BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF X891 ^j^m ailMjif 9724 Cornell University Library Sound .. 3 1924 031 253 614 olin,anx The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031253614 CAMBRIDGE PHYSICAL SERIES SOUND AN ELEMENTARY TEXT-BOOK FOR SCHOOLS AND COLLEGES CAMBRIDGE UNIVERSITY PRESS iLonBon : FETTER LANE, E.G. C. F. CLAY, Managek (Etriniurgi): 100, PBINCKS STREET a-onHon: H. K. LEWIS, 136, GOWEB STREET, W-C. WILLIAM WESLEY & SON, 28, ESSEX STREET, STRAND JScrlin: A. ASHEB AND CO. !Lei?jig: V. A, BROOKHAUS Iftth) Horlt: G. P. PUTNAM'S SONS BomliBg anB dakutta: MACMILLAN AND CO., Ltd. aCotonto: J. M. DENT AND SONS, Ltd. STofigD: THE MARUZEN-KABUSHIKI-KAISHA All riyhts reserved SOUND AN ELEMENTARY TEXT-BOOK FOR SCHOOLS AND COLLEGES ■ by J. W. CAPSTICK, M.A. (Camb.), D.Sc. (Vict.) " Fellow of Trinity College, Cambridge :•■■ fit Cambridge : at the University Press 1913 A^>ss^7^ PRINTED BY JOHN OLAT, M.A. AT THE UHIVEKSITY PRESS PREFACE THIS book is intended primarily for students of Physics, but enough has been included of Helmholtz's Theory of Consonance to make it adequate also for students of Music. Those who wish tp limit themselves to the more purely physical parts of the subject may omit everything after § 263, with the exception of §§ 279 and 280. As the later chapters will have little interest for readers with no acquaintance with Music, it has not been thought necessary to define all the musical terms that are used. Chapter xvi includes many details of the construction and use of musical instruments that cannot be regarded as necessary to be known either by students of Physics or of Music. The greater part of the chapter is intended mainly for those who play in orchestras, and who wish to know some- thing of the principles underlying the construction of their instruments. Certain sections of the book are marked with an asterisk to indicate that they are of a rather higher standard than the remaining sections. They can be omitted by readers who wish to acquire only an elementary know- ledge of the subject, but in most cases the result reached should be known, even though the proof of that result is not read. The writer of an elementary Text-Book on Soimd suffers from the disadvantage that many of the most ordinary phenomena cannot be explained adequately without the use of mathematics of a somewhat advanced type. Con- sequently the argument cannot be presented in a logically VI PREFACE continuous form, and the reader is frequently called on to accept statements the reason for which cannot be given. The best the writer can do is to endeavour to provide experimental evidence or arguments from analogy, when the theoretical explanation passes beyond the limits of the mathematical attainments that can be assumed to be possessed by his readers. I hope that in the present work I have at least made it clear where such gaps occur, and have not left the reader in doubt as to whether any statement is to be taken as a deduction from what precedes, or is to be accepted without proof My aim has been to make the least possible use of mathematical methods, in order that the reader may not be led to evade the mental effort required for the appreciation of the physical connexion between the phenomena described. It is hoped that the book may serve as a useful intro- duction to the more analytical treatises, such as those of Prof E. H. Barton and Lord Rayleigh. I am greatly indebted to Mr T. G. Bedford, the general editor of the Cambridge Physical Series, for the valuable help he has given me throughout the prepara- tion of the book. I have also to express my thanks to Prof E. H. Barton for permission to make certain extracts from his Text-Book of Sound, to Mr D. J. Blaikley for much information concerning musical instruments, and to the authorities of the Universities of Cambridge, London and Dublin and the National University of Ireland for per- mission to make use of questions set in the examinations of those Universities. J. w. c. November, 1913. CONTENTS CHAP. PAGE I Nature of Sound 1 II Elasticity and Vibrations 13 III Transverse Waves 35 IV Longitudinal Waves 57 V Velocity op Longitudinal Waves ... 79 VI Reflection and Refraction. Doppler's Principle 90 VII Interference. Beats. Combination Tones . 109 VIII Resonance and Forced Vibrations . . . 128 IX Quality of Musical Notes 145 X Organ Pipes 165 XI Rods. Plates. Bells 181 XII Acoustical Measurements 196 XIII The Phonograph, Microphone and Telephone . 216 XIV Consonance 224 XV Definition of Intervals. Scales. Temperament 239 XVI Musical Instruments 250 Questions . . 274 Answers to Questions 289 Index 291 CHAPTER I NATURE OF SOUND 1. Meanings of the word Sound. The word Sound has two meanings in everyday life. When we say we hear a sound, we refer to the sensation. When we say that sound travels faster with the wind than against it, we refer to some physical phenomenon external to ourselves. The two meanings of the word rarely lead to ambiguity, and no attempt will be made to distinguish the two ideas by different expressions. 2. Velocity of Sound. Many of the facts relating to sound can be readily deduced by observation of what is going on around us, without the need for special appliances. It is a matter of common observation that sound travels much more slowly than light. The flash of a gun a mile away is seen about 5 seconds before the report is heard, and there is often a considerable interval between a flash of lightning and the resulting thunder-clap. The methods of measuring the velocity of sound will be discussed later. It will then be seen that the velocity in air at ordinary temperatures is about 1100 ft. per second, and knowing this it is easy to form an estimate of the distance of a thunderstorm, by observing the interval between the lightning and the thunder. The velocity of light is so great compared with that of sound, that for the purpose of this observation the time taken by the light in travelling from the electric discharge to the observer may be neglected. 3. Medium by which Sound is carried. If sound is carried from its source to the observer by material sub- stances, it is clear that air must be one of these substances ; for when a rocket explodes in the air, or an aeroplane passes by, the sound is heard, though there is no material substance between the source of sound and the observer, except the air. C. 8. 1 2 NATURE OF SOUND [CH. I It might be argued that we can also see the rocket explode, and it is obvious that the light by which we see it does not need the presence of air for its propagation, for we can also see the stars across space, which we believe to be devoid of air. A simple experiment, however, will shew that air or some other form of matter is esseirtial for the conduction of sound. 4. Experiment with an exhausted bell-jar. Place an electric bell or a small alarm clock on the plate of an air- pump, and cover it with a bell-jar. If the air is now pumped out, the sound heard will be somewhat weaker, shewing that the air took at least some part in the propagation of the sound. Next repeat the experiment, but place a pad of soft felt under the bell, or suspend it by india-rubber cords, and it will be found that now the sound is scarcely audible, when the air is pumped out. From this it appears reasonable to conclude that the ether cannot conduct sound, but that the air can. The sound cannot be made to disappear entirely, for the bell must be supported by something, and all material substances carry sound, though with varying degrees of facility. When the clock rested directly on the metal plate of the pump, the sound was conveyed quite readily to the air outside, whilst the pad of felt cut it off almost entirely. 5. Propagation of Sound in Solids. Experiments to shew the propagation of sound through solids are easily devised. Lay a watch on one end of a long rod of wood or metal. The ticking of the watch will be heard when the ear is placed near the other end of the rod, but not when it is some distance away ; from which we conclude that the sound heard in this case is conducted through the rod, and not directly through the air. This experiment must not be taken as proving that the material of the rod is a better conductor of sound than air. The sound communicated directly to the air by the watch spreads out freely in all. directions, and rapidly grows weaker as it travels farther from the watch, whereas little of that which passes into the rod emerges until it reaches the end. 3-8] NATURE OF SOUND 3 6. Velocity of Sound in difiTerent media. A similar experiment will shew that sound travels with different velo- cities in different media. Stand near an iron rail, whilst someone strikes the rail a few hundred yards away. Two sounds will be heard, one carried by the iron, and one by the air. It will be noticed that the sound which arrives first becomes louder if the ear is placed close to the rail, whilst the other is not affected by the change of position, so that we may conclude that sound travels more quickly through iron than through air. The velocity is in fact about 16 times as great in iron as in air. Sound is also carried readily by liquids. A person swimming under water can hear sounds made on the bank, and if two stones are struck together under water, a sound is heard in the air above the water. 7. Musical Notes and Noises. Sounds are usually divided into two classes, musical notes and noises, though there is no sharp distinction between the two. The essential difference is that musical notes have a recognizable pitch, whilst noises have not ; but few noises are entirely devoid of musical pitch, and few musical notes are devoid of unmusical noise. Drop two pieces of wood of different shape on the floor. What is heard would be classed as noise, yet there would be a difference between the sounds made by the two pieces, and a person with a trained musical ear would be able to distinguish between them from the admixture of musical note with the noise. Again, we can generally by close atten- tion detect some slight hissing of wind, or scraping, in the sound of a musical instrument. The ear is in general able to distinguish musical notes of different pitches sounded together, but if the notes are very numerous and close together in pitch, they cease to be dis- tinguishable, and the resultant sound is described as noise. It is probable that noise is usually such a mixture, where no one of the constituent notes is so prominent as to give a definite sensation of pitch. 8. Sound due to Vibrations. A little consideration shews that sound always takes its rise from some body which is vibrating. The ordinary method of making a tuning-fork give out its note by striking the prongs shews that what is 1—2 4 ■ NATURE OF SOUND [CH. I required is to make the prongs vibrate. Look closely at them when the fork is sounding, and from the haziness of their outline it will be seen that they are vibrating. Touch them very lightly with the fingers, and the vibrations will be felt. Press the fingers more firmly on them so as to stop the vibra- tions, and the sound will cease. Similarly it is easily seen that a violin or harp produces sound only when a string is vibrating. The vibrations which give rise to the sound are not so obvious in the case of a wind instrument, such as a flute or a trombone, for here it is the column of air which vibrates, and not the material of the tube. In the case of a large- organ-pipe the air can be felt to be vibrating, if the hand is held near the mouth, and in other cases where the vibrations are too small to be felt, they can be detected by appropriate methods. If for instance a small paper tray with a little sand on it is lowered by a thread into an open organ-pipe with a glass panel in its side, the sand will be seen to dance about, as soon as the pipe is made to sound. 9. Sound is a Wave Motion. We have seen then that sound takes time to travel from the vibrating body to the hearer, and that it requires some material substance to carry it. It is clear that when the sound is travelling through the air it does not consist of a bodily transference of the air. If one stands in front of a trombone that is being blown loudly, no blast of air is felt, nor does a gale of wind spread outwards from a cannon when it is fired. These facts, coupled with the observation that sound always takes its rise from some body which is vibrating, suggest that what travels through the air is a series of waves of some kind. This conjecture is consistent with all the observed pro- perties of sound, and is in particular supported by the existence of interference, that is, by the quenching of sound at certain points by the superposition of another sound . Hold a vibrating tuning-fork to the ear and turn it round slowly with the fingers. In certain positions the sound will be almost in- audible. Whilst it is in one of these positions, get someone to slip a paper tube over one of the prongs, being careful not to touch the prong atid stop its vibrations. The sound will be 8-11] NATURE OF SOUND 5 heard to swell out again. This experiment shews that in certain positions the sound coming from one of the prongs is able to neutralize that coming from the other, and this clearly could not happen, if sound were merely a current of air. The phenomenon will be discussed more fully in a later chapter. It is mentioned here as being the kind of evidence on which we rely for our belief that sound consists of wave motion. 10. Characteristics of a Musical Note. Two musical sounds may differ fi'om one another in three, and only three, ways. They may differ in loudness, in pitch, and in quality. By quality we mean the characteristic which enables the ear to distinguish, for instance, between a note played on a flute and a note of the same pitch and loudness played on a violin. We shall discuss very briefly the way in which these characteristics of a musical note are related to the features of the vibrations that give rise to the note. What will be said must not however be regarded as a proof of the relation- ships, but rather as a synopsis of what will be more fully proved in the succeeding chapters. 11. Loudness. Strike a tuning-fork strongly, so as to make it give out as loud a note as possible. The extent of vibration of the prongs can be seen from the extent of the haziness at their ends. _It will be noticed that this amplitude gradually diminishes, and that at the same time the sound grows weaker. Thus it appears that the greater the extent of vibration of the fork, the louder is the sound that it gives out. A similar conclusion may be drawn from the vibrations of a stretched string, such as that of a harp. The farther it is drawn aside before being let go, the greater will be the amplitude of its vibrations, and the louder the sound it will give out. It is reasonable to suppose that the more widely the prongs of a fork vibrate, the greater will be the amplitude of the air waves sent out, and we may assume that the loudness of the sound depends in some way on the amplitude of the air waves — the greater the amplitude, the louder being the sound. NATURE OF SOUND [CH. I We need not discuss here the exact nature of the air waves, nor the exact meaning of the term amplitude as applied to them. The analogy of the familiar waves on water will serve our present purpose, though these differ in important respects from sound waves. In the case of water waves we mean by amplitude the height of a crest above the mean level of the water. In Fig. 1 AB is the am- plitude of the upper wave, and the two waves differ in no other respect than in amplitude. They represent, in a way that will be explained later, two sounds which have the same pitch and quality, but differ in loudness. 12. Pitch. A simple experiment for finding what feature of the vibrations of a body characterizes the pitch of the note emitted may be made with Savart's Toothed Wheel. A metal disc has saw-teeth round its circumference, and can be rotated at various rates by a handle or a small .electric motor. Turn the wheel at a gradually increasing rate, whilst a card is held against the teeth. At first separate taps are heard, as the card drops from tooth to tooth, but as the rate of rotation increases, the taps blend into a harsh note, whose pitch rises, as the wheel is made to rotate more rapidly. In this experiment the card is made to vibrate by the teeth, and the number of vibrations it makes per second depends on the rate of rotation of the wheel ; whence we conclude that the pitch of a note is determined by the number of vibrations per second of the body which gives out the note. The apparatus could be used to find how many vibrations per second correspond to any given note, such as that of a particular tuning-fork. Turn the wheel at such a rate that the card gives out the same note as the fork,. and count the number of turns made by the wheel in a minute. Suppose there are 100 teeth on the wheel, and it is necessary to turn it 120 times in a minute to give a note of the same pitch as that of the fork. Then the number of taps is evidently 200 per 11-13] NATURE OF SOUND 7 second, and this is the number of vibrations per second required to produce the note in question. This method, however, is not capable of any great accuracy. We shall shew in a later chapter thah, when the sound travels from the vibrating body to the ear through air or other medium, the number of waves that reach the ear per second is equal to the number of vibrations per second of the vibrating body; that waves of all lengths travel with the same velocity in any one medium such as air, and that therefore the diflFerence between a sound of high pitch and one of low pitch may be said to be that the wave-length of the former is less than the wave-length of the latter. Fig. 2 represents two waves of the same am- plitude but of different wave- lengths. The lower curve corresponds to a higher note Pig 2 than the upper. 13. Iiimits of Audibility. It should be mentioned here that a musical note is heard only when the number of vibrations per second is between certain limits, which are different for different persons. If the vibrations are slower than about 30 per second they do not blend into a note, but are heard separately. Even when they are somewhat above 30 per second, they do not give the sensation of a note, unless they are of the kind known as Simple Harmonic Vibrations, to be described in the next chapter. When the vibrations are very rapid> they cease to produce any impression on the ear. The sensitiveness of the ear to high notes falls off with advancing age. Children can generally hear notes with 20,000 vibrations per second, elderly people cannot generally hear anything above 15,000. We conclude then that within these limits the pitch of a note depends on the number of vibrations executed per second by the body which gives out the sound. This number is called the Frequency of the vibrations. 8 NATURE OF SOUND [CH. I 14. Intervals. It is often required to express numeri- cally the relationship between the pitches of two notes, and we must consider what is the most convenient method of measuring the interval between the notes. In deciding what method to use the chief fact of which we have to take account is the existence of certain intervals such as the octave, the fifth, etc., which give a constant and recognizable mental impression at all parts of the musical scale. A fifth, for instance, is the same interval to the ear, whether it be low in the bass or high in the treble, and whatever measure we adopt for an interval, it must be such as will give the same numerical value at any part of the scale to what the ear judges to be the same interval. We must first then find experimentally what feature of the relationship between the frequencies of two notes forming such an interval as a fifth is constant. This is readily done by the use of the Disc Siren. 15. The Disc Siren. A circular disc of cardboard or metal is mounted on an axle, so that it can be rotated rapidly. The disc has several circles of holes pierced through it, and a jet of air from the mouth or bellows can be directed on one of the circles by a narrow glass tube. Whenever a hole comes in front of the tube a pufi^ of air passes through the disc, so that we have a rapid succession of puffs, which will be found to blend into a musical note, and the pitch of the note will depend on the rate of rotation of the disc, and on the number of holes in the circle. Fig. 3 Let us suppose there are four circles of holes with 40, 50, 60, and 80 holes in the respective circles. Whilst the disc is turned at a steady rate, direct the jet of air first at the 80 circle, and then at the 40 circle. The interval between the two notes heard will be recognized as being an octave. Turn the disc more quickly, and both notes will rise in pitch, but the interval between them will still be an octave. Whatever 14-17] NATURE OF SOUND 9 may be the rate of rotation of the disc, the frequencies of the two notes must be in the ratio 80 to 40 or 2 to 1, whence we conclude that this ratio, which is called the Vibration Ratio of the interval is characteristic of the octave, and the fraction 2/1 may be taken as the measure of the interval of an octave. If now we make a similar experiment with the 60 circle and the 40 circle, we find the interval is always a fifth, whence we infer that the vibration ratio of a fifth is 3/2. If the disc is turned at such a rate that the 40 circle gives the middle C of the Pianoforte, the four circles will give the i -5 — • If the disc is turned a little common choi-d of C faster they will give the chord of D, and so on. 16. The Measure of Intervals. We may conclude therefore that, if we have any two notes with frequencies m and n, the interval between the notes may be measured by the fraction mjn, for however m and n may vary, the interval will be judged by the ear to remain the same, provided their ratio mjn remains constant. 17. Consonant Intervals. This is true of any interval whatever, but certain intervals are found to have an effect so pleasing to the ear that they are classed as Consonant Intervals, and have special names assigned to them. The consonant intervals within the limits of an octave are given below with their vibration ratios. The first five and the last of these ratios can be verified directly with the disc siren as described above. 2 6 Octave T Minor Third ■= 1 5 Fifth I Major Sixth | 4 8 Fourth g Minor Sixth -= o 5 Major Third j We shall make frequent use of these intervals, and the student should make himself familiar with their vibration ratios. 10 NATURE OF SOUND [CH. I 18. The Sum of two Intervals. To find the vibra- tion ratio of the interval obtained by adding together two intervals we multiply together the ratios of the two con- stituent intervals. Suppose we have three notes, which we will call p, q, and r, and suppose p has the lowest, and r the highest pitch of the three. Let us find the vibration ratio of the interval between p and r, when the interval between p and 5' is a fourth, and the interval between q and r is a major third. The vibration ratio of a fourth is 4 to 3 and that of a major third is 5 to 4. If then the frequency of p is taken as unity, that of q will be 4/3. The frequency of r is 5/4 times as great as that of q, and is therefore 4/3 x 5/4 or 5/3. Thus we find that the frequency of r is 5/3 times as great as that of p, or the vibration ratio of the interval ^ to r is 3 : 5, which corresponds to a major sixth. Similarly a major third and a minor third make 5/4 x 6/5 or 3/2, which is the vibration ratio of a fifth. It is evident that the rule holds generally, and we may say therefore that if we add the intervals whose vibration ratios are mjn and m! jn' we get an interval whose ratio is mm'/nn'. 19. The Difference of two Intervals. We can deduce at once from this the converse proposition. If we take the interval whose ratio is m/n from the interval whose ratio is m'jn', the difierence is an interval whose ratio is m'/n' -T-m/n; for by the preceding rule the interval »n/w added to the interval m'jn' -h m/n or m'n/m,n' gives the interval m,/n X m'njmn' or m'jn'. Thus, to add intervals we multiply their vibration ratios together, and to subtract we divide the ratio of the larger interval by that of the smaller. 20. The Diatonic Scale. It will be convenient to give here the vibration ratios that define the intervals between each note of the ordinary Diatonic Musical Scale and the lowest note or tonic of the scale. The reasons for the choice of these particular intervals will be given in Chapter XIII. Since the intervals remain the same whatever note is taken as the tonic, it is of no consequence what note we 18-21] NATURE OF SOUND 11 choose as the tonic. We shall take the scale of the white keys of the pianoforte, which have C as their tonic. C D E F G A B c 9 5 4 3 5 15 8 4 3 2 3 8 The meaning of the table is this. Whatever may be the frequency of C, that of c, an octave higher, will be twice as great ; that of E will be one and a quarter times as great, and similarly for the other notes. The fractions below the notes are proportional to the frequencies of the notes, in whatever octave on the pianoforte we may take them. The scale can be extended indefinitely upwards and downwards. In order to rise an octave we double all the frequencies, and to fall an octave we halve them. It will be found by reference to the table in Par. 17 that the interval between C and c is an Octave and G is a Fifth C and F „ Fourth Cand E A and c O and A E and c Major Third Minor Third Major Sixth Minor Sixth 21. Pitch Notation. It is often convenient to be able to define the pitch of a note in the scale so extended, by using a different notation in each octave. We shall use for this purpose the notation introduced by Helmholtz. The letters of the table just given denote the octave from C near the middle of the Bass Clef to the middle c of the pianoforte. The octave below this is denoted by capital letters with the suffix 1, as Ci, Di, etc. The octave next below this has the suffix 2, as Cg, Dj, etc. Going upwards from the middle c of the pianoforte we have first an octave with small letters c, d, e, etc., next an octave with the affix 1 as c\ d\ etc., then c^, d', etc., and so on. Thus the successive C's of the pianoforte scale are O2, Oj, C, c, c\ c^ c^ c\ 12 NATURE OF SOUND [CH. I 22. Quality. We have now only the quality of the note to consider. This is less simple than the loudness and pitch, and the experiments shewing the relation between the nature of the vibrations and the quality of the resulting note are too complex to be given at this stage. We can however make a conjecture as to what feature of wave motion is likely to affect the quality of the corresponding note. Two trains of waves may differ in three, and only three, ways. They may have different amplitudes, different wave- lengths, and different shapes. What is meant by shape is most easily shewn by a diagram. Fig. 4 represents two waves, which we may regard as travelling towards the right. They have the same amplitude and the same wave-length, but different shapes. A is sym- metrical on the two sides of the crest, whilst B is steeper in front of the crest than it is behind. It is clear that we may have an infinite variety of shapes whilst keeping the same ampli- tude and wave-length. Now amplitude and wave- length have already been appropriated as characterizing loud- ness and pitch, and so we have nothing left but shape to characterize quality. The manner in which quality depends on shape will be considered in Chapter IX. Fig. 4 CHAPTER II ELASTICITY AND VIBRATIONS 23. Origin of Sound. We saw in the preceding chapter that sound always takes its rise from some vibrating body. It is possible to make a body vibrate by means of a system of cogwheels and levers, without making use of the elasticity of the parts, but vibrations so produced are of little importance for our purpose. Sound almost always arises from vibrations which are due to the elasticity of the vibrating body. Even when the vibrations are not originally elastic vibrations, they become so as soon as they are communicated to the air. Hence it is necessary to give some consideration to the nature of elasticity, and to the vibrations arising from elasticity. 24. Nature and Limits of Elasticity. A body is said to be elastic, if on being deformed in any way by an amount not too great, it tends to return to its original state. Stretch a spiral spring for instance. It tends to return to its original length, and does so as soon as the stretching force is removed. Bend a thin metal rod, and it tends to straighten itself again. Twist the same rod, and it tends to untwist. Close the outlet of a bicycle pump, and press down the handle. The air in the barrel is compressed, but as soon as the pressure is removed, it returns to its original volume. In most cases of elastic deformation of solids there are limits beyond which the deformation must not go, if the body is to recover its original state. If, for instance, a spiral spring is stretched to a very great extent, it may be that it will acquire a certain amount of permanent stretch, and will not return to its original length. It is then said to have been deformed beyond the limits of elasticity. 14 ELASTICITY AND VIBRATIONS [CH. II The limits differ very much for different substances, and for different kinds of deformation. A spiral spring of large diameter and made of steel of good quality might be stretched to double its length without acquiring a permanent set. A straight wire of the same length and material as the spring could be stretched only a very little without suffering permanent deformation, or breaking ; whilst a similar wire made of lead would have extremely narrow limits of elasticity, either for bending or stretching. Nevertheless, lead is elastic within its narrow limits, for if a long lead pipe or rod is tapped at one end, the sound is carried along the material to the other end, and we shall see later that sound invariably travels through a body by virtue of the elasticity of the body. 25. Imperfectly elastic solids and viscous liquids. Some substances are elastic if the deforming force acts only for a short time, but acquire a permanent set if the force is maintained. Take, for instance, a rod of pitch or sealing wax, and fix it horizontally by clamping one end. If a small weight is hung on the other end the rod will be bent down a little, and if the weight is soon removed, the rod will recover its original position. If however the weight be allowed to remain, the end of the rod will gradually sink down, and will not return when the weight is removed. It is not, in fact, necessary to put any weight on the end, for the weight of the rod itself will cause it to sink gradually. A lump of pitch placed on a flat horizontal surface will in the course of a few weeks spread out into a thin cake. It behaves in this case as a very viscous fluid, whilst, if the deforming force acts for a very short time, as it does in the case of sound vibrations, the substance behaves as an elastic solid. There are many substances that behave like pitch, and it has even been suggested that no substance is really solid, but that every body gives way gradually to forces however small, if long enough maintained. This is mere speculation, for the great majority of the bodies which we regard as solid shew no signs whatever of such gradual deformation, even though the forces have lasted for centuries. There is no indication that the Pyramids are flattening under the action of gravity. 24-27] ELASTICITY AND VIBRATIONS 15 nor that metals shew any such effect, for ancient coins still shew their inscriptions sharply marked. A distinction must be drawn between very viscous liquids such as pitch, and solids with narrow elastic limits such as lead. For forces acting for a short time, pitch has wider elastic limits than lead, yet lead is a real solid. There is no reason to suppose that very small forces give it a permanent deformation, however long they act on it. 26. Elasticity of Liquids and GS-ases. Liquids and gases can be deformed in only one way, namely by alteration of their volume. They offer in general no permanent resistance to changes of shape, though, as we have seen, a viscous liquid may offer resistance to rapid changes of shape. They cannot be said to have any limits of elasticity, for, however great the pressure applied to a liquid or gas, the volume will return to its original value when the pressure is removed, provided the other original conditions, such as the temperature, remain the same, or are regained when the pressure is removed. This statement is not strictly true of liquids, for it has been shewn that in exceptional cases a liquid can be made to break under tension. The exception has no practical bearing on Acoustics. 27. Relation between the Deformation and the Force which causes it. We must next consider the relation between the magnitude of the de- forming force and the amount of deformation. For the experimental determination of this relation a spiral spring is convenient, as its limits of elasticity are wide. Hang up the spring by one end, and at the other end fix a scale pan, and a pointer with a graduated scale behind it. If it is found that any of the coils of the spring are in contact with each other, put such a weight in the pan as is sufficient just to 16 ELASTICITY AND VIBRATIONS [CH. II separate all the coils, and take the reading of the pointer on the scale. Now put such a weight in the pan as will lengthen the spring by an amount that is easily measured. The weight required will depend on the length and stiffness of the spring. Read the new position of the pointer on the scale. Add another similar weight, and read the position again, and so on, repeating the operation several times. Suppose, for instance, that a weight of 10 gm. lengthens the spring by 1 cm., then the second 10 gm. will be found to lengthen it by another centimetre, and so on, or in other words the lengthening is proportional to the added weight. This proportionality of the deformation to the force is strictly true only for small deformations. In the case of a spiral spring it holds only so long as the coils are approximately horizontal. For a spring a foot long and an inch in diameter, for instance, there would be no great deviation from the law up to an extension of 3 or 4 inches, but if the extension were consider- able, say a foot or more, it would be found that the force increased more rapidly than the elongation, even though the elastic limit were not passed. The law holds not only for the extension of a spring, but also for its compression. If a weight of 10 gm. placed in the pan stretches it 1 cm., an upward force equal to the weight of 10 gm. applied to the lower end of the spring will shorten it by 1 cm. 28. Hooke's Law. 'The law of proportionality of the deformation to the force applied holds for all small distortions of elastic solids, and is known as Hookers Law. The law may be stated thus : — Any small distortion of an elastic body in proportional to the distorting force. As further illustra- tions we may take the following cases. Fix a rod horizontally by clamping one end in a vice, and hang weights from the free end. The depression of the end will be found to be proportional to the weight applied. Stretch an elastic string or wire horizontally between two points, and hang weights to its middle point; the deflexion will be proportional to the weight. Fix a rod at one end, and apply a couple to the other end, so as to twist it ; the angle through which the end is twisted will be proportional to the couple. 27-29] ELASTICITY AND VIBRATIONS 17 Hooke's Law as originally stated applied only to the deformations of solids, but there seems little reason why it should not be taken to include the compression of liquids and gases also, for in the case of these too tlje change of pressure is proportional to the change of volume, when this change of volume is very small. It is easily seen that when a gas is greatly compressed the added pressure is not proportional to the diminution of volume, for Boyle's Law states that for a gas at constant temperature the product of pressure and volume is constant. Suppose a column of gas is enclosed by a piston in a cylinder a foot long, and is at the ordinary atmospheric pressure of 15 lbs. per sq. in. Press down the piston 3 in. The gas is now reduced to three quarters of its original volume, and its pressure is consequently 15 x 4/3, or 20 lbs. per sq. in. Press down the piston another 3 in., and the pressure becomes 2 x 15, or 30 lbs. per sq. in. Thus the first 3 in. requires an added pressure 5, whilst the second 3 in. requires 10, or the pressure increases more quickly than the diminution of volume. The law of proportionality of added ' pressure to diminution of volume can in the case of gases be assumed only for very small compressions. We shall see later that the divergence from the law, when the compression is not very small, gives rise to Combination Tones. The law of proportionality may be taken as holding generally for liquids, for in their case the resistance to com- pression is so great that the compression is always small. 29. Forces of Restitution. If an elastic body is deformed, as, for instance, when a rod clamped at one end has the free end drawn aside from its position of rest, the deformation gives rise to internal forces, which tend to bring it back. These are called forces of restitution. It is evident that if the rod is held at rest with any given amount of de- flexion, the force required to cause this deflexion will be balanced by an equal and opposite force due to the elasticity of the rod. Hence the force exerted by the rod is proportional to the displacement, and is in the opposite direction to the displacement. 18 KLASTICITY AND VIBRATIONS [CH. H 30. Potential energy of a Deformed Body. Work has to be done on the rod to displace it, and the rod acquires potential energy equal in amount to this work. ^°|''5 "°?^® by a constant force is measured by the force multiplied by the distance through which it acts. In the case of an elastic body the force increases with the deformation, and hence we must take the average force, and multiply by the total de- formation to get a measure of the work. Where the force increases proportionally to the deformation, the average force is the force for half the final deformation. Hence if a is the displacement, the work is ^ka x a or lka\ where k is the coefficient of elasticity for the particular kind of deformation we are considering, that is, the force which will give the unit displacement. We see then that the potential energy of an elastic body, which has been deformed by forces appropriate to the kind of deformation in question, is proportional to the square of the displacement. 31. Vibrations due to Elasticity. Now release the body, and the forces due to its elasticity at once begin to draw it back to its equilibrium position with, increasing velocity, thus transforming the potential energy into kinetic. When the body reaches its equilibrium position, there is momentarily no deformation and no elastic force, and the potential energy has been entirely converted into kinetic. The momentum the body now possesses causes it to pass through its equilibrium position, and to swing out to a distance a on the other side ; when its kinetic energy will again have been converted into potential. In this position it has no kinetic energy, but is momentarily at rest, and so falls back, and if there were no dissipation of energy, it would continue to vibrate between the limits + a and —a. There will however in general be losses of energy. Part of the energy of vibration will be transferred to the air in the form of waves of compression and rarefaction, as we shall see later; part will be spent in warming the body itself in consequence of internal viscosity, which acts similarly to friction ; and there may be other causes of loss of energy, such as skin-friction, or if the vibrating body happens to be magnetised, there may be 30-33] ELASTICITY AND VIBRATIONS 19 production of electrical currents in neighbouring bodies. The result of these losses is that the vibrations gradually die down in such a way that the amplitude of any one elongation is in a constant ratio to the amplitude of the next. 32. Isochronism. Hooke's Law leads to the im- portant result that the vibrations of a body due to its elasticity are isochronous; that is, the time of performing one complete oscillation is the same whatever the extent of the oscillation. A simple instance of a body that performs isochronous vibrations is the pendulum. The force that acts on the pendulum is gravity, and not an elastic force, but if the arc of vibration is small, the relation between the force and the displacement is the same as for elastic forces, and the vibra- tions are consequently isochronous. A familiar instance, where the vibrations are due to elasticity, is the balance wheel of a watch. Here the vibrations are due to the elasticity of the hair spring, and are maintained by the main spring acting through the train of wheels and the escapement. If for any reason such as increased friction, or diminished force in the main spring through the watch being pearly run down, the arc of vibration of the balance becomes smaller, the time of vibration is not appreciably altered. The balance continues to vibrate at the same rate whatever its amplitude, and the rest of the watch is merely a contrivance for maintaining the vibrations and counting and recording their number. 