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Full text of "The dynamical theory of sound"

THE 

DYNAMICAL THEOKY 

OF 

SOUND 



BY 



HORACE LAMB, SC.D., LL.D, F.R.S. 

PROFESSOR OF MATHEMATICS IN THE VICTORIA UNIVERSITY OF MANCHESTER ; 
FORMERLY FELLOW OF TRINITY COLLEGE, CAMBRIDGE 



LONDON 
EDWARD ARNOLD 

1910 
[All Eights reserved] 



PREFACE 

A COMPLETE survey of the theory of sound would lead into 
-^^- many fields, physical, physiological, psychological, aesthetic. 
The present treatise has a more modest aim, in that it is 
devoted mainly to the dynamical aspect of the subject. It is 
accordingly to a great extent mathematical, but I have tried to 
restrict myself to methods and processes which shall be as 
simple and direct as is possible, regard being had to the nature 
of the questions treated. I hope therefore that the book may 
fairly be described as elementary, and that it may serve as a 
stepping stone to the study of the writings of Helmholtz and 
Lord Rayleigh, to which I am myself indebted for almost all 
that I know of the subject. 

The limitation of methods has involved some sacrifices. 
Various topics of interest have had to be omitted, whilst 
others are treated only in outline, but I trust that enough 
remains to afford a connected view of the subject in at all events 
its more important branches. In the latter part of the book 
a number of questions arise which it is hardly possible 
to deal with according to the stricter canons even of mathe- 
matical physics. Some recourse to intuitional assumptions is 
inevitable, and if in order to bring such questions within the 
scope of this treatise I have occasionally carried this license 
a little further than is customary, I would plead that this is 
not altogether a defect, since attention is thereby concentrated 
on those features which are most important from the physical 
point of view. 

257816 



iv PBEFACE 

Although a few historical notes are inserted here and there, 
there is no attempt at systematic citation of authorities. The 
reader who wishes to carry the matter further will naturally 
turn in the first instance to Lord Rayleigh's treatise, where 
full references, together with valuable critical discussions, will 
be found. I may perhaps be allowed to refer also to the 
article entitled " Schwingungen elastischer Systeme, insbeson- 
dere Akustik," in the fourth volume of the Encyclopddie der 
mathematischen Wissenschaften (Leipzig, 1906). 

I have regarded the detailed description of experimental 
methods as lying outside my province. I trust, however, that 
no one will approach the study of the subject as here treated 
without some first-hand acquaintance with the leading pheno- 
mena. Fortunately, a good deal can be accomplished in this 
way with very simple and easily accessible appliances; and 
there is, moreover, no want of excellent practical manuals. 

I have to thank Mr H. J. Priestley for kind assistance in 
reading the proof-sheets. 



H. L. 



January, 1910. 



CONTENTS 



INTRODUCTION 

AKT. PAGE 

1. Simple Vibrations and Pure Tones 1 

2. Musical Notes 3 

3. Musical Intervals. Diatonic Scale . .5 



CHAPTER I 

THEORY OF VIBRATIONS 

4. The Pendulum 8 

5. Simple- Harmonic Motion 9 

6. Further Examples 11 

7. Dynamics of a System with One Degree of Freedom. Free 

Oscillations 12 

8. Forced Oscillations of a Pendulum 16 

9. Forced Oscillations in any System with One Degree of 

Freedom. Selective Resonance 20 

10. Superposition of Simple Vibrations 22 

11. Free Oscillations with Friction ...... 24 

12. General Dissipative System with One Degree of Freedom. 

Effect of Periodic Disturbing Forces .... 27 

13. Effect of Damping on Resonance 32 

14. Systems of Multiple Freedom. Examples. The Double 

Pendulum 34 

15. General Equations of a Multiple System . . . . 41 

16. Free Periods of a Multiple System. Stationary Property . 44 

17. Forced Oscillations of a Multiple System. Principle of 

Reciprocity 47 

18. Composition of Simple-Harmonic Vibrations in Different 

Directions 48 

19. Transition to Continuous Systems 52 

20. On the Use of Imaginary Quantities 53 

21. Historical Note 58 



CONTENTS 



CHAPTER II 

STKINGS 

ART. PAGE 

22. Equation of Motion. Energy ...... 59 

23. Waves on an Unlimited String 61 

24. Eeflection. Periodic Motion of a Finite String ... 64 

25. Normal Modes of Finite String. Harmonics ... 68 

26. String excited by Plucking, or by Impact .... 72 

27. Vibrations of a Violin String 75 

28. Forced Vibrations of a String 80 

29. Qualifications to the Theory of Strings .... 81 

30. Vibrations of a Loaded String ...... 82 

31. Hanging Chain . . . .... . . 84 



CHAPTER III 
FOURIER'S THEOREM 

32. The Sine-Series . ... . . . . . 87 

33. The Cosine-Series 92 

34. Complete Form of Fourier's Theorem. Discontinuities . 92 

35. Law of Convergence of Coefficients ... . . . 94 

36. Physical Approximation. Case of Plucked String . 96 

37. Application to Violin String . . . . . . 98 

38. String Excited by Impact 99 

39. General Theory of Normal Functions. Harmonic Analysis 101 



CHAPTER IV 

BARS 

40. Elementary Theory of Elasticity. Strains . . . . 106 

41. Stresses 108 

42. Elastic Constants. Potential Energy of Deformation . . 110 

43. Longitudinal Vibrations of Bars . . ... . . 114 

44. Plane Waves in an Elastic Medium . . . . . 118 

45. Flexural Vibrations of a Bar . . . . . .120 

46. Free-free Bar . . . . . . . .... 124 

47. Clamped-free Bar 127 

48. Summary of Results. Forced Vibrations . . ... 130 

49. Applications . 131 

50. Effect of Permanent Tension . ... . . . . 132 

51. Vibrations of a Ring. Flexural and Extensional Modes . 133 



CONTENTS 



CHAPTER V 

MEMBRANES AND PLATES 

ABT. PAGE 

52. Equation of Motion of a Membrane. Energy . . . 139 

53. Square Membrane. Normal Modes . . . . . 142 

54. Circular Membrane. Normal Modes .'"''/ . . . 144 

55. Uniform Flexure of a Plate ...'.... 150 

56. Vibrations of a Plate. General Results . . k /. 152 

57. Vibrations of Curved Shells . 155 



CHAPTER VI 

PLANE WAVES OF SOUND 

58. Elasticity of Gases . . . ... . ,- . . 157 

59. Plane Waves. Velocity of Sound . -. . . . . 160 

60. Energy of Sound- Waves . . . ... ... 163 

61. Reflection . . . . 168 

62. Vibrations of a Column of Air . . . , . . 170 

63. Waves of Finite Amplitude . / ... . . 174 

64. Viscosity . . 183 

65. Effect of Heat Conduction . . . ; . . . . 187 

66. Damping of Waves in Narrow Tubes and Crevices . . 190 

CHAPTER VII 

GENERAL THEORY OF SOUND WAVES 

67. Definitions. Flux. Divergence 197 

68. Equations of Motion . 200 

69. Velocity-Potential 201 

70. General Equation of Sound Waves 204 

71. Spherical Waves 205 

72. Waves resulting from a given Initial Disturbance . . 212 

73. Sources of Sound. Reflection 214 

74. Refraction due to Variation of Temperature . . . 216 

75. Refraction by Wind 219 

CHAPTER VIII 

SIMPLE-HARMONIC WAVES. DIFFRACTION 

76. Spherical Waves. Point-Sources of Sound .... 223 

77. Vibrating Sphere 228 

78. Effect of a Local Periodic Force . 233 



viii CONTENTS 

ART. PAGE 

79. Waves generated by Vibrating Solid 236 

80. Communication of Vibrations to a Gas .... 237 

81. Scattering of Sound Waves by an Obstacle . . . 240 

82. Transmission of Sound by an Aperture .... 244 

83. Contrast between Diffraction Effects in Sound and Light. 

Influence of Wave-Length . . . . . "''.'!. 248 



CHAPTER IX 

PIPES AND RESONATORS 

84. Normal Modes of Rectangular and Spherical Vessels . 254 

85. Vibrations in a Cylindrical Vessel 259 

86. Free Vibrations of a Resonator. Dissipation . . . 260 

87. Corrected Theory of the Organ Pipe 266 

88. Resonator under Influence of External Source. Reaction 

on the Source ... . . . t v, . . 270 

89. Mode of Action of an Organ Pipe. Vibrations caused by 

Heat . . . . 276 

90. Theory of Reed-Pipes . . . . . . . . 278 

CHAPTER X 

PHYSIOLOGICAL ACOUSTICS 

91. Analysis of Sound Sensations. Musical Notes . . . 284 

92. Influence of Overtones on Quality 286 

93. Interference of Pure Tones. Influence on the Definition 

of Intervals . . . 287 

94. Helmholtz Theory of Audition . . . . . . 289 

95. Combination-Tones 292 

96. Influence of Combination-Tones on Musical Intervals ' . 297 

97. Perception of Direction of Sound . . . ' . . 298 

INDEX . 301 



THE DYNAMICAL THEOEY 
OF SOUND 

INTRODUCTION 

1. Simple Vibrations and Pure Tones. 

In any ordinary phenomenon of sound we are concerned, 
first with the vibrating body, e.g. a string or a tuning fork or 
a column of air, in which the disturbance originates, secondly 
with the transmission of the vibrations through the aerial 
medium, next with the sensations which the impact of the 
waves on the drum of the ear somehow and indirectly produces, 
and finally with the interpretation which, guided mainly and 
perhaps altogether by experience, we put upon these sensations. 
It is in something like this natural order that the subject 
will be discussed in the following pages, but the later stages 
involving physiological and psychological questions can only be 
touched upon very lightly. 

As few readers are likely to take up this book without 
some previous knowledge of the subject we may briefly re- 
capitulate a few points which will be more or less familiar, with 
the view of fixing the meaning of some technical terms which 
will be of constant occurrence. Many of the matters here 
referred to will of course be dealt with more fully later. 

The frontier between physics and physiology is reached at 
the tympanic membrane, and from the physical standpoint it is 
to the variations of pressure in the external ear-cavity that we 
must in the last resort look, under normal (as distinguished 
from pathological) conditions, for the cause of whatever sensations 
of sound we experience. These variations may conveniently 
be imagined to be exhibited graphically, like the ordinary 
variations of barometric pressure, by a curve in which the 
abscissae represent times and the ordinates deviations of the 

L. 1 



2 , DYNAMICAL .THEOKY OF SOUND 

pressure on one side or other of the mean, the only difference 
being that the horizontal and vertical scales are now enormously 
magnified. 

The variety of such curves is of course endless, and it is 
impossible to suppose that a distinct provision is made in the 
ear for the recognition of each, or even of each of the numerous 
classes into which they might conceivably be grouped. It is 
therefore necessary to analyse, as far as possible, both the 
vibration- forms and the resulting sensations into simpler 
elements which shall correspond each to each. 

As regards the vibration-forms, there is one mode of 
resolution which at once claims consideration on dynamical 
grounds. The fundamental type of vibration in Mechanics is 
that known as "simple-harmonic," which is represented graphic- 
ally by a curve of sines (Fig. 3, p. 10). This is met with in 
the pendulum, and in all other cases of a freely vibrating body 
or mechanical system having only one degree of freedom. It 
can moreover be shewn that the most complicated oscillation of 
any system whatever may, so far as friction can be neglected, be 
regarded as made up of a series of vibrations of this kind, each 
of which might be excited separately by suitable precautions. 
The reason for the preeminent position which the simple- 
harmonic type occupies in Mechanics is that it is the only type 
which retains its character absolutely unchanged whenever it 
is transmitted from one system to another. This will be ex- 
plained more fully in the following chapter. 

The analysis of sensations is a much more delicate matter, 
and it was a great step in Acoustics when Ohm* in 1843 
definitely propounded the doctrine that the simplest and 
fundamental type of sound-sensation is that which corresponds 
to a simple-harmonic vibration. This implies that all other 
sound-sensations are in reality complex, being made up of 
elementary sensations corresponding to the various simple- 
harmonic constituents into which the vibration-form can be 
resolved. The statement is subject to some qualifications, in 
particular as to the degree of independence of elementary 

* G. S. Ohm (17871854), professor of physics at Munich 184954, known 
also as the author of " Ohm's Law " of electric conduction. 



INTRODUCTION 3 

sensations very near to one another in the scale, but these need 
not detain us at present. It may be regarded as in the main 
fully established, chiefly in consequence of the labours of 
Helmholtz*. The sensation corresponding to a simple-harmonic 
vibration is called a " simple tone " or a " pure tone," or merely 
a "tone." The sound emitted by a tuning fork fitted with 
a suitable resonator, or by a wide stopped organ pipe, gives the 
best approach to it. 

Since the form of the vibration -curve is fixed, the distinction 
between one simple tone and another can only be due to 
difference of frequency or of amplitude. The " frequency," i.e. 
the number of complete vibrations per second, determines the 
" pitch," greater frequency corresponding to higher pitch. The 
lower and upper limits of frequency for tones audible to the 
human ear are put at about 24 and 24,000 respectively; the 
range employed in music is much narrower, and extends only 
from about 40 to 4000. As between tones of the same pitch, 
the amplitude, or rather its square, determines the rate of 
supply of energy to the ear and so the relative "intensity," 
but it will be understood that it is physical rather than 
subjective intensity that is here involved. Between tones of 
different pitch only a vague comparison of loudness is possible, 
and this may have little relation to the supply of energy. Near 
the limits of audibility the sensation may be feeble, even though 
the energy-supply be relatively considerable. 

2. Musical Notes. 

From the chaos of more complex sounds there stands out a 
special class, viz. that of musical "notes." The characteristic of 
such sounds is that the sensation is smooth, continuous, and 
capable (at least in imagination) of indefinite prolongation 
without perceptible change. The nature of the corresponding 
vibrations is well ascertained. If we investigate any contrivance 

* Hermann Helmholtz (1821 94), successively professor of physiology 
(Konigsberg 1849), anatomy (Bonn 1855), physiology (Heidelberg 1858) and 
physics (Berlin 1871). Reference will often be made to his classical work : Die 
Lehre von den Tonempfindungen als physiologische Grundlage filr die Theorie der 
Musik, Brunswick, 1862. An English translation from the third edition (1870) 
was published by A. J. Ellis under the title Sensations of Tone, London, 1875. 

12 



4 DYNAMICAL THEOEY OF SOUND 

by which a note of good musical quality is actually produced, 
we find that the vibration can be resolved into a series of simple- 
harmonic components whose frequencies stand to one another 
in a certain special relation, viz. they are proportional to the 
numbers 1, 2, 3, .... Individual members of the series may be 
absent, and there is practically a limit on the ascending side, 
but no other ratios are admissible. It is evident from the 
above relation that the resultant vibration-form is necessarily 
periodic in character, recurring exactly at intervals equal to the 
period in which the first member of the series goes through its 
phases. It must be remembered, however, that the ear has no 
knowledge of the periodic character as such, and it must not be 
supposed that every periodic vibration will necessarily produce a 
sensation which is musically tolerable. The superposition of 
simple-harmonic vibrations to produce periodic vibration-forms 
is illustrated by some of the diagrams given below in 
Chapter III. 

One musical note may differ from another in respect of 
pitch, quality, and loudriess. The pitch is usually estimated 
as that of the first simple-harmonic vibration in the series, viz. 
that of lowest frequency, but if the amplitude of this first 
component be relatively small, and especially if it fall near the 
lower limit of the audible scale, the estimated pitch may be 
that of the second component. 

By " quality " is meant that unmistakable character which 
distinguishes a note on one instrument from the note of the 
same pitch as given by another. Every musical instrument 
has as a rule its own specific quality*, which is seldom likely to 
be confused with that of another. Everyone recognizes for 
instance the difference in character between the sound of a 
flute, a violin, a trumpet, and the human voice, respectively. 
It is obvious that difference of quality, so far as it is not due to 
adventitious circumstances f, can only be ascribed to difference 
of vibration-form, and so to differences in the relative amplitudes 
and phases of the simple-harmonic constituents. According to 

* French timbre ; German Klangfarbe. 

f Such as the manner in which the sound sets in and ceases ; this is different 
for instance in the violin and the piano. 



INTRODUCTION 5 

Helmholtz the influence of phase is inappreciable. This has 
been contested by some writers, but there can be no doubt that 
in most cases the difference of quality is a question of relative 
amplitudes alone. 

Comparisons of loudness can only be made strictly between 
sounds of the same quality and about the same pitch. 

It follows from the preceding that, so far as Ohm's law is 
valid, the sensation of a musical note must be complex, and made 
up of the simpler sensations, or tones, which correspond to the 
various simple-harmonic elements in the vibration-form. This 
doctrine has to contend with strong and to some extent 
instinctive prepossessions to the contrary, and some preliminary 
training is usually necessary before it is accepted as a fact of 
personal experience. We shall return to this question later; at 
present we merely record that that element in the sensation 
which corresponds to the gravest simple-harmonic constituent 
is called the " fundamental tone," and that the others are termed 
its " overtones " or " harmonics." 

3. Musical Intervals. Diatonic Scale. 

There are certain special relations, familiar to trained ears, 
in which two notes or two simple tones may stand to one 
another. These are the various consonant and other "intervals." 
Physically they are marked by the property that the frequencies 
corresponding to the respective pitches are in a definite 
numerical ratio, which can be expressed by means of two small 
integers. The names of the more important consonant intervals, 
with the respective ratios, are as follows : 

Unison 1 : 1 Octave 1 : 2 

Fifth 2 : 3 Fourth 3 : 4 

Major Third 4 : 5 Minor Sixth 5 : 8 

Minor Third 5 : 6 Major Sixth 3 : 5. 

The ear has of course no appreciation of the numerical 
relations themselves ; but each interval is more or less sharply 
" defined," in the sense that a slight mistuning of either note is 
at once detected by the beats, and consequent sensation of 
roughness, which are produced. The explanation of these latter 
peculiarities must be deferred for the present. 



6 DYNAMICAL THEORY OF SOUND 

The names given to the various intervals are in a sense 
accidental, and refer to the relative positions of the notes on 
the ordinary " diatonic scale." This is based on the " major 
chord," which is a combination of three notes forming a Major 
and a Minor Third ; i.e. their frequencies are as 4 : 5 : 6. If we 
start from any arbitrary note, which we will call C, as keynote, 
the two notes which lie a Fifth above and below it are called 
the " dominant" (G) and the " subdominant " (F,) respectively. 
If we form the major chord from C we get the notes E = f C, 
and G = | C. Again if we form the major chord from G we get 
the notes B = f G = - 1 /- C, and d = f G = f C. The latter falls 
outside the octave beginning with C ; the corresponding note 
within the octave is D = f C. Lastly, forming the major chord 
from F, we get A, = f F = f x f C = f C, the octave of which is 
A = -| C, and C itself. We thus obtain the scale of seven notes 
whose frequencies are proportional to the numbers here given : 
C D E F G A B 

i I I f f S 

24 27 30 32 36 40 45 

This is continued upwards and downwards in octaves ; the same 
letters are repeated as the names of the notes, but the various 
octaves may be distinguished by difference of type, and by 
accents or suffixes. The precise pitch of the key-note is so far 
arbitrary; it determines, and is determined by, that of any 
other note in the scale. Among musicians the standard has 
varied in different places and at different times, the general 
tendency being in the direction of a rise. German physical 
writers, including Helmholtz, have followed a standard which 
assigns to a certain A a frequency of 440*. On this basis we 
have the following frequencies for a certain range of the 
scale : 



- 






















<j 


f^> 


~~Z3 ^~ 


1 




"Z?" 


-& 


cs 
















c 


D 


E 


F 


G 


A 


B c 


d 


e 


/ g 


a b 


66 


74 


82$ 


88 


99 


110 


123| 132 


148$ 


165 


176 198 


220 247$ 



* This makes c' = 264. Physical instrument makers now often take c' = 256, 
which is convenient on account of its continued divisibility by 2. 



INTRODUCTION 



V d' J f tf a' V d' d" e" f" g" a" b" 
264 297 330 352 396 440 495 528 594 660 704 792 880 990 

Underneath the ordinary musical symbols we have placed 
the convenient literal notation employed by German writers. 
This may be continued upwards by means of additional accents 
(c'", c*, ...), and downwards by suffixes (C,, C,,, ...). 

If in the construction of the scale we had used, instead of 
the major, the minor chord, which consists of a Minor and a 
Major Third in ascending order, the frequencies being as 
10 : 12 : 15, we should have required three notes not included 
in the above scheme. And if, starting from any note already 
obtained (other than C) as a new key-note, we proceed to 
construct a major or a minor scale, further additional notes are 
required. In the case of the violin, or of the human voice, or of 
some other wind-instruments which allow of continuous varia- 
tion of pitch, this presents no difficulty. But in instruments 
like the piano or organ the multiplication of fixed notes beyond 
a moderate limit is impracticable. It is found, however, that 
by a slight tampering with the correct numerical relations the 
requirements of most keys can be fairly well met by a system 
of twelve notes in each octave, which are known as 
C C* D D* E F F* G GS A A3 B. 

This process of adjustment, or compromise, is called "tempera- 
ment"; on the usual system of " equal " temperament the 
intervals between the successive notes are made equal, the 
octave being accordingly divided into twelve steps for each of 

which the vibration-ratio is 2 T X Thus the ratio of G to C is 
made to be 2 T *= T4983 instead of 1-5. 



CHAPTEE I 



THEORY OF VIBRATIONS 

4. The Pendulum. 

A vibrating body, such as a string or a bar or a plate, 
cannot give rise to a sound except in so far as it acts on the 
surrounding medium, which in turn exerts a certain reaction 
on the body. The reaction is however in many cases so slight 
that its effects only become sensible after a large number of 
oscillations. Hence, to simplify matters, we begin by ignoring 
it, and investigate the nature of the vibrations of a mechanical 
system considered as completely isolated. 

The theory of vibrations begins, historically and naturally, 
with the pendulum. With this simple apparatus 
we are able to illustrate, in all essentials, many 
important principles of acoustics, the mere differ- 
ences of scale as regards amplitude and period, 
enormous as they are, being unimportant from the 
dynamical point of view. 

A particle of mass M, suspended from a fixed 
point by a light string of length I, is supposed 
to make small oscillations, in a vertical plane, 
about its position of equilibrium. If the inclina- 
tion of the string to the vertical never exceeds 
a few degrees, the vertical displacement of the 
particle may (to a first approximation) be neg- 
lected, and the tension (P) of the string may be 
equated to the gravity Mg of the particle. Since the horizontal 
displacement (x) is affected only by the horizontal component 
of the tension, we have 

M = - P - = - Ma - m 




Fig. 1. 



THEORY OF VIBRATIONS 



If we put n* = gll, (2) 

d?x 

this becomes rf^ +w2a? = ^' ^ 

and the solution is 

x A cos nt + B sin nt, (4) 

where the constants A, B may have any values. That this 
formula really satisfies (3) is verified at once by differentiation ; 
and since it contains two arbitrary constants A, B, we are able 
to adapt it to any prescribed initial conditions of displacement 
and velocity. Thus if, when < = 0, we are to have #=o? , 
dxjdt = u Q , we find 



Un . 

cos nt + sin nt. 
n 



.(5) 



It is of course necessary, in the application to the pendulum, 
that the initial conditions should be such as are consistent with 
the assumed " smallness " of the oscillations. Thus in (5) we 
must suppose that the ratios x /l and u /nl are both small. In 
virtue of (2) the latter ratio is equal to */(u */gl), so that u 
must be small compared with the velocity " due to " half the 
length of the pendulum. 

5. Simple-Harmonic Motion. 
If in 4 (4) we put 

A ~D /"I \ 

as is always possible by a suitable choice of a and e, we get 

The particular type of vibration represented by this formula 

is of fundamental importance. 

It is called a "simple-harmonic," 

or (sometimes) a "simple" 

vibration. Its character is best 

exhibited if we imagine a 

geometrical point Q to describe 

a circle of radius a with the 

constant angular velocity n. 

The orthogonal projection P of 

Q on a fixed diameter AOA' 

will move exactly according to 




10 DYNAMICAL THEORY OF SOUND 

the formula (2), provided it be started at the proper instant. 
The angle nt+e(=AOQ) is called the "phase"; and the 
elements a, e are called the " amplitude " and the " initial 
phase," respectively. The interval Zir/n between two suc- 
cessive transits through the origin in the same direction is 
called the " period." In acoustics, where we have to deal with 
very rapid vibrations, it is usual to specify, instead of the 
period, its reciprocal the " frequency " (N), i.e. the number of 
complete vibrations per second ; thus 



In the case of the pendulum, where n = *J(g/l), the period 
is 2ir^/(l/g). As in the case of all other dynamical systems 
which we shall have occasion to consider, this is independent 
of the amplitude so long as the latter is small (. 

The velocity of P in any position is 

............ (3) 



as appears also by resolving the velocity (na) of Q parallel 
to OA. 




Fig. 3. 

In all cases of rectilinear motion of a point the method of 
graphical representation by means of a curve constructed with 

* The want of a separate name for the angular velocity n in the auxiliary 
circle is sometimes felt. In the theory of the tides the term "speed" was 
introduced by Lord Kelvin. As an alternative term in acoustics the word 
"rapidity" may perhaps be suggested. 

f This observation was made by Galileo in 1583, the pendulum being a 
lamp which hangs in the cathedral of Pisa. 



THE OK Y OF VIBRATIONS 11 

the time t as abscissa and the displacement x as ordinate is 
of great value. This is called the " curve of positions," or the 
" space-time curve." In experimental acoustics numerous 
mechanical and optical devices have been contrived by means 
of which such curves can be obtained. In the present case 
of a simple-harmonic vibration, the formula (2) shews that the 
curve in question is the well-known " curve of sines." 

6. Further Examples. 

The governing feature in the theory of the pendulum is 
that the force acting on the particle is always towards the 
position of equilibrium and (to a sufficient approximation) 
proportional to the displacement therefrom. All cases of 
this kind are covered by the differential equation 



and the oscillation is therefore of the type (2) of 5, with 
n z = K/M. The motion is therefore simple-harmonic, with 
the frequency 



determined solely by the nature of the system, and independent 
of the amplitude. The structure of this formula should be 
noticed, on account of its wide analogies. The frequency 
varies as the square root of the ratio of two quantities, one 
of which (K) measures the elasticity, or the degree of stability, 
of the system, whilst the other is a coefficient of inertia. 

Consider, for example, the vertical oscillations of a n 
mass M hanging from a fixed support by a helical 
spring. In conformity with Hooke's law of elasticity, 
we assume that the force exerted by the spring is 
equal to the increase of length multiplied by a certain 
constant K, which may be called the "stiffness" of 
the particular spring. In the position of equilibrium 
the tension of the spring exactly balances the gravity 
Mg\ and if M be displaced downwards through a 
space x, an additional force Kx towards this position 
is called into play, so that the equation of motion is of 



12 DYNAMICAL THEORY OF SOUND 

the type (1). The inertia of the spring itself is here 
neglected*. 

Again, suppose we have a mass M attached to a wire which 
is tightly stretched between 
two fixed points with a ten- 
sion P. We neglect gravity 
and the inertia of the wire Flg> 5 ' 

itself; and we further assume the lateral displacement (x) to 
be so small that the change in tension is a negligible fraction 
of P. If a, b denote the distances of the attached particle 
from the two ends, we have 




which is of the same form as 4 (3), with n? = P (a + b)/Mab. 
The frequency is therefore 



ab 

This case is of interest because acoustical frequencies can 
easily be realized. Thus if the tension be 10 kilogrammes, 
and a mass of 5 grammes be attached at the middle, the 
wire being 50 cm. long, we find N = 63. 

7. Dynamics of a System with One Degree of Freedom. 
Free Oscillations. 

The above examples are all concerned with the rectilinear 
motion of a particle, but exactly the same type of vibration 
is met with in every case of a dynamical system of one degree 
of freedom oscillating freely, through a small range, about 
a configuration of stable equilibrium. 

A system is said to have "one degree of freedom" when 
the various configurations which it can assume can all be 
specified by assigning the proper values .to a single variable 
element or "coordinate." Thus, the position of a cylinder 
(of any form of section) rolling on a horizontal plane is defined 
by the angle through which it has turned from some standard 
position. A system of two particles attached at different points 
of a string whose ends A, B are fixed has one degree of freedom 

* A correction on this account is investigated in 7. 




THEORY OF VIBRATIONS 13 

if it be restricted to displacements in the vertical plane through 
A, B, for the configuration may be specified by the inclination 
of any one of the strings to the horizontal. Again, the con- 
figuration of a steam-engine and of the whole train of machinery 
which it actuates is defined by 
the angular coordinate of the 
flywheel. The variety of such 
systems is endless, but if we 
exclude frictional or other dis- 
sipative forces the whole motion 
of the system when started ^7 

anyhow and left to itself is 
governed by the equation of energy. And in the case of 
small oscillations about stable equilibrium, the differential 
equation of motion, as we shall see, reduces always to the 
type 6 (1). 

We denote by q the variable coordinate which specifies 
the configuration. As in the case of Fig. 6, this may be 
chosen in various ways, but the particular choice made is 
immaterial. From the definition of the system it is plain 
that each particle is restricted to a certain path. If in 
consequence of an infinitesimal variation Bq of the coordinate 
a particle ra describes an element Ss of its path, we have 
8s = a$q, where a is a coefficient which is in general different 
for different particles, and also depends on the particular 
configuration q from which the variation is made. Hence, 
dividing by the time-element St, the velocity of this particle 
is v = adq/dt, or in the fluxional notation *, v = aq. 

Hence the total kinetic energy, usually denoted by T, is 

T=&(m*) = la#, (1) 

where a = 2 (ma 2 ), (2) 

the summation X embracing all the particles of the system. 

The coefficient a is in general a function of q ; it may be 
called the " coefficient of inertia " for the particular configura- 
tion q. For example, in the case of the rolling cylinder referred 

* The use of dots to denote differentiations with respect to t was revived by 
Lagrange in the Mecanique Analytique (1788), and again in later times by 
Thomson and Tait. We write q for dqjdt and q for 



14 DYNAMICAL THEORY OF SOUND 

to above, it is the (usually variable) moment of inertia about 
the line of contact with the horizontal plane, provided q 
denote the angular coordinate. 

The potential energy of the system, since it depends on the 
configuration, will be a function of q only. If we denote it 
by V, the conservation of energy gives 

%aq*+ F=const., ..................... (3) 

provided the system be free from extraneous forces. The 
value of the constant is of course determined by the initial 
circumstances. If we differentiate (3) with respect to t, the 
resulting equation is divisible by q, and we obtain 



which may be regarded as the equation of free motion of the 
system, with the unknown reactions between its parts elim- 
inated. In the application to small oscillations it greatly 
simplifies. 

In order that there may be equilibrium the equation (4) 
must be satisfied by q = const. This requires that d V/dq = ; 
i.e. an equilibrium configuration is characterised by the fact 
that the potential energy is "stationary" in value for small 
deviations from it. By adding or subtracting a constant, we 
can choose q so as to vanish in the equilibrium configuration 
which is under consideration, whence, expanding in powers of 
the small quantity q, we have 

F= const. + c 2 +..., .................. (5) 

the first power of q being absent on account of the stationary 
property. The constant c is positive if the equilibrium con- 
figuration be stable, and V accordingly then a minimum*. It 
may be called the " coefficient of stability." 

If we substitute from (5) in (4), and omit terms of the 
second order in q, q, we obtain 

aq + cq = Q, ........................ (6) 

where a may now be supposed to be constant, and to have the 
value corresponding to the equilibrium configuration. 

* In the opposite case the solution of (6) below would involve real exponen- 
tials instead of circular functions, indicating instability. 



THEORY OF VIBRATIONS 15 

Since (6) is of the same type as 6 (1), with 

n* = c/a, (7) 

the variation of q is simple-harmonic, say 

q = C cos (nt + e), (8) 

with the frequency 

AT 



Moreover, since the displacement of any particle of the 
system along its path, from its equilibrium position, is pro- 
portional to q (being equal to aq in the above notation), we see 
that each particle will execute a simple-harmonic vibration of 
the above frequency, and that the different particles will keep 
step with one another, passing through their mean positions 
simultaneously. The amplitudes of the respective particles are 
moreover in fixed ratios to one another, the absolute amplitude, 
and the phase, being alone arbitrary, i.e. dependent on the 
particular initial conditions. 

The kinetic and potential energies are respectively 

T = i of = \ n*aC* sin 2 (nt + e), ) 
V= I c(? = IcC 2 cos 2 (nt + e), j 



the sum being 

T+ V=\n\iV* = \cG\ ............... (11) 



in virtue of (7). Since the mean values of sin 2 (w + e) and 
cos 2 (nt -he) are obviously equal, and therefore each =J, the 
energy is on the average half kinetic and half potential. 

The application of the theory to particular cases requires 
only the calculation of the coefficients a and c, the latter being 
(in mechanical problems) usually the more troublesome. In 
the case of a body attached to a vertical wire, and making 
torsional oscillations about the axis of the wire, a is the moment 
of inertia about this axis, and c is the modulus of torsion, 
i.e. cq is the torsional couple when the body is turned through 
an angle q. 

Again in the case of a mass suspended by a coiled spring 
(Fig. 4), if we assume that the vertical displacement of any 
point of the spring is proportional to its depth z below the 



16 DYNAMICAL THEOBY OF SOUND 

point of suspension in the unstrained state, the kinetic energy 
is given by 



..................... (12) 

if p be the line-density, I the unstretched length, and q the 
displacement of the weight. The inertia of the spring can 
therefore be allowed for by imagining the suspended mass to be 
increased by one-third that of the spring. 

8. Forced Oscillations of a Pendulum. 

The vibrations so far considered are " free," i.e. the system 
is supposed subject to no forces except those incidental to its 
constitution and its relation to the environment. We have 
now to examine the effect of disturbing forces, and in particular 
that of a force which is a simple-harmonic function of the time. 
This kind of case arises when one vibrating body acts on 
another under such conditions that the reaction on the first 
body may be neglected. 

For definiteness we take the case of a mass movable in a 
straight line, the subsequent generalization ( 9) being a very 
simple matter. The equation (1) of 6 is now replaced by 



(1) 



the last term representing the disturbing force, whose amplitude 
F, and frequency p/Zir, are regarded as given*. If we write 

f, .................. (2) 



we have -j+n?x=fcospt ................... (3) 

The complete solution of this equation is 

x = A cos nt + B sin nt 4- -~ - cos pt t ...... (4) 

7i 2 p z 

as is easily verified by differentiation. 

The first part of this, with its arbitrary constants A } B, 
represents a free vibration of the character explained in 5, 

* The slightly more general case where the force is represented by F cos (pt + a) 
can be allowed for by changing the origin from which t is reckoned. 



THEORY OF VIBRATIONS 



17 



with the frequency H/^TT proper to the system. On this is 
superposed a " forced vibration " represented by the last term. 
This is of simple-harmonic type, with the frequency p/2?r of the 
disturbing force, and the phase is the same as that of the force, 
or the opposite, according as p $ n, i.e. according as the imposed 
frequency is less or greater than the natural frequency. 





The above theory is easily illustrated by means of the 
pendulum. If the upper end of the string, instead of being 
fixed, is made to execute a horizontal motion in which the 
displacement at time t is (Fig. 7), the equation of motion (1) 
of 4 is replaced by 



.(5) 



or 



.(6) 



This is the same as if the upper end were fixed, and the bob 
were subject to a horizontal force whose accelerative effect is 
?i 2 f . If as a particular case we take 

acospt, ........................ (7) 

The annexed Fig. 8 repre- 
sents the forced oscillation in the two cases of p < n and p > n y 
respectively. The pendulum oscillates as if C were a fixed 
L. 2 



we get the form (3), with /= ?i 2 ct. 



18 DYNAMICAL THEOKY OF SOUND 

point, the distance CP being equal to the length of the simple 
pendulum whose free period is equal to the imposed period 



This example is due to Young*, who applied it to illustrate 
the dynamical theory of the tides, where the same question of 
phase arises. It appears from this theory that the tides in an 
imagined equatorial belt of ocean, of a breadth not exceeding 
a few degrees of latitude, and of any depth comparable with 
the actual depth of the sea, would be "inverted," i.e. there 
would be low water beneath the moon, and high water in 
longitudes 90 E. and W. from it, the reason being that the 
period of the disturbing force (viz. 12 lunar hours) is less than 
the corresponding free period, so that there is opposition of 
phase. 

The arbitrary constants in the complete solution (4) are 
determined by the initial conditions. Suppose, for example, 
that the body starts from rest in the zero position at the instant 
t = 0. We find 

x -j4 a (cos nt cos pt), ......... . ..... (8) 

as may be immediately verified. 

When the imposed frequency p/2?r is nearly equal to the 
natural period, the last term in (4) becomes very large, and it 
may be that the assumption as to the smallness of x on which 
the equation (1) is usually based (as in the case of the pendulum) 
is thereby violated. The result expressed by (4) is then not to 
be accepted without reserve, but we have at all events an indica- 
tion of the reason why an amplitude of abnormal amount ensues 
whenever there is approximate agreement between the free and 
the forced period. 

In the case (p = ri) of exact coincidence between the two 
periods, the solution (4) becomes altogether unmeaning, but an 
intelligible result may be obtained if we examine any particular 

* Dr Thomas Young (1773 1829), famous for his researches on light, and 
other branches of physics. The elementary theory of free and forced oscilla- 
tions was given by him in an article on " A Theory of the Tides, including the 
consideration of Resistance," Nicholsons Journal, 1813 ; Miscellaneous Works, 
London, 1855, vol. n., p. 262. 



THEORY OF VIBRATIONS 19 

case in which the initial conditions are definite. Thus, in the 
case of (8), the formula may be written 
/ sin \ ( p n) t . 



and as p approaches equality with n this tends to the limiting 
form 



(10) 



This may be described (roughly) as a simple vibration 
whose amplitude increases proportionally to t. For a reason 
just indicated this is only valid as a representation of the earlier 
stages of the motion. 

The case of a disturbing force of more general character 
may be briefly noticed. The differential equation is then of 
the form 



+ *=/() ................... (11) 

The method of solution, by variation of parameters, or 
otherwise, is explained in books on differential equations. The 
result, which may easily be verified, is 

x = - sin nt I f(t) cos nt dt cos nt I f(t) sin nt dt. (12) 

It is unnecessary to add explicitly terms of the type 
A cos nt + B sin nt, which express the free vibrations, since 
these are already present in virtue of the arbitrary constants 
implied in the indefinite integrals. 

If the force f(t) is only sensible for a certain finite range of 
t, and if the particle be originally at rest in the position of 
equilibrium, we may write 

x = - sin nt I f(t) cos nt dt -- cos nt I f(t) sin nt dt, (13) 
n J -ao n J _QO 

since this makes x 0, dx/dt = for t = - oo . The vibra- 
tion which remains after the force has ceased to be sensible is 
accordingly 

x = A cos nt + B sin nt, ............... (14) 



where 



=-- 1 f(t)smntdt, B = -T f(i)cosntdt. (15) 
nj -oo nj _aj 4 



22 



20 DYNAMICAL THE OB Y OF SOUND 

For example, let 

* ..................... < 16 > 



this represents a force which is sensible for a greater or less 
interval on both sides of the instant t = 0, according to the 
value of r, the integral amount or impulse being //,*. By 
making r sufficiently small we can approximate as closely as 
we please to the case of an instantaneous impulse. Since 

cosntdt_7r r smntdt_ m 

~ ~ 



LL6 ** T 

we have x= - sin nt ................... (18) 

The exponential factor shews the effect of spreading out 
the impulse. This effect is greater, the greater the frequency 
of the natural vibration. 

9. Forced Oscillations in any System with One Degree 
of Freedom. Selective Resonance. 

The generalization of these results offers no difficulty. When 
given extraneous forces act on a system with one degree of 
freedom, whose coordinate is q, the work which they perform in 
an infinitely small change of configuration, being proportional to 
8q, may be denoted by QSq. The quantity Q is called the 
"force" acting on the system, "referred to the coordinate q." 
For instance, if q be the angular coordinate of a body which can 
rotate about a fixed axis, Q is the moment of the extraneous 
forces about this axis. 

It follows that in any actual motion of the system the rate 
at which extraneous forces are doing work is Qq. The equation 
of energy now takes the form 

j t (T+V)=Qq, ..................... (1) 

whence, inserting the value of T from 7 (1), we have 

' 



* The graph of this function is given, for another purpose, in Fig. 14, p. 33. 
t The former of these integrals is evaluated in most books on the Integral 
Calculus. 



THEORY OF VIBRATIONS 21 

When dealing with small motions in the neighbourhood of 
a configuration of equilibrium we may neglect terms of the 
second order as before. Hence, substituting the value of V 
from 7 (5), we find 

aq+cq = Q ......................... (3) 

When Q is of simple-harmonic type, varying (say) as cos pt, 
the forced oscillation is given by 



which is of course merely a generalized form of the last term in 

8 W- 

Two special cases may be noticed. When p is very small, 
(4) reduces to q = Q/c. This may be described as the "equili- 
brium" value* of the displacement, viz. it is the statical 
displacement which would be maintained by a constant force 
equal to the instantaneous value of Q. In other words, it is 
the displacement which would be produced if the system were 
devoid of inertia (a = 0). Denoting this equilibrium value by 
q, we may write (4) in the form 



where, as in 7, n denotes the speed of a free vibration. 

When, on the other hand, p is very great compared with n, 
(4) reduces to 

q = -Q/p*a, ..................... (6) 

approximately. This is almost the same as if the system were 
devoid of potential energy, the inertia alone having any sensible 
influence. 

When two or more disturbing forces of simple-harmonic 
type act on a system, the forced vibrations due to them may be 
superposed by mere addition. Thus a disturbing force 

Q =/ cos ( Pl t + d) +/ 2 cos (pj, + oj + ...... (7) 

will produce the forced oscillation 

a 2 )H-.... (8) 



* The name is taken from the theory of the tides, where the equilibrium 
tide-height is denned as that which would be maintained by the disturbing 
forces if these were to remain permanently at their instantaneous values. 



22 DYNAMICAL THEOKY OF SOUND 

It will be observed that the amplitudes of the various terms 
are not proportional to those of the corresponding terms in the 
value of Q, owing to the difference in the denominators. 

This is an illustration of a remark made in 1 that the 
simple-harmonic type is the only one which is unaltered in 
character when it is transmitted, the character of the composite 
vibration represented by (8) being different from that of the 
generating force. In particular if one of the imposed speeds 
p lt p z , ... be nearly coincident with the natural speed n, the 
corresponding element in the forced vibration may greatly 
predominate over the rest. This is the theory of selective 
"resonance," so far as it is possible to develop it without 
reference to dissipative forces. 

10. Superposition of Simple Vibrations. 

The superposition of simple-harmonic motions in the same 
straight line has many important applications. For instance, 
the height of the tide at any station is the algebraic sum of a 
number of simple-harmonic com- 
ponents, the most considerable 
(at many stations) being those 
whose periods are half a lunar 
and half a solar day, respectively. 

The composition of two 
simple vibrations may be illus- 
trated by the geometrical 
method of Fig. 2. If two 
points Q lt Q 2 describe concentric 
circles with the angular velo- 
cities TH, n^ their projections 

on a fixed diameter will execute simple-harmonic vibrations 
of the forms 

#! = aj cos (nj + e^, # 2 = a 2 cos (n + e 2 ), (1) 

where Oj , a 2 are the radii of the two circles, and ej , e 2 are the 
initial inclinations of the radii OQi, OQ 2 to the axis of x. The 
result of the superposition is 

(2) 




THEORY OF VIBRATIONS 23 

and it appears that the value of x is the projection of OR, the 
diagonal of the parallelogram determined by OQ l} OQ 2 . 

If Tij = 7i 2 , the two component vibrations have the same period, 
the angle QiOQ 2 is constant, and the resultant vibration is 
simple-harmonic of the same period. 

But if Wj, 7*2 are unequal, the angle QiOQ 2 will vary between 
and 180, and OR will oscillate between the values c^ a 2 . 
In Lord Kelvin's "tidal clock," the "hands" OQ l} OQ 2 revolve 
in half a lunar and half a solar day, respectively, and the sides 
QtR, Q 2 R of the parallelogram are formed of rods jointed to 
these and to one another. The projection of R then indicates 
the tide-height due to the superposition of the lunar and solar 
semidiurnal tides. 

If the periods Sir/r^, 27r/n z are very nearly though not 
exactly equal, the angle QiOQ 2 will vary very little in the course 
of a single revolution of OQ l or OQ 2 , and the resultant vibration 
may be described, in general terms, as a simple vibration whose 
amplitude fluctuates between the limits c^ a z . The period 
of a fluctuation is the interval in which one arm OQ l gains four 
right angles on the other, or STT/^ n 2 ). Inverting, we see 
that the frequency of the fluctuations is the difference of the 
frequencies of the two constituent vibrations. We have here 
the reason for the alternation of " spring " and " neap " tides, 
according as the phases of the lunar and solar semidiurnal 
tides agree or are opposed. In acoustics we have the important 
phenomenon of " beats " between two tones of slightly different 
pitch. The contrast between the maximum and minimum 
amplitudes is of course greatest when the amplitudes a lf a^ of 




Fig. 10. 



the primary vibrations are equal. We then have 
x = o^cos (n^t + ej + a? cos (nj -f e 2 ) 

= 2acos {(7*! - n^) t + (e l - e 2 )} cos {J (n^ + w 2 ) t + J (e l -I- e 2 )}. (3) 
This may be described, in the same general manner as before, 



24 DYNAMICAL THEORY OF SOUND 

as a simple vibration whose period is 27T/-J- (% + w 2 ), and whose 
amplitude oscillates between the limits and 2a, in the time 
7T/-| (n^ n z ). This is illustrated graphically, with x as ordinate 
and t as abscissa, in Fig. 10, for the case of n^ : n z = 41 : 39. 

11. Free Oscillations with Friction. 

The conception of a dynamical system as perfectly isolated 
and free from dissipative forces, which was adapted provisionally 
in 4 10, is of course an ideal one. In practice the energy of 
free vibrations is gradually used up, or rather converted into 
other forms, although in most cases of acoustical interest the 
process is a comparatively slow one, in the sense that the 
fraction of the energy which is dissipated in the course of a 
single period is very minute. 

To represent the effects of dissipation, whether this be due 
to causes internal to the system, or to the communication of 
energy to a surrounding medium, we introduce forces of resist- 
ance which are proportional to velocity. The forces in question 
are by hypothesis functions of the velocity*, and when the 
motion is small, the first power only need be regarded. 

The equation of free motion of a particle about a position of 
equilibrium thus becomes 

,, d z x rr dx 

M M = - Kx - R di< .................. < J > 

where R is the coefficient of resistance. If we write 

k, .................. (2) 



weget 



The solution of this equation may be made to depend on 
that of 4 (3) by the following artifice f. We put 



//IX 
(4) 



* We shall see at a later stage (Chap. VIII) that the resistance of a medium 
may introduce additional forces depending on the acceleration. These have 
the effect of a slight apparent increase of inertia, and contribute nothing to 
the dissipation. It is unnecessary to take explicit account of them at present. 

t Another method of solution is given in 20. 



THEORY OF VIBRATIONS 25 

and obtain, on substitution, 



We have now three cases to distinguish. If the friction be 
relatively small, more precisely if k < 2n, we may put 

r^ = 7i 2 -J#, ..................... (6) 

and the solution of (3) is 

y = A cos n't + B sin n't, ............... (7) 

whence x = e " **' (A cos n't + B sin n't) ............. (8) 

Changing the arbitrary constants, and putting 

r = 2/&, ........................... (9) 

we have x = ae~^ T cos(n' + e) ................ (10) 

This may be described as a modified simple-harmonic vibration 

in which the amplitude (ae ~ ^ r ) sinks asymptotically to as t 
increases. The time T in which the amplitude is diminished 
in the ratio l/e is called the " modulus of decay." The 
relation between x and t is exhibited graphically in Fig. 11, 




Fig. 11. 

where the dotted lines represent portions of the exponential 

curves x = ae ~ . For the sake of clearness the rapidity 
of decay is here taken to be much greater than it would be in 
any ordinary acoustical example. 



DYNAMICAL THEOEY OF SOUND 



We have seen that a true simple-harmonic vibration may 
be regarded as the orthogonal projection of uniform motion in 
a circle. An analogous representation of the modified type (10) 
is obtained if we replace the circle by an equiangular spiral 
described with constant angular velocity ri about the pole 0, in 
the direction in which the radius vector r decreases*. The 
formula (10) is in fact equivalent to # = rcos#, provided 

r = ae~ il \ 6 = n't + (11) 

Eliminating t we have 

r=ae~ f "> (12) 

where = (n'r)~ l , a. ae^ . This is the polar equation of the 
spiral in question. The curve in Fig. 12 corresponds in scale 
with Fig. 11. 

In most acoustical applications the fraction k/2n, or 1/nr, is 
a very small quantity. 
In this case, the dif- 
ference between n and 
ri is a small quantity 
of the second order, 
and may usually be ig- 
nored ; in other words, 
the effect of friction on 
the period is insensible. 
It may be noted that 
the quantityl/nr, whose 
square is neglected, is 
the ratio of the period 
27T/71 to the time 2-7TT 
in which the amplitude is diminished in the ratio e~ * or -^. 

If k be greater than 2n the form of the solution of (3) is 
altered, viz. we have 

*, (13) 




Fig. 12. 



y 



whence 
if 



Ae 



-at 



(14) 



* This theorem was given in 1867 by P. G. Tait (18311901), Professor of 
Natural Philosophy at Edinburgh (18601901). 



THEORY OF VIBRATIONS 27 

The particle comes asymptotically to rest but does not oscillate; 
in fact we may easily see that it passes once at most through 
its zero position. This type of motion is realized in the case of 
a pendulum swinging in a very viscous liquid, and in "dead-beat" 
galvanometers and other electrical instruments, but it is ' of 
little interest in acoustics. 

If k = 2n, exactly, the solution of (3) is of the form 

x = (A +Bt)e~ nt , .................. (16) 

as to which similar remarks may be made. 

12. General Dissipative System with One Degree of 
Freedom. Effect of Periodic Disturbing Forces. 

The effect of dissipation on the free motion of any system 
having one degree of freedom is allowed for by the assumption 
that there is a loss of mechanical energy at a rate proportional 
to the square of the generalized velocity, so that in the notation 
of 7 

J <#) = -&?, ............... (1) 



whence aq + bq + cq = Q ................... (2) 

This is of course the same as if we had introduced a frictional 
force Q = - bq in 9 (3). 

The equation (1) has the same form as 11 (3), and the 
results will correspond if we put 

n 2 = c/a, r = 2a/6 ................... (3) 

When the dissipation is small, the rate of decay of the 
amplitude can be estimated by an independent method, due to 
Stokes*, which we shall often find useful. The period being 
practically unaffected by vicosity, a considerable number of 
oscillations can be fairly represented by 

q = G cos (nt + e), ..................... (4) 

provided C and e be gradually changed so as to fit the altering 
circumstances. The average energy over such an interval will 
be Jrc 2 a(7 2 , approximately, by 7 (11); and the rate of dissipa- 
tion will be 

bcf = iw 2 60 2 (1 - cos 2 (nt + e)}, 

* Sir George Gabriel Stokes (18191903), Lucasian Professor of Mathematics 
at Cambridge (18491903). 



28 DYNAMICAL THEOEY OF SOUND 

the mean value of which is ^ri*bC\ Equating the mean rate of 
decay of the energy to the mean dissipation, we get 



-ifrtC', .................. (5) 






< 6 > 



or tf=<7 <T t/T , ........................ (7) 

if T = 2a/6, as in (3). 

When there are given extraneous forces in addition to the 
dissipative influences, the equation of energy takes the form 

& ............ (8) 



whence aq + bq + cq = Q ...................... (9) 

As in 9 we consider specially the case of a disturbing 
force of simple-harmonic type, say 

Q = Ccospt ...................... (10) 

A particular solution of (9) is then obtained in the form 

q = Fcospt + Gsinpt, ............... (11) 

provided the constants F, G are properly chosen. The necessary 
conditions are found on substitution to be 



-- . 

If we put 



Rcosa, pb=Rsina, ......... (13) 

we find F=cosa, = sina, ............... (14) 



c 

whence* q = -^ cos (pt a) (15) 

The values of R and a are determined by 

/nTi 

(16) 



R is to be taken positively, and a may be assumed to lie 
between and TT. 

* A more rapid way of obtaining this solution is explained in 20. 



THEORY OF VIBRATIONS 29 

The equation (9) is still satisfied if we add to (15) terms 
representing a free oscillation; and these added terms are 
necessary in order to constitute a complete solution capable of 
adjustment to arbitrary initial conditions. The free vibration 
dies out, however, asymptotically, so that after the lapse of a 
sufficient time the forced vibration (15) is alone sensible. 

The circumstances which affect the amplitude and phase of 
this forced vibration require careful attention. The amplitude 
is a maximum when R? is least, i.e. when 



(-!-. a) 

and the maximum amplitude is accordingly 



In most cases of interest 6 2 /oc is a small quantity of the 
second order; the maximum is then C/nb, and occurs when 
p = n, very approximately. 

Again, it appears from (15) and (16) that the phase of q lags 
behind that of the disturbing force by an angle a, which lies 
between and JTT, or between TT and TT, according as p 2 is less 
or greater than c/a, i.e. according as the imposed frequency 
is less or greater than the natural frequency. If, keeping p 
constant, we diminish the dissipation-coefficient b, a. tends to 
the limit or TT, respectively, in accordance with 8, where we 
found exact agreement or opposition of phase in the absence 
of resistance. But if, keeping b constant, we make p approach 
the value n ( = \/(c/a)) which determines the frequency in the 
absence of dissipation, a tends to the limit ^TT, and the phases 
of q and Q differ by an amount corresponding to a quarter-period. 
This means that the maxima of the disturbing force are now 
synchronous with the maxima of the velocity q. 

Some light is thrown on these relations if we examine 
the case of a pendulum whose bob receives equal positive 
and negative instantaneous impulses alternately at regular 
intervals. It is seen at once from Fig. 13 that an impulse in 
the direction of motion accelerates or retards the phase of an 
otherwise free vibration, according as it precedes or follows 



30 



DYNAMICAL THEORY OF SOUND 



(within the limits of a quarter-period) the instant of maximum 
velocity. Thus if when the particle is at P, on its way to 0, 
the velocity be increased in the ratio of PQ to PQi, the phase 
is accelerated by the angle QOQ l} whilst a similar impulse at P' 
would retard the phase by the angle Q'OQ\. 

In order that no effect may be produced on the phase it 
is necessary that the impulse be delivered at the instant of 
passing through 0. If we imagine that a small assisting 
impulse is given at every such passage, as in the case of the 
ordinary clock escapement, we have an illustration of the 
circumstances of maxi- 
The 



mum resonance, 
period of the disturbing 
force is exactly equal to 
the natural period, and 
the force synchronizes 
with the velocity. The 
amplitude is deter- 
mined by the considera- 
tion that the work done 
by the impulses must 
balance that lost by 
friction. The result is 
not essentially different 
if the impulse be dif- 




Fig. 13. 



fused symmetrically about 0, as in the case of a simple- 
harmonic force, since the acceleration of phase on one side of 
is cancelled by the retardation on the other. 

Next suppose that the assisting impulses are given 
each time the bob passes the symmetrically situated points 
P, P' inwards. There is an acceleration of phase at each 
impulse, and the period is shortened. This illustrates the case 
of a disturbing force whose period is less than the natural 
period, and whose maxima and minima precede the maxima and 
minima of the velocity. If on the other hand the impulses are 
given as the bob passes the points P and P' outwards, there is 
a repeated retardation of phase, and the period is lengthened. 
This corresponds to the case of a disturbing force whose period 



THEORY OF V1BKATIONS 31 

is greater than the natural period ; the maxima and minima 
of the force now follow those of the velocity. The reader is 
recommended to follow out in detail the argument here sketched, 
and to examine the effect of substituting a continuous simple- 
harmonic force for the series of disconnected impulses. An 
explanation may also be found, on the same principles, of the 
fact that a small frictional force varying as the velocity has no 
sensible effect on the free period. 

We return to the analytical discussion. A difference of 
phase between the force and the displacement is essential in 
order that the disturbing force may supply energy to compensate 
that which is continually being lost by dissipation. When, as 
in 9, there is complete agreement (or opposition) of phase 
between q and Q, the force is, in astronomical phrase, "in 
quadrature " with the velocity q, that is, the phases differ by \ir, 
and the total work done in a complete period is zero. Under 
the present circumstances the disturbing force is at any instant 
doing work at the rate 

Qq = -^D~ s i n (pt ~ cos^tf 

= g {sin a -sin (2^- a) j, ............ (19) 

the mean value of which is 



(20) 



The same expression is of course obtained as the mean value 
of bq 2 , since the energy supplied by the disturbing force must 
exactly compensate, on the average, that which is continually 
being lost by dissipation, the mean energy stored in the system 
being constant. 

It follows from (16) and (20) that the dissipation is greatest 
when OL^TT, or p = n, i.e. when the imposed frequency coincides 
with that of the free vibration in the absence of resistance. 
The maximum value is %C*/b, being greater, of course, the 
smaller the value of b. 



32 DYNAMICAL THEORY OF SOUND 

13. Effect of Damping on Resonance. 

The abnormal amplitude and dissipation which ensue 
whenever the imposed period is equal, or nearly equal, to the 
natural period constitute the phenomenon of "resonance," 
already referred to in 8, of which we shall have many 
acoustical examples in the sequel. It may be illustrated 
mechanically by giving a slight to-and-fro motion of suitable 
period to the point of suspension of a simple pendulum, or 
better by means of a double pendulum ( 14), i.e. an arrange- 
ment in which two weights are attached at different points to 
a string hanging vertically from a fixed point. If the upper 
weight (M ) be considerable, whilst the lower one (ra) is relatively 
small, M will swing almost exactly like the bob of a simple 
pendulum, the reaction of ra being slight. Under these 
conditions the motion of ra is practically that of a pendulum 
whose point of suspension has an imposed simple-harmonic 
vibration ( 8), and if the length of the lower portion of the 
string be properly adjusted, a violent motion of ra may ensue. 

