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THEORY OF
VIBRATING SYSTEMS AND SOUND

THEORY OF

VIBRATING SYSTEMS
AND SOUND

By
IRVING B. CRANDAT.T,. P H .D.

Member of the Technical Staff
BELL TELEPHONE LABORATORIES, INC.

NEW YORK
D. VAN NOSTRAND COMPANY

EIGHT WARREN STREET
1926

D. VAN NOSTRAND COMPANY

Printed in th< U. S. A.

PREFACE

THIS treatment of the Theory of Sound is intended for the
student of physics who has given a certain amount of attention
to analytical mechanics, and who desires a sufficient acquain-
tance with the theory of sound and its recent applications to
bring it into balance with his studies in other branches of me-
chanical science. To this end a few fundamentals of theory are
emphasized, principally for the purpose of introducing other
(and less abstract) matters. With the background thus assumed
on behalf of the reader, it does not seem necessary to include
such developments as the proof of Lagrange's Equations, the
theory of elastic deformations, or the theory of electrical net-
works, with which he will have become familiar through reading
collateral literature; but on the other hand, there are (for exam-
ple) certain analogies between the theories of mechanical and
electrical oscillations which might well be indicated, since they
have become so important a feature of acoustical technique.
In a general way it is hoped, herein, to supplement rather than
to replace the accepted treatises on Sound.

From the nature of things, there are many references in this
text to the classical works of Lord Rayleigh and Prof. Horace
Lamb. My obligations would be even greater, if (departing
from the spirit of the text) I had attempted to treat anew the
general theories of strings, bars, and plates, and the theory of
Harmonic Analysis, which occupy so much space in the classical
literature. The reader may take what he needs of these theories
from the standard treatises. Coming to later sources, the pub-
lished work of C. V. Drysdale on various problems in the
mechanics of fluids has furnished much useful special informa-
tion. It will also be recognized that Chapter V is to a degree
dependent on the original studies in Architectural Acoustics
recorded in W. C. Sabine's Collected Papers, as well as on the

vi PREFACE

more recent work of Dr. P. E. Sabine, which he has kindly
placed at my disposal with many helpful suggestions. I take
it for granted that the research worker in acoustics will sup-
plement what he may find in the present volume by copious
reading of these and other sources referred to in the text, and
in Appendix B, which is intended as a guide to the newer lit-
erature.

In the recent years great progress has been made in applied
acoustics, and a number of contributions have originated in
Bell Telephone Laboratories. The studies on which this book
is based were made entirely on this atmosphere, and the mate-
rial was first presented by the writer in one of the " out-of-
hour" courses in the Laboratories. Through these courses the
members of its technical staff obtain from each other theoret-
ical training and first-hand knowledge of new methods. It is
to be expected that, from time to time, similar presentations
of other subjects of importance in our work will appear in
style uniform with the present volume.

Much of the present book has also received class presenta-
tion in a course given at the Massachusetts Institute of Tech-
nology during the spring of 1926 at the invitation of the Depart-
ment of Electrical Engineering.

It is impossible to acknowledge in detail the helpful criti-
cisms and contributions 1 have received from many of my
associates; I am also indebted to others, in the academic sphere,
for friendly suggestions on various matters in which they were
specially interested. This cooperation has undoubtedly made
possible a better book. In conclusion, I may express my thanks
to the staff of the Technical Library of our Laboratories, whose
services in furnishing literature and bibliographical data have
been indispensable; and to Mr. L. A. MacColl, M.A., who has
read all the proofs for correctness, both as to text and as to
mathematical style.

IRVING B. CRANDALL.

NEW YORK,
June i, 1926.

CONTENTS
CHAPTER I

SIMPLE VIBRATING SYSTEMS

ART. PAGE

1. Introduction; The Principle of Superposition I

2. Equation of Motion of a Simple System 4

3. Natural Oscillations 6

4. Periodic Driving Force 8

5. Complete Solution for Forced and Free Oscillations 10

6. Initial Conditions Under Periodic Driving Force II

7. Periodic Driving Force with Variable Frequency 12

8. Physical Nature of the Constants of the System 17

9. Equilibrium or Low Frequency Theory; Circular Membrane 20

10. General Theory of the Circular Membrane; Bessel's Functions .... 21

n. Air Damping; Piston System; Ber and Bei Functions 28

12. Equivalent Piston; Mean Velocity; Diaphragms 36

PROBLEMS i-io 39

CHAPTER II

GENERAL THEORY OF VIBRATING SYSTEMS; RESONATORS AND FILTERS

20. Generalized Coordinates 42

21. System of Two Degrees of Freedom; Natural Oscillations in General . . 43

22. Steady State Theory, for Two Degrees of Freedom and in General ... 46

23. Lagrange's Method and Equations of Motion 50

24. Resonators 53

25. Resonator Coupled to a Diaphragm 59

26. The Problem of the Loaded String; Filters 64

27. Acoustic Filters 73

28. Finite String; Normal Coordinates; Normal Functions 77

PROBLEMS 11-20 82

CHAPTER III

THE PROPAGATION OF SOUND

30. Properties of the Medium; Equation of Wave Motion 85

31. Properties of Plane Waves of Sound 89

32. Sound Transmission in Tubes 95

33- Approximate Theory of Resonance in Tubes and Pipes 103

vii

viii CONTENTS

ART. PAGE

34. General Discussion of the Physical Factors Affecting Transmission . . . 107

35. General Theory of Sound Waves in Three Dimensions 113

36. Spherical Waves of Sound; the Point Source n6

37. The Pulsating Sphere as a Generator of Sound 120

38. Reactions of the Surrounding Medium on a Vibrating String 124

PROBLEMS 21-30 133

CHAPTER IV

RADIATION AND TRANSMISSION PROBLEMS

40. General Considerations; Single and Double Sources 135

41. High Frequency Radiation from a Piston; Diffraction 137

42. Radiation from a Piston into a Semi-Infinite Medium 143

43. End Corrections for a Tube; Impedance of a Circular Orifice 149

44. Characteristics of Horns; Conical Horns 152

45. Flaring Horns of Exponentially Varying Section 158

46. The Finite Exponential Horn 163

47. The Effect of Sound Waves on a Simple Vibrating System 174

48. Acoustic Radiation Pressure 178

PROBLEMS 31-40 180

CHAPTER V

THE ACOUSTICS OF CLOSED SPACES; ABSORPTION, REFLECTION
AND REVERBERATION

50. Architectural Acoustics 182

51. Reflection and Absorption 185

52. Layers of Absorbing Material 192

53. Reverberation in a Closed Tube 200

54. Reverberation in Three Dimensions 205

55. Standing Wave Systems; Focal Properties of an Enclosure; Acoustic

Difficulties 214

56. The Reaction of an Enclosure on a Source of Sound 222

PROBLEMS 41-50 226

APPENDICES

A. RESISTANCE COEFFICIENTS FOR CYLINDRICAL CONDUITS 229

B. RECENT DEVELOPMENTS IN APPLIED ACOUSTICS 242

INDEX OF NAMES 261

INDEX OF SUBJECTS 264

LIST OF SYMBOLS

, > , displacement, velocity, acceleration.

to, o> > maximum values; e.g., = /'"'.

r, w, Sy resistance, mass and stiffness coefficients, in one

degree of freedom.

*i> **> applied forces; e.g., * = V e iut .

Z, Z ky Z jk y impedances; defined as needed.

/, Wy frequency; w = inf.

fo,fk, n y n ky natural frequency; n = 271/0, n k = 27r/t.

A, A*, damping coefficient or inverse modulus of decay.

' = Vw 2 -A T > frequency of natural oscillations X 27r.

X, A*, A/, roots of auxiliary equation.

a jk , b]k> Ofcj mass, resistance and stiffness coefficients in
general theory.

T y ^, Fy kinetic energy, potential energy and dissipation

functions.

D y LAGRANGE'S determinant of coefficients of equa-

tions of motion.

ajy *; ft, ft; factors of D.

Ty tension.

p, . density. (As applied to lines, surfaces or

volumes.)

Py p y total pressure, mean pressure.

py Spy excess pressure with respect to mean.

Ry R ky radiation resistance.

Ry Riy . resistance coefficients per unit length, con-
stricted conduits.

/i, viscosity coefficient.

v = -> kinematic viscosity.

p
7, ratio of specific heats for a gas,

x LIST OF SYMBOLS

*, /, coefficient of cubic elasticity; * for a gas = yp c

Sy A, condensation and dilatation; 5 = A.

Cy X, velocity of sound, wave length; X/ = c.

Sy Siy area of equivalent piston; area of wall surfac

Ky K}y conductivity of an orifice.

f y Fy volume or capacity of a resonator; volume <

a room.

A y strength of small source of sound; A = S\ Q .

phase constant in a non-absorbing medium.

0)
- >

== -; phase constant in an absorbing medium.

c'y phase velocity; in general, c f ^ c.

cty attenuation coefficient in wave transmission.

, rjy f, fluid displacements parallel to #, y y z, axes.

<t>, velocity potential.

Ey energy density in medium.

, intensity. - = E-c-, W = T + V.
at at

V 2 , LAPLACE'S operator.

12, solid angle.

Ty amplitude reflection coefficient.

R ='T 2 y energy reflection coefficient.

ty tj^ transmission coefficients.

Ay energy absorption coefficient. (A + R = 1.)

a = ZAjSjy total absorbing power of area s/.

T y reverberation time.

Ky SABINE'S constant for reverberation.

CHAPTER I

SIMPLE VIBRATING SYSTEMS

i. Introduction; the Principle of Superposition

The physical bases of the Theory of Sound may be reduced
to three primary phenomena. The first of these is that sound
waves are produced whenever a vibrating body is placed in
contact with an elastic substance. Next in order is the trans-
mission of sound by the elastic substance or medium: the
velocity of propagation being greater, the greater the ratio of
stiffness of the medium to its density. Sound, in undergoing
transmission, has all the characteristics of wave-motion. The
parts of the medium which are traversed execute periodic
motion; the volume elements in the medium undergo periodic
expansion and contraction; or what is the same thing, peri-
odic changes in density and pressure. Lastly we observe that
when suitably constructed apparatus is immersed in a field of
sound waves, parts of the apparatus will yield to the momen-
tum of the particles of the medium, or to the alternating excess
pressures at a given point in the medium, with the result that
the apparatus is driven into a state of vibration and so becomes
a detector or meter of the sound energy which falls on it.

By a judicious application of mechanical theory, supported
by experimental research, the science of Acoustics has been
developed into a wide field of interesting phenomena with
many useful applications. Some of these are purely mechan-
ical, as for example, the use of high-frequency vibrating sys-
tems in submarine signalling; or again, the use of horns as an
aid to sound radiation in loud-speaking apparatus. Some
relate to physiology; mechanical theory has unquestionably
been a valuable aid in the study of the mechanism of speech,
and of hearing; and some applications are of psychological or

2 THEORY OF VIBRATING SYSTEMS AND SOUND

esthetic aspect, as they concern the power of the human ear to
distinguish between ordered sounds such as music or speech
and the disordered sounds which we call noise. At the present
time it is possible to analyze and classify sounds of all kinds;
and conversely, apparatus is available for generating and de-
tecting sounds of almost any degree of complexity. By means
of electrical transmitting apparatus sounds may be amplified
or recorded with high precision. Sound, as a wave motion, is
capable of interference and diffraction; these effects have been
used to advantage to accomplish directive emission and detec-
tion. Even friction, or the dissipation of mechanical energy, has
been turned to account to control resonance in vibrating sys-
tems, and the reverberations due to reflecting surfaces.