33. Proof of the Isochronism of Elastic Vibra- tions. It is easy to see in a general way why elastic vibrations are isochronous. A' o o C B A Fig. 6 Suppose we have two particles, one of which can vibrate about an equilibrium position o, and the other about an equilibrium position 0. Suppose also that the two particles 2—2 20 ELASTICITY AND VIBRATIONS [CH. H have the same mass, and that the same elastic forces act on them when they are at the same distances from o and respectively. Draw the first particle aside to a and release it. It will vibrate between the limits a and a', where oa = oa'. Draw the second particle aside to A, where OA = 2oa. It will -vibrate between A and A'. The vibrations of the two particles may then be taken to represent two vibrations of the same particle, one of the vibrations having double the amplitude of the other. Divide oa and OA each into the same number of equal parts. If the number of parts is very large, each of the parts will be very small, and the force of restitution may be taken as constant over any one part, and equal to its average value over that part. Let/ be the average value of the force over the part ah. Then if m is the mass of the particle, and « its acceleration, we know that f= ma or, if the force is proportional to the displacement, the acceleration is so also. Also, since we are assuming f is constant over the part ah, we may assume that a is constant. Now it is known that if a, body starts from rest with constant acceleration a, and moves for a time *, it will pass over a space s given by the equation s = \at^, or if ah = s, the — , and when it reaches b it will have a velocity at or Jllas. Now consider the particle that was drawn out to A before being released. Since OA = 2oa the acceleration will now be 2a and AB will be 2oa or 2s. Hence the particle will reach ^ in a time . / — ^r — or . / — , the same as in the case of the particle drawn out to A. Its velocity on reaching B will be 2a< or 2v2as. Thus it follows that the two particles reach the end of their first stage in the same time, but the particle with the greater amplitude arrives with dbuble the velocity. Next consider the second stage he. Let V be the velocity with which the first particle reaches b and /3 its average 33-34] ELASTICITY AND VIBRATIONS 21 acceleration over he. Then the second particle reaches B with a velocity 2 V, and during the stage BG it has an acceleration ip, since the centre of GB is twice as far from as is the centre of cb from o. We know that if a body has an initial velocity F, and moves over a space s with uniform acceleration /8 in time t, then s = F< + |/8<^. From this equation we can find the time the first particle takes to pass over the stage be. If we form a similar equation for the second particle, we have to replace s by 2s, F by 2 F and /3 by 2/3, which makes no change in the equation, as we have merely multiplied both sides by 2. Thus each of the particles will take the same time in traversing the second stage, and the process can evidently be continued until they reach o and respectively. OA has been taken as being twice oa merely for the sake of simplifying the equations. The result would be the same whatever multiple OA is of oa ; the particles would reach their equilibrium positions in the same time whatever their ampli- tudes. It is obvious from symmetry that the time the particle takes in moving from 4 to is one quarter of the time it takes in going from A to A' and back again to A, and there- fore the time of a complete vibration is the same whatever the amplitude, or the vibrations are isochronous. 34. Simple Harmonic Vibrations. Such vibra- tions are called Simple Harmonic Vibrations, and are of great importance in the theory of Sound, as a vibration of this kind, and of this kind only, gives the sensation of a pure tone of definite pitch with no admixture of tones of other pitches. Any other vibration than a Simple Harmonic Vibration gives rise to a note, which can by suitable appliances be resolved into two or more tones of different pitch. The note arising from a Simple Harmonic Vibration cannot be so resolved. We shall return to this point later. « ■?. ^ 17?. If elastic vibrations had not been isochronous, music in its present form would have been impossible. Suppose, for instance, that the law connecting amplitude and number of vibrations per second had been that the one was proportional to the other. We have seen that pitch is determined by the number of vibrations per second, and loudness by the 22 ELASTICITY AND VIBRATIONS [CH. II amplitude. We should therefore have the j-esult that the louder a note, the higher its pitch. It would be impossible to keep the instruments in the orchestra in tune with each ; other, and a crescendo would mean a rise in pitch of the whole orchestra. 35. Geometrical Illustration of Simple Har^ monic Vibration. The nature of a Simple Harmonic Vibra- tion can be shewn by the following useful geometrical method. Suppose a point P moves with uniform speed round the circumference of a circle of radius a. Drop a perpendicular B ' '■ from P on any diameter ^-^-^ I ""^^.P AA' ; then we can shew that N, the foot of the perpen- dicular, describes simple har- monic vibrations in the line AA'. The radius OP revolves with uniform angular velo- city, which we will denote by 0). The acceleration of P is therefore aw^ in the direction PO, and the ac- celeration of N is the com- ponent of this in the direction AO, or aco^cos^, where 6 is the angle POA. Since OJV=acos^, we can write the ac- celeration of N in the form m^ON, or the acceleration of N is proportional to its distance from 0. If iV is a particle of mass m vibrating in the line A A' in consequence of a force directed towards 0, the force required to give this accelera- tion must also be proportional to ON, since F = ma. This relation of force to displacement is in accordance with Hooke's Law, and N describes simple harmonic vibrations. The period of vibration of N is the time taken by the radius OP to make one complete turn, or 27r/a). If F is the elastic force when the displacement is unity, the force for displacement ON is F x ON, and the acceleration is - V ON. The geometrical method gives o)= x ON as the 34-36] ELASTICITY AND VIBRATIONS 23 acceleration. We can therefore use Fig. 7 to represent the vibrations of a particle of mass m, vibrating with amplitude a under the action of an elastic force of magnitude F for unit displacement, if we describe a circle of radius a, and make the radius OF revolve with such an angular velocity that H^ 9 u? — - — ; and further, since r = — , we see that the period of m (0 vibration of the particle will be 27r . /et- This expression is independent of the radius. It is the same whatever the radius of the circle, provided co has the IF . value / — . Hence this method of treating simple harmonic vibrations leads also to the conclusion that their period is independent of their amplitude, or they are isochronous. 36. Method of Calculating the Period of Vibra- tion. If we know F and m in any particular case, we can calculate the period from the expression t = 2?? / -ef • We must be careful to use a consistent system of units in the calculation. If, for instance, we use the c.G.s. system, m will be the mass of the vibrating body in grammes and F will be the force in dynes required to give a displacement of one centimetre. We have taken the simplest case, where the particle vibrates in a straight line, and nothing that is moving has any inertia except the particle itself. We cannot secure this exactly in practice, but a mass suspended by a light spiral spring approximates to it. Suppose the mass of the pan and the body placed in it is M. gm., and suppose an additional mass m gm. causes it to sink n cm., then »n/ri gm. would depress it 1 cm. The force with which m\n gm. is attracted to the earth is tngln dynes, where g\& 981 cm. per second per second. Consequently the period of vibration of the mass is V mg If the period calculated in this way is compared with the period observed directly, it will be found to be a little too 24 ELASTICITY AND VIBRATIONS [CH. IX small, as we have uiiderestimated M. The spring is moving, and adds something to the inertia. The lowest part of the spring moves as much as the suspended body and the highest part does not move at all. It is plain therefore that we ought to add something less than the mass of the whole spring to the mass of the body. It can be shewn that one third the mass of the spring should be added. The force required to give unit deformation of any kind to any elastic body is called the coefficient of elasticity for the particular body and the particular kind of displacement. The expression we have found for t will give the period of elastic vibrations of any kind, if F and M are suitably expressed. F will not always be a simple force, and Jf will not always be a mass. Suppose, for instance, a body is hung by a wire, and is turned round so as to twist the wire a little. When it is released, it will perform rotational vibrations ; the wire twisting first in one direction and then in the other, whilst its axis remains at rest. In this case the coeflScient of elasticity ^with which we are concerned is the couple that will twist the end of the wire through the unit angle ; and the inertia term is the moment of inertia of the suspended body about its axis of rotation. 37. Method of tuning an Elastic Body. In all cases the greater F is, or the ' stifFer " the body is to displace, the less t will be ; and the greater the inertia, the greater T will be. It is useful to remember this when we have to tune a vibrating body. A tuning-fork gives a good instance. In this case the bending which gives rise to the elastic forces is chiefly at the base of the prongs, and the motion is chiefly at the free ends. If then we scrape or file the prongs near the base, we shall diminish F without making much change in M, and so shall lower the pitch. If on the other hand we file the prongs near the free ends, we shall diminish the inertia without altering, the elasticity much, and shall raise the pitch. This is the usual method of tuning a fork. If we wish to lower the pitch of a fork temporarily, we can do it by sticking a little wax on the ends of the prongs, and so increasing the inertia, 36-39] ELASTICITY AND VIBRATIONS 25 38. Period. Amplitude. Phase. The three main characteristics of a simple harmonic vibration are its period, its amplitude and its phase. We have already defined the period as the time occupied by one complete to and fro vibration. The amplitude is one half the extreme range of the vibra- tion, or the distance between the equilibrium position and either of the points at which the vibrating particle is momentarily at rest. According to this definition the amplitude of # in Fig. 7 is OA or a. The phase of the vibration at any moment is the state of the vibrating particle as regards its position and its direction of motion at that moment. Whenever, for instance, the foot of the perpendicular in Fig. 7 is passing through a particular point, and is moving say from right to left, it is in the same phase. The radius OP rotates uniformly in the same direction, and the position and direction of motion of N is known, if the position of OP is known, and hence the phase can be measured by the angle 6. The term phase is most commonly used in speaking of the difference of phase between two points vibrating with the same period. If they are imagined both to be vibrating in the line A A', though not necessarily with the same amplitude, each will have its rotating radius ; and since the particles have the same period, the rotating radii must complete one circuit in the same time making a constant angle with each other, and this angle measures the difference of phase. The fraction this angle is of the whole circuit is the fraction of a period that one particle is behind the other. If for instance the angle is 90°, we may say that one particle is a quarter of a period behind the other, and this is the most usual way of expressing difference of phase. If expressed as an angle the difference of phase would be said to be 7r/2. If the particles pass through their equilibrium position at the same moment but in opposite directions, they differ in phase by half a period, or, as it is often expressed, they are in opposite phase. 39. The Sine Curve. The position of the particle N at any time can be shewn by a curve as follows. Divide the 26 ELASTICITY AND VIBRATIONS [CH. II circumference of the circle into any number of equal parts beginning at £, and going round the circle in the direction BAB' A'. Take a straight line of any length and divide it into the same number of equal parts. As P moves with uniform speed, each of the sections into which the circumference is divided will be traversed in the same time, and the points marking the ends of the divisions may be regarded as marking a time scale. If a perpendicular be drawn to AA' from the end of each division, and the distance from of the foot of the perpendicular be measured, we shall have the displacement of N at the ends of a series of equal intervals of time. Now transfer the displacement corresponding to each dividing point on the circle to the corresponding dividing point on the straight line. If iV" is to the right of 0, draw an ordinate equal in length to this displacement upwards from the corresponding dividing point on the straight line. If iV is to the left of 0, draw the ordinate downwards. Draw a smooth curve through the ends of the ordinates and we shall get a curve such as that shewn in Fig. 8. Fig. 8 The curve is drawn only for one complete vibration, but it could evidently be continued indefinitely to the right to represent any number of vibrations. The curve enables us to find the position of the point at any time, for distances measured along the horizontal line from A are proportional to the time elapsed from the moment when the particle was passing through its equilibrium position to the right, and the ordinate at the point corresponding to any given moment shews the displacement at that moment — to the right if the ordinate is above the axis, to the left if it is below. The curve is known as the Sine Curve, for if the maximum ordinate is of unit length, and the length ^^ is taken to 39-41] ELASTICITY AND VIBRATIONS 27 represent 360°, the ordinate at any point in the line will give the sine of the angle corresponding to that point — the sine being positive when the ordinate is above the axis, and negative when it is below. 40. Relation of Velocity to Displacement. The velocity of the vibrating particle when passing through any point of its path can also be shewn by a sine curve. Referring to Fig. 7, it will be seen that, at the moment for which the figure is drawn, the velocity of N is the com- ponent of the velocity of P in the direction JVO, or v sin 6. Hence, since v is constant, the velocity of JV is propoftional to sin and so can be shewn by a sine curve. The curve will be displaced a quarter of a period to the left as compared with the displacement curve, for the velocity is a maximum when the displacement is zero, and vice versa. The two curves are shewn in Fig. 9, where points in the same vertical line correspond to the same moment. In the lower curve ordinates above the axis denote velocities to the right, and ordinates below the axis denote velocities to the left. Fig. 9 41. Composition of Simple Harmonic Motion with uniform motion in a straight line. We shall next discuss the curve traced out by a particle which executes harmonic vibrations, and has at the same time some other motion impressed on it. The simplest instance is the case where the particle executes its harmonic vibrations in one line, and is at the same time carried with uniform velocity in a direction at right angles to that of the vibrations. We can realize this case and trace the required curve in the following way. Fix a bristle or wire to the prong of a tuning-fork by a small piece of wax, and, whilst the fork is vibrating, hold the end of the bristle against a sheet of paper, which has been smoked with burning turpentine or camphor. The bristle 28 ELASTICITY AND VIBEATIONS [CH. II will trace on the paper a straight line, whose length shews the range of vibration of the fork. Now draw the fork uniformly in the direction of its own length, and instead ot a straight line we shall get a sinuous curve on the paper. Fig. 10 From what has been said before it is clear that, if the fork is moved with uniform velocity, the curve will be a sine curve. The straight line that would have been traced by the fork, if the prongs had not been vibrating, is the axis of the curve. Equal distances along the axis correspond to equal intervals of time, and the ordinate drawn from any point of the axis to the curve gives the displacement of the prong at the moment determined by the position of the foot of the ordinate on the axis. This method of converting a simple harmonic vibration into a sine curve forms the basis of one of the most accurate methods of finding the frequency of a tuning-fork, as will be seen later. 42. Composition of two Simple Harmonic Vibra- tions at right angles. Let us next find what curve we shall get by compounding two harmonic vibrations in lines at right angles to each other. We can construct a curve that is approximately compounded of two simple harmonic vibrations by means of the apparatus shewn in Pig. 11 and known as the Harmonograph. A and B are two pendulums suspended so as to swing in planes at right angles to each other. Each pendulum is continued for a few inches above its point of support, and at the top of each a light horizontal lever is attached by a flexible joint. These levers C and D are joined together at E 41-42] ELASTICITY AND VIBRATIONS 29 by a flexible joint, and a pen is attached to the joint. If the •pendulum A is alone set swinging in the plane of the paper, ^ Fig. 11 the joint E will execute approximately harmonic vibrations from right to left. They will not be strictly harmonic for the top of the pendulum B is at rest and the joint E must therefore describe small arcs of a circle round it, but if the levers C and D are 10 or 12 inches long and the arc of vibration of the pendulum is small, the motion of E is nearly enough harmonic for our purpose. If B is set swinging in a plane at right angles to the plane of the paper and A is at rest, the pen will vibrate harmonically in a direction at right angles to its former direction. If now both pendulums are set swinging, the motion of the pen will be compounded of the two motions, and a curve will be traced out on the table. If we give an extended meaning to the word " sum,'' we may say that the displace- ment of the pen at any moment is the sum of the displacements due to the two separate vibrations at that moment. There is only one sense in which we can "add" two displacements not in the same straight line, and that is in 30 ELASTICITY AND VIBRATIONS [CH. II Fig. 12 accordance with the Parallelogram Law used in compounding two forces. If a particle, when vibrating in the line OP, would have at some moment a displacement OP, and a vibration in the direc- tion OQ would give it at the same moment a displacement OQ, then its actual displacement at that moment must be OB, if each of the components has its full effect. We may imagine the particle to vibrate in the line OP, and at the same time the line OP to vibrate in the direction OQ always remaining parallel to itself. The Parallelogram Law, with its extension the Polygon Law, holds not only for the displacements but also for the velocities of vibrating particles. 43. Composition of two Simple Harmonic Vibra- tions of equal periods. Let us return now to the Harmono- graph. The pendulums are generally made with bobs that can be clamped at any point of the rods, so that the times of vibration can be varied. Adjust the bobs to such positions that the time of swing is the same for both pendulums. Now it is clear that E will descrilae some kipd of oval or circular curve. We can shew that a circle is a possible curve. If P travels round the circle in Fig. 13 with uni- form speed, and PN, PIT are the perpendiculars from f on two diameters at right angles, then K and N' describe Simple Harmonic Vibrations in their respective diameters. Their displace- ments at the moment for 'S- 13 which the figure is drawn are ON and ON' and OP is the a' p \ * o r J 42-43] ELASTICITY AND VIBRATIONS 31 displacement got by compounding ON and ON' by the Paral- lelogram Law. This evidently holds for every position of P, and therefore the motion of P may be regarded as obtained by compounding the vibrations of N and N'. These vibrations have the same amplitude and period, but di£Fer in phase by a quarter of a period. Whenever we have these relations between two perpendicular simple harmonic vibrations their resultant is uniform motion in a circle. If the amplitudes are not the same, it can be shewn that the curve is in general an ellipse, the direction of whose axes depends on the relation between the phases of the constituents. In the particular cases in which the phases are either the same, or differ by half a period, the ellipse degenerates into a straight line. If AA' and BB' represent the directions and amplitudes of vibration of the two constituents, the curves in Pig. 14 shew five forms of path of the pencil. In No. 1 the two constituents may be said to be in the same phase. When the pencil is passing through towards the right under the influence of one Fig. 14 vibration, it is passing upwards through under the influence of the other. In No. 2 the difference in phase is between zero and a quarter period. No. 3 shews the case where the differ- ence of phase is a quarter period. The ellipse is then symmetrical about the lines of vibration of both constituents. In No. 4 the difference of phase is between a quarter and a 32, ELASTICITY AND VIBRATIONS [CH. II half period, and in No. 5 the two vibrations differ in phase by half a period. 44. Composition of two Simple Harmonic Vibra- tions of nearly equal periods. If the periods are very nearly, but not quite equal, the curve described during a single complete period of the vibration will be approximately an ellipse ; but one of the vibrations will slowly gain on the other, and the difference in phase will change slowly. The curve will then pass through the series of forms in Fig. 14. Beginning say with No. 1, the line will slowly open out into an ellipse, the curve will pass through the forms 2, 3 and 4 to the straight line 5, and then it will pass through the series in the opposite direction until it reaches No. 1 again. If the amplitudes remain constant the curve will always touch the four sides of a rectangle with sides equal and parallel to A A' and BB'. When the curve has passed from 1 through 2, 3 and 4 to 5, and back again to 1, the phases have returned to their original agreement. Consequently one of the pendulums must have gained exactly one vibration on the other. This gives us a means of comparing their times of swing, for if we find, for instance, that the curve changes from 1 to 5 and back again to 1 in one minute, we know that one pendulum makes one complete vibration per minute more than the other, or the frequency of one is one-sixtieth of a vibration greater than that of the other. This method has been much used in comparing the frequencies of tuning-forks, as will be seen later. 45. Iiissajous' Figures. The curves of Fig. 14, and' those obtained when the frequencies of the constituents have other aliquot ratios than 1 ; 1, are known as Lissajous' Figures. We will take as our second illustration the case where the frequencies are in the ratio 2:1. We then get in general a figure of 8, degenerating into a parabola for certain relations of phase of the constituents. The relationship between the phases cannot in this case be expressed shortly, as is the case when the ratio of the frequencies is 1:1, for even though the ratio is exactly 2 : 1 the difference of phase is not constant. A vibration in one direction is completed in half the time of a 43-45] ELASTICITY AND VIBRATIONS 33 vibration in the direction at right angles to it, and therefore, even though the two vibrations start in the same phase, they immediately come into phases differing from each other. The term difference of phase can in fact hardly be said to be applicable to two vibrations, unless they have the same or very nearly the same period. Fig. 15 Fig. 15 shews five forms of Lissajous' Figures for the ratio of frequencies 2:1. The parabolic form of the first and fifth arises when the phases are such that the vibrating point comes to the end of its swing in each of the two perpendicular directions at the same moment. If the ratio is not exactlj' 2 : 1 the curve will change gradually back and forwards through the series of figures of Fig. 15. Fig. 16 Fig. 16 shews the symmetrical and the degraded form for the ratio 3 : 2. The vibration ratio of the constituents of any of these figures can be found by inspection. In the first curve of C. s, 3 34 ELASTICITY AND VIBRATIONS [CH. II Fig. 16, for instance, the curve touches a horizontal side of the rectangle three times and a vertical side twice. It follows that the tracing pen makes three vibrations in a vertical direction in the time it takes to make two in a horizontal direction, or the ratio is 3:2. When the method is applied to one of the degraded forms, it is to be remembered that the complete path consists of the curve described twice, first in one direction and then in the other. Where the degraded curve comes to a corner of the rect- angle, it must be regarded as touching each of the adjacent sides once, and when it has a side of the rectangle as a tangent, it must be regarded as touching that side twice. 46. Optical method of compounding Simple Harmonic Vibrations. Lissajous' Figures can be produced optically by means of two tuning-forks. Each fork has a small mirror on the side of one prong. One fork A has its prongs vertical, and the other B has its prongs horizontal, as shewn in Fig. 17. A beam of light from a small source S Pig. 17 strikes the mirror on the prong of the vertical fork, is reflected to the prong of the horizontal fork, and thence to a screen C where it is brought to a focus by the lens L. If the fork A alone is vibrating, the spot on the screen describes harmonic vibrations so rapidly that only a bright vertical line of light is seen. If the horizontal fork B alone vibrates, the spot describes a horizontal line. If both forks vibrate, the two vibrations of the spot are compounded, and if the periods of the forks bear some simple ratio to each other, one of Lissajous' figures is seen on the screen. CHAPTER III TRANSVERSE WAVES 47. Introductory. In Chapter II we discussed the motion of a single particle executing Simple Harmonic Vibra- tions. In the present Chapter we shall consider the way in which wave motion arises from the simultaneous vibration of a series of particles. We shall take as simple a case as possible, and the possibility of the motion to be described must be assumed. It can be shewn theoretically to be a possible form of motion of, for instance, consecutive short sections of a stretched string, but the method of proof is beyond our scope. At a later stage in the chapter experiments will be described which are consistent with our conclusions, but cannot be taken as a complete proof of the correctness of the assumptions. This chapter must be regarded rather as a description of the properties of transverse waves than as a logical deduction of those properties from the laws of elasticity. 48. Waves arising fi-om the Harmonic Vibration of a series of particles. Let us suppose we have a series of particles placed at equal distances from each other along a straight line, and each capable of vibrating harmonically in a direction at right angles to that line, the lines of vibration all lying in the same plane. 3—2 36 TRANSVERSE WAVES [CH. Ill Imagine the particles to be vibrating with equal periods and amplitudes, and in such phases that each is some constant fraction of its period behind the particle on its left. There is then a constant difference of phase between any two con- secutive particles, the retardation of phase increasing as we go from left to right along the series. Suppose, for instance, that each is one eighth of a period behind its left-hand neighbour, and that at the moment we are considering particle 1 is pass- ing downwards through its equilibrium position. Then since 2 is one eighth of a period behind 1, it will not have reached its equilibrium position, but will be moving downwards towards it. Its position can be found graphically from a diagram similar to Fig. 7. Describe a circle with radius* equal to the amplitude of vibration of each of the particles, and draw a diameter P^Pj. P./-^^^ N ^)P- \/ \y Pe'C ?pz Fig. 19 The motion of any particle of the series is given by the motion of the foot of the perpendicular drawn to the diameter P^P^ from the end of a radius which revolves with uniform velocity. The particle 1 is passing downwards through its equi- librium position and is therefore represented by the foot of the * (Fig. 19 is drawn on a larger scale than Fig. 18 for the sake of 48-49] TRANSVERSE WAVES 37 perpendicular from P^, the radius OP^ revolving in the direction of the hands of a clock. Particle 2 is an eighth of a period behind 1. Its radius is therefore OP^, which is an eighth of a complete revolution, or 45°, behind OPj, and ON is the distance of 2 above its equilibrium position. Particle 3 is 4-5° behind 2. It is therefore at a distance OP^ from its equilibrium position, and is momentarily at rest. Particle 4 is at a distance ON above the axis and is moving upwards. Particle 5 is moving upwards through its equilibrium position. Particle 6 is at a distance ON' below, and is moving upwards, and so on. When the radius reaches /\ again, we begin a second round of the circle; P^ is at P^, Py, at -Pg, etc. All the particles between 3 and 7 are moving upwards, and all those between 7 and 11 are moving downwards. 3, 7 and 1 1 are at the ends of their swings and momentarily at rest. Thus it appears that at the moment we are considering the particles will lie along a curve such as is shewn in Fig. 18, and it is evident that this is the sine curve, for the method by which we have drawn it is the same as that by which we drew the sine curve. 49. Velocity of the Waves. Next let us consider where the particles will be at a moment one eighth of a period later. 1 will have moved downwards below its equilibrium position. 2 will have reached its equilibrium position and be moving downwards through it. 3 will have moved downwards from the end of its swing and will be as much above the axis as 2 was one eighth of a period earlier ; 4 will have reached the end of its swing, and so on. Each particle will have come into a position corresponding to that occupied by its left-hand neighbour one eighth of a period earlier. Particles 4 and 12 will now be on the crests of the waves instead of 3 and 11, and 8 will be at the bottom of the trough instead of 7. In fact the whole curve has moved to the right by an amount equal to the distance between two particles. Thus we see that, when a series of particles execute similar simple harmonic vibrations with constant difference of phase as we pass along the series, the result is a wave in the form 88 TRANSVERSE WAVES [CH. Ill of a sine curve travelling in the direction in which we find a gradual retardation of phase as we pass along the particles. If in passing along the particles in the order 1, 2, 3, 4, etc. we had found a gradually advancing pliase, the result would have been -similar. The particles would lie as before along a sine curve, but the waves would travel towards the left. Let us next find with what velocity the wave travels. We saw that in one eighth of a period the crest moves from 3 to 4. In what time will it travel over one whole wave-length ? We take as a wave-length the distance between any particle and the nearest particle that is in a corresponding position. 1 and 9 for instance are a wave-length apart, as are also 3 and 11. We must not take 1 and 5, for, though they are both passing through their equilibrium position, 1 is moving downwards and 5 upwards. Let us choose in particular 3 and 11. It is evident from what we said above that the crest which is now occupied by 3 will be occupied by 11 after one whole period, and the wave will travel one wave-length in the time taken by any one particle to execute one complete vibration. 50. RelationbetweenWave-LengthandVelocity. Now the velocity of the wave is the distance it travels in one second. Let t be the period of vibration and X the wave- length, then the number of vibrations a particle makes in one second is 1/t, which we denote by n, and the distance the wave travels in one second is n\, whence we have the formula v = nX. The quantity n is what we have called the frequency, and denotes either the number of vibrations executed by any one particle in a second, or the number of waves passing a given point in a second. The formula v = n\is frequently used in acoustical calcula- tions, as it gives the relation between three important characteristics of a wave train. It is sometimes more con- venient to use the equation in the form \ = vt. 49-52] TRANSVERSE WAVES 39 51. Non-Harmonic Waves. In finding the wave form of Fig. 1 8 we have supposed that each of the particles is vibrating harmonically. We have done this because as has already been said, the simple harmonic vibration must be regarded as the fundamental form in the theory of Sound. It is not however necessary to limit ourselves to Simple Harmonic Vibrations. We shall get a progressive wave — though not in the form of a sine curve — if the particles all vibrate in the same way, whatever that way may be. The vibrations must be periodic, that is, each particle must continue repeating the same succession of movements, and there must be a regular change of phase as we pass along the series. As we cannot represent a non-harmonic vibration as the projection of uniform circular motion, we cannot express the difference of phase in this case by an angle, but we may express the relations of phase by saying that each particle must pass through its equilibrium position some fixed fraction of a period later than its left-hand neighbour. For the present we shall confine ourselves to Simple Harmonic Vibrations and leave the more complicated forms until we have discussed Fourier's Theorem. Waves such as have been described are called Transverse Waves, because each particle vibrates in a line transverse to the direction in which the wave travels. In Chapter IV we shall describe a form of waves called Longitudinal Waves, where the line in which the particle vibrates is coincident with the direction in which the wave travels. 52. The Medium does not travel with the Waves. It should be noted that in these, as in all other waves, the vibrating particles do not travel with the waves. Every particle of air, or water, or string, or whatever it may be that transmits the waves, travels repeatedly over the same limited path, and what passes on is a shape or arrangement of the particles. When waves are passing over the surface of water, floating bodies do not travel with them. A cork merely describes a small closed curve as the waves pass it, from which we see that -it is not the water itself which travels onwards, but the shape of the surface. 40 TRANSVERSE WAVES [CH. Ill *53. The Equation for Simple Harmonic Wave Motion. The equation of the Simple Harmonic Wave of Fig. 18 can be obtained in the following way : We shall represent the motion of any particle of the series as before by the motion of the foot of the perpendicular drawn from the end of a uniformly- rotating radius on a fixed diameter of a circle. In order to bring the equation into its most usual form we shall measure the angle 6 from the diameter AA', but drop the perpendiculars on the diameter BB'. We have then the displacement OiV^=asin^, and as 6 increases uniformly with the time, we may write 6 = u>t, whence y = a sin mt, where y is the displacement of a particular particle at any time t. Every other particle executes exactly similar vibrations, but the phase is uniformly retarded as we pass to the right along the row of particles. That is to say, if X is the distance measured towards the right from the particle whose motion is given hy y = a sin wt to some other particle, the angle 6 for this other particle will be smaller by an amount proportional to x, say kx, where A is a constant, and therefore this second particle's motion will be given by ^ = a sin {lot — kx). This equation applies to any particle of the series, including that for which a; = 0, and is therefore the general equation giving the displacement of any particle at any time, or the equation of the wave. 53-54] TRANSVERSE WAVES 41 The period of vibration t of any particle is the time taken by the radius OP in making one revolution, and is therefore 27r/a). It follows that but there are other methods such as that of Clement and Desormes, which are independent of the velocity of sound, and it is found from these that y has the value 1-41 for air. When Newton's formula is corrected by replacing p by yp, it gives a value for the velocity of sound which is in agreement with the results of experiment. It might be thought that though there may not be enough transference of heat from the hot to the cold places to keep the temperature constant, there might be some transference, and that we ought therefore to use a value for the coefficient of elasticity intermediate between p and yp. It has been shewn however by Stokes that, if there were any appreciable trans- ference of heat short of what is required to make the changes isothermal, the sound would be rapidly stifled. Sound could not travel such distances as are actually observed, unless the compressions are either isothermal or adiabatic, and Laplace's calculation enables us to say that they must be adiabatic. We shall now use the formula r= a /— to find how the V P velocity of sound in a gas depends on the nature of the sound waves, and on the pressure, density and temperature of the gas. 105-109] VELOCITY OF LONGITUDINAL WAVES 85 106. Velocity almost independent of Amplitude. We see in the first place that neither the wave-length nor the amplitude of the waves appears in the expression, and therefore to a first approximation waves of all lengths and amplitudes travel with the same velocity. This is true only when the amplitude is small, as is generally the case. It will be remembered that in proving the coefficient of elasticity to be -fp we neglected certain small terms. If these terms were retained the coefficient would be a little greater. We have assumed for gases a law similar to Hooke's Law for solids, and it is not strictly correct to make this assumption except when the amplitude is infinitely small. For all ordinary sounds it is sufficiently accurate to take the velocity as independent of the amplitude, but very loud sounds, such as the sound of a cannon, do in fact travel appreciably faster than ordinary sounds. 107. Velocity independent of Pitch. Since the velocity of sound is the same for all wave-lengths and ampli- tudes provided the amplitude is not very great, we see from the equation v = n\ that the wave-length is inversely proportional to the frequency ; that is to say, the higher the pitch of a note the less is its wave-length. The lowest audible note has a frequency 30. If we assume the velocity of sound is 1090 ft. per sec. we find from the equation above that the wave-length of the note is 36-3 ft. The highest audible note has a frequency about 20,000, whence its wave-length is f inch. A man's ordinary speaking' voice has a wave-length of about 8 feet, and a woman's about 4 feet. 108. Velocity independent of Pressure. It is clear that a change in the pressure of the air has no effect on the velocity of sound, for the density changes in the same proportion as the pressure and ^/p is unchanged. Hence a rise or fall of the barometer makes no change in the velocity, and sound travels with the same velocity at high altitudes as at the sea level. 109. Velocity in different gases. If we compare the velocity in any one gas with that in another of different density, we see that the velocities are inversely proportional to the square roots of the densities, provided y has the same 86 VELOCITY OF LONGITUDINAL WAVES [CH. V value for the two gases. Oxygen and hydrogen, for instance, have the same y, but oxygen is 16 times as dense as hydrogen, and therefore the velocity of sound in hydrogen is 4 times as great as in oxygen. We shall see later, when we treat of organ pipes, that a whistle blown with hydrogen would give a note two octaves higher than the same whistle blown with oxygen at the same temperature, for the pitch of the whistle depends on the number of times the sound can travel the length of the pipe in a second. The change of pitch due to change of density is easily shewn by blowing a whistle first with air and then with coal gas. Blaikley has pointed out (Cantor Lecture, 1904) that, if the earth's atmosphere had consisted of hydrogen instead of air, a Piccolo would have needed to be a yard long, and a contra-bass Saxhorn as long as a cricket pitch, to give notes of the same pitch as at present. 110. Effect of Temperature on Velocity. In order to find the effect of temperature we may write the expression for the velocity in the form \lypv. "We know from Charles' Law that pv = kt, where A is a constant for any one gas, and^ is the absolute temperature, and therefore we have V=s/ykt or the velocity of sound in a gas is proportional to the square root of the absolute temperature of the gas. If C is the temperature on the Centigrade scale V=^/y^O + WS) and if F is the temperature on the Fahrenheit scale r=\/yA;(i?'