good example of a being available at 256, one a semitone higher was substituted. Fig. 104 is from this plate of Bevier's in the Physical Eeview, Nov. 1902, and shows curves for the five vowels 624 SOUND OHAP. XI in question, at two pitches for each. The vowels are indicated by a special notation at the left naargin, tlie numbers near, 1, 4, 6, and 7, refer to the voice by which the vowel was sung, while the numbers 272, 512, 256, etc., at the right margin indicate the frequencies'. 633. The Telephone. — The salient points in the history of the telephone are briefly as follows : — In 1876, Graham Bell, of Edinburgh, Montreal, and b A-l E "^4 vvxnArVWvVvrWWW 612 \7 /\fU\/\/\A/\lV\/\JUW^Af^ •266 J AAA~/\/\AJV\AJlvvJ\AA-AAA^AAA-AAA~A/\AVWvy\A/\~'V\A512 Fig. 104. — Bevier's Vowel Cueves. Boston, patented in the United States the speaking telephone. The invention was favourably reported on by Lord Kelvin at the British Association in 1876, and the telephone itself was exhibited at the Association's meeting at Plymouth in 1877. At that time Bell's telephone acted both as a transmitter and as a receiver. That is to say, the instruments at each end of the line were identical. What is termed the transmitter is the one which transmits to the line the signals corresponding to the sounds spoken into it. While the other instrument which receives these 633-635 EECOEDERS AND EEPEODUCEES 625 is called the signals and reconverts them into sounds receiver. 634. Sir William Preece says, " So far as the receiver is concerned, the telephone has remained virtually the same as it is described in Bell's patent ; alterations have been made, but in essential principle every suc- cessful receiver hitherto introduced is covered by Bell's invention. " It is, however, quite a different matter with, the trans- mitter. The original Bell instrument, which was identical with the receiver, has been almost completely superseded as a transmitter. In its place some form of carbon trans- mitter is now generally used." The first carbon transmitter was constructed by Edison in 1877, who ascribed its action to a varia- tion of electrical resistance of the carbon due to pressure. In 1878 Hughes discovered the micro- phone, and showed that the effect of Edison's carbon transmitter was dependent on loose contact. 635. The action of Bell's telephone both as transmitter and receiver may be under- stood from Figure 105, which shows a section of one of the later forms. In this Figure NS is the permanent bar magnet of steel with one pole towards the vibrating plate or diaphragm D, which is of very thin iron. On the end of the magnet next the diaphragm is the coil of wire C , whose ends are connected to the outside terminals TT'. In order to telephone, two such instruments may lie connected by a pair of wires constituting what is known as a metallic circuit, or one terminal of each instrument may be put to earth, and the remaining terminals of the instruments connected by a single wire, 2s Fig. 105. — Bell's Telephone. 626 SOUND CHAP, xi 636. Consider first the action of the transmitter, that is, the instrument which receives the spoken sounds and transmits the signals to the line. The waves falling upon the saucer-shaped mouthpiece M are concentrated on to the diaphragm D and set it in vibration. Each change in position of the disc changes the magnetic field between it and the magnet NS, and each change in this field sends a transient induced current through the coil and the wires connected with it, called the " line." Thus, an approach of the diaphragm strengthens the field and sends a current one way, a recession of the diaphragm weakens the field and sends a current the opposite way. Moreover, these currents are approximately proportional to the motions of the diaphragm to which they are due. Thus the features of the spoken sounds are represented by the undulatory currents started by the transmitter. 637. Let us now trace out what happens to these currents. They are propagated very rapidly along the line, whether it consists of a pair of parallel wires or a single wire overhead and the conducting earth below. On their arrival at the other end we are concerned with the action of the receiver. These undulatory currents, by passing through the coil of the receiving instrument, serve according to their direction to increase or decrease the magnetisation of its magnet, and thus cause the diaphragm to be more strongly or less strongly attracted than when no current passed. It is thus set in vibration, and accordingly generates in the air sound-waves which are the approximate counterpart of those which originally fell upon the diaphragm of the transmitter. Thus the transmitter acts like a tiny dynamo of a special form, in that currents are generated by it from the mechanical motion of the diaphragm. Further, the receiver is a tiny electro-motor, in that on receipt of electric currents it produces the approximate motions of the diaphragm. 636-639 EECOEDEES AND EEPEODUCEES 62V In the arrangement just described, since the transmitter is itself a generator, no battery is needed. 638. Sensitiveness. — The ordinary explanation of the action of the receiver as outlined above is known as the " push and pull theory." But some physicists have objected to this, and have supposed that some special molecular action occurred, believing that the forces otherwise available are inadequate to produce in the diaphragm sufficient motion to generate audible sounds. But Lord Eayleigh's experiments on this point support the ordinary received theory. He also points out that the minimum amplitude for audibility is extremely small (see Arts. 582-587). Thus it is not so surprising to find that a Bell receiver will respond to a current of the order 4 '4 x 10"^ ampere as determined by Eayleigh. Indeed, Tait found 2 x 10"^^ ampfere, and Preece 6x10"^^ ampere as the currents to which a telephone receiver can respond. •639. Permanent Field Indispensable. — In the tele- phone receiver it is important to note the necessity explained by Heaviside for the field due to the permanent steel magnet. Suppose the field due to the magnet near the diaphragm is denoted by H, and let a very small change in it due to currents in the coil be denoted by ± li. Now the induced magnetism in the diaphragm is proportional to this field, and the attraction is proportional to the pro- duct of the two. Hence we have an attraction varying between the limits k{H^ lif, where k is some constant. Thus the difference of attractions available for motion of the diaphragm is 4.1 Hh. That is, the force available is proportional to the product of the permanent field and the change which can be produced in it. Hence the sensitive- ness of the apparatus is much increased by the presence of a strong permanent field, provided it is not suQh as to lessen the possible changes in it due to the currents received. Here it must be noticed that there is a practical limit to the intensity of the permanent field which it is 628 SOUND CHAP. XI wise to use. For, if the magnet is nearly saturated, very little change could be made in the field. Hence, in that case, the gain in making H large would be balanced by a consequent diminution of h for a given current. But this limit is not reached with ordinary steel magnets, hence their value in the instrument. 640. Edison's Carbon Transmitters. — The first carbon transmitter constructed by Edison in 1877 passed through various stages, and afterwards received the form shown in Fig. 106. When using for telephony a carbon transmitter, a battery is needed to generate the current, and it is the T t' Fia. 106. — Edison's Caebon Tkansmittee. function of the transmitter, by the variation of its own resistance, to vary the current thus independently produced. How it does this may easily be seen from Fig. 106. The sound-waves pass through the mouthpiece M, fall upon the vibrating plate D, and set it in motion. This vibratory motion acts upon the rounded ivory button B and the adjoining platinum plate, which thus makes a variable contact with the disc of carbon 0. The electrical circuit through the instrument is from the terminal T through the spring S to the platinum plate, thence through the carbon and the case of the instrument to the second terminal T'. Hence a downward motion of the diaphragm improves the contact with the carbon, lessens the electrical 640-642 RECOEDEES AND EEPEODUCEES 629 resistance, and therefore increases the current. An upward motion makes the carbon contact worse, increases its resistance, and hence decreases the current. 641. Suppose now that for sounds of a given intensity the change of resistance in the transmitter is one per cent of the total resistance in the circuit. Then a one per cent change in the current is produced and the telephone receiver affected accordingly. Next, let a longer line be used so that the total resistance in the circuit becomes ten times its former value. Then if the same battery is used, and sounds of the same intensity fall upon the diaphragm of the transmitter, we have two changes to notice. First, the current is only one-tenth of its original value ; and, second, the changes which the transmitter can produce are only about one-tenth per cent of that reduced current, because the changes in resistance of the carbon contact remain of the same absolute value as at first, while the total resistance is made tenfold. Thus to maintain the changes in current at the receiver the same when the total resistance in circuit is made tenfold, we need to increase the voltage of the battery to one-hundredfold. Or, to generalise : when the line changes in length in order to maintain unimpaired the efficiency of the circuit, we should need to make the voltage of the battery vary as the square of that length. And this would be quite out of the question in practice on a large scale. 642. To obviate this necessity Edison passed the current from the transmitter through the primary of an induction coil, whose secondary was connected to the line. The transmitter then acts in connection with a small resistance due to the battery, the transmitter itself, and the primary wire of the induction coil. Hence the variations of resist- ance of the transmitter have a considerable relative magni- tude, and produce correspondingly large relative changes in the current through the primary of the induction coil. And these changes, by induction, give rise to corresponding 630 SOUND CHAP. XI periodic currents in the secondary of the induction coil, their voltage (or E. M. F.) being very high. Hence the line may be fairly long without prejudice to the action of the circuit. 