A firm foundation of mechanical theory is essential if we are
to deal consistently with these varied phenomena. Lord Ray-
leigh's treatise, published thirty years ago, is still the authori-
tative statement l of the Principles; but there have been many
contributions since (some due to Rayleigh himself), and the
introduction of impedance methods, according to the modern
practice, has been of the utmost advantage to the student of
Acoustics. Hence the objective of the present text, which is to
develop in its current form the indispensable minimum of estab-
lished theory and show how effectively it lends itself to practical
applications. The problems which we shall consider are as far
as possible representative, but by no means do they cover the
whole field of applied Acoustics. They have been selected
from classical and more recent sources primarily for the purpose
of illustrating, or providing a working substance for the theory.

The reader is supposed to have a fair acquaintance with the
general principles of Physics, and but little need be said regard-
ing the special facts which relate particularly to elementary
experimental Acoustics; these matters are well treated else-

'The references to Rayleigh 's "Theory of Sound" in the present volume are to
the second edition (i vols.), London, 1894 and 1896. Equally indispensable to the
student, and a model of compactness in arrangement and style, is Prof. Lamb's "Dy-
namical Theory of Sound." Rayleigh's treatise has recently been reprinted, and a
second edition of Lamb's "Sound" (London, 1925) has just appeared. The refer-
ences to Lamb in what follows are to this work unless otherwise stated.

PRINCIPLE OF SUPERPOSITION 3

where. 1 The Analysis of Musical Sounds is doubtless familiar
to the reader through the work of D. C. Miller. 2 An exhaustive
collection of the data of Acoustics is soon to be available with the
publication of the International Critical Tables; here will be
included, for example the extended data on Speech and Hearing
which have come from the staff of the Bell Telephone Labora-
tories. 3 The Critical Tables will also include the data of W. C.
Sabine and later workers in the field of Architectural Acoustics.

In what follows we shall take for granted the usual assump-
tions of mechanics, but one principle, which is applied more
frequently in Sound than in any other branch of Physics, de-
serves a special statement. This is the Principle of Superposi-
tion, according to which we may find the total displacement of
a system acted on by a number of forces, by solving the prob-
lem for each applied force separately, and adding together all
the resulting solutions. This follows from the fact that the dif-
ferential equations of motion of such systems as we shall meet
are linear; hence linear combinations of solutions of a given
equation are solutions. Kinematically we shall sometimes have
occasion to compound a number of vibrations of different peri-
ods to obtain the complete solution of a given problem. In
some problems, the successive natural frequencies of the system
are harmonically related, and to these cases the methods of
Fourier's Series apply. But in other equally important prob-
lems, the component vibrations do not necessarily have fre-
quencies which are simple multiples of one another; and the
reader is warned that there is no limitation as to the relation
between the component frequencies in applying the principle
of superposition. The fact is that in the problems we shall
encounter, there is always a solution in terms of a series of
normal Junctions, each term of which represents a possible
mode of vibration of the given system, and iys usually sufficient

l Vf. H. Bragg, "The World of Sound"; E. H. Barton, "Textbook of Sound";
Poynting and Thomson, "Textbook of Physics," Vol. II, "Sound." See also the
Literaturverzeichniss in A. Kalahne's " Grundziige der Akustik," Leipzig, 1910-1913.

2 D. C. Miller, "The Science of Musical Sounds."

3 See also the notes on Speech and Hearing in Appendix B.

4 THEORY OF VIBRATING SYSTEMS AND SOUND

to investigate only one typical term of the series, in order to
understand the behavior of the system. 1

2. Equation of Motion of a Simple System

We have observed that the first major problem in the theory
of sound is that of producing sound waves by a "generator"
or "vibrator" placed in contact with the sound-transmitting
medium. To solve this problem in a general way we must de-
termine the properties of the vibrating system in itself, the reac-
tions of the medium upon it, and the properties of transmission
in the medium. It is logical therefore to begin our studies with
a treatment of the simplest type of vibrating system.

A particle of mass m is fastened to a stiff spring so that
small motions take place in a horizontal line and the effect of
gravity on the system can be neglected, also for the moment,
any frictional "forces." Using the notation outlined at the
beginning, the force required to produce a small displacement
from the equilibrium position is s- . The potential energy of
the system for any displacement is therefore

y - w,

and the kinetic energy of the system is

T - \rn\\
The total energy of the system at any instant is

w = r+r = K^ 2 +^ 2 ), (0

in which m and s are constant, and , , functions of time. As
there is no dissipation of energy due to friction or other causes,
we can apply the "energy principle," whence we obtain

aw .a\ , as , x

^-**2P- '<*-<>* <*>

or in the convenient notation we have chosen, since ~r = >

m\ +j = o, (2)

1 The subject of normal functions, of which the terms in a Fourier Series form a
particular example, will recur in 28.

EQUATION OF MOTION OF A SIMPLE SYSTEM 5

which is the equation of motion of the system, under the action
of no external forces.

If there is an external force (/) acting on the system (that
is on the mass m) its rate of doing work is clearly *(/) and to
apply the energy principle in its general form we have, instead
of equation (2)

iH + jH, (2')

whence the more general equation of motion

(2'*)

Consider now the effect of frictional forces, such as for example,
the air friction on the mass m, as it oscillates back and forth.
It is convenient, and often sufficiently accurate, to take the
frictional force as directly proportional to the velocity of motion,
i.e.:

frictional force = r-,

in which r is the "resistance constant." Now, as the net effect
of the frictional force r- is to oppose the action of the external
force *(/) we have finally

ml+rs +st = *(/), . (3)

for the complete equation of motion of the "system of one de-
gree of freedom."

This equation specifies the behavior of the system under
any applied force'*(/), and with any initial conditions we choose
to impose, so long as m, r and s remain constant. In practice
this means that the displacement-coordinate must not be
forced to unduly large values; the "constants" of the system
are then no longer constant and embarrassing discrepancies
occur between the simple theory and the (more complicated)
observed facts. In these cases it must be borne in mind that it
is not equation (3) which is at fault: the simple theory is simply
not applicable.

The analogy between equation (3) and the general equation

6 THEORY OF VIBRATING SYSTEMS AND SOUND

of the electric circuit containing inductance, resistance and
capacity, namely

Lq + Rq + ^ = (/), (l = q) (3')

will not be labored here. It will doubtless interest the student
who has mastered alternating-current theory to note all such
analogies as we proceed; and we may commend Rayleigh, Vol.
I, Chap. XB., and Maxwell, Vol. II, Chap. V-VII, as suitable
collateral reading in this connection.

3. Natural Oscillations

We have to discover the important properties of the motion
of the system by solving the linear* differential equation (3).
The simplest case is that in which an arbitrary displacement
and velocity is given to the system which is then let go. In this
case *(/) = o and the solution of this equation contains two
arbitrary constants, thus:

= A** + Be, (4)

in which Xi, X2, are the roots of the auxiliary equation. Forming
the auxiliary equation from (3) and (4),

and solving,

r I 7* 2 s

X = V o >

2m ^^m* m
letting

r s

= A and = 2 , *
im m

we have:

Xi = - A + iVn 2 '-'* 2 s - A + in'.

(5)

\2 A IV n 2 A 2 s A |V;
so that

+ Be-**). (6)

NATURAL OSCILLATIONS 7

This solution can be written

= c~"[(A + B) cos n't + i(A - B) sin '/]

= e-"(A f cos n't + B f sin '/),
or more concisely,

= ^-* sin (n't + 0), (6')

in which A 9 and are the arbitrary constants to be adjusted to
the given boundary conditions. The reader may easily con-
vince himself that all possible initial conditions can be fitted by
(6'), and he may also derive the corresponding equation for the
velocity:

= nAc~" cos ('/ + e + e).

,A l

e = tan" 1 -- ss tan"" 1

' V 2 - A 2

Geometrically, (6') is the equation of a sinusoidal curve
whose amplitude diminishes with increasing time. The mea-
sure of this decay in amplitude is the constant A, called the
"damping coefficient." The rate of decay depends on the ratio

of resistance to mass, ( j but the rate of dissipation of energy

(r 2 ) in the system due to friction depends only on the resist-
ance factor r. If there were no dissipation in the system, the
frequency of the oscillations, known as free or natural oscilla-

tions in this case, would be/ = ; i.e., this is the frequency in

2/ff

which the system tends to oscillate, when suddenly excited.
This frequency is diminished somewhat by the damping; the

T ___ _

actual frequency of the oscillations is/' = V n 2 A 2 . If A is

2?r

made large enough, e.g., ^ n (by increasing r or diminishing m>
and with it s) y Vw 2 A 2 may be made zero or imaginary, and
free oscillations no longer are possible. To discuss this last case
let = and =o, when / = o; then applying (6) in the form

8 THEORY OF VIBRATING SYSTEMS AND SOUND
and calculating the constants A and 5, we have

_^_-
2V A 2 - n 2

-J-

Thus the motion is the sum of two motions whose amplitudes
decay at unequal, but usually rapid rates. It will be observed
that this solution is not valid for exact aperiodicity (or critical
damping), for then A = ; the solution for this case is left to
the reader as a part of problem 2, at the end of this chapter.

To summarize, we have found the natural or characteristic
frequency of the system, that is, the frequency in which it
tends to oscillate for an indefinite period of time; but we
observe that for all practical purposes, the natural oscillations
may be made to decay very rapidly, or to disappear altogether,
if the damping factor is made very large.

4. Periodic Driving Force

Consider now the simple vibrating system under the action
of a periodic force *(/) = "9 e iut . The solution of the equation

ml + r$ + jf - V e M , (3)

will contain two parts, a "transient" and a "steady state"
term. The former, obtained in the preceding section, expresses
the temporary reaction of the system to any suddenly applied
force, and in the long run its amplitude becomes very small.
The steady state term expresses the tendency of the system to
follow 'as well as it can, the driving force, and after the initial
transient has disappeared the steady state term is a sufficiently
complete description of the motion, unless further change in
driving force is made. (At this stage we prefer to speak from
the standpoint of physical experience, solving the problem step
by step but it must be noted that in doing this we sacrifice a
certain amount of rigor, for the sake of obtaining a more
concrete picture.)

Neglecting any transient conditions, we may assume a

PERIODIC DRIVING FORCE 9

steady state motion = Ce iut to be a particular solution of the
equation (3) above. Then, since = i{ and =

and

= T%^ wr in which Z = r + iYiw - -V (8)

/Z \ w/

In the algebra of complex quantities (which is indispensable in
dealing with problems such as this),

I a ib i . , r i ^ / x

g"'*, if </ = tan" 1 -. (o)

Hence the amplitude and velocity are respectively

and

t- = f/.it = ? p<(w* 0)

(10)

in which

-, ^

( *> - - )
\ co/

= tan-H -/ and Z = Vr 2 +

The reader need not be confused by the use of the symbol
Z interchangeably for the complex impedance or its absolute
value; it is always evident from the context which is meant.

Before going further note:

(a) The 90 difference in phase between and ;

(b) The phase lag <t> between driving force and resultant

velocity;

has a minimum value Z = r when

2, 3].
We shall retui^T to a more detailed discussion of these later.

io THEORY OF VIBRATING SYSTEMS AND SOUND

5. Complete Solution for Forced and Free Oscillations

To find the most general behavior of the system we must
add together the solutions for free and forced vibrations and
make them fit such boundary conditions as we desire to impose.
Usually the forced vibrations are the more important in prac-
tical calculations so consider these first. In practice the driving
force is usually specified by a series whose typical term is

C\ cos wi/ or Di sin <*>i/,

so we sacrifice little by using (say) only the real component of
the rotating unit vector e iut , e.g., letting *=* cos w/ for the
forced vibrations, whence:

= cos / -

and

sin (/ -

(100)

It is clear that the imaginary components, used consistently
all the way, would have equally sufficed, the result being
merely an interchange of sine for cosine, with due regard to
signs.