+459). In order to get a numerical estimate of the effect of temperature let us suppose the temperature rises from 60° F. to 61° F., then we have Velocity at 60° _ v/5l9 1039 Velocity at 61° ~ ^520 ~ 1040 "^^^^^y- As the velocity of sound at 60° is not far from 1039 feet per second, it follows that the velocity of sound in air increases by about one foot per second for each Fahrenheit degree rise of temperature. 109-111] VELOCITY OF LONGITUDINAL WAVES 87 m. Velocity in Mixtures of Gases. If we have a mixture of two or more gases, we must use the actual density of the mixture in calculating the velocity of sound. If the y of the constituents is not the same, the appropriate value to he used must be calculated from the values for the separate constituents*. The mixture with which we are most usually concerned is moist air, which is merely a mixture of water vapour and air. Water vapour is lighter than air in the ratio of 9 to 14"4; hence the more moisture there is in the air the greater is the velocity of sound. As an illustration of the use of the formula let us find the velocity of sound in dry air at 20° C, assuming the velocity in dry air at 0° C. to be 332 metres per sec. We have velocity at 20° \/293 velocity at 0° ^273 /293 .-. velocity at 20° = ^ ^yg 332 = 344. Next suppose the air is saturated with moisture at 20°, and that the height of the barometer is 760 mm. The maximum vapour pressure of water vapour at 20° C. is 1 7-4 mm., hence the partial pressure of the water vapour is 17 '4 and that of the air 742-6 mm. If the density of hydrogen be taken as 1, that of air is 14 '4, and that of water vapour 9 under the same conditions of temperature and pressure. The density of the moist air will then be in the same units 17-4 X 9 + 742-6 X 14-4 , , „_ 760 °'^^'^^- * H P is the pressure and V the ratio of the specific heats of the mixture it can be shewn that ^=^+^ +etc., r-1 7i-i 72-1 ■where pi ;p2 < etc. are the partial pressures, and 7i , 72 > etc. the ratios of the specific heats of the constituents of the mixture. 88 VELOCITY OF LONGITUDINAL WAVES [CH. V The value of y for water vapour is about 1 -3, whilst that for air is 1-41, but as the amount of water vapour in the air is so small, the effect of this difierence is inappreciable, and we may assume that y for the wet air is the same as for dry air. We may therefore write velocity of sound in wet air at 20° _ ^14'4 velocity of sound in dry air at 20° ;^14'28 Or the velocity in wet air at 20° / 14-4 = A / -r-r-^K 344 = 345 -3 metres per sec. V 14-28 ^ Thus the effect of saturating the air with moisture at 20° C. is to make sound travel faster by 1 '3 metres per sec. 112. Velocity of Sound in Liquids. Longitudinal -waves can travel through liquids in the same way as they travel through air, and the expression for their velocity is the same as that for air. Liquids require very great forces to compress them, or in other words their coefficient of volume elasticity is very great. This large coefficient of elasticity is more than enough to compensate their greater density, and it is found that the velocity of sound in liquids is in general greater than in gases. The velocity of sound in water, for instance, is about 1450 metres per sec, which is nearly 5 times as great as the velocity 113. Velocity of Sound in Solids. We can have longitudinal waves in solids, but the expression for their velocity is not so simple as in the case of gases and liquids. A solid has two fundamental coefficients of elasticity, a volume coefficient and a rigidity coefficient, and both are involved in the expression for the velocity. When a longitudinal wave travels along a thin rod of metal, there is lateral expansion of the rod at regions of compression, and shrinkage at regions of rarefaction. In this case the coefficient of elasticity concerned is Young's Modulus, the ratio of the force used to stretch a rod of unit section to 111-114] VELOCITY OF LONGITUDINAL WAVES 89 the resulting extension of unit length of the rod. If longi- tudinal waves travel through a mass of the same metal the lateral expansion and shrinkage cannot take place, and the coefficient of elasticitj' is greater than in the case of a rod. It follows that the waves travel more slowly through a rod than through an extended mass of the same material. The methods used for measuring the velocity of sound experimentally will be given in Chapter XIT. 114. Velocity of Sound in Various Media. The annexed table gives the velocity of sound in several media. The last five must not be taken as more than rough approxi- mations. Different samples of the materials vary -greatly in their properties, and different observers have found very different values for the velocity of sound. Air at 0° C 332 metres per second Hydrogen 1270 Carbon Dioxide 257 Water 1450 Iron 5000 Oak 3400 Pine 5200 Glass 5500 Vulcanized indiarubber 43 CHAPTER VI REFLECTION AND REFRACTION. DOPPLER'S PRINCIPLE 115. Reflection of Spherical Waves. We have already dealt in Chapter IV with the reflection of sound at the closed or open end of a pipe. We must turn now to the more general case of reflection of waves in the open air. Let A (Pig. 44) be a source of sound from which spherical waves spread out in every direction, and let the circles drawn with A as centre be the sections of those spherical wave-fronts which shew the positions of the surfaces of maximum com- pression at the moment for which the figure is drawn. The radius of each of these circles increases by about 1100 feet per second, if the air has its ordinary temperature, and the distance between any two consecutive circles measured along a radius is the wave-length of the sound. Let MN be a rigid wall and the point where the perpendicular from A meets the wall. At the moment for which the figure is drawn the wave-front 6 touches the wall at 0. A little later the part of the wave-front that is not hindered by the wall will have come into the position 7, but the part between G and D will have been reflected towards the left. The part near reached the wall first and was reflected first, the parts farther from were reflected a little later, whilst the parts at G and D have just come into a position to be reflected. Consequently the wave-front between G and D will have taken some such shape as the curve GEU ; that is to say, the wave-front which had the form of the sphere 606' has now been bulged inwards between G and D, and the bulge is travelling towards the left. As the original wave-front 115] REFLECTION 91 continues to spread outwards from A the bulge will get deeper and wider, and will occupy the successive positions shewn in the figure. It can be shewn that the bulges are spheres spreading outwards from B as centre, where £ is as far behind the wall as A is in front of it, and their velocity is the same as that of the original wave-fronts. 92 REFLECTION [CH. VI 116. Sound Images. These spheres form the reflected wave-fronts ; and we see therefore that the sound after reflection appears to radiate from the point B. Using the analogy of light we may say that B is the sound image of A. The construction for finding B is the same as would be employed if it were required to find the image oi A, HA were a source of light, and MN were a mirror. In Optics it is often more convenient to solve problems by making use of rays instead of wave-fronts, and we may use the same method here. We should then look on A as sending out rays of sound in every direction ; any ray such s,s AC being reflected by the wall in such a direction that the incident and reflected rays make the same angle with the normal to the wall. If AG and OF make the same angle with the normal the triangle GOB formed by continuing FG backwards until it meets A produced is equal in all respects to the triangle AGO, and therefore OB is equal to AG. The same applies to every ray from A that strikes the wall, and consequently all the rays will after reflection appear to have come from B, if our assumption that the incident and reflected rays make the same angle with the normal is correct. We shall return to this point presently. 117. Echoes. We have now the explanation of the simplest kind of echo. Suppose a person situated at A (Fig. 45) makes a sharp sound, such as that produced by clapping the hands, or firing a gun, and there is a listener at F. The sound reaches F by two paths, one of which is the direct line AF, and the other is the broken line AGF. These paths are of different lengths and therefore two sounds will reach F, the first by the path AF, and the second by the path AGF. The interval of time between them will be the time sound takes to travel the difference between the lengths of the two paths. Suppose for instance that AF is half a mile 116-118] REFLECTION 93 and AG + GF is a mile and a half. The second sound has to travel a mile farther than the first, and so will reach F nearly 5 seconds later. It is possible to make a rough determination of the velocity of sound by making use of an echo. Stand about 100 yards from a high wall or the side of a house and clap the hands. An echo will be heard, and if by some means the interval between the clap and the arrival of the echo is measured, we shall know the time taken by the sound to travel to the wall and back along a line perpendicular to the wall. The velocity could not be measured very accurately in this way, as in the case given the interval between the clap and the arrival of the echo would be only a little over half a second, which is too short to measure without special appliances. Moreover, in the open air most of the sound spreads out sideways and is lost, so that the echo is faint unless the sound is loud. The experiment can be carried out more easily in a tunnel with a closed end or in a cloister. Stand near one end of the cloister and clap the hands at regular intervals, timing the intervals to one second by means of a watch, or a metronome which ticks seconds. Now move slowly away from the wall, and presently an echo will be heard following each of the claps. Continue to move from the wall until the echoes fall exactly halfway between the claps, which can be judged with considerable accuracy by the ear, and measure the distance from the wall when this happens. It is clear that the sound now takes half a second to travel to the wall and back, and if, for instance, the distance of the observer from the wall is found to be 275 ft., the velocity of sound is 1100 ft. per second. 118. Reflection of Sound by Spherical Mirrors. We have assumed that sound is reflected according to the same laws as hold for light, namely : — (1) The incident ray, the reflected ray, and the normal to the reflecting surface are in the same plane. (2) The incident ray and the reflected ray make equal angles with the normal. It is shewn in books on Optics that, if these laws are true, a beam of parallel rays falling normally on a spherical mirror 94 REFLECTION [CH. VI will come together at the principal focus of the mirror after reflection ; the principal focus being the centre of a radius of the mirror drawn in the direction in which the incident beam travels. Conversely, if a source of light is placed at the principal focus, the rays diverging from it so as to strike the mirror will be reflected as a parallel beam. A similar result is obtained experimentally when a source of sound is used instead of a source of light, and the experiment may be taken as a verification of the two laws stated above. Fig. 46 Place two large spherical mirrors facing each other, and 20 to 30 feet apart. At the focus of one of the mirrors place a watch that ticks fairly loudly. The sound waves which strike the mirror A will be reflected in a parallel beam, and will therefore travel without loss of intensity to the mirror B, where they will be reflected again and converge to the focus. There will consequently be a concentration of the sound at the focus of the second mirror, and the ticking of the watch will be heard clearly if a small funnel connected with a rubber tube leading to the ear be placed at the focus, but will be inaudible if the funnel is moved a little distance away. The mouth of the funnel must be pointed towards the mirror B, for the sound is heard by means of rays that have been reflected from the two mirrors, and not by means of rays that come directly from the watch to the funnel. These latter rays are divergent, and are too weak to affect the ear. The experiment can be shewn in a striking manner to a number of persons at once by using a sensitive flame instead of a funnel and tube. To make a sensitive flame draw out a piece of glass tubing to a fine point, and connect it to a gas bag 118-119] REFLECTION 95 of coal gas. Light the gas issuing from the narrow tube, and gradually increase the weights on the gas bag until the flame is on the point of roaring. The flame will now be steady and a foot or more in height according to the size of the jet, but if any sound of high pitch is made in its neighbourhood, it will become much shorter and will roar. The rattle of a bunch of keys or a hiss will make it drop at once, and if one talks in its presence, it will drop whenever the letter S is pronounced. If such a sensitive flame is placed with the point of the glass tube at the focus of the mirror £, the flame will dance in time with the ticks of the watch, but the dancing will cease if the jet is moved a little to one side, for it is only at the focus that the sound is strong enough to affect the flame. 119. Concentration of Sound by the Walls of a Room. One occasionally meets with an instance of this concentration of sound by curved surfaces in a large room. If the opposite ends of the room are curved, a person speaking at the focus of one end is heard distinctly by a person standing at the focus of the other end. As the curved wall is generally part of a cylinder and not part of a sphere, the focus is not a point but is spread out into a line. The concentration of the sound is therefore not so great as it is when the reflecting surfaces are spherical. It is not essential that the source of sound should be at the principal focus of the first mirror. It must be on or near the common axis of the two mirrors, but there will be a con- centration of the sound at some point, wherever the source is situated along the axis. This follows from the analogy of the corresponding optical phenomenon. It is not even necessary that there should be two mirrors. A Ught placed anywhere on or near the axis of a spherical mirror, and farther from the mirror than the principal focus, will have a real image somewhere, the light and its image being situated at a pair of conjugate foci; and the same is true of sound. There are stories of churches where one can hear at certain points what is said in distant confessional boxes. If the stories are true, it is probable that the sound of the voices is collected by some curved surface and concentrated at a distant point. 96 REFLECTION [CH. VI 120. Whispering Galleries. Thewell-known Whisper- ing Gallery of St Paul's owes its peculiar acoustical properties to the reflection of sound by the walls. The gallery is in the form of a circle running round the base of the inside of the dome. Pig. 47 If a person at any point A puts his head close to the wall and whispers, the whisper can be heard at any part of the gallery, if the listener also places his head near the wall. This is not a case of the concentration of the sound at a focus, for the whisper can be heard almost equally well at any part of the gallery, and the listener can easily convince himself that the sound creeps round the dome in a thin sheet close to the wall. If the dome were not present, the sound from A would spread out equally in all directions, and would fall off in intensity inversely as the square of the distance from A. The effect of the dome however is to prevent much of the spreading. A ray AB^ for instance, which starts in a direction that does not make a large angle with the tangent to the wall near A, will take the path ABCD, etc., and so remain near the wall, and many other rays follow similar paths, so that the sound 120-122] REFLECTION 97 remains strong enough to be audible at any point of the circumference. 121. Musical Echo from a Row of Faling^s. If a sharp sound is made near the end of a row of palings, the echo sometimes takes the form of a musical note. Each of the palings reflects the sound in turn, and the further they are from the listener, the longer the sound takes to return, so that the result is that we have a series of echoes following each other in rapid succession, and if the palings are sufficiently close together, the echoes will blend into a musical note. 122. Experimental demonstration of Stationary Vibrations in Air. We saw in Chapter IV that, when a train of waves is reflected from a solid obstacle, the direct and reflected trains combine to gi^'e a series of stationary vibrations with nodes half a wave-length apart, the reflecting surface being at a node. This phenomenon can be conveniently investigated experimentally with the aid of a sensitive flame. A note of high pitch is needed for the experiment, for the flame is not affected by notes of medium or low pitch. A whistle that gives a note suitable for the purpose can be made in the following way. Take a brass tube six inches long and an inch in diameter, and close one end with a very thin plate pierced with a hole ^\j- in. in diameter. Take a second tube eight to ten inches long, and of such a diameter that it slides tightly over the shoi'ter tube, and fix at its centre a thin plate also pierced with a hole ^^ in. in diameter. Slide the wider tube over the closed end of the narrower tube, until the two pierced plates are half an inch or less from each other, and connect the open end of the narrower tube with a gas bag full of air. The whistle will be found to give a high note which has a powerful effect on the sensitive flame. The pressure on the gas bags supplyi;ng the whistle and flame should be regulated so as to give the best effect, and should be kept constant as the pitch of the note depends on the pressure. Place the whistle with its mouth several feet from a vertical board, and bring the sensitive flame between the whistle and the board. As the flaime is moved along the normal from. the whistle to the board, there will be found a 0. a 7 98 REFLECTION [GH. VI series of equidistant points at. which it is undisturbed. These points are the nodes of the stationary waves, and the distance between any two consecutive nodes is half a wave-length. The experiment gives us the "means of finding the pitch of the whistle, for we know the velocity of sound, and can iind the wave-length by doubling the distance between two consecutive nodes, and so we can calculate the frequency ii from the equation v = n\. Fig. 48 The wave-fronts here are spheres with the mouth of the whistle as' centre, and therefore the circumstances are not the same as in the case where stationary waves are formed in a tube as explained in Chapter IV". Since the flame is moved along that radius of the spheres which is normal to the board, the tangents to the wave-fronts at points on the path of the flame are all parallel to the board. The only rays with which we are concerned lie very close to the normal, the oblique rays being reflected away from the flame, and we may therefore regard the wave-fronts as being small parallel planes. The nodes will not all be equally well marked, for the sound falls off ill intensity as it gets farther from the whistle. The farther a node is from the board, the greater will be the difference between the intensities of the direct and the reflected ■waves, and the less well marked will be the node. Measure- ments should therefore be made as close to the board as possible. If accurate measurements are desired some means must be adopted for preventing the reflection of waves from 122-124] REFEACTION 99 the table on which the apparatus stands, for these waVfes alter the positions of the nodes. A sheet of cotton wool or felt will destroy the greater part of the reflection. 123. Refraction of Sound. We have a further analogy with light in the refraction of sound. When plane sound-waves cross the boundary separating two media in which their velocity is different they are deflected, their direction of propagation making a smaller angle with the normal to the separating surface in the medium in which they have the smaller velocity. The reason for the refraction need not be given here as it is the same as for light, and can be found in any treatise on optics. The laws of the refraction of sound too are the same as for light. The incident ray, the normal to the refracting surface, and the refracted ray are in the same plane ; and the sine of the angle between the incident ray and the normal bears a constant ratio to the sine of the angle between the refracted ray and the normal, this . ratio being equal to the ratio of the velocities of sound in the two media. 124. Total Internal Reflection of Sound. The case of Total Internal Reflection has an interesting application in acoustics. Let AB be the surface separating two media of which the lower is that in which the ' sound travels more slowly, and let CO be a ray of sound incident „ at 0. By the second Law of ;^ ~ Iv B Refraction we have sin COS _ «! .where v-^ and v^ are the velocities of sound in the two media. If CO just grazes the surface, '^' COE= 90° and sin COE= 1, whence sin DOF= v^/v^. Now the path of any ray, whether of light or sound, is reversible, and therefore' if BO is the incident ray, the refracted ray will graze the surface on emergence into the upper medium. If 7—2 100 EFFECT OF WIND [CH. VI the incident ray makes any greater angle than FOD with the normal it cannot emerge at all into the upper medium, for if it did emerge it would violate the second law. It is then totally reflected at the separating surface, and remains in the lower medium. The critical angle beyond which total reflec- tion takes place is the angle which has a sine equal to v^\v-^. Let us suppose the lower medium is air and the upper is pine wood. The velocities of sound in the two media are 332 and 5200 metres per second respectively. Consequently the sine of the critical angle is -^^^, and the angle is about 3^°1 Thus we see that unless the sound strikes the surface very nearly normally none of it will get into the wood, and conversely, if a source of sound is inside a mass of wood, the rays which emerge into the air will nowhere make an angle of more than 3|^° with the normal at the point where they emerge. We see too why sound loses so little in intensity when it travels along a pipe, for it cannot get out unless it strikes the walls nearly normally. 125. Effect of Wind on Sound. It is well known that sounds travelling with the wind are heard better than those travelling against it. This is due to a phenomenon analogous to refraction. Let us suppose that both sound- and wind are travelling from left to right, and A (Fig. 50) represents a wave^front at any moment. The wind travels more slowlynearthegroundthanhigher up, and the upper part of the wave-front is helped forward more than the lower part. Con- sequently, as the wave-front moves on, it will tilt forward into the positions B, C, D, etc. Now the sound always travels in a direction at right angles to the wave-front, and therefore any ray drawn at right angles to all the wave-fronts takes a path w^hich curves dowjiw8.rds towards the ground like the curved line in Fig. 50, _ .- ^ - ;. ::' . 124-126] EFFECT OF WIND 101 A person when speaking sends out rays of sound in every direction. If the air is still the rays are straight, and so Fig. 51 most of them go above the listener and are lost. If however the wind blows from the speaker to the listener, the rays that travel with the wind curve downwards and many that would otherwise be lost are able to reach the listener. The effect is specially marked when there are obstacles between the speaker and the listener, for the curvature of the rays enables the sound to rise over the obstacle and come down on the other side. Fig. 52 When the sound travels against the wind the effect is reversed. The upper part of a wave-front is retarded more than the lower part. The wave-front therefore tilts back- wards, and the sound rays curve upwards as in Fig. 52. The result of this is that at some little distance from the speaker there are no sound rays Temaining near the ground, and the sound is inaudible. 126. Effect of Varying Temperature of the Air. -A similar effect is observed when there is a gradjial ehaiige in the temperature of the air from the ground upwards. 102 SOUND SHADOWS [CH. VI Suppose, for instance, the temperature rises as we go upwards. The warmer the air, the greater is the velocity of sound in it, and therefore the result will be the same as when sound travels with the wind. The upper parts of the wave-fronts will travel faster than the lower parts, and the sound rays will curve downwards. Conversely if the tempera^ ture of the air gets lower from the ground upwards, the sound rays will be deflected upwards and will be lost in the upper air. 127. Sound Shadows. There is one marked difference between light and sound which merits a little explanation, though the complete explanation is beyond the scope of this book. Sound and light both consist of wave motion, and we have seen that many of the phenomena of light have their counterpart in the case of sound. In the matter of the casting of shadows, however, there appears at first sight to be a differ- ence. If an opaque obstacle is placed in the path of a beam of light, no light in general gets round to the back of the obstacle, but we have a sharply defined shadow. If an obstacle is placed in the path of a beam of sound, this effect is generally almost absent. The sound appears to be able to get round corners quite freely, and is heard almost as well behind the obstacle as in front of it. The apparent anomaly arises from the difference in the ratio of the wave-length to the dimensions of the obstacle in the two cases. It can be shewn that a sharp shadow is formed only when the obstacle is large compared with the wave-length, whether the waves be those of light or sound. The wave-length of light varies between -^^xsis il- ^"*^ B'TSW^T ^^- according to its colour. Any ordinary obstacle is much greater than this in diameter, and therefore light in general casts sharp shadows. If however the obstacle is very small, such as a fine wire, it is found that some light gets round to the back, and the shadow is not sharply bounded. The wave-length of a man's ordinary speaking voice is 8 to 10 feet, and that of a woman's voice is 4 to 5 feet. Now the obstacle must be at least 50 wave-lengths in diameter to cast a shadow and even then the shadow would be badly defined, so that we cannot expect ordinary objects such as walls or houses 126-128] SOUND SHADOWS 103 to have much effect in cutting off the sound of the human voice. Practically the only weakening effect such objects have aiises from their compelling the sound to take a longer patli from a point in front to a point behind them. When the pitch of the note is high and the obstacle large, the sound shadow may be very marked. The writer has met with a striking instance of this on Pilling Moss in North Lancashire. In the Spring the sea-gulls resort in large numbers to the Moss to lay their eggs, and when the young birds are able to fly, the air is tilled with their shrill screams. There is a road at a little distance from the nests, and by the side of the road there is sometimes a row of stacks of peat. The length of one of these stacks is many times as great as the wave-length of the screams of the birds, and consequently a good sound shadow is formed. As one walks along the road the alternations of sound and silence are very marked. Opposite the gap between two stacks the sound is unpleasantly loud ; opposite the stack itself there is almost complete silence, and the change from sound to silence is quite sudden. . A similar effect can be obtained in the laboratory, though it is less marked. If a card two fetit square is held between the ear and the high pitched whistle described earlier in this chapter, the sound is perceptibly weakened. If a medium pitched tuning-fork is used instead of the whistle, the inter- position of the card has little or no effect in weakening the sound. It should be mentioned here in anticipation of what will be said in the chapter on Quality that sounds are generally composite, consisting of tones of various pitches, and the quality of a sound depends on the number, pitch, and intensity of the tones that are present. Some of these additional tones may be of high pitch, so that a given obstacle may be able to cast a sound shadow for the higher constituents, whilst having little effect on the lower constituents. Consequently at the back of the obstacle such a sound will be found to have a different quality, from the weakening of the higher con- stituents relatively to the lower ones. 128. Doppler's Principle. When a source of sound is moving to or from a stationary listener, the pitch of the 104 doppleb's principle [CH. VI note heard by the listener is not the same as when the source is stationary. Similarly a listener moving to or from a stationary source of sound hears a note of different pitch from that which he would hear if he were not moving. This may often be observed at a railway station. If a passing train whistles as it goes through the station, the pitch of the note given by the whistle falls just as it passes the listener. When the train is approaching, the pitch is higher than it would be if both train and listener were at rest, and when the train is receding, the pitch is lower. If the train is travelling at 40 miles per hour, the fall of pitch is nearly a tone. The effect can be shewn in the laboratory by holding in the hand a vibrating tuning fork mounted on a resonance box, and swinging it rapidly in a circle. The person who is swinging the fork will hear no change of pitch, because the fork is always at approximately the same distance from his ear, but any other person, who places himself in the plane in which the fork swings, will hear that the note is higher when the fork is moving towards him than it is when it is moving from him. We may say in general terms that when the source and the listener are approaching each other the pitch of the note is raised ; when they are getting farther apart the pitch is lowered. It is not, liowever, a matter of indifference whether it is the source or the listener that moves. If the whistle and the observer are approaching each other -with' a given velocity, the observer will hear a note of rather higher pitch if he is at rest and the whistle is moving towards him than he would hfear if the whistle were at rest and he were moving towards it. The explanation of these phenomena given in §§ 129-132 is known as Doppler'a Principle. • 129. Source in motion and Listener at rest. Let us suppose first that the listener L (Fig. 53) is at rest, and tTie source S is moving' towards hira. The source is producing waves at a definite rate n per second, n being, the frequency of the note that would be heard if, source and..listener_ were, at rest. These waves travel 128-130] doppler's principle 105 towards L with a velocity v, and the source follows them with a velocity V. At the end of one second S will have reached /S", where 8&' = F, and the wave sent out at the S S' O L Fig. 53 beginning of the second will be at 0, where SO = v. Now the source has given out n waves during the second, and these n waves must all be between and 6'^ is also an octave below the lower of the two notes, or is eb. The same method can be applied to find Summation Tones. The first Summation Tone of No. 4 and No. 5, which are a major third apart, is No. 9, which is a ninth above the lower primary. It is a useful exercise to sound c' on the harmonium with each of the notes in the octave below it. As the lower note rises from c^ through dj^, e^ etc. the difference tone is heard falling from c^ through 6'f, g^ etc., and the tones heard can be compared with those obtained by the method of calculation just given. It should be mentioned that the notes of a harmonium as ordinarily tuned are not quite in accordance with the ratios given by the harmonic series, and some of the difference tones will be found to be out of tune. The reason for this will appear when we speak of temperament. 156. Beats caused by Difference Tones. We shall have more to say about difference tones when we come to the subject of musical concords, but one instance of their use may be given here. A tuning-fork mounted on a reson- ance box gives a note which is practically a pure tone, that is, the vibrations produced are sensibly simple harmonic vibra- tions. Take two such forks an octave apart, and put one a little out of tune by sticking wax on its prongs. If the forks are now sounded together beats will be heard. The notes of the forks differ too much in pitch to beat directly, and we must look for another cause. Suppose one of the forks has a frequency 100 and the other 198. The first difference tone will have a, frequency 98, and this gives two beats a second with the note of the lower fork. Two forks can be tuned to an exact octave by making use of these beats. If the forks are adjusted until the beats disappear, the ratio of the frequencies must be exactly 2 to 1. CHAPTER VIII RESONANCE AND FORCED VIBRATIONS 157. Free and Forced Vibrations. The vibrations of sounding bodies discussed in the preceding chapters are all of the kind known as free vibrations. They are the vibra- tions which a body executes if it is made to vibrate and then left to itself, and their period depends only on the dimensions and elastic constants of the body. The period of vibration of a body in such circumstances is called its Free Period. We must now consider the case where a body is maintained in a state of vibration by a periodic- force, which has not necessarily the same period as the free vibrations of the body. When the period of the force is not the same as the free period of the body, the body ultimately vibrates in time with the force, and its vibrations are called Forced Vibrations. In the special case where the period of the force and the free period of the body are the same, we have the phenomenon known as Resonance. As the latter case is the simpler, we shall take it first. 158. Resonant Vibrations of a Pendulum. Make a simple pendulum by suspending a heavy bob by a string of such a length that the centre of the bob is 39 inches below the point of suspension. This pendulum will have a period of about two seconds, that is, at intervals of two seconds it will be found in a particular phase, say at the end of its swing to the right. Fix two light threads to the bob. Take one in each hand and, starting with the bob at rest, pull very gently to the right for one second, then to the left for one second, then to the right, and so on. By this means we apply to the pendulum bob a force which may be regarded as roughly representing a harmonic force, and it is clear that the 157-160] RESONANCE AND FORCED VIBRATIONS 129 bob will soon swing with considerable amplitude. The first pull will draw the bob a little to the right, the second pull will draw it to the left, the third will draw it rather farther to the right than before, the fourth still farther to the left, and so on. As the force changes its direction every second, and the pendulum changes the direction of its motion every second, the force will always act in the direction in which the bob would move if left to itself, and so the 'whole of the work done by the force will be used in increasing the amplitude of the swing, except such small amount as is required to com- pensate the loss of energy from air-friction, etc. 159. Forced Vibrations. If the pulls have a period that is not the same as the free period of the pendulum, the amplitude will not continually increase. Suppose, for instance, that the period of the force is a little less than that of the pendulum. For the first few swings the force will be nearly in time with the pendulum, and the amplitude will increase, but as the force gets more and more in advance of the pendulum in phase, a time will come when it is half a period in advance, and then it will act to the right when the pendulum is moving to the left, and vice versa, so that, instead of the force doing work on the pendulum and in- creasing its swing, the pendulum does work on whatever exerts the force, and for several swings the amplitude gets less. These rises and falls of amplitude will continue for some time, but it can be shewn mathematically that the pendulum will ultimately settle down to a vibration that has the same period as the force. Such a vibration is called a Forced Vibra- tion. We shall defer further consideration of forced vibrations to a later stage of this chapter, merely stating here that their amplitude is generally small, except when the period of the force is nearly the same as the free period of the body. 160. Vibrations produced by a Non-Harmonic Force. Suppose next that, instead of giving alternating pulls each lasting a second, we give every two seconds a pull Fig. 62 of short duration and always in the same direction. The force is again periodic with a period c. s, 9 130 RESONANCE AND FORCED VIBRATIONS [CH. VIII of two seconds, but it bears little resemblance to a simple harmonic force. A simple harmonic force would be represented graphically by a sine curve, whereas the force we have just described would be represented by some such curve as that of Fig. 62. The result however will be the same as before. If the pendulum starts from rest it will gradually acquire a large arc of swing, for the force will always act at the right moment to increase the swing. If the short pull is given every four seconds, we shall again have vibrations of large amplitude, and similarly if the interval between the pulls is any multiple of two seconds. We see then that either a harmonic or a non-harmonic force can excite large vibrations in a body ; but there is a difference between the two oases. A harmonic force excites resonant vibrations in a body only when its period is the same as the free period of the body, or one of its free periods if, as is generally the case, the body has more than one possible mode of vibration, each with its own period. The non- harmonic force may excite resonant vibrations when its period is not the same as the free period of the body. It is not correct to say that it mil excite such vibrations, when its period is a multiple of that of the body, for it is only in special cases that it will do so. The discrimination between the cases in which a non-harmonic force of period different from that of the body does excite resonant vibrations and those in which it does not must be left until Fourier's Theorem has been explained. 161. Instances of Resonant Vibrations. Resonant vibrations are a common occurrence in everyday life. Every child knows that by certain motions of the body he can increase the arc of an ordinary swing, and he soon finds that his motions must keep time with the swing, if they are to be effective. If one leans first to one side and then to the other of a heavy boat, a considerable roll can be set up, if the motions of the body have the same period as the natural swing of the boat, A suspension bridge has been known to give way through the large swing produced by the regular tramp of soldiers, when the tramp happened to keep time with the natural swing of the bridge. The writer once 160-162] RESONANCE AND FORCED VIBRATIONS 131 experienced a similar resonant swing on a seaside pier. A large number of people were walking along it to the strains of a band, thus keeping in step with each other, and an oscillation was produced that was great enough to cause some persons to fall. Fortunately the alarm was so great that the people rushed about regardless of the band, and the oscillation quickly subsided. Resonance can be shewn by making use of the monochord. Tune one of the strings to a tuning-fork. Make the fork vibrate, and hold its stem against one of the bridges. The string will quickly take up the vibration, as can be shewn by placing a paper rider on it. It is not necessary for the fork to be in tune with the fundamental of the string. It will give resonance if it is in tune with any one of the overtones. If it is, for instance, an octave above the fundamental, the string will vibrate in two sections with a node between them. This is a convenient method of shewing the various modes of vibration of a stretched string. 162. Helmholtz's Resonators. The air in a hollow body with a narrow neck, such as a bottle, has a definite period of vibration, as is shewn by blowing across the neck, when a note of recognizable pitch is heard. If a tuning-fork of the same pitch as this note is held near the mouth, the sound swells out greatly, through the resonant vibrations excited by the fork in the air in the bottle. Resonators similar in principle to this are of great use in acoustical investigations. They were used by Helmholtz in his A A work on the quality of musical notes, and are therefore usually called Helmholtz's Resonators. Two of the more common forms of these resonators are shewn in Fig. 63. In each case A is the open mouth and 5 is a short neck which is inserted in the ear, so ^ig- 63 that, if resonant vibrations are excited iu the contained air, they can be heard even though 9—2 132 RESONANCE AND FORCED VIBRATIONS [CH. VIII faint. The second form has the greater part of the body cylindrical and double, so that the volume of air can be adjusted by sliding the outer part over the inner. These resonators are of such importance that we must consider their action in some detaiL 163. Nature of the vibrations in a Helmholtz Resonator. We saw in Chapter II that the period of vibration of an elastic body is of the form 1 , he held the resonators to his ear in turn and recorded his estimate of the relative strengths of the various harmonics. In the same way he found what harmonics are present, and with what intensities, in the notes of other instruments. The next step was to try to imitate the quality of the note of any particular instrument, by producing pure tones cor- responding in pitch and intensity to those found by analysis in the note of the instrument, and sounding them together. 