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 http://www.archive.org/details/cu31924031295623 
 
arV178S2^°'"^' University Library 
 Acoustics. 
 
Clronkn ^xm Bifm 
 
 ACOUSTICS 
 
 D O NK I N 
 
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 MACMILLAN AND CO. 
 
 PUBLISHERS TO THE UNIVERSITF OF 
 
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€hxmliavc ^r^ss Sm^s 
 
 ACOUSTICS 
 
 THEORETICAL 
 PART r 
 
 Wr'F. PONKIN, M.A., F. R. S., F. R. A. S. 
 
 SAVILIAN PROFESSOR OF ASTRONOMY, OXFORD 
 
 AT THE CLARENDON PRESS 
 M.DCCCLXX 
 
 y AV ritrhta reserved 1 
 
CORNEi 
 UNIVERSITY! 
 x^ LIBRARY 
 
ADVERTISEMENT. 
 
 As this is the only portion of a treatise on Acoustics, 
 intended to comprise the practical as well as the the- 
 oretical parts of the subject, which will proceed from 
 the pen of its Author, a few words are required to 
 explain the circumstances under which it now appears. 
 
 The Author, the late Professor Donkin, has passed 
 away prematurely from the work. It was a work he 
 was peculiarly qualified to undertake, being a mathe- 
 matician of great attainments and rare taste, and taking 
 an especial interest in the investigation and application 
 of the higher theorems of analysis which are necessary 
 for these subjects. He was, moreover, an accomplished 
 musician, and had a profound theoretical knowledge 
 of the Science of Music. 
 
 He began this work early in the year 1867 ; but he 
 was continually interrupted by severe illness, and was 
 much hindered by the difficiilty, and in many instances 
 the impossibility, of obtaining accurate experimental 
 results at the places wherein his delicate health com- 
 pelled him to spend the winter months of that and the 
 following years. He took, however, so great an interest 
 in the subject, that he continued working at it to within 
 two or three days of his death. 
 
 The part now published contains an inquiry into 
 the Vibrations of Strings and Rods, together with an 
 explanation of the more elementary theorems of the 
 
V' ADVERTISEMENT. 
 
 subject, and is, in the opinion of its Author, complete in 
 itself; his wish was that it should be published as soon 
 as possible ; and he was pleased at knowing that the 
 last pages of it were passing through the Press im- 
 mediately before the time of his death. It is the first 
 portion of the theoretical part. 
 
 It was intended that the second portion should con- 
 tain the investigation into the Vibrations of Stretched 
 Membranes and Plates ; into the Motion of the Mole- 
 cules of an Elastic Body ; and into the Mathematical 
 Theory of Sound. Professor Donkin did not live long 
 enough to complete any part of this section of the 
 work. 
 
 The third portion was intended to contain the prac- 
 tical part of the subject ; and the theory and practice of 
 Music would have been most fully considered. It is 
 exceedingly to be regretted that the Professor did not 
 live to complete this portion ; for the combination of the 
 qualities necessary for it is seldom met with, and he 
 possessed them in a remarkable degree. Not even a 
 sketch or an outline is found amongst his papers. He 
 had formed the plan in his own mind and often talked 
 of it with pleasure.- It can now never be written as 
 he would have written it. 
 
 BARTHOLOMEW PRICE. 
 
 1 1, St. Giles', Oxford, 
 Feb. i6, 1870. 
 
CONTENTS. 
 
 PAGE 
 
 CHAP. I. GENERAL INTRODUCTION . . I 
 
 „ II. MISCELLANEOUS DEFINITIONS AND 
 
 PROPOSITIONS . . . .13 
 
 „ III. COMPOSITION OF VIBRATIONS . . 29 
 
 „ IV. THE HARMONIC CURVE ... 47 
 
 „ V. VIBRATIONS OF AN ELASTIC STRING 6"] 
 
 „ VI. VIBRATIONS OF A STRING . . 85 
 
 „ VII. ON THE TRANSVERSE VIBRATIONS 
 OF AN ELASTIC STRING (DYNA- 
 MICAL THEORY) . . . .106 
 
 „ VIII. ON THE LONGITUDINAL VIBRATIONS 
 
 OF AN ELASTIC ROD . . . I44 
 
 „ IX. ON THE LATERAL VIBRATIONS OF 
 
 A THIN ELASTIC ROD . . . 160 
 
CHAPTER I. 
 
 GENERAL INTRODUCTION. 
 
 1. The sensation of sound, like that of light, may be pro- 
 duced in exceptional and extraordinary ways. But the first step 
 in the usual process consists in the communication of a vibra- 
 tory motion to the tympanic membrane of the ear, through 
 slight and rapid changes in the pressure of the air on its outer 
 surface. 
 
 The ear may be considered as consisting essentially of two 
 parts, of which one is the organ of communication with the 
 external world, and the other of communication with the brain. 
 
 2. The former part is a tube of irregular form, divided into 
 two portions of nearly equal length by the tympanic membrane, 
 which is stretched across it somewhat obliquely as a transverse 
 diaphragm. The shorter and wider part of the tube is outside 
 the tympanic membrane, and ends at the orifice of the external 
 ear. This part is called the Meatus. 
 
 The part of the tube immediately within the tympanic mem- 
 brane is called the Tympanum, and the remainder the Eustachian 
 Tube. 
 
 The Eustachian tube leads into the pharynx, that is, the 
 cavity behind the tonsils and uvula, into which the nostrils also 
 open. But the orifice of the Eustachian tube, though capable 
 of being opened, is usually closed. It is opened involuntarily 
 in the act of swallowing, and can be opened voluntarily by 
 a muscular effort not easily described but easily made ; and the 
 opening is accompanied by a slight sensation in the ear, due to 
 a temporary change in the pressure of the air in the tympanum. 
 Thus the meatus on the one hand, and the tympanum and 
 
a Structure of the Ear. 
 
 Eustachian tube on the other, always contain air, but under 
 different conditions. The air in the meatus is liable to be 
 directly aflfected by every change, however slight and rapid, in 
 the pressure of the external air, with which it is always in free 
 communication ; but that within the tympanic membrane, being 
 only occasionally put into communication with the external air 
 by the opening of the Eustachian tube, is not liable to be 
 directly affected by slight and rapid changes, though it takes 
 part in the slower fluctuations shewn by the barometer. 
 
 3. The second, or interior, part of the ear is contained 
 within a cavity which is called the bony labyrinth, because it is 
 of a complicated form, and is surrounded by bone except in 
 two places. These two places may be compared to windows, 
 looking into the tympanum, but completely closed by mem- 
 branes, so that' neither air nor fluid can pass through them. 
 One of these is called the mal and the other the round window. 
 
 The interior of this bony labyrinth is filled with fluid, in 
 which are suspended membranous bags, following nearly the 
 same form, and themselves containing fluid. 
 
 The terminal fibres of the auditory nerve are distributed over 
 the surfaces (or parts of the surfaces) of these membranous 
 bags, and there are special arrangements of which the object 
 appears to be the communication to these nervous fibres of 
 any agitation affecting the fluid. 
 
 4. The tympanic membrane is connected with that which 
 closes the ^oval window' by a link-work of small bones con- 
 tained in the open space of the tympanum, in such a manner 
 that when the former membrane is bulged inwards or outwards 
 by an increase or diminution of the pressure on its external 
 surface, a similar movement is impressed on the latter; and 
 although the fluid within is probably as incompressible as water 
 (of which it chiefly consists), the membrane of the ' round 
 window' allows it to yield by expanding outwards when that 
 of the oval window is forced inWards, and vice versd. 
 
 Thus the motion impressed on the tympanic membrane by 
 the external air is communicated to the fluid contained in the 
 labyrinth, and from that to the fibres of the auditory nerve, by 
 
 'iii 
 
Curve of Pressure, 3 
 
 means of the apparatus mentioned above, which need not be 
 further described at present ^ 
 
 It is probable also that motion is partly propagated from the 
 tympanic membrane through the air in the tympanum to the 
 membrane of the round window, and so to the fluid. 
 
 5. The ' pressure' of the air at any point must be understood 
 to mean, as usual, the pressure which would be exerted on a 
 unit of surface by air of the same density and temperature as 
 at the point in question. 
 
 When the pressure at any point varies with the time, the 
 variation may be graphically represented by means of a ' curve 
 
 of pressure,' in which the abscissa M oi any point P is pro- 
 portional to the time elapsed since a given instant, and the 
 ordinate MP to the excess of pressure above a standard value, 
 which may be taken arbitrarily. A negative ordinate (as at P) 
 represents of course a defect of pressure below the standard 
 value. 
 
 As we shall chiefly have occasion to consider cases in which 
 the average (or mean) pressure remains unaltered, it will be 
 convenient to assume that average as the standard value repre- 
 sented by the axis of abscissse X. 
 
 Changes of density may evidently be represented in the same 
 way by a ' curve of density,' in which positive ordinates repre- 
 sent condensation, and negative ordinates dilatation. The curves 
 of pressure and of density will differ slightly in form, because 
 
 ^ A full description, with illustrations, of the structure of the ear, so far ~ 
 as it is known at present, will be found in Helmholtz, p. 198, &c., and in 
 Huxley's ' Lessons in Elementary Physiology,' p. 204, &c., and other recent 
 works on Anatomy and Physiology. But the reader is recommended to 
 study the subject, if possible, with the help of anatomical preparations or 
 models. For the purposes of this treatise, however, nothing is absolutely 
 necessary to be known beyond what is stated here or hei-eafter in the text. 
 
4 Noises and Notes. 
 
 pressure is not in general strictly proportional to density; the 
 reasons and consequences of this fact, however, do not concern 
 us at present. 
 
 Although it is convenient to use the word ' curve,' it must 
 be understood that the lines representing changes of pressure 
 or density are not necessarily curved in the ordinary sense, but 
 may consist either wholly or in part of straight portions. 
 
 6. Those slight and rapid changes in the pressure of the air 
 in contact with the tympanic membrane, which cause (Art. 1) 
 the sensation of sound, do not in general alter the average 
 pressure ; so that they would be represented by a wavy curve 
 including upon the whole equal areas above and below the 
 axis of abscissae. 
 
 A wavy curve may or may not be periodic. A periodic curve 
 consists of repetitions of a single portion, thus : 
 
 Fig. 2. 
 
 The period or wave-length of such a curve is the smallest dis- 
 tance A B which, measured from any arbitrary point along the 
 axis, has the ordinates at its extremities always equal : in other 
 words, it is the projection, on the axis, of the smallest portion 
 which is repeated. 
 
 7. Sounds are usually divided into two classes : namely, un- 
 musical sounds, or noises; and musical sounds, or noks. 
 
 As regards sensation, the distinction between these two classes 
 of sounds is said to be that notes have pitch and noises have 
 not; and, as regards the mode of their production, that the 
 curve of pressure (Art. 6) is periodic in the case of a musical 
 sound, and non-periodic in the case of a noise. But these 
 statements require more explanation and correction than might 
 at first sight be expected. 
 
 In the first place, it is obvious to common observation that 
 few, if any, noises are perfectly unmusical, that is, absolutely 
 without pitch. Two noises of the same general character often 
 
Generation of Musical Notes. 5 
 
 differ from one another in a way which we describe by calling 
 one of them more acute or sharp, and the other more grave or 
 flat ; for example, the reports of a pistol and of a cannon. 
 
 On the other hand, few, if any, sounds are perfectly musical, 
 that is, absolutely unmixed with noise. 
 
 Hence the question presents itself whether there is after all 
 a real distinction in kind between noises and notes, and if there 
 is, in what it consists. 
 
 8. Before attempting to answer this question, we must notice 
 two facts of fundamental importance. 
 
 The first is, that any sound whatever, if repeated at equal ■ 
 and sufficiently short intervals of time, generates a note, of' 
 which the pitch depends upon the frequency of repetition of 
 the original or elementary sound. This is easily verified by 
 simple experiments. For instance, if the point of a quill pen, 
 or the edge of a card, be held against the teeth of a wheel 
 which is turned slowly, the passage of each tooth produces a 
 sharp noise or ' click.' But if the velocity of rotation of the 
 wheel be gradually increased, the clicks gradually cease to be 
 heard separately, and are replaced by a sound which gradually 
 acquires a continuous character, and a pitch which rises as the 
 velocity increases. The connection between the frequency of 
 repetition of the elementary sound, and the pitch of the resultant 
 note, will be considered afterwards. 
 
 The second fact is, that more than one sound can be heard 
 at once. This is familiar to every one. But the following 
 considerations will shew that it is very remarkable. 
 
 9. In the case above supposed, each passage of a tooth of 
 the wheel across the quill or card produces a disturbance in the 
 air, which is propagated in all directions (unless some obstacle 
 intervene) in the form of a wave. 
 
 The nature of sound-waves in air will be considered here- 
 after. At present it is sufficient to say that though they are 
 essentially different in most respects from ordinary waves in 
 water, they have one important property in common, namely, 
 that different sets of waves can be propagated at the same time, 
 either in the same or in different directions, without destroying 
 
6 Composition and Resolution 
 
 one another. And it results from the mechanical theory that 
 when the waves are very small, the different sets are simply 
 superposed; that is, the disturbance produced at any point, and 
 at any given time, by the combined action of waves belonging 
 to different sets, is the sum of the disturbances which would 
 have been produced by the waves of each set separately. 
 
 In this proposition the word sum is to be understood, accord- 
 ing to circumstances, either in the ordinary algebraical sense, 
 or in the extended sense in which the diagonal of a parallelo- 
 gram is called (in symbolical geometry) the sum of two con- 
 tiguous sides. 
 
 Thus, in the case of sound-waves in air, the disturbance at 
 any point may be considered either as a displacement of par- 
 ticles, or as an alteration of pressure and density. And the 
 whole change, either of pressure or density, is the algebraical 
 sum of the changes which would have been produced by waves 
 of each set separately ; while the displacement of any particle 
 is the sum of the separate displacements in the sense just 
 explained. 
 
 Thus, suppose A is the undisturbed position of a particle 
 which, at a given instant, would be 
 displaced to P by the action of a 
 wave belonging to one set, and to 
 Q by that of a wave belonging to 
 another set. Then the actual dis- 
 placement at the same instant will 
 be io R, AR being the diagonal of 
 ■^'g- 3- the parallelogram constructed upon 
 
 AP, AQ. 
 This law of the composition of displacements, which is iden- 
 tical in form with that of the composition of forces in Mechanics, 
 may be stated in another manner thus : 
 
 Each separate cause of displacement acts independently of other 
 causes; a proposition which is to be understood as follows : — 
 
 The actual displacement AR\% the same as if the particle 
 were first displaced by one cause along A P, and then by the 
 other along a line P R, equal and parallel to the displacement 
 
of Disturbances. 7 
 
 A Q, which the latter cause would have produced if acting 
 alone. 
 
 It is important to recollect that this law of the ' superposition 
 of displacements ' is not a universal law in the same sense as 
 that of the composition of forces. It depends, in fact, upon 
 the condition that the force which tends to restore a displaced 
 particle to its undisturbed position is directly proportional to 
 the displacement; and this condition, in most cases in which 
 it subsists at all, subsists rigorously only for infinitely small 
 displacements. 
 
 The law, however, is sensibly true, so far as most of the 
 phaenomena of sound are concerned, in the case of the greatest i 
 disturbances produced by ordinary causes. 
 
 10. This being premised, let us consider a continuous noise, 
 lasting say for a small fraction of a second, in which the ear 
 can recognise no definite pitch. And suppose the curve of 
 B 
 
 Fig. 4. 
 
 pressure (that is, the curve representing in the way above ex- 
 plained (Art. 5) the changes of pressure close to the tympanic 
 membrane) to be the black line A B C D, Fig. 4. 
 
 Suppose a different noise, of the same duration, to have for its 
 curve of pressure the dotted line in the same figure. Then, if 
 
 Fig- 5' 
 
 the causes producing such noises both act at once, the resultant 
 variation of pressure will be represented, as in Fig. 5, by a 
 
8 Power of Resolution 
 
 curve, of which the ordinates are the algebraical sums of the 
 corresponding ordinates in Fig. 4. 
 
 -Now in such a case the two noises do not in general 
 coalesce into one, but are heard distinctly, though simul- 
 taneously. It is only when the two curves of Fig. 4 are nearly 
 similar, that is, when the two noises are very like one another, 
 that the ear does not easily resolve the resultant sound into its 
 two components. 
 
 On the other hand, there is nothing in the curve of Fig. 5, 
 which can suggest to the eye the process of composition by 
 which it was generated. It looks quite as simple as either 
 of its components ; and although it might be arbitrarily re- 
 solved into two in an infinite variety of ways, there is nothing 
 in its appearance to indicate one way as more natural than 
 any other ^. 
 
 It is evident then t)iat the ear has a power of resolution 
 which the eye has not ; or rather, that the ear resolves accord- 
 ing to some law peculiar to itself, whereas the eye either does 
 not resolve at all, or resolves arbitrarily. 
 
 11. This phsenomenon is still more remarkable when the 
 component curves are periodic. Let AB CDEFG, Fig. 6, 
 
 B E 
 
 -At 
 
 C C ^ F 
 
 Fig. 6. 
 
 represent two periods of a periodic change of pressure; the 
 curve consisting of repetitions of the portion ABCD. Simi- 
 larly let abcdefg represent three periods of another periodic 
 change, occupying the same time as two periods of the former. 
 
 1 The curves in the above figures are represented as made up of strairfit 
 hues, merely for convenience of drawing. Whether their forms be or be 
 not such as could really occur, is quite immaterial to the argument. The 
 pomt essential to be understood is this : that whatever be the forms of the 
 component curves, the eye cannot in general distinguish them in the 
 resultant curve. 
 
pecttliav to the Ear. 9 
 
 The resultant curve consists of repetitions of the portion 
 P Q, Fig. 7, and is therefore also periodic. 
 
 Fig. 7. 
 
 If the periods be sufficiently short, each of the curves in 
 Fig. 6 will correspond to a musical note ; and the curve in 
 Fig. 7 represents the variation of pressure which takes place 
 when the two notes are heard simultaneously. Now, it is 
 a well-known fact that in such a case the ear in general dis- 
 ting^shes both notes ; that is, it resolves the resultant curve of 
 Fig. 7 into the two components of Fig. 6. But the eye sees 
 nothing more in Fig. 7 than a curve having a period repre- 
 sented by the abscissa P Q, and entirely fails to distinguish the ' 
 different periods of the component curves. Here again, then, I 
 we find that the ear resolves according to a law of its own. ' 
 
 12. But a new question now presents itself. If a curve like 
 P Q is resolved by the ear into two components, why is not 
 each component similarly resolved ? If the repetition oi P Q 
 represents two simultaneous but distinguishable notes, why may 
 not the repetition oi AB C D represent two notes in the same 
 way; and why may not each component of ABCD be itself 
 again resolved, and so on ad infinitum P ( 
 
 The answer is, that all this does in general really happen; 
 and that a perfectly simple musical tone, that is, a tone such as , 
 the ear cannot resolve, is rarely heard except when produced! 
 by means specially contrived for the purpose. 
 
 Thus the sound produced by a vibrating string, of a piano- 
 forte or violin for instance, is in general compounded of 
 simple tones, theoretically unlimited in number. Only a few 
 of them, however, are loud enough to be actually heard. These 
 few constitute a combination which is always heard from the 
 same string under the same circumstances ; hence we acquire 
 
lo Wkai is a Noise? 
 
 a habit of associating them together and perceiving them as 
 a single note of a special character ; and it requires an effort 
 of attention, which is difficult when first attempted by the 
 unassisted ear, to analyse the compound sensation. In the 
 case of a string this difficulty is increased by the circumstance 
 that the audible component tones form in general an agreeable 
 consonance. When, however, two strings, not in unison, 
 vibrate at once, we distinguish their notes perfectly; partly 
 because there is in this case no habit of association, and partly 
 because the component tones of one do not in general foiin 
 a consonant combination with those of the other. 
 
 The sound of a large bell is compounded of many simple 
 tones, some of which combine agreeably and some produce 
 harsh dissonances. In this case every one perceives easily the 
 complex character of the sound. Still the habit of association 
 prevents us from mistaking the sound of one bell for the sound 
 of two ; and on the other hand, when two bells are struck at 
 once, we distinguish the two compound sounds more or less 
 perfectly ; notwithstanding the very confused combination of 
 simple tones. 
 
 13. We can now give at least a partial answer to the ques- 
 tion. What is a noise? It is in general a combination of 
 a number of musical tones too near to one another in pitch 
 to be distinguished by the unassisted ear. 
 
 The effect produced by ringing all the bells of a peal at 
 once, or striking all the twelve keys of an octave on a piano- 
 forte at once, shews how a confused combination of tones 
 tends to become a noise. And it may be easily conceived that 
 the change would be much more complete if, for example, 
 twelve notes intermediate between C and C J were heard at 
 once. 
 
 It appears, then, that a noise and a simple tone are extreme 
 cases of sound. The former is so complex that the ordinary 
 powers of the ear fail to resolve it. The latter is incapable 
 of resolution by reason of its absolute simplicity. 
 
 14. It is evident therefore that the question, Whal is an 
 absolutely simple tone ? or rather. What is the character of the 
 
Notes and Tones. ii 
 
 sound-waves which produce such tones, and of the correspond- 
 ing curves of pressure ? is of fundamental importance. 
 
 It is evident also that some correction is required to the 
 statement referred to above (Art. 7), that musical notes have 
 pitch. Strictly speaking, only an absolutely simple tone has 
 a single determinate pitch. When we speak of the pitch of the 
 note produced by a string or an organ-pipe, we mean in fact 
 the pitch of the gravest simple tone in the combination. This 
 particular component is in general louder than any of the 
 others, and is that on which attention is fixed. The others 
 become associated with it by habit, and seem only to modify 
 its character, without destroying the unity of its pitch. 
 
 15. As it will often be necessary to distinguish between 
 simple and compvound musical sounds, we shall always use 
 (as in the preceding articles) the words tone and note for this 
 purpose. 
 
 Objections may easily be made to both these terms. In fact 
 the word tone is often used with express reference to that very 
 complexity which it is here intended to exclude ; as when we 
 say that the tone of a violin is different from that of a clarinet, 
 or that any instrument has a good or bad tone. Again, note 
 properly signifies the written mark which indicates what musical 
 sound is to be played or sung, arid not the sound itself. 
 
 But it may be answered that tone (Gr. ToVor) really means! 
 tension, and the effect of tension is to determine the pitch of the 
 sound of a string. Hence the word may naturally be used 
 to denote a sound with reference only to its pitch ; and there- 
 fore, in particular, to denote a simple sound which has a single 
 pitch, and has (as will be seen hereafter) no other distinctive 
 quality except loudness or softness. 
 
 And with respect to note, it may be answered that the trans- 
 ition from the written mark to the thing signified is in fact 
 habitually made, as when we say that a person sings wrong 
 notes. 
 
 Again, the written note is a direction to sing or play, not 
 a simple sound, but that particular complex sound which is 
 produced by the voice or by the instrument intended to be 
 
I a Notes and Tones. 
 
 used; and the particular sound produced is in fact a note 
 or characteristic mark of the kind of instrument. In this sense 
 we speak of the note of the blackbird or of the nightingale. 
 
 16. For these reasons we shall henceforth denote a simple 
 musical sound by the word tone, and the elementary sound 
 of any instrument, such as that of a violin string, an organ-pipe, 
 or the human voice, by the word note. 
 
 Thus a note will be in general a complex sound, though 
 it may accidentally, through the peculiarity of a particular 
 instrument, be simple, or nearly so'. 
 
 ' Helraholtz uses the words Klang and Ton to signify compound and 
 simple musical sounds. We have followed him in adopting the latter 
 term. But such a sound as that of the human voice could hai-dly be called 
 in English a clang, without doing too much violence to estabUshed usage. 
 
CHAPTER II. 
 
 MISCELLANEOUS DEFINITIONS AND PROPOSITIONS. 
 
 The contents of this Chapter will consist chiefly of de- 
 finitions and statements of fact, which may conveniently be 
 introduced here, though partly belonging to a later stage of 
 the subject. 
 
 17. The word vihration may be used to denote any periodic 
 change of condition ; especially when, relatively to appropriate 
 standards of comparison, the change is small, and the period 
 short. 
 
 When a change is caUed periodic, it is in general implied 
 that a period is an interval of time of constant length, and that 
 whatever condition exists at any instant, is restored after the 
 lapse of a period. 
 
 Thus the series of changes which happen between any two 
 instants separated by a period, constitute a cycle which is con- 
 tinually repeated. 
 
 The particular stage of change which has been reached at 
 any instant is called tht phase of the vibration. And the vibra- 
 tion is said to go through all its phases in one period. 
 
 The period is also called the time of a vibration. It may, 
 and frequently does happen, that the changes in successive 
 periods grow successively less, though remaining similar in 
 character, while the period continues sensibly unaltered. Thus 
 when a string is put into a state of vibration and then left 
 to itself, the sound gradually dies away, but retains sensibly both 
 the same pitch and the same character as long as it is heard at 
 all. On the other hand, the period may change, as when the 
 
14 Pitch Dependent on Period. 
 
 tension of a string is increased or diminished while it is 
 vibrating. 
 
 But when nothing is said to the contrary, it is always to 
 be understood that the period is supposed to be constant. 
 
 18. When a musical note (Art. 16) is heard in the ordinary 
 way, the air in contact with the tympanic membrane of the 
 ear is in a state of vibration. But it has been already seen 
 that the ear in general resolves a series of vibrations into 
 several component series having different periods, each of 
 which produces a simple tone with a determinate pitch. 
 The tone produced by the component vibrations of longest 
 period, being the lowest in pitch, and also in general the 
 loudest, is called the fundamental tone, and its pitch is usually 
 spoken of as the pitch of the note. 
 
 We shall, however, suppose at present that the vibrations are 
 of that kind which the ear does not resolve, so that only one 
 tone is heard; and proceed to state certain facts relative to 
 the connection between the pitch of the tone and the period 
 of the vibrations. 
 
 To avoid circumlocution, we may call the vibrations which 
 produce any particular tone the vibrations of the tone ; and the 
 period of the vibrations may be called the period of the tone. 
 
 19. If the period of the vibrations be too long, no tone 
 is perceived, but only a succession of distinct impulses which 
 affect the ear with a peculiar sensation different both from 
 noise and tone. If, on the other hand, the period be too 
 short, no sensation at all is produced; the sound is simply 
 inaudible. 
 
 Different observers have made various statements as to the 
 period of the slowest vibrations which produce a tone. But 
 it is certain that to all ordinary ears the perception of pitch 
 begins when the number of vibrations in a second is some- 
 where between eight and thirty-two. In experiments on this 
 point there is an uncertainty arising from the doubt whether 
 the tone heard is really fundamental, or only one of the 
 component tones having a shorter period than that of the 
 compound vibration. For it is certainly difficult and perhaps 
 
Comparison of Intervals. 15 
 
 impossible so to arrange the experiment as to be quite sure that 
 the vibration is of that kind which the ear does not resolve. 
 
 The extreme limit in the contrary direction is known to 
 be diiferent to diflferent ears^- In general it is probable that 
 no tone is heard when the number of vibrations in a second 
 exceeds 40,000. 
 
 20. Difference of period causes a difference of pitch which 
 we describe by calling tones of shorter period higher, or more 
 acute, and those of longer period lower, or more grave. 
 
 The change of pitch which takes place when the period 
 is gradually altered, strikes us as having an analogy to the 
 gradual change of position of a moving point. Hence we say 
 that there is an interval between tones of different pitch. 
 
 Suppose P, Q, Ji are three tones, taken in order of pitch. 
 Then we say that the interval from F to Ji is the sum of the 
 intervals from F to Q and from Q to Ji; or briefly, that' 
 PJi = FQ+QJi. 
 
 So far as this we might go in many other cases of con- 
 tinuously varying sensation. For instance, we might speak of 
 the interval between two pains of the same kind, but of dif- 
 ferent intensities, and call one interval the sum of two others. 
 But in the comparison of tones we can make a further step 
 which we cannot make in the comparison of pains. For we 
 can compare with precision the magnitudes of two intervals. 
 The ear decides whether any interval P Q \% greater than, 
 equal to, or less than any other interval RS. How far this 
 faculty of comparing intervals is an absolutely simple and 
 ultimate property of the sense of hearing, it is not very easy 
 to decide, for reasons which will appear hereafter. 
 
 21. But however this may be, it is certain that the judgment 
 is made ; and it is a fact ascertained by experiment that any 
 interval P Q '\& judged to be equal to another interval R S, 
 whenever the periods (Art. 18) of the two tones F, Q, are to 
 
 ' This appears to have been first observed by Wollaston. See his 
 memoir ' On Sounds inaudible by certain Ears,' Phil. Trans, for 1820, 
 p. S06. On the audibility of high tones, see Savart in Ann. de Cbim. el 
 de JPhys., t. 44, p. 337 ; on low tones, Helmholtz, p. 263. 
 
i6 Measure of Intervals. 
 
 one another in the same ratio as the periods of the tones R, S. 
 The same proposition may of course be expressed by saying 
 that the intervals are equal when the ratios of the numbers of 
 ^ vibrations in a given time are equal; for the periods are in- 
 1 versely proportional to the numbers of vibrations. Thus, if 
 P, Qhe produced by 200 and 300 vibrations in a second, and 
 Ji, S by 600 and 900, then the intervals P Q, R S are equal, 
 for 200 : 300 : : 600 : 900. 
 
 22. In general, if a tone P be produced by / vibrations in 
 
 a second, and another tone Q \>y q vibrations, the ratio - 
 
 P 
 determines the magnitude of the interval PQ; for it follows 
 
 from the proposition of Art. 21 that no interval can be equal 
 
 to P Q unless the numbers of vibrations of its tones have 
 
 the same ratio. 
 
 But although this ratio determines the interval, it cannot be 
 taken as a measure of the interval, as we shall now shew. 
 
 Suppose Q is higher than P, and let 72 be a third tone 
 higher than Q, having r vibrations in a second. Then the 
 interval P R is the sum of the intervals P Q, QR; and there- 
 fore any number taken as a measure ol P R ought to be the 
 sum of the numbers taken by the same rule as the measures of 
 
 q r r 
 
 P Q, QR. Now the ratios -, -, -, which determine the three 
 
 p q p 
 
 intervals P Q, QR^ PR, do not fulfil this condition, for 
 
 or. , r 
 
 -H — IS not equal to -. 
 P S P 
 
 But if we take for the measure of an interval, not the ratio 
 of the numbers of vibrations, but the logarithm of that ratio, 
 
 f Of 
 
 the required condition is satisfied ; for - = - x -, and there- 
 
 r P P 9 
 
 fore the logarithm of - is the sum of the logarithms of - 
 , r P P 
 
 and -. 
 
 23. If then p, q be the numbers of vibrations, in a given 
 time, of two tones P, Q, the logarithm of - may properly be 
 taken as a numerical measure of the interval P Q. And in 
 
Symbolical Names of Intervals. 17 
 
 order to compare the magnitudes of different intervals we must 
 compare, not the corresponding ratios, but the logarithms of 
 those ratios. The base of the system of logarithms may be 
 any whatever. In fact the choice of the base merely de- 
 termines what interval shall be represented numerically by 
 unity, since the logarithm of the base is i in every system. 
 Thus, if we took 2 for ^ the base, we should have log 2 = 1; 
 and therefore i would be the measure of the interval which 
 has 2 (or f) for the ratio of the vibrations of its tones. In 
 other words, the octave (see Art. 31) would then be the unit- 
 interval. There would be some advantage in this; but prac- 
 tically it is convenient to use the common logarithms, of which 
 the base is 10. 
 
 24. Though the ratio - cannot be used as a measiure, it 
 
 P 
 is often convenient to use it as the name of the interval P Q. 
 
 When it is so used we shall inclose the fraction in brackets 
 -) ;' the brackets being intended to remind 
 the reader that the measure of the interval is not -, but 
 log t. 
 
 25. The interval between two tones P, Q, may, like the 
 interval between two points in a line, be reckoned in two 
 opposite ways, namely, from P to Q, or from Q io P ; and this 
 difference will henceforth be indicated by calling the same 
 interval P Q ox QP accordingly. 
 
 And if we introduce the signs + and — in the same way 
 as in modern elementary Geometry, it is evident that the rules 
 for the addition and subtraction of straight lines (in the same 
 direction) may be applied at once to the addition and sub- 
 traction of intervals. Thus, QP=-PQ,oxPQ + QP = 0. 
 And if P, Q, R be any three tones whatever, then 
 PR = PQ + QR = QR- QP; 
 ^•nd PQ+ QR + RP = 0, &c. 
 
 26. In designating any interval by the corresponding ratio 
 as a name, we shall put that number in the denominator which 
 
 c 
 
1 8 Addition and Subtraction. 
 
 is proportional to the number of vibrations producing the tone 
 from which the interval is reckoned. 
 
 Thus, ' the interval (|) ' will mean P Q, while ' the interval 
 (-)' will mean QP. 
 
 Then also log ^ being taken as the measure of P Q, log ^ , 
 P ? 
 
 will be the measure oi QP with its proper sign; for log 
 
 ^- = -logf. , 
 
 The equations - = ix-, - = --=--, compared with 
 
 PR = PQ+ QR, PR = QR— QP, shew that addition 
 and subtraction of intervals correspond to multiplication and 
 division of ratios ; the words addition and subtraction being no 
 longer restricted to their arithmetical sense, but used in the 
 same way as in the geometry of a straight line. 
 
 In fact this is only another way of saying that the logarithm 
 of the ratio is a proper measure (both as to magnitude and 
 sign) of the interval. 
 
 It follows, from the conventions above made, that those 
 intervals are to be considered ^positive which are reckoned from 
 a lower to a higher tone ; since the logarithm of a ratio greater 
 than unity is positive. 
 
 27. It will be a convenient abridgment to call the ratio 
 which determines an interval the ' ratio of the interval,' though 
 the expression is in itself unmeaning. 
 
 28. The interval between the lowest and highest audible 
 tones is theoretically capable of unlimited subdivision by the 
 interposition of intermediate tones, though there is a limit 
 to the power of the ear to distinguish nearly coincident tones, 
 as there is to the power of the eye to distinguish nearly co- 
 incident tints. 
 
 A series of tones at finite intervals, selected aceording to 
 some definite law, is usually and appropriately called a scale ; 
 for the selected tones are the steps of a ladder by which we 
 ascend from a lower to a higher pitch. 
 
 A scale formed by taking an unlimited succession of tones, 
 
Harmonic Scale. 
 
 19 
 
 produced by vibrations of which the numbers (in a given time) 
 are proportional to i, 2, 3, 4, &c. is called the scale ol Natural 
 Harmonics. We shall henceforward refer to it simply as ' the 
 harmonic scale.' 
 
 The different notes which can be produced from a simple 
 tube, used as a trumpet, belong to a scale of this kind. And 
 each note of the trumpet, of the human voice, of a vibrating 
 string, in short every musical note produced in any of the most 
 usual ways, is compounded of simple tones also belonging to 
 a harmonic scale. The explanation of these facts on me- 
 chanical principles will be given afterwards; at present we 
 state them as a reason for giving here a general view of this 
 primary and fundamental scale. 
 
 29. The absolute pitch of the lowest or fundamental tone 
 may be any whatever. If we choose the thirty-third part of 
 a second for the period of this tone, it will have the same pitch 
 as the lowest C of a modern pianoforte, according to a standard 
 now very commonly adopted ^- (If we chose a different period 
 we should of course merely transpose the scale without altering 
 the intervals.) 
 
 The series of tones will then begin as follows : 
 
 1234567 89 
 
 s 
 
 m 
 
 I22Z 
 
 d' 
 
 C, 
 
 10 
 
 G 
 
 11 
 
 
 12 
 
 IS 
 
 15 16 
 
 -fri^- 
 
 Bs^- 
 
 IC2 &c — : 
 
 * 
 
 g' 
 
 * 
 a' 
 
 * 
 
 bb' 
 
 b' 
 
 The numbers written above the notes are the numbers of 
 vibrations in the thirty-third part of a second. 
 
 ' This is the German pitch, 
 standard. 
 
 In -England there is at present no uniform 
 
 C 2 
 
20 Names and Ratios 
 
 Those notes which are marked with an asterisk do not 
 exactly represent the corresponding tones, but are the nearest 
 representatives which the modern notation supplies. 
 
 30. The letters below the notes are, with a slight alteration, 
 those used by German writers. In this • system of notation, C, . 
 D, E, F, G, A, B, represent the seven notes beginning with the 
 second C (reckoning upwards) of a modern pianoforte, or the 
 lowest note of a violoncello. 
 
 The octaves above these notes are represented by the small 
 letters Cj d, e, f, g, a, b. Thus, c is the lowest note of a viola, 
 and the lowest c of a tenor voice ; and g is the lowest note of 
 a violin. 
 
 Higher notes than b are represented by putting accents above 
 the small letters; and lower notes than C by putting accents 
 below the capitals. Each accent above a small letter raises the 
 tone by an octave ; and each accent below a capital lowers the 
 tone by an octave. 
 
 Thus, g" is the highest g of an ordinary soprano voice, C, is 
 the lowest C of a modern pianoforte, and C„ is the so-called 
 32-foot C of an organ. The compass of the newest piano- 
 fortes is from A„ to a"", &c. 
 
 As we shall use this notation for the future, the reader is 
 recommended to become familiar with it. 
 
 (German writers use h instead of our b, and b instead of 
 ourt'b.) 
 
 31. The intervals between the several tones of the harmonic 
 scale have in a few cases received names derived from the 
 places of the tones in the diatonic scale (Art. 35). 
 
 The most important of these intervals, with their ratios, are : 
 
 Octave C C 
 
 Fifth c' _ G 
 
 Fourth G c 
 
 Major third c e 
 
 Minor third e g 
 
 g — bb* 
 
 bb* — c' 
 
 T 
 3 
 
 4 
 
 'S 
 
 5 
 
 T 
 
 7 
 8 
 
 T 
 
of Important Intervals. 2 1 
 
 Major second c' — d' 
 
 Minor second d' — =- e' 
 
 Diatonic semitone . . , . b' — c" 
 
 8 
 
 is. 
 
 15 
 
 The two intervals (|^), (f) are not formally recognized in 
 modern music, though they are probably often used both by 
 singers and by players on instruments not restricted to a finite 
 number of notes. 
 
 The major and minor second are often called major and 
 minor tones; but it is desirable to avoid the use of the word 
 tone in two senses. Other intervals (such as the sixth, &c.), 
 formed by addition and subtraction of those in the above list, 
 may be omitted for the present. 
 
 32. The second and all following tones of the harmonic 
 scale are often called the ' harmonics' of the first or funda- 
 mental tone. The second tone is the first harmonic, the third 
 tone is the second harmonic, and so on. 
 
 If we take any tone of the harmonic scale as a new funda- 
 mental tone; the whole series of its harmonics will be found in 
 the original scale. 
 
 Thus the series 3, 6, 9, 12 . . . ., gives the harmonic scale of 
 the third tone ; and in general the series 
 
 n, 2w, 3«, 4«, , 
 
 gives the harmonic scale of the «th tone. 
 
 33. Omitting for the present any discussion of the character 
 of other intervals, we may here notice a peculiar property which 
 distinguishes the first interval in the harmonic scale, the octave. 
 When we compare two tones which differ by an octave, we are 
 affected by a certain sense of sameness, which we do not feel in 
 the case of any other interval unless it be a multiple of an 
 octave. A supposed explanation of this property will be dis- 
 cussed hereafter; but, whether it be. explicable or not, it entitles 
 the octave to be regarded as a natural unit (see Art. 23) with 
 which other intervals may be compared. At the same time it 
 gives a periodic character to the scale : every tone which has 
 occurred once seems to occur again and again at equal in- 
 tervals. 
 
22 Bisection of Intervals. 
 
 34. If we compare the intervals into which any two succes- 
 sive octaves of the harmonic scale are divided, we see that 
 every interval between consecutive tones in one octave is divided 
 into two intervals in the next higher octave. 
 
 The law of this subdivision is worth observing. The ratio 
 of the interval between consecutive tones is always of the form 
 
 «_±i. „ow ^+^ = ^^±^ = ^^^ X ^^^'; in other 
 
 y.« + In. . , . , 
 
 words (see Art. 26) the interval \ „ ) »s the sum of the 
 
 two intervals ( \ ( ,^ , A and it is in fact divided 
 
 into these two in the next higher octave; for in that octave 
 occur three consecutive tones corresponding to the numbers 
 zn, zn + 1, zn + z, of vibrations. 
 
 Thus, the first octave, C, — C, is undivided. The second 
 octave, C — c, is divided into a fifth and a fourth, 
 
 2 _ 4. _ 3 V *- 
 T - ■^ - 2, '^ 'S- 
 
 In the third octave the fifth, c — g, is divided into a major 
 and a minor third, f = f = f >^ t > 
 
 .whilst the fourth, g — c', is divided into two interyals, (•^), (f), 
 which have not received names. In the fourth octave the major 
 third, c' — e', is divided into a major and a minor second, 
 
 5 _ 10 _ 9 V 10 
 
 1 - S ~ 'S ■^ ■ff • 
 Thus every interval is divided into a major and minor half (if 
 the word half may be so used), according to a law which may 
 be called the law of natural bisection, the major half being 
 always the lower. 
 
 35. The theory of artificial scales cannot be discussed here ; 
 but it will be useful to state, without reference to theory, the 
 actual construction of the modern diatonic major scale. If we 
 take two tones at an interval of a fifth, and the intermediate 
 tone which bisects the fifth naturally (Art. 34), for example, 
 c, e, g, we obtain three tones which when sounded together 
 produce a triad or common chord. And if we take three such 
 triads one above another, so that the highest tone of the first 
 is the lowest of the second, and the highest of the second the 
 
Diatonic Scale. 23 
 
 lowest of the third, we obtain seven tones, rising one above 
 another by alternate major and minor thirds, thus : 
 
 F — A — c — e — g — b — d' 
 6. e. 6 e 5 8 
 
 4 6 T 6 ¥ T 
 
 Lastly, if we take the lowest tone (c) of the middle triad as 
 the so-called ionic or first tone of the scale, and bring all the 
 other tones within the compass of an octave by substituting f, 
 a, d, for F, A, d', we obtain the seven tones of the diatonic 
 scale, to which an eighth (viz. c', the octave of the tonic) is 
 usually added, thus : c, d, e, f, g, a, b, c'. 
 
 Returning, however, to the above system of triads, let us find 
 the ratios of the intervals from the lowest tone, F, to each of 
 the other tones. This will be done by successive multiplication 
 of the ratios |, f, &c. (Art. 26), and thus we obtain the fol- 
 lowing ratios for the intervals : 
 
 F, A, c, e, g, b, d'. 
 
 1 6 3 1594S27 
 T> T» ■?> F > T> Tff) ¥ • 
 
 n. 
 
 Here the ratio written under each tone is that of the interval 
 from F to that tone. 
 
 If we reduce all these fractions to the least possible common 
 denominator, viz. 16, the numerators will be the smallest whole 
 numbers proportional to the numbers of vibrations of the cor- 
 responding tones, thus : 
 
 F, A, c, e, g, b, d'. 
 16, 20, 24, 30, 36, 45, 64. 
 
 Recollecting now that the number of vibrations of f is double 
 that of F, &c., we obtain the following series for the diatonic 
 scale : 
 
 c, d, e, f, g, a, b, c'. 
 24, 27. 3o> 32, 36, 40, 45> 48- 
 
 36. The tones of the diatonic scale have all received technical 
 names, of which it is sufficient to mention three, viz. the tonic 
 or first tone of the scale, the dominant or fifth tone, and the 
 subdominant or fourth tone. Thus, in the above scale c is the 
 tonic, g the dominant, and f the subdominant. 
 
24 Relation of Diatonic 
 
 37. The tones of a diatonic scale, having for their actual 
 fnumbers of vibrations those given at the end of Art. 35, belong 
 ) evidently to the harmonic scale (Art. 28), of which the funda- 
 mental tone has one vibration. This fundamental tone is five 
 octaves below the subdominant, for \^ = (f )°. 
 
 ~ Hence, neglecting the difference between tones which differ 
 by a whole number of octaves, we may say that the diatonic 
 scale is selected from the harmonic scale of its subdominant. 
 
 This proposition is to be understood merely as the statement 
 of a mathematical fact, and not as involving any theory of the 
 actual derivation of the scale. 
 
 The diatonic scale can only be represented in whole num- 
 bers when the number of vibrations of its tonic is divisible both 
 by 3 and 8. 
 
 Hence we may say that any harmonic scale contains the 
 tones (or their octaves) of the diatonic scales of all those 
 amongst its tones which correspond to multiples of 3, and of 
 no others. 
 
 The series of multiples of 3, viz. 3, 6, 9, 12, ... ., gives a 
 harmonic scale which has for its fundamental tone the third 
 tone of the original scale. 
 
 Thus, every harmonic scale may be said to contain the 
 diatonic scales of its third tone, and of all the harmonics of 
 that tone. 
 