One very important point remains to be mentioned. As the 
interval p/n between the forced and the natural frequencies 
diverges from unity (on either side), the dissipation falls off 
from its maximum the more rapidly, the smaller the value of 
the frictional coefficient b. In other words, the greater the 
intensity of the resonance in the case of exact coincidence of 
frequencies, the narrower the range over which it is approxi- 
mately equal to the maximum. For example, a tuning fork, even 
when mounted on a "resonance box," requires very perfect tuning 
in order that it may be excited perceptibly by the vibrations of 
another fork in the neighbourhood, whereas the column of air 
in a nearly closed vessel (e.g. a bottle or an organ pipe) will 
respond vigorously to a much wider range of frequencies. To 
elucidate the point, we notice that the expression (20) of 12 
for the dissipation may be written 



,, 
26 S1 



where = %nb/c = I/WT, ........ . ............ (2) 



THEORY OF VIBRATIONS 33 

in the notation of 12 (3). The second factor has its maximum 
value 1/fi when p = n, and evidently diminishes more rapidly, 
as p/n deviates from unity, the smaller the value of 0. The 
question may be conveniently illustrated graphically by con- 
structing a curve which shall shew the dissipation corresponding 
to different frequencies. As regards the abscissa, it would in 
strictness be most proper to take, not the ratio p/n, but its 
logarithm, since equal intervals (in the musical sense) then 
correspond to equal lengths of the axis of x. We might 
therefore write 



,(3) 



but when, as usually happens, the sensible resonance is confined 
to a small range of p/n, we may use the simpler formulae 

ft 



Hr* 



1 



.(4) 



The curve represented by the latter equation is symmetrical 
about the axis of y, and approaches the axis of x asymptotically 
as x increases. It is evident that if $ be increased in any 




ratio, the new curve is obtained by increasing all the abscissae in 

that ratio, and diminishing the ordinates in the inverse ratio, 

the area (TT) included between the curve and the axis of x being 

L. 3 



34 DYNAMICAL THEORY OF SOUND 

unaltered. The intensity sinks to one-half its maximum when 
a? = fp, or 

-ll (5) 

n nr 

Thus if the damping be such that a free vibration would have 
its amplitude diminished in the ratio l/e in 10, 100, 1000 
periods*, respectively, the corresponding values of the interval 
p/n at which the dissipation would be reduced to one-half the 
maximum would be 1 '016, 1 '0016, 1 '00016. The curve 
in (4) is shewn in Fig. 14. 

The above argument deals with the dissipation, which is the 
most important feature. The consideration of the square of the 
amplitude, or of the energy stored in the system, leads to very 
similar results, especially when the damping is slight. 

14. Systems of Multiple Freedom. Examples. The 
Double Pendulum. 

We approach the consideration of systems having any finite 
number of degrees of freedom. A system is said to have ra 
such degrees when m independent variables, or " coordinates," 
are required and are sufficient to specify the various configura- 
tions which it can assume. The notion, first brought into 
formal prominence by Lord Kelvin f, has a wide application 
in mechanism and in theoretical mechanics. In the case of 
the telescope of an altazimuth instrument or of an equatorial 
we have m = 2; in the gyroscope, or (more generally) in any 
case of a rigid body free to turn about a fixed point, m = 3 ; 
for a rigid structure or frame movable in two dimensions 
m = 3; for a rigid structure freely movable in space m = 6. 
The choice of the coordinates in any particular case can be 
made in an endless variety of ways, but the number is always 
determinate. Thus in technical mechanics we have the pro- 
position that a rigid frame movable in one plane can be fixed by 

* In an experiment by Lord Eayleigh, the number of periods for a particular 
tuning fork of 256 v.s. was about 5900. When a resonator was used the number 
fell to 3300. Theory of Sound, vol. n., p. 436. 

t William Thomson, afterwards Lord Kelvin (18241907), Professor of 
Natural Philosophy at Glasgow 184699. The matter is explained in Thomson 
and Tait's Natural Philosophy, 2nd ed., 195201 (1879). 



THEORY OF VIBRATIONS 35 

means of three links connecting any three points of it to any 
three fixed points in the plane*. Similarly any rigid three- 
dimensional structure can be anchored firmly by six links 
connecting six points of it with six points fixed relatively 
to the earth. 

Proceeding to the vibrations of a multiple system about 
a configuration of equilibrium, we begin as before with the 
examination of a few particular cases. 

Take first the oscillations of a particle in a smooth bowl of 
any continuous shape. By means of suitable constraints, the 
particle may be restricted to oscillate in any given vertical 
plane through the lowest point 0, e.g. by confining it between 
two frictionless guides infinitely close to one another. In 
general there will be a lateral pressure on one or other of these 
guides, which will however vanish if the plane in question 
passes through either of the principal directions of curvature 
at 0. Hence two modes of free simple-harmonic vibration, in 
perpendicular directions, are possible, with speeds 

nx-Vto/A), n, = ^(g/R,\ (1) 

where B^, R 2 , are the radii of curvature of the principal sections 
at 0. On account of the assumed smallness of the motion, 
these vibrations may be superposed. The result is, if x, y be 
horizontal rectangular coordinates through 0, 



x = A l cos n t t + A 2 sin rz^,] 
y = Bj cos n^t + B 2 sin nj.) 



Since this contains four arbitrary constants, we can adjust 
the solution to given initial values of x, y, x, y. 

This case is very neatly illustrated by Blackburn's pen- 
dulum ( (Fig. 15). A weight hangs by a string CP from a point 
C of a string A CB whose ends A , B are fixed. The strings being 
supposed destitute of inertia, the point P will always be in the 
same plane with A, B, C. Under this condition the locus of 
P is the ring-shaped surface generated by revolving a circle 

* Provided the directions of the three links be not concurrent (or parallel). 
There is a proviso of a more complex character in the case which follows ; but 
such details need not occupy us here. 

t H. Blackburn, Professor of Mathematics at Glasgow 184979. 

32 



36 



DYNAMICAL THEORY OF SOUND 



with centre C and radius CP, in the plane ACB, about AB as 
an axis; and the principal radii of curvature at the lowest point 
are E l - CO, R z = EO, where E is the point of A B vertically 
above 0. The corresponding directions of vibration are re- 
spectively in and perpendicular to the plane ABO. 




Fig. 16. 



PC 

Fig. 15. 




Fig. 17. 



Another very simple case is that of two equal particles M 
attached symmetrically at distances a from the ends of a tense 
string, whose total length is, say, 2 (a + 6), so that 26 denotes 
the length of the central portion. One obvious mode of 
simple-harmonic vibration is that in which the deflections of the 
two particles are always equal and of the same sign (Fig. 16). 
If P be the tension of the string, the equation of motion of 
either particle is then 

*5s >?. (8) 



and the speed is therefore 



(4) 



In another mode the two deflections are equal in magnitude 
and opposite in sign, so that the middle point of the string is 
stationary (Fig. 17). The circumstances are therefore exactly 
the same as in 6, and the speed is 

'P a + b\ 

^r)' (5) 

greater, as we should expect, than 7^. If we denote the 



THEORY OF VIBRATIONS 



37 



deflections of the two particles by x, y, the superposition of 
the two modes gives 

x = A cos (nj + eO + B cos (n + e 2 ),) 

\ (b) 

y = A cos (%< 4- 61) B cos (n z t -4- e 2 ),| 

where the four constants JL, 5, e 1} e 2 are arbitrary. 

In the case of three attached particles the nature of the 
various modes is not so immediately obvious, even in the case 
of symmetry. We will suppose that the masses are equal, 
and that they divide the line into four equal segments a. 
Denoting the deflections by x, y, z, we have 



-P 



dt~ a a ' ( 

dt 2 a, a ) 

If we put, for shortness, fi = P/Ma, these may be written 



.(7) 



.(8) 



To ascertain the existence of modes of vibration in which 
the motion of each particle is simple-harmonic, with the same 
period and phase, we assume, tentatively, 

x = A cos (nt -f e), y = B cos (nt + e), z C cos (nt -f e). (9) 

It appears, on substitution in (8), that the equations will be 
satisfied provided 



(10) 



These three equations determine the two ratios A : B : C 
and the value of ri 2 . Eliminating the former ratios we have 

............. (11) 



38 



DYNAMICAL THEORY OF SOUND 



This is a cubic in n\ One root is nf = 2/z, and we find on 
reference to (10) that this makes B l = 0, A l = - C 1} and there- 
fore 

x = A l cos (njt 4- 6j), i/=0, z- A 1 cos(n l t + e l ). ...(12) 
This mode might have been foreseen, and its frequency 
determined at once, as in the preceding example. The 
remaining roots of (11) are 



and it appears from (10) that these make 

A 2 =C 2) B 2 = -</2A 2 , and A 3 =C 3 , B 3 
respectively. The corresponding modes are therefore 
x = A 2 cos (n + 6 2 ), y = - V2 A 2 cos (n 2 t + e 2 ), 

z A 2 cos (n 2 t + 6 2 ), 
and 

x A 3 cos (n s t + e 3 ), y = V2 A 3 cos (n 3 t + e 3 ), 

^ = J. 3 cos (n 3 ^ + e 3 ). 
These are shewn, along with the former mode, in Fig. 18. 
The complete solution of the equations is obtained by super- 
position of (12), (13) and (14), and contains the six arbitrary 
constants A lf A 2 , A S) e l} e 2 , e 3 . 



.(13) 



.(14) 




Fig. 18. 

We conclude these illustrations with the case of the double 
pendulum, where we are entirely dependent on general method. 
A mass M hangs from a fixed point by a string of length a, 
and a second mass ra hangs from M by a string of length b. 
For simplicity we suppose the motion confined to one vertical 






THEORY OF VIBRATIONS 



plane. The horizontal excursions x, y of M, m respectively 
being supposed small, the tensions of the upper and lower 
strings will be (M + m)g and mg, approximately. The equa- 
tions of motion are therefore 



d*x 



m ^ = ~ m 9 



x 



(15) 



To find the possible modes of simple-harmonic vibration we 
assume 

x = A cos (nt + e), y B cos (nt + e) ....... (16) 

The equations are satisfied provided 



...(17) 



.(18) 



where /ji = 

Eliminating the ratio A : B, we find 



.(19) 



which is a quadratic in n z . The condition 
for real roots, viz. 



r 
ab 



.(20) 



is obviously always fulfilled. It is further 
easily seen that both roots are positive, so 
that n also is real. 

The problem includes a number of inter- 
esting special cases, but we will only notice 
one or two. If the ratio /*, = m/(M 4- m), 
be small, the two roots of (19) are nf^g/a, 
nf=g/b, approximately. In the former 
case M oscillates like the bob of a simple 
pendulum of length a, whilst m executes 
what may be regarded as a forced oscillation 




40 DYNAMICAL THEOKY OF SOUND 

of the corresponding frequency ; this case has already been 
referred to in 13. In the second mode the ratio A : B is small, 
as appears from the second of equations (17); M is then nearly 
at rest, whilst m oscillates like the bob of a pendulum of 
length b. 

Since the expression on the left-hand side of (20) cannot 
vanish, the two frequencies can never exactly coincide, but they 
become approximately equal if a = 6, nearly, and //, is small. 
A curious phenomenon may then present itself. The motion 
of each mass, being made up of two superposed simple-harmonic 
vibrations of nearly equal period, may fluctuate greatly in 
extent, and if the amplitudes of the two vibrations are equal 
we have periods of approximate rest, as explained in 10. The 
motion then appears to be transferred alternately from m to M, 
and from M to m, at regular intervals*. 

If, on the other hand, M is small compared with m, p is nearly 
equal to unity, and the two roots of (19) are ?i 2 = g/(a + b) and 
n* = mg/M . (a -f b)/ab, approximately. The former root makes 
B/A = (a + b)/a, nearly, so that the two masses are always 
nearly in a line with the point of suspension, m now oscillating 
like the bob of a pendulum of length a + b. In the second 
mode the ratio B/A is small, so that m is approximately at 
rest ; the motion of M is then like that of a particle attached 
to a string which is stretched between fixed points with a 
tension mg (cf. 6). 

Another case of interest is obtained if we make a infinite. 
One root of (19) then vanishes, and the other is 



which makes A/B = - m/M. This indicates that if the support 
of a simple pendulum yield horizontally, but without elasticity, 
the frequency is increased in a certain ratio which is of course 

* The influence of dissipation is of course here neglected. If m be subject 
to a frictional resistance, and especially if the modulus of decay be less than 
the period of the fluctuation given by the above theory, the phenomena are 
modified, and the illustration of the theory of resonance ( 12) is improved. 
There is now a continual, though possibly a slow, drain on the original energy 
otM. 



THEOKY OF VIBKATIONS 41 

smaller the greater the inertia of the support. This is however 
more easily seen directly. 

15. General Equations of a Multiple System. 

The general theory of the small oscillations of a multiple 
system can only be given here in outline. In the case of one degree 
of freedom ( 7) it was possible to base the theory on the equation 
of energy alone, but when we have more than one dependent 
variable this is no longer sufficient, and some further appeal 
must be made to Dynamics. For brevity of statement we will 
suppose that there are two degrees of freedom, but there is 
nothing in the argument which cannot at once be extended to 
the general case. 

We imagine, then, a system such that every configuration 
which we need consider can be specified by means of two 
independent geometric variables or "coordinates" q l9 q 2 . If in 
any configuration (q l} q^) the coordinate q 1 (alone) receive an 
infinitesimal variation 8q lt any particle ra of the system will 
undergo a displacement 88 1 = a 1 8q 1 in a certain direction. 
Similarly if q z alone be varied m will be displaced through a 
space Bs 2 = *q* in a certain direction, different in general from 
the former. The resultant displacement Bs when both variations 
are made is given by 

8s* = Bs, 2 + 2&! &? 2 cos 6 + &? 2 2 

= di 2 Bqi* + 2a 1 2 cos 6 8q 1 Bq^ + c^ 2 fy 2 2 , (1) 

where 6 denotes the angle between the directions of 8s lt Bs 2 . 
If we divide by Bt*, we obtain the square of the velocity v 
of the particle m, in any motion of the system through the con- 
figuration (q lt q 2 ), in terms of the generalized "components of 
velocity" q lt q 2 , thus 

v z = ct^ 2 + 2^0, cos Oq-fa + a 2 2 ? 2 2 (2) 

The total kinetic energy of the system is therefore given by 

2T = 2 (mi; 2 ) = a u tf + 2a 12 ^ 2 4- o^ 2 , (3) 

where 



a n = 2, (rav), a 12 
the summation 2 extending over all the particles m of the 



42 DYNAMICAL THEOBY OF SOUND 

system. The coefficients a n , a 12 , a^ are in general functions of 
q lt q 2 ; they are called the "coefficients of inertia" for the par- 
ticular configuration considered. 

Next, let FI denote the total force acting on m, resolved 
in the direction of s ly and let F 2 have the corresponding meaning 
for the direction of Bs 2 . The work done on the system in any 
infinitesimal displacement will therefore be 

2 (FM + F 2 Ss 2 ) = 2 (F&) % + 2 (F^) 8q 2 . ..... .(5) 

If there are no extraneous forces, this work is accounted for 
by a diminution in the potential energy V of the system. When 
extraneous forces act we have in addition the work due to these, 
which we may suppose expressed in the form 



The coefficients Q lt Q 2 are called, by an obvious analogy, the 
generalized "components of (extraneous) force." Hence 

. . .(6) 
whence 




In the application to small oscillations we assume that q lt q 2 
are small quantities vanishing in the configuration of equi- 
librium, and for consistency we must also suppose that the 
disturbing forces Q lt Q 2 are small. The quantities a l9 2 and 
therefore also a n , a l2 , a& may now be treated as constants. 
The velocity of the particle m is made up of components a^, 
a 2 <?2 in the directions & x and 8s 2 , respectively; and if we neglect 
the squares of small quantities its acceleration is made up in 
like manner of components a^, ,#.,*. Hence resolving in the 
direction of Bsi the forces acting on m we have 
m (,& + ct 2 q 2 cos 6) = F l ,| 
and similarly m (a^ cos 6 + o^q 2 ) = F 2 .) 

* The former of these two quantities is (to the first order) the acceleration 
calculated on the supposition that q\ alone varies, and the latter is the accelera- 
tion when #2 alone varies. It is only on the hypothesis of infinitely small 
motions that the resultant acceleration is obtained by superposition of these. 



THEORY OF VIBRATIONS 43 

If we multiply the former of these equations by ^ and the 
second by a^, and sum for all the particles of the system, we 
find, with the notation of (4), 

dV . 






.(9) 



and similarly a^ + a&q 2 = 5 h Q 

oq% 

where 021 is of course identical with a 12 . 

When there are no extraneous forces these equations are by 
hypothesis satisfied by q^ = 0, q a = 0. The configuration of 
equilibrium is therefore characterized by the property that 

i- =0 ' f-- (10) 

in other words, the potential energy is stationary for all infini- 
tesimal displacements therefrom. Hence if V be expanded in 
powers of q lt q 2 , the terms of the first order will be absent, and 
we may write with sufficient approximation 

2 V = C-aq? + 2c 12 <? 1 <7 2 + Ca^jj 2 , (11) 

a constant term being omitted. The quantities c u , c 12 , c^ are 
called the " coefficients of stability." 
Hence (9) may be written 



a^i + ^22^2 
where c 21 = c 12 . 

If we look back to any of the special problems of 14 we 
shall recognize that the equations of motion are in fact of this 
type. For example, in the case of the double-pendulum we 
have 

...(13) 



The formulae therefore correspond if we put 

ft = ar, q* = y, \ 

a n = M, 12 = 0, 022 = m, V ...(14) 

c n = (M + m) g/a + mg/b, c u = - mg/b, C& = mg/b.J 



44 DYNAMICAL THEOKY OF SOUND 

The general case of m degrees of freedom hardly differs 
except in the length of the formulae. We have then m equations 
of the type 

a*i?i + a# 2 + . - + a sm q m + c gl ^ + c^ + . . . 4- C 8m q m = Q s , (15) 
where s is any one of the integers 1, 2, 3,...m. 

16. Free Periods of a Multiple System. Stationary 
Property. 

In the case of free vibrations we have Qi = 0, Q 2 = Q, and 
the solution of 15 (12) then follows exactly the same course 
as in the particular examples already given. We assume 

q 1 = A l cos(nt + e), q 2 = A 2 cos(nt + e), (1) 

and obtain (c u - n z a u ) A, -f (c 12 - n 2 a l2 ) A 2 = 0,1 
(c 21 - ?i 2 (A 21 ) A l + (c sst - n^a^) A^ = 0.) 
Eliminating the ratio A l :A 2) we obtain 

| c u -w 8 a n , C 12 -/i 2 a 12 
C 21 -rc 2 a 21 , CJB 71*022 ~ 

where (it is to be noticed) the determinant is of the " sym- 
metrical " type. This equation gives the two admissible values 
of n*. Adopting either of these we obtain a solution in which 
the ratio of A l to A 2 is determined by either of the equations 
(2). The mode of vibration thus ascertained involves therefore 
two arbitrary constants, viz. the absolute value of (say) A 1} and 
the initial phase e. The second root of (3) leads to another 
solution of like character. 

The extension of the method to the general case is obvious, 
but it may be well to state the results formally. In any 
conservative system of m degrees of freedom there are in 
general m distinct " normal modes " of free vibration about 
a configuration of stable equilibrium, the frequencies of which 
are given by a symmetrical determinantal equation of the mih 
order in n 2 , analogous to (3), and so depend solely on the con- 
stitution of the system. In each of these modes ihe t system 
oscillates exactly as if it had only one degree of freedom, the 
coordinates q^,q^, ... q m being in constant ratios to one another, 
and the description of 7 therefore applies. The directions of 
motion of the various particles and the relative amplitudes are 



THEOKY OF VIBRATIONS 45 

in any one mode determinate, though usually different for 
different modes, the only arbitrary elements being the absolute 
amplitude and the phase-constant. 

The equations of motion being necessarily linear, since 
products and squares of the coordinates and their differential 
coefficients with respect to the time are expressly excluded, it 
follows that the different solutions may be superposed by 
addition of the corresponding expressions. This has been 
sufficiently illustrated in the preceding examples. By super- 
posing in this way the m normal modes, each with its arbitrary 
amplitude and phase, we obtain a solution involving 2m 
arbitrary constants, which is exactly the right number to 
enable us to represent the effect of arbitrary initial values of 
the coordinates q l} q^, ... q m and velocities q lt q z , ... q m . In 
other words, the most general free motion of the system about 
a configuration of stable equilibrium may be regarded as made 
up of the m normal modes with suitable amplitudes and initial 
phases. This principle dates from D. Bernoulli* (1741). 

In particular cases it may happen that two (or more) of the 
natural periods of the system coincide. There is then a corre- 
sponding degree of indeterminateness in the character of the 
normal modes. The simplest example is furnished by the 
spherical pendulum, or by a particle oscillating in a smooth 
spherical bowl. The normal modes may then be taken to 
correspond to any two horizontal directions through the position 
of equilibrium. From the theoretical standpoint such coinci- 
dences may be regarded as accidental, since they are destroyed 
by the slightest alteration in the constitution of the system 
(e.g. if the bowl in the above illustration be in the slightest 
degree ellipsoidal), but in practice they often lead to interesting 
results. Cf. 53 below. 

An important characteristic of the normal modes, first 
pointed out by Lord Rayleigh in 1883, has still to be referred 

* Daniel Bernoulli (17001782), one of the younger members of the 
distinguished family of Swiss mathematicians. Professor of mathematics at 
St Petersburg (172533), and of physics at Bale (175082). His chief work 
was on hydrodynamics, on the theory of vibrating strings, and on the flexure 
of elastic beams. 



46 DYNAMICAL THEORY OF SOUND 

to. If, by the introduction of frictionless constraints which do 
no work, the system be restricted to vibrate in a mode only 
slightly different from one of these, the period will be altered 
only by a small quantity of the second order. In other words 
the periods of the several normal modes are " stationary " when 
compared with those of slightly different constrained modes. 
Suppose, for instance, that the normal mode in question is such 
that in it the coordinate q l alone varies. We have, then, in (2), 
a 12 = 0, c 12 = 0, and the natural frequency is determined by 
n* = c u /a 11 . If the constraint be expressed by q 2 = \qi, the 
condition that the constraining forces shall do no work, viz. 
ftfc + Q2fc = 0, or ft + XQ 2 = 0, leads to 

(a 11 + X s toa)& + (c 11 + X s c B )g 1 = 0, ............ (4) 

and the speed (p) is accordingly given by 






When X is small, this differs from n 2 by a small quantity of 
the second order. The proof, although limited to two degrees, 
is easily generalized. Owing to our liberty of choice of the 
coordinates, we can always arrange that q l shall be the only 
coordinate which varies in the mode in question, and that 
the constraint shall be expressed by a system of relations of 
the type q 2 = \q l} q 3 = fj,q lf q 4 = vq lt .... 

For an obvious illustration we may have recourse again 
to the particle on a smooth surface. If the constrained path 
be a vertical section through the lowest point, the period is 
%wJ(Rlg\ where R is the radius of curvature of the section, and 
it is known that R is a maximum or minimum for the principal 
sections. 

The equation (5) shews further that the constrained period is 
(as in the particular case) intermediate between the two natural 
periods ; this property can also be generalized. 

It follows that even when it is not easy to ascertain the 
precise character of a particular normal mode, a close approxi- 
mation to the frequency can often be obtained on the assumption 
of an assumed type which we can judge on independent grounds 
to be a fairly good representation of the true one. And in the 



THEORY OF VIBRATIONS 47 

case of the gravest natural mode the frequency thus obtained 
will be an upper limit. Take, for instance, the case of three 
equal particles attached at equal intervals to a tense string 
( 14), and consider an assumed type of symmetrical vibration 
in which x = z \y. The kinetic energy is then given by 

2T = M(d? + y*+ &) = M(I + 2\*)f, ......... (6) 

so that the inertia-coefficient is M(l + 2X 2 ). For the potential 
energy we have 



2 , (7) 
a a 

as is found by calculation of the work required to stretch the 
string (as in 22), or otherwise. The coefficient of stability is 
therefore P/a . (4\ 2 - 4\ 4- 2). For the speed ( p) we then have 

P 4X*-4X + 2 



This is stationary for X = + ^ \/2, and the corresponding speeds 
are as in 14. In this case it was evident beforehand that the 
assumed type would include the true natural modes of sym- 
metrical character. 

It is unnecessary for the purposes of this book to discuss in 
detail the theory of dissipation in a multiple system. The 
general effect is the same as in 12 ; the free vibrations 
gradually die out, but if the dissipative forces be relatively 
small the periods are not sensibly affected. 

17. Forced Oscillations of a Multiple System. Prin- 
ciple of Reciprocity. 

The theory of forced oscillations is sufficiently illustrated if 
in 15 (12) we assume that Qi varies as cospt, whilst (? 2 = 0. 
The equations will be satisfied if we assume that q l and q t both 
vary as cospt, provided 

qi + (cv-pdu) ? 2 = ft, 



ft, I 
0. J 

These determine the (constant) ratios of q 1 and q 2 to Q l ; thus 



48 DYNAMICAL THEOEY OF SOUND 

where A (p 2 ) is the determinant on the left-hand side of 16 
(3), with p 2 written for n 2 . The general conclusion is that when 
a periodic force of simple-harmonic type acts on any part of 
the system, every part will execute a simple-harmonic vibration 
of the same period, with synchronism of phase, but the 
amplitude will of course be different in different parts. When 
the period of the forced vibration nearly coincides with that 
of one of the free modes, an abnormal amplitude of forced 
vibration will in general result, owing to the smallness of the 
denominator in the formulae (2). For a complete account of 
this matter we should have to take dissipative forces into 
consideration, as in 12. 

A remarkable theorem of reciprocity, first proved by Helmholtz 
for aerial vibrations, and afterwards greatly extended by Lord 
Rayleigh, follows from (2). If we imagine a second case of 
forced vibration (distinguished by accents) in which Q/ = 
whilst Q 2 ' varies as cos pt, we shall have 



Comparing with (2), we see that 

,:ft = 9,':ft'. ..................... (4) 

The interpretation is most easily expressed when the "forces" 
Q 1 and Q 2 ' are of the same character, e.g. both ordinary statical 
forces, or both couples, in which case we may put Qi = Q 2 ', and 
obtain q 2 = qi'. In words: The vibration of type 2 due to a 
given periodic force of type 1 agrees in amplitude and phase 
with the vibration of type 1 due to an equal force of type 2. 
An example from the theory of strings will be found in 28. 
The above proof is easily extended to the general case of 
ra degrees of freedom. 

18. Composition of Simple-Harmonic Vibrations in 
Different Directions. 

We recur to the subject of composition of simple-harmonic 
vibrations which, though not so important as in Optics, claims a 
little further attention. If in a freely vibrating system we fix 
our attention on a particular particle, the directions in which it 



THEORY OF VIBRATIONS 49 

oscillates in the several normal modes will in general be different. 
The superposition then takes place of course according to the 
law of geometrical or vector addition. 

It will suffice to consider the case of two degrees of freedom, 
where we have independent simple-harmonic vibrations in the 
directions corresponding to the Bs l} &s 2 of 15. The result is a 
plane orbit, usually of a complicated character. For instance, 
in the case of Blackburn's pendulum ( 14), we have 

x = A cos (nj + ej), y = E cos (n z t + e 2 ), (1) 

where x, y are rectangular coordinates. The orbit is here 
contained within the rectangle bounded -by the lines x A, 
y = E. If n lt HZ are commensurable, the values of x, y and 
x, y will recur after the lapse of an interval equal to the least 
common multiple of the two periods, and the path will be 
re-entrant. The resulting figures, obtained in this and in other 
ways, are associated with the name of Lissajous*, who has had 
many followers in a region which is very attractive from the 
experimental point of view. 

The simplest case is that of rij = n^ If we eliminate t in 
(1) we then obtain 

s(e 1 -e 2 ) + |^ = sm 2 (e 1 - 2 ) (2) 

This represents an ellipse which, if the initial phases e lt e 2 coincide, 
or differ by TT, degenerates into a straight line (Fig. 20). The 
simplest mechanical illustration is furnished by the spherical 
pendulum. When the relation is that of the octave (r^ 2n?) 
we have a curve with two loops, which may degenerate into one 
or other of two parabolic arcs (Fig. 21). The curves in these and 
in other cases of commensurability are easily traced from the 
formulae (1) with the help of tables. A simple geometrical 
construction is indicated in Fig. 22, where the circumferences of 
the auxiliary circles are divided into segments corresponding to 
equal intervals of time in the two simple-harmonic motions 
which are to be compounded. If we start at a corner of any 

* J. A. Lissajous (182280). Professor of physics at the Lyce"e St Louis 
1850 74; rector of the Academy of Chambe'ry 1874 5, and of Besancon 
1875 9. His chief memoir, Etude aptique des mouvements vibratoires, was 
published in 1873. 

L. 4 



50 



DYNAMICAL THEORY OF SOUND 











THEORY OF VIBRATIONS 



51 



one of the rectangles in the figure, and proceed diagonally, we 
pass through a succession of points, equidistant in time, on a 
curve of the system. 




Fig. 22. 

Another conception of these figures, also due to Lissajous, 
may be mentioned. If we write 6 for nj, and adjust the origin 
of t, the formulae (1) are equivalent, on the hypothesis of 
commensurability, to 



y 



-(# a), 



.(3) 



where p/q is a fraction in its lowest terms. These equations, 
when combined with 

z asinO, ........................ (4) 

represent a curve of sines traced on the surface of the circular 
cylinder 

x * + 2 * = a * ........ ................ (5) 

and going through its period p times in q successive circuits 
of the cylinder. The Lissajous curve (3) is the orthogonal 
projection of this curve on a plane (z = 0) through the axis of 

42 



52 DYNAMICAL THEORY OF SOUND 

the cylinder. This is illustrated by Fig. 22, where the dotted 
branch may be regarded as the projection of that part of the 
sine-curve which lies on the rear half of the curved surface. A 
change in the relative phase in (1) is equivalent to a change in 
the angle a, and may be represented by a rotation of the cylinder 
about its axis, of corresponding amount. This, again, may be 
illustrated from Fig. 22 by starting the curve one step further to 
the right or left. When the ratio of the periods is nearly, but 
not exactly, that of two integers, the orbit gradually passes 
through the various phases of the commensurable case, in a 
recurring cycle*. Thus in the case of approximate unison, or 
approximate octave, the cycle includes the phases shewn in 
Fig. 21 or 22, followed by the same in reverse order. The same 
result is obtained by a continuous rotation of Lissajous' 
cylinder. 

19. Transition to Continuous Systems. 

The space which we have devoted to the study of dynamical 
systems of finite freedom is justified by the consideration that 
we here meet with principles, in their primitive and most easily 
apprehended forms, which run through the whole of theoretical 
acoustics. In the subsequent chapters we shall be concerned 
with systems such as strings, bars, membranes, columns of air, 
where the number of degrees of freedom is infinite. Mathematic- 
ally, it is sometimes possible to pass from one of these classes to 
the other by a sort of limiting processes when D. Bernoulli (1732) 
discussed the vibrations of a hanging chain as a limiting form 
of the problem where a large number of equal and equidistant 
particles are attached to a tense string whose own mass is 
neglected. In any case, there can be no question that the 
general principles referred to retain their validity. The main 
qualification to be noticed is that the normal modes are now 
infinite in number. It is usual to consider them as arranged 

* In Lissajous' method the vibrations which are optically compounded are 
those of two tuning forks. The figures obtained when the tones sounded by the 
forks form any one of the simpler musical intervals give a beautiful verification 
of the numerical relations referred to in 3. In the case of unison, when the 
tuning is not quite exact, the cycle of changes synchronises with the beats 
which are heard ; see 10. 



THEORY OF VIBRATIONS 53 

in ascending order of frequency ; the mode of slowest vibration 
may still be called the " fundamental," and is generally the 
most important. 

Before leaving the general theory it may be desirable to 
emphasize once more the importance of the simple-harmonic 
type of vibration from the dynamical point of view. We have 
seen that it is the characteristic type for a frictionless system of one 
degree of freedom, or (more generally) for a system oscillating 
as if it possessed only one degree, as in the case of the normal 
modes. It is also the only type of imposed vibration which is 
accurately reproduced, on a larger or smaller scale, in every 
part of the system. If a force of perfectly arbitrary type act at 
any point, the vibrations produced in other parts of the system 
have as a rule no special resemblance to this or to one another; 
it is only in the case of a periodic force following the simple- 
harmonic law of variation with the time that the induced 
vibrations are exactly similar, and keep step with the force. 
Moreover it is only in so far as the disturbing force is simple- 
harmonic, or contains simple-harmonic constituents, that it is 
capable of generating a forced vibration of abnormal amplitude 
when a critical frequency is approached. It is in these circum- 
stances that Helmholtz found the clue to his theory of audition, 
to which we shall have to refer at a later stage. 

20. On the Use of Imaginary Quantities. 

The treatment of dynamical equations can often be greatly 
simplified by the use of so-called "imaginaries." As we shall 
occasionally have recourse to this procedure, it may be convenient 
to explain briefly the principles on which it rests. 

The reader will be familiar with the geometrical representa- 
tion of a "complex" quantity a+ ib, where a, b are real and i 
stands for \/( I), by a vector drawn from the origin to the 
point whose rectangular coordinates are (a, b), and with the 
fact that addition of imaginaries corresponds to geometrical 
addition (or composition) of the respective vectors. The 
symbol a + ib when applied as a multiplying operator to any 
vector denotes the same process by which the vector a + ib may 
be supposed to have been derived from the vector 1, viz. it 



54 



DYNAMICAL THEORY OF SOUND 



alters the length in a certain ratio r, and turns it through a 
certain angle a. These quantities are defined by 

rcosa = a, r sin a. = b, (1) 

the quadrant in which a lies being determined by the sign 
attributed to cos a or sin a by (1). We have then 

a + ib = r (cos a -I- i sin a) (3) 

Hence a symbol of the form cos a + i sin a denotes the opera- 
tion of turning a vector 
through an angle a without 
alteration of length; in par- 
ticular the symbol i denotes 
the operation of turning 
through a right angle in the 
positive (counter-clockwise) 
direction. 

The symbol 
w = cos + i sin (4) 
may be represented by a unit 
vector OP drawn from in 
the direction 6. If we regard Fi g . 23. 

this as a function of 0, and if 

w + Bw be represented by OP', the angle POP' will be equal 
to BO. The vector PP' which represents Bw will therefore have 
a length BO, and since it is turned through a right angle 
relatively to OP, its symbol will be iBO.w. Hence 




.(5) 



It is easily shewn that the only solution of this equation 
which fulfils the necessary condition that z = 1 for 6= 0, is 

w = e, (6) 

where e ie is to be taken as denned by the ordinary exponential 

series. Thus 

e* = cos0 + *sin0 (7) 

We may add that the "addition-theorem" of the exponential 
function can now be derived immediately from the geometrical 
representation. 



THEORY OF VIBRATIONS 55 

It has been thought worth while to recapitulate these ele- 
mentary matters because they have interesting illustrations in 
the present subject. Thus if x, y be rectangular coordinates, 
and we write 

z = x + iy, ........................... (8) 

the equation z = Ce int , ........................... (9) 

where C may of course be complex, expresses that the vector C 
is turned in a time t through an angle nt in the positive direc- 
tion. It therefore represents uniform motion in a circle, with 
angular velocity n, in the positive direction. The radius of this 
circle is given by the "absolute value" of G, which is often 
denoted by | G ; thus if C = A + iB, where A and B are real, we 
have C \ = \I(A Z + -B 2 ). In the same way the equation 

z =G' e -int ........................ (!0) 

represents uniform motion in a circle, with angular velocity n, 
in the negative (or clockwise) direction. 

We come now to- the application to linear differential 
equations with constant coefficients. From our point of view 
the simplest case is the equation 



of 4. In order that every step of the work may admit of 
interpretation, we associate with this the independent equation 

0, ................. -....(12) 



as in the theory of the spherical pendulum. The two may be 
combined in the one equation 

-' +*-. ..................... < 13 > 



which may indeed be regarded as representing directly, without 
the intermediary of (11) and (12), the law of acceleration in the 
spherical pendulum and similar problems. To solve (13) we 
assume z = Ce**, and we find that the equation is satisfied 
provided X 2 -f n 2 = 0, or X = in. Since different solutions can 
be added, we obtain the form 

int , .................. (14) 



56 DYNAMICAL THEORY OF SOUND 

with two complex arbitrary constants (7, G f . These can be 
determined so as to identify z and z, at the instant t = 0, with 
the vectors which represent the initial position and velocity of 
the point (x, y). It appears from (14) that the most general 
motion of a point subject to (13) may be obtained by the 
superposition of two uniform circular motions in opposite direc- 
tions. The same problem (virtually) has been treated in 18, 
where the path was found to be an ellipse. This resolution 
of an "elliptic harmonic" vibration into two circular vibrations 
in opposite directions has important applications in Optics. 

The solution of the equation (11) may be derived from (14) 
by taking the " real " part of both sides, i.e. by projecting the 
motion on to the axis of x. Since <7, C' are of the forms 

C=A+iB, C'=A' + iB', (15) 

it might appear at first that the result would involve four 
arbitrary constants. These occur, however, in such a way 
that they are really equivalent only to two. Thus we find 

x = (A+A')cosvt-(B-B')smnt (16) 

The kinematical reason for this is that, as regards their 
projections on a straight line, right-handed and left-handed 
circular motions are indistinguishable. An important practical 
corollary follows. We should have obtained equal generality, 
so far as the solution of (11) is concerned, if we had contented 
ourselves with either solution of (13), for example 

z=Ce int , (17) 

and taken the real part 

x A cosnt Bsmnt (18) 

This conclusion is obviously not restricted to the particular 
differential equation (11) with which we started. The use of 
an adjunct equation such as (12) has only been resorted to 
in order to remove the suspicion of anything that can truly 
be called " imaginary " in the work. Such assistance can 
always be invoked mentally, but it is as unnecessary as it 
would be tedious always formally to introduce it. If in any 
case of a linear differential equation between x and t, with 
constant real coefficients, we seek for a solution of the type 
x = Cte xt , the imaginary values (if any) of X will occur in 



THEORY OF VIBRATIONS 57 

conjugate pairs of the form m in, and we may assert that 
the part of the solution corresponding to this pair of roots 
will be given with sufficient generality if we make use of one 
only of these, writing, for instance, 

x=Ce (m + in}t , ..................... (19) 

and taking the real part. 

We may apply these considerations, for example, to the 
equation 



of resisted motion about an equilibrium position ( 11). If we 

put x = Ce , we have 

A. 2 + &X + /4 = ...................... (21) 

Hence \ = -\kiri, ..................... (22) 

where TO' = VO"-i& 2 ), ..................... (23) 

provided k 2 < 4//,. On the above principle a sufficient solution 



or, in real form, 

x=e~* kt (Acosn't-Bsmn't\ ......... (24) 

which is equivalent to 11 (8). 

The same method can be followed with regard to the 
equation of forced oscillations, say 



?SBS /<8jtf ............. (25) 

Instead of this we take the equation 

g + ** + / u.^ ................ (26) 

the implied adjunct equation being of the type (25) with 
fsinpt instead of /cos pt on the right hand. A particular 
solution is 

z=Ce ipt , ........................ (27) 

provided (/z - p 2 + ikp) G =/. .................. (28) 

fj pt 
Hence * = - ^ sr- ................... (29) 

* 



58 DYNAMICAL THEORY OF SOUND 

If we put /ju-p* = Rcos a, kp = Rsina, (30) 

this becomes z = ^e i(pt ~ a \ (31) 

the real part of which is 

x = ^cos(pt-a) (32) 

This may be compared, for brevity, with the process of 12. 

21. Historical note. 

The theory of vibrations has a long and rather intricate 
history, in which Pure Mathematics and Mechanics have 
reacted on one another with great advantage to the progress 
of both sciences. Various special problems of great interest 
had been solved by the Bernoullis, Euler*, and other mathe- 
maticians, but it is to Lagrange -|- that we owe the general 
theory of the small oscillations of a system of finite freedom 
treated by means of generalized coordinates. The work of 
Lagrange was purposely somewhat abstract in formj; the 
full dynamical interpretation was reserved for Thomson and 
Tait (Natural Philosophy, 1867), to whom we also owe the 
now current terminology of the subject. The theory has 
been very greatly extended by Lord Rayleigh, and systematic- 
ally applied to acoustics as well as other branches of physics, 
in various writings, most of which (down to the year 1896) 
are incorporated in his Theory of Sound . 

* Leonhard Eoler, born at Bale 1707, died at St Petersburg 1783. He wrote 
extensively on most branches of mathematics and mechanics, and fixed to 
a great extent the notations now in use. 

t Joseph Louis Lagrange, born at Turin 1736, died at Paris 1813, "the 
greatest mathematician since the time of Newton." 

J "On ne trouvera point de Figures dans cet Ouvrage. Les m^thodes que 
j'y expose ne demandent ni constructions, ni raisonnemens ge'ome'triques ou 
me'chaniques, mais seulement des operations alge"briques, assujeties a une 
marche reguliere et uniforme." (Preface to the Mecanique Analytique, 1788.) 

1st ed. London 1877, 2nd ed. London 1894 6. See also his Scientific 
Papers, Cambridge 18991902. 



CHAPTER II 

STRINGS 

22. Equation of Motion. Energy. 

We proceed to the more or less detailed study of the 
vibrations of various types of continuous systems. Amongst 
these the first place must for many reasons be assigned to 
the transverse vibrations of a uniform tense string. Historically, 
this was the first problem of the kind to be treated theoretically. 
The mathematical analysis is simple, and various points of the 
general theory sketched in the preceding chapter receive 
interesting illustrations, which are moreover easily verified 
experimentally. Again, the sequence of the natural periods 
of free vibration has the special "harmonic" relation which 
has long been recognized as in some way essential to good 
musical quality, although the true reason, which is ultimately a 
matter of physiology, has only in recent times been investigated. 
The mathematical theory has further suggested some remarkable 
theorems, as to the resolution of a vibration of arbitrary type 
into simple-harmonic constituents, which are of far-reaching 
significance. Finally it is to be noted that in the propagation 
of a disturbance along a uniform string we have the first and 
simplest type of wave-motion. 

The string is supposed to be of uniform line-density p, 
and to be stretched with a tension P. The axis of x is taken 
along the equilibrium position, and we denote by y the trans- 
verse deflection at the point x, at time t. It is assumed that 
the gradient dy/dx of the curve formed by the string at any 
instant is so small that the change of tension may be 



60 DYNAMICAL THEOEY OF SOUND 

neglected. Under these conditions the equation of motion 
of an element Bx is 



(1) 



where i|r denotes the inclination of the tangent line to the 
axis of x. The right-hand side is, in fact, the difference of 
the tensions on the two ends of the element, when resolved in 
the direction of y. In virtue of the assumption just made we 
may write sin \|r = tan i/r = dy/dx, so that (1) becomes 



where c 2 =P/p ............................ (3) 

It is easily seen that the constant c has the dimensions of 
a velocity. 

The kinetic energy of any portion of the string is given by 

T=lpjfdx ..................... (4) 

taken between the proper limits. The potential energy may 
be calculated in two ways. In the first place we may imagine 
the string to be brought from rest in its equilibrium position 
to rest in any assigned form by means of lateral pressures 
applied to it. For simplicity suppose that at any stage of the 
process the ordinates all bear the same ratio (k) to their final 
values y, so that the successive forms assumed by the string 
differ only in amplitude. The force which must be applied to 
an element Sx to balance the tensions on its ends is 

(P sin i|r) &c, 
ox 

where sin -^ is now to be equated to kdy/dx', and the displace- 
ment when k increases by &k is y 8k. The total work done on 
this element is therefore 



where the accents indicate differentiations with respect to x. 
The potential energy is accordingly 

v (5) 



STKINGS 61 

In the alternative method we calculate the work done in 
stretching the string against the tension P. The increase in 
length of an element &e is 



approximately, so that 

F=4P/y'=<fo. ..................... (6) 

The formulae (5), (6) lead to identical results when applied to 
the whole disturbed extent of the string. For by a partial 
integration we have 

-Syy"dx = -[yy r l+Sy'*-dx, ............... (7) 

where the first term refers to the limits. It vanishes at the 
extremities of the disturbed portion, since y is there = 0. 

23. Waves on an Unlimited String. 

The solution of 22 (2) is 

y=f(ct-x) + F(ct + x) ................ (1) 

where the functions /, F are arbitrary. It is easily verified 
by differentiation that this formula does in fact satisfy the 
differential equation, and we shall see presently that by means 
of the two arbitrary functions which it contains we are able to 
represent the effect of any given initial distribution of displace- 
ment (y) and velocity (y\ It was published by d'Alembert* 
in 1747. 

The two terms in (1) admit of simple interpretations. 
Taking the first term alone, we see that so far as this is 
concerned the value of y is unaltered when x and ct are 
increased by equal amounts ; the displacement therefore which 
exists at the instant t at the point x is found at a later instant 
t 4- r in the position x 4- CT. Hence the equation 

y=f(ct-x) ..................... (2) 

represents a wave-form travelling unchanged with the velocity 
c in the direction of ^-positive. The equation 

y = F(ct + x) ..................... (3) 

represents in like manner a wave travelling with the same 
velocity in the direction of ^--negative. And it appears that 

* J. le Kond d'Alembert (171783), encyclopaedist and mathematician ; he 
made important contributions to dynamics and hydrodynamics. 



62 DYNAMICAL THEORY OF SOUND 

the most general free motion of the string may be regarded as 
made up of two such wave-systems superposed. 

The form of the expression \/(P/p) for the wave-velocity is 
to be noticed. As in all analogous cases the wave- velocity 
appears as the square root of the ratio of two quantities, one 
of which represents (in a -general sense) the elasticity, and the 
other the inertia, of the medium concerned. 

A simple proof of the formula for the wave-velocity has 
been given by Prof. Tait*. Imagine a string to be drawn with 
constant velocity v through a smooth curved tube, the portions 
outside the tube being straight and in the same line. Since 
there is no tangential acceleration the tension P is uniform. 
Also the resultant of the tensions on the ends of an element 
Ss, at any point of the tube, will be a force PSs/R in the 
direction of the normal, where R is the radius of curvature. 
This will balance the "centrifugal force" p$s.v 2 /R (fv z =P/p. 
Under this condition the tube may be abolished, since it exerts 
no pressure, and we have a standing wave on a moving string. 
If we now impress on everything a velocity v in the opposite 
direction to the former, we have a wave progressing without 
change of form, on a string which is otherwise at rest, with the 
velocity \/(P/p). It will be noticed that this investigation does 
not require the displacements to be small. 

The motion of an unlimited string consequent on arbitrary 
initial conditions 

y=4>(x), y = + (x\ [* = 0], ............ (4) 

may be deduced from (1), but it will be sufficient to write down 
the result, viz. 



rx+ct 

+(z)dz. (5) 

J x-ct 



This may be immediately verified. 

If the initial disturbance be restricted to a finite extent 
of the string, the motion finally resolves itself into two 
distinct waves travelling without change in opposite directions. 
In these separate waves we have 



* Encyc. Brit. 9th ed. Art. " Mechanics." 




STRINGS 63 

as is seen at once by considering two consecutive positions of the 
wave-form. Thus if in Fig. 24 the curves A, B represent the 
positions at the instants t, t + St, we have PQ = c&t, RP = ySt, 
RP/PQ = y', whence the former 
of the relations (6). The same 
thing follows of course from 
differentiation of (2). Con- 
versely, it is easily seen from 
(5), or otherwise, that if the 
initial conditions be adjusted so that either of the relations (6) 
is everywhere satisfied, a single progressive wave will result. 

When the string is started with initial displacement, but 
no initial velocity, the formula (5) reduces to 

y = i((a-c*)+(0 + cO} 00 

The two component wave-forms resemble the initial profile, but 
are of half the height at corresponding points. It is easily seen 
without analysis that this hypothesis satisfies the condition of 
zero initial velocity. 

It appears from (6) that in any case of a single progressive 
wave the expressions (4) and (6) of 22 for the kinetic and 
potential energies are equal. Lord Rayleigh has pointed out 
that this very general characteristic of wave motion may be 
inferred otherwise as follows. Imagine the wave as resulting 
from an initial condition in which the string was at rest, and 
the energy E therefore all potential, in the manner just 
explained. The two derived waves have half the amplitude (at 
corresponding points) of the original form, and the potential 
energy of each is therefore J E. Since the total energy of each 
wave must be ^ E, it follows that the kinetic energy of each 
must be \E. 

In mathematical investigations it is not unusual to find the 
effect of dissipation represented by^jt he hypothesis that each 
element of the string is resisted by a force proportional to its 
velocity, so that the differential equation takes the form 



dt - 

As regards the theory of stringed instruments this particular 



64 DYNAMICAL THEOKY OF SOUND 

correction has no importance, the direct influence of the air 
being quite insignificant; but the solution of (8) when k is 
small is of some interest from the standpoint of wave-theory, 
and may therefore find a place here. If the square of k be 
neglected, the equation may be written 



This is of the same form as 22 (2), and therefore 

y = e~* kt f(ct-x) + e-* kt F(ct+a;) ....... (10) 

This represents two wave-systems travelling in opposite 
directions with velocity c; but there is now a gradual diminu- 
tion of amplitude in each case as time goes on, as is indicated 
by the exponential factor. Again, since the functions are 
arbitrary, we may replace f(ct x) and F(ct + x) by 

e W-*to f(ct- x ) and e* k(t+xlc) F(ct+x), 
respectively, so that the solution may also be written 

y = e - lkxlc f(ct-x) + e* lixlc F(ct + x) ....... (11) 

This form is appropriate when a prescribed motion is maintained 
at a given point of the string. Thus if the imposed condition 
be that y = (f)(i) for x = 0, the waves propagated to the right 
of the origin are given by 

(12) 



The exponential shews the decrease of amplitude as the waves 
reach portions of the string further and further away from the 
origin. 

24. Reflection. Periodic Motion of a Finite String. 
If a point of the string, say the origin 0, be fixed, we must 
have y at this point for all values of t. Hence, in 23 (1), 

f(ct) + F(ct) = 0, or F(z) = -f(z). 
The solution therefore takes the form 

y=f(ct-x)-f(ct + x) ................ (1) 

As applied for example to the portion of the string which 
lies to the left of 0, this indicates the superposition of a direct 



STRINGS 65 

or "incident" wave represented by the first term, and a "re- 
flected" wave represented by the second. The amplitude of 
the reflected wave is equal, at corresponding points, to that 
of the incident wave, so that there is no alteration in the 
energy, but the sign of y is reversed. It is otherwise obvious 
that if on an unlimited string we start two waves which are 
antisymmetrical with respect to 0, in opposite directions, the 





| Fig. 25. 

point of the string which is at will remain at rest, even if 
it be free. Hence by the crossing of the waves the circum- 
stances of reflection at a fixed point are exactly represented. 
It will be noticed that a lateral force is exerted on the fixed 
point during the process of reflection. 

In the case of a finite string whose ends are (say) at the 
points x 0, x = I, we have the further condition that 

f(ct-l)-f(ct + l) = (2) 

for all values of t. If we write z for ct I, this becomes 

/(*)-/(* + SJ) (3) 

so that f(z) is a periodic function, its values recurring when- 
ever z increases by 2/. It follows that the motion of the string 
is periodic with respect to t, the period 2l/c being the time 
which a wave would take to travel twice the length. It is 
otherwise evident that a disturbance starting from any point 
P of the string, in either direction, will after two successive 
reflections at the ends pass P again, in the same direction as 
at first, with its original amplitude and sign. 

L. 5 



66 DYNAMICAL THEORY OF SOUND 

When the initial data are of displacement only, i.e. with 
zero initial velocity, the successive forms assumed by the string 
in the course of a period can be obtained by a graphical con- 
struction. We suppose the initial form y </> (x), where <f> (as) is 
originally defined only for values of x ranging from to I, to be 
continued indefinitely both ways, subject to the conditions 
4>(_#) = -</>(tf), (Z + aj) = -(Z-aj) (4) 

If we imagine curves of the type thus obtained to travel 
both ways with velocity c, and if we take at each instant the 
arithmetic mean of the ordinates, in accordance with 23 (7), it 
is evident that the varying form thus obtained will represent 




Fig. 26. 

a possible motion on an unlimited string, in which the points 
x = 0, x = I, x = + 21, . . . remain at rest. The portion between 
x = and x = I will therefore satisfy all the conditions of the 
question. The process is illustrated in the annexed Fig. 26 ; 
the initial form here consists of two straight pieces meeting at 
an angle, and the result after an interval l/8c is ascertained. 

In this way we might trace (after Young) the successive 
forms assumed by a string excited by " plucking," one point of 
the string being pressed aside out of its equilibrium position, 
and then released from rest, but the actual construction can in 
such a case be greatly simplified. It is easily seen that the 
form of the string at any instant consists in general of three 
portions; the outer portions have the same gradients as the 
two pieces into which the string was initially divided, whilst 
the gradient of the middle portion is the arithmetic mean of 



STRINGS 



67 



these, account being taken of sign. The line of this middle 
portion moves parallel to itself, with constant velocity, back- 
wards and forwards between the two corners of the parallelogram 
of which the initial form constitutes two adjacent sides. 




Fig. 27. 

In the annexed Fig. 27, which corresponds with Fig. 26, the 
plucking is supposed to take place at a distance of one-fourth 
the length from one end, and the phases shewn follow one 
another at intervals of one-sixteenth of a complete period, the 
successive forms being APB, AQ&B, AQ^B, AQ 3 R 3 B, AQ 4 R 4 B, 
and so on. It is evident on inspection of the figure that any 
point of a plucked string moves backwards and forwards with 
constant velocity between two extreme positions, in which it 
rests alternately during (in general) unequal intervals. The 
space-time diagrams of the middle point, and of the point 
plucked, under the conditions of Fig. 27, are given in Fig. 28. 





Fig. 28. 



52 



68 DYNAMICAL THEOKY OF SOUND 

In the latter case one of the intervals of rest vanishes*. 

It is of course with the vibrations of a finite string that 
we are chiefly concerned in acoustics. The string is usually 
stretched with considerable tension between the two points which 
limit the vibrating portion. At one at least of these points the 
string passes over a bridge resting on a sounding-board, whose 
function it is to communicate the vibrations to the surrounding 
air. The direct action of the string in generating air-waves 
is quite insignificant, but by the alternating pressure on the 
bridge the whole area of the sounding-board is set into forced 
vibration. This implies of course a certain reaction on the 
string itself, which is however, in the first approximation, 
usually negligible, for the reason given in 4. 

For experimental purposes an arrangement called a " mono- 
chord " is used. The sounding-board here forms the upper face 
of a rectangular " resonance chamber." The distance between 
the bridges can be varied and measured, and the tension, being 
produced by a weight attached to one end of the wire, which 
passes over a smooth pulley, can be regarded, at all events 
approximately, as known. For purposes of comparison one or 
more additional wires may be stretched alongside the former, 
their tension being adjusted, as in the pianoforte, by means of 
pegs at the extremities. 

25. Normal Modes of Finite String. Harmonics. 

The preceding investigations have been given on account of 
their historical importance, and for the sake of the analogies with 
other types of wave-motion which we shall meet with later. 
From the purely acoustical point of view they are however of 
secondary interest. The ear knows nothing of the particular 
geometrical forms assumed by the string, and is concerned 
solely with the frequencies and intensities of the simple- 
harmonic constituents into which the vibration can be resolved. 

* The theoretical vibration-forms have been verified experimentally by 
Krigar-Menzel and Kaps, Wied. Ann., vol. L., 1893, so far as the initial stages 
of the motion are concerned. After a few vibrations the form is seen to be 
undergoing a gradual change. This is attributed to a slight yielding of the 
supports of the string, in consequence of which the normal frequencies are not 
exactly commensurable, and the resulting motion therefore not accurately 
periodic. The construction in Fig. 27 is also due to these writers. 



STRINGS 



69 



To ascertain the normal modes of vibration of a finite string 
we may have recourse to the general procedure explained in 
Chapter I. In any such mode y will vary as a simple-harmonic 
function of the time, say cos (nt + e). This makes y = ri*y, 
and the equation (2) of 22 therefore assumes the form 



.(i) 



(2) 



The solution of this, exhibiting the time-factor, is 

/ . nx -T. . nx\ f . ^ 
y = I A cos -- 1- B sin 1 cos (nt -f e). ... 
\ c c / 

The fixed ends of the string being at x = 0, # = I, we must 
have A=Q, sin (nl/c) = 0, whence 

nl/7rc = l, 2,3, ...................... (3) 

This gives the admissible values of n. In any one normal mode 
we have, therefore, 

n . STTX /STTCt \ ,.. 

y = C 8 sm-- cos(-- + 6j, ............... (4) 



where 5 is an integer, and the amplitude C g and initial phase 
s are arbitrary. The gravest, or fundamental mode, which 
determines the pitch of note sounded, corresponds to s = l. 





The string then oscillates in the form of the curve of sines 
between the two extreme positions shewn in the upper part of 
Fig. 29. The frequency is 



70 DYNAMICAL THEOKY OF SOUND 

and so varies inversely as the length and as the square root of 
the line-density, and directly as the square root of the tension. 
These statements, which were formulated as the result of 
experiment long before the mathematical theory had been 
developed, are known as Mersenne's laws*. The determination 
of absolute pitch by the formula (5) does not admit of very 
great accuracy owing to the difficulty in measuring the tension, 
which is apt (owing to friction) to be slightly different on the 
two sides of a bridge. 

The principles that the frequency diminishes with increase 
of length and with increase of line-density have a familiar 
illustration in the pianoforte, where longer and intrinsically 
heavier strings are used for the graver notes. If the relation 
of pitch were adjusted by length alone the strings corresponding 
to the lower notes would have to be at least 100 times as long 
as those belonging to the highest. In order to secure a suffi- 
ciently low pitch within practical limits of length, and with 
a sufficient degree of tension, the string is loaded with a coil of 
wire wrapped closely round it. This has the effect of increasing 
the inertia without seriously impairing the flexibility, which is 
an essential point. The influence of tension, again, is illustrated 
in the process of tuning, which consists in tightening up the 
wires when these have stretched, or the pegs have yielded, so 
that the instrument has fallen in pitch, or become " flat." 

In the next normal mode after the fundamental the middle 
point x \l is at rest (Fig. 29). And in the 5th mode, whose 
frequency is by (3) s times that of the fundamental, there are 
s 1 internal points of rest, or " nodes," in addition to the 
ends. Midway between these we have the points of maximum 
amplitude, or " loops." Each segment into which the string is 
divided by the nodes vibrates as in the fundamental mode of a 
string of 1/sth the length. 

As already stated ( 2) the sequence of simple vibrations 
with frequencies proportional to the natural numbers 1,2,3,-..., 
which we here meet with, has important properties, musically 

* M. Mersenne (15881648), a Franciscan friar, was a schoolfellow and 
lifelong friend of Descartes, and maintained an extensive correspondence with 
him and other men of science of the day. 



STRINGS 71 

and physiologically. Its occurrence in vibrating systems is of 
course quite exceptional. Even in the present case, if the 
string deviate appreciably from uniformity or from perfect 
flexibility, the above scale of frequencies is at once departed 
from *. 

We were led in 16 to the conclusion, on physical grounds, 
that in any system of finite extent the effect of the most 
general initial conditions consistent with its constitution may 
be obtained by superposition of the several normal modes, with 
suitable amplitudes and phase-constants. We infer that the 
most general motion of a finite string can be represented by 
the formula 

/?x 

............ (6) 



provided the constants C 8 , e g be properly determined, the 
summation S extending over all integral values of s. An 
equivalent form is 

STTCt n . S7TCt\ . STTX /h _ x 

,cos-j- +4am-j-J sm-p, ...... (7) 

where A 8 = C 8 cose 8) B 8 =-C 8 s'm s ............. (8) 

If the string start from rest in a given position at the 
instant = the coefficients B 8 will vanish; if it be started 
with given velocities from the equilibrium position (y 0) 
the coefficients A 8 will vanish. 