187. Electrically maintained Tuning-forks. The most convenient way of producing a tone approximately 156 QUALITY OF MUSICAL NOTES [CH. IX pure is by holding a tuning-fork of the proper pitch before a resonator, and this was the method employed by Helmholtz, c Fig. 73 but as the vibrations of a tuning-fork soon die away, an electrical method was used for maintaining them. A tuning-fork has attached to the prong B a bent wire which just touches the surface of the mercury in the vessel M, when the fork is not vibrating. The ends of the prongs of the fork are between the poles C and D of an electromagnet. When the wire makes contact with the mercury at S, an electric current flows from the battery through the coils of the electromagnet, from this to the mercury in H, and so by way of the fork back to the battery. The electromagnet is thus excited by the current, and the prongs of the fork are attrgxited outwards towards the poles, but as soon as the upper prong has moved a little way the contact breaks' at £!, the circuit is broken, and the magnet loses its magnetism, allowing the prongs to fall back. The circuit is then closed again and fhe prongs of the fork drawn apart as before, and this process is repeated continually, the fork being thus maintained in a state of vibration. Whilst the vibration is going on, the spark produced when the current is broken makes a good deal of noise, and so this fork cannot be used for our present purpose. It will be seen that the current in the circuit is interrupted once during each vibration of the fork, and this periodic current can be made to drive other forks placed so far from the interrupter as to be out of hearing of the crackle of the spark. 187] QUALITY OF MUSICAL NOTES 157 The fork whose vibrations are to be maintained is placed with its prongs between the poles of an electromagnet, and the intermittent current produced by the interruptor is passed through the coils. The fork will now have a periodic force acting on it, and will be made to execute forced vibrations with the period of the force. If the natural period of the fork is the same as that of the current, or in other words the interruptor fork and the maintained fork have the same pitch, there will be one impulse for each vibration, and by the principle of resonance the vibrations wDl soon become large. Fig. 74 Similarly, if the maintained fork has twice the frequency of the interruptor fork there will be one impulse for each two vibrations, and in this case also large resonant vibrations will be produced, and so on ; any fork whose frequency is an exact multiple of that of the interruptor having large vibrations produced in it. The action of the intermittent current on the forks may be explained in another way. When the interruptor fork has reached a steady state, the current must be periodic, and therefore can be regarded as made up of a series of currents which vary harmonically in intensity, and have frequencies 1, 2, 3, etc. times that of the interruptor. If the maintained forks are tuned to these frequencies 1, 2, 3, etc., each will' be in tune with one of the harmonic constituents of the force 158 QUALITY OF MUSICAL NOTES [CH. IX acting on it, and so will be maintained in active vibration. This is a case of the general principle that a periodic force will produce large vibrations in a body on which it acts, provided one of the harmonic constituents of the force has the same period as one of the natural vibrations of the body. This principle should be specially noted as it has important applications to the theory of musical instruments. The make and break of the current is fairly sudden; consequently the curve by which it could be represented has points of great curvature at the make and break. From what was said before about the Fourier analysis of curves with great curvature, it follows that a long range of terms of the harmonic series will be required to express the periodic force, and therefore forks with pitches high in the harmonic series can be maintained by it. If the make and break were quite sudden, and always occurred at the same point in the swing of the prongs, the interrupter fork would not be maintained in vibration; for the prong would have the same force acting on it in its outward and inward journeys, and over the same range in each, and so would lose as much energy in one half of its vibration as it gained in the other. Both make and break however are delayed a little by two causes. The self-induction of the circuit, which is considerable on account of the presence of the electromagnet, causes a spark at the break, which prolongs the current a little ; it also prevents the immediate rise of the current to its full strength at the make. Further, the wire sticks to the mercury at the break, and draws it up a little before the wire and surface part company, and at the make the surface is not broken and contact made until a dimple has been formed in the surface. The result is that the work done by the magnetic force while it acta in the direction of motion of the prong is a little greater than that done while it acts against the motion, and so, on the whole, energy is communicated to the fork. 188. Helmholtz's Synthesis of Complex Vibra- tions. Helmholtz used 8 forks tuned to the first 8 terms of the harmonics of Bt?. Behind each fork was a resonator, which could be placed at different distances from the fork, so 187-189] QUALITY OF MUSICAL NOTES 159 as to intensify the sound by any desired amount, and each resonator had a shutter by which its mouth could be closed (Fig. 74). The electromagnets of the 8 forks were all placed in series in the electric circuit of the interrupter fork and so all the forks were kept in vibration. This method of driving the forks not only permits the noisy spark to be put where it will not be heard, but has the further advantage that it ensures the frequencies of the notes given out being exactly in the ratios 1, 2, 3, etc. If one of the forks is a little out of tune when vibrating freely, the current forces it into the proper pitch, and the only harm that is done is that the amplitude of its vibrations is not quite so great as it would be if the fork were exactly in tune. Since a tuning- fork without a resonator is almost inaudible, very little sound is heard when the shutters of all the resonators are closed. Helmholtz found by analysis with his resonators that the note of a particular organ pipe consisted mainly of three members of the harmonic series, No. 1 strong, No. 3 moderate, and No. 5 weak ; the other members being inaudible. He then tried to build up this note with his forks. The shutters of all the resonators were closed except those of Nos. 1, 3, and 5, and these three resonators were separately adjusted to such distances from their forks as to give tones of the same relative intensities as had been found in the analysis of the note of the organ pipe. On allowing the three forks to sound to- gether with this adjustment of intensities, it was found that the quality of the note of the organ pipe was reproduced very closely. In the same way the notes of the horn, clarinet, and some other instruments were imitated. The notes of the hautboy and violin could not be reproduced with so few forks as 8. The penetrating quality of these instruments arises from the prominence of high harmonics in their notes, and many of these harmonics lay beyond the range of Helmholtz's forks. 189. Relation between the quality of a note and the phases of its constituents. Helmholtz used this set of forks also to investigate the connexion between the relative phases of the constituent harmonics and the resulting quality of note produced. 160 QUALITY OF MUSICAL NOTES [CH. IX It was stated in Chapter VIII that, if a resonator be put. slightly out of tune with a fork, the resonance will be weakened, and at the same time the phase of the resonator vibrations will be altered. We have seen that the pitch of a resonator can be lowered by making the mouth smaller, and this was the method used by Helmholtz. He partly closed the mouth of a resonator, thus putting it out of tune with its fork. This altered the phase of the vibrations, and at the same time diminished the resonance. He then moved the resonator nearer to the fork, and so restored the intensity of the sound. The harmonic corresponding to this fork had then the same pitch and intensity as before, but a different phase in relation to the phase of the current. He found that no difference was made in the quality of the compound note by altering one or more of the constituents in this way, and therefore concluded that the phase of the harmonics has no effect on the quality. This conclusion is not accepted by all physicists, and much has been written on the subject since Helmholtz's time. It would perhaps be correct to say that at the present time the prevalent opinion is that Helmholtz's theory is right as a first approximation, but that change of relative phase of the harmonics of a note is not quite without effect on its quality. 190. Speech. Diffferences in the quality of sounds play an important part in ordinary speech. Consonants are in many cases merely methods of beginning and ending vowel sounds. They are only passing sounds and not continuous. The vowels on the other hand are musical sounds which can be maintained indefinitely. Take tlie word bad as an instance. If in the course of a song this word has to be sung on a particular note, and the ;, note has to be maintained, say over a semibreve, what happens i is as follows. At the beginning of the semibreve the voice begins with a special kind of explosion which represents the 6. It then settles down to the vowel a, and maintains it to the end of the semibreve, when the note comes to an end with aaiother kind of explosion, which represents the d. Neither the sound of 6 nor that of d can be maintained. It is only a 189-191] QUALITY OF MUSICAL NOTES 161 vowel that can be maintained in singing. Consonants are in most cases mere noises, whilst vowels are musical notes. Moreover, a vowel maintains its characteristics so long as it is maintained. It is not necessary to hear the beginning or end to decide what vowel it is. It is clear then that our recognition of any vowel must be due to its quality. We distinguish between the vowel a and the vowel u in the same way as we distinguish between the sound of a violin and that of a flute. 191. The Vocal Organs. The voice is produced by forcing air from the lungs through the opening between a pair of stretched membranes, each of which is able to cover half of the larynx or passage from the lungs to the mouth. These membranes are called the Vocal Chords, and when not in use for speaking or singing, their free edges are widely separated, so as not to interfere with the breathing. Fig. 75 By means of muscles these membranes can be stretched and their edges brought together. If air is now forced between them, their edges are set in vibration, and the air issues in a series of puffs, which give rise to a musical note. The pitch of the note is varied mainly by altering the tension of the chords, the changes of tension being brought about by muscles attached to the larynx. The pitch and quality of the. note, can probably, also be altered by changes in the distribution of the mass of the chords. Qn their under c. 8, n 162 QUALITY OF MUSICAL NOTES [CH. IX surface there is a layer of membrane, which can be moved towards or from the edge, thus weighting the vibrating part to a greater or less extent, and so altering the period and nature of the vibrations. The adjacent edges of the chords probably touch each other in the course of each vibration, and so make the stream of air discontinuous. The result of this discontinuity is that Fourier's analysis gives a long series of harmonics in the note produced. It is possible to detect as many as 15 or 16 in the note sung by a powerful bass voice. The sound has to pass through the mouth on its way to the outer air, and the mouth and its adjoining cavities have natural periods of their own. Consequently such harmonics as approximate in pitch to any of the natural periods of the mouth will be strengthened by resonance, and the quality of the note will be altered. The pitch of the mouth regarded as a resonator can be altered at will, either by altering its volume, or by altering the size of its opening. Changes of volume can be brought about either by moving the tongue, or by opening the jaws more or less widely, and changes of opening by means of the lips. It will be seen therefore that we can make changes in the quality of the voice by altering the shape and size of the mouth, and it is by such changes that the different vowels are produced. 192. Vowel Theories. There are at present two theories as to the cause of the dififerences between the vowels, each of the theories having its adherents. All are agreed that a vowel sound contains a long series of harmonics, some one of which is strengthened by the resonance of the mouth. In certain cases the mouth cavity is divided into two by the arch of the tongue, and in these cases two harmonics are strengthened, for each of the parts of the mouth cavity has its own natural period. The point at issue is whether the strengthened tone is fixed in pitch, whatever may be the pitch of the note on which the vowel is sung, or whether it moves up and down with the pitch of the note, always remaining at the same interval above the fundamental. The two theories are called the fixed pitch and the relative pitch theories respectively. Helmholtz believed the fixed pitch theory to be true. 191-193] QUALITY OF MUSICAL NOTES 163 Taking the vowel o, for instance, as in the word note, he found the strongest resonance was always at 6^t?, whose vibra- tion number is 466, whatever might be the pitch of the note to which o was sung. He gives the following scheme for the maximum resonance due to the mouth cavity when different vowels are sung. The vowels are to be given their German pronunciation. 8^ i s -rjr- qouOaAe I otr Fig. 76 According to the fixed pitch theory then, the mouth is set to a definite shape for each vowel, and retains that shape unchanged when a scale is sung. This theory is more generally held than the relative pitch theory at the present time — at least as giving the main cause of the difference between the vowels. The relative pitch theory states that for a given vowel the harmonic which is strengthened by the mouth resonance is a certain definite member of the harmonic series, and so moves up and down with the fundamental. This view assimilates the vowel characteristic to what we have called the quality of the notes of a musical instrument. According to this theory the quality of the vowel is the same for all pitches of the fundamental, as the relative strength of the harmonic con- stituents remains the same. Hence if we sing a rising scale on some one vowel, the mOuth must be altered at each step of the scale, so that its resonance pitch may rise, as the pitch of the fundamental rises. We shall have more to say on the vowel theories when we have described the Phonograph. 193. Harmonic constituents of the notes of Musical Instruments. We shall conclude the chapter 11—2 164 QUALITY OF MUSICAL NOTES [CH. IX with a brief account of the distribution of the harmonics in the notes of a few instruments. The Pianoforte has the second harmonic nearly as strong as the fundamental. The third, fourth, etc. fall off rapidly in intensity, until above the sixth or seventh the harmonics are very faint. The Violin has a long series of harmonics falling off gradually in intensity. They fall off somewhat rapidly as far as the fourth, and then more slowly. A Stopped Organ pipe has only the odd harmonics. If the pipe is wide the note is practically pure ; if it is narrow, the third harmonic is strong and the fifth is perceptible. An Open Organ pipe has the full series so far as they extend. A wide pipe has the octave fairly strong, and one or two others perceptible. A narrow pipe has a series of gradually diminishing intensity as far as the sixth or seventh. The Flute gives a nearly pure tone. The octave is faintly audible, but no other harmonics can be heard. The Clarinet has the third, fifth and seventh harmonics fairly strong. The fourth, sixth and eighth are also very faintly audible. The Hautboy, the Brass Instruments, and the Human Voice are alike in having a full series of harmonics falling off gradually in intensity, but they differ in the height to which the series extends. The note of the French Horn has no appreciable harmonics above the sixth, the Trumpet and Trombone have harmonics quite perceptible as far as the eighth or ninth, whilst the Hautboy and the Human Voice extend to the' sixteenth or higher before becoming inaudible. CHAPTER X ORGAN PIPES Organ pipes may be divided into two main classes, Flute or Flue Pipes and Reed Pipes, but within each of these classes there are many varieties differing in shape and material. • 194. Action of the mouth of a Flue Pipe. Fig. 77 is a section of a typical Flue Pipe. Air is blown into the pipe at A. It issues from the long narrow slit 5 in a thin sheet, which strikes the sharp edge G of the front of the pipe, and sets up vibrations in the body D. The pipe may be made of metal or of wood, its ■ section may be circular or square, and its upper end may be open or closed. The manner in which the sheet of air excites vibrations in the pipe is uncertain, but it is probable that something of the following kind takes place. We shall assume that the pipe is closed at the upper end as in the figure. The sheet normally strikes the edge C, but some accidental circumstance, such as a slight movement of the air, may deflect it inwards. A puff of air then enters the pipe and causes a condensation, which travels to the closed upper end, is reflected there, and returns to the mouth. The rise of pressure resulting from the arrival of the condensation at the mouth deflects the sheet of air outwards. As the mouth is to be regarded as an open end, the condensation is now converted by reflection into a rarefaction, and the amount of rarefaction is increased by the action of the sheet 166 ORGAN PIPES [CH. X of air, which is now blowing across the outside of the mouth, and by a well-known hydrodynamical principle lowers the pressure inside the pipe*. The rarefaction then travels to the top, and down again to the mouth. On its arrival at the mouth it sucks, so to speak, the sheet of air inwards, a condensation is produced, and the whole process is repeated. Thus, if the vibration is once started, it will be maintained by the oscillations of the sheet of air, for, whenever a condensation arrives at the mouth and is converted by reflection into a rarefaction, the conversion is helped by the sheet passing outside the pipe, and similarly the change from a rarefaction to a condensation is helped by the sheet passing inside the pipe. 195. Period of vibration of a closed pipe. The account we have given of the action of the blast serves also to shew that the period of vibration of a closed pipe is the time taken by a pulse in travelling twice up and down the pipe, for a pulse starting as a condensation makes two journeys to the top and back before starting again as a condensation, or, it is only after two journeys that it completes its cycle. The distance a wave travels in one period is one wave-length, and therefore the wave-length of the tone given out by a closed pipe vibrating in the manner described is four times the length of the pipe. We shall see presently that the pipe can vibrate in other modes, which give a different relation between the wave-length and the length of the pipe. The mode we have described is that corresponding to the lowest or fundamental tone of the pipe. 196. Period of vibration of an open pipe. Let us consider next the case of a pipe which is open at the top as well as at the mouth. A condensation started at the mouth travels to the open top, and is returned thence as a rarefaction. On arriving at the mouth it again changes phase, and starts its second journey up the pipe as a condensation. Thus it always starts from the mouth in the same phase, and therefore * This phenomenon can be shewn by holding a glass tube with its lower end in water, and blowing strongly across its upper end. The water will rise a little in the tube, shewing that the pressure has been reduced by the current of air. 194-198] OBGAN PIPES 167 the time it takes to travel from the mouth to the top and back is one period, and the wave-length is twice the length of the pipe. If then we have two pipes of the same length, one open and the other closed, the open pipe will give a tone whose wave-length is half that of the tone given by the closed pipe, or the open pipe will sound an octave higher than the closed pipe. 197. Correction for an open end. This result is not strictly true. We saw in a former chapter that the reflection from an open end must be regarded as taking place a little way beyond the end, and consequently the open pipe, having two open ends, must be regarded as being acoustically a little longer than the closed pipe, which has only one open end. Twice the length of the open pipe with its two additions is more than one half of four times the length of the closed pipe with its one addition, and therefore the open pipe is rather less than an octave above the closed pipe. Blow an open organ pipe and then cover its end. The note will be heard to fall a little less than an octave. The addition that has to be made to the length of a pipe with an open end is called the correction for the open end. The smaller the bore of the pipe is in proportion to the wave- length, the smaller is the correction. For the present we shall suppose the pipe is narrow and not very short, and shall neglect the correction. 198. Overtones of flue pipes. Another method of deducing the mode of vibrations of pipes may usefully be given, as it leads more directly to the relations between the overtones. We may regard the blast at the mouth as sending a train of waves along the pipe. These waves are reflected from the other end — open or closed, as the case may be — and by the superposition of the reflected train on the direct train stationary vibrations are produced in the pipe, in the manner explained in Chapter IV. We may then imagine the pipe to enclose any number of segments of a train of stationary waves, subject to two con- ditions : — 168 ORGAN PIPES [CH. X (1) An open end must be situated at an antinode, for the change of pressure must be a minimum at an open end. (2) A closed end must be at a node, for there cannot be any motion at a closed end. Pig. 78 Let Fig. 78 represent the displacement curve for a series of stationary waves at two instants. The full curve shews the maximum displacements of the various particles of air in one direction, and the dotted curve shews their maximum displacements in the opposite direction half a period later. The nodes are situated at E, F, Q, H and J and the antinodes at A, B, G and J), and any two consecutive nodes, or any two consecutive antinodes, are half a wave-length apart. As an open pipe must have an antinode at each end, we may regard it as enclosing the length AB, or AG, or AD of the train. In the first ease it encloses half a wave, in the second case a whole wave, in the third case three half waves, and so on. 199. Overtones of an open pipe. If we are con- sidering the various modes of vibration of an open pipe of fixed length, we must imagine the wave-length of the train to shrink step by step, so as to allow 1, 2, 3, etc. half waves to fill the pipe, always beginning and ending at an antinode. We can now find the wave- lengths and modes of vibration of the various possible tones of the pipe. The fundamental may be re- presented diagram matically by No. 1 of Fig. 79. There is a node in the middle of the pipe, and an antinode at each end, and the wave-length is double the length of the pipe, as we saw before. Pig. 79 198-200] ORGAN PIPES 169 Admitting that we have stationary vibrations in the pipe, and that the ends are antinodes, it is clear that we cannot have any mode of vibration that will give a lower note than this, for there must be at least one node between any two antinodes. If we have more than one, as in No. 2 or No. 3, the -wave-length must be less, and the pitch higher. The mode of vibration that gives the first overtone is shewn in No. 2. Here there are two nodes, each one quarter of the length of the pipe from an end. The pipe contains one ■wave, and therefore the wave-length is one half, and the frequency twice what it was for the fundamental. Similarly the third mode has three nodes, one of which is in the middle and each of the others one-sixth of the length of the pipe from an end. The wave-length is one-third and the frequency three times that of the fundamental. Evidently this process can be continued indefinitely, and we conclude that an open pipe can give a series of tones whose frequencies are proportional to the series of numbers 1, 2, 3, 4, etc. — the same series of tones as we found for a stretched string. 200. Overtones of a closed pipe. The case is different for a closed pipe. Here the closed end is a node and the open end an antinode. The lowest note for which these conditions are fulfilled is one for which one quarter wave- length fills the pipe as in No. 1. The next higher note pos- sible is one which takes in the part from J" to 5 of Fig. 78. This mode is shewn in No. 2 of Fig. 80. There is a node at Q. PQ is half a wave-length and QR is a, quarter of a wave- length, and therefore the node Q is one-third of the length of the pipe from the open end. No. 1 contains one quarter-wave and No. 2 contains three quarter-waves ; and therefore the second mode has three times the frequency of the first. Fig. 80 170 ORGAN PIPES [CH. X The third mode has two nodes in addition to that at the closed end. One of these nodes is one-fifth the length of the pipe from the open end. The pipe contains five quarter-waves and the frequency is five times that of the fundamental. This process again can be continued indefinitely, and thus we see that the possible tones of a closed pipe have frequencies proportional to the series of odd numbers 1, 3, 5, etc. 201. Experiments on the overtones of pipes. An open pipe then can give the full series of harmonics shewn in Fig. 31 whilst a closed pipe gives only every alternate tone beginning with the lowest. An ordinary tin whistle is merely an open organ pipe, whose length can be altered by uncovering the holes in turn. ] f all the holes are kept covered, the first three oi- four tones of the series can be produced without much difficulty. Blow gently, and the fundamental is heard. Blow harder, and the note jumps an octave to the first overtone. Blow still harder, and the note goes up to a twelfth above the fundamental. To shew the overtones of a closed pipe a small closed organ pipe may be used. The first overtone, a twelfth above the fundamental, is easily produced by vigorous blowing. The next is two octaves and a major third above the fundamental. It will probably be found difficult to blow the pipe hard enough to produce this note. It is easier to produce the overtones on a narrow pipe than on a wide one. The manner in which the air vibrates in a pipe can be illustrated by a spiral spring similar to that described in Chapter IV. Hang such a spring vertically, draw down its lower end slowly, and then release it. The coils of the spring will execute vibrations similar to the vibrations of the successive layers of air in a closed pipe giving its fundamental. The upper end of the spring corresponds to the closed end of the pipe where there is no motion, and the free end corresponds to the open end of the pipe. Any oue coil of the spring describes harmonic vibrations, the amplitude of the coil situated at the free end being the greatest, and that of the coil at the fixed end being zero. At any moment all the coils of the spring are moving in the same direction, either closing in on the fixed end, or spreading outwards from it. 200-202] ORGAN PIPES 171 and the velocity of a coil is greater, the farther that coil is from the fixed end. Any other mode of vibration of a closed or open pipe can be illustrated by imagining two or more such springs joined end to end. For an open pipe giving its fundamental imagine two such springs with their fixed ends joined together, and the phases of the vibrations so adjusted that each of the two halves has its greatest extension at the same moment. In this case the state of condensation or rarefaction is the same at any moment at points equidistant from the centre of the pipe. The student will find it a useful exercise to work out one or two other cases in the same way. The positions of the nodes and -loops in an open pipe can be found experimentally by lowering into an organ pipe with one side made of glass a small paper drum, on which a pinch of sand is sprinkled. When the drum is anywhere near an antinode, the motion of the air makes the sand dance, whilst when it is at a node, the sand does not move. By this means it can be shewn that, when the pipe is sounding its funda- mental, there is a node at the middle point, and when it is overblown so as to sound its first overtone, there are two nodes, each a quarter of the length of the pipe from an end. It will be seen also that in all cases the sand is violently agitated when the drum is near either end of the pipe. 202. The Manometric Capsule. Eonig's Mano- metric Capsule may also be used for shewing the positions of the nodes. Pig. 81 172 ORGAN PIPES [CH. X The manometric capsule consists of a small box divided into two parts by a thin flexible membrane. The space on one side of the membrane has a short tube A, which can be inserted in the wall of the pipe. The space on the other side of the membrane has two openings. Through the lower opening B gas is admitted, and the upper opening is provided with a pinhole burner, where the gas bums with a small flame. If the tube A is inserted near a node of the pipe or, as is more usually the case, the left-hand half of the capsule is omitted, and the diaphragm forms part of the wall of the pipe, the variations of pressure will make the membrane vibrate, and the flame will rise and fall in time with the vibrations. The dancing of the flame is too rapid to be detected by the unaided eye, but is easily seen with the help of a revolving mirror. A cubical box has mirrors on its four vertical faces, and is made to rotate about a vertical axis. The image of the flame seen in the mirror travels so quickly across the field of view, that it appears as a band of light. If the flame is at rest the band is continuous ; if it is dancing, the upper edge of the band shews a series of teeth. If the capsule is placed at the middle of an open organ pipe, which is giving out its f\indamental, the band will shew teeth, for we have seen that there is a node at the middle of the pipe. If now the pipe is blown more strongly so as to give out its next higher tone, the teeth will almost disappear, for the capsule is now at an antinode. The teeth will not disappear completely, for there are always harmonics present, some of which have nodes at or near the middle of the pipe. 203. Methods of tuning flue pipes. There are several methods of tuning open pipes. One method is t© make a hole in the side of the pipe near the open end and cover this with a flap of lead fixed to the pipe by one edge. If the flap is pulled away slightly from the pipe, the pipe is virtually shortened, and the pitch is raised. Another method is to make the upper part of the pipe slide telescopically over the lower part, so that the length can be altered. Many of the flue pipes in an organ consist of cylindrical 202-204] OEGAN PIPES 173 tubes of soft metal. These can be tuned by altering slightly the size of the opening at the top of the pipe. A hollow cone of wood pressed over the end closes the opening a little and lowers the pitch. A. solid cone pressed into the opening enlarges it and raises the pitch. Metal pipes often have two flaps standing out, one on each side of the mouth. If these are bent a little towards each other, the pitch is lowered, and vice versa. The principle of these last two methods will be understood, if the pipe be regarded as a resonator, for we have seen that enlarging the mouth of a resonator raises its pitch. Stopped pipes generally have the upper end closed by a tightly fitting plug, and a pipe can be tuned by pushing the plug in or pulling it out, so as to alter the length. 204. Determination of the correction for the open end. "We must return now to the correction for the open end of a pipe, and explain in the first place how it has been determined, and afterwards how its existence affects the pitches of the proper tones of the pipe. The theoretical determination of the amount that must be added to the open end of a pipe to give the effective length has hitherto proved to be too difficult for mathematicians, except in the case of a narrow cylindrical pipe with an infinitely large flange at the end. In this case the correction is found to be '82 R, when R, the radius of the pipe, is small compared with the wave-length. Unfortunately flanged pipes are of little practical import- ance, and we are compelled to rely on experiment for finding the value of the correction for such pipes as are actually used in musical instruments. Rayleigh determined the correction for an unflanged pipe by finding the change in the pitch when the flange was removed. Two pipes of nearly the same pitch were blown together, one of the pipes having a flange which could be removed. The number of beats per second was counted, first when the flange was in position, and next when it was removed. The difierence between the numbers of beats per second in the two cases is the change in frequency caused by the removal of the fiange. 174 OfiGAN PIPES [CH. X It is clear from what was said of the mouth of resonators in Chapter VITI that the flange must act in the direction of hindering the free egress of air from the pipe, as it confines the stream lines into a smaller space. Hence removing the flange has the same effect as enlarging the mouth of a resonator, that is to say, it raises the pitch of the pipe. The pipe used by Rayleigh had a frequency 242, and it appeared that the effect of the flange was to reduce the frequency by IJ. Rayleigh took the velocity of sound at 60° F. to be 1123 ft. per sec. ; the effective length of the pipe was therefore about 28 inches. 'The radius was 1 inch. Thus the correction due to the flange is the same fraction of 28 in. as 1^ is of 242, or about '2 H. Since the correction for a flanged end is known to be '82 iJ, that for the open end of an unflanged pipe is -62 2i. Blaikley determined the correction by immersing the lower part of a thin brass tube in water, and finding the length of the unimmersed part when the tube resounded most strongly to a fork of known pitch. The water surface forms a closed end to the pipe, and therefore the pipe gives maximum resonance when its length with the correction added is I' 3j,5j,etc. Blaikley measured the two shortest lengths of pipe which gave resonance. Call these Zj and l^, then ^a — ^ = „ > for no correction is to be added to the half-wave from the bottom of the pipe to the node v.T in the second mode of vibration. Also li + c = JX, or &2 — 3fcj M Blaikley found "58 R as the mean value of the correction. 205. Effect of the correction for the open end on the overtones. If the correction depended only on the >h Fig. 82 204-206] OKGAN PIPES 175 width of the pipe, and not on the wave-length of the note produced, its existence would not affect the relative pitches of the natural tones of a pipe. The pipe would merely have to be regarded as longer by an amount c, and what has been said as to the pitches of the natural tones would still hold, each tone being a little flatter than it would be if no correction were needed. This is not generally the case, for the correction varies with the wave-length. It is found that with open pipes of wide bore the overtones are all sharper than the harmonics of the fundamental, and the" divergence is greater for the overtones of higher order. The effect is less marked with narrow pipes, and for these the lower proper tones are fairly concordant with the harmonic series. 206. Effect of the correction for the open end on the quality. The existence of the correction for the open end has an effect on the quality of organ pipes. The pipe may be regarded as a resonator, which . is excited by the vibrations of the sheet o# air at its mouth. These vibrations are maintained unchanged whilst the pipe is sounding, and are therefore periodic, and can be expressed by the harmonic series. It has been stated that a resonator will respond appreciably to a periodic force only when there is a harmonic constituent of the force having a period near that of one of the proper modes of vibration of the resonator. A narrow pipe has proper tones which are nearly in accordance with the harmonic series, and therefore fall near the constituents of the force. For such a pipe therefore the resonance will extend to a considerable number of the lower members of its modes of vibration, and the note given out by the pipe will contain many harmonics. It should be noted that the constituents of the note of the pipe are hannOnics and not the proper tones of the pipe. If a particular proper mode of vibration has a period near a harmonic of the mouth vibration, the vibration excited is not in this natural period of the pipe but is " forced " into the period of the harmonic. The nearer the proper tone and the harmonic are to each other in pitch, the greater will be the intensity of the corresponding harmonic constituent of the note. Wide pipes have proper tones which depart markedly 176 ORGAN PIPES [CH. X from the harmonic series. Such pipes therefore give notes which contain few harmonics, and these harmonics are low in the series. If a wide closed pipe is blown gently, it gives a note which is almost a pure tone. The note of a narrow pipe is rich and full, that of a wide pipe is smooth and rather dull. 207. Conical Pipes. All the pipes of which we have spoken up to this point have been assumed to be of uniform bore throughout their length. Conical pipes are used to some extent in musical instruments, "and should be mentioned here. Their theory is less simple than that of cylindrical pipes. It has been shewn by Helmholtz that a pipe consisting of a cone closed at the narrow end has a series of proper tones whose frequencies are in the ratios 1, 2, 3, etc., or a- closed conical pipe has the same series of proper tones as an open cylin- drical pipe. The hautboy, bassoon and daring all consist of tubes with a reed at one end, and, as we shall see later, the reed end is to be regarded as a closed end. The hautboy and bassoon have conical tubes with the reed at the narrow end and therefore rise an octave when overblown, whilst the clarinet has a cylindrical tube and therefore rises a twelfth. 208. Reed Pipes. We must now pass to the second group of organ pipes — the Reed Pipes. In these the vibration is caused by a tongue of metal, which vibrates in front of an opening in a metal plate, thus allowing a series of puffs of air to pass through the opening. Pig. 83 is a section of one form of reed pipe. Air from the bellows of the organ is introduced into the box B through the tube A. The pipe D is fixed with its lower end inside the box. a This end consists of a metal tube with one side pig. 83 flattened, and pierced with a long narrow opening at G. Over this opening is a rectangular tongue of 206-210] OfiGAN PIPES 177 metal called the reed, fixed at its upper end, but free for the rest of its length. The tongue is a little larger than the opening, so that when pressed down it cuts off completely the supply of air to the pipe. It is not quite flat, but bent outwards a little in a curve, and so, when it is at rest, air can pass from the box to the pipe. The curvature of the reed also makes the closing less sudden — the reed rolling over the hole, so to speak, and closing it gradually. 209. Free reeds and beating reeds. A reed such as that described in § 208 is called a beating reed, since at each vibration it strikes the plate in which the opening is cut. There is another kind of reed called a free reed, which is slightly smaller than the opening, and so does not cut off the supply of air so completely at each vibration as does the beating reed. Fig. 84 shews the two forms of reed. The full line is the vibrating tongue and the dotted line is the opening. '^ J 13 Fig. 84 In the cases where the reeds are associated with pipes, as in the organ, free reeds have been superseded by beating reeds almost completely. Many modern organs have no pipes with free reeds. When however the reeds are not associated with pipes, as in the harmonium and concertina, free reeds are used, since beating reeds give too harsh a tone, if they are not mellowed by the addition of pipes. 