Adding the transient terms (equations 6', 6a y 3), we have:

sin ('/ + 0) + -^ sin (f - *),

" Z 9

= nAe~" cos (n't + 6 + e) + -~ cos (w/
I . /y

Equations (u) with due adjustment of yf and to fit boundary
conditions, describe the motion of the system under the excita-
tion of a simply periodic force. Applying the methods of the
Fourier Series and Integrals a sum of such solutions is always
possible, which will fit any boundary conditions and a driving
force of any nature whatsoever.

INITIAL CONDITIONS

6. Initial Conditions Under Periodic Driving Force

A good enough understanding of the behavior of the system
under periodic excitation can be had if we study equations (n)
first as functions of time, and second as functions of the fre-
quency with which the system is driven.

Considering frequency If = ) constant, let the system
start from rest, i.e.,

when

For these conditions,

o = A sin 6 H sin (0) (amplitude),

I (A/j

o = nAcos (6 + e) + cos (0) (velocity).

(12)

The rigorous solution of these equations for A and B is much
more cumbersome than it appears offhand, and to make prog-
ress with the matter approximations must be made, e is usu-
ally small /sin e = -, 3 j and may be sacrificed in the interest
of symmetry. We then have, squaring and adding

// 2 (sin 2 e + cos 2 0) = -~ /sin 2 </> + 2 cos 2 <t>\ = -y^- Q 2 .
Choosing the negative root, as applicable to the conditions,

A r ~

= \ sin2

CO 2

cos2

i . o> i

sm B = ^ sin 0, cos 6 = - -=. cos

(13)

12 THEORY OF VIBRATING SYSTEMS AND SOUND

Thus for co = n (i.e.,/ =/ ) is the negative of <, Q = I and
the amplitude /L at / = o, is the negative of the steady state

amplitude * of the forced vibration which is finally attained.

coZ/

But in general,

coZ

sin (ut - <f>] - Qe~* sin (n't + 0) ,
i = y cos (w/ -</>)- (2^~ A< cos (n't + + c) ;

the indicated solution for from (13) being understood. The
history of the system from / = o is as follows: starting from rest,
the amplitude (or velocity, if you prefer) can be considered first
as the sum of a "transient" amplitude and a "steady state"
amplitude equal in numerical value but opposite in sign. They
are exactly opposed, strictly speaking, only at the start; for in
general n ^ n' ^ co. During the early stages of the motion, the
transient vibration, steadily weakening in amplitude, is beating
with the forced vibration, and the observable result of this
interference is a gradual asypmtotic rise in the total ampli-
tude, reaching its steady state value when the transient has
disappeared. During this rise in amplitude the (apparent)
frequency of the vibration is neither that of the transient nor
that of the impressed force; it is a variable composite of the
two which approaches in value the steady state frequency as the
transient disappears. In listening, for example to a receiver
diaphragm, the ear can often distinguish the transient tone
when a periodic driving force is suddenly applied, particularly
if the driving force has a frequency which differs consider-
ably from the principal natural frequency of the diaphragm.

7. Periodic Driving Force with Variable Frequency

Consider the steady state of vibration established and let
the frequency of the driving force be varied. The velocity and

PERIODIC DRIVING FORCE, VARIABLE FREQUENCY 13

amplitude (equations (TO), (n)) are:

= -~ sin (co/- <);

in which

= tan

^ /2_l/ S \ 2

Z = \r 2 + (mu I ,

x \ co/ r

Or, focussing the attention on maximum values only

(16)

For constant * > the following table gives values for the various
factors from which the reader may infer the way in which the
behavior of the system depends on the driving frequency.

\z\

Vir

i
nr

7T
4

* Approximately; valid when A 2 is small as compared to w 2 .

i 4 THEORY OF VIBRATING SYSTEMS AND SOUND

The reader may check these values, and plot and against
frequency. If the frequency scale is logarithmic, or by octaves,
the curve for the velocity will be symmetrical about the point

R(f-f.)
i(moo-J)

FIG. i. CIRCLE DIAGRAM OF IMPEDANCE.

f =f . This is the frequency of maximum velocity, or in the
vernacular the "resonance frequency." If the vertical scale for
the amplitude is made -times as great as that for the velocity,

PERIODIC DRIVING FORCE, VARIABLE FREQUENCY 15

the maxima of the two curves at/ = /> will nearly coincide; but
for o </ ^/ the curve for amplitude will lie above the velocity
curve, and conversely for points above resonance, it will lie
below the velocity curve. It is not symmetrical with respect
to/ ; and the amplitude at/ =o (compare the velocity) is

Xp . .

, as is evident from "static" or "equilibrium" theory.

The difference between the frequencies for which 4 = -

27T 4

is the commonly accepted measure of the sharpness or blunt-
ness of the resonance curve. For responding well to a wide
variation in frequency in the driving force, bluntness of reson-
ance or tuning in the system is desirable; this is obtained by
making the damping large, with the collateral advantage of
minimizing the interference due to natural vibrations or transi-
ents hence the utility of such systems. Sharply tuned sys-
tems have a definite place in single-frequency apparatus, such
as for example tuning forks and resonators generally.

Experimentally, the damping of a bluntly tuned system is
best obtained from the shape of the resonance curve of velocity;
for the sharply tuned system, the damping should be deter-
mined from the rate of decay of the natural oscillations.

The phase relations between force, impedance and velocity
are clearly shown in the circle diagram l first used by Kennelly

1 Proc . Am. Acad. Arts and Sci. y 48, 1912, p. 113. When the telephone receiver
is driven by alternating current a counter alternating e.m.f. is generated by the oscil-
lations of the diaphragm. This counter e.m.f. is proportional to the velocity of the
diaphragm, and hence to the driving current, at any given frequency. It manifests
itself as a component of the impedance of the machine, which component is the dif-
ference between the impedance when the moving member is free to oscillate, and the
impedance when motion is prevented. Kennelly and Pierce applied the term " motional
impedance" to this component and showed how it could be experimentally determined;
they also showed how, by a transformation of dimensions and phase, the motional im-
pedance measures the velocity of the vibrating member, as a function of frequency.

The circle diagram is in wide use in the current literature of electrical vibrating
apparatus, many references to which are given in Appendix B. For example, see
Kennelly and Taylor, Proc. Am. Phil. Soc., 55, 1916, p. 415; Hahnemann and Hecht,
Phys. Zeit., 20, 1919, p. 104 and ibld^ 21, 1920, p. 264; also R. L. Wegel, J.A.I.E.E.,
1921, p. 791.

16 THEORY OF VIBRATING SYSTEMS AND SOUND

and Pierce, in their study of the telephone receiver. In its
simplest form this is shown in Fig. I. In vector notation:

i.e., if

Z = r + / wco -- )>

to/

I I

z'jzT

and to simplify matters, let us study for a steady alternating
force, the ratio of velocity at frequency/, to velocity at reson-
ance (/ ), i.e.,

y.V.e ------ tT- <'7>

r + / 1 mw --

A distance OR = - = unity is laid off on the #-axis and this is

taken as the velocity vector at resonance, measured in arbitrary
units. On this as a diameter, construct the circle, and draw the
tangent BRA parallel to the y axis. Then for any line OF\Yi y
OFi is the reciprocal of OY\ y a familiar property of inversion
between points on the line and points on the tangent circle, of
unit diameter. For some frequency / </ lay off the vector

*" Zi

RY% equal to the ratio of reactance to resistance. Then OY* = -
^ r

and OFi (the mirror image of OFa) is equal to -. Thus plotting

the reactances for all values o < / ^ oo the end-point F^of the
velocity vector will travel around the circumference of the
circle, passing through the point R when/ =/ . The points F&
and F' A will be recognized as those corresponding to the fre-

quencies / -- and /> H -- , respectively.

J*n J*TT

PHYSICAL NATURE OF THE CONSTANTS 17

A somewhat similar diagram could be plotted for the ampli-
tude vector, but it must be noted that the locus of its end point
is not a circle. This curve, which is nearly closed, must be

turned about through an angle - to allow for the lag of

the amplitude with respect to the velocity. For further dia-
grams of amplitude and velocity vectors the reader may refer
to Kennelly's "Electrical Vibration Instruments" (Macmillan,

The circle diagram sums up the 'kinematics of the steady
state theory of the simple vibrating system.

8. Physical Nature of the Constants of the System

Turn now from kinematical considerations (in which m, r,
and s are mere parameters in an equation) to a more physical
view of the nature of these "constants." Practically, each of
these factors is an "effective" or mean value, which to a certain
degree of approximation replaces the total effect of inertia or
resistance or stiffness of the system, each factor being summed
up or averaged with respect to the coordinate in terms of
which the motion of the system is to be described. Thus a vari-
ety of geometrical and mechanical considerations enter into the
determination of each.

Much progress has been made in the study of the telephone
diaphragm, for example, by assuming that such an essentially
complicated system can be replaced by an equivalent simple
system with constant m y r, and s. Logically this assumption
has no foundation whatsoever, because in order to determine
m and s a priori the form which the diaphragm takes in bending
must be determined, and this, particularly when the stress is
not uniformly applied over the surface of the diaphragm is in
itself a mechanical problem of great difficulty. The diaphragm
is really a system of a large number of degrees of freedom and
consequently its form of vibration will vary with frequency;
there is no such thing as a constant mass factor for a diaphragm.
The full significance of these statements will appear more

1 8 THEORY OF VIBRATING SYSTEMS AND SOUND

clearly when we come to consider systems of several degrees
of freedom; it is sufficient for our present purpose to point out
the limitations of the simple theory. The methods sketched
in this chapter for dealing with actual systems often suffice to
give a fairly clear picture of the behavior of the system, and
this is their sole justification.

In the case of resistance, the matter is even more compli-
cated. The resistance, or friction coefficient in the case of
bending a rod or plate must include not only the internal fric-
tion, due to the sliding of all the elements of the structure with
respect to each other, but air-resistance factors as well, or in
more general terms the effect of immersing the vibrating system
in the medium. In the first place sound energy is radiated from
the system, whenever the system is in contact with the me-
dium, and this represents a steady drain on the energy supplied
to the system. The rate at which the system works on the
medium by virtue of radiation is, as we shall see later, 7? 2 in
which R has the dimensions of resistance. R is an important
constant characteristic of the medium; it may be called the
radiation resistance.

In addition to radiation there is usually a pure-friction
term due to lamellar motion in the surrounding air, in which
the viscosity of the medium comes into play. The resistance
in the case of the simple system we have studied was due en-
tirely to this effect, and it is possible, in case great damping
is required so to arrange the system that air friction or viscosity
is utilized especially for this purpose. If we consider the fric-
tional air resistance to be included in the inherent resistance of
the system, then the total resistance against which the system
works is

f == ^internal i -^radiation >

the rate of expenditure of energy being

r^-(r, +R){*. (18)

The electrical analogy is of course evident.

PHYSICAL NATURE OF THE CONSTANTS 19

It is an old contention that the opposing force due to fric-
tion may not be strictly proportional to the velocity of motion.
Experimentally we believe that the assumption of the linear
relation is fairly well substantiated, for the small velocities we
deal with in sound and vibrating systems. It is true that for
rapidly moving bodies in air (e.g., artillery projectiles) there
are large amounts of energy dissipated in eddies (i.e., in kinetic
energy effectually abstracted from the energy of the projectile)
and the corresponding dissipative reaction or force in this case
is proportional to the square of the velocity. But as long as we
are dealing with pure stream-line motion and small velocities
we are correct in writing

Frictional force = Const. -n'%. (p = viscosity coefficient.)