Since the value of every term in (6) or (7) recurs whenever 
t is increased by 2/c, the vibration is essentially periodic, as 
already proved in | 24. In all other respects the motion of the 
string when started in an arbitrary manner is, from the present 
point of view, of a complex character, being made up of an 
endless series of simple-harmonic vibrations. The resulting 
note is accordingly made up of a series of pure tones, consisting 
(in general) of a fundamental, its octave, twelfth, double octave, 
and so on. 

It is not altogether easy to excite a string in such a way 

* The fact that a particular sequence of notes, musically related to one 
another, is associated with lengths of string proportional to the quantities 
1 i> i> i> was known to the Greeks, and was the origin of the name 
"harmonic" as applied to the numerical series. 



72 DYNAMICAL THEOEY OF SOUND 

that the resulting motion shall be strictly simple-harmonic, 
and the sensation accordingly that of a pure tone. But, as 
will be shewn more fully in 39, it is possible to suppress 
all the tones below any assigned rank (s) by checking the 
vibration at a node of the 5th mode, as, for instance, by 
contact with a camel-hair pencil. The remaining nodal points 
of this constituent are then points of rest, whilst half-way 
between them there is vigorous vibration. The experiment, 
which is very striking, is easily made with the monochord. 

The energy in any normal mode is easily calculated. We 
find 

...... (9) 

...(10) 

The coefficients are equal, in virtue of 22 (3), and the total 
energy in this mode is 

T+v 




It is further easily proved that the whole energy of the 
string is the sum of the energies corresponding to the various 
normal modes, viz. 

T+V-^-S.*Of-^*f(4*+Bf) ....... (12) 

This is a general property of the normal modes of a vibrating 
system. The proof, in the present case, depends on the fact 
that 

[ l . STTX . STT3C ..''* /-, ox 

I sin j- sm = dx = 0, ............ (13) 

JO * ' 

if s, s be any two unequal integers. See 32 (4). 

26. String excited by Plucking, or by Impact. 

The relative amplitudes of the various modes is of course 
a matter of importance, as on it the quality of the note 
depends ( 2). Usually a string is excited in one of three 
ways, viz. by plucking (as in the harp, zither, &c.) f by striking 
with a hammer (pianoforte), or by bowing (violin, violon- 
cello, &c.). 



STRINGS 73 



If the string be pulled aside through a small space , at 
a distance a from the end a? = 0, and then be released, the 
values of the coefficients in 25 (7) are found to be 



/ ,~ . srra . STHZ? STTC 

whence y , y, - x 2 - sin =- sin = cos -- *- . . . .(2) 
7r 2 a(/ a) ,s* I I I 



The mode of calculation will be explained in the next chapter 
(see 36). We notice that the harmonic of order 8 will be 
altogether absent if sin (sirajl) 0, i.e. if the point of plucking 
be at one of its nodes; this was remarked by Young (1841). 
Thus if the string be plucked at the centre, all the harmonics 
of even order will be absent. The formula (1) combined with 
25 (12) shews that, apart from a trigonometrical factor which 
lies between and 1, the intensities of the successive harmonics 
will vary as 1/s 2 . The higher harmonics are therefore relatively 
feebly represented in the actual vibration of the string. 

The effect of the impact of a hammer depends on the 
manner and duration of the contact, and is more difficult to 
estimate. The question is indeed, strictly, one of forced 
vibrations ( 28); but in the somewhat fictitious case where 
the duration is so small that the impact has ceased before 
the disturbance (travelling with the velocity c) has had time 
to spread over any appreciable fraction of the length, we 
may treat the problem as one of free motion with given initial 
velocity concentrated on a short length. The result is 



where a is the distance from the origin to the point struck, 
and //. represents the total momentum communicated by the 
impact. Hence 

2u _., 1 . Sira . STTX . Sirct 
y = -- 2 - sin y- sin = sin = .......... (4) 

TTpC Sill 

As in the previous problem, the sih mode is absent if the 
origin be at one of its nodes. Apart from the trigonometrical 
factor on which this circumstance depends, the intensities of 



74 DYNAMICAL THEOBY OF SOUND 

the successive modes are, according to 25 (12), now of the 
same order of magnitude. The unreal character of the pre- 
ceding hypothesis betrays itself in this result ; but we may at 
all events infer that in the case of a very brief impact the 
higher harmonics are relatively much more in evidence than 
in the former problem. 

In reality the impact, even in the case of a metallic 
hammer, is far from instantaneous, the time of contact, though 
very short as measured by ordinary standards, being at all 
events comparable with the period of vibration of the string*. 
The effect of an impulse of finite duration has been calculated 
by Helmholtz, to whom most of the present theory is due, on 
the supposition that the pressure begins at the instant t = 0, 
and lasts for a time T, during which it rises from zero to a 
maximum and falls to zero again, according to the law sin (TT/T). 
A somewhat simpler result is obtained if we imagine the law 
of pressure to be 



where /* represents the time-integral of the force from t = oo 
to t + oo . This law, whose graphical representation has the 
form of the curve in Fig. 14, p. 33, has the defect that there is 
no definite instant of beginning or ending, but as the true law 
is in any case unknown, it may serve for purposes of illustration. 
The interval of time during which the force is sensible is 
comparable with r, and can be made as narrow as we please 
by diminishing T. The details of the calculation will more 
conveniently find a place in the next chapter ( 38). The 
result is 

(6) 



When T is infinitesimal this agrees with (3). In other cases 

the intensities of the higher harmonics vary as e ~ 8irCT ' , if we 
omit the trigonometrical factor. 

Although the pressure is thus rendered less abrupt as 
regards its variation with the time, it is still assumed to be 

* Kaufmann, Wied. Ann., vol. LIV. (1896). 



STKIJJGS 75 

concentrated at a point. If we were to imagine it distributed 
continuously over a short length of the string this would 
further increase the relative weight of the lower harmonics 
(see 38). 

According to a general principle, which is here exemplified, 
and which will be further referred to in the next chapter, 
the higher harmonics are excited in greater relative intensity, 
the more abrupt the character of the originating disturbance. 
From a musical point of view the harmonics after about the 
sixth are to be discouraged, since they come sufficiently 
near to one another in the scale to be mutually discordant. 
In the pianoforte the hammers are covered with layers of softer 
material, so that the variation of pressure during the impact 
is rendered more gradual. 

The point at which the blow is delivered is also a matter 
of importance. To obtain a note of rich musical quality the 
lower harmonics should be present in considerable force, and 
the middle regions of the string are on this account to be 
avoided. On the other hand, the harmonics of higher order 
than the sixth are prejudicial, as already stated. Both re- 
quirements are met by fixing the striking point at a distance 
of about one-seventh of the length from one end. The partial 
tones which have nodes at or near this point will then not 
be excited at all, or only with comparatively feeble intensity. 

27. Vibrations of a Violin String. 

The theory of the vibrations of a string when excited by 
bowing is somewhat difficult, but the main features have been 
elucidated by Helmholtz. Since the pitch is found to be that 
natural to the string, the vibrations are to be regarded as in 
a sense " free," the function of the bow being to maintain the 
motion by supplying energy to make up for the losses by 
dissipation. In the case of the violin &c., where the strings 
are of light material and pass over a bridge resting on a very 
sensitive surface (of the resonance cavity), these losses may 
be relatively considerable. The mode of action of the bow 
appears to be that it drags the string with it for a time by 
friction, until at length the latter springs back ; after a further 



76 DYNAMICAL THEOEY OF SOUND 

interval the string is carried forward again, and so on*, the 
complete cycle taking place in the period of vibration. 

In order to obtain data for mathematical analysis Helmholtz 
began by an experimental study of the character of the vibration 
at various points. The device was an optical one, of the kind 
employed by Lissajous ( 18), by which the rectilinear vibration 
of the point examined is compounded with an independent 
vibration at right angles, whose period is commensurable, or 
nearly so, with that of the string. A microscope whose axis is 
horizontal is directed to the point to be studied, the string 
itself being vertical. The eye-piece of the microscope is fixed, 
but the objective is carried by one of the prongs of a tuning 
fork and vibrates in a vertical direction. When the fork alone 
vibrates the image of a bright point on the string is drawn 
out into a vertical line; when the string alone vibrates the 
appearance is that of a horizontal line. When both vibrations 
coexist the result would be a closed curve if the periods were 
exactly commensurable. For example, if the period of the fork 
were exactly commensurable with that of the string, and if the 
vibration of the point examined were simple-harmonic, the 
result would be one of the corresponding series of Lissajous 
figures ( 18); whilst if the relation between the periods were 
inexact, the curve would pass in succession through the various 
phases of the series. In the actual circumstances the forms of 
the curves are modified, and it is possible from the result to 
make inferences as to the true nature of the vibration studied. 




Fig. 30. 

The interpretation is facilitated by the ideal representation 
of the successive phases as orthogonal projections of a curve 
traced on a revolving cylinder. It was found that the space- 

* In order that work may be done it is necessary to suppose that the 
frictional force is greater in the first stage than in the second. This is 
consistent with the known law that friction of (relative) rest is greater than 
friction of motion. The remark is due to Lord Rayleigh. 



STRINGS 77 



time diagram of the point of the string 
under examination has the simple form 
shewn in Fig. 30. The nature of the 
modification in the Lissajous figures may 
be illustrated in the case of unison 
between fork and string. If the portion 
of the broken line in Fig. 30 which lies 
between A and B be wrapped round a 
cylinder whose circumference is equal to 
A B y its projections on planes through 
the axis will include such forms as are 
here shewn (Fig. 31)*. 

The period of vibration of the point 
examined is made up of two intervals, 
usually of unequal duration, during 
which the point moves backwards and 
forwards, respectively, with constant but 
(in general) unequal velocities. The 
ratio of the two intervals is further 
ascertained to be equal to that of the 
two segments into which the string is 
divided by the point. These results have 
been confirmed by subsequent observers 
who have obtained the space-time dia- 
gram in a more direct mannerf. In 
order that they may come out clearly 
some precautions are necessary. Some- 
thing depends on the skill with which 
the bow is used, and apparently on the 
quality of the instrument. In order, also, 
that the diagram should be free from 
minor irregularities the bow should be 

* In the actual experiments of Helmholtz the 
frequency of the string was four times that of the 
fork. The circumference of the cylinder in the 
above mode of representation then includes four 
periods of the zig-zag line in Fig. 30. 

t Erigar-Menzel and Raps, Wied. Ann., vol. 
XLIV. (1891). 






78 DYNAMICAL THEOKY OF SOUND 

applied at a node of one of the harmonics, and the point 
observed should be at another node of the same. 

Except at the two instants in each period when the velocity 
suddenly changes, the acceleration of the point (P) examined is 
zero. It follows from 22 (2) that the curvature of the string 
in the neighbourhood of P vanishes, and that the form of the 
string at any instant is accordingly made up of straight pieces. 




Fig. 32. 

It appears that all the conditions of the problem can be satisfied 
if we assume that the form is always that of two such pieces 
meeting at a variable point Q. In Fig. 32 let AB (= I) be the 
undisturbed position of the string, and let a (= AN) and /3 
(= NQ) be the coordinates of Q referred to A as origin and AB 
as axis of abscissae. The equations of the two portions of the 
string are 

yi = M> y* = P(i-x)l(i-*), ......... (i) 

and the difference of the velocities near Q on the two sides 
is accordingly 



In the time St a length d8t of the string is traversed by the 
point Q, so that a mass pd&t has its velocity increased by the 
above amount. This is the effect of the transverse force 



where P is the tension, acting for the time St. Equating the 
change of momentum to the impulse of the force we find 

* ...................... (4) 



STRINGS 79 



The point of discontinuity Q (of the gradient) must therefore 
travel right or left with the velocity c. 

Let us suppose that Q starts from A at the instant t = 0, 
and that & is at first positive. The observations of Helmholtz 
shew that the velocity at a point x, viz. 



is during an interval x/c constant, whence 

= (7a(Z-a), ..................... (6) 

no additive constant being admissible, since ft must vanish with 
a. This is the equation of a parabolic arc passing through A, B. 
The conditions of the problem are therefore all fulfilled if we 
imagine Q to travel backwards and forwards along two such 
arcs, with velocity c, in the manner indicated in Fig. 32. In 
terms of the maximum displacement j3 we have G = 4/8 /Z 2 , 
and the equations of the two portions of the string at any 
instant are therefore 

y,-^(i-), -^<l-*> ...... (7) 

It only remains to resolve this motion into its simple- 
harmonic constituents. The details of the calculation are 
given in 37. The result is 

STTCt 



where the summation embraces all integral values of s. Com- 
paring with 25 (7) we have 

A = 0, B.-3| ................... (9) 

These results, and indeed the whole investigation, take no 
account of the position of the point to which the bow is applied. 
It is plain, however, that the position of the bow must have 
some influence on the character of the vibration; and it is 
found in fact that those normal modes are absent which have a 
node at the point in question. It is for this reason that the 
somewhat idealized vibration-form which is adopted as a basis 
of calculation is only obtained in its purity at corresponding 
nodes. 



80 DYNAMICAL THEORY OF SOUND 

28. Forced Vibrations of a String. 
The simplest case of forced vibration is where a given 
simple-harmonic motion 

y = 0coa(pt + a) .................. (1) 

is imposed at a point (x = a). The portions of the string on the 
two sides of this point are to be treated separately. The 
results are 



. px 
sin 




cos (pt -fa) [0 < x < a] , 



sm 



sn 



for these satisfy the general differential equation 22 (2), they 
make y 1 = for x = 0, and y z = for x = I, and they agree with 
(1) when x = a. The amplitude of y l or y 2 becomes very great, 
owing to the smallness of the denominator, whenever pa/c or 
p(l a)/c is nearly equal to a multiple of TT, i.e. when the 
imposed period 2ir/p approximates to a natural period of a 
string of length a or I a, respectively. To obtain a practical 
result in such cases we should have to take account of dissipative 
forces. 

The case is illustrated by pressing the stem of a vibrating 
tuning fork on a piano string. The sound swells out powerfully 
whenever the portion of the string between the point of contact 
and either end has a natural mode in unison with the fork. 
This plan is recommended by Helmholtz as a means of producing 
pure tones, since the higher modes of the fork, not being 
harmonic with the fundamental, are not reinforced. 

When a transverse force of amount Y per unit length acts 
on the string, the equation (2) of 22 is replaced by 



In general Y will be a function both of x and t. 

The case of a periodic force F cos (pt + a) concentrated on an 
infinitely short length of the wire at x a may be deduced from 



STRINGS 81 

the formulae (2). The value of ft in terms of F is found from 
the consideration that the force must just balance the pull of 
the string on this point, i.e. 

F C os(pt + ct)=P yi '-Py 2 ' ............... (4) 

for x = a. This leads to 

. px . p(l a) 
sm sin-- - 



sn 

c c 

The formula for y z differs only in that the letters x and a are 
interchanged ; we have here an instance of the reciprocal theorem 
of 17, according to which the vibration at a point x due to a 
periodic force at a must be the same as the vibration at the 
point a due to an equal force (of the same period) at x. 

The amplitude becomes as a rule great when sm(pljc) is 
small, i.e. when the imposed period approaches a natural period 
of the whole string. An indeterminate case occurs when 
sin (pale) and sin (pile) = simultaneously, the point x = a 
being then a node. 

29. Qualifications to the Theory of Strings. 

We have in 26, 27 considered the relative amplitudes of 
the different harmonics when a string is excited in various ways, 
but we must not assume that the corresponding relative inten- 
sities are accurately reproduced in the resulting sound-waves, 
which are started indirectly through the sounding board. 
If we neglect the reaction on the string, which may for a 
considerable number of vibrations be insensible, we may regard 
the string as exerting on each bridge a force proportional to 
the value of dy/dx there*, as given by the respective formula. 
The differentiation introduces a factor s in the coefficient of 
the 5th harmonic, and so increases the importance of the 
higher modes. On the other hand, the amplitude of vibra- 
tion of the sounding board due to a simple-harmonic force 
of given amplitude, will vary somewhat with the frequency, 

* Thus in the case of the plucked string it appears from Fig. 27 that 
the pressure on each end alternates between two constant values of opposite 
sign. 

L. 6 



82 DYNAMICAL THEOKY OF SOUND 

on the general principle illustrated in 9*. This is probably 
to the relative advantage of the lower modes. The effect of 
yielding of the bridges in modifying the natural frequencies 
of the string has been discussed by Rayleighf; it is probably 
in practice very slight. 

Another cause which must be mentioned as affecting our 
results to some extent is the imperfect flexibility of the string, 
or wire. In the case of the higher normal modes the segments 
into which the string is divided may be so short that flexural 
couples come into play, and tend to raise the frequency by 
increasing the potential energy of a given deformation. This 
will be referred to later ( 50). A further point is that the 
abrupt forms postulated in the theory of plucked or bowed 
strings are not exactly realized, and that such investigations as 
those of 26, 27 are to be viewed as approximations, which are 
however quite adequate so far as the determination of the ampli- 
tudes of the graver and more important harmonics is concerned. 

30. Vibrations of a Loaded String. 

We conclude this chapter with the discussion of one or two 
problems which, besides being of some interest in themselves, 
may serve to remind us again that the harmonic scale of fre- 
quencies is after all an exceptional phenomenon, even in the 
case of strings. 

Take first the case of a string, otherwise uniform, loaded 
with a mass M at its centre. It is obvious that those normal 
modes of the unloaded string which have a node at this point 
are unaffected. Leaving these on one side, we consider only 
those vibrations in which there is at every instant complete 
symmetry with regard to the centre. If the lateral displacement 
of M be {3 cos (nt + e), we have, for the first half of the string, 




(1) 



* Some interesting experiments bearing on these questions have been made 
by Barton and Garrett, Phil. Mag. (6), vol. x., 1905. See also Barton, Text- 
Book of Sound, London, 1908, 361. 

t Theory of Sound, 135. 



STRINGS 



The equation of motion of M is 



(2) 

where after the differentiations we must suppose x = I. This 
gives 

nl nl I 

2c 2c = b' (3) 

where b is written for M/p, i.e. b is the length of string whose 
mass would be equal to that of the attached particle. The 
frequencies are therefore determined by 

where x 1) # 2 , x z , ... are the roots of the transcendental equation 

x tan x = l/b (5) 

Equations more or less of this character occur in many branches 
of mathematical physics, and can often be solved approximately 
by graphical construction. Thus in the present instance if we 
trace the curves 

b 
2/ = cot#, y= I ., (6) 

the abscissae give the roots. If 6 be relatively small these 
fall a little short of JTT, |TT, |TT, ..., respectively, and the 



b-l 




Fig. 33. 



65 



84 DYNAMICAL THEOKY OF SOUND 

frequencies are therefore slightly lower than in the symmetric 
modes of an unloaded string. As b increases the frequencies 
all diminish, the physical reason being of course the increased 
inertia. Finally, when M is very large compared with the mass 
pi of the string, l/b is small, and the lowest root of (5) is given 
approximately by a? = 1/b, whence 



n = 



in agreement with 6 (4). 

31. Hanging Chain. 

The contrast with previous continuous systems is still more 
marked in the case of the small oscillations of a uniform chain 
hanging vertically from its upper end, which is fixed. This 
has no immediate acoustic importance, but it is interesting 
historically*, and is, from the standpoint of general theory, 
instructive in various ways. 

We take the origin at the equilibrium position of the free 
end. The tension at a height x above this point will be 
P = gpx, the vertical motion being neglected as of the second 
order. Hence if y denote the horizontal deflection, we have 



or 



a* 

3 / dy 
I*-K 



(*y\ 

rat/ 1 



To ascertain the normal modes we assume that y varies as 
cos (nt + e), and obtain 

i( x d /] + n -y = Q ................... (3) 

dx \ dxj g * 

This can be integrated by a series, but the solution assumes 
a somewhat neater form if we introduce a new independent 
variable in place of as. The wave-velocity on a string having 
a uniform tension equal to that which obtains at the point x 
would be *J(Plp) or *J(gx). Hence if r denotes the time which 

* It appears to have been the first instance in which the various normal 
modes of a continuous system "were determined, viz. by D. Bernoulli (1732). 
The Bessel's Function also makes its first appearance in this connection. 



STKINGS 85 

a point moving always with this local wave-velocity would take 
to travel from the lower end to the point x, we have 



In terms of r as independent variable the equation (3) becomes 

0- ... ............... (5) 



For the present purpose we do not require the complete 
solution, but only that solution which remains finite when 
T = 0. This is 



where G is arbitrary, as may be verified by actual differentia- 
tion, and substitution in (5). The function defined by the 
series in brackets presents itself in many physical problems; 
it is called the "Bessel's Function of Zero Order," and is 
denoted by J Q (nr)*. Hence, inserting the time-factor, 

y = CJ (riT)cos(nt + 6) (7) 

The value of T corresponding to the upper end (x I) is 

T 1 = 2V(%), (8) 

and the condition that this end should be fixed gives 

Jo (nil) = (9) 

This determines the admissible values of n. The first few 
roots are given by 

WTl /7r = -7655, 1-7571, 2-7546,..., (10) 

where the numbers tend to the form s J, s being integral. In 
the modes after the first, the values of T corresponding to the 
lower roots give the nodes. Thus in the second mode there is 
a node at the point T/T, = -7655/1-7571, or a;/l = T*lT l * = '190. 
The gravest period is 2?r/w = 5'225 \J(llg\ whereas the period 
of oscillation of a rigid bar of the same length is 5130 ^(Ijg). 
The comparison verifies a general principle referred to in 16, 

* Elaborate numerical tables of the Bessel's Functions, calculated by 
Meissel and others, are given by Gray and Mathews, Treatise onBessel Functions, 
London, 1895. A convenient abridgment is included in Dale's Five-Figure Tables 
of Mathematical Functions, London, 1903. 



86 



DYNAMICAL THEORY OF SOUND 



according to which any constraint has the effect of quickening the 
gravest oscillation. The first two modes are shewn (on different 
scales) in Fig. 31, the two nodal points representing the point of 
suspension in the two cases. 




Fig. 34. 



CHAPTER III 

FOURIER'S THEOREM 

32. The Sine-Series. 

The study of the transverse vibrations of strings has 
already suggested a remarkable theorem of pure mathematics, 
to which some further attention must now be given. The 
theory of the normal modes has led us ( 25) to the conclusion 
that the free motion of a string of length I, started in any 
arbitrary manner, can be expressed by a series of the form 

STTCt n . S7TCt\ . STTX 

8 cosj- + 5 f sm Jsin-j-, ...... (1) 

where s = 1, 2, 3, ..., provided the constants A 8) B 8 be properly 
determined. In particular if the string be supposed to start 
from rest at the instant t = in the arbitrary form y = f(x), it 
should be possible to determine the coefficients A s so that 



(2) 



for values of x ranging from x = to x = I. This is a particular 
case of "Fourier's Theorem*." Since I is at our disposal we 
may conveniently replace it (for general purposes) by TT, and 
the statement then is that an arbitrary function f(x) can be 
expressed, for values of x ranging from to TT, in the form 

f(x) = A l sin x + A 2 sin 2# -f . . . + A 8 sin sx + ....... (3) 

* J. B. J. Fourier (17681830). The history of the theorem is closely 
interwoven with that of the theory of strings, and of the theory of heat- 
conduction. Fourier's own researches are expounded in his Theorie de la 
Chateur, Paris, 1822. An outline of the history is given in Prof. Carslaw's 
book cited on p. 96. The subject is treated most fully by H. Burkbardt in his 
report entitled Entwickelungen nach oscillirenden Funktionen..., Leipzig, 1908. 



88 DYNAMICAL THEOKY OF SOUND 

The reasoning by which we have been led to this result is 
of a physical rather than a mathematical nature, and we have 
moreover not referred to the restrictions which physical con- 
siderations alone would impose on the character of the arbitrary 
function f(x). Leaving such points for the moment, and as- 
suming the theorem provisionally, we proceed to the deter- 
mination of the coefficients. If we multiply both sides of (3) 
by sin sx, and integrate from x to x = ir y we get on the right 
hand a series whose general term is 

A r I sin rx sin sx dx 
Jo 

= %A r l {cos (r s) x cos (r 4- s) x} dx. ...(4) 
J o 

When the integers r, s are unequal this vanishes, since each 
cosine goes through its cycle of values, positive and negative, 
once or oftener within the range of integration. But when 
r = s, the first cosine is replaced by unity, and the result is 
Hence 

2 f v 
A g = -\ f(x)$msxdx ................ (5) 



The process may be illustrated by a few examples. Take, 
first, the case of 

/(*) = (-*), ..................... (6) 

which is represented by an arc of a parabola. We find, after 
a series of partial integrations, 

2 C w 4 

A 8 =-\ x (TT x) sin sx dx = -(1 COSSTT). ...(7) 

7T J o TrS"* 

This is equal to or 8/?rs 3 , according as s is even or odd. The 
theorem therefore becomes 

x (TT x) - ( sin x + sin 3# + - sin 5x + . . . ) . -(8) 

7T \ O O / 

If we put x J TT in this we obtain the formula 



32"" 3 3 5 3 '"' ' 

which is known on other grounds to be correct. The equality 
in (8) may also be tested graphically. It is found that the 
discrepancy between the graph of x (TT x) and that of the 



FOURIER'S THEOREM 



function represented by the first three terms on the right hand 

is so slight that it would be barely perceptible on a scale 

suited to the pages of this book. 

In the next example the graph of f(x) consists of two straight 

lines through the points x = 0, x = TT, respectively, meeting at 

an angle at the point x = a. If we assume the ordinate at the 

latter point to be unity, we have 

/(*) = #/ [0 <*<], j 

/(#) = (TT - at) / (IT -a) [*<*<*].) 

We find, after some reductions, 

2 f a 2 f 71 " 

A 8 xsinsxdx-\ -- -. -- ^1 (TT x) sin sx dx 
Tra.'o 7r(7r-a)J a 

1 

-sinsa. ...(11) 






a(7r-a) V 





Fig. 35. 



90 DYNAMICAL THEOKY OF SOUND 

Thus 

2 / 1 

f(x) = -j- I sin a sin x 4- ~- 2 sin 2a sin 2x 

+ i 2 sin3asin3#+...y ...(12) 

As a check on this result we may put a = JTT, x \ir\ this 
gives 



which is known to be right. This example is of interest in 
connection with the theory of the plucked string ( 26, 36). 
Fig. 35 shews the graph of f(x) together with that of the 
function represented by the first eight terms of the series on 
the right hand of equation (12), in the case of a = f?r. The 
fourth and eighth terms contribute nothing to the result in 
this case, since they correspond to modes having a node at the 
point plucked. 

Again, let /(#) = TT a? ...................... (14) 

2 f 77 2 

We find A s = \ (TT #)sins#c& = - ............. (15) 

TTJO s 

The theorem therefore asserts that 

TT - x = 2 (sin x + \ sin 2# -f sin 3# +...). . . .(16) 
If we put x \ TT, we obtain 



which is Euler's formula for the quadrature of the circle. 
The formula (16) also verifies obviously for #=TT; but if we 
put x we see that there is some limitation to its validity. 
The necessary modification is stated in 34. The series is 
moreover much more slowly convergent than in the preceding 
case; this is illustrated by Fig. 36, which shews the graph 
of TT x together with that of the function represented by 
the first eight terms of the series. For any value of x other 
than we can obtain an approximation as close as we please, 
provided we take a sufficient number of terms, but the smaller 
the value of x the greater will be the number of terms required 
to attain a prescribed standard. 



FOUKIER'S THEOREM 



91 




Fig. 36. 



The preceding illustrations, with the diagrams, afford at 
all events a presumption in favour of the theorem in question, 
but shew at the same time that it is subject to some restric- 
tions. The theorem admits of independent mathematical proof 
under certain conditions as to the nature of the "arbitrary" 
function f(x). We shall, however, not enter upon this, but 
shall content ourselves with the following formal statement : 

If we form the sum of the first m terms of the series (2), 
and write 



fm (#) = AI sin x + A z sin 2x + ... + A m sin mx, (18) 



where 



A 8 = I f(x)sinsxdx, ............ (19) 



it may be shewn that, for any assigned value of x in the range 
from to TT, the sum f m (x) will tend with increasing m to the 
limit f(x), provided the function f(x) is continuous throughout 
the above range, has only a finite number of maxima and 
minima, and vanishes for x = and x = IT. 

It will be noticed that the conditions here postulated are 



92 DYNAMICAL THEOKY OF SOUND 

fulfilled as a matter of course by any function which it is 
natural to assume as representing the initial form, or the 
initial velocity, of a tense string. We also see that the 
difficulty met with in the case of (16) can be accounted for 
by the fact that the function does not vanish for x = 0. 
An extension of the statement to meet such cases will be 
given presently ( 34). 

33. The Cosine- Series. 

The theory of the longitudinal vibrations of rods, or of 
columns of air, leads, in addition, to a similar theorem relating 
to the expansion of an arbitrary function in a series of cosines. 
The formal statement is now as follows : 

If we write 

fm (#) = A + A l cos x + AZ cos 2% + . . . + A m cos mx, (1) 



where 

whilst for s > 



2 
= -l f(x)co$sxdx, (3) 



it may be shewn that as m increases the sum f m (x) will tend to 
the limit /(#), provided f(x) is continuous throughout the 
range from to TT, and has at most a finite number of 
maxima and minima. There is now no restriction as to the 
values of /(O) and /(TT). 

If the determination of the effect of special initial conditions 
in a longitudinally vibrating bar which is free at both ends 
were as interesting a problem as it is in the case of strings 
we should have recourse to the cosine-series. 

34. Complete Form of Fourier's Theorem. Discon- 
tinuities. 

The question arises as to what is represented by the 
sine-series or the cosine-series, supposed continued to infinity, 
when x lies outside the limits and TT. The answer is supplied 
by the consideration that both series are periodic functions 
of x } the period being 2?r, whilst the former is an odd, the 



FOURIER'S THEOREM 93 

latter an even function of x*. This is illustrated by the 
annexed graphical representations, in which f(x) is given 
primarily only for the range TT, but is continued in one 
case as an odd and in the other as an even periodic function 
of x. It will be noticed that in the former case the stipulation 
that f(x) is to vanish for x = and x = TT is necessary if 
discontinuities are to be avoided. 




Since any function f(x) given arbitrarily for values of x 
ranging (say) from TT to TT can be resolved into the sum of an 
even and an odd function, viz. 

/(*) = i !/(*)+/(-*)} +i {/(*)-/<- *)). -<i) 

we derive the more general theorem that the sum 

f m (x) A + A! cos x + A z cos 2# + . . . + A m cos mx 

4- B! sin x + B 2 sin 2x + ... + B m sin mx, ...(2) 
where 



* An "odd" function is one which is simply reversed in sign with x, 
like x 3 or sinx. An "even" function is one which is unaltered in value 
when the sign of x is changed, like x- or cos x. 



94 DYNAMICAL THEORY OF SOUND 

whilst for s > 

If* 1 1 f" \ 

A s = - I {/(#) +/( a))} cos sxdx = I f(x) cos s#cfo, 

TTJo " TTJ - n I ^ 

i t w i r* 1 

B 8 =-\ \f(x) f(- x)} sin s#cfo? = - I /(#) sin s#d#, 

TTJo * TTJ _ ff 

tends with increasing m to the limit /(a?), provided f(x) is 
continuous from X TT to # = TT and has at most a finite 
number of maxima and minima, and provided also that 
/( TT) =/(TT). For values of x outside this range the limit 
represents, under these conditions, a periodic function of 
period 2?r. This is the complete form of Fourier's Theorem, 
and includes the others as special cases. 

We should be led directly, on physical grounds, to this form 
of the theorem if we were to investigate the "longitudinal" 
vibrations of the column of air in a reentrant circular tube. 

We have so far supposed the function /(#) to be continuous, 
as well as finite, even when continued beyond the original 
range as a periodic function. But the theorems hold, with 
a modification to be stated immediately, even if f(x) have 
a finite number of isolated discontinuities. In such a case 
the series f m (x) still converges, with increasing m, to the value 
of /(#), except at the points of discontinuity. But if a be 
a point where f(x) abruptly changes its value, the sum f m (a) 
tends to the limit 



where f(a 0) and/(a + 0) represent the values of f(x) at 
infinitesimal distances to the left and right, respectively, of the 
point a. For example, in the case of the sine-series 32 (3), 
if f(x) does not vanish when x = or when X = TT, there is 
discontinuity at these points in the periodic function, and 
the series / m (0), for example, has the limit 0, which is 
the arithmetic mean of the values of the continued function 
on the two sides of the point #=0. This is illustrated in 
Fig. 36. 

35. Law of Convergence of Coefficients. 
It remains to say something as to the law of decrease of 
the successive terms. It is evident at once that under the 



FOURIER'S THEOREM 95 

litions laid down the values of the coefficients A 8 and B 8 
must ultimately diminish indefinitely as s increases, owing to 
the more and more rapid fluctuation in sign of cos so; and 
sin sx, and the consequent more complete cancelling of the 
various elements in the definite integrals of 32 (5) and 
33 (3). 

More definite results have been formulated by Stokes. 
The following statement must be understood to refer to the 
function as continued in the manner above explained; and 
care is necessary, in particular cases, to see whether discon- 
tinuities of f(x) or its derivatives are introduced at the 
terminal points of the various segments: 

If f(x) have (in a period) a finite number of isolated 
discontinuities, the coefficients converge ultimately towards 
zero like the members of the sequence 

1, 2> 3> 4> ! 

This is exemplified by 32 (16) and Fig. 36. 

If f(x) is everywhere continuous, whilst its first derivative 
f'(x) has a finite number of isolated discontinuities, the con- 
vergence is ultimately that of the sequence 

ill 1 

' 2 2 ' 3 2 ' 4 2 ' * 

This is illustrated by 32 (12) and Fig. 35. 

If f(x\ f(x) are continuous, whilst /"(#) is discontinuous 
at isolated points, the sequence of comparison is 

l > %3> ^3> #> 

as in the case of 32 (8). And, generally, if f(x) and its 
derivatives up to the order n 1 inclusive are continuous, whilst 
the nth derivative has (in a period) a finite number of isolated 
discontinuities, the convergency is ultimately as 

1 1 1 



The nature of the proof, which is simple, may be briefly 



96 DYNAMICAL THEORY OF SOUND 

indicated for the case of the sine-series. We have, by a partial 
integration, 

2 f 77 
A s I f(x) sin sx dx 



1 F2 "1 2 C n 

= -- -/O)coss# + I f(x)w$sxdx, ...(4) 



where the integrated term is to be calculated separately for 
each of the segments lying between the points of discontinuity 
of /(#), if any, which occur in the range extending from x = to 
# = TT inclusively. For example, if as in 32 (14) the only 
discontinuity is at x 0, its value is 2/(0)/S7r. In any case 
there is, for all values of s, an upper limit to the coefficient of 
l/s in the first part of (4); we denote this limit by M. The 
definite integral in the second term tends ultimately to 
zero, as s increases, owing to the fluctuations in sign of cos sx. 
Hence A 8 is ultimately comparable with M/s. If there is no 
discontinuity of /(#), even at the points x = 0, x TT, the first 
term in the above value of A 8 vanishes, and continuing the 
integration we find 

A s = -- \- f O)sin sx\ - -f- fjT<*) sin ** dx. . . .(5) 
r \_Tr J s "^J Q 

In the first part, regard must be had to the discontinuities of 
/'(#), if any. Denoting by M the upper limit of the coefficient 
of l/s 2 , we see that A s is ultimately comparable with M/s*, the 
second term in (5) vanishing in comparison, by the principle 
of fluctuation. The further course of the argument is now 
sufficiently apparent. 

36. Physical Approximation. Case of Plucked String. 

It has been thought worth while to state Fourier's theorem 
with some care, although we do not enter into the details of the 
mathematical proof, which is necessarily somewhat intricate, 
owing to the various restrictions which are involved*. 

From a physical point of view the matter may be dealt with, 
and perhaps adequately, in a much simpler manner. To explain 
this, it is best to take a definite problem, for instance that of 

* The most recent English treatise on the subject is that of Prof. 
H. S. Carslaw, Fourier's Series and Integrals, London, 1906. 



FOURIER'S THEOREM 97 

the plucked string ( 26). The differential equation, and the 
terminal conditions, are satisfied by the finite series 



. TTX Tret . fax 

y=A l sm -j- cos -j- 4- A z sin j- cos -. \- 



. mirct 

+ A m sm j- cos j , ...(1) 

each term of which represents a normal mode of vibration. 
This makes the initial velocity zero, whilst the initial form is 

. TTX . fax . rrnrx 

y = A 1 sm -j- 4- A 9 mn ; + ... + A m smj . ...(2) 

II V 

The question we now have to consider is, how to determine the 
coefficients A lt A z , ... A m so that (2) may represent, as closely 
as may be, a prescribed initial form 

y-/() ......................... (3) 

There are many reasons why, from the physical point of 
view, we may be content with an approximate solution of the 
problem. Leaving aside such questions as the resistance of the 
air and the yielding of the supports at the ends of the string, we 
have still to remember that in substituting a mathematical line 
of matter, capable only of exerting tension, we have considerably 
over-idealized the circumstances. In the higher normal modes, 
at all events, the imperfect flexibility, and the uncertainty as to 
the true nature of the terminal conditions, render this representa- 
tion somewhat inadequate, so that a solution which professes to 
determine these modes accurately is open to the criticism that 
it attempts too much. Again, the assumed initial form in 
which two straight pieces meet at a point, is one which can 
only be approximately realized ; if we go too far in this direction 
we should produce a permanent bend, or kink, in an actual 
wire. 

The determination of the coefficients in the finite series 
(2) will depend on the kind of approximation aimed at. For 
example, we might divide the length of the string into ra + 1 
equal parts, and choose the coefficients 'so that the functions (2) 
and (3) should be equal at the m dividing points. The curves 
represented by these equations will then intersect in m points in 
addition to the ends. Another method is to make the sum of 

L. 7 



98 DYNAMICAL THEORY OF SOUND 

the squares of the errors involved in the substitution of (2) for 
(3) as small as possible. Thus if, for shortness, we replace I by 
TT, we have to choose the coefficients so as to make the integral 



/(#) ( A l sin x + AZ sin 2# 4- . . . + A m sin mx)} z dx (4) 
a minimum. If we differentiate with respect to A 8 we get . 
)(A 1 smx+A z sin2x + ...+A m smmx)}smsxdx=(), (5) 

or, by 32 (4), 

2 /**' 
A 8 =- \ f(x)smsxdx ................ (6) 

TTjQ 

Hence this method of least squares, applied to the expression 
(2) consisting of a finite number of terms, gives precisely the 
values of the coefficients which were obtained by Fourier's 
process*. Each coefficient is determined by itself, and the 
effect of adding more terms to (2) is to improve the approxima- 
tion, without affecting the values of the coefficients already 
found. If we revert to general units, the formula (6) is 
replaced by 

2 l / s S7TX 
S= ~ 



In the case of the plucked string, the form to which we 
endeavour to approximate is 

y = Pxla [0<x<a], y = # (I -x)l(l- a) [a<x<l\ (8) 

The result is obtained at once from 32 (11) if we write jrx/l 
for x, and therefore irajl for a, I for TT, and introduce the factor fi. 
Thus 

, 2/9Z 2 . STTO, 



as stated in 26 (1). The nature of the approximation is 
illustrated in Fig. 35. 

37. Application to Violin String. 

To apply the method to the problem of the violin string 
( 27), we take as origin of t the instant when the point Q in 

* This theorem is due to A. Toepler (1876). 



FOURIER'S THEOREM 



Fig. 32 starts from A to describe the upper parabolic arc. At 
this instant we have y = 0, everywhere, whilst 



(i) 



We therefore begin with the finite series 



. 7TX . TTCt . 

y A l sm-j- sin -j- + A^ sm j- sm -j I- ... 

. mirx . mirct /sn 

4- A in sm sm . ...(2) 



This satisfies the differential equation, and makes y = for t = 0. 
It only remains so to determine the constants that the series 



. D . D . mirx /ON 

1 sm -j- + J5 2 sm j~ + . . . + .O TO sin y , ...(3) 

where B t = --A s , ..................... (4) 



may represent the initial distribution (1) of velocity, as nearly 
as possible. The determination of B 8 has virtually been made 
in 32 (15). With the necessary modifications of notation we 
find 



as stated in 27. The graph of the initial velocity, and the 
approximation attained by taking the first eight terms of the 
series (3), are shewn in Fig. 36. 

It will be noticed that our approximation has even an 
advantage over the result obtained by carrying the series to 
infinity. In the latter case, the initial velocity, as represented 
by (1), is discontinuous when x = 0, being zero for # = 0, but 
equal to 4y9 c/Z when x differs ever so little from 0. The 
idealized representation of the motion in 27 is in this 
respect imperfect ; the parabola in Fig. 32 should be slightly 
modified so as to touch the line AB at its extremities. 

38. String Excited by Impact. 

As a final example we take the case of a string started by 
an impact, as in 26. We begin with the case of a force 

72 



100 DYNAMICAL THEORY OF SOUND 

distributed continuously in -space and in time, the differential 
equation being 



as in 28. Suppose, in the first place, that 

y * /*\ rjrx /\ - ^Tne .c *+\ m7nr . . 

- =/j (t) sm -j- +/, (0 sin -j- + . . . +f m (t) sin -y- , (2) 

the coefficients being known functions of t. The equation (1) 
is then satisfied by 



7TO) . ?? mirx 

-- + 77 2 sm-- + ...+7; m sin - , ...(3) 



provided ***/. ................... (4) 

The solution of this equation has been given in 8. If we 
assume that r) s = 0, 17, = for t = oo , and that/^ (t) is sensible 
only for a finite range of t, the resulting value of rj 8 is 



57TC^ f 00 ,. ,.. . STTCt 



If as a particular case we put 



we have ^ = e-^^sin, ........... (7) 

STTC L 

by 8 (18). As a function of t, Y now follows the special law 
indicated by the last factor in (6), at every point of the string, 
but we have not yet made any special assumption as to the 
distribution of the force over the length. Its time-integral is 
given by 



1 T 00 

P J - 



TT7 , ~ . ~ . ^ . mirx 

Ydt= Ojsm-y- + O 2 sm- 7 - + ... + C m sm y- . (8) 



We may now seek to determine the values of the coefficients 
so that this expression may be sensible only in the neighbour- 
hood of the point x a. We assume, then, that 



PJ -oo 



FOURIER'S THEOREM " *101 

where <f> (x) vanishes except between the limits a e and a + e, 
say. The formula (7) of 36 then gives 

2 [ l . , N . STTX , 2 f a+t , , . . S7TX , , . 

C g = j <f> O) sm r <te = 7 # W sm ~T~ * ( 10 ) 

& ./ ^ *J a-t 

If be small, the series thus obtained converges at first very 
slowly, and a great many terms might have to be taken to 
secure a reasonable approximation. In the terms of lower 
order we have 

2 . STra f a+e . . , 2u . sira 
0,-jgUi-y-J <0r)<fo=-^sm-y-, ...(11) 

/a+c 
<f>(x)dx, (12) 
a 

i.e. /Lt represents the total impulse. The corresponding term in 
the value (3) of y is 

2/4 1 .. . STra . STTX . sirct /1Q , 

.- e-*"* 11 . sm -j- . sm =- . sm = . . . .(13) 
PTTC s III 

But however small e may be, so long as it is not evanescent, 
the value of G 8 given by (10) will ultimately tend to zero with 
increasing 5, owing to the more and more complete mutual 
destruction of positive and negative elements under the integral 
sign. This shews the effect of diffusing the impulse over a 
small but finite portion of the string. 

The case of an instantaneous local impulse is obtained by 
putting r=0(cf. 26). 

39. General Theory of Normal Functions. Har- 
monic Analysis. 

The space which has been devoted to Fourier's theorem is 
no more than is warranted by its importance, especially in rela- 
tion to the theory of strings, but it is well to remember that 
from the standpoint of the theory of vibrations the theorem is only 
one out of an infinite number which can be based on the same 
kind of physical considerations. Every vibratory system has its 
own series of "normal functions," as they are called, which 
express the configuration of the system in the various normal 
modes. In the case of a uniform string, or of the doubly-open 
organ pipe, these functions happen to have he simple form 



102 DYNAMICAL THEORY OF SOUND 

sin (STTX/I), or cos (sirx/l), respectively. More complicated forms 
will be met with when we come to the theory of transverse 
vibrations of bars, and to that of membranes ; and even in the 
cases just mentioned the simplicity of type would at once 
disappear if the uniformity of line-density, or of cross-section, 
respectively, were departed from. In some problems, indeed, 
of considerable interest, e.g. that of the vibrations of a rectangular 
plate, the precise form of the functions has still to be discovered. 
But in any case the functions theoretically exist ; and on the 
principle that any free motion whatever of the system consists 
of some combination or other of the various normal modes, it 
must be possible to express any arbitrary initial state, and 
therefore any arbitrary function of position in the system, by a 
series of normal functions. Such preeminence as attaches to 
Fourier's theorem is, from the present point of view, due merely 
to the fact that in it we have the simplest exemplification of 
this principle in the case of a continuous system, and the one 
where the physical induction has been most fully corroborated 
by independent mathematical proof. It may also be added that 
it is only in the case of strings that the calculation of the 
effect of particular initial conditions has any great interest. 

There is however another point of view from which the 
resolution of a function into a series of sines or cosines of the 
variable is of peculiar importance, viz. when we are dealing with 
functions of the time. The dynamical reason for this has 
already been dwelt upon ( 19). 

When a function f(t) is known to be periodic, of period T, 
its resolution by Fourier's theorem is 



/v,x r 

j (t) = A + A l cos -- f- A z cos -- h A a cos -- [- . . . 



-f B l sin - - + J5 2 sin h B 3 sin ^-^ + . . . , (1) 

T T T 

where A =*( T f(t)dt, (2) 

whilst for s > 0, 

/A p nH fl4 T) / f(t\ oin /O\ 

^t^ oos ac, xj, - i y ^c^ sin . v* 5 / 



FOURIER'S THEOREM 103 

This is of course merely a restatement of the theorem of 
34, with the necessary changes of notation. It will be noticed 
that A represents the mean value of the function. 

We have already been led to formulae of the type (1) as 
expressing the motion at any assigned point of a freely vibrat- 
ing string, the period r being equal to 2l/c. Another important 
acoustical application is to the analysis of a periodic current of 
air, as in the siren, or the reed-stops of an organ ( 90). Again, 
in the case of electromagnetically driven tuning forks, a periodic 
current can powerfully excite, not only a fork in unison with 
itself, but also others whose natural frequencies are respectively, 
twice, three times, ... as great. This is due to the fact that the 
disturbing force is of the type (1), the selective resonance taking 
place according to the principles of 9. 

Various mechanical contrivances for resolving a given 
periodic curve into its simple-harmonic constituents, and 
conversely for compounding a number of independent sine- 
and cosine-curves whose periods are as 1, , J, ..., have been 
devised by Lord Kelvin and others. From the standpoint of 
the present subject the most remarkable of these is perhaps 
the machine constructed by Prof. A. A. Michelson, in which 
provision is made for as many as 80 constituents*. 

It is hardly necessary to say explicitly that the resolution 
of a periodic function of t in the form (1) can only be effected 
in one way, the values of the coefficients as given by (2) and (3) 
being determinate. In particular, a series of the above type 
cannot vanish for all values of t unless its coefficients severally 
vanish. Thus in a freely vibrating string, if the motion at any 
given point x be prevented, as by touching with a camel-hair 
pencil, the coefficients of cos(s7rct/l) and sin(s7rct/l) in the 
general formula (7) of 25 must be zero, i.e. we must have 

. ' . STTX -. . STTX _ 
A 8 smj- = Q, B 8 sm-j- = (4) 

for all values of s. Unless x be commensurable with I this 
requires that A a = Q, B a = Q, and the whole string will be 

* Phil. Mag. (5), vol. XLV. (1898) ; this paper contains a number of most 
interesting examples of results obtained. The construction is also explained in 
his book On Light Waves and their Uses, Chicago, 1903. 



104 DYNAMICAL THEORY OF SOUND 

reduced to rest. In the excepted case the conditions (4) are 
satisfied independently of the values of A s and B s whenever 
sin (STTX/I) = 0, i.e. those normal modes remain unaffected 
which have a node at the point touched. 

A question arises as to the effect of non-periodic forces on 
a dynamical system. For the reason already so often insisted 
upon, it is convenient, whenever possible, to resolve the force 
into a series of terms of the type 

A cos pt + B sin pt ................... (5) 

Each element A cospt or Bsmpt then produces throughout 
the system its own effect, viz. an oscillation of the same type 
and period, the configuration and its amplitude depending on 
the speed p. In some cases the resolution presents itself quite 
naturally, as for example in the theory of the tides. The 
disturbing effect of the sun and moon, when account is taken 
of their varying declinations, and of the inequalities in their 
orbital motions, can be sufficiently represented by a series of 
terms of the type (5). It follows that the tide-height at any 
particular place must be expressed by a series of like character, 
in which the values of p are known. The theoretical determina- 
tion of the coefficients is out of the question for the actual 
ocean, with its variable depth and irregular boundaries, but 
their values can be inferred a posteriori with more or less 
accuracy from a comparison of the formula with observation, 
and when once ascertained can be used for prediction*. 

When the disturbing force is perfectly arbitrary in character, 
without any obvious periodic elements, the question is more 
complicated. There is a form of Fourier's theorem specially 
appropriate to this case, but its application is usually difficult, 
and it is simpler to have recourse, as in 38, to the formula (12) 
of 8. The objection that this implies a knowledge of the 
whole previous history of the system is met if we introduce 
the consideration of damping, which is in reality always present. 
The equation 

.................. (6) 



* For an elementary account of the matter see Sir G. H. Darwin, The Tides, 
London, 1898. 



FOURIER'S THEOREM 105 

may be written 



provided n' 2 = n 2 -JA; 2 ...................... (8) 

Hence, by the formula referred to, 

cos n'tdt 



~~ **' 



e~ **' cos n't je* kt f() sin n'tdt. . . .(9) 



If we have x 0, x for t oo , the limits of integration 
are oo and t. For instance, the value of x when t = 
becomes 

x = -l,r e* kt f(i)sinn'tdt. (10) 

Owing to the presence of the exponential factor it is only for 
a certain range of negative values of t that the function under 
the integral sign has as a rule an appreciable value. In other 
words, the effects of the action of the force prior to a certain 
antecedent epoch have practically disappeared. 



CHAPTER IV 

BARS 

40. Elementary Theory of Elasticity. Strains. 

We require a few elementary notions from the theory of 
elasticity. As regards the purely geometrical study of de- 
formations, or "strains," it is usual to begin with the con- 
sideration of a body in a state of uniform or "homogeneous" 
strain. This is sufficiently defined by the property that any 
two lines in the substance which were originally straight and 
parallel remain straight and parallel, although their direction 
relative to other lines in the substance is usually altered. 
A parallelogram therefore remains a parallelogram, and it 
easily follows that the lengths of all finite parallel straight 
lines are altered in the same ratio ; this ratio will however 
usually be different for different directions in the substance. 

It can be shewn that there are three mutually perpendicular 
directions in the substance which remain mutually perpen- 
dicular after the deformation; these are called the "principal 
axes" of the strain. It is unnecessary, for our purposes, to 
give the formal proof, as the existence of such axes will be 
in evidence in such simple cases as we shall meet with. 
It follows from this theorem that any originally spherical 
portion of the substance is deformed into an ellipsoid whose 
axes are in the directions of the principal strain-axes. 

If PQ, P'Q' denote any straight line in the substance, 
before and after the strain, the ratio of the increase of length 
to the original length, viz. 



PQ 



BARS 



107 



is called the "extension"; it will in general be different for 
different directions of PQ. In the theory of elastic solids, e is 
always a very minute fraction. We denote by e 1} e 2 , 6 8 , the 
extensions in the directions of the principal axes. 

The ratio of the increase of volume to the original volume 
is called the " dilatation." Denoting it by A, and considering 
the change of volume of a cubical block whose edges are along 
the principal axes, we have 



or A = 6! +6 2 + 6 3 , (2) 

the products of small quantities being neglected. 

There are two special types of homogeneous strain which 
require notice. First, suppose 6! = 6 2 = 6 3 , = 6, say. Any origin- 
ally-'^pherical portion of the sub- 
stance then remains spherical, and 
the extension is therefore the same 
in all directions. The strain may 
accordingly be described as one of 
uniform extension; and we note 
that = JA. 

Again, take the case of e l = 6 2 , 
= 6, say, whilst 6 8 = 0, and therefore 
A = 0. A square whose diagonals 
AOC, BOD are parallel to the axes 
1, 2 is converted into a rhombus A'B'C'D ', and since 




to the first order, the lengths of the sides are unaltered. Also 

...(3) 



tan A' OB' = i-t? = tan (J TT + e), 



1-e 

so that the angles of the rhombus are \ IT 2e. Another view 
of this state of strain is obtained if we imagine the rhombus 
A'B'C'D' to be moved in its own plane so that A'B' coincides 
with AB. This is legitimate, since no displacement of the 
body as a whole affects the question. We then see that any 
two planes of the substance parallel to AB and the axis 3 are 
displaced relatively to one another, without change of mutual 



108 



DYNAMICAL THEORY OF SOUND 



distance, by an amount proportional to this distance. This 
kind of strain is called a " shear," from the fact that it is of the 
type which tends to be set up by the action of the two edges 
of a pair of shears. The " amount " (77) of the shear is specified 
by the relative displacement per unit of mutual distance, i.e. by 
the ratio DD'/AD, or 2e, in the first part of Fig. 39. Again, 
by moving B'C' into coincidence with BC, we might prove that 
the strain is also equivalent to a shearing of planes parallel to 
BC and the axis 3, in the direction of BC. This is shewn in 
the second half of the figure. 



D D' 



C C A A 



D D 



A B B C 

Fig. 39. 

41. Stresses. 

The name " stress " is applied to the mutual action which 
is exerted across any ideal surface S drawn in a body, between 
the portions of matter immediately adjacent to $ on either side. 
We are here concerned with molecular actions sensible only over 
an exceedingly short range, so that the portions of matter in 
question are confined to two exceedingly thin strata, whose 
common boundary is S. The resultant force on a small portion 
of either stratum may then be taken to be ultimately pro- 
portional to its area, and the intensity of the stress is 
accordingly specified by the force per unit area. This force 
may be of the nature either of a push or a pull, and may be 
normal or oblique, or even tangential to the area. 

For simplicity, it is usual to begin with the notion of a 
state of uniform or " homogeneous " stress, i.e. the stress over 
any plane is assumed to be uniform, and the same in direction 
and intensity for any two parallel planes. It will of course in 
general be different for planes drawn in different directions. 
It may be shewn that there are then three mutually perpendic- 



BARS 



109 



ular sets of planes such that across each of these the stress is 
in the direction of the normal ; but for a reason already indicated 
we need not stop to prove this theorem. The planes in question 
are called the " principal planes " of the stress, and the corre- 
sponding stress-intensities are called the "principal stresses." 
They are usually reckoned as positive when of the nature of 
tensions; we denote them by Pi,p 2 ,p 3 . 

There are certain special types of stress to be noticed. 
First let j9 1 =j9 2 = PS- The stress across every plane is then in 
the direction of the normal, and of uniform intensity, as in 
hydrostatics. 

Next, let^! = pt,=p, say, whilst p 3 = 0. Consider a unit 
cube whose faces are parallel to the principal planes. The 
portion included between the faces represented by AB, DA, in 
the figure, and the diagonal plane represented by BD, is in 
equilibrium under three forces. Two of these forces are 
parallel and proportional to DA and A B, viz. the forces on AB 
and DA, respectively. The third force is therefore along and 





Fig. 40. 



Fig. 41. 



proportional to BD\ and its amount (CT) per unit area is p. 
A similar result holds with respect to the diagonal plane A C. 
A cube four of whose faces are parallel to these diagonal planes 
is in equilibrium under tangential stresses, in the manner 
shewn. This type is accordingly called a "shearing stress." 
Its amount (CT) is specified by the tangential force per unit 
area on the planes in question. 



110 DYNAMICAL THEOEY OF SOUND 

In general the states of strain and stress in a body are not 
uniform, or " homogeneous," but vary continuously from point 
to point; but the above notions are still applicable to the 
infinitely small elements into which the body may be conceived 
to be divided. 

42. Elastic Constants. Potential Energy of Defor- 
mation. 

The theory of strains is a matter of pure geometry ; that of 
stresses one of pure statics. When we come to connect the two 
we require some physical assumption. The usual hypothesis, 
known as " Hooke's law," * is that the stresses are linear 
functions of the strains. This law ceases to hold, even approxi- 
mately, when the strains exceed certain values called the 
"elastic limits"; but for the purposes of acoustics it may be 
adopted without hesitation, on account of the excessive minute- 
ness of the varying strains with which we are concerne^f. 

In an "isotropic" substance, i.e. one in which there is no 
distinction of properties between one direction and another, the 
principal axes of strain must evidently coincide with those of 
stress. Moreover the principal stress p^ must involve the 
principal strains e 2 , e 3 symmetrically, and so on. The most 
general assumption consistent with this requirement, and with 
Hooke's law, is of the form 



(1) 



where X, /j, are constants depending on the nature of the 
material]:. It will be noticed that e l5 e a , e 3 are pure ratios and 
that the dimensions of \, p are therefore those of stress, or force 

* Robert Hooke (1635 1703), professor of geometry at Gresham College 
16651703. 

t If Hooke's law were sensibly departed from, the frequencies of the normal 
modes of a vibrating bar would no longer be independent of the amplitude. 
Since the ear is very sensitive to variation of pitch, this would easily be detected. 
This remark is due to Stokes. 

J There is a great diversity of notation as regards these constants. The 
above symbols are those introduced by G. Lame" (1795 1870), professor of 
physics at the Ecole poly technique 1832 44. 



BAKS, 111 

divided by area, viz. [ML^T^] if [M], [L\ [T] denote the units 
of mass, length, and time. 

There are various combinations of the constants X, p which 
are important in physics, as well as in technical mechanics. In 
a uniform dilatation we have p 1 = p 2 = p 3 ( = p, say), l = e 2 = e s 
(= JA), whence 

p = (X + |^)A ...................... (2) 

Hence if we write * = \ + ft, ........................ (3) 

K will denote the " volume-elasticity " or " cubical elasticity " of 
the substance, i.e. the ratio of the uniform stress to the dilatation 
which it involves. 

Next suppose that l = - e 2 = e, e 3 = 0, and therefore 
Pi = p^=p, p 3 = 0, which is the case of a pure shear, 
involving a shearing stress. According to the investigations 
of 40, 41 the shearing stress is vr = p, and the shear is 
rj = 2e. Hence, from (1), 

*r = /"7, ........................... (4) 

i.e. fj. denotes the ratio of the stress to the strain (appropriately 
measured) in a pure shear. It is called the " rigidity " of the 
substance. 