 38. Returning now to the series of whole numbers which 
 represent the diatonic scale (Art. 34), we find for the intervals 
 Detween successive tones of the scale the following ratios : 
 
 ff = f • (major second). 
 
 1^ =~V' (minor second). 
 
 14 = TT (diatonic semitone), 
 
 f I = f (major second). 
 
 If = V* (minor second), 
 
 rf = f (major second). 
 
 rf = XT (diatonic semitone). 
 
 39. In this scale the octave is divided into two so-called 
 tetrachords, c, d, e, f ; g, a, b, c', separated by the major second 
 
 c . . 
 
 ■ d. 
 
 d . . 
 
 • e, 
 
 e . , 
 
 •f, 
 
 f . . 
 
 • g. 
 
 g • • 
 
 . a. 
 
 a . . 
 
 ■ b. 
 
 b . . 
 
 T._ a.1_ 
 
 • c, 
 
to Harmonic Scale. 25 
 
 f g. These tetracLords are nearly, but not exactly, alike; 
 
 for the intervals in the lower and upper tetrachords are : 
 
 (lower) major second, minor second, semitone ; 
 
 (upper) minor second, major second, semitone. 
 
 Hence, the upper tetrachord of the scale of c is not exactly 
 identical with the lower tetrachord of the scale of g ; for the a 
 in the latter scale is a major second above g, and in the former 
 a minor second only. The^ a of the scale of c is therefore 
 flatter than that of the scale of g by the difference between 
 a major and a minor second. This difference is called a 
 comma; and the ratio of a comma is (Art. 26) 
 
 9 _j_ 10 _ 81 
 "5" • ■9 - F(7- 
 
 A comma is nearly equal to the fifth part of a diatonic semi- 
 tone ; for (f^)' is nearly equal to ^. 
 
 These details belong more properly to another chapter ; but 
 ■ they have been given here in order to shew at once, by a simple 
 example, the imperfection of the ordinary musical notation for 
 all but practical purposes. The letter a and the note (m); '^— 
 are made to serve for the scales both of c and g, to ^^ 
 say nothing of other scales; whereas in fact the letters, as 
 ordinarily used, and the notes on the stave, are capable of 
 representing accurately one diatonic scale and no more. 
 
 The whole structure, however, of modern music is founded 
 on the possibility 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. 
 
 40. The relation between intervals and numerical ratios 
 may be illustrated by the curve called the logarithmic (or equi- 
 angular) spiral. This curve (see Fig. 5) consists of convolu- 
 tions, of which the number reckoned from any point, either 
 inwards or outwards, is infinite. Inwards, the curve approaches 
 continually nearer to a point (called the pole), which it never 
 actually reaches; Outwards, it recedes from the pole without 
 limit. These two properties are common to many spirals. 
 But the curve in question has this particular property, that if 
 straight lines be drawn from the pole to any two points on the 
 
26 
 
 Graphic Representation 
 
 curve, the logarithm of the ratio of their lengths is proportional 
 to the angle between them. It follows (Art. 23), that if the 
 lengths of any two such lines, as A, OJB,he taken to repre- 
 sent numbers of vibrations in a given time, the angle A OS 
 may be taken as a measure of the interval between the tones 
 produced by those vibrations. 
 
 'J}. 
 
 v5' 
 
 y' 
 
 T?" 
 
 Fig. 5- 
 
 The curve can be so drawn, and is so drawn in the figure, 
 that a complete revolution' doubles the distance from the pole. 
 Thus, Oa is double of OA, and Od of Oa, Sec; so that a 
 point, starting from A and following the curve outwards, doubles 
 its distance from whenever it crosses the line A produced, 
 
of Scale. 27 
 
 that is, whenever it completes a revolution. But, when the 
 number of vibrations is doubled, the tone is raised an octave. 
 Hence the angle described in a complete revolution, that is, 
 four right angles or 360°, represents an octave; and any frac- 
 tion or multiple of four right angles represents the same fraction 
 or multiple of an octave. 
 
 We may, therefore, consider a point, following the curve 
 outwards, to represent a tone continually rising in pitch ; and 
 successive passages of the point through any given line drawn 
 from the pole will represent passages of the tone through suc- 
 cessive octaves. Thus the geometrical periodicity of the curve 
 presents to the eye a sort of picture of the periodicity perceived 
 by the ear in a continuously rising tone. 
 
 The angles representing the intervals used in the diatonic 
 scale cannot (with the exception of the octave) be exactly 
 expressed in degrees, minutes, and seconds ; but, to the nearest 
 second, they are as follows^ : 
 
 Octave 360 o o 
 
 Fifth 210 35 II 
 
 Fourth 149 24 49 
 
 Major third "S S3 38 
 
 Minor third 94 41 33 
 
 Major second 61 10 22 
 
 Minor second 54 43 16 
 
 Diatonic semitone . . . . 33 3^ ^^ 
 (Comma) 6 27 6 
 
 ^ The equation to the curve is r = a . 2 ^ " ; hence, if r, r' , be two radii nec- 
 tores in the ratio oim.n, we have 
 
 m 2jr . 
 
 — = 2 y 
 
 n 
 
 log m — log n 
 hence «-"='" T^T^ = 
 
 which gives the angular measure of the corresponding interval. Thus, for 
 the fifth, — = - ; hence the angle (in degrees) is 
 
 log 2 
 The angles being known for the fifth and the fourth, the rest of those which 
 occur in the diatonic scale can be found by addition and subtraction. 
 
28 Division of Octave into Atoms. 
 
 41. In Fig. 5 the letters represent the tones of three octaves 
 of the diatonic scale. It may be observed that the same figure 
 might be used to represent a descending instead of an ascend- 
 ing scale. The distances from the pole would then be pro- 
 portional to the periods of the tones, or (as will be s^ewn 
 hereafter) to the lengths of the strings, of given kind and given 
 tension, which produce them. : The angles would then have to 
 be taken in the reverse orden 
 
 42. A more usual mode of representing intervals graphically 
 is by parts of a straight hne, of which the whole is assumed to 
 represent an octave. When this method is adopted it is con- 
 venient to divide the octave line into equal parts, of which the 
 number is approximately the product of some power of i o by 
 the logarithm of 2. Thus, if we divide it into 3.01 parts (301 
 being 10' x log 2 nearly) the logarithm of the ratio defining 
 the interval measured by n divisions will be, approximately, 
 
 ft 
 ■ . Conversely, the number of divisions representing any 
 
 interval will be 1000, multiplied by the logarithm of the ratio 
 of that interval. Thus, a comma will be represented by 
 
 1000 X logf^= 5 divisions nearly. 
 A division on this scale corresponds to something more than 
 a degree on the scale described in Art. 40. Dr. Young divided 
 the octave into 30,103 parts, and Mr. De Morgan has proposed 
 to call each of these parts an ' atom.' An atom would be very 
 nearly equal to 43" on the scale of Art. 40. (See De Morgan, 
 ' On the Beats of Imperfect Consonances,' Camb. Phil. Trans. 
 vol. X. p. 4.) 
 
CHAPTER III. 
 
 COMPOSITION OF VIBRATIONS. 
 
 43. The simplest type of periodic motion is afforded by a 
 point describing a circle with a constant velocity. For this is 
 the only kind of motion in which the velocity and the change 
 of direction are both uniform. 
 
 Such a motion may be considered as the simplest possible 
 vibration of a point (Art. 17). We shall call it, for the present, 
 a simple circular vibration. 
 
 44, The vibration of a point, in its most general form, may 
 be defined as motion in a curve which returns into itself, with 
 a velocity which is always the same at the same point of the 
 curve. 
 
 Suppose the curve to be plane, such, for instance, as in 
 Fig. I. Draw arbitrarily any 
 two axes, OX, OF. (It is im- 
 material whether they meet 
 the curve or not). From P, 
 the position of the moving 
 point at any time, draw P M, 
 PN, parallel to OF, OX. 
 As P moves round and round 
 the curve, the point M will 
 move backwards and forwards 
 in the line O X with a pe- 
 riodic rectilinear motion; that 
 is, .^will perform rectilinear 
 vibrations. In the same way N will perform rectilinear vibra- 
 tions in OF. 
 
30 Composition and Resolution 
 
 The curvilinear vibration of P is said to be compounded of the 
 rectilinear vibrations of M and N. 
 
 In fact, if we consider P as a point displaced from 0, the dis- 
 placement OP is compounded of the two displacements M, 
 N, in the sense explained in Art. 9. It is evident that any 
 given vibration of a point in a plane may in this manner be 
 . resolved into two component rectilinear vibrations, in an infinite 
 variety of ways ; and the character of each component depends 
 in general upon the directions of both the axes. 
 
 When nothing is said to the contrary, it is assumed that the 
 axes are rectangular. In this case P M is perpendicular to 
 O X ; and the vibration of M (the orthogonal projection of P) 
 is then called absolutely the rectilinear component, in the direction 
 O X, of the vibration of P. 
 
 45. Any vibration of a point, whether plane or not, may be 
 similarly resolved into three component rectilinear vibrations by 
 taking three arbitrary axes, OX, OF, O Z. The point M, in 
 O X, is then determined by drawing P M parallel to the plane 
 YO Z. The projections of P on J^ C Z, are to be found in 
 the same way, mutatis mutandis. 
 
 46. We will now return to the case of a simple .circular 
 vibration (Art. 43). Such a vibration is completely determined 
 when four things are given : 
 
 (i) Th& period, or time of describing the whole circle. 
 
 (2) The radius of the circle. 
 
 (3) The position of the moving point at some one given 
 
 instant. 
 
 (4) The direction of motion (whether right-handed ^^^^, or 
 
 left-handed i^^"^). 
 
 The phase of the vibration is defined by the angle between 
 an arbitrary fixed radius and the radius of the moving point at 
 any time. 
 
 Hence, the third of the above data may be expressed by 
 saying that the phase is given at a given instant. The fourth 
 datum will seldom have to be referred to. 
 
 Through the centre of the circle (Fig. 2) draw any two 
 
of Circular Vibrations. 
 
 31 
 
 rectangular axes, OX, OF, and from P, the position of the 
 moving point at any time, draw P M, P N, perpendicular to 
 the axes. (The origin is taken at the centre for convenience, 
 but it might be any- 
 where.) Then, as P de- 
 scribes the circle with a 
 uniform motion, M vi- 
 brates in the line A A', 
 and iVin ££', and it is 
 evident that these two- 
 rectilinear vibrations are 
 perfectly similar. 
 
 Vibrations of the kind 
 performed by M or N, 
 that is, rectiUnear com- 
 ponents of simple cir- 
 cular vibrations, are dis- 
 tinguished by many re- 
 markable properties, and may properly be considered as the 
 simplest kind of rectilinear vibrations. This will appear more 
 clearly as we proceed. 
 
 47.' Let an arbitrary radius, K, be taken as a fixed direc- 
 tion from which to measure the angle 6 (01 K O P) described 
 by OP, and suppose the direction of the motion to be such 
 that 6 increases with the time. Then, putting a for the radius 
 of the circle, and a for the angle K A,\iq have 
 
 ON = a sin {6 + a). 
 
 Suppose the time, /, is reckoned from the instant of a pas- 
 sage of P through K, and let t be the period of the vibration, 
 or time of describing the whole circumference. Then the time 
 
 a 
 
 of describing the arc ^P is — t, and when the moving point 
 
 e 
 
 is at P the value of i must be /= — 7- + «r, n being some 
 
 2 IT 
 
 whole number ; whence we have 6 = 2 « jr. Intro- 
 
32 Rectilinear Vibrations. 
 
 ducing this value of 6 in the above value of N, and putting 
 N = y, we find 
 
 J/ = a sin ( ha) ( ■ ) 
 
 as the equation which determines the position of N at the 
 time /. 
 
 In like manner, putting M = j;, w6 should find 
 
 '2 TT / 
 
 X = a cos 
 
 (^S a); (.) 
 
 and, if the latter equation be put in the form 
 
 '2 tt/ 
 
 X = a^vx 
 
 +- + a), 
 
 T 2 ■^ 
 
 it is seen that both (i) and (2) represent vibrations, of the same 
 kind, but that x andj' take the same values at different times. 
 In fact, the value of x at time t is the same as that of^ at time 
 
 f 
 
 i -\ — , so -that the first vibration may be said to be a quarter 
 
 4 
 of a period behind the second. 
 
 48. Vibrations of the kind considered in the last article, 
 namely, rectilinear vibrations in which the displacement of the 
 moving point at the time / can be represented by an expression 
 of the form 
 
 "z IT i 
 
 (i:Li+„), (3) 
 
 a sm 
 
 ' T 
 
 may be conveniently called rectilinear harmonic vibrations. 
 
 The constant a is called the amplitude, because its value is 
 that of the greatest displacement. 
 
 The angle 1- a is called the phase of the vibration. 
 
 The constant a is therefore known if the phase at any given 
 time be given. 
 
 49. A rectilinear harmonic vibration can be resolved into 
 two others of the same kind in an infinite variety of ways. 
 Usually the directions of the two components are taken at right 
 angles to one another. In this case the component of (3) 
 
Composition of. Vibrations. 33 
 
 (Art. 48), in a direction making the angle a with the direction 
 of (3), is evidently 
 
 h ay 
 
 Composition of Vibrations. 
 
 50. Rectilinear harmonic vibrations may be compounded 
 according to the general law of the superposition of displace- 
 ments explained in Art. 9. The two cases of most importance 
 are those in which the two vibrations to be compounded are, 
 (i) in the same direction, (2) in directions at right angles to 
 one another. 
 
 First, then, let the vibrations be in the same direction; in 
 this case the resultant displacement is simply the algebraical 
 sum of the component displacements. If the component vibra- 
 tions be represented by the expressions 
 
 (Ivt V ,2 IT i ^ 
 h «J , b sin y—, h P) , 
 
 the resultant displacement wiU be represented by the sum 
 
 ,2 TT / V /lit t « 
 
 a sin (-^ + °) + 3 sin (-^^ + /3). 
 
 This expression is periodic if the periods r and t be com- 
 mensurable ; for its value will then evidently be unaltered if / 
 be increased by any common multiple of t and t'. But it does 
 not in general admit of any useful reduction. 
 
 If, however, the periods r, t', of the component vibrations be 
 equal, the resultant motion is itself a harmonic vibration having 
 the same period. For in this case the above expression may 
 
 be written in the form 
 
 27r/ .. »•^^ iv t 
 (a cos a -H 3 cos /3) sm h (« sm a -|- 5 sm /3) cos ; 
 
 T T 
 
 and this is identical with 
 
 provided A and B be so taken that 
 
 A cos B = a cos a ■\- h cos 3, 
 .(4 sin 5 = a sin u + 3 sill ^, 
 
 D 
 
34 Composition of Vibrations 
 
 which equations are satisfied by 
 
 A = v/a^ + i5^ + 2 a 3 cos (a — ^), 
 
 „ , « sin a + 3 sin ^ 
 
 B = tan-i —7 ~, 
 
 a cos a -\- b cos j3 
 
 where we may suppose the square root to be taken posi- 
 tively. 
 
 The value of A, the amplitude of the resultant vibration, 
 depends both on the amplitudes a, I, of the component vibra- 
 tions, and also on the angle a~*~^, which is the difference of their 
 phases. 
 
 If a — /3 = o, the component vibrations are always in the 
 same phase, and A = a -^ b; ox the amphtude of the resultant 
 is the sum of the amplitudes of the components. 
 
 If a -»- ;3 = !r, one of the component vibrations is half a 
 period behind the other, and A = a~^ b; or the amplitude of 
 the resultant is the difference of amplitudes of the components. 
 In this latter case, if a = 3, then A = o, and the component 
 vibrations completely destroy one another. 
 
 51, Next, let the component vibrations be at right angles to 
 one another, and let their periods be r, t; putting for conve- 
 
 2 TT 2 !r , , , 1 
 
 nience — = », — ^ = « , we may represent them by the equations 
 
 T T 
 
 X = a sin (ni + a), y = b sin («7 + /3), 
 
 and these are the co-ordinates, at the time t, of the moving 
 point (see Art. 48). 
 
 The equation to the locus of the point would be found by 
 eliminating / between these two equations. The elimination 
 can, theoretically, be performed so as to lead to an algebraical 
 equation between x and y, whenever t and t are commen- 
 surable, though the process is impracticable except in simple 
 cases. 
 
 The simplest case of all is that in which the periods of the 
 component vibrations are equal. We have, then, to eliminate 
 t between the two equations 
 
 X = asia. {nt + a), y = i sin (»/ + /S) ; (4) 
 
at Right Angles. 35 
 
 from them we have 
 
 X 
 
 cos a sin nt + sin a cos ni = — , 
 
 a 
 
 y 
 
 COS ^ sm «/ + sm (3 cos nt =^ ; 
 
 
 
 and if the values of sin nt, cos «/, be found from these equa- 
 tions, and the sum of their squares be equated to 1, the 
 result is 
 
 ? ■''i^ ~ ^ 3 ^°^ (° - ^) - sin^ (a - ^) = o. (6) 
 
 52. This equation always represents an ellipse, which may 
 however degenerate into a circle (when the axes are equal), or 
 a straight line (when the length of one axis is o). 
 
 The dimensions and position of the ellipse depend only on 
 the amplitudes {a, b) and the difference of phases (a -»~ |3) of 
 the component vibrations. If the component vibrations are 
 always in the same phase, that is, if a — /3 = o, the equation 
 becomes 
 
 and the ellipse degenerates into a straight line, or, more strictly, 
 into two coincident straight Unes. Again, if a -»~ ^ = jr, that 
 is, if one of the component vibrations be halfz. period behind 
 the other, the equation becomes 
 
 /X Vsf 
 
 and the locus is again a straight line. 
 
 If a -»- 13 = -, or — , that is, if one component be a quarter 
 
 of a period behind the other, the equation becomes 
 x^ y^ 
 
 arid the axes of the ellipse coincide with the directions of the 
 component vibrations. 
 
 53. It is evident from equations (4) that the ellipse described 
 
 D 3 
 
36 
 
 Case of Equal Periods. 
 
 by the vibrating point is always inscribed in a rectangle, of 
 which the sides are 2 a, 2 b, since the extreme values of x and 
 
 J/ are + a, + ^ (see Fig. 3). This may be formally proved 
 from equation (5) ; for if in that equation we put y = h, the 
 result is 
 
 2 
 
 (- — cos (a — /3)) = O, 
 
 shewing that the ellipse meets the line_y = 3 in two coincident 
 points (as at Q), of which the abscissa is a cos (a — ^). Simi- 
 larly, by putting x = a,^Q find b cos (a — ^) for the ordinate 
 of the point of contact R. 
 
 When u = ft ^ and R coincide at C, and the ellipse dege- 
 nerates into the diagonal CC; and when a~^ ^ = jt, it degene- 
 rates in hke manner into the diagonal DD. 
 
 An ellipse inscribed in a' given rectangle is completely deter- 
 mined if one point of contact be given. Hence, it follows that 
 if the angle a — /3 varied continuously from o to tt, the ellipse 
 would pass continuously through every form capable of being 
 inscribed in the rectangle DC D C, including the two dia- 
 gonals as extreme cases. 
 
 54. Let us now suppose B B (Fig. 3) to be the axis of a 
 cylinder, of which DC, CD are the circular ends seen edge- 
 wise by an eye placed at a distance in a line through per- 
 
General Case. 37 
 
 pendicular to the plane of the paper. Suppose also the cylinder 
 to be transparent, and to have marked on its surface the trace 
 of a section made by a plane touching' the circular edge of 
 each end. This section would be an ellipse, which, as seen by 
 the eye, would be orthogonally projected into another ellipse 
 inscribed in the rectangle D C D C . And if the cylinder were 
 turned uniformly about its axis, this inscribed ellipse would 
 evidently go through all its possible forms in such a manner 
 that the distance B Q would be equal to a multiplied by the 
 cosine of a uniformly varying angle. 
 
 Comparing these results with those of the last article; we see 
 that this construction gives a perfect representation of the 
 change undergone by the ellipse which is the resultant of two 
 rectilinear harmonic vibrations at right angles to one another, 
 when the difference of phase of the component vibrations varies 
 uniformly ; the angular velocity of the rotating cylinder being 
 equal to the rate of variation of the difference of phase, (For 
 BQ = aco%{a-^)). 
 
 55. A similar construction may be employed in the much 
 more general case of any two rectilinear vibrations of which 
 the periods are commensurable, and of which one, at least, is 
 harmonic ; a case which could rarely be treated algebraically 
 (Art. 50). 
 
 For the sake of clearness, let us suppose the direction of the 
 harmonic vibration to be horizontal, or in the axis of x. This 
 vibration will therefore be represented (Art. 47) by an equa- 
 tion of the form 
 
 X = asm ( + a) • (6) 
 
 Let / denote any periodic function, which is always finite, 
 and such that/"(3) =/{z + 2,ir). Then the vertical vibration 
 may be represented by an equation of the form 
 
 . y-/C-^ + ^). (7) 
 
 Let us also suppose the ratio of the periods t, t to be such that 
 fiT = MT, m and n being integers. 
 
 Now let a curve be drawn in which the abscissa of any point 
 
38 Representation by 
 
 is proportional to the time /, and the ordinate is the corre- 
 sponding value of _y givep by equation (7). (Fig. 4, in which 
 AB = T, may be taken to represent two periods of such a 
 curve.) And suppose the unit-line to have been so chosen 
 
 Fig. 4. 
 
 that a line equal to n x AB shall be equal to m times the 
 circumference of a circle of which the radius is a. If, then, 
 a rectangular slip of paper were cut out, containing exactly 
 n periods of the curve, it could be rolled m times round a 
 cylinder of radius a. We should thus obtain a complicated 
 curve on the ..surface of the cylinder. Suppose the whole to 
 become transparent, and to be viewed in the manner explained 
 in Art. 53; then if a point P were to describe the curve in 
 such a manner that the projection of F on the base of the 
 cylinder should describe the circumference of the circle uni- 
 formly, the horizontal motion of P, as seen by the eye, would 
 be a harmonic vibration, while its vertical motion would be 
 identical with that defined by equation (7). Moreover, m hori- 
 zontal vibrations would be completed in the same time as « 
 vertical vibrations. 
 
 Hence the apparent co-ordinates of P would be always iden- 
 tical with the values of x and y in equations (6) and (7), pro- 
 vided the cylinder were turned (if necessary) about its axis into 
 such a position as to give x and y corresponding values at any 
 one instant ; and thus we should get, as before, a perfect repre- 
 sentation of the resultant vibration. 
 
 56. An alteration in the value of either of the angles a or /3 
 (equations (6), (7)) would alter the time at which the corre- 
 sponding vibration passes through a given phase, and cause 
 different values of x andj/ to become contemporaneous. And 
 
Transparent Cylinder. 39 
 
 the same effect would be produced to the eye by turnmg the 
 cylinder about its axis into a new position. 
 
 Thus, if a were changed into a + c, the effect would be the 
 same as if the cylinder were turned backwards (that is, with a 
 motion contrary to that of the projection of P) through^ an 
 angle e, since in either case the passage of the horizontal vibra- 
 tion through a given phase would be accelerated by a time — r. 
 
 A retardation of the vertical vibration would have the same 
 effect as an equal acceleration of the horizontal vibration ; now, 
 
 if /3 were changed into ^ — c', the retardation would be — /, 
 
 and this would be equal to the former acceleration, if e'r' = c r, 
 
 T 
 
 or f = - f . 
 
 T 
 
 Hence, a change of /3 into |3 + e' is equivalent to a change 
 of a into a H — e', that is, to a turning of the cylinder backwards 
 
 T 
 
 through an angle - c', or - e'. 
 r n 
 
 57. Since the curve rolled on the cylinder consists of n 
 
 similar portions, the angular distance between corresponding 
 
 . 2 n- 
 pomts on two consecutive portions is ^. 
 
 Hence, if the cylinder were turned continuously, .the pro- 
 jected curve seen by the eye would go continuously through 
 all its possible forms during a rotation of the cylinder through 
 
 2 TT 
 
 the angle — . 
 n 
 
 We see also that a uniform variation either of a, or of /3, 
 
 would have the same effect as a uniform rotation of the 
 
 cylinder^. 
 
 58. The results obtained in Arts. 52-55 may be exhibited 
 to the eye. But before explaining the mode of doing so, we 
 
 > This mode of representing stereoscopically tlie jiomposition of vibrations 
 is due to M. Lissajous. See his memoir ' Sur I'Etude optique des Mouve- 
 ments vibratoires,' 4nn. (fe C*. «/ £?e Piys., t. 51, p. 147. 
 
40 Case in which the Periods 
 
 will consider the case in wHich the ratio of the periods of the 
 two vibrations is nearly, but not exactly, expressed by a given 
 numerical fraction. 
 
 Suppose then, as before, that t, t are given constant quan- 
 tities, such that niT = m' (the ratio m : n being in its lowest 
 terms). And suppose that the vertical vibration is the same 
 as before (see equation (7), Art. 54), but that' the horizontal 
 (harmonic) vibration is now defined by the equation 
 
 X = a sin( — (i + k)t + a\, 
 
 so that its period is no longer t, but , which will differ 
 
 I + « 
 
 slightly from t, if ^ be a small quantity, positive or negative. 
 If now we put a H / = a , the above equation becomes 
 
 X = a sm( 1- a\, 
 
 and the result of eliminating / between this last equation and (7) 
 would be the same as before, except that a would be changed 
 into a, that is, into a uniformly varying instead of a constant 
 quantity. We may, therefore, consider that the vibrating point 
 still describes the curve (8), but that the curve is disturbed by 
 the variation of u, just as we consider a planet to describe an 
 ellipse disturbed by the variation of one or more of its ele- 
 ments. 
 
 59. It has been seen (Art. 57) that a uniform variation 
 of a would have the same effect as a uniform rotation of the 
 cylinder with an angular velocity equal to the rate of variation 
 
 of a, that is, in the case now supposed, to ; and also that 
 
 the curve, as seen by the eye, would go through all its forms 
 
 while the cylinder turned through an angle — . Hence the 
 
 n 
 
 curve would go through all its forms in a time equal to 
 — -«- , or -T. Suppose that M,N, are the actual num- 
 
 n T ft fi 
 
are not exactly Egtial. 
 
 41 
 
 bers, in a unit of time, of the horizontal and vertical vibrations ; 
 
 , .. 1 + k , ,- I n 
 
 then M = , and iV^ = - = — . 
 
 T T mr 
 
 Now the number of complete cycles of change of the curve 
 in the unit of time is — , and this is equal to 
 
 T 
 
 nM — mN. (9) 
 
 This expression is worthy of remark, especially in connection 
 with the theory of the so-called beats of imperfect consonances, 
 as will appear afterwards. 
 
 The sign of nM— mN, being the same as that of k, is 
 positive or negative according as the ratio M : N \s greater or 
 less than m : n. In the former case, the rotation of the cylinder 
 is backwards ; in the latter, forwards. 
 
 In the particular case in which the periods of the vibrations 
 are nearly equal, or m = n = \, the expression (9) becomes 
 simply M — N. Or the number of cycles of change in a unit 
 of time is the difference of the actual numbers of vibrations. 
 
 60. The two Figures, 5, 6, illustrate the case in which both 
 vibrations are harmonic and have the same amplitude, while 
 
 Fig. 5. 
 
 Fig. 6. 
 
 the periods are such that two horizontal vibrations occupy the 
 
 same time as three vertical. Suppose 3T, 2t, are the two 
 
 periods ; then the curve in Fig. 5 is defined by the equations 
 
 . ini . "" t 
 
 X = a^va. — , V = fl sm — , 
 
 3r T 
 
42 Particular Case discussed. 
 
 while in Fig. 6 the first equation is changed into 
 
 . /2 IT t 7r\ 
 
 X = a sin ( I . 
 
 V 3T 6^ 
 
 (The origin is in each case at the centre of the square). 
 
 These two curves are particular cases of the appearance 
 
 presented to the eye by a transparent cylinder having a curve 
 
 traced upon its surface in the manner above described, that is, 
 
 in the present case, as follows. Let three complete ' waves,' or 
 
 periods of the curve j/ = a sin - — , be drawn upon a rectangular 
 
 slip of paper, so that the axis of x is parallel to a side of the 
 rectangle; the length of a rectangle which will just contain 
 them will be 4 tt a, and such a rectangle can therefore be rolled 
 twice round a cylinder of radius a. Suppose this to be done, 
 and the whole to become transparent. Then, if the cylinder be 
 held vertically at a distance from the eye, and turned about its 
 axis, the curve will appear to go through a series of forms of 
 which two are represented in the figures. Fig. 5 is changed 
 into Fig. 6 by a rotation through 30°; a second rotation through 
 30° brings back Fig. 5, with those parts of the curve va. front of 
 the cylinder which were at the back, and vice versd ; a third 
 similar rotation produces Fig. 6 reversed right and left, and a 
 fourth reproduces the original Fig. g. Thus a whole cycle of 
 
 forms is completed in a rotation through 120°, or — (see 
 
 3 
 Art. 57). A greater number of the forms belonging to this 
 and other cases of composition of vibrations may be seen 
 figured in Tyndall's ' Lectures on Sound,' p. 319. They were 
 originally given in the memoir of Lissajous above cited. 
 
 61. If the periods of the component vibrations were nearly, 
 but not exactly, in the ratio of 3 to 2, then (as in Art. 58) the 
 locus of the vibrating point might be represented by the equa- 
 tions 
 
 . (Iirt ,\ . tt/ 
 
 X = fl sm 1 ha), _>< = a sin — , 
 
 3'' T 
 
 where a is an angle which varies slowly and uniformly with 
 the time. 
 
Wkeatstonis Kaleidophone. 43 
 
 In this case the path of the point never actually coincides 
 with any curve corresponding to a constant value of d, just as 
 the path of a disturbed planet is never actually an ellipse ; but 
 if the variation of d be suflSciently slow, the path during one or 
 more repetitions of the period 3 t will be sensibly the same as 
 if d were constant, and during a longer time will appear to 
 undergo a continuous change through all the forms corre- 
 sponding to constant values of d, that is, through all the form$ 
 presented by the curve on the rotating cylinder. 
 
 62. These phaenomena may be exhibited by various con- 
 trivances, of which the simplest is the > kaleidophone. This 
 instrument, invented by Wheatstone and improved by Helm- 
 holtz, consists of two thin and narrow rectangular slips of steel, 
 or other elastic material, joined together in such a manner that 
 their longitudinal central axes are in the same straight line, 
 while their planes are at right angles to one another. Thus 
 a compound elastic rod is formed, of which one part can easily 
 vibrate in one direction, and the other part in a direction at 
 right angles to the former. The whole is fixed in an upright 
 position by clamping one of the slips in a stand, and a small 
 bright object (such as a silvered bead) is attached to the top 
 of the other. 
 
 If now the rod be disturbed in any manner, so that both its 
 parts are bent, and then left to itself, the motion of the free 
 end will be compounded of vibrations which (when the deflec- 
 tion is not too great) are sensibly rectilinear and harmonic, and 
 at right angles to one another. The period of one component 
 is fixed, depending on the length of the upper part of the rod ; 
 but the period of the other component may be altered within 
 certain Umits by clamping the lower part at different points. 
 
 If the two periods are commensurable, suppose them to be 
 viT, nr, where the ratio »« : « is in lowest terms. Then mnr 
 (the least common multiple of ot r and n t) is the period of the 
 resultant vibration, and the path of the free end is a re-entrant 
 curve, described in this period. If then otmt be sufficiently 
 small, e.g. not greater than about one-tenth of a second, the 
 impression on the retina made by the bright bead in any posi- 
 
44 Vibration Microscope. 
 
 tion has not time to fade away before the bead comes into the 
 same position again, and the eye sees a continuous bright 
 curve, more or less complicated according as the numbers m 
 and n are greater or smaller. 
 
 If the periods of the component vibrations are incommen- 
 surable, or if their ratio is expressible only by high numbers; 
 then the real path of the bead is a non-re-entrant curve, or a 
 very complicated re-entrant one. But, in either case, if the 
 ratio is approximately expressible by low numbers, the actual 
 appearance to the eye will be that of a curve gradually going 
 through all the forms which would be possible if the approxi- 
 mate ratio were exact, in the manner already explained (Arts. 
 56-59). 
 
 63. Another method, due to M. Lissajous, consists in re- 
 ceiving upon a screen a beam of light which has been success 
 sively reflected by two small mirrors fixed to the ends of two 
 tuning-forks vibrating in planes at right angles to one another. 
 It must be observed that the effect in this case depends, not 
 on the motion of translation of the mirrors, but upon theif 
 angular motion, which, though small, impresses on the reflected 
 beam a sufficient deviation to produce a considerable displace- 
 ment of the spot of light on a distant screen. 
 
 64. The methods described in the last two articles are useful 
 for illustration; but, for the purpose of exact observation, the 
 following, also devised by M. Lissajous, is much better adapted. 
 The object-glass of a microscope is attached to one prong of 
 a tuning-fork, so as to vibrate with it, the other prong being 
 counterpoised : the axis of the object-glass is at right angles to 
 the plane of the vibrations. The eye-piece is fixed. When 
 the fork vibrates, the image of any stationary luminous point, 
 formed by the object-glass, performs also vibrations which are 
 very approximately linear and harmonic; thjs vibrating image 
 is viewed through the fixed eye-piece, and (the vibrations being 
 sufficiently rapid) appears as a continuous straight line. But, 
 if the luminous point itself, instead of being stationary, performs 
 rectilinear vibrations at right angles to those of the fork, and in 
 the same plane, the image will appear as a curve, either inva- 
 
Imaginary Unrolling. 45 
 
 riable in form, or changing according to the conditions ex- 
 plained in Arts. 55, &c. Let us suppose, for clearness, that 
 the vibrations of the object-glass are horizontal, and those of 
 the luminous point vertical. Then the horizontal component 
 vibration of the image being harmonic, and of given period, 
 the form of the curve will depend upon the character of the 
 vertical component, and upon the ratio of its period to that of 
 the horizontal component ; hence, if the ratio of the periods be 
 known, the character of the vertical vibrations (i. e. the actual 
 vibrations of the luminous point) can be inferred from the 
 observed form of the curve. 
 
 In fact, it has been seen that in order that the curve may be 
 distinctly observable, the periods of the two component vibra- 
 tions must be to one another either exactly, or very nearly, in 
 some simple ratio. The curve may then be supposed to have 
 been formed by rolling on a transparent vertical cylinder a 
 plane curve in which the (horizontal) abscissa is proportional 
 to the time, and the (vertical) ordinate is equal to the actual 
 displacement of the vibrating point, at the given time, from its 
 mean positioji. We have, therefore, to reproduce this plane 
 curve, which completely defines the actual vibration of the point, 
 by imagining the cylinder to be unrolled. It has been re- 
 marked by Helmholtz that it is easier to see what the result of 
 this unrolling would be, when the ratio of the periods is not 
 quite exact, than when it is ; because then the cylinder appears 
 to turn about its axis (Art. 57) so that the observer is enabled 
 to see it, so to speak, on all sides, and to disentangle from one 
 another those parts of the curve which are on the front from 
 those which are at the back, through the contrary directions of 
 their motion. 
 
 65. In reference to this subject, the student will find the 
 following a useful exercise. Draw on a slip of paper two com- 
 plete waves of a harmonic curve (Art. 66), choosing the wave- 
 length so that the double wave can just be wound once round 
 a glass cylinder (e.g. a common lamp-chimney). Cut the paper 
 along the curve, so as to obtain a slip of which one edge has 
 the form of the curve ; and then, having rolled this on the glass 
 
46 Experiment described. 
 
 cylinder, mark the glass along the edge of the paper with a 
 glazier's diamond pencil. This is very easily done, and the 
 cm-ve on the glass is distinctly visible both in front and at the 
 back. If now the cylinder be held vertically at a moderate 
 distance from the eye, and turned about its axis, the series of 
 forms will be seen which would be produced if, in an observa- 
 tion of the kind described in the last article, the luminous point 
 performed simple harmonic vibrations with a period nearly 
 equal to half that of the vibrations of the microscope. 
 
 Then vary the experiment by substituting for the harmonic 
 curve on the paper either one or more waves of any other kind, 
 and particularly of a zigzag formed by portions of straight 
 lines, thus: 
 
 Fig- 7- 
 
 The same thing may be done with a wave-length so chosen 
 that the paper will go two, three, or more times round the 
 cylinder. The curve on the glass then becomes more and 
 more complicated, and the marking with the diamond pencil is 
 not so easy, on account of the over-lapping of the paper. 
 
 The object of the exercise is to practise the eye in the 
 imaginary unrolling, which is necessary in order to infer the 
 form of the plane curve on the paper from that of the curve 
 as seen on the cylinder. (See Helmholtz, p. 139.) 
 
CHAPTER IV. 
 
 THE HARMONIC CURVE. 
 
 66. If the motion of a point be compounded of rectilinear 
 harmonic vibrations, and of uniform motion in a straight line 
 at right angles to the direction of those vibrations, the point 
 will describe a plane curve which is called the harmonic curve. 
 
 Let the straight line be taken for axis of x, and let v be the 
 velocity of the motion along it; then we may suppose the 
 origin to be so taken that the abscissa of the moving point at 
 any time / shall be given by the equation x = vt. The ordi- 
 nate _>/ will be given (Art. 47) by the equation 
 
 . flirt . \ 
 y = asmC h a\, 
 
 where a and t are the amplitude and period of the harmonic 
 vibrations. 
 
 Eliminating / between these two equations, and then putting 
 z/ T = X, we obtain 
 
 for the equation of the harmonic curve. (It was formerly com- 
 monly called the ' curve of sines.') 
 
 If a wheel were to turn imiformly about a horizontal axis, 
 and at the same time to slide uniformly along it, the projection 
 of any point in the wheel upon a horizontal plane would de- 
 scribe such a curve. Or if a wooden cylinder, terminated at one 
 end by an oblique plane section, were smeared with printer's ink, 
 ,and then rolled over a sheet of white paper, the line bounding 
 
48 
 
 Harmonic Curve. 
 
 the blackened part of the paper would be a harmonic curve. 
 (The proof of this proposition, which has in fact been already 
 implied (Art. 53) may be left to the reader.) 
 
 If in equation (i) we put x ± z'X instead of x {i being any 
 integer), the value of j/ is unaltered. The curve, therefore, 
 consists of an infinite series of similar waves, thus, — 
 
 Fig. I. 
 
 which are divided symmetrically into upper and under portions 
 by the axis of x. The distance between corresponding points 
 of two consecutive waves is X, which is called the wave-length; ' 
 and the constant a, which is the greatest value of the ordinate, 
 is- called, as before, the amplitude. 
 
 The value of a has no effect on the form of the curve, but 
 determines its position; so that a change in a would shift the 
 whole curve along the axis of x. 
 
 67. A harmonic curve is most easily drawn in practice by 
 determining a number of points and drawing the curve through 
 them by hand. The points may be found by erecting ordinates 
 at (arbitrary) equal distances, and making them equal to the 
 ordinates of points on a circle of arbitrary radius, which divide 
 the circumference into an arbitrary number of equal parts. 
 The radius of the circle determines the amplitude, and the 
 distance between two consecutive ordinates of the curve. 
 
Composition of Harmonic Curves. 49 
 
 multiplied by the number of parts into which the circle is 
 divided, is the wave-length. 
 
 Composition of Harmonic Curves. 
 
 68. The formation of a resultant curve, in which the ordi- 
 nate of any point is the algebraical sum of the corresponding 
 ordinates of the component curves, has been already explained 
 (Art. 10, &c.). The case in which the component curves are 
 harmonic is specially important. 
 
 Two harmonic curves which have equal wave-lengths can 
 always be compounded into another harmonic curve with the 
 same wave-length. 
 
 For let the component curves be 
 
 >/ = flsin(2 7r- + a), y = h ^Wi.iiTt - ■\-^ , 
 
 A A 
 
 the value of j/ in the resultant curve is the sum of the values 
 given by these two equations, and this can be put in the form 
 
 y = csm{2w^ + y), 
 
 if c and y be determined (see Art. 50) so as to satisfy the 
 equations 
 
 f cosy = a cos a + ^cos^, 
 
 f siny = a sin a + ^sin/3. 
 
 The value of c, the amplitude of the resultant curve, is 
 a/s^ + P + 2 (7 3 cos (a — ^), 
 which may vary from a + b to a~»~b, according to the value 
 a — /S, which determines the relative position of the compo- 
 nents. 
 
 In the particular case in which the amplitudes of the com- 
 ponents are equal, and one of them is half a wave-length before 
 the other, so that cos (a — /3) = — i, the value of c is o ; or 
 the resultant curve degenerates into a straight line coinciding 
 with the axis of x, the components completely neutralizing one 
 another. 
 
 69. It evidently follows that any number of harmdnic curves 
 
50 Composition of Harmonic Curves. 
 
 having equal wave-lengths may be compounded into a single 
 harmonic curve with the same wave-length, and of which the 
 greatest possible amplitude is the sum of the amplitudes of the 
 component curves. 
 
 70. If the component curves have different wave-lengths, 
 they can no longer be compounded into a single harmonic 
 curve ; though, if the wave-lengths be commensurable, the re- 
 sultant curve is periodic. 
 
 Suppose X to be the least common multiple of the wave- 
 lengths, so that their actual values are — , -, &c., where m,n,... 
 
 m n 
 
 are integers. The equation to the resultant curve is then 
 2 77— V a-) -\- I sin(2 7r— 1- ^) + . . . . 
 
 which does not admit of reduction ; but we see that the value 
 oiy is unaltered by putting x -\-\iox x, so that the period or 
 wave-length of the resultant curve is X, that is, the least com- 
 mon multiple of the wave-lengths of the components. 
 
 If the component wave-lengths are incommensurable, they 
 have no finite common multiple, so that the period of the 
 resultant curve is infinite : in other words, the resultant is non- 
 periodic. 
 
 The forms of the component harmonic curves depend only 
 on their amplitudes and wave-lengths ; but ihtir positions depend 
 upon the constants u, /3, &c. A variation in any one of these 
 shifts the corresponding curve along the axis ; and any such 
 shifting will evidently alter the form of the resultant curve, and 
 the positions of its points of intersection with the axis, without 
 altering its wave-length. 
 
 71. If the wave-length only of the resultant be given, the 
 wave-lengths of the components may be all possible aliquot parts 
 of it, including the whole as one case of an aliquot part; 
 and the number of the possible components is therefore un- 
 limited. 
 
 Thus every curve which could be constructed in this manner, 
 so as to have a given wave-length X, would be found amongst 
 those produced by placing along the same axis an unlimited 
 
Fourier s Theorem. 51 
 
 number of harmonic curves as components, with wave-lengths 
 X, \\, IX, &c. 
 
 It is evident that by varying arbitrarily the amplitudes of the 
 components, and shifting them arbitrarily along the axis, an 
 infinite number of resultants could be produced, having all the 
 same wave-length X. But it could not be assumed without 
 proof that every possible variety of periodic curve could be so 
 produced. 
 " This, however (with a limitation to be mentioned below), is 
 true, and constitutes the celebrated theorem of Fourier. Before 
 giving a formal enunciation of it, we will define precisely the 
 meaning of the word axis as used above. Corresponding 
 points of a periodic curve lie upon a straight line parallel to 
 a fixed direction. Any straight line parallel to that direction 
 may be called an axis of the curve ; but it is convenient to call 
 the axis that line which cuts off equal areas from the curve on 
 its opposite sides. Thus, the axis of x is the axis of a har- 
 monic curve, as defined by the equation y = as\a(nx + a). 
 This being premised, we proceed to enunciate 
 
 Fourier's Theorem. 
 
 72. If any arbitrary periodic curve be drawn, having a given 
 wave-length X, the same curve may always be produced by 
 compounding harmonic curves (in general infinite in number) 
 having the same axis, and having X, ^X, iX, .... for their wave- 
 lengths. 
 
 The only limitations to the irregularity of the arbitrary curve 
 are, first, that the ordinate must be always finite ; and secondly, 
 that the projection, on the axis, of a point moving so as to 
 describe the curve, must move always in the same direction. 
 
 These conditions being satisfied, a wave of the curve may 
 have any form whatever, including any number of straight 
 portions. 
 
 Analytically the theorem may be expressed as follows : 
 
 It is possible to determine the constants C, C^ C^ &c., 
 > E 2 
 
5^ Enunciation of 
 
 Oj, a^, &c., so that a wave of the periodic curve defined by the 
 equation 
 
 y=C-^ CiSin(-Y- + ^) + C^sinl^a-^ -\- a^)+ ... 
 
 or jc- = C + 2,.^i Q ®"^ V^X" + "*-)' 
 
 shall have any proposed form, subject to the conditions men- 
 tioned above. 
 
 By a change of notation, we may write the above equation 
 in the following more convenient form, viz : 
 
 y = A,+ 2 2^^^ ^iCOS -^^ + 3 2;=! ^iSm-^^. (2) 
 
 (For a demonstration of the theorem, see the Appendix to 
 this chapter.) 
 
 73. The demonstration of the theorem just enunciated 
 includes a determination of the values of the constants. 
 But we may observe here that, assuming the truth of the 
 theorem, we can obtain the expressions for these values at 
 once, by means of the following simple propositions. 
 