A good example of viscosity-damping will be given in a suc-
ceeding article. 1

Finally, to complete our discussion of the nature of resist-
ance and stiffness, we note the effects of forcing the system to
oscillate at such large amplitudes that elastic hysteresis is
brought into play. In this case both r and s depend on the
amplitude of the motion, and the simple theory is of no avail;
power series r = /i() and^ =/2() must be substituted for the
simple constants and the solution of the problem is so difficult
that special methods are required. The treatment of such cases
is beyond the scope of the present outline; the reader may refer
to Rayleigh (67, 68) for suggestions as to theory, for the sim-
ple system. In Rayleigh, Appendix to Chapter V (II, p. 480),
some of the properties of non-linear vibrations in compound
systems are investigated. 2

ir rhe reader interested in viscosity, hydrodynamical resistance, etc., may consult
Chaps. Ill, IV, V of "The Mechanical Properties of Fluids," Drysdale et als., Van
Nostrand, 1924. This is a very useful book and will be referred to again.

2 Following Rayleigh (67, 68), the reader may refer to Part II of a paper by E.
Waetzmann, Phys. Zeit., XXVI, 1925, p. 740, on "Modern Problems of Acoustics."
This part of the paper deals with non-linear oscillations, difference tones, etc., and
gives references to Waetzmann's earlier work.

20 THEORY OF VIBRATING SYSTEMS AND SOUND

9. Equilibrium or Low-Frequency Theory; Circular Membrane

The static or equilibrium theory (more logically speaking,
the low-frequency theory) is often of considerable utility in
studying such problems as the vibration of diaphragms and the
like. We shall illustrate its application to an interesting prob-
lem, and incidentally gain some notion of the determination
of mass and stiffness constants.

A thin circular membrane, with fixed boundary, may vi-
brate in a number of modes. Of those that are symmetrical
with respect to the axis we shall investigate the fundamental
or gravest mode. Obviously if we can gain a fair idea of the
shape of the membrane in its position of maximum distension
we can make approximate computations of mass and stiffness
factors, and so determine the natural frequency of this mode
of vibration.

For a circular membrane of no inherent stiffness and loaded
with a uniform force P per unit area, it is trivial to show that,
for small displacements,

in which a is the radius of the fixed boundary, r the tension,
and the central or maximum displacement. Let us assume
that this paraboloidal form of displacement is substantially
maintained up to the frequency of the gravest mode; and de-
termine m and s in terms of f and . Summing up the annular
elements of mass iirprdr we have for the kinetic energy of the
system

T =

Thus the mass constant is one third of the total mass of the
diaphragm. The potential energy can be calculated from the
work done by the total force in distending the diaphragm, tak-
ing care to allow for the variation of P with . Since

GENERAL THEORY OF CIRCULAR MEMBRANE 21

that

> 2

2 whence j = 2,rr. (21)
The natural frequency is then (neglecting damping),

7 1.22

- 2 =

This is a surprisingly accurate result as will appear by com-
parison with that obtained more rigorously in the next section.

10. General Theory of the Circular Membrane; Bess el's Functions

This problem of the symmetrical vibration of the circular
membrane is of some importance and its general solution in-
volves mathematical devices which are among the most inter-
esting and useful in analytical mechanics. Neglecting damping
as before, we adapt to our notation (9) the straightforward
treatment of Lamb (Sound, 54):

Ci t

The stress across a circle of radius r has a resultant r-i^r --

normal to the plane of the undisturbed membrane and the dif-
ference between the stresses on the edges of the annulus whose
inner and outer radii are r and r + dr gives a force

Equating this to p-27rrdr- which is the rate of change of mo-
mentum of the annulus, we have

22 THEORY OF VIBRATING SYSTEMS AND SOUND
Now if we assume for the solution of (23)

the factor depending on / can be removed from the differential
equation, and we obtain

in which k 2 = This is Bessel's equation (the simplest form)
and if we try solutions of the type

and determine the coefficients by substitution in (24) we find

*(*r) = .

which function is known as the Bessel's function of the first
kind, of zero order. The general solution of (24) has two arbi-
trary constants and is written

= [AJ Q (kr) + BK (krW nt , (26)

in which K is one of the Bessel's functions of the second kind
of zero order. K (kr) becomes infinite for r = o and obviously
does not satisfy the boundary conditions, therefore B = o. The
solution of the problem (since = </"' for r = o and = o for
r a) depends on those values of k which satisfy the equation

^ = * /o(**) = o, i.e., J (ka) = o. (27)

By reference to tables (see Gray and Mathews, "Bessel Func-
tions," or Rayleigh, Vol. I, p. 321) we find, for the roots of

(27)

ka ,

= .766, 1.757, 2.755.

Thus using the value k\ = .766 - for example and letting o < r < a

BESSEL'S FUNCTIONS

we can determine the shape of the distended membrane for the
lowest natural frequency from the equation

07*)

and the natural frequency in this mode of vibration is

l ki T 1.20

This result is in close agreement with that obtained previously
(eq. 22, 9), assuming (what is substantially correct) a para-

1.00

-40
FIG. 2. FORM OF VIBRATING CIRCULAR MEMBRANE IN TERMS OF BESSEL'S FUNCTIONS.

boloidal form for the distended membrane in the fundamental
mode of vibration.

In order to get a better idea of the behavior of the mem-
brane, there have been plotted, in Fig. 2, the three functions

JL T (i r \ k ,

J OV*-' / ) ** rt-l) K-Zy H-)

for values (o ^ r <* a). A dotted line indicates the section of the
paraboloid which is nearly equivalent to J Q (kir). For further

24 THEORY OF VIBRATING SYSTEMS AND SOUND

graphs of the Bessel's functions reference may be had to the
"Funktionentafeln" of Jahnke and Emde.

The discussion now takes a more general aspect, to account
for the introduction of damping into the system. Let a resist-
ance constant R per unit area account for dissipation by fric-
tion and radiation; then instead of (23) we have

Assuming
we have

in which
Thus (cf. 3),

in which

r dr

^ + - f + *)*(r) - o,
Qr 2 r Qr /

P X 2 + \R + k 2 r = o.

(24*)

A + in', \2 = A in'\

n r =

. o 2 72

A , n = K
p

(260)

Now suppose the membrane given an initial distortion of the
form Z Q J(kjr) y and let go; then it is clear that with the proper
adjustment of A and B (cf. 6 ; ) we have for the amplitude

(29)

in which (kjr) is a root of the / as before. The natural fre-
quencies (as modified by damping) are now determined by

2ir

etc.

(28*)

If, instead of being distended into a shape corresponding
to one of its normal modes the membrane is initially given an

GENERAL THEORY OF CIRCULAR MEMBRANE 25

arbitrary distension (as for example the form of a cone) it is
possible (as can be proved in a way analogous to the proof of
the Fourier synthesis) to build up a series

{ = Ai J (ki r) *- A * cos (m t + ej

+ Aljofafy-** COS (0 2 / + fe) + . . (29)

which will satisfy all the conditions of the problem. The vibra-
tion in this case, until it is completely extinguished by damping,
is thus a composite of all the possible symmetrical natural
vibrations of the system, the relative amplitudes A\ y A^ and
phases 61, 62, being chosen so as to build up, by superposi-
tion, the arbitrary initial shape of the membrane.

Suppose now that the membrane is driven by a force per
unit area lwl ; we wish to determine the general properties
of the motion of the membrane as the driving frequency is
varied. Instead of equation (230), we have

and if = / and = or (steady state theory),

in which

~ o I ^ I * S * )

3r 2 r 9r T

pco 2 /a?/?

If" -

(30

The right-hand member of the equation (31) is not a function
of the radius r\ and if the right-hand member of the equation
were zero, then AJ Q (itr) would be a solution of the equation.
We therefore conclude that the solution of (31) is

26 THEORY OF VIBRATING SYSTEMS AND SOUND

Determining A for the boundary condition (a) = o,

the amplitude is

and the velocity is (cf. eq. (10)):

In this expression for the velocity, the impedance per unit area
is

Z(r) = _J__ ___J" (33 )

/CO I --^ - I - - f -r

Jo(Ka)] L Jo(K#)l

To fully discuss equations (33-33^) and so determine the be-
havior of the membrane when the driving frequency is varied
is a difficult matter because of the complex variable nr which
is the argument of the Bessel's function. It can be seen at once
that for zero frequency (since K Q = o and/ (o) = i) Z = oo and
the velocity is zero. If it were not for the resistance factor R
which we have introduced into * 2 , we should have J O (KO) = o
whenever co = n\, #2, etc. (n\ y n<* . . . each being 2?r times a
resonant frequency of the membrane) and for these frequencies
Z would vanish, giving an infinite velocity. Without rational-
izing Z and computing its absolute value, it is evident that Z
tends to show a series of minimum values corresponding to
these points, and these minimum values rise as R increases.
It will also be noted that as the mode of vibration changes
with frequency, mass and stiffness constants must vary with
frequency. The curve of Fig. 3, showing the variation of the
mass constant is due to R. L. Wegel, and was made in connec-
tion with his study of the telephone receiver.

GENERAL THEORY OF CIRCULAR MEMBRANE 27

Equation (33^) was obtained to fit the boundary condition
Z = oo when r = a. An interesting test of (33^) can be made by
computing for low frequencies the impedance per unit area at
the center of the diaphragm^ for which ]J^r) = I . Since we know
that the membrane takes a paraboloidal form under uniform

.300

v
"o

.aoo

.100

FIG. 3. MASS COEFFICIENT IN TERMS OF TOTAL MASS FOR CIRCULAR MEMBRANE.

pressure at low frequencies, and departs only slightly there-
from even at the first resonant frequency, we need use only two
terms of the Bessel's function, i.e.,:

and we have

T / \ K U

J<,(Ka) = I ,

cf. (25)

(34)

28 THEORY OF VIBRATING SYSTEMS AND SOUND

Thus the impedance appears in the standard form, and we
recognize at once a stiffness constant, s reso = ;, a result con-
sistent with equation (19). This is to say that if a small por-
tion at the center of the membrane could be made to vibrate,
piston-like, with the mass and stiffness constants that appear

in (34) its natural frequency would be \ T - y which is not quite

TTfl * p

the natural frequency of the membrane as a whole, in its funda-
mental mode. The impedance per unit area is a minimum at
the center, rising to oo at r = a and the stiffness constant per
unit area increases while the mass constant per unit area de-
creases as r increases from zero to the value a.

The complete discussion of all the modes of vibration of the
circular membrane is readily available in the classical literature.
Together with the theory of the circular plate, it is of funda-
mental importance in the specialized study of the telephone
receiver diaphragm.

For the theory of Bessel's Functions, the reader may consult
Chap. XVII of "Modern Analysis," by E. T. Whittaker and
G. N. Watson (Cambridge, 1920), in addition to references
previously given.

1 1. Air Damping; Piston System; Ber and Bei Functions

In the case of the symmetrical vibration of a membrane
it was noted that a small portion at the center could be con-
sidered as a piston which, in its motion to and fro, was
always perpendicular to the axis of symmetry. In many prob-
lems it is convenient to replace the whole active portion of
a diaphragm by an imaginary "equivalent piston/' this device
being legitimate only for those frequencies for which all parts
of the diaphragm move strictly in phase with one another.
Thus in the low frequency theory we virtually replaced the
membrane ( 9) by a piston whose mass was one-third the total
mass of the membrane, and whose total stiffness constant
(27rr) was one-half the stiffness constant at the center of the

AIR DAMPING; PISTON SYSTEM 29

/. AT\

membrane (i.e., ) multiplied by the area of the diaphragm.