643. Hughes' Microphone. — In 1878 Professor Hughes introduced what he called the microphone, which is really a form of telephone transmitter. Its action depended on a loose contact whose resistance varied with the sounds incident upon it. The changes in resistance thus produced affected the currents derived from a battery whose circuit included both the microphone and a telephone receiver. One of the earliest forms of the apparatus consisted of two FiQ. 107. — Hdqhes' Micbophone. nails laid side by side, but not in contact, across which a third nail was laid. The effect is, however, better when carbon pencils are used, and the apparatus employing carbon may be regarded as the standard type of the microphone, and is with modifications still retained. One form of it consists of a small pencil of gas carbon A, Fig. 107, with pointed ends resting lightly in small circular holes in the two pieces of carbon B, C, so that the pencil takes up a vertical position between them. The pieces B and C are fixed to a thin sounding board or diaphragm D, fitted into a frame and mounted on a solid base F F. A battery E and a telephone receiver E are included in the circuit with the pieces B and 0. The instrument, though apparently so 643-645 KECOEDEES AND EEPEODUCEES 631 rough, is of surprising delicacy, the movement of a fly on the diaphragm serving to produce in the receiver audible effects. 644. The true nature of the action at the loose contact is perhaps not yet fully understood, though of late years much research has been done upon the contacts in the case of the coherers used in wireless telegraphy. Possibly the phenomena in the two cases are somewhat analogous. For the loose contacts of the microphone Shelford Bidwell considered carbon is the best material, because ib is unoxidisable and infusible, a poor conductor, and has a lower resistance when heated. The difference between Edison's carbon transmitter and Hughes' microphone does not seem great, but the Edison form has disappeared, and the carbon transmitters now generally in use may be regarded as modifications of Hughes' microphone. The details of these various instruments and all the intricacies of current practice are outside the scope of this work. The reader interested in them should consult one of the technical treatises, such as the Manual of Telephony by Preece and Stubbs. 645. Trunks and Transformers. — We may, however, with advantage mention here the following devices used for long-distance telephony: — For transmission to any consider- able distance, free from the disturbances due to induction, the double wire or metallic circuit is imperative. It is therefore invariably used for the so-called trunk lines connecting towns at any great distance apart. But suppose a subscriber at one end is on a single wire circuit only, and desires to communicate with the distant town reached by the metallic circuit of the trunk-line. To admit of this, transformers or translators are used at the ends of the trunk or double line between the two central stations. These translators are induction coils of special construction. The National Telephone Co. have used coils of 290 ohms 632 SOUND CHAP. XI resistance for the trunk, and 140 ohms for the local sections. The core is of the softest iron, and the coils wound closely and regularly. In the most approved forms the cores are more than double the length of the coils, and after the completion ' of the winding, the projecting ends of the iron wires constituting the core are folded back over the windings of the coil. Since the use of a translator involves some loss of effect it is desirable to avoid having more than two in use at any one time in any given speaking circuit. Indeed, in some districts it has been the rule with the National Telephone Co., that of the two subscribers using a trunk wire one must have a metallic circuit, so that only one translator is in use. 646. Vibrations of a Telephone-Membrane. — In 1902 E. Kempf-Hartmann ^ experimented on the vibrations of a telephone -membrane, obtaining many interesting results. Vibration curves were photographically produced by the light reflected to a moving film from a mirror fixed on the membrane of a telephone receiver. It was thus shown that the membrane very quickly takes up the vibrations impressed upon it, one-thousandth of a second in some cases sufficing for the steady state to be reached. Curves are given in the original paper for vowels and consonants at different pitches and intensities. The effect of the frequency natural to the membrane is also shown in a special set of curves. 647. Rhythm Electrically perceived. — Prof. M'Ken- drick, pursuing the researches mentioned before (Arts. 626- 629), arranged the following combination of apparatus: — To the glass plate diaphragm in the reproducer of the phonograph was attached a tube connecting it with a microphone transmitter. This transmitter was connected in series with a battery and the primary of an induction coil. The wires from the secondary of this induction coil ' Ann. d, Physik. viii. 3. pp. 481-538, Juue 1902. Science Abstracts, p. 27, 1903. 646-648 EECOEDEES AND EEPEODUCEES 633 ended in platinum plates dipped in weak salt solution. The phonograph was then set going and the fingers put into the beakers containing the salt solution. By this means the intensity of every note could be felt; indeed the variation of intensity, the rhythm, and even the ex- pression of music were all felt. M'Kendrick writes, " This experiment suggests the possi- bility of being able to communicate to those who are stone deaf the feeling, or, at all events, the rhythm of music. It is not music, of course, but, if you like to call it so, it is music 071 one plane and %uitliout colour. There is. no apprecia- tion of pitch or colour or of quality, and there 'is no effort at analysis, an effort which, I believe, has a great deal to do with the pleasurable sensation we deriye from music. In this experiment you have the rhythm which enters largely into musical feeling. On Saturday last (Dec. 1896) through the kindness of Dr. J. Kerr Love, I had the opportunity of experimenting with four patients from the Deaf and Dumb Institution, one of whom had her hearing up till she was eleven years of age, and then became stone deaf. The girl had undoubtedly the recollection of music, although she does not now hear any sound. She wrote me a little letter, in which she declared that what she felt was music, and that it wakened in her mind a conscious something that recalled what music was. The others had no concep- tion of music, but they were able to appreciate the rhythm, and it was interesting to notice how they all, without exception, caught up the rhythm, and bobbed their heads up and down, keeping time with the electrical thrills in their finger tips." 648. The Speaking Arc. — In 1898, H. Simon showed that the continuous current electrical arc could be used as a telephone receiver. His experiment, as quoted by Duddell in 1900, was as diagrammatically represented in Fig. 108. Such arrangements are now- referred to as the speaking arc. 634 SOUND CHAP. XI In this figure M denotes the microphone, E the battery, A and B the primary and secondary of an induction coil. When the microphone is spoken into, its varying resistance changes the current in A, and thus produces in B induced currents which are superimposed on the main current in the arc derived from its source of supply. Thus the current through the arc suffers periodic variation. This variation is accompanied by corresponding changes in the vapour column between the carbons, and so produces sounds which are the duplicate of the original ones incident upon the microphone. In order to obtain louder and clearer speech, Duddell, who brought this subject before the Institution of Arc + - Fig. 108. — Simon's Speaking Akc. Electrical Engineers in 1900, prefers the arrangement shown in Fig. 109. 649. In this arrangement M and E denote as before the microphone and its battery ; A and B are the primary and secondary of the induction coil by which the microphone affects the current in the arc. But the secondary B is now placed in shunt with the arc instead of being in series with it. The main current of the arc is prevented from passing round B by the presence of the condenser S in the shvmt circuit. Also, the alternate currents induced in B are practically prevented by the inductance L from flowing through the cells which supply the arc. Thus in the arc, and in the arc only, have we the superposition of the steady current supplying it, and the periodic currents due to the microphone and the sounds falling upon it. The variations 649-651 EECOEDEES AND EEPEODUCEES 635 of current in the arc are consequently greater than if the microphone had to affect the currents flowing through a steadying resistance, such as E, and any inductances that may occur in the main circuit of the arc, as in Simon's arrangement. 650. The details of Duddell's arrangement of speaking arc are given by him as follows : — "The microphone M was supplied by the National Telephone Co. and was intended for long-distance transmis- sion. E shows two accumulators used in series with it. The induction coil A E had an iron wire core about 15 mm. diameter. A had 600 turns, resistance 1-52; Fig. 109. — Duddell's Speakinq Arc. B 400, resistance 1-53. Mutual induction 0-0253 henry. The condenser S had a capacity of 2 or 3 microfarads, for the arc cored carbons were used 11 to 13 mm. diameter, the current being 10 to 12 amperes, and the arc length 20 to 30 mm. To obtain these long arcs with ease the carbons must be cored, or some other means taken to introduce foreign bodies, such as salts of potassium and sodium, into the arc. These salts may be introduced by soaking the carbons in their solutions instead of using them as cores." Duddell also described an arrangement by which the arc could be used as a telephone transmitter, an ordinary telephone receiver being used in conjunction with it. 651. The Musical Arc. — Because of its close connection with the speaking arc, we mention here the musical arc, 636 SOUND CHAP. XI though perhaps it is not strictly within the scope of this chapter as a recorder or reproducer of sound. It seems to be rather a case in which the sound originates owing to the extreme excitability or sensitiveness of the arc, and the presence of conditions favourable to the continuance of the sound when once produced. It seems to have been first observed by Elihu Thomson in 1892, but was independently discovered and brought forward by Duddell in 1900, whose arrangement for producing it is shown in Fig. 110. 