Again, suppose we have a bar stretched lengthways, but free 
from lateral stress. We put, then, in (1), p% = 0, p 3 = 0. This 
leads to 

pi = Ee l9 .............. . ............ (5) 



where ^ ................ 

X-f p 



This ratio of the longitudinal stress to the corresponding 
extension is called "Young's modulus" of elasticity; its 
technical importance is obvious. We also find 

2 = e 8 = -o- 1 , ........................ (7) 



Where 



This fraction accordingly measures the ratio of lateral con- 
traction to longitudinal extension under the circumstances 
supposed ; it is known as " Poisson's ratio." * 

* 8. D. Poisson (17811840). His chief contributions to acoustics relate to 
the vibrations of membranes and plates, and to the general theory of sound- 
waves in air. 



112 DYNAMICAL THEORY OF SOUND 

By solving the equations (1) we can express e l} e 2 , e 3 as linear 
functions of p lt p z , p a . It is obvious, however, that the formula 
for e l must involve p z and p s symmetrically; and from this 
consideration, and from the physical meanings of the constants 
E and cr, it follows immediately that the result must be 
equivalent to 

E t *>p l -#(p t +p t ), ; 

............... (9) 



Of the various elastic constants and their combinations, 
one or other may appear specially important, according to the 
nature of the question in view, and this may account for the 
great diversity of notations which has arisen. In any case two 
independent quantities are necessary and sufficient to define the 
elastic behaviour of an isotropic substance. From a physical 
standpoint K and p, might appear to be the most fundamental ; 
whilst as regards facility of direct measurement preference may 
be given to E and //,, whence K and cr can be derived by the 
formulae 

fiE E 

" = 9 P -3E' ^V" 1 ' ............ (1 

which follow easily from (3), (6) and (7). On a particular 
hypothesis as to the ultimate structure of an elastic solid 
Poisson was led to the conclusion that the two elastic constants 
are not independent, but are connected by an invariable relation, 
which in our notation is expressed by X = /JL. This makes 

=/*, E=ln, cr = J ................ (11) 

On experimental grounds Wertheim (1848) proposed the 
relation X = 2/A, which makes 

* = f ft E = K, <r = J ................ (12) 

More accurate methods of measurement, introduced by 
Kirchhoff* and others, support the view, which has been con- 
sistently held by English physicists f that there is no necessary 

* G. E. Kirchhoff (182487), professor of physics at Heidelberg 185475, 
at Berlin 1875 87 ; famous for his share in the discovery of spectrum analysis, 
but the author also of important memoirs on the theory of elasticity and its 
applications to the vibrations of bars and plates. 

t Notably by Green (17931841), Stokes, and Lord Kelvin. 



BARS 



113 



definite relation between X and fj,, and consequently no universal 
value of cr. We may note that in an absolutely incompressible 
medium we should have 

* = oo, E=%n, <r = i (13) 

The following table gives the results of a few determinations 
by Everett (1867). The second column gives the volume- 
density in grammes per cubic centimetre. The next three 
columns give the respective elastic constants, in dynes per 
square centimetre. These are followed in the last column by 
the corresponding values of a. The last two rows illustrate the 
fact that the elastic constants may vary appreciably in different 
specimens of nominally the same substance. 





P 


E 


/* 


K 


ff 


Steel 


7-849 


2-139 x 10 12 


8-19 xlO 11 


1-841 XlO 12 


310 


Iron (wrought) 


7-677 


1-963x1012 


7-69 xlO 11 


1-456 XlO 12 


275 


Iron (cast) 


7-235 


l-349x!0 12 


5-32 x 10 11 


964 xlO 12 


267 


Copper 


8-843 


1-234 xlO 1 * 


4-47 x 10 11 


1-684 xlO 12 


378 


Glass (1) 


2-942 


603 x 10 12 


2-40 x 10 11 


415 x 10 12 


258 


Glass (2) 


2-935 


574 x 10 12 


2-35 x 10" 


347 xlO 12 


229 



For technical purposes the elastic constants E, K, fj, are 
often expressed in gravitation measure, e.g. in grammes per 
square centimetre. The corresponding numbers in the above 
table are then divided by g. Another mode of specification, 
employed by Young, is in terms of the length of a bar of the 
particular substance, whose weight per unit area of cross-section 
would be equal to the modulus in question when expressed in 
gravitation measure; this is called the "length-modulus." Thus 
if L be the length-modulus of extension of a bar free to contract 
laterally we have 

E= ffP L (14) 

Taking g 981, the above table gives, in the case of steel, 
L = 278 x 10 6 centimetres. 



114 DYNAMICAL THEORY OF SOUND 

The potential energy (W) per unit volume of a strained 
isotropic substance may be found by calculating the work done 
by the stresses on the faces of a unit cube, on the hypothesis 
that the strains increase from zero to their final values keeping 
their mutual ratios unchanged. The average stresses are then 
one-half the final stresses. 

Thus in the case of a uniform dilatation A we have 



In the case of a pure shear 77, 

Tr=4^ = J^ ................... (16) 

In the extension of a bar, with freedom of lateral con- 
traction, 

W=J ft e, = itf,' ......... . ......... (17) i 

In the general case we have 



, + 2 + 3 ) 2 + p (ef + 6 2 2 + 6 3 2 ) 

+ J /* {(e 2 - 6 3 ) 2 + (e 3 - 6l ) + (e t - 6 2 ) 2 ). . . .(18) 

This shews that in order that the potential energy may be 
a minimum in the unstrained state K and p must be positive. 
It is otherwise obvious from the meaning of the symbols that 
if either of these were negative the unstrained state would be 
unstable. 

43. Longitudinal Vibrations of Bars. 

We take the axis of x along the bar, and denote by x + f 
the position at time t of that cross-section whose undisturbed 
position is a?, so that f denotes the displacement. An element of 
length is then altered from Sac to S(#+ f), or (1 + ')Sa;, where 
the accent denotes differentiation with respect to x. Equating 
this to (1 + e) 8x, we have 

.-!* ............................ CD 

dx 

The tension across the sectional area (&>) is therefore Eeco. 
The acceleration of momentum of the mass included between 
the two cross-sections corresponding to x and x + Sx is p&Sx . %. 



BARS 115 

Equating this to the difference of the forces on the two ends, 
we have 






If the section be uniform, this reduces to 

^-^ (3) 

~** 



where c* = E/p ........................... (4) 

It will be noticed that in this investigation it is not 
necessary to assume the substance of the bar to be isotropic, 
provided the proper value of the Young's modulus be taken*. 

The result is also unaffected if the bar, or wire, be subject 
to a permanent longitudinal tension, since by Hooke's law 
the stress due to the extension (1) may be superposed on the 
permanent tension, so long as the limits of perfect elasticity 
are not transgressed. 

As in 23 the general solution of (3) is 

g = f(ct-x) + F(ct+x), ............... (5) 

representing two wave-systems travelling unchanged in opposite 
directions with the velocity c given by (4). In terms of the 
length-modulus, we have by 42 (14) 

c = V(<?); ........................ (6) 

this is the velocity due to a fall from rest through a height J L. 
Some numerical values of c are given in the last column of the 
table on p. 119. 

The application to particular problems may be treated very 
briefly. The various cases that arise present themselves in 
a more interesting form when we come to the vibrations of 
columns of air. 

In the case of a rod or wire fixed at both ends, we have 
f = for x = and x = I (say) ; and the mathematical theory 

* In an "aelotropic" or crystalline solid the values of E will be different for 
bars cut in different directions from the substance. 

82 



116 DYNAMICAL THEORY OF SOUND 

is exactly the same as in the case of the transverse vibrations of 
a string. The frequencies of the various modes are given by 

N = sc/2l, 00 

where s = l, 2, 3, .... The result is unaffected by permanent 
tension in the wire. 

When the rod is free, the condition of zero stress at the 
ends gives f ' = for x = and x = I. Introducing this condition 
in (5) we find 

F' ( ct ) = f (ct), F'(ct + l)=f(ct-l), (8) 

for all values of t. The former of these gives on integration 

F(ct)=f(ct), (9) 

no explicit additive constant being necessary since it may be 
supposed included in the value of f(ct). The second relation 
then gives 

f(ct + l)=f(ct-l)+C. (10) 

The constant G is connected with the total momentum of the 
bar in the direction of its length. We have, from (9) and (10), 

f f dx=c \\f\ct - x) +f(ct + x)} dx = cC. . . .(11) 

Jo Jo 

Since nothing essential is altered if we superpose any uniform 
velocity in the direction of the length, we may assume the 
mass-centre to be at rest, in which case (7=0. The formula 
(10) then shews that the residual motion is periodic, since 
everything recurs when t increases by 2l/c. 

In the analytical process for ascertaining the normal modes 
we assume that f varies as cos (nt -f e), whence 



and 



(WIT W 'T'N 
4 cos +sin ) cos (nt + e) (13) 
c c / 

The conditions that 9j-/3a? = for x and x = I require B 0, 
sin (nl/c) = 0, whence 

nJ/c = 6-7r, (14) 

where 5 = 0, 1, 2, 3, ..., the scale of periods being harmonic. 
The nodes (f = 0) are given by cos (sirx/l) = 0, and the loops, 



BARS 117 

or places of zero stress, by sin (STTX/I) = 0. In the gravest mode 
(s 1) we have a node at the centre*. 

On the principles explained in 16, 32 the most general 
free motion of the bar, under the present conditions, may be 
expressed by a series 

STTCt D . S7TCt\ S7TX 

cos j- + B 8 sin =- \ cos -y-, ...... (15) 

where 5 = 0, 1 , 2, 3, .... Thus if the bar be started from rest 
in the state of strain defined by 

= /<*) [ = 0], .................. (16) 

we have B 8 = Q; and we infer that it must be possible to 
determine the coefficient A a so that 



^ / A 

= 2,(A a 



(17) 



for values of x ranging from to /. This is the result referred 
to by anticipation in 33. 

The longitudinal vibrations of bars or wires have hardly 
any practical application of importance, except in some primitive 
forms of telephone. As regards bars, the pitch is very high 
compared with that of the transverse vibrations, which it is 
difficult to avoid exciting simultaneously. Again, if we compare 
the frequencies of longitudinal vibration of a tense wire with 
those of the corresponding transverse modes, the ratio will be 
that of the wave-velocities, i.e. of *J(E/p) to ^(P/pco), where P 
denotes the permanent tension. If e be the extension due 
to P, we have P = Ee .(o, and the ratio is l/\/e , which is 
usually very great f. Longitudinal vibrations may be elicited 
on the monochord by rubbing the wire lengthwise with a piece 
of leather sprinkled with resin ; the resulting note is very 
shrill. 

It is assumed in the preceding theory that the extension 
and the accompanying stress are at any instant uniform over 
the cross-section ; in other words, we have assumed that the 

* The case * = needs, in strictness, separate examination. It leads to 
= AQ (l + at), which may be interpreted as an oscillation of infinitely long 
period. If the mass-centre be at rest we have A =0. 

t This comparison is due to Poisson (1828). 



118 DYNAMICAL THEOEY OF SOUND 

lateral contraction adjusts itself instantaneously through the 
thickness. This is not quite exact, as there is a certain degree 
of lateral inertia, but the error is insignificant so long as the 
wave-length is large compared with the diameter. In the 
modes of very high order it might become sensible, but these 
are in any case of no importance from the point of view 
of acoustics. ~~ A correction has been investigated by Lord 
Rayleigh. 

44. Plane Waves in an Elastic Medium. 

The theory of plane waves in an unlimited isotropic elastic 
medium is so closely analogous to that of longitudinal waves 
in a rod that it may be briefly noticed here. It is assumed 
that the state of things is at any instant uniform over any 
plane perpendicular to the direction of propagation (#). 

Such waves may be of two types, which are distinguished 
as " dilatational " or " longitudinal," and " distortional " or 
" transversal," respectively. In the former class the displace- 
ment is wholly in the direction of propagation. Denoting it 
by f, we have, in the notation of 42, 

6! = dg/diK, 6 2 = 0, 3 = 0, 

and therefore p l = (\ + 2/i) l = (K + f /*) d/dx .......... (1) 

Considering the portion of matter corresponding to unit area 
of a stratum of thickness 8x, we have 



whence = 



< 2 > 



if tt = (* + iri)lp ...................... (3) 

Some numerical values of the wave-velocity a are given on 
the next page, and it will be observed that they are in all 
cases greater than the corresponding values of c, as was to be 
expected, since the potential energy due to a given extension 
8f/3# is now greater owing to the absence of lateral yielding. 

In the second type of plane waves the displacement is 
everywhere at right angles to the direction of propagation. 
It may be resolved into two components parallel to y and z, 
respectively, which may be treated separately. Considering 



BARS 



119 



the former component (17) alone, we see that the strain at 
any point consists in a shear of amount drj/dx. The consequent 
stress across any plane perpendicular to Ox is parallel to Oy, 
and its intensity is fidrj/dx. Hence forming the equation of 
motion of a portion of matter defined as before we have 



or 



if V=dp (5) 

Some values of the wave-velocity b are tabulated below. 

Wave-velocities (metres per second). 





a 


b 


c 


Steel 


6-11 xlO 3 


3-23 x lO 3 


5-22 x 10 3 


Iron (wrought) 


5-68 x 10 3 


3-16x103 


5-06 x 10 3 


Iron (cast) 


4-81 x 10 3 


2-71 x 10 3 


4-32 x 10 3 


Copper 


5-08 xlO 3 


2-25 xlO 3 


374X10 3 


Glass (1) 


5-00 xlO 3 


2-86 x 10 3 


4-53X10 3 


Glass (2) 


474 x 10 3 


2-83 x 10 3 


4-42 x 10 3 



It may be shewn that any local disturbance in an unlimited 
elastic medium breaks up into two waves, diverging with the 
velocities a and 6, which tend ultimately to assume the 
" longitudinal " and " transverse " characters, respectively. 
The theory is historically important in relation to Optics, 
but in our present subject great caution is necessary in 
drawing inferences as to the propagation of waves in limited 
solids. We have already seen that in a cylindrical or prismatic 
rod the velocity of longitudinal waves is quite distinct from a, 
and the theory becomes altogether different in the case of 
flexural vibrations, to be referred to presently. In these cases 
a modification was of course to be expected, since the wave- 
length is understood to be large compared with the dimensions 



120 DYNAMICAL THEOBY OF SOUND 

of the cross-section. But even in the other extreme, when all 
the dimensions of the body are large compared with the wave- 
length, the circumstances may be profoundly modified by the 
existence of a free boundary. A new type of waves, called 
after the discoverer the " Rayleigh waves " (1885), make their 
appearance, and under some conditions may become, from the 
observational point of view, predominant. These are surface 
waves in which the agitation penetrates only to a relatively 
small depth. Their velocity is somewhat less than that of 
the distortional waves ; thus for an incompressible solid it 
is *9554 b, whilst on Poisson's hypothesis (cr = J) it is '91946. 
In modern observations of the tremors due to distant earth- 
quakes three phases of the disturbance are often recognized. 
The first is interpreted as due to the arrival of the dilatational 
waves, propagated directly through the substance of the earth, 
the second as due to that of the distortional waves, also 
propagated directly, and the third to that of the Rayleigh 
waves, which have travelled over the surface and are therefore 
delayed more than in proportion to the difference of wave- 
velocity*. The latter waves as they spread over the surface 
are less attenuated than the former, which diverge in three 
dimensions. It has even been attempted to deduce estimates 
of the volume-elasticity and rigidity of the materials of the 
earth from the various wave-velocities, as inferred from the 
seismic records f. 

45. Plexural Vibrations of a Bar. 

We proceed to the transverse jv&F 

vibrations of a bar naturally straight. 
To avoid unnecessary complications 
we will suppose that the bar has a 
longitudinal plane of symmetry, and 
that the flexure takes place parallel 
to this plane. We will also assume 
for the present that the total longi- 
tudinal stress on any section is zero. 
The resultant stress at a section there- Fig. 42. 

* E. D. Oldham, Phil. Tram. A, 1900. 

t Prof. A. E. H. Love, Phil. Trans. A, vol. ccvu., p. 215 (1908). 




^-% 



( M+bN 

^ *r 



121 



fore reduces to a, transverse " shearing force " F, and a couple 
or " bending moment " M. These will be functions of x, the 
longitudinal coordinate. If rj denote the lateral displacement, 
parallel to the plane of symmetry, then, resolving transversally 
the forces acting on an element Sx of the length, we have 



Y) 

or P &) o^ = o~ ...................... (1) 

dt z dx 

Again, if K denote the radius of gyration of the area of the 
cross-section G> about an axis through its centre of gravity, 
normal to the plane of flexure, the element of mass is 
ultimately a disk of area o>, thickness 8x, and moment of 
inertia pwSx . K 2 *. Since the axis of this disk has been turned 
through a small angle drj/dx from the position of equilibrium, 
the equation of angular motion is 



whence pmK = + F . .................. (2) 

Tf we eliminate F between (1) and (2) we have 



provided the sectional area 6> be uniform. 

We have next to express M in terms of the deformation 
of the bar. Consider in the first instance the case of a bar 
uniformly bent, so that its axis becomes an arc of a circle. 
It is evident from symmetry that the shearing force F now 
vanishes, and it hardly needs calculation to shew that the 
strain in any part of the cross-section will be proportional to 
the curvature. Hence by Hooke's law the resultant couple M 
will also vary as the curvature, or 

M=WIR, ........................ (4) 

where R is the radius of curvature, and 23 is a constant 
depending on the shape and size of the cross-section, and on 
the elastic properties of the material. 

* The symbol K is not required at present in its former sense as an elastic 
constant. 



122 



DYNAMICAL THEORY OF SOUND 



J 



J 




The value of 23 is found as follows. We take rectangular 
axes Gy, Gz in the plane of a 
section, the origin being at the 
centre (i.e. the centre of gravity 
of the area), and the axis of z 
normal to the plane of flexure. 
Assuming the axis of the bar, 
i.e. the line through the centres 
of the sections, to be unex- 
tended, we see that if R denote 
the radius of the circle into 
which it is bent, the length 
of a longitudinal linear element 
whose distance from the plane 
xz is y is altered in the ratio 
of R + y to R, and that the 
extension is accordingly y/R. 
The corresponding stress per 
unit area of the section is JEy/R, 
where E is the appropriate Young's modulus. The total 
longitudinal tension is therefore 

E 






Fig. 43. 



This justifies the provisional assumption that the axis (as 
above defined) is on the present hypothesis unextended. For 
the bending moment we have, taking moments about Gz, 

E 



Except in the special case just considered, viz. that of a bar 
bent statically into an arc of uniform curvature, there will be 
a shearing of cross-sections relative to one another, and also 
a warping of the sections so that these do not remain accurately 
plane. An exact investigation is out of the question, but 
enough is understood of the matter to warrant the statement 
that the additional sfam'rcs thus introduced are as a rule small 
compared with those taken account of in the preceding calcula- 
tion. We therefore adopt the formula (5) as sufficiently 



BARS 123 

accurate in all cases, provided R denote the radius of curvature 
at the point considered. 

In the present application drj/dx is a small quantity, so that 
we may put R~* = &rj/dx?, and therefore 



(6) 
Substituting in (3) we obtain 

(7) 



For most purposes this equation may be simplified by the 
omission of the second term, as we shall see immediately. 
The kinetic energy of the bar is 



The second term, which represents the energy of rotation of the 
elements, is usually negligible. 

The potential energy is found, in accordance with 42 (16), 
by integrating the expression %Ee z , =\Ey' i lJ&, first over the 
area of the cross-section, and then over the length; thus 

. .............. .(9) 



Consider for a moment the propagation of a system of waves 
of simple-harmonic profile along an unlimited rod, assuming 

v)=Ccosk(ct-x) ................... (10) 

Since everything here recurs whenever # is increased by 2-Tr/fc, 
the constant k is connected with the wave-length A. by the 
relation 

fc=27T/X ......................... (11) 

On substitution the equation (7) is found to be satisfied 
provided 



This gives the wave-velocity c, which is seen not to be a definite 
quantity fixed by the constitution of the rod, but to depend also 
on the wave-length. To trace the progress of a wave of any 
type other than (10), it would be necessary to resolve the wave- 



124 DYNAMICAL THEORY OF SOUND 

form into simple-harmonic functions of x. Each of these would 
travel with its own velocity, so -that the resultant wave-profile 
would continually alter. For this reason it would be hopeless 
to look for a general solution of (7), or even of the modified 
form (13) below, of the same simple character that we met 
with in the theory of strings ( 23), and again in that of the 
longitudinal vibrations of rods. 

A further remark is that when we substitute from (10) 
in (7), the second term is of the order &V as compared with 
the first. When the wave-length is large compared with the 
dimensions of the cross-section this is a very small quantity, 
and the term in question, which arose through taking account 
of the rotatory inertia of the elements of the bar, viz. in 
equation (2), may be neglected. It is easy to see, and it may 
be verified a posteriori, that the same simplification is legitimate 
in discussing the vibrations of a finite bar, at all events so long 
as the distance between successive nodes is large compared 
with K. We accordingly take the equation 




P 

as the basis of our subsequent work, together with the formulae 

M = E**%{, FW-E^. ...(14) 

3# a doc W 

46. Free-Free Bar. 

To ascertain the normal modes of a finite bar we assume as 
usual that 77 varies as cos (nt + e). The equation (13) of the 
preceding section then reduces to 



where m* = ri i p/K 2 E. ........................ (2) 

It is to be noted that m is of the nature of the reciprocal of 
a line. The solution of (1) is 

77 = A cosh mx + B sinh mx -f- C cos mx + D sin mx, (3) 
the time-factor being for the present omitted. The three ratios 
A:B:C:D, and the admissible values of m, and thence of n\ 
are fixed by the four terminal conditions, viz. two for each end. 



BARS 



125 



Take first the case of a perfectly free bar, of length I, say. 
If we take the origin at the middle*, these conditions are, by 
45(14), 

," = 0, ,'"=0 [c-i<] (4) 

The normal modes fall naturally into two classes ; in one of 
these 77 is an even, in the other an odd function of x. For the 
symmetrical vibrations we have 

77 = A cosh mx + C cos mx, (5) 

with the terminal conditions 

A cosh J ml G cos ^ml = 0, j 

A sinh ^ml + Csin |w = 0,J 
whence tanh ^ml= tan ^ml (7) 



+ 1 




-1 



Fig. 44. 

The roots of this equation are easily found approximately by 
graphical construction, viz. as the abscissae of the intersections 
of the curves y = tan #, y = tanh x, the latter of which is 

* This improvement on the ordinary procedure is due to Sir A. G. Greenhill, 
Mess, of Math. vol. xvi., p. 115 (1886). 



126 DYNAMICAL THEOKY OF SOUND 

asymptotic to the line y = 1. The figure shews that we have 

approximately 

) .................. (8) 



where 6? = 1, 2, 3, ..., and a s is small. It follows from (2) that 
the frequencies of the successive normal modes of symmetrical 
type are approximately proportional to 3 2 , 7 2 , II 2 , .... For a 
more exact computation of the roots we have 



_ __ m _ 

ia *--- 



where s = e~ sir ...................... (10) 

Hence a 8 = ten-> (&' 2a >) = ^ s e~^-^e~^+ .... (11) 

Since ? g is small, even for s = 1 (viz. & = "00898), this is easily 
solved by successive approximation. 
In the asymmetric modes we have 

rj = Bsinhnuc + Dsmmx, ............... (12) 

with the terminal conditions 

B sinh J ml D sin \ml 0,j 

cosh imJ-D cos JmZ = 0,J 
whence tanh \ml = tan \ml ................... (14) 

The roots of this are given by the intersections of the curves 
y = tan x, y = tanh X, the latter of which is asymptotic to the 
line y = 1 ; see Fig. 44. It appears that 

JmJ = ( + J)ir-&, .................. (15) 

where 5 = 1, 2, 3, ..., and ft is small. The corresponding 
frequencies are approximately proportional to 5 2 , 9 2 , 13 2 , .... 
For the more exact calculation we have 



1+ tan mn. tanh 
where ^ = e~ 2s7r ~ i7r ........ , .......... ...(17) 

Hence ft = tan- 1 (^ 2ft )= ?^ 8 -K^~ 6 ^ + ....... (18) 

Since i = "00039, the approximation is very rapid, even for 



BAES 127 

Combining the results for the two classes it is found that 
wZ/7T = 1-50562, 2*49975, 3'50001, ..., ...(19) 
where the values for the symmetric and asymmetric types 
alternate. The subsequent numbers are adequately represented 
by s + |. The fact that the frequencies are approximately 
proportional to 3 2 , 5 a , 7 2 , ... was ascertained, from observation 
alone, by Chladni*. 

To examine the form assumed by the bar in any normal 
mode, we require the ratio of the arbitrary constants, as deter- 
mined by (6) or (13). Thus in the case of symmetry we have 
rj = C (cos | ml cosh mx + cosh ml cos mx) cos (nt + e), (20) 
where m is a root of (7). The curve may be traced with the 
help of a table of hyperbolic functions, and the positions of the 
nodes found by interpolation. The form assumed in the 
gravest mode is shewn in Fig. 45. The nodes here are at a 
distance of '224 of the length from the ends. 




Fig. 45. 

The corresponding formula for the asymmetric modes is 
77 = G (sin J ml sinh mx + sinh ml sin mx) cos (nt 4- e), (21) 
where m is determined by (14). 

47. Clamped-free Bar. 

The next most interesting case is that of a bar clamped at 
one end and free at the other. Here also there is an advantage 
in taking the origin at the middle point of the length )-. The 
terminal conditions then are 

^=0, v = o [* = -jq (i) 

* E. F. F. Chladni, born at Wittenberg 1756, died at Breslau 1827. 
Distinguished by his experimental researches in acoustics. These are recorded 
in his book Die Akustik, Leipzig, 1802. 

t Greenhill, 1. c. 



128 DYNAMICAL THEORY OF SOUND 

at the clamped end, and 

," = 0, ,'" = O = JZ] (2) 

for the free end. In one class of vibrations we have 

77 = A cosh mx -f D sin mx, (3) 

with the conditions 

A cosh \ml D sin \ml 0,^1 ... 

A sinh ^ml + Dcos ^ ml = 0, 1 

whence coth \ml = tan \ml (5) 

This is solved graphically by the intersections of the curves 
y tan cc, y = coth x, the latter of which has y 1 as an 
asymptote ; see Fig. 44. We have, approximately, 

iwiZ = (* + i)7r + a; (6) 

where s = 0, 1, 2, 3, ..., and a/ is small. This leads to 

tana/^- 2 "'', (7) 

where f f = e~ 2 "- iir , (8) 

whence a/= ^~ 2a<! '-i ?.'*~ 6 ; + , (9) 

which can easily be solved by successive approximation, except 
in the case of the first root (s = 0). For this special methods 
are necessary*. In the remaining type of vibrations we have 

77 = B sinh mx + Ccosmx, (10) 

with Bsinh ml + 



.(11) 
B cosh \ ml + C sin | ml = 0, 1 



whence coth \rnl- tan \ml ................ (12) 

The intersections of the curves y = tan ,r., y = coth x are also 
shewn in Fig. 44. The roots of (12) are given by 

imJ = (*-i)9T-&', ............... (13) 

where s = 1, 2, 3, .... Hence 

? s 6 2 ^, .................. (14) 



where f. = e"" ir , .................. (15) 

and therefore 

A' = &*' - J ?. w + ............. (16) 

* One such method will be indicated later in connection with the radial 
vibrations of air in a spherical vessel ( 84). Another very powerful method is 
explained in Rayleigh's treatise. 



BAES 129 

The frequencies of the whole series of normal modes, after the 
first, are approximately proportional to 3 2 , 5 2 , 7 2 , . . . , as found 
experimentally by Chladni. The accurate solution gives, to 

five places, 

ml/7r = -59686, V49418, 2-50025, (17) 

In the modes which follow the first we have respectively one, 
two, three, ... internal nodes. The annexed figure shews the 
gravest mode. 



Fig. 46. 

Other problems, which are however of less interest, may be 
obtained by varying the terminal conditions. We will only 
notice the case where both ends are " supported," i.e. fixed in 
position but free from terminal couples. The conditions then 
are, by 45 (14), 

77 = 0, 7/"=o |>=iq (is) 

In the symmetrical class we have 

77 = G cos mx . cos (nt -f e), (19) 

with cos ^ml = 0, whence 

mll7r = l, 3, 5, (20) 

In the asymmetric class 

77 = G sin mx . cos (nt + e), (21) 

with raZ/7r = 2, 4, 6, (22) 

The frequencies are, by 46 (2), proportional to the values of 
m 2 , and so to the squares of the natural numbers. 

The foundations of the theory of the transverse vibrations 

were laid by D. Bernoulli (1735) and Euler (1740). The 

latter also gave the numerical solution of the period equation 

in a few of the more important cases. In more recent times 

L. 9 



130 DYNAMICAL THEORY OF SOUND 

the calculations, including the determination of the nodes &c., 
have been greatly extended by Lissajous (1850), Seebeck* (1848) 
and Lord Rayleigh. 

48. Summary of Results. Forced Vibrations. 

In any one of the preceding cases, and in any particular 
mode, ra varies inversely as I, and therefore, by 46 (2), the 
period 27r/n will for bars of the same material vary as Z 2 //c. 
Hence for bars which are in all respects similar to one another 
(geometrically) the period will vary as the linear scale. For 
bars of the same section the period is as the square of the 
length. As regards the shape and size of the cross- section, 
everything depends on the radius of gyration K\ thus for bars of 
rectangular section the frequency varies as the thickness in the 
plane of vibration, and is independent of the lateral dimension. 
This latter statement needs, however, some qualification ; it is 
implied that the breadth is small compared with the length of 
the bar, or (more precisely) with the distance between con- 
secutive nodes. When this condition is violated the problem 
comes under the more complex theory of plates ( 55). 

It is of interest to compare the frequencies of transverse and 
longitudinal vibration of a bar in corresponding cases. For a 
bar free at both ends we have, in the gravest transverse mode, 

?i 2 = / ^(m0 4 = ^ / ^x(l-50562) 4 , (1) 

whilst in the gravest longitudinal mode 

(2) 



Hence -, = 7'122y (3) 

71 I 

This explains the relative slowness of the transversal modes. 
The comparison is due to Poisson. 

We pass over the question of determining the motion 
consequent on arbitrary initial conditions, by means of the 
normal functions. In the case of the free-free bar, for example, 
these are given by the expressions in brackets in equations (20) 
and (21) of 46. 

* L, F. W, A, Seebeck (180549), professor of physics at Leipzig. 



BARS 131 

The theory of forced vibrations again, is of little acoustical 
interest, although it has some technical importance. A simple 
example is furnished by the coupling rod which connects the 
wheels of a locomotive. Attending only to the vertical com- 
ponent of the motion, and treating the bar as uniform, we have 
to solve the equation (13) of 45 subject to the conditions 

where n is the angular velocity of the wheels, and & is the 
vertical amplitude. The forced oscillation is evidently of 
symmetrical type, and we therefore assume 

rj = ( A cosh mx + C cos mx) cos (pt + a) (5) 

This satisfies the differential equation, provided 

m*=p*p/K*E', (6) 

whilst the terminal conditions give 

A cosh^ml + Ocosra = /8, ) ^ 

A cosh \ml-G cos \ml = 0, J 

the latter equation expressing the absence of terminal couples 
(" = 0). Hence 



2 cosh ml' 



The oscillations would become dangerously large if c 
were small, i.e. if the imposed frequency (>/2?r) were to ap- 
proximate to that of one of the symmetrical free modes of the 
bar when " supported " at the ends ( 47 (20). 

49. Applications. 

The use of transverse vibrations of bars in music is re- 
stricted by the fact that the overtones are not harmonic to the 
fundamental. If a flat bar, otherwise free, be supported at the 
nodes of the fundamental (Fig. 45), and struck with a soft 
hammer, the production of overtones is, however, in some 
measure discouraged, and musical instruments of a kind (such 
as the " glass harmonica ") have been constructed on this plan. 

The most important application is in the tuning fork. 

92 



132 DYNAMICAL THEOKY OF SOUND 

Theory and observation alike shew that the effect of curving 
a bar is to lower the pitch of the gravest mode and to make the 
nodes approach the centre. It was found by Chladni that 
when the bar takes the form of an elongated U, the nodes are 
very close to the bend. The amplitude of vibration at the 
centre of the bend will therefore be small compared with that 
at the end of the prongs. The circumstances are somewhat 
modified by the attachment of the stem, but the transmission 
of energy is comparatively slow, and the vibrations have con- 
siderable persistence. A fork may also be compared to a couple 
of bars each clamped at one end, and the formula (2) of 46, 
with ml/7r = '59686, may be used to estimate the frequency 
theoretically. If this analogy were exact there would of course 
be no loss of energy of the kind just referred to. 

Massive forks are usually set into vibration by means of 
a violoncello bow applied to one prong near the free end. The 
production of overtones having nodes in this neighbourhood is 
thus discouraged. The fundamental is further reinforced re- 
latively to the other modes if the stem be screwed into the 
upper face of a resonance box of suitable dimensions. 

When a fork is excited in this or in other ways, it often 
happens that the motion is not in the first instance symmetrical 
with respect to the medial plane. In that event the vibration 
may be regarded as made up of a symmetrical and an unsym- 
metrical component. These will in general have slightly 
different frequencies, and beats may be produced. But unless 
the stem be very firmly fixed the vibrations of the latter class 
are rapidly dissipated by being communicated to the support, 
since they involve an oscillation of the centre of mass of the 
fork. 

The first overtone of a fork may be elicited in considerable 
intensity by bowing one of the prongs near the bend ; the note 
produced is very shrill. 

50. Effect of Permanent Tension. 

In the theory developed in 45 it was assumed that the 
longitudinal tension, when integrated over the area of the 
cross-section, vanishes. It is easy to see that the effect of 



BARS 133 

a permanent tension P is merely to add a term P?/' to the 
equation (13) of 45, so that 






where c z -P/pa> (2) 

This equation has been employed to estimate the effect of 
stiffness of a piano-wire on the sequence of proper tones, but 
the matter is complicated by the uncertainty as to the nature 
of the terminal conditions. A wire, where it passes over a 
bridge, cannot be quite accurately regarded either as merely 
"supported" or as "clamped." The question will perhaps be 
sufficiently illustrated if we consider a wave-system 

7] = Ccosk(ct-x) (3) 

on an unlimited wire. We find, on substitution in (1) 

c 2 = c 2 +cr (4) 

E 

where c^ = .k z /c', (5) 

i.e. Cj is the velocity of transverse waves of length 2-7T/& on a bar 
free from tension. We have seen that in the case of a piano- 
string E/p is large compared with c 2 ; on the other hand K is 
usually an exceedingly minute fraction of the wave-length. In 
the graver modes of a piano-string this second influence pre- 
dominates, and (Cj/Co) 2 is small ; the wave-velocity is practically 
unaffected by stiffness, and the harmonic sequence is not 
disturbed. It is only in the case of the modes of very high 
order, where the length is divided into a large number of 
vibrating segments, that a sensible effect could be looked for. 
It has already been stated that in the pianoforte such modes are, 
so far as may be, discouraged on independent grounds. In any 
case it appears from (4) that the effect of stiffness is relatively 
less important, the greater the value of c , i.e. the tighter the 
wires are strung. 

51. Vibrations of a Ring. Flexural and Extensional 
Modes. 

The theory of the vibrations of a circular ring is important 
as throwing light on some later questions which can only be 



134 DYNAMICAL THEORY OF SOUND 

dealt with imperfectly in this book, owing to the difficulties of 
an exact investigation. As various points of interest arise, we 
treat the matter somewhat fully. 

The ring is supposed to be uniform, and the section is 
assumed to be symmetrical with respect to a plane perpen- 
dicular to the axis. We further consider only vibrations 
parallel to this plane. Let u, v be the displacements of an 
element of the ring along and at right angles to the original 
radius vector, so that the polar coordinates of the element are 
changed from (a, 0) to (a + u, 6 +- v/a). We require expressions 
for the extension, and for the change of curvature. In con- 
sequence of the assumed smallness of the displacements, we 
may calculate the instalments of these quantities which are due 
to u and v separately, and add the results. The radial displace- 
ment by itself changes the length of an element from a&O to 
(a + u) 80, and so causes an extension u/a. The transverse 
displacement obviously contributes dv/adO. The total extension 
is therefore 



.(1) 



Again, in consequence of the radial displacement alone the 
normal to the curve is rotated backwards so as to make an 
angle du/adO with the radius, and the mutual inclination of the 
normals at the ends of an element a BO is accordingly diminished 
by fru/adfr . S0. Dividing the angle between the normals by 
the altered length (a + u) BO we get the altered curvature, thus 

(BO- -Tr^e>0)-Ma 



a a 2 \80 2 

Since the transverse displacement v by itself contributes 
nothing, the increase of curvature is 



a? 



The resultant stress across any section may be resolved into 
a radial shearing force P, a tangential tension Q, and a bending 
moment M. On the principles of 43, 45 we have 



BARS 



135 



the bending moment being now proportional to the increase of 
curvature. 

Resolving along and perpendicular to the radius vector the 
forces on a mass-element pwaSO, we have (see Fig. 47) 
P a>aS0. u = SP- QSe, pvaW.v = BQ + PS0 ; 
and, taking moments about a normal to the plane of the ring, 



the rotational inertia being neglected as in the case of a straight 
bar ( 45). Thus 



8 2 w 8P n 

a* = ;S|-C. P^ a ^ = 



dt* 80 



.(4) 



and 



80 



= Pa. 



-(5) 



P+bP 



These, together with (3), are 
the equations of our problem. 
It is easily seen that they cannot 
be satisfied on the assumption 
that the tension Q vanishes, and 
that accordingly some degree of 
extension is involved in any 
mode of vibration. This is 
readily accounted for, a stress 
of this kind being necessarily 
called into play by the inertia of 
the different portions swinging 
in opposite directions. It may be shewn however that in the 
" flexural " modes to be referred to presently the corresponding 
strains are small compared with those involved in the change of 
curvature. 

Eliminating P, Q, M between (3), (4), and (5), we find 

E ( dv 



(6) 




Fig. 47. 



, 

** + 

_ L _ u 

pa 2 + ~ + = 



To ascertain the normal modes we assume that u and v vary 
as cos (nt + e). Again, the ring being complete, u and v are 



136 DYNAMICAL THEOKY OF SOUND 

necessarily periodic functions of 6, the period being 2?r, and 
can accordingly be expanded by Fourier's theorem in series of 
sines and cosines of multiples of 6 ; moreover it is easily 
proved that the terms of any given rank in the expansion must 
satisfy the equations separately. We find, in fact, that a 
sufficient assumption for our purpose is 

u = A cos s6 . cos (nt + e), v = B sin sB . cos (nt + e), (7) 
where 5 is integral or zero. This leads to 



(8) 

- 2 )5 = 0, 

/ 
where {3 = ri*a?p/E. (9) 

Hence 



Since tc/a is small, the sum of the roots of this quadratic in ft is 
s 2 + 1, approximately, whilst the product s 2 (s 2 I) 2 /c 2 /a 2 is small. 
The two roots are therefore 

<?2 / Q 2 _ 1 \2 ,.2 

0-*+i, *-'-^J, ............ (ID 

approximately. 

On reference to (8) we see that the former root makes 
B = sA nearly. The corresponding modes are closely analogous 
to the longitudinal modes of a straight bar, the potential 
energy being mainly due to the extension ; and the frequencies, 
which are given by 

2 = (* 2 + l) / . .................. (12) 

are, for similar dimensions, of the like order of magnitude. The 
case s = is that of purely radial vibrations. 

The vibrations corresponding to the second root are more 
important. We then have, from (8), A + sB = 0, nearly ; thus 

A 

u = A cos s& . cos (nt + e), v = -- sin s& . cos (nt + e), (13) 

s 



with n'--. ...(14) 

2 4 



BARS 



137 



It follows from (1) that the extension is negligible, and the 
energy mainly flexural. The frequencies are in fact comparable 
with those of transverse vibration of a bar. In the mode of 
order s there are 2s nodes, or places of vanishing radial motion, 
but these are not points of rest, the tangential motion being 
there a maximum*. In the case 5=1 the circle is merely 
displaced as a whole, without deformation, and the period is 




Fig. 48. 

accordingly infinite. The most important case is that of s = 2, 
where the ring oscillates between two slightly elliptical extreme 
forms. The arrows in the annexed figure shew the directions of 
motion at various parts of the circumference at two epochs, 
separated by half a period, when the ring passes through its 
equilibrium position. The dotted lines pass through the nodes 
of the radial vibration. 

One farther point is to be noticed. Owing to the assumed 
uniformity of the ring the origin of 6 is arbitrary, and other 
modes, with the same frequencies, are obtained by adding a 
constant to 0. In particular we have the flexural mode 

u = A sin sO . cos (nt + e), v = cos sd . cos (nt + e), (15) 

s 

with the same value of n 2 as in (14). We have here an instance 
of the kind referred to in 16, where two distinct normal modes 

* This point is illustrated by the vibrations of a finger-bowl when excited by 
drawing a wetted finger along the edge. The point of rubbing is a node as 
regards the radial vibration, and the crispations on the contained water are 
accordingly most conspicuous at distances of 45 on either side, where the radial 
motion is a maximum. 



138 DYNAMICAL THEOKY OF SOUND 

have the same frequency, and the modes themselves accordingly 
become to some extent indeterminate. The case would be 
altered at once if the ring were not quite uniform, e.g. if it were 
slightly thicker at one point. The normal modes in which 
there is a node or a loop respectively, of radial vibration, at 
this point would differ somewhat in character, and have slightly 
different frequencies. Accordingly when both modes are excited 
we should have beats between the corresponding tones. This 
is a phenomenon often noticeable in the case of bells (and 
finger-bowls), the inequality being due to a slight defect of 
symmetry. 

The vibrations of a ring in its own plane were first investi- 
gated by R. Hoppe (1871) ; a simplified treatment of the flexural 
modes was subsequently given by Lord Rayleigh. The theory 
of vibrations normal to the plane is more intricate, since torsion 
is involved as well as flexure. The problem has been solved by 
J. H. Michell (1889), who finds, in the case of circular cross- 
section, 



s 2 + 1 + a- ' pa 4 ' 
where a is Poisson's ratio. 



CHAPTER V 

MEMBRANES AND PLATES 

52. Equation of Motion of a Membrane. Energy. 

The vibrations of membranes are not very important in 
themselves, and the conditions assumed for the sake of mathe- 
matical simplicity are, moreover, not easily realized experi- 
mentally. The theory is however, for a two-dimensional system, 
comparatively simple, and the results help us to understand in 
a general way the character of the normal modes in other cases 
where the difficulties of calculation are much greater, and 
indeed often insuperable. 

The ideal membrane of theory is a material surface such that 
the stress across any line-element drawn on it is always in the 
tangent plane. We shall consider only cases where the surface 
in its undisturbed state is plane, and is in a state of uniform, or 
"homogeneous," stress; i.e. it is assumed that the stresses across 
any two parallel and equal lines are the same in direction and 
magnitude. We further suppose, for simplicity, that the stress 
across any line-element is perpendicular to that element. It 
follows, exactly as in hydrostatics, from a consideration of the 
forces acting on the contour of a triangular area, that the stress 
(per unit length) is the same for all directions of a line-element. 
This uniform stress is called the " tension " of the membrane ; 
we denote it by P. Its dimensions are those of a force divided 
by a line, or [MT~*]. 

We take rectangular axes of x, y in the plane of the 
undisturbed membrane, and denote by f the displacement 
normal to this plane. The surface-density (i.e. the mass per 



140 DYNAMICAL THEOEY OF SOUND 

unit area) is assumed to be uniform, and is denoted by p. To 
form the equations of motion we calculate the forces on the 
sides of a rectangular element BxSy having its centre at (x, y). 
In the displaced position, the gradient of a line parallel to x is 
3?/da?, and that of a line parallel to y is d/dy. Hence the stress 
across a line through the centre of the element parallel to By, 
when resolved in the direction of the normal to the plane xy, is 
Pd^/dx . Sy. The corresponding components of force on the two 
edges % of the rectangle are 



where the upper signs relate to the edge whose abscissa is 
x + -J&z?, and the lower to the edge x ^x. The sum of these 
gives P9 2 f/3# 2 . SxSy. A similar calculation for the two edges &z? 
gives Pd*/dy 2 . &x$y. The resultant force on the rectangle is 
therefore 



< 



The above may be compared with the investigation by 
which, in the theory of Capillarity, the tensions across the 
boundary of an element 8S of a soap-film are shewn to be 
equivalent to a normal force 



where B lf R 2 are the principal radii of curvature of the surface. 
It is shewn in books on solid geometry that, if f denote distance 
from the plane xy, we have 



R, R, 8# 2 a^ 2 

at points where the inclination of the tangent plane to xy is 
small. 

Equating the expression (1) to the acceleration of momentum 
of the element, viz. p&xSy . , we obtain the equation of motion 



This is due to Euler (1766). 



MEMBRANES AND PLATES 141 

The kinetic energy is given by 



taken over the area of the membrane. 

The potential energy is found most easily as the work 
required to stretch the membrane. As in the theory of 
capillarity this is equal to the tension P multiplied by the 
increase of area. Now if a prism be constructed on a rectangular 
element Sx&y of the plane xy as base, this will cut out from the 
displaced membrane a nearly rectangular portion whose sides 
are 



~ VK-S'h 

and whose area is therefore, to the second order, 






The same expression is obtained by calculating, from the 
expression (1), the work done by normal pressures applied 
(as in 22) to deform the membrane into its actual shape, the 
ratio of f to its final value being, at any stage of the process, 
the same all over the membrane. The result is 



The reader who is familiar with the theory of attractions will 
recognize that this is equal to 



where in the first term the integration extends over all the 
elements Ss of the contour, and 8n is an element of the normal 
to &s drawn inwards, in the plane of the membrane. Since 
at a fixed edge f = 0, the formula agrees with (5). 



142 DYNAMICAL THEOKY OF SOUND 

53. Square Membrane. Normal Modes. 
To ascertain the normal modes of a limited membrane we 
assume as usual that f varies as cos (nt 4- e), so that 



where k* = n*p/P ......................... (2) 



At a fixed boundary we must have f = 0. It is found that 
the solution of (1) subject to this condition is possible only 
for a series -of definite values of k, which determine, by (2), the 
corresponding frequencies. 

In the case of a rectangular membrane, we take the origin 
at a corner, and the axes of x, y along the edges which meet 
there. The equations of the remaining edges being, say, 
x = a, y = b, the equation (1) and the boundary condition 
are satisfied by 

f=0sin sin^cos(ri$ + e), ......... (3) 

CL (J 

where s, s' are integers, provided 



It may be shewn, by an easy extension of Fourier's theorem, 
that (3) is the only admissible type of solution in the present 
case ; it was given by Poisson in 1829. 

In any normal mode for which s or s' > 1, we have nodal 
lines parallel to the edges. It appears from (4) that if the 
ratio a? : b 2 is not equal to that of two integers, the frequencies 
are all distinct, and the nodal lines are restricted to these 
forms. But if a 2 : 6 2 is commensurable, some of the periods 
coincide, and the corresponding modes may be superposed 
in arbitrary proportions ( 16). The nodal lines may then 
assume a great variety of forms. The simplest instance is 
that of the square membrane (a = 6), when 

) ...................... (5) 



MEMBKANES AND PLATES 



143 



Thus by superposition of the modes for which s = 2, s' = 1 and 
s = 1, s' = 2, respectively, we get 



oe sin 



.Try . irx . 

sm - -- + X sin sm 

a a a a 



. TTX . iry ( TTX Try\ , . 

sm sm - - cos h Xcos -1 , (6) 

a a \ a a J 



where X may have any value. For example, in the cases X= 1 
the diagonals a; + y = a, x y = 0, respectively, are nodal lines. 
The figure shews the cases X = 0, X = , X = 1, which 
give a sufficient indication of the various forms that may 
arise. 



A"i 

Fig. 49. 



Again, by superposition of the cases s = 3, s f = 1 and s 
s' = 3, we get 



. OTTX . fry . TTX . tTry 

sm sm - -f X sm sm 

a a a a 

a a ( a V a )) ' 

The cases X = 0, X = J, X = + 1 are shewn in Fig. 50 ; 
intermediate forms are readily supplied in imagination. 

A still greater variety is introduced by the fact that a 
number which is the sum of two squares can sometimes be 
so resolved in more than one way. For example, the modes 
for which 

s = 4, 7, 1, 8,| 

s = 7, 4, 8, 1,} 
respectively, have all the same frequency. 



144 



DYNAMICAL THEOEY OF SOUND 






Fig. 50. 

54. Circular Membrane. Normal Modes. 

In the case of the circular membrane we naturally have 
recourse to polar coordinates, with the origin at the centre. 
The differential equation may be obtained by transformation 
of 52 (3), but a more direct process is preferable. 

Take first the case of the symmetrical vibrations where 
is a function of r, the distance from 0, only. The stress across 
a circle of radius r has a resultant P . ZTTT . 9f/9r normal to the 
plane of the undisturbed membrane, and the difference of the 
stresses on the edges of the annulus whose inner and outer 
radii are r and r + Sr gives a force 



, 

Equating this to p . 2?rr5r . f, which is the acceleration of 
momentum of the annulus, we get 



If f varies as cos (nt -f e), this reduces to 



where & 2 = n 2 p/P, as before. 



(2) 



MEMBRANES AND PLATES 



145 



If we assume, as is necessarily the case when the origin is 
included within the region to which (2) applies, that f can be 
expanded in a series of ascending powers of r, the coefficients 
(after the first) may be found by substitution in (2), and we 
obtain 

(3) 



provided 



2 2 .4 2 



.(4) 



This is the Bessel's Function* of zero order, "of the first kind," 
which we have already met with in 31 ; it is represented 
graphically in Fig. 51. If a be the radius of the boundary, 




Fig. 51. 

supposed fixed, the admissible values of k and thence of n are 
determined by the equation 

J (ka) = 0,_ ,...(5) 

viz. we have 

kal-n- = 7655, 17571, 2'7546, 3'7534, (6) 

the numbers tending to the form ra J, where m is integral. 
The first of these roots corresponds to the gravest of all the 
normal modes of the membrane. In the rath mode there are 
in 1 nodal circles, in addition to the edge, whose radii are 
given by the roots of lower order. Thus in the case of the 
second root we have for the nodal circle kr/7r = '7655, whence 
r/a = *4356. The characters of the various normal modes will 
be understood from Fig. 51, which may be taken to represent 
a section through the centre, normal to the plane of the 
membrane. 



* F. w. 

181046. 

I*. 



(17841846), director of the observatory at Konigsberg 

10 



146 DYNAMICAL THEOEY OF SOUND 

The complete solution of the differential equation (2), 
which is of the second order, would consist of the sum of 
two definite functions of kr, each multiplied by an arbitrary 
constant; but the second solution, which is called a Bessel's 
Function " of the second kind," becomes infinite for r = 0, and 
is therefore inapplicable to a complete circular area. In the 
case of an annular membrane, however, bounded by concentric 
circles, both solutions would be admissible, and both would be 
required in order to satisfy the conditions at the two edges*. 

The theory of the symmetrical vibrations of a circular 
membrane was given by Poisson (1829), who also calculated 
approximately a few of the roots of the period-equation (5). 

When the vibrations are not symmetrical we may begin by 
calculating the forces on a quasi-rectangular element of area 
bounded by two radii vectores and two concentric circles, the 
sides being accordingly $r and rB0. The stresses on the 
curved sides give a resultant 



normal to the plane, whilst the stresses on the straight sides 
produce 



Equating the sum of these expressions to pr$0Sr . , we obtain 

p^pJil^U- 8 ^ (7) 

p tf (r9rV 8rJ r 2 90 2 j' 

or, in the case of simple-harmonic vibrations, 



with the same meaning of & 2 as before. 

* On account of the frequent occurrence of the Bessel's Functions in 
mathematical physics, especially in two-dimensional problems, great attention 
has been devoted to them by mathematicians. The difficulty in investigating 
their properties is much as if we had to ascertain all the properties of the 
cosine-function from the series 



and were ignorant of its connection with the circle. 



MEMBRANEiS AND PLATES 147 

Since f is a periodic function of 0, of period 2-Tr, it can be 
expanded (for any particular value of r) in a series of sines and 
cosines of multiples of 0, thus 

f = R Q + R l cos 6 + $ sin + ... 

+ .RsCoss0-f &sms0+ ...-, ...(9) 

by Fourier's theorem; and this formula will apply to the 
whole membrane if the coefficients be regarded as functions 
of r. Moreover on substitution in (8) it appears that each 
term must satisfy the equation separately. Thus we have 
a typical .solution 

.cos(nt+e), ............ (10) 



provided + i ! + *_ fi. _0. ...(11) 

2 z 



r 



The solution of this, which is finite for r = 0, can be found in 
the form of an ascending series. In the accepted notation we 
have R 8 = A 8 J 8 (kr), where the function /, is defined by 



This is known as the Bessel's Function of the sth order, of the 
first kind. As in the case of (2) there is a second solution 
which becomes infinite for r = 0, but in the case of the complete 
circular membrane this of course is inadmissible. We have then 
the normal modes 

=AJ s (kr)coss0.cos(nt + e), ......... (13) 

where k is determined by 

J 9 (ka)=0 ...................... (14) 

Similarly, taking a term 8 g sin sO from (9) we should have been 
led to the modes 

=BJ 8 (kr)sms0.cos(nt + e), ......... (15) 

with the same determination of k. Owing to the equality of 
periods the normal modes are to some extent indeterminate. 
Thus, for any admissible value of k, we may combine (13) 
and (15) in arbitrary proportions, and obtain 

?= CJ, (kr) cos (80 + a) . cos (nt + e) ....... (16) 

102 



148 



DYNAMICAL THEOKY OF SOUND 



We have here s nodal diameters, given by 
s0 + a= \ir, f?r, ..., 

and accordingly arranged at intervals of TT/S. Again for every 
value of k after the lowest we have one or more nodal circles 
whose radii are given by the roots of lower order. In the case 
s = l, where there is one nodal diameter, we have 

ka/7r= 1-2197, 2'2330, 3'2383, 4'2411, (17) 

the numbers tending to the form ra + J . The characters of 
the corresponding modes may be gathered from the annexed 



10 



10 



Fig. 52. 

graph of the function J l (z) ; this may be supposed to represent 
a section through the centre, normal to the nodal diameter. In 
the second of the above modes, the radius of the nodal circle is 
given by 

r/a = 1-2197/2-2330 = '546. 

Fig. 53 shews in plan the configuration of the nodal lines 
in the first three modes of the types s = 0, 5 = 1, s 2, re- 
spectively. The + and signs distinguish the segments 
\vhich are at any instant in opposite phases of vibration. 

Whatever the form of the boundary, the value of f in the 
neighbourhood of any point of a membrane must admit of 
expression in the form (9), with 

R 8 = A 8 J.(kr), S 8 = B 8 J 8 (kr), (18) 

the factor cos (nt + e) being of course understood. If be on 
a nodal line we must have f = for r = 0, and therefore A = 0. 
The form of the membrane near is then given by 

^ = (A 1 cosO + B 1 sme)J l (kr), (19) 

ultimately, and the direction of the nodal line at is accord- 



MEMBRANES AND PLATES 



149 



ingly given by tan 6 = A l /B l . If all the coefficients of order 
less than s vanish, we have, for small values of r, 



(20) 

The node has then s branches passing through 0, making equal 
angles TT/S with one another, their directions being given by 
tan s0 = AgjBg. This is illustrated in the preceding diagrams ; 
for instance the cases s = 2, s = 3, s = 4 all occur in Fig. 50. 












Fig. 53. 

According to a general theorem stated in 16 it must be 
possible by combination of the various normal modes of a 
membrane in suitable proportions, and with proper relations 
of phase, to represent the effect of arbitrary initial conditions. 
We do not enter into this; and the theory of the forced vibrations 
must also be passed over except fora simple example. 



150 DYNAMICAL THEORY OF SOUND 

When a force Z per unit area acts on a circular membrane, 
the equation (1) is replaced by 



it being supposed, for simplicity, that there is symmetry as 
regards the distribution of Z and the consequent displacements 
If, further, Z vary as cos (pt + a), we have 

Z 



(23) 

If Z be independent of r, so that the impressed force is 
uniform over the membrane, the solution of (22) is obviously 

t=-lp + CJ t (kr), ............... (24) 

and determining the constant C so that f =0 for r = a, we find 

(25) 



(ka) 

The amplitude becomes very great whenever fca approximates 
to a root of (5), i.e. whenever the imposed frequency approaches 
that of one of the symmetrical free modes. When, on the 
other hand, the imposed vibration is relatively slow, ka is 
small, and we have by (4) 



(26) 



approximately. This is the statical deflection corresponding to 
the instantaneous value of the disturbing force. 

55. Uniform Flexure of a Plate. 

The theory of the transverse vibrations of plates stands in 
the same relation to that of bars as the theory of membranes 
to that of strings. The reader will understand from this com- 
parison that the mathematical difficulties are considerable, arid 
will not be surprised to learn that some of the most interesting 
and, at first sight, simple problems remain unsolved. On the 
other hand the subject readily admits of experimental illustra- 
tion. If the plate be horizontal, and fixed at one point, the 



MEMBRANES AND PLATES 151 

configuration of the nodal lines can be exhibited by means 

of a little sand previously strewn on the surface. When any 

particular normal mode is excited, the sand is shaken away 

from the places of vigorous motion, and accumulates in the 

neighbourhood of the nodal lines. Usually the plate is set 

into vibration by bowing at right angles to the edge, and the 

desired mode is favoured by touching the edge with the fingers 

at one or more nodal points. If, as in the case of a rectangular 

plate fixed at the centre, the point of support is a nodal point 

>f several normal modes, a great variety of beautiful figures 

nay be obtained. An extensive series of diagrams of results 

obtained in this way were given by Chladni; many of these are 

eproduced in the current manuals of experimental acoustics. 

In the theoretical treatment it is assumed that one of the 
principal axes of strain and stress is normal to the faces of the 
plate, and that the corresponding stress vanishes. Putting, 
then, p s = in the formulae (9) of 42, we find, for the 
remaining principal stresses, 



)> p> = E'( > + a ei ) t ......... (1) 

where E f = E/(l - <r 2 ) ...................... (2) 

If R l , R 2 be the principal radii of curvature at any point of the 
plate, when bent, we have, by an investigation similar to that 
of 45, 

C^Z/R!, e 2 = z/R 2 , .................. (3) 

where z denotes distance from the medial plane. If we consider 
a rectangular element of the plate bounded by lines of curvature, 
and denote by h the half-thickness, this leads to bending 
moments 




per unit length of the respective edges, in the planes of the two 
principal curvatures. 