Thus a piston of a certain area, having the mass and stiffness
constants as given, and oscillating according to the equation
= cos nt well represents the motion of the membrane, at
frequencies below the first natural frequency. At higher fre-
quencies this substitution is no longer valid. The basis on
which the area of the equivalent piston should be determined
is set forth in the next section.

lembreme

Air Film/ \ Damping Plate

^Insulating Material

FIG. 4. SECTION OF CONDENSER FIG. 40. EQUIVALENT PISTON

TRANSMITTER. SYSTEM.

Let us now consider a problem in variable stiffness and
resistance over the surface of a membrane due to certain con-
straints: the investigation being simplified by the use of an
equivalent piston whose area is arbitrarily fixed.

The structure of a simple form of condenser transmitter is
shown in section in Fig. 4. The diaphragm is a thin stretched
membrane whose stiffness, already great (due to the tension),
is further increased by the compression of the air between the
membrane and the "damping plate " below it, with the result
that the first resonant frequency is very high. Thus there is
a wide working range of frequency in which the diaphragm
vibrates very nearly in its first normal mode. Due to the static
charge on the system as a condenser, the membrane is attracted
by the damping plate and takes in this equilibrium position a
nearly paraboloidal form; such small oscillations as are pro-
duced in it, by the impact of sound waves, are virtually in-
creases or decreases of this paraboloidal form of distension, as
long as the frequency is within the working range. It seems

30 THEORY OF VIBRATING SYSTEMS AND SOUND

reasonable, therefore, to replace the central or active portion
of the membrane by a piston (as in Fig. 40), and as the curva-
ture of the membrane is small in this region, the piston can be
taken as a plane disc which, in its mean position, is separated
by a certain distance d from the damping plate. Motion of the
piston produces in the thin air film a radial displacement of the
air, which is impeded by its viscosity. At low frequencies, the
air having ample time to escape, there is a maximum amount
of dissipative or frictional reaction, and a minimum of elasticity
or stiffness due to accumulated pressure on the film. At high
frequencies this situation is reversed, little air can escape, and
the stiffness reaction increases rapidly to a maximum value.

The piston is given a small displacement = cos co/. As-
suming that the temperature of the air remains constant (i.e.,
that the air remains in thermal equilibrium with its metallic
bounding surfaces) the excess pressure due to simple compres-
sion \s pi = B(-J) m which B is the atmospheric pressure. From

this must be subtracted the loss in pressure due to radial dis-
placement (TJ) in order to obtain the excess pressure at any
point in the film. For an annulus the radii of whose bounding
cylinders are r and r+dr the volume of contained air is

and the net loss in contained air due to radial displacement is

9 '- '-r-Wr,

so that the resulting loss in pressure due to radial displace-
ment is

(35)

AIR DAMPING; PISTON SYSTEM 31

The excess pressure at any point in the film is then p = p\ p%.
Since -~-l = -~ J is the pressure gradient, we have, analo-
gously to Ohm's law, for the velocity of flow

1 dp'* u * Cdptj* i f\

~T whence r,= R \-^t } (36)

in which R is a resistance coefficient, the inertia of the moving air
being neglected as t\ is small. In treatises on fluid motion (Lamb,
"Hydrodynamics," 4th ed., p. 576; also Drysdale, et als.,
"Mechanical Properties of Fluids," p. 116) it is shown l that
for a fluid of viscosity M flowing between parallel walls sepa-
rated by a very small distance d^

'37)

Substituting in (35) the value of jj from (36), and differentiating
with respect to time, we have

R 3j>2 d 2 p2 i 3/>2
B ~dt = -&- + r V

or, since p2 must vary as e iut , i.e., p2 = i<*p2>

Here again on account of the axial symmetry of the structure,
we encounter Bessel's equation, and the Bessel's functions of
zero order are required for its solution. The pressure must be

1 The resistance coefficient in this case is analogous to Poiseuille's coefficient for the
case of a very narrow cylindrical tube. This latter is derived in Appendix A, eq. (I).
See also problem 26, following Chapter III.

32 THEORY OF VIBRATING SYSTEMS AND SOUND

finite when r = o, so as in 12 we discard the function K (kr)
because it becomes infinite for r = o. We thus have

p 2 = 4J (kr)e tut in which k' 2 = - r-> (39)

and the complete solution is

p= pl -p 2 = |[ u - 4J Q (kr)V. (40)

In order to evaluate this expression, note the form of J (kr)

when k 2 is imaginary. Letting a = v-^ -, we have, instead
of equation (25)

r // \ r / / ~~ \ - a2r2 a4 ^ 4

/o(*r) = /o(V-ir) = i + i-^- - ^- -

or,

/ r) = ber r+ / bei

using the terminology established by the electricians in solving
the mathematically related problem of alternating-current re-
sistance and inductance in a cylindrical conductor. (See A.
Russell, Phil. Mag., April, 1909, p. 524; or "Alternating Cur-
rents," V. I, Chap. VII). Tables have been computed for the
ber and bei functions from the series

ber or = i - + - .r^To.,-

02 . A 2 <j2. A 2.h2.ft2

bei a;

a 2 r 2

2 2.

(41*)

These functions, and the corresponding functions of the second
kind (ker or and kei r) are the key to the reactions in cylin-
drical structures when periodic driving forces are applied.
We began with the assumption that = cos w/. We must

BER AND BEI FUNCTIONS

therefore have p\ = , cos co/ and similarly retain only real

quantities in the final solution. Writing for //, CV'*, and sub-
stituting (410) in (40), for the real portion of the solution,

p = - f o cos co/ C[ber ar-cos (co/ + </>) bei r-sin (co/ + <)].

(42)

To determine C and we note that, for all values of /, p = o
when r a because of free communication with the atmos-
phere. Then, if

ber aa = X cos x y bei aa = X sin AT, X 2 = ber 2 + bei 2 aa,
we must have

-y COS CO/ = CX COS (co/ + + #);

hence,

1 i - J 1 A> ^>

= x y cos = ^>ber #, sin = ;>bei aa, C = -7^1
A A dX

and the final solution is therefore

ber ar ber a^ + bei or bei

_ 5^ F

5= _^_. J -

a L

ber j + bei"

QQg

5f ft fber or bei # bei r-ber OLO\ .

-- o -- - TIT^ --- sm

^/ I ber 2 aa + bei 2 aa J

In this solution for the pressure the coefficient of the amplitude
<> cos co/ represents the stiffness per unit area as a function of r.
From (41^) ber (o) = i and bei (o) = o; thus the stiffness at the

center is -y i -- ~~ , falling to zero when r = a. The sine
term is in phase with the velocity (f = co sin co/), hence the
resistance per unit area is -- times the coefficient of sin co/ in

34 THEORY OF VIBRATING SYSTEMS AND SOUND

The resistance is therefore .

0>/7

B bei ota

at the center, and

(42*).

also falls to zero when r = a. Thus the vibrating piston with a
variable distribution of impedance over its surface, should now
be considered to bend slightly, if the argument is to proceed
rigorously. This refinement, however, is beyond the practical
necessities of the case, and we continue the discussion on the
basis of the piston system of one degree of freedom.

10 5 X1.6

1.4

to:

.S3
*co

\ Resistance at Center

Stiffness at Center

8.0X10*

4.0

2.0

2OOO 400O 6000 800O

Frequency

FIG. 5. REACTIONS IN AIR DAMPING FILM: PISTON SYSTEM.

In Fig. 5 there are plotted curves showing roughly the fre-
quency variations of resistance and stiffness per unit area at the
center of the piston, and mean resistance and stiffness per unit
area\ all taken for a piston and damping disc of radius 1.63 cm.
separated by an air film 2.2 X io~ 3 cm. thick. From these curves
a good idea can be gained of the typical reactions in the sim-
plest air damped system. The data are based on some old

AIR DAMPING; PISTON SYSTEM 35

calculations for an early form of condenser transmitter, 1 the
mean values of the constants being obtained by integration
over the surface of the piston.

This theory has been tested, by application to a condenser
transmitter of simple design, making certain modifications to
allow for the shape of the membrane and it has been found that
a good quantitative explanation of the behavior of the instru-
ment is given. An interesting special test has also been made of
the resistance formula for low frequencies, setting up a con-
denser transmitter l having a diaphragm weighted at the center
with a heavy disc (to insure piston motion and lower the natural

frequency), and measuring the quantity A = ~-- from an elec-
trically obtained graph of the natural oscillations. In this case
(since ber (o) = i and bei (a.r) = \c?r* as or = o) we have, per
unit area

i B\c?a* - 2 r 2 ] R f 2 2 , , ,

r'o = ~ , = ~> 2 - r 2 ), (43)

co a L 4 J 4

and the total resistance, since R = ,-, by (37), is

27T

or, per unit area, on the average

(44*)

This gives the point at zero frequency for the mean resistance
shown in Fig. 5.

1 Phys. Rev. y XI, 1918, p. 449. The integrated values for total stiffness and total
resistance over the piston are, respectively,

ira*B[ 2 ber a bei 'a* ber' aa benaa

d L (xa ber 2 a* + bei 2 a* J'

and

/ - 2 ^^[ b ei ota bei'aa + beraa ber'aal
toad I ber 2 tf + bei 2 atf J

36 THEORY OF VIBRATING SYSTEMS AND SOUND

For the technology of condenser transmitter design, the
reader may refer to recent papers by E. C. Wente l in addition
to the reference already given. In recent practice the tendency
has been to make use of the resistance properties of the air film
to a high degree, and the problem has been considered here, in
a simplified form because the instrument is an outstanding
example of the application of viscosity damping to the vibrating
system.

12. Equivalent Piston; Mean Velocity; Diaphragms

We now consider the principles and methods used in deter-
mining the area of the equivalent piston for an actual vibrating
system. If the system radiates energy into the medium (i.e.,
into a compressible fluid) it is required to know the amount of
fluid displaced in unit time in terms of the motion of the system.
This constant, known as the "strength of the source" or the
"rate of emission of fluid at the source" is best given as the
product of a certain area and the mean or average velocity over
that area: A = %S. If A is known we can compute the amount
of radiation from the system and hence its radiation resistance.
Both and S are, within reasonable limits, at our disposal, as
long as their product is made equal to A.

In practical problems it is usually of advantage to adjust S
so that, for purposes of calculation, the radiation conditions of
the problem are most simply taken into account. For example,
suppose a positively driven diaphragm of area S\ is fitted to a
tube of slightly larger cross-section 2. In this case the sensible
procedure is to replace the diaphragm (of unequal motion over
its surface) by a piston of area 62 and consider the piston to

_ O _ __

have a mean velocity 2 = TTI> in which \\ is the mean velocity

of the diaphragm over the area S\. The area of the equivalent
piston being so determined, the inherent mass, stiffness and re-
sistance properties of the system may be related to the piston

1 For these and other references to Acoustic Devices, see Appendix B.

EQUIVALENT PISTON; MEAN VELOCITY 37

in the way which best represents the internal structure of the
system.

To determine A in a given case we must compute the mean
velocity. A typical example of determining the mean velocity
over the working area of a (simplified) telephone diaphragm
will be given. The determination of the shape of a disc clamped
at the edges, and vibrating in its normal modes, is a difficult
problem, involving a differential equation of the fourth order
and we shall only quote the result. It is shown in Rayleigh
( 217-2210) that the amplitude of such a disc is

* = C[/ (*r) + X/ (i*r)] cos (/ - e), \
in which

(45)

(ih = thickness; p = density; E = Young's modulus, <r =
Poisson's ratio; n =2?r times any natural frequency.)