652. In this arrangement the arc consists of solid carbons, and there is simply a shunt circuit to the arc containing a condenser S and an inductance L. Now it is -C=i Fig. 110. — Duddbll's Musical Abc. well known that in electrical matters an inductance and a capacity are respectively analogous to a mass and the reciprocal of a spring in mechanical matters. Thus in this shunt circuit we have the possibility of electrical oscillations of a definite frequency and period (viz. T=27r^Z*S^as shown in article 656). Duddell, commenting on this arrangement, writes : " It must be remembered that although we have an alternate current through the condenser and self-induction, the source of supply is not an alternating one, and that it is the arc itself which is acting as a converter and transforming a part of the direct current into alternating, the frequency of ivhich can he varied hetioeen. very wide limits hy altering the self-induction and capacity. The upper limit I find to be about 10,000 v*^ per 652-654 EEOOEDEES AND EEPEODUCEES 637 second, and the lower limit, if such exists, is well below 500 >-^ per second." 653. In the discussion on Duddell's paper the close analogy of this interesting electrical phenomenon to some acoustic ones was very aptly brought out as follows by Prof. Ayrton : — -Mr. Duddell "has shown us that an ordinary so-called perfectly silent arc supplied with' current from accumulators is, if the carbons be solid, like the mouth- piece of a flageolet or flute but not blown. The application of a shunt to that arc, consisting of a capacity in series with a self-induction, performs two operations. It starts vibrations in the arc, just as blowing a flute gives rise to vibrations of many different rates. Just as one of these rates of vibration is picked out and reinforced in the case of a flute or flageolet by the form of the resonance chamber dependent on the position of your fingers or keys, so in this musical arc the particular one of the many vibrations that are probably started which is picked out and reinforced depends on the capacity of the condenser and the value of the self-induction which is in series with it." 654. Oscillatory Discharge of a Condenser. — We now pass to the electrical theory which is applicable to the phenomena dealt with, but not found in the more elementary text-books on electricity. Some of this theory, besides being useful for its application to tele- phony and other examples of electrical oscillations, will also be found interest- ^'"^ ■^pjscg^„|,E. ing as affording analogies to some of the acoustic phenomena already dealt with, such as free and forced vibrations. We commence with the oscillatory discharge of a condenser, the mathematical theory of which was given by Kelvin in 1853. Let a condenser of capacity S and charge Xg be connected at time t = with a coil of resist- ance B and inductance L, as shown in Fig. 111. 638 SOUND CHAP. XI Also at time t let the charge on the condenser be x and the current u. Then the current u= —x and the electrostatic electromotive force or voltage driving it is xjS. But this voltage must equal the sum of Bu and the back E.M.F. of self-induction which is Lu. We thus obtain the equation of motion for the system, which may be written Lx + Ex + x/S^O (1). 655. This equation has two types of solution, the one or other applying according as B is above or below a certain critical value. If B is above that value, the corresponding solution indicates a gradual subsidence of the charge, and with this case we are not here concerned. If B is below this critical value the general solution expresses oscillations, and may be written x = Ae'''* cos (nt + a), where n and q have definite values, A and a being dependent only on the initial conditions. This may be verified by differentiation and comparison with (1). To fit the given initial conditions we may write the solution in the form X = XQe''-^'! cos nt + - sin nt ) (2), where q = B/2L, n^^J IC- -\-I?p''. This quantity has been called by Heaviside the impedance of the coil for the frequency pj^ir. He defines impedance as " the ratio of the amplitude of the impressed force to that of the current when their variations are simple harmonic." 660. Another method of representing the results contained in (11) by means of the imaginary quantity ■i = x/ — 1 is sometimes more convenient. It may be written as follows : — Let the impressed E.M.F. be given by y = the real part of 2/o*'^*' then, the current is expressed by u = the real part of /o"^- (12). JR + Lip On rationalising these expressions for E.M.F. and current, 2 T 642 SOUND CHAP. XI they will be found equivalent to those in (10) and (11) above. 661. Resistance and Inductance modified by Alterna- tions and Damping. — It is further to be noticed that not only must the impedance be substituted for the ordinary ohmic resistance in the case of alternating currents in a coil of appreciable inductance, but also that the values of the resistance and inductance to steady currents become themselves modified when the currents are alternating. When the current is steady it is uniformly distributed throughout the cross-section of the conductor. But when the current is simple harmonic it is more concentrated at the surface than at deeper or more internal parts of the cross-section. This causes the resistance to increase and the inductance to decrease. Following on the theory of Maxwell, this case for a straight cylindrical conductor has been worked out by Lord Eayleigh ^ who obtained the following results : — ""O ^ 12 E- 180 B' ^ '^ ■^ „2;2,,2 '^\2 48 M' 8640 B* '^\48 B' 8640 B^ (14). In thSse equations B and L are the ordinary values of the resistance and inductance for steady currents, B' and L' their special values when the currents are simple harmonic of period 2Tr/p, I is the length of ,the wire, /* its magnetic permeability. The mathematical analysis by which the above results are derived is far too long for introduction here, especially as the matter is only on the border-line of the subjects treated. 662. When the alternations are damped simple harmonic, 1 Pha. Mag., May 1886. 661-663 EECOEDEES AND EEPEODUCEES 643 instead of being uniformly sustained, a further change sets in. Both resistance and inductance are increased by the damping. Indeed, the inductance may rise to a higher value than that for steady currents. Let the values of the equivalent resistance and inductance for this case be denoted by B" and L" when the currents vary as e~'^^^ COB pt. Then it has been shown ^ that for a straight cylindrical conductor the following relations hold : — S!' = Ii(l + i+il 2,2^2^2 ^ ^0-+^) ^«„,y //' = L + l/j, 1 2 ^ 1 24 1- ■2k'- ■ 3^■* fay 180 (e pafji 1 - 3k^ 48 p^aY /<1- A C -k') p^ay (15). (16), where a = IjB is the conductivity for steady currents of unit length of the wire. Prof. W. B. Morton and the writer have shown^ that the criterion for the oscillatory discharge of a condenser is slightly modified when these new values of the resistance and inductance are introduced, the condition that corre- sponds to the critical state when R and L are used becoming an oscillatory discharge with the values of R" and L". And further, they showed ^ that in the case of damped or decaying currents we may have axial concentration instead of the svirface concentration due to harmonic currents of sustained amplitude. 663. Alternating Currents in Parallel. — Consider now the case of two currents in parallel, each possessing resistance and self-induction, their mutual induction, however, being negligible. And let an harmonic E.M.F. ' E. H. Barton, Phil. Mag., May 1899. 2 PMl. Mag., July 1899 ; Fhys. Soc, March 1899. 3 Phil. Mag., July 1899 ; Phys. Soc, May 1899. 644 SOUND CHAP. XI be applied to the system. The problem of determining the currents has been treated by Prof. J. A. Fleming in his Alternate Current Transformer, from which the following is derived. The circuits are represented in Fig. 112. The main current, before dividing at A into the two coils, and again after reuniting at B, is denoted by Usm pt. Let u denote the current at time t through the upper coil of resistance E and inductance L. Similarly let w denote the current at the same instant in the lower coil of resistance S and inductance N. It is accordingly required to find u and w in terms of U, p, and the constants of the coils. 664. By tracing through the upper coil we see that .u.„,t ^ g^<>'^'^ Uempt ..i-^B 5- N D Fig. 112. — Paballel Inductive Cieouits. the potential difference between A and B at time t is expressed by Ru + Lit. At the same instant a correspond- ing expression holds for the lower coil. But this gives the same potential difference, namely, that between A and B. We thus have the equation LS' B + , M = ?7 sin «(! and «' = Usmpt (24). If, howe%'er, the reverse holds, the inductances or frequencies being so high that the terms involving the resistances are negligible in comparison with the inductive terms, we then find u=U\ smM^-l cos«<^ (25), J L . ^ , LS-BN .] ,_., and ir = Ui sm«i+ cos My (26). 666. These equations express the subdivision of the current according to the inductances of the circuits, which ha\'i' now become paramount. Lord Eayleigh has treated the case where the two branches of the circuit have mutual as well as self- induction. This brings out the striking result that the current amplitude in each branch may exceed that of the main current. Thus, in a case cited, a main current of amplitude unity may divide into currents of 3/5 and 2/5 in the two branches, their phase difference being small or zero. Or, the main current, still of amplitude unity, may di^'ide into currents of amplitudes 3 and 2 in the branches, their phases being opposite. Or, to express both the amplitudes and phases algebraically, we may say that in the first case + 1 = + 3/5 + 2/5 ; whereas in the second case, the subdivision of the main current is expressed by + 1 = -f 3 — 2. In the first case the two coils are connected so that steady currents would circulate in them in opposite directions, M the mutual induction being then negative. In the second case the opposite state of things holds good and M is positive. Such a case of the branch currents exceeding tlie main current has been experi- 666-668 EECOEDEKS AND REPEODUCEES 647 mentally verified by Lord Eayleigh and shown to an audience. 