 If z,j, be integers, all the integrals 
 
 /azVj; si'trx , ^ . 2i'n x . 2Jitx , 
 
 cos — r — cos ^^ — ax, I sm — - — sm — ^ ax, 
 
 I' 
 
 . 2 11? X ZiTTX, 
 
 sm — - — cos -^ — ax. 
 
 taken between the limits o and X, vanish unless y = z. lij = i 
 the third integral still vanishes, while the first two have the 
 
 common value -, or X, according as t is different from, or 
 
 equal to, o. 
 
 Hence, if we multiply both sides of the equation (2) by 
 
 2tnX J , . 2l7rX , , . . , 
 
 cos — - — ax, or by sm — - — dx, and mtegrate m each case 
 
 between the above Umits, we obtain, for all values of i, in- 
 cluding o, 
 
 , I /"^ 2{'n-X , „ I A . 2inX . 
 
 * " X / -^ ^°^ "~x~ •*' * " x7 -^ ^''^ ~ir" ■ ' 
 
Fourier's Theorem. 5 3 
 
 74. Thus, whatever the function _/(^) may be (the proper 
 limitations being always supposed), the expression 
 
 \\y(x) dx + I XZ cos "-^f/{x) COS ^ dx 
 
 , 2 fca, . 21T7X [^., . . 211TX , . . 
 
 •represents a periodic function of which the value coincides with 
 that oi /{x) for all values of x between o and X, but not for 
 other values, unless the function y(^) be itself periodic, and 
 have X for its period, so thaty(j*r + X) =/{x); in which last 
 case alone the expression (3) may be taken without limitation 
 as equivalent io /{x). 
 
 75. When the actual values of the coefficients A., B., are 
 required, they have to be found by evaluating the definite in- 
 tegrals (Art. 72) by which they are expressed. 
 
 Suppose, as in Art. 73, that_y = f(x) from x = o to x = \. 
 This equation may subsist in two different senses, which it is 
 important to distinguish. 
 
 (1) y may be a given function of x in the ordinary alge- 
 braical sense ; that is, it may be possible to assign a rule by 
 which the value oi y can be calculated for any assumed value 
 of X. Or (still using the word function in the same sense), j' 
 may be a given function from x = q Xa x = a, another given 
 function from x = a \.o x = b, &c., a, b, &c., being given 
 values between o and X. 
 
 In these cases the ordinary methods of the integral calculus 
 are appKcable to the evaluation of the definite integrals. But 
 
 (2) y may be a given function of x only in the more general 
 sense (including the former as a particular case) in which the 
 word function should always be understood in mathematical 
 physics ; viz. that for every value of x there is a determinate 
 value of ^, though it may be impossible to assign any rule for 
 calculating it, in which case it is only to be ascertained by 
 actual observation or measurement. 
 
 Thus, if we draw by hand upon paper an arbitrary curve 
 between two points, we can measure as many ordinates as we 
 
54 Illustrations of 
 
 please, but we can give no rule for calculating the value oi y 
 for an assumed value of x. 
 
 In such a case the values of the definite integrals can only 
 be found approximately, by measuring a sufficient number of 
 values of J/, and applying the method of quadratures. We can 
 thus obtain, by means of equation (2), an approximate rule for 
 calculating the value oi y for any value of x between ,given 
 limits. The rule would be exact if the coefficients A ., B., were 
 exactly known; but, in order to calculate them exactly, we 
 must be in possession of such a rule to begin with. 
 
 Since, however, the coefficients actually exist, though we may 
 not be able to ascertain their values rigorously, the abstract 
 truth of the theorem is in no way interfered with ; and its great 
 value consists partly in this, that it furnishes an analytical ex- 
 pression (within finite limits) for any function whatever, whether 
 it be a function in the ordinary algebraical sense, or only in 
 the physical sense explained above. 
 
 When the coefficients A., B., are considered as having arbi- 
 trary values, the expression (2) (Art. 71) evidently represents 
 a completely arbitrary periodic function of x. 
 
 76. The following is a simple and useful example of the 
 application of the theorem. 
 
 B X 
 Fig. I. 
 
 Let OA, AB,\)& two straight lines, cutting the axis of x at 
 the origin O and at a point B, such that OB = X. 
 
 It is required to determine the coefficients so that the ex- 
 pression (3), Art. 73, shall give the value of the ordinate at any 
 point of the ' curve' OAB, from x = o to x = \. 
 
 Suppose a, b, are the coordinates of A. Then /(.ar) \% -x 
 
 a 
 
 from X = o to X = a, and is {x — X) from .^ = a to 
 
Fourier's' Theorem. 55 
 
 :*: = X. Hence, putting, for the present -r— = n, we have 
 
 (Art. 72) 
 
 h ['■■ ' b C^ 
 
 \A. = - xcosnxdx -\ -/ (x - \)cosnx dx 
 
 * ajf, a — \J„ ^ ' 
 
 3 /■« ^ A 
 
 \B. = - I xsinnx dx -] / (x — \) sin n x dx. 
 
 ' aj,^ a - \Ja "^ ' 
 
 Performing the integrations, we find, after easy reductions, 
 
 . b a{i — cos « X) — X (i — cos n a) 
 
 i n^Xa X — a 
 
 ^ b X sin « a — a sin « X 
 
 B, = -s- — • r . 
 
 2 n \a X — a 
 
 The value of ^^ is most simply obtained directly from the 
 
 I Z*^ 
 expression - / /{x) dx, which gives at once 
 XJo 
 
 ^d = - X (area OAB) = -. 
 X 2 
 
 But it may also be found by evaluating in the usual way the above 
 expression for A ., when i (and therefore n) = o. Only it is to 
 be observed, that for this purpose we must not assume i to be 
 an integer before putting i = o. The same process gives, of 
 
 course, B^ = o. If now we introduce the value of « ( = -— - ) 
 
 X 
 we find, for values of t from i to 00 , 
 
 . b ^^ / . a \ 
 
 A . = .o 2 • -A \ { cos 2 ITT - — I ) 
 
 B. = ^ „ ■ -TT T Sm 2 Z 77 -. 
 
 These expressions (with A = -) are to be introduced in the 
 
 equation (2), Art. 71, which then becomes the equation to a 
 periodic curve, of which one wave is OAB. 
 
 The result is easily reducible to the following form : 
 
 2t1T{x ) 
 
 b b>? ■^r.i=a> I . lira . 2 
 
 y = - + -9 — 7; s2j- I "aSm— - ■ 
 
 OK 1^,^=00 \ . ma . 2 /»\ 
 - + -5 — 7; r^- ■■ ToSm— — sm , W 
 
 1 7T^ a (\ — a) ^i=l t^ \ X 
 
56 
 
 Illustrations of 
 
 It may be observed that all the periodic terms on the right 
 
 of (^) vanish {ox x = -, and also for x = ; from which 
 
 it follows that every one of the harmonic curves represented 
 by the several terms passes through the two points C, D, which 
 bisect the lines OA,AB, since the abscissae of these points 
 
 h_ 
 2 
 
 are -, , and their ordinates are both 
 
 2 2 
 
 It is also 
 
 evident that the ' axis' (Art. 70) of the whole locus represented 
 by (3) (consisting of repetitions of AE) is the line C D 
 indefinitely produced. 
 
 77. The following simpler example is also instructive. It 
 is required to find a periodic function, of which the value shall 
 
 coincide with that of m(x ) from x 
 
 ^ 2 
 
 o to .a; = X. In 
 
 other words, to find the equation to a periodic curve consisting 
 
 T 
 
 ^X 
 
 Fig. 2. 
 
 of repetitions of the straight line A B (Fig. 2), in which OC = \ 
 and ta.n BMC = m. 
 In this case we have 
 
 2 ijrx . 
 cos — ;^ — ax 
 
 . m f^ , X, 
 
 A. = - (x ) 
 
 » XJo ^ 2' 
 
 Tf m ['^ X. . 2 iirx , 
 
 and it is easily found that A^= o for all values of i, while 
 « = 77^- Hence equation (2), Art. 71, gives, 
 
Fourier's Theorem. 57 
 
 y ^ ^i-^ -■ sin ^ — ' 
 
 or 
 
 m\( 1 . 2Ti X I. 2 1TX I. 27rj;, ")/x 
 
 J' = ^ - sin — h - sin 2 --— + - sin 3 ^-- + ...\ (4) 
 
 7rt.I A 2 A 3^ J 
 
 which is the required equation. 
 
 Here we may observe that, if the locus be considered as 
 consisting of the detached lines A B, BE, &c., the value of^ 
 
 undergoes a sudden alteration from m - to — m - when x 
 
 2 2 
 
 passes through o or any multiple of X, while for these critical 
 values of x the equation (4) givesj/ = o, that is, the arithmetic 
 mean of the two values just mentioned (see Appendix to this 
 chapter). On the other hand, since every term in the series (4) 
 is continuous, it is impossible that the sum of the series can 
 undergo an absolutely sudden change of value, without passing 
 through the intermediate values. This subject cannot be fuUy 
 discussed here; but the true view seems to be that while x 
 varies from a value infinitely near to a critical value, but less, 
 to a value infinitely near, but greater, j/ passes instantaneously 
 
 through all values from m- to — m-. Or, geometrically, the 
 
 locus of -the equation (4) ought to be considered as including 
 the portions A A, B B, &c.., which are inclined at an infinitely 
 small angle to the true perpendiculars drawn through 0, C, Sec, 
 and cut them in those points. 
 
 Assuming that we may differentiate equation (4) we obtain 
 
 2 m 
 ax 
 
 (ZttX 2it X 21T X , 1 
 
 cos — V cos 2 — 1- cos 3 — r- V ..-> 
 
 Now, considering the series 
 
 1 + 2 (cos 5 + cos 2 5 4- cos 3 fl + . . .) 
 
 as the limit of 
 
 I + 2 (c cos 5 + f ^ cos 2 5 + . .), 
 
 I — f^ 
 
 that is, of the fraction 3— — -„, when c (increasing) 
 
 ' I — 2 f cos 5 + f* 
 
 becomes = i, we see that it must be taken as representing o- 
 
58 Illustrations of 
 
 for all values of B except the critical values o, 2 is, Sec, which 
 give cos 6 = I, in which case it becomes 00 . And we avoid, as 
 before, the difficulty of attributing a sudden change of value to 
 a series of which all the terms are continuous, by considering 
 that it passes through all values from o to 00 , and back again 
 from 00 to o, while x varies from being infinitely little less to 
 being infinitely little greater than a critical value. And it fol- 
 lows that the series within brackets in the above expression 
 
 dy dy 
 
 for -J- has in general — \ for its value, so that -7— = m; but, 
 
 while X passes through the critical values o, X, &c., the value 
 of the series passes instantaneously through all values from — ^ 
 to 00 and back again ■'. 
 
 Hence A',A,£,£', &c. in Fig. 2, are to be considered as 
 points at which the tangent changes its direction, not with 
 absolute suddenness, but by turning round those points. So 
 that if we suppose a moveable point P to be describing the 
 locus A B B', &c., the tangent coincides with A B until P 
 approaches infinitely near to B, and, while P is passing through 
 B, turns through all directions between A B and B B', with 
 which last it coincides as soon as P has passed through B. 
 In other words, the two lines AB, B B', are to be considered 
 not as making an angle at B, but as being connected by an 
 infinitely short curved arc, of which the radius of curvature 
 
 ' We purposely avoid here all discussion of the legitimacy of differen- 
 tiating (4), and of the logical validity of reasoning founded upon the 
 properties of the series (5). What is certain is, that it is impossible to have 
 clear notions of the true nature of Fourier's series, especially in its applica- 
 tion to the representation of discontinuous functions, without some such 
 illustrations as those in the text. For a view of the various methods which 
 have been proposed in order to treat the subject with perfect rigour, and 
 of the theoretical questions connected with it, the reader is referred to 
 
 Stokes, On the Critical Values of the Sums of Periodic Series. (Camb. 
 Phil. Trans., vol. viii.) 
 
 De Morgan, Diff. and Int. Cat, p. 605, &c. 
 
 Price, Infinitesimal Calculus, vol. ii. § 197. 
 
 Thomson and Tait, Nat. Phil., vol. i. § 75. 
 
 Boole, On the Analysis of Discontinuous Functions. (Trans, of R.I. A., 
 vol. xxi. pt. I.) 
 But Fourier's original work, Thiorie analytique de la Chaleur, which unfor- 
 tunately is now rare, should be consulted by all students who can obtain 
 access to it. 
 
Pourier's Theorem. 59 
 
 is CO at the extremities and infinitely small at the middle. This 
 remark is of course equally applicable to the angular points in 
 the locus, Fig. i. In general, we may say that Fourier's theo- 
 rem evades the difficulty of expressing analytically the abrupt 
 changes of value which may, and do, occur in nature, by sub- 
 stituting for them continuous, but infinitely rapid, changes. 
 
 78. Any physical condition (such as density, pressure, velo- 
 city, &c.) which is measurable in magnitude or intensity, and 
 which varies periodically with the time, is expressible as a 
 function of the time by means of Fourier's series. For in the 
 case of actual physical changes, the conditions which make the 
 theorem applicable are necessarily fulfilled. In other words, 
 every actual vibration can be resolved mathematically into har- 
 monic vibrations. If j/ represent the magnitude in question, 
 and T the period of its vibration, then y is expressible by an 
 equation of the form 
 
 where _;/ is the mean ^ value oly, and each of the variable terms 
 represents by itself a harmonic vibration, of which the period 
 is an aliquot part of the whole period t. 
 
 79. Thus every periodic disturbance of' the air, and in par- 
 ticular such vibrations as excite the sensation of sound, can be 
 so resolved. Now we know as a fact that a vibration of the 
 air excites in general the sensation of a musical note, which is 
 not a simple tone (Art. 12, &c.), but a combination of tones cor- 
 responding in general to vibrations of which the periods are 
 aliquot parts of the period of the original vibration. (The ex- 
 ceptions to this statement are apparent rather than real. The 
 so-called vibration of a tuning-fork, for example, is not a single 
 ' vibration' in the strict sense of the term, but is compounded 
 of vibrations of which the periods are incommensurable. Hence 
 
 I /■'■ 
 ' The value of ^ , viz. - / ydt,is the average of all the values which y 
 
 ° '^Jo 
 
 has, at instants separated by infinitely small equidistant intervals, during 
 
 the period t. 
 
6o Demonstration of 
 
 the whole motion is not really periodic. In this and other~ 
 similar cases, component tones are heard which do not belong 
 to the harmonic scale of the fundamental tone.) 
 
 The ear, therefore, resolves a note into simple tones after 
 the same manner in which Fourier's theorem resolves a vibra- 
 tion into harmonic vibrations ; and the question naturally arises 
 whether each simple tone perceived by the ear is really caused 
 exclusively by the corresponding harmonic component of the. 
 complex vibration. 
 
 We shall soon be able to assign a conclusive reason for 
 believing that this is so ; and we shall thus obtain an answer 
 to the question suggested in Art. 14. 
 
 APPENDIX TO CHAPTER IV. 
 
 FOURIERS THEOREM. 
 
 The equation 
 
 
 I — 2 C COS {x — a) + c 
 which may be also written 
 
 y 
 
 (i — r) sec* 
 
 2 
 
 (x-cf^ (i +f)2tan2-^^^ 
 
 a and c being any constants, represents a periodic curve of 
 which the ordinate is always positive if c be numerically less 
 than I. In what follows this condition will be supposed, and 
 also that c is positive. 
 
 Then v will have , — ■ — for maximum and minimum 
 
 Values, corresponding to x = a + iiir, x = a + (2 ?' + i)jr. 
 
Fourier's Theorem. 
 
 6i 
 
 i being any integer; and the distance between the ordinates 
 of corresponding points on successive waves of the curve 
 
 is 2 TT. 
 
 Fig- 3- 
 
 Fig. 3 represents a portion of the curve in the case in which 
 £■ = f . AQ, BR, &T& two of the maximum ordinates, and 
 P„ Mf, a minimum ordinate. It is evident that if the value of 
 c be increased, tending towards i as a limit, the maximum 
 and minimum ordinates will tend towards oo and o as limits. 
 At the same time if a fixed point M, or M', be taken, however 
 near to A, the ordinate MP or M' P" will tend to o as a 
 limit. 
 
 Hence, if the curve be considered as described by a point P, 
 the motion of P tends, as the value of c approaches i, to 
 become that of a point which moves along X except when 
 'mfinitely near to one of the points A, B, ho.., but passes those 
 points by going up the left side of the ordinate to an infinite 
 distance, and down again on the right side. 
 
 The values (a + 2 zV) of x at A, B, &c., will be called 
 critical values. 
 
 The area included between the curve, the axis of x, and two 
 consecutive corresponding ordinates (as P M, P' M", or A Q, 
 BR) is 3 jr. This is easily found by integrating directly the 
 expression J' dx; but it is most simply obtained by first deve- 
 loping j/ in a series : thus, 
 
 y = 1 + 2{CC0S{X — a) -\- C^C05 2{X — a) + .. '.), 
 
62 Demonstration of 
 
 from which, observing that 
 
 fa: + 2 7r 
 
 COS i {x — a) dx 
 
 r 
 
 for all values of i except o, we find at once 
 
 Cx + 2tr 
 
 y dx = 2 IT. 
 
 J X 
 
 The area between a maximum and next following minimum 
 ordinate, (as Q A M^ P^, is of course = jr. The following are 
 obvious inferences : — 
 
 If MP, M' P be any two fixed ordinates, including one 
 maximum ordinate A Q between them, the area MP Q M' 
 tends to 2 n- as a limit when the value of c approaches to i. 
 This is true however near either or both ordinates may be to 
 A Qi so long as neither of them coincides with A Q absolutely. 
 Each of the areas MP QA, AQP' M tends to become = n. 
 
 And if M' P', M"P" be any two fixed ordinates, both in- 
 cluded between two consecutive maximum ordinates, the area 
 M P' P" M' tends at the same time to o. And this is true 
 however near M P', M" P" may be to the maximum ordinates 
 AQ, BR, so long as there is not absolute coincidence with 
 either. 
 
 And, however small the fixed quantities AM, AM', M'B 
 may be, it is possible to take c so nearly equal to i that the 
 areas MPQA,AQP'M', M'P"RB, shall each differ from 
 TT, and the area M P' P" M" from o, by quantities less than any 
 assigned quantity. ft 
 
 These conclusions may be expressed analytically as follows. 
 If Xo, x^, be two values of x, including between them only one 
 critical value (say a), with which neither of them absolutely 
 coincides, and if f, ^ be any positive constants such that a + e, 
 and a — ^ are also both included between x^ and x^, then the 
 four integrals 
 
 Co. - el Co. ra + e rxi 
 
 / y<ix, / ydx, i ydx, / ydx, 
 
 have for their limiting values o, tt, w, o, when c approaches to i, 
 however small e and e^ may be. The sum of the four integrals 
 
 IS y dx, and, supposing Xj — x„ to have the greatest admis- 
 
 . ° 
 sible value (2 jr), this is equal to 2 tt ; and in any case has 2 n 
 
 for its limit when c approaches to i. The sum of the two 
 ' y dx, and this has 2 tt for its limit. 
 
/: 
 
 •' Xa 
 
 .1: 
 
 Fourier's Theorem. 63 
 
 Let us now suppose that x^, x^ have any values such that 
 jf J — x^ is not > 2 JT ; and let /{x) be any function which is 
 finite for all values of x from x^ to ^j inclusive. Then {y 
 having the value (i) as before) the integral 
 
 y{x)y dx, (2) 
 
 since _>■ is always positive, is equal to the product of the integral 
 ~''\ 
 
 ydx,hy some quantity intermediate between the (algebrai- 
 -^0 «- 
 
 cally) least and greatest values which /"(jtr) takes while x varies 
 from Xf, to Jfj. 
 
 There are three cases to be considered. 
 
 (1) There may be no critical value (a + 2 iir) from x = x^ 
 
 to x = ^j inclusively. In this case the limiting value of I y dx, 
 
 J Xo 
 
 and therefore of the integral (2), is o when c approaches to i. 
 
 (2) There may be a critical value, say x - a, between x^ 
 and x^, but not coinciding with either of those limits. 
 
 In this case the integral (2) may be considered as the sum 
 of the three integrals 
 
 J(-a - e» /"a + e T^l 
 
 /{x)ydx, / /{x)ydx, / /{x)ydx, 
 
 Xq •/a — e^ %/a + e 
 
 of which the first and third have o for their limiting values (by 
 the first case), while the second is equal to ydx multiplied 
 
 J a — e^ 
 
 by some quantity M intermediate between the least and greatest 
 values which /{x) takes while x varies from a — e' to a + t. 
 But e and e^ may be as small as we please ; they may therefore 
 be taken so small that M shall differ infinitely little from/(a). 
 
 ydx is 2 JT, as was shewn above. 
 
 -. — el 
 
 Hence we infer that in this case the limit of the integral (2), 
 when c approaches to i, is 
 
 3 5r/(a). (3) 
 
 (3) One or both of the limits, j:„, x^, may coincide with a 
 critical' value. Suppose, for instance, Xg = a, x^= a -{- zv. 
 Then by considering the integral (2) as the sum of 
 
 /(x)ydx, /{x)ydx, /{x)ydx, 
 
 it will be seen without difiiculty that the limiting value is 
 "■/W + V^W ; that is, JI- (/(a) +/(a + 2 tt)). 
 
64 Demonstration of 
 
 If only one of the limits x^, x^, coincided with a critical 
 Value a, the result would evidently be ir/{a). 
 
 The above conclusions require modification in the case in 
 which the functiony(j;) is such as to undergo a sudden finite 
 change . of value when x passes through the critical value. If 
 a sudden change take place for any value of x not absolutely 
 coinciding with the critical value, however near to it, the rea- 
 soning is not affected, because we may take e and t^ so small 
 that the value in question shall not lie between a -)- e and 
 
 But suppose (in case (2)) \hs.\. /{x) is = a when x is infi- 
 nitely little less than a, and = b when ;*: is infinitely little greater 
 
 than a. Then, considering the integral / f(x)ydx, as the 
 
 sum of / f{_x)y dx and / /{x)y dx, we see that the first of 
 
 J a.— t?- J a. 
 
 these will have for its limit ■^/{a), and the second !r/"(3) ; so 
 that the limiting value of (2) will be 
 
 ^{/{a) +/(J>)), 
 
 that is, half the sum of the values given by the general rule for 
 the two values oi/{x). 
 
 We may enunciate these results in the form of the following 
 
 TJteorem. 
 
 If x^, x^ be two values of x such that Xf,<a< x^, and 
 x^ — Xg be not > 2 tt, and ii /{x) be any function which is 
 finite for all values of x from x = x^Ui x = Xj^ inclusively, then 
 the value of the integral 
 
 /■■^i 1 — c^ 
 
 / /{^) 7 T"! — 2 ^x, (2) 
 
 when c (increasing) approximates indefinitely to i, has in general 
 ■for its limit 2 7r/(a). But if /(;*:) undergoes a sudden change 
 of value from a to b when the value of x passes through a, then 
 the limit is 7r(/(<2) +/((5)). 
 
 If either x^ = a, or x^ = a, while x^ — x^<2ir, then the 
 limit is ir/{a). 
 
 But a X^= a,X^= a + 2ir, then it is n {/(a) +/{a + 2n)). 
 
 On these last suppositions we may provide for the case of 
 sudden changes of value of the function by understanding (if 
 necessary) /{a) to mean /(a + f), and /{a -f- 2 jr) to mean 
 /(a + 2 TT — f), where e is an infinitely small ~ positive quan- 
 tity. 
 
Fourier's Theorem. 65 
 
 Since the value of a must not transgress the limits x^, x^, and 
 ATj — x„ must not be > 2 tt, we shall give the greatest possible 
 extent to the theorem by supposing x,^— Xg= 2 tt. 
 
 The notation used above was convenient in the course of 
 demonstration ; but in actual applications it is better to change 
 it by writing a for x and x for a, so that a becomes the variable 
 in the integration. The principal result may be then stated as 
 follows : 
 
 The limit of the integral 
 
 Jr»i I — f " 
 ' y'C") } r~; — 3 '^"i 
 ^■^ w J _ 2 ^gQg (^ _ a) ^ ^ 
 
 where a^ — a^ = 2 jr, and x has any value between ii„ and Uj, is 
 in general 2 7r/{x), when c approximates indefinitely to i. 
 The special cases are of course to be stated as before, mutatis 
 mutandis. 
 
 If now in the above integral we substitute for the fraction its 
 development, viz. 
 
 I + 2 (c cos {x — a) -\- c'' cos 2 (^ — o) + . . . .), 
 we obtain the series 
 
 / f{a) da + 2 2jir ^' / y^(") cos i{x — a) da, 
 
 and we may therefore aflHrm, that when c approximates to r, 
 the limit of this series is in general 2 ■nfix) for any value of x 
 between a„ and a^, but is tt (<? + V) for a value of x correspond- 
 ing to a sudden change from a to i5 in the value oif{pc), and 
 
 T (/("o^ +/K)) for X = a^QXX = a^. 
 
 Assuming, then, that when c =\ the value of the series 
 becomes equal to its limit', we may write the result as follows 
 (excluding of course the exceptional cases) : 
 
 2 ■Kfipc) = 2l=!„ / /(») cos i(x - a) da, 
 
 for values of x between o^ and a^ 
 
 . Zttx' 2 jra' . 
 
 If now we write for x, and ^r— for a, assummg also 
 
 \ A 
 
 c<j = o, aj = 2 TT, the above equation becomes 
 
 ,,2irx\ l.^i=m f^ ,27ra\ 2tn{x'—a') 
 
 A-ir) = -x2i=-.j/(^)'=°« — K — '^^ ' 
 
 1 This assumption appears to be the only point in the demonstration 
 which is open to objection. But we cannot here discuss the proposition, 
 ' what is true up to the limit is true in the limit.' On the convergence of 
 the series (3"), see the demonstrations of Tait and Thomson and of Stokes, 
 referred to above (Art. 77). 
 
66 Fourier's Theorem. 
 
 or, omitting accents, and writing/(j»r) instead '^^/(^'~\~)y 
 
 .. , I ^i=oo p ,, . 2 lit {x — a) J 
 f{x) = - 2,.= _„y^/(«) cos :^ da, 
 
 ■which is now true for values of x between o and X. 
 
 This equation may, by an obvious transformation, be written 
 thusr 
 
 ^(^)=xio-^^"^'^" + X^*=^ COS-^^/(a)eOS-j^<fa 
 , 2 _,j=a, . Zlnxf^ . 2 1jra 
 
 + x2,=, sm-^^/(a)sm-^</a,(3) 
 
 in which form it Is usually most convenient. 
 
 A further transformation, which gives an expression fory^(x) 
 by means of a double definite integration, and which is also 
 often referred to as ' Fourier's Theorem,' is not required for the 
 purposes of this treatise. 
 
 If we had taken above a^= — tt, Oj = + tt, we should have 
 found in the same way a series only differing from (3) in having 
 
 , + - for limits in the integrations, instead of o, X. Sup- 
 
 2 2 
 
 pose this alteration made, and then suppose further thaty(.r) 
 is an odd function, that is, that /{— x) - —/{x}... In this 
 
 case it, is easily seen that the integral / _/"(a)cos — r — da 
 
 J-^ X 
 
 vanishes- for all' values of i, including r = o,, smce it may Be 
 -divided into pairs of equal elements,, bjit with ©pposke. signs. 
 
 But r\ • r\ • 
 
 Its ,/ V . 2tna l^ jr/\ • 2Z7r(i. 
 
 / /(a) sm — - — da= 2 I r{a) sm.— rr — da 
 J_^ X Jo A 
 
 since the integral on the left may be divided into pairs of equal 
 elements with the same sign. 
 
 Hence, if we put - =; /, we obtain instead of (3) 
 
 , , 2.^1=00 . i'^x P . . . tira 
 fix) = - 2^^j sm -^j^ /{a) sm — da,. (,4) 
 
 which is, true from x <= — I to x == + I for any {urteiian/{x) 
 which is odd between those limits. And it will evidently be 
 true from x = o to .«: = / for any function. 
 
 'M/{x) besides being odd, is periodic, with period (or wave- 
 length) 2/, then (4) will be true without limitation. 
 
 "The case in which'^(d;) is an even function {/(js). =^J~{.— x)) 
 may be left to the reader. 
 
CHAPTER V. 
 
 VIBRATIONS OF AN ELASTIC STRING. 
 
 80. Op the various modes in which musical sounds may be 
 produced, one of the most usual in practice, and simple in 
 theory, consists in the vibration of elastic strings. 
 
 The term 'elastic string' is to be understood as implying 
 ideal qualities, which do not belong absolutely to any actual 
 string; namely, non-resistance to flexure, and extensibility 
 according to the law that extension is proportional to tension. 
 
 According to this law, if / be the natural length of the string 
 (or its length when not subjected to any tension), and /' the 
 length when it is stretched by a force T, then 
 
 T 
 
 where ^ is a constant, of which the value depends on the 
 nature of the string. 
 
 Since the homogeneity of the above equation requires that 
 
 T 
 
 -=, should be an abstract number, ^ is a quantity of the same 
 
 kind as T, that is, a force. And since the supposition /' = 2t 
 gives E = T', we see that E may be defined as the force which 
 would be required to stretch the string to twice its natural 
 lengtli. 
 
 No actual string is perfectly flexible, nor indefinitely exten- 
 sible, according to the above law. But when the changes of 
 length are small, it is proved by experiment' that they are sen- 
 sibly proportional to the changes of tension. And when the 
 tension is considerable, and the thickness of the string very small' 
 
 F 2 
 
'68 Vibrations of 
 
 . in comparison with its length, the resistance to ilexure is very 
 small in comparison with the resistance to extension. 
 
 Under these conditions the phsenomena with which we are 
 concerned are so nearly the same as they would be in the case 
 of the ideal string, that for most purposes they may be con- 
 sidered identical. The case in which imperfect flexibility must 
 be taken into account will be considered apart. 
 
 81. Supposing the two ends of an elastic string to be fixed 
 at points of which the distance from one another is greater 
 than the natural length of the string, its form in the condition 
 of equilibrium will be a straight line, and its tension will be the 
 same at all points'; namely, that with which the string would 
 have to be pulled at any point in order to maintain the equi- 
 librium, if it were cut there. 
 
 Suppose now the string to be slightly disturbed by any forces 
 whatever, which, at a certain instant, cease to act. The sub- 
 sequent motion of the string will depend upon the positions 
 and velocities of its particles at that instant ; and the mechanical 
 theory shews that it will vibrate, that is, will go through a series 
 of periodic changes, which, theoretically, would never cease. 
 The vibration actually does cease because the string gradually 
 gives up its motion to the surrounding air, to the bodies which 
 sustain the tension of its ends, and, in a different form, to its 
 own molecules. 
 
 The condition of the string at the instant when the disturb- 
 ing forces cease to act is called its initial condition. 
 
 When the initial displacements of all the particles of the 
 string are lateral, that is, at right angles to its original direction, 
 and when the initial velocities are lateral also, the vibrations 
 are sensibly lateral. 
 
 When the initial displacements and velocities are longitu- 
 dinal, the vibrations are necessarily longitudinal, and the form 
 of the string remains a straight line. 
 
 Vibrations of both kinds can coexist, but it is best to con- 
 sider them separately. Longitudinal vibrations of a string are 
 
 ^ The effect of gravity is neglected, being insensible in all cases of tlie 
 kind here considered. 
 
a Siring. 69 
 
 of the same kind as those of a straight rod, which will be 
 treated of in another chapter. In the case of a string they are 
 practically unimportant. At present, therefore, we shall only 
 examine the lateral vibrations. 
 
 82. Lateral initial displacements and velocities, if they are 
 not in one plane, may be resolved into components in two 
 arbitrary planes at right angles to one another ; and each set of 
 components gives rise to vibrations in its own plane, which 
 coexist independently, and of which the actual vibration is the 
 resultant. 
 
 It is sufficient, therefore, to suppose that the initial displace- 
 ments and velocities, and therefore the subsequent vibrations, 
 are in one plane. 
 
 83. The true nature of the vibrations of a finite string may 
 be best understood by considering them under that aspect 
 which is suggested by the dynamical theory (see Chap. VII) ; 
 in which the string is regarded as infinitely long, and out of 
 the various possible forms of motion of an infinitely long string, 
 those are selected which are characterised by the existence of 
 nodes, that is, of motionless points. Any two such motionless 
 points of the' string might evidently become fixed without dis- 
 turbing the motion; and then all the string not included be- 
 tween them might be removed, so as to leave a finite string, 
 with fixed ends, which would continue to have the same motion 
 as it would have had under the original conditions. We shall 
 therefore begin with the case of an infinite string. 
 
 84. It must be always understood that though the following 
 propositions, in so far as they are merely geometrical, are true 
 without limitation, it is only when the displacements are infi- 
 nitely small that they are rigorously true mechanically; but they 
 are sensibly true within limits wide enough for the most im- 
 portant practical applications. 
 
 85. The simplest form of motion of an infinitely long elastic 
 string consists in the transmission of a single wave. 
 
 Let AB represent part of the string in its undisturbed con- 
 dition, and let CD be a line parallel to AB, of which any arbi- 
 trary portion QR is bent into an arbitrary curve. Imagine CD 
 
7p Transmission of Waves 
 
 to be moving with a constant velocity, in the direction of its 
 length, either towards the right or towards the left, an4 that 
 part of the string AB which at any time is opposite to QR, tq 
 
 D 
 
 Fig. I. 
 
 be always bent into the same form by a lateral displacement of 
 its particles, the rest being always straight. Then the string 
 A B will be transmitting a wave. When a single wave is thus 
 transmitted, any particular particle of the string is disturbed 
 during the passage of the wave, and is at rest at all times before 
 and afterwards. 
 
 86. It is necessary, however, to explain how such a wave 
 could originate. 
 
 Suppose a portion of the string, which we may. call PQ^ to 
 be bent into any arbitrary form, and a determinate lateral velo- 
 city to be then commun,icated to each particle, of P Q, according 
 to any arbitrary law consistent with the continuity of the string. 
 The subsequent motion will depend on the form of P Q, and 
 on these initial velocities of its particles. 
 
 Now the form of PQ being given arbitrarily, it is possible, to 
 assign the initial velocities in one way so that a single wave of 
 the form PQ shall be transmitted to the right, and in another 
 way so that a similar ■yyave shall be transmitted to the left. 
 
 But if the initial velocities were assigned arbitrarily, the wave 
 PQ would in general be resolved into two components, of 
 which one would travel to the right, and the other to the left. 
 
 At present, however, we assume that the initial velocities have 
 been such as to give rise to a single wave. 
 
 Any number of similar pr dissimilar, waves may be trans^, 
 mitted at the s,ame time, either in the same or in, contrary 
 
along a String, 
 
 71 
 
 directions. It is convenient to distinguish the direction of 
 transmission of waves, by calling them positive or negative 
 according as they are transmitted from left to right, or from 
 right to left. 
 
 87. The velocity of transmission depends only on the nature 
 and tension of the string, and not on the form or length of the 
 wave. 
 
 88, Let us now suppose two waves of equal length, but 
 otherwise of arbitrary forms, to be transmitted in contrary 
 directions so as to meet. After passing one another they will 
 proceed in their original forms. But during the passage a part 
 of the string will be disturbed by both waves at once, and the 
 displacements of its particles at any instant will be the sums of 
 the displacements due to the separate waves. This part of the 
 string will evidently have for its length the length of a wave, 
 and for its middle point the point at which the ends of the 
 waves first meet. 
 
 Fig. 2. 
 
 Let PQ, PQ (Fig. 2) represent the positive and negative waves 
 before meeting, and ~A the point at which they will first meet. 
 And let us now further suppose that each wave is (as in the 
 Figure) a reversed copy of the other, so that the figure would not 
 be altered by taming the paper upside down. (We may express 
 the same condition by saying that every straight line through A 
 which cuts one wave, cuts the other also at a point equidistant 
 from^ .4.) Two such waves may be called contrary waves. 
 
 Now \tXpni,p'n{ be any two- ordinates equidistant from A. 
 The corresponding points p, p of the two waves will arrive 
 opposite to A at the game instant, and the displacement of A 
 at that instant will be the algebraical sum oi pm wi^ p' rri , that 
 is aero, since / m, p'ni are equal in length, but opposite in 
 
72 
 
 Nodes. 
 
 direction. 
 
 sP 
 
 Fig. 3- 
 
 Hence, since the same thing is true for every pair 
 of corresponding points, the point A will not 
 be displaced at all; in other words, the string 
 will have a node at A, 
 
 Thus, when an infinite string transmits two 
 contrary waves in opposite directions, there is 
 a node at the point at which they first meet. 
 
 89. Let us now inquire what conditions are 
 necessary in order that there may be two nodes. 
 Let A be the place of the first node and B of 
 the second; and suppose that a positive wave 
 PA, and a contrary negative wave A Q, are just 
 beginning to meet at A. The wave PA, pass- 
 ing on to the right, will after a certain time arrive 
 at the position P'B, and will then begin to dis- 
 turb B unless it meets at that instant a contrary 
 wave BQ' moving towards the left. But this 
 new negative wave BQ', when it arrives at the 
 position AQ, will begin to disturb A unless it 
 meets a new positive wave. And, continuing 
 the same reasoning, we see that there must be 
 an infinite series of positive waves meeting an 
 infinite series of negative contrary waves. 
 
 There will then be nodes at A and at B; but 
 it is evident that there will be also an infinite 
 number of other nodes at equal intervals. Hence 
 an infinite string cannot have more than one 
 node without having an infinite number. 
 
 The distance between corresponding points of 
 consecutive positive (or negative) waves is evi- 
 dently twice A B. For when the positive wave 
 which at any instant is at PA shall have arrived 
 at P'B, the next following one will not be at 
 PA, but so far behind that position as to meet 
 the negative wave B Q' when the latter arrives 
 sXAQ. 
 
 90. Let us, however, fix our attention upon 
 
Reflection of Waves. 73 
 
 the portion of string included between two consecutive nodes 
 A, £, supposing the rest of it to be hidden from view, and con- 
 sider its condition at the instant when the positive and negative 
 waves have just passed one another at A. The form of the 
 visible string will then be this, — 
 
 Fig. 4. 
 
 and the positive wave, of which the left-hand end is now at A, 
 will be transmitted unaltered until its right-hand end reaches B. 
 It then begins to meet the negative wave coming from the 
 invisible part of the string, and to be compounded with it ; by 
 this composition the disturbed part of the visible string under- 
 goes a change of form, until the positive wave has passed com- 
 pletely out of sight, and is replaced by the negative wave, 
 thus, — 
 
 Fig- 5- 
 
 which is transmitted to A, where it meets a new positive wave, 
 and a converse series of changes takes place ; and so on inde- 
 finitely. Thus, a wave will appear to pass backwards and 
 forwards between A and B, and to be reflected at those points. 
 Each reflection consists in a shortening of the wave to half its 
 length, followed by an equal lengthening; and during this 
 shortening and lengthening its form is changed into that of 
 the contrary wave. 
 
 91. During motion of this kind, both the whole string and 
 the portion A B are in a state of vibration, that is, undergoing 
 a periodic change of condition ; and the period of a vibration, 
 
74 Nodes and Waves. 
 
 er the interval of time between the recurrence of similar con- 
 ditions, is evidently the time required for the transmission of 
 a wave through twice the distance {AE) between consecutive 
 nodes. 
 
 92. Hitherto we have considered a wave as consisting only 
 of the bent or curved portion of the string. But we will now 
 call the- whole portion included between two consecutive similar 
 poiiits a wave, as we have always done before in treating of 
 periodic curves, so that a wave-length is now twice the distance 
 between consecutive nodes. In the preceding illustrations we 
 have supposed only a small portion of this wave to be curved, 
 in order to make the process of reflection clearly intelligible; 
 but since the length attributed to the curved portion was arbi- 
 trary, we may suppose it to occupy, as it generally does in fact, 
 the whole wave-length. 
 
 Let ^C be a wave-length, the figure representing part of 
 a series of positive waves, and also of the contrary negative 
 waves, in the position which they have at a given instant. The 
 
 T 
 
 Fig. 6. 
 
 actual form of the string at that instant (which is not drawn) 
 would be found by compounding the two curves. Thus (he 
 middle point of any ordinate Pp would be a point on the 
 string. 
 
 The nodes- are the points A, B, C, D, &c. on the axis, which 
 at this instant, as at aU others, bisect the ordinates drawn 
 through them. Now suppose Pp, P'p' are. any two ordinates 
 equidistant from any node A; it is evident that the middle 
 points of these ordinates lie upon a straight line which is 
 bisected at A; in other words, every straight line drawn through 
 a node at any instant and terminated both ways by the string, is 
 bisected by th» node. 
 
Finite String. 75 
 
 93. Hence, considering a whole waveJength of the resultant 
 curve, GGGupying two nodal intervals, we see that the half wave 
 on one side of the middle node is always contrary in form 
 (Art. 88) to the other half. 
 
 Thus the whole infinite string will always have the form of 
 a series of similar waves divided into contrary halves by alter- 
 nate nodes ; and the two halves will have exchanged their forms 
 after every half period. 
 
 , The string will therefore appear to oscillate, but there will be 
 in general no visible appearance of transmission^ of waves in 
 either direction, the positive and negative- series completely 
 disguising each other except in the ease in which a consider- 
 able portion of each wave is straight: in this case only, the 
 curved portions will appear to be transmitted in contrary direc- 
 tions ; or, if only a portion of string between two consecutive 
 nodes be looked at, to be reflected backwards and forwards 
 from node to node in the manner explained in Art. 90. 
 
 94. It has been already explained that, in order to pass from 
 an infinite to a finite string, we must suppose two nodes to 
 become fixed. But these may, of course, include any number 
 of nodes between them. Hence the most general form of the 
 vibration of a finite string, fixed at both ends, consists in an 
 oscillatory motion, with nodes dividing the string into equal 
 aliquot parts. The form of the part included between any 
 two consecutive- nodes will always be contrary to that of the 
 adjacent intervals; and any two adjacent intervals exchange 
 forms after half a period. The wave-length is twice the dis- 
 tance between consecutive nodes ; and the period of a vibration 
 is the time occupied by the transmission of a wave over this 
 double distance. Hence the period is known if we know the 
 length of the string, the number of nodes, and the velocity of 
 transmission. / 
 
 When there are no nodes between the extreme points, the 
 wave-length is twice the length of the string, and the period is 
 the time of transmission over this double length. This is called 
 simply 'the time of vibration' of the string. 
 
 ' Hence such waves are often called stationary. 
 
76 Time of Vibration. 
 
 If / be the length of the (stretched) string, PTits weight, T 
 the tension by which it is stretched (expressed in terms of the 
 same unit as W), and g the so-called accelerating force of 
 gravity (that is, the velocity acquired by a falling body at the 
 end of a unit of time), then the velocity of transmission of a 
 
 lateral wave is v -C?^ i and the time of a vibration is there- 
 
 I IW 
 fore 2 V —Tf, (see below. Art. 123). 
 
 95. It may be here said, once for all, that the term ' vibration' 
 is always to be understood as implying a complete cycle of changes. 
 
 In many of the most usual cases (as in that of a string) the 
 vibration may be divided into two parts, equal in duration and 
 converse in character; and it has been the practice of some 
 (especially of French) writers to call each half a single vibration, 
 and the two together a double vibration; or, what is much worse, 
 to use the term vibration without distinctly explaining in which 
 sense it is to be understood. 
 
 In the case of a common pendulum the habit of giving the 
 name vibration to what is only half a cycle, namely, a swing in 
 one direction, has become inveterate. 
 
 In this work, however, the term will always mean what 
 Prof. De Morgan has proposed to call a swing-swang. 
 
 Thus, the time of vibration of a so-called seconds' pendulum 
 is two seconds. 
 
 It must be remembered also that a vibration in general is 
 not necessarily divisible into a ' swing' and a ' swang' of equal 
 duration and opposite character. 
 
 96. We will now proceed to the analytical expression of the 
 results obtained in the preceding articles. 
 
 li f{x) be any function which has one real and finite value 
 for every value of x, the equation j/ =/{x — vt) will represent 
 the form of a string which is always bent so as to follow that 
 of a curve J/ =/{x), supposing the latter to be moving in the 
 positive direction of the .^-axis with a constant velocity v. (For 
 if in the former equation we remove the origin along the axis 
 to a distance vt, the equation becomes,;' =/{x)) 
 
Analytical Expressions. tj 
 
 Similarly _y = F{x -\- vt) represents the transmission of the 
 form y = F {x), in the negative direction, with an equal velo- 
 city. Hence the equation 
 
 y =/{x ~vf) + F{x + vt) 
 represents the form of a string in which the displacement of 
 any particle at the time i is the sum of the displacements due 
 to both these causes. And ii /(x), F{x) are both periodic 
 functions, the last equatioii will represent the transmission of 
 two sets of waves in opposite directions. 
 