At the boundary (r = a) there are two conditions to observe;

both and -7- must vanish. These two conditions determine X
dr

and k\ and we have (assuming X known)

cos / -

For the first natural frequency kia = 3.2, \i = T ( ., < -056,

J o

the natural frequency being [cf. (45)],

(3.2)2 VE- h

= =

(Graphs of the Bessel's functions used, and of the shape of the
diaphragm in this mode are given in Kennelly, "Electrical
Vibration Instruments," pp. 304-305.) The average velocity,
in terms of the maximum (central) velocity is:

- ^ rw - r" u&gjm,.*. (47)

T?a 2 J Q k 2 a?J 1 + X

38 THEORY OF VIBRATING SYSTEMS AND SOUND
The integral is readily calculated since

The function iji(ix) which appears on integrating the second
term in the numerator is known as h(x) and is a real quantity,
if x is real; tables of h(x) are available. The calculation (47)
gives

t~= -306 { , (48)

which is the mean velocity sought. This value, applying to the
gravest symmetrical mode is sufficiently accurate for many prob-
lems in which radiation from telephone diaphragms is consid-
ered, provided the frequency is less than twice the first natural
frequency, and the vibrations are small.

The reader may compare the integration (47) with that
given in Kennelly (loc. cit.), in which the square of the velocity
is integrated over the diaphragm for the purpose of determining
the mass constant. It may also be noted that the function

( x
(49)

represents the static deflection under uniform load P per unit
area, the elastic constants being taken as before. (See Love,
Elasticity, Third Ed., 1920, p. 494.) If this function is inte-
grated over the area, for the purpose of obtaining a mean static
deflection we find = i [cf. eq. (20)]. Thus in terms of the
velocity at the center, the mean velocity of the diaphragm does
not vary greatly over the frequency range from zero to the
first natural frequency. In view of certain departures from the
theoretical mode of vibration in the actual form of vibrating
diaphragms, due to lack of symmetry in the driving force, over
driving, etc., it is not worth while to be too precise in the deter-
mination of such "constants" as the mean amplitude. The
example has been given merely to illustrate the principle, and
incidentally record a few facts concerning the vibration of a
clamped circular plate.

PROBLEMS

The experimental study of diaphragms, i.e., of the variation
of their constants under the constraints imposed in various
structures belongs to the domain of engineering development
and design with which we are not primarily concerned. The
purpose of this chapter has been to present, in reasonably
concrete terms, the essential properties of the simple vibrating
system, and if this has been accomplished we have reached a
point of departure. 1 In the next chapter a discussion will be
given of general methods applicable to systems of several de-
grees of freedom. In later chapters we shall deal with the pro-
duction and transmission of sound radiation.

PROBLEMS

1. (a) Given the following constants of a telephone diaphragm (in
terms of motion at the center) for frequencies near resonance:

m = .675 gram, s = 2.67 X io 7 dynes/cm.
r = 270 dyne. sec. /cm.

Determine the resonant frequency, /"<,; the damping coefficient, A;
the frequency of the natural oscillations.

(b) The center of the diaphragm is given an arbitrary displace-
ment of .01 cm. from the equilibrium position and let go. Write the
equation of motion.

(r) Write the equation of motion of the center of the diaphragm
for an arbitrary initial velocity of 60 cm. /sec., the system being
started from the equilibrium position.

2. (a) A constant force F is suddenly applied to a simple vibrating
system. Discuss the growth of the displacement with time for the
periodic case. Find the value of / for the first point of inflection (or

1 Some readers may desire at this point to consider in detail such matters as the
elastic constants of solids, and the vibrations of bars and plates. For these, the refer-
ences are to the classical theory, e.g., Lamb, Chaps. IV and V. It is planned to avoid,
wherever possible, mere restatement of material already available.

40 THEORY OF VIBRATING SYSTEMS AND SOUND

point of zero acceleration) of the growth curve; show also that at
that point

qs = r{ if q = - - .

(V) A constant force F is suddenly removed from a critically
damped (i.e., aperiodic) system. Discuss the decay of the displace-
ment, noting that the two roots of the auxiliary equation are equal
in this case, and the solution must be in the form

= (A 1 + B'i)e-".

Determine A' and B'\ is there any point of inflection in the decay
curve?

3. A force <> cos w/ acts on the diaphragm. Considering the equiv-
alent simple system to have the constants as given in Ex. i, find the
r.m.s. values of the amplitude and velocity at resonance, for ty = 10
dynes. Find the r.m.s. values for the amplitude and velocity for the
frequencies 30 per cent above and below resonance; for the frequencies

(n A); also for zero 'frequency. Plot all these data to show the
frequency response of the system.

4. Discuss analytically the motion of the central point of the
diaphragm, when following steady state conditions the driving force
o cos / is suddenly removed.

^?o

5. Taking the velocity as = e itat in the steady state theory of

the simple system, show that the instantaneous rate of dissipation in
the system is 2 r.

6. Derive the simplest expression you can for the energy stored in
the simple system, in the steady state. Does this fluctuate? Is the
maximum kinetic energy ever equal to the maximum potential energy,
and if so, when ?

7. Derive equation (19).

8. A circular membrane of sheet steel, 4 cm. in diameter, and .005
cm. thick is stretched to a certain tension. Under -a uniform static
pressure of half an atmosphere applied to one side of the membrane,

PROBLEMS 41

the central deflection is .01 cm. What are the first three natural fre-
quencies for the normal symmetrical mode of vibration ?

9. A pistonlike air-damped system was set up with piston of
radius 1.63 cm. and mass 41.9 grams. The observed damping con-
stant of the system was 3.4 X io 3 , for a mean separation of 2.9 X io~ 3
cm. between piston and damping plate. From these data what would
you determine for the viscosity constant of the damping fluid, on the
basis of low frequency theory ?

10. Show that for low frequencies, the mass coefficient for a circu-
lar plate, clamped at the edge, is, in terms of the motion at the center,
of the total mass of the free portion of the plate. Show also that the
corresponding stiffness coefficient, for low frequencies, is

, = '- 28 - r-^-Vv t cf " (49)1,
9 (i cr;<^

and on this basis show that the first natural frequency of the plate is

8 A_ 111X7,

which is closely equivalent to the more rigorously obtained result of

CHAPTER II

GENERAL THEORY OF VIBRATING SYSTEMS; RESONATORS AND

FILTERS

20. Generalized Coordinates

In a system of several degrees of freedom, that is, a system
in which more than one independent variable (1, fe, 3 . .) is
required in order to adequately describe the motion, it is gen-
erally impossible to deduce the equations of motion from the
Energy Principle, as was done in 2 for case of a single degree
of freedom. The reason for this is, that while the energy prin-
ciple is necessarily valid, it can give no information regarding
interchanges of energy between the inner parts of the system
on which external forces do not act directly. The equations of
motion of the system must take account not only of the external
forces, but also of the mutual reactions within the system.

To begin with, the independent variables (fi . . . w ) repre-
sent only those displacements which are allowed to take place,
by virtue of the constraints inherent in the system. If we pro-
ceeded according to the Newtonian Method, the forces due to
all these constraints would be resolved each into 3 rectangular
components and, with the applied forces similarly resolved, we
should require ^m ordinary mechanical equations to discuss
the motion. But if we resolve the applied forces, so that only
those components each of which acts in the direction of a possi-
ble displacement (/ are considered, and if we describe the con-
stitution of the system only in terms of these displacements
(1 . . . m) and of certain mass, stiffness and resistance factors
which take into account the internal reactions, then by the
application of one of the broader dynamical principles (such as
that of D'Alembert, or of Least Action) it is possible to obtain

TWO DEGREES OF FREEDOM 43

m equations which fully state the motion of the system. The
particular machinery which we shall use to bring all this about
is the method of Lagrange; the simplified coordinates (i . . . m)
are known as generalized coordinates; the forces l which tend
to increase these displacements are called generalized forces;
and as a result of the application of the method, the equations
of motion are almost automatically obtained in concise and
convenient form.

This method is fundamental to mechanics generally and
indepensable in dealing with, the typical problems presented in
vibrating apparatus. While we cannot take time to derive it,
we can become familiar with its mechanism and its underlying
principles. This we propose to do by an inductive method,
first considering an easy problem in two degrees of freedom.

21. System of Two Degrees of Freedom; Natural Oscillations in
General

A thin string of length j/, stretched to tension r, has at-
tached to it equal masses at two points distant / from either end.
The masses (i) and (2) are allowed to vibrate in a vertical
plane; their displacements from the equilibrium position of the
system are respectively 1 and 2- Friction is taken into account
by equal resistance constants r in each degree of freedom. The
component of force on either particle, due to tension in the end-
portion of the string and resolved in the direction of motion is

v. It is evident that there are two types of motion possible in

the system, i.e., one with the displacements 1 and 2 having the
same sign, and the other with 1 and 2 of opposite sign. In
either case the component of the force on the first particle (for
example), due to the tension in the mid-portion of the string,

and resolved in the direction of 1 is ^(1 2). Letting c = -.

1 Strictly, the generalized force S^ corresponding to any generalized coordinate &
is defined as the ratio of the work done to the displacement, when & is varied by an
infinitesimal amount 5&, all of the other coordinates remaining constant.

44 THEORY OF VIBRATING SYSTEMS AND SOUND
the equations of motion are therefore

2<r& -

in which *i, %, are the impressed forces acting on particles I,
2, in their respective directions of motion. It appears that
the interest in the problem will center chiefly on the effect of
the elastic connection or "coupling" between the two vibrating
masses.

The complete solution, as in the case of one degree of free-
dom, must contain expressions for both steady state and tran-
sient phenomena. Considering the latter first, and assuming
motions of the type = X ' we have, since *i = 2 = o for free
oscillation,

= o, (0 = \*m . , ._,,

(5 2 )

- Ch +

=0.

In these two simultaneous equations, for consistency, we must
have

-c

- c ,

or
or

*- 0(0 + 0=o,

L = + C y /32 = C.

We thus obtain two equations in X, namely

\ 2 m + Xr + c = o and \ 2 m + \r + y = o,
whose solutions are

(53)

(54)

= _^ , x/ __

r . he
X 2 = i V^ -

A dz

- A db

(54^)

taking ' = Vw 2 A 2 as in 3. Now it is apparent that there
are two types of oscillation which are normal for the system,

i.e., one of frequency/i = y and one of frequency/2 = \/3/i.

NATURAL OSCILLATIONS

On account of the symmetry there is no reason for associating
one of these motions more closely with particle (i) than with
the particle (2); and it must be that both types of motion are
simultaneously present in both degrees of freedom. Thus for
the transient oscillations we must have (cf. eq. 6'),

1 = Aie~" sin (n\t + 0i) + A 2 e'^ sin (n' 2 t + 2 ), 1

^ = A\e~" sin (n\t + 0'i) + A'*c~" sin (' 2 / + 0' 2 )J 55

Now all the 8 constants are not independent; only 4 are required
for the general solution of 2 simultaneous second order equa-
tions. Letting /3i = c y which corresponds to the lower-frequency

oscillation (/i = n'\\ we must have 1 = 2 (eq. 52) and

\ 2?r /

similarly if 2 = c y for the other mode, i = 2. The only
solution consistent with these conditions is

& = [Ai sin (n'it + Oi) + A 2 sin (n' 2 t +
fe = [^i sin (w'i/ + ft) - ^ sin ( f a / +

the numerical equivalence of the coefficients being due, of
course, to the physical symmetry in the system. A\ y A 2y 0i, 02,are
to be determined in terms of the initial displacements and
velocities assigned to the two masses (i) and (2).