210. Action of the reed. The explanation of the action of a reed pipe is somewhat similar to that of a flue c. s. 12 178 ORGAN PIPES [CH. X pipe, where the sheet of air might be regarded as resembling a reed in some respects, though, as we shall see, there is the important difference that the mouth of a flue pipe is an open end, whereas a reed pipe must be regarded as closed at the reed end. When a reed pipe is blown, air passes the sides of the reed into the opening. This causes a condensation to run up to the open end of the pipe, and return as a rarefaction. Meanwhile the reed has been closed by the rush of air, and is held closed by the arrival of the rarefaction. Consequently the pulse is still a rarefaction when it starts on its second journey. It travels to the open end, changes sign, and returns to the reed end as a condensation, where its pressure added to the elasticity of the metal tongue causes the reed to open, and the whole process is repeated. Thus we see that the period of vibration is the time taken by the pulse in travelling four times the length of the pipe, or the wave-length is four times the length of the pipe. We have used this somewhat artificial explanation once more, because it shews why the reed end is to be regarded as a closed end, in spite of the fact that it is at this end that the air and the energy enter the pipe. The reed is always closed when it is at a centre of rarefaction, and to a great extent because it is at a centre of rarefaction. It is always open when it is at a centre of compression, but it does not then behave as an open end, for the moment it opena it admits a stream of air at high pressure, which increases the condensation already existing. 211. Effect of the reed on the pitch of the pipe. When the reed is very flexible, like the thin cane reed of a clarinet, the note produced depends solely on the length of the pipe, the reed being constrained to vibrate in unison with the pipe. If the reed were sufficiently stiff and heavy, it would force the vibrations of the pipe, and the pitch of the note would depend solely on the natural period of the reed. The reeds used in organ pipes are of thin metal, and there is reciprocal constraint. The reed forces the vibrations of the pipe, and the pipe modifies the natural period of the reed. It is necessary in practice that the reed and the pipe should have 210-213] ORGAN PIPES 179 nearly the same pitch, for, if this were not the case, the note would have a poor quality, and the pipe would not " speak " readily; that is to say, it would not begin to sound the moment the air was admitted. 212. Method of tuning a reed pipe. A reed pipe is tuned, not by altering the length of the pipe, but by altering the free period of the reed. A wire E passes through the top of the box B (Fig. 83), and its bent end presses against the reed, so that the part of the reed from its upper end to the place where it is touched by the wire is held against the seating, and only the part below the wire can vibrate. Hence, if the wire is pulled up, the reed is lengthened, and the pitch lowered, and vice versa. The reed is able to control the pipe within the range required for ordinary tuning, but if it is shortened by a considerable amount, the note jumps to a pitch somewhere near one of the higher proper tones of the pipe. The reeds could, if desired, be used without pipes, and, as we have said, free reeds are so used in the harmonium. The chief reason for adding the pipe is that the tone is thereby greatly improved and strengthened. A beating reed alone gives a note of harsh and unpleasant quality from the strength of its high harmonics. When a pipe is added, such har- monics of the reed as fall near proper tones of the pipe are strengthened, and the others are unaflfected. If the pipe is not very narrow, it is mainly the lower harmonics that are strengthened, and the note is therefore made less piercing and disagreeable. The higher harmonics are always present to a greater or less extent, and are the cause of the characteristic penetrating quality of reed pipes. 213. Effect of temperature on Organ Pipes. Flue Pipes and Reed Pipes are both altered in pitch by change of temperature, but to a different extent. ^Rise of temperature affects a flue pipe in two ways. The pipe gets a little larger from the expansion of the wood or metal of which it is made, and the density of the air within it is diminished. The increase of size of the pipe is so small as 12—2 180 ORGAN PIPES [CH. X to have no appreciable effect on the pitch, but the change in the density of the air alters the pitch by an amount that is quite perceptible. We saw that the velocity of sound in a gas is proportional to the square root of the absolute temperature, and the frequency of the vibrations of a pipe is vjX, where A. is constant, being twice or four times the length of the pipe according as it is open or stopped. The frequency of the pipe is therefore proportional to the square root of the absolute temperature. Let us suppose the temperature of the organ rises from 60° F. to 75° F. in the course of a performance. If then a flue pipe has a frequency 512 at 60°, it will have a frequency — 512, or 519, at 75°. This is nearly a quarter of v/459 +60 ^ ^ a semitone sharper. The pitch of a reed pipe is dependent on the pitch of the reed as well as on that of the pipe, and these are altered in opposite directions by rise of temperature. The reed becomes a little softer and therefore flatter, whilst the pipe becomes sharper. The effect on the reed is in general less than that on the pipe, and therefore the pitch on the whole rises, though to a less extent than the pitch of the flue pipes. It follows that the different sections of an organ can be accurately in tune with each other only at one temperature. CHAPTER XI RODS, PLATES AND BELLS We shall conclude the discussion of the vibrations of different classes of bodies with a brief account of the modes of vibration in several miscellaneous cases. Some of these have only a theoretical interest, and nearly all require advanced mathematics for their complete investigation. 214. Longitudinal vibrations of Rods. Elastic rods can vibrate in many different ways. The vibrations may be longitudinal, transverse, or torsional ; the ends of the rods may be fixed, supported, or free ; and the rods may be of any section. Only a few of the possible modes need be mentioned. The longitudinal vibrations of a rod are closely analogous to those of the column of air in an organ pipe. The fixed end of the rod corresponds to the closed end of a pipe or to a node, and the free end corresponds to an open end of a pipe or to an antinode. We may have waves of compression and rarefaction running along the rod, and stationary vibi'ations produced by their reflection at the end, exactly as we had in the case of a column of air. The chief difference between the two cases is that the rod needs no correction corresponding to the correction for the open end of the pipe. The velocity of waves in a rod is expressed by the usual /J V D' formula , / -= , where E is the elastic constant and D the inertia term involved in the particular "kind of wave that is being considered. In the case of longitudinal waves E is Young's Modulus, and D is the density of the material of which the rod is made. The formula shevsrs that the velocity 182 RODS, PLATES AND BELLS [CH. XI is independent of the tension in the cases where the rod is fixed at both ends and stretched. It is also independent of the thickness, provided the material remains the same. Doubling the cross section doubles the inertia of any layer, but it also doubles the forces of restitution, and so leaves the velocity unchanged. 215. Iiongitudinal vibrations of a rod fixed at both ends. The longitudinal vibrations in a rod fixed at both ends are not analogous to the vibrations of the complete column of air in any organ pipe, for a pipe cannot have both ends closed. They resemble the vibrations of the column bounded by two nodes in one of the higher modes of a pipe. They are easily produced in a wire 6 or 8 feet long, firmly fixed at its two ends, and provided with some appliance by which it can be drawn tight. If such a wire is rubbed near its middle with a piece of leather dusted with resin, it will give its fundamental tone. If the wire is held at its middle point between the finger and thumb, and one of the halves is rubbed, the note produced will be an octave higher than before. It is needless to enter into details of the various possible modes of vibration. The fixed ends are always nodes, and there may be any number of nodes between them. It follows as in the case of the transverse vibrations of a string that the natural tones must form the full harmonic series. 216. Longitudinal vibrations of a rod fixed at one end. A rod fixed at one end has modes of vibration similar to those of the air in a closed pipe. The fixed end of the rod is always a node, and the free end an antinode. The natural tones of the rod have frequencies proportional to the odd numbers 1, 3, 5, etc., and the positions of the node for each tone is the same as for the corresponding tone of a closed pipe. The tones are easily produced by rubbing with resined leather a thin metal rod clamped in a vice at one end. 217. Longitudinal vibrations of a rod firee at both ends. The longitudinal vibrations of a rod free at both ends are analogous to the vibrations in an open organ 214-218] RODS, PLATES AND BELLS 183 pipe. They have a special interest from their being the kind of vibrations used in Kundt's apparatus to be described iii Chapter XII. Hold a glass rod or tube five to six feet long at its middle, and rub it lengthways near the end with a piece of wet flannel. The fundamental tone of the rod will be produced without much diificulty. The point held in the hand is a node, and the two free ends are antinodes, and therefore the wave-length in the glass is twice the length of the rod. In order to produce the first overtone hold the rod one quarter of its length from one end, and rub the shorter section. We have now two nodes, each a quarter of the rod's length from an end, and the note produced is an octave higher than the fundamental. It is theoretically possible to prpduce the full series of harmonics, but difiicult in practice to get anything above the second with a rod of reasonable length. The tones produced are very powerful. They are occasionally so powerful as to shatter the glass. The student should avoid making the mistake of supposing that the wave-length in air is, for instance, four times the length of the rod, when a rod fixed at one end gives out its fundamental. When we spoke of organ pipes we assumed that the vibrating substance was the same inside and outside the pipe, and the wave-length was the same inside and outside. With rods the case is different. The period of vibration of the rod is the same as the period of the vibrations produced by it in the air, but the wave-length is diSerent. It is evident from the equation vt = X that, if t is the same for the rod and the air, the wave-length in the rod is to the wave-length in air, as the velocity of waves in the rod is to their velocity in air. The velocity of sound in an iron rod is about 15 times as great as in air, and therefore an iron rod 4 ft. long fixed at one end would produce air waves a little over a foot in length. * 218. Transverse vibrations of rods. There are many modes in which a rod can vibrate transversely, according to the way in which it is supported. It may have 'one or both ends free, one or both ends firmly fixed, or both ends resting on supports. When both ends are free it may be supported or fixed at one or more intermediate points. 184 RODS, PLATES AND BELLS [CH. XI No. ] No. 2 Fig. 85 If both ends are- supported, the rod takes a form such as No. 1 of rig. 85, where it can change its direction at the ends*. If both ends are fixed, it takes the form of No. 2, where the direction of the ends cannot change. The elastic forces resisting bend- ing are clearly greater iil the second case, and therefore the pitch is higher, if the two rods are of the same length and thickness. Only two cases of transverse vibrations need be considered, namely, that in which one end is fixed and the other free, and that in which both ends are free, the rod being supported at two intermediate nodes. The former, which we may term a iixed-free vibration, is the more important. 219. Transverse vibrations of a rod fixed at one end. The tongues of reeds, the vibrators of musical boxes, and the prongs of tuning forks may be regarded as fixed-free rods. The first three modes of vibration of a fixed-free rod are shewn in Fig. 86, where A, B, and C are nodes. The modes of vibration bear some re- semblance to those of a closed pipe, but there are important differences. The natural tones of a closed pipe have fre- quencies in the ratio 1 : 3 : 5 : etc., whilst those of a fixed- free rod do not follow the harmonic series, but rise in pitch miich more rapidly. The Fig. 86 * The rod could not be made to vibrate with its ends supported as shewn in the figure unless the amplitude were small. If it were large the rod would jump off its supports. 218-220] RODS, PLATES ANB BELLS 185 relative frequencies for the first three modes are approximately 1, 6i 17J. The nodes too are not in the same positions as in a closed pipe. The node in the second mode is a little more than a fifth of the length of the rod from the free end ; those in the third mode are about one eighth and one half respectively. The length BC is not the same as CD, though the two segments have, of course, the same period. The reason is that at D the direction of the rod is fixed, whilst at B and G the direction can change. The greater length of CD compensates the greater elastic forces due to the constraint at D. The vibrations are easily produced in a visible form in a stout brass or steel wire fixed in a vice at its lower end. The wire should be of such a length that the vibrations in the first mode are quite slow — say one a second — otherwise those in the higher modes will not be easily seen. In order to produce the second mode hold the wire between the finger and thumb about one fifth of the length from the top, and pluck below the hand. The vibrations will be seen to continue in this mode when the fingers are removed. The third mode can be produced in a similar way. The frequencies of the fundamentals of rods of the same thickness and material but different lengths are inversely proportional to the squares of the lengths. Rods of the same length and of rectangular section, the direction of vibration being parallel to one of the sides of the rectangle, have frequencies proportional to the length of the side of the rectangle in the plane of vibration, and independent of the dimensions of the rectangle at right angles to the direction of vibration. The latter relation is easily seen to be true, for if a rod is doubled in width in the direction at right angles to the plane of vibration, both the inertia and the elastic forces are doubled, and therefore the period is unaltered. If the thickness in the plane of vibration is doubled, the inertia is doubled but the elastic forces are increased eightfold, and therefore the period is halved. 220. Wheatstone's Kaleidophone. It will be seen then that a rod of rectangular section fixed at one end has different periods of vibration in two directions at right angles 186 RODS, PLATES AND BELLS [CH. XI to each other. If the free end is drawn aside in a direction not parallel to either of the sides of the rectangle, and is then released, both vibrations will be executed simultaneously, and the motion of the free end will be that got by compounding the two. If the dimensions of the rod are such that the periods of the two vibrations bear a simple ratio to each other, the free end will describe one of Lissajous' Figures. Such a rod with a small bright bead fixed to the free end to permit the figure to be seen is known as Wheatstone's Kaleidophone. The Kaleidophone as more commonly constructed consists of two thin strips of steel joined end to end, the plane of one of the strips being at right angles to that of the other. One end of the Kaleidophone so formed is clamped in a vice. Each strip then vibrates only in a direction at right angles to its own plane and the vibration of the free end is compounded of the vibrations due to the two parts separately. The advantage of this form of the instrument is that the period of vibration of the lower strip can be varied by clamping the strip at different points and so altering its length. It is thus possible to vary the ratio of the frequencies of the two component vibrations and to produce several of Lissajous' Figures with the same instrument. 221. The musical box. The vibrator of a musical box consists of a comb cut from a steel plate. The teeth of the comb form a series of fixed-free rods, and are graduated in length so as to give a musical scale. The teeth are set vibrating by small pins fixed in the surface of a revolving drum in such positions that they pluck the ends of the rods at the proper moments. 222. Method of tuning a rod. The pitch of a fixed- free rod can be adjusted by scraping. If a little is scraped off near the free end, the inertia is diminished, whilst little change is made in the elastic forces, and therefore the pitch is raised. If the rod is scraped near the fixed end, the elastic forces are weakened without much change in the inertia, as there is little motion at this part of the rod, and therefore the pitch is lowered. The free reeds of harmoniums are fixed- free rods, and they are tuned by scraping away the metal at 220-224] RODS, PLATES AND BELLS 187 the free end or the base, according as it is desired to raise or lower the pitch. 223. Transverse vibrations of a rod ft-ee at both ends. A rod free at both ends takes the forms shewn in Fig. — ~ . ^ : =— — ^ ^ ^ — 87 when it gives out its funda- mental. A and B are nodes, Fig. 97 and the bar may be supported at these points by wedges of s6me soft material such as india- rubber or felt without interference with the vibrations. Each node is -22 Z from an end, where I is the length of the rod. The higher modes have 3, 4, 5, etc. nodes, and the fre- quencies of the vibrations in the various modes bear no simple relation to those of the harmonic series. Bars free at both ends are used in the harmonicon. The bars consist of flat strips of glass, metal, or wood cut to such lengths as will give the musical scale. The lengths are inversely proportional to the square roots of the required frequencies. The strips are supported at their nodes on strings, or on strips of wood covered with felt, and are made to vibrate by being struck with a hammer. 224. The tuning-fork. The tuning-fork may be regarded as two fixed-free rods on the same base. It is an instrument of the greatest value in acoustical experiments from its constancy as a standard of pitch. It is little affected by external conditions, and if kept free from rust, and not subjected to excessive ill treatment, it retains its pitch for years without appreciable change. The only correction needed is for change of temperature, and this is so small as to be negligible for ordinary purposes. A rise of temperature increases the size of the fork and diminishes its elasticity, but the latter change has much the greater effect on the pitch. It is found that the frequency of a fork diminishes by one ten thousandth of its amount for a rise of one Centigrade degree. Hence it would require a rise of 20° C. or 36° P. to lower the frequency from 512 to 511. The overtones of a fork do not belong to the harmonic series. Their pitch depends on the shape of the fork, but 188 RODS, PLATES AND BELLS [CH. XI they are always much higher than the fundamental. The first is 2 to 3 octaves, and the second 4 to 5 octaves above the fundamental. They are usually audible for only a few seconds after the fork is struck, as they die away more quickly than the fundamental. A fork mounted on a resonance box gives a note that is practically a pure tone. The box, like the fork, has a series of natural tones, but in general none of the tones of the box lie near the tones of the fork, except the fundamental, and therefore only the fundamental is strengthened. A tuning-fork is sometimes regarded, not as two fixed-free rods, but as a single free-free rod bent at its middle. This second way of regarding the vibrations is useful, as it gives a clearer idea of the manner in which the vibrations in the fork cause ' ' vibrations in the resonance box. Fig. 88 shews three stages in the bending of a straight rod into a fork. There are two nodes in each case, those in the fork being at the base of the prongs. It will be seen on considering the successive stages that the short piece between the nodes at the base of the fork vibrates up and down, when the prongs vibrate outwards and inwards, and the up and down motion is com- municated to the top of the reson- ance box, and so to the air. ^ig- 88 The method of tuning a fork is similar to that used for tuning reeds. The prongs are filed near their free ends to raise the pitch, and near their bases to lower the pitch, as was explained in § 37. 225. The Triangle. The Triangle used in orchestras is a steel rod bent into a shape convenient for suspending and striking. It is struck with a steel rod, and its note contains a multitude of powerful constituents lying so close together that no definite pitch can be recognized. It is an extreme 224-226] RODS, PLATES AND BELLS 189 case of the quality of note that is called " metallic," if indeed it can be said to give a note. 226. Vibrations of Plates. Chladni's Figures. The vibrations of flat plates have not much practical im- portance, but they are rendered interesting by the simplicity and beauty of the method devised by Ohladni for shewing the nodal lines in the various modes of vibration. The plates used for producing Chladni's Figures may be square or round, and may be fixed at any point We shall take as an illustration a square plate fixed at its centre. The plate is generally of glass or brass, 12 to 15 inches square, and is held firmly at its centre by a screw passing through a hole in the plate, or by a clamp. It should be of the same thickness everywhere, as otherwise the figures produced will be irregular. Most plates shew some irregularities, and these may be ascribed to irregularities in the thickness or elasticity of the plate. Scatter a very little fine sand evenly over the plate, hold the point A at the centre of one side with the finger and thumb, and apply a violin bow near the corner £. The plate will then give out the lowest note it is capable of producing, and the sand will gather along two diameters parallel to the sides of the square. Some practice is needed to produce the notes easily. The bow should be held nearly vertical, and pressed fairly firmly against the plate. It should not be in contact with the plate at the moment of changing from an up stroke to a down stroke. In each stroke get the bow in motion before it touches the plate, and remove it before it comes to rest for the next stroke. The lines of sand mark the lines on the plate which remain at rest, or the nodal lines. The quarters of the plate vibrate in such a way that when 1 is coming up, 2 is going down, 4 is coming up, and 3 is going down. It is a general rule in all Chladni's Figures that the sections divided from eanh other by any nodal line are always moving in opposite directions. The line clearly could not be a node, if this were not the case. 1 2 3 4 A Fig. 89. 190 RODS, PLATES AND BELLS [CH. XI In the case we are considering 1 and 4 move together, and 2 and 3 move together, but in the opposite direction to 1 and 4. If then 1 and 4 are at any moment moving upwards and sending out a condensation, 2 and 3 must be moving downwards and sending out a rarefaction. The waves from the different quarters must therefore interfere to some extent. This can be proved by holding a card over each of 2 and 3 whilst the plate is vibrating, when the sound will be heard to be strengthened. The next higher note that can be produced gives the figure No. 1 of Fig. 90. Hold the plate at a corner and bow at the centre of one side to produce this figure. Nos. 2 and 3 are two other patterns that are easily produced. In each case bow at the centre of a side, and hold the plate at the point a where a nodal line meets that side. Circular plates fixed at the centre give two classes of nodal lines. The first class consists of radial lines dividing the plate into an even number of sectors, and the second consists of circles concentric with the plate. The nodal circles are most easily produced, if the plate is not fixed at the centre, but supported on three small cones of wood placed so as to lie on a nodal circle, and is then tapped at the centre with a soft hammer. When there is only one nodal circle, it is about two-thirds of the radius from the centre. 227. Vibrations of Bells. Bells have modes of vibration somewhat similar to those of circular plates. The nodal lines are of two classes, radial lines running from the point of support of the bell to its edge, and dividing it into an 226-227J RODS, plates and bells 191 even number of equal sections, and nodal circles running round the bell at various heights above the mouth. Let us consider the mode in which there are four nodal lines from the summit to the rim and no nodal circles. This is the mode corresponding to the deepest tone that the bell can give out. Pig. 9 1 represents the mouth of the bell when it is vibrating in this mode; A, B, G, D are the ends of the nodal lines. Between these points there are segments where the rim vibrates rapidly in and out. If at any moment it is outside its normal position at E and G, it will be inside at F and H. This is a similar statement to that made about Chladni's Figures. On crossing a node the direction of motion and the displacement are reversed. It is evident that the blow of -ttie^clapper is suited to set up this form of vibration, but it can also be produced in another way. When for a moment the bell has the shape in which the segment AB is inside its normal position the length of rim between A and B must be less than that between A and D ; and a little later, when the segment AD is inside its normal position, and the segment AB is outside, the segment AB will be longer than AD. It follows that during each vibration 192 ROPS, PLATES AND BELLS [CH. XI a little of the rim is transferred through A from the segment AD to the segment AB and back again, or there is a tangential vibration of the rim at each of the nodes A, B, G, D. We might say that this tangential vibration has nodes at E, F, G, H and antinodes at A, B, C, D, but we must not treat it as a separate mode, for we cannot have it apart from the radial vibration. If we start the tangential vibration, it must of necessity be accompanied by the radial vibration. The well-known method of making a wine glass sing by drawing a wet finger round its edge is an instance of this. The motion of the finger excites tangential vibrations, and these give rise to radial vibrations. This is easily seen by partly filling the glass with water, when the surface is seen to be disturbed at four points at equal distances from each other round the circumference. These points of greatest disturbance are the antinodes of the radial vibrations. They follow the finger round the glass, since the finger is always at a point of maximum tangential motion, and therefore always half way between two points of maximum disturbance of the water. The note of a bell contains many and powerful constituents, which may or may not be harmonious with the main tone, and the art of the bell founder is largely devoted to finding by experiment the shape of bell which brings the more important constituents into harmonic relations with each other, as this conduces to sweetness of tone. The best bells of the present day have the notes shewn in Fig. 92 as the most prominent constituents, assuming the note of the bell to be C. Bells are an exception to the general rule that the lowest tone present is that which characterizes the pitch as a whole. The pitch of the bell whose main constituent tones are shewn in Fig 92 would not be taken to be the pitch of No. 1 but of No. 2. The tone No. 1 is called by bell founders the Hum Note. It '^" " is more persistent than the others, and is often heard alone as the sound dies away. No. 3 is generally strong, and con- sequently a chord of a major third or a major tenth sounds bad when played on two bells. The second or the first tone ^ IC2I V IS2I 227-229] RODS, plates and bells 193 of the bell of higher pitch is then a semitone above the third tone of the lower bell, and an unpleasant effect is produced by the clashing of the two tones with each other. A minor third or minor tenth is much better. Bells are not generally accurately in tune when first cast. The maker puts them in tune and brings the subordinate tones into consonance with the main tone by placing the bells on a lathe and turning metal from the inside. Experience alone can shew from what part the metal must be taken to have the desired effect. 228. Vibrations of stretched membranes. A stretched membrane, such as the head of a drum, can vibrate in various modes somewhat similar to those of a plate. It is not necessary to enter into any details of the modes and nodal lines, as they are of no practical consequence. The great drum and side drums give mere noises, and are used only to accentuate the rhythm of the music. Orchestral drums have round their circumference a number of screws by which the parchment forming the drum head can be tightened and the pitch altered. *229. The Method of Dimensions. We shall con- clude the discussion of the modes of vibration of different classes of elastic bodies by giving the proof of an important law which holds for vibrations of any type. If two elastic bodies made of the same material and performing vibrations of the same type have the same shape, and differ only in size, their periods of vibration will be proportional to their linear dimensions. The theorem is most readily proved by what is known as the method of dim,ensions. Assuming that the proportions of the body remain the same whatever its size may be, the period of vibration may depend on any linear dimension I, the density of the material p, and the elastic coefficient of the material e. The period then can be expressed as a function of I, p, and e, and this function can in general be expanded into a series, which we may represent by "SiAPp"^, where .4 is a numerical constant. The equation t = ^AfpV must be homogeneous ; that is to say, every term must be of the same dimensions in c. s, 13 194 BODS, PLATES AND BELLS [CH. XI mass, length and time. If this were not the case, a change in the size of the fundamental units as, for instance, the change from a foot to an inch as the unit of length, would render the two sides of the equation unequal. On one side of the equation we have only t, which is a time, and has the dimensions T. On the other side ? is a length and has dimensions L, and p is M the mass of unit volume with dimensions -7^ . There are two fundamental coefficients of elasticity, either or both of which may be involved according to the type of vibration. The coefficient of volume elasticity is, as we saw in Chapter V, of the form hp-h — and has therefore the dimensions of a pressure M or y^ . The coefficient of rigidity is the shearing force per unit of area which gives the unit angle of shear, or lores Ji/ =- angle, whence its dimensions are also -^r7^„ . Hence area * ' LT^ we have in the expression for t a series of terms with dimensions of the form V'i^A (^r7n-A or L'-^v-^ Mv+'' T'^ -mm where x, y and z must be so related that the dimension of each term of the series is T. We have therefore, equating the indices of the units of length, mass and time respectively on the two sides of the equation, = a; — 3y - «, = y + z, 1 = - 2«, which give x=\, y = \, «= — i^ as the only possible set of values that will satisfy the conditions, and the series reduces to the single term Al \/P . We see then that the period of vibration is proportional to the square root of the density of the body, and inversely proportional to the square root of the coefficient of elasticity — two conclusions which we reached by another method in Chapter II. We have also the result that the period is proportional to the linear dimensions. If we 229] RODS, PLATES AND BELLS 195 have, for instance, two tuning-forks which are geometrically similar in shape, and are made of the same material, but one is double the size of the other, the larger fork will be an octave lower than the smaller in pitch. Similarly if two spherical Helmholtz Resonators are so made that one has double the diameter of the other, and has also a mouth of double the diameter, the larger will resound to a note an octave lower .than that to which the smaller resounds. 13—2 CHAPTEE XII ACOUSTICAL MEASUREMENTS In the preceding chapters we have described a few such methods of measurement as fell naturally into the line of our argument. In the present chapter we shall gather together a number of other methods which have been used by investigators. The aim will be not so much to provide a complete list, as to describe representatives of the various types of measurement. We shall deal first with the measurement of the pitch. 230. Measurement of pitch by Cagniard de la Tour's Siren. In Chapter I it was shewn that the frequency Fig. 93 230] ACOUSTICAL MEASUREMENTS 197 of a note can be found by means of Savart's Toothed Wheel or the Disc Siren. These instruments are capable of giving only a rough approximation to the frequency. A better instrument of the same class, where the number of vibrations per second is counted directly, is Oagniard de la Tour's Siren shewn in Fig. 93. This siren consists of two circular discs nearly in contact, of which the lower forms the fixed top of a wind chest, and the upper can rotate freely on a spindle. Each of the discs is pierced with a circle of holes, the holes in one disc corre- sponding in number and position with those in the other. The wind chest is supplied with air from bellows with a pressure regulator. Suppose now the upper disc is rotating. Every time the holes in the two rows coincide a jet of air will escape from each hole in the upper disc, and if there are, for instance, 20 holes in each circle, there will be 20 puflfs for each turn of the disc. Each puff will of course consist of 20 separate jets, but as they take place simultaneously they may be regarded as a single puff. The rotation is maintained by the pressure of the air in the wind chest, for the holes are cut obliquely as shewn in Fig. 94, so that the stream of air from / / — the lower hole strikes the side \ \ of the upper hole. The greater „. „ . the pressure of the air, the more rapid is the rotation of the upper plate, and therefore the higher the pitch of the note given out. The spindle of the upper plate is connected with a counter which can be put in or out of action as desired, and shews on a dial the number of rotations in a given time. In order to use the instrument for finding the frequency of the note of an organ pipe, for instance, start the siren by admitting wind from the bellows, and wait until its note ceases to rise. Then increase or diminish the pressure, until the siren and pipe are in unison. The final adjustment is most easily made by making the note of the siren a little too high, and bringing it down to that of the pipe by slightly reducing the air supply by means of a clamp on the supply tube. When the two notes have been brought to the same pitch, throw the counter into gear for "a definite period, such 198 ACOUSTICAL MEASUREMENTS [CH. XII as a minute, and find from the dial the number of revolutions in that period. This number multiplied by the number of holes in the circle gives the number of air vibrations per minute, or 60 times the frequency. The determination cannot be made with any great accuracy for several reasons. Throwing in the counter reduces the speed of the disc a little, and puts the notes out of tune. This source of error can be avoided to some extent by leaving the counter in gear all the time, and when the tuning has been adjusted, taking the time of, say, 1000 revolutions by means of a stop watch. It is not possible to keep the pitch of the siren quite constant during the experiment. The tuning is effected by making the beats between the two notes become gradually slower, until they disappear. If, as is generally the case, they reappear whilst the experiment is going on, they must be removed as quickly as possible, and this is not easy to do, for one cannot tell whether the siren is now too high or too low. The pitch can be lowered by touching the spindle gently with a piece of paper or a feather. If this quickens the beats, the pitch was already too low, and it must be raised by opening the clamp on the air tube a little. Por further refine- ments of the method the reader is referred to Barton's Text Book of Sound. 231. Measurement of pitch by the Monochord. We saw in Chapter III that the frequency of the vibrations of the string of a Monochord giving out its fundamental note \ fr yri. \pj' Kl^ rj Fig. 108 Fig. 109 Take next the Minor Tenth for comparison with the Minor Third. Referring to Fig. 107 we see that the dissonance has been somewhat increased by widening the interval. In the case of the Minor Tenth the 2nd harmonic of the higher note is within a semitone of the 5th of the lower note, whereas in the case of the Minor Third it was the 4th of the higher which formed a semitone with the 5th of the lower. The 2nd harmonic is generally stronger than the 4th, and the increased roughness due to this has more effect than the gain in smoothness arising from the absence from the Tenth of the tone interval found in the Third. It will be found that most intervals change in consonance when extended by an octave, some becoming more smooth and others less smooth. Helmholtz makes the general statement that, if the smaller number is even, when the vibration ratio of an interval is expressed by the smallest possible integers, the consonance will be improved by widening the interval by an octave, and if the smaller number is odd, the consonance will be made worse. The Fifth whose ratio is 2 : .S belongs to the first class, and the Minor Third whose ratio is 5 : 6 belongs to the second. Helmholtz's rule is merely an extension of the general statement made earlier in this chapter that the smaller the integers expressing the vibration ratio, the smoother is the interval. Suppose we widen the interval by putting the lower note down an octave. If its number was originally 232 CONSONANCE [CH. XIV even, no fractions are introduced by halving it, and one of the integers expressing the new vibration ratio is smaller than before whilst the other is unchanged. If the lower number is odd, both numbers expressing the vibration ratio must be doubled before we can halve the lower without introducing a fraction. Thus in this case the lower integer remains un- changed and the higher is doubled. 262. Consonance of an interval formed by two notes which have not both the full series of har- monics. All the conclusions come to in §§ 260 and 261 rest on the assumptions (1) that each of the notes has the full series of harmonics, and (2) that the harmonics above the sixth are so weak as to be negligible. If these conditions are not satisfied, the results may be different. If for instance the notes are sounded on two stopped pipes, which have only the odd harmonics, the consonance is generally better than when the full series is present. The student can easily verify tliis for himself. If one of the notes has the full series, and the other has only the odd members, there may be a difference in smoothness according to which of the notes is the higher. We will give one instance of such a difference. The hautboy has a conical tube and gives the full series of harmonics, whilst the clarinet has a cylindrical tube, and gives only the odd members of the series. Suppose the two instruments play two notes making an interval of a Major Third, first with the hautboy above the clarinet and secondly with the positions reversed. Fig. 110 shews the two cases, and it is clear that there is a great difference in consonance. Wben the hautboy takes the higher note there is no clashing within the first six harmonics, •whilst when the clarinet is above the hautboy, there are two pairs of notes making the interval of a semitone. The student may test his knowledge of the method by finding the best position for the Fig. 110 ^ lEZ 22r ZZ21 CI H CI 261-264] CON-soNANOE 233 instruments when the interval is a fourth. He will find that in this case the clarinet should play the upper note. The eflFect is not so striking in practice as might appear from what has been said, for in the case of the clarinet, and still more of the hautboy, the higher harmonics are so strong, that those above the sixth cannot be ignored. 263. Effect of combination tones formed by the harmonics. The harmonics of two notes will, if powerful enough, generate Combination Tones with each other, and it might be thought that this would introduce new tones, and so modify the conclusions. The harmonics are generally too weak to generate any but the first Difference Tones, and even these are not strong enough to have any practical effect on the consonance. Helmholtz has shewn, moreover, that, when the full series of harmonics is present, the first Difference Tones cannot generate beats, except when beats of the same frequency are already present from the clashing of the harmonics them- selves, and therefore there can only be a slight increase in the strength of the beats. If however both notes contain only the odd harmonics, the Difierence Tones may introduce the even terms, and so have some slight efi'ect on the consonance. We are for the present not taking account of the Difference Tones produced by the fundamentals. We shall see later that they have some effect in modifying the character of a consonance, though they may not cause beats. 