The potential energy per unit volume is 

ef) .......... (5) 



152 DYNAMICAL THEOBY OF SOUND 

If we substitute from (3), and integrate over the thickness, we 
find for the potential energy per unit area of the plate 



The formulae (4) may be applied to the case of a flat bar of 
rectangular section, uniformly bent by two opposing couples 
MJ), where b denotes the breadth. Along the free edges we 
have M 2 = 0, and therefore 

JZr' = -rJZr' (7) 

The bending moment is accordingly 

M 1 b=%Ebh 3 /R l) (8) 

by (4). This agrees, as it must, with 45 (5), since o> = 2bh, 
K * == *h?. The formula (7) shews that when a bar of rectangular 
section is bent in a plane parallel to one pair of faces, an opposite 
or " anticlastic " cur- 
vature is produced in 
the plane of the cross- 
section, the ratio of 
the curvatures being 
identical with Pois- 
son's ratio &. This 
circumstance has 
been made the basis 

of practical methods Fig. 54. 

of determining <r, by 

Cornu* (1869) and Mallock (1879), the curvatures being 
measured by optical or other means. 

It follows from the above that a perfectly free rectangular 
plate cannot vibrate after the manner of a bar, with nodal lines 
parallel to one pair of opposite edges, since couples would be 
required, about the remaining edges, to counteract the tendency 
to anticlastic curvature. 

56. Vibrations of a Plate. General Results. 
In a vibrating plate the directions and amounts of the 
principal curvatures will in general vary from point to point. 

* A. M. Cornu (1841 1902), professor of physics at the Ecole Polytechnique 
1871 1902. Famous for his experimental determination of the velocity of light, 
and for other important contributions to optics. 




MEMBEANES AND PLATES 153 

Shearing forces will also be called into play normal to the plane 
of the plate. The circumstances are somewhat complicated, 
but the deduction of the equation of motion for the body of the 
plate is a straightforward matter, and presents no real difficulty. 
A more serious question arises when we come to the conditions 
to be satisfied at a free edge. It appears that the simple 
condition of strain which has been postulated as the basis of 
the formulae (4) of 55 cannot be assumed to hold, even 
approximately, right up to the edge. In the immediate neigh- 
bourhood of the edge, i.e. to a distance inwards comparable 
with the thickness, a peculiar state of strain in general exists, 
one remarkable result of which is a shearing force on sections 
perpendicular to the edge, of quite abnormal amount. 

For the further development of the subject reference must 
be made to other works*. We merely quote a few of the more 
important results which have been obtained, relating chiefly to 
plates whose edges are free. 

It is found that for a plate of given lateral dimensions the 
frequency (n/2?r) of any particular normal mode is given by 

*-'.#.*, (1) 

where, as in 46, m is a constant, of the nature of the reciprocal 
of a line, given by a certain transcendental equation, and p 
denotes the volume-density. For plates with geometrically 
similar boundaries the frequency accordingly varies as the 
thickness, and inversely as the square of the lateral dimensions. 
In the case of a perfectly free circular disk the nodal lines 
are circles and equidistant 
diameters. In the symmetrical 
modes, which were investigated 
to some extent by Poisson 
(1829), we have nodal circles 
alone. Thus in the gravest 
mode of this type we have a 

nodal circle of radius '678a, where a is the radius of the disk ; 
in the next mode there are two nodal circles of radii '39 2a 

* See Lord Kayleigh. Theory of Sound, chap. 10 ; Love, Theory of Elasticity, 
Cambridge, 1906, chap. 22. 




154 



DYNAMICAL THEORY OF SOUND 



and '842a, and so on, the numbers varying slightly however 
with the value adopted for or. According to Poisson, the 
values of m for the above modes are given by 

ra 2 a 2 = 8-8897, 38-36, (2) 

on the hypothesis that a- = J. 

The complete theory of the free circular plate was worked 
out by Kirchhoff in a celebrated memoir (1850). It appears 
that the gravest of all the normal modes has two nodal 
diameters, and no nodal circle. Its frequency is 

5234. /(- 




according as we adopt the value cr \ or a = % of Poisson's 
ratio. The figure shews the 
configuration of the nodal lines 
in the simplest cases of one and 
two nodal diameters. 

The theory of a circular 
plate clamped at the edge has 
been treated by Poisson and 
others. In the first two symmetrical modes it is found that 

ra 2 a 2 = 10-2156, 39'59, (3) 

respectively. In the second of these modes there is a nodal 
circle of radius '38 la. The theory has been applied by Lord 
Rayleigh to calculate the natural frequencies of a telephone 
plate. 



Fig. 56. 




Fig. 57. 

In the case of a square plate we have to depend almost 
wholly on observation, there being at present no exact theory. 
As in the case of the square membrane ( 53), the nodal lines 



MEMBRANES AND PLATES 155 

may assume a great variety of forms, owing to the superposition 
of different modes having the same frequency. The gravest 
mode of a free plate is that in which the nodal lines form a 
cross through the centre, with arms parallel to the sides. 
The figure shews other cases in which possible forms can be 
assigned to the nodal lines from considerations of symmetry. 

57. Vibrations of Curved Shells. 

When we proceed to the vibrations of curved plates, or 
shells, we meet with further complications due to the fact that 
no absolutely sharp line can be drawn between flexural and 
extensional modes. This has been already exemplified in the 
case of the ring ( 51). It appears, however, that as the thick- 
ness is (in imagination) reduced the normal modes tend to fall 
into two distinct categories. In one of these the frequencies 
tend to definite limits, the deformations being mainly 
extensional, and so analogous to the longitudinal vibrations 
of a bar, where the dimensions of the cross-section were found 
to be immaterial. In the second category the frequencies 
diminish without limit, being ultimately proportional to the 
thickness, as in the flexural vibrations of a bar or plate. 

It will be understood that, acoustically, the flexural vibra- 
tions are alone of real interest. When the shape is one of 
revolution about an axis, the nodal lines will evidently be 
parallels of latitude and equidistant meridians. As in the case 
of 51 these are not lines of absolute rest, the tangential motion 
being there relatively at its greatest. This has an application 
to bells. A theoretical calculation of the frequencies of an 
actual bell is of course out of the question ; but it is somewhat 
remarkable that no systematic experimental study appears to 
have been made until the subject was taken up by Lord 
Rayleigh in 1890. Some unexpected results were obtained. 
To quote a typical case, the normal modes of a particular bell, 
when arranged in ascending order of frequency, were found to 
have the following numbers of nodal meridians and parallels, 
and the pitches indicated : 

(4,0) (4,1) (6,?) (6,?) (8,?) 
e c" f" + &"(, d'". 



156 DYNAMICAL THEORY OF SOUND 

Of these the only one which has any relation to the nominal 
pitch (d"} of the bell is the fifth in order, and this is out by an 
octave. A mistake of an octave in judging pitch is not 
uncommon, for physiological reasons, but it is surprising that 
the presence of the lower dissonant tones should be so easily 
disregarded. It is conceivable that the mode of striking may 
be in some degree unfavourable to the production of the more 
discordant elements. 

The vibrations of an elastic solid whose dimensions are all 
of the same order of magnitude are from our present point of 
view of subordinate interest. The only case which has been 
worked out is that of the sphere. In the most important 
mode one diameter extends and contracts whilst the perpen- 
dicular diameters simultaneously contract and expand, respec- 
tively. The frequency of this mode is, for such values of a- as 
are commonly met with, 



(1) 



about, where a is the radius. This is the lowest of all the 
natural frequencies. For a steel ball one centimetre in radius, 
this makes N= 136000. 



CHAPTER VI 

PLANE WAVES OF SOUND 

58. Elasticity of Gases. 

In any fluid there is a definite relation between the pressure 
p, the density p, and the temperature 0, and any two of these 
quantities accordingly serve to specify the physical state of 
the substance. It is often convenient to use in place of p its 
reciprocal v, the volume of unit mass. 

In thermodynamical investigations the two quantities 
usually chosen as independent variables are p and v. In 
Watt's " indicator diagram " these are taken as rectangular 
coordinates, p being the ordinate and v the abscissa. Any 
particular state is then represented by a point on the diagram, 
and any succession of states by a continuous line. We may 
imagine the unit mass of the fluid to be enclosed in a deform- 
able envelope, and that an infinitesimal change of volume is 
produced by a displacement of the boundaiy in the direction 
of the normal, whose amount is (say) v for any given surface- 
element 8S. The work done by the contained gas in this 
process is 2 (p&S . v), or pSv, since 2 (v8S) = Sv. Hence the 
work done in any succession of changes, represented by a curve 
on the diagram, will be given by fpdv, i.e. by the area included 
between the curve, the axis of abscissae, and the first and last 
ordinates. This area is of course to be taken with its proper 
sign, according as the work is positive or negative. 

There are two kinds of successions of states which are 
specially important. In the first of these the temperature does 
not vary, and the representative lines are therefore called 
" isothermals." By means of a system of isothermal lines 
drawn at sufficiently small intervals the properties of the 



158 DYNAMICAL THEOKY OF SOUND 

substance can be completely mapped out. The other suc- 
cessions referred to are those in which there is no gain or loss 
of heat to the substance, as if it were enclosed in a vessel (o: 
variable volume) whose walls are absolute non-conductors. The 
corresponding lines are therefore called " adiabatics." 
In a perfect gas we have 

p = R P 0, or pv = R0, ............... (1) 

where 6 is the absolute temperature on the gas thermometer 
and R is a constant depending on the nature of the gas. The 
isothermal lines pv = const, are therefore rectangular hyperbolas 
asymptotic to the coordinate axes. As regards the adiabatics 
the heat required to increase the pressure by &p when the 
volume is constant will be given by an expression of the form 
PSp. If c denote the specific heat (per unit mass) at constant 
volume, this must be equal to c&0, where $0 is the corre- 
sponding change of temperature. Now when $v = we have 
Sp/p = $0/0, whence, comparing, P = c0/p. Again, the heat 
required to' augment the volume by Iv when the pressure is 
constant- may be denoted by QSv, which must be equal to c'&0 
where c' is the specific heat at constant pressure. Since, when 
fy = we have &v/v = $0/0, we find Q = c0/v. The heat ab- 
sorbed when both pressure and volume are varied infinitesimally 
is therefore 

............... (2) 



and the differential equation of the adiabatics is therefore 

$M* = 0. ...(3) 

p c v 

The ratio c'/c. of the two specific heats is practically constant. 
Denoting it by 7, .we have 

\ogp 4- 7 log v = const., 
or pv? = const., ........................ (4) 

as the equation of the adiabatic lines. The value of 7 as found 
by direct experiment is about 1*41 for air, oxygen, nitrogen, 
and hydrogen. The figure shews the isothermal and adiabatic 
lines through a point P of the diagram, the latter curve being 
the steeper. 



PLANE WAVES OF SOUND 



159 



When the pressure and volume vary in any connected 
manner, the ratio of 
the increment &p of 
the pressure to the 
" compression," i.e. the 
negative dilatation 
Sv/v, may be called 
the " elasticity of 
volume." Its value 
will depend not only 
on the particular state, 
but on the manner in 
which the variations 
from that state are 
supposed to take place, 
i.e. on the direction 
of the corresponding 




p. 



curve on the diagram. 

If the tangent at the 

point P meet the axis of p in U, and NU be the projection of 

P U on this axis, we have 



dv 



.(5) 



this projection therefore represents the elasticity under the 
particular condition. On the isothermal hypothesis, to which 
these letters refer in the figure, the elasticity is equal to the 
pressure p, as follows at once from (1), or from the fact that 
the tangent to a rectangular hyperbola is bisected at the 
point of contact. If the variations are subject to the adiabatic 
law, the elasticity, as deduced from (4), is yp, and so greater 
than in the former case. This is represented by NU' in the 
figure. Even in the case of solid and liquid bodies we ought, 
in strictness, to discriminate between isothermal and adiabatic 
coefficients of elasticity, but the differences happen not to be 
very important. 

The work done by unit mass of a gas in expanding between 
any two adjacent states is easily read off from a diagram as 
o-v), or Po(v -v) + %(p-p )(v -v), (6) 



160 



DYNAMICAL THEORY OF SOUND 



correct to the second order of small quantities. 
states are a finite distance apart we 
require, to know the manner of transi- 
tion. For changes along an isothermal 
line pv = p v we have 



When the two 



dv=p Q v \og ...(7) 
For variations along an adiabatic 



V % 

Fig. 59. 



59. Plane Waves. Velocity of Sound. 

The theory of plane waves of sound is very similar to that 
of the longitudinal vibrations of rods ( 43). We assume that 
the motion is everywhere parallel to the axis of x, and is the 
same at any given instant over any plane perpendicular to this 
axis. We denote displacement from the equilibrium position 
by f . The symbols p, p, % are supposed to refer at the time 
t to that plane of particles whose undisturbed position is 
x] they are therefore functions of the independent variables 
x and t. The constant equilibrium values of p, p are dis- 
tinguished as PQ, p . 

The dilatation A was defined in 40 as the ratio of the 
increment of volume to the original volume, viz. 



(1) 



In the present branch of the subject it is usual to introduce 
a symbol s to denote the " condensation," i.e. the ratio of the 
increment of density to the original density ; thus 



(2) 



Since v=l/p, we have 



.(3) 



The stratum of air which was originally bounded by the 
planes x and#+& is at time t bounded by the planes x+% and 



PLANE WAVES OF SOUND 161 

x + f -t- &B + Sf , and its thickness is therefore changed from 8x 
to &c -f 8f , or (1 + 9f/9#) 8#, and the dilatation is accordingly 

- ' A - *= ...................... <*> 



Hence, in the case of infinitely small disturbances, we have, 

by(3)> 



In forming the equation of motion we assume that the 
pressure varies with the density according to some definite law. 
We have then, for small values of s, 

p=p Q + KS, ........................ (6) 

where K is a coefficient of cubic elasticity. t Considering the 
acceleration of momentum of unit area of a stratum originally 
bounded by the planes x and x -f &c, we have 



where 8p represents the excess of pressure on the anterior face. 
Hence, by (5) and (6), 

**-*** (7) 

dP'^da?' 

where c = V(*/?o) ......................... (8) 

The solution of (7) is as in 23, 43 

%=f(ct-x) + F(ct + x), ............... (9) 

and represents two systems of waves travelling in opposite 
directions with the velocity c*. 

If we assume, as Newton -f- did, that, the expansions and 
contractions of a gas, as a sound-wave passes, take place 
isothermally, i.e. without variation of temperature, the relation 
between p and p is given by Boyle's law, viz. p/p Q = p/p = 1 4- s, 
whence K=p , as already proved. This makes 



Now for air at C. we may put, as corresponding values, 
p = 76 x 13-60 x 981, p 9 = '00129, 

* The analytical theory of plane waves of sound is due to Euler (1747) and 
Lagrange (1759). 

t The investigation is given in Prop. 48 of the second book of the Principia 
(1726). 

L. 11 



162 DYNAMICAL THEORY OF SOUND 

in absolute centimetre-gramme-second units, whence c = 280 
metres per second. This is considerably less than the observed 
velocity. 

The discrepancy was first fully accounted for by Laplace 
and Poisson*. When a gas is rarefied or condensed the 
temperature tends to fall or rise, except in so far as the 
process is mitigated by the supply or abstraction of heat. In 
ordinary sound-waves the condensation s changes sign so fre- 
quently, and the temperature consequently rises and falls so 
rapidly, that there is no time for sensible transfer of heat 
between adjacent portions of the gas. The flow of heat has 
hardly set in from one element to another before its direction is 
reversed, and the conditions are therefore practically adiabatic. 
The formula 



becomes, for small values of $, 

p=p Q (I+ys), ..................... (12) 

whence K = yp 0) as in 58, and 

c = V(7po/Po) ...................... (13) 

Putting 7=1*41 we find that the Newtonian velocity of sound 
in air must be increased in the ratio 1'187, whence c = 332 
metres per second at 0C. This is in good agreement with 
direct observation. 

As there is now no question as to the soundness of this 
explanation, and as the direct determination of 7 is a matter 
of considerable difficulty, the formula (13) is often used in the 
inverse manner, as a means of deducing the value of 7 for 
various gases from the observed velocities of sound-waves in 
them. For example, it was in this way that in 1895 the value 
of 7 for the newly discovered gas argon was found by Lord 
Rayleigh to lie between 1'6 and 1-7. The experimental method 
(due to Kundt) is referred to in 62 below. 

Since p /p = R6 , the velocity of sound as given by (13) is 
independent of the actual density, but will vary as the square 
root of the absolute temperature. Also, so far as 7 has the 
same value, the velocity of sound in different gases will vary 

* About, or before, the year 1807. 



PLANE WAVES OF SOUND 163 

inversely as the square root of the density, provided the com- 
parison be made at the same pressure. These conclusions are 
in agreement with observation. 

The formula (8) will of course apply to any fluid medium, 
provided the proper value of K be taken. In liquids the 
difference between the isothermal and adiabatic elasticities 

3 may be neglected. For water at 15 C. we may put 
K = 2-22 x 10 10 , p = l, in C.G.S. units, whence c = 1490 metres 

; per second. The number found by Colladon and Sturm (1826) 

I by direct observation, in the water of the lake of Geneva, was 

! 1435, at a temperature of about 8 C. 

Another formula for the velocity of sound may be noticed. 
If H denote the height of a " homogeneous atmosphere," i.e. of 
bi column of uniform density p Q whose weight would produce the 
actual pressure p per unit area, we have p = gpoH, and the 
Newtonian formula (10) becomes 

c = V(<7#); ..................... (14) 

cf. 43 (6). The velocity is accordingly that due to a fall from 
rest through a height %H. It appears from 58 (1) that for 
a given gas, and at a given place, H depends only on the 
temperature. The corresponding adiabatic formula is 

(15) 



60. Energy of Sound- Waves. 

The kinetic energy of a system of plane waves is, per unit 
area of the wave-fronts, 






where the integration extends over the space which was occupied 
by the disturbed air in the equilibrium state. 

The work done by unit mass in expanding through a small 
range was found in 58 to be given accurately, to the second 
order, by the expression 



where the suffix refers to the final state. If we form the sum 
of the corresponding quantities for all the mass-elements of the 
system, the first term disappears whenever the conditions are 
such that the total change of volume is zero. Again, in the 

112 



164 DYNAMICAL THEORY OF SOUND 

second term we may put, with sufficient accuracy, pp = KS, 
V Q V = V O S, and obtain ^KS Z .V O . The expression %/cs* is there- 
fore to be integrated over the volume occupied in the undis- 
turbed state. So far nothing is stipulated as to the hypothesis 
to which K relates; but it is only in the case of adiabatic 
expansion that the result can be identified with the potential 
energy in the strict sense of this term. We then have 

V = ^ K fs 2 dx } ..................... (3) 

where K = <yp 0) per unit area of wave-front. If K refer to the 
isothermal condition, the expression on the right hand is what 
is known in thermodynamics as the " free energy." 

It is unnecessary to repeat what has been said in 23 as to 
the resolution of an arbitrary initial disturbance into two 
wave-systems travelling in opposite directions. In a single 
progressive wave-system, say 

?=/((*-), ..................... (4) 

we have by 59 (5) = cs, ........................... (5) 

where denotes the particle- velocity in the direction of propa- 
gation. Since f has the same sign as s, an air-particle moves 
forwards (i.e. with the waves) as a phase of condensation passes 
it, and backwards during a rarefaction. It appears moreover, 
from (1), (3), and (5), that the total energy is half kinetic 
and half potential. This also follows independently from the 
general argument given in 23. 

The case of a simple-harmonic train of progressive waves is 
specially important. The formula 

.................. (6) 



represents a train of amplitude a, frequency n/2?r, and wave- 
length X = 27rc/ft. We find 



n (t - -] 



sin 2 n t - - dx 



(7) 



The mean value of the second term under the integral sign is 
zero, and the average kinetic energy per unit volume is therefore 
and the average value of the total energy 



PLANE WAVES OF SOUND 165 

Since iia is the maximum particle-velocity, we see that the 
energy in any region including an exact number of wave-lengths 
is the same as the kinetic energy of the whole mass when 
animated with the maximum velocity of the air-particles. If s l 
be used to denote the maximum condensation, we have 81 = na/c, 
and the average energy per unit volume may therefore also be 
expressed by ^potfsf. 

We can also estimate, incidentally, the nature of the approxi- 
mation involved in the derivation of the equation of motion 59 
(7). The approximation consisted in neglecting the square of s, 
or 9f/9a?. Since 81 = 27ra/\, this means that the amplitude a is 
assumed to be small compared with X/27T, a condition which is 
abundantly fulfilled in all ordinary sound-waves. 

So far we have traced the course of waves regarded as 
already existent, without any reference to their origin. As an 
example, though a somewhat artificial one, of the manner in 
which waves may be supposed to be generated, imagine a long 
straight tube, of sectional area , in which a piston is made 
to move to and fro through a small range, in any arbitrary 
manner. The origin of x being taken at the mean position 
of the piston, the forced waves in the tube, to the right, 
due to a prescribed motion 

?=/(*) (8) 

of the piston, will evidently be given by 



In particular, if % = acosnt, (10) 

we have = acosnU ) (11) 



The rate at which work is being done by the piston on the au- 
to the right is 



4f/i 2/Y 2 

= p 9 <ona sm nt -\ -- o>sm*nt. ......(12) 

c 

The mean value of the first term is zero, whilst that of the 

second is 

(13) 



166 DYNAMICAL THEORY OF SOUND 

This is exactly the mean energy contained in a volume we of 
the space occupied by the wave-train (11). The result may 
perhaps at first sight appear to be a mere truism. It may be 
argued that in each unit of time fresh waves are generated 
which occupy a length c of the tube, and that the piston must 
as a matter of course supply the corresponding amount of 
energy. It must be remembered, however, that an infinitely 
long train of waves of the type (11) would take an infinite time 
to establish, and that in the case of a finite train the suggested 
line of argument would require us to examine into what is 
taking place at its front. In the present instance the result 
would, it is true, be unaffected, but the case would be altered 
if the wave-velocity were different for different wave-lengths, 
as it is for example in dispersive media in optics, in deep-water 
waves in hydrodynamics, and in the case of flexural waves on a 
long straight bar ( 45). There is then a distinction between 
the wave- velocity (for a particular wave-length) and the "group- 
velocity" which determines the rate of propagation of energy. 

In the above problem force must be applied to the piston in 
order to maintain the vibration (8) against the reaction of the 
air. If the piston be free, the store of energy which it origin- 
ally possessed will be gradually used up in the generation of 
air-waves. Suppose, for example, that the piston is attached 
to a spring, and that in the absence of the air the period of its 
free vibrations would be ^TT/U. Under the actual conditions, 
its equation of motion will be of the form 

M(Z + n^) = -(p- Po )co, (14) 

where the variable part of the pressure alone appears, since the 
constant part merely affects the equilibrium position. From 
the general theory of progressive waves we have 

p-p^KS^tcg/c, (15) 

and the equation (14) becomes 

5 + ?+^ = o (16) 

This is of the form discussed in 11, and the solution is 

f .-=(7<?-'/ T cos(rc'+ e), (17) 



PLANE WAVES OF SOUND 167 

provided T = ZMc/tca = ^Mjp^c, n' z = n*-l/r*. i ..... (18) 

When nr is large the effect on the period may be neglected. 
The condition for this is that 2M* must be large compared 
with p o>X/27r, where \ is the wave-length. The inertia of the 
piston must therefore be great compared with that of the air 
contained in a length X/2-n- of the tube. The same law of 
decay would be given also by the indirect method explained 
in 12. 

We have seen in (13) that the rate of propagation of energy 
across unit area of wave-front in a progressive system of waves 
of simple-harmonic type is Jp ri 2 a a c, or J/^oC 3 ^ 2 , if s l denote the 
maximum condensation. The result was obtained for plane 
waves, but will hold for all kinds of wave at a sufficient distance 
from the source. Consequently if W denote the total emission 
of sonorous energy per second from a source near the ground, 
the value of s a , at a distance r, will be given by the relation 



W =i/3 c 3 5 1 2 x 27rr 2 = 7rp c 3 r 2 5 1 2 .......... (19) 



This formula was applied by Lord Rayleigh to estimate the 
limit of audibility of a sound of given pitch. The value of W, 
as inferred from the power spent in actuating the source 
(a whistle), is the product of the current into the pressure, and if 
r be the distance at which the sound is just audible, the formula 
will give a value of s lf which is necessarily, however, greater 
than the true limit, since the value of W is too high, not all 
the energy being spent in sound. In this way it was ascer- 
tained that sounds could be heard in which Sj_ was certainly less 
than 4 x 10~ 8 . The corresponding amplitude as deduced from 
the formula ncu cs^ was 8 x 10~ 8 cm. By an independent 
method, in which the above source of uncertainty was avoided, 
the limit of audibility was fixed at about s l = 6 x 10" 9 . Subse- 
quent experiments by Wien (1903) and Rayleigh f indicate an 
increase of sensitiveness with rise of pitch, for tones near the 
middle of the ordinary musical scale. 

* The factor 2 would disappear if the piston were supposed to generate waves 
on both sides. 

t Phil. Mag. (6), vol. xiv. (1907). 



168 DYNAMICAL THEORY OF SOUND 

61. Reflection. 

When there is a fixed barrier at the origin the general 
solution is replaced, as in 24, by 

g = f(ct-a;)-f(ct + x) (1) 

Considering, for example, the region to the left of the origin, 
the first term may be interpreted as representing a primary 
wave-system approaching the barrier; the second term then 
represents the reflected system. The latter has the same 
amplitude at corresponding points ; the velocity j is reversed, 
but the condensation s (= di~/dx) has its sign unchanged. We 
have here, in its simplest form, the explanation of echoes. 

There is another case of reflection which it is important to 
consider. Suppose that at one point (say x = 0) the condition 
of unvarying pressure (s = 0) is imposed. We must have then, 

in 59 (9), 

F'(et)=f(ct), (2) 

which shews that the functions f, F must differ only by 
a constant. Since this constant would merely represent a 
displacement common to the whole mass, which is without 
influence on the question, it may be ignored. We have then 

f = f(ct"x)+f(ct + x) t (3) 

where as before the first term may be taken to represent an 
incident, and the second a reflected wave-system, in the region 
lying to the left of 0. The velocity f is here reflected un- 
changed, but the sign of s is reversed. The conditions would 
be realized if the air were in contact at the plane as = with 
a medium capable of exerting pressure, but destitute of inertia. 
This is of course an ideal case, but the condition of invariable 
pressure is approximated to in some degree at the open end of 
a pipe. The present investigation has also an application to 
the reflection of longitudinal waves at the free end of a rod 
( 43). 

The general problem of (direct) reflection at the common 
boundary of two distinct fluid media is hardly more complicated. 
The origin being taken in the boundary, a wave-system ap- 
proaching from the left will give rise to a reflected wave on the 
left and a transmitted wave on the right. We distinguish 



PLANE WAVES OF SOUND 169 

quantities relating to the incident and reflected wave by the 
suffixes 1 and 2, respectively, whilst those relating to the 
transmitted wave are indicated by (grave) accents. Since the 
velocity and the pressure must be the same for the two media 
at the origin, we have 

?i + ? 2 = i\ *s 1 + *s 2 = *Y [>=0], ......... (4) 

the equilibrium pressure p being necessarily the same. Now 
^ = 0$!, J 2 = cs 2 , c> whence 

c(s l s 2 ) = cs\ ic (! + * 2 ) = *V [# = 0] ....... (5) 

^ 

Hence S " 



These formulae relate in the first instance to the state of 
things at the origin, on the two sides; but it is easily seen that 
they will also represent the ratios of amplitudes at correspond- 
ing points in the respective waves. If the inertia of the second 
medium were infinite, we should have c = 0, and therefore 
$2 = $!, as in the case of reflection at a rigid barrier. On the 
other hand, if the inertia of the second medium were evanescent, 
we should have c = oo and s 2 = i, as above. 

The energies of corresponding portions of the various waves 
are proportional to KS^C, KSC, #W, since the lengths occupied 
by these portions will vary as the respective wave-velocities. 
The conservation of energy therefore requires 

KSi*C = KS 2 *C + KS"*C', .................. (7) 

this is easily verified from (6). 

If we put K = p c z , K = poC*, we have, from (6), 



*i A> c + p c 

As an example, take the case of air- waves incident normally 
on the surface of water. We have p /p x = '00129, c/c' = '222, 
about; whence s z /s l = '99943. There is therefore almost com- 
plete reflection, with hardly any transmission. 

In the case of two gaseous media having the same ratio of 
specific heats, and therefore the same elasticity (K = yp ), the 
formulae simplify ; thus 

* = ^, S -=^ (9) 

S, C + C 5 X C + C 



170 DYNAMICAL THEOKY OF SOUND 

These are identical with Fresnel's formulae for the amplitudes 
of reflected and transmitted light in the case of normal inci- 
dence on the common boundary of two transparent media. 

62. Vibrations of a Column of Air. 

When we come to the free oscillations of the air contained 
in a pipe of finite length, the question definitely arises as to the 
condition to be satisfied at an open end. There is here a 
transition, more or less rapid, from plane waves in the tube 
to diverging spherical waves in the external space, which it is 
difficult to allow for exactly. In the usual rudimentary theory, 
which dates from D. Bernoulli, Euler, and Lagrange, it is 
assumed that the variation of pressure in the tube, at the open 
end, may be neglected. As already stated, this would be 
accurately the case if the external air were replaced by a 
substance capable of exerting pressure (p ) but devoid of 
inertia. There would then be no loss of energy on reflection 
at the open end ( 61), and the vibrations in the tube, once 
excited, would be persistent. The hypothesis is obviously 
an imperfect representation of the facts ; the condition s = 
can only be approximately fulfilled, and energy must con- 
tinually be spent in the generation of waves diverging outwards 
from the mouth, so that the vibrations if left to themselves will 
be sensible only for a very limited time ; this may however 
cover hundreds of periods. We shall return to these questions 
later (Chapter IX) ; at present we content ourselves with 
tracing out the consequences of the approximate theory. 

The periodic character of the motion in a finite pipe can be 
inferred from the theory of waves, exactly as in the case of 
strings ( 24). Suppose for example that a wave ,of limited 
extent is started in either direction from a point P of a tube 
AB. After two reflections, at A and B, the wave will pass P 
again in the same direction as at first. If both ends be closed, 
the sign of s is unaltered at either reflection, whilst that of j is 
twice reversed. Hence after the interval 2l/c, where l AB, 
the initial circumstances are exactly reproduced. The same 
result holds if both ends be open, since there have now been 



PLANE WAVES OF SOUND 171 

two reversals of s and none of f in the interval in question. 
But if one end be closed and the other open, the signs of s and 
f at P have each undergone one reversal only in the interval 
2l/c, and a further interval of like duration must elapse before 
the original state of things at P is restored. 

The foregoing theory explains one or two important points 
in the theory of organ-pipes. Thus the frequency, in the 
gravest mode, is inversely proportional to the length, and is 
lower by an octave for a "stopped" pipe, i.e. a pipe closed 
at one end, than for an " open " pipe, i.e. one open at both ends, 
of the same length. It is, again, directly proportional to the 
velocity of sound, and so increases with rise of temperature. 

In the analytical method for determining the normal modes 
we assume as usual that f varies as cos (nt + e). The equation 
59 (7) then becomes 



the solution of which is 

/ . nx D . nx\ /ON 

t=\A cos h B sin cos (nt + e), (2) 

\ c c J 

as in 25. The corresponding wave-length of progressive waves 
in free air is X = 2?rc/n. Hence in any system of standing waves 
there is a series of nodes ( = 0) at intervals of X, and a series 
of loops, or places of zero condensation, (df/dx = 0), half-way 
between these. 

For a tube closed at both ends (x = 0, x = I) we have 

-4=0, sin(y/c) = 0, (3) 

and therefore 

~ . rmrx frmrct \ 
I f = w sm j- cos!-^ + m ), (4) 

\ 6 / 

where m 1, 2, 3, ..., the normal modes forming a harmonic 
series. 

For a pipe open at both ends, the condition that s=d^/dxQ 
for x = and x = I gives 

= 0, sin(^/c) = 0, (5) 

and the typical solution is 



n mirx frmrct \ 
= G m cos -j cos j + e m , (6) 

t \ 6 / 



172 DYNAMICAL THEORY OF SOUND 

where ra = l, 2, 3, .... Here, again, the sequence of normal 
modes is harmonic. The figure illustrates the cases m=l, 
m = 2. The arrows shew the direction of motion at the loops, 
whose position is indicated by the dotted lines, in two opposite 
phases ; the nodes are indicated by the full transverse lines. 



Fig. 60. 

In the case of a pipe closed at # = and open at x=l, 

we have 

.4=0, cos(ttZ/c) = 0, .................. (7) 

whence nl/c ^mjr, the integers m being odd. We thus obtain 

~ mirx (rmrct 
=---- 



where m= 1, 3, 5, .... The absence of the harmonics of even 



Fig. 61. 

order determines the characteristic " quality " of stopped pipes 
( 91). The figure shews the cases m = 1, m = 3. 

The formula (2) can be applied also to the case of forced 



PLANE WAVES OF SOUND 173 

vibrations of given frequency (n/27r). Thus if a prescribed 
vibration 

f = J. cos(nt + e) .................. (9) 

be maintained at x = 0, and if the tube be closed at x = I, the 
motion of the gas is given by 



sin - cos (nt + e). . . .(10) 

c 

The amplitude becomes abnormally great, even when we take 
account of dissipative forces, if sin (nl/c) = 0, or I = %m\, where 
m is integral. This is the principle of a method due to Kundt 
(1868) by which the velocity of sound in various gases can be 
compared by small-scale experiments. The wave-lengths are 
found by measuring the distances between the nodes, whose 
position is indicated by the heaping up of lycopodium powder 
previously scattered in the tube. The vibrations are excited 
in the two tubes (containing the two gases to be compared) by 
disks fitted to the two ends of a longitudinally vibrating rod. 
If the end x= I is open, the formula (10) is replaced by 

g= ^008^=^ 008 (lit + 6), ,..(11) 

cos (nl/c) c 

and the condition of strongest resonance is cos (nl/c) = 0, or 
/ = Jm\, where m is an odd integer. 

The preceding investigations would apply also to the 
vibrations of a column of water, or other liquid, contained 
in a tube, provided the material of the tube were absolutely 
rigid. In practice, however, the yielding of the walls has 
an appreciable effect; the potential energy corresponding to 
a given strain (dg/dx) of the fluid is diminished, and the wave- 
velocity is lowered. The fact was observed by Wertheim (1847), 
but the true explanation is due to Helmholtz (1848). The 
question has been further investigated by Korteweg (1878) 
and the present writer. Owing to the much greater velocities 
( 44) of elastic waves in solids such as glass or steel, as 
compared with the sound- velocity in water, the stresses in 
the walls adjust themselves so rapidly that it is legitimate 
to assume that the deformation of the tube has the statical 
value corresponding to the instantaneous distribution of 



174 DYNAMICAL THEOEY OF SOUND 

pressure in the liquid. If c be the theoretical velocity of 
sound in the liquid, as given by 59 (8), c the actual velocity, 
it is found that in the case of a tube of small thickness h 

...(12) 



where a is the internal radius, K is the volume-elasticity of 
the liquid, and E is the value of Young's modulus for the 
material of the tube. Thus in the case of water (K = 2'22 x 10 10 ) 
contained in a glass tube (^=6*03 x 10 11 ) whose thickness is 
one-tenth of the radius, we find c '759c . Even in the other 
extreme, when the walls are very thick, it is found that 

where /A is the rigidity. The value of /z, for glass is, roughly, 
about 10 times the value of K for water; this would give 
a diminution of about 5 per cent, in the wave-velocity. 

63. Waves of Finite Amplitude. 

The laws of sound propagation, as they are investigated in 
this and succeeding chapters, are subject to some qualifications 
which may best be considered in relation to plane waves, where 
the theory is simplest. 

In the first place, it has been assumed that the conden- 
sation s may be treated as infinitely small. This hypothesis 
is adequate for most purposes, but there are certain "second 
order " effects which are of some theoretical importance. 

It is easy to shew that a progressive wave of finite (as 
distinguished from infinitely small) amplitude cannot be pro- 
pagated without change of type, except on the hypothesis 
of a certain special relation between pressure and density. 
Assuming, for a moment, that a wave of permanent type is 
in progress, we may in imagination impress on the whole 
mass of air a velocity equal and opposite to that of the 
wave. In this way we obtain a condition of " steady motion " 
as it is called, in which the velocity, pressure, and density at 
any point of space are constant with respect to the time. 
For definiteness we may fix our attention on the air contained 
in a long straight tube of unit sectional area. The velocity u 



PLANE WAVES OF SOUND 175 

being now a function of the space-coordinate x only, the 
acceleration of the air-particles will be given by udu/dx as 
in ordinary dynamics. Hence, considering the acceleration of 
momentum of the mass which at the instant considered lies 
between the planes x and x + 8x, we have 

du dp 
pu- r = -/- ...................... (1) 

dx dx 

Also, since the same amount of matter crosses each section in 
unit time, we have 

pu = const. =??i, ..................... (2) 

say. Hence mdu/dx dp/dx, and 

p= C mu, ........................ (3) 



or p-p = m(u -u) = m* ---, ......... (4) 

\Po pJ 

where the symbols p Q , p , u refer to the parts of the medium 
which in the original form of the question were undisturbed. 
This gives the special relation referred to. In terms of the 
volume per unit mass we have 

p-p =m*(v Q -v), .................. (5) 

which is the equation of a straight line on the indicator 
diagram. A relation of this type does not hold for any 
known substance, whether under the adiabatic or the iso- 
thermal condition, and could in any case only apply to a 
limited range, since the volume would otherwise shrink to 
nothing under a certain finite pressure. 

If, however, the range of density be small, the equation (5) 
can be identified with 59 (6) provided m z =Kp . Since m=p ^o> 
where u is the wave- velocity in the original form of the problem, 
this gives u Q 2 = ic/p , in agreement with 59 (8). The process 
is equivalent to choosing m so that the straight line (5) shall 
be a tangent at the point (v , p ) to the curve which on the 
indicator diagram gives the effective relation between p and v. 

The condition (5) was obtained in different ways by 
Earnshaw (1860) and Rankine* (1870). 

To ascertain the character of the continual change of type 

* W. J. M. Rankine (1820 72), professor of engineering at Glasgow, 
185572. 



176 DYNAMICAL THEORY OF SOUND 

which must take place in sound-waves propagated in actual 
fluids, we must have recourse to accurate equations of motion. 
On the plan of 59 we have 



and P = po/(l + A) = p /(l -f ^ (7) 

Hence, on the adiabatic hypothesis that 

P/Po ~ (p/po) 7 > (8) 

we find by elimination of p and p 

2\Y+1 

, (9) 



where c 2 = ypo/po as before. 

For illustrative purposes it is sufficient to consider the 
isothermal case, which is derived from the above by putting 
7=1, so that 



We have seen in 60 that on the hypothesis of infinitely 
small vibrations there is a definite relation between particle- 
velocity and condensation in a progressive wave. Following 
Earnshaw, we assume (tentatively) that the same thing holds 
in the general case, and write accordingly 

(11) 



where the form of the function is to be determined. From this 
we deduce 



da?' 

and therefore = f ................ (IS) 

Hence (10) is satisfied provided 






PLANE WAVES OF SOUND 177 

no additive constant being necessary if we assume that f = 
in the parts of the medium not affected by the wave. This 
may also be written 

?= + clog(l+s), .................. (16) 

by 59 (3). Another form is 

P/P* = e^' c ...................... (17) 

When s is infinitesimal the formula (16) reduces to f = + cs, in 
agreement with 60. 

To find the rate at which any particular value of s is 
propagated, in either of these cases, we note that the value of 
9f/9a? which is associated with the particle x at the instant t 
will have been transmitted to the particle x + $>x at the instant 
t + $t, provided 

|f 8M.ga.-o, 

oxdt da? 
i.e. by (12) and (14), 

&c c(l+s)& = ................ (18) 

The phase s is therefore propagated with the velocity 

J = + c(l + S ) .................. (19) 

relative to the undisturbed medium. To find the rate of 
propagation in space we have to take account of the total 
variation of x + , which is 



The required velocity is therefore 

('+DM S ........ ; 

The lower sign relates to a wave travelling in the direction of 
#- positive. It appears from (16) that positive values of f are 
then associated with positive values of s, as in the approximate 
theory of 60 ; but the formula (20) shews that the velocity 
of propagation is greater, the greater the value of s. The parts 
of the wave where the density is greater therefore gain con- 
tinually on those where it is less. Thus if the relation between 
s and x be exhibited graphically, the curve A in the annexed 
L. 12 



178 



DYNAMICAL THEORY OF SOUND 



figure takes after a time some such form as B*. The wave 
becomes, so to speak, continually steeper in front, and slopes 
more gradually in the rear, until a time arrives at which the 
gradient at some point becomes infinite. After this stage the 
analysis ceases to have any real meaning. 





Fig. 62. 

The adiabatic hypothesis leads to results of the same 
general character. The reader will find no difficulty in verifying 
the following statement. The formula (16) is replaced by 



and the velocity of propagation of a particular value of s is 

Tc(l+*)* (7 + 1) .................. (22) 

relative to the undisturbed medium, or 



in space. In the latter formula the particle-velocity is added 
to the velocity of sound proper to the actual density, which is 
on the adiabatic hypothesis dependent on the degree of con- 
densation and consequent change of temperature. The general 
conclusions are as before. 

* It is not very important here whether the coordinate x be supposed (as in 
the previous part of this investigation) to refer to the undisturbed medium, or 
to be an ordinary space-coordinate. In either case the tendency is the same. 



PLANE WAVES OF SOUND 179 

It must be remembered that since the equation of motion 
(9) is not linear, distinct solutions, such as those representing 
waves travelling right and left, respectively, which we have 
just been considering, cannot be superposed by mere addition. 
It may however be remarked that, as a result of a more 
complete investigation, Riemann* found (1860) that a localized 
arbitrary initial disturbance does eventually resolve itself into 
two waves of the above kinds, travelling in opposite directions. 

To follow exactly the career of waves of finite amplitude 
generated in any given manner is a problem of considerable 
difficulty; but some indications may be obtained by methods 
of approximation. This procedure was adopted by Airy f (1845) 
in his work on the dynamical theory of the tides, where similar 
questions arise with respect to tides in shallow seas and 
estuaries. 

Suppose, for instance, we have a long straight tube in which 
a piston (at x = 0) is made to move in an arbitrary manner 

?=/(<) ....................... ..(24) 

The equation (9) becomes, if we neglect terms of the third 
order in the derivatives of f, 



If we omit the last term, we have as in 60 the first 
approximation 

(26) 



Substituting this value of f in the small term of (25) we 
obtain 



The solution of this which is consistent with (26) is 

....... <2S> 



as is easily verified. The correction to the first approximation 

* Bernhard Riemann (1826 66), professor of mathematics at Gottingen 
185766. 

t Sir George Biddell Airy (1801 92), Plumian professor of astronomy at 
Cambridge 182835, astronomer royal 183581. 

122 



180 DYNAMICAL THEORY OF SOUND 

(26) is proportional to x, and to the square of the ratio of the 
velocity of the piston to the velocity of sound. This latter ratio 
may in practice be exceedingly small, but as we travel to the 
right the correction continually increases in importance, until at 
length the neglect of terms of the third and higher orders 
would no longer be justified. This is what we should expect 
from the results of Earnsnaw's investigation. 

When the motion of the piston is simple-harmonic, say 

/(0 = acosnt, (29) 

the formula (28) gives 

g = a cos n (t- -} + (v + V n9a * % \ _ C os2rc (t- -}l. (30) 

\ C/ oC \ C/j 

The displacement of any particle is no longer simple-harmonic, 
but consists of a part independent of t together with two 
simple-harmonic terms, one having the frequency of the 
imposed vibration (29), and the other a frequency twice as 
great. This illustrates the implied limitation to infinitely 
small motions in the usual theory of forced oscillations ( 17). 

Again, if the given vibration of the piston be made up of 
two simple-harmonic components, say 

f(t) = ! cos nj + a 2 cos n 2 t, (31) 

we find 

f = ttj cos nj (t - - j + a 2 cos n z (t -J 

f 1 ( / #\ 

- x \ n?a? + n 2 2 a 2 2 nfaf cos 2w x It 
kr ( V cj 

n 2 W cos 2n 2 (t 

V c 

-f 2/1^2 Oi a z cos (/*! n 2 ) { t - 



2n 1 n 2 a 1 o 2 cos (^ + ^2) It JY . 

.(32) 

We thus learn that in addition to the vibrations of double 
frequency, other simple-harmonic vibrations whose frequencies 
are respectively the difference and the sum of the primary 
frequencies now make their appearance. In acoustical language, 
two simple vibrations of sufficient amplitude may give rise not 



PLANE WAVES OF SOUND 181 

only to the corresponding pure tones, but to their octaves, as 
well as to certain "combination- tones," whose occurrence 
reminds us again, that the principle of superposition is no 
longer valid. We shall have occasion to refer to this investi- 
gation at a later period (Chap. X). 

The analogous phenomenon in tidal theory is the production 
of "over- tides," which are in fact appreciable, and have to be 
provided for in the Harmonic Analysis referred to in 39. 

We have seen that the main effect of finite amplitude is 
that in a progressive wave the gradients, both of pressure and 
of density, tend to become infinite. This has suggested the 
question whether a wave of discontinuity might not finally be 
established, analogous to a "bore" in water-waves. To examine 
into the possibility of such a wave we take the question in its 
simplest form, and assume that the circumstances are everywhere 
uniform, except for the sudden transition at the plane of dis- 
continuity. Further, by the superposition of a certain uniform 
velocity, we reduce the problem to one of steady motion in 
which the plane in question is fixed. 

The symbols p , p , u will then be supposed to refer to the 
region to the left of this plane, whilst the values of the corre- 
sponding quantities on the 
right are denoted by p, p, u. 
Since in every unit of time 
the same mass (m) of fluid 
crosses any unit area normal 
to the direction of flow, we have 

pu = p u =m, or u = mv, u Q = mv (33) 

Again, since in unit time a mass m has its velocity changed from 
UQ to u, the momentum of the portion of air included between 
two planes in the positions indicated by the dotted lines in 
Fig. 63 is increasing at the rate m (u - u ), whence 

p -p = m(u-u Q \ (34) 

or, in virtue of (33), 

p p = m?(v v ), (35) 

in agreement with (5). If we now superpose a uniform velocity 



182 DYNAMICAL THEOEY OF SOUND 

MO, we get the case of a wave advancing into a region 
previously at rest. The wave-velocity is given by 



V Q -V p-p po 

as first found by Stokes (1848), and afterwards independently 
by Earnshaw, Riemann, and Rankine. A difficulty, first pointed 
out by Lord Rayleigh, arises, however, as to the conservation of 
energy. The rate at which work is being done on the portion 
of air above considered is p u Q pu, whilst that at which the 
kinetic energy is increasing is \ m (u? u<?). The difference is 

p u -pu-\m (u 2 - uf) = \m ( p + p) - v). . . .(37) 

If the two points (v, p), (v , p ) on the indicator diagram be 
denoted by P, P , respectively, the expression (37) is m times 
the area of the trapezium bounded by the straight line P P, 
the axis of v, and the ordinates p 0) p. If the transition be 
effected without gain or loss of heat, the points P , P will lie 
on the same adiabatic, and the gain of intrinsic energy will be 
represented by the area included between this curve, the axis 
of v, and the same two ordinates. Since the adiabatics are con- 
cave upwards, the latter area is (in absolute value) less than the 
former. It appears on examination of the signs to be attributed 
to the areas that if v > v the work done is more than is accounted 
for by the increase of the kinetic and intrinsic energies, whilst if 
VQ < v the work given out would be more than is equivalent to 
the apparent loss of energy. 

It is evident that no complete theory of waves of discon- 
tinuity can be attempted without some reference to viscosity 
and to thermal conduction, since at the point of transition 
the gradients of velocity and temperature are infinite. 

It does not appear probable that under ordinary conditions 
the modifications due to finite amplitude are of serious im- 
portance. In equation (30), for instance, the ratio of the 
amplitude of the vibration of the second order to that of the 
primary vibration is comparable with tfax/c 2 , or with n*a/g . x/H, 
where H is the height of the homogeneous atmosphere. With 
ordinary amplitudes a, and ordinary distances x, this ratio will 
be very small. In three dimensions the effect must be very 



PLANE WAVES OF SOUND 183 

much less, owing to the diminution of amplitude by spherical 
divergence. 

64. Viscosity. 

The essence of viscosity is that in a moving fluid the stresses 
differ from a state of pressure uniform in all directions about a 
point, by quantities depending on the rates of deformation. It 
is usually assumed that these quantities are linear functions of 
the rates of strain ; from our present standpoint this is 
sufficiently justified by the fact that the strain- velocities are 
regarded as infinitely small. As in 40 there will at any 
instant, and at any given point, be three principal axes of the 
deformation which is taking place, and these will naturally be 
the principal axes of the corresponding stress. We therefore 
write, by analogy with 42 (1), 



............... (1) 

where lt e 2 > *s are the principal strain- velocities, and 

A = 1 + 2 + 3 ...................... (2) 

By the same kind of proof as in 41, // is recognized as the 
coefficient of viscous resistance to a shearing motion in parallel 
planes; viz. if TJ denote the rate of shear, and TS the corre- 
sponding stress, we have 

r = //i) ......................... (3) 

The value of // has been determined with considerable accuracy 
for a number of fluids, gaseous as well as liquid. 

It will be noticed that the meaning of the symbol p, and 
consequently the value of V, is so far indeterminate, since 
nothing is altered in the shape of the formulae (1) if we 
incorporate in p any constant multiple of A. In the case of 
liquids it is in fact usual so to incorporate the second terms in 
(1). In the application to gases it is convenient to regard p as 
defined by the gaseous laws (p = DpO). There is at present no 
experimental evidence as to how far the mean stress about a 
point, viz. 

i (Pi +p* +p,) = -P + (*' + I A*') A, 



184 DYNAMICAL THEORY OF SOUND 

differs, in a moving gas, from p, as thus fixed ; but from 
considerations based on the kinetic theory of gases Maxwell* 
inferred (1866) that the two things are identical, and that 
accordingly 

V }/*' W 

As we are interested chiefly in the order of magnitude of 
the effects, the precise determination of X' is not of much 
consequence to us; accordingly Maxwell's view is adopted for 
simplicity in what follows. 

The dimensions of // are those of a stress multiplied by 
a time, or [ML~ 1 T~ 1 ]. It is found that /*' is independent of 
the density, but (in gases) increases with rise of temperature. 
Its value for air at C. is about '000170 in absolute c.G.s. 
units. It will appear however immediately that the effect of 
viscosity in modifying motion depends not so much on the 
value of fA as on its ratio to the inertia of the fluid. This 
ratio 

v = p/po (5) 

is therefore called by Maxwell the " kinematic " coefficient of 
viscosity; its dimensions are [L 2 T~~ 1 ]. For air at C. its value 
is about '132 c.G.s. 

The rate at which the stresses on the faces of a unit cube 
are doing work in changing its size and shape is given by 

X'A 2 + 2fl' (tf + 6 2 2 + 6 3 2 ) 
' {( 2 - 6 3 ) 2 + (63 - <0 2 + (^ - 6 2 ) 2 }. . . .(6) 

The term p& represents the rate at which the intrinsic 
energy is increasing. The remaining terms, which are essenti- 
ally positive, indicate a dissipation of energy at the rate 

f/{fe-*s) 2 + (- *0 2 + (, -<U 2 } (7) 

per unit volume. The mechanical energy thus lost is converted 
into heat. It will be noticed that (7) vanishes in the case of 
uniform expansion (e t = e 2 = 6 3 ) ; this is a necessary consequence 
of our previous assumption as to the value of the constant X'. 

* James Clerk Maxwell (1831 79), professor of experimental physics at 
Cambridge (187179) ; author of the electromagnetic theory of light. 



PLANE WAVES OF SOUND 185 

In the case of a pure shearing motion (17), the formula (6) 
takes the shape 

w^ = /*V ......................... (8) 

In plane waves of sound we have e 2 = 0, e 3 = 0, and therefore 
from (1) and (4) 

p l ^-p-\-^fJi l = -p -KS + ^fJL / l .......... (9) 

Moreover, in the notation of 59, 

2- *- ............... <'> 

The equation of motion, viz. 

o^- 9 ^ (11) 

*!*- ..................... l " J 

therefore becomes 



* 

To obtain a solution appropriate to the case of free waves 
we put 

f=Pcosfcp, ..................... (13) 

where P is a function of t, to be determined. We find that 
(12) will be satisfied, provided 



<"> 



This has the form of 11 (3), and the solution is therefore 

P = ta-<<' T cos(7tf-He), ............... (15) 

provided r = 3/2i/& 2 , n? = & 2 c 2 - 1/r 2 ............. (16) 

In all cases of interest cr is a considerable multiple of the 
wave-length (X = 2?r/A?), so that n = kc, practically, the friction 
having as usual no appreciable effect on the period. Thus 

f = Ce~ tir cos (kct + e) . cos kx .......... (17) 

This represents a system of standing waves with fixed nodes 
and loops. There is a similar solution in which cos kx is 
replaced by sin&#, and by superposition of the two we can 
construct a progressive wave-system 

% = Ce-*l r wak(ct-x) ................ (18) 

Putting i/ = -132 for the case of air, we find T = '288X 2 , the 
units being the second and the centimetre. 



186 DYNAMICAL THEOEY OF SOUND 

The solution of (14) may also be effected concisely by means 
of imaginary quantities. Thus in investigating forced simple- 
harmonic vibrations of prescribed frequency we assume that 

fc _ ftgi nt+ mx (19) 

whence, on substitution, 

n 2 



(20) 



c 2 -I- 1 ivn 

The ratio vn/c* is usually very small; thus for n = 1500 its value 
is, with previous data, about 1*8 x 10~ 7 . Hence 



Taking the lower sign, which corresponds to waves travelling in 
the direction of ^-positive, and rejecting the imaginary part of 
(19), we find 

............... (22) 



provided l=3(?/2vn* ...................... (23) 

This represents a system of waves generated to the right of 
the origin by a prescribed motion f = a cos nt at this point (as 
by a piston in a tube if we neglect the friction at the sides). 
The waves advance, with (sensibly) the usual velocity c, but 
diminish exponentially in amplitude as they proceed*. The 
linear magnitude I measures the distance over which the waves 
travel before the amplitude is diminished in the ratio \\e. In 
terms of the wave -length we have 

I = (Sc/STr 2 !/) . X 2 , .................. (24) 

or, with previous data, I = 9'56X 2 x 10 3 . The effect of viscosity 
in stifling the vibrations is therefore very slight except in the 
case of sounds of very high frequency and consequently short 
wave-length. Even for \ = 10 cm. the value of / is nearly 10 
kilometres. When we come to the discussion of three- 
dimensional waves it will be clear that the effect of viscosity 
may for most purposes be ignored in comparison with the 
diminution of intensity due to spherical divergence. It is, 
however, of some interest to observe that there is a physical 

* This calculation was first made by Stokes (1845). 



PLANE WAVES OF SOUND 187 

limit to the frequency of vibrations which are capable of 
propagation for more than a very moderate distance. 

The viscosity being small, the rate at which work is done 
per unit area by the piston in maintaining the wave-system 
(22) must have sensibly the value |p n 2 a 2 c found in 60. Since 
the energy in the medium to the right is now finite and on the 
average constant, this must be equal to the rate of dissipation 
of energy by viscosity. The equality is easily verified. The 
dissipation is, by (7), 



= J // ^rf V** cos 2 "(*-*) dx, . . .(25) 

approximately, if we keep only the most important term. 
Writing 



and taking the mean value with respect to the time, we obtain 



by (23). 

65. Effect of Heat Conduction. 

A further cause of dissipation of energy is to be found in 
the thermal processes consequent on the alternate expansions 
and rarefactions of the air. If indeed these succeed each other 
with sufficient rapidity, the variations are almost accurately 
adiabatic, as explained in 59 ; but, as was first pointed out by 
Kirchhoff (1868), the residual conduction of heat is in any case 
of equal importance with viscosity. On the kinetic theory of 
gases the coefficients of " thermometric " conductivity (v) and 
of kinematic viscosity are in fact of the same order of magnitude ; 
according to Maxwell the relation is v=-^v. For this reason 
the preceding calculations of the effect of viscosity on air- waves 
must not be looked upon as more than illustrative. A complete 
investigation, in which both influences are taken into account, 
shews that the effect is equivalent to an increase in the 
kinematic viscosity, but the order of magnitude is unaffected. 



188 DYNAMICAL THEORY OF SOUND 

If on the other hand the alternations of density were to 
take place with extreme slowness, as in the case of very long 
waves of simple-harmonic type, there would be time for 
practical equalization of temperature, and the dissipative 
influence of conduction as well as viscosity would again be 
insignificant. Since the expansions are here nearly isothermal, 
the wave-velocity will approximate to the Newtonian value 

(I 59 (10))- 

In intermediate cases the theory shews that the wave- 
velocity would no longer be constant, but perceptibly dependent 
on the frequency. Since no such effect is observed, we infer 
that in all ordinary cases the conditions are practically adiabatic. 
It appears also that in such intermediate cases the dissipation 
would be very greatly increased. The investigation of Stokes 
(1851), which is here referred to, relates to the effect of 
radiation ; the extension to conduction was made independently 
by Kirchhoff and Lord Rayleigh. It is probable that the 
effects of radiation alone are of subordinate importance. 

The detailed calculation must be passed over, but the 
general explanation of the manner in which thermal processes 
may operate to produce dissipation of energy has been stated 
with such admirable clearness by Stokes that it is worth while 
to reproduce the passage in question. The explicit reference is 
to radiation, but the same principles are involved in the case of 
conduction also. 

" Conceive a mass of air contained in a cylinder in which an 
air-tight piston fits, which is capable of moving without friction, 
and which has its outer face exposed to a constant atmospheric 
pressure ; and suppose the air alternately compressed and 
rarefied by the motion of the piston. If the motion take place 
with extreme slowness, there will be no sensible change of 
temperature, and therefore the work done on the air during 
compression will be given out again by the air during expansion, 
inasmuch as the pressure on the piston will be the same when 
the piston is at the same point of the cylinder, whether it be 
moving forwards or backwards. Similarly, the work done in 
rarefying the air will be given out again by the atmosphere as 
the piston returns towards its position of equilibrium, so that 



PLANE WAVES OF SOUND 189 

the motion would go on without any permanent consumption 
of labouring force. Next, suppose the motion of the piston 
somewhat quicker, so that there is a sensible change of tempera- 
ture produced by condensation and rarefaction. As the piston 
moves forward in condensing the air, the temperature rises, and 
therefore the piston has to work against a pressure greater than 
if there had been no variation of temperature. By the time 
the piston returns, a good portion of the heat developed by 
compression has passed off, and therefore the piston is not 
helped as much in its backward motion by the pressure of the 
air in the cylinder as it had been opposed in its forward motion. 
Similarly, as the piston continues its backward motion, rarefying 
the air, the temperature falls, the pressure of the air in the 
cylinder is diminished more than corresponds merely to the 
change of density, and therefore the piston is less helped in 
opposing the atmospheric pressure than it would have been had 
the temperature remained constant. But by the time the 
piston is returning towards its position of equilibrium, the cold 
has diminished in consequence of the supply of heat from the 
sides of the cylinder, and therefore the force urging the piston 
forward, arising, as it does, from the excess of the external over 
the internal pressure, is less than that which opposed the piston 
in moving from its position of equilibrium. Hence in this case 
the motion of the piston could not be kept up without a 
continual supply of labouring force. Lastly, suppose the piston 
to oscillate with great rapidity, so that there is not time for any 
sensible quantity of heat to pass and repass between the air and 
the sides of the cylinder. In this case the pressures would be 
equal when the piston was at a given point of the cylinder, 
whether it were going or returning, and consequently there 
would be no permanent consumption of labouring force. I do 
not speak of the disturbance of the external air, because I am 
not now taking into account the inertia of the air either within 
or without the cylinder. The third case, then, is similar to the 
first, so far as regards the permanence of the motion; but there 
is this difference ; that, in consequence of the heat produced by 
compression and the cold produced by rarefaction, the force 
urging the piston towards its position of equilibrium, on 



190 DYNAMICAL THEOKY OF SOUND 

whichever side of that position the piston may happen to be, is 
greater than it would have been had the temperature remained 
unaltered. 