667. Electric Waves along Parallel Leads. — One of the most important of telephonic phenomena is that of the propagation of the currents along the line, or, in other words, the propagation of electric waves along its parallel leads. These parallel leads may, of course, be the two parallel wires constituting the metallic circuit now almost always used, or they may be the nearly obsolete system of a single wire and the earth's surface. We now often refer to these phenomena as the propagation of electric waves, since by Maxwell's theory and the experimental confirmation of Hertz and his successors, we know that the currents in the wires are only a small portion of the general phenomena of the electro-magnetic field between and round them, and its propagation along the line with the speed of light. It is fairly obvious that the resistance of the line will cause the waves and currents to diminish as they proceed along it. Further, since for periodic currents we have the ohmic resistance replaced by the impedance is/ E^ + L''i)^, it would appear that the diminution or attenuation of an alternating current would depend upon ■p, that is upon its frequency. 668. Thus, if no special pains are taken to avoid it, we should expect the various partial tones composing any note or vowel to be diminished in different ratios by their passage over the same length of the line. This selective diminution is called distortion, since it distorts the relation between the partials of any composite tone, and by which it is characterised, and so alters its quality, or if a vowel changes it to some slightly different vowel. Accordingly both attenuation and distortion are to be avoided as far as possible in the currents passing along a telephone line. This matter has been very fully treated by Oliver Heaviside, who showed mathematically how to make a line which would not involve any distortion of the 648 SOUND CHAP, xi signals sent along it. This he called a distortionless circuit. Heaviside showed that if in a line without resistance or leakage a resistance be inserted, then, at the spot in question there is a reflection of the incident waves in which the current is reversed. If, however, this resistance be removed, and a conducting bridge be placed across the pair of leads constituting the line, then, in the reflection which occurs at the bridge it is the potential difference of the wave that is reversed instead of the current. Now let there be both a resistance inserted, and a conducting bridge placed across, and let them be so proportioned that the waves reflected in virtue of each are equal as well as opposite, the reflected wave is accordingly abolished. Part of the original wave is absorbed in the resistance and bridge, and the rest passes on without distortion. This explains the pith of the matter in words. We now pass to the mathematical theory, following Heaviside's method, though with slightly modified notation. 669. Heaviside's Distortionless Circuit. — Let the line consist of two parallel conducting leads, of which R, L, K, and S are respectively the resistance, inductance, leakance (or leakage conductance), and permittance (capacity), all reckoned in electro-magnetic units per unit length of the line. At the time t and place z along the line let y denote the transverse voltage, and u the loncdtudinal current. Then the potential gradient along the line expresses the E.M.F., and may be equated to the usual function of the current when resistance and inductance are present. Thus Again, the space rate of decrease of current would express the time rate of increase of charge if leakage were absent. Accordingly, making the requisite correction for leakage in the present case we may then equate to the 669-671 EECORDEES AND EEPEODUCERS 649 product of capacity and rate of increase of potential. We thus obtain an equation symmetrical with the first, namely du dti -T.=^y+'rt (28). 670. We have now to eliminate u between these two equations. Thus on differentiating (27) to a we find (fy _ j^du d^u dz^ dz dtdz And, substituting in the right hand side of this from (28) differentiated where necessary, we obtain '^^ = KRy + {KL + B^^ + L^^ (29). dz- dt dt" We now introduce Heaviside's condition for the dis- tortionless circuit, namely •Let BlL = KlS=q,ss.j (30). Also for brevity let LSv' =1 (31). Then (29) becomes 671. Now take a new variable Y defined by y = e-^'Y (33). Equation (32) now reduces to V r = -T^ (34). dz' dt' ^ ' And this is the familiar equation of undisturbed wave propagation, whose solution is F=/i(2-^0+/2(« + *0 (35)- . .. Hence by reference to (33), we have as the solution sought, y = .-«'{,A(a - vf) +/o(a + vt)) (36). 660 SOUND From equations (28), (30), and (31), we find that the corresponding current is given by 672. Or, in words, the current in the positively travelling waves is given by potential -f-Zv, and the current in the negative wave by potential divided by ( — Lv). Equations (36) and (37) express waves proceeding right and left at speed V without distortion, but with logarithmic attenuation. The symbols /i and /^ denote arbitrary functions which must be chosen to fit any given initial conditions. It is not to be expected that the distortionless circuit can be attained in actual telephone practice, but it is the ideal to be kept in view and approached as nearly as possible. As to the value of v, the speed of propagation, its limiting value may be found thus. Let the line consist of parallel wires of radius a and at a relatively large distance apart &. Then we have approximately, L = /jl4: log/bja), and (S' = /c/4 F^ log.^ (&/a), where /u, and k are respectively the magnetic permeability and dielectric constant of the medium between the wires, and V is the speed of light in vacuo. Thus by (31) we obtain or, for air, ij, — k= 1, and v= V, the speed of light, say 3x10'" cm. per second. 673. Reflections at Terminal Bridges. — An important problem in connection with telephonic practice is the determination of the reflection occurring when the waves along the line reach a bridge of any kind across the two leads. This bridge may be considered as having resistance only, resistance and inductance, capacity only, or all three. Also, the bridge may be at the end of the line or at some intermediate position. Consider first a bridge without inductance or capacity, 672-674 RECOKDEES AND EEPEODUCEES 651 but of resistance B^, and situated at the end of the line. Let the transverse voltage of the incident wave be expressed by 2/1 and that of the reflected wave by t/^, the corresponding currents being respectively u^ and m,. Then by (.SV) we have "^1 = VilLv and Mj = - y.jLv (38). Since the bridge is devoid of capacity there is no accumula- tion of electricity there. Hence, applying Ohm's law to the current through it, we have «i + i^2 = (2/i+2/2)M (39). Thus, on substituting from (38) in (39) we obtain yi-yi _ yi + y-i Lv B, ' , I M, — Lv whence y^/yi = ^^jy^ C^"^)- 674, This equation applies to waves of any form, and is not restricted to those of simple harmonic type. Three special cases of terminal bridge reflection call for notice. These .are as follows: — (i.) Ends of line insulated, ^j = 00 , 2/2/2/1 = + ^ U/i 1 \ (current reversed by reflection) J^ (ii.) Ends of line short circuited, B^ = 0, 2/2/2/1 = — ^XrAO^ (voltage reversed by reflection) J '''' (iii.) Bridge of critical resistance, B^ = Lv, 2/2/2/1 = ]/ao\ (incident wave entirely absorbed) J '^' Cases (i.) and (ii.) are obviously analogous to the re- flections occurring at the stopped and open ends of organ pipes. These results were predicted by Heaviside in treating waves of telephonic frequency. For high-frequency waves from condenser discharges as introduced by Hertz, the first two cases were experimentally confirmed by V. Bjerknes in 1891. For oscillator waves of about thirty- five million per second, the third case of total absorption 652 SOUND ■ CHAP, xi was experimentally confirmed in 1896 by the writer and Dr. G. B. Bryan.i 675. As Heaviside points out, it is this property of total absorption that should be possessed by the telephone receiver which constitutes a special form of terminal bridge. Such a telephone at the end of a line which is itself one of Heaviside's distortionless circuits would afford an ideal arrangement. But a telephone receiver is not a pure resistance with- out either capacity or inductance. The theory accordingly needs extension to include such bridges. Heaviside has shown that in the case of simple harmonic waves incident upon a bridge the single symbol R^, in equation (40) for a pure resistance, may be replaced by a certain function of all the constants concerned for a bridge of the most general type. 676. Generalised Bridge. — Thus let the bridge have resistance R^, inductance L^, and permittance (capacity) ;S^i, and let the incident waves be of frequency pjiir, so that y^ is the real part of yo**^') where ■i=v—l. Then R^ in (40) is replaced by the generalised resistance operator Z, expressed by Z=R^ + L^ip + llS-,ip (44). The reason for R^ being replaced by iJj + L^ip is seen from equation (12). Further, the fact that capacity opposes inductance has been pointed out at the end of article 658. Tlius equation (40) becomes in the general case 2/2 = the real part of yo*'*" (45), Z+ Lv or, with the understanding that only real parts are taken, this may be abbreviated into y.2^ Z-Lv y^ Z+Lv ' Phil. Mag., January 1897. 675-677 EECOEDEES AND EEPEODUCERS 653 la either form Z is given by (44) ; of course if any of the quantities represented by the right side of (44) are absent, their symbols are to be omitted from the expression for Z, which holds good for any one or more of them. It should be noticed that if L^ or 8-^ occur, then a change of phase is introduced in the reflected wave. 677. Intermediate Bridges. — Consider now an inter- mediate bridge whose resistance operator is Z defined by equation (44). Suppose the incident waves and con- sequently all the others are simple harmonic. Let the transverse voltages of the incident, reflected, and transmitted waves be denoted respectively by y^, y^, and y^, and let the corresponding currents be u-^, u^, and u^. Then by equation (37) we have Ml = y^lLv, M2 = - y^jLv, wg = y^jLv (46). At the bridge itself we have «i -f- Mj - Mj = (3/1 + y^jZ (47). Lastly, we see that 2/3 = 2^1 + 2/2 (48). Hence (46) in (47) gives 2/1-2/2-2/s 2/l + ?/2 Lv Z (49). Whence by (48) we obtain -23/2 2/1+2/2 Lv Z ' . • 1 • 1 1 2/2 - -^^ (50). wmcuyi^xu^ y^-2Z+Lv Then by (48) and (50) we have y3_ 2^ (51). 2/1 2Z+Lv These last equations give the reflected and transmitted waves in terms of the initial and the constants of the bridge and line; ih using them y/^' should be written for 654 SOUND CHAP. XI y^, and, finally, real parts ouly retained, i.e. the abridged notation following (45) is here employed. 