264. Consonant Triads. The intervals less than an octave which are admitted as consonant in music are the following : — Interval Vibration Eatio Fifth 3:2 Fourth 4:3 Major Third 5:4 Minor Third 6:5 Major Sixth 5:3 Minor „ 8:5 Let us see how we can add these intervals in pairs, so as to give chords of three notes, each of which forms a consonant 234 CONSONANCE [CH. XIV interval with both the others, and such that the interval between the highest and the lowest notes of the chord is less than an octave. Such a chord is called a Consonant Triad. Take, for instance, two notes which we may call p and q, making one of the consonant intervals with each other, and a third note r, making one of the consonant intervals with q, p being the lowest and r the highest in pitch of the three. We have then a chord of three notes p, q and r, where p to q and q to r are consonant intervals. If it prove that p to r is also consonant, and less than an octave, the three notes form a Consonant Triad. We can make the following table giving all the pairs of consonant intervals whose sums are less than an octave : 1 Minor Sixth + Minor Third gives - x - = - 6' 2 Fifth + Major Third 3 Fifth + Minor „ 4 Fourth + Fourth 5 „ + Major Third 6 „ + Minor „ 7 Major Third + Major Third 8 „ „ + Minor „ 9 Minor „ + „ ,, , We find that only three of the resulting intervals are consonant, the 5th, 6th and 8th, which are a Major Sixth, a Minor Sixth and a Fifth respectively. Each of these combinations gives two Consonant Triads, for the Major Sixth, for instance, can be made up of a Major Third above 5-^5 ~25 3 a 2^*4^ 15 ~ 8 3 6 2'^5 9 "5 4 4 16 " 9 4 5 5 "3 4 6 8 "5 5 5 4*^4 = 25 "16 5 6 3 "2 6 6 36 "25 264-265] CONSONANCE 235 a Fourth, or of a Fourth above then the following chords : a Major Third. We have Major Third above Fourth Fourth „ Major Third Minor Third „ Fourth Fourth „ Minor Third Minor Third „ Major „ Major „ ,, Minor „ 265. Derivation of the Consonant Triads from two fundamental forms. In musical notation the triads can be written as follows : &^^^^ Fig. Ill These are the only groups of three notes within the compass of an octave that form consonant intervals with each other. They can of course be transposed into any other key. We are concerned here only with the relations which the notes in any triad bear to each other. The first four of the triads can be derived from the 5th and 6th in a simple way. Raise the lowest note of 5 an octave and we get ^•\ ^ which is the same as 4, since it is a Fourth above a Minor Third. Now raise the lowest note i of this triad an octave, and we get ^-^ g : which is the ^ same as 1. Similarly raising the lowest note of 6 an octave gives 2, and again raising the lowest note we get 3. Hence we may write the Triads in two groups in accord- ance with this method of derivation. The two groups are termed Major and Minor Triads respectively, because in the first or fundamental form of the 236 CONSONANCE [CH. XIV iirst group we have a Major Third between the two lowest notes, whilst in the second group we have a Minor Third. ISI W i 12=^ 5 tm-^i Fig. 112 In each group the first chord is called the Common Chord — Major or Minor as the case may be — of the lowest note of the chord. The second chord in each group is called the First Inversion of the Common Chord, and the third is called the Second Inversion. 266. Consonance of the triads. The consonances of these chords can be investigated in the same way as the chords of two notes, It is found that the Second Inversion of the Major Triad is the smoothest, and the Second Inversion of the Minor Triad is the roughest of the six chords. The scheme for these is given in Fig. 113. Fig. 113 The difference between the two chords is very marked. There are only two semitone intervals in the first group, and these are fairly high in the series, whilst the second group contains five. The student should be able to work out the other four triads in the same way without difficulty. We can estimate the consonance of the triads still more simply by considering what chords of two notes enter into them, and making use of what has been found earlier with regard to the consonance of chords of two notes. • The first Triad of Fig. Ill contains a Fourth, a Major 265-266] CONSONANCE 237 Third and a Major Sixth ; the third contains a Fourth, a Minor Third, and a Minor Sixth. The Fourth is common to the two, the Major Third is not greatly different from the Minor Third in smoothness, but the Minor Sixth is decidedly worse than the Major Sixth, and therefore the third group is worse than the first. If we estimate the consonance of the fundamental forms of the Major and Minor Triad in this way, we find no difference between them, for each contains a Major Third, a Minor Third and a Fifth ; yet there is no doubt that the Minor Triad is less harmonious than the Major Triad. Helm- holtz ascribes the difference in harmoniousness to the Difference Tones formed by the three fundamentals of each triad taken in pairs. Fig. 114 shews the positions of the First Difference Tones i =1= Ed=^ r=m- m± &. ^P Fig. 114 for each of the six chords. The chords are shewn in Minims, and the difference tones in Crotchets. A difference tone written with two tails can be obtained in two ways. Thus, for instance, the relative frequencies of the three notes in the first chord are 4:5:6. The difference tone of the two lowest notes has a frequency 5-4, a:nd that of the two upper notes has a frequency 6-5. The two Difference Tones therefore coincide at a frequency 1, which is two octaves below the lowest note of the triad. The highest and lowest notes of this first chord give a difference tone of frequency 6-4, which is one octave below the lowest note of the chord. The difference tones of the remaining five chords are easily obtained in a similar way. 238 CONSONANCE [OH. XIV The difference tones of the Major Triads introduce no notes extraneous to the chords. They merely double in a different octave notes already present. The difference tones of the Minor Triads are, it is true, not within beating distance of each other or of the notes of the chords, but some of them fall quite outside the harmony. They are not strong enough to give the character of dissonance, but they disturb the harmoniousness of the chords, and give the Minor Triad its peculiar veiled and mysterious effect. In former times it was not uncommon to use a major chord as the concluding chord of a composition, that was elsewhere in the minor key. This is not often done now. It was no doubt due to musicians of past times regarding the Minor Triad as hardly deserving the name of a consonance, and as not being sufficiently satisfying to the ear to be worthy to take its place as the final chord. It should be reiterated that these conclusions are true only of just intonation. When, as is more usual, tempered intonation is used, the difference between the major and minor chords is to a great extent obscured by other causes due to the mistuning of the intervals. CHAPTER XV DEFINITION OF INTERVALS. SCALES. TEMPERAMENT 267. Definition of intervals. We must now return to chords of two notes, and find to what extent an interval may be mis tuned without introducing unpleasant elements into the concord. In all systems of Temperament some of the intervals are a little out of tune, and it is important, when devising such a system, to know which intervals must be correct or nearly so, if dissonance is to be avoided, and which intervals can be modified without serious ill-effect. We shall find that in most cases beats are produced when an interval is a little out of tune, and the stronger these beats are, the more accurately must the interval lie tuned. An interval for which the beats due to mistuning are strong is said to be sharply defined. 268. Definition of intervals formed by pure tones. The beats of mistuning generally arise from the harmonics, but in some cases they are caused by combination tones. In the case of pure tones there are no harmonics above the first, and therefore such definition as exists must arise from the combination tones. We have already seen, when discussing the consonance of intervals formed by pure tones, that, as the interval is gradually increased, we have first powerful beats due to imperfect unison. These beats get more rapid, until at an interval of about a Minor Third they cease to be perceptible as beats. They reappear when we are getting near a Fifth, and get gradually slower, until at the interval of an exact Fifth, they disappear. A Fifth between two pure tones then is defined to some extent, since beats are introduced by mistuning, but aS the beats arise from the clashing of a first and a second difference tone, they are very faint, and therefore the definition of the interval is slight. 240 DEFINITION OF INTERVALS [CH. XV Near an Octave the beats are stronger, for they arise from the proximity of the first difiference tone to the lower of the two primaries. The Octave then is fairly well defined, even when the notes are pure tones. It is easy to adjust the interval between two tuning-forks on resonance boxes to an exact octave by raising or lowering the pitch of one of them until the beats disappear. It is difficult to tune a Fifth in this way, as the beats are too faint. The remaining consonant interv9,ls of pure tones within an octave give no perceptible beats when mistuned, and there- fore cannot be said to be defined at all. 269. Definition of intervals formed by complex notes. We turn next to the intervals formed by notes having the full series of harmonics, and limit ourselves as befoi'e to the first six harmonics. It is seen in Fig. 103 that, when the two notes make a true Octave, every harmonic of the upper note coincides with a harmonic of the lower. In particular, the lowest harmonic or fundamental of the upper note coincides with the 2nd harmonic of the lower note. These are generally very strong, and therefore, if one of the notes is put a little out of tune, powerful beats will be heard. The Octave is in fact so sharply defined that it is not possible to put it out of tune to the slighest extent without causing conspicuous beats. Consequently, in any system of tuning all Octaves must be true. The Fifth (Fig. 104) is also well defined, though not so well as the Octave. The 2nd harmonic of the higher note coincides with the 3rd of the lower, and these beat if the interval is not exactly in tune. The beats are easily heard, and the interval can be accurately tuned by making use of them. They are not however strong enough to be very obtrusive, and if the mistuning is slight, so that they are slow, they do not have any serious eSect on the consonance. In the modern system of tuning keyed instru- ments all the Fifths are very slightly flatter than the true Fifth. The Fourth (Fig. 105) is still less sharply defined than the Fifth, as the harmonics which coincide are now the 3rd of one series and the 4th of the other. 268-270] SCALES, temperament 241 In the case of the Major and Minor Tliirds and Sixths the intervals are very weakly defined, as the lowest coinciding harmonics are too high in the series to be of much con- sequence. These intervals can be mistuned to a considerable extent without suifering much in consonance, and on modern instruments they do in fact diflfer appreciably from the true intervals. We see then that not only does the quality of a note depend on the presence of its harmonics, but also that the precision with which intervals must be tuned to avoid beats depends on these harmonics. 270. Tonic relationship. Modern music depends largely for its effect on the relation of the various notes to some one note which is called the Tonic. The tonic is what one might term the centre of gravity of the music. After a few notes of a melody have been sung, or still more noticeably after a few chords of harmony have been played, one feels that the music centres round some tonic. The melody usually comes to an end on the tonic, and the last chord is almost invariably a major or minor chord with the tonic as its lowest note. This is so very commonly the case, that one is hardly ever led astray by taking the last note of the bass as the tonic of the piece. The close is more marked and feels more restful when both the melody and the bass part end on the tonic. If the composer adopts some other ending, it may generally be taken to be due to his wishing to secure a less restful finish. A melody sometimes ends on the third or fifth above the tonic, and the peculiar effect of such an ending accentuates the feeling of tonic relationship. Since then the tonic relationship occupies such a prominent place in modern music, it may be anticipated that anything which weakens one's appreciation of intervals will detract from the effect of the music. This is the case when pure tones are used. The tonic is identified by the intervals between it and the other notes of the scale, and the appreciation of these intervals is less precise when the higher harmonics are weak or absent. There is an instrument called the Ocarina which produces notes exceptionally free from harmonics. Anyone who, like c. s. 16 242 DEFINITION OF INTERVALS. [CH. XV the writer, has had the opportunity of hearing a quartet of four Ocarinas, will understand why such instruments are not used in the orchestra. The general eifect is soft and smooth and for a short time is pleasant, but it very soon becomes monotonous. The harmony is quite colourless, and does not seem to be made much worse when, as is often the case, the notes are out of tune. A similar effect is noticed when a hymn tune is played on the organ on some stop such as a Stopped Diapason, which gives notes that are nearly pure tones. The music sounds woolly and indefinite, not so much from the special quality of the notes, as from a feeling of want of definiteness in the intervals. In the Mixture Stops of an organ each note has one or more additional pipes, which are tuned to the octave, twelfth, etc. of the note. Such ^ stop cannot well be used alone, but is most useful with the Full Organ, and especially in accompany- ing congregational singing, for by strengthening some of the harmonics it accentuates the relationships of the notes to each other, and makes it easier for the singers to keep in tune. 271. Advantages of the diatonic scale. In Chapter I we described the musical scale known as the True or Diatonic Scale, and gave the vibration ratios for the intervals included in the scale. This scale, or a scale approximating to it, has been in use amongst European nations for many centuries, and its origin is unknown. We have seen that the intervals which are found in the scale can be obtained by the subdivision of a string, and it is possible that the scale took its rise from this fact, for the ancients were acquainted with the natural tones of vibrating strings. Whatever may have been its origin, the scale has survived to the present day, and there can be no doubt that the reason for its survival is that no other scale is so well fitted to provide harmonious combinations. The following method of deriving the scale hy combining the most consonant intervals shews its advantages from the point of view of hiirmony, but it is not suggested that it took its rise from any Such considerations. Suppose we wish to build up the scale on the note C. 270-271] SCALES. TEMPERAMENT 243 Add to C the notes which make the best consonances with it, namely, the Fourth, the Fifth, and the Octave, and call these F, G and a respectively. We have now the following notes in ascending order of pitch : C F G c 1 1 ? 2 3 2 If the frequency of the lowest note is taken as unity, the frequencies of the other notes will be given by the numbers placed below them. We saw that the best combination of three notes is the Major Triad. Let us then place a Major Triad on each of the notes C, F and G, and see what new notes are introduced. It does not matter which form of the Major Triad we use. The result will be the same whether we use the fundamental form or either of its inversions. Taking the fundamental form, and remembering that the frequencies of the three notes are in the ratio of 4:5:6, we get for the triad on C the frequencies 1, 5/4 and 3/2. For the triad on F we get 4/3, 5/3 and 2, and for the triad on G we get 3/2, 15/8 and 9/4. Now arrange all these notes in order of frequency, bringing the note 9/4 down an octave, so as to bring it between C and c. We have then the following scale : c D E F G A B 1 9 5 4 3 5 15 8 4 3 2 3 8 "9 10 16 9 10 "g" 16 8 9 15 8 Fig. 9 115 8 15 and this is the ordinary diatonic scale. The note is called the tonic of the scale, and the fraction immediately below any note is the vibration ratio of the interval formed by that note and the tonic. The lower row of fractions is got by dividing each fraction in the upper row by that on its left, and gives the vibration ratios of the intervals between the pairs of consecutive notes of the scale. 16-2 244 DEFINITION OF INTERVALS [CH. XV This scale, modified somewhat to meet the exigencies of modern music and instruments, is in almost universal use by civilized nations. One or two other scales, such as that of the Scotch Bagpipes, have survived from former times, but they are quite exceptional, and perhaps owe such charms as they possess to the contrast they present to the superior sweetness of the more familiar scale. So long as music is used only for melody, it is of no great consequence what scale is used. Some music-loving though half-civilized nations have in fact no fixed scale at all. Their vocal music is only an accentuated form of speaking, the voice rising and falling not by fixed steps, but by a continuous glide. When once harmony is introduced, a definite scale is required, and the diatonic scale was generally adopted long before Helmholtz shewed why it is so much better fitted than any other to provide consonant harmonies. The intervals between consecutive pairs of notes are of three kinds ; the Major Tone 9/8, the Minor Tone 10/9 and the Semitone 16/15, and the intervals must be arranged in the order shewn in Fig. 115 to give a true diatonic scale. Each of the notes has received a name which indicates its relationship to the tonic. These names are as follows : C Tonic G Dominant D Supertonic A Submediant E Mediant B Leading Note F Subdominant c Octave. These names are given not to notes of specified absolute pitch, but to notes making specified intervals with the tonic. If a diatonic scale is built up on the note D as tonic, the subdominant, for instance, will now be G, a fourth above D. 272. Modulation. Defects of the Diatonic Scale. A characteristic of modern music is that it frequently modu- lates or changes its tonic. A composition may begin with C as tonic, and presently change into the key of G ; that is to say it now requires a diatonic scale withG as the tonic. Let us find whether the diatonic scale of C, when extended in both directions by raising or lowering all its notes by one or more octaves, will provide the notes required for a diatonic 271-272] SCALES. TEMPERAMENT 245 scale with G as tonic. Keeping to the scale of frequencies in which the frequency of C is represented by unity, we have 3/2 for the frequency of G, and this must be multiplied by each of tlie fractions in the middle line of Fig. 115 in turn to give the frequencies of the notes required for the Diatonic scale of G. Without needing to perform the multiplications, we can see at once that some extra notes will be needed. The interval from tonic to supertonic is 9/8, whereas the interval from G to A is 10/9, which is a smaller interval than 9/8. Hence for the supertonic of G we need a new note a little sharper than A. The note B will serve as the mediant of G, since it is a true Major Third above G. Also c will serve as the subdominant, as it is a fourth above G. Similarly, if D and E are raised an octave, they will serve as the dominant and submediant of G ; but F is much too flat to serve as the leading note. It should be 9/8 above the submediant, whereas it is only 16/15 above. Thus the transposition to G as tonic requires the addition of two new notes to the scale. Similarly to give a true diatonic scale on F as tonic we shall require again two new notes, one of them a semitone above A, and the other a little flatter than D. If we proceed in the same way to take all the other notes in the scale of C as tonics, we shall introduce other new notes. It is found that eleven such notes have to be added to those belonging to the key of C in order to provide a diatonic scale with each note of the key of C as tonic. Nor is this the end of the matter. The Minor Keys have to be provided for, and this again requires additional- notes, for the Minor Scale of C for instance requires a minor third, a minor sixth and a minor seventh. Further, modern music often requires "accidental" notes, a semitone above or below the notes of the scale, and the same note will not serve both for CJf and DtJ, for instance, for two semitones of the diatonic scale do not make a tone. Moreover it must be possible to take any one of these accidentals as the tonic of a diatonic scale. Thus it will be seen that to provide for all these con- tingencies a very large number of notes will be required in each octave. This causes no diificulty with unaccompanied vocal music, for the voice can adjust its pitch so as to give 246 DEFINITION OP INTERVALS [CH. XV true intonation in any key, and the same is true of such instruments as violins and trombones, which have no fixed keys, the pitch of each note being determined by the per- former. Instruments such as the pianoforte, the organ, and most of the orchestral wind instruments, which have fixed keys giving notes of definite pitches, are in a different position. The performers on such instruments cannot adjust the pitch of each note to true intonation. They must take the pitch provided for them by the instrument maker or tuner, and it is plainly impracticable to have a great number of keys in each octave. 273. Temperament. Consequently a compromise has to be effected. The notes in each octave are limited to such a number as is found practicable, and some of the intervals are altered a little from the true diatonic intervals, so as to make it possible to modulate without departing greatly from the diatonic scale. The number of notes to the octave is twelve for all instruments in ordinary use at the present time. On the pianoforte the white keys give a scale not much different from the diatonic scale, and five black keys are added, which give notes dividing the intervals of a tone into two semitones, differing a little from the true diatonic semitones. In discussing methods of tuning it is convenient to make use of a smaller interval than any we have employed hitherto. The interval generally used is called the Comma, and is defined as the difference between the Major Tone 9/8 and the Minor Tone 10/9. Its vibration ratio is therefore 9/8-=- 10/9 or 81/80. This is about a fifth part of a semitone. A still smaller interval named the Cent is also often used. The cent is the 1200th part of an octave, or, since there are on the usual system of tuning 12 semitones in an octave, it is the 100th part of a semitone. Only two methods of tuning, or Temperaments, as they are called, have been used at all extensively, and one of these is now practically extinct. 274. Mean -tone Temperament. The Mean-Tone Temperament was in common use in organs until 50 years 272-275] SCALES, temperament 247 ago. As it has now been displaced by Equal Temperament, it is not necessary to give any lengthy account of it. If we tune upwards four true fifths from 0, we reach a note which is a comma above the true E. Each rise of a fifth increases the vibration ratio by the factor 3/2. Hence if the frequency of C is called unity, that of a note four fifths above C is 1 x 3/2 x 3/2 x 3/2 x 3/2 or 81/16. To reach the note E which is nearest to the note four Fifths above C we must rise two octaves and a major third. Hence the frequency of this note E is 1 x 2/1 x 2/1 x 5/4 or 20/4. We find then that the interval between the untrue E and the true E is 81/16 -=-20/4 or 81/80, which is a Comma. The untrue E can be made to coincide with the true E by reducing each of the fifths by a quarter of a comma, and this flattened fifth is the basis of the Mean Tone system. Briefly stated, the scale is obtained by rising or falling repeatedly by two of the flattened fifths, and returning by a true octave, until a sufficient number of notes have been obtained within the compass of an octave. The main feature of the scale is that, , if only keys not far removed from C are used, such as G, F, Bt', D, the fifths are all a quarter of a comma flat, the major thirds are true and the minor thirds are a quarter of a comma flat. These divergences from true intonation are not very noticeable, and therefore the scale has a good effect in these keys. When however we modulate into keys such as Fjf or B, which are remote from 0, in that they require the use of many of the black keys of the pianoforte, the intonation is so untrue, that such keys are called " wolves," and cannot be used. The Temperament receives its name from the fact that each tone in the scale is the same, and is the mean of the major and minor tones. 275. Equal Temperament. Modern music demands free access to all keys, and therefore the Mean -Tone Tempera- ment, which permits of modulation into only a few keys, has given place to Equal Temperament. In this system the octave is divided into twelve equal intervals called Equal Tempera- ment Semitones. It is clear, then, that whatever may be the defects of the scale of C, they will be exactly the same in the key of Cjt or any other key, for to reach any key we raise each 248 DEFINITION OF INTERVALS [CH. XV note of the key of by the same number of equal semitones. Thus all the keys are equally good or equally bad, and modulation makes no change in the consonance. The Equal Temperament Semitone is an interval which gives an octave when added to itself twelve times. If therefore its vibration ratio is pjq, the fraction pjq multiplied by itself 12 times must make 2, or {p/qy^ = 2. This is easily solved by the use of logarithms, and it is found that p/q or 5^2 is a little less than 1'06, or expressed as ratio it is nearly 106:100. On the piano or any instrument tuned to Equal Tempera- ment, twelve fifths make seven octaves. In true intonation the vibration ratio for an interval made up of twelve fifths is {3/2y\ and that of an interval of seven octaves is (2/1)'. The latter of these is greater than the former in the ratio of 531441 to 524288, and therefore the Equal Temperament fifth is flatter than the true fifth by one twelfth of the interval defined by this ratio. Expressed in terms of the comma the Equal Temperament fifth is one eleventh of a comma flat. Similarly three Equal Temperament major thirds make an octave, whereas three true major thirds make the interval (5/4)' or 125/64, which is less than an octave. Hence the Equal Temperament major third is sharper than the true major third. The difference is 7/11 comma. The divergences of the Consonant Intervals in Equal Temperament from those in true intonation are shewn below. Octave True Minor Third ^Oor ama flat Major „ 7 11 „ sharp Fourth 1 11 » >j Fifth 1 11 flat Minor Sixth 7 11 >) >J Major „ 8 n J, sharp 275] SCALES. TEMPERAMENT 249 It is seen then that in present-daj' music no. intervals are true except the octaves ; and the thirds and sixths differ quite conspicuously from the true intervals. Hence according to Helmholtz's theory the adoption of Equal Temperament must have introduced into music dissonance which would not be present if true intonation were used. This conclusion deserves a little consideration. In the first place Equal Temperament tampers with just those intervals which can best stand it. The octave is very closely hedged in by harmonics, which beat strongly, if there is mistuning. Consequently, every Temperament is compelled to keep the octaves true. A mistuned octave would not be tolerated in music. The fifths aie the next in order of closeness of definition, and in Equal Temperament they are only 1/11 comma flat. The beats due to this mistuning are not so rapid as to be seriously unpleasant. It is the thirds and sixths which sufier most in Equal Temperament, and we saw that these intervals are only feebly defined, so that we can tolerate more mistuning in their case than we could in the case of the fourths and fifths. Helmholtz's Theory is a physical theory which aims at explaining the physical property of smoothness of concords, and does not necessarily arrange the concords in their order of desirability from an aesthetic point of view. The major third on his theory is much inferior to the fifth or octave ; yet musically it is more agreeable. The absence of a third in a chord makes it thin and uninteresting, and it is a general though not invariable rule in Harmony to include a third in every chord. Moreover, actual dissonances such as the major and minor ninth are often introduced with excellent effect. Thus we may conclude that smoothness is not the only desirable feature of a chord in music. Our appreciation of Tempered Intonation is no doubt largely a matter of education. As Donkin says, "the whole structure - of modern music is founded on the possibilit)' of educating the ear not merely to tolerate or ignore, but even in some degree to take pleasure in slight deviations from the perfection of the diatonic scale. " CHAPTER XVI MUSICAL INSTRUMENTS 276. Classification of Orchestral Instruments. We shall not attempt to give a complete account of the construction and use of musical instruments, but shall merely touch on such points as afford illustrations of the Acoustical Theories developed in the preceding chapters. The instruments in common use in the Orchestra may be divided into four main classes, Strings, Wood-Wind, Brass, and Percussion. We shall treat them in this order. The Strings may be again subdivided into (1) the Pianoforte and Harp, where the vibrations are produced by striking or plucking the strings and (2) the Violin class, where the strings are bowed. 277. The Pianoforte. In the pianoforte the strings are stretched on a wooden or metal frame, and a thin wooden sound-board is fixed to the frame. As has been said pre- viousl}', if the strings were fastened to a rigid frame without a sound-board, very little sound would be given out. The sound-board must not be regarded as a resonator. Its vibrations are forced and it owes its efficiency merely to its large surface. It has of course natural tones of its own, but the damping is so great that after a very few vibrations its natural vibrations become inappreciable, and nothing remains but the forced vibrations. Each note has from one to three strings. When there are more than one, the strings are tuned in unison with each other. We saw in Chapter III that the pitch of a string can be altered by changing its mass, length or tension and all three methods are employed in the pianoforte. The bass strings are several feet long, whilst those at the treble end 276-277] MusrcAL instruments 251 are onl}'- two or three inches long. Further, the bass strings are weighted by one or more layers of wire twisted round them. This is better than merely using a thicker string, as it does not interfere so much with the flexibility. The strings are made longer and heavier in the bass in order to equalize the tension. If the strings were of the same length and densitj' throughout, the tension in the treble would have to be much greater than in the bass. A pianoforte has generallj' a range of about 7 octaves. With this range the highest note has a frequency 128 times as great as the lowest, and if this were to be secured merely by difiference of tension the highest string would have 16,384 times the tension of the lowest. This is an impracticable range, for, even though the upper strings had such a tension that they were on the point of breaking, the lower strings would be so slack that they would give a very poor and weak note. The strings are struck with hammers covered with felt. We saw that, when a string is struck, no harmonic is present that requires a node at the point struck, and also that a string gives a more metallic tone when struck near the end than when struck near the middle. It has often been stated that pianoforte strings are struck one seventh of their length from the end in order to get rid of the seventh harmonic, which is the lowest that falls out of the musical scale. It is in practice seldom that a string is struck so far from its end as one seventh ; a more usual position is one eighth or one ninth from the end. The treble strings are struck still nearer the end, for their stiflfhess hinders the formation of the higher harmonics, and their quality would be different from that of the lower strings, if the production of the higher harmonics were not encouraged by the nearness of the point struck to the end. The quality of the note is also influenced by the shape and hardness of the hammer. A hard hammer and a narrow striking surface both favour the production of high harmonics ; consequently narrower hammers' are used for the treble in order to equalize the quality. Each maker adopts the point of striking and the kind of hammer that he has found by experience give the quality he 252 MUSICAL INSTRUMENTS [CH. XVI desires. The differences between the instruments of different makers depend largely on the choice of hammers and striking points. 278. The Harp. Much of what has been said of the pianoforte applies also to the harp. The bass strings are weighted, so that the tensions niaj' be equalized. The sound- board is much smaller than in the pianoforte and the sound is consequently weaker. The strings are usually plucked not far from their middle points and the note is therefore soft and somewhat dull from the weakness of the higher harmonics. *279. Motion of a Violin stringy. The Vibration Microscope. The Violin, Viola, Violoncello and Double Bass are alike in principle, differing merely in size. In all these instruments the string is made to vibrate by being bowed. The motion of a violin string under the action of the bow has been investigated experimentally by Helmholtz by the use of the Vibration Microscope. Briefly stated, the principle of the vibration microscope is as follows. A tuning- fork, whose pitch is the same as that of the string, has a lens fixed to one of its prongs, and is placed so that the vibrations of the lens and those of the string are in directions at right angles to each other. A grain of starch is fastened to the string. If now the fork is made to vibrate, the image of the grain will vibrate, and a person looking at the grain through the lens will see it drawn out into a short line. If the string is vibrating and the fork at rest, the grain will be seen drawn out into a line at right angles to the former. If both fork and string vibrate the two separate motions of the image are compounded with each other, and a Lissajous' Figure is seen. The vibrations of the lens are simple harmonic, those of the string are periodic but not simple harmonic. Consequently the figure will not resemble any of those shewn in Chapter II, since those figures were drawn for the case in which both vibrations are simple harmonic. The general form of the. figures seen by Helmholtz was as shewn by the curve at the top of Fig. 1 16. Here the fork is describing simple harmonic vibrations in a horizontal direction, and the string is vibrating in a vertical direction, the fork and string being in unison. It remains to deduce the relation 277-279] MUSICAL INSTRUMENTS 253 between the displacement of the string and the time. This is readily done, as the known mode of vibration of the fork provides us with a time scale in the horizontal direction. At a cer- tain moment, for instance, the displacement of the string at the point we are observing is ab up- wards, and the moment at which the displacement has that value can be determined from the displacement oa of the fork at the same moment. Draw a circle with its centre anywhere in the vertical line through 0, the centre of the Lissajous' Figure, and with a radius equal to the amplitude of vibration of the fork. Divide the circumference of the circle into any number of equal parts, and through each of the points of division draw a vertical line. Fig. 116 In the figure there are 16 sections, the points of division being numbered from 1 to 16. The point numbered 1 is for convenience chosen so as to lie vertically under one of the points where the Lissa,jous' Figure cuts the horizontal line through 0. The vertical lines will mark a scale of equal time intervals on the horizontal line through 0. The numbers on the curve shew the positions of the tracing point of light at the ends of successive equal intervals of time. Now take a straight line divided into 1 6 equal parts as in Fig. 117, and at each dividing point draw an ordinate equal to the ordinate of the Lissajous' Figure at the point with a corresponding number, and draw a smooth curve through the ends of the ordinates. The curve is found to consist of straight lines meeting at angles with each other. This cur^'e does not shew the shape of the string, but the displacement of one particular point of the string at different times. The tangent of the angle between the curve and the 254 MUSICAL INSTRUMENTS [CH. XVI axis at any point is the ratio of a change of displacement to the time interval in which that change takes plaCe, or is the velocity of the point of the string. Consequently a straight line represents uniform velocity, and we see that any point of the string moves with uniform velocity downwards, then changes its direction of motion suddenly and moves with uniform velocity upwards, and so on. The to and fro velocities are the same only at the centre of the string ; they Fig. 117 differ frona each other the more, the nearer the point observed is to the end of the string. Helmholtz found that at any moment the string takes the form of two straight lines meeting at an angle. The angle is not always at the same point of the string, but travels back and forwards along a flat curve, which passes through the ends of the string.' When the angle is travelling in one direction it is above the equilibrium position of the siring, and when travelling in the other direction it is below it. 280. Action of the Violin Bow. The action of the bow in exciting vibrations in the string depends on the difference between static and kinetic friction. When the string and bow are at rest relatively to each other, the friction is greater than when there is relative motion. The bow moves with uniform velocity, and carries the string forward with the same velocity. Presently the force of restitution becomes so great that the string breaks away from the bow, and Helmholtz's experiment shews that it returns also with uniform velocity. When it reaches the end of its swing and 279-282] MUSICAL instruments 255 stops, another part of the bow grips it, and carries it forward again, and so on. 281. Quality of the note of the Violin. It will be seen that the vibrations are not at all like simple harmonic vibrations. The string does not slow down gradually as it reaches the end of its swing, but changes suddenly from an outward uniform velocity to an inward uniform velocity. A vibration of this kind requires a large number of terms of the Fourier series to express it, and the note of a violin has therefore a large retinue of harmonics. Helmholtz considers that the cutting character of the note is due to the strength of the sixth to the tenth harmonics as compared with those of other instruments. The nature of the vibrations is not much aflFected by the position of the point that is bowed. 282. Production of the Scale on a Violin. The lower strings of a violin are heavier than the higher strings, for the reasons given when we spoke of the pianoforte. The strings of the violin like those of all other stringed instru- ments are tuned by alteration of their tension. They are tuned to the notes g, d\ a}, e', making fifths with each other, and it is to be noticed that if the fifths are true, it is not possible to play an equally tempered scale on the instrument making use of the open strings, for all the fifths on the tempered scale are 1/11 comma flat. The point has merely a theoretical interest, for 1/11 comma is too small an interval to be of practical consequence. The notes intermediate between those to which the strings are tuned are obtained by pressing the strings against the finger-board with the fingers, and so shortening the strings by the required amount. On some stringed instruments, such as the banjo, small raised strips of metal or ivory called Frets are fixed across the finger-board at the points to which the strings are to be shortened for the various notes of the scale. These frets make the instrument easier to play, but they have the disadvantage that if one of the strings is out of tune, all the notes produced from that string must be out of tune, whereas if one string is out of tune on the violin, the player can adjust the point of stopping, so as to bring all except the 256 MUSICAL INSTRUMENTS [CH. XVI open note into tune, and the note that should be given by the open string can be obtained from the string below. The peculiar shape of the body of the violin is beyond the reach of theory. It was arrived at by experience and has remained practically unchanged for 200 years. 283. The Wood-Wind Instruments. The Wood- Wind consists of two classes. The first class contains the Flutes and the second contains the Reed Instruments, such as the Hautboy, Clarinet and Eassoon. 