Important conclusions which may now be reached, for this
symmetrical system of two degrees of freedom are: (i) that the
squares of the two natural frequencies are harmonically dis-
posed with respect to the square of the natural frequency of
either particle when the other is held in its equilibrium position,
and (2) all possible free oscillations of the system are accounted
for by compounding one motion in which the two particles are
strictly in phase, and another in which they are strictly opposed
in phase throughout the course of the motion; thus the ampli-
tudes of corresponding transients in the two degrees of freedom
are numerically equal.

For the more general system of m degrees of freedom we
conclude: (cf. Lamb, p. 44).

46 THEORY OF VIBRATING SYSTEMS AND SOUND

1. There are m natural frequencies, and hence m compo-
nent natural oscillations occurring simultaneously in all parts
of the system. These are the "normal" oscillations of the
system.

2. To the m natural frequencies correspond m damping
factors (Ai . . . Am) and these natural frequencies and damping
factors are given by a determinantal equation of the mth order
in X 2 , analogous to (53) in the foregoing example.

3. In each of the m modes of vibration the system oscillates
exactly as if it had only one degree of freedom, the coordinates
(i> 2 . . m) having constant ratios as far as amplitude is con-
cerned; the only arbitrary constants being the scale of the am-
plitude, and the phase constants.

The general equation embodying these conclusions, for the
transients in the rth degree of freedom is:

f f - S Bjdrjc-** sin (n'jt + 0,),

(55*)

in which the im constants Bj and fy are arbitrary, or assigned
factors, dependent on initial velocities and amplitudes; while
the ratios of the constants A T j are determinate, since they
depend on the equations

. (56)

. . . A\ m = unity. For example,
B 2 = A^ A\\ = yfi2 = i, A 2 \ = i,

22. Steady State Theory, for Two Degrees of Freedom and in
General

To determine the behavior of the system in the steady state,
and so complete the solution, we let one particle be driven by

^21 fe

^22 fe

Aim

XT; -I | C-i

|}aKj ---ll SI

..";*; i

n -n m dlm

^ fm

__ -^mm

1 .! A\\ 1

n-n m 1m

We can take A\\ = A\*
in (550) we have B\ = A
= i.

STEADY STATE THEORY

a force *i = * <u * (eq. 51), and simplify the procedure by
taking ^2 = o. This entails no sacrifice in essentials, on account
of symmetry. We then have, for motions of the type 1 = ol e i(at

/>co

(57)

The form of these equations shows at once the simplification
to be gained, if we speak in terms of impedance and velocity,
as in the dynamical theory of electrical networks. The set (57)
is equivalent to

__ _ n

. 2
/CO

r + /

(57*)

and, solving for the velocities, we have

and &j

/CO

-, (58)

1 <V 7<1 """ *~( C M 4. M

a. - -.-)( a + .- I I a -.-)( a + -.- I

V /CO/\ /CO/ \ /CO/\ /CO/

or, letting

(a )=ai =r + /(wco -) and 2 = r + n wco -),

\ /co/ \ co/ \ co/

c* I

These equations might be rationalized and studied in detail for
the case a real applied force *o cos co/, acting on particle (1)-
Practically, in view of the similar study made in (7) for the
simple system this laborious operation is not necessary, in
order to understand the behavior of the system. Note first

48 THEORY OF VIBRATING SYSTEMS AND SOUND

the quantities of impedance dimensions, giving them special
designations:

aw D iu /co

*'ii = - = ; ^12 = via* = D. (Co)

J *

The first maybe called a " driving-point impedance " or "appar-
ent impedance at the driving point"; the second a "transfer
impedance" or "apparent impedance at point (2) for force
applied at point (i)." Both Zn and Z i2 have minima for the
two natural frequencies (cf. iy 2 ) found in the "transient"
solution, 21 ; at these frequencies very large forced oscillations
occur. At the intermediate frequency for which = o, a = r\ Z\\
is a maximum and particle (i) moves with small amplitude; but
at this frequency particle (2) has a larger amplitude, in the ratio

. . c may be considered a "coupling factor," increasing with

l(,T

increasing tension in the string, or decreasing distance / between
the particles. Thus for the frequency range between/i = \/

and/2 = v 3/1 the apparatus, driven at point (i), and with am-
plitude or velocity observed at point (2) behaves like a crude
filter, the sharpness of declines in amplitude for/</i and/ >/2,
and the height of the minimum amplitude between f\ and/a both
being increased as the resistance constant r is diminished; the
intermediate minimum amplitude being farther increased with
increasing coupling factor c. The general properties of the anal-
ogous electrical system, consisting of two identical meshes,
coupled through a mutual capacity, are so well understood that
we shall not pursue this limited problem further.

Some general deductions with regard to the steady state
behavior of the system of m degrees of freedom are now war-
ranted.

( r) When a periodic force of type *<> cos w/ acts on any part
of the system, every part of the system executes a vibration of
the same period, the velocities and amplitudes of which are
given by equations of the type (580). In these equations there
always appears in the denominator of the right-hand member

STEADY STATE THEORY 49

the same determinant />(i, <*2, . . . <* w ) which when placed
equal to zero [D(pi-p 2 . . . ft)> (eq. 53)] determines the natural
frequencies and damping constants of the system. Thus
maxima in the forced oscillations occur for those values of the
quantities ft which give D(a\, 2, . . . am) minimum values.

(2) The relative phases and amplitudes of the motions in
different parts of the system will vary as the frequency is
varied. But if there were no dissipation in the system^ the
oscillations would all be in constant phase relation. [Note

-;i f (5&OJ
2 c

(3) In general, for a system of m degrees of freedom, for
a single force *, applied to the^'th degree of freedom,

*.-**, and fc-* (60)

in which D = aia<2 . . . a m , the determinant of the coefficients
of the general equations corresponding to set (57), and M* is
the minor of D obtained by suppressing the &th column and the
jth row of/), and multiplying by ( i) J+k .

(4) The general solution of the problem is obtained by com-
bining equations of the type (60) with corresponding equations
of type (55^) and determining from the initial conditions the
im arbitrary constants.

Finally, to the reader who has pondered the matter, it must
be evident that since the determinant D plays such an impor-
tant part in the analytical discussion of both transient and
steady state conditions, it must provide a key wherewith to
obtain equations (60) directly, if equations (55^) are known, and
vice versa. This is indeed true: it must be so, in order to trans-
late the intimate physical relations between steady state and
transient phenomena in a given system into adequate mathe-
matical terms. This was realized by Heaviside and the idea has
been further developed by later students of oscillation theory.
Those interested may refer to papers by J. R. Carson (Phys.
Rev. y IX, 1917, p. 217; also the series on "Electric Circuit

50 THEORY OF VIBRATING SYSTEMS AND SOUND

Theory, etc./' Bell Sysf. Tech. Jour., 1925-1926) and T. C.
Fry (Phys. Rev.,XlV y 1919, p. 117) for the analytical discussion,
which is too specialized to include the present outline.

In view of what has just been said, it usually suffices to
consider only the transient state of oscillation in order to under-
stand the mechanism of a given system.

23. Lagranges Method and Equations of Motion

To proceed with the general theory, consider first the
Potential Energy, V. In mechanical problems it is usually an
economy to derive such forces as depend only on the position
of a body, by differentiating a certain Potential Energy function
(if one exists) with respect to the coordinate in the direction
of motion. Suppose that we had written, for the forces due to
elastic reactions, in equation (51)

- 02 = ~,- = 2f fa -

O?2

Integrating (61), we have

V =

or equally

V =

Since for consistency </>i(i) = c%i 2 and #2(&) = & 2 there must
be for this particular problem, a potential function

V = c(tf ~ fife + fe 2 ). (6ia)

The coefficients in the right-hand member are all numerically
equivalent to c, due to the accident of equal spacing of the
masses in loading the string.

Now consider the Kinetic Energy. For the discrete masses
mi = mz of the problem we write at once

T - lm(ji + &0, (62)

LAGRANGE'S METHOD; EQUATIONS OF MOTION 51

there being no mutual reactions due to inertia. This is gener-
ally true in purely mechanical problems. (In the dynamical
theory of electrical circuits mutual inertias (mutual induc-
tances) do occur: the coupling between different degrees of
freedom is accomplished more often in this way than other-
wise). The advantage in setting up the quadratic expression
for Kinetic Energy and obtaining the rate of change of momen-
tum in any part of the system by successive differentiation of
this function is evident. In the general case the homogeneous
quadratic function of the velocities contains cross product
terms mtji& to allow for the mutual inertia reactions and
we have for the force due to inertia, in any degree of freedom
of the system

,, x

(63)

To take account of frictional forces, or more correctly, of
energy dissipated in friction Lord Rayleigh introduced another
homogeneous quadratic function of the velocities, with resist-
ance factors as coefficients, which function when differentiated
with respect to a given velocity gives the frictional force against
which work is done in the given degree of freedom when motion
takes place. This function, known as the Dissipation Func-
tion, would be written for the simple problem we have been
considering

F - 4K*i 2 + *2 2 ), (64)

but in general it contains cross-product terms as well, though
not in the problems we shall encounter. The function F is
one-half the total rate of dissipation of energy in the system as
a whole.

According to the method of Lagrange (ably presented by
Rayleigh, I, Ch. IV, 80-90), the equations of motion in
terms of T y V and F are

T =

F =

V = J('lll 2 + *22fa 2 + . + fl2*lfe + 0,

52 THEORY OF VIBRATING SYSTEMS AND SOUND

and so on for the other coordinates. It is evident that, with
7", V) and F known (eqs. 6i#, 62, 64) for the problem of
the preceding sections, we may by applying (65) obtain at once
the equations of motion previously given (eq. 51).

In the classical notation, the three quadratic functions are
written:

(66)

the various factors a jky b^ Cjt being respectively the mass,
resistance and stiffness constants. In the case of the constants
indicating coupling, a jk = a^ etc.

From an algebraic standpoint it is usually possible to choose
the coordinates so that a jt = b^ = o* = > eliminating cross-
product terms; then, the new coordinates l 'i . . . %' m become
"normal" coordinates. These have interesting properties; the
m equations of motion then contain each only a single variable,
&, the mutual reactions between the different degrees of free-
dom being eliminated. One reason for the name " normal"
is that, in replacing any actual coupled system by an equiva-
lent normal system, the natural oscillations which occur sepa-
rately in the different degrees of freedom of the normal system
are of the same frequencies as the natural or normal oscillations
which occur, simultaneously in all the degrees of freedom of
the original system. The idea is of interest mathematically,
and is occasionally of practical use as will appear in a later
section. We note in passing that in studying the normal modes
of a circular membrane ( 10) the idea of normal coordinates
is implied; one such coordinate being used for each mode of
motion of any part of the membrane.

1 See Whittaker, " Dynamics," Article 94, for practical details of the transforma-
tion; also Rayleigh, I, 96-1 oo.

RESONATORS 53

24. Resonators

With the needs of the next section in mind, digress for an
interval to consider resonators. In acoustics the term resonator
has come to mean a simple vibrating system consisting of com-
pressible fluid contained in an enclosure which communicates
with the external medium through an opening of restricted area
in one of the walls. Typical cases are illustrated in Figs. 6 and
6a. In either case, the stiffness in the system is due to the

volumetriccompression ~ within the enclosure, the vibrating

mass being the average quantity of fluid which surges to and fro
in the neighborhood of the opening. The theory of resonators
has been given in detail by Helmholtz (who invented them) and
by Rayleigh: here only a simplified treatment is required.