678. It may be noticed that for a bridge of pure re- sistance i2j, the above become y., _ - Lv ya _ 2i?i y, 2B, + Lv y, 2R^ + Lv ^ >■ Again for a bridge of permittance S^, the resistance and inductance being both negligible, Z becomes IjS-^ip, and we accordingly obtain y.i — Lv — LvS{ip (2 — ZvS^ip) y, " {'lJs'^p) + Lv " {-f+LrS.ip) (2^~L^Jp) _ -(LvS,pf-2iLvS,p and ^3^ 2/^l^j7 _ 2{2 - LvS.ip) Vi ( - 1^1 il>) + Lu (2 + LvS^ip) (2 - LvS^ip) 4:~2iLvSip 679. It is noteworthy that the reflected and transmitted waves differ in phase by a quarter of a period whatever be the capacity of the condenser, although their respective phase differences from the incident waves depend on the value of that capacity. This may be seen as follows : If the expression for y^jy^ be put in the form A + iB, the change of phase being called e^, we have tan e., = B/A = ^^1^ (55 ). — Lvb^p Similarly the change of phase for the transmitted wave being 63, we have from (.':i4) tan €3= -X,;,S>^2 (56), whence, apart from sign, e^-e^ = 7r/2. 678, 679 EECORDERS AND EEPEODUCEES 655 Experiments with high frequency waves by the writer and Dr. Louis Lownds as to the amplitude and phase of reflection and transmission at an intermediate bridge con- sisting of a condenser only' gave results in harmony with the above theory of Heaviside. ^ Phil. Mag., October 1900. EXAMPLES ON CHAPTER I 1. Describe and explain the three requisites for the production of sound waves. 2. Distinguish between the propagation of waves and the actual transit of matter, giving experimental illustrations. 3. Enumerate the three features possessed by any musical tone, and explain the physical basis of each. 4. Prove experimentally that pitch depends upon frequency, and that interval depends upon and is fixed by ratio of frequencies. 5. Establish mathematically that it is only when measuring pitches logarithmically that a resultant or composite interval is measured by the sum of the measures of its component intervals. 6.. Calculate for each of the other six notes of the scale the number of logarithmic cents up from the keynote C. Ans. 204, 386, 498, 702, 884, 1088. 7. If the keynote C is made of frequency 66 per second, determine the frequencies of the other notes of the scale. Alls. 74-25, 82-5, 88, 99, 110, 123-75. 8. Write a short essay on the production and propagation of sound. EXAMPLES ON CHAPTER II On Arts. 13-31. 1. Define Simple Harmonic Motion, and find expressions for the velocity and acceleration of a point executing it. 2. If a point executes a simple harmonic motion of period 657 2 U 658 SOUND 47r seconds and amplitude 3 cm., find its maximum velocity and that at half its full displacement, also the acceleration at its turning-points and when the displacement is 1 cm^ Alls. Velocities ±3/2 and ± 3 ■^S/i cm./sec. Accelerations ± 3/4 and ±1/4 cm./sec.^ 3. Plot to scale a displacement curve or "graph" for the motion of Question 2, and verify by its aid the results previously obtained. (This and other like exercises may be done on squared paper.) 4. Enunciate the chief characteristics of transverse progressive waves of simple harmonic type. 5. Draw a displacement curve for longitudinal progressive waves and explain the significance of each feature of the curve. 6. Define the phase of a vibration, and explain and illustrate by equations the various ways of measuring it. 7. Write equations for the following progressive waves : — (a) Amplitude O'l cm., period 1/64 sec, wave length 450 cm. (b) Amplitude r27 x 10~^ cm., frequency 256, velocity of propagation 33,000 cm./sec. 8. Find the maximum " condensation " s in Question 7 (h). Ans. s=6 X 10-». 9. Compound two collinear simple harmonic motions of equal periods, their amplitudes being 2 and 3 em. and their epochs 7r/4 and tt/S respectively. 10. Compound two simple harmonic motions of equal ampli- tudes, their frequencies being as 15 : 16, and exhibit graphically the resultant for eight of the shorter periods. 11. By the method of displacement curves compound two simple harmonic vibrations of periods and amplitudes each as 2: 1. 12. AVrite a short essay on progressive waves. On Arts. 32-50. 13. By means of a ball hung by a thread fixed at its upper end, illustrate experimentally the composition of rectangular vibra- tions of equal periods, their phase differences being successively zero, half, and quarter. 14. Compound analytically or graphically two rectangular EXAMPLES ON CHAPTER II 659 vibrations of equal amplitudes, both vibrations starling with no displacement and their periods being as (a) 2 : 1 and (b) 2 : 3. 15. By experiments with a Blackburn's pendulum make sand traces to verify the results found for Example 1 4. 16. State the chief characteristics of stationary waves, show- ing particularly how they differ from progressive waves, and writing the equations for each. 17. Enunciate and explain Huyghens' Principle of Wavelets and Envelopes. 18. Apply Huyghens' principle to the reflection of plane waves at a plane surface, and show that the angles of incidence and reflection are equal. 19. In the case of refraction at a plane surface treated by Huyghens' principle prove that the known laws of refraction of light are confirmed. 20. State and explain the conclusions to which Huyghens' principle leads as to the passage of waves through small openings and large ones. 21. How may the principle of Huyghens be experimentally illustrated ? On Arts. 51-59. 22. Express y as, a, function of t by Fourier's theorem given that frorn t = Qio t/2, y = and that from t = t/2 to t, y = 1h. 4k ( sin wt sin 3u>t sin Sat } Jus. y = ^---|-3-+-^- + ^-+ . . .| 23. From t = to a, i/=ft-h, and from t = a to t, y^fj(T-f)-h\ expand y by Fourier's theorem. ^ ^^ ' Ans. See art. 348. 24. From Example 22 plot to scale the first three terms of the Fourier series, compound them, and compare with the original conditions. , . 25 In Exg,mple 23 put a = T/3, and plot curves to scale tor the first four terms of the Fourier series, comparing the result of composition with the terms of the question. 26 Plot the equiangular spiral r cm. = 4i-», choosing h so that the radius vector shrinks to half in each complete revolution. {Note. — h = 1-1166 nearly.) 27 If a point describes the spiral of Example 26 at the rate 660 SOUND of 50 revolutions per second, what are its linear velocity and accelerations at the start ? Ans. Velocity = 1264 cm./sec, accelerations are 399,424 centrally, and 87,974cm./sec.2 opposite to velocity . 28. The motion of the point in Example 27 is projected upon a fixed straight line parallel to the initial radius vector. Write the equation of this piojected point, and find expressions for its velocity and acceleration at time t. Ans. See equations (11) and (12) of art. 58, in which write ffi=4 cm., k = .34-8 per sec. nearly, 2 = 31416 radians per sec, and ^=316 radians per sec. On Arts. 60-70. 29. Explain Doppler's principle, and show that, in the absence of wind, a given speed of approach of source raises the apparent pitch more than the same speed of approach of the recipient. 30. A cyclist riding at 15 miles per hour meets a carriage with bells proceeding in the opposite direction at 5 miles per hour. How many logarithmic cents will the pitch appear to fall on passing 1 (Take speed of sound 1100 ft./sec.) Ans. 9 2 '3 4 cents nearly. 31. Make a diagram showing the refraction of sound by a wind whose speed is 100 miles per hour, the original wave front being inclined at 45°, and distinguish between the refraction of the wave front and the refraction of the direction of propagation. 32. Establish the law of refraction of sound by an abrupt change of wind speed, and account for a possible total reflection. 33. Define Strain, Strain Ellipsoid, and Simple Shear. Obtain the fractional change of volume consequent upon a general strain. 34. Show that one uniform dilatation and two simple shears are able to build up any specified general strain. Hence resolve the strain (0'03, 0'02, 0"01) into a uniform dilatation and two shears. 35. Eesolve into axial strains each of the three following strains : — (a) The simple elongation (O'Ol, 0, 0); (b) The uniform dilation (0-02, 0-02, 0-02) ; and (c) The simple shear (0-03, -0-03, 0). Ans. See equations (7), (8), and (9) of art. 69. EXAMPLES ON CHAPTEE Til 661 36. Explain fully with diagrams that a simple shear may be regarded as a progressive sliding of undistorted planes. Also show that the amount of an indefinitely small shear is double of its fractional elongation or contraction. EXAMPLES ON CHAPTER III 1. Explain the nature of Elasticity, and give an equation showing how it is measured. 2. Draw up a tabular statement of the chief elasticities, showing by symbols the stresses and strains characteristic of each, and also the class of substance to which each applies. 3. Define Kigidity. Explain by aid of a diagram that a second view of the corresponding stress may be taken, and find the relation between the two. 4. Express Young's modulus and~ Poisson's ratio for an iso- tropic substance in terms of the volume elasticity and rigidity. Ans. See Table V. 5. Find the value of the elongational elasticity of a solid in terms of (a) its Young's modulus and Poisson's ratio, and (b) its volume elasticity and rigidity. Ans. See Table V. 6. Express in terms of the volume elasticity and rigidity of an isotropic solid the values of (a) the Eatio of elongation to tension, and (b) the Eatio of lateral contraction to tension. Ans. See Table V. 7. Experimentally verify Hooke's law by tension of a vertical wire provided with scale and vernier, plot a curve with loads and elongations as co-ordinates, and deduce Young's modulus. 8. Eepeat the experiment of Example 7 by one of the finer methods, and note the higher accuracy obtained, and express it by retaining more figures in the answer. 9. Treat mathematically the torsion of a cylinder of isotropic substance, and obtain an expression for the equal and opposite couples applied at the end faces. 10. Determine experimentally the Young's modulus, rigidity and volume elasticity of the substance of the given short piece 662 SOUND of wire or fibre (glass or quartz), only the ordinary apparatus of a physical laboratory being available. Method. — First for Young's modulus by bending. Let the fibre rest on supports a distance I cm. apart, and be observed microscopically to be depressed a depth a cm. at the middle in consequence of a load of m gm. suspended there. Then if the radius of the fibre or wire is r and the Young's modulus q, it may be shown that q = mgP/l27rr^a (1). Second for Pdgidity by torsional oscillations. Prepare a base board of wood about 10 cm. long, 2 cm. wide, and 2 mm. thick. Make a hole at the centre to admit one end of the wire or fibre, which is then fastened off underneath, the main part of the fibre standing at right angles to the board. Clip the far end of the fibre so that it hangs vertically with the board horizontal at the bottom, and free to oscillate by torsion of the fibre. On the board set two ordinary laboratory weights, each of M gm. (say 50 gm.) at a distance 2ij apart, centre to centre, and suppose the period of complete torsional oscillations is found to be Ty Next, place the weights nearer, a distance 2h^ apart say, and let the period of oscillation be T^. Then if the length of fibre in use is L and the radius r, it may be shown (see arts. 81 and 85) that the rigidity is given by i^Z3/_ (V-fc,^) " r* (T^-Ti) Third, having the Young's modulus and rigidity, the Volume Elasticity is found by the relation k = nq/(9n - 3q) in Table V. The above is the method referred to at the end of art. 83. It has been carried out by students at Nottingham, one of whom (Mr. T. J. Eichmond) obtained for a fibre of soda glass q= 5'61 X 10^' dynes per sq. cm. m = 2'33 X IQii dynes per sq. cm. whence /.