 Supposing, then, these periodic functions to be expressed 
 by means of Fourier's series (Art. 71), and to have the same 
 wave-length 2 /, the equation will take the form 
 
 = 2,=i Qsm( ^ + ''<) + 2<=i CiSm( ^ +a,y 
 
 (The constant term is omitted, because it could be got rid of, if 
 necessary, by removing the origin along the axis ofy.) 
 
 97. We have not yet introduced the condition of the exist- 
 ence of nodes. Let us now suppose that there is a node at the 
 origin ; that is, that when x = o,y = o for all values of /. When 
 X = o, the terms of the order t in the above series may be 
 written 
 
 . ITSVt , „ „, , ^ 
 
 sm— T— (— CjCOSOj + CjCOSaJ 
 
 -f COS — ^ (Cj sin a,. + Cj sin a'j), 
 
 and it is evident that the coefficient of each must vanish sepa- 
 rately ; that is, 
 
 Ci cos a^ — C"j cos a'j = o, 
 
 Cj sin n^ + C'i sin d^ = o. 
 These equations are satisfied by assuming C ^ = C;, d^ = — a^ ; 
 and if we introduce these conditions in the value of j/ at the end 
 of Art. 96, and put 2C^co^a^ = A^, 2C^%m.a^ = B^, 2l=VT, 
 we obtain 
 
 y = 2,.=i sm -^ (^j cos —^ -h Bi sm — ;- ) , (i) 
 in which t is the period of vibration. Since y vanishes not 
 
78 Initial Circumstanced. 
 
 only when x = o, bat also when x is any multiple of /, for all 
 values of /, we see that the condition of the existence of a node 
 is satisfied not only at the origin, but alsd at an infinite number 
 of points separated from; one ariother by an interval / (or half 
 i, wave-length). 
 
 Heiice this equation Represents in the most general manner 
 the oscillatory motion of an infinite string described in Art; 98. 
 And if we confine our attention to values of x between o and /, 
 it represents the vibration of a finite string of length /, fixed at 
 its two ends. 
 
 98. If the initial form of the finite string, and the initial velo- 
 cities of its particles, are gi'<fen, the values of the constants A^ 
 B^ are determined. Suppose, for instance, that when / = o, j/ is 
 
 J dy 
 to be a given functiony(.«') frorrl x = o to x = I, and ■-^" another 
 
 given function ^{x) within the same -limits. Putting, theny 
 / = o in the equation (i), and in- its differential eoefiieient with 
 respect to t, we must have 
 
 2f=r^iSin'-^=/(4, 
 
 2t = oo .7, . tit jC T J / V 
 
 t£ism-j- = --<l>ix), 
 
 * — A / 2 77 
 
 from ^ = o to X = I. Hence, multiplying each side of these 
 
 , . i'ttx - . . . . , 
 
 equations by sm — y- ax, and integrating from j; = o to x = 1, 
 
 we obtain 
 
 '■ ° IJ ■^^^^ ^^"^ "7" ' * " TiTij * ^"^^ ^'" ~7~ 
 
 (It should be observed thaty(.:v) is necessarily expressible 
 by means of a series of sines only (see Appendix to Chap. IV, 
 ad fin.); for the forni of the string is at any time half a wave 
 of a series in which the alternate values are contrary (Aft. 93)', 
 so that the function f{x)\ besidies satisfying the condition's 
 /{ml) = o for all integer values of pi including o, must also' satisfy 
 the conditions /{x + 2 ml) =/{x) arid /(2 ml— x) = —/{x) ; 
 
 and these are satisfied by every such term as sin — y-, but not- 
 
Particular Cases. 79 
 
 by cos —J-. The same remark applies to ^ {x), ^hich, as is 
 
 easily seen h priori, ought to satisfy the same conditions.) 
 
 A case frequently useful is that in which the string is initially 
 at rest, and its initial form is given by an equation 
 
 In this case <^ {x) = o, and_/'(x) is the series just given for _>» ; 
 
 2 l^ i IT X 
 
 and therefore Bf-o, and Ai= ■=■/ j'sin — —dx = Cf, hence 
 
 the subsequent vibration is given by the equation 
 ■^-,=00^ . iitx ziirt 
 
 y = ^ ._, Qsm —J- cos , 
 
 where r = — as before. 
 
 V 
 
 99. Another useful case is that in which the initial form of 
 the string is a bent line A QB, where AB = /, and a, h are the 
 coordinates of Q reckoned from A. 
 
 Fig, ?. 
 
 Suppose further, that the initial velocities of all points are 
 zero. This is the case- of a String' made to vibrate by being 
 pulled aside at one point and then left to itself; 
 
 Here <^ {x) is o, and f{x) \s, - x from sc = a Xa x = a, and 
 
 h ^ 
 
 — -- Jx—l) from X = aio X = I'; and therefore Bi = 6, and 
 2& i C<^x . iirx ,, , I- P, „ . iitx J. \ 
 
 .. iito- 
 „ sin — 7— 
 
 2bP I 
 
 ^a{i-a) i" 
 
 (The. reader should, obseive that the problem here solved- 
 
8o Variations of Form. 
 
 diflfers from that in Art. 76 in this respect, that the bent Une is 
 here half a wave, whereas there it was a whole wave.) 
 Introducing this value of ^^ in equation (i), we obtain 
 
 „ sin —f- . . 
 
 2 or --,1=00 / . tTTX 2nrt , 
 
 ->' = ;?^(73^2,=i-7^sm — cos-^, (.) 
 
 which gives the form of the vibrating string at the time /. (On 
 a particular case of this formula see below, Art. 118.) 
 
 100. It is easy to trace geometrically the variations of form 
 during a vibration. The equation 
 
 y=/{x-vf)+F{x + vt), 
 and its differential coefficient with respect to /, 
 
 -£= — »/'(•* - ^t) + vF'{x + vi) 
 
 give, when / = o, 
 
 y-/{.x)+F{x) 
 
 ^£=-v{f{x)-F{x)\ 
 
 and these expressions must coincide, from x = o\.o x = 1, with 
 the given, functions which define the initial positions and velo- 
 cities. Hence, supposing the initial value of _y to be </> (x), and 
 
 of -^ to be » ^'. {x), we have 
 
 , f(x)^-F{x) = <^{x\ 
 f (x) - F' i^x) = - V'' (.x\, 
 whence f{x) — F {x) = C — ■^ {x). 
 
 These equations give (the arbitrary C being taken = o) 
 f\x) = i (<^ (^) - V' {X) ), F{x) = \ \^ {x) + ^\x)), 
 
 so that_/"(j;) and F^x) are known from x = o to x = l. That 
 is, one half of the positive and one half of the negative wave 
 are given in the position in which they produce by composition 
 the initial form of the string. And since, in order to maintain 
 the nodes at its extremities, the half of the negative wave to the 
 right of the string must be contrary in form (Art. 88) to the 
 given half of the positive wave, while the half of the positive 
 wave to the left of the string must be contrary to the given half 
 
Particular Case. 8i 
 
 of the negative wave, a whole wave of each is determined. And 
 we have only to shift the positive wave to the right, and the 
 negative to the left, through equal distances, and compound 
 those parts which fall within the limits of the string, in order to 
 find the form at the time corresponding to that amount of 
 shifting. 
 
 101. The case treated in Art. 99 affords an easy and inte- 
 resting example. The initial velocities being zero, we have 
 •^ {x) = o, and we may therefore take i/^ {x) ■- o ; hence 
 
 /{x)=F{x) = k<l>{x); 
 that is, the given halves of the positive and negative wave coin- 
 cide, and the ordinate at any point of each is half the ordinate 
 of the corresponding point of the string in its initial form . 
 Hence, completing the waves as in the last article, we have the 
 following forms and initial positions : 
 
 Fig. 8. 
 
 A PB represents the halves of the positive and negative waves, 
 coinciding when t=o. ap A is the other half of the positive, 
 and B qb oi the negative wave. A, B are, as before, the ends 
 of the string, and the initial form is that given by compounding 
 the two ' curves' which coincide in APB; that is, by doubling 
 the ordinate, at every point. 
 
 After the lapse of a quarter of a period, the waves will have 
 been shifted (one to the right and the other to the left) through 
 a distance equal to half the length {AB) of the string; and 
 that part of the figure which is within the hmits A B will have 
 been changed into the following : 
 
 Fig. 9. 
 o 
 
82 ■ Sudden Changes 
 
 And from this we obtain by composition the following as the 
 form of the string at that instant : 
 
 Fig. lo. 
 
 At the end of half a period the form will be evidently con- 
 trary to the initial form, the two half waves again coinciding ; 
 and at the end of three-quarters of a period, the form will be 
 that of the last figure reversed right and left (as it would be 
 seen through the paper from the back). 
 
 If the original p>oint of flexure be at the middle of the string, 
 
 the form at the end of the first, third, &c. quarter of a period 
 
 will be a straight line. But in other cases the string will always 
 
 be divided into either two or three straight portions, viz. three 
 
 in general, but only two at the instants when the positive and 
 
 negative waves coincide, and also at the instants when a summit 
 
 of each is opposite the same end of the string, — in all, four 
 
 times in each vibration. 
 
 dy 
 The value of -r— at either end of the string varies discon- 
 
 tinuously in a remarkable way. This may be traced geome- 
 trically without difficulty, but the following mode of proof 
 affords a good example of the use of Fourier's series. 
 
 dy 
 Putting X = o in the value of -^ derived from (a), we have, 
 
 at that end of the string, 
 
 dy 2 hi ,^i=oo I .lira ziirt 
 
 -5—= 77 r^. , -. sm — r-cos . (3) 
 
 dx Trail— a) '*'»=i t I T 
 
 Now the series 2,1" C'^ cos evidently represents a pe- 
 riodic function of which the period is r, and which satisfies the 
 condition /{t) =f(T — /) ; and any such function can, with 
 the addition if necessary of a constant term, be represented by 
 such a series. We may include the constant term by extending 
 the summation to ?' = o . 
 
of Direction. 83 
 
 Consider, then, a function/" (/) of which the value is 
 c from / = o to / = a ; 
 k from /=ato t = T — a; 
 c from t = T — a to /=T. 
 
 Assuming/(/) = 2^-"" Cj cos ^-^ , we shall have 
 
 c„ 
 
 Ci = - f(t) cos a/, 
 
 for all values of i except o. Performing the integrations in 
 separate parts (as in finding ^^ in (2)), we find 
 
 T 
 
 /-, 2 (c — k) . 2 {na , 
 
 Cj = — : sm when i is not o. 
 
 Z IT T 
 
 Hence the discontinuous function /{i) is represented by the 
 series 
 
 k-\ {c — k) + -^ i^r-sin cos . 
 
 which will be identical with (3) if we assume a, k, and c so that 
 
 k^- --{c-K) = o, — = -, c-k = —- — - , 
 T T I a (I — a) 
 
 from which we find 
 
 5 5 ar 
 
 a I— a 2 / 
 
 and it follows that the string, at the point A (Fig. 7, Art. 98), 
 
 d T 
 
 maintains its initial direction A Q from / = o to / = — -, (that is, 
 
 2 / 
 
 during the time of wave-transmission along a distance a), it 
 then suddenly becomes parallel to QB and maintains that direc- 
 tion until t = (i ^) r, when it resumes the first direction 
 
 until the end of the period t, and so on successively. The 
 representation of these changes by a diagram in which the 
 
 G 2 
 
84 Sudden Changes of Direction. 
 
 abscissa and ordinate are proportional to / and to the value of 
 
 dy 
 
 -3— at ^ (or_/(/)) is obvious, and may be left to the reader. 
 
 (See Helmholtz, p. 93, where a greater number of interme- 
 diate forms of the string is given, and a diagram representing 
 
 dv 
 the values of -i- at one end, which is taken as the curve of 
 dx 
 
 pressure (or rather of variation of pressure) on a bridge sup- 
 posed to support the string at that end. The variations of 
 
 dy 
 pressure are sensibly proportional to those of -j-, as will be 
 
 seen in Chap. VII.) 
 
CHAPTER VI. 
 
 VIBRATIONS OF A STRING {continued). 
 
 102. The most general form of the infinitesimal vibrations 
 (in one plane) of a string is given by the equation (i) of 
 Art. 97, which, by a transformation inverse to that employed 
 in Art. 72, may be written thus : 
 
 «=i CiSm^-cos{-^ + a,), (13) 
 
 where x is the distance of any point in the string from one 
 end, and J/ is the lateral displacement of that point at the time /. 
 This displacement is therefore the sum of the displacements, in 
 general infinite in number, represented by the several terms of 
 the series ; and the vibration of the whole string may accord- 
 ingly be said to be compounded of the vibrations represented 
 by those terms. 
 
 'Let us then consider separately the vibration represented by 
 the term of the order t. Supposing this term to exist alone, 
 we have, instead of (13), 
 
 y = Ci sm —J- cos \^-^ + a^) , 
 
 and the form of the string at any time is therefore a harmonic 
 curve cutting the axis in fixed points or nodes, which divide 
 the whole length into i equal parts, while the amplitudes of the 
 waves of the curve vary periodically with the time, and every 
 individual point (except the nodes) performs harmonic vibra- 
 
 •tions with the same period -. Thus, the motion of the whole 
 
 string is an oscillation with nodes, of the kind described in 
 
86 Harmonics of a String. 
 
 Art. 93, but with this distinction, that the waves are of the 
 harmonic form. 
 
 Now if, returning to the general equation (13), we suppose 
 all the coefficients preceding Q to vanish, the rest remaining 
 arbitrary, there will still be the same number of nodes, and the 
 
 period of the vibration will still be - ; but the form of the waves 
 
 may be any whatever, as in Art. 93. 
 
 If Cj is not zero, there are no actual nodes (except the fixed 
 ends), and the first component of the vibration consists in an 
 oscillation of which the period is t and the wave-length is twice 
 the length of the string. 
 
 103. When a string, vibrating without nodes, produces an 
 audible note, the lowest component tone heard is in general 
 that of which- the pitch corresponds to the period t of the whole 
 vibration, and this is called the fundamental tone of the string. 
 But other tones, belonging to the harmonic scale of this funda- 
 mental tone, are in general heard also. (If, however, the period 
 T is so long that the fundamental tone does not fall within the 
 limits of audibility, the lowest tone heard will of course be one 
 of the higher components.) 
 
 If there is one node, the period becomes -, and the lowest 
 
 2 
 
 tone heard is the octave of the fundamental tone. 
 
 And in general, if there are i — \ nodes, the period is -, and 
 
 the lowest tone heard is the {i— i)"^ harmonic of the funda- 
 mental tone. 
 
 But in each case higher harmonic components are in general 
 heard, so that the sound is a compound note. 
 
 The notes produced by a string vibrating with one or more 
 nodes, are called by musicians the harmonics of the string. 
 
 104. Now, when the string vibrates without nodes, so as to 
 produce what is called its fundamental note, the series of har- 
 monic component tones is in general complete so far as it can 
 be traced by the ear; and a practised ear, properly assisted 
 (see below. Art. 115), can easily distinguish ten or more. But 
 we are able, as we shall see presently, to make a string vibrate 
 
Vibrations how Propagated. 87 
 
 in such a manner that for any proposed value of i all the 
 coefiScients C^, C^^, Cg,-, &c. in the series (13) shall vanish; so 
 that the component harmonic vibrations of which the periods 
 
 T T 
 
 are -., — ., &c. are extinguished. And it is found, as a fact, 
 
 that when this is done, iAe corresponding tones become either quite 
 or nearly inaudible. 
 
 105. From the facts stated in the last two Articles it is 
 obvious to infer that each component tone actually heard, is 
 produced exclusively, or at least mainly, by the corresponding 
 component harmonic vibration of the string ^- 
 
 But to appreciate the force of this conclusion, we must con- 
 sider the phsenomena more precisely. 
 
 106. The vibrations in the ear which ultimately produce the 
 sense of sound, are very remotely derived from those of the 
 string. In the first place, the sound-waves in the air are 
 excited in a very slight degree by the string itself. This may 
 be shewn by stretching a violin or pianoforte string between 
 two very firmly fixed supports — for instance, iron pegs in a 
 wall — when it will be found impossible to make it yield a note 
 of any considerable strength. In all actual stringed instru- 
 ments, therefore, the supports of the string are so arranged as 
 to communicate a state of forced vibration to a considerable 
 surface of wood. Then this vibrating surface originates waves 
 in the air ; and these, being propagated to the tympanic mem- 
 brane of the ear, put that membrane itself into a state of forced 
 vibration, which is further communicated, by means of the link- 
 work of small bones mentioned in Art. 4, to the membrane of 
 the oval window; and finally, from that, through the fluid of 
 the labyrinth, to those parts of which the vibrations ultimately 
 affect the auditory nerve. 
 
 Evidently, therefore, it is essential to inquire, by what law 
 the form and period of vibrations excited at any part of a 
 material system by given vibrations maintained at any other 
 part, are connected with the form and period of the latter. 
 
 ' For a discussion of the question, What sort of vibrations produce simple 
 tones ? see the Memoirs of Ohm and Seebeck, in Poggendorf, vol. lix. p. 49 7 ; 
 Ix. 449 ; Ixii. I. 
 
88 Law of Forced Oscillations. 
 
 107. The answer to this question is contained in a statement 
 of the law of forced oscillations. (See Appendix to this chapter.) 
 
 If a. material system, acted on by a conservative system of 
 forces S be very slightly disturbed from a configuration of stable 
 equiUbrium, and then left to itself after having had very small 
 velocities (or none) impressed upon any of its particles, it will 
 continue for ever to execute small oscillations; that is, every 
 particle (except such as may remain at rest) will describe a 
 path in which it will always be very near to the position which 
 it had in the condition of equilibrium. The motion may or 
 may not be a vibration, in the proper sense. That is, the 
 system may, or may not, pass again and again at equal intervals 
 of time through the same configuration. 
 
 108. If, however, the displacements and velocities are always 
 so small that their squares and products may be treated as 
 insensible, then the motion is sensibly either a true vibration, 
 or else is compounded of vibrations each of which might subsist 
 by itself, but of which the periods are in general incommen- 
 surable; so that by their superposition they produce a non- 
 periodic motion, or more properly, a vibration of infinitely long 
 period. 
 
 We may call these component vibrations the natural \'ibr&- 
 tio'ns of the system, to distinguish them from the forced vibra- 
 tions which are now to be considered. 
 
 109. In addition to the suppositions just made, let us now 
 further suppose, either that certain points of the system are 
 subjected to small obligatory periodic motions, or else to the 
 action of small periodic forces; that is, forces of which the 
 intensity is expressed by the product of two factors, one of 
 which may be either constant or dependent on configurations, 
 and the other is a periodic function of the time. 
 
 In either case, provided that all the displacements and velo- 
 cities continue to be of the order of magnitude above supposed, 
 the whole motion is compounded of two sets of vibrations: 
 
 ^ That is, a system in which the mutual action between any two particles 
 is independent of the velocities of those and of all other particles. (Thom- 
 son and Tait, § 2"]!.) 
 
Hypothesis of Resistance. 89 
 
 one which as before may be called natural, of which the periods 
 are independent of the imposed motions or forces, and which 
 might be entirely extinguished by a proper choice of the dis- 
 posable initial displacements and velocities; and another set 
 which are forced by the imposed motions or forces, and which 
 are permanent, and in no way dependent on initial circum- 
 stances. 
 
 And it can be shewn that no forced vibration can have any 
 harmonic component of a period which does not exist amongst the 
 periods of the harmonic components of the imposed motions or forces. 
 
 110. We have so far supposed the original system of forces 
 to be conservative, so that the natural vibrations, if once begun, 
 would continue for ever. But in all actual cases there are 
 resistances of various kinds, which sooner or later extinguish 
 the natural vibrations. There is thus an apparent destruction 
 of energy ; but the energy is not really destroyed, but changed 
 into other forms, which can in general be assigned, such as 
 heat, &c. ' 
 
 When, however, we only wish to take account of the energy 
 of the system in the ordinary mechanical sense, we have to 
 introduce these resistances under the fictitious form of forces 
 of the non-conservative class, that is, forces which are not inde- 
 pendent of velocities; and on a particular hypothesis, which 
 there is reason to believe gives sensibly correct results for small 
 velocities, namely, that the motions of the particles are resisted 
 by forces directly proportional to their velocities^, no addi- 
 tional difficulty arises in the mathematical treatment of the 
 problem, but the result is modified in the following manner. 
 
 ' It must be understood that the word ' resistance' is here used to denote 
 any cause which tends to extinguish that particular kind of motion which 
 constitutes the vibration considered. In the case of a vibrating string, for 
 instance, the resistance of the air is one such cause ; another is the com- 
 munication of motion from the ends of the string to the bodies which 
 support its tension ; and a third is probably the conversion of part of its 
 energy into heat. The hypothesis is, that the combined effect of these 
 causes may be represented by assuming a retarding force to act on each 
 particle, directly proportional to its velocity ; and it is at any rate certain 
 that results calculated on this hypothesis agree in general much better with 
 experience than those obtained by neglecting resistances altogether. Any 
 other law of resistance would introduce insuperable difficulties in.to the 
 mathematical treatment of most cases. 
 
go Second Approximation. 
 
 The periods of the natural vibrations are altered (in general 
 slightly), without ceasing to be constant; but their amplitudes 
 diminish rapidly, so that the system is soon brought sensibly 
 to rest, if there are no obligatory motions or periodic forces. 
 
 If, however, there are, as we have above supposed, small 
 periodic forces or obligatory periodic motions, the system soon 
 assumes a permanent condition of motion, consisting of vibra- 
 tions (which we will c^iSi forced, as before) of which the periods 
 are connected with those of the imposed motions or forces 
 according to the law already stated ; and there is no trace of 
 the natural vibrations except this : that the amplitudes and phases 
 of the forced vibrations depend upon the relative magnitudes 
 of their periods and those which the natural vibrations would 
 have if they existed. 
 
 111. If any harmonic component of one of the imposed 
 motions or forces have a period nearly equal to that of a har- 
 .monic component of any one of the natural vibrations, then 
 there will be a corresponding forced vibration with a large 
 amplitude. On the supposition of no resistances, and of abso- 
 lute equality of periods, the amplitude in question would go on 
 increasing with the time, so as soon to violate the supposition 
 of small motions. But in all actual cases the effect of the 
 resistances is to limit the increase of the amplitude to a definite 
 magnitude, which, however, may still be larger than is con- 
 sistent with the supposition referred to. 
 
 112. In fact, in a great numlser of cases, the supposition that 
 the squares and products of displacements and velocities may 
 be neglected leads to results which agree well, as a first ap- 
 proximation, with experiment, but fail to explain phaenomena 
 of a delicate but still perceptible kind, which can be accounted 
 for by a second approximation. 
 
 The result of this second approximation shews that there are 
 in general forced vibrations of which the amplitudes are small 
 magnitudes of the second order, and of which the harmonic 
 components have periods which are either the halves of the 
 periods of harmonic components of the imposed motions or_ 
 forces, or are such that the numbers of vibrations in a given , 
 
Experiments. 91 
 
 time are the sums or differences of the numbers of vibrations 
 of the latter taken two and two. 
 
 113. We shall shew afterwards how this result accounts for 
 the remarkable phsenomena of so-called ' combination-tones.' 
 At present we see that, so far as the first approximation goes, 
 we are entitled to assume that the vibrations which ultimately 
 affect the auditory nerve have no harmonic components differ- 
 ing in period from those of the vibrations of the body from 
 which the sound originates. And when we compare this theo- 
 retical conclusion with the observed fact that the extinction of 
 any harmonic component of the vibration of a string, extin- 
 guishes (very nearly, if not entirely) the sensation of the cor- 
 responding tone, the inference appears unavoidable that the 
 sensation of simple tone is produced by simple harmonic 
 vibration. 
 
 114. We will now describe some simple experiments which 
 exhibit the accordance of the theory of vibrating, strings with 
 facts. 
 
 In the first place, the isochronism of small vibrations (that 
 is, the independence of their periods on their amplitudes) is 
 shewn by the familiar fact that a given string produces a note 
 of sensibly the same pitch, whether it be made to sound loudly 
 or sofdy, provided the variations of loudness do not much 
 exceed the limits usually allowed in music. 
 
 The point next in importance is the verification of the com- 
 pound character of the note usually produced. 
 
 The following is an easy method of making some of the 
 principal harmonic component tones sensible to an unpractised 
 ear ; or, rather, of making the ear conscious that it hears them. 
 *Strike any note of a pianoforte rather strongly, say c, and 
 hold the key down so that the vibrations may not be stopped 
 by the damper. (The two or three strings belonging to the 
 note should be tuned well in unison.) Immediately afterwards 
 strike very gently any note belonging to the harmonic scale 
 of c, holding the key also down. Then, if the attention be 
 fixed upon the sound of this latter note as it dies away, it will 
 be heard to remain as a component of the note first struck j 
 
92 Resonators. 
 
 and so distinctly, that it will often appear quite surprising that 
 what is now a conspicuous phsenomenon- should have entirely 
 escaped observation before attention was thus directed to it^. 
 
 In this way eight or ten harmonic component tones may 
 generally be distinguished. An ear which has been musically 
 trained will soon acquire great facility in tracing these harmonic 
 tones, up to a certain number varying with circumstances, 
 without any assistance. But, in order to distinguish the higher 
 and fainter ones, it is necessary to put the ear in communica- 
 tion with resonators, the action of which may be here briefly 
 explained. 
 
 115. They are usually made of glass or brass, in the shape 
 of nearly spherical bottles. The neck of the bottle is short, 
 and so formed that by coating it with sealing-wax it may be 
 made to fit closely into the outer part of the meatus of the ear. 
 There is another orifice, opposite to the neck. When such 
 a resonator is applied to the ear, it forms, with the meatus as 
 far as the tympanic membrane, a cavity with one opening ; and 
 the air in such a cavity is capable of vibrating with a deter- 
 minate period, which depends on the form and size of the 
 cavity and of the opening. 
 
 Suppose now that there is an external vibrating body in the 
 neighbourhood, the air in the cavity wiU be put irito a state of 
 forced vibration, of which the component periods will be those 
 of the harmonic components of the vibrations of the external 
 body, but of which the amplitudes will in general be incon- 
 siderable. But if the period of any one of these harmonic 
 components coincide exactly, or vej-y nearly, with that of the 
 natural vibration of the cavity, the amplitudes of the forced 
 vibrations will be large (Art. Ill), and the ear will hear that 
 particular component with great distinctness, and indeed often 
 with unpleasant loudness. In order to obtain this effect in the 
 highest degree, the other ear should be closed. 
 
 116. We shall have to refer afterwards to the general prin- 
 ciples of resonance, and to the use of these resonators in 
 particular. 
 
 » Helmholtz, p. 86. 
 
Resonators. 93 
 
 Imperfect substitutes for them may be made of paste-board, 
 and the following is an easy way of roughly illustrating their 
 action. If a stiff paste-board tube, of about i;|^ inch in dia- 
 meter, and of any length, from three or four inches upwards, 
 be pressed with one end closely upon the ear, the tube and ear 
 together form a cavity open at one end; and the note cor- 
 responding to the natural vibrations of this cavity is easily 
 ascertained by tapping the outside of the tube with the ends of 
 the finger-nails or with a pencil. The sound of the taps is not 
 a mere noise, but has a determinate pitch, which, however, an 
 unpractised ear is liable to estimate an octave too low. If, now, 
 the corresponding note be struck on a pianoforte, and the 
 coincidence of pitch be nearly exact, the effect of the tube in 
 strengthening the fundamental tone of the string is very con- 
 spicuous, and may be made still more striking by alternately 
 removing and replacing the tube. And if any other note be 
 struck of which one of the stronger harmonic components has 
 the pitch corresponding to the natural vibrations of the cavity, 
 the strengthening of that component may be made more or 
 less strikingly sensible in the same way. 
 
 By tilting the tube, so that it ceases to touch the ear all 
 round, the pitch of the natural vibrations is raised, and can 
 therefore be brought into coincidence with that of a higher 
 tone. In this manner, by different angles of tilting, the same 
 tube may be made to strengthen several harmonic components 
 of the same note ; and so, by tilting it backwards and forwards 
 between complete contact and a considerable opening, this 
 series of tones may be heard upwards and downwards several 
 times before the vibrations of the string once struck cease to 
 produce audible sound. This experiment requires a little care 
 and practice, but when successful is very striking. 
 
 117. The vibration of a string with nodes is easily shewn on 
 any instrument of the violin species, or on a horizontal piano- 
 forte. In the latter case, if a key be struck and held down, 
 while a finger is lightly applied at a nodal point, the string will 
 sound the corresponding harmonic instead of its fundamental 
 note. Or the finger may be applied after the key is struck, in 
 
94 Experiments on 
 
 ■which case the fundamental tone, which is heard at first, is 
 extinguished, and the harmonic remains and is heard alone. 
 Thus, if the fundamental note be c, and the finger be applied 
 at one-third of the distance from either end of the string, the 
 harmonic note g' will be heard. In this case there are two 
 nodes, and the existence of that one which is not touched by 
 the finger may be shewn by placing on the string a small bent 
 strip of paper : if it be placed at any point except the node, it 
 will be shaken when the key is struck ; but if at the node, it 
 will remain undisturbed. 
 
 The harmonic notes of a harp string may be produced in the 
 same way, and the first of them (or octave) is sometimes used 
 by harp players. 
 
 The production of harmonic notes on the violin or violon- 
 cello, by touching the string lightly with the finger at the place 
 of a node, is familiar to players on those instruments. 
 
 It is obviously essential to the production of harmonics that 
 the point at which force is applied to make the string vibrate 
 (whether by a hammer, the finger, or a bow) should not be 
 a node. 
 
 118. It was stated above (Art. 104) that a string may be 
 made to vibrate in such a manner that any proposed harmonic 
 component vibration, with all those whose periods are aliquot 
 parts of the period of that one, shall be extinguished. 
 
 This follows from the formula (2) of Art. 99 ; for if in this 
 
 formula we suppose a = —I (— being a proper fraction in 
 
 lowest terms), that is, if we suppose that the string is made to 
 vibrate by being plucked at any one of the points which divide 
 it into n equal parts, all those terms in the series vanish for 
 which i is a multiple of n, and therefore the component vibra- 
 
 T T T 
 
 tions of which the periods are - , — , — , &c. do not exist. 
 
 « 2 « 3« 
 
 It was also stated that when this is done, the corresponding 
 series of harmonic tones becomes nearly or quite inaudible^ 
 
 ' This fact was discovered by Dr. T. Young. See his ' Experiments and 
 Inquiries respecting Sound and Light,' Phil. Trans, for 1800, p. 138. 
 
Harmonics of String. 95 
 
 119. To shew this experimentally, it is only necessary to pass 
 the point of the finger very lightly across a, pianoforte or violin 
 string. This should be first done at a point not coinciding 
 with a node in some proposed division, for instance, at a point 
 not dividing the string into three equal parts, and the attention 
 directed to the corresponding harmonic (in this case the third 
 component, or 'twelfth' above the fundamental tone), so that 
 it may be heard distinctly. Then, if the finger be passed 
 across a node, the absence of the same harmonic will be 
 unmistakeable. 
 
 The following is an easy and striking way of making this 
 experiment. Pluck the string alternately, with a finger of each 
 hand, at its middle point and at a point one-third of its length 
 from either end. These points must be taken very exactly, 
 and the fingers passed lightly across the string. Then along 
 with the fundamental tone will be heard very distinctly the 
 twelfth when the string is plucked at its middle point, and the 
 octave when it is plucked at the other point. Thus, if the 
 third ^ or d' string of a violin be used, the tones a", d", will be 
 heard alternately. 
 
 ^ The fourth string, which is covered with wire, does not answer so well. 
 The wire covering appears to have the effect of immediately re-establishing 
 the component vibrations which at the first instant were extingviished by 
 the pluclc. 
 
APPENDIX TO CHAPTER VI. 
 
 ON FORCED OSCILLATIONS. 
 
 (The following is to be taken only as a slight sketch of the 
 subject, in which many points of interest are omitted. For 
 more complete details on that part which relates to natural 
 oscillations, see Thomson and Tait, § 343. The process here 
 given only differs from that employed by those authors in 
 modifications of detail, and in the extension to the case in 
 which the system is subjected to obligatory motions.) 
 
 If X, y, z be the coordinates of any point of a material 
 system referred to fixed rectangular axes, the differential equa- 
 tions which define the motion of the system under the action 
 of given external forces are to be derived from the formula 
 
 ^m{x"hx-\-y"ty-^^'hz) = ^{Xh x + Yhy -^r Zhz), (i) 
 
 where accents signify total differentiation with respect to /, and 
 the summation extends, on the left hand to every particle m, 
 and on the right hand to every point at which an external force 
 is applied. X, V, Z are the components of the force applied 
 at the point x,y, z. 
 
 This formula expresses the proposition (known as D'Alem- 
 bert's theorem) that the system is at every instant in a con- 
 figuration of equilibrium with respect to the applied forces and 
 the resistances to acceleration arising from the inertia of the 
 particles. When the system is not entirely free, the possible 
 motions of the particles are limited by equations of condition ; 
 and hx, hy, &c. in (i) represent any arbitrary infinitesimal 
 alterations which could, at the time /, be made in the coordi- 
 nates X, y, &c. without violating those equations. In other 
 words, the equation (i) must at every instant subsist for values 
 of 8x, &c. corresponding to any arbitrary infinitesimal dis- 
 placement which the system could at that instant undergo 
 
Forced Oscillations. gj 
 
 without violating the conditions which limit the freedom of 
 its motion. 
 
 When the equations of condition contain the time / explicitly, 
 the expression 'configuration of equilibrium' at any proposed 
 time is to be understood as meaning that which would be a 
 configuration of equilibrium, if i, in the equations of condition, 
 became constant, with the value which it has at the instant in 
 question. (Thus, if a particle be constrained to move on a 
 surface which is continually changing its form, it is in a con- 
 figuration of equilibrium at any instant if the force applied to 
 it is in the direction of a normal to the surface at that instant.) 
 
 We may call the equilibrium relative when the equations of 
 condition contain / explicitly, and absolute when they do not. 
 
 Our present object is to consider the case in which the equa- 
 tions of condition contain / explicitly, only because given points 
 of the system are subject to obligatory motions, so that the 
 coordinates of those points are given functions of t. This, it 
 should be observed, is only a generalization of the common 
 case in which given points of the system are fixed. In either 
 case the terms referring to those points disappear from both 
 sides of the formula (i), because, whether a point is fixed, or 
 subject to an obligatory motion, it could not, at any proposed 
 time, receive any displacement without violating the condition 
 imposed upon it ; hence hx - o, &c. for all such points. 
 
 Suppose, now, we refer the system to a set of independent 
 
 coordinates Ij, ?2) o^ ^1^7 kind, that is, quantities of which 
 
 the values at any time would determine the configuration of the 
 system at that time, but could be all assumed arbitrarily without 
 violating the equations of condition. 
 
 The transformation to the Lagrangian form* of the difi"eren-> 
 tial equations is in no way affected by the suppositions now 
 made, so thai we shall have as many equations of the second 
 order as there are independent coordinates, viz. : 
 
 fdTs' dT 
 
 ^df)-JJ, = ^- ^'- (^) 
 
 on which equations, however, the following observations are to 
 be made. 
 
 T'is the expression, in terms of the new variables |j, &c. of 
 
 the summation extending to all coordinates which appear on 
 the left-hand side of (i). The coordinates of those points 
 
 ' See Price, Inf. Cat., vol. iv., § 302. 
 H 
 
pS Forced Oscillations. 
 
 which are subject to obligatory motions therefore do not appear 
 in ^as expressed in terms of x,y, z, &c. 
 
 Let us denote by x, y, z, &c. the original coordinates of those 
 points which are subject to obligatory motions. Then the 
 relations between the old and new variables will enable us to 
 express each of the other coordinates x,_}>, z, &c. as a function 
 of ^1, 4) • • • ^nd X, y, z, &c. ; suppose, for instance, 
 
 x = ^ (^1, 4. ■ • • X, y, z, • ■ ■) ; 
 
 , ax ,, ax ,, ax , 
 
 and when these values are substituted in the original expression 
 for T, the result is evidently a function which is homogeneous, 
 and of the second degree, with respect to ^\, ^'^ + . . . x', /, . . . 
 Now X, y, . . . x', /, . . . are given functions of /; and when 
 their values are introduced T is no longer homogeneous with 
 respect to ^'j, ^'g, ... and also contains t explicitly. 
 
 (This new value of T is merely the kinetic energy (or half 
 vis-viva) of the whole system expressed in terms of the inde- 
 pendent coordinates.) 
 
 For the present, however, we will suppose T to be expressed 
 in terms of ^'j, . . ., x', . . . ., so that we may assume 
 
 + ^f j; + . . . + T^\ x' + . . + i7x>' + . . . (3) , 
 
 where the coeflScients P, Q, &c. are given functions which may 
 contain all the coordinates, but not their differential coeiEcients 
 
 I'l, &q- 
 
 The functions Hj, S^, ... in equations (2) are to be found! 
 by means of the equation 
 
 Xbx + rby + Zhz + ... = Si^b^-^ + S^hi^ + ..., (4) 
 
 which gives 
 
 </|l iSf^j rf|i 
 
 so that X, y, ... being given functions of the original coordi- 
 nates, Sj, &c. are known functions of the new coordinates, con- 
 taining also in general x, y, &c. 
 
 We have now to introduce the supposition that the obligatory 
 motions consist of small vibrations. Let us assume, for in- 
 stance, that the point .^x, y, z) is subject to a vibration, so that 
 
 x = x„-t-aa!, y = yo + °^, z = z„ + aw; 
 
 where x^, y^, z„ are constants, u, v, w are given periodic func- 
 
Forced Oscillations. 99 
 
 tions of /, and a is a constant which may be considered a 
 small quantity of the first order. We may call (x„, y„, z^) the 
 mean position of the point (x, y, z) ; (the point (x, y, z) would 
 become fixed at (x^, y„, Zj) if a were =0). 
 
 Next we will suppose that the system is always nearly in 
 a configuration of absolute stable equilibrium ; that is, that the 
 values of the coordinates ^j, ^2, ... only differ by quantities of 
 the first order from values (which we will denote by (IJ &c.) 
 which would belong to a configuration of stable equilibrium if 
 a were = o. 
 
 The excursions of the particles being of the first order, we 
 make the further assumption (which must be justified^ a pos- 
 teriori) that the differential coefficients ^\, &c. are also of the 
 
 // T 
 first order. Hence T is of the second order. But -j-rr , &c. 
 
 are of the first order, while -77-, &c. are of the second (since 
 
 ^1, &c. only occur in the coefficients P, &c.). Hence, if we 
 retain only terms of the first order, we must reject the terms 
 
 -JJ-, &c. in the equations (2). 
 
 Moreover, it is evident that in the coefficients P, Q, &c. we 
 may put x„ for x, &c. and the equilibrium values for Ij, &c., so 
 that those coefficients will receive constant values. Hence we 
 may take 
 
 T= k [i> i] i\' + i b, 2] I? + . . . + [1, 2] r, f , + 
 
 + «! i\ x' + 3j I'l y' + fj I'j z' 
 
 + terms in x', /, . . ■ only, which will disappear in forming 
 equations (2). 
 
 Hence the left-hand member of the first of equations (2) 
 becomes 
 
 [i,i]ri+[i,3]n+[i,3]r3 + ... + «iX" + 3,y"+.,z", 
 
 and of the second equation — 
 
 [2, i] f 'i + [2, 2] k\ + [2, 3]f' 3 + . .+ fl^ x" + 3^ y" + c^ z", 
 
 and so on. In these equations the symbols [r, i], &c. as well as 
 flj, &c. merely denote given constants, and [i, 2] = [?,i], &c, 
 
 ' The justification is this : The motions of the particles consist of isochro- 
 nous vibrations, from which it follows that the velocities may be diminished 
 indefinitely by diminishing the amplitudes. The supposition of small velo- 
 cities therefore merely implies a certain limit to the allowable magnitude 
 of the amplitudes. 
 
 H 3 
 
lOO Forced Oscillations, 
 
 With respect to the right-hand members of (2) we observei 
 that, ptitting as before (li), &c. for the equilibrium values of 
 ^, &c. we shall have, by Taylor's theorem, as far as terms of 
 the first order, 
 
 »,-w+a-(i,))(g')+«.-«.))(g;)+- 
 +'(-(S)+'('^)+"(S)). 
 
 where brackets signify values corresponding to equilibrium 
 values of the coordinates. 
 
 Now, when the variables have equilibrium values 
 
 2 (Z 8 jt + FSj/ + Z 8 z) = o, 
 and therefore 
 
 But since 8|j, &c. are independent and arbitrary, this implies 
 (Sj) = o, (g,) = o, &c. 
 
 Further, it is evident that we may assume the zeros of the 
 coordinates Ij, &c. in such a manner that their equilibrium 
 values (li), &c. shall = o. Hence, if we denote the constants 
 
 (-7^^ , &c. by letters A, B, &c. the first of equations (2) will 
 
 become 
 
 [1,1] i\ + [1,2] i\ + . . . + a^ X" + 3, y" + q f 
 
 = A^^^ + AJ^ + ... + a{/u+gv + hw). 
 
 Now X ='x„ + au, Ssc. so that x" = a «", &c. We have also 
 supposed that u, v, w are periodic functions of /, and therefore 
 each of them may be expressed as a series of simple harmonic 
 terms by means of Fourier's theorem. Suppose sin(«/ + /3) 
 occurs in the value of x, then it will also occur in x"; and we 
 
 now see that, putting D for -^, we may write the above equa- 
 tion in the form 
 ([i,i]Z)^-^,)li+([i,2]Z)2-^,)f, + ...=>Jsin(«/+^) + ..., 
 
 the right-hand side consisting entirely of harmonic terms of 
 different periods, multiplied by small coefficients. The second 
 equation will be of the form 
 
 ([2,i]Z»^.-5,)|,+ ([2,2]Z)2-^,)4 + ... = /sin(«/-f-y) + ..., 
 and so on. Thus we shall have a set of simultaneous differen- 
 
Forced Oscillations. loi 
 
 tial equations of the second order, as many in number as there 
 are independent coordinates. 
 
 We will now, however, introduce the further supposition of 
 small resistances varying directly as the velocities of the par- 
 ticles. This does not add any difficulty to the integration of 
 the equations, and leads to results more in accordance with 
 experience. 
 
 This supposition is equivalent to writing jf — t-j-, &c. 
 
 instead of X, &c. in some or all of the terms of the original 
 formula, e being a small constant; and it is evident that the 
 effect of this will be, when terms of the second order are 
 neglected, to add to 3^, &c. linear functions of i\, I'g, &c. with 
 constant coefficients; and transposing these terms to the left- 
 hand side of the differential equations, we shall have (using 
 (2j, &c. in a new sense), — 
 
 = >4sin(«/+/3)+. . 
 
 ([2. i] i5H3i i?-A) li+([3. 3] -OH^, Z»-^.) ^2 + - • • 
 
 = /sin(«/-)-y)-|-. . 
 
 which are the differential equations of the problem. 
 
 This system of equations may be treated in several different 
 ways, of which the following appears most convenient for our 
 present purpose. 
 
 Adopting abbreviations for the operative symbols, we rnay 
 write the equations thus: 
 
 [a fl] li + [a 3] I2 + . . . = >5 sin («/-)- 0) +, . 
 [3 a] Ij + [3 <5] ^, + . . . = / sin (»/ + y) + . . 
 
 (where it is to be observed that [<z J] is not the same as [3 a]). 
 Let V be the determinant 
 
 \aa'\, 
 \ba\ 
 \ca-\, 
 
 \ci\. 
 
 \_ac-\... 
 lhc-\... 
 \cc-\... 
 
 Now if we operate upon the differential equations respectively 
 with the minor determinants 
 
 dv dv d V 
 
 d\ad\ d\bd\ d[caf] 
 
 just as in a .system of algebraic linear equations, ^^, ^3,. Sec. will 
 
102 Forced Oscillations. 
 
 be eliminated; and, observing the effect of differentiation on 
 the periodic terms in the right-hand members, we see that the 
 result will be 
 
 V li = ^sin («/ + £) + . . <5) 
 
 and in like manner 
 
 K, K' L, &c. being known constants, and K, K' , . . . small. 
 
 To integrate these equations we first suppose their right- 
 hand members to be absent, so that all the variables |j, i^, ... 
 satisfy the same differential equation, which we may write 
 
 V « = o, 
 
 and which is of the order 2 w, if ot be the number of the inde- 
 pendent coordinates li, &c. 
 