"Now the first case is analogous to that of the sonorous 
vibrations of air when the heat and cold produced by sudden 
condensation and rarefaction are supposed to pass away with 
great rapidity. For we are evidently concerned only with the 
relative rates at which the phase of vibration changes, and the 
heat causing the excess of temperature passes away, so that 
it is perfectly immaterial whether we suppose the change of 
motion to be very slow, or the cooling of heated air to be very 
rapid. The second case is analogous to that of sound, when we 
suppose the constants q* and n comparable with each other; and 
we thus see how it is, that, on such a supposition, labouring force 
would be so rapidly consumed, and the sound so rapidly stifled. 
The third case is analogous to that of sound when we make the 
usual supposition, that the alternations of condensation and 
rarefaction take place with too great rapidity to allow a given 
portion of air to acquire or lose any sensible portion of heat by 
radiation. The increase in the force of restitution of the piston, 
arising from the alternate elevation and depression of tempera- 
ture, is analogous to the increase in the forces of restitution 
of the particles of air arising from the same cause, to which 
corresponds an increase in the velocity of propagation of 
sound." 

66. Damping of Waves in Narrow Tubes and Crevices. 

A somewhat greater effect of viscosity may be looked for 
when the air is in contact with a solid body, as at the walls of 
a pipe or resonator, owing to the practically infinite resistance 
which the surface opposes to the sliding of the fluid immedi- 
ately in contact with it. It seems in fact to be well-established 
that the relative velocity vanishes at the surface, whereas in 
our theoretical investigations we assume for the most part that 
sliding takes place quite freely. A closer examination shews 
however that in the case of rapid vibrations, such as we are 
concerned with in acoustics, the effect is mainly local, being 

* [q is a constant of radiation.] 



PLANE WAVES OF SOUND 191 

confined, practically, to a very thin layer of air near the 
surface, and is except in very narrow spaces unimportant. 

The matter may be sufficiently illustrated by a very simple 
case. Suppose that the fluid above the plane y = is acted on 

by a periodic force 

X = fcosnt, (1) 

per unit mass, parallel to Ox, the plane forming a rigid 
boundary. The consequent motion being everywhere parallel 
to Ox and independent of the coordinate x, there is no variation 
of density, and the deformations which are taking place are of 
the nature of shearing motions parallel to y = 0. Denoting the 
velocity f by u, the rate of shear will be 



and the shearing stress on a plane parallel to y is accordingly 
pdu/dy. The stratum bounded by the planes y and y + By 
therefore experiences a resultant force 

a 



per unit area, parallel to x, and the equation of motion is of the 
form 

du d*u 



We have to solve this under the condition that u = Q for 
y = 0. For conciseness we put X = fe int , and reject (in the 
end) the imaginary part of our expressions. The equation is 
then satisfied by 

u = (l n + Ae'y)e i ', .................. (4) 

provided m 2 = in/v, or 

m=(l+0& ..................... (5) 

where = vW 2 ") ......................... ( 6 ) 

Since we are looking for a solution which shall be finite for 
y = oo we take the lower sign. Also, the condition that a = 
for y = requires that A = f/in. Hence 



192 DYNAMICAL THEORY OF SOUND 

or, keeping only the real part, 

(8) 



Tl fit 

a result which is easily verified. When fty is moderately large 
the value of u reduces practically to the first term, which is 
the same as if there had been no friction. The rigid boundary 
accordingly acts as a drag only on a thin stratum ; for example 
when y Sir/ft the velocity falls short of its value at a great 
distance from the surface by about one part in 535. 

In actual problems of acoustics (relating for example to 
vibrations in pipes) the force pX per unit volume is replaced 
by the negative pressure-gradient dp/da, and we have of course 
changes of density to take into account, but the results have 
a similar interpretation. The linear magnitude 

h = 2ir/l3 = )/(4eirp.2irln) ............... (9) 

may be taken to measure the extent to which the dragging 
effect penetrates into the fluid. With the previous data its value 

in centimetres is about 1'29/N*, where N is the frequency; thus 
for N = 256 we find h = "80 mm. 

We may apply the above investigation to obtain an estimate 
of the effect of viscosity on the wave-velocity in a tube, on the 
supposition that the diameter is small compared with the 
wave-length but large compared with the quantity h. The 
tangential stress on the fluid at the boundary y = is, in the 
case of (7), 



by (9), the time-factor e int being understood. The total tan- 
gential force exerted by the walls of a cylindrical tube of radius 
a on the contained air may therefore be equated to 



per unit length, where p denotes the mean pressure over the 
section (?ra 2 ). Hence if u be the mean velocity, we have, 
calculating the forces on the air contained in an element Bx 
of the length, 

9 du -dp . /n ., , dp 

7rp a? = - ira? + $(I-i)ha^-, 

r dt dx * ^ dx 



PLANE WAVES OF SOUND 193 

1 dp L ,- ~ h 



,- ~ 
or ^ = -- f 11 (!-{)-= } ............. (11) 

dt ' 



To this we must add the relations 

..................... (12) 



The elimination of p and s between these equations leads to 

............. < 14 > 



It is already assumed that the time enters through a factor 
e int ; and the solution of (14) is therefore of the type 

u=Ce int+mx , ...(15) 



with ra 2 = 



or m 



approximately, on account of the assumed smallness of h/a. 
For waves propagated in the direction of ^-positive we take 
the lower sign, and write 

m = in/c' a, (18) 



where c' 



and a = nh/4s7rac (20) 

We have, then u=Ce-* .&<- xlc '\ (21) 

or, in real form, u Ce""* cos n It >j (22) 

The wave-velocity is therefore diminished in the ratio given 
by (19). The exponential factor in (22) expresses the law of 
decay of the waves as they advance. If I be defined as in 
64 (23) it will be found that al is of the order \*/ah. The 
rate of decay is therefore much greater under the present 
conditions than in the case of sound waves in the open. 

A formula equivalent to (19) was published without demon- 
stration by Helmholtz in 1863. The above proof is a variation 
of that given by Lord Rayleigh in his Theory of Sound. 

L. 13 



194 DYNAMICAL THEOKY OF SOUND 

A more complete investigation was instituted by Kirchhoff 
(1868) in which thermal processes are considered, as well as 
viscosity. The effects are thereby increased, as already explained, 
but remain of the same order of magnitude. 

As already stated, it is implied in the above calculation that 
the diameter of the tube greatly exceeds the quantity h. When 
on the other hand the diameter is comparable with, or less 
than h, the walls have relatively a much greater hold on the 
vibrating mass, and the character of the motion is entirely 
altered by the friction. In particular, when h is large com- 
pared with the width the mere inertia of the fluid ceases to 
have any appreciable influence, the mean velocity over a 
cross-section being determined by an approximately statical 
equilibrium between the pressure-gradient (in the direction of 
the length) and the friction of the walls. We have, then, 



(23) 



where R is a coefficient of resistance, depending on the nature 
of the fluid, and on the shape and size of the cross-section. 
Again, by Boyle's law, 

=.pb(l + 5), ..................... (24) 

the isothermal hypothesis being adopted as now the most 
appropriate, since, owing to the assumed narrowness of the 
tube, transfer of heat can take place freely. Eliminating p and 
s between (13), (23), and (24), we find 

du_p d*u 

dt~ RW 

This has the same form as the equation of linear conduction of 
heat. Assuming that 

u=Ce int+rn ^, .................. (26) 

we have m* = inR/p , and therefore 

m=(l+*). .................. (27) 

if ^ = ^nR/ Po ................... (28) 

Taking the lower sign we obtain 

M-Ck-ws+^ne-wa^ ............... (29) 

or, in real form, u = Ce~ wx cos (nt ^x) ............. (30) 



PLANE WAVES OF SOUND 195 

The value of R will be sensibly the same as if the fluid were 
incompressible. Its determination is therefore the same as in 
the case of the steady flow of a liquid under pressure through 
a capillary tube. In this case, if the section be circular, the 
shearing stress per unit length on a coaxial cylindrical surface 
of radius r is 27n* . ^"dujdr, and the resultant of the longitudinal 
forces on the two curved faces of a cylindrical shell of thick- 
ness 8r is therefore 



per unit length. The sectional area of the shell being Zirrdr, 
the requisite pressure-gradient is 



, (31) 

drj 

which is independent of x. There being no radial motion, we 
have dp/dr = 0, so that p, and therefore dp/dx, is also independent 
of r. The equation (31) is then satisfied by u = A+Br !t t 
provided B be properly determined. The constant A is fixed 
by the consideration that there is no slipping at the wall 
(r = a) of the tube. In this way we find 



The mean velocity over the area of the section is therefore 

?ra 2 .' o 9# &P>' ' 

Hence, for a circular section, 

12 = V/a (34) 

The formula (33) contains Poiseuille's* law of efflux of liquid 
through a capillary tube, viz. that the discharge per second 
varies as the pressure-gradient and as the fourth power of the 
diameter. It may be made the basis of an experimental method 
of determining p. 

* J. L. M. Poiseuille (1799 1869), a practising physician in Paris, who was 
interested in the capillary circulation of the blood. The date of the memoir 
referred to is 1844. 

132 



196 DYNAMICAL THEORY OF SOUND 

The case of an elliptic section can be solved in a similar 
manner. The result, first given by Boussinesq (1868), is 
E = V(a 2 + 6 2 )/a 2 & 2 , (35) 

where a, b are the semi-axes. If we put a = oo we get the 
case of a narrow crevice, bounded by parallel planes, the 
breadth being 26, viz. 

E = V/6 2 (36) 

This can of course be obtained more easily by an independent 
process. 

The formula (30), when combined with (34) or (36), agrees 
with the result of the more complete investigation given by 
Lord Rayleigh (1883). It appears that u goes through its 
cycle of phases in a distance 27r/r, but that within this space 
the amplitude is diminished in the ratio e~ 2n = 1/535. In the 
case of circular section we have 

OT 2 = ifi'n/pda*, (37) 

by (28) and (34). Hence when the circumstances are such that 
the ratio v/na? is large, the distance in question is small com- 
pared with the wave-length (X = ^TTC/U) in the open ; for we 
have 

(Xs7/27r) 2 - *7 2 c 2 /tt 2 = 4>v/na 2 (38) 

Hence in a sufficiently narrow tube the waves are rapidly 
stifled, the mechanical energy lost being of course converted 
into heat. 

The investigation has been employed by Lord Rayleigh to 
illustrate the absorption of sound by porous bodies. When 
a sound-wave impinges on a slab which is permeated by a large 
number of very minute channels, part of the energy is lost, so 
far as sound is concerned, by dissipation within these channels, 
in the way just explained. The interstices in hangings and 
carpets act in a similar manner, and it is to this cause that the 
effect of such appliances in deadening echoes in a room is to be 
ascribed, a certain proportion of the energy being lost at each 
reflection. It is to be observed that it is only through the 
action of true dissipative forces, such as viscosity and thermal 
conduction, that sound can die out in an enclosed space, no mere 
modification of the waves by irregularities being of any avail. 



CHAPTER VII 

GENEKAL THEORY OF SOUND WAVES 

67. Definitions. Flux. Divergence. 

In respect of notation it is convenient now to take a point 
of view somewhat different from that adopted in the preceding 
chapter. We denote by u, v y w the component velocities, 
parallel to rectangular axes, considered as functions of position 
(x, y, z) and of time t. With each point of space there is 
accordingly associated, at any given instant, a vector (u, v, w), 
and the whole assemblage of such vectors gives an instantaneous 
picture of the distribution of velocity*. On the other hand 
the variations of u, v, w with the time, for given values of 
x, y, z, give the history of what goes on at a particular place f, 
but supply in the first instance no information as to the 
careers of the various particles which (so to speak) successively 
cross the scene. 

When we proceed to calculate the component accelerations 
of the particle which at the instant t is in the position (x, y, z) 
we have to take account of the fact that after the lapse of a 
time however short its velocities u, v, w will be given by the 
respective functions of the altered position as well as the altered 
epoch. Suppose that at two successive instants ^, ^ a particle 
occupies the positions P and P', respectively, and that the 
corresponding values of the ^-component of the velocity are 

* M. Marey and others have taken photographs, of short exposure, of a two- 
dimensional current of water carrying suspended motes. The image of each 
mote is drawn out into a short line, which indicates the direction and magnitude 
of the corresponding velocity. 

t As if we were to view the surface of a stream through a narrow tube, 



198 DYNAMICAL THEOEY OF SOUND 

u l} u 2 at P and w/, u% at P'. The as-component of the 
acceleration of this particle will be the limit of 



The limit of the first term on the right is du/dt, the rate of 
change of u at P. Again u% u z is the difference of simul- 
taneous velocities at the points P, P', so that, ultimately, 

*,), ............ (2) 



where du/ds is a space-differentiation in the direction PP', and 
q is the resultant velocity \/(u 2 + v* + w z ). The final expression 
for the acceleration parallel to x is therefore 

du du , 

Similar values are obtained in like manner for the other 
components. If (I, m, n) be the direction-cosines of PP', 
we have 

du du dx du dy du dz 
ds dx ds dy ds dz ds 

j du du du 
= t^+ra + 7i , (4) 

dx dy dz 

Philst u=lq, v = mq, w = nq (5) 

!ence we may write (3) in the form 
du du du du ._. 

dt* U dx +V ty+ W 3z> 

which is familiar to students of Hydrodynamics. 

It has been thought worth while, as a matter of principle, 
to accentuate the changed point of view, but in the application 
to motions which are treated as infinitely slow the distinction 
loses its importance. The second term in (3) is then of the 
second order in the velocities, and the component particle- 
accelerations may be identified with du/dt, dv/dt } dw/dt. The 
extent of the error here involved, in acoustical questions, may 
be estimated as in 60 by a reference to plane waves of sound. 
If 

= a cos k (ct x), (7) 



GENERAL THEORY OF SOUND WAVES 199 

the ratio of the maximum value of udu/dx to du/dt is ka. The 
restriction to " infinitely slow " motions therefore means that 
the amplitude must be small compared with X/2-7T. 

If we fix our attention on any geometrical surface, open or 
closed, drawn in the region occupied by the fluid, the expression 

(lu + mv + nw) BS . Bt, 

where (/, m, n) is the direction of the normal drawn from an 
elementary area BS of the surface, towards one side, measures 
the volume which in the infinitely short time Bt crosses BS. 
The coefficient of Bt in this expression is called the "flux" 
across BS, and its integral 

(lu + mv + nw) dS, . (8) 

taken over the surface, is called the total flux across the latter 
towards the side on which the normals are supposed drawn. It 
measures the rate at which fluid is being carried across the 
surface, expressed in terms of volume per unit time. 

To calculate the flux outwards across the boundary of an 
elementary rectangular region BxByBz having its centre P at 
the point (x, y, z), we note that the average velocities parallel 
to x, over the faces ByBz, being equal to the values of u at the 
centres of these faces, will be 



respectively. The difference of the fluxes, from left to right, 
across these faces is accordingly du/dx .BxByBz. Adding the 
corresponding terms for the other pairs of faces, we obtain the 
result 



C r 

(c 



ou dv dw\ ~ . 

~- + 5- + )Ba:ByBz (9) 

dy oz 



The expression in brackets gives a sort of measure of the rate 
at which the substance in the neighbourhood of P is on the 
whole flowing away from P. It is therefore called the 
" divergence " of the vector (u, v, w), and is denoted by 
div (u, v, w) ; thus 

. du dv dw 
, W ) = + + (10) 



200 DYNAMICAL THEOEY OF SOUND 

By dividing any finite region into rectangular elements we 
see that the total flux outwards across the boundary must be 
equal to the volume-integral of the divergence, or 



This can of course be proved mathematically without attributing 
any kinematical meaning to the symbols. 

68. Equations of Motion. 

To form the dynamical equations, we fix our attention on 
that portion of matter which at the instant t occupies the 
rectangular space &xyz. On the hypothesis of infinitely 
slow motion its acceleration of momentum parallel to x is 
p Sx &y$z .du/dt, where p is the density. The mean pressures 
on the respective faces may be taken to be the pressures at the 
centres of those faces, and the total pressures on the two faces 
perpendicular to x are therefore 



The difference gives a force dp/dx. Sx&y&z in the direction of 
^-positive. Equating this to the acceleration of momentum, we 
obtain the first of the following system of equations : 

du _ dp dv _ dp dw _ _dp ,-. 

p dt~~d~x' p dt~~dy' p ~dt~~dz' 

Since the variations of p when multiplied by du/dt, ..., ... may 
be neglected, we may replace p by its equilibrium value p , but 
it will not always be necessary to preserve the suffix. 

As in 59 we write 

} ........................ (2) 



where s denotes the condensation (p po)/po, and K is the cubic 
elasticity of the fluid. If we further write 

c 2 = */po, ........................ (3) 

as before, we obtain 

du_ 8s dv_ _ 8s <^__ c2 ds ( ft 

dt~ dx> dt~ d' dt~ dz ....... W 



GENEKAL THEOEY OF SOUND WAVES 201 

If A denote the dilatation of volume of the fluid which at 
the instant t fills the space SasSySz, as compared with its 
equilibrium condition, we evidently have 

-^- = div (u, v, w\ (5) 

or since, in the case of small motions, s = A, 



The equations (4), (6) are fundamental in the present branch of 
our subject. The purely kinematical relation (6) is sometimes 
called the " equation of continuity." 

69. Velocity-Potential. 

If we integrate the equations (4) of 68 with respect to t 
we obtain 

,1 

OX j || v_i/ j u i / 1 \ 



where U Q , v 0t W Q are the values of u, v, w at the point (a?, y, z) at 
the instant t = 0. In a large class of cases, these initial values 
of u, v, w can be expressed as the partial differential coefficients 
of a single-valued function of (x, y, z), thus 



Throughout any region to which this statement applies, the 
values of u, v, w at any subsequent instant t can be similarly 
expressed; thus, from (1), 



- ' ......... 

rt 
where <^> = c 2 I sdt+fa ................... (4) 

J o 

This function <f> is called a " velocity-potential," owing to its 
analogy with the potential-function which occurs in the theories 
of Attractions, Electrostatics, &c. It was introduced into 
hydrodynamics by Lagrange. 



202 DYNAMICAL THEORY OF SOUND 

The instantaneous configuration of the " equipotential sur- 
faces" </> = const, indicates at once the distribution of velocity, 
as regards both magnitude and direction. 
Suppose two consecutive surfaces to be 
drawn, for which the values of < differ by 
S<. Let PP' be drawn normal to these, and 
PP l parallel to x\ and let PP'=v. Ac- 
cording to (3) the velocity at P, resolved Flg< 64 ' 

in the direction PP ly is 

3 PP' 8 




ultimately, if I denote the cosine of the angle which the normal 
PP' makes with Ox. From this, and from the analogous forms 
of v, w t it is seen that the velocity at P is normal to the equi- 
potential surface passing through that point, and is equal in 
magnitude to the limiting value of </>/*/. Hence if a system 
of surfaces be drawn corresponding to values of <f> which differ 
by equal infinitesimal amounts, the velocity is everywhere 
orthogonal to these, and inversely proportional to Sv, the distance 
between consecutive surfaces. More precisely, the velocity is 
everywhere in the direction in which < decreases* most rapidly, 
and is equal in absolute value to the gradient of <. 

If we draw a linear element PQ (= Bs) in any other direction, 
the velocity resolved in the direction of PQ is equal to the limit of 



-. ............... (6) 

or - d(f>/ds. 

The cases in which a velocity-potential exists include all 
those where, in the region considered, the fluid was initially at 
rest, for we may then put </> = 0, simply, and the subsequent 
value is 

4>=c 2 f sdt ........................ .(?) 

J o 

This will hold whenever the motion has been originated by the 
vibration of solid or other bodies. 

* It should be mentioned that in many books is taken with the opposite 
sign; thus u = d<J>jdx, &c. 



GENERAL THEORY OF SOUND WAVES 203 

The real meaning of the property which differentiates the 
present type of motion from all others is most clearly expressed 
in terms of the "circulation" round a closed curve. If we divide 
the curve into infinitesimal linear elements, and multiply the 
length of each element by the tangential component of the 
velocity, estimated always in the same direction round the 
curve, the result is the "circulation" referred to. It may be 
denoted by 

\( u-^- +v -^ -\-w-j-Jds, or \(udx + vdy + wdz). ...(8) 

On the present hypothesis the tangential velocity is d<f>/ds, and 
the integral of this, taken round the circuit, is zero, the first and 
last values of < being the same. The circulation is therefore 
zero in every circuit which can be drawn in the region in 
question. For a reason which may be understood by reference 
to the case of an infinitesimal circuit, the type of motion now 
under consideration is called "irrotational." The name has the 
advantage of calling attention to a geometrical property rather 
than to an analytical form of expression. 

A dynamical interpretation can also be given to the 
velocity-potential. The equations (3), when written in the 
forms 

p u = pfi^/dx, p v = - pdd<f>/dy, p w = pfifydz, (9) 

shew that < is the potential per unit mass of a system of 
extraneous impulsive forces which would generate the actual 
motion of the fluid instantaneously from rest. 

The theorem as to the persistence of the irrotational character 
is most important; but it is necessary to observe the restrictions 
under which it has been proved. It was implied, in the first 
place, that the fluid was frictionless, and this is essential. 
Again the medium has been supposed free from extraneous 
forces, but the restriction is easily removed in the case of forces 
which, like gravity, have a potential (per unit mass). Finally, 
the assumption has been made that the motion is infinitely 
small. This simplifies the proof, and covers most cases which are 
of interest in acoustics. A more rigorous investigation would 
shew that the circulation is (under the above condition) still 



204 DYNAMICAL THEOKY OF SOUND 

constant round any circuit, provided we imagine the circuit to 
move with the fluid. If initially zero for every circuit which 
can be drawn in a finite portion of the fluid, it will remain zero 
for every such circuit. 

70. General Equation of Sound Waves. 

We postulate henceforth the existence of a velocity potential, 
at all events in the case of a uniform medium, to which we 
confine ourselves for the present. We have then, from 
68 (6) 



This symbol V 2 is called the "Laplacian operator," from its 
constant occurrence in the analytical theory of attractions as 
first developed by Laplace. Again, by differentiation of 69 (4) 
with respect to t we get 



Finally, by elimination of s, 

w 



This may be regarded as the general differential equation of 
sound waves in a uniform medium. If a solution can be 
obtained which gives prescribed initial values to </> and s 
(or dQ/dt), and satisfies the other conditions of the problem, the 
subsequent value of s is given by (3), and the values of u, v, w 
by 69 (3). 

We may stop for a moment to notice the form assumed by 
the equations when the fluid is incompressible. This may be 
regarded as an extreme case, in which c is made infinite, whilst 
s is correspondingly diminished, in such a way that c 2 s, which 
= (p p )/p , remains finite. The equation of continuity, 68 
(6), takes the form 

(5) 



GENEKAL THEOEY OF SOUND WAVES 205 

which is otherwise obvious from the meaning of "divergence." 
In the case of irrotational motion, this becomes 

V 2 < = 0, ........................... (6) 

which is identical with "Laplace's equation" in the theory of 
attractions. The same equation occurs in the theory of steady 
electric (or thermal) conduction in metals. If, for example, $ 
denote the electric potential, the formulae (3) of 69 give the 
components of current, provided the specific resistance of the 
substance be taken to be unity. This analogy will be found 
useful in the sequel. 

The theory of the motion of incompressible fluids is capable 
of throwing more light, occasionally, on acoustical phenomena 
than might at first sight be anticipated. We are apt to forget 
that the velocity with which changes of pressure are propagated 
in water is after all only four or five times as great as in air, 
and that the visible (or at all events easily imaginable) motions 
of water, under circumstances where the compressibility has 
obviously little influence, may supply a valuable hint as to the 
behaviour of a gaseous substance under similar conditions. This 
remark will have frequent illustration in the following chapters. 

The kinetic energy of a system of sound waves is 



. 

The potential energy, as given by the argument of 60, is 

...(8) 
The integrations extend over the region affected. 

71. Spherical Waves. 

In the case of plane waves with fronts perpendicular to Ox 
the equation (4) of 70 reduces to 

**-<-** m 

dt*~*dtf' 

whence <f> -f(ct x) + F(ct + x) ................ (2) 

This need not be further discussed. 



206 DYNAMICAL THEORY OF SOUND 

The case which comes next in importance is that of 
symmetrical spherical waves. If <j> be a function of the 
distance r from the origin and of t, only, the velocity is d<t>/dr 
outwards, in the direction of the radius, and is uniform over 
any spherical surface having the origin as centre. 

Instead of applying the general equation to the present 
circumstances it is simpler to form the kinematical relation 
corresponding to TO (1) de novo. The flux outwards across 
a sphere of radius r is d<f>/dr . 4-Trr 2 , and the difference of flux 
across the outer and inner surfaces of a spherical shell of thick- 
ness Br is accordingly 



The volume of the shell being 47rr 2 8r, this must be equal to 
A . 4?rr 2 Sr or s . 47rr 2 Sr, whence 



dt c 

bmce c2s== ^i (4) 

dt 

i i. <P& c 2 d / 96\ 
as usual, we have : = - 5- r 2 ^- (5) 

dc r dr \ d/*/ 

This may also be written 

The solution of this equation, viz. 

represents the superposition of two wave-systems travelling 
outwards and inwards, respectively, with the velocity c. In 
the case of a diverging wave- system 

r<t>=f(ct-r) (8) 

we have, by (4), crs=f(ct r) (9) 

Any value of rs is propagated unchanged ; the condensation A- 
therefore diminishes in the ratio l/r as it proceeds, and the 
potential energy per unit volume diminishes as l/r 2 . For the 
particle-velocity we have 

3JL 1 1 

-r) (10) 



GENERAL THEORY OF SOUND WAVES 207 

The law of dependence on distance is here more complicated, 
but as the wave spreads outwards the first term ultimately 
predominates ; the velocity at corresponding points of the wave 
then varies as 1/r, and the kinetic energy per unit volume 
as 1/r 2 . 

In a diverging wave-system we have, from (9), 

an = -l(r+), .................. (11) 

and similarly, in a converging wave-system. 



These relations correspond to (5) of 60, which is indeed a 
particular case, since as r increases our spherical waves tend to 
become ultimately plane. 

The general argument of 23 can be adduced to prove that 
in a diverging (or a converging) wave-system by itself the 
energy is half kinetic and half potential. 

The solution (7) can be applied to a region included 
between concentric spheres, or to a region having only one 
finite spherical boundary, internal or external. In any case, 
the conditions to be satisfied at the boundaries, whether finite 
or infinite, must be given in order that the problem may be 
determinate. In particular, even when the region is otherwise 
unlimited, the point r = is to be reckoned as an internal 
boundary ; . this point might for instance be occupied by a 
" source " of sound ( 73). When there is no source there, the 
flux across a small spherical surface surrounding must vanish, 
i.e. we must have 



(13) 

r =0 \ v 

When applied to (7) this condition gives 

f(ct) + F(ct) = 0, .................. (14) 

for all values of t, and the general solution therefore takes the 
shape 

r$ = F(ct + r)-F(ct-r) ............. (15) 

This formula may be used to determine the motion con- 
sequent on arbitrary initial conditions which are symmetrical 



208 DYNAMICAL THEOKY OF SOUND 

about 0, in an unlimited medium. Suppose that when t = 
we have 



(16) 



The former of these functions determines the initial distribution 
of velocity, and the latter that of condensation. The function 
F must now satisfy the conditions 

F(r)-F(-r)=r^(r) ................ (IV) 



(18) 



It is to be noted that the variable r is essentially positive ; this 
explains why two equations are necessary to determine F for 
positive and negative values of the argument. 

Suppose, for example, that there is no initial velocity 
anywhere, but only an initial condensation, so that </> (r) = 0. 
From (17) and (18) we deduce 

F' (r)=-J"( -r) = \ r - x .(r) .......... (19) 

The condensation at time t is given by 

Ity F'(ct+r)-F'(ct-r) 

= c *tt- ~^r 

This takes different forms according as ct is less or greater than 
r. In the former case 

-<*)), ...(21) 



and in the latter 



As a particular case, suppose we have an initial condensation 
which is uniform ( = s ) throughout the interior of a sphere of 
radius a, and vanishes for r > a ; and let us examine the 
subsequent variations of s at points outside the originally 
disturbed region. Since % (r) vanishes by hypothesis for r > a, 
the first part of the solution (21) or (22) disappears in the 



GENERAL THEORY OF SOUND WAVES 209 

present case. So long as ct < r - a, the second part of (21) will 
also vanish, but when ct lies between r a and r we shall have 



(23) 



When ct > r, the "second formula (22) applies, and we find 
that, so long as ct < r + a, the result (23) will still hold. 
Finally, when ct > r + a we have again 5 = 0. The results 
are shewn graphically in the following figure which exhibits 
the variation of s with t at a particular point, and the space- 



cb-a, 




ct+a 



Fig. 65. 

distribution of s at a particular instant, respectively. It 
appears that after the lapse of a certain time (2a/c) we have a 
diverging wave in the form of a spherical shell of thickness 2a, 
and that s is positive through the outer half, and negative 
through the inner half of the thickness. The changes in the 
velocity may be inferred by means of the formula q = d<j>/dr. 
For values of t between (r a)/c and (r + a)/c, i.e. during the 
time of transit of the wave across the point considered, we find 

......... (24) 



whilst for other values of t we have <f> = 0. Hence within the 
aforesaid limits of time we have 

(25) 



When r is large compared with a this changes sign for t = r/c, 
approximately, the velocity being directed outwards in the 
outer half, and inwards in the inner half of the shell. At the 
boundaries of the disturbed region, where r = ct a, we have 
q = + cas /2r. As the diverging wave reaches any point the 
velocity suddenly rises from zero to the former of these values, 
and as it leaves it the velocity falls suddenly from the latter 
(negative) value to 0, The origin of the discontinuities in this 
L. 14 



210 DYNAMICAL THEORY OF SOUND 

solution is to be sought of course in the discontinuity of the 
initial distribution of density. Any difficulty which may be 
felt on such grounds may in general be removed by substituting 
in imagination an initial distribution in which the discontinuity 
is replaced by a very rapid but continuous transition. 

The solution of (6) in terms of the general initial con- 
ditions (16) may be investigated in a similar manner, but it 
must suffice to quote the results. It may easily be verified 
that they satisfy all the conditions of the question. They are 

n/> = i (r + ct) fa(r + ct) + $(r- ct) fa (r - ct) 

1 fr+ct 

+ il %<*)<* -(26) 

&&J r-ct 

for ct < r, and 

r$ = | (ct + r) fa(ct + r)-(ct- r) fa(ct-r) 

1 rct+r 

+ 2-J -()*... (27) 

*C J ct-r 

for ct > r. 

Since the origin evidently occupies an exceptional position 
in the theory of spherical waves it is desirable to calculate the 
value of (f> there, more especially as the result will be of service 
presently when we come to the solution of the general equation 
70 (4) of sound waves. The result may be deduced from 
(27), or more directly from (15). We find 

...(28) 



and therefore from (17) and (18) 



(29) 



For example, in the special problem above considered, where 
fa (r) = 0, whilst % (r) c\ or according as r a, we find 
$ = c\t or according as t a/c. The consequent value of 
s at is s for t < a/c and zero for t > a/c, whilst at the instant 
t = a/c it is negative infinite. To escape this result we must 
slightly modify the data, replacing the original distribution 
of density by a continuous one. The figure is an attempt to 



GENERAL THEOEY OF SOUND WAVES 



211 



shew an initial distribution of s which varies rapidly but 
continuously from s to in the neighbourhood of r = a, 
together with the consequent time-variation of s at 0. 




cb 



Fig. 66. 

The problem which we have discussed exhibits a marked 
contrast with the theory of plane waves, in that the wave 
resulting from an arbitrary disturbance contains both con- 
densed and rarefied portions, even when there is no initial 
velocity and the initial disturbance of density has everywhere 
the same sign. The statement is easily generalized by means 
of equations (1) of 69. If we take the integral of the value 
of s at any point P over a time which covers the whole transit 
of the wave, so that the values of u, v, w vanish at both limits, 
we find that its space-derivatives are all zero. The integral 
has therefore the same value for all positions of P. And by 
taking P at an infinite distance, so that s becomes infinitely 
small by spherical divergence, we see that the value is in fact 



zero, i.e. 



sdt = 0. 



.(30) 



The mean value of s at any point is therefore zero. This result 
is of course not limited to the case of spherical waves. 

142 



212 DYNAMICAL THEOKY OF SOUND 

72. Waves resulting from a given Initial Disturbance. 

We have next to trace the effect of initial conditions in an 
unlimited region, in the general case. We suppose that at the 
instant t = we have 



where the functions are arbitrary. To deduce the effect at any 
subsequent instant, at any assigned point P, we consider in the 
first instance the average value of < over a sphere of radius 
r described with P as centre. This will be denoted by 



(2) 



if So> represent the elementary solid angle (SS/r 2 ) subtended at 
P by any elementary area $S of the sphere. In the same way 
we write 



* 47T 



//* (3) 



This, like (2), will be a function of the variables r and t only. 
If in 70 (3) we multiply both sides by 8&)/4?r, and integrate 
over the aforesaid sphere of radius r, we find 



It is also evident that the average normal velocity over the 
sphere will be d(j>/dr. The argument by which the rate 
of change of s was in 71 inferred from the consideration 
of the total flux out of the region bounded by the spheres 
r and r + &r can then be applied to prove that in the present 
case 



_ 

B; t*Sr 

Eliminating s, we have 

Sf-^ 
dt' 



which is identical in form with (5) of 71. We recognize then 
that < is the velocity-potential of the system of spherical waves 



GENERAL THEORY OF SOUND WAVES 213 

which would result from initial distributions of velocity and 
condensation expressed by 

. ............... (7) 



these functions of r being the average values of < (a, y, z) and 
^o (x, y, z) taken over the aforesaid sphere. It follows from 
71 (29) that the value of <f> at P is given by 



(8) 



This gives a rule for calculating the value of < for a point P at 
any given instant t. It may be stated in words as follows : 

To find the part of < due to the given initial distribution 
of condensation, we describe about P a sphere of radius ct, and 
calculate the average of the given initial values of d<f>/dt, i.e. of 
the function ^o (#> y> z\ at the points of space through which 
this surface passes, and multiply by t. To find the part due to 
the initial velocities we replace the average of the given values 
of d<f>/dt by the average of the given initial values of <, i.e. of 
the function < (x, y, z), and differentiate the result, as thus 
modified, with respect to t. 

The theorem contained in (8) was given by Poisson (1819); 
the actual form (8) and the interpretation are due to Stokes 
(1850). It will be seen that the result, as thus stated, is in 
reality very simple, if regard be had to the great generality of 
the circumstances which are taken into account. 

To trace the sequence of events at P we employ a series of 
spheres whose radii (ct) increase continually from zero. If P 
be external to the region which is the locus of the initial 
disturbance, no effect is produced so long as the spheres do 
not encroach on this region. If r lt r 2 be the least and greatest 
distances of P from the boundary, the disturbance at P will 
begin after a time r^c, will last for a time (r 2 rj/c, and will 
then cease. 

If with the various points of the boundary of the originally 
disturbed region as centres we describe a series of spheres of 
radius ct, the outer sheet of the envelope of these spheres will 
mark out the boundary of the space which has been invaded by 



\ 



214 DYNAMICAL THEORY OF SOUND 

the disturbance up to the instant t. The envelopes corre- 
sponding to successive values of t will form a series of what are 
known in geometry as "parallel surfaces"; in other words, the 
boundary of the disturbed region spreads everywhere normal 
to itself with the constant velocity c. 

As a simple application of the formula (8) we may take the 
problem already discussed in 71, where an initial uniform 
condensation s was supposed to extend throughout the interior 
of a sphere of radius a 
having the origin as 
centre. When a spherical 
surface of radius ct, de- 
scribed with P as centre, 
intersects the boundary of 
the originally disturbed 

region, as in the figure, 

,1 P ,1 ,. Fig. 67. 

the area of the portion 

included within the latter is 2?r . PQ Z (1 - cos OPQ), and the 
average of the given initial values of s over the whole surface 
(47r . PQ 2 ) is therefore 




where r = OP. Hence, by the rule, 

in agreement with 71 (24). 

73. Sources of Sound. Reflection. 

The very useful conception of a "point-source" was 
introduced into the subject by Helmholtz. We may imagine 
(with Maxwell and Lord Rayleigh) that at such a point fluid 
is introduced or abstracted at a certain rate, and that the 
"strength" of the source is measured by the volume thus 
introduced per unit time. The wave-train due to a source of 
strength f(t) at the origin is accordingly represented by 

(1) 



GENERAL THEORY OF SOUND WAVES 215 

since this makes 



If we differentiate the general equation of sound waves ((4) 
of 70) with respect to x or y or z, we recognize that if <j> is a 
solution so also is d<t>/dx, or d(j>/dy, or d<f>/dz. Thus from (1) we 
derive the solution 



which satisfies the general differential equation except at the 
singular point r = 0. The value of <f> thus obtained may be 
interpreted as the velocity-potential of a " double source " due 
to the juxtaposition of two simple sources which are always in 
opposite phases. This will be explained more fully in 76, in 
the particular case where the variation with time is simple- 
harmonic. 

The problem of reflection of sound by a rigid infinite plane 
is readily solved by the method of "images." If with every 
source P of sound on the near side of the boundary we associate 
a similar source at the geometrical image P' of P with respect 
to the plane, it is obvious that the condition of zero normal 
velocity over the plane would still be fulfilled if the boundary 
were abolished. Hence, in the actual case, the motion on the 
near side will be made up of that due to the given sources P 
and of that due to the images P'. It may be mentioned that 
the present case of a rigid plane boundary is the only one where 
the physical " image " of a point-source is itself accurately a 
point-source. 

The problem of reflection at the plane boundary of two 
distinct fluid media has been discussed in 61, in the case of 
direct incidence. The case of oblique reflection was solved by 
Green (1847). The results are chiefly of interest for the sake 
of the optical analogies, but one curious point, noticed by 
Helmholtz, may be mentioned. Owing to the greater velocity 
of sound in water, the conditions for total reflection may occur 
when the waves are incident from air on water (in fact when- 
ever the angle of incidence exceeds about 13), but not in the 
converse case. This is of course the reverse of what holds with 
regard to light. 



216 DYNAMICAL THEORY OF SOUND 

74. Refraction due to Variation of Temperature. 

Questions relating to wave-propagation in heterogeneous 
media can only be discussed in a general way, and with the 
help of conceptions borrowed from geometrical optics. If at 
any surface there is an abrupt change of properties the law of 
propagation is of course altered. If the dimensions of the 
surface, and its radii of curvature, are large compared with 
the wave-length, we have phenomena of regular reflection and 
refraction, as in optics. Cases of absolute discontinuity are of 
course not met with in the atmosphere, but the theory would 
be practically unaffected if the change of properties were 
effected within a space which is small compared with the 
wave-length. 

When on the other hand we have a continuous variation 
such that the change of properties within a wave-length is 
negligible, the case is analogous to that of atmospheric 
refraction of light, which is discussed in books on optics and 
astronomy. In an atmosphere of the same gas, at rest, a 
variation in the velocity of sound can only arise through a 
variation of temperature ( 59). The refraction due to varia- 
tion of temperature with altitude was first discussed by Osborne 
Reynolds (1876). Suppose that, as usually happens, the 
temperature diminishes upwards. Since the velocity of sound 
varies as the square root of the absolute temperature, the lower 
portions of a wave-front will be propagated faster than the upper 
ones, so that a front which was originally vertical gets tilted 
upwards more and more as it proceeds. The sound will there- 
fore, for the most part, pass over the head of an observer at 
a sufficient distance, such residual effects as he perceives being 
referable to diffraction. On the other hand, whenever the 
temperature increases upwards the waves will be tilted down- 
wards, and the effect at a distance will be greater than if the 
temperature had been uniform. This latter condition of the 
atmosphere sometimes prevails on a clear night following a 
warm day, when, owing to the cooling of the ground by 
radiation, the lower strata of the atmosphere are reduced in 
temperature relatively to the upper ones. 

The theory has been further developed by Lord Rayleigh, 
by means of the conception of rays of sound. The surfaces of 



GENEKAL THEORY OF SOUND WAVES 



217 



equal wave- velocity being supposed to be horizontal, each ray 
will travel in a vertical plane. The cur- 
vature of a ray may be calculated directly 
by a method due to Prof. James Thomson*. 
If R be the radius of curvature, the two 
wave-fronts passing through the extremities 
of an element 8s of the path will be 
inclined at an angle 8s/R, and if 8s be the 
length intercepted on an adjacent ray in 
the same vertical plane, we have 




are 



(2) 



.(3) 



where 8n denotes the distance between 
the two rays, the standard case being 
that shewn in the figure. Since the elements 
described in the same time we have 

8s _ 8s 
iT+8c"~"c' 
whence, by comparison with (1), 

1 1 dc 

IR ~ ~cdn' 

When the temperature diminishes upwards, 9c/9n is negative 
and the curvature l/R is positive, as in the figure, and the rays 
are curved upwards. But if the temperature increase upwards, 
the curvature is downwards, so that an observer at the level of 
the source may hear sounds which would otherwise have been 
intercepted by obstacles. 

The formula (3) leads to the ordinary law of refraction. If 
>Jr be the inclination of the ray to the horizontal we may write 

dc dc dc dc . 

5 = -j- cos y, = -j-sm-\lr, (4) 

on dy ds dy 

if y be the vertical coordinate. Hence, along the course of 
a ray, 

i?*lr 1 1 tJ.r. 

.(5) 



ds 



1 
^D 

R 



Idc 
-y- 

cds 



* James Thomson (1822 92), professor of engineering at Belfast 1857 72, 
and at Glasgow 1872 89. 



218 DYNAMICAL THEOEY OF SOUND 

or c sec -Jr = const., ..................... (6) 

which is the law in question. Conversely, from (6) we can 
derive the formula (3). When c is known as a function of y 
the equation (6) determines the paths. 

The simplest hypothesis is that the temperature decreases 
(or increases) upwards with a uniform gradient. This includes 
the particular case of an atmosphere in " convective equili- 
brium " under gravity, where the gradient is 

- 7 " 1 - (7) 

'IP 

H being the height of the homogeneous atmosphere ( 59) 
corresponding to the temperature 6*. This is at the rate of 
about 1C. per 100 metres. If a law of uniform decrease were 
to hold without limitation, we should at a certain altitude 
meet with a zero temperature (absolute). If for a moment 
we take the origin at this level, and draw the axis of y 
downwards, the temperature will be proportional to y t and the 

wave- velocity c to y?. Hence by (6) we have, along any ray, 

...................... (8) 



The paths are therefore cycloids, the generating circles of which 
roll on the under side of the line y = 0. If on the other hand 
the temperature increases upwards with a uniform gradient, the 
paths are the cycloids whose generating circles roll on the upper 
side of the line which corresponds to the zero of temperature. 
In any practical case we are concerned only with the portions 
of the curves near the vertices. The arcs may therefore be 
taken to be circular, with a radius double the distance below 
(or above) the level of zero temperature. In the extreme case 
of upward diminution to which the formula (7) refers, this 
radius will therefore be (roughly) 2 x 273 x 100 = 54600 metres, 
for a temperature of C. 

* It was pointed out by Lord Kelvin (1862) that this is the condition into 
which the atmosphere would be brought by the free play of convection currents 
alone, without conduction or radiation. It is therefore one of neutral equilibrium. 
If the temperature diminish upwards at a greater rate the equilibrium becomes 
unstable. 



GENERAL THEORY OF SOUND WAVES 219 

75. Refraction by Wind. 

Another interesting question is that of refraction by wind. 
A uniform motion of the medium introduces of course no 
complication, the relative motion of the sound waves being 
exactly the same as if the medium were at rest. Usually, 
however, the wind- velocity near the ground is less than above, 
the motion of the lower layers of air being obstructed. Hence 
when a wave-front travels with the wind, the upper portions 
are propagated (in space) somewhat faster than the lower, the 
velocity of the wind being superposed on that of sound. The 
front is therefore continually being tilted downwards. For a 
similar reason a wave-front travelling against the wind gets 
tilted upwards, so that the sound tends to pass over the 
head of an observer at a distance. This explanation of the 
familiar fact that sound can be heard better, and further from 
the source, when this lies to windward than when it is to 
leeward of the observer, was first given by Stokes (1857). 
The only previous suggestion had been that a sound which 
has travelled a certain distance with the wind has really 
traversed a shorter length of air. and has consequently become 
less attenuated by spherical divergence, than if the wind had 
been absent. Owing to the smallness of wind- velocities in 
comparison with that of sound, this cause is quite inadequate 
to explain the very marked effects which are observed. The 
true theory was discovered independently by Reynolds (1874), 
and confirmed by a number of interesting experiments. 

If we proceed to apply optical methods to the question, 
it is necessary to dis- 
tinguish, as in the theory 
of aberration, between the 
direction of a ray and that 
of a wave-normal. Let S l 
represent the position of a 
wave-front at time t, S' 
the position at time t + 8t 
of those particles which 
were on S l} and 8 2 the 
new position of the wave- 




220 DYNAMICAL THEOKY OF SOUND 

front. Let P l be any point on Si, and P' the corresponding 
point on S f , so that P-f is the path of a particle of the 
medium in the time Bt. On the principles of optics, the new 
position $ 2 of the wave-front is obtained as the envelope of 
a system of spheres of radius c$t, described with the various 
points P' of S' as centres. If P 2 be that point on the 
envelope which corresponds to P', P^P Z will be an element of 
a ray, and P'P Z an element of the wave-normal. Also since 
Pf U8t, where U is the velocity of the medium, the " ray- 
velocity" (PiPJfo) is the resultant of the wave- velocity and 
the velocity of the medium. 

In the present question the velocity U is horizontal, and 
a function of the altitude (y) only. If i/r, < denote the 
inclinations to the horizontal of the ray and the wave-normal, 
respectively, we have 

n V9 T)~ c s Q r> () 

or <f> = T|T H simjr, (2) 

c 

if U/c be small, as will usually be the case. 

To ascertain the law governing 
the change of direction of the ray, 
consider first the case of refraction at 
the common horizontal boundary of 
two uniform currents U, U'. If <, <' 
be the inclinations of the wave-normal 
on the two sides of the plane of 
discontinuity, we have 

c sec < 4- U = c sec </>' + U', (3) 

each side expressing the horizontal velocity of the trace of 
the wave-front on the plane in question. Since a continuous 
variation of U can be approximated to by a series of small 
discontinuities, we infer that (3) will still hold if <, U and 
<f>', U' refer to any two positions on the same ray. This gives 
the altered law of refraction. Lord Rayleigh points out that 
since sec $ <%. 1, <f>' will become imaginary if 

(U'- Z7)/c> sec </> - 1 (4) 




GENERAL THEORY OF SOUND WAVES 221 

There is therefore total reflection, at the stratum to which the 
accents refer, of all wave-fronts whose initial inclination (0) to 
the vertical falls short of a certain limit. 
Along any one ray we have 

sec < H = const., (5) 

c 

or, by (2), sec i/r + sec 2 ^ = const., ....(6) 

provided ^ be not too great. If we differentiate this with 
respect to the arc s, and put d-^/ds = 1/R, dy/ds = sin i|r, we 

find 

1 /, 20" .\ IdU /lyx 

r . 1 + -- sec \Ir =__ (7) 

E\ c r J c dy 

The ray is therefore curved downwards or upwards, according 
as dU/dy is positive or negative, i.e. according as the ray is 
travelling with or against the wind. If the gradient dU/dy 
be uniform, the rays have 

all the same uniform curva- ri 

ture, approximately, owing 

to the smallness of U/c, 

unless indeed the inclination Fig 71 

vjr becomes considerable. It 

will be noticed that in this problem the path of a ray is 

not reversible. 

This is a convenient place for a reference to what is 
known as " Doppler's principle " *. Suppose, for instance, 
that a periodic source of sound is approaching a stationary 
observer. The number of maxima of (say) the condensation s 
which strike the ear of the latter in a second is increased, and 
the pitch is therefore raised. The diminution in the period is 
to the period when the source is at rest in the ratio of the 
velocity of approach to the velocity of sound. When the 
source recedes from the observer, this ratio is negative, and 
the pitch is lowered. When the motion of the source is 
oblique to the rays by which the sound is heard, the com- 

* Christian Doppler (1803 54), an Austrian mathematician, professor of 
physics at Vienna 1851. 



222 DYNAMICAL THE OK Y OF SOUND 

ponent of its velocity in the direction of the ray is alone 
effective. Analogous effects are produced when the source is at 
rest and the observer in motion. The principle is exemplified 
in the apparent change of pitch of the whistle of a locomotive 
as a train dashes through a station ; but its most striking and 
fruitful applications are met with in the theory of radiation. 



CHAPTER VIII 

SIMPLE-HARMONIC WAVES. DIFFRACTION 

76. Spherical Waves. Point-Sources of Sound. 

From this point it is convenient to consider specially the 
case of simple-harmonic vibrations. In problems relating to 
the impact of sound waves on obstacles, or their transmission 
by apertures in a screen, and so on, the results will vary in 
character with the pitch, the determining element being the 
relation between the wave-length and the linear dimensions of 
the obstacles, &c. 

It will be desirable, for the sake of conciseness, to use 
imaginary quantities somewhat more freely than in the pre- 
ceding chapters. Thus we assume that the velocity-potential 
< varies as e int , or e ikct , where 



..................... (1) 

if X be the wave-length of plane waves of the same period 2?r/n. 
The general equation of sound waves ( 70 (4)) therefore be- 
comes 

Vty + Afy = ...................... (2) 

In the case of plane waves whose fronts are perpendicular 
to the axis of x, we have 

+"*-<> ..................... < 3 > 

the solution of which may be written 

<f> = Ae- ikx + Be ikx , ..................... (4) 

or <f> = C cos kx + D sin kx, ............... (5) 



224 DYNAMICAL THEOBY OF SOUND 

the time-factor e int being understood. Thus a train of simple- 
harmonic waves travelling in the direction of ^-positive is 
represented by 



When we proceed to calculations of energy it is of course 
necessary to revert to real forms. Thus, taking the real part 
of (6), we have 

<I> = A cos k(ct x). '. ................. (7) 

The mean energy per unit volume, as given by 70 (7), (8), 
is ^pk*A 2 , and the mean energy transmitted per unit time, 
per unit area of the wave-front, is 

%pk 2 cA 2 , or $pn*/c.A* ................ (8) 

We may call this the " energy-flux " in the wave-system (7). 

The equation of symmetrical spherical waves, 71 (6), now 
takes the form 

^ + *W)-0, .............. ....(9) 

and the solution is 

r<t> = Ae~ ikr + Be ikr , .... .............. (10) 

or r<f> G cos kr + D sin AT, ............ (11) 

the time-factor being understood as before. The two terms 
in (10) correspond to waves diverging from, or converging to, 
the origin, respectively. In particular, the diverging waves 
due to a source Ae ikct at the origin are represented by 



(12) 



or, in real form, <t> = ~ A oosnff -] ............ (13) 

\ c) 



This is of course a particular case of 73 (1). 

The maintenance of such a source in an unlimited medium 
requires a certain expenditure of energy. The work done per 
unit time at the surface of a sphere of radius r, on the fluid 
outside, is the product of the pressure, the area, and the 
outward velocity, or 

. (14) 



SIMPLE-HAKMONIC WAVES. DIFFRACTION 225 

It is evident that p contributes nothing to the average effect, 
since the mean value of d<f>/dr at any point is zero. If we 
substitute from (13) we find that the average of the remaining 
part is 



This quantity W is independent of r, as was to be anticipated, 
since the mean energy in the space included between two 
concentric spheres is constant. It measures the emission of 
energy (per unit time) by the source. The formula may also 
be inferred from the consideration that at a great distance the 
waves may be regarded as plane. If in (8) we replace A by 
the A/4>7rr of (13), and multiply by 4nrr*, we obtain the result 
(15). 

It must be remembered that this calculation of the energy 
emitted applies only to an isolated source in free space. A 
source placed in an enclosure with rigid walls does no work on 
the whole, since the energy of the gas is constant. Even in an 
open space the emission of energy may be greatly modified by 
the neighbourhood of an obstacle. Thus in the case of a source 
P close to a rigid plane boundary the amplitude of vibration at 
any point is doubled by the reflection as from the image P' 
( 73); the intensity is quadrupled, and the emission (on one side) 
is therefore twice that of an equal source in free space. 

The equation -j^ ****>, .................. (16) 

7? / v\ 

or, in real form, </>= ^ - cos n U + -), 

^rirT \ c/ 



may likewise be interpreted as representing a " sink " of sound, 
i.e. a point where energy is absorbed, under similar conditions, 
at the rate ri 2 pB*l&Trc. This conception is however of no great 
assistance in acoustics. 

The notion of a simple source, valuable as it is for theoretical 
purposes, is seldom realized even approximately in practice. 
A vibrating body such as a membrane, or either prong of a 
tuning fork, is tending at any instant to produce a condensation 
of the air in contact with it on the one side and a rarefaction 

L. 15 



226 



DYNAMICAL THEORY OP SOUND 



on the other, and is therefore more adequately represented, in 

the simplest cases, by a combination of two simple sources near 

together but in opposite phases. Idealizing this a little further 

we are led to the mathematical conception of a "double source." 

We begin with a simple source of strength m at a point 0, 

and a simple source of strength -f m at an adjacent point 0', 

the signs indicating the oppo- 

sition of phase. If we next 

imagine m to become infinitely 

great, whilst the distance 00' 

becomes infinitely small, in such 

a way that the product m.OO' 

remains finite, we have the ideal 

" double source " of theory. The 

direction 00' is called the 

" axis," and the limit of m.OO' 

is called the " strength." The resulting motion is evidently 

symmetrical about the axis. 

If the direction 00' be that of the axis of x t and be 
taken as origin, the velocity-potential at P due to simple 
sources + m at ' and 0, respectively, will be given by 




p -ikr\ 

- V)' 



where r= OP, r' = O'P. If we draw PP' equal and parallel to 
O'O, we have /= OP', and the expression in brackets is equal 
to the change of value of the function e~ ikr /r caused by a 
displacement of P to P'. Hence, ultimately, if P'P = 8x, 



Putting m&x=I, we deduce the formula for a unit double 
source at 0, having its axis along Ox, viz. 



(20) 



this is a particular case of 73 (3). When x alone is varied, 
whilst y and z are constant, it appears from the figure that 



SIMPLE -HARMONIC WAVES. DIFFRACTION 227 

Sr = cos &c, where 6 denotes the inclination of OP to Ox. 
Hence d/dx = cos 6 9/9r, and 



(21) 



Performing the differentiation, we find 

(22) 



For small values of kr, i.e. within distances from which are 
small compared with X/2w, this becomes 

...................... (23) 



On the other hand, for large values of kr, 

,,-ikr 

tfc- - cos0, (24) 



r 

so that along any one radius vector the condensation (s = 
varies ultimately as 1/r. The radial and transverse components 
of the velocity are to be found by the formula (6) of 69 ; 
viz. they are 9$/9r and 9</>/r90, respectively. It appears 
that near the origin these are of the same order of magnitude, 
whilst at a great distance the lateral velocity is less than the 
radial in the ratio 1/Ar. 

Introducing the factor Ce int in (24), and taking the real 
part, we find that the velocity-potential due to a double source, 
of strength G cos nt y at a great distance, is 



bC! / r\ 

-- -sin n (*-- cos ............. (25) 

4?rr \ cj 



The waves sent out in any direction are therefore ultimately 
plane, of the type (7), provided A=kCcos6/4<7rr, the mere 
difference of phase being disregarded ; and the flux of energy 
(across unit area) will therefore be pk*cC 2 cos* ^/32?r 2 r 2 . Multi- 
plying by 27rrsin 6. rSO, which is the area of a zone of a 
spherical surface of radius r bounded by the circles whose 
angular radii are and 6 -f 0, and integrating from 6 to 
6 = TT, we find that the total emission of energy by the double 
source C cos nt is 

<> 

152 



228 DYNAMICAL THEOEY OF SOUND 

It will be noticed that as the wave-length X is increased, 
and k accordingly diminished, the fundamental equation (2) 
tends to assume the form 

W> = ........................ (27) 

which is met with in the dynamics of incompressible fluids, and 
in the theories of attractions and of electric and thermal con- 
duction. This assimilation may come about in two ways, either 
through a diminution in the frequency (n/Zir), or by an increase 
in the elasticity of the medium and consequently in the wave- 
velocity. Under the same condition the formula (12) 
approximates to the form 



which is the expression for the potential of a magnetic pole, or 
for a source of electricity, and so on ; whilst in the case of the 
double source (21) the limiting form is (23), which is recognized 
as the potential of an infinitely small magnet. 

A further remark of great importance is that within any 
region, free from sources, whose dimensions are small compared 
with X, the configuration of the equipotential surfaces <= const. 
is at any instant sensibly the same as if the fluid were incom- 
pressible. For the value of </> due to an external source differs 
from its value in the case of incompressibility chiefly by a 
factor e~ ikr , where r denotes distance from the source. If 
6 denote the greatest breadth of the region, this factor can at 
most vary in the ratio e~ ikb , which differs very little from unity 
when kb is small. 

77. Vibrating Sphere. 

By means of the fiction of a double source, of suitable 
strength, at the centre, it is possible to calculate the waves 
generated in the surrounding air by a vibrating solid sphere of 
any radius. As this is almost the only problem of the kind 
which can be completely solved we devote some space to it. The 
work is simple, and the results throw a good deal of light on 
other cases. 

For reasons just referred to, it is instructive to look first at 
the case where the fluid is incompressible. We take the origin 



SIMPLE-HAKMONIC WAVES. DIFFRACTION 229 

at the mean position of the centre of the sphere, and the axis of 
x along the line of its vibration ; and we denote its velocity 
by U. The velocity of the fluid in contact with the sphere at 
any point P, resolved in the direction of the normal, must be 
equal to the normal component of the velocity of the point P of 
the sphere itself, i.e. to UcosO, where is the angle POx. 
This gives 

- d ^=Ucos0 [r=a], ................ (1) 

if a be the radius. The velocity due to a double source at in 
an unlimited mass of incompressible fluid is of the form 

sfl; ..................... (2) 



and in order that this may be consistent with (1) we must have 

C=27ra*U. ........................ (3) 

With this determination of C the effect of the sphere on the 
fluid is exactly that of the double source, and the solution of 
our problem is 

*=g- 3 cos0. ..................... (4) 

This depends only on the instantaneous value of U, as we should 
expect, since under the present hypothesis disturbances are 
propagated with infinite velocity. It should also be noted that 
there is so far no assumption that U is small. 