= 3-15 x lO'i dynes per sq. cm. These are very fair values for such a simple method, capable of being worked in an hour or two, and without any special apparatus. 11. "Write an essay on Strains, Stresses, and Elasticity. EXAMPLES ON CHAPTER IV 663 EXAMPLES ON CHAPTER IV On Arts. 84-90. 1. Give a general expression for the period of a vibrating mass. Show that a weight at the middle of a stretched string has a period in accord with this. What must be the tension in a string a yard long in order that a mass of 4 lb. at the middle vibrates with a period of 2 seconds 1 Ans. 29'58 poundals, or 0'92 lb. weight nearly. 2. What is the total energy of the mass in Example 1 while the amplitude is 3 inches ? Ans. 1-23 ft. poundals, or 00382 ft. lb. wt. 3. Calculate the total energy possessed by a cubic centimetre of steel at the end of a steel tuning-fork where the amplitude is 2 mm., the frequency being 64 jier second. (Take the density of steel as 8 gm. per cc.) Ans. 25,870 ergs, or 26-37 cm. gm. wt. 4. In the case of a mass M suspended from an elastic cord or spring of mass m find the period of vertical oscillations, correcting for the mass of the spring and establishing the expressions used. 5. Obtain, either with or without the calculus, the motion of a particle subject to a resistance proportional to its velocity as well as a restoring force proportional to its displacement. 6. Plot a displacement- time curve showing the damped vibration of a particle under conditions such that the friction lowers its pitch by one logarithmic cent to the value 40 per second, the initial amplitude being one centimetre and velocity zero. Ans. The recpiired curve is y = e"^* cos iQtrt nearly. 7. If in Example 6 the particle in question is a cubic milli- metre near the end of a large steel bar, calculate the values of the restoring and frictional forces acting upon it. (Take the density of steel as 8 gm./cc.) Ans. Restoring force = 506 dynes per cm. Frictional force = 0-144 dyne per unit speed {i.e. 1 cm./sec); this gives a maximum resistance of 36 dynes as the particle passes through its equilibrium position with a speed of about 260 cm. per second. 664 SOUND On Arts. 91-111. 8. Treat the problem of the vibrations forced upon an elastic system by an impressed harmonic force, using either the calculus or an elementary method. 9. Show how the vigour of forced vibrations depends upon the closeness of tuning between the frequency of the force and that natural to the system. What prevents their indefinite increase ? 10. Discuss the phenomenon known as the "sharpness of resonance," and explain upon what quantity it depends. 11. An elastic system under the influence of an harmonic impressed force often exhibits two sets of vibrations, the "forced" and the "natural." Explain exactly how each set arises, and how one may be withdrawn or fail to appear. 12. If an impressed harmonic force begins to act upon an elastic system at rest, what motion immediately ensues, and how is it modified as time goes on ? 13. A small bullet is hung by a fine silk thread a metre long to a large mass which is itself suspended by a strong wire 110 centimetres long. All being initially at rest the large mass is struck horizontally by a hammer so that it begins to oscillate with an amplitude of 1 cm. Describe in general terms Avhat happens to the bullet. Also obtain an equation expressing its displacement as a function of the time, supposing it is subject to resistances which would reduce the amplitude of its own vibra- tions to one-half in 100 vibrations. 14. Set up apparatus in experimental illustration of Example 1 3, and confirm or disprove the conclusions theoretically reached. 15. Write an essay on Forced Vibrations. On Arts. 112-116. 16. Write and solve the equations of motion of an elastic system in which the value of the restoring force in-^'olves the square of the displacement. 17. Explain carefully under what conditions the octave and the twelfth of an original tone may be produced Avhen the amplitude is large. 18. If an asymmetrical elastic system is under the sim- ultaneous action of two periodic impressed forces what is the nature of its response 1 Prove your statements mathematically, 19. Write a short essay on the Principle of Superposition, EXAMPLES ON CHAPTEE IV 665 pointing out the limitations of its application and giving a numerical example. On Arts. 117-129. 20. Establish the expression for the speed of waves along a stretched cord, and explain how it may be experimentally confirmed. 21. Derive a general expression for the speed of sound in a gas, also discuss the formulae due to Newton and to Laplace. 22. How does the speed of sound in gases vary with pressure and temperature 1 Establish an approximate formula for the speed in air at various temperatures. 23. Discuss the possibility of waves of sound of permanent type advancing through the air, and show why, in the case of intense sounds, the condensed portions overrun the rarefied portions. 24. Obtain the speed of longitudinal waves in a solid rod, and show that no available tension will make transverse waves go so fast. 25. Derive the differential equation for small transverse disturbances along a stretched cord. 26. Establish the differential equation for plane waves in a gas. 27. Treat, by the calculus, the case of small longitudinal disturbances in a solid prism and in an extended solid. 28. Write an essay on the propagation of small disturbances, treating typical cases in detail. On Arts. 130-144. 29. Solve the differential equation for small disturbances, and apply the solution to the case of any initial disturbance without velocity. 30. Treat the case of the propagation of small disturbances in an elastic medium with an initial motion but no displacement, the differential equation being assumed known. 31. Obtain expressions for the reflections occurring at a " fixed " and at a " free " end. 32. Make a series of diagrams showing the state of things at various instants in a stretched cord ten metres long, in which there was initially a small displacement without velocity between the points distant three and four metres respectively from one end. 666 SOUND 33. Treat, both analytically and graphically, the motion of a stretched string pulled aside by a sharp point at one-seventh of its length and then let go. 34. Solve the differential equation for small disturbances in a manner directly applicable to a stretched string with fixed ends and without explicit mention of waves and their reflection. 35. Give a solution by Fourier's series for a plucked string, making a diagram showing the decreasing law followed by the different overtones. 36. Discuss the motion of a struck string, using Fourier's theorem, and treat the convergence of the partials and its de- pendence upon the place of excitation. On Arts. U5-155. 37. Obtain expressions for the energy (per vmit volume) and energy current (per unit area) of plane sound waves in air. 38. In the case of a musical tone of frequency 256 per second, the sound waves in the air being plane and having an amplitude of one -thousandth of a millimetre, find (a) the energy of the radiation per cubic centimetre, and (b) the energy current per square metre, the speed of propagation being 33,200 cm. per second. (Take the density to be 0"00129 gm. per c.c.) Jns. (a) 0-0000167 erg/c.e. nearly. (b) 5540 ergs per square metre per second. 39. Obtain the general, differential equation for small vibratory disturbances in a light compressible fluid medium in space of three dimensions. 40. Transform the general differential equation for vibrations in a compressible fluid (obtained in answer to Example 39) so as to express spherical radiation. 41. Discuss the reflection of spherical waves at the origin. On Arts. 156-163. 42. Obtain expressions for the reflection and transmission coefficient of sound waves on reaching the surface of separation of two gases. 43. If a sound wave originates in hydrogen gas and passes by normal incidence into air, what fraction of the initial energy emerges 1 (See art. 161 for method.) 44. If a wave disturbance travels along a thick stretched cord till it reaches a junction with a thinner cord, show what happens EXAMPLES ON CHAPTER V 667 and that the result is in harmony with the conservation of energy. 45. Establish expressions for the partial reflection and trans- mission of longitudinal waves along solid rods at a junction where some change in dimensions or properties suddenly occurs. Examine and justify by general considerations the changes of phase, if any, in the reflected wave (a) when the Young's modulus does not change at the junction, and (b) when the density does not change ; the cross section being constant for the two rods under each supposition. Ans. (a) Displacement reversed by reflection from denser rod in which waves travel slower. (6) Displacement not reversed by reflection from rod in which Young's modulus is lower and waves travel slower. 46. A longitudinal wave is started in a solid rod whose cross section, density, and Young's modulus are respectively s-^, p^, and 5j. It then reaches a place at which the rod changes suddenly in cross section and material being now characterised by s^, p^, and q^ Show that the fraction of incident amplitude reflected at the junction may be represented by 6 ^ ^1 ^^P ig i - % '^Pi ig'i = S2 Vpafe EXAMPLES ON CHAPTER V On Arts. 164-181. 1. Determine the notes which may be emitted by a stretched string of given length, diameter, and density. 2. Experimentally verify the laws of vibrations of strings by a monochord and suitable tuning-forks. (See art. 166.) 3. Discuss the modifications introduced into a string's vibra- tions by its own stiffness or the lack of fixity of its ends. 668 SOUND 4. Obtain expressions for the possible tones yielded by rods in longitudinal vibration when fixed at the middle. How may they be elicited ? 