 If we put for a moment v =f{p), and call a^, u^, a,, 
 . . . a^m the roots of the equation f{x) = o, then we know that 
 the general solution of v « = o is 
 
 Cj, C2, ... C^m being arbitrary constants'. Hence the ■values' 
 of the variables Ij, |,^, ... are of this form, differing only in the 
 values of the constants Cj, &c. ; but since the complete solution 
 of the system can only contain 2 m arbitrary constants, those 
 in the values of Ij, &c. must be expressible as functions of those 
 in li. The relations necessary for this purpose would have to 
 be ascertained by substituting the expressions for $, , &c. in the 
 original system of equations of the second order. We have no 
 occasion, hoWever, to perforni this operation actually. 
 Considering now the value of any coordinate, say 
 
 ^i=CieM+ ■, 
 
 we see that the motion cannot consist of permanently small 
 oscillations, unless the roots of the equation /{x) = o consist 
 of imaginary pairs, and the real part of each pair be negative, 
 so that the value of |j may be put in the form 
 
 k = ^i^""!' sin (w,/+ ^i) + C.f — s^sin (wj/+ fi,)+... (6) 
 
 (where Oj &c. are used with a new meaning). The factors 
 €-"1' &c. are introduced solely by the resistances, as is evident 
 if we observe that the differential equations would only contain 
 even powers of D if there were no resistances. 
 
 The above values of I,, &C. determine the natural vibrations 
 of the^ system, such as^ could exist if the obligatory vibrations' 
 
Forced Oscillations. 103 
 
 were suppressed by fixing the points subject to them ; and they 
 are compounded of harmonic vibrations, each of which could 
 subsist alone. We see also that the effect of the resistances 
 would be gradually to diminish the amplitudes of the vibrations 
 so that the motion would be ultimately extinguished. 
 
 We have thus obtained the complete solution of the system 
 of equations (5) &c., on the supposition that K, K', &c. are 
 all o. And the solution of those equations in their actual form 
 may now be found as follows : 
 
 Assume, for the complete value of ^^, the terms in the ex- 
 pression (6) 
 
 + ^sin(«/ + Z) + -5 cos {nt + Z)+... 
 where A, B ... are constants to be determine'd. 
 
 (A similar pair of terms is to be assumed for every term on 
 the right-hand side of fs) ; but as the following process would 
 merely have to be repeated for each pair, we shall attend only 
 to the first.) 
 
 Substituting this value of l^ in (5), and observing that the 
 operation v destroys the part (6), we obtain the condition 
 
 A V sm.(nt-\-L)+B v cos {»/+£) =^sin (ni+Z) (7) 
 
 for the determination of A and B. 
 
 Now the operation v , consisting partly of even and partly of 
 
 odd powers of D f or — j , may be put in the form 
 
 ^ = <p {!)')+ 1) X(D% 
 
 where the second term is introduced entirely by the resistances, 
 and if they are considered as small quantities of the first order 
 the terms depending on them in (p (-0^) will be of higher orders, 
 so that (j> (D^) may be considered as what v would be reduced 
 to if there were no resistances. The result of the operations on 
 the left-hand side of (7) is easily seen to be 
 
 {A <i> {-nF)-Bnx (-«')) sin {nt^L) 
 
 -f {Anx{.-n^) +B^{—n^)) cos («/+Z), 
 and this is to be identical with ^sin (ni+L) ; hence we must 
 have A<t>(-n')-Bnx(-n'')=K, 
 
 Anx(-n^)+, B<j> (-«') = o; 
 from which we find 
 
 A -B K 
 
 and when the values -of A and Bjhns given are introduced in 
 
I04 Forced Oscillations, 
 
 the expression assumed for li, it becomes (after an obvious 
 reduction) 
 
 |, = Cje-'i«sinK/+^i)+--- 
 
 + —. — Tsin(»/+Z + e)+..., (8) 
 
 6 being a constant of which we need not write the actual value. 
 
 Thus we see that the effect of every harmonic term, in the 
 obligatory vibrations is in general to add to the value of each 
 coordinate a term with the saihe period ; and these added terms 
 represent the/breed vibrations of the system. The other terms, 
 ^hich are the same as \i K = o, &c. give the natural vibrations, 
 but these in general soon become insensible through the di- 
 minution of the factors e-"i', &c. introduced by the resistances, 
 while the forced vibrations are permanent. 
 
 It is important to observe that it follows from the nature of 
 the whole process that no periods will be introduced in the 
 forced vibrations which do not exist in the harmonic com- 
 ponents of the obligatory vibrations, so long as terms of the 
 first order only are considered, because no periodic terms are 
 ever multiplied together. 
 
 With respect to the amplitudes of the forced vibrations, we 
 see that those will be large of which the periods are such that 
 the denominators of the' coeflScients in (8) are small. 
 
 Now in the expression 
 
 the second term is introduced by the resistances, and is there- 
 fore small ; hence the vibration, (of which the period is — J, 
 
 will have a large amplitude if n be such that i^ ( — «^) = o ; but 
 the roots of the equation <^ {x^') = o give the values of 
 
 + OTjV — I, &c. 
 
 on the supposition that there are no resistances, so that on that 
 supposition ^ {—m^) = b, &c. 
 
 Hence it follows that the amplitude of any component of the 
 forced vibration will be large if its period coincide with that 
 which would belong to a component of the natural vibrations if 
 there were no resistances. And since the effect of the resist- 
 ances in altering the periods of the natural vibrations is in 
 general small, we may say in general terms that there will be 
 forced vibrations of large amplitude if amongst the harmonic 
 components of the obligatory vibrations there exist any with 
 periods equal to those of natural vibrations. 
 
Forced Oscillations. 105 
 
 Lastly, we see from (8) that though the periods of the forced 
 vibrations at all parts of the system are the same as those of 
 the obligatory vibrations which give rise to them, the phases are 
 in general different. 
 
 The process to be used in the case of periodic forces is 
 nearly identical with that which has been just explained, but 
 somewhat simpler, and the results are exactly of the same kind. 
 It is therefore omitted. 
 
CHAPTER VII. 
 
 ON THE TRANSVERSE VIBRATIONS OF AN ELASTIC 
 
 STRING. {Dynamical Theory^ 
 
 120. The rigorous differential equations which define the 
 motion of an elastic string under given conditions, can in 
 general be formed without diflSculty, but cannot be integrated. 
 It is only when the circumstances of the problem are such that 
 certain quantities involved in the equations may be neglected 
 without sensible error, that an integrable approximate form is 
 obtained. For our present purpose it will not be necessary to 
 form the rigorous equations; we shall introduce ah initio the 
 conditions of the actual problem, and neglect the small quan- 
 tities mentioned above, so as to obtain the approximate equa- 
 tions directly. 
 
 Let us then consider an elastic string, of which the ends are 
 fixed at two points A, B, zX z, distance / from one another, 
 greater than the fiatural length of the string. 
 
 In the condition of equilibrium the form of the string will be 
 a straight line of length /, and it will have a tension, T, con- 
 stant throughout its length, depending on the amount of the 
 extension to which it has been subjected. 
 
 If now the string be slightly disturbed and then left to itself, 
 its motion will be such that no particle will ever be much 
 displaced from its position of equilibrium, and we make the 
 following assumptions : — 
 
 (i) The original disturbance is such that the vibrations are 
 sensibly transversal ; that is, the projection of any 
 particle on the line AB may be regarded as a fixed 
 point. 
 
Vibrations of a Sirifig. 107 
 
 (2) The inclination of every part of the string to the line 
 
 AB is always an angle so small that the square of 
 its sine or tangent may be neglected. 
 
 (3) The extension of any portion of the string due to its 
 
 change of form may be neglected; or the ratio of 
 the actual length to the length in equilibrio differs 
 insensibly from unity. (This is evidently a con- 
 sequence of (2).) 
 
 (4) The tension is sensibly constant, and may therefore be 
 
 taken as always equal to T. (This is a consequence 
 of (3)-) 
 . 121. Now let the position of any element of the string be 
 fefeiTed to rectangular axes having their origin at A, and the 
 axis of X coinciding with A B. 
 
 It follows from (i) that, for a given element, x is to be 
 regarded as constant, and from (2) that dx = ds. Consider now 
 an element of which the length is ds, or dx, and let x,y, Z 
 be the coordinates of that end of it which is nearest to A, and 
 x-\-dx, _y+dy, z + dz of the other end. 
 
 If no external forces are taken account of, the element is only 
 acted on by the tensions at its extremities ; and the components 
 of these, in the directions of the coordinate axes, are 
 
 -T^, -7^. -r^ at the end next ^, 
 ds ds ' ds 
 
 at the other end; so that the whole components are 
 
 ^(4f). ^(4)' A^l)- 
 
 But since T is constant and ds = dx, these become 
 Hence if dm be the mass of the element, we have 
 
lo8 Vibrations of a Siring. 
 
 Let p be the longitudinal density of the string, or the mass of a 
 unit of length in its actual state of extension. (The string 
 being supposed uniform, p will be constant.) Then 
 
 dm = pds = p dx, 
 
 T 
 
 and if we put — = a^, and take dx constant, we obtain from 
 
 P 
 (i) the equations 
 
 df ~ dx"' df ~ dx^'~ ^ ' 
 
 The integrals of these equations will determine y and z as 
 functions of the two independent variables x and /; that is, 
 they will give the position, at the time /, of any proposed particle. 
 (The value of x defines the particle, and the values of j>, z 
 determine its displacement from the position of equilibrium.) 
 
 122. Since the first of equations (2) does not contain ?, and 
 the second does not contain y, it follows that the motion of the 
 projection of the string on the plane of xy is independent of 
 that of its projection on the plane oiyz; so that it will be 
 sufiicient to discuss one of these equations ; and we will suppose 
 for simplicity that the motion corresponding to the other does 
 80t exist. This is evidently equivalent to supposing that the 
 displacements and velocities of the particles produced by the 
 original disturbance were all in one plane, which we will take 
 to be that of xy. 
 
 We have therith^ single equation 
 
 ^ = a^^ (3) 
 
 d/' " dx' ' ^^' 
 
 of which the general solution is 
 
 y = /{x~at)+F{x^^af). (4) 
 
 This equation represents the transmission of two arbitrary 
 forms along an unlimited line, with the same velocity a, (cor- 
 responding to the V of Art. 06), in contrary directions; and we 
 have already shewn (Arts. 88, &c.) what must be the general 
 character of these forms in order that the resultant curve may 
 have nodes, so that a portion of the infinite string contained 
 between any two nodes may be considered as a finite string 
 
therefore 2 A / -^ . For a string of given material and 
 
 Solution of Equations, 109 
 
 fixed at its two ends. The mode of obtaining the same con- 
 clusions from equation (4) in a more analytical manner may be 
 seen in treatises on mechanics. (See, for example, Poisson, 
 Traits de Me'caniqw, t. ii. chap. viii. Price, Inf. Cal. vol. iv. 
 § 281.) 
 
 It was also shewn how the most general solution of (3) in 
 which the functions / and F satisfy the conditions imposed by 
 the problem, may be expressed in the form 
 
 y = 2i=i sm -J- [Ai cos — — + Bi sm -^;—) , (5) 
 
 where ar-il, 
 
 T 
 
 123. The value of a^ is — ; let W be the weight of the string, 
 
 P 
 
 then W=gpl, and a^ = -^ , and r, the time of a vibration, is 
 
 'Wi 
 
 Tg- 
 
 thickness, W ool, and therefore t varies directly as the length 
 
 and inversely as the square root of the tension. If the material 
 
 only be given, W is proportional to the length and to the square 
 
 of the thickness, so that t varies as the length and the thickness 
 
 directly, and the square root of the tension inversely. 
 
 If c be the length of string of which the weight would equal 
 
 2/ 
 the tension T, then T=gpc, and therefore t = • Hence if 
 
 Vgc 
 
 g be expressed in the usual manner, the number of vibrations in 
 
 a second is —: vgc. 
 2/ 
 
 124. The form (s) may be arrived at directly as follows : 
 
 The form of equation (3) suggests at once as a particular 
 
 solution 
 
 y = sinmx{A cos amt-\-B sin ami) 
 
 + cosmx {Ccosamf-\-Dsmamf), 
 
 in which, as long as no particular conditions are specified, all 
 
 the constants m,A,B, C, D are arbitrary. But in order that 
 
 this solution may represent the motion of the finite string, we 
 
no Vibrations resisted 
 
 must have, for all values oi t, y - o when x = o, and also when 
 X = 1. The first of these two conditions gives C = o, Z) = o ; 
 and the second gives 
 
 sinOT/= o 
 
 whence ml=iTr, i being any positive or negative integer. Thus 
 we obtain 
 
 . tTTX f . iirai „ . iTsaL 
 y = sm — y- \A cos — — h B sm — y— j 
 
 as a particular solution ; and since the sum of any nurnber of 
 
 particular solutions is a solution of a linear differential equation, 
 
 we obtain (putting lar^l) the form (5) as a more general 
 
 solution. 
 
 The coefficients A^, B^2lXt all arbitrary unless the initial 
 
 circumstances of the motion are given ; but it has been shewn 
 
 already (Art. 98) that they can be so determined as to give the 
 
 dy 
 required initial values toy aud — for every part of the string. 
 
 The form {5) therefore is the most general solution of the dif- 
 ferential equation applicable to this simple case of the lateral 
 vibration of a string. (Longitudinal vibrations will be considered 
 in connection with those of a rod.) 
 
 125. The problem becomes little more complicated if we 
 introduce the supposition of a retarding force acting at every 
 point of the string, and proportional to the velocity \ The first 
 of equations ( i ) then becomes 
 
 dm.%^Tdd^ycdx.'^, 
 
 di^ ^ax^ dt 
 
 c being a constant. And instead of the first of (2) we shall 
 
 have d^ dy^ ^ d^ 
 
 dt^^"^" dt'" dx-"' ^^ 
 
 where 2 ^ is put for — • 
 P 
 
 Multiplying (2') by c**, and using the theorem 
 ' See Art. 110, notf. 
 
by Retarding Forces. 1 1 1 
 
 we obtain 
 
 Assuming as a solution of this equation 
 
 6*'j/ = sin (mx + a) sin (//+^), 
 we find the condition 
 
 or / = {a^ m^— k^)^ ; hence 
 y = e-T't sin {mx-\-a) sin ((a^ m^ - k'^)^ /+ /3) = o 
 is a solution. 
 
 The ends of the string being fixed, we must have (for all 
 values of /)_)/ = o when x = o, and when x = 1; and therefore 
 
 sin a = o, sin {ml-\-a) = o. 
 It is easily seen that we lose no generality by taking a = o ; then 
 
 ml=iiT, ox m= — . Fmally, therefore, the most general so- 
 lution of (2') appropriate to the problem is 
 
 in which Ai, |3^ are arbitrary constants, to be determined as 
 usual by initial circumstances. 
 
 From the above expression we see that the amplitude of the 
 vibrations will progressively diminish, and ultimately become, 
 insensible, through the diminution of the factor <:"'*'. 
 
 Also that the period of vibration of the ?*!» tone is increased 
 
 / PP si 
 
 by the resistance in the ratio of i to (^i — -5-vi) ' 
 
 If the value of k were so great that, for any values of i, 
 
 k > —J- , the value of _y would contain non-periodic terms. 
 
 But we shall not discuss this case, as it does not concern us 
 practically. 
 
 128. We proceed now to some problems of a less simple 
 kind, but of more or less practical interest. 
 
112 Forced Vibrations of Siring. 
 
 Problem i. To find the motion of the string when a given point 
 of it is subject to a given obligatory transverse vibration. 
 
 We suppose, in the first instance, that there is no resistance. 
 
 Let b be the distance of the given point from the end A of 
 the string ; and suppose that at this point the value of j/ is to be 
 k sin nt. (We have no occasion to consider the external forces 
 necessary to maintain this obligatory motion ; we suppose them 
 to be applied, whatever they may be.) 
 
 It is evident that the portions of string on the two sides of 
 the given point may move independently of one another, except 
 that the value of j' must be the same for both when x = b. 
 
 Let us assume then that from x-= oXo x = b, 
 y = ^yumx (A cos mat+Bsmmat). 
 This satisfies the differential equation, and also the condition 
 that y = o when x = o. 
 
 We have next to satisfy the condition that when x = b, 
 y = ksmnt. This gives 
 
 sin mb {A cos mci,t-\-B sin mat) = k sin nt, 
 
 which can only be true for all values of t on the supposition 
 that A = o, ma = n\ hence 
 
 . nt) 
 Sm — 
 a 
 
 and the value oiy becomes (for this part of the string) 
 
 k . nx . 
 
 y = 7 • sm — . sm nt. (6) 
 
 . nh a 
 
 sm — 
 a 
 
 Tiiis however is only a particular solution, since it contains no 
 arbitrary constant. But it is evident that we may still satisfy 
 the differential equation, without violating the prescribed con- 
 dition, if we ad4 the terms which would give the natural vibration 
 of this part of the string, if the given point were fixed. Thus 
 we get 
 
 k . nx . 
 
 y = 5 sm — sm nt 
 
 . nb a 
 sm — 
 a 
 
 , .^i=x . i'^x ^ aint . aint^ , , 
 
 + 2^=1 ^^^~ V^i ^°^ —^ + BiSm ~j-), (6) 
 
 and the series of arbitrary constants A^, B^, will enable us to 
 satisfy the initial conditions relative to this part of the string, 
 
Forced Vibrations of String. 113 
 
 so that the above equation gives the general solution of the 
 problem. 
 
 The motion of the other part may be found exactly in the 
 same way, and will be given merely by writing l—x and l—b 
 instead of x and b in (6'). 
 
 127. Attending for the present only to the first portion of 
 the string, we see that the value oi y in (6') consists of two 
 parts, the first of which depends only on the imposed obligatory 
 motion, and ' determines the forced vibration of that portion. 
 The other part is independent of the obligatory motion, and 
 represents the natural vibration. 
 
 In any actual case the natural vibration is soon extinguished, 
 because the string is constantly giving up some of its mo- 
 mentum to the air and to the bodies which support the tension 
 of its ends. But the forced vibration will continue as long as 
 the obhgatory motion is sustained. 
 
 If we call T the period of this forced vibration, so that 
 
 T = — , and neglect the natural vibration, the equation (6) may 
 
 71 
 
 be written ^ . 2^ . 2,7/ 
 
 ar 
 
 
 sm 
 
 ar 
 
 in which it is to be remembered that k is the amplitude of the 
 vibration imposed upon a point at the distance d from one end, 
 so that 5 is the length of the portion of string now considered. 
 For any given point at a distance x from the fixed end" the 
 amplitude of the forced vibration is therefore 
 
 . 2n-j? 
 sm- 
 
 ar . 27ri5 
 
 sm 
 
 ar 
 
 which expression becomes 00 (except for x^o) if sm = o. 
 
 This cannot of course actually happen ; in fact, the whole of 
 the preceding reasoning fails if the excursions of any part of 
 the string become large. All that we can really infer therefore 
 
 is that when sin is o, or very small, the amplitude of the 
 
 ar 
 
 forced vibrations will be much greater than under other con- 
 
 ditions. Now sm = o gives = z n-, or 20 = tar. 
 
 ar ar 
 
 Suppose \ the length of a portion of the string which would 
 
 I 
 
114 Forced Vibrations of Siring. 
 
 have T for the period of its natural vibration ; then 2 X = at 
 (Art. 94), and therefore the above condition is equivalent to 
 ^ = z'X ; that is, 
 The amplitude of the forced vibrations in this portion of the 
 string becomes large when its length is any multiple of that 
 which would vibrate naturally in the same period ; or, which is 
 the same thing, when the tone corresponding to the period of 
 the forced vibration belongs to the harmonic scale of tones 
 which would be given by the natural vibrations of this portion 
 if both ends were fixed, 
 
 128. The points of minimum disturbance will be those at 
 
 which sin vanishes ; that is, at which sm — - = o ; or, on 
 
 the supposition now made, sin —7— = o ; that is, at points which 
 
 divide the length h into i equal parts, so that there will be quasi 
 nodes at these points'. Similar conclusions may be deduced for 
 the other portion of the string ; and it is easily seen that if the 
 obligatory vibration be imposed at any one of the points of 
 division of the whole string into i equal parts, and if its period 
 
 be - th of the period of the natural vibrations of the whole 
 I 
 
 string, the forced vibrations will be large, and the tone will be 
 the z'* of the harmonic scale of the string. 
 
 Thus we learn that in order to produce strong forced vibra- 
 tions in a string, the obligatory vibration must be imposed at 
 what would be a node in the case of natural vibrations of the 
 same period; a conclusion which- may appear strange. It 
 might have been conjectured that a point of greatest motion 
 ought to have been chosen. The explanation is simple, 'and 
 may be left to the reader. 
 
 129. We have so far supposed, for simplicity, that the 
 obligatory motion was a simple harmonic vibration. But if it 
 were a compound vibration consisting of the superposition of 
 any number of harmonic vibrations of different periods, phases, 
 and amplitudes, it would be easily shewn that every one of these 
 components would produce a corresponding term (analogous to 
 (7)) in the expression for the forced vibrations of the string. 
 
 Thus, if the imposed vibration required that, when x = b, 
 
 (. Iirt „ . 27r/>, 
 A^ cos- h £i sin ) , 
 
 the forced vibration of the string, from x = o to x = b, would be 
 expressed by the equation 
 
Experimental Illustrations. 115 
 
 :2 
 
 . 21! X 
 
 sm 
 
 . a 
 sm 
 
 — - (A, COS H ^,- sm ) 
 
 7r£ \ Tj T^ J 
 
 2/ 
 In this case, if r:= -^ . that is, if the period of every com- 
 la ' 
 
 ponent of the imposed vibration is some aliquot part of the 
 
 natural period of the whole string, then the above expression is 
 
 not altered by changing x and b into l—x and l—b; and 
 
 therefore it holds good for the whole string. 
 
 130. These results may be approximately verified by ex- 
 periment as follows : If a tuning-fork be struck, and the end of 
 its stalk be then placed on the string of a pianoforte, violin, or 
 violoncello, at any point, it may be considered approximately 
 as imposing an obligatory vibration on that point. And it will 
 be found that in general the only sound heard is the note of the 
 tuning-fork, very weak. If, however, the point of application 
 be such that the portion of string intercepted between it and 
 either end could vibrate naturally so as to give, either as a 
 fundamental tone or as a harmonic, one of the component 
 tones of the tuning-fork, then that portion of the string is 
 thrown into strong vibration, so as to give the corresponding 
 tone very distinctly. 
 
 The fundamental tone of a tuning-fork is by far the strongest 
 of its component tones. The higher components proper, are 
 very high tones with incommensurable periods, which are hardly 
 heard after a few seconds. But there is also a harmonic tone, 
 an octave above the fundamental tone, which is weak, but per- 
 sistent, and this, as well as the fundamental tone, may be pro- 
 duced from a string in the manner just described ^- 
 
 For instance, if a c" tuning-fork be placed on the second 
 string of a violin at the point where the finger would be placed 
 in playing c", that tone will be heard, and /" may, by attention, 
 be distinguished as sounding with it. But if the fork be placed 
 on the first string at the proper point for the finger in playing 
 /", that tone will be distinctly heard, while /' will be weak '. 
 
 ^ This harmonic octave in the sound of the tuning-fork is a phEenomenon 
 of the second order, of the kind mentioned in Art. 112. 
 
 ' The sound actually heard is caused by waves in the air originating not 
 directly from the vibrations of the string, but indirectly from those com- 
 municated from the string, through the bridge, to the sound-board. An 
 investigation of the effect of placing a tuning-fork on the string, in which 
 this circumstance is taken into account, is given by Helmholz {Beilage III). 
 The practical results agree with those of the simpler process in the text. 
 
 I 2 
 
ii6 '. Forced Vibrations 
 
 These are particular cases of the general phaenomenon of 
 resonance, of which we shall have to speak hereafter. 
 
 131. The expression (6), or its equivalent (7), becomes in- 
 applicable, as was shewn in Art. 125, when the obligatory 
 vibration is imposed at such a point that its period is any aliquot 
 part of the natural period of vibration of the portion of string 
 considered. 
 
 This inconvenience may be avoided by introducing the hy- 
 pothesis of resistance, under the form already employed in 
 Art. 125. And as the result will be useful in a subsequent 
 problem, we shall give the process as briefly as possible. We 
 shall neglect altogether the natural vibrations, which are soon 
 extinguished, and with which we shall have no concern in the 
 application to be made hereafter. 
 
 Let / be the length of the whole string. "Then, putting c 
 instead of zk in the differential equation (2') of Art. 125, we 
 
 and we are to obtain a solution of this equation satisfying the 
 conditions 
 
 y=p sinni+^cosni when x = 5, andj' = o when x = o. 
 For this purpose we may assume 
 
 j' = u sin ni+v cos «/, 
 u and V being fuiictions of x to be determined. 
 
 Substituting this value of j/ in the differential equation, and 
 equating to o the coefficients of sin«/ and cos «/, we find the 
 two conditions 
 
 {a^ iy' + n')u + cnv = 0,1 „\ 
 
 {a^B' + n')v-cnu = oJ W 
 
 in which D stands for -p- • 
 dx 
 
 Eliminating v we obtain 
 
 {ia'n' + nY + c^n'}u = o; 
 and therefore 
 
 where a^, a^, a^, a^ are the four roots of the equation 
 
 that is, the four values of the expression 
 
modified by Resistance. 117 
 
 To put this result in a convenient form, assume 
 
 C 1/' 
 
 — = tan ^, tan - = jS, then 
 n 2 
 
 I = r (cos -^ + V — I sin il^) ; 
 
 cost//' 
 whence, employing De Moivre's theorem, and observing that 
 
 cos \^ _ I 
 
 \/cos\(f a/i— /3^ 
 we find for the values of the expression (a) those of 
 
 and the above valu e of a becomes (after putting A, B instead of 
 
 « = sin 5 (^eP« + ^€-P») + cos5 (CeP« + i3E-P«), 
 
 m which, 6 = — , • 
 
 0^/1-0" 
 
 The differential equation for v is of the same form as that for 
 u. Hence the value of v will only differ from that of u in 
 having different constants A', B', C, D' instead of ^, ^, C, D. 
 But these eight constants cannot be all independent of one 
 another, since the solution of the simultaneous equations (I) 
 cannot contain more than four arbitrary constants. In fact, on 
 substituting the values of u and » in those equations, we obtain 
 the relations A' = C, B' =-D, C'=-A, D' = B. 
 Moreover, since y = o when x = 0, for all values of /, we must 
 have « = o and v = when 6 = 0; hence 
 
 C+D = o, and C"+Zy=o, 
 from which it follows that A = B. Finally, therefore,- changing 
 A and C into ^ A and 5 C, we may write the values of u and v 
 thus: y = {A(rsm6+Cbcose)smnt 
 
 ->r(C<Tsm.6— Ah COS 6) cos nt; where 
 
 (T = } S = • 
 
 2 2 
 
 it remains to determine A and C so that the above expression 
 
 may become identical with 
 
 / sin«/+^cosM/ 
 
 nb 
 when X = b, that is, when 6 = — -. • 
 
1 1 8 Exact and Approximate Results. 
 
 Putting therefore for this particular value of 6, and tr„, 8, 
 for the corresponding values of <j, 8, we have the two equations 
 
 ^ (Tj sin ^ + C 8(, cos ^ =/, 
 C o-(| sin ^ - ^ 8o cos ^ = q, 
 
 whence 
 
 /I (o-/ sin^ ^ + 8/ cos^ <^ = p(r^sin(f>— g 8„ cos 0, 
 C {a-^ s\n^ (j) + 8(,^ cos^ ([>) = ^ cr„ sin ^ + / 8„ cos <^ ; 
 
 and the values of A and C thus determined are to be introduced 
 in the above expression fox y. 
 
 The result may be conveniently expressed thus : 
 
 £fo-„sind) +/8„cos(i ^ 
 
 put ^ °-. ^ ^J' ^ = tan*; 
 
 "^ /i(r„sin<^ — ^8jCOS9 
 
 then j/= \f ' ^ ^ -{(7 sin 5 sin («/+*) 
 
 (<r,2sin'''<^ + 8/cos2# 
 
 + 8 cos fl cos («/+*)}. (II) 
 
 This gives the exact solution of the problem ; that is, it de- 
 terihines the motion of the string from x = o to x = 6. The 
 motion of the remaining part of the string will be given by 
 putting !—x for x, and 1—6 for 6, in (II). The radical in the 
 denominator must evidently be understood to have the same 
 sign as sin<^, in order that this value ofj' may agree with that 
 found, as in Art. 126, when resistance is neglected, or /3 = o. 
 
 The period of the obligatory vibration is — , and that of the 
 
 natural vibration of the part of the string from .»: = o to .^ = ^ is 
 
 — ; hence, if the former period be any aliquot part of the latter, 
 
 or - = — , the value of <i ( = — : ^ becomes — ; , so 
 
 that, /3 being supposed very small, and 8„ being of the same 
 order as /3, the denominator in the expression (II) becomes 
 very small ; but it cannot vanish for any value of 0, and there- 
 fore this expression always gives a finite value foxy. 
 
 The expression (II) is however too complicated for use in 
 the problem for the sake of which we have obtained it. (See 
 Art. 138.) We shall therefore neglect /3 in every part of that 
 expression except where it is essential to retain it. Now = o 
 
 gives o- = I, o-j = I, 8 = o, 8=0, tan * = - . But we must retain 
 
Case of Periodic Pressure. 119 
 
 the terln h^ cos^ ^ in the denominator, in order that y may not 
 become infinite when sin <^ = o. We tlius obtain 
 
 . , psinnf+a cosni -tttn 
 
 y = smd : £ {i-i-i-) 
 
 (sin-<^ + Vcosl0)* 
 
 as an approximate expression, which sensibly coincides with 
 that obtained on the supposition of no resistance, for all but very 
 small values of (j>, but agrees with experiment in giving a large 
 
 but not infinite value for v when sin ^ = o, or ^ = . (It must 
 
 n 
 
 be remembered that, in (III), &= — and <b = — ^ • 
 
 If the vibration imposed at the point where x = h were not of 
 the simple harmonic kind, then instead of the right-hand mem- 
 ber of (III) we should have a series of analogous terms, as in 
 Art. 129. 
 
 132. Problem 2. To find Ike motion of the string when a given 
 point of it is subject to a given finite periodiq pressure, in a direction 
 at right angles to AB, resistance being neglected. 
 
 Suppose that the pressure is applied at a distance b from the 
 end A, and that at the time / it is / sin nt,p being a given finite 
 constant force. 
 
 To solve this problem we must recur to the fundamental 
 equation (see (i). Art. 121) 
 
 dt^ ^dx' 
 
 of which the meaning is that the force on any element dm is 
 the difference of the transverse componefits of the tension at 
 its two ends. 
 
 Now in general the change of direction of the string is con- 
 tinuous, so that d ( -r- ) is infinitesimal. But if the element dm 
 ^dx' 
 
 contain the point at which the finite pressure is applied, there 
 
 wiU, or at any rate may, be a sudden change of direction at that 
 
 dv 
 point, so that the values of -p at two points on different sides 
 
 of it, however near, may differ by a finite quantity. Hence, 
 
 (dy-^ 
 -7-) instead of 
 
 d(^\ ; and since it is acted on by the given pressure in ad- 
 
I20 Case of Periodic Pressure. 
 
 dition to the difference of component tensions, we must have,. 
 for that element, 
 
 and this equation must subsist, however small we' take dm; 
 hence in the limit it becomes 
 
 r.A(i^)+/sin«/=o, (8) 
 
 which is a condition that must be satisfied for the point at which 
 X = b. 
 
 At all other points the usual differential equation (3) must be 
 satisfied. 
 
 Now we may satisfy (3) with the conditions that_y = o for 
 
 X = and X = /, and at the same time allow the possibility of a 
 
 dy " 
 
 finite change in the value of — when x passes through the 
 
 value b, by assuming "'■'^ 
 
 y = €Ya.mx [A cos mat-^B sin maf) from x = 0X0 x = b, 
 y = %Ya.m {l—x) (^'cos mat -^B' sin mat) from x = bt.o x = l. 
 
 The condition that these values ofj/ must coincide when x = b, 
 
 gives Asmmb = A'5mm (/- b), 
 
 B %m. mb = B' sin m (l—b). 
 
 Moreover it is evident that it will be impossible to satisfy (8) 
 without taking ma = n. 
 Hence we may put 
 
 fl (I q\ fix ' 
 
 v= sin— ^^ sin — (Ccos«/+Z)sin«/), 
 
 a a ^ ' 
 
 from X = o to X = b, and 
 
 flu fl (I — 3C\ 
 
 v = sin — sin — ^^ (C cos «/+ i? sin nt) 
 
 a a ^ ' 
 
 from x = h\a x = l; 
 
 where C and D are constants to be determined by the condition 
 
 (8)- dv 
 
 Now if we put ;t; = 3 in the values of -f- derived from the two 
 
 dx 
 
 expressions above given, and then subtract the first result from 
 
 the second, we obtain 
 
 /dy\ n nl 
 a(^— )= sm — (CcOS«/+Z)sin«/); 
 
Pianoforte String. 121 
 
 so that, in order to satisfy (8), we must have C = o, and 
 
 2? = - . — — — • ; whence 
 
 '^ rsin^ 
 a 
 
 . n (I— S) , nx 
 
 sin— ^ '- sin — 
 
 a p a a 
 
 y - — ~ni' ■ ' — sin nl (j? = o to ;i; = b), 
 
 '"■ J- . nl 
 
 sm — 
 
 . nh . nU-x) r(9) 
 
 sin — sin— ^ - 
 
 a p <^ a . . , ,, 
 
 _>' = -—• sm «/ {x = o \.o x = I). 
 
 sin — 
 a 
 
 These expressions determine the_/or«(/ vibration of the string; 
 and ' it is evident that we may add the terms representing the 
 natural vibration, viz. 
 
 2fci sin -J- I ^ . cos -J- + B^ sin — ^ | . (qO 
 
 (The above expressions for the forced vibration lead to a 
 
 result which at first sight appears paradoxical. Suppose, namely, 
 
 nh 
 that sin — = 0, which will happen if the period of the forced 
 
 vibration coincide with that of the natural vibration of the first 
 portion of the string, considered as fixed at both ends. Then 
 the value oi y in the other portion will be always o, or that 
 portion will remain at rest, if the constants A^, B^ be all o. 
 
 The explanation is jnerely that in this case the periodic 
 pressure is equal to that which the string would exert on a fixed 
 point at the same place, if the portion on one side of it were 
 vibrating naturally, and the other portion at rest.) 
 
 133. We have introduced this problem for the sake of the 
 use which may be made of it in finding approximately the 
 character of the vibrations excited in a pianoforte string by the 
 blow of the hammer. For this purpose we shall adopt the 
 hypothesis proposed by Helmholz, (the reason for which is 
 explained below, Art. 134,) namely, that the pressure of the 
 hammer on the string during contact may be represented by an 
 expression of the form / sinw/, but that it lasts only during 
 
 half a period, viz. from/=o to /= -. The breadth of the 
 
 . n 
 
 hammer is neglected. 
 
 Now if, in the solution of the above problem, we suppose 
 
122 Pianoforte String. 
 
 the constants A^, B^ to be so determined that at some one 
 
 instant when the pressure vanishes, say when /= o, the values 
 
 dy 
 oiy and of -7- shall vanish at all points of the string, then the 
 a/ 
 
 subsequent motion will be the same as if the pressure had only 
 
 begun to exist, and to disturb the string, at that instant. If then 
 
 dy ' 
 we find the values of j/ and -j- at the end of the first half-period 
 
 of pressure, 'that is, when / = -, these will give the initial data 
 
 n 
 
 for calculating the natural vibrations which would follow if the 
 
 pressure then ceased to exist, as we suppose it to do in the case 
 
 of the pianoforte. 
 
 Now referring to the general expression (9) + (9') obtained 
 
 for y, we see that the condition y = o when / = o gives ^^ = o 
 
 for all values of z ; and the condition -y = gives 
 
 a . _ . z'lrx 
 TT-'SiBiSm — 
 
 . n{l—b) . nx 
 sm sm — 
 
 = — ^ ; {_x = o to x = b) 
 
 , sm — 
 a 
 
 . nh . n(l—x) 
 sm — sm 
 
 - {x = b to x = l); 
 
 . nl 
 
 sm — 
 
 a 
 
 whence, multiplying each side by sin — -— dx, and integrating 
 from X = o to X = I, yf& have 
 
 iTtT , nl ^ . nil—V) 
 
 sm— ^. = sm 
 
 2p a a 
 
 Jf* . nx . tirX , 
 sm — sm — — dx 
 a I 
 
 ^ . nb p , n{l-x) . tiTX , , . 
 
 + sm — / sm— ^ sm——dx. (10) 
 
 The value of the right-hand member of this equation will be 
 found to be 
 
 naP . nl . zV3 
 
 sm — sm - 
 
 n^P-d-i^'^'' a I 
 
 and therefore 
 
Pianoforte String. 123 
 
 2p naP . itrb 
 
 Now if, in the general expression (9) + (9') for jy, we put 
 Ai = o, we find, for the instant when nt = tr, 
 
 V = S jCfj sin — ;- sm — - , 
 
 dv an . _ . inx iir^a , 
 at I I nl 
 
 where the value of_/ is 
 
 . n(l—b) . nx 
 
 sin sin — 
 
 af a a. c ^ , 
 
 — -^ from X = o to X = 0, 
 
 T . nl 
 
 sin — 
 a 
 
 . nl . n(l—x) 
 sm — sin — ^^ ^ 
 
 and — -^ from x = 5 to x - I; 
 
 T , nl 
 
 sin — 
 
 ■a 
 
 and these are the initial values of y and -f- with which we are 
 
 dt 
 
 to calculate the subsequent natural vibrations. 
 
 If then we begin to reckon / afresh from this instant, and 
 
 assume 
 
 J/ = 2 sm — (Q cos -^ + A sin -j—); (12) 
 
 and compare the above initial values with those derived from 
 this expression, we find 
 
 C, = B, sin 
 
 nl 
 
 '^"■^ •T^ • '^'''X air .„ t'n^a . inX 
 
 —-^tJJiSin -Y- = --2iB,cos — — sin—;- +/; 
 / I I nl I 
 
 fnX 
 
 and multiplying the last equation by sin — — dx, and integrating 
 from x = oto x = l, (observing that (10) gives 
 
124 Law of Pressure of Hammer 
 
 and therefore the value (12) oi y becomes, after a slight re- 
 duction, 
 
 . i^-,J=a> _ iiP'a . t'lrx . lira ,, , t,. , . 
 
 ^ = 22,.i ^iCOS^^sm— sin — (/+-); (13) 
 
 which equation determines the motion of the string after the 
 pressure has ceased to act. 
 
 The amplitude * of the component vibration which gives the 
 
 z'th harmonic tone is therefore 2^^ cos =-; or, B: being 
 
 replaced by its actual value (11), the amplitude is 
 4/ nap . iirb iir^a 
 
 (This expression will be seen to agree with the result obtained 
 in a different manner by Helmholz, {Beilage IV. equation 
 (1212),) if it be observed that/, T, i, n in (14) correspond to 
 Helmholz's A, S, n, m; and 'that Helmholz's A^ is the am- 
 plitude only of the negative wave, and is therefore half the 
 amplitude of the complete vibration.) 
 
 134. In order to understand the results deducible from the 
 expression (14) we must recur to the hypothesis made above 
 concerning the law of pressure during the contact of the 
 hammer with the string. That hypothesis is founded on the 
 assumption that the impact may be assimilated to that of an 
 elastic body upon a hard fixed obstacle. On thi's supposition 
 let X be the length of the hammer, fi its mass, ft k'^ its moment 
 of inertia about the axis on which it turns, 6 the angle through 
 which it has turned from rest at the time /, 6^ and ^'^ the values 
 
 d6 
 of 6 and — at the instant when contact begins, and at which 
 
 we will suppose /=o; then' the pressure will be \g{d—e^) 
 during contact, g being a constant depending on the elasticity 
 
 * The meaning of 'amplitude' has been before defined (Art. 48) with 
 reference to the harmonic vibration of a poinl. In the case of the harmonic 
 vibration of a string, expressed by the equation 
 
 . iirx . Altai ,,, 
 
 or by the equivalent form 
 
 y = sm -p (Ccos — — + Bsm —7-); 
 
 the amplitude may be defined as the maximum displacement from the 
 position of equilibrium. This maximum displacement is evidently equal to 
 i, or to (C^ + Z)^)*, and occurs at points which bisect the nodal intervals. 
 
 >:iM.iMl 
 
on Helmhol^s Hypothesis. J25 
 
 of the material of which the head of the hammer is made ; and 
 we shall have, during contact, 
 
 dH _ _ xy {6- 6„) _ 
 
 and if this be integrated in the usual way, and the constants 
 
 determined by the conditions 6 = 6^, -— = 6\ when t=o, the 
 
 result is q ' 
 
 6—e„=-^ sin«/ 
 
 where « = t x/ > 
 
 and the pressure during contact is therefore 
 
 B^^k'Jiiq .smnt, 
 and the duration of contact is 
 
 7T k /H 
 
 n \ ^ q 
 
 Hence the value of/ in (14) is ^^k^/ixq, which depends on 
 the velocity of the hammer at the beginning of impact, as well 
 as on its weight, material and form. But the value of n, which 
 determines the duration of contact, depends only on the latter 
 circumstances, and not on the velocity. 
 
 135. Referring now to the expression (14) we see that it 
 
 vanishes if sin — j— = o ; that is, if ib =ml, m being any integer. 
 
 This shews that if the blow of the hammer be applied at any 
 one of the points which divide the string into i equal parts, all 
 the harmonic component tones which would have nodes at 
 those points are extinguished. (We found a similar result on 
 the supposition that the sound was excited by plucking the 
 string at a given point.' See Art. 119.) 
 
 In general the quality of the note produced, which is determ- 
 ined by the comparative strength of its different component 
 tones, is independent of the momentum of the blow, (which 
 only affects the value of the coefficient p,) but depends upon 
 
 the two ratios — and — , that is, upon the place at which the 
 
 blow is struck, and upon the ratio of the duration of contact to 
 
 the period of the fundamental tone of the string (viz. — -\ If 
 
126 Intensity of Tone. 
 
 we put V for this latter ratio, the expression (14) may be written 
 in the form 
 
 The value of v depends, caeteris paribus, on the coefficient of 
 elasticity q, and becomes very small if ^ be very great, that is, if 
 the hammer is of a liard unyielding material. But the product 
 pv IS, seen, on reference to the values of / and v, to be in- 
 dependent of q, so that the above expression for the amplitude 
 
 becomes A . inb , ■ ■, 
 
 -.^^-^-^^sm— cos (...); 
 
 where A depends upon the weight, form, and velocity of the 
 hammer, but not upon its elasticity. If we suppose the hammer 
 
 absolutely hard, or v = o, then the expression becomes — sin — - . 
 
 t i 
 
 A table of results calculated on different suppositions as to 
 the place of the blow and the value of v may be seen in Helm- 
 holz, p. 135- 
 
 It appears that in general the effect of diminishing v is to 
 increase the strength of some of the higher harmonies as com- 
 pared with that of the fundamental tone. 
 
 In numerical applications it must be remembered that the 
 intensity of the tone is supposed to be proportional (for each 
 harmonic component) to the square of the amplitude multiplied 
 by i^. See Problem 3. 
 
 136. Peoblem 3. To find the energy of a string vibrating 
 naturally. 
 
 First, suppose the vibrations are in one plane, and such that 
 
 the note produced is simply the z* harmonic component. Then 
 
 the form of the string at any time may be represented by the 
 
 equation i„x i^at v , > 
 
 J/ = ^ sm -J- sm (^ +°j- (15) 
 
 The total energy at any time consists, as we know from general 
 principles, of two parts, kinetic and potential, of which the sum 
 is constant. The kinetic energy is that due to the motion of 
 the string, and is measured by half the vis-viva. The potential 
 energy is that due to the deviation of the string from the form 
 of equilibrium, and is entirely converted into kinetic energy 
 whenever the string is passing through the position of equi- 
 librium, that is, whenever 
 
 . /it! at \ 
 sm(^ — - — ^ + aj = 0. 
 
Energy of Vibrating String. 127 
 
 Hence the total energy at any time is equal to the kinetic 
 energy at any one of those particular instants. 
 
 Now (15) gives, when sin(— ^ l-a) = o, 
 
 dy ."■« . . iiTX 
 
 --=±.-Jsm — ; 
 
 so that, if p be the mass of a unit of length, the kinetic energy 
 at that instant is 
 
 which is the required value of the total energy at any time. We 
 may transform this expression as follows : let M be the mass of 
 the whole string, and t^ the period of the vibration. Then 
 
 , , , , 2 / ta 2 
 
 (tl=Al, and aT^= —r , or — - = — ; 
 
 t I rj 
 
 hence the above expression becomes 
 
 n'M~; (16) 
 
 from which we learn that in this case the energy is proportional 
 to the product of the mass of the whole string, the square of the 
 amplitude, and the square of the number of vibrations in a unit of 
 tiTne. 
 
 Next, suppose the vibrations still in one plane, but of the 
 most general kind ; then, at any time, 
 
 . z'ttx r , irrat _ . inai\ 
 
 J/ = S sm —J- (Ai cos — ^ +Bi sm ~j~} ' 
 
 In this case the string, in general, never passes through the 
 form of equilibrium, and the potential energy is therefore never 
 entirely converted into kinetic. 
 
 Let us consider the string at the instant when / = o : at that 
 
 instant _ ^Vx , - 
 
 j/ = 2j.sm — , (17) 
 
 and the total energy is jr A- P 
 
 where K is the kinetic energy due to the motion expressed by 
 (18), and P is the potential energy due to the form expressed 
 by (17). 
 