284. The Flute. The Flute is made in various sizes, the best known varieties being the orchestral flute, the military flute and fife, and the piccolo. The piccolo is the instrument of the highest pitch used in music. Each of these instruments consists of a tube closed at one end. In the side of the tube near the closed end is a hole across which the player sends a sheet of air from his lips and so produces the sound, the vibrations being set up in the same way as in the flue pipes of an organ. The flute then is analogous to an open organ pipe, and gives the full series of harmonic overtones. Flutes were formerly made with the bore of the half of the tube nearest to the open end slightly conical, the narrowest part of the cone being at the open end. The conical bore is still used in military flutes, but orchestra] flutes are now always made with a cylindrical bore. The bore near the mouth hole is commonly slightly contracted in the form of a paraboloid. 285. The Finger Holes. In the half of the tube farthest from the mouth are six holes which are used for forming the scale. In the so-called eight-keyed concert flute these holes were covered with the fingers. In the Boehm flute and others of modern make they are covered by padded keys, which are pressed down with the fingers. When all the holes are closed, the flute gives the note of an open pipe whose length is the distance between the mouth-hole and the open end. If the holes were as large as the bore of the tube, they would reduce the effective length of the pipe to the distance between the mouth-hole and the highest hole left open, and the distances of the holes from the mouth would 282-285] MUSICAL instruments 257 have to be inversely proportional to the frequencies of the notes of the scale. It is not practicable to make the holes so large as this. If they are above a certain size they are not easily covered, and the notes produced are unmanageable. In the case of the eight-keyed concert flute the holes were much smaller than the bore. When the holes are small the notes of the instrument are weak, and it is largely for this reason that in modern flutes the holes are covered by keys, and can therefore be made larger than is possible when they are covered by the fingers. It is claimed also for the Boehm and other similar flutes that the holes can be placed more nearly in their theoretically correct positions, as the fingers need not be directly over the holes, but can open and close them by means of levers. A simple experiment will shew that the part of the tube below the highest hole open is not without effect on the pitch of the note. The experiment can be made with a tin whistle. Cover the two highest holes and blow the whistle. Now cover also the three lowest. The highest hole open is the same as before, yet the pitch is lowered a little. Thus it appears that the tube is not completely cut off at the highest open hole. A warning should be given that the experiment will not generally succeed if the highest hole is left open and the remaining five alternately opened and closed. In this case closing the lower holes can with great care in blowing be made to lower the pitch, but it will generally raise it a semitone. . When all the holes are open, the pipe between the mouth and the highest hole gives its fundamental. When all the holes except the highest are closed, we shall almost certainly get the first overtone of the full length of the pipe, which is a semitone above the note given out when all the holes are open. The first overtone has an antinode in the middle of the pipe, and a node a quarter of the length from each end. If all the holes are open, we cannot have a node in the lower half of the pipe. The only possible position for a node is in the closed upper half. When all the holes but the highest are closed, it is almost impossible to prevent the formation of an antinode at the middle of the pipe, since that point is connected with the open air through the highest hole. As it is now possible for a node to form in each half of the pipe, the first overtone of the whole pipe is produced. c. e, 17 258 MUSICAL INSTRUMENTS [CH. XVI 286. The Flute regarded as a Resonator. As a first approximation we can regard the flute as a pipe whose length can be varied, but this will not explain the whole of the details of the construction. The holes are not arranged so that their distances from the mouthpiece are inversely proportional to the frequencies of the notes produced, and they are not always all of the same size. We can get a step farther in the explanation by looking on the flute as to some extent analogous to a Helmholtz Resonator, whose pitch can be raised by enlarging the opening. Uncovering the holes of a flute is equivalent to enlarging the opening of a resonator, and the larger a hole is, the greater is its effect in raising the pitch. The part of the tube in which the holes are open is not quite freely open to the air, and is therefore not quite without effect on the pitch of the note given out. This explains why it is not a matter of indifference whether the holes below the highest open hole are open or closed. Closing them reduces the connexion of the whole interior with the open air, and so lowers the note. It is similar in effect to shading the open end of an open organ pipe. We can now understand why the holes maj', within limits, be made in any positions convenient for the fingering. If when so arranged they do not give a true scale, they can be altered in size so as to correct the errors. With merely six holes as described, the flute will give a diatonic scale extending over about three octaves. For the lowest octave the fundamental of the pipe is used, for the second octave the first overtone an octave higher is used, and for the third octave the overtone two octaves above the fundamental is used. The semitones intermediate between the notes of the scale are produced by means of holes covered by keys, which can be opened when required. An open hole prevents the formation of a node in its neighbourhood, but favours the formation of an antinode. Cross fingering is an application of this principle to the production of certain high overtones. The holes near the points where nodes are situated in the particular form of vibration required are closed, whilst the holes near the antinodes are left open. 286-288] MUSICAL instruments 259 The note of the flute is almost free from the higher harmonics. The octave is faintly audible, but no others. Consequently the note has a smooth quality which contrasts well with that of the violins and reed instruments. 287. The Ocarina. In an earlier chapter we men- tioned an instrument called the Ocarina. It is not used in the orchestra, but is worth a short description, as it is a simple resonator in principle, with none of the characteristics of a pipe. The Ocarina is a hollow pear-shaped instrument generally made of metal. ^ is a flat tube by which a sheet of air is Kg. 118 blown across a hole not seen in the figure, and the instrument made to give out a note. On the front are eight holes which can be covered by the fingers, and on the back are two holes for the thumbs. The scale is produced by uncovering the holes one after the other. The interesting point about the insti'ument is that the positions of the holes are of no con- sequence. All that matters is their size or more strictly their conductivity. Choose two holes of the same size, and open first one and then the other. They will be found to have the same effect in raising the pitch wherever they are situated. The instrument has not a vibrating column of air with nodes and antinodes, but merely a mass of air which is alternately compressed and rarefied, and the pitch depends only on the volume of the air and the total conductivity of the openings. 288. The Clarinet. We come next to the reed instruments, comprising the Clarinet, Hautboy and Bassoon, 17—2 260 MUSICAL INSTRUMENTS [CH. XVI with some others in less general use, such as the Basset BLom, Cor Anglais, Double Bassoon, Saxophone, etc. The Clarinet has a cylindrical tube spreading out into a small bell at one end, and closed at the other end by a single reed of cane, beating on a rectangular opening in the side of the mouthpiece. We have already explained the action of a reed in Chapter X, and have shewn that the reed end of a pipe is to be treated as a closed end. The clarinet therefore, being a closed cylindrical pipe, gives only the odd harmonic overtones 1, 3, 5, etc. When overblown its note rises a twelfth, unlike that of the flute which rises an octave. The notes of the scale are produced by opening in turn a series of holes in the side of the tube, but as it is necessary to cover a range of a twelfth before beginning again with the first overtone, keys are provided for giving a few notes below and above those obtained by the use of the holes. A clarinet in Bt' gives the note F when all the finger holes are closed, but E, EP, and D can be got by the use of keys, which cover other holes below the lowest of the finger holes. Similarly when all the finger holes are open the note given out is F an octave above the former, and keys above the highest hole enable the player to produce Fjf, G and Gjp. Thus by the use of the keys and holes the tube, whilst always sounding its fundamental, can be altered in length so as to give a scale extending over a semitone less than a twelfth. Now return to the fingering used for the lowest D, but make the tube give its first overtone, and we get the note A a twelfth higher. We then get by repeating the former fingering a scale in which all the notes are a twelfth higher than in the lower register. Throughout the upper register a key called the Speaker Key is used to facilitate the formation of the first overtone. In the lower register there is a node at the reed and an antinode somewhere near the highest hole that is open. In the higher register there is a second node between the antinode at the highest open hole and the reed, and a second antinode between this second node and the reed. The speaker key opens a small hole near this second antinode and encourages its formation. As a single speaker key has to serve throughout the register, it cannot be exactly at the antinode for every 288^290] MUSICAL instruments 261 note of the register, but it is sufficiently near to ensure the production of the first overtone of the tube. Most of what was said of the production of the scale on the flute applies also to the clarinet. The intermediate semitones are provided by means of additional keys, and cross fingering is employed for the highest notes for reasons already explained. As the proper tones of the clarinet form the odd series of harmonics, none but the odd harmonics are conspicuous in its note, and this is the main cause of its characteristic quality. The even harmonics are not quite absent, for the periodic current of air from the reed contains the whole series, and the even members set up weak forced vibrations in the column of air. The pitch of the clarinet depends almost entirely on the length of the tube. The reed has a natural frequency de- termined by its mass and elasticity, but its mass is so slight that it is constrained to vibrate with the frequency proper to the column of air. Clarinets are made in a variety of pitches. Those in common use in the Orchestra are in A, BP, C. The Basset Horn, a fourth below the Bi' Clarinet, and the Bass Clarinet, an octave below the Bb, are also sometimes used. 289. The Saxophone. The Saxophone is a brass instrument resembling the Clarinet in the form of its reed, but it has a conical tube, and therefore has the full series of harmonic overtones. Its fingering is similar to that of the Flute. It is seldom heard in this country, but is in common use in military bands on the Continent. 290. The Hautboy. The Hautboy has a conical bore, and the sound is produced by a double reed. Two thin pieces of cane slightly curved are bound together with their concave faces towards each other, so that there is a narrow lenticular opening between their edges, when they are at rest. When they are made to vibrate by the pressure of the air in the player's mouth, the lenticular space alternately opens and closes, thus admitting pufis of air into the tube. The reed is placed at the vertex of the cone formed by the bore of the 262 MUSICAL INSTRUMENTS [CH. XVI tube, the base of the cone being at the open end of the instrument. In consequence of its conical bore the hautboy gives the full harmonic series of overtones. Hence the arrange- ment of the holes and keys and the fingering are the same as for the flute. The notes of the hautboy are very penetrating in quality on account of the strength of their higher harmonics. 291. The Bassoon. The Bassoon is merely a Bass Hautboy. Its length is so great that for the convenience of the performer it is doubled on itself, and the reed is placed at the end of a short side tube. In consequence of the great length of the tube the holes are bored obliquely through the wood, so that whilst their outer ends are close enough together to be easily reached with the fingers, their inner ends are widely enough separated to give the notes required. In other respects the bassoon resembles the hautboy. The reed is double and the bore of the tube is conical. The overtones form the full harmonic series. 292. Tuning the Wood-Wind. The range within which it is possible to alter the pitch of an instrument of the wood-wind class to bring it to the pitch of a pianoforte or other instrument is very limited. The pitch of a flute, for instance, can be lowered by drawing out the joints a little, but this puts the notes of the scale out of tune with each other. The spacing of the holes is arranged to give the proper intervals when the tube has its normal length with all the joints pushed close. When the pitch is lowered, the section between every two holes should be lengthened in the same proportion as the whole tube is lengthened, if the relative dimensions of the instrument and the relative pitches of the notes are to remain unchanged. When the flute is flattened by drawing out the head joint, it is plain that the holes will be a little too close together to give the correct intervals with the increased length. There is no difficulty in tuning the strings and brass by any amount that is likely to be needed and therefore it is usual to tune the orchestra to the wood-wind instruments, when the pitch is not fixed by the inclusion of an organ or pianoforte. The hautboy is generally chosen for this purpose, 290-293] MUSICAL INSTRUMENTS 263 but probably the clarinet would be better. The flute and bassoon are what may be termed flexible instruments, that is to say, their notes can be modified a little in pitch by the player, and so brought into tune. The flute player adjusts the pitch by covering the mouth hole to a greater or less extent with his lips, and the bassoon player by varying the pressure on the reed. The hautboy and clarinet are less flexible, and perhaps the clarinet is the less flexible of the two. Consequently these instruments suffer most in intonation when their joints are drawn out, and it is for this reason that one of them is chosen to give the pitch to the rest of the orchestra. 293. Standards of Pitch. There are two Standard Pitches in use at the present time. Up to the year 1896 the High or Philharmonic Pitch had been in use for many years. This had a vibration number 452'4 for the note A at a temperature of 60° F. In 1896 the Philharmonic Society changed their standard to what is known as the Low Pitch, and has a vibration number 439 at 68° F. In specifying a standard of pitch it is necessary to specify the temperature, for the standard is fixed mainly for the guidance of the instrument makers, and instruments vary in pitch with variation of temperature. In fixing the Low Pitch as stated above, the intention is that the makers should construct the instruments in such a way that at a temperature 68° F. the note A shall ha'ste a vibration number 439. At any other temperature the instruments will not only have a different pitch but will also differ from each other, as different instru- ments vary in different ways with change of temperature, and they will therefore have to be brought into tune with each other by the use of the tuning appliances appropriate to the various instruments. The temperature was fixed at 68° F. as this is an average temperature for a concert room, and so entails the smallest amount of adjustment of the instruments. The Low Pitch has been adopted by all the great London Orchestras, but the High Pitch is still used in military bands and in most provincial orchestras. The difference between the two pitches ' is not great, yet it is enough to make it impossible to use the same clarinet or hautboy for both. Con- sequently, players who are in the habit of using both pitches 264 MUSICAL INSTRUMENTS [CH. XVI are compelled to have two instruments, one for each pitch. If it were not for this difficulty, it is probable that the Low Pitch would now be universal in this country. Unfortunately the military bands have retained the High Pitch, as the expense of providing new instruments of Low Pitch is too great. As provincial orchestras are often dependent on the local military band for filling the gaps in their ranks, it would appear that we shall not have one standard of pitch in orchestras, until the military authorities can face the expense of providing new instruments. 294. The Brass Instruments. The Brass Instru- ments all consist of metal tubes of a more or less conical bore provided at the narrow end with a cup-shaped mouthpiece. The lips of the player are placed against the mouthpiece and by their vibrations produce the sound in the same way as the voice is produced by the vocal chords. In all these instruments the aim of the maker is to provide the full series of harmonic overtones. The overtones form the basis of the scale as in the case of the wood-wind, but much higher members of the series are used. The horns and trumpets, for instance, go as high as the sixteenth harmonic. 295. Shape of the Brass Instruments. Though the bore is in general conical, it is by no means so regular a cone as the bore of a hautboy or a conical organ pipe. The bugle is a fairly regular cone, but ■ the trombone is cylindrical for two-thirds of its length, and spreads only in the lowest third. All the instruments widen rapidly at the open end to form a bell. The positions of the nodes and the pitches of the overtones of such tubes cannot be calculated theoretically, and the makers have evolved by experiment the shapes that are found to give overtones in accordance with the harmonic series. Their efforts are not always successful, and the difference between a good instru- ment and a bad one is largely a difference in the accuracy of the pitch of the overtones. The nodes vary in position according to the overtone that is sounded. If there is a constriction in the tube at some point, any overtone which requires a node at that point is a little sharp, and any over- tone which requires an antinode there is flat. A bad bruise 293-298] MUSICAL instruments 265 in the tube may therefore have the effect of putting some of the notes out of tune. 296. Effect of shape on Fitch and Quality. As the shape of the bore of a brass instrument is not amenable to theoretical treatment, we can only mention a few experi- mental conclusions bearing on the relation of the shape to the quality and pitch of the notes that can be produced. When the tube is wide relatively to its length, as in the euphonium, tuba, and other instruments of the Saxhorn group, the lower members of the harmonic series including the fundamental are easily produced, and are of good quality. "When the bore is narrow, as in the horn and trumpet, the fundamental is difficult or impossible to blow, but the higher members are easy. The best range of the horn is from about the fourth to the twelfth harmonic. A wide spreading bell, as in the horn, makes the tone smooth. A small bell, as in the trombone, conduces to a brighter quality. Widening the bell beyond the size for which the tangent at the edge makes an angle of about 45° with the axis has no eifect on the pitch, though it continues to make the notes smoother in quality. The shape of the mouthpiece has a great effect on the quality of the notes. A shallow cup-shaped mouthpiece like that of the trombone gives a bright tone. A deep conical mouthpiece narrowing gradually from the rim, like that used with the horn, gives a smoother tone. 297. The Bugle. Some instruments, such as the Bugle, Post Horn, and the French Cor de Chasse, have no other notes than those of the harmonic series. All the military Bugle Calls are formed from the following notes ; Some higher notes are possible but difficult. The Regulation Bugle is in Bt', and therefore the actual sounds are all a tone lower than those shewn. 298. The Cor de Chasse and Hand Horn. The Cor de Chasse is similarly limited, but, the tube being long 266 MUSICAL INSTRUMENTS [CH. XVI and narrow, the higher tones are more easily produced, and, as the harmonics lie closer together the higher we ascend in the series, the melodic possibilities of the Cor de Chasse are greater than those of the bugle. The horn used in orchestras or, as it is often called, the French Horn, had until recently no mechanism for filling the gaps in the harmonic series. Its scale of open notes was therefore limited in the same way as that of the Gor de Chasse. The performer, however, placed his hand inside the bell, and by closing the opening to a greater or less extent could flatten each of the notes by a tone or more, and so produce a complete chromatic scale over a range of about two octaves from the third harmonic upwards. The notes so flattened were called stopped notes, and were greatly inferior in quality to the open notes. The horn was provided with a set of lengthening pieces or " crooks " by which it could be put into the key of the music to be played, the number of stopped notes required being thereby reduced. Horns of this kind are called Hand Horns. Horns are now always pro- vided with valves, by means of which a complete chromatic scale is obtained over the whole range of the instrument without the use of stopped notes, as will be explained in § 301. 299. The Ophicleide Class. The earliest method in general use for filling the gaps between the successive harmonics was similar to that used in the wood-wind in- struments. Holes, sometimes covered by keys, were made in the side of the tube, and by opening these in turn a scale was produced in the same way as on the flute. The once popular Key Bugle was an instrument of this class, as was also the Ophicleide. Their chief defect was the inequality in the notes. A note that was produced with all the holes closed had the advantage of the softening effect of the bell, whilst for the other notes the vibrations did not extend to the bell, and the quality was less satisfactory. The last survivor of the class was the Ophicleide, which has now been superseded by the Tuba. 300. The Trombone. The second method of forming the scale is by the use of a slide, as used on the Trombone. There are three Trombones in general use, the Alto Trombone 298-300] MUSICAL INSTRUMENTS 267 in Eb, the Tenor in Bt', and the Bass in G. They diflfer only in size and a description of the Tenor will serve for the three. About two-thirds of the tube nearest to the mouthpiece is cylindrical, and the remaining third is conical. The cylin- drical part is bent in the middle so that its two halves are parallel to each other and is made double, the outer part sliding telescopically over the inner. By drawing out the outer part the tube is lengthened in the same manner as the right-hand branch of the interference tube shewn in Pig. 57. When the slide is closed, the Tenor Trombone gives the note Bb as its fundamental. If the slide is drawn out a little way, the fundamental is lowered to A, if a little farther still to A'', and so on, until with the greatest extension possible EH is reached. Thus there are seven positions of the slide giving all the semitones from B^ down to Etf. In each of these positions we can produce not only the fundamental but also its harmonic overtones, and so we have a harmonic series on each of seven consecutive notes at intervals of a semitone. The range of the trombone in practice extends upwards as far as the eighth harmonic. An expert player can produce a few higher notes, but they are seldom demanded by composers. In the following table are shewn all the notes that can be produced in the various positions of the slide. The fundamental is difficult to blow in any but the first three positions, and is therefore omitted in the rest. The seventh harmonic is omitted throughout, as it does not coincide with any note of the scale with the fundamental as tonic. Position 12 3 4 5 6 7 Fig. 119 268 MUSICAL INSTRUMENTS , [CH. XVI It will be seen that from Etf, the second harmonic in the seventh position to B^, the eighth harmonic in the. first position, we have a complete chromatic scale, and three lower notes in addition. The slide Jias enabled us to bridge com- pletely all the gaps above the second harmonic in the first position, and to bridge the upper half of the gap between the first and second harmonics. Some of the higher notes can be obtained in more than one position of the slide, and this is an advantage to the player. He -chooses such positions as require the least movement of the slide in passing from note to note. The trombones, like the violins, can be played in true intonation, since the pitch of each note is under the control of the player. 301. Valved Instruments. The third method of producing the scale on brass instruments is by the use of valves. The valve is a piston which, on being pressed down by the player, lowers the pitch of the instrument by throwing in an extra length of tube. There are generally three valves, the first of which lowers the pitch of the instrurnent two semir tones, the second lowers it one semitone, and the third lowers it three semitones. As the valves can be used separately or together, they enable the player to lower the pitch by any number of semitones up to six, and thus answer the same purpose as the slide of a trombone. If we have an instrument in BtJ provided with three valves. Fig. 119 can be made to represent the notes obtainable with the various combinations of valves in the following way : Position 1 corresponds to no valve „ 2 „ valve 2 1 1 )> " >j jj ■■■ „ 4 „ „ 3 or 1+2 „ 5 „ valves 2 + 3 „ 6 „ „ 1 + 3 7 „ ,,1 + 2 + 3. Here again the scale is complete from a note six semitones below the second harmonic. This is generally all that is needed, for on most brass instruments the fundamental is not 300-301] MUSICAL INSTRUMENTS 269 used. The tuba and some other bass instruments of the Saxhorn class form exceptions. Their fundamental is of good quality, and these instruments have therefore a fourth valve, which lowers the pitch by a fourth or five semitones. With the help of this valve the pitch can be lowered by any number of semitones up to eleven, and therefore the gap between the fundamental and its octave can be bridged. The introduction of valves has greatly increased the facility with which brass instruments can be played. Rapid passages can be played on them which would be quite im- possible on a slide. They suffer however from the defect that, whenever two or more valves are used together, the note produced is a little sharp. The first valve, for instance, is tuned to lower the pitch a tone when used alone. This requires the length of the tube to be increased in the ratio of 8 to 9, or the valve must add to the tube one eighth of its length. Similarly the second valve used alone adds about one fifteenth to the length. If now the two valves are used together, the first increases the length by one eighth, but the second increases it by only one fifteenth of its original length, which is less than one fifteenth of its length when already increased by the first valve. Thus, thoiigh the two valves used separately give a true tone and semitone respectively, the two together give a lowering of pitch less than three semitones. In most cases the player corrects the error as well as he can with his lips. The error is not very great with three valves, and it is possible to force the note down a little by relaxing the pressure of the lips. The horn player is in a better position, for he can flatten the note by closing the tube a little more with his hand, whenever he uses two valves at once. The error is more serious in instruments with four valves, and in their case compensating valves are often used to correct it. Brass instruments can often be put into several keys by the use of lengthening pieces or crooks. The Cornet, for instance, is generally in B^, that is to say, it gives the note BI? and its harmonics when no valves are used, but it is sometimes put into the key of A, by the insertion of a piece of tube between the mouthpiece and the instrument. If the valve tubes are adjusted tp add the right lengths to the Bt' 270 MUSICAL INSTRUMENTS [CH. XVI cornet, they will be too short for the A cornet, and must be readjusted, when a change is made from BP to A. Each of the tubes has a slide like that of a trombone, and it is with these slides that the adjustment is made. All brass instruments used at the present day have valves with the exception of the trombones. Attempts have been made to introduce valves on these also, but without success — at least in this country. The solemn and dignified quality of tone of the trombone is not suited to rapid passages, and the gain in facility by the addition of valves does not make up for the loss in purity of intonation. Valve trombones are however common in Continental military bands. 302. The Trumpet. The Trumpet is the Treble representative of the trombone family. It had formerly a short slide by which the pitch could be lowered one or two semitones. The slide has now gone out of use, and its place has been taken by valves. The Trumpet is normally in F and has crooks with which it can be put into any key down to Bt'. Its tube is narrow in proportion to its length, and consequently it can be made to give the higher -members of the harmonic series. The range within which it is most commonly used in modern music is from the second to the twelfth members of the harmonic series. The cornet is often admitted into the orchestra as a substitute for the trumpet, and it will be of interest to compare the two instruments. 303. Comparison of the Trumpet and Comet. The Trumpet when lowered by a crook to B^ has a tube about 9 ft. long, and its harmonics can be used as high as the twelfth. The Cornet in Bt? has a wider tube than the Trumpet, and is half the length. In consequence of its wider bore the harmonics above the sixth are difficult to produce. It follows that' the pitch of the highest practicable note is about the same on the two instru- ments. The actual sounds of the open notes which can be easily pro- fc SE ~CP~ ->n- m. -C3- ^ duced are shewn in Fig. 120, the Fig. 120 301-303] MUSICAL INSTRUMENTS 271 first series being the notes of the trumpet and the second those of the cornet. The fundamental is omitted in each case, as it is not used. The seventh and eleventh are enclosed in brackets, as they are out of tune. Both instruments give with the help of their valves a complete chromatic scale over a range of two octaves' or more, but there is a great difference in the certainty with which the notes can be produced. The player presses his lips more tightly together the higher he wishes to rise in the scale, and in order to produce a given note the pressure has to be adjusted to suit that note. Suppose the note Bf in the middle of the treble clef is to be sounded. The trumpet player has an open note a tone above the Bt', and another rather more than a tone below, and if he adjusts the pressure a little wrong, he gets C or Ab, when BI? is wanted. The Bt? of the Cornet is not so closely hedged in by other open notes, the nearest being a major third above and a fourth below. Consequently, the cornet player does not need the same accuracy of pressure as the trumpet player, and is not so likely to blow a wrong note. This is the main reason why an inferior cornet player is less likely to cause a catastrophe than an inferior trumpet player. The note of the trumpet is so brilliant and piercing, that, even when played softly, it is easily heard through the rest of the Orchestra, and, if the player makes a mistake, everyone hears it. In the past few years instruments with the same length of tube as the cornet but the shape of the trumpet have been gradually making their way into the orchestra, and they have almost displaced the orchestral trumpet in F. These new trumpets in BP have much of the certainty of the cornet and the brilliancy of the trumpet. The Trumpet parts written by Bach and Handel are so high that they are almost impossible on modern trumpets, and a form of instrument called the Bach Trumpet has been devised specially for these jiarts. The Bach Trumpet is merely a straight Coach Horn with two or three valves added. The French Horn presents the same difficulties as the trumpet and for the same reason. Its best range is from the fourth to the twelfth harmonic, where the open notes lie 272 MUSICAL INSTRUMENTS [CH. XVI close to each other. The bubbling sound' made by the horn players when feeling about for their note at the beginning of a phrase is a Hot unfamiliar sound in the Orchestra. 304. The Saxhorn Class. The Cornet is the smallest member of a large family of instruments named Saxhorns from their inventor Saxe. The other members are the Tenor Saxhorn, the Baritone, the Euphonium, and the Tuba or Eb Bombardon. The Tuba is the only member regularly used in the Orchestra. The Cornet and Euphonium are sometimes used, the former as a rule only when a Trumpet cannot be obtained. The Saxhorns are essentially Military Band instruments. They are wide in bore, and therefore in the larger forms the fundamentals are easily blown, and are of good quality. From the facility with which the notes are produced they are capable of greater execution than any other brass instruments. 305. Tuning the Brass Instruments. The brass instruments are affected by temperature in the same way as are the flue pipes of an organ, but an additional com- plication is introduced by the breath and hands of the player. A small instrument such as the cornet is soon warmed throughout its length to nearly the temperature of the breath and is not greatly afTected by changes in the temperature of the surrounding air. A large instrument such as the tuba contains such a great volume of air that it is not much affected by the breath of the player, and so is free to respond to chauges in the temperature of the room. All the brass instruments have a short telescopic tuning slide, like the slide of a trombone, by which their pitch can be varied to bring them into tune with the rest of the orchestra. If the pitch were lowered considerably by the use of the slide, it would be necessary to adjust also the tuning slide of the valves, in order that the additional length of tube thrown in by any one valve might continue to bear its right proportion to the total length of the instrument. In practice this is not necessary, for any small defect in intonation caused by the use of the tuning slide is easily corrected by a slight alteration in the pressure of the lips on the mouthpiece. 303-306] MUSICAL INSTRUMENTS 273 306. The Drums. The remaining class of instruments, the percussion instruments, present few features of interest that have not been already referred to in the preceding chapters. With the exception of the Kettle Drums they are not tuned to any particular notes, and are merely used to mark the rhythm. The Kettle Drum consists of a hemi- spherical shell of metal with a skin of parchment stretched over its open end. The tension of the parchment can be varied by a number of screws distributed round the rim, and thus the pitch of the note can be altered. The practicable range of pitch for any one drum is a fifth. If the parchment is too slack the tone is bad, and if it is too tight there is a risk of its being torn. It is most usual to have two drums tuned to the tonic and dominant of the music, but other numbers of drums and other notes are often used. 18 QUESTIONS Many of the following questions are taken from Examination Papers set in the Universities of Cambridge, London and Dublin and in the National University of Ireland. Questions below the line in each section are somewhat more difficult than those above the line. CHAPTER I 1. Can sound be propagated through a vacuum ? Describe an experiment bearing on this question. 2. What characteristics distinguish a musical sound from a noise 1 3. Describe an experiment from which you conclude that sound is due to the vibrations of the sounding body. Do all vibrations give rise to audible sounds ? Illustrate your answer by examples. 4. On what characteristics of an air-wave do the intensity, pitch and quality respectively of the corresponding sound depend ? 5. What experimental evidence leads to the belief that sound is propagated by wave motion ? 6. How would you prove experimentally that the musical interval between two notes can be measured by the ratio of the vibration numbers of the two notes ? 7. Explain how to use the Disc Siren with circles of 40, 50, 60 and 80 holes respectively to find the vibration ratio of a major sixth, assuming that the interval between the note G and the note e above it is a major sixth. 8. Shew that the measure of the difference between two intervals is obtained by dividing the vibration ratio of the larger interval by the vibration ratio of the smaller. 9. By what interval do two intervals whose vibration ratios are f and | differ from each other ? What are these intervals and what is the interval got by adding them together ? 10. Shew that if the interval between two notes is measured by the logarithm of the ratio of the frequencies of the two notes, the sum of the measures of two intervals will be the measure of the sum of the intervals. QUESTIONS 275 CHAPTER II 1. What is meant by a perfectly elastic body and by limits of elasticity ? Distinguish between a soft solid and a very viscous liquid. 2. State clearly the properties a medium must have in order that it may serve as a sound carrier. 3. State Hooke's Law and mention a few cases in which the law holds. Does it hold for gases ? 4. What is meant by a Simple Harmonic Vibration ? How is the displacement related to the acceleration in such a vibration ? 5. What is meant by Isochronism ? Give instances of iso- chronous vibrations. 6. Shew that in the ease of the Simple Harmonic Vibration of a small mass the potential energy is proportional to the square of the displacement of the mass from its equilibrium position. 7. Shew that if a radius of a circle rotates with uniform angular velocity, the foot of the perpendicular from its end on any fixed diameter will perform simple harmonic vibrations. 8. Prove that a mass supported at the end of a spiral spring will execute Simple Harmonic Vibrations when slightly displaced. Calculate the periodic time on the assumption that the mass of the spring may be neglected. 9. Explain how the variations of the displacement and the velocity respectively with the time for a body executing simple harmonic vibrations may be represented by Sine Curves. What is the relation of the two curves to each other as regards their phases ? 10. Define the terms period, amplitude and phase of a vibrating particle. Shew how the difference between the phases of two particles vibrating with the same period can be expressed by an angle. 11. Prove that in the case of simple harmonic motion the period of vibration is equal to 2ir divided by the square root of the acceleration of the vibrating body when it has unit displacement. 12. Find the period of vibration of a mass of 1 kgr. attached to a spiral spring of such stiffness that an extra load of 15 gm. produces an extension of 1 cm. 13. Describe and explain a method of adjusting the pitch of a tuning-fork. 18—2 276 QUESTIONS 14. Describe the changes that take place in the curve traced out by a point which has simultaneously two simple harmonic vibrations in directions at right angles, the periods of the two vibrations being nearly, but not quite, equal. 15. A particle vibrates harmonically with a period of 2 sec. Find its amplitude if its maximum velocity is 10 cm. per sec. 16. Find the total energy of the particle in Example 15, assuming its mass to be 20 gm. 17. A string two feet long has its ends fixed at two points one foot apart and in the same horizontal line. Another string is attached to the middle point of the first string and has a bob at its lower end. Shew that when this second string has certain definite lengths' the bob will describe Lissajous' Figures when set swinging. Find the length of the second string when the bob is capable of describing the 2 : 1 figure. CHAPTER III 1. Shew that when a train of transverse waves is passing along a series of particles the time taken by the waves to travel one wave-length is equal to the period of vibration of a particle. 2. From the relation in Question 1 prove the equation v=nK. 3. Shew graphically that when two similar trains of waves travel in opposite directions along a string, stationary vibrations are produced. 4. Construct a diagram shewing the result of compounding two trains of waves one of which has double the wave-length and double the amplitude of the other. 5. Assuming that the velocity of waves on a stretched string is s^Tjp, find the period of vibration of a string when givihg out its fundamental note. 6. How would you cause a stretched wire to emit its difiForent harmonic overtones ? What relations have these overtones to the fundamental of the wire ? 7. How may the velocity of a wave in a string be deduced from a knowledge of the frequency of the vibrations and the positions of the nodes ? 