5. Establish the frequencies of the possible tones of open and stopped parallel pipes. 6. How may the vibrations of organ pipes be experimentally demonstrated ? 7. AVrite an essay on organ pipes, dealing especially with the modifications introduced into the elementary theory by the necessary corrections for the open end and mouth. On Arts. 182-197. 8. Obtain equations expressing the stationary vibrations in conical pipes, and apply them to the cases of a complete cone with base closed and base open. 9. Find the positions of the nodes and antinodes in a complete open cone whose corrected length is one metre, when responding to a tuning-fork whose pitch is the fifth partial of the cone. Ans. Measuring from the vertex throughout : — Nodes, 0, 28-61, 49-18, 69-42, and 8949 cm. Anti- nodes, 20, 40, 60, 80, and 100 cm. (See Table VHI. and art. 188.) 10. Establish the differential equation for small disturbances of a stretched membrane. 11. Treat the vibrations of a square membrane dealing with at least three of its possible tones and sketching some of the possible nodal systems. On Arts. 198-220. 12. Calculate the bending moments required to bend a given bar to a specified curvature, illustrating your answer by two numerical examples. 13. Obtain the relations between applied forces, shearing stress and bending moments in a bar at rest. 14. Derive the differential equation of transverse motion of an elastic bar, also the conditions which apply at the ends under various circumstances. 15. Assuming the differential equation for the lateral motion of a bar, solve the equation and show how the frequencies of the possible tones of the bar depend upon its dimensions and properties. EXAMPLES ON CHAPTEE VI 669 16. Discuss tlie lateral vibrations of a free-free bar, giving a graphical solution of the equation found for the series of partials of virhich it is capable. 17. Show that a slight modification of the graphical method referred to in Example 16 vrill solve the equation defining the sequence of tones which may be elicited from a fixed-free bar vibrating laterally. 18. Experimentally obtain the lateral vibrations of a bar free at both ends, and of a bar fixed at one end and free at the other. (See arts. 210 and 212.) 19. How has it been shown experimentally that the simple static- theory of Lord Rayleigh gives a close approxima,tion to the period of lateral vibrations for a fixed-free bar ? 20. Describe the change in position of nodes which pro- gressively occurs as a bar is bent more and more from the straight form to a U-shaped bar. 21. The temperature variation of a steel tuning-fork being shown by Koenig to be minus 0'000112 per 1° C, and the linear expansion being "0000 12 per 1° C, show that the temperature variation of the Young's modulus for steel is m,iiius 0'"000236 per rC. (See art. 220.) On Arts. 221-231. 22. Describe what are known as Chladni's figures, giving sketches of a few for a square plate with free edges. 23. Obtain experimentally four or more Chladni's figures with the plate provided. 24. Explain by diagrams and general reasoning the derivation of a simple Chladni's figure. 25. Discuss the vibrations of a ring, a cylindrical shell, and a bell. EXAMPLES ON CHAPTER VI On Arts. 235-244. 1. Enumerate at least ten examples of resonance and allied phenomena, classifying them in tabular form according to the nature of the interactions occurring. 2. Experimentally tune the given flask to resonance with each of the forks provided. 670 SOUND 3. Determine experimentally by the adjustable water resonator the relative frequencies of the prime and second tone of the given fork. 4. Give a simple theory of a resonator, obtaining an ex- pression for the frequency of its prime tone. 5. Write a short essay on resonance, pointing out how the subject may be experimentally illustrated. On Arts. 245-258. 6. Describe the two forms of Melde's experiment, and give a specimen set of loads to exhibit the various numbers of segments. 7. Experimentally obtain the vibrations of a thread attached to a tuning-fork as in Melde's experiment, one, two, and three segments being exhibited in each of the two modes. 8. Explain the action of Dr. Erskine-Murray's phonoscope. 9. Describe by aid of diagrammatic sketches the salient physical features of the human ear. 10. Follow in detail the various chief occurrences in the act of hearing in the case of simple and compound tones. 11. What reasons have we for believing that the vibrations received at the outer ear are modified as they pass through the train of aural mechanism ? On Arts. 259-278. 12. Describe the action of a violin bow and also that of Goold's generators. 13. Write an essay on the theory of singing flames. 14. Explain the paradox that a vibration maintained by a harmonic impressed force is most encouraged if the force is at its maximum when the displacement is zero, whereas a vibration maintained by sudden heating and cooling is most encouraged if these efiects occur when the displacements are at their maximum. 15. Explain the arrangements for driving a tuning-fork electrically, showing particularly how it is that the electro- magnetic action helps the vibrations one way mgre than it hinders it the other. On Arts. 279-287. 16. Describe several forms of sensitive jets and flames, pointing out the special advantages of each. 17. Explain the phenomenon of the setting of a disc normally to motions in the air, and describe how to demonstrate this effect. EXAMPLES ON" CHAPTEE VII 671 18. Give the theory underlying the production of Kundt's dust figures, and particularise some of the precautions essential to success. 19. Experimentally obtain a set of Kundt's dust figures. 20. Explain the occurrence of the striations in Kundt's dust figures. EXAMPLES ON CHAPTEE VII On Arts. 288-296. 1. Describe several methods of experimentally demonstrating the phenomena of interference. 2. Explain precisely the distinction between interference and beats. 3. State how the law of the frequency of beats may be established. 4. How may the phenomena of beats be demonstrated to a single observer, and how to a large audience 1 On Arts. 297-306. 5. What is meant by the term combinational tones ? Give two or more methods by which they may be experimentally demonstrated. 6. Draw up a table showing the pitches of the differentials and summationals produced by given generating tones. 7. Do you suppose the vibrations corresponding to combina- tional tones to occur first in the ear itself, or to occur inde- pendently of the ear altogether 1 Give reasons for your belief. On Arts. 307-321. 8. How may tones of higher *rder arise in an elastic system from the influence of a loud simple tone ? Account mathe- matically for the production of some one higher partial in this manner. 9. Give an experimental example of the octave being heard from a vibrating body which has no parts whose frequency is that octave. Discuss the possible ways of production of this upper partial. 10. Summarise the researches of Koenig on beat notes, and give Bosanquet's views on the subject. 672 SOUND 11. Explain the methods and results of the researches of Riicker and Edser on combinational tones. 12. Describe the recent work by E. Waetzmann on combina- tional tones. 13. State the theory put forward by Everett to account for resultant tones. 14. Write an essay on the objectivity of combinational tones. EXAMPLES ON CHAPTER VIII On Arts. 322-363. 1. Analyse the essential parts of a musical instrument re- garded from the view-point of the physicist. 2. Give a tabular classification of the chief musical instru- ments showing the approximate compass of each. 3. Discuss the quality of tone from plucked strings. Can the vibrations be represented by a Fourier expansion ? 4. Compare the qualities of tone from strings struck (a) with a sharp, hard edge, and (b) with a soft rounded hammer. 5. Enumerate and explain the chief points of scientific in- terest in instruments of the violin family. 6. How has the problem of bowed strings been treated by Helmholtz ? 7. State in general terms the nature of the motion, as determined by Helmholtz, of a well-bowed string when emitting a note of good musical quality. 8. Describe the researches of Krigar-Menzel and Raps on bowed strings. 9. How may the characteuistic motion of a bowed string be optically demonstrated to an audience ? 10. Does the motion of the string alone of a stringed instru- ment determine the quality of the sound heard ? If not, state why not, and describe any work undertaken to throw light on the subject. On Ark. 364-380. 11. Discuss the arrangement and tuning of organ pipes with and without reeds. EXAMPLES ON CHAPTEE VIII 673 12. Explain how organ pipes without reeds "speak," and account for the different qualities of tone of various pipes. 13. Describe the typical " wood-wind " instruments, explain their action, and account for their characteristic qualities of tone. On Arts. 381-398. 14. How is the sound produced in a brass instrument ? Explain also how the various possible notes are obtained on the French horn (without valves) and on the slide trombone. 15. Explain the production of the various notes of the chromatic scale on a brass instrument with three valves. 16. Show that the intonation of the ordinary three-valved brass instruments must be faulty, and explain its improvement by compensating pistons. On Arts. 399-404. 17. Explain the production of musical tones by the human voice. 18. Give a classification of the chief vowels, and explain the modifications of the mouth cavity and opening on which they depend. 19. Explain the ordinary view as to what characteristics in a compound tone impart to it the vowel quality, and state the resonance pitches for the leading vowels. On Arts. 405-415. 20. Discuss the experimental results that have been obtained as to the pressures used in playing wind instruments. 21. How are the pitches of orchestral and other instruments affected by temperature changes ? 22. Compare the special qualities of tone from various wind instruments, and state how you could demonstrate their indi- vidual composition by experimental synthesis. 23. Describe the analysis of the voice by a set of Helmholtz resonators. 24. Experimentally examine by Koenig's manometric capsule and flame a series of vowels, each sung at several different pitches. 25. How may vowels and consonants sung or spoken be analysed and their characteristic vibrational forms exhibited to an audience? (See arts. 414, 2.50, and 251.) 2x 674 SOUND EXAMPLES ON CHAPTER IX On Arts. 416-428. 1. Give an account of Helmholtz's theory of concord and discord. 2. State several ways in which beats may arise, and describe experiments to illustrate two or more of these cases. 