128 Energy of Vibrating String. 
 
 The value of K is easily found, for 
 
 and if we suppose the square of the series within brackets to be 
 developed, we see at once that all the terms will be destroyed 
 by the integration except those comprised in the series 
 
 s(.-^5/(sini^)); 
 
 and these give, on integration, ^ / s z^ B^. Hence we have 
 
 K=\pl{^)\i^B?. (19) 
 
 To find the value of P, we may proceed as follows : 
 Suppose the string to be put in the form (i 7), and then left to 
 itself without any initial velocity. After the lapse of any time /, 
 its total energy will still be equal to P, but will have been partly 
 converted into kinetic, so that 
 
 P-K^ + Pr, 
 
 where K^ is the kinetic energ)' at the instant in question, and 
 Pj is the potential energy due to the form at the same mstant. 
 Suppose then the string to be brought to rest in that form, and 
 again left to itself for a time f ; we should have in like manner 
 
 Pi = K^ + P^, 
 
 and so on successively ; thus 
 
 P = K^ + K^^-K^+... adinfin., 
 
 where K^, K^, ... are the kinetic energies which the string 
 
 would have after successive intervals of time equal to /, if, 
 
 beginning with the form (17), it were left to .itself for a time /, 
 
 then brought to rest in its actual form, and left to itself again, 
 
 and so on successively. 
 
 Now the string, left to itself at a given instant in the form 
 
 (17), will vibrate (see Art. 97) so that at any time (reckoned 
 
 from that instant) 
 
 . , inx iw a t 
 ;/ = 2 ^i sm — cos— ^ ; 
 
 and at the end of any time /', if we put — — = 6, we shall have 
 
 y = 'SAiC0si6.sm—j-; (20) 
 
Energy of Vibrating String. 129 
 
 dy ■na ..... , itrX , , 
 
 ■^= — -Sz^^sinz^.sin — ; (21) 
 
 and ^1 is the kinetic energy obtained from (21) in the same 
 way as K was from (18) ; hence 
 
 ^l = i/'^(^)'2^■M/(sin^•5)^ 
 
 To find K^ we have to proceed in the same way, nierely assum- 
 ing (30) instead of (17) as the initial form; thus 
 
 K. 
 
 2-4 
 
 \pl (^) S e Af (cos i6)^ (sin i6f ; 
 
 and so on successively, the value of ^„+i being always deduced 
 from that of K^ by changing ^^ into ^^ cos i6. Therefore 
 
 = \pl(^-j) S e Al (sin ifff(\ + (cos i6f + (cosV^)*. .. ad inf.) 
 
 and, the series within brackets being equivalent to 
 
 I I 
 
 I - (cos /^)^" (sin 2-^)2' 
 
 we have finally " _. 2 
 
 P = lpl(^-f)^i^Ai. 
 
 Adding the value of P thus found to that of ^ (19), we obtain 
 the required expression for the total energy (E) of the string, 
 
 VIZ. 
 
 E=\pl{^)^z^A?^Bl) 
 /( 2 I n2 
 
 ' The representation of P by an infinite series corresponds to the physical 
 fact that it would require an infinite number of operations of the Icind 
 described in the text to bring the string into the condition of equilibrium. 
 It may be observed' that, if the arbitrary be taken incommensurable with 
 ir, the series within brackets cannot become divergent, though for infinitely 
 large values of ; it may approach infinitely near to divergence ; but this will 
 be compensated by the factor (sinifl)' becoming infinitely small. If we 
 took / equal to half a period (or 6 = tt), it is evident that the operations 
 described would never bring the string to rest. In this case the factor 
 (sin i By would vanish, and the series within brackets become oc . But we 
 arrive at a true result by interpreting the product as representing i, for this 
 as for all other values of 9. 
 
13© Energy of Vibrating String. 
 
 ■where, as before, M\& the mass of the string, and T^ the period 
 of the ?'*h harmonic component vibration. 
 
 Now, observing that A^+Bl is the square of the amplitude 
 of the component vibration, and comparing this result with 
 (i6), we see that the total energy E is the sum of the energies 
 due to the several harmonic components. 
 
 Lastly, we will take the most general case, in which the 
 vibration is not in one plane. The displacement of any point 
 in the string at a distance x from one end is then compounded 
 of two displacements y and z in planes at right angles to one 
 another, and the whole vibration is compounded of two repre- 
 sented by equations 
 
 . iirX /■ , in at _ . tirat^ 
 
 y = 2 sm —J- \^i COS —J— +Bi sin — 7—) ' 
 
 . iizX f ., iirat - _, ., i'nai\ 
 z = -s,%\n.—j-\A jcos — — VB ^ sm— — -j ; 
 
 the square of the velocity at any point of which the abscissa is 
 X, is now, when / = o, 
 
 (y) {(^ ' ^i ^"^ -r) + (^ ' ^' '''' ~r) ^ ' 
 
 and the process is the same as before, with obvious modifi- 
 cations which may be left to the reader. 
 The result is 
 
 ^^^^,,4^f^^M7±^. 
 
 and, as before, the total energy is the sum of the energies due 
 to the several harmonic component vibrations in both planes. 
 
 It will be observed that the numerator in the above expression 
 is for each harmonic vibration the sum of the squares of the 
 amplitudes of its components in the two planes ; and this sum 
 may, by an extension of meaning, be called the square of the 
 amplitude of the actual vibration, which, for a given point, is in 
 general elliptic. 
 
 137. The value of E found in the last article, is the equi- 
 valent of the work which would have to be done, if the string 
 were at rest, in order to put it into its actual form and state of 
 motion. And it appears natural to take this as the measure of 
 the strength or intensity of the note produced. But the pro- 
 priety of this definition cannot be absolutely demonstrated by 
 experiment, because, although the ear can judge with great 
 accuracy which of two notes is the louder, when both have the 
 
Vibrations of Violin String. 131 
 
 same quality and the same pitch, it cannot form a precise judg- 
 ment as to the ratio of intensity. On the other hand, when two 
 notes have the same quality but differ moderately in pitch, the 
 ear can still decide with some certainty whether they are or are 
 not of equal intensity, and if not, which is the louder ; and it 
 might perhaps be possible to arrange an experiment in which 
 a series of notes should have the same quality and equal in- 
 tensities according to the theoretical measure, and the ear would 
 judge whether equality of loudness subsisted at the same time. 
 (The definition of quality will be discussed in another chapter.) 
 
 138. Problem 4. To examine the motion of a violin string 
 under the action of the bow. 
 
 This problem is much more difiScult than that of the piano- 
 forte string, because the force exercised by the bow upon the 
 string is determined by circumstances which seem to defy 
 calculation, and we can hardly make any plausible hypothesis^ 
 a priori. We are obliged therefore to have recourse to ob- 
 servation, and endeavour to determine experimentally some 
 characteristics of the motion from which the analytical repre- 
 sentation of it may be deduced. 
 
 In the first place then it may be easily verified by any one 
 with a practised ear, that when the bow is drawn across the 
 string at any point of aliquot division, no component tone 
 which would (if existing alone) have a node at that point, is 
 heard in the note produced. (In order however to extinguish 
 these tones, it is necessary that the coincidence of the point of 
 application of the bow with the node should be exact. A very 
 small deviation reproduces the missing tones with considerable 
 strength.) The other facts to which we shall have to refer are 
 ascertained not by the ear but by the eye. The character of 
 the vibration of any point of the string may be observed by 
 means of the ' vibration-microscope', the principle of which 
 was explained in Art. 65, and in this way Helmholz has 
 arrived at results of which the following are the most im- 
 portant : 
 
 (a.) When the bow is applied at a point of which the distance 
 from the bridge is an aliquot part of the string, and the point 
 observed is one of the other nodes of the same division, the 
 curve obtained by the imaginary unroUing of the cylinder 
 (Art. 65) reduces itself to a zigzag line, so that a complete 
 vibration is represented thug — 
 
 K 3 
 
IS2 
 
 Vibrations of Violin String. 
 
 Fig. X. 
 
 A B represents the period (t) of the vibration, and the ordinate 
 P M oi any point P in the line D Misrepresents the displace- 
 ment of the observed point at the time t { = A M) reckoned 
 from the instant of greatest negative displacement. It appears 
 to be implied that AB^CE, or that the excursions on op- 
 posite sides of the position of equilibrium are equal. 
 
 It follows evidently that the velocity of the observed point is 
 constant throughout each of the two parts (the ' swing' and the 
 ' swang') of the vibration, but is not in general the same in 
 each. When, however, the observed point is at the middle of 
 the string, it is found that AC = CB, and the velocities are 
 therefore equal. 
 
 (I.) If at any point we call the ' swing' that part of the 
 vibration which is performed while the point is moving in 
 accordance with the bow, then the velocity of the swing is less 
 than that of the swang, if the observed point is in the same half 
 of the string as the point of application of the bow, and greater 
 in the contrary case. Thus, in Fig. i, iJM would represent the 
 swing and E F 'Cos swang at a node in the contrary half to that 
 in which the bow is applied. It appears probable that at the 
 point of application the string is dragged by the bow with its 
 own velocity during the swing. 
 
 (c.) When the observed point is not one of the nodes, the 
 vibration is still represented approximately but not exactly by 
 Fig. I. In this case the lines DE, EF, instead of being per- 
 fectly straight, consist of a series of ripples or wavelets, though 
 maintaining their average directions. 
 
 When the bow is applied at a point which is not a node, the 
 character of the vibrations has not been satisfactorily made out. 
 (Helmholz, p. 139, &c.) (The reader will observe that we are 
 here using the word node to signify not an actual node, but a 
 point which would be a node if the corresponding component 
 vibration existed alone.) 
 
Vibrations of Violin String. 133 
 
 These results have been confirmed by Professor Clifton, who 
 observed the vibration-curves of points on the string by means 
 of revolving mirrors. (On the principle of this method of ob- 
 servation, see Note at the end of this chapter). 
 
 139. Assuming the facts above stated, let us suppose that 
 the length of the string is /, and that the bow is applied at a 
 point which we will call Q, at a distance b from the bridge. 
 Now, whatever be the character of the actual vibration of Q, 
 we know that it can be expressed by means of Fourier's theorem 
 
 in the form 2,^/ „ . 22^^ , , 
 
 S^^Qcos VD^%va. ); (22) 
 
 where r is the period of the vibration, which must be that of 
 the natural vibration of the string, since we assume that the 
 
 fundamental note is produced : hence t = — . 
 
 Moreover it is evident that if the actual vibration of Q were 
 known, we might suppose it to become oUtgatory, and the 
 motion of the rest of the string would remain unaltered. 
 
 We shall therefore in the first place assume that the series of 
 coefficients C^, D^ are known, and that (22) is the obligatory 
 value of J/ at Q. 
 
 Then (see equation (III), Art. 131, and the remark at the 
 end of that article) the vibration at any point of the string from 
 X = o to X = b will be approximately represented by the 
 equation 
 
 _ ^i=m sin 6^ 
 
 ^'■^^-^ (sin^<^, + ./cos^ <#.,)* ^ 
 
 iCs cos 2 zV - + Ds sin 2 zV ~\; (23) 
 
 where, in the present case, 
 
 ziiTX iirx inb 
 
 r a I I 
 
 and e/ is a small quantity depending on the resistance. And 
 since this formula is not altered by changing x into l—x and 
 b into l—b, it will hold good for the whole length of the string. 
 
 140. The facts above stated ((a) (3) (c)) have been ascer- 
 tained only in the case in which the point Q, at which the bow 
 is applied, is a node. 
 
 We must therefore assume this ; and in order to determine 
 Ci, -Di, we shall further assume that the vibration of Q, re- 
 presented by (22), is of the same kind as that observed at other 
 
134 Vibrations of Violin, String. 
 
 nodes; so that (22) must give the value of the ordinate at any 
 point /* in a line such as DEF (Fig. i), if the abscissa AM 
 be taken proportional to t. 
 
 We shall have then AB = t; let ^C = 7-„, and CE=^, so 
 that is the amplitude of the vibration at Q, t^ is the duration 
 of the ' swing,' and t—t^ of the ' swang,' at the same point. 
 
 Now the problem of representing a locus such as.DEEhy 
 means of a periodic series with period r, has been already 
 solved in Art. 76. In order to make use of equation (3) of 
 that Article, it is evidently only necessary to omit the constant 
 term, and change ^^ ^^ ^^ ^ 
 
 into 2^, T, T„, /. 
 We thus obtain for the ordinate the value 
 
 -2 — r 2 -^ sm — 5 sm (/ -) ; (24) 
 
 and the values of C,-, D^ are to be so taken that the expression 
 (22) shall be identical with this. It is unnecessary to write 
 down these values, as it is easily seen that when they are in- 
 troduced in (23) that equation will become 
 
 „ 2 sm z 77 — sm5,- 
 
 141. In this equation it is to be remembered that 0f, <t>i are 
 abbreviations for — — , — y- . In order however that it may 
 determine completely the value of j^ at every point of the string, 
 it is necessary that the value of the ratio — should be known. 
 
 r 
 
 Now it has been already stated as a fact of experiment (Art. 
 138) that every component tone is extinguished which would 
 have a node at the point of application of the bow ; that is, 
 every component of which the period is an aliquot part of 
 the period of vibration of a string of length b. Hence the value 
 
 2 3 T 
 
 of _y (25) ought to vanish when — is a multiple of - ; now 
 
 a t 
 
 b = ~ , and t = — , so that y ought to vanish when . ' is a 
 in 1 a in a 
 
 multiple of -r-, or when tp^ is a multiple of jt. Therefore 
 sin ITT -^ ought to Vanish when sin ^^ vanishes, and the simplest 
 
 T 
 
 hypothesis we can make is that 
 
Vibrations of Violin String. 135 
 
 Sin = ± sin 9i = ± sm —p- • 
 
 This may be satisfied either by 
 
 h T, l-h 
 
 V ""' f = — = 
 
 but if / (in Fig. i) is reckoned from an instant at which Q 
 begins to" follow the bow, so that the positive direction oi y 
 is that of the motion of the bow, the latter supposition must be 
 adopted, because t„ ought to be greater than r— t„. Then we 
 shall have 
 
 sm 
 
 . ITT 6 
 
 sm — J— 
 
 -COS?Jr, 
 
 J . . . tTTX . ln(l—x) 
 
 and since — cos 2 n- . sin — j- = sm , 
 
 if we now agree to measure x from the other end of the string, 
 which is equivalent to changing l—x into x, (25) will be 
 reduced to the form 
 
 J' = -i — 7 t2- t —^ -^ — r sin (/—-). (26) 
 
 T 1 • ■ 1 r sin (bj , , 
 
 In this equation the factor — — j is very nearly 
 
 (sin'(/)i + e/cos='(^i)^ 
 
 equal to i, for all values of z' except those which make sin ^; = o 
 or very small. If we substitute i for this factor, we obtain the 
 approximate equation 
 
 y = -s—j ^v 2i=i -^ sin -— sin (/- -^) ; (27) 
 
 and, comparing this with (24), we see that for any particular 
 
 value of X, that is, for any particular point of the string, it gives 
 
 a vibration-curve of the same kind. For if we take a quantity 
 
 / X 
 r such that - = -1 , (27) may be put in the form 
 
 y= -5 — -, ;2,-i ^ sm sin — it ^ •"), (28) 
 
 in which the part under the sign of summation can be reduced 
 to the same form as in (24) by changing the arbitrary instant 
 from which / is reckoned. Hence it represents a zigzag like 
 
1 36 Vibrations of Violin String. 
 
 Fig. I, but with a different amplitude; and the phase of the 
 
 vibration at any'given time varies with /, that is, with x. 
 
 Now T is the duration of a whole vibration, and / of the first 
 
 / X 
 part of it, or ' swing ;' hence the equation — = -r expresses that 
 
 at any point the durations of the 'swing' and of the 'swang' 
 are proportional to the lengths of the two parts into which that 
 point divides the string; and Helmholz has ascertained, by 
 observing the ratio oi AC to CB in Fig. i, that this relation 
 actually subsists, so that the hypothesis assumed above is 
 justified. 
 
 The equation (27) agrees with the approximate formula which 
 Helmholz has obtained in a somewhat different manner. It 
 fails to represent two of the observed facts, namely, (i) the 
 extinction of those component tones which have nodes at Q, 
 and (2) the existence of ripples in the vibration-curve when the 
 observed point is not a node. 
 
 The more accurate formula (26) represents these facts ex- 
 actly, if we consider the factor to be = i (as 
 
 ^ (sin>, + e/cosVi)* ^ 
 
 it is very nearly) except when sin (/)j = o, in which last case it 
 is = o. For we have already seen that Ae vanishing of this 
 factor causes the extinction of the component tones in question ; 
 and the Vibration-curve (27) will be modified by the disappear- 
 ance of the corresponding component curves ; and the effect of 
 their disappearance will evidently be to change the zigzag of 
 straight lines represented by (27) into a zigzag of rippled 
 lines *. 
 
 If we put j3' for the amplitude of the vibration at any par- 
 ticular point, then, neglecting f^, we have 
 
 whence /3' = 
 
 't„(t — To) ir'T'(T— /) 
 
 /3/(r-/) ^X{l-x) T^ 
 
 ^0 ('■—To) '0 ('"— To) l^ 
 
 let P be the amplitude at the middle point of the string, then 
 
 P = \ , and therefore 
 
 ToCt-To) „ 
 
 ^=^x{l-x\ 
 
 ' Helmholz's Fig. 25 (p. 144) represents the vibration-curve of a point 
 so near the end of the string, that one side of the zigzag is too steep to 
 have ripples. But Professor Clifton has found that they are seen on both 
 sides when the observed point is nearer to the middle of the string. 
 
Vibrations of Violin String. 
 
 137 
 
 which gives the ratio of the amplitude at any point to that 
 at the middle point. 
 
 Introducing the above value of P in (26), we obtain 
 
 8P 
 
 «y I . iTtX . 2Z7r/, r. \ 
 
 (29) 
 
 which agrees with Helmholz's equation (3^) {Beilage V). 
 
 This equation (or (26)), considered as an equation between 
 X and_>', determines approximately the form of the whole string 
 
 T T 
 
 at any time. When / — ^ is o, or any multiple of - ,y vanishes 
 
 for all values of x, so that the whole string is straight at these 
 instants. 
 
 142. At all other times the two portions of the string between 
 its extremities and the point of greatest displacement, are 
 straight. This is easily shewn as follows. It was proved in 
 Art. 98 that the equation to a locus consisting of two straight 
 portions AC, CB (Fig. 2) 
 
 Fig. 2. 
 
 is (from x = o\.o x = l), 
 
 '%'x^{l-Xa) i 
 
 2 ^r sin ■ 
 
 ITt x^ 
 
 sm 
 
 m X 
 
 (30) 
 
 where AB = l and x^, y^ are the coordinates of C (Fig. 2), 
 reckoned (like x and.y) from ^. And it is evident that this 
 equation may be made identical with (29) at any determinate 
 time /, by taking .r^ and^j so that 
 
 ^yJ' 
 
 8P 
 
 _2^ (1^ ^\" * ^ ' 
 
 ■,i^x^[l-x^ 
 
 sm- 
 
 mx^ 
 
 ± sm 
 
 2 iTT 
 
 (^-^); 
 
 the same sign being taken in both. 
 
 The iirst of these equations shews that the locus of the vertex 
 C of the string consists of two parabolic arcs passing through 
 
138 
 
 Vibrations of Violin String. 
 
 its extremities A, B, and on opposite sides of AB, belonging 
 to the parabolas of which the equations are 
 
 ^= ^^x{l- 
 
 ■x). 
 
 The second determines the position of the veftex at the 
 time / ; and writing it in the form 
 
 Xf. , , t -H T,, 
 
 sm 2 TT - 
 
 ± sin ? TT - 
 
 \t 
 
 we see that (beginning with the instant when t = \T^ it is 
 satisfied by supposing x^ to vary uniformly from o to / and 
 then from / to o and so on successively, the time occupied 
 by each of these successive changes being \ t, and the upper 
 and imder sign being taken alternately. (This will be most 
 clearly seen by examining the case of t = i.) 
 
 Fig- 3. 
 
 The string therefore vibrates in the following manner : 
 
 It is always divided into two straight portions, as A C, CB, or 
 A C, CB ; and the vertex C describes the two parabolic arcs 
 alternately in such a manner that the foot of the ordinate, M, 
 moves backwards and forwards between A and B with a con- 
 stant velocity. (Helmholz, Beilage V.) 
 
 Hitherto we have supposed the bow to be applied in the 
 usual manner, so as to produce the fundamental note of the 
 string. But if the point of application be taken gradually 
 nearer to the bridge, while the bow is drawn with a somewhat 
 quicker motion and lighter pressure, the fundamental tone 
 becomes weaker ; and ultimately a node is established at the 
 middle of the string, the fundamental tone is extinguished, and 
 the note produced is the octave, or first harmonic. 
 
 These changes in the character of the note are accompanied 
 by a corresponding series of changes in the vibration-curve, 
 which passes from the original zigzag, through a series of 
 intermediate forms, into a similar zigzag of half the period and 
 smaller amplitude, (See Helmholz's Fig. 26, p. 145). Thi^ 
 
Vibrations of a Loaded Siring. 139 
 
 phsenomenon has also been observed by Professor Clifton. But 
 it has not been submitted to mathematical analysis. 
 
 143. Problem 5. To examine the vibration of a string which is 
 loaded with a finite mass at a given point. 
 
 We shall assume that the weight of the mass is insignificant 
 compared with the tension of the string, so that the vibration is 
 modified only by its inertia ; and also that its dimensions are so 
 small that the consideration of its motion relatively to its own 
 centre of inertia may be neglected. In other words, we shall 
 consider it as a small but finite mass, concentrated at a point. 
 
 Let then ft be this mass, and suppose / is the length of the 
 
 string, and b the distance of the point at which /i is attached 
 
 from that end of the string from which x is reckoned. 
 
 dy 
 As in Problem 2, we must suppose that the value of — may 
 
 undergo a sudden change when x passes through the value b ; 
 and at that point the equation 
 
 must be satisfied. The rest of the string is subject to the usual 
 differential equation „ „ 
 
 dt"" dx'' ' 
 
 Now we may satisfy the latter equation, together with the 
 conditions that y = o when x = o and when x = I, and that 
 the value oiy must not change suddenly when x = b, exactly in 
 the same way as in Problem 2. We shall therefore assume 
 
 y = smm Q—b) sin mx (A cos amt+B sin ami), (32) 
 
 from X = o to X = b, and 
 y = sinmb sin m {i—x) {A cos amt+B sinamt), (33) 
 ' from X = b to x = I. 
 Either of the above equations gives, for x = b, 
 
 ^=—a^m'' sinmb sin m (l—b) {A cos amt+B sin ami) ; 
 dt^ , 
 
 a-nd, taking the difference of values of ^ given by the two 
 
 equations when x = b,-we find, as in Problem 2, 
 
 ^(Sf) = _OT sin ml. (A cos amt+B sin amt) ; 
 ^dx 
 
 ^'J'^-iB (3.) 
 
140 Vibrations of a L oaded String. 
 
 hence, in order to satisfy (31), we must have 
 
 \i.(^m sin mh sin m {l—b) = Tsin ml. 
 
 This last equation determines m, while A and B remain arbitrary 
 unless the initial circumstances of the motion are given. 
 
 T 
 
 The value of a^ is — (Art. 123), where p is the longitudinal 
 
 density of the string. If then we put ^ = pX, so that X is the 
 'length of string which would have the same mass as ^, the 
 above equation will become 
 
 w X sin OT 3 sin m {l—b) = sin m I. (34) 
 
 It will evidently have an infinite number of roots, and if we 
 denote them by m^, m^, . . . . the complete solution of the pro- 
 blem will be given by the equation 
 
 y = 2 -Zr ^i {A cos amil + B.i sin a w^ /) ; (35) 
 
 in which 
 
 y _ / sin OTj {l—b) sin m^x f.a? = o to j? = b), 
 * ~ \sm m^b sin m^ {l—x) {x = b to x = I), 
 
 and the coefficients A^, B^ are all arbitrary, unless determined 
 by initial conditions. 
 
 144. The complete vibration therefore consists, as in the case , 
 of the unloaded string, of simple harmonic vibrations super- 
 posed.. But the values of m, which determine the periods of 
 these component vibrations, are not in general commensurable 
 numbers, so that the component tones do not belong to a 
 harmonic scale, and can only be improperly called ' harmonics.' 
 
 If in equation (34) we suppose X = o, or b = 0, or i5 = /, we 
 get the condition foran unloaded string, namely, sin ot / = o, as 
 in the original investigation of that case. 
 
 If, on the other hand, we suppose fi = 00 (which we are at 
 liberty to do if we also suppose gravity not to act), then X = c», 
 and the second member of (34) is insignificant in comparison 
 with the first, and the condition for determining m becomes 
 
 gin mb sin w (/— 3) = o ; 
 
 so that either sin w ^ « 0, or sin m {l—b) = o. 
 
 Either of these equations gives y^o when x = b; that is, the 
 point at which /* is attached remains fixed, as it evidently 
 ought. 
 
 The periods of the component vibrations are now those 
 which belong to the separate portions of the string ; and equa- 
 
Vibrations of a Loaded String. 141 
 
 tion (35) shews that each can only exist in its own portion. 
 But it is evident that we may now, without violating any pre- 
 scribed condition, take sin w 5 = o in one portion and sin m 
 {l—b) = o in the other; thus the motion consists in general 
 of the natural vibrations of the two portions, existing inde- 
 pendently. The infinite attached mass is simply equivalent to 
 a fixed point. 
 
 145. In general the roots of the transcendental equation (34) 
 could only be found by troublesome approximations. Two 
 special cases however deserve attention. 
 
 The first is that in which the mass ju is attached at a node, so 
 
 that b = -^l, j,f being integers. (34) then becomes 
 
 /'_/ _ ;•' 
 
 mX sinmd sin - — r^ m5 = sm—r m5: 
 / J 
 
 and it is evident that this is satisfied by taking for m6 any 
 multiple ofy TT. Thus we shall get a series of roots (not all the 
 roots) by giving t integer values from i to co in 
 
 t'l'w z'/'tt 
 
 The period of the i^^ component tone given by this series is, 
 as we see from (35), 
 
 277 2 / 
 
 ami "■^J 
 
 2 / 
 Now -—r, is the period of the vibration of the unloaded string, 
 
 aj 
 when it is vibrating so as to give the lowest harmonic tone 
 which has nodes at the points of division of the string into j' 
 equal parts. 
 
 We see therefore that when the mass is attached at a node, 
 all the component tones which have nodes at that point remain 
 unaltered. But the fundamental, and other component tones, 
 will be changed. 
 
 This may be verified by attaching a small lump of wax to 
 one of the points of aliquot division of a violin or pianoforte 
 string. . 
 
 The other case is that in which the attached mass is so small 
 
 that the square of the ratio - may be neglected. 
 
 Since (34) is satisfied by ml=in when X = o, we may assume 
 
142 Vibrations 0/ a Loaded String. 
 
 that when X is small it will be satisfied by ml=Z7r + e, where e is 
 of the same order as j ■ 
 
 Substituting therefore — j^- for ot in (34), we obtain 
 
 - (iTT + e) sin - (z'TT + f) sin^i — -) (zV + «) = sin (t'lr + e) ; 
 
 and, terms of the second order being neglected, this becomes 
 
 . X . . ^ . X. .6. 
 ZTT - sm ZTT - sm(z JT— ZTT 7) = E cos z JT ; 
 z z ^ z ' 
 
 whence we get j^ 3 
 
 — z'tt y sin^ z'tt y = e; 
 z z 
 
 so that we may take 
 
 zV + e iTT /■ X . 3v 
 
 ^,= -^ = -^(i--sm^.-). 
 
 The period of the corresponding tone is ^; or the number 
 
 am- '^^i 
 of vibrations in a unit of time is — -; hence the number of 
 
 2 77 
 
 vibrations is diminished by the load in the ratio of 
 
 I — y sm'^ZTTy to I, 
 
 and the fundamental tone, as well as the higher components, 
 are all lowered ; moreover the components belong nearly but 
 not exactly to a harmonic series, so that the compound note 
 will sound slightly discordant. The examination of particular 
 cases may be left to the reader. 
 
 NOTE. 
 
 On the Principles of the Use of Revolving Mirrors, 
 
 (See Art. 138.) 
 
 If a plane mirror revolve about a fixed axis AB vci its own plane, the 
 path of the image of any stationary point Q_is a circle which passes through 
 Qj and has its centre at the point where a perpendicular from Q^meets AB. 
 And this path is f he same whether one side only, or each side, of the mirror 
 be a reflecting surface. 
 
 If the axis of rotation, AB, be not in, but parallel to, the plane of the 
 .mirror, tl\en the path of the image of Q_is a curve of the 4th degree, having 
 
Revolving Mirrors. 143 
 
 a double point at Q_, and two loops, one -within and the other without the 
 circle described as above. The inner loop is the path of the image formed 
 by reflection at the outer surface (reckoning from A B) of the mirror, and 
 the outer loop of that formed by tlie inner surface. 
 
 A usual arrangement is to join four mirrors together so as to form four 
 sides of a cubical box, with tie axis of rotation passing through the centre 
 of the box, parallel to their planes, and equidistant from them all. The 
 outer surfaces of course alone reflect, and the images formed by them all 
 describe the same path. 
 
 But an eye placed at any determinate point will only see one image at 
 one time, and only while it describes a small portion of its path ; and if the 
 velocity of rotation be sufficiently great, this small portion of the path of the 
 image of a stationary and continuously illuminated point will appear to the 
 eye as a continuous and stationary line. If however the point, while con- 
 tinuously illuminated, have a vibratory motion of sufficiently short period, 
 parallel to the axis of rotation, and if the velocity of rotation of the mirrors 
 be so adjusted that one quarter of its period is equal to, or a multiple of, 
 the period of vibration, then the passage of each mirror through any given 
 position will always happen when the vibration is in the same phase ; and 
 consequently the visible portion of the path of the image will appear as one 
 or more waves of a continuous and stationary vibration curve, formed by 
 compounding the two motions along and perpendicular to the line before 
 mentioned. 
 
 On the other hand, if the point be stationary, but illuminated only at 
 -instants separated by sufficiently short intervals of time, it will appear, when 
 viewed by the eye directly, as a continuously illuminated point ; but when 
 seen by reflection from the revolving mirrors, it, will appear, not as a con- 
 tinuous line, but as a row of points, which will be stationary if one quarter 
 of the period of rotation of the mirrors be equal to, or a multiple of, the 
 interval of time between successive illuminations. 
 
 Some of the most usual applications of revolving mirrors depend upon 
 these principles. They appear to have been first used, for purposes of 
 observation, by Wheatstone. 
 
CHAPTER VIII. 
 
 ON THE LONGITUDINAL VIBRATIONS OF AN 
 ELASTIC ROD. 
 
 146. The vibrations of a uniform elastic rod may be either 
 transversal or longitudinal, and both kinds may, when small, 
 coexist without sensibly modifying each other. We may there- 
 fore study them separately ; and we shall begin with the theory 
 of longitudinal vibrations, as being the simplest. We might, 
 as we did in the case of the string, first consider the subject 
 kinematically, assuming the law of wave-propagation in a rod of 
 infinite length; but we prefer to proceed at once to the dy- 
 namical theory. 
 
 147. We suppose then the motion of all the particles to be 
 in directions parallel to a fixed straight line in space, with which 
 the axis of the rod always coincides. By the axis of the rod is 
 rheant a line passing through the centres of inertia of its trans- 
 verse sections. 
 
 Further, we suppose that all the particles which at any one 
 instant are in a plane at right angles to the axis, continue to be 
 so at all times. In other words, the velocities of all the particles 
 in the same transverse section are equal. 
 
 The first of these suppositions cannot be rigorously true, 
 because we know that a longitudinal extension of any part 
 of the rod is in general accompanied by a lateral contraction, 
 and vice versd. But when the vibrations are small these lateral 
 displacements may be neglected without sensible error. 
 
Longitudinal Vibrations^ 145 
 
 148. We shall first investigate the- conditions of equilibrium, 
 and then deduce the equations of motion from them by the help 
 of D'Alembert's principle. 
 
 The usual law of elasticity is assumed, namely, 
 
 where ^ is a constant (the modulus of elasticity), and 7' is the 
 force, per unit of area, which must be applied, in contrary 
 directions, to any two transverse sections, in order to produce 
 an extension (or compression) t. If Tis a tendon, or pulling 
 force, the effect will of course be positive extension, or elonga- 
 tion ; and if 7" be a pushing force (or negative tension) the 
 effect will be negative extension, or contraction. The definition 
 of extension, which includes both cases, is 
 actual length 
 
 natural length 
 
 O A 
 
 \B X 
 
 Fig. I. 
 
 Let AB (Fig. i) be the axis of the rod, coinciding with a line 
 OX fixed in space. And let us suppose that the rod is actually 
 in equilibro under the action of given forces, of which the 
 directions are all parallel to X. Let | be the actual abscissa 
 of any transverse section Pp reckoned from the fixed origin 0, 
 and X the natural (or unextended) distance of the same section 
 from the end A. (By the same section is meant the section con- 
 taining the same particles.) And let !„ be the value of ^ at ^. 
 
 We have then to consider the conditions of equilibrium when 
 the following external forces are applied : 
 
 (i) A force F^ per unit of surface applied to the end A, 
 and a force F^ of the same kind at the end B. The 
 usual rule of signs being adopted, F^ will be a pushing 
 force if it is positive, and a pulling force if it is nega- 
 tive, while the converse will be true of F^. 
 
 L 
 
146 Longitudinal Vibrations 
 
 (2) A force X per unit of mass applied throughout the 
 
 infinitesimal slice between two sections of which the 
 
 actual abscissae are | and ^ + d^. 
 
 149. Let o> be the area of the section, and / the natural 
 
 length of the rod. The equilibrium being established, if the 
 
 part of the rod between Pp and £ were cut off, in. order to 
 
 maintain the equilibrium of the remainder it. would be necessary 
 
 to apply to the surface of the section Pp some force J^ per unit 
 
 of area, and the condition of equilibrium would be 
 
 I*^co + <of pXd^+Fm = o; (i) 
 
 •'ia 
 
 ■where p represents the actual density in the section of which 
 the abscissa is ^, so that paXd^ is the whole force upon the 
 interior mass of the slice. 
 
 To find an expression for F we observe, that the natural 
 thickness of any slice being dx, and the actual thickness d^, 
 
 the extension is -; 1, and the force per unit of area, on each 
 
 dx 
 
 face, required to produce this extension, is therefore 
 
 {d^ N 
 
 supposing no forces (such as X) to act on the interior mass 
 of the slice. Now when the thickness of the slice is diminished 
 without limit, the forces on its faces remain finite, being pro- 
 portional to areas, whereas the interior forces, if there are any, 
 being proportional to volume, diminish without limit also, and 
 are therefore negligeable in comparison with the forces on the 
 
 faces. Hence q a (- i\ is the force which must be ap- 
 plied to the surface of the section to maintain the existing state 
 of extension ; this therefore is the value ofF. Thus (i) becomes 
 
 ^o + JUdn,{^-.y-o; 
 which, since x == o when ^ = !„, may be written 
 
of a Rod. 147 
 
 Differentiating this equation with respect to x, we obtain 
 
 d^ dH 
 P^d^+^d^^"- 
 
 Let p„ be the natural density of the rod; then, since p^^adx 
 and pad^ both express the mass of the same slice, we have 
 
 d^ 
 
 so that the last equation becomes 
 
 150. To deduce from this the equation of motion, in the 
 case in which no forces are actually applied except on the sur- 
 faces of the ends, we have merely to substitute for the force 
 ap^Xdx supposed to act on the slice, the resistance to ac- 
 
 celeration arising from its inertia, namely, — oj p^ ^a: . — ^ ; that 
 
 d'^i 
 
 is, to write yt instead of X. Thus we obtain, putting 
 
 dt 
 
 Po IF^" ^' ^^^ 
 
 an equation of the same form as that to which we were led by 
 the problem of vibrating strings. 
 
 The solution of this equation gives ^ as a function of the two 
 independent variables x and /. That is, it gives the position, 
 at any time /, of any proposed section defined by the value of 
 X, its natural distance from the end A. 
 
 The solution is 
 
 ^^ = ^{x-af)^-f{x + at); (5) 
 
 and the two arbitrary functions have to be determined by the 
 initial displacements and velocities, together with the given 
 conditions relative to the extremities. 
 
 151. If in equation (2) we put for p— its value p^, we get 
 
 (6) 
 
 
 
 F.^P^lxdx^.d- 
 
 -.)-»; 
 
 and. 
 
 putting 
 
 .r = in this. 
 
 \SdxK=^ 
 
 L 2 
 
 ■)• 
 
148 Longitudinal Vibrations 
 
 If we had considered the equilibrium of the other part of the 
 rod, we should have found in like manner 
 
 ^.='((s)„.-)- <« 
 
 These two equations merely express that the condition of 
 extension at each end of the rod is always such as corresponds 
 to the force applied there. 
 
 152. The differential equation (4) is satisfied by such a 
 value as | _ ^ ^^ (^^ _,. „) gjn (»?«/+ ;3) ; 
 
 but in order to satisfy the given conditions in all cases, we shall 
 find it necessary to add a non-periodic term such as 
 
 l-\-c{x—a{)-^c'(x-\- af)y 
 which is obviously of the general form (5) and satisfies (4). 
 But the part b-\-{c' —c)at of this term merely signifies a uni- 
 form motion of translation of the whole rod. Such a motion 
 may, if the terminal conditions admit it, exist ; but as it in no 
 way modifies the relative motion of the different sections, which 
 is all we are concerned with in studying the vibrations, we may 
 neglect it, and assume only a term kx in addition to the periodic 
 part of |. 
 
 153. We proceed to consider the most important cases. 
 And first we will suppose the rod entirely free,, and acted on by 
 no forces. Then F^ = 0, F^ = o, and the two equaitjons (6), 
 (7), give ^^x 
 
 dx ' 
 both when x = and when .a? = /, as the terminal conditions. 
 Assuming then 
 
 ^ = kx-\-A sin {mx+a) sin {mai+^), 
 the terminal conditions are 
 
 k + mA cos (mx + a) sin {mai+^) = i, 
 when X = and when x = I, for all values of /. 
 Hence we must evidently have 
 
 k=I, COSa = 0, cos {ml+ a) = 0; 
 
 of which equations the last two are satisfied hy a = — , ml =iir, 
 (?■ being any integer), and therefore 
 
of a Rod. 149 
 
 . . iir X . fii! at \ 
 § = Jf + j4 cos — r— sm(^ — f-jSj 
 
 is a solution, A and B being arbitrary ; and making a slight 
 change of form, as in former cases, we may take 
 
 ,^8=00 inx , iirat . inat\ ,„, 
 
 | = ^+2i=iCos-^(^iCos— J— +^ism-y-) (8) 
 
 as the general solution. 
 
 (If we included z' = o in the summation, we should merely add 
 a constant term to |, which would be equivalent to an alteration 
 of the fixed origin from which | is measured). 
 
 154. To understand the equation (8) we must recollect that 
 I is the distance from a fixed origin, at the time t, of the par- 
 ticles in a plane section of which the natural distance from the 
 end A is x. The value of x therefore depends only on the 
 particular set of particles considered, and is independent of the 
 origin of ^. 
 
 Let us now find the position of the centre of inertia of the 
 rod. Its abscissa | is given by the equation 
 
 IJP^^-^J. 
 
 P^d^, 
 
 the integrations being extended from one end to the other. 
 Now we have (Art. 149), 
 
 hence if we put, in the above equation, — dx for d^, so that the 
 limits of the integrations are .ar = o and x = l,\X becomes simply 
 
 'll=f\dx; 
 ■Jo 
 
 n p 
 
 but from (S) we have / $dx = — ; hence 
 Jo 2 
 
 or the centre of inertia remains fixed (as we know a priori that 
 it must do under the supposed conditions), and is at a distance 
 
 - from the origin of f It must not however be inferred from 
 
150 Longitudinal Vibrations 
 
 this that the section which, in the natural condition, contains 
 the centre of inertia remains fixed ; that will only happen when 
 Ai, B^ are o for all even values of /. In general, no section 
 remains fixed, the place of the centre of inertia being occupied 
 by different particles periodically. 
 
 155. If the vibrations ceased, the centre of inertia still main- 
 taining its position, the periodic part of (8) would disappear, 
 and we should have ^ = x 2A all points of the rod ; hence the 
 periodic part, which is the actual value of ^—x, gives the dis- 
 placement, at the time /, of the section defined, as before 
 explained, by the value of ;*:, 
 
 The density at any point is given (Art. 149) by the equation 
 
 P Po d^ 
 Hence the general equation (5) gives 
 
 ^ = cj)'(x-at) +/'{x + a(); 
 
 which represents the transmission of two j/a/M of density in 
 contrary directions with the same constant velocity a relatively 
 to the matter of the rod. This is not rigorously the same thing 
 as a constant velocity relatively to fixed space, because x, in the 
 state of motion, is not the actual abscissa of the particles in a 
 given section, reckoned from a fixed point, but -differs from it 
 by the small periodic displacement due to the vibration. 
 
 This being understood, we may say that the vibration con- 
 sists in the transmission, in contrary directions, of waves of 
 condensation and dilatation ; just as the lateral vibration of a 
 string consists in the transmission of waves of lateral displace- 
 ment ; and the waves appear to be reflected from the ends in 
 both cases. 
 
 156. The periodic part of (8) does not in general vanish for 
 any value oix, so that there are in general no nodes, or sections 
 of no displacement. But there will be n nodes, at sections for 
 
 which X is any odd multiple of — , provided A^, S^ vanish for 
 
 all values of i except odd multiples of n. Thus the rod may 
 have any number of nodes, of which those next the ends arg 
 
of a Rod. 151 
 
 distant from the ends by half the distance between any two 
 nodes. 
 
 From (8) we have also — = 3^ 
 p ax 
 
 IT . . tnx / . iirat „ . itTat\ 
 = I — y 2 ? sin —j- \Ai cos — h-ffj sin ~j-) \ 
 
 hence — = i when x = o and when x = l. That is, there is no 
 
 variation of density at the free ends. But there will in general 
 be variation of density at all other points. If A^, B^ vanish 
 
 except when i is a multiple of n, the variable part of ^- vanishes 
 , ax 
 
 when j; is a multiple of — . Hence, when there are nodes, the 
 
 71 
 
 sections in which there is no variation of density ,are those which 
 bisect the nodal intervals in the state of equilibrium, and these 
 sections of no variation of density are also sections of greatest 
 displacement, as will be seen on inspection of (8), since 
 
 cos — J- is ± I for values of x which make sin — r— = o. 
 
 157. The vibration represented by (8) consists as usual of 
 the superposition of an infinite number of simple harmonic 
 vibrations, each of which might subsist by itself; the i^ com- 
 ponent vibration would have i nodes ; and in this case, as in 
 the case of the string, the tones corresponding to the com- 
 ponent vibrations form in general a complete harmonic series. 
 
 2/ 
 The period of the i^ component tone is t- , and the period of 
 
 . 2 I 
 the fundamental tone is — . 
 
 7 ^ 
 
 Since — would be the time of transmission of a wave over 
 a 
 
 the distance 2 /, we infer, exactly as in the case of a string, that 
 the wave-length is twice the length of the rod. Since the value 
 
 of a (Art. 150) is (—) , the number of vibrations in a unit of 
 
 a . Po 
 
 time, or — ^ , is i ^x 
 
152 Longitudinal Vibrations 
 
 and is therefore, cateris paribus, inversely proportional to the 
 length. It is, as it evidently ought to be, independent of the 
 thickness. 
 
 158. The most general case in which there is a node at the 
 middle of the rod is that in which cos -y- vanishes, for all 
 
 / 
 
 values of t included in the series (8), when x = - . In order that 
 
 2 
 
 this may be the case, A^, Bi must vanish for all even values of 
 
 /. The gravest component tone is then the fundamental tone 
 
 of the rod, but the higher tones of even orders disappear. 
 
 Thus the first upper tone will be at an interval of a twelfth 
 
 (octave + fifth) above the fundamental tone. 
 
 Now in this case the middle section of the rod might become 
 absolutely fixed without disturbing the motion, and either half 
 might then be taken ^way, so as to leave a rod of half the 
 original length, with one end fixed and the other free. 
 
 Hence we infer that the fundamental tone of a rod, with one 
 end fixed, is the same as that of a rod of twice the length, with 
 both ends free. But the component tones of the rod with a 
 fixed end do not form a complete harmonic series, containing 
 only the tones of odd orders. The wave-length is four times 
 the length of the rod. We shall arrive at the same conclusions 
 afterwards in a more direct manner. 
 
 159. We will next suppose the terminal sections of both ends 
 to be fixed. 
 
 Let /' be the distance between the planes of the fixed ends. 
 Then, if /' be different from the natural length /, the rod, when 
 at rest, is in a state of uniform elongation or contraction, main- 
 tained by the tensions or pressures on the fixed ends. We will 
 suppose, for clearness, that /' > /, so that the rod is elongated. 
 