8. Explain how two bridges should be placed in order to divide a stretched string 100 cm. long into three segments whose fundamental frequencies are in the ratio 1:2:3. QUESTIONS 277 9. Explain how you would use the monochord to prove that the frequency of vibration of a stretched string is proportional to ^IT. A certain string has a frequency 100. Find its frequency when both its tension and length are doubled. 10. Two strings of the same material and 12 and 15 in. in length respectively are stretched over a sounding board. If the tensions of the strings are produced by weights of 64 lbs. and 36 lbs. respectively, what will be the interval between the notes produced when the strings are plucked ? H. Two strings of the same length give the same note, but the tension of one is double that of the other. Compare the masses of the strings. 12. Two wires one of aluminium and the other of steel, identical in shape and volume, are subjected to equal tensions. What will be the ratio of the frequencies of the notes emitted when each wire is sounding its fundamental ? The specific gravity of aluminium is 2-65 and that of steel is 7-8. 13. Prove that the velocity of the waves in a string is •JTjp. 14. What is the frequency of a string whose length is 100 cm. and whose weight is 1 gm., when stretched by a weight of 20 kgm. ? 15. If an addition of 25 lbs. to the tension of a string raises its pitch a fifth, what was the original tension ? 16. Shew by a method similar to that of § 53 that the velocity of a particle at any moment is given by the equation „ 2jraw 257 , ^ , F= — r — cos ^ {vt - X) . A A CHAPTER IV 1. What do you understand by longitudinal and transverse waves ? A number of particles are arranged in a straight line at equal distances apart. Shew on a diagram the relative positions of the pai-ticles at a particular instant when (a) a longitudinal wave and (6) a transverse wave is passing along the row of particles. 2. Shew how the displacement at any instant of successive layers of air through which sound-waves are being propagated may be graphically represented. Indicate on your diagram the layers having maximum and minimum velocity respectively, and those at maximum and minimum pressure respectively. 3. Sound travels with a speed of 1120 ft. per sec. at 60° F. What are the wave-lengths of notes with frequencies 32 and 256 ? What is the frequency of a note whose wave-length is 1 in. 1 278 QUESTIONS 4. state the law of variation of the intensity of sound at a point with the distance of the point from the source of sound, giving a general explanation of the cause of the change of intensity. Does the wave-length or the amplitude differ at dift'erent distances from the source ? 5. State and shew by diagrams the way in which condensation is related to displacement in a progressive wave and in a stationary vibration in air. 6. Describe the motion of the air in two adjoining segments of a train of stationary vibrations. 7. What is a wave-front ? How is it related to the direction of propagation of the sound ? 8. State generally the nature of the reflection of sound at the closed and open end of a pipe respectively, and give the reason for the difiference. 9. Give a general account of the distribution of energy in (a) a progressive wave and (6) a stationary vibration. 10. Shew that if sound travels along a tube which has a sudden change of bore at some point, there wUl be reflection of the sound at that point. 11. Shew that if curves similar to those of Fig. 41 be drawn for waves travelling towards the left, the convention as to the meaning of upward and downward ordinates remaining the same, the velocity curve will not occupy the position shewn in Fig. 41. CHAPTER V 1. When a regiment of soldiers is marching behind a band, it is seen that the men are not all in step with each other but a kind of wave travels backwards along the column. Why is this ? 2. A band is playing at the head of a procession 1080 ft. long, and the men step 128 paces to the minute exactly in time with the music as they hear it. Those in the rear are exactly in step with those in front. What is the velocity of sound ? 3. Devise an experiment to shew that the velocity of sound in air varies with the temperature. 4. At what temperature is the velocity of sound in air 1200 ft. per sec, assuming that at 0° C. its velocity is 1090 ft. per sec. ? 5. How could you shew experimentally that the velocity of sound in air is independent of the pressure ? QUESTIONS 279 6. Find the wave-length in hydrogen of a note whose frequency is 200. Assiime that the velocity of sound in air is 1100 ft. per sec. and that the density of air is 14'4 times that of hydrogen. 7. A sound generated in water has a wave-length of 580 cm. in water. If the velocity of sound in water is 145,000 cm. per sec. and in air 34,000 cm. per sec, find the frequency and wave-length of the note heard by an observer in air. 8. Explain the nature of the error made by Newton in cal- culating the velocity of sound, and the nature of Laplace's correc- tion. 9. The temperature of air through which sound-waves are propagated is supposed to be subjected to changes of an alternating character. Describe the nature of these changes and give an explanation of them. 10. The sound of a gun from a distant fort was heard at a certain house 20 sec. after the appearance of the flash. On another day the interval was 21 sec. What causes may have contributed to the difference ? 11. Equal volumes of hydrogen and oxygen are mixed with each other. What is the ratio of the velocity of sound in the mixture to that in oxygen ? Assume that the density of oxygen is 16 times that of hydrogen. 12. What evidence could you give that the velocity of sound is practically independent of the amplitude and frequency of the air vibrations ? 13. A heavy piston moves without friction in a closed cylinder containing air and is set vibrating. Shew that the work done on the air in the cylinder when the piston describes its inward path is equal to the work done by the air when the piston describes its outward path, provided the compressions are either isothermal or adiabatic, but that the two quantities of work are not equal if there is a transference of heat between the air and the walls of the cylinder less than is required to make the compressions isothermal. Shew that in the latter case the vibrations of the piston will be damped and thence deduce Stokes' conclusion that the com- pressions and rarefactions in a sound-wave must be either iso- thermal or adiabatic. CHAPTER VI 1. Explain the formation of echoes. A person standing between two high parallel walls makes a sharp sound. Account for the series of echoes he will hear. Consider the case where the observer is half way between the walls and the case where he is nearer to one wall than to the other. 280 QUESTIONS 2. Two men are equidistant from the face of a plane vertical clifF and are 1000 ft. apart. On one of them firing a pistol the other hears the echo one second after hearing the' direct report. The velocity of soimd being 1100 ft. per sec, find the distance of the men from the cliff. Would the interval between the two sounds be longer on a hot or a cold day ? 3. A person standing at the end of a row of posts one foot apart makes a sharp sound which he hears "reflected from each of the posts in succession. What is the frequency of the note resulting from the series of echoes ? 4. A vibrating tuning-fork is placed in front of a wall. How could you find the positions of the nodes formed between the fork and the wall, and how could you use your observations for cal- culating the frequency of vibration of the fork ? Why are the nodes sharpest near the wall ? 5. Explain why a whisper at one point in a large room can sometimes be heard plainly at some other point. 6. A source of sound is situated under water. Give a diagram shewing the change in the direction of the rays when they emerge from the water. 7. Account for the difference between light and sound as regards the formation, of shadows. 8. How is the apparent pitch of a sound affected by (1) motion of the source, (2) motion of the observer, (3) wind ? Two horns on a moving motor car sound a perfect fifth. WiU a stationary observer hear the same or a different interval ? 9. The note sounded by the horn of a motor car falls a whole tone in pitch as it passes a stationary observer. Shew that the car must be travelling about 45 miles per hour. 10. Devise an experiment for proving directly the laws of reflection of sound. 11. A train 'is approaching a hiU from which a weU-defined echo can be heard. The engine driver sounds his whistle. What will be the character of the echo heard (1) by the engine driver and (2) by an observer who is stationary ? 12. A train approaches a stationary observer, the velocity of the train being one-twentieth the velocity of sound, and a sharp blast is blown with the whistle of the engine at equal intervals of a second. Find the interval between the successive blasts as heard by the observer. QUESTIONS 281 13. Shew that in Example 1 the observer will hear simul- taneously three series of equidistant echoes, the interval between the rnenibers of a series being the same in each of the three series. 14. A train whistles as it passes a stationary observer with velocity v. Find the interval between the notes heard by the observer when the train is approaching him and when it is leaving him, and shew that the total fall of pitch would be the same if the observer were in the train and the whistle were stationary. CHAPTER VII 1. State what you understand by the term Interference. Mention several instances. 2. Describe the method of finding the velocity of sound by Seebeck's Tube, and mention any advantages the method possesses as compared with those carried out in the open air. 3. Shew that when two trains of similar waves cross each other at an angle there will be changes of velocity and displacement at the crossing point, but there may be no change of pressure. 4. Two tuning-forks have nearly the same pitch. Explain how to use the method of beats to find which is the higher and to find the difference between their frequencies. 5. Shew that the number of beats per second of two notes a semitone apart is diflferent at difierent parts of the scale. Find the frequency of the beats when the lower of the two notes has frequencies 50, 250 and 1000 respectively. 6. What are Combination Tones and how are they produced ? Find the frequencies of the first and second DiflFerence Tones and of the first Summation Tone of two notes whose frequencies are 100 and 150. 7. Find the first difference tone for each of the consonant intervals within the limits of an octave. 8. Whilst the middle C of a harmonium is strongly sounded, the notes A, G, F and E from the octave below are successively sounded with it. What succession of notes will the first Difierence Tones give ? 9. If two organ pipes in exact unison be sounded at opposite ends of a large room, what will be heard by a man who walks from one of the pipes to the other ? 10. If a vibrating fork is rapidly moved towards a wall, beats may be heard between the direct and reflected sounds. Account for them and calculate their frequency if the fork makes 512 vibra- tions per second and approaches the wall with a velocity of 300 cm. per second. 282 QUESTIONS 11. What is the velocity of sound in a gas in which two waves of length 1 and I'Ol metres respectively produce 10 beats in 3 seconds ? 12. A wire stretched by a weight of 4 kilogrammes made 3 beats per second when sounded with a fork, and 5 beats per second with the same fork when the stretching weight was increased by 400 grammes. An addition to the stretching weight of 100 grammes made the beats slower in the first case and quicker in the second. Find the frequency of the fork. CHAPTER VIII 1. Explain the meaning of the terms Free Vibration, Forced Vibration and Resonant Vibration. State in general terms the effect of a periodic force in producing vibrations in an elastic body. 2. A tuning-fork is struck smartly and held with its base on a sound-board. After the sound has died away the fork is again struck with the same intensity and held in the hand away from the sound-board. Which arrangement gives (1) the louder sound and (2) the longer duration of sound 'I Give reasons for your answer. 3. Why must two tuning-forks be very nearly in unison to shew resonance, whilst two strings on the same sound-box give resonance when they are only approximately in unison ? 4. A fork making 256 vibrations per second is held over a tall narrow jar filled with water, and the water is allowed to run out gradually through a tap at the bottom until maximum resonance is obtained. What is the depth of the water level below the top of the jar ? Find two other notes to which the same column of air would resound. (The correction for the open end may be neg- lected.) 5. A tube closed at one end is 10 inches long and 1 inch in diameter. Find the frequency of the note to which it resounds. 6. A tube 2 inches in diameter and open at both ends resounds to the note g. What is its length ? (c = 261.) 7. Prove that the frequency of a Helmholtz Resonator is approximately inversely proportional to the square root of its volume. State in general tenus how the pitch is affected by variations in the size and shape of the mouth. 8. Find the vibration ratio of the interval between two Reson- ators which are of the same size, but one of which has only a single opening while the other has three, the openings being all of the same 'size and shape. What would be the nature of the effect on the interval if' the three openings were close together ? QUESTIONS 283 CHAPTER IX 1. State Helmholtz's Theory of the cause of the differences of quality of the notes of different instruments. 2. How would you prove that the note given by plucking a stretched string is not a simple tone ? Describe and explain the differences of quality due to plucking at different points and with different instruments. 3. Describe three different methods by which you could shew that the note of a pianoforte contains a harmonic a twelfth above the fundamental. State how one or more of your methods would be interfered with if the pianoforte were badly out of tune. 4. How can an approximately pure tone be produced ? 5. A key on a pianoforte is held down and the next adjacent octaves above and below are separately struck staccato. Describe and explain the result in the two cases. 6. Two strings are stretched side by side on a monochord and tuned to C and c. If either string is sounded and then silenced, the other will be found to be sounding c. Explain this. 7. Shew why the resonance of a massive body such as a tuning-fork, which, when set in vibration, goes on for some time, is a more precise indication of the existence in a sound of a component of definite frequency than the resonance of a body of less mass, such as a column of air. 8. What is meant by a Periodic Curve ? State generally the nature of Fourier's analysis of such a curve. What is the application of the analysis to a complex musical note ? 9. How does the mouth modify the quality of singing ? 10. State the rival theories as to the nature of vowel sounds. 11. Compare the eye and the ear as regards their power of analysing complex vibrations. What are the limitations to the resolving power of the ear? 12. State Ohm's Law as to the physical basis of pure tones. What is the nature of the evidence on which it rests? 13. Give a general account of Helmholtz's method of synthe- sizing complex notes. 14. How did Helmholtz shew that the relative phases of the constituents of a complex note have no effect on the quality of the note? 284 QUESTIONS CHAPTER X 1. Shew that the wave-length of the fundamental of a closed organ pipe is four times the length of the pipe. 2. Shew that if a closed organ pipe is so narrow that the correction for its open end may be neglected, the proper tones of the pipe will have frequencies in the ratio 1, 3, 5 etc. 3. Describe the motion of the air in an open organ pipe which is giving out its first overtone. What change will take place in the note and the mode of vibration, if a hole is bored in the pipe near its centre 1 4. How do the motions of the particles of air within a sounding organ pipe differ from the motions of the particles of air outside the pipe by which the sound is transmitted to a distance ? 5. Assuming that the velocity of sound in air is 1100 ft. per sec, find the approximate vibration number of an open pipe 4 ft. long, neglecting the correction for the open ends. 6. Explain why the note falls nearly but not quite an octave, when you close the open end of an open organ pipe. 7. Explain why the end of a reed pipe at which the reed is situated must be regarded as a closed end. 8. Describe and explain the various methods of tuning organ pipes. 9. How can the mode of vibration of the air in an open organ pipe be shewn experimentally ? 10. An open pipe has a manometric capsule at its middle point, and another a quarter of its length from one end. The pipe is made to give its first four tones in succession. Describe the effect on the flames in each case. 11. Give a general statement of the efiect of a change of temperature on the pitch of different classes of pipes. 12. Shew that two flue pipes of different pitches will rise in pitch by the same interval for a given rise of temperature. 13. A flue pipe gives the note C at 15° C. At what temperature will it give the note Cjf, a semitone higher ? 14. Discuss the effect of the correction for the open end on the relative frequencies of the proper tones of a pipe. 15. Explain why the note of a wide pipe contains fewer harmonics than the note of a narrow pipe. 16. Why are the higher harmonics of a reed pipe stronger than those of a flue pipe ? Why do the reeds in an organ always have pipes associated with them ? QUESTIONS 285 17. A whistle gives a note whose frequency is 500 when blown with air at 20° C. When the same whistle is put in a furnace and again blown with air, it is found to give a note whose frequency is 1200. What is the temperature of the furnace ? CHAPTER XI 1. A rod is fixed at one end and has the other end free. Describe in general terms its possible types of vibration. 2. Explain the analogy between the longitudinal vibrations of rods and the vibrations of the air in organ pipes. 3. What are the advantages of a tuning-fork as a standard of pitch? 4. Explain the principle of Wheatstone's Kaleidophone. 5. How could you shew that when a plate is vibrating two adjacent sections separated by a nodal line are always moving in opposite directions ? 6. Two tuning-forks have the same proportions, and are made of similar material. One of the forks is a fifth higher than the other. What is the ratio of their weights? 7. Explain why the frequency of the transverse vibrations of a . rod fixed at one end is independent of the dimensions of the rod at right angles to the plane of vibration, but is not independent of the dimensions in the plane of vibration. 8. Shew that when a rod fixed at one end vibrates transversely, the nodes cannot be equidistant. 9. Shew that when a bell is giving its lowest proper tone, there must be both radial and tangential motion at the rim. How is this fact made use of in the ordinary method of making a wine glass sing by rubbing a wet finger round the rim ? CHAPTER XII 1. Mention the chief objections to the earlier methods of finding the velocity of sound in the open air by observing the interval between the flash and report of a cannon. 2. Describe some laboratory method of finding the velocity of sound in air. 3. Give a brief outline of two methods by which the velocity of sound in carbon dioxide could be found. 28C QUESTIONS 4. Describe some form of the Siren aad explain how it could be used to find the frequency of vibration of an electrically main- tained fork. 5. Describe a method by which you could determine as directly as possible the ratio of the frequencies of two tuning-forks. 6. How is the Tonometer used for finding the frequency of vibration of a tuning-fork ? 7. Being given several forks of known frequencies and a resonance tube, what experiments would you make to shew that different notes travel in air with equal velocities ? 8. State what change, if any, would be made in the dust figures in Kundt's experiment by a change in (1) the length and (2) the thickness of the vibrating rod. 9. State the advantages of the Lissajous' Figure method of comparing the frequencies of two forks as compared with the method of beats. 10. Two tuning-forks are known to be slightly more than a fifth apart. They are used for producing a Lissajous' Figure, and it is found that the figure completes its cycle of changes in 15 seconds. Assuming the frequency of the lower fork to Jbe 261, find the frequency of the higher. CHAPTER XIII 1. Explain the principle of the Phonograph; 2. A violin solo is reproduced on a phonograph, but the phono- graph is run twice as fast as when the record was taken. Will the solo sound in tune ? 3. Explain how the phonograph has been used to test the theories of the origin of vowel sounds. 4. Explain the action of the Bell Telephone. 5. Describe some form of Carbon Transmitter. CHAPTER XIV 1. Describe and explain the sensations produced when the pitches of two notes originally in unison are gradually varied until they differ by a major third. In what way does the experiment depend on whether the notes are of high or of low pitch ? 2. Two tuning-forks nearly an octave apart and free from overtones give beats when sounded together. What is the cause of the beats 1 QUESTIONS 287 3. On what grounds do we suppose beats to be an important factor in our sensations of consonance and dissonance 1 4. Use Helmholtz's method for comparing the consonance of a major third (4 : 5) and a major tenth (2 : 5). 5. A major sixth is played by an open and a stopped organ pipe. What difference in the effect will there be in the two cases when (1) the open pipe plays the lower note and (2) the stopped pipe plays the lower note ? 6. A major third is given (1) on two clarinets, (2) on two haut- boys and (3) one note on a clarinet and one on a hautboy. Explain the different degrees of dissonance. 7. What do you understand by a Consonant Triad ? Find what consonant triads are possible within the compass of one octave. 8. Compare the consonances of the second inversion of a major triad and the second inversion of a minor triad, taking account of the Difference tones as well as the Harmonics. CHAPTER XV 1. Explain how the existence of higher harmonic tones accom- panying the fundamental in the sounds of a musical instrument assists the ear in estimating exactly a consonant interval. 2. An octave and a twelfth are both perfect consonances, yet a mistuned octave is worse than a mistuned twelfth. Why is this ? 3. Two pure tones are to be tuned to a fifth. Show that the timing is facilitated if the octave of the lower note is also sounded. 4. Why is it necessary that in any system of temperament the octaves must be true, whilst the minor thirds may be considerably different from true minor thirds 1 5. What is the true diatonic scale and what are its advantages over other scales ? 6. How could the monochord be used to tune eight notes on a pianoforte to a true diatonic scale ? 7. Shew that if the scale of C on a pianoforte is tuned with true intonation, the note A so obtained will be out of tune, if used as the supertonic of the scale of G. 8. Why is it impracticable to have the musical intervals perfectly true on a pianoforte? 288 QUESTIONS 9. Find the interval between successive semitones in the equally tempered scale. 10. On the pianoforte twelve fifths make seven octaves. Find from this the error of a fifth. 11. Shew that an equal temperament minor third is about -^ comma flat by using the fact that four equal temperament minor thirds make an octave. CHAPTER XVI 1. Why do the strings of a pianoforte differ in thickness and length ? 2. State generally how the quality of the note of a pianoforte is affected by (1) the point struck, (2) the shape of the hammer and (3) the hardness of the hammer. 3. Explain the method of producing the scale on a Brass Instrument with valves. If the valves are tuned correctly when used separately, the note produced when two valves are used together will be sharp. Why is this? 4. Explain how and why a rise of temperature affects the pitch of the wind instruments in the orchestra. If the velocity of sound is 1120 ft. per sec. at 60° F. and 1140 ft. per sec. at 77° F., how much would a trumpet player have to alter the length of his instrument in order to keep to his original pitch, if the temperature of the instrument rose from 60° F. to 77° F. ? (The tube of the trumpet in F is 6 ft. long.) 5. Would a given rise of temperature affect the pitch of a bass trombone and of a cornet to the same extent? Give reasons for your answer. 6. Why are the notes of a flute or clarinet put out of tune with each other, when the joints of the instrument are pulled out to lower its pitch ? 7. Which of the instruments used in the orchestra can play in true intonation ? 8. Which musical instruments have the same overtones as a stretched string? 9. Explain the principle and use of the Vibration Microscope. 10. Explain the action of a Violin Bow and describe the mode of vibration of the string of a Violin. ANSWERS TO QUESTIONS CHAP. I. 7. The vibration ratio of a major sixth is obtained by doubling the lower of the two integers which give the vibration ratio of a minor third. 9. Difference of intervals is i^-. Sum of intervals is f . The two intervals are a fifth and a major sixth and their sum is a major third plus an octave or a major tenth. II. 12. T=2,ry^^-i5^=:l-64 sec. nearly. 15. Max. vel. = aw = 10, t = — = 2, .-. a. = — . 0} TT 16. Total energy = Jm x (max. vel.)2=4 x 20 x 102 = 1000 ergs. 17. If X is the length required, we have the relation x:a; + 6\/3 = P:'22 whence a;=2\/3in. The arrangement described is known as Blackburn's Pendulum. III. 8. The lengths of the segments are 54/^ cm., 27tV c™- ^.nd 18yt cm. respectively. 1 /^p 1 /07' 9. We have 100=^ a/ - and « = — » /— , whence a; = 50 \/2. Ai \ p 4t V /5 1 /64 1 /36 ^ 10. «i:n2 = 23^^-:2^^^-,whence«i:n,=5:3or the interval is a major sixth. 11. The mass of the string having the greater tension is double that of the other. 12. If "1 is the frequency of the aluminium string and n^ the frequency of the steel string, then mi : ri2 = V'V-S : v/2-65. J. /20,000 X 981 xlOO V ' 14- "=2-^V ' -01 =^'''- 15. 5 = ,-^^ , whence 2*= 20 lbs. 2 V J' IV. 3. Use the equation v = n\. The wave-lengths are 35 ft. and 4f ft. The frequency is 13,440. 0. S. 19 290 ANSWERS TO QUESTIONS CHAP. V. 2. The sound travels the whole length of the column in the time the men take to make 2 steps, i.e. it travels 1080 ft. . 60 X 2 , 1080 X 128 , , .„ ,, m sec. ; whence v= — gjr — = — = 1152 ft. per sec. . 1200 s/x° + 27S° , ,_„„ 4. TPion = — / ' whence x = 58° C. 1090 \/273° 6. 20-9 ft. 7. Frequency 250. Wave-length 136 cm. 11. 1-37:1. VI. 2. The path taken by the echo must he 1100 ft. greater than the distance between the men, whence by geometry we find the distance of the men from the wall is 923 ft. The interval is greater on a cold day. 3. The period of the note heard is the time taken by the sound in travelling twice the distance between two adjacent posts. Taking the velocity of sound to be 1100 ft. per sec, the frequency is 550. 12. The interval is '95 sec. VII. 5. The ratio of the frequencies is in each case 15 : 16, whence the differences of the frequencies are 3-3, 16"7 and 66'7 respectively. 6. First Difference Tone 50. Second Difference Tones 50 and 100. Summation Tone 250. 10. If the person who carries the fork is the listener, he will hear 9'2 beats per sec. A stationary listener in the line of motion of the fork hears no beats when the fork is ap- proaching him, and 9-2 per sec. after it has passed him. 11. 336f metres per sec. 12. If n is the frequency of the fork, n' that of the string with 4000 grm. and n" that of the string with 4400 gm. we have n'=n-S and n"=n+S, whence ■ m-3 x/4000 == , and ™=166-9. JH-5 V4400 VIII. 4. 1-07 ft. 256 X 3 and 256 x 5. 5. The effective length of the pipe is 10-3 in. Hence the wave- length is 41-2 in. and if the velocity of sound is taken as 1100 ft. per sec. the frequency is 320-4. 6. 15 6 in. 8. n:?i' = v/r:v/3 = l: 1-73. X. 5. 137-5. 13. 54-6° C. 17. 1416° C, XI. 6. 27:8. XII. 10. 391-57. XVI. 4. If in. INDEX The figures refer to the pages Adiabatic coefficient of elasticity of a gas, 83 Air-waves, general description of propagation, 59 Amplitude defined, 25 Analysis of complex vibrations, 154 Antinode defined, 49 Appun's tonometer, 202 Associated curves, 64 Bassoon, 262 Beating reeds, 177 Beats, general explanation, 121 , pitch of note heard, 123 , experimental illustrations, 123 , due to combination tones, 127, 227 , measurement of pitch by, 201 , effect on the ear, 225 , effect on consonance of pure tones, 225 Bell-jar experiment, 2 Bells, vibration of, 190 , overtones of, 192 , tuning of, 193 Bell telephone, 219 Blaikley, on length of wind instru- ments, 86 , on correction for open end, 174 , on velocity of sound, 209 Brass instruments, 264; shape of , 264; crooks of, 264 Bravais and Martin, on velocity of sound at high altitudes, 206 Bugle, 265 Cagniard de la Tour's siren, 196 Carbon transmitter, 222 Chladni's figures, 189 Clarinet, 232, 259 Clement and Desormes, on ratio of specific heats, 84 Closed end of tube, reflection at, 69 Closed organ pipe, period of, 166; overtones of, 169 Colladon and Sturm, on velocity of sound in water, 206 Combination tones, 124; theories of origin, 125 ; method of finding pitch, 126 ; formed by harmonics, 233 Common chord, 236 Comparison of intensities of sounds, 214 Complex vibrations, analysis of, 154 ; synthesis of, 158 Composition of simple harmonic motion with uniform motion, 27 of two simple harmonic motions, 28, 30, 32 of harmonic vibrations pro- duced optically, 34, 185, 252 Concentration of sound by spherical mirror, 93 ; by walls of room, 95 Condensation in air-waves, 62 , absolute, in organ pipes, 213 Conical pipes, overtones of, 176 Conjugate foci, 94 Consonance, 224 , Helmholtz's theory of, 227 of intervals of pure tones, 227 of intervals of complex tones, 228 292 INDEX Consonance of notes which have not the full series of harmonics, 232 of triads, 236 Consonant intervals, defined, 9 ; relative smoothness of, 224 ; analysis of relative smoothness of, 228; effect of widening the intervals by an octave, 230 Consonant triads, defined, 233; derivation of, 235; effect of difference tones, 237 Cor de chasse, 265 Cornet, 270 Correction for open end, 167; method of finding, 173; effect on overtones, 174; effect on quality, 175 Crooks of brass instruments, 269 Crova's disc, 65 Curves of displacement of vibrating particles, 25 ; relation to velocity curve, 27; by Lissajous' Figures, 32; in transverse wave motion, 35, 71 ; in stationary vibrations, 48; in air- waves, 64, 71, 74; in beating notes, 122; in complex vibrations, 148 ; in organ pipes, 168 Definition of intervals, 239; of pure tones, 239 ; of complex notes, 240 Diatonic scale, defined, 10 ; deri- vation and advantages of, 242; defects of, 244 Difference of phase defined, 25 Dimensions, method of, 193 Disc siren, 8 Displacement. See Curves of dis- placement Dissonance due to beats, 226 Dominant, 244 Doppler's principle, 103; applied to light, 107 Drums, 193, 273 Bar, described, 143 , resolving power of, 145 , limitations of 146 The figures refer to the pages Ear trumpet, 69 Echoes from plane surfaces, 92 from palings, 97 Elastic deformation, relation to deforming force, 15 Elastic vibrations, general descrip- tion of, 18; isochronism of, 19; calculation of period of, 23 Elasticity, nature and limitations of, 18 , imperfect, 14 of liquids, IS, 17 of gases, 15, 57 , coefficient of, 24, 82, 83 Energy transmitted along train of stationary waves, 50 of air-waves, 68 Equal temperament, 247; errors of intervals in, 248 Equal temperament semitone, vibra- tion ratio of, 248 Plue-pipes, described, 165 , period of, 166 , overtones of, 167 , methods of tuning, 172 , conical, 176 Flute, 256 Fogsignals, zones of silence near, 120 Forced vibrations, 129; amplitude of, 187; phase of, 139; initial stages of, 139; used in musical instruments, 140 Fourier's theorem, 150; applica- tions of, 152 Free reeds, 177 Free vibrations, defined, 128 French horn, 265, 271 Frequency, defined, 7 Frets, 255 Fundamental defined, 55 Graphic method of measuringpitch, 199 Hand horn, 265 Harmonic constituents of notes of musical instruments, 163 INDEX 293 The figures refer to the pages Harmonic series, 55, 151, 224 Harmonics, defined, 55, 153 , given by Fourier's theorem, 151 , quality dependent on, 154 in the voice, 162 of organ pipes, 175 , influence on consonance, 228 , influence on definition of intervals, 240 of violin, 164, 255 of flute, 164, 259 of clarinet, 164, 261 of hautboy, 164, 262 Harmonium, 177 Harmonograph, 28 Harp, 252 Hautboy, 232, 261 Helmholtz, resonators, 131; see also Resonators , theory of quality, 154 , analysis of complex vibra- tions, 1S5 , synthesis of complex vibra- tions, 158 , theory of consonance, 227 , on the vibrations of violin string, 252 Hooke's Law, 16 Hughes's microphone, 221 Hunning's transmitter, 222 Intensity of sound, variation with distance, 99; near interfering sources, 115 ; absolute measure- ments of, 211 Interference, meaning of the term, 109 near two sources, 115 shewn by branched tube, 116 ; by tuning fork, 117 ; by Seebeck's tube, 118 ; near fog signal, 120 Intervals, 8 , measurement of, 9 , consonant, 9 , sum of, 10 , difference of, 10 , definition of, 239 Intervals. See also Consonant in- tervals Inversions of triads, 235 Isochi-ouism, 19 Isothermal coefficient of elasticity, 82 Ealeidophone, 185 Key bugle, 266 Kohlrausch, on sensation of pitch, 203 Konig's manometric capsule, 171 Krakatoa eruption, 206 Kundt's method of measuring velo- city of sound, 209 Laplace's correction of Newton's calculation of the velocity of sound, 82 Leading note, 244 Limits of audibility, 7 of elasticity, 13 Linear dimensions , relation to pitch , 194 Lissajous' figures, 32; optical method of producing, 84; pro- duced by kaleidophone, 186; used for measuring differences of frequency, 202 Longitudinal vibrations of air par- ticles, 60 Longitudinal waves, 57 ; illustrated by spiral spring, 58; properties of, 61, 64; condensation in, 62; shewn by Crova's disc, 65; velocity of, 79 ; superposition of, 109- Loudness of sound, 5 Manometric capsule, 171 Mayer, on comparison of intensities, 214 Mean tone temperament, 246 Mediant, 244 Melde's experiment, 142 Membranes, vibrations of, 193 Microphone, 221 Mixture stops, 242 294 INDEX The figures refer to the pages Modulation, 244 Monochord described, 51 , resonance shewn by, 131 used for measuring pitch, 198 Mouth of organ pipe, action of, 165 Musical box, 186 Musical echo from palings, 97 Musical instruments, described, 250 ; qualities of notes of, 163 Musical notes, characteristics of, 3, 5 Natural modes of vibration, 153; see also Overtones Newton's calculation of velocity of sound, 81 ; Laplace's correction of, 82 Nodes defined, 49 Noise, 8 Nomenclature of stationary vibra- tions, 49 of complex notes, 153 Non-harmonic force, effect in pro- ducing vibrations, 129, 158 Non-harmonio waves, 39 Note defined, 153 Ocarina, 241, 259 Ohm's Law, 125, 147 Open end of tube, reflection at, 75 ; con-ection for, 78, 167, 173 Open organ pipe, period of, 166; overtones of, 168 Ophicleide, 266 Organ pipes, flue, 165; reed, 176; amplitude of vibration of air in, 213 Overtones, defined, 55 of strings, 55 , harmonic and inharmonic, 153 of organ pipes, 167 of rods, 182 of tuning-forks, 187 of plates, 189 of bells, 192 of clarinet, 260 of hautboy, 262 Overtones of brass instruments, 264 Partials, defined, 56 Periodic curves, 149 Personal equation, 205 Phase, defined, 25 of forced vibrations, 139 , effect on quality, 159 Phonograph, described, 216 used to test vowel theories, 218 Pianoforte, 250 Pipes of variable bore, 76 , velocity of sound in, 207 , see also Organ pipes Pitch, measured by frequency, 6 notation, 11 of note given out from moving source, 104 of note heard by moving listener, 105 , dependence on dimensions of sounding body, 194 , measurement by siren, 196 ; by monochord, 198; by graphic method, 199; by tonometer, 200 , sensation of, 203 , standards of, 263 Plane waves, constancy of intensity of, 69 Plates, vibrations of, 189 Potential energy of deformed body, 18 Proper modes of vibration, 153 Pure tones, consonance of, 225, 227 Quality, characterized by wave form, 12 , meaning of the term, 145 , Helmholtz's theory of, 154 , dependence on phase of constituents, 159 Ratio of specific heats, 84; deter- mined by Kundt's method, 211 Eayleigh, on correction for open end, 173 INDEX 296 The figures refer to the pages Rayleigh, on absolute measurement of intensity, 211 , on comparison of intensities, 214 Eeciprooal firing, 204 Eeed pipes, 176 , method of tuning, 179 Eeeds, free and beating, 177 , effect on pitch of pipes, 178 Eeflection of waves at end of string, 45 • of waves at closed end of pipe, 69 of waves at open end of pipe, 75 at surface of changingdensity, 77 of spherical waves, 90 • of sound by spherical mirrors, 93 , total internal, 99 Refraction, 99 Regnault, on measurement of sound in open air, 205 ; in pipes, 207 Resonance, 128 — , effect of mistuning on, 137 Resonance box of fork, 136 Resonant vibrations, of pendulum, 128; instances of, ISO; shewn by monochord, 131 Resonators, Helmholtz's, 131 , nature of vibrations, 182 , pitch of, 133 ■ , conductivity of mouth of, 134 with several mouths, 135 Restitution forces, 17 Rods, longitudinal vibrations of , 181 , transverse vibrations of, 183 Riicker and Edser, on combination tones, 126 Savart's toothed wheel, 6 Saxhorn, 272 Saxophone, 261 Scheibler's tonometer, 200 Seebeck's tube, 118, 208 Sensitive flame, 94 Simple harmonic vibrations, 21 ; geometrical illustration of, 22; period of, 23 Sine curve, 25 SUde of trombone, 267 Sound, simpler properties of, 1 vibrations superposed on molecular motions, 65 propagated at right angles to wave-front, 68 images, 92 shadows, 102 Soundboard of musical instruments, 140 Speaker key, 260 Speech, 160 Spherical waves , general description of , 67 ; variation of intensity with distance, 69; superposition of, 113 Spiral spring, law of stretching of, 15; illustrating longitudinal waves, 58; illustrating overtones of pipes, 170 Standards of pitch, 263 Stationary vibrations, of string, 48; experimental demonstration of, 49, 97; nomenclature, 49; in closed tube, 70; properties of, 72; compared with progressive waves, 75; in open tube, 77; effect on ear, 111 Stretched string, waves on, 41; velocity of waves on, 42 ; super- position of waves on, 44, 47; frequency of, 50 ; laws of vibra- tion of, 51 ; experimental test of laws, 52; nodes on, 54; effect of imperfect flexibility, 56 Submediant, 244 Superposition of waves on string, 44,47 of trains of waves, 109 of oblique trains of waves, 112 of spherical waves, 113 , limits of law of, 124 Supertonic, 244 Synthesis of complex notes, 158 296 INDEX Tlie figures refer to the pages Telephone, 218 Temperament, 246 , mean tone, 246 , equal, 247 Temperature, effect on velocity of sound, 86 , effect on direction of pro- tion, 101 I effect on organ pipes, 179 , effect on pitch of brass instruments, 272 Threlfall and Adair, on velocity of sound in water, 207 Tone, definition of, 153 Tonic relationship, 241 Tonometer, 200 Topler and Boltzmann, on ampli- tude of vibration of air, 213 Total internal reflection of sound, 99 Transmitter, 222 Transverse waves, 35; velocity of, 37, 42; equation of, 40; on a string, 41 ; superposition of, 44, 47 ; not possible in a gas, 57 Triads. See Consonant Triads Triangle, 188 Trombone, 266 Trumpet, 270 Tuba, 272 Tube, reflection at closed end, 69 • , reflection at open end, 75 of variable bore, 76 Tuning an elastic body, 24 flue pipes, 172 reed pipes, 179 wood wind, 262 brass instruments, 272 drums, 273 Tuning-fork, on resonance box, 136 , electrically maintained, 155 , general description, 187 , effect of temperature on, 187 , overtones of, 187 Two adjacent sources of sound, 113 Valves of brass instruments, 268; defects of, 269 Velocity of particles, relation to displacement, 27 Velocity of sound, nearly' inde- pendent of amplitude, 85; inde- pendent of pitch and pressure, 85; in different gases, 85; effect of temperature, 86; in mixtures of gases, 87; in liquids, 88, 206, 207 ; in solids, 88 ; in open air, 204; at high altitudes, 206; in pipes, 207 Velocity of sound measured, by echo, 93; by resonance tube, 208; by Seebeck's tube, 208; by organ pipes, 209; by Kundt's tube, 209 Velocity of transverse waves, 37; relation between velocity and wave-length, 38 Velocity of elastic waves of any type, 81 Ventral segment, defined, 49 Vibration microscope, 252 Vibrations. See Elastic vibrations Violin, 252 , motion of string of, 254 , action of bow, 254 r, quality of, 255 Viscous liquids, 14 Vocal organs, 161 Vowels, 160 Vowel theories, 162, 218 Wave-front, 67 Waves. 5eeLongitudinal waves and Transverse waves Wheatstone's kaleidophone, 185 Whispering gallery, 96 Wind, effect on direction of pro- pagation of sound, 100 , effect on pitch, 107 , effect on velocity measure- ment in open air, 204 CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PBBSS CAMBRIDGE PHYSICAL SERIES The Times. — "The Cambridge Physical Series... has the merit of being written by scholars who have taken the trouble to acquaint themselves with modern needs as well as with modern theories." Modern Electrical Theory. By N. R. Campbell, Sc.D. Second Edition, largely rewritten. Demy 8vo. pp. xii + 400. 9^. net. 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