3. Explain the absence of obtrusive beats between two tuning-forks whose interval is a mistuned major third. On Arts. 429-436. 4. Describe, with passages in musical notation and a graph, how the degree of harmoniousness varies for the chief intervals within an octave, the tone being of violin quality. 5. How may the variation of harmoniousness of two notes within an octave be experimentally demonstrated in confirmation of Helmholtz's theory 1 6. Give an example of a concord between instruments of different quality of tone, which is changed in degree of con- sonance if the instruments interchange parts. Why is this so 1 7. Write down a few of the major triads in the scale of C, give some of their best positions and some of their less perfect positions, accounting for the difference. 8. Explain why minor triads are usually less harmonious than major triads. 9. Write an essay on concord and discord. On Arts. 437-450. 10. Enumerate the chief reasons for temperament in the tuning of musical instruments with fixed notes and keys. 11. It is often stated that human voices, instruments of the violin family, and slide trombones can easily perform in just intonation. Critically discuss this statement. 12. What difficulties would be entailed if just intonation were demanded from instruments with fixed notes like the piano and organ 1 On. Arts. 451-464. 13. Explain the derivation of mean-tone temperament, its tuning, its ad^-antages and disadvantages. EXAMPLES ON CHAPTER X 675 14. What pairs of notes in the Pythagorean tuning become fused in the equal temperament and why 1 State also what advantage is fully obtained in the equal temperament and what drawbacks are involved. 15. Show precisely how the effect of the " comma " (22 cents) may be illustrated harmonically on the violin by sounding successively a fourth and a major sixth. 16. What was the tuning adopted in Bosanquet's cycle of fifty-three, and what were the advantages aimed at by it ? How do you account for its failing to attain any vogue ? On Arts. 465-472. 17. How may the special musical character of certain keys be scientifically explained ? 18. State the frequency ratios of the following intervals, also their measure in logarithmic cents. Just Intonation : the comma, diatonic semitone, trumpet seventh. Mean-tone Temperament : whole tone, diatonic semitone, fifth. Equal Temperament : semitone, major third, fifth. (See Table XLVI. in art. 471.) 19. Trace the changes of pitch which have occurred in the course of musical practice. (See Table XLVH. in art. 472.) 20. Write an essay on Consonance and Temperament. EXAMPLES ON CHAPTER X On Arts. 473-488. 1. Give an account of the classic determinations of the velocity of sound in free air, and state the results obtained. 2. How have the velocities of sounds in water been obtained by large-scale experiments 1 3. Describe Regnault's celebrated researches on the velocity of sound in pipes. What was the view of Rink respecting this work 1 4. What has been done to test the dependence of speed of sound on pressure and pitch ? 676 SOUND On Arts. 489-508. 5. How was the speed of sound in iron found by Biot ? 6. Describe Dulong's experiments with organ pipes and various gases, indicating the results obtained by him. 7. What did AVertheim find as to the speed of sound in different-sized organ pipes ? 8. Explain the experiments by Wertheim as to the speed of sound in liquids, also the erroneous views held by him and exposed by Helmholtz, Kundt, and Lehmann. 9. Give an account of Kundt's researches by means of his method of dust figures, and state the chief results obtained by him. 10. Explain what other determinations can be made by Kundt's tube besides the relative speeds of sound in air and other gases. 11. Experimentally obtain the speeds of sound in brass, oak, pine, etc. by Kundt's tube. 12. Obtain an experimental determination of the values of Young's modulus for various solids by Kundt's tube. 13. How may the value of the mechanical equivalent of heat be deduced from acoustic and thermal data 1 On Arts. 509-532. 14. Describe Hebb's telephone method for the speed of sound in air. 15. How may the speeds of sound in soft bodies be deter- mined ? 16. Write a risumi of the theoretical and experimental work which has been done on the dependence of speed of sound in pipes upon pitch and diameter. 17. Describe Wiillner's researches on the speed of sound in various gases and at different temperatures. 18. Explain carefully the reasons which led Blaikley to adopt a bulbous tube for his experiments on the speed of sound, and state the results thus obtained by him. 19. What do you know of recent work as to the speed of sound in hot gases and in gases at liquid-air tempers tures ? 20. Give an account of Blaikley's experimental determination of the correction for an open end, and state how a simple repeti- tion of this may be carried out iu any laboratory. EXAMPLES ON CHAPTER X 677 On Arts. 533-568. 21. Describe the vibration microscope, and state the uses to which it has been put. 22. How may Lissajous' figures be optically projected and utilised to determine the relative frequencies of two forks 1, 23. Experimentally determine the ratio of the frequencies, of two forks by the monochord. 24. Explain the method of tuning and the manner of using the set of forks which constitute Scheibler's tonometer. 25. How did Lord Eayleigh determine the absolute pitch of a note on a harmonium ? 26. Determine experimentally the pitches of the given fork and organ pipe by use of the siren of Cagniard de la Tour. 27. By means of the fall plate experimentally determine the frequency of the given fork, the value of " g " being assumed. 28. By experiments with the vertical monochord obtain the absolute frequency of the given tuning-fork. 29. Assuming the speed of sound in air at 0° C, experimentally determine the pitches of the forks provided. On Arts. 569-58L 30. Explain the methods adopted by Hartmann-Kempf in examining the variation of pitch and decrement of a fork with amplitude, and state also the results obtained. 31. When a resonator responds to a fork, in what way is the pitch of the fork itself modified ? 32. Give an account of the various experiments as to the lowest pitch which constitutes a humanly-audible sound. 33. What do you know as to the highest pitches audible, and how would you test any person's upper limit of audition ? 34. Give a general explanation of harmonic echoes, the sound returned being apparently an octave (or more) above the original sound. 35. How may a musical sound arise by the successive echoes of a very sudden sound of extremely short duration ? 36. Upon what conditions does the pitch of an ^Eolian tone depend ? On Arts. 582-593. 37. Explain, with equations, liayleigh's fork method for determining the minimum amplitude audible. 678 SOUND 38. Describe Dr. Shaw's electrical determination of the minimum amplitude audible. 39. Enumerate and explain the various deflections and losses suffered by sound in its transit over long ranges. On Arts. 594-606. 40. Explain generally why sound waves incident upon a surface produce a pressure, and describe the experimental confirmation of this phenomenon. 41. Establish Larmor's radiation pressure theorem. 42. Discuss Poynting's conception of the momentum of radia- tion, adding an outline of the work of Rayleigh on this subject. 43. Write an essay on the pressure of sound. On Arts. 607-619. 44. Discuss the dependence of the quality of a musical sound upon the phase relation of its component simple tones. 45. Give the evidence for and against the iixed-pitch theory of vowels. 46. Explain the work of Lord Eayleigh on our perception of sound direction. 47. Describe the work of Marage, and that of Sabine, upon architectural acoustics. EXAMPLES ON CHAPTER XI On Arts. 620-632. 1. Describe the construction and action of some form of phonograph. 2. Which theory as to the characteristic quality of vowels is favoured by Hermann's experiments with the phonograph ? 3. Review M'Kendrick's researches with the phonograph, point- ingout the difficulties met with and howthey were finally overcome. 4. Give a brief outline of the results obtained by M'Kendrick as to the character of the vibrations present in the sounds from various instruments and the voice. 5. Explain the methods followed by Bevier in his phonograph analysis, and give a risumi of the chief results thus obtained. On Arts. 633-647. 6. Give an account of the introduction of telephony and a sketch of Bell's receiver. EXAMPLES ON CHAPTEE XI 679 7. Explain the action of a telephone receiver, and account for its high sensitiveness. 8. Prove that in the ordinary telephone receiver a permanent magnetic field is indispensable. 9. Describe, with a sectional sketch, the construction and working of an Edison carbon transmitter. 10. Show that to insure constant efficiency with a telephone circuit, consisting of line, carbon transmitter, battery, and receiver, the voltage would have to vary as the square of the length of the line. How is this drawback obviated 1 11. Describe a simple form of the device known as Hughes' microphone. 12. Explain the use of transformers in long-distance telephony. 13. How have some of the elements of music been electrically communicated to patients who are stone deaf 1 On Arts. 648-653. 14. Explain with sketches the electrical arrangements which constitute the " speaking arc." 15. How does the "musical arc" act, and to what familiar musical instrument may it be compared 1 On Arts. 654-666. 16. Discuss mathematically the oscillatory discharge of an electrical condenser, and show that charge and current are each damped harmonic functions of the time. 17. Describe in general terms what happens when an electric condenser is allowed to discharge through a circuit containing both resistance and inductance. To what may the various electri- cal quantities be likened in the case of a mechanical vibration 1 18. An electrical condenser of capacity a hundredth of a microfarad ■'■ is charged, insulated from the battery, and then connected to a circuit whose resistance is one thousand ohms and inductance one henry ; determine the ensuing phenomena. Ans. The period t is 0'00063 second nearly, and the damping factor q (of art. 655) is 500 per sec, so that the amplitude is reduced to less than two-fifths in three periods. 1 The analysis in the text (art. 654 et seq.) is valid for t. ) Tenor Trombone (B|>) Bass Trombone (G) Bass Tuba (in C) Soprano (Cornet) (El>) Cornet (in Bt>) Tenor Horn (Eb) Euphonium (Bl>) Bombardon (El>) .Double Bass (Bb) C, Plucked 1 i Guitar . J Harp \ Mandolin l?5 Struck . 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