 Assuming, as in Art. 153, 
 
 ^ = kx->rA sin (mx + a) sin (mai+P) 
 
 (in which equation x has its original meaning), we have to satisfy 
 
 d$ 
 the conditions ^ = o when x = and when x = I, for all values 
 
of a Rod. 153 
 
 of t. Hence we must have sin(w^ + a) = o when j? = o and 
 when x = l, which conditions are satisfied by taking 
 
 a = o, ml=iiT; 
 moreover, since the ends are now fixed, we may assume the 
 origin from which ^ is measured to coincide with one end, so 
 that ^ = o when x = o; then k must be such that i = l' when 
 
 x^l, or k = — . Thus the expression above assumed for ^ 
 becomes 
 
 S = jA- + ^ sm— sm(-y— + ^); (9) 
 
 now let x' be the distance of the section defined by x from 
 the end of the rod when at rest in its actual condition of ex- 
 
 tension, then x' = — x; and if we put a' - -- a, and take, as 
 
 before, the sum of the particular solutions for the general 
 solution, we obtain 
 
 . , . ITTX f . it: at „ . l'!!dt\ , „ 
 
 1 = ;); + 2sm-^(^^iCos-^ + ^jSm-^). {(f) 
 This equation evidently expresses a vibration in which the 
 velocity of wave-transmission is «' = -r- a ^. Thus the tension to 
 which we have supposed the rod subjected increases the velocity 
 
 ' This value of the velocity of wave-transmission might be obtained 
 directly thus : let T be the tension in the state of rest, T the actual tension 
 iX any point ; then 
 
 r' = ,(£-x) and r=,(f-x); 
 from which we find by eliminating d», 
 
 r=r'+(y+r)(g-i); 
 
 if we now investigated directly the differential equatioji, we should find 
 
 ^=a^^ where a^ -^, 
 
 p' being the density of the rod at rest. But /)„ being the unextended density, 
 we have —r = y = n ' 
 
 'hence «'-.^^ = (^)'f = (i)'^'- 
 
154 Longitudinal Vibrations 
 
 of transmission in proportion to the extension. But the period 
 
 2l' 2l . ^ 
 
 — r = — , IS the same as if there was no tension. 
 a a 
 
 160. Comparing (9') with (8), we see that the periods of the 
 fundamental and other component tones are the same in the 
 rod with both ends fixed as in that with both ends free. But 
 when there are nodes they are not at the fiame places. The 
 rod with fixed ends has always two nodes, namely, the fixed 
 ends themselves ; the i"^ harmonic component would have i— i 
 nodes (besides the ends) dividing the rod into z equal parts. 
 The mode of division in the free rod was explained in Art. 156. 
 
 161. The theory of the longitudinal vibrations of a rod ex-» 
 tended by tension at its ends, is evidently applicable at once to 
 the case of a string similarly extended, in so far as the assumed 
 relation between tension and extension may be supposed to 
 subsist. 
 
 The longitudinal vibrations of a pianoforte string may be 
 ■excited by gently rubbing it longitudinally with a piece of india- 
 rubber, and those of a violin string by placing the bow obliquely 
 across the string, and moving it along the string longitudinally, 
 keeping the same point of the bow upon the string. The note 
 is unpleasantly shrill in both cases. (The relation between the 
 fitch of lateral and longitudinal vibration will be considered 
 afterwards,) 
 
 If the peg of the violin be turned so as to alter the pitch of 
 the lateral vibrations very considerably, it will be found that the 
 pitch of the longitudinal vibrations has varied very slightly. 
 The reason of this is that in the case of the lateral vibrations 
 the chstnge of velocity of wave-transmission depends chiefly on 
 the change of tension, which is considerable. But in the case 
 of the longitudinal vibrations, the change of velocity of wave- 
 transmission depends on the change of extension, which is 
 comparatively slight. For experiments on the longitudinal 
 vibrations of rods, it is convenient to use rods of deal, or of 
 steel, or glass tubes. One end may be fixed in a stand ; or the 
 rod may be held lightly io the fingers at the place of a node. 
 
of a Rod. 155 
 
 The vibrations may be excited by rubbing the glass with a wet 
 cloth, and the rods with powdered rosin on a dry glove. 
 
 162. If, instead of supposing both ends of the rod fixed, we 
 supposed a constant force applied at each end, the terminal con- 
 
 ditions would be i^„ = -F^ = constant. Hence (Art. 151) ~— r 
 
 ax 
 
 must have the same given value, say ^— i, at both ends. As- 
 suming then, as before (Art. 153), 
 
 ^ = kx-^A sin {m.x-\-a) sin {mat+^), 
 we must have 
 
 ^—x+mA cos (mx+a) sin (mat+^) =e—j 
 both when x = and when x = I, and therefore 
 
 k = e, cos a = o, cos {ml+a) = o ; 
 whence, as in Art. 153, 
 
 a = -, , ml = lir; 
 . , •^:,i=<» i'ttX / . iirat _ , tnat\ 
 
 i = ex+ ^.^^ cos ~ {^Ai COS -y— + £i sm -—) ; 
 
 and, putting ex = x', el=l', ea = a', as in Art. 159, we should 
 have 
 
 A r . ■^-.J=m inx f . ITrai , „ . lirc^t. 
 
 i = x + 2^=1 cos—^{Ai COS —^ + BiSm —p—) I 
 
 in which x' now signifies the distance of any section from the 
 end A of the rod, supposing it to be at rest under the action of 
 the terminal forces. 
 
 Comparing this result with equation (8) we see that the 
 
 (2/' 2/x 
 = —^ = — J is the same 
 
 in both cases, and the nodes are similarly situated. 
 
 We infer then that in the three cases, (i) both ends free, 
 (2) both ends fixed, (3) both ends pulled or pushed by equal 
 constant forces, the series of component tones is the same. 
 But the distribution of the nodes, which is the same in (i) and 
 (3), is different in (2). 
 
 The case (3) cannot be realized in practice, because it is 
 impossible by any mechanical contrivance to apply a constant 
 force at the ends. In case (2) the force supplied by the fixed 
 
] 56 Longitudinal Vibrations 
 
 supports of the ends is a periodically varying quantity, of which 
 the actual value is that of — ?(-^ — i) at the end A, and 
 q( A at the other end, and the mean value is 
 
 163. The only remaining case of practical interest is that in 
 which one end of the rod is fixed, while the other end is either 
 entirely free, or loaded with a given finite mass. 
 
 We shall shew how the solution of the problem in these cases 
 may be obtained by means of the more general supposition that 
 both ends of the rod are loaded, but otherwise free. The con- 
 dition of fixity at either end can then be introduced by sup- 
 posing the mass attached at that end to be infinite, and the 
 condition of perfect freedom by supposing it to be nothing. 
 
 Suppose then masses M^, M^ to be attached to the two ends. 
 The forces F^, F^ (Art. 151) will then be the resistances to 
 acceleration arising from the inertia of these masses ; and the 
 terminal conditions will therefore be 
 
 whence (see equations (6), (7)), 
 
 dk M, dH , 
 
 I = ' — - -j-r when x = o, 
 
 dx qa> dr 
 
 di M^ dH , . 
 
 1 = -r^ when X = I. 
 
 dx qa dt' 
 
 d^ d^^ 
 And if in these equations we substitute the values of — » -j-^ > 
 
 from the assumed equation 
 
 ^ = kx + Asm{mx + a)sm{mai+p), (10) 
 
 we see that, in order to satisfy them for all -values of /, we must 
 
 have k= I and 
 
 M 
 cot (mx + a) = —mc^ — - when j; = o, 
 qa 
 
 2 -^1 1. 1 
 
 ma'' — ~ when x = l. 
 q a 
 
 ijet -- — z = (jji, -- — i = ^j, then these conditions give 
 
 /^o) ^'" Iq 
 
 cot (inl-\-a) = ji^in 
 
 cot a= —n^ml, 1 ,j ^ 
 
of a Rod. 157 
 
 from which, by eliminating a, we find 
 
 i^ — l^olhi^^?) ta.nml+iiJ,^ + ii^)ml=o. (12) 
 
 164. Suppose m^, m^, &c. are the values of m which satisfy 
 the equation (12). To each value Wj will correspond a value 
 of a, say oj, which can be found from (11). Then. (10) will 
 give the form 
 
 1 = jt + Ssin (miX + ai) {A^ cos m,iai+B^ sin m^ a i) ; (13) 
 where A^, B^ are arbitrary constants. 
 
 This would give the solution of the problem if the values of 
 m^ were known. These values in general could only be found 
 by troublesome approximations. We see however that (13) 
 expresses a vibration compounded of simple harmonic vilsra- 
 tions, of which the periods are inversely proportional to the 
 values of wz^, so that the component tones do not in general 
 belong to a harmonic scale. 
 
 165. If we suppose f,, = o and /Hj = o, then both ends of the 
 rod are perfectly free, and equations (11) give 
 
 cot a = O, cot (ml+a) = 0; 
 
 or a = - , ml=t'!r, as we found before. 
 2 
 
 Again, if /*„ = 00 and /n^ = 00 , then both ends are fixed, and 
 we have ^Qt a = 00, cot (m l+a) = 00, 
 
 or a = o, ml = ITT, 
 
 as we also found before in the case of fixed ends without 
 extension. 
 
 But if /id = 00 , fij = o, then the end A is fixed and the other 
 end free ; and we have from (11), 
 
 cot o = 00, cot (m l+a) = o, 
 
 or a = 0, ml={2z + l)-r, 
 
 so that (13) becomes 
 (2 t + i)nx 
 
 ^ = x + -s. 
 
 2/ 
 
 ,, (2t+i]7rai „ '. (2t + i)nai\ , ^ 
 
 Here the periods of the component vibrations are the values of 
 
 —, and the numbers of vibrations in a unit of time are 
 
 (2 t+ i) a 
 
 therefore proportional to the odd numbers i, 3, 5, . . . . Thus 
 
158 Longitudinal Vibrations 
 
 the component tones form a harmonic scale with alternate 
 tones (namely, the octave of the fundamental tone with all its 
 harmonics) left out. The wave-length of the fundamental tone 
 is 4 /, and its pitch is the same as that of a rod of length 2 /, 
 fixed at both ends or free at both. (See Art. 158,) 
 
 166. Lastly, we shall consider the case of a rod fixed at one 
 end, and loaded with a small mass at the other. Then we 
 shall have 
 
 Fo = °°. /'i = f (a small quantity) ; 
 and therefore, from (i i), 
 
 cot a =00, cot (ml+d) = e ml: 
 the first of these is satisfied by a = o, and the second becomes 
 cos m I = e m I sin m I. 
 
 Now if € were = o the solution of this would be ml= (az'+ i)- ; 
 
 2 
 we may therefore assume 
 
 mfl=(2t+i)- + S, 
 
 2 
 
 6 being a small quantity of the same order as e. Then 
 
 cos ((2 «■+ 1)- +^) = e ((2 i+ \)- + 6) sin ((2/+ i)- + e) ; 
 
 or, quantities of the second order being neglected, 
 
 — 6 sin (2 «■+ i) - = e (2 i+ i) - .sin (2 i+ 1) - ; 
 2 2 2 
 
 whence 6= —e (zt+i)- t 
 
 2 
 
 7r 
 
 and mJ=(2i + i)-(i—e). 
 
 2 
 
 This shews that the effect of the small load is simply to 
 diminish the number of vibrations of every component tone 
 in the ratio i — e: i. Each tone is therefore lowered by the 
 same interval, and the whole series still belongs to a harmonic 
 scale with alternate tones omitted ^. 
 
 '^ The following result is found by carrying the approximation one step 
 further. 
 
 Let « mean the ratio of the attached mass to the whole mass of the rod 
 and the attached mass. Then 
 
 where »i is the number of vibrations, in a unit of time, of the unloaded rod. 
 
of a Rod. 159 
 
 167. Since r? = —(Art. 150), the values of i^^, jj.^ (Art. 163) 
 are Po 
 
 how Ip^a is the mass of the rod; hence ju.„, jj,^ are simply the 
 ratios of the attached masses to the mass of the rod, and e in 
 the last problem has the same meaning. 
 
 168. The results of this chapter afford a method of determ- 
 ining experimentally the modulus of elasticity by observing the 
 tones produced by longitudinal vibrations. Thus, taking the 
 case of the rod with one end fixed and the other free, we have 
 for the period of the fundamental tone (Art. 165), 
 
 a S 
 
 and therefore if n be the number of vibrations in a unit of time, 
 
 i6n^P = — and ^ a>=i6n^l.Ip.a> ; 
 Po 
 
 now if a second be the unit of time, the weight of the rod 
 expressed in theoretical units of force is Ip^tog; hence, calling 
 this PF, we have i6«^/ 
 
 Qod = . W. 
 
 g 
 
 Thus the pulling force which would double the length of the 
 rod, if the law of extension held good without limit, is 
 
 X weight of rod. 
 
 g 
 The, value of n can be ascertained with great accuracy by 
 methods of which the principle will be explained afterwards. 
 
 Hence the higher component tones become unharmonic. For instance, the 
 ratio of the interval between the fundamental tone and the first upper tone 
 is (to the same approximation) 3 (i +f w^e'), which exceeds a twelfth by an 
 interval of which the ratio is i + ^ir^t^. This would be a diatonic semitone 
 
 if g-T'e'=-j^, or t^ = — 3, which gives ^ nearly for the ratio of the 
 
 attached mass to the mass of the rod. 
 
 The interval by which the first upper tone is put out of tune relatively to 
 the fundamental tone, being measured by the logarithm of i + -f tt" «^ varies 
 as ^ nearly. 
 
CHAPTER IX. 
 
 ON THE LATERAL VIBRATIONS OF A THIN 
 ELASTIC ROD. 
 
 169. The theory of the lateral vibrations of a rod becomes 
 susceptible of tolerably simple mathematical treatment when the 
 following assumptions are made. 
 
 The rod is supposed to be homogeneous. In its undisturbed 
 condition it is straight, and all its transverse sections are similar, 
 equal, and similarly situated, 
 
 A line passing through the centres of inertia of all transverse 
 sections may be called the axis of the rod. 
 
 We suppose the vibrations to be small, and such that 
 (i) One principal axis of every section remains in a fixed 
 plane. 
 
 (2) No part of the axis of the rod undergoes any elongation 
 
 or contraction. 
 
 (3) The particles which in the undisturbed state are in any 
 
 transverse plane section, remain always in a plane 
 normal to the axis of the rod. 
 
 (4) The principal axis mentioned in (i) is small. (This is 
 
 what is meant by calling the rod thin.) 
 The plane which always contains the principal axis mentioned 
 
 in (i) may be called the plane 0/ vibration. 
 
 It follows evidently from the above assumptions that the 
 
 condition of the whole rod at any time is determined by the 
 
 position and form of its axis. 
 
 170. Taking rectangular coordinate axes fixed in space, we 
 
Lateral Vibrations. i6i 
 
 ■will suppose, for clearness, that the axis of x is directed hori- 
 zontally from left to right, and the axis of y vertically upwards. 
 Thus the plane of zx is horizontal. We will also suppose that 
 the axis of the rod in its undisturbed condition coincides with 
 the .r-axis, and that the plane of vibration is vertical, coinciding 
 with the plane of xy. Thus the small principal axis of every 
 section is always in the vertical plane of xy, and the other 
 (which may or may not be small) is always horizontal. Calling 
 the axis of the rod AB, we will suppose that in the undisturbed 
 condition the left-hand end A coincides with the origin. 
 
 Let X be the abscissa of any given particle in the axis of the 
 rod in the undisturbed condition. We suppose (as in the case 
 of the string) that the vertical displacement of any particle in 
 the axis is so small that its horizontal displacement may be 
 neglected. Hence we may consider that x remains constant 
 for a given particle in the axis, and is the same as the abscissa 
 of that particle reckoned from the fixed origin. Then, ifjcbe 
 the (vertical) ordinate of the same particle, y is always a small 
 quantity. Also, if ds be an element of length of the axis, we 
 
 dx dy dy _,, 
 
 may put — - = i, 'j~ = ~i- ^ as in the case of the strmg. The 
 CIS CIS ax 
 
 problem is to express j/ as a function of the two independent 
 
 variables x and /. 
 
 171. In the undisturbed condition let p be the density of the 
 rod, a the area of the section, and / the actual length of the 
 axis (which remains constant). 
 
 If either one or both ends of the axis are free, / is the natural 
 length of the rod. But if both ends are fixed at a distance 
 different from the natural length, the axis is in a state of 
 permanent extension or contraction, and / is not the natural 
 length. 
 
 When either end of the axis is fixed, the whole of the terminal 
 face at that end may or may not be fixed. When both ends of 
 the axis are so fixed that there is extension or contraction, then 
 if either terminal face be not entirely fixed, we must suppose 
 normal tensions or pressures applied at all points of its surface 
 such as would, in the undisturbed condition, maintain all the 
 
 M 
 
1 62 Conditions of Equilibrium 
 
 longitudinal filaments of the rod at the same length as the 
 axis. 
 
 172. We must first investigate the conditions of equilibrium 
 of the rod under the action of such forces as could produce a 
 displacement of the kind supposed to exist at any time during 
 the motion. 
 
 The usual rule of signs will be observed with respect to 
 forces ; and moments will be considered positive which tend to 
 produce rotation /rozw the axis of x towards that of^. 
 
 The slice included between two plane sections cutting the 
 .r-axis at distances x, x + dx from the origin, always contains 
 the same matter, though its faces are not in general parallel in 
 the disturbed condition. We shall suppose 
 
 (i) A vertical force i^per unit of mass, constant throughout 
 the same slice (so that i^ is a function of x). 
 
 (2) Forces parallel to the plane of vibration, acting on the 
 
 particles of the slice, and reducible to a couple, in that 
 plane, of which the moment is 
 
 Z X (mass of slice) = L pa dx, 
 L being a function of x. (If there were such forces 
 not reducible to a couple, they~ would in general tend 
 to produce extension or contraction of the axis.) 
 
 (3) A force T^ per unit of area, applied at every part of the 
 
 terminal face A, at right angles to its plane. 
 
 (4) Tangential forces in the plane of the same face, parallel 
 
 to the plane of vibration, and reducible to a single, 
 force F^ a, in that plane, applied at its centre. 
 
 (5) Forces applied to the surface of the face A, reducible to 
 
 a couple in the plane of vibration, of which the mo- , 
 ment is G^ a. 
 
 (6) Analogous forces appUed at the face B, and denoted by 
 
 •^1. ^11 <^i- 
 In the condition of equilibrium these' forces are balanced by 
 the forces of elasticity called into action by the state of strain 
 which they produce in the rod. 
 
of Elastic Rod. 
 
 163 
 
 Suppose then the equilibrium to subsist. It would not be 
 disturbed if the part of the rod included between any two trans- 
 verse sections became rigid. And if the rest of the rod were 
 then removed, in order to maintain the equilibrium of this part 
 it would be necessary to apply to it certain forces. We proceed 
 to ascertain what these forces are, on the supposition that the 
 part we have supposed to become rigid is an infinitesimal slice, 
 and to deduce the differential equation which expresses the con- 
 dition of equilibrium. 
 
 173. In Fig. I suppose the plane of the'paper to be the plane 
 of vibration, and let ah be an infini- 
 tesimal portion of the axis of the rod, 
 and FG the section of an infinitesimal 
 slice contained between transverse sec- 
 tions cutting the axis in a and b. 
 
 Let X be the abscissa of P, the middle 
 point of ah, and x—\ dx, x-\-\dx the 
 abscissae of a, h. 
 
 Suppose the transverse sections 
 through a, h meet in C; then (quantities 
 of the third order being neglected) P C 
 is the radius of curvature of the axis at 
 P, which we wiU call R. 
 
 Suppose the slice to be made up of 
 
 longitudinal filaments having da for the 
 
 area of their section; and let a//3 be 
 
 the projection of such a filament on the plane of vibration. 
 
 Then, if Pp = jj, it is plain that 
 J. /^ 
 
 Now the state of extension of any such filament may (since its 
 length is infinitesimal) be considered to vary uniformly from 
 one end to the other, so that we may obtain the state of 
 extension at its middle point by calculating it as if it had the 
 same value at all its points, that is, by the formula 
 
 actual length 
 
 natural length 
 
 M 3 
 
 Fig. I. 
 
164 Conditions of Equilibrium 
 
 Let then (fx^ be the natural length of dx or al. This is also 
 the natural length of a/3, so that the extension at/ is 
 
 (77 \ dx 
 
 Now 1 is the extension" of the axis (which is constant), so 
 
 dx^ 
 
 that if we put ^^ x „ 
 
 where q is the modulus of elasticity, T will be the value of a 
 constant tension, per unit of sectional area, due to the per- 
 manent extension. 
 
 Then, calling T' the actual tension in the filament o^. We 
 have 
 
 Hence the pulling force exercised by the filament upon any 
 element da of the section DE (on either side of it) is 
 
 (7'+-|-(^+7'))</o,. 
 Consider all these forces on one side (say the left) of the 
 section. They are reducible to a resultant force = — / T' da ap- 
 plied at P, and a couple of which the moment is / 1) T'da, the 
 
 integrations being extended over the whole section. Now since 
 the ordinate tj is reckoned from a horizontal axis in the plane of 
 the section, passing through its centre of inertia P, we have 
 
 / I) da - O, and / rj^ da = MK^j 
 
 where « is the radius of gyration of the area of the section about 
 the horizontal principal axis in its plane. Thus the forces 
 acting on the left-hand side of the section, due to extension, are 
 equivalent to a resultant force — Ta perpendicular to its plane, 
 applied at P, and a couple in the plane FG of which the mo- 
 
 . ■ ?+^ 2 
 ment is -^=r—Kra, 
 
of Elastic Rod. 165 
 
 - 174, We can now find the forces which must be applied to 
 the elementary slice in order to maintain its equilibrium when 
 the rest of the rod is supposed to be removed, the slice having 
 become rigid. 
 
 If we call Q the moment (which we have just determined) 
 of the couple due to extension acting on the left-hand side 
 of the section DE, then the moment of that on the left-hand 
 face of the slice will be 
 
 and of that on the right-hand face 
 
 The sum of these is 
 
 -f^.= -(,+ r).^.^(^)... (a) 
 
 Moreover, we found a resultant piflling force Ta on each side 
 ofP- Hence on each face of the slice there is a resultant 
 piilling force, applied perpendicularly at a and b respectively. 
 The horizontal components of these forces may be considered 
 equal (in opposite directions). But the vertical components 
 
 (smce ^ = ^1 are 
 V ds dx' 
 
 these are equivalent to 
 
 a vertical resultant force = Ta> j— 2 dx, {b) 
 
 (Py 
 ' dx^ 
 and no couple. (For the moment of the couple resulting from 
 
 the horizontal components is 
 
 — Tady, =—Ta>j-dx, 
 
 while that from the vertical components is 
 
 dy 
 (diflference of forces) x \ (distance between them) = ^<» ^ dx). 
 
1 66 Differential Equation 
 
 But we must not assume that the only forces lost by the re-- 
 moval of the other parts of the rod are the couple and resultant 
 force just found; for the parts removed will in general have 
 exercised tangential forces in the planes of the faces of the slice, 
 reducible to resultant forces in the plane of vibration and applied 
 at the centres of the faces. 
 
 Suppose then icFw is. the value of this tangential resultant 
 force on the left and right sides of the section DE; then, on 
 the left-hand face of the slice, it will be 
 
 and on the right-hand 
 
 » 2 
 
 These expressions will also give (as we neglect (;f^) ) the 
 
 vertical components ; and therefore the forces in question are 
 equivalent to 
 
 dF 
 a resultant vertical force — <» ^— dx, (.c) 
 
 dx ' 
 
 and a couple of which the moment is 
 
 —aFdx. (d) 
 
 (The horizontal components, being of the order of (t-) , are 
 
 neglected.) 
 
 175. Now in the actual condition of equilibrium all these 
 forces are balanced by those which we have supposed to act on 
 the interior mass of the slice, namely, a vertical force p a Ydx, 
 and a couple of which the moment is p <» Z dx. Hence we 
 must have (j) + (^ ) + p „ Ydx = o, 
 
 (a) + (d) + p a Z, dx = o. 
 
 Introducing the actual values of (a), (b), (c), (d), and ob- 
 
 /rfvx" I d^y 
 
 servmg that, smce we neglect \-f-) , we may put —^ = —j-^ > 
 
 we find, after dividing by a> dx, 
 
(^+^)«^"^.-^^. + p"^-P^=o (3) 
 
 of Eqtdlibrium. 167 
 
 (?+7')«=g3-i?'+pZ = o. (2) 
 
 In order to eliminate' the unknown F, we have only to sub- 
 tract, after differentiating (2). We thus obtain, finally, 
 
 i^_r^+ — - 
 dx'^ dx^ dx 
 
 as the differential equation which must be satisfied in the con- 
 dition of equilibrium. 
 
 176. We have no occasion to integrate the equation (3) ; but 
 it is essential to ascertain the conditions which would serve to 
 determine the arbitrary constants contained in its general so- 
 lution. These are to be obtained from the data of the problem 
 relative to the ends of the rod. 
 
 Now the forces acting on the surface of the left-hand face 
 are only the given external forces (Art. 172), and the interior 
 tensions arising from extension, and these must balance one 
 another. Hence we must have 
 
 i?'^=o + ir =0, 
 and (see end of Art. 173) 
 
 The first of these, combined with' (2), gives 
 
 / (^+r)«^(^)^^^+i^„+p(Z)^=o = o. (5) 
 
 These ■ equations ((4) and (5)) will furnish the required con- 
 ditions. (It is evident that similar equations must subsist at 
 the other end of the rod.) 
 
 177. In order to form the differential equation of motion, we 
 have now only to substitute in (3) the forces arising from the 
 resistances of the particles to acceleration, instead of those sup- 
 posed to act on the interior of the mass. 
 
 Hence, instead of the vertical force paFdx, supposed to act 
 
 d^y , 
 on the mass of a slice, we must substitute —^a—^dx or 
 
 — v4 instead of Y, 
 dx^ 
 
1 68 Differential Equation of 
 
 -And instead of the supposed couple paL dx, we must sub- 
 stitute one which is found as follows : 
 
 Since the particles in a plane transverse section remain in 
 
 a plane section, and since the inclination of the plane of the 
 
 dv 
 section to the vertical is -^ , if ij be the ordinate of a particle 
 
 drn, reckoned (as in Art. 173) in the plane of the section from 
 , , its horizontal principal axis, the angular velocity of the plane 
 
 being -— - (^^ (in the direction of positive rotation), the linear 
 at ^ax-' 
 
 velocity of dm (estimated from right to left) is ij -^ v- , and 
 
 therefore its resistance to acceleration (in the same direction) is 
 
 — rj (-j-\ -^ . dm, and the moment of this resistance, with its 
 \dt^ ax V 2 J 
 
 proper sign, is — "J^T-r) -r- • d^- Now, considering dm as the 
 
 mass of an element of an infinitesimal slice, we have dm = p.dadx ; 
 hence the above moment is 
 
 -" -dPd^'"^""^''' 
 and taking the sum of all such moments for the whole slice, 
 we have ^^ r a^y 
 
 -piFd-x^^r^""=-''''''-d^x^=' 
 
 as the expression to be substituted for paLdx. Hence we 
 
 must put -K^ , ■ , instead of Z. 
 ^ dr dx 
 
 Making these substitutions in (3) we obtain 
 as the differential equation required ^. 
 
 ' See Klebsch, Theorie der Elasticilal fester K'drper, § 61, where this 
 equation (with a different notation) is deduced, as a particular case, from 
 the general theory of elastic solids. The equation usually given in ele- 
 
 d*y 
 mentary works does not contain the term ^ ^ , which arises from the 
 
 angular motion of the sections of the rod. (See, for example, Poissou, 
 Traiti de Mecanique, tom. ii. § 5.) It may in fact be neglected without 
 sensible error in ordinary cases. 
 
Motion formed and integrated. 1 69 
 
 178. This may be put in a somewhat more convenient form 
 as follows : 
 
 Put q-\-T=I? p, T=a^p, and it becomes 
 
 (In order to see the homogeneity of this equation it is de- 
 sirable to observe the meanings of a and b. Ta> is the actual 
 tension, and p the actual density, in the axis of the rod. ^ <» is 
 the tension which would have to be applied to the rod in its 
 natural state in order to double its length, if the law of ex- 
 tension held good without limit. Hence Ta> and q o> are forces, 
 and can be represented by weights, say by the weights of 
 lengths X and \' of the rod, taken at its actual density p. Then 
 Ta =g p\a>, qa =g p X'ffl, and therefore 
 g{\ + \')^b\ g\ = a?; 
 so that a^, P are the half squares of the velocities which would 
 be acquired by a heavy body falling vertically down distances X, 
 X -)- X'. Hence a and b are of one dimension in space and — i 
 in time.) 
 
 179. In order to find particular integrals of {6') we assume 
 
 V = « cos — / -1- » sm — /, (7) 
 
 K K 
 
 u and V being functions of x to be determined. 
 
 Substituting this value of j/ in (6'), we find an equation of 
 
 the form »2 . , tz ■ ^ . 
 
 U cos — i+ K sm — / = o, 
 
 K K 
 
 in which £7 and Fdo not contain /; so that in order to satisfy 
 
 the equation we must have separately 
 
 U= o, F= o. 
 
 These equations are exactly similar in form, and we therefore 
 
 need only consider one of them. The first is 
 
 „ „ rf*« , „ „, d^u m^ ,„. 
 
 ..,._ + (^3_,.)____, = ,. ^ (8) 
 
 which, being linear with constant coefficients, can be integrated 
 in the usual way. The" general solution is 
 
1 70 Determination of Constants. 
 
 where k^, k^, k^, k^ are the foots of the equation 
 
 ^bn^^{m^-a')k^-%^o; (9) 
 
 which gives ,„ i , 
 
 ^'=2-^2{'^'-«'±((«''-^'T + 4'«^3^)*}. (9') - 
 
 We may represent the four values of k in the form 
 
 ± ^1 {m), ± <^2 {ni) ; 
 
 and, making for convenience a change in the meaning of the 
 constants A,B, C, D, we may put the values of u and v in 
 the form 
 
 « = ^ 1-5 
 
 2 2 
 
 + C +D ; (10) 
 
 2 2 ' ^ ' 
 
 » = a similar expression with different constants, say A', B', 
 C',U. 
 
 180, Introducing these values of u and v in (7), we obtain a 
 value of_>', which is a particular integral of (6'), and in which 
 all the constants, including m, are arbitrary, and would remain 
 so if we had only to satisfy (6'). But in every actual problem 
 we have also to satisfy the terminal conditions ; and these con- 
 ditions lead to an equation of which the roots are the only 
 admissible values of m, besides other equations which partly 
 determine the constants A, B, &c. 
 
 Before investigating these conditions however in particular 
 cases, we will examine more closely the nature of the four 
 values of k. Since the last term of (9) is negative, one value 
 of W' is necessarily positive, and the other negative. Suppose 
 we call the positive value c? and the negative value — /3^, then 
 we shall have 
 
 ^= ± a, k= i: 3v — I 
 
 for the four values of k ; and thus we may put (10) in the form 
 
 ,aa:_i_-— aa: -aa; — aiB 
 
 u = A + B h CcosjS.r + Z'sin^x, (11) 
 
 2 2 ' \ / 
 
 in which it is important to remember that a, ^ are determinate 
 functions of m, given by (9), while the value of m itself has still 
 to be determined. 
 
Case of Fixed Ends of Axis. 171 
 
 181. We will take now the case which is in some respects 
 the most simple, namely, that in which the ends of the axis 
 of the rod are fixed, but the terminal faces are subject to no 
 other constraint. (The tension T, which must be supposed to 
 be applied to them on every unit of surface, will evrdently leave 
 the directions of the faces free.) This case is of practical in- 
 terest, because it may be taken to represent that of a wire 
 (e. g. a pianoforte string) of which the vibrating part is de- 
 termined by stretching it over bridges. 
 
 Now referring to the terminal equation (4) (Art. 176) we see 
 that on the suppositions now made we must put G^ = o, and 
 consequently ^2 
 
 dx'^ ~ 
 at both ends of the rod. (The equation (5) gives no condition. 
 It would merely serve to determine the value of the pressure 
 Ff^a supported by the point to which the end of the axis is 
 fixed.) 
 
 But the fixity of the ends of the axis gives us two more 
 conditions, namely, y = o 
 
 at both ends. We have then to put in (7) the value (11) for «, 
 and a similar value for v, with A', B', &c. instead of A, B, &c., 
 and then express the conditions that 
 
 both when x = o and when x = l, for all values of t. These 
 conditions, relatively to .^i? = o, give, as will be easily found, 
 
 AJrC=o, A'+C' = o, 
 
 a^A-^C=o, ",M'-^='C" = o; 
 which it is impossible to satisfy otherwise than by 
 
 A = o, C=o, A'=o, C"=o. 
 The conditions relatively to x = / thus become simplified, and 
 are 
 
 4- i? sin |3 / = o, 
 -/32Z'sin(3/=o, 
 
172 Final Equation in this Case. 
 with similar equations for B' and If. These give 
 B— -=o, Z?sin/3/=o; 
 
 now the factor multiplied by B cannot vanish, since a is real. 
 Hence we must have 
 
 B = o, B' = o, sin^/=o. 
 Hence D and U remain arbitrary, while |3 must satisfy the 
 equation sin /3 / = o, which gives 
 
 182. The values of «, v (see equation (11)) are now reduced 
 to u = D sm. ^x, V = ly sin ^x, and therefore from (7) 
 
 • ^ /t. ^^ TV • '>ni\ 
 y = sin.^x (Dcos 1- Z' sm — )j 
 
 i 7T 
 
 in which j3 may have any of the infinite series of values — 
 
 obtained by giving all values to the integer z from i to 00 , and 
 to each value of /3 corresponds a value of m obtained from (9) 
 by putting k^ = — ^^. Hence, taking the sum of all the par- 
 ticular values of _j/, we have for the general solution of (6') 
 appropriate to this problem 
 
 y = ^i=i sm-^(Qcos-^ + Asm-i-); (12) 
 
 where 0^, 2?^ are arbitrary constants in the usual sense, that is, 
 depend only on initial displacements and velocities. (It will be 
 easily seen that it is useless to include negative values of i, 
 since w^ (see next article) only depends on P) 
 
 183. Solving equation (9) for w", we find, after slight re- 
 ductions, ^2 d'-K^Pk'' 
 
 bi • 
 
 2 "" » 2 A2 » 
 
 K I — K « 
 
 and since, for real values of /3, 
 the above formula gives 
 
 72 -2 1 .-2 _2 .2 12 
 
 (13) 
 
 m? Pi^ l^a^+i^K^i 
 
 /2 /2 + «V2/c2 
 
Application to Metallic Wire. 173 
 
 Now (12) shews that the vibration is compounded of simple 
 
 harmonic vibrations of which the periods are the values of ; 
 
 mi 
 
 or the number of vibrations, in a unit of time, of the i"^ com- 
 ponent tone, is • — — . Calling this number n^, we have from (13), 
 
 a TT (C 
 
 i ^Pa^ + r-7r\H\i 
 
 This shews that in general the component tones do not belong 
 to a harmonic scale '. 
 
 184. Let us however examine some special cases. 
 If we suppose the rod infinitely thin we may neglect k al- 
 together. The differential equation (6') then reduces itself to 
 ^_ <f> 
 df" '"■ dx'' 
 the ordinary equation for a perfectly flexible string; and (14) 
 
 t (Z 
 
 gives «j = — -J, the value found before. 
 
 But if we consider the rod as very thin, without being in- 
 finitely thin, so that —^ is a very small fraction, the value (14) 
 will be applicable to the case of a metallic string or wire. In 
 this case, neglecting the square of -j^, and assuming the 
 section of the wire to be a circle with radius r, so that t^ = j r^, 
 
 id 
 hence if we put — r = iV^ (the number of vibrations calculated on 
 
 the supposition of infinite thinness or perfect flexibility ^), and 
 put for b"^ and c^ their values (Art. 178), we have 
 
 ' The process which has been given in Arts. 179-183 is substantially 
 the same as that of Klebsch, § 6i. 
 
 ^ Strickly speaking, the supposition of infinite thinness ought to be dis- 
 tinguished from that of perfect flexibility. We can imagine a thick string 
 of which only the central infinitely thin axis should resist extension or con- 
 traction. Such a string might be regarded as perfectly flexible. But the 
 
 we find 
 
174 Correction for Rigidity. 
 
 •.=*.hT?4|' 
 
 which gives what is called the correction for rigidity. 
 
 This correction may be put in another form thus: from 
 (14) we have 
 
 "< ■ 77 ('■' + '■' -"^ («'- "'))* o="iy. 
 
 so that for a given value of i, that is, for a tone of given 
 order, the number of vibrations corresponding to any actual 
 tension T may be calculated as if the string were perfectly 
 flexible, by substituting for 7' a fictitious tension 
 
 TT^ 
 
 r"" q 
 
 41- 
 
 The term thus added is sensibly independent of T, since / (the 
 actual length between the bridges) is constant, and r is sensibly 
 invariable, at least for moderate variations of T. If the tension 
 be supplied by a weight W, then W= Ta. Suppose Q is the 
 weight which. would double the length of the string if the law of 
 extension held good indefinitely, then Q = qa>. Hence the 
 fictitious weight to be substituted in calculation for the actual 
 weight is „2 „z 
 
 W+i^j^Q. 
 
 It would be difficult to calculate the value of the added term 
 
 a priori, because the values of the very small ratio \^) and of 
 
 the very large weight Q, could hardly be obtained with sufficient 
 accuracy; but it is easily ascertained experimentally by com- 
 
 inertia of the outer parts would introduce the term . „ . „ in the differential 
 
 equation, though the term — ^ , which arises from the resistance of the outer 
 
 parts to extension or contraction, would disappear. But the strings used 
 for musical purposes never approximate to this character, though the con- 
 verse arrangement is common, e. g. in- guitar strings made by winding fine 
 wire upon a silk core. 
 
Case of no Tension. 175 
 
 paring the tones produced by two different weights. The tones 
 corresponding to other values of W can then be calculated, 
 and they are found on trial to agree very exactly with those 
 actually produced. 
 
 185. We will next suppose that, the ends of the axis being 
 still fixed, the distance between them is the natural length of the 
 rod, so that T= o; hence a = o, and (14) becomes 
 I ., K h 
 
 where d^ = — ■ 
 P 
 
 In this case, since k is small, the values of n^ are, for moderate , 
 values of t, sensibly proportional to the series of square num- 
 bers i'^, 2'^, s"*, &c., so that the component tones are the i^t, 
 4tii, 9th &c. of a harmonic scale. Thus the first of the upper 
 tones is two octaves above the fundamental tone; and the 
 second is one octave and a major second above the first ; &c. 
 
 The supposition now made may be approximately realized in 
 several ways, of which the simplest consists in merely laying 
 a rod (e. g. a bar of steel) upon two bridges placed as close as 
 possible to its ends. 
 
 186. If, instead of merely supposing the ends of the axis 
 fixed, we had supposed the planes of the terminal faces of the 
 rod to be fixed, then, instead of the simple formula (13) which 
 gives the values of m, we should have found a very complicated 
 transcendental equation. The same thing happens if one 
 terminal face be fixed and the other entirely free, or if both be 
 entirely free. In the two latter cases this equation is always 
 somewhat simplified by the circumstance that T== o, and there- 
 fore a = o. But it becomes much more simplified if we neglect 
 
 d*y 
 the term ^ in the differential equation, which we may 
 
 generally do without sensible error. We shall therefore intro- 
 duce this simplification in what follows. 
 
 187. Since the differential equation was founded on the 
 hypothesis that the particles which in the undisturbed state are 
 in a plane at right angles to the axis continue to be so at all 
 times, if a terminal face of the rod be fixed, the axis at that end 
 
176 Simplified Differential Equation 
 
 must always be at right angles to it ; and as the end of the axis 
 itself is fixed, we must have, at that end, 
 
 dy 
 
 These therefore are the terminal conditions for a fixed face. 
 
 But if a terminal face be entirely free, we must obtain the 
 terminal conditions from equations (4) and (5) (Art. 176). 
 Now, at a free end, G^ (or G^ and F^ (or F^ are both o. 
 Also L, in (5), arises (see Art. 177) from the angular motion of 
 the planes of the elementary slices, the effect of which we are 
 now going to neglect; hence these equations give 
 
 d^y _ ^_ 
 dx'~°' d^~° 
 as the terminal conditions at a free end. 
 
 We have already seen that the conditions are 
 
 d^y 
 ^ = °' ^^=° 
 
 at an end where only the extremity of the axis is fixed, so that 
 the direction of the plane of the face is free. 
 
 There are altogether six possible combinations, of which, we 
 •have already considered one. Of the remaining 'five we shall 
 only examine the three which are of most importance, namely, 
 both faces fixed, both free, one fixed and the other free. 
 
 188. The equation (6'), if we omit the second term, and put 
 fl = o (since we suppose T = o), becomes 
 
 ^ + «'^ = o. (.5) 
 
 To find particular solutions of this equation, we may con- 
 veniently assume 
 
 y =u cos -p- m't-\-v sm -j^ m'i; (16) 
 
 in which u, v are to be functions of x, I is the length of the 
 rod, and m a constant to be determined. Substituting this 
 value of>' in (15), we find that in order to satisfy that equation 
 for all values of /, we must have 
 
 d'^u _ w* ^v_ n^ 
 
 The general solution of the first of these equations may 
 evidently be written in the form 
 
Case of both Ends free. 177 
 
 , mx „ . mx 
 M = ^ cos — - — h B sin — — - 
 
 mx mx mx mx 
 
 ~T I r "T r 
 
 + ^^^^- + ^^-i^; (x7) 
 
 and » will be given by a similar equation, with other constants 
 A', B', C, D'. 
 
 It will save much trouble to adopt the following abbreviated 
 notation. 
 
 Let _J__ = .(5), ___ = a(5). 
 
 Then we shall evidently have, 
 
 (7(5) = ,t(-5), 8(5) = _8(-^), ,r(o)=i,. 
 
 8 (o) = o, —a-{n6) = nh{n ff), —-h(n6) =n<T {n 6). 
 
 189. Thus the equation (17) becomes 
 
 . mx ^ . mx „ rmx^ ^^rmx-^ , „. 
 « = ^cos-^+5,sm-y-+C<r(— ) + Z»8(— ); (18) 
 
 and we have now to find the values of the constants which will 
 satisfy the terminal conditions in each case. 
 
 First, then, let us suppose both ends entirely free. The con- 
 ditions (see Art. 187) are 
 
 dx'' ~°' dx'~°' 
 
 both when x == o and when x = I ; and since these are to be 
 satisfied for all values of /, it is evident that we must have 
 d^u d^v d^u d^v 
 
 'd^^°' dx^^°' 1^'^°' d^^^°' 
 
 Putting then x = and x = I successively in the values of 
 these differential coefficients deduced from (18) and from the 
 corresponding expression for v, we find (for x = o) 
 
 -A + C = o, -B + D = o; 
 
 so that 
 
 . / mx /mx^\ _/ . mx ^/mx\\ 
 u = A ^cos— + <r(— )j+5^sm-y-+S(— )]; 
 
 and then the conditions relative to x = l become 
 
 A[— cos 711-^ (T {m)) + B {—sm m-\-b{m)) = o, 'l , . 
 
 a\ sin ;«-t-S (ot)) + -5(-cos»z-|-<7 (w))-o ;j ^ ' 
 
178 Case of Fixed Terminal Faces 
 
 from which, eliminating A and B, we have 
 
 (o- {m) — cos nif = (8 (jriff— sin^ wz ; 
 now, by the definition of a {m) and (8 (»«)), 
 
 hence this equation becomes 
 
 a- {m) cosm = 'i, or 
 
 cosw=i. (20) 
 
 The roots of this equation are the admissible values of m ; and 
 if we denote them by m^, m^, &c., and call A^, B^ the corre- 
 sponding values oi A,.B, either of the equations (19) gives the 
 ratio A^ : B^. We may therefore take 
 
 ^i= Q(sin »2^-8(»z^)), 
 
 ^i= -Ci (cos m^ - <r (w^)), 
 
 where Q is arbitrary. Thus we shall have from (18) 
 
 Ui=CiXi, where 
 
 Z, = (sm»2i-8K)) (cos-^- + <r(-i-)) 
 
 -(cos»?i-aK)) ^sm-j- +8(-^)j; (21) 
 
 and v^ = D^X^, where ZJj is another arbitrary constant. The 
 general value oi y, which is the sum of all particular values, 
 will then be 
 
 y = -S,Xi {Ci cos '^ m,H+Di sin "-^ w/ /); (22) 
 
 and this is the equation expressing the vibration of a rod free at 
 both ends. 
 
 The constants Cj, iP^ are determined by the initial displace- 
 ments and velocities, in a manner which will be explained 
 afterwards. 
 