The directions of motion at various points of the field may 
be shewn by tracing the " lines of motion," which are lines 
drawn from point to point, always in the direction of the 
instantaneous velocity. In the case of small vibratory motion, 
which we have especially in view, each particle oscillates 
backwards and forwards through a short distance along the line 
on which it is situate. If Sr, rS0 be the radial and transverse 
projections of an element of such a line, these quantities must 
be proportional to the radial and transverse components of 
velocity, viz. d<f)/dr and -d<l>/rd6, respectively. Hence 

Sr rS0 
cose Jsinfl' ' 
the integral of which is 

........................ (6) 



DYNAMICAL THEORY OP SOUND 



where 6 is a parameter which varies from one line of motion 
to another. The curves, which are identical in form with the 
lines of force due to a small magnet, are shewn in Fig. 73. 




Fig. 73. 



To calculate the reaction on the sphere we divide the surface 
into zones by planes perpendicular to Ox. The area of a zone 
being 27ra 2 sin0S0, the resultant force on the sphere in the 
direction of ^-positive is 



(7) 



The constant part of the pressure contributes nothing to the 
resultant. The variable part is, if terms of the second order in 
the velocities be neglected, 



.(8) 



SIMPLE-HAKMONIC WAVES. DIFFRACTION 231 

since t enters only through U. Substituting we find 

(9) 



The remarkable point here is that the force is independent 
of the velocity, and depends only on the acceleration of the 
sphere. If the mass of the sphere be M, and if it be subject to 
other extraneous force X, its equation of motion will be 



(10) 



or (*+f/^~Z ................... (11) 

This is the same as if the fluid were abolished, and the inertia 
of the sphere were increased by Trpa 3 , i.e. by half that of the 
fluid which it displaces. It was shewn by Stokes (1843) that 
this conclusion is accurate even when the restriction to small 
motions is abandoned. 

There is, as we shall see ( 79), nothing peculiar to the 
sphere in the general character of the above result, but the 
apparent addition to the inertia will vary of course with the 
shape as well as the size of the solid, and will usually be 
different for different directions of motion, as e.g. in the case of 
an ellipsoid. The theory here touched upon has had a great 
influence on recent physical speculations, and is responsible 
ultimately for the suggestion that the apparent inertia of 
ordinary matter may be partly or even wholly due to that of a 
surrounding aetherial medium. 

Turning now to the acoustical problem, let the velocity of 
the sphere be expressed symbolically by 

U = Ae int ...................... (12) 

The surface-condition will have the same form (1) as before. 
The velocity-potential of a double source Ce in * at is 



by 76 (21), the time-factor e int being omitted. The ratio of C 
to A is then determined by (1). 



232 DYNAMICAL THEOKY OF SOUND 

The most interesting case is where the radius a of the 
sphere is small compared with X/2?r, where X is the wave-length. 
In the immediate neighbourhood of the sphere kr will then be 
small, and the formula (13) is, for this region, practically 
identical with (2). It follows that 

<7=2iraM, ..................... (14) 

nearly, and further that the lines of motion near the sphere 
will have sensibly the configuration shewn in Fig. 73. The 
apparent addition to the inertia of the sphere has very 
approximately the same value f Trpa? as before. On the other 
hand, at distances r which are comparable with, or greater 
than, X, the motion of the fluid is altogether modified by the 
compressibility. At sufficiently great distances we have, by 
(13) and (14), 

<t> = ^ika s A e ^cos0, ............... (15) 

or, in real form, 

S ff, ......... (16) 



corresponding to a velocity 

U = Acosnt ..................... (17) 

of the sphere. The amplitude now varies ultimately as 1/r, 
instead of 1/r 2 , as in the case of (4). 

The investigation so far discloses nothing analogous to 
a frictional resistance, whereas we know that owing to the 
generation of waves travelling outwards a continual abstraction 
of energy must take place. To calculate either the dissipative 
resistance, or the work done, at the surface of the sphere, we 
should have to use the complete formula (13); but the emission 
of energy may be ascertained independently from the formula 
(26) of 76. The strength of the equivalent double source 
being given approximately by (14), we find 

W = %7rpk*a 6 cA 2 ................... (18) 

If p' denote the mean density of the sphere, its energy when 
vibrating under the influence of (say) a spring will be 






SIMPLE -HARMONIC WAVES. DIFFRACTION 233 

If, following a procedure explained in 11, we equate the rate 
of decay of this energy to W, we find 



and therefore A = A e- T , ..................... (20) 



8 o 
provided T= /y ...- ................... (21) 

p 



The ratio (nr/^Tr) of the modulus of decay to the period is 
therefore usually very great. 

78. Effect of a Local Periodic Force. 

Corresponding results can, with the help of more or less 
intuitive considerations, be obtained for other forms of vibrating 
solid, but the work is much simplified by a preliminary theorem, 
which has also an independent interest. This relates to the 
effect of a periodic extraneous force concentrated about a point 
in a gaseous medium. 

An elementary proof can be derived at once from the pre- 
ceding investigation. The result will obviously be the same if 
the force be imagined to act on an infinitely small sphere 
having the same density as the surrounding fluid. The effect 
is therefore that of a double source ; and if we now denote the 
concentrated force, supposed acting parallel to x, by Pe int , we 
find, putting M = f Trpa 3 in 77 (11), 

P=2i7rpkca 3 A, ..................... (1; 

and therefore, by 77 (15), for large values of kr, 

p p -ikr 

<t> = T~ -cos<9 ................... (2) 

4f7TpC T 

The following investigation is of a more formal character; 
but it involves mathematical processes more intricate than 
those which are employed in other parts of this book. The 
work depends on the solution of the equation 

V0+#* = <l>, ..................... (3) 

where < is a given function of x, y, z which vanishes outside a 
certain finite region R. In the theories of attraction, and of 
thermal and electric conduction, we meet with the equation 

*, ........................ (4) 



234 DYNAMICAL THEORY OF SOUND 

where <I> represents a distribution of density (p = <l>/47r), or 
of sources of heat, &c. The solution of (4) appropriate to 
infinite space (when there are no sources at infinity) is known, 
viz. it is 



where 4>' denotes the value of <3> at (of, y', z / ), r denotes distance 
from this point to the point P, or (x, y, z), for which the value 
of < is required, and the integration extends over all space for 
which <E> differs from 0. For example, if we put <!>' = 4-Trp', 
we get the ordinary expression for the gravitation potential of 
a continuous distribution of matter. 
The analogous solution of (3) is 



(6) 



This represents a distribution of simple sources through R, the 
strength per unit volume being 4>, and it is therefore obvious 
at once that the equation V 2 < + A^ = is satisfied at all points 
P external to R. The only question of any difficulty arises 
when P is inside R. We then divide R into two regions R l 
and R 2) of which R 2 encloses P and is ultimately taken to be 
infinitely small in all its dimensions. The parts of at P due to 
the sources in R l and R% , respectively, may be denoted by fa and 
< 2 - Since P is external to ^ we have V 2 ^ + A; 2 ^ = as before. 
Within R 2 we may ultimately put e~ ikr l t and <f> 2 then 
approximates to the gravitation potential of matter of density 
4>/47r restricted to the space R 2 . We have then, ultimately, 
on known principles, V 2 < 2 = <I> and < 2 = 0. Hence (1) is satisfied 
by <f> = </>! + </> 2 . It is further evident that (6) is the solution 
of (1) consistent with the condition that there are no sources of 
sound except at the points to which <l> refers. 

When forces X, Y, Z per unit mass act at the various 
points of a gaseous medium, the equations (4) of 68 are 
replaced by 

9 <> +z , g = _ c ,^ + r , *?__<-* + *. ...(7) 

dt dx dt dt dt dz 



SIMPLE-HARMONIC WAVES. DIFFRACTION 235 

If we eliminate u, v, w by the kinematical relation (1) of 70 
we obtain 

(8) 



If X, Y, Z, s all vary as e ikct , this becomes 

V*s + k*s = \div(X, Y,Z), ............ (9) 



c 

in the notation of 67. This is of the form (3), and the 
solution is therefore 

1 niftX' ar dZ'\e- ik ' ,, ,,, 

8 = - A ; -*-T + -5-7 + *-i) - dxdy dz' 
4>7rc z JJj\dx dy dz J r 



The transformation is effected by partial integration of the 
several terms, the integrated portions vanishing at infinity if 
X', Y 7 , Z' do so. Also, since 

r = J[(x - xj +(y- yj + (z- 



we have 
whence 



- <> 



From this the value of ^> follows by the relation 

(12) 



As a particular case, suppose that Y' = 0, Z' 0, and that 
X' differs from only in a small region about the origin, 
and put 

pjjjX'dx'dy'dz' = P ................ (13) 

We have < = - r ^- j- (} , ............... (14) 

dx\ r J 



or, for large values of kr, 

P p-ikr 

cos(9, ............... (15) 



r 

as before. Comparing with 76 (24) we see that a concentrated 
force Pe int has the effect of a double source of strength iP/pkc. 



236 DYNAMICAL THEORY OF SOUND 

79. Waves generated by Vibrating Solid. 

We return to the problem of investigating the waves 
generated by a vibrating body. In order not to complicate the 
question too much we will assume that the body has some sort 
of symmetry with respect ta an axis ; thus it may be a form 
of revolution about this axis, or it may have two mutually 
perpendicular planes of symmetry meeting in this axis, or 
(again) a single plane of symmetry perpendicular to the axis. 
In any case this axis is taken to be the direction (Ox) of 
vibration. 

The dimensions of the solid being supposed small in com- 
parison with \/2-7r, the motion of the fluid in the immediate 
neighbourhood will be sensibly the same as in the case of 
incompressibility, and the principal effect on the body will there- 
fore be equivalent to an increase of inertia. To establish this 
latter point in a general manner, we note that the (irrotational) 
motion of a frictionless liquid due to the motion of a solid in it 
will have the velocity at every point in a determinate ratio 
to the velocity U of the solid, and that the total kinetic energy 
of the fluid may therefore be expressed by ^pQ'U 2 , where Q' is 
a constant, of the nature of a volume, depending only on the 
size and shape of the solid and the direction of its vibration. 
Hence if M be the mass of the body, the equation of energy 
takes the form 



(l) 



where the right-hand member represents the rate at which 
work is being done by the extraneous force X. Thus 

X, .................. (2) 



which shews that the inertia of the body is apparently increased 
by the amount pQ'. An equivalent statement is that the 
reaction of the liquid is equivalent to a force pQ'dU/dt. 

In the actual case of the gaseous medium, it is plain that if 
the solid were removed, and its place supplied by fluid, the 
motion at a distance would be very approximately the same as 
would be produced by a suitable periodic force from without, 



SIMPLE-HARMONIC WAVES. DIFFRACTION 237 

acting on the substituted matter. Since this force has to 
produce an acceleration of momentum pQdU/dt, where Q is 
the volume displaced by the solid, as well as to balance the 
reaction just referred to, its amount would be 



, ...... (3) 

if U=Ae int ......................... (4) 

By 78 (2), the velocity-potential at a great distance r will 
therefore be 



cos 6 = Ae-^ cos 6. ...(5) 

- 



4-TT/oc r 4-Trr 

Comparing with 76 (24) we see that the effect of the vibrating 
solid is equivalent to a double source of strength C = (Q+ Q')A, 
and that the emission of energy is accordingly 



(6) 



by 76 (26). In the case of the sphere we have Q = %7ra?, 
Q'= iQ, and the result accordingly agrees with 77 (18). It 
can be shewn that for a circular disk of radius a, moving 
broadside on, Q' = ^ira 3 , whilst Q of course =0. 

80. Communication of Vibrations to a Gas. 

The circumstances which govern the efficiency of a vibrating 
body in generating sound waves, and the comparative effects in 
different gases, were elucidated by Stokes in a classical memoir 
" On the Communication of Vibrations from a Vibrating Body 
to a surrounding Gas*." The starting point of the investigation 
was an observation by Prof. J. Leslie (1837), who found that the 
sound emitted by a bell vibrating in an atmosphere of hydrogen 
was extremely feeble as compared with the effect in air. No 
satisfactory explanation of this phenomenon was forthcoming 
up to the time of Stokes' paper. The essence of the matter is 
conveyed in the following quotation : 

" When a body is slowly moved to and fro in any gas, the 
gas behaves almost exactly like an incompressible fluid, and 

* Phil. Trans. 1868. The passage which follows below is from the 
'* abstract " in the Proc. Roy. Soc. 



238 DYNAMICAL THEORY OF SOUND 

there is merely a local reciprocating motion of the gas from the 
anterior to the posterior region, and back again in the opposite 
phase of the body's motion, in which the region that had been 
anterior becomes posterior. If the rate of alternation of the 
body's motion be taken greater and greater, or, in other words, 
the periodic time less and less, the condensation and rarefaction 
of the gas, which in the first instance was utterly insensible, 
presently becomes sensible, and sound waves (or waves of the 
same nature in case the periodic time be beyond the limits of 
audibility) are produced, and exist along with the reciprocating 
flow. As the periodic time is diminished, more and more of the 
encroachment of the vibrating body on the gas goes to produce 
a true sound wave, less and less a mere local reciprocating flow. 
For a given periodic time, and given size, form, and mode of 
vibration of the vibrating body, the gas behaves so much the 
more nearly like an incompressible fluid as the velocity of 
propagation of sound in it is greater ; and on this account the 
intensity of the sonorous vibrations excited in air as compared 
with hydrogen may be vastly greater than corresponds merely 
with the difference of density of the two gases." 

These remarks are exemplified in the results of 77 (13), 
(14). If we fix our attention on a point at a distance from the 
sphere, supposed vibrating with the velocity 

U=Acoant, (1) 

the motion there is given, when the period is sufficiently long, 
by the formula 

A a 3 
<f> Y7 cos Q . cos nt, (2) 

as if the fluid were incompressible. But when the frequency is 
increased until the wave-length is small compared with the 
distance r from the centre, the appropriate formula is 

ka?A / r 



?A A . f.r\ /ox 
cos . sin n(t j , (3) 



and the amplitude is accordingly greater than in the former 
case in the ratio kr, or 2irr/\. For the same frequency, the 
amplitude, which depends on k/c or n/c 2 , will in different gases 






SIMPLE-HARMONIC WAVES. DIFFRACTION 239 

now vary inversely as the square of the wave-velocity. Again, 
the emission of energy is, by 77 (17), 

W = lirpkWcA* = lirpnW/c?.A*, (4) 

and so varies (for the same gas) as the fourth power of the 
frequency. The emission in different gases will (for the same 
frequency) vary inversely as the fifth power of the wave- 
velocity, if we assume ( 59) that the latter varies inversely as 
the square root of the density. For instance it will be about 
1000 times less in hydrogen than in oxygen. 

In order further to illustrate the effect of the lateral motion 
of the gas, near the surface of the sphere, from the hemisphere 
which is at the moment moving outwards to that which is 
moving inwards, in weakening the intensity of the waves 
propagated to a distance, we may calculate what the emission 
would be if this lateral motion were prevented. For this 
purpose we may (after Stokes) imagine a large number of fixed 
partitions to extend radially outwards from near the surface. 
In any one of the narrow conical tubes thus formed, the motion 
will be of the same character as in the case of symmetrical 
spherical waves. Now a uniform radial velocity C cos nt over 
the surface of a sphere would be equivalent to a simple source 
4-Tra 2 C cos nt, and the corresponding emission per unit area 
would be %tfa?pc C*, by 76 (15). If we now put G= A cos 6, 
and integrate over the surface, we get the total emission in our 
system of conical tubes. The result is 

W ' = 1-rrlcWpcA*, (5) 

since the average of cos 2 6 for all directions in space is J . If 
we compare this with (4), we see that the effect of the lateral 
motion is to diminish the emission in the ratio J& 2 a 2 . 

When, as for example in the case of a plate or a bell, the 
surface is divided by nodal lines into a number of compart- 
ments vibrating in opposite phases, the opportunity of lateral 
motion is increased, and the emission of energy correspondingly 
weakened. For facility of calculation Stokes took the case of 
a spherical surface, with various symmetrical arrangements of 
nodal lines. In the problem of the oscillating sphere we have 
one such line, viz. the great circle 6 = \ TT, and the emission, as 



240 DYNAMICAL THEORY OF SOUND 

we have just seen, is diminished by the lateral motion in the 
ratio k' 2 a?. For a spherical surface with two nodal great circles 
meeting at right angles the effect is much greater, the ratio 
being -fakta*. And as we increase the number of compartments 
into which the sphere is divided, the ratio, already very small, 
decreases with enormous rapidity. 

For the sake of simplicity it has been assumed in the 
preceding statements that the perimeter 2-n-a of the sphere is 
small compared with X. The influence of lateral motion is 
however not confined to this case, but will make itself felt 
whenever the dimensions of the compartments referred to are 
small compared with X. In the case of the oscillating sphere 
there is no difficulty in working out the result without any 
restriction to the value of ka, starting from the formula (13) of 
77. 

Stokes has also investigated mathematically the case of a 
cylinder vibrating at right angles to its length, where the same 
cause is of course operative. In this way an estimate is 
obtained of the direct effect of a vibrating string in generating 
air-waves. This involves the ratio of the perimeter of the 
cross-section of the string to the length of the air-waves, and 
is in any practical case extraordinarily minute. As explained 
in 24, almost the whole of the sound given out when a piano 
string is struck comes from the sounding board. 

81. Scattering of Sound Waves by an Obstacle. 

We have next to consider the disturbance produced in a 
train of sound waves by a rigid obstacle whose dimensions are 
small compared with the wave-length. The scattered waves 
which are sensible at a distance are due mainly to two causes. 
If the obstacle were absent the space which it occupies would 
be the seat of alternate condensations and rarefactions. The 
effect of the obstacle in refusing to execute the corresponding 
contractions and expansions of volume is, at a distance, 
approximately the same as if in a medium otherwise at rest its 
volume were to undergo a periodic change of just the opposite 
character. The result is equivalent to a simple source. On 
the disturbance thus produced there is superposed a second 



SIMPLE-HARMONIC WAVES. DIFFRACTION 241 

wave-system, which is due to the immobility of the obstacle. 
If the latter were freely movable, and if it had moreover the 
same density as the surrounding air, it would swing to and fro 
with the air-particles, and the second wave-system would be 
absent. This system is accordingly the same as would be 
produced if the obstacle were constrained to oscillate with 
a motion exactly equal and opposite to that of the air in the 
primary waves when undisturbed. The effect is, as we have 
seen in 79, that of a double source. It might appear, at first 
sight, that the former of these disturbing influences would be 
much less important than the second, but in its effect at a 
distance it becomes comparable, owing to the greater attenuation 
by lateral motion of the waves proceeding from a double source. 
If Q be the volume of the obstacle, the strength of the 
simple source due to the first cause is 



where s, <f> refer to the primary waves. In the case of a system 
of plane waves 



incident on a small obstacle at 0, this gives a velocity-potential 



As regards the second cause, we will assume for simplicity 
that the obstacle has the degree of symmetry postulated in 
79 with respect to the direction (Ox) of the vibration in the 
air-waves. If the wave-system (2) were undisturbed, the 
velocity of the air-particles at would be represented 
symbolically by ikC, and the strength of the double source due 
to the obstacle moving with this velocity reversed would be 
ik(Q + Q') (7, in the notation of 79. The scattered waves at 
a distance, due to the immobility, are therefore represented by 



(4) 



by 76 (24). The complete result is given by < = fa + < 2 . 
It follows that the amplitude of the scattered waves at any 
L. 16 



242 DYNAMICAL THEORY OF SOUND 

distant point is, for similar forms, directly proportional to the 
volume of the obstacle and inversely proportional to the 
square of the wave-length. This latter particular might have 
been foreseen without calculation. The ratio to the original 
amplitude must necessarily vary directly as the volume Q, 
and inversely as the distance r, and in order that the result 
may come out a pure number we must divide by X 2 , since X 
is the only other linear magnitude involved. The emission 
of energy, being proportional to the square of the amplitude, 
will therefore vary as X~ 4 . This law of the inverse fourth 
power holds also in optics, and for a similar reason, with respect 
to the scattering of light by particles whose dimensions are 
small compared with the dimensions of the light-waves. The 
blue of the sky, for instance, is attributed to the relative 
preponderance of the shorter waves in the light scattered by 
the molecules of air, and possibly by other particles ; in the 
transmitted light, on the other hand, the longer waves pre- 
dominate. The theory is due to Lord Rayleigh, who has also 
pointed to an acoustic illustration in what are called " harmonic 
echoes." If a composite musical note, consisting of a funda- 
mental tone with its octave, &c., be sounded near a grove 
of trees, for example, the ratio of the intensity of the octave to 
that of the fundamental will in the scattered sound be 16 times 
what it was in the original note. The scattered sound may 
therefore appear to be raised in pitch by an octave. 

The actual scattering of energy is found by adding the 
results due to the simple and the double source. This may be 
proved by calculating the work done at the surface of a sphere 
of large radius r. The terms due to the combined action of the 
two sources contain a factor cos 9, and so disappear when 
integrated over the surface. Hence, by 76 (15), (26), 



The energy-flux in the primary waves being %pk 2 cC 2 , by 
76 (8), the ratio which the energy scattered per second bears 
to this is 

(6) 



SIMPLE. HARMONIC WAVES. DIFFRACTION 243 

In the case of the sphere we found Q / = JQ = | 7 ra 3 , and the 
expression (6) therefore reduces to 

J(*a).iro' ......................... (7) 



In other words, the sphere scatters only the fraction | (&a) 4 of 
the energy which falls upon it. For example, if the wave- 
length be a metre (which corresponds to a frequency of about 
332), and the diameter of the sphere 1 mm., the fraction is 
roughly 7 '6 x 10~ n . In the case of the circular disk, where 
Q' = f Tra 8 , Q = 0, the ratio of the scattered to the incident 
energy is ^(Ara) 4 . 

The mathematical theory of the scattering by cylindrical 
obstacles is more difficult. We will merely quote the result, 
based on Lord Rayleigh's calculations, that when plane waves 
are incident on a circular cylinder of radius a the fraction of the 
incident energy which is scattered is f Tr^&a) 3 , approximately, 
it being assumed as usual that ka is small. For a wire of 
diameter 1 mm., and a wave-length of a metre, this 

= 1-15 x 10- 7 . 

It is to be observed however that in the case of very minute 
obstacles the order of magnitude of the results may be con- 
siderably modified by viscosity. The determining element here 
is the ratio of the diameter of the obstacle to the quantity 
h which was introduced in 66 as a measure of the thickness of 
the air-stratum, at the surface of the obstacle, whose motion 
is appreciably affected by the friction. When the ratio in 
question is moderately large the influence of viscosity on the 
results will be very slight. 

The distribution of velocity in the immediate neighbourhood 
of the obstacle will be sensibly the same as in the case of a 
uniform current of incompressible fluid flowing past the body. 
In the case of the sphere it can be determined completely, but 
the following approximation will be sufficient. We assume 



(8) 



where the first term represents the incident waves, and the 

162 



244 DYNAMICAL THEOEY OF SOUND 

second is the form which the velocity-potential of a double 
source assumes ( 76) when kr is small. This makes 



- ik r + cos0, ......... (9) 

and the condition of zero normal velocity for r = a is therefore 
approximately satisfied provided B= l%ika?C. Hence in the 
neighbourhood of the sphere we have 



(10) 

nearly. The velocities are therefore nearly the same as if the 
fluid were incompressible. The pressure is given by 

p=p + p<j>=p + inp(f> ............. (11) 

This differs from the pressure (p + inpG) which would obtain 
at the origin if the obstacle were absent by a term which 
is small, of the order kr, in comparison. At points whose 
distance r is a moderate multiple of a, whilst still small 
compared with X, the pressure approximates even more closely 
to that due to the incident waves alone. 

82. Transmission of Sound by an Aperture. 

In discussing the transmission of sound waves by an aperture 
in a thin screen we will suppose, in the first instance, that the 
dimensions of the aperture are small compared with the wave- 
length. This is of course the most interesting case from an 
acoustical point of view. 

The screen being supposed to occupy the plane x 0, and 
the origin being taken in the aperture (S), let a wave- train 
represented by 



be incident from the left. If we distinguish the functions 
relating to the two sides of the screen by the suffixes 1 and 2, 
we should have, if the screen were complete, 

fr^Ce-^ + Ce^*, < 2 = o, ............ (2) 

the second term in <f> 1} which represents reflected waves, being 
chosen so as to make dfa/da; = for x = 0. 

In the actual problem the disturbance due to the aperture 
will be confined mainly to the immediate neighbourhood of S, 
and may be taken to be very small at distances from which, 



SIMPLE-HARMONIC WAVES. DIFFRACTION 245 

though large as compared with the linear dimensions of S, 

are small compared with X. Let two surfaces be drawn, on 

the two sides, at some such distance from 0, each abutting 

on the screen in the manner indicated by the dotted lines 

in the figure. Within the region thus bounded, the fluid 

oscillates backwards and forwards almost as if it were in- 

compressible, and the total flux ( 67) through the aperture 

will therefore bear a constant ratio to the difference of the 

velocity-potentials at the two surfaces. This will perhaps be 

understood more clearly if we have 

recourse to the analogy of electric 

conduction. Suppose we have a / 

large metallic mass, severed almost / 

in two by a non-conducting parti- / 

tion occupying the place of the ;,' 

screen. If this mass form part of \ 

an electric circuit, there will be \ 

little variation of potential in it \ x 

except in the neighbourhood of the Xv 

narrow neck which connects the two 

portions. The electric potentials Fig" 74 

at a distance on the two sides being 

<>! and <f>. 2 , the current through the neck will be 

*(*.-&) ......................... (3) 

where K may be called the " conductivity " of the neck, the 
specific conductivity of the substance being taken to be- unity. 
In the hydrodynamical question, also, the quantity K may 
appropriately be called the conductivity of the aperture. It is 
easily seen that it is of the nature of a length. 

At the two surfaces shewn in the figure we have ^ = 2(7, 
<f> 2 0, approximately, and the total flux through the aperture 
is therefore 2KC. If an equal flux were directed symmetrically 
from the aperture on the left-hand side, the combination would 
be equivalent, in an unlimited medium, to a simple source of 
strength 4>KC. Hence, by 76 (12), 






246 DYNAMICAL THEOEY OF SOUND 

The corresponding velocity-potential on the near side is 
evidently 

e- ikr . .-.(5) 



The energy ( W) transmitted by the aperture per second is 
by the above reasoning one-half that due to a simple source 
at 0, whence, by 76 (15), 

W = fccKWir ...................... (6) 



The energy-flux in the primary waves (1) being %pk*cC' 2 , the 
ratio of W to this is 2^T 2 / 7r - ^ ^ s t be n ted that this is 
independent of the wave-length X, so long, of course, as X is 
large compared with the linear dimensions of S. 

The exact calculation of K for various forms of aperture is 
naturally a matter of some difficulty. For a circular aperture 
of radius a it is found that K = 2a ; for other forms differing 
little from a circle the value is sensibly the same as for a 
circular aperture of the same area, the circle being evidently 
a " stationary " form, in the sense in which this term is used 
in the theory of maxima and minima. It appears then that 
a circular (or nearly circular) aperture transmits the fraction 
8/7T 2 , or '816, of the energy propagated across an equal area 
(?ra 2 ) in the primary waves. This is, under the present 
limitation as to size, very great compared with the energy 
intercepted by a disk of the same dimensions ( 81). The 
figure opposite gives the shapes of the surfaces of equal 
pressure (< = const.), drawn for equidistant values of </>, in the 
immediate neighbourhood of a circular aperture, and shews 
how rapidly these tend to assume the spherical form. The 
directions of vibration of the air-particles are of course normal 
to these surfaces. 

With regard to further problems of the kind we must 
content ourselves with a few statements of results. In the 
case of an aperture in the shape of a long narrow slit, whose 
breadth is small compared with X, the energy transmitted is 
again comparable with, and may even considerably exceed, 
that corresponding to an equal area of wave- front in the 
primary waves. In the case of a grating composed of equal, 



SIMPLE-HARMONIC WAVES. DIFFRACTION 247 

parallel, and equidistant slits in a thin screen, the fraction of 
the total incident energy which is transmitted is found to be 
1/(1 + A?/ 2 ), where k = STT/X, as usual, and 

, a + b , Trb 
U __, ogsec ___ (7) 

where a denotes the breadth of an opening, and b that of each 
intervening portion of the screen. As a numerical example, 




Fig. 75. 

suppose the wave-length to be ten times the interval a + 6 
between the centres of successive apertures ; then even if the 
apertures form only one-tenth part of the whole area of the 
screen, 88 per cent, of the sound will get through. In the 



248 DYNAMICAL THEORY OF SOUND 

case of a grating formed by equidistant bars of circular section, 
the corresponding value of / is 

l=7rb*/a, (8) 

where b is the radius of the section, and a the distance between 
the axes of consecutive bars. It is implied, however, that the 
ratio b/a must not exceed (say) J. 

83. Contrast between Diffraction Effects in Sound and 
Light. Influence of Wave-Length. 

In the investigation of 82 an aperture was found to act as 
a simple source from which sound diverges on the farther side 
uniformly in all directions. This is in striking contrast with 
what is usually observed in the case of light. We have so far 
no indication of anything of the nature of beams or rays of 
sound, just as when sound waves were incident on an obstacle 
we found nothing of the nature of a sound-shadow. The 
difference in the results is due to the fact that the dimensions 
of the aperture (or obstacle) have been supposed small in 
comparison with the wave-length, whereas with light the 
relation is usually the reverse. 

We have avoided trespassing on the domain of Optics, but 
as the dynamical conditions are in the present subject perfectly 
definite, it may be permissible to examine this question of the 
influence of wave-length a little more fully. 

Consider the region of space lying to the right of the plane 
x = 0. If this plane were a fixed boundary, and if there were 
no sources of sound in the region, any disturbance would 
ultimately pass away. Any steady periodic motion in the 
region must therefore in the absence of internal sources be due 
to motion of the boundary, and will be determinate when the 
value of the normal component of the velocity at every point of 
the latter is given. It can, moreover, be expressed in terms of 
this distribution of normal velocity, as follows. The flux out- 
wards from an element 8$ of the plane is d<f>/dn . &S, if &n 
denote an element of the normal drawn inwards from &S, and 
if in imagination we associate with this an equal flux in the 
opposite direction on the other side, the result is equivalent 



SIMPLE-HARMONIC WAVES. DIFFRACTION 249 

to a source - 29</3n . 8S in infinite space. The corresponding 
velocity-potential at a point P is 



where r denotes the distance of 8S from P. Integrating over 
all the elements &S of the plane, we have 

1 rr^^ a -ikr 



which is the required formula. 

The motion to the right of the plane x = is also determinate 
when the value of </> at every point of the plane, and thence 
the pressure, is given, these two quantities being connected by 
the relation p p + p<j> = p Q + ikcp<f>. Suppose for a moment 
that in an otherwise unlimited medium we have a thin massless 
membrane occupying the plane x = 0, and that on each element 
of this a normal force X per unit area is exerted, which is 
adjusted so as to produce the actual periodic pressure, and 
therefore the actual value of $, on the positive face of the 
membrane. By the theorem of 78 (15), the effect for an 
element BS, will be equivalent to a double source, and the 
corresponding velocity-potential at a point P will be 



(2) 



The variable parts of the pressures on the two faces of the 
membrane, viz. + p(f> = ikcp<f>, must balance the force X, so 
that X = %ikcp$. Substituting in (2), and integrating over the 
plane x = 0, we obtain 



* * 



The structure of the integrals in (1) and (3) recalls the 
process by which " Huygens' principle " is applied in optics to 
find the disturbance at any point P in terms of "secondary waves" 
supposed to issue from the various elements of a wave-front. 
There was at one time much discussion as to the exact character 
to be assigned to these secondary waves, more especially as to 
the law of intensity in different directions. We now recognize 



250 DYNAMICAL THEORY OF SOUND 

that the problem has mathematically more than one solution ; 
either of the above formulae will lead to an exact result, and 
we might even use a combination of the two, in any arbitrary 
proportions. This resolution of a historic controversy is due to 
Lord Rayleigh. 

As a verification of (3), suppose that the value of <j> at x = 
is that due to a train of plane waves <f> = e~ ikx . Let OT denote 
the distance of BS from the orthogonal projection of the point 
P on the plane x 0, so that r 5 = # 2 4- w 2 . For the aggregate 
of elements 88 forming a certain annulus of the plane we 
may write 27ror&nr = 27rr&r. We have also dr/dx = a;/r. The 
formula (3) therefore gives 



r 9 fe~ ikr \ 
-x ^ e )dr = e- ik *. ...(4) 



In the case of waves transmitted by an aperture in a plane 
screen (x = 0), we have, in (1), 3</>/3n = except over the area of 
the aperture. If, further, the dimensions of the aperture S are 
small compared with X, then at a point P whose distance r is 
large compared with X, the function e~ ikr /r will have sensibly 
the same value for all the elements of S, and we may write 



(5) 



where the first factor represents the total flux through S. 
Under these circumstances the aperture acts like a simple 
source, as in 82. 

It is understood of course that the expression d<f)/dn in 
(1) or (5) represents the normal component of the velocity, as 
modified by the action of the screen. When as in the case just 
considered the aperture is relatively small, the distribution of 
normal velocity over it will differ considerably from that due to 
the primary waves alone. This distribution can be ascertained 
approximately, in the case of plane waves incident directly on 
a circular opening, from the electrical analogy of 82. The 
lines of flow have the same configuration as the lines of force 



SIMPLE-HARMONIC WAVES. DIFFRACTION 251 

due to an electrified disk*, and the normal velocity has the 
distribution 

_d$ = B 

dn V( 2 -^ 2 )' 

where tn- denotes the distance of any point of the aperture from 
its centre. The velocity becomes very great near the edge, and 
is mathematically infinite ( at the edge itself (r = a), but it 
appears on integration that the parts of the area near the edge 
contribute little to the total flux, which is 



(7) 

If the incident waves be represented by 



(8) 

the same flux will as in 82 be expressed by 2KC, or 4a(7. 
Hence, comparing, 

In the other extreme, where the wave-length is only a 
minute fraction of the dimensions of the aperture, the effect of 
the screen in modifying the distribution of normal velocity over 
the latter is practically confined to a distance of a few wave- 
lengths from the edge, and the corresponding part of the integral 
in (1) is quite unimportant. In this case, the incident waves 
being still expressed by (8), we can put d(f>/dn = ikC with 
sufficient accuracy over the whole area of the aperture, whence 

*-(f^ .-do) 



For the methods of approximating to the value of this integral, 
by the use of Huygens' or Fresnel's " zones," or otherwise, we 
must refer to books on Optics. It is found that the amplitude 
is nearly uniform within the space bounded by a cylindrical 
surface whose generators are normal to the screen through the 
edge of the aperture, and is nearly zero in the surrounding 
region. Near the cylindrical boundary, on either side, we have 

* See Fig. 75, p. 247, which represents the configuration of the equipotential 
surfaces. 

f The awkwardness of this conclusion may be avoided by giving the screen 
a certain thickness, and rounding the edges. 



252 DYNAMICAL THEOKY OF SOUND 

the diffraction effects which are especially studied in the theory 
of Light. 

The question of the impact of waves on a plane lamina can 
be treated in a similar manner. For this purpose the formula 
(3) is most convenient. The lamina being in the plane x = 0, 
and the primary waves being represented by (8), we may write 
<t> = Ce~ ik * + x , ..................... (11) 

where % is the velocity-potential due to a vibration of the 
lamina normal to its plane with the velocity ikC, equal and 
opposite to that in the primary waves. It is evident that the 
values of this function at any two points which are symmetric- 
ally situated with respect to the plane x = will be equal in 
magnitude but opposite in sign. We have then, to the right 
of the lamina 



This only requires a knowledge of the value of % at the positive 
face of the lamina, the value at all other points of the plane 
x = being obviously zero. The case where the dimensions of 
the lamina are small compared with X has been noticed in 
81 ; the scattered waves have then a much smaller intensity 
than those transmitted by an aperture of the same size and 
shape. In the opposite extreme, the value of % near the 
positive face is, except near the edge, the same as in the case 
of an infinite vibrating plate, viz. % = Ce ikx , so that we have 
with sufficient accuracy 



A detailed study of this integral would indicate, in the complete 
solution expressed by (11), the existence of a sound-shadow to 
the right of the lamina. For large values of kr the formula 
(13) may be replaced by 

ikr 

............ (14) 



and for small obliquities 6 we may further put cos 6 = 1. The 
formula then becomes, except as to sign, identical with (10), 
shewing that the disturbance produced by the lamina is, under 



SIMPLE -HAKMONIC WAVES. DIFFRACTION 253 

the conditions postulated, exactly opposite to that transmitted 
by an aperture of the same dimensions. This is a familiar 
fact in Optics ; but the preceding considerations shew that it 
may be utterly wide of the mark when the wave-length is no 
longer small compared with the linear dimensions concerned. 

It need hardly be said that there are acoustical phenomena 
where, as in the case of large reflecting or obstructing surfaces, 
optical relations are approximated to. The results are then 
analogous, the resemblance being more complete the higher 
the pitch of the note sounded. By the use of a source of very 
high pitch, and of a sensitive flame as a detector, Lord Rayleigh 
has succeeded in imitating some of the most delicate phenomena 
of physical optics. 

In the above theoretical investigation we have been obliged 
to rely to some extent on intuitive considerations, as e.g. in the 
assumed distribution of velocity over the area of an aperture 
when the wave-length is relatively small. It is therefore 
desirable that such assumptions should be tested if possible by 
exact calculation. The only instance, at present, where this has 
been successfully carried out is that of waves incident on a 
plane screen with a straight edge. The reflection by the screen, 
the transmission past the edge, the formation of a shadow 
behind the screen, and the diffraction phenomena near the 
boundaries of the respective regions, all come out in practical 
accordance with the usual theory. The investigation was 
published by Sommerfeld in 1895*. 

* A simplified version is given in the Proc. Lond. Math. Soc. (2), vol. iv. 
(1906). 



CHAPTER IX 

PIPES AND RESONATORS 

84. Normal Modes of Rectangular and Spherical 
Vessels. 

The main object in this chapter is to develop the laws of 
vibration of air contained in cavities, such as those of resonators 
and organ pipes, which are in communication with the external 
atmosphere. A little space may however be devoted in the 
first instance to some problems relating to the vibrations of air 
in spaces which are completely enclosed by rigid walls. These 
will at all events supply some interesting examples of the 
general theory of normal modes ( 16). 

The analytical process consists in finding solutions of the 
equation 

V 2 </> + #ty = ........................ (1) 

consistent with the condition 



which expresses that the component of the fluid velocity in 
the direction of the normal (n) vanishes at the boundary. It 
appears that, as in former analogous problems, this is only 
possible for a certain sequence of values of k, which determine 
the nature and the frequency of the respective normal modes. 
In the case of a rectangular cavity we take the origin at a 
corner, and the coordinate axes along the edges which meet 
there. If the lengths of these edges be a, b, c, the condition 
(2) is fulfilled by 



TTTZ 
</>= (7 cos- cos -j. cos -- 

d G 



PIPES AND RESONATORS 



255 



where p, q, r are any integers; and the equation (1) is also 
satisfied provided 

(4) 



If we put <? = 0, r = 0, the case degenerates into that of the 
doubly closed pipe ( 62). 

A more interesting case is that of a spherical cavity. The 
symmetrical radial vibrations come under the methods of 71, 
76. The formula (15) of 71, which implies that there is no 
source at the origin, gives, in the case of simple-harmonic 
vibrations, 

^Ller ttr 

(5) 

...(6) 



_ 



or, say, 



sin kr 




256 



DYNAMICAL THEOEY OF SOUND 



The condition (2) requires that d(f>/dr = for r a, the radius of 
the cavity. Hence 

tan kaka ......................... (7) 

This is a transcendental equation to find k, and thence n (= kc). 
The roots are obtained graphically (see Fig. 76) as the abscissae 
of the intersections of the lines y = tan x, y=-x y the zero root 
being of course excepted as irrelevant. We have, approximately, 
&a=(ra+i)7r, where m= 1, 2, 3, .... More accurate values of 
the first three roots are 

ka/ir = 1-4303, 2-4590, 3*4709 ............. (8) 

The numbers give the ratio of the diameter 2a of the cavity to 
the wave-length. In the modes after the first there are internal 
spherical nodes (i.e. surfaces of zero velocity) whose relative 
positions are indicated by the roots of inferior rank. In the 
higher modes the nodal surfaces tend, as we should expect, to 
become equidistant, since the conditions, except near the centre, 
approximate to those of plane waves. 

Equations of somewhat similar structure to (7) occur (as 
we have seen) in various parts of our subject, as well as in 
other branches of mathematical physics, and processes of 
numerical solution have been de- 
vised by Euler, Lord Rayleigh 
and others. There is one method, 
of very general application, which 
is so elegant, and at the same 
time so little known, that it may 
be worth while to explain it. It 
is given by Fourier in his Theorie 
de la Chaleur (1822). Starting 
with a rough approximation, say 
= 0?!, to a particular root of (7), 
we calculate in succession the 
quantities x z , x s , 0? 4 , ... determined 
by the relations 

# 2 = tan" 1 0?i , # 3 = tan" 1 # 2 
The figure illustrates the manner in which these converge 
towards the desired root as a limiting value, no matter from 




Fig. 77. 



tan" 1 



(9) 



PIPES AND RESONATORS 257 

which side we start. Some fairly obvious precautions are 
necessary in using the method, and it is easily seen that the 
convergence will be slow if the two curves have nearly the 
same inclination (in the same or in opposite senses) to the axis 
of x. Expressed as multiples of TT, the successive approximations 
obtained in this way to the first root of (7) are* 

1-5, 1-433435, T430444, 1-430304, 1*430297, .... 

The same analysis can obviously be applied to the theory of 
vibrations in a conical pipe whose generating lines meet in 0. 
If the tube extend from the origin to r = a, the usual approxi- 
mate condition (s = 0) to be satisfied at the open end gives 

sin&a = 0, (10) 

the same as for a doubly open pipe of length a ( 62). For the 
case of a tube extending from r = a to r = b, and open at both 
ends, we require the complete solution 

r<f> = A cos kr + B sin kr. (11) 

The conditions give 

Acoska + Bsinka = Q, Acoskb + Bsinkb = Q, (12) 
whence sin k (b a) = 0, (13) 

as in the case of a doubly open pipe of length b a. 

If x be any solution of the general equation (1), it appears 
on differentiation throughout with respect to x that the equation 
is also satisfied by </> = d%/3# . We have already had an example 
of this in the general double source of 73. From (6) we 
derive in this way the solution 

d /sinferX ( . 

* =G Wx()> (14) 

or, if x = r cos 6, 

ri d/smkr\ - C ,, , 7 \ Q /- ~\ 
6 = C=- - - cos 6 - (kr cos kr sin kr) cos 6. (lo) 
or \ T / r 1 

This leads to another series of normal modes of the air con- 

* In calculations of this kind, and for the purposes of mathematical physics 

generally, trigonometrical tables based on the centesimal division of the 

quadrant are most convenient. A four-figure table of this type is included in 

J. Hoiiel's Eecueil des Formules et des Tables Numfriques, 3rd ed., Paris, 1885. 

L. 17 



258 



DYNAMICAL THEOEY OF SOUND 



tained in a spherical cavity. The condition 3</>/3r = is satisfied 
for r = a, provided 



tan ka = 



2 - 



,(16) 



The solution can be carried out as in the case of (7). The 
annexed diagram of the curves y = cot x, y = (2 # 2 )/2#, shews 
that the roots tend after a time to the form mnr. Approximate 
values of the first few roots are 

5, 1-891, 2'930, 3'948, 4*959, ...(17) 




Fig. 78. 

the first of which alone gives any trouble. This root corresponds 
to the gravest of all the normal modes of the cavity. The air 
swings from side to side, much as in the case of a doubly closed 
pipe, and the wave-length is X = 2-7T/A; = T509 x 2a. The 
forms of the equipotential surfaces, to which the directions of 
vibration of the air-particles are orthogonal, are shewn in Fig. 79. 
In the next mode the radial velocity vanishes over the sphere 
r/a= '6625/1-891 ='350. 

The study of the more complicated normal modes of vibra- 
tion in a spherical vessel would lead us too far. The problem 
is fully discussed in Lord Rayleigh's treatise. 



PIPES AND RESONATORS 



259 




Fig. 79. 

85. Vibrations in a Cylindrical Vessel. 

The theory of the purely transversal vibrations of the air 
enclosed by a circular cylinder is very similar. As in 54, the 
equation 

v (p (Z> j i f\ /-i \ 

-^- -j- fcP(p = ( 1 ) 

03j Vll 

where x, y are Cartesian coordinates in the plane of a cross- 
section, becomes in polar coordinates 



and the typical solution, when there is no source at the 
origin, is 



x 
GJ m (kr) cos W0.+<k ) ............... (3) 



The admissible values of k are determined by the condition that 
d<f>/dr = for r = a, or 

J m '(ka)=0 ......................... (4) 

For the radial vibrations (ra = 0) the earlier roots are given by 

ka/ir = 1*2179, 2-2330, 3-2383,..., ......... (5) 

172 



j 



260 DYNAMICAL THEOEY OF SOUND 

the limiting form being integer + J. In the case m = 1, which 
includes the gravest mode, 

&a/7r=-586, 1-697, 2717,..., ............ (6) 

the limiting form being integer J . 

The purely longitudinal modes of a closed circular cylinder 
come under 62. There remain the vibrations of mixed type. 
The equation (2) has now to be modified by the inclusion of 
a term 9 2 </>/d 2 , where z is the longitudinal coordinate. It is 
found that the equation is satisfied by 



(7) 



provided k* = /3 2 + m'V 2 /^, .................. (8) 

the origin being taken at the centre of one end. The condition 
of zero normal velocity (d(f>/dz) at the other end (z = I) is 
satisfied if m' be integral. The corresponding condition at the 
cylindrical surface requires that /3 should be a root of 

J m '(l3a) = ...................... (9) 

86. Free Vibrations of a Resonator. Dissipation. 

The foregoing examples are of theoretical rather than 
practical interest, since the vibrations of a mass of air enclosed 
by rigid walls would be completely isolated. For acoustical 
purposes the vibrating mass must have some communication 
with the external atmosphere ; on the other hand it is essential 
that the communication should be so restricted that the frac- 
tion of the energy which is used up in a single period in 
the generation of diverging waves shall still be very small. 
Otherwise the free vibrations could hardly be regarded as 
approximately simple-harmonic, and might even resemble 
the "dead-beat" type (11). 

The theory is simplest in the case of " resonators " such as 
were employed by Helmholtz in his researches on the quality 
of musical notes. These are nearly closed vessels, with an 
aperture, and are used to intensify, by sympathetic vibration of 
the enclosed air, the effect of a simple tone produced in the 
neighbourhood. The precise form is not important; it may 
be spherical or cylindrical, or almost any shape, so long as 



PIPES AND RESONATORS 261 

the least diameter considerably exceeds the dimensions of 
the aperture. In his synthetic work on the vowel sounds 
Helmholtz used cylindrical resonators having a circular opening 
at the centre of one end. When the object was to detect and 
to isolate a particular overtone in a complex sound, he used 
the more convenient form shewn in Fig. 80. The small open 
nipple opposite the mouth is inserted into the ear cavity, so 
that the tympanic membrane becomes part of the internal wall 
of the resonator. 





Fig. 80. Fig. 81. 

The theory of resonators was treated mathematically for the 
first time by Helmholtz in 1860, and was afterwards greatly 
simplified by Lord Rayleigh (1871). Suppose in the first place 
that we have a vessel with a narrow cylindrical neck which is 
occupied by a plug or piston freely movable to and fro (Fig. 
81). Let Q denote the capacity of the vessel, I the length of 
the neck, o> its sectional area, p the density of the piston. We 
will assume that the period of vibration is so long that the 
corresponding wave-length (X) in air is large compared with 
the diameter of the vessel. Under this condition the con- 
densation s will at any instant be almost uniform throughout 
the interior, and we may put s = cox/Q, where x denotes the 
small displacement of the piston outwards from its mean 
position. The resulting excess of pressure on the base of the 
piston is /5C 2 so), or pc 2 (0 2 x/Q, and the equation of motion of 
the system is, approximately, 

(1) 



DYNAMICAL THEORY OF SOUND 



The motion is accordingly simple-harmonic, with a period 27r/n, 
provided 



The nature of the piston is of little importance, provided 
its mass be sufficiently small. We may even replace it by air, 
if the length I be small compared with X, for under this 
condition the column of air in the neck will behave almost as if 
it were incompressible. We have then p' = p, and 

(3) 



Even in the case of a resonator whose mouth consists of 
a mere opening in the wall, without a neck, the theory is not 
very different. It is only a question of obtaining a proper 
measure of the inertia of the mass of air in the immediate 
neighbourhood of the mouth, inside and outside, which takes 
the place of the piston in the above problem. The flow 
through the aperture at any instant is still regulated, ap- 
proximately, by the same laws as that of an incompressible 
fluid, or of electricity in a uniform conductor. There being 
little motion in the interior, the 
value of $ there will be sensibly 
uniform ; we denote it by fa. Out- 
side, at a short distance beyond the 
mouth, we shall have <j> = 0, nearly. 
If q denote the volume of air which 
has passed through the aperture 
outwards up to time t, the current, 
or flux, outwards at this instant will 
be q, and we have, by the electric analogy, 

........................ (4) 




where K is the " conductivity " ( 82), which depends, of course, 
on the shape and size of the aperture and the configuration of 
the wall in its neighbourhood. It is to be observed that this 
relation (4) is purely kinematical ; from the point of view of the 
generalized dynamics of a system of one degree of freedom 



PIPES AND RESONATORS 263 

( 7), it expresses the momentum (which may be symbolized by 
pfa) in terms of the velocity q. The dynamical equation 

c 2 s=<i ........................... (5) 

of 70 (3) may in like manner be interpreted as expressing 
the relation between change of momentum and force. If the 
zero of q correspond to the equilibrium state, we have 

s = -q/Q ......................... (6) 

Eliminating s and <, between (4), (5), and (6), we obtain 



The motion is therefore of the type 

q = Ccos(nt + e), ..................... (8) 

provided n* = Kc?/Q ............ . ............ (9) 

If we write n = kc, this gives 

t? = K/Q, X = 2nV(Q/#) ................ (10) 

The wave-length depends, as we should expect, solely on the linear 
dimensions of the resonator and its aperture. For resonators 
which are geometrically similar in all respects, it varies directly 
as the linear dimension. This is in accordance with a general 
principle which may be inferred from the differential equation 
(2) of 76, or otherwise. The formula (9) indicates further that 
the pitch of the resonator is lowered by contracting or partially 
obstructing the aperture, whilst it is raised by diminishing the 
internal capacity. 

The kinetic energy, being mainly resident in the neighbour- 
hood of the mouth, may be calculated from the principles 
applicable to an incompressible fluid. If the actual motion 
were generated instantaneously from rest, the work required 
would be the sum of half the products of the impulses into 
the corresponding velocities. The equations (9) of 69 shew 
that the requisite impulsive pressure is p^ hence 



The potential energy is, by 70 (8), 

F=4 / >cVQ = |( / >cVQ).3 2 ............. (12) 

The coefficients in these expressions being known, the speed 
n of the oscillations can be inferred at once by the general 



264 DYNAMICAL THEORY OF SOUND 

formula (7) of 7. It was under this form that the theory was 
presented by Lord Rayleigh. It is to be noticed that the 
inertia-coefficient is proportional to the " resistance " of the 
aperture (in the electrical sense), whilst the coefficient of 
stability, or elasticity, varies inversely as the capacity Q. 

The preceding theory applies only to the gravest mode of the 
resonator. In the higher modes the internal space is divided 
into compartments by one or more " loop surfaces " (i.e. surfaces 
of constant pressure, where <j> = 0), and the frequencies are 
much greater. The wave-length is then at most comparable 
with the linear dimensions, as in the problems of 84. 

As already stated ( 82) the calculation of K is usually difficult. 
For a circular aperture in a thin wall K is equal to the 
diameter, and for any form differing not too much from a circle 
we may put K=2 V(w/7r), approximately, where to is the area. 

The frequency, as determined by (9), will then vary as a^/Q . 
It is remarkable that this law was arrived at empirically by 
Sondhauss at a date (1850) anterior to the theory. When the 
aperture is fitted with a cylindrical neck, the conductivity is 
limited mainly by the neck itself, and we may put K = w/l, 
approximately, where I is the length. The formula (9) then 
agrees with (3). It is implied that I is small compared with X, 
and at the same time large compared with the diameter of the 
channel. 

We have in the above theory allowed for the inertia of the 
external atmosphere, but not for its compressibility, and the 
vibrations as given by (8) are accordingly persistent. In other 
words, we have neglected the apparent* dissipation of the 
energy of the resonator due to air-waves diverging outwards 
from the neighbourhood of the mouth. This will have, in 
general, no appreciable influence on the period, but will 
manifest itself by a gradual decay of the amplitude. 

The effect can be estimated with sufficient accuracy in- 
directly. The flux outwards at the mouth is, by (8), 

q=-nCsm(nt + e) (13) 

* True dissipative influences such as viscosity and thermal conduction are 
ignored in the present investigation. They probably play as a rule a wholly 
subordinate part. 



PIPES AND RESONATORS 265 

If the resonator were practically isolated in space, then on 
account of the assumed smallness of its dimensions as com- 
pared with \, the effect of the flux at a distance would be that 
of a simple source of strength nC, and the rate of emission of 
energy would accordingly be 

W=n 4 pC*/Sirc, ..................... (14) 

by the formula (15) of 76. The energy E of the motion, 
being equal to the potential energy at its maximum, is, 
approximately, 

(15) 



by (12). Equating, on the principles of 11, the rate of decay 
of this energy to the emission W, we find 



and therefore q = G e~ tlr cos (nt + e), ............... (17) 

provided r = $7r(?/n'Q = S-rrQ/K^ ............ (18) 



in virtue of (9). The ratio of the modulus of decay to the 
period (%7r/kc) is given by 



(19) 



Since K is at most comparable with the mean breadth of the 
aperture, this ratio is usually very great, and the preliminary 
assumptions implied in the above process are amply justified. 

If the mouth of the resonator were furnished with an 
infinite flange, i.e. one whose breadth is large compared with X, 
the equivalent source would, as explained in 82, have double 
the strength above assumed, and the emission of energy, now 
operative in one half of the surrounding region, would be twice 
as great. The modulus (18) would accordingly be halved. 

As a numerical illustration of the theoretical results, take 
the case of a spherical vessel 10 cm. in diameter, with a circular 
aperture 1 cm. in radius, so that Q = 523*6, K 2. The wave- 
length, calculated from (10), is 101*6 ; and the frequency there- 
fore about 327. The modulus of decay, as given by (18), is 
about one-tenth of a second. 



266 DYNAMICAL THEOEY OF SOUND 

87. Corrected Theory of the Organ Pipe. 

The same principles can be applied to obtain a correction 
to the imperfect theory of the open pipe which was given in 
62. We may begin by a brief examination of the slightly 
simpler problem of reflection at an open end of an infinitely 
long pipe ( 61). 




Fig. 83. 

Near the open end there is a certain region, whose dimensions 
are small compared with the wave-length, within which the 
transition takes place from plane waves within the tube to 
diverging spherical waves outside*. We take the origin inside 
the tube, near the mouth, but in the region of plane waves, and 
the positive direction of the axis of x along the tube. For the 
region of plane waves we may write 

<f> = Ae ik * + Be- ik *, ....(1) 

where the first term may be taken to represent a train of waves 
approaching the end, from the right, whilst the second term 

* The figure, which is based on formulae given by Helmholtz in another 
connection, relates to the two-dimensional form of the problem. In three 
dimensions the transition to a state of uniform radial flow outwards from the 
mouth would be still more rapid. 



PIPES AND RESONATORS 267 

represents the reflected waves. The outward velocity at is 
therefore represented by ik (A - B), and the flux is 

q = ika>(A-B\ ..................... (2) 

where a> is the sectional area. The velocity-potential at is 
A + B. The " resistance " between the section x = and the 
external region to the left may be specified as equivalent to 
that of a certain length a of the pipe, and is accordingly 
denoted by a/to. Hence, by the electrical analogy, 



to 



(3) 



u ^ lika , 

whence = - - ...................... (4) 

A I+ika. 

If we put ka tan & fi, ..................... (5) 

this may be written B/A = - e -* k t ...................... (6) 

Hence < = A {e ikx - e~* ***> ) ................ (7) 

The reflected train is therefore equal in amplitude to the 
incident one, as was to be expected, since the inertia only of the 
external air is so far taken into account ; but there is a difference 
of phase. In the theory of 61 the condition to be satisfied at 
an open end was s = 0, or <f) = 0. Hence if we write (7) in the 
form 

= 4*^ {***> -e-*^} ............ (8) 

we recognize that the circumstances are the same as if the pipe 
were prolonged to the left for a length ft, and the reflection at the 
mouth were to take place according to the rudimentary theory. 
The wave-length being assumed to be large compared with the 
diameter of the pipe, ka. will usually be small, so that ft = a, 
nearly. But if the pipe be very much contracted or obstructed 
at the mouth, ka may be considerable, and k/3 will in that case 
approach \ir. We then have B = A, nearly, and the circum- 
stances approximate to those of reflection at a closed end. 

The actual determination of a is a problem in electric 
conduction which has at present only been solved, even 
approximately, in a very few cases. Lord Rayleigh estimates 
that for an accurately cylindrical tube fitted with an infinite 
flange the value of a is about "82 of the sectional radius. For 



268 DYNAMICAL THEOEY OF SOUND 

an unflanged cylindrical tube experiment seems to indicate a 
value of about '6 of the radius. 

We will next suppose the pipe to be of finite length, and to 
be closed at x = I, the origin being chosen as before, near the 
mouth, in the region of plane waves. For this latter region we 

may assume 

<f> = A cos k(l x), .................. (9) 

since d<f>/dx must vanish for x = l. The flux outwards at 
the mouth is therefore 

q = codcj>/dx = kco A sin kl, .................. (10) 

and the potential at is A cos kl. Hence with the same 
meaning of a as before we have 

A cos kl-x kco A sin kl, 

CO 



or cotkl = ka ...................... (11) 

This equation determines the wave-lengths (^Tr/k) of the 
various normal modes. Usually, ka. is small, and the solution 
of (11) is then 

kl = (m + J) TT ka, 

or fc(/ + ) = (w + 4)9r, ............... (12) 

where m is integral. The character of the normal modes is there- 
fore the same as on the rudimentary theory ( 62), provided we 
imagine the length of the pipe to be increased by the quantity 
a. In particular, the frequencies are as the odd integers 
1, 3, 5, ... , so long as the wave-length remains large compared 
with the diameter. 