FIG. 6. FIG. 6a.

RESONATORS.

Take first the case of the "bottle/' Fig. 6. If represents
a small motion of the "plug" of air in the neck of the bottle,
we have, for the force acting on the area <? of the neck,

Q = - s - - vr. = - t, (67)

V O * o r o

. L . , dV. , . . . . . n

in which 7/>orr is the increase in air pressure, originally p oy

due to adiabatic compression -- ; we borrow from 30 of the

following chapter the useful relation

Vpo = pc 2 , (*') from (1050) and (108)

54 THEORY OF VIBRATING SYSTEMS AND SOUND

in which p is the density and c the velocity of sound. The mass
of the (nearly cylindrical) plug of moving air is pSl, if / is the
length of the neck. Thus we have for effective mass and elastic
constants

(68)

the equation of motion being a\\ + c\% = o;
whence

i \cl c f~ S" c \~K ,, ,

re = T^y. = T^V. (69)

for the natural frequency. The quantity K = -.- is called the con-

ductivity of the neck, on account of its dimensions.

The natural frequency of any simple resonator is best given
in terms of the " conductivity" of its aperture, its volume, and
the constant c which depends on the medium in which the
resonator is immersed. In the case of resonator of Fig. 6a with
no neck there is an unequal distribution in fluid velocity at dif-
ferent points in the opening; the effective mass of the moving
air in the circular aperture in the thin wall has been determined,

in terms of a conductivity K as - ,/-, the quantity K being equal

to the diameter of the opening. (See Lamb 82, 86, Rayleigh,
Vol. II, 306.) Now imagining an equivalent piston of area S
and mean amplitude of motion , (c\ being unchanged) we
obtain equation (69) for the natural frequency as before.

The notion of conductivity, in addition to leading to a sim-
ple formula for the natural frequency, facilitates the determina-
tion of total conductivity, when two or more conductivities are
connected in series or in parallel; the formulae are identical
with those used in computing electrical conductivities. Sup-
pose for example it is desired to "correct" the conductivity
formula for a short neck, to allow for the divergence of the air
stream at the end of the neck: this is equivalent to adding the

RESONATORS 55

reciprocal conductivity for the neck itself to that for the ter-
minal orifices, for which, taken together, the formula K = 2r
applies. In this case we have

TT S *r 2 v

Ki - - - - K 2 = 2r,

" 7r

These results will find application in the problem of acoustic
filters.

Thus far no account has been taken of the influence of dis
sipation on the behavior of the resonator. Dissipation due to
radiation is usually of much greater importance than dissipa-
tion due to friction in the neck of the resonator, provided the
resonator is so situated that it can radiate. (In a coupled
filter system, for example, it is not designed to radiate.)

We have tacitly assumed in the preceding treatment that
the resonator was small as compared with the wave length; it
is therefore reasonable to make the same assumption with re-
gard to the aperture. Making use again of results to be obtained
later, we have, due to radiation,

d75)
and as

The form in which b\ and Ai are written here emphasizes their
dependence on the dimensions of the resonator. The derivation
of the formulae for b\ and a\ will be considered in Chapters III
and IV. For the present the reader may merely note their
dimensional correctness.

Viscosity damping, while serious only in the case of resona-

56 THEORY OF VIBRATING SYSTEMS AND SOUND

tors with small apertures, or long necks, should at least be con-
sidered. The mass of moving air in the latter case is plS. To
find the frictional resistance we make use of a relation developed
in Appendix A which embodies Helmholtz's coefficient for the
resistance offered to the oscillations of a viscous fluid in a tube:

= + R/% in which R = - Vip/zw , (69^)

~ is the pressure gradient, r the radius of the tube, and /x the
viscosity coefficient. For a tube of section S, we have,
force due to friction = Sdp = R/S- |,

and therefore

fa = R/S; (6 9 d)

thus for the component of damping due to viscosity,

A 2 = =

2p/S

It will be noted that dimensionally, the r in this formula
merely offsets other linear dimensions in the definition of JJL and
p; the damping coefficient due to friction does not depend on
the area <?, except as undue constriction in S = irr 2 enhances
friction according to Helmholtz's coefficient. Thus for n similar
necks to a resonator, if all the motions were in phase, the ratio
A 2 would be the same as for one alone; but of course the energy
dissipated (b* | 2 ) would be proportional to n. Herein is sug-
gested a means for varying the natural frequency of a resonator
without changing its damping, if the damping is due entirely
to friction in the neck; for the frequency will rise with , or
with the number of orifices used.

The raison d'etre of a resonator is to select and amplify a
sound component of given frequency, in the same way that an
alternating current is selected and amplified by tuning an elec-
trical circuit. (The classic example of this procedure is the work

RESONATORS 57

of Helmholtz on the analysis of the vowel sounds, but there are
important modern applications of resonators, as will be noted
in Appendix B.) To illustrate, we shall show how the particle
velocity and the excess pressure of an incident sound wave are
amplified when the resonator is tuned to the frequency of the
incident wave. As before, we borrow from Chapter Ilia neces-
sary mechanical relation; there it is shown that for plane waves

(30

in which dp is the excess pressure, |i the particle velocity and
pc the radiation resistance of the medium.

The driving force on the mouth of the resonator is (to first
approximation) S-dpe" 1 , hence the velocity in the mouth of

the resonator at resonance is , ; and if the opening is not

so constricted that it brings viscosity into play, the resistance
b\ will be due entirely to radiation. Making use of equations
(69), (175)* and (1170) above, we have for the velocity in the
orifice

in which 1 is the maximum particle velocity of the incident

sound wave. The quantity -- = , since it is the ratio T-

oro TTO 1

may be termed the velocity amplification coefficient of the reso-
nator; and if we had a velocity measuring device (e.g., a hot
wire microphone) set up in the orifice, we should expect a gain
of this order of magnitude in using the resonator to amplify the
original particle velocity of the wave. To illustrate, suppose
that a resonator is tuned to 1000 cycles, and has an area of ori-

fice S = - sq. cm. (as in the problem of 25). We then have,
4

since - = 5.3, a velocity amplification ratio of 450, which would

obviously be worth while. We remark that since the velocity

58 THEORY OF VIBRATING SYSTEMS AND SOUNI

amplification varies inversely as the area of the orifice,
small orifices usually imply losses due to viscosity, one sh<
not expect to realize in practice the large values of ampli
tion which the simple theory would indicate.

Following Helmholtz, many have used the resonator
a small tube leading from the back of the resonator to
ear, so that the pressure changes within the resonator act
tually, with little loss, directly on the ear. Hence the
ception of pressure-amplification, that is the ratio of maxir
pressure within the resonator to the maximum driving pres
of the incident sound wave. Since the excess pressure in

resonator is, from (67), - -, we have,

ATTC 3 . A.TTC

putting in terms of |i from (6yf). The quantity ^^ is no\

pressure-amplification coefficient of the resonator. In gen
this is less than the velocity amplification coefficient, for
quantities are in the ratio

Pressure Amplification __ w 6* __ iirS

Velocity Amplification ~~ c K ~~ \K*

in which X is the wave length of the sound. In the example

viously used, from 15 (since - = --- (at 1000 cycles)

c 5-3

Y? = -) the pressure amplification is the less in the ratio c

K . 4 .

but it is evident that the resonator is still of advantage in

plifying the sound as it affects the ear.

A problem in sound amplification for a resonator of
degrees of freedom is set for the reader, at the end of this cl
ter. Some further consideration of the behavior of a reson
in a field of sound waves will be given in 47, Chap. IV.

It must be noted that the simple theory of this section

RESONATOR COUPLED TO A DIAPHRAGM

plies only to the gravest mode of the typical resonator. In the
higher modes the motions within the resonator are not all in
phase, and various patterns of nodes and loops are set up
within the enclosure. These depend on the shape of the en-
closure as well as the frequency; the calculations become much
more complicated and except in the case of organ pipes the
higher modes are of little practical importance. The theory of
organ pipes can rest until certain other problems have been
considered.

25. Resonator Coupled to a Diaphragm

Consider now a problem in two degrees of freedom, one of
the elements being a resonator. The system is sketched in
Fig. 7; one wall of the cylindrical enclosure being a telephone

FIG. 7. DIAPHRAGM AND COUPLED RESONATOR.

diaphragm i.e., a clamped circular plate. We desire to find
the effect of coupling the resonator to the diaphragm; to add
interest to the problem we shall design the resonator before-
hand so that its natural frequency, as a simple system, is equal
to that of the diaphragm alone.

The following constants relate to the diaphragm: mass
constant #1 = .50 gram, in terms of velocity at center; area

60 THEORY OF VIBRATING SYSTEMS AND SOUND

= 15 sq. cm. = area of the cylinder section; average amplitude
= .30 of amplitude at center (cf. I2);/ = 1000, whence ^(cen-
tral stiffness coefficient) = 1.97 X io 7 c.g.s.; resistance (central)
b\ = 200 c. g. s.

For the resonator, take the length of the cylinder L = 1.86
cm.; the volume is then V* = 27.9 cu. cm. and for a diameter of
orifice K = i cm. we shall have/ = 1000 since c = 3.32 X io 4
cm. /sec. The constants of the resonator are then, since K = i.o,

a 2 = P S' 2 (= 8 X io~ 4 ),
2 = -~ 2 ( = 7.6x10-2),

2 02

c 2 = P ~(= 3.16 X io'),

" o

in which S is the area of the equivalent piston, or approxi-

mately the area of the orifice, --.

Now to deal with the diaphragm onanequivalentpistonbasis,
let^' = 15 sq. cm. (area of diaphragm) and A = (.3 X 15) = 4.5
sq. cm. Letting 1 stand for motion of the center of the dia-
phragm, and 2 stand for the motion in the orifice (piston) we
have, for the excess pressure in the cylinder

The expression in the parentheses is a net equivalent piston
area multiplied by a certain amplitude, say A-%. Letting this
be a "normal coordinate" for the moment, the potential energy
due to compression is

v =

*/O

PC 2

RESONATOR COUPLED TO A DIAPHRAGM 61

Now adding the potential energy due to the diaphragm itself,
we have for the total potential energy of the system,

V =

or, if we take a new set of constants, viz.,

22 = C2 y
^

in which & = - = 5.74, we have
o

V = i(^1ll 2 - 2*i 2 {lfa + ^22^2 2 ). (70)

The Kinetic Energy and Dissipation Functions are:

7*, ^, and -F are now in the form of (66) and applying the La-
grangian operation (65), letting applied forces equal zero, we
have the following equations of motion for natural oscillations:

+ ^2^2 + ^22^2 ^12^1 = O.

0i& - a 2 = o,
in which pi = \ 2 a\ + X^i + 01,

Following the usual method [ 2r, eq. (52)] the determinantal
equation in X takes the form

(72)

This gives the following equation in X:

+ (aifa + b\a^) X 3 + (^fe + ^11^2 + ^22^1) X 2

f!2 2 ) = O. (72*)

62 THEORY OF VIBRATING SYSTEMS AND SOUND

The development has been carried rigorously thus far as a mat-
ter of principle; but it is evident that, due to the essential
asymmetry between the two degrees of freedom and due to the
inclusion of dissipation constants, a rigorous solution of (72*0)
for the two pairs of roots Xi, X'i; X 2 , X' 2 ; is a laborious matter.
The most practical method to follow to obtain a fair approxi-
mate solution of (720) is