 190. If instead of supposing the ends of the rod entirely 
 free,, we suppose both the terminal faces entirely fixed, the 
 terminal conditions are (Art. 187) 
 
 dy 
 ■^ = °' ^ = °' 
 both when x = and when x = I. 
 
 Assuming then (16) and (18) as before, and proceeding 
 exactly as in the last Article, we find the same equation (20) for 
 the determination of the values of ot^, but 
 
and of Permanent Tension. 179 
 
 A^ = C^ (sin w^— 8 (w?;)), 
 
 and ^ = Si'i (C.cos '^»z// + A sin*^ »///), (23) 
 
 where i'; = (sin w^- 8 (w^)) ^cos ^ - o- (^)) 
 
 -(cos»2i-o-(»zJ)(sin^-8(-j-)]. (24) 
 
 Comparing the expressions (22), (23), we see that the com- 
 ponent tones have the same pitch, whether the terminal faces 
 be both free or both fixed. For the values of m^ are the roots 
 of the same equation (20) in both cases, and the number of 
 vibrations in a unit of time, for the tone of the t^^ order, is 
 
 The constant 6 depends (Art. 178) only on the material of 
 which the rod is made, and ntf is an abstract number, inde- 
 pendent both of material and dimensions. Hence, when the 
 material is given, the number of vibrations, for a tone of given 
 order, varies inversely as the square of the length of -the rod, 
 and directly as the radius of gyration of the sectional area about 
 that diameter which is at right angles to the plane of vibration. 
 
 If the section is elliptic or rectangular, then k is simply pro- 
 portional to the thickness measured in the plane of vibration. 
 
 191. In the last Article it was supposed that both the terminal 
 faces were fixed, but that there was no permanent tension, so 
 that the natural length of the axis was maintained. 
 
 The supposition of permanent tension, with fixed terminal 
 faces, leads to much more complicated equations, but they may 
 be treated in an approximate manner in the only case of 
 practical importance, namely, that in which the thickness of the 
 rod is very small compared with its length. The result may 
 then be considered as giving the correction for rigidity for a 
 wire, or for a long and thin lamina, not stretched over bridges, 
 but firmly clamped at the ends. 
 
 We may take in this case the equations (7), (9), and (11), as 
 
 in Arts. 179 and 180. But we shall suppose the term ^ ^ 
 in (6') to be neglected, so that instead of (9') we get the simpler 
 form 2(c2^^/l'' = a2±(«* + 4»«'^^)*; 
 
 N 2 
 
i8o Application to Wire 
 
 and therefore, since a^ and — j3^ are the two values of 1^, we may 
 write the value of c? thus : 
 
 2K -o- 
 
 and i3^ will be given by changing + 1 into — i in the last term 
 of the numerator. 
 
 Now in the case of a metallic wire or lamina, -— is a large 
 
 number (see Art. 178) since T is small compared with q. But 
 
 K 
 
 1 
 
 is very small ; and the legitimacy of the following approxi- 
 
 mation depends upon the assumption that -^ is so small that 
 
 -j2 2" ^^ ^^^° ^^"^y small. From this assumption it follows that 
 
 a / is very large, since <? P is expressed by a fra9tion in which 
 the numerator is > 2 and the denominator is the small fraction 
 
 'LI 
 
 dv 
 Yi.Qiy^ the terminal conditions are _y = o, — = o, at both ends ; 
 
 and from these, proceeding as in Art. 189, we find from 
 equation (ii) ^ + C = o, aB-\-^D = o, 
 
 J o- (a /) + 5 8 (a /) + C cos /3 /+ Z> sin /3 / = o, 
 a(^S(a/)+-5<r(a/))+^(Z»cos^/-Csin^/) = o; 
 
 and hence, eliminating A, B, C, D, and reducing by means 
 of the identity (o- (a /))^- (8 (a l)Y = i, we find, finally, 
 8 (a /) sin (3 / , 2 a ^ 
 
 I — o- (a/) COSfil 
 
 and if in this equation the values of a and ^ given above were 
 introduced, we should obtain an equation in m, of which ihe 
 roots would be the values of m^, m^, &c. 
 
 Now the values of a', PP give, as will be found at once without 
 
 difficulty, -TZT^ = ~r ■ ^^'^^ ^'^^^^ " ^ ^^ very large, we have, 
 neglecting e-"^, o-(a/) = S (a/) = •|€+'^', and the equation irn) 
 becomes i£»^sin/3/ 2mb _ 
 
 i-Aeo^cos^/"*" a^ ~°' 
 
with Clamped Ends. i8i 
 
 or, <=-»^ being again neglected, 
 
 tan /3 / = — ;— . 
 w' 
 
 Now the value of ^ gives 
 
 P ^^ = -5-2 nearly 
 (by developing the binomial as far as the second term) ; hence 
 
 , ^ ml 
 
 //3 = — nearly. 
 Ka ' 
 
 But the number of vibrations in a unit of time is ; and 
 
 since the case differs very little from that of an infinitely thin 
 
 Z CI lYll 
 
 string, this number differs very little from — , so that — differs 
 
 2 / K.a 
 
 very little from iir, or ;3/= zV + 5, where is very small; hence 
 
 2mb . 2mb 
 
 — ^ = tan /3 / = tan fl is very small ; and we may take — ^ = 6, 
 
 and therefore „ , . 2mb 
 /3/= zttH =- J 
 
 , ... ml 
 
 and, equatmg this to — , we have 
 Ka 
 
 / / 2 i5v 
 
 m 
 
 or, introducing the subscript index to distinguish the different 
 values of m, 
 
 m, lira . K d\~'- lira f k b\ . 
 
 T= — (^-^7^) = — (' + ^7a) •^""'■^^- 
 
 Let »; be the number of vibrations, in a unit of time, of the 
 z'* tone, and N^ the number calculated on the supposition of 
 infinite thinness ; then 
 
 717- ^'^ J % 1. 
 
 iV, = -, and «,= ^^; hence 
 
 «, = i\^,(i+2i|) = i\^, (1 + 2^(1 + -^f). in) 
 
 Comparing this with the corresponding expression deduced 
 in Art. 184 on the supposition that the directions of the terminal 
 faces were free, viz. 
 
 ■N^ 
 
 . / I' it' k' g \ , ,, 
 
1 8 2 Case of one Fixed End. 
 
 ■we see that they differ essentially, especially in this respect, that 
 in the case («) oi fixed faces the pitch of all the component 
 tones ' is raised, by the rigidity, through the same interval, so 
 that they do not cease to form a harmonic series ; whereas 
 in the other case («') each tone is raised through a greater 
 interval than the next lower one, and the series is therefore no 
 longer strictly harmonic. 
 
 An expression equivalent to («), and obtained by nearly the 
 same process, was given by Seebeck''-, and found by him to 
 agree with experiment when the ends of the wire were clamped. 
 
 In the case of a wire stretched over bridges, the form («') 
 has been found to agree with experiment, in the manner 
 mentioned at the end of Art. 184. But the deviation of the 
 upper tones from the harmonic scale is probably too small 
 to be made sensible to the ear. 
 
 192. The last of the cases which we proposed to examine 
 is that in which one terminal face (suppose that at which x = o) 
 is fixed and the other free. The conditions then are (Art. 187) 
 
 y -o, -f- = o, when x = o, 
 
 ax 
 
 <Py d\y 
 
 ^ = °' ^ = °' ^^^" ^ = ^- 
 
 Again then, assuming (i6) and (i8), we find in the first 
 place A + C=o, B + I) = o, and then 
 
 A (cos m + <r(m)) + B {sin m+ 8 (m)) = o, 
 
 A (sin m— 8 {m))—B (cos m + a- (m)) = o ; 
 
 eliminating A and £ we obtain, after reduction, 
 
 a-{m) cosm = — I, or 
 
 ^m _i_ p— m 
 
 COS w = — I (25) 
 
 2 
 
 as the equation for determining the values of m ; and we may 
 take ^ _ c, (sin m, + 8 (m,)), B.^-C^ (cos m, + a («,)) ; 
 so that the' value of j^ will be 
 
 J/ = 2 Zi (C, cos ~- %'/+ A sin "-ji m? i) ; (26) 
 
 in which Cj, Z>j are arbitrary, and 
 
 ' See the memoir referred to below (Art. 205). 
 
Periods of Tones. 183 
 
 Zi = {%mm^-\-h{m>i) (cos -^ " \rr) 
 
 -(cos»Zi + <rK)) (sm -j ^("i"))- (^^^ 
 
 Since the equation (25) is not the same as (20), the periods 
 of the component tones will not be the same as in the two 
 former cases. But the law of their variation with the length 
 and sectional area of a rod of given material is still the same as 
 that stated at the end of Art. 190. 
 
 193. To complete the solution of the problems considered 
 in Arts. 187-192, we should have first to find the roots of the 
 equations (20) and (25), which determine the periods of the 
 component tones, and then to find the values of x which satisfy 
 the equations X^ = o, i^^ = o, for each root of (20), and Z^^o 
 for each root of (25), in order to ascertain the positions' of the 
 nodes corresponding to each tone. The required calculations, 
 for small values of i, which belong to the most important tones, 
 are troublesome; especially those which relate to the nodes. 
 And we shall only give a sufficient specimen of them to enable 
 the reader, who may be, so disposed, to verify the results which 
 will be given below. 
 
 First, then, we have to find the values of m^y which are the 
 roots of the two equations (see (20) and (25)), 
 
 cos X = ± x\ (28) 
 
 3 
 
 where the upper sign corresponds to the case of both ends 
 fixed or both free, and the lower to that of one fixed and the 
 other free. 
 
 It is evident on inspection that if m be any root of either 
 equation (28), then — w, and ± « \/— i are also roots; now 
 observing that 
 
 co^ {—m6) = co%m6, a- {—m 6) = a- (m 6), 
 
 cos {±md\/—i) = <r(md), (T{±m6\/ — i) =zo%m6, 
 sin(— OT^) = — sinOT^, h{ — m6) = —b(m.6), 
 
 sin( -twi^V — j) = ± V — 1 .8(»2 5), 
 
 8 ( ± OT 5 V — i) = ± V — I . sin OT 5, 
 
 we see, on examining the forms of the functions X^, F^, Z^, 
 that the effect of changing any one of the four values ± tn^, 
 ±mi\/—i into any other, will in every case be merely to 
 
184 Periods of Tones found 
 
 multiply the function by one of the factors ±1, ± v — i ; and 
 consequently all the four terms in the value o/Ly givea by the 
 four roots, can be united into one term of the form (22), (23), 
 or (26), according to the case in question. It is therefore only 
 necessary to consider the positive real roots of (28). 
 
 194. The' position of the roots of (28) may be most clearly 
 exhibited by a graphic construction. If we draw the curve 
 of which the equation is 
 
 e=" + €-'^ 
 
 V = cos X, 
 
 2 
 
 it will cut the positive axis of x at distances 
 
 222 
 from the origin, and the distances from the axis oi y at which 
 it cuts the two lines>'= +1, will be the positive roots of (28). 
 The curve itself will consist of a series of unsymmetrical waves, 
 of which the amplitudes increase without limit. In Fig. 2, the 
 
 
 
 Oj 
 
 P^ 
 
 
 ^^ 
 
 
 
 A\ 
 
 B 
 
 C 
 
 
 B 
 
 t 
 
 'j 
 
 F 
 
 3 
 
 
 Fig. 2. 
 
 lines P P^, P^Pi> A A represent portions of the curve in- 
 cluded between the lines >■= ± i, so that P p^, P p^ are two 
 roots corresponding to the upper sign in (28), and QP-^, QP^, 
 Q /"a are three roots corresponding to the lower sign. 
 
 Since — ■- increases indefinitely with x, it is evident that 
 
 2 
 (28) requires ± cos x to diminish indefinitely with x, so that 
 for large values of i the values of m^ must approximate without 
 
 limit to ±(22-1-1)-. At the points A, B, C, &c., where 
 
in two Cases. 185 
 
 2 2 2 
 
 the values of — are alternately negative and positive, increasing 
 
 numerically without limit. Hence the roots corresponding to the 
 
 upper sign are alternately greater and less than — j — > &c., 
 
 2 2 
 
 and those corresponding to the lower sign alternately greater 
 
 and less than - , ^^— , &c. On the scale to which the figure is 
 22 
 
 drawn, the portion of curve p^ P^ is quite undistinguishable from 
 
 the ordinate at C. 
 
 We will now shew how to calculate the values of Q P^, QP^, 
 
 Pp.- 
 
 195. Suppose that in either of the equations 
 
 e^ + e-^ 
 
 cos X = ± I 
 
 2 
 
 we have found an approximate value of x, say x = m. Then, 
 assuming m + a as the true value, we have 
 
 f m+a _j_ f — m— a 
 
 cos (m + a) = ± I . 
 
 2 ^ ' 
 
 Developing this, and neglecting powers of a above the first, 
 we find (using the notation explained in Art. 188) 
 
 cos m . or (m) =F I / \ 
 
 a = -, ^—^ . (29) 
 
 sm m . o- (m) — 8 (m) cos m 
 
 If the value m + a thus found is not sufficiently exact, then it 
 must be assumed as an approximation, and the process repeated, 
 and so on as often as may be necessary. 
 
 We will take as an example the case of the fundamental tone 
 of the rod with one end fixed. We then have to find the least 
 positive root of (28), taking the lower sign; and we have seen 
 
 (Art. 104) that this is somewhat greater than - . Assummg 
 
 then x = - + a as a first approximation, and puttmg m = - m 
 2 z 
 
 (20) we have i „ , 
 
 ^ ^' a = = 0.398 nearly. 
 
 Assuming therefore 
 
 X = - + 0.398+ a', 
 
1 86 Numerical Results. 
 
 we shall find the valte of a by putting 
 
 m = - + 0.398 
 2 
 
 in (29) (taking the lower sign in the numerator). This gives 
 a' =—0.089 ^iid .r=i.88 nearly. 
 The next approximation gives 
 
 .r= 1.8751, 
 which is sufficiently accurate. 
 
 196. For the higher tones of the rod with one end fixed, and 
 for all the tones of the rod with both ends fixed or both free, 
 the approximation is more rapid. The following are the results 
 in the two cases 
 
 I. II. 
 
 One end fixed and one free. Both ends fixed or both free. 
 »Zj 1.8751 4-7300 I 
 
 ^1 4-6940 7-8532 x:\i> 
 
 ^3 7-8548 10.9957 ^-^^ 
 
 OT, 10.9955 14-1372- Q_ 
 
 For still higher tones the formula i ■i.'!i -^' 
 
 OT. = (2^TI)- 
 may be used without sensible error, the upper sign belonging to 
 case I. and the lower to case II. 
 
 The numbers of vibrations, being proportional to the values 
 of m^, are, for the higher tones, sensibly proportional to the 
 squares of the odd numbers. 
 
 197. To find the interval between any two tones we may 
 proceed as in the following example. The interval between the 
 fundamental tone and the first upper tone, of the rod with one 
 
 end fixed, is defined by the ratio (-^— ^ — ) , of which the 
 
 logarithm is 0.79704; and observing that log 6 = 0.77815, we 
 find that this ratio is equal to 6 x 1.0445 nearly. Now 
 
 ^•^445 = i+^^&c., 
 
 of which the first three convergents are ti ff> tt > hence 1-0445 
 exceeds ff by a fraction less than -^, and the ratio of the 
 interval is therefore a very little less than 
 
 6V 23 _4 y S V 23 
 
Position of Nodes. 187 
 
 It follows that the interval is = two octaves + fifth + (ff ) nearly ; 
 and the interval (ff) is a little less than f'^'s of a diatonic semi- 
 tone, for (if)* = fj- nearly. 
 
 Hence, if the fundamental tone were C, the first upper tone 
 would be flatter than b a' by a little more than a quarter of a 
 diatonic semitone '. 
 
 In this way we find the following to be the four first tones of 
 a rod in the two cases, supposing the fundamental tone in, each 
 case to be C. 
 
 I. 
 
 
 II. 
 
 One end fixed and one 
 
 free. 
 
 ^Both ends free or both fixed. 
 
 c 
 
 
 c 
 
 ^d - 
 
 
 tf/- 
 
 b d'" + 
 
 
 /' + 
 
 b </(*) + 
 
 
 d"- 
 
 The sign + signifies that the actual sound is somewhat 
 sharper, and the sign — that it is somewhat flatter, than the 
 tone indicated by the letter. 
 
 We find also that the ratio of the interval between the fund- 
 amental tone of a rod with one end fixed, and of the same rod 
 with both ends free or fixed, is 6 x 1.106 nearly; so that the in- 
 terval is a very little less than two octaves -F fifth-)- minor second. 
 Thus, if the fundamental tone were C in the first case, it would 
 be a little flatter than a' in the second. 
 
 198. We shall now shew how to determine the position of 
 the nodes in the several cases, and we will take first that of the 
 rod with one end fixed. 
 
 Referring to equations (26), (27), (Art. 192), we see that 
 since at a nodeji^ = o for all values of /, the values of x, or the 
 distances of the nodes from the fixed end, must be in this case 
 roots of the equation Z^ = o, namely, those positive roots which 
 are less than /. 
 
 1 A more systematic way of defining a small interval is to assign its ratio 
 to the semitone of the ' equal temperament,' which is the twelfth part of an 
 octave, and which we may call the ' mean semitone.' An interval of which 
 
 the ratio is r contains — -, — mean semitones. Thus the intei-val (1.0445) 
 
 contains °^ ^'"'^"^^ ^ 0.787 such semitones nearly. (See Art. 23.) 
 Jg log 2 
 The octave contains 10.74 diatonic semitones nearly; and the mean 
 semitone is about 0.89 diatonic semitones. (Compare the table of intervals 
 on p. 27.) 
 
1 88 Position of Nodes 
 This equation (subscript indices being omitted) is 
 [smm + S (m)) (cos— "^("7"}) 
 
 — {co%m + <T(m))\wx— 8(^— jj=o, (30) 
 
 in which m is a determinate root (w^) of the equation 
 
 <r(»?) cos »z = — I. (31) 
 
 The equation (30) may be transformed as follows : 
 From (31) we have o- {m.^ = —sec ot^, and therefore 
 
 (8(«2,))==(^K))2-i=tan»z«,. 
 
 Now we have seen (Art. 194) that m^ is greater or less than 
 
 (2 z'— i) - according as i is odd or even, so that we may put 
 
 (i = o being excluded) 
 
 % = (2 2-l) '^-(-)'aj (32) 
 
 2 
 
 where a^ is a small positive quantity, which diminishes in- 
 definitely for increasing values of i. 
 
 Hence cos 1n^ is always negative, and sin m^ has the same 
 
 sign as sin(2z'— i)-, that is, ( — )'"'"■'• Consequently, since 
 2 
 
 S {m>) is necessarily positive, we must have 
 
 S {"'■d = ( — )' t^'ii w» = cos in tan % ; 
 and therefore 
 
 sin ra^ + S (»Zj) sin Wj + cos z'tt tan W; cosWj + coszV 
 cos m^ + cr (mf) cosm^— sec w^ sin « ^ + sin z' n- 
 
 mi + tir 
 
 = — cot - ; 
 
 2 
 
 and (30) is therefore easily seen to become (indices being 
 omitted) 
 
 /tnx m + zir^ /■mx-. m + iTr 
 cos (-- ) — iri—r-) cos 
 
 
 2 
 
 which is reducible, by means of the identities 
 
 cos 5 + sin 5= \/2 .sinr^+ -), cos^^sinfl= •y2.cos(5+ -), 
 
• investigated. 189 
 
 to the form 
 
 mx 
 
 1— /m X M + tTT-. —r- , ,m + zir 7r\ 
 
 V2.cos(-- )— 6 sm ( 1--) 
 
 \/ 2 ' ^2 .<-' 
 
 mx 
 
 4' 
 — e ' COS( 1 — ) =0. (30) 
 
 N 2 4^ 
 
 Now let the origin of abscissae be removed to the middle of the 
 
 axis of the rod, by writing x + - instead of x, so that x will 
 
 2 
 
 now mean the distance of a node from the middle point. The 
 
 above equation then becomes 
 
 mx m , . 
 
 rmx Zttn — — . fm + ttt 7r\ 
 
 V2.cos(—; )— e • ^^ sm ( 1 — ) 
 
 ^ I 2 ■' - '^ 2 4' 
 
 fm-^rlTT TTv 
 
 
 cos I 
 
 Now from equation (31), which is 
 
 = — sec m, 
 
 2 
 
 we get (by adding and subtracting i) 
 
 m m 
 
 ^ (e^ +£ 2)2=— sec m sin^ ■ — 1 
 
 2 
 
 m m 
 
 ^(e^ — c ')^= -secwcos^ — ; 
 and since sec m is always negative, whilst the sine and cosine of 
 
 — have the same signs as the sine and cosine of (2 z'— i) - 
 2 4 
 
 (as is evident from (32)), these equations give 
 
 mm 
 
 fS-i-e 2 =2V— secOT.sm — sm(2z — i)-i 
 
 2 4 
 
 - — / m TT 
 
 e2_g 2 =2V— secOT.cos C0S(2Z— i) — , 
 
 2 ^ '4 
 
 and thence, by addition and subtraction, 
 
 — , /■m in 7r\ 
 
 f2 = V— £cc»z.cos( h-l 
 
 ^ 2 2 4^^ 
 
 -— / rm in n-v 
 
 _e 2 = V— sec»i.cos( 1 )• 
 
 V 2 2 4' 
 
1 90 Position of Nodes 
 
 Introducing these values in (33), and observing the identities 
 sin 6 cos ^ = \ (sin (^ + ^) + sin (6 — cj))), 
 
 cos 5 cos ^ = I (cos (^ + <^) + cos {6 - (}))), 
 
 •we find, after obvious reductions, 
 
 V2.C0S(-- j + ^V — COSOT. (« ' — e 'cosz!r) = o; 
 
 but from (32) it is evident that 
 
 COS mf=— sin a^ ; 
 hence the equation becomes 
 
 m^x mix 
 
 cos (— ^ '\ + ^V^Smaf.(e' —e ^ COS zV) = O. (34) 
 
 Another form is obtained from (33) by substituting for m^, 
 under the sine and cosine in the last two terrns, the value (32). 
 This will easily be found to give 
 
 V2.cos(-j ) + e ^' •^''sm^Oj 
 
 — e ^« ■<" COS Z TT cos ^ Oj = O. (34) 
 
 199. To find the places of the nodes when both ends are 
 free, we have (see Art. 189) to solve the equation 
 
 (smm—s{m)) (cos— ^ — '"'^(~7^)) 
 
 - (cos OT- <T {m)) y sm -j- + 8 (-^) j = o ; (35) 
 
 in which w is a determinate root {m^ of the equation 
 
 o- {m) = sec m ; 
 and (Art. 194), z'= o being excluded, 
 
 TT 
 
 mi = {2t+i)~-{-y^i, (36) 
 
 2 
 
 where p^ is a small positive quantity, which diminishes in- 
 definitely for increasing values of t. 
 
 In this case cos m^ is always piositive, and = sin /3j ; and, 
 proceeding as in the last Article, we find 
 
 8 (m^) = tan m^ cos zV ; also 
 
investigated. 191 
 
 sin m,— h (m,) m,- + iir 
 
 1-^ = tan —^ , and 
 
 cos m^~ a- (nii) 3 
 
 e ^ = «/sec»Zi COS (— ^ + (^ 1 — )\; 
 
 hence (35) becomes 
 
 V2sin(— ) + (! cosf + -^ 
 
 ^ i 2 ' ^24' 
 
 mx , • 
 
 — =- . ,m-\-nt 7r\ 
 
 — f ' sm ( h — ) = o : 
 
 \ 2 4^ 
 
 and, transferring the origin of abscissse to the middle point of 
 the axis as before, we obtain finally, instead of (34) and (34'), 
 the two equations 
 
 iriiX 
 
 -y---Y)+ivisin/3i.(e' -e «coszV) = o; (37) 
 
 V2sm(-i )+f ^^ ^''sini/3. 
 
 — € ^i ' cos z n- COS ^ /3j = o. (37) 
 
 200. The equations (34) or (34') and (37) or (37') de- 
 termine, in the two cases, the positions of the nodes for each 
 value of t, that is, for each component tone ; the value z' = i 
 belonging to the fundamental tone in both cases. The values 
 of X which lie between —^l and + 5 / give the distances of the 
 nodes from the middle point. 
 
 The numerical values of m^ in the two cases have already- 
 been given. The values of a^, (3^ are the differences (taken 
 
 positively) between these numbers and the values of (2 z'+i) -, 
 and are as follows : ^ 
 
 0^ = 0.3043, /3^ = 0.0176, 
 
 02 = 0.0184, /Sj = 0.0008, 
 
 03 = 0.0008, |33 = O.OOOT. 
 
 (For values of t above 3, there is no significant figure in the first 
 four decimal places.) 
 
 201. We will first consider the case of the rod free at both 
 ends, which is the simpler of the two. 
 
 It is evident that equation (37) is satisfied hjx = o when 
 / is even ; and that, in all cases, if x' be a root, then — x' is a 
 root also. 
 
192 Position of Nodes 
 
 Hence the nodes are symmetrically distributed with respect 
 to the middle point, as might be foreseen a priori;, and when 
 i is even, that is, for the i°^, 4*'', &c. component tones, there is 
 a node at the middle point. 
 
 For values of i is not greater than 3, the actual numerical 
 values of vl^ and ^^ must be introduced, and the equation (37) 
 or (37') solved by approximation. 
 
 Since the second term of (37) is essentially positive, the first 
 term must be negative ; and from this condition it may be 
 shewn (but more easily by making a graphic construction for 
 one or two particular cases) that the number of roots between 
 \ I and —\l, that is, the number of nodes, is z'+ i. Thus the 
 fundamental tone has two nodes, &c. (see Art. 205). 
 
 For greater values of i, it is evident on inspection of the 
 values of /3; given in Art. 200, that the second term of (37) will 
 
 DC 
 
 be insignificant when - is numerically small. Hence, for the 
 
 higher component tones, and for nodes not near the ends of the 
 rod, the values of x will be such as make the first term vanish, or 
 niiX . TT 
 
 — ; Z - = ± « TT, 
 
 / 2 
 
 n being an integer ; and putting in this equation the approximate 
 
 value (2 z'+ i) - for »«;, we get 
 
 X i ± 2n 
 
 I 2 z'+ I 
 
 Thus the nodes which are not near the ends are distributed at 
 sensibly equal distances, the interval between any two con- 
 
 2 
 
 secutive nodes being — — — /. 
 
 21 + 1 
 
 202. But for values of i greater than 3, and for nodes near 
 the ends, we may proceed as follows. The last term of (37') 
 
 will be insignificant when x is positive, and y not a small 
 
 fraction. In the second term we may put for sin 1 0^ an ap- 
 proximate value derived from the equation 
 
 I 
 
 (T (mi) = sec Mi = —. — T- 
 
 (see Art. 199), which gives 
 
 2 _^ 
 
 sin/" 
 
 e^i + e""* 
 
investigated. 193 
 
 (the square of e-'»i being neglected), and therefore 
 
 sin i ^^ = e-»i, 
 since /3j is very small. Hence the second term of (37') becomes 
 
 ^ I 2 
 
 and if in this term we put the approximate value 
 
 (22 + 1)- for — i , 
 4 2 
 
 the equation becomes 
 
 V3sm(^ ) + € ' ^ 4=0. 
 
 / 2 ' 
 
 Now let T- = 3, then 
 
 2 / 
 
 /- -^-- 
 
 V 2 sind = e 4 ; (38) 
 
 and this equation will have a determinate series of positive 
 roots, say Si, S2) ••■ 3>-, •■•, which can be found by approxi- 
 mation. The values of x will then be given by the equation 
 
 Xf in 
 
 or, if m-i be replaced by its approximate value, 
 X ■ iir— 2 $!j 
 
 I (2z'-|-l)?r 
 
 (39) 
 
 This formula gives the distances, from the middle point, of 
 the nodes towards the positive end. And we know that the 
 nodes in the negative half of the rod are respectively at the 
 same distances from the middle point. 
 
 It is easily found (by roughly drawing the two curves 
 
 TT 
 
 J' = V 2 sin .ar, and y = i 4) 
 
 that, for increasing values of/, $ij tends rapidly to become 
 
 X • 
 {j— i)it; so that the above expression for -j- tends to assume 
 
 the same form as that given in Art. 201 for nodes near the 
 middle. The numerical results will be given below. (Art. 205.) 
 203. When one end of the rod is fixed, the nodes are not 
 symmetrically distributed, and the positions of those near the 
 two ends must be found separately. For values of i not greater 
 
 
 
194 Position of Nodes 
 
 than 3, the equation (34) or (34') must be solved by approxi- 
 mation, after the numerical, values of a^ have been introduced. 
 
 X 
 
 But for greater values of i, and for small values of - (that is, 
 
 for nodes not near the ends), we may neglect the second term 
 of (34), and we then find (in the same way as at the end 
 of Art. 201), 
 
 Xj t± (2«+l) _ 
 T " 2i-l ' 
 
 from which it follows that, near the middle, the interval between 
 
 2/ 
 any two consecutive nodes is sensibly equal to — -. , and 
 
 that when i is odd (being greater than 3) one node (namely, 
 the middle node, if the fixed end be reckoned as one) is sensibly 
 at the middle of the rod. 
 
 For nodes near the firee end, since — is positive, we may 
 
 neglect the last term in (34') ; and in the second term we may 
 introduce the approximate value of a^ derived from the equa- 
 tions (Art. 198) 
 
 "■ ("^») = — sec OTj, cos w,i = — sin a^ ; 
 
 which give approximately (as the corresponding equations in 
 Art. 202) sinla, = e-'".-; 
 
 so that (34') is reduced to 
 
 /- ImiX iirX , mi 
 
 V2C0s(-^--j+e 
 
 
 TT 
 
 or, since OTj= (22 — 1) - approximately (see (32)), 
 
 \/2C0s(-^ T/ ' 
 
 SO that, if we put 
 
 /tt ^iX ^^ 
 
 2 I 
 
 we have V2cosfl + e 1=0; 
 
 and if we call 6 ,6^^ . . . 6j,. . . the roots of this equation, then 
 
 Xi ITT . 
 
 mi-f = —-^j, or 
 
investigated. i p g 
 
 Xj in — I 6 J 
 
 T=(23--I)^' (4°) 
 
 for increasing values of/ it is easily seen that 6^ tends to be- 
 come (2 7— !)-• 
 
 2 
 
 204. For nodes near the fixed end x is negative, and there- 
 fore the second term of (34'), which is (see last Article) ap- 
 proximately *_^i 
 
 is small. Neglecting it, and putting i for cos — in the last 
 
 2 
 term, we have 
 
 V2C0Sf-r ^—6 » »cosz'7r = o; 
 
 m, I IT IT 
 or, smce — = approximately. 
 
 V 2 cos \—^ — j - € ' 2 -t COS zV = o ; 
 
 and if we now put 
 
 m.x tiT 
 
 the equation is reduced to 
 
 V 2 cos <p — € 4=0. 
 Let (p^, (f)^,. . . (j}y, ... be the roots of this last equation ; then 
 Xi /I'lr 
 
 m. 
 
 
 X,- iir — 2 tb- I . 
 
 or -T^--!—. r^; (41) 
 
 while, for increasing values of 7', ^j tends to become (27+1)77. 
 
 205. The following numerical results have been given by 
 Seebeck ^, who, however, has treated the fundamental equations 
 (30) and (35) in a somewhat different manner. 
 
 Case I. (One end fixed and the other free.) 
 
 ' In a memoir on the transverse vibrations of rods. {Abhandlungen d. 
 Math. Phys. Classe d. K. Sachs. GeselUchaft d. Wissenscbaften. Leipzig, 
 1852.) 
 
 O 2 
 
196 Theory of Transverse 
 
 Distances of nodes from the free end, the length of the rod 
 being taken as unity : 
 
 i^-^ tone, 0.2261, 
 
 3rd 
 
 O.I321, 0.4999, 
 
 4* 0.0944, 0.3558, 0.6439, 
 
 it\ ^-3222 ^ 4-9820 ^ 9-0007 ^ 4/- 3 
 
 42— 2 4 z'— 2 4 i— 2 42—2 
 
 4 z- 10. 9993 4 2- 7-0175 
 
 -. > : • 
 
 42—2 42 — 2 
 
 The last i*ow in this table must be understood as meaning that 
 4 ;'— 3 
 
 — : may be taken as the distance of the /tl> node from the 
 
 42—2 -' 
 
 free end, except for the first three and last two nodes. 
 Case II. (Both ends free.) 
 Distances of nodes from nearest end : 
 
 i^t tone, 0.2242, 
 
 
 2H<i 0.132 1, 
 
 0.5, 
 
 3rd 0.0944, 
 
 0-3558, 
 
 .., 1.3222 
 
 4.9820 
 
 9-0007 ^ 4^-3 
 
 42+2' 42 + 2 42+2 42+2 
 
 206. To complete the theory of the transverse vibrations of 
 a rod, it is necessary to shew how the form of the axis, at any 
 time during the motion, is determined by the initial displace- 
 ments and velocities of its points. 
 
 On reference to Art. 179, &c. it will be seen that in all the 
 cases which have been considered, the equation to the axis at 
 the time / is of the form 
 
 y = X-ir "i (^i <=°s n^t-\-B^ sin n^t) ; (42) 
 
 in which ft^, n^, . . . n^, . . . are determinate constants, depending 
 upon the roots of an equation, in general transcendental, and 22^ 
 is a determinate function of x and of «,-, which satisfies a 
 differential equation such as (8), in which the coefiicients depend 
 (through m) upon n^. We shall consider only the case in which 
 there is no tension, so that a = o, and shall also neglect, as 
 
 , ,- , o ^« , - , . r- , '^ty - 
 
 before, the term m' -r— ^ , which arises from the term ,„ , „ in 
 dx^ dr dx^ 
 
Vibrations completed. 197 
 
 (6'), introduced by taking account of the angular motion of the 
 transverse sections of the rod. 
 
 With this simplification we may write equation (8) thus for 
 any two different roots of the transcendental equation referred 
 to above : 
 
 ^+^*"^ = °' 
 df^Uf 
 
 in which p^^pj are two constants, of which we only require to 
 know that they are different. 
 From these equations we have 
 
 d'^U; d^u, , . 
 
 Now if we multiply this by dx, and integrate from ^ = o to 
 X = 1, the result is 
 
 {Pi-Pj) / «iUj 
 
 dx ■- 
 
 for the first two terms of the equation, multiplied by dx, are 
 the differential of 
 
 d^Uf d^Uj d'^Uj d^u^ 
 
 "^ :dx^~"''d^^'''i^~''-'j^' 
 
 of which every term vanishes at both limits, on every suppo- 
 sition as to the terminal conditions. (See Art. 187.) 
 It follows therefore that when/ is different from i 
 
 ^0 
 
 u^ Uj dx = o, (43) 
 
 but / ufdx 
 
 f 
 
 will be a determinate constant, depending upon/ 
 
 Now the initial displacements and velocities being supposed 
 to be given, we have, when t=o, 
 
 where/ (.x:) and ^ {x) are functions of which the value is given 
 for all values of x from o to /. Hence, from (42), 
 
 2 Ai Ui =f{x), ^niBiUi = <l> {x) ; 
 
iqS Torsion Vibrations. 
 
 and if these equations be multipled by Ujdx, and integrated 
 from J? = o to ;«: = /, the result (by (43)) is 
 
 Aj / «/ dx= f(x) Uj dx, 
 Jo Jq 
 
 fij Bj / uf dx= \ <^ {x) Uj dx ; 
 Jo Jo 
 
 so that Aj, Bj are determined, and the form of the axis of the 
 rod at any time t is then given by equation (42). 
 
 Torsion Vibrations. 
 
 207. Torsion vibrations may be properly included in the 
 general class of lateral vibrations; but as they are of little 
 practical importance we shall discuss them briefly. Such vi- 
 brations will be produced when a uniform elastic rod is left 
 to itself after undergoing a slight disturbance by forces reducible 
 to couples in planes perpendicular to its axis. 
 
 If the rod were in equilibrio under the action of such forces, 
 it would be in a state of torsion or twist ; and the twist may be 
 called simple when the particles in any transverse plane section 
 are not displaced relatively to one another, and the distance 
 between any two sections remains unaltered. 
 
 Suppose a cylindrical rod, whether solid or hollow (as a 
 tube), to be twisted by equal and opposite couples applied only 
 in the planes of its ends ; then, if there is no relative displace- 
 ment of particles in either of these terminal sections, it is 
 evident that there will be none in any other transverse section ; 
 and it is known that the length of the rod remains unaltered, or 
 rather is altered only by a quantity of the second order, when 
 the twist is small. Under the action of such forces the rod, 
 when in equilibrio, will be in a state of uniform simple twist. 
 It is probable that such a condition cannot be realised in 
 practice except in the case of cylindrical rods, though it may 
 subsist, more or less approximately, for other forms. In what 
 follows we shall assume that the form is cylindrical. 
 
 208. When the twist is uniform, the rate of twist is defined 
 by the angle through which any transverse section is turned 
 relatively to any other, divided by the distance between the two 
 sections. And the limit of this ratio, when the distance between 
 the two sections is diminished indefinitely, is" the rate of twist in 
 that section with which they ultimately coincide, whether the 
 twist be uniform or not. 
 
 When the twist is uniform, all the particles which, in the 
 
Condition of Equilibrium. 199 
 
 untwisted state, lay upon any straight line parallel to the axis, 
 will lie upon a hehx. And the inclination of a tangent to this 
 helix, at any point, to the axis, will be directly proportional to 
 the rate of twist and to the distance from the axis. When this 
 inclination is small where it has its greatest value, that is, on 
 the exterior surface, the twist may be called small. It is evident 
 that a small twist is consistent with a large relative angular dis- 
 placement of the terminal sections, if the radius of the cylinder 
 be small compared with its length. 
 
 209. When equilibrium subsists under the action of couples 
 applied only in the planes of the terminal sections, it is evident 
 that the moments of these couples must be equal and opposite ; 
 and it is known from experiment that when the tvsdst is small, 
 the rate of twist is, for a rod of given material and section, 
 proportional to the moment of the couples. 
 
 If, besides the terminal couples, there are twisting forces 
 acting on the interior matter of the rod, the conditions of equi- 
 librium are easily found as follows. Let x be the distance of 
 any transverse section from one end (A) of the rod, and 6 the 
 
 angular displacement of that section. Then — (Art. 208) is 
 
 the rate of twist in a section at the distance x from A ; and the 
 moment of the couple which would have to be applied in the 
 plane of that section in order to maintain equilibrium, if the rod 
 
 were cut there, would be C ^— , C being a constant which we 
 
 ax 
 
 shall consider more particularly below. 
 
 210. Let us consider then an infinitesimal slice contained 
 between two sections at distances x — \dx, x-\-\dx from ^. 
 If all the rest of the rod were removed, it would be necessary, in 
 order to maintain the equilibrium of the slice, to apply in its 
 two faces couples of which the moments are 
 
 ~^\dx~^^' )' ^^dx^^dx''^^) 
 (those couples being considered positive which tend to in- 
 crease 6). 
 
 Hence, if Lpadx be the moment of the twisting forces 
 acting on the mass of the slice, where p, a> are the density, and 
 area of section of the rod, the condition of equilibrium will be 
 
 d'^e 
 Lpa>dx+C -j—i dx f o, 
 
 and the differential equation of motion will be obtained from 
 
200 Isotropic Rod. 
 
 this as usual, by substituting for Z p a dx the sum of the mo- 
 ments of the resistances to acceleration of the particles of the 
 shce, namely, ^2^ - 
 
 where da is an element of area of the transverse section, and 
 r the distance of da from the axis. If then we put k for the 
 radius of gyration of the area of the section about the axis 
 of the rod, so that 
 
 \r^da = k^a>, 
 
 we shall have d'^e _ C d'^e 
 
 'dF'WJZJ^ ^^' 
 
 for the equation of motion to be satisfied at all parts of the 
 rod. 
 
 The terminal conditions will be 
 
 6 = at a fixed end, and 
 
 de 
 
 ^— = o at a free end. 
 
 dx 
 
 (The latter condition is evident if it be observed that at a free 
 end the rate of twist must be o, since there is no couple in the 
 terminal face.) 
 
 211. On the supposition that the material of the rod is 
 isotropic {Tail and Thomson, § 676), and therefore equally 
 elastic in all directions, the constant C can be expressed in 
 terms of q, the modulus of elasticity (Art. 148), and of another 
 constant 11, the meaning of which we will now explain. 
 
 If a uniform bar, of any section, be extended by forces 
 applied uniformly to the surfaces of its ends only, it is known 
 that the transverse linear dimensions are contracted. Let t be 
 the longitudinal extension (see Art. 148), and ^i e the transverse 
 or lateral contraction ; then, the extensions and contractions 
 being always supposed small, /i is a constant for a given ma- 
 terial, and moreover must have a value between o and \, if, 
 as is the case with all ordinary substances, the volume of the 
 bar is increased under the circumstances supposed. 
 
 It can be shewn that in the case of a cylindrical, solid or 
 hollow, rod, the value of the constant C in the last Article is ^ 
 
 k'^aq 
 
 C = 
 
 2(1+^)' 
 
 ' The demonstration of these propositions is elementary, but we cannot 
 afford space for it here. See Klebscb, §§ 2, 3, 92. 
 
Result of Integration. 201 
 
 so that equation (i) becomes 
 
 (2) 
 
 dt'^ 2{i+n)p dx^' 
 
 212. We need not repeat the process of integrating (2), since 
 It IS exactly analogous to that which has been applied in former 
 problems (see Art. 122) and can offer no difficulty. We shall 
 merely give the result in two cases. 
 
 (i) If both ends of the rod are free, then 
 
 „ ■^,1=00 . iTv X . /mat \ 
 
 ^ = :^i=i ^i cos -^ sm (-^ + a;) • 
 
 (2) If the end from which x is measured is fixed, and the 
 other end free, then 
 
 . •^,!=oo . . (2 2'4- 1) jT jf . /'(2 z'+ i) 77 a/ \ 
 6 = 2,=„ A, sm ^ -I— sm (^-2_J + „^) . 
 
 where A^, a., in each case, represent arbitrary constants, to be 
 determined by initial circumstances, and a is defined by the 
 equation 
 
 a' = " 
 
 2(l+/i)p 
 
 The period of the- z'th tone is therefore 
 
 2/^2 (i+^)p.4 . 
 
 — ( ) mease (I), 
 
 4I /2 (H-^)pNi . 
 
 — : ( I m case (2). 
 
 2Z + I ^ g f ^ ' 
 
 Now comparing these with the periods of longitudinal vi- 
 brations of the same rod under the same terminal conditions, 
 we see that the tone of given order produced by torsion vi- 
 brations is lower than that of the same order produced by 
 longitudinal vibrations, by an interval of which the ratio is 
 
 (2(i+f))*- 
 
 213. The value of the constant \i, is probably different for 
 different substances. Navier and Poisson, by reasoning now 
 generally admitted to be illegitimate, deduced a priori the value 
 fi = i for all substances. 
 
 Wertheim found experimentally ^ ju = -J for glass and brass. 
 
 Kirchoff ^ found values differing sensibly from this for steel 
 bars, and for a drawn brass bar in which the longitudinal 
 elasticity differed from the lateral. 
 
 ' Ann. de Chim. el de Phys. 3rd series, vol. xxiii. p. 54. 
 ' Poggendorf, vol. cviii. p. 369. 
 P 
 
202 Value of ft. 
 
 The results of the last Article would afford an experimental 
 means of determining /x, if it were possible to be assured that 
 the rods used were isotropic, and to observe with sufficient 
 precision the intervals between tfie tones given by longitudinal 
 and torsion vibrations. 
 
 Chladni asserts that this interval is always a j^//k. If this 
 were so, or rather, for substances in which it is so, we must 
 have 2 (i +;ii) = |, or /i = -I- If the value of /x were -I, the ratio 
 of the interval would be (f )* = 1.632. 
 
 It is impossible however, or at any rate very difficult, to 
 observe with great exactness the interval between the two tones, 
 and a small error in the ratio of the interval may evidently 
 produce a considerable error in the value of /i. Hence this 
 constant must be determined by other methods. 
 
 214. Torsion vibrations may be excited in a cylindrical rod 
 by friction with the same substances as would excite longitudinal 
 vibrations in the same rod. 
 
 Thus, if a piece of stout glass tube, four or five feet long, be 
 gently but firmly clamped in a table-vice at its middle, after 
 winding a piece of broad tape about it at that part to protect it 
 from the vice, and if a wet piece of the same tape be passed 
 once round the tube not far from its middle, and the ends 
 rather lightly and quickly pulled backwards and forwards at 
 right angles to the tube by the two hands, the torsion vibrations 
 will be easily produced. 
 
 When the rod is not cylindrical the friction of a bow (charged 
 as usual with powdered rosin) should be used. Thus the torsion 
 vibrations of a rectangular deal rod may be excited by clamping 
 one end in a vice and drawing the bow across one of its edges 
 at right angles to the rod, at a point distant from the fixed end 
 about a fourth of the length.