If the aperture be contracted the value of a is increased, 
and the result tends to become less simple. In particular, the 
harmonic relation of the successive frequencies is violated, as 
may easily be seen from a graphical discussion of the equation 
(11). When the pipe is almost closed, a is relatively great, and 
the solution of (11) is kl = 1/ka, or k* = 1 /la. This agrees with 
the formula (10) of 86, if we put col = Q, co/a = K. 

In the case of a pipe open at both ends the period equation 
is found to be 

tan &Z = -( + '), ............... (13) 

where a, a' are the corrections for the two ends, but the calcula- 



PIPES AND RESONATORS 269 

tion implies that ka. and ka! are small. It is, however, only on 
this condition that the conductivities at the two ends can, as a 
rule, be estimated independently of one another. The equation 
is then equivalent to 

sin &( + + a 7 ) = 0, ............... (14) 

and the frequencies are therefore those which are assigned to 
a pipe of length I + a + a! by the rudimentary theory. The 
harmonic relation between the various normal modes is pre- 
served, but it must be remembered that the approximation is 
the more precarious, the higher the order of the harmonic. 

The wave-lengths of the proper tones are in all cases fixed 
by the linear dimensions, but the frequencies, which vary as 
the velocity of sound, will rise or fall with the temperature. 
An "open" organ pipe is tuned by means of a contrivance 
which increases or diminishes the effective aperture at the open 
end, i.e. the end remote from the " mouth " proper. The pitch 
of a " closed " pipe is regulated by adjusting the position of a 
plug which forms the barrier. 

To calculate the rate of decay of the free vibrations it will 
be sufficient to take the case of the stopped pipe. The kinetic 
energy corresponding to 

(f) = A cos k (I x) cos nt ............... (15) 

is given with sufficient accuracy by 



T = Ipto dx = pk*c0l . A* cos 2 nt t . . .(16) 

if ka. be small, since cos kl = 0, nearly. A more careful calcula- 
tion, taking account of the transition region between the plane 
and the spherical waves, replaces I by I + a, approximately, in 
this formula, but the correction is not important. The total 
energy, being equal to the kinetic energy at its maximum, is 

accordingly 

E = \pteu>lA'> ...................... (17) 

If the mouth be unflanged it acts, in relation to the external 
space, as a simple source of strength kcoA sin kl, or kaA, nearly, 
and the consequent emission of energy per second is accordingly 
W = pfrco'cAt/STr, .................. (18) 



270 DYNAMICAL THEOEY OF SOUND 

by 76 (15). Equating the rate of decay of the energy to W, 
we are led to the equation 

dA k*a>c A _ 

"3* + 



and the modulus of decay is therefore 

r = 47rZ/A; 2 6>c ...................... (20) 

The ratio of this to the period (2?r/A;c) is 21/kco, or (in the 
gravest mode) ffi/ira), nearly. Since the moduli of the various 
normal modes are proportional to the squares of the respective 
wave-lengths, the decay is the more rapid the higher the order. 
For a flanged pipe the result (20) would be halved. 

88. Resonator under Influence of External Source. 
Reaction on the Source. 

The theory of forced vibrations due to an external source 
of sound, to which we now proceed, involves some rather 
delicate considerations, and is often misunderstood. That 
the mass of air contained in a resonator or an organ pipe 
should be set into vigorous vibration by a source in approxi- 
mate unison with it is intelligible enough; but it is further 
desirable to have some estimate of the amplitude of the forced 
vibration, and in particular to understand why the sound which 
is apparently emitted by the resonator should under certain 
conditions enormously exceed that which would be produced 
by the original source alone. 

For simplicity we will suppose that this source is main- 
tained at constant amplitude by a suitable supply of energy, 
so that the vibration of the air is everywhere steady. It is 
evident at once that under this condition no work is done, 
on the average of a whole period, at the mouth of a resonator 
on the contained air, the energy of the latter being constant, 
and consequently that no work can in turn be done by the 
reaction of this mass on the external atmosphere. Any 
increased propagation of sound to a distance must be due to 
the changed conditions which the action of the resonator has 
introduced in the neighbourhood of the original source. If 
this source be not maintained constant, but merely started 
with an initial fund of energy (as in the case of a tuning 



PIPES AND RESONATORS 271 

fork), this fund will under the influence of the resonator be 
more rapidly consumed. 

In order to treat the question in a form free from unessential 
details, which may vary from one case to another, we take the 
case of a resonator of the type considered in 86, whose 
dimensions are small compared with the wave-length. 

The theory is simplest when the frequency of the source 
is very nearly equal to the natural frequency of the resonator, 
as determined by 86 (9), so that the forced vibration in the 
latter is at its strongest. It will perhaps make the matter 
clearer if we imagine in the first instance that the resonator 
has a short cylindrical neck in which a thin massless disk, 
almost exactly fitting it, can be made to move to and fro by 
a suitable application of force. Suppose then that the disk 
is made to execute a vibration such that the volume swept 
over by it outwards up to time t is 

q = C cos nt ; (1) 

and let the extraneous force which must be applied to the 
disk to compensate the difference of the air-pressures on the 
two sides be denoted by 

A cos nt + B sin nt, (2) 

this expression being (say) positive when the force is outwards. 
The component Acosnt which keeps step with the displace- 
ment is required to control the inertia of the air. From the 
general theory of forced vibrations ( 9, 12) it appears that the 
coefficient A can be made to have one sign or the other 
by adjusting the value of n, the sign being the same as that 
of C when the imposed vibration is relatively slow, and the 
opposite when it is relatively rapid. We may therefore 
suppose n to be so adjusted that A = 0. The circumstances 
are then very nearly those of a free vibration, and the required 
value of n is given by 

n* = Kc*/Q, (3) 

very approximately. The second component of the force (2), 
which keeps step with the velocity (q), is required to maintain 
the emission of energy outwards, which is, by 76 (15), 

(4) 



272 DYNAMICAL THEORY OF SOUND 

This must be equal to the mean value of pq, where p is the 
pressure at the outer face of the disk. Hence by comparison 
we find that p must have the form 

p = p + D cos nt -r sin nt ............. (5) 



The corresponding pressure on the inner face will be 

p = p Q + D cos nt, ..................... (6) 

simply, since no work is done, on the whole, on the air 
contained in the resonator. 

We may now invoke the action of the external source. 
If this be such as would produce the pressure 



P = P<> + ~A~ s ^ n nt .................. CO 



at the mouth of the resonator if the disk were at rest, then 
in the motion which is compounded of that due to the source 
and that due to the disk no extraneous force will be required, 
and the disk may therefore be annihilated without causing 
any appreciable change in the conditions. If </> 2 be the 
velocity-potential due to the source alone, at the mouth of 
the resonator, we must have, in this case, 



since (7) must be identical with p =p Q + p^. 

The hypothesis of a rigid disk vibrating in a cylindrical 
space was merely introduced for facility of conception, and 
is in no way essential to the argument. The disk may, if 
we please, be replaced by a flexible and extensible membrane 
enclosing the aperture of the resonator, and abutting on the 
external wall in the region of diverging waves. 

Comparing (1) with (8) we see that to a disturbing potential 
whose value at the mouth is 

< 2 = Jcos(nt-e) .................. (9) 

will correspond a vibration 






PIPES AND RESONATORS 273 

under the condition of maximum resonance, when n is given 
by (3) approximately. The corresponding flux is 

g = -g-sin(?rt-e) ................... (11) 

The emission of energy is best calculated from a con- 
sideration of the circumstances at a great distance. The 
velocity-potential will be compounded of that due to the 
original source and that due to the flux q, and under certain 
conditions the latter component may greatly preponderate. 
The emission of energy is then 

W=27rpcJ*, ..................... (12) 

approximately, by 76 (15). 

Thus if < 2 be due to a simple source A cos kct at a distance b 
from the aperture, we have 

&) ................ (13) 



Hence J=A/4nrb, 

and q = j-j sin k (ct b) ................... (14) 

This is equivalent to a source whose amplitude is to that of 
the primary source in the ratio l/kb. If b be small compared 
with X/27T this ratio is large ; and the emission of energy 
exceeds that due to the original source in the ratio l/k?b z . 
In the case of a double source B cos kct we may write, 
if kb be small, 

D 

< 2 = 7 Tg c s a cos k(ctb), ............ (15) 

by 76 (23), if a denote the angle which the axis of the source 
makes with the line drawn from it to the aperture. Hence 
/ = B cos a/4?r6 2 , and the emission, as given by (12), is 

TF = pc J B 2 cos 2 a/87r& 4 ................ (16) 

The emission due to the original (double) source alone would 
be ptfcIP/ZiTr, by 76 (26). The ratio in which the emission 
is increased is therefore 3 cos* a/fab*. Since the mean value 
of cos 2 a is J, the mean value of this ratio, for all directions 
of the axis of the double source, is 1/frb 4 . That the ratio 
L. 18 



274 DYNAMICAL THEORY OF SOUND 

should, under the given conditions, be so much greater than 
in the preceding case is due to the relatively smaller efficiency 
of a double source, as compared with a simple one, in propagating 
energy outwards ( 80). 

It may be well to insist again that the increased output 
of energy is an indirect consequence of the presence of the 
resonator, which itself does no work. The whole energy is 
supplied by the original source, where the motion takes place 
against an augmented component of pressure in the same phase 
with the velocity. The velocity-potential due to the flux q 
outwards from the resonator, as given by (11), is 

<>, = sin (nt kr e), (17) 

kr 

and the resultant pressure is 

P=PQ-\ cos (nt kr e). (1$) 

r 

In the case of a simple primary source we had J=A/4>7rb ) 
e = kb; hence, putting r = b, we find that the consequent 
pressure in the neighbourhood of this source is 

(19) 



Since the imposed outward flux is A cos nt, the mean rate of 
work against this part of the pressure is 



The output is therefore greater than it would be in the 
absence of the resonator, in the ratio cos 2kb/k 2 b 2 . This agrees 
with the former result, obtained on the hypothesis that kb is 
small. 

The energy stored in the resonator under the conditions of 
maximum vibration is, by 86 (15), 

E = 8 



This varies directly as the capacity Q, and is for apertures of 
similar form inversely proportional to the area. 

The effect of a resonator under the influence of a distant 



PIPES AND RESONATORS 275 

source in unison with it may be sufficiently illustrated on the 
assumption that the incident waves are plane. If 

fa = Jcosk(ct-x), ............... (22) 

the ratio of the energy scattered by the resonator, which is 
given by (12), to the energy-flux in the primary waves, viz. 
JpAr'cJ" 2 , is 47T/& 2 , or X 2 /7r. The energy diverted per second, 
at its maximum, is therefore equal to '318 of that which in 
the primary waves is transmitted across a square area whose 
side is the wave-length. It may be added that a similar 
law is met with in the theory of selective absorption of 
light. 

When approximate agreement between the frequency 
(n/2?r) of the source and the natural frequency (w /27r) of the 
resonator is no longer assumed, the external pressure which is 
required to maintain a steady vibration (1) through the aperture 
will consist of two parts. In the first place we have a component 
keeping step with the displacement, which is required in order 
to control the inertia of the air. This is easily found by an 
extension of the method of 86. If $1 denote the velocity- 
potential in the interior of the resonator, </> 2 that at a short 
distance outside the aperture, in the region of approximately 
spherical waves, we have 

q-Kfa-M .................. (23) 

in accordance with the electrical analogy. In the interior we 
have 5 = q/Q, c 2 s = ^, as before. Hence 

q + n<?q = -K<f> 2 , .................. (24) 

where n 2 = ^Tc 2 /Q ...................... (25) 

This gives, for the external pressure, 



(26) 



The second part, which is in the same phase as q, is needed in 
order that there may on the average be no gain or loss of energy 
to the air contained in the resonator, and is accordingly given 
by (7). Hence we have, altogether, 

-smnt); ...(27) 
182 



276 DYNAMICAL THEORY OF SOUND 

and the complete expression for the disturbing velocity -potential 
near the mouth must be 

kK \ . 

nt --^ cosnt \ ....... (28) 

In the problem as it actually presents itself the value of </> 2 
at the mouth is prescribed, say 

</> 2 = Jcos(nJ-e); ............... (29) 

and in order to identify this with (28) we must have 

TcK nC r . /- n 2 \ nC 
_._, / sin = (l-_j)^. (30) 



Hence 



This determines C in terms of J. If r denote the modulus of 
decay of free vibrations, as given by 86 (18), the formula may 
also be written 

7"2 1 ( f n 2\2 A M 2^ 

^_ 2 = J_|h _M +JL.1LI (32) 

Except in the case of approximate synchronism the second 
term within the brackets will be small compared with the 
first. Hence for a given value of J, the value of nC (which is 
the amplitude of the flux q) will be greatest when n = n , 
approximately. Moreover, for a given deviation of the ratio n/n 
from unity the intensity of the resonance falls short of the 
maximum in a greater proportion the greater the value of W O T, 
i.e. the greater the ratio of the modulus of decay to the free 
period. In other words, the smaller the damping of free 
vibrations, the more sharply defined is the pitch of maximum 
resonance. This is in accordance with the general theory 
of 13. 

The vibrations of a resonator under the influence of an 
internal source of sound are discussed in 90 with special 
reference to the theory of reed-pipes. 

89. Mode of Action of an Organ Pipe. Vibrations 
caused by Heat. 

Although the loss of energy in a single period may be small, 
the free vibrations of the column of air contained in an organ 




PIPES AND RESONATORS 277 

pipe are practically dissipated in a fraction of a second ; this is 
owing to the small inertia as compared with that of a piano- 
wire. For musical purposes some device for sustaining the 
note is required. In the ordinary " flute pipe," 
the lower part of which is shewn in section in 
Fig. 84, a thin stream of air is driven by pressure 
from a wind-chest so as to strike against the 
bevelled lip of the aperture. Under these cir- 
cumstances a very slight cause will make the jet 
pass either wholly inside or wholly outside the 
pipe. The precise mode of action is obscure, but 
there can hardly be any doubt that in its main 
features it is analogous to that of a clock-escape- 
ment. Periodic impulses are given by the jet, 
alternately inwards and outwards, to the air near 
the mouth, always in the direction in which the 
air is tending ; whilst the vibrating column itself 
mainly determines the epochs at which these impulses shall 
occur. The circumstances are accurately periodic, so that 
the driving force can be resolved by Fourier's theorem into 
a series of harmonic components whose frequencies are as 
1, 2, 3, .... The relative amplitudes with which these are 
reproduced in the vibrating column will depend on the 
closeness of their frequencies to the natural frequencies. Thus 
in a " closed " pipe, i.e. one closed at the upper end, the 
harmonics of odd order are alone excited. Again the theory 
of 87 indicates that in a sufficiently wide pipe the natural 
frequencies may deviate sensibly from the harmonic relation, 
in which case only the lower harmonics (after the funda- 
mental) will be sensible; in particular, a wide closed pipe 
gives almost a pure tone. On the other hand a pipe which is 
narrow in comparison with the length may give a note rich in 
harmonics. Indeed, if such a pipe be blown with sufficient 
force, the fundamental is not sounded at all, the period becoming 
that of the first harmonic; if the strength of the blast be 
further increased the note may jump to the next member of 
the series, and so on. An explanation is probably to be found 
in the sort of dynamical elasticity possessed by the jet. 



278 DYNAMICAL THEORY OF SOUND 

Metal pipes are richer in harmonics than wooden pipes of 
the same dimensions. This may be partly due to the greater 
fineness of the lip, which introduces a greater degree of abrupt- 
ness in the action of the jet, and so favours the amplitude of 
the terms of higher order in the Fourier series which expresses 
the driving force. Another source of the contrast in quality 
may be found in the smaller rigidity and imperfect elasticity 
of the walls of a wooden pipe, which may tend to absorb the 
energy, especially in the case of the higher harmonics. 

The " speaking " of a resonator of any kind, when a jet of 
air is blown across its mouth, is to be explained on the same 
principles. In a resonator of the usual type the normal modes 
after the first are far removed in pitch from the fundamental, 
and are not sensibly excited by the essentially periodic impulse. 
The note obtained is therefore a pure tone. 

The vibrations of a column of air may also be excited by the 
periodic application of heat, as in the well-known experiment 
of the " singing flame," where a jet of hydrogen burns within an 
open cylindrical pipe. For the maintenance of the vibration it 
is necessary that heat should be supplied at a moment of con- 
densation, or abstracted at a moment of rarefaction. To explain 
how the adjustment is effected, it would be necessary to take 
account of the fact that the vibrating system includes the gas 
contained in the supply tube of the jet, as well as the column 
of air in the pipe. The matter is thus somewhat intricate, 
but a satisfactory theory has been made out, which accounts 
clearly for the several conditions under which the experiment 
is found to succeed or to fail*. 

90. Theory of Reed-Pipes. 

The mechanism of the reed stops of the organ is quite 
different. The current of air issuing from the wind-chest is 
made intermittent by its passage through a rectangular 
aperture in a metal plate, which is periodically opened and 
closed (partially) by a vibrating metal tongue, or " reed." The 
period is accordingly determined mainly by the elasticity and 
inertia of the tongue itself. The vibrations of the latter were 

* Lord Rayleigh, Theory of Sound, 322 h. 



PIPES AND RESONATORS 279 

found by Helmholtz, by direct observation, to be of the simple- 
harmonic type, but the fluctuations in the current of air are 
necessarily of a more complex character. If the periodic 
current be expressed by a Fourier series 

C + Cicos(n -{-!)+ C 2 cos(2nt + 2 ) + ..., ...(1) 

the coefficients C z , G 3 , ... are usually by no means insensible 
as compared with C lt and accordingly if the sound is heard 
directly it has a very harsh and nasal character. In practice, 
the reed is fitted with a suitable resonator, or " sound-pipe," 
which specially reinforces one or more of the lower elements in 
the harmonic series (1). 

For the purposes of mathematical treatment we may 
idealize the question somewhat, and imagine that at a given 
point in the interior of the resonator we have a simple source 
of the type corresponding to one of the terms in (1). It 
appears from the elementary theory of 62 that in the case of 
a cylindrical pipe, with the source at one end, the frequencies 
of maximum resonance are very approximately those of the 
free vibrations when that end is closed. Hence a reed fitted 
with a cylindrical sound-pipe of suitable length will emit 
a series of tones whose frequencies are proportional to the odd 
integers 1, 3, 5, .... In a conical pipe, on the other hand, with 
the source near the vertex, we have the complete series of 
harmonics with frequencies proportional to 1, 2, 3, 4,... (see 
84). But in either case the harmonics of high order are 
discouraged by the increasing deviation of the frequencies of 
maximum resonance from the harmonic relation which neces- 
sarily holds in the expression for the essentially periodic current 
of air. 

As the question is instructive in various ways it may be 
worth while to examine more in detail the case of a cylindrical 
sound -pipe (of any form of section), applying the correction 
for the open end, and allowing for the dissipation due to the 
escape of sound outwards. The plan of the investigation is 
similar to that of 87, the difference being that we now have 
a source Ce int (say) at the end x = I. For simplicity we will 



280 DYNAMICAL THEOKY OF SOUND 

assume this source to be distributed uniformly over the cross- 
section, so that 



Let us suppose for a moment that we have a flux 

2 = A cos nt ........................ (3) 

outwards from the mouth. The pressure at will consist of 
two components. We have first the part necessary to control 
the inertia of the 'air near the mouth ; the corresponding part 
of the velocity-potential just inside is 

6 - q = cos nt, .................. (4) 

CO * ft) 

where a has the same meaning as in 87. Next we have the 
part which is effective in generating diverging waves outside. 
On the principles of 88 this is found to be 

7 A cos nt, ..................... (5) 

kA 
corresponding to 4>-^r~ sm nt > ..................... (6) 



since k = n/c. The total velocity-potential at 0, corresponding 
to (3), is therefore 

............. (7) 



Generalizing this, we may say that to a flux 

q = Ae int ........................ (8) 



corresponds <f> = o - - e int , . ...(9) 

\ 47T/ ft) 

the expression (7) being in fact the real part of (9), when A 
is real. The correspondence will hold even if A be complex, 
since this is merely equivalent to a change in the origin of t. 
We now assume, for the region of plane waves, 

<t>=r~ {B cos k (I - x) - C sin k (I - x)}e int , . . .(10) 



PIPES AND RESONATORS 281 

where the constants have been adjusted so as to satisfy (2). 
Comparing with (8) and (9) we find 



j, -vn- *. 

Hence 

B = A {sin -H ( ka - - ) cos kl\ , 

(12) 

!/ ik 2 (o\ ) 

cos & ( &a ) sin &Z [ . 
V 47T/ J / 

The latter equation gives A in terms of C. Considering only 
absolute values we have 

= (cos kl - ka sin kVf + (^)* sin2 kl - -( 13 ) 

Since fcco is usually a small fraction, the emission of energy, 
which varies as \A*\, will be greatest for a given source (7cosn 
when 

cos kl = kassmkl, (14) 

nearly, i.e. when the imposed frequency approximates to that 
of one of the normal modes of the pipe when closed at x = I, 
as determined by 87 (11). In the case of the reed-pipe, 
therefore, the tones which are specially reinforced consist of 
the fundamental and the harmonics of odd order. 
When (14) is satisfied, we have by (12) 

C 



This determines the relation between the flux outwards at the 
mouth, and that constituting the source. The former now greatly 
exceeds the latter in amplitude, and the factor i shews that it 
differs in phase by a quarter-period. 
Again, from (10) and (11) we have 

</> = T- \ sin kx + ka cos kx cos kx ! e int . . . .(16) 



282 DYNAMICAL THEOEY OF SOUND 

When (14) holds, this reduces to 

x) -r sin kl cos kx e int 



'kcoSmkir 47T 

{ l"i 



,*"? 011 \ cos A; (Z - a) - V- sin M cos ^ r e * n '- (17) 
) 2 sm 2 ^[ y 4?r j 



The real part gives 

<f> = = -T =-; -^ cos k(l x) sin w^ : sin &Z cos ^a? cos w^ [ , 
A^<w 2 sm 2 A;Z ( 4?r j 

(18) 

corresponding to a source Ccosnt. The variable part of the 
pressure at the source (x = I) is 

-r sin H cos 



The first part of this is by far the more considerable ; it is, 
moreover, the only part which is effective in doing work. The 
mean rate of work done at the source, i.e. the mean value of 
pCcosnt, is 

ZTTpcC* 

"Jfc"Bin"' 

It may easily be verified that this is equal to the work spent in 
generating waves at the mouth, where, by (15), 

tf ................ (21) 



7 

k*o> sin kl 

It appears further from (19) that the maximum of pres- 
sure at xl synchronises almost with the maximum influx 
of air, following it however by a short interval. There is 
therefore a tendency slightly to lower the pitch of the reed, 
which is, in the instruments here referred to, of the " in- 
beating type," i.e. the passage is opened when the reed swings 
inwards, towards the wind -chest. The fact that the resultant 
force on the reed is approximately in the same phase with the 
displacement indicates that the reed is vibrating with a 
frequency somewhat less than that natural to it ( 12). 



PIPES AND BESONATOBS 283 

The reed-stops of an organ fall in pitch as the temperature 
rises, owing to the diminished elasticity of the metal tongues ; 
this is the opposite of what happens with regard to the flute- 
pipes ( 62). A reed-pipe is tuned by a contrivance which 
alters the effective length of the vibrating tongue. 

It should be mentioned that there is another class of 
instruments in which the " reed " has a much smaller elasticity 
and is mainly controlled by the reaction of the resonant 
chamber, its own natural frequency being relatively low. The 
reed is then of the "out-beating type," the aperture being 
widest when the reed swings outwards, i.e. with the wind. 
The human larynx comes essentially under this class. 



CHAPTER X 

PHYSIOLOGICAL ACOUSTICS 

91. Analysis of Sound Sensations. Musical Notes. 

The vibrations of elastic bodies and the propagation of 
waves through the atmosphere are subject to well-ascertained 
mechanical laws, and the inferences drawn from these can be 
controlled by more or less decisive experiments. But when 
we approach the field where the human mechanism comes into 
play, we are met by the peculiar difficulties which are inherent 
in the observation and study of subjective phenomena. In 
particular, when we endeavour to analyse a familiar complex 
sensation into its elements, we are attempting a task for which 
the experience of daily life has peculiarly unfitted us. Thus 
we may have been accustomed to interpret the sensation in 
question as indicating the presence of a particular object, or 
the occurrence of a particular kind of event, in a particular 
place. The elements of which it is made up give individually 
little or no information; it is the combination which is significant, 
and attention to the details would only distract from what is of 
immediate practical interest. To use a rough and indeed an 
utterly inadequate illustration, it is as if we were to insist upon 
spelling every word we read. 

The theory of sense-perception, especially in relation to 
optics and acoustics, is a fascinating subject, but it cannot be 
dealt with here. The student who is unversed in it may be 
referred to the writings of Helmholtz*. 

* The theory is explained in its acoustical bearings in the Tonempfindungen, 
already cited (p. 3). It is also discussed from the optical point of view in his 
Handbuch der physiologischen Optik, 2nd ed., Hamburg and Leipzig, 1896. 
Elementary expositions will be found in the two volumes of his Vortrage und 
Reden, Brunswick, 1884, of which there is an English translation by E. Atkinson, 
with the title: Popular Lectures on Scientific Subjects, 2nd ed., London, 1893. 



PHYSIOLOGICAL ACOUSTICS 285 

There are one or two questions, however, relating principally 
to Ohm's Law ( 1), to which some reference is necessary. The 
first point on which the student should satisfy himself is that 
the various simple-harmonic vibrations which are as a rule 
combined in the production of a musical note are really 
represented by independent elements in the resulting sensation ; 
that the latter can in fact be resolved into a fundamental tone 
and a series of harmonics. For this a slight course of education 
is necessary. A series of resonators of the type shewn in 
Fig. 80, p. 261, tuned to the overtones which it is desired to 
detect, are of great service for this purpose*. But such 
assistance is not indispensable, and a good deal can be effected 
with the piano or monochord. Take for instance the note c, 
whose harmonics are c', g f , c", e", g ', If on the piano one 
of these, say g' } be gently sounded, and the key then released, 
so that the vibration is stopped, and if immediately afterwards 
the note c be struck with full intensity, it is not difficult to 
recognize in the compound sensation the presence of the 
element previously heard. This is often more perceptible as 
the sound dies away, the overtones being apparently extinguished 
more slowly than the fundamental. A more immediately 
convincing series of experiments can be made with the 
monochord, or with a piano whose strings are horizontal and 
therefore easily accessible. If a string be set into vibration 
whilst damped at a nodal point of one of the harmonics by 
contact with a hair-pencil, the fundamental tone and all the 
harmonics of lower rank may be reduced in intensity or 
altogether extinguished, according to the degree and duration 
of the pressure applied. In this way a whole series of types 
of vibration can be produced in which the harmonic in question 
is accompanied by a varying admixture of the fundamental, &c. 
The occurrence throughout of the corresponding sensation as 
an independent element in the resulting sound is in this way 
easily appreciated. The piano also lends itself readily to the 

* It may be noted that the external ear-cavity is itself a resonator, 
responding most intensely to a certain tone, which varies for different 
individuals but is usually in the neighbourhood of ff* g ir . The aperture 
being relatively large, the damping and consequently the range of resonance 
is considerable. 



286 DYNAMICAL THE OK Y OF SOUND 

analysis of compound notes by resonance. If the string c', for 
example, be freed from its damper by holding down its key, 
whilst c is sounded for a moment, the harmonic c is taken up 
and continued by the first-mentioned string. If on the other 
hand, the string c be free from its damper, whilst c is sounded 
for a moment, the tone c' is taken up as a harmonic of the 
lower string. These simple experiments, which (with others) 
are recommended by Helmholtz, can of course be varied in 
many ways. Again, when the ear has learnt to distinguish the 
partial tones in a complex note, it is easy to note the absence 
of a particular tone of the series when the corresponding simple- 
harmonic vibration is not excited. For instance, when a string 
is struck at its middle point, the harmonics of even order are 
wanting ( 26). 

92. Influence of Overtones on Quality. 

The quality of a musical note is determined ( 2) by the 
number and relative intensities of the various tones which 
compose it. The kind of influence which overtones of different 
ranks exercise on the quality is summarised by Helmholtz, 
somewhat as follows: 

1. Pure tones like those of tuning forks with resonance 
boxes, or of wide stopped organ pipes, are soft and pleasing, 
smooth, but wanting in power. 

2. Notes which contain a series of overtones up to the 
fifth or sixth in rank are richer and more musical, and are 
perfectly smooth so long as no higher overtones are sensible. 
The notes of the piano and of open organ pipes are examples, 
whilst those of the flute, and of the flute-stops on the organ 
when softly played, approximate more to the character of pure 
tones. In the "mixture" stops of the organ the lower 
harmonics are expressly provided in greater intensity by 
auxiliary pipes which are played automatically along with that 
which gives its name to the note. 

3. When the harmonics of even order are absent, as in the 
case of a stopped organ pipe, or a piano string struck at the 
middle point, the note has a hollow, and even a nasal character, 
if the odd harmonics are numerous. 



PHYSIOLOGICAL ACOUSTICS 287 

4. The sound may farther be described as "full," if the 
fundamental tone be predominant, and as "empty" if it be 
relatively feeble. This is exemplified in the difference of 
quality between the sound of a piano-wire when struck with 
a soft or a hard hammer, respectively ( 26, 38). 

5. When harmonics beyond the sixth or seventh are 
present in considerable intensity, the sound is harsh and 
rough, owing to the discords which these higher overtones 
make with one another. If, however, the higher harmonics, 
though present, are relatively weak, as in the case of the 
stringed instruments of the orchestra, reed-pipes, and the 
human voice, they are useful as giving character and expression 
to the sound. Brass instruments, on the other hand, with 
their long series of powerful overtones, are as a rule only 
tolerable in combination with others, or for the sake of 
particular effects. 

The analysis of the sounds of the human voice is naturally 
a more difficult matter. In particular, the constitution of the 
vowel sounds has been much debated, without any very definite 
conclusion. The same vowel may be sung on a wide range of 
notes, but preserves its peculiar character throughout ; and the 
question arises, does this special quality depend solely on the 
relative intensities of the various partial tones, or on the 
predominance of one or more overtones of, or near to, a 
particular pitch ? It will be remembered that the vibration of 
the larynx is periodic, and that particular harmonics may be 
reinforced by the resonance of the mouth-cavity, as in the case 
of a reed-pipe ( 90). The balance of authority appears to 
incline, though not very decisively, to the " fixed-pitch " theory, 
which is the second of the two alternatives above stated. A 
review of the subject down to the year 1896 will be found in 
the concluding chapter of Lord Rayleigh's treatise. 

93. Interference of Pure Tones. Influence on the 
Definition of Intervals. 

It has so far been assumed that the sensations due to 
two coexistent simple-harmonic vibrations are produced quite 
independently of one another. This appears to be in fact the 



288 DYNAMICAL THEOBY OF SOUND 

case when the interval between the two tones is sufficiently 
great ; but when the interval is small we have " interference," 
as we should expect from the analysis of 10, and the sensation 
is in whole or in part intermittent. The phenomenon of" beats" 
hardly needs description ; it is often met with in mistuned pairs 
of piano-wires, in the vibrations of finger-bowls, and so on. For 
methodical study two pure tones are required of equal intensity, 
as e.g. from two tuning forks (with resonators), or two stopped 
organ pipes, which can be made to differ in pitch by a variable 
amount. As unison is departed from, the beats (whose 
frequency is always equal to the difference of the frequencies 
of the primary tones) are at first slow and easily counted. As 
the interval widens they become more rapid, and a sensation of 
roughness or discord is experienced ; moreover, the primary 
tones are now heard along with the beats. Finally, as the 
interval is continually increased, the beats and the consequent 
roughness gradually cease to be perceptible. 

The intervals at which roughness begins and ceases, vary in 
different parts of the scale. For the same interval the rough- 
ness is less, the higher the pitch ; on the other hand for a given 
number of beats per second the roughness is greater in the 
higher octaves. 

In the case of two (or more) compound musical notes, we 
may have beats and eventual roughness between any constituent 
tones which are sufficiently near in the scale. We may even 
have interference between the higher overtones of the same 
note ; and it is for this reason that harmonics of higher order 
than the sixth are prejudicial to good musical quality. 

It is through the interference of pairs of overtones that 
deviations from the consonant intervals ( 3) usually make 
themselves felt. Thus in the case of the Octave cc' we have 
tones with the frequencies 

c = 132, 264, 396, 528, 660, 792, ..., 
c' = 264, 528, 792, ..., 

and if this be mistuned all the harmonics of c' are interfered 
with by the even harmonics of c. 



PHYSIOLOGICAL ACOUSTICS 289 

In the case of the Fifth eg we have 

c = 132, 264, 396, 528, 660, 792, ..., 

g= 198, 396, 594, 792, ..., 

and if this be mistuned the second tone of g beats with the 
third tone of c, and so on. When the ratio of the vibration 
numbers of the fundamentals is less simple, the harmonics 
which can interfere are of higher order. Thus in the case of 
the Major Third, where the ratio is 4:5, the first pair of 
interfering overtones consists of the fifth tone of the lower note, 
and the fourth of the higher. Since in many musical instruments 
the fifth tone is very feeble, this consonance is less well defined 
than the preceding ones. On the other hand the fundamentals 
may fall, in the lower parts of the scale, within beating distance 
(for example c= 132, e 165), so that this consonance is to be 
reckoned also as less perfect than the former ones. Similar 
remarks apply with greater force to such cases as the Minor 
Third (5 : 6) and the Minor Sixth (5 : 8). 

94. Helmholtz Theory of Audition. 

The connection between primary sensations and simple- 
harmonic vibrations has still to be accounted for. The problem 
is a physiological one; but the theory which Helmholtz has 
framed to explain Ohm's law, so far as it holds, and the various 
deviations from it, is in its essentials so simple, and is so 
successful in binding together the facts of audition into a 
coherent system, that a brief statement of it may be attempted. 

In its simplest form the theory postulates the existence, 
somewhere in the internal ear, of a series of structures each 
of which has a natural period of vibration, and is connected 
with a distinct nerve-ending. For brevity we will speak of 
these structures as "resonators," since that is their proper 
function. A particular resonator is excited whenever a 
vibration of suitable frequency impinges on the ear; the 
appropriate nerve is stimulated ; and the sensation is com- 
municated to the brain. In this way the resolution of a 
musical note into its constituent tones is at once accounted 
for. 

It is necessary to suppose that the resonators are subject 

L. 19 



290 



DYNAMICAL THEOEY OF SOUND 



to a considerable amount of damping. If it were not so, 
each resonator would go on vibrating, and the corresponding 
sensation would persist, for an appreciable time after the 
exciting cause had ceased. A similar interval of time would 
elapse before the sensation reached its full intensity when the 
cause first sets in. The effect would be that the sensations 
due to a sufficiently rapid succession of distinct notes would 
not be altogether detached from one another in point of time. 
From considerations of this kind Helmholtz estimated that 
the degree of damping must be such that the intensity (as 
measured by the energy) of a free vibration would sink to 
one-tenth of its initial value in about ten complete vibrations. 
It follows, as explained in 13, that each resonator will 
respond to a certain range of frequencies on each side of 
the one which has maximum effect. It is assumed, further, 
that the difference of pitch of adjacent resonators is so small 
that the same simple-harmonic vibration will excite a whole 
group, the intensity falling off from the centre on either side. 




This is illustrated by the annexed figure, repeated from 13, 
which may now serve to exhibit the distribution of intensity 
over a continuous series of resonators under the influence of 
a given simple-harmonic vibration. The abscissa is p/n 1, 
where p is now taken to represent the natural frequency of a 
resonator, and n that of the imposed vibration. The horizontal 
scale depends on the value of fi, or 1/rw, where r is the 



PHYSIOLOGICAL ACOUSTICS 291 

modulus of decay of a free vibration. On the above estimate 
of Helmholtz we shall have 



whence = '018. The intensity is therefore one-half the 
maximum for 

2 = 1+ -018. 

It will be observed that on the above view we ought in 
strictness to speak of "simplest" rather than of "simple" 
sensations of sound, absolutely simple sensations, in the strict 
physiological meaning, being impossible to excite. 

When two simple-harmonic vibrations, sufficiently far apart 
in the scale, are in operation, the two groups of resonators 
which are affected will be practically independent, and the 
two sensations (of pure tones) will coexist. But when the 
interval between the frequencies is sufficiently small, the two 
groups will overlap, and the energy of vibration of those 
resonators which are common to them will fluctuate in the 
manner explained in 10. The excitation of the corresponding 
nerve-endings will therefore be intermittent, with a frequency 
equal to the difference of those of the two originating vibrations. 
This is, on the theory, the explanation of beats. As the interval 
is increased, the beats become more rapid. The " roughness " 
which is ultimately perceived, in spite of the diminishing 
amplitude of the fluctuations, has a more remote physiological 
explanation. According to Helmholtz, there is here an analogy 
with the painful effect produced by a flickering light, and in 
other cases where a nerve is stimulated repeatedly at intervals 
of time which are neither too great nor too small. When the 
intervals are sufficiently long, the nerve has time to recover 
its initial sensibility, and so experiences the full effect of each 
recurring stimulus. When on the other hand the intervals 
are sufficiently short, the sensation tends to become continuous. 
It is for this reason that beats exceeding, say, 132 per second 
cease to produce the sensation of roughness, even although 
the interval between the beating tones be such as would be 
perceptibly discordant in a lower part of the scale. 

192 



292 DYNAMICAL THEOKY OF SOUND 

The student of dynamics cannot fail to admire the beauty 
of a theory which lends itself readily to the explanation of so 
many complicated relations; but it is with the physiologist 
and the anatomist that in the last resort it lies to decide 
whether a mechanism of the kind postulated is really to be 
found in the internal ear. In the original form of the theory 
(1862) the resonators were identified with the structures known 
as "Corti's rods," which are found arranged, some 3000 in 
number, along the basilar membrane in the spiral cavity 
of the cochlea. A disturbing discovery by Basse that these 
structures do not occur in the ears of birds, to whom we can 
hardly deny the perception of pitch, led to a modified form 
of the theory. In the third edition of the Tonempfindungen 
(1870) Helmholtz propounded the view that the resonators 
consist of the various parts of the basilar membrane itself. 
This membrane varies in breadth from one end to another, 
like a very acute-angled triangle, and the tension appears to 
be very much less in the direction of length than in that 
of breadth. On this view the different parts could be set 
into sympathetic vibration, much as in the case of a series 
of strings of variable length placed side by side, except that 
the independence of adjacent parts would be approximate 
instead of absolute. For a full description of the complicated 
structure of the internal ear, and for further speculations as 
to the functions performed by its various parts, we must refer 
to books on physiology. 

95. Combination-Tones. 

In one important respect the theory as so far developed 
is inadequate. The explanation of consonant intervals outlined 
in 93 assumes that one at least, and generally both, of the 
notes concerned is complex, and contains one or more overtones 
in addition to the fundamental. It was in fact through the 
interference of two tones, one at least of which is an overtone, 
that departure from the exact relation of pitch was stated to 
make itself manifest. When both tones are pure this means 
of definition is wanting, and on the theory of audition sketched 
in the preceding section there appears to be no reason why 



PHYSIOLOGICAL ACOUSTICS 293 

the octave (for example) should be distinguished by any 
character of smoothness from adjacent intervals on either 
side, the two groups of sensations being in any case quite 
independent. Since the more consonant intervals at all events 
are as a matter of fact easily recognized by the ear, even in 
the case of apparently pure tones, and are thoroughly well 
defined, the difficulty is a serious one. To meet it, Helmholtz 
developed his theory of "combination-tones," which are assumed 
to supply the function of the missing overtones. 

In most of our investigations it has been assumed that the 
amplitude of the vibrations may be treated as infinitely small, 
so that disturbances due to different sources may be super- 
posed by mere addition. In the theory now in question this 
assumption is abandoned ; the vibrations are regarded as 
small, but not as infinitely small, and the interaction of the 
disturbances due to different causes is, to a certain degree of 
approximation, investigated. 

We have already had an indication in 63 of the manner 
in which two imposed simple-harmonic disturbing forces of 
small but finite amplitude, with frequencies N lf N 2 respectively, 
may generate in the air other simple-harmonic vibrations 
whose frequencies are 

2N lt 2# 2 , Ni-N,, Ni + N*, 

and whose amplitudes involve the squares or product of the 
amplitudes of the two primaries. If the approximation were 
continued we should meet with further vibrations whose fre- 
quencies are of the type p l N l piN^ where p lt p 2 are integers. 
In acoustical language, two simple-harmonic vibrations can, if 
of sufficient intensity, give rise not only to the pure tones 
usually associated with them, but also to a series of other pure 
tones of higher order. The fact that a single harmonic vibration 
can by itself give rise to a pure tone together with its octave, &c. 
is itself of some importance, but the most interesting result is 
due to the interaction, viz. the " difference-tone " (N-^ N 9 ). 
The existence of difference-tones was observed, apart from 
all theory, by Sorge (1745) and Tartini (1754). The "sum- 
mation-tone " (Ni + N 2 ) is more difficult to hear, and its 



294 DYNAMICAL THEORY OF SOUND 

existence has even been denied. It has however been objectively 
demonstrated by Riicker and Edser*, by its effect on a tuning 
fork of the same frequency. 

Difference-tones due to the causes just considered are most 
easily perceptible where we have a mass of air which is subject 
to the joint and vigorous action of the primary vibrations, as 
in the harmonium and the siren; they can then, like other 
tones, be reinforced by suitable resonators. 

There is however a way in which combination tones may 
conceivably be originated in the ear itself. To explain this 
it is necessary briefly to consider the forced vibrations of an 
unsymmetrical system. When a particle, or any system having 
virtually one degree of freedom, receives a displacement x, the 
force (intrinsic to the system) which tends to restore equilibrium 
is a function of x, and may be supposed expressed, for small 
values of x, by a series 

An example is furnished by the common pendulum, where 
the force of restitution is proportional to g sin 6, or 



but here, on account of the symmetry with respect to the 

vertical, the force changes sign with 6, so that 

only odd powers of 6 occur. The correction for 

small finite amplitudes depends therefore on the 

term of the third order in 6. But if the system 

be unsymmetrical, as in the case of a pendulum 

hanging from the circumference of a horizontal 

cylinderf, the term of the second order comes in, 

and the correction is more important. Helmholtz Fi 

lays stress on the fact that in the slightly 

* Phil. Mag. (5), vol. xxxix. (1895). 

f If a be the radius, and I the length of the free portion of the string when 
vertical, the potential energy is 







where .9 is the arc described by the bob from the lowest position. The restoring 
force is therefore 

dV_mg a lmga^ t 



PHYSIOLOGICAL ACOUSTICS 295 

funnel-shaped tympanic membrane and its connections we 
have precisely such an unsymmetrical system, the restoring 
force being somewhat greater for inward than for outward 
displacements of the same extent*. If we keep only the first 
two terms in (1), the equation of motion is of the type 

x+/juc = -auc*+X, .................. (2) 

where X represents the disturbing force f. The joint action 
of two simple-harmonic forces will be represented by 

X =f 1 cosn l t+f z cosnj ................ (3) 

Neglecting, for a first approximation, the square of #, we have 

x = l .. cos nj -\ -- - cos n 2 t, ......... (4) 

/n-n,* fjL-nJ 

the terms which represent the free vibrations being omitted, 
since these are rapidly destroyed by dissipation. If we sub- 
stitute this value of x on the right hand of (2), and write for 
shortness 

AAfc -%*)** / 2 /(/*-^ 2 )=# 2 , ......... (5) 

we obtain the differential equation 

x + x = X - a + 2 2 - af cos 2nJ - a. 2 cos 



- <*9i92 cos (HJ - O t - ag,g z cos fa + w 2 ) t, . . .(6) 
correct to the second order of /i,/ 2 - The terms written in full 
on the right hand may be regarded as a correction to the 
disturbing force X. The solution of (6) gives, in addition to 
(4), the terms 



The first term merely indicates a shift of the mean position 

* It may be noted that the same element of asymmetry is present in the 
investigation of 63. When we proceed to the second order of small quantities, 
the changes of pressure due to condensations s are no longer equal in 
amount. 

f It is unnecessary to take account of the variability of inertia, since this 
can be got rid of by a proper choice of the coordinate x. In any case 
it will not alter the general character of the results obtained in the second 
approximation. 



296 DYNAMICAL THEOEY OF SOUND 

about which the oscillations take place. For the rest, we have 
octaves of the primary tones, together with a difference- and 
a summation-tone. If the approximation were continued we 
should obtain combination-tones of higher order, as in the 
former case. 

When, as in the case of the tympanic membrane, the 
free period 2?r/\//A is relatively long, the most important 
combination- tone is the difference-tone (n^ n 2 ), on account of 
the relative smallness of the corresponding denominator in (7). 

The theory of combination-tones here reproduced has not 
been accepted without question. The difference-tones, as 
already mentioned, were known as a fact since the time of 
Tartini, and a plausible explanation had been given by 
Thomas Young (1800). According to this view the beats 
between the two tones, as the interval increases, ultimately 
blend, as if they were so many separate impulses, into 
a continuous tone having the frequency of the beats. The 
difficulty of this explanation is that the actual impulses 
during a beat are as much positive as negative, so that 
it does not appear how any appreciable residual effect in 
either direction could be produced, if the vibrating system 
be symmetrical. It is true that if we turn to the figure on 
p. 23, it is apparently periodic, with the period of the in- 
termittence ; but from the point of view of Fourier's theorem 
the lower harmonics are all wanting, and the only two which are 
present are precisely the two which are used in constructing the 
figure. On the Helmholtz theory of audition the intermittent 
excitation of a particular resonator m times a second is a wholly 
different phenomenon from the excitation of an altogether 
distinct resonator whose natural frequency is m. Young's 
view appears indeed to be inadmissible on any dynamical 
theory of audition, at least in the case of infinitely small 
vibrations. On the other hand it is true, as we have seen, 
that given a finite amplitude, and an unsymmetrical system, 
a vibration of the type shewn in Fig. 10, p. 23, does actually 
generate (among others) a vibration whose period corresponds 
to the fluctuations there shewn. The distinction between the 
two theories might therefore, from a merely practical point of 



PHYSIOLOGICAL ACOUSTICS 297 

view, be held to be almost verbal, were it not that Young's 
theory fails to give an explanation of combination-tones other 
than the first difference-tone. 

96. Influence of Combination-Tones on Musical In- 
tervals. 

A brief indication of the way in which combination-tones 
may assist in defining the consonant intervals is all that can be 
attempted here. Take first the case of (primarily) pure tones. 
In the case of a slightly mistuned Octave, say ^ = 100, 
N 9 = 201, we have N z N 1 = 101, which gives a difference-tone 
making 1 beat per second with N l9 

For the Fifth, let N, = 200, N 2 =301. We have 



giving combination-tones with 2 beats per second. 
For the Fourth, let ^ = 300, N 2 = 401. Then 



2^-^=199, 2^-2^ 

and the corresponding tones make 3 beats per second. 

For the Major Third, let ^=400, JV 2 = 501. We have 
2^-2^ = 202, 3^-2^=198, giving 4 beats per second. 

We might proceed further in the list, but it will already 
have been remarked that combination-tones of increasingly 
high order are being invoked. This is quite in conformity 
with the observed fact that the beats are, in all cases after the 
octave, very faint unless the primaries be especially vigorous. 

A more effective part is played by the combination-tones 
when the notes concerned have one or two overtones, but not a 
sufficient range of them to account for the definition on the 
principles of 93. Take for instance the case of the Fifth, 
when each note has a first harmonic in addition to the 
fundamental. If the interval be slightly mistuned, we have say 
the primary tones: 200, 400; 301, 602. These give the two 
difference-tones 301 - 200 = 101, 400 - 301 = 99, which inter- 
fere with one another. 

The combination- tones have an influence again, in the case 



298 DYNAMICAL THEOKY OF SOUND 

of consonant triads, especially of simple tones, but enough 
has been said to shew their importance from the musical point 
of view. For further developments reference must be made to 
the work of Helmholtz*. 

97. Perception of Direction of Sound. 

One important question of physiological acoustics in which 
dynamical principles are involved remains to be mentioned. 
An observer, even when blindfolded, and with no adventitious 
circumstances to guide him, is in general able to indicate with 
great accuracy the direction from which a sound proceeds. In 
the case of pure tones the discrimination between back and 
front is indeed lost, as was to be expected, considering the 
symmetry with respect to the medial plane of the head, but 
right and left are clearly distinguished. For tones of small 
wave-length this may be accounted for by the difference of 
intensity of the sensation in the two ears, since the head acts 
to some extent as a screen, as regards the further ear. But 
when the wave-length of the sound much exceeds the peri- 
meter of the head the investigation given near the end of 81 
shews that this difference must be very slight. According to 
the most recent investigations of Lord Rayleigh^, the in- 
terpretation depends on the relative phase of the sounds as they 
reach the two ears, a difference of even a fraction of a period 
being effective. He found that if the same tone be led by 
different channels to the two ears, and all extraneous dis- 
turbances be excluded, the sound can be made to appear to 
come from the right or left at will, by adjusting the relative 
phase. The origin of the sound was always attributed to that 
side on which the phase is in advance (by less than half a 
period). The result, which has been arrived at independently 
by other observers, is at present unexplained. It has been 
suggested that the phenomena may really be due to a differ- 
ence of intensity. A fraction of the sound may be transmitted 
from each side to the opposite internal ear, through the bones of 

* See also Sedley Taylor, Sound and Music, London, 1873. 
t Phil. Mag. (6), vol. xm. (1907). 



PHYSIOLOGICAL ACOUSTICS 299 

the head, in which case the original difference of phase would 
produce a slight difference of intensity on the two sides owing 
to interference between the direct and transmitted vibrations*. 
It is impossible to suppose, however, that this difference could 
be other than exceedingly minute. 

* Myers and Wilson, Proc. Roy. Soc. vol. LXXX. A, p. 260 (1908). This 
hypothesis is discussed by Lord Kayleigh, Proc. Roy. Soc. vol. Lxxxm. A, p. 61 
(1909). 



INDEX 



[The numerals refer to the pages] 



Absorption of sound, 196 

Adiabatic lines, 158 

Air-waves, general theory of, 204 

see also Sound waves 
Amplitude, minimum audible, 167 
Analysis of sound sensations, 2, 284 
Anticlastic curvature of a flat bar, 152 
Approximate solution of period- 
equations, 83, 125, 126, 128, 256, 
258 
Audibility, range of frequency for, 3 

least amplitude for, 167 
Audition, Helmholtz theory of, 289 

Bars, longitudinal vibrations, 114 
flexural vibrations, 120, 130 

Beats, 23, 132, 138 
relation of, to dissonance, 288 

Bells, 155 

Bessel's functions, 85, 145, 147, 259 

Blackburn's pendulum, 35 

Chain, vibrations of hanging, 84 
Circular vibrations, 55 
' Circulation ' denned, 203 * 
Clamped-free bar, transverse vibrations 

of a, 127 

Combination-tones, 181, 292, 294| 
Communication of vibrations to a gas, 

237 

' Condensation ' defined, 160 
Conduction of heat, effect of, on sound 

waves, 187 

' Conductivity ' of an aperture, 245 
Conical pipe, normal modes of a, 257 
Consonant intervals, 3, 283, 297 
Cosine-series, 92 
Curved shells, vibrations of, 155 
Cylindrical vessel, normal modes of a, 

259 

Damping of vibrations, 25, 27, 57 
effect of, on resonance, 32 
of air- waves by viscosity, 185, 186, 

190 

of a resonator, 265 
of an organ pipe, 269 



Degrees of freedom of a dynamical 

system, 12, 14 
Diatonic scale, 5 

Diffraction of sound, 240, 244, 248 
'Dilatation' defined, 107 
Direction of sound, perception of, 

298 

Discontinuity, waves of, 181 
Dissipation of energy by friction, 27, 

184 
Dissipation (apparent), by generation 

of air-waves, 166, 225, 227, 232, 264, 

269 

'Divergence' defined, 199 
Doppler's principle, 221 
Double pendulum, 38 
' Double source ' of sound, 215, 226 

Elasticity, elementary theory of, 106 

coefficients of, 110, 113 

of gases, 159 
Elliptic vibrations, 49 
Emission of energy, by a simple source, 
225 

by a double source, 227 

by a resonator, 265 

by an open pipe, 269 
Energy, of a simple-harmonic vibra- 
tion, 15 

of a string, 60 

of an elastic solid, 114 

of a bar, 123 

of a membrane, 141 

of a bent plate, 151 

of air-waves, 163, 205 
'Extension' defined, 107 
Extensional vibrations of a rod, 114 

of a circular ring, 136 

Finite amplitude, air-waves of, 174 
Flexure, uniform, of a bar, 121 

of a plate, 150 
Flexural vibrations, of a bar, 12Q 

of a ring, 136 

of a plate, 153 
'Flux,' defined, 199 
< Flux of energy, '165, 224 



302 



INDEX 



[The numerals refer to the pages'] 



Forced oscillations, 16, 20, 47 
effect of friction on, 28, 57, 104 

Fork, tuning, 132 

Fourier's theorem, 87, 92 

influence of discontinuities in, 92 
law of convergence of coefficients in, 
94 

Freedom, degrees of, 12, 34 

Free- free bar, transverse vibrations of 
a, 124 

Free oscillations, 12 
with friction, 24 
general theory of, 44 

Frequency, range of, for audibility, 3 

Friction. See Dissipation 

Gas, elasticity of a, 159 

isothermal and adiabatic lines of a, 

157, 158 
Graphical solution of period-equations, 

83, 125, 126, 128, 256, 258 
Grating, transmission of sound by a, 

247 

Harmonic analysis, 101 
Harmonics, 5 

Heat, vibrations caused by, 278 
Heat-conduction, effect of, on sound 

waves, 187 

Hooke's law of elasticity, 11, 110 
Huygens' principle, 249 

Imaginaries, use of, 53 

Impact, vibrations of a string due to, 

73, 99 

Indicator diagram, 157 
Inertia, coefficients of, 42 
' Irrotational ' motion defined, 203 
Interference of simple-harmonic vibra- 
tions, 23 

of pure tones, 287 
Intervals, musical, 5, 288 

degree of definition of, 297 
Isothermal lines, 157 

Laplace's equation, 205 

Leslie's experiment, 237 

Lines of motion, 229 

Lissajous' figures, 49 

Loaded string, normal modes of a, 36, 

37,82 
Local periodic force, effect of, in a 

gaseous medium, 233 
Longitudinal vibrations, of bars, 
114 

of columns of air, 170, 266 
Loops, on a vibrating string, 70 

in a pipe, 171 



Membrane, transverse vibrations of a, 
139 

normal modes of a rectangular, 
142 

of a circular, 144 
Mersenne's laws, 70 
'Modulus of decay,' defined, 25 

of air-waves, 185 

of a vibrating sphere, 233 

of a resonator, 265 

of a pipe, 270 
Modulus, Young's, 111 
Multiple system, equations of motion 
of a, 41, 44 

normal modes of a, 44 

forced vibrations of a, 47 

Nodal lines, of a membrane, 143, 144, 
148 

of a plate, 153, 154 
Nodes, in a vibrating string, 70 

in a bar, 116, 127 

in a pipe, 171 

' Normal functions,' 101, 130 
Normal modes of vibration, 44 
Notes, musical, 3, 285 

Ohm's law, 2, 285, 289 

Organ pipe, normal modes of, 171 

corrected theory of, 266 

mode of action of, 276 
Overtones, 5 

influence of, on quality, 286 

on the definition of consonant 
intervals, 288 

Pendulum, 8, 16 
Blackburn's, 35 
double, 38 

Period-equations, graphical solution 
of, 83, 125, 126, 128, 256, 258 

Permanency of type, condition for, in 
air-waves, 174 

Pipe, normal modes of a, 171 
modulus of decay of a, 270 
velocity of sound in a narrow, 193 

Plane waves in an elastic medium, 118 
in air, 160, 174, 223 

Plate, transverse vibrations of a cir- 
cular, 153 ; of a square, 154 

Plucked string, theory of, 66, 98 

Poiseuille's law, 195 

Poisson's ratio, 111 

'Quality' of musical notes, 4 
influence of overtones on, 286 

Beciprocity, principle of, 47, 81 



INDEX 



303 



[The numerals refer to the pages] 



Rectangular vessel, normal modes of 

a, 254 

Reed-pipes, theory of, 278 
Reflection of waves, 64, 168, 215, 

267 

Refraction of sound, due to variation 
of temperature, 216 

to wind, 219 

Resonance, 18, 20, 22, 32, 270 
Resonator, 261 

free vibrations of a, 262 

forced vibrations of a, 270 
Ring, normal modes of a, 133 

Scattering of sound waves by obstacles, 

240 

Sensations, analysis of, 2, 284 
Shearing strain, 107 

stress, 109 

Shells, vibrations of curved, 155 
' Simple source' of sound, 214, 224 
Simple-harmonic vibrations, 2, 9 

energy of, 15 

superposition of, 22, 48 
Sine-series, 87 
Sound, velocity of, in air, 161, 162 

in water, 163 
Sound waves, plane, 160 

spherical, 20o, 224 

general, 204, 212, 214 

of finite amplitude, 174 
Sounding board, function of, 68, 81 
Source of sound, simple, 214, 224 

double, 215, 226 

'Speed' of a simple vibration, 10 
Sphere, waves produced by oscillating, 
228 

vibrations of an elastic, 156 
Spherical vessel, normal modes of a, 

255, 258 

Stability, coefficients of, 43 
Stationary property of normal modes, 

45 
Stiffness of piano-wire, effect of, 82, 

133 

Strains, 106 
Stresses, 108 



String, transverse vibrations of a, 59 

waves on a, 61, 64 

normal modes of a finite, 68 

forced vibrations of a, 80 
String excited by plucking, 66, 72, 98 

by impact 73, 99 

by bowing, 75, 98 
Superposition of vibrations, 22, 48 

Temperament, equal, 7 
Temperature, effect of unequal, on 

propagation of sound, 216 
Tension, effect of permanent, on the 

vibrations of a bar, 132 
Tones, pure, 1 

interference of, 287 
Transmission of sound by an aperture, 
244 

by a grating, 247 
Transverse vibrations, of strings, 59 

of bars, 120 

of membranes, 139 

of plates, 152 
Tuning fork, 132 

Velocity of sound, 161, 162 

in a narrow pipe, 193 
'Velocity-potential,' 201 
Violin-string, 75, 98 
Viscosity, 183 

effect of, on air- waves, 185, 186, 

190 

on waves in a narrow pipe, 193, 194, 
196 

Water, velocity of sound in, 163 
vibrations of a column of, 173 
Watt's indicator diagram, 157 
Waves, on a string, 61, 64 
in a bar, 115, 123 
in an elastic medium, 118. See 

also Sound waves 

Wind, influence of, on sound propa- 
gation, 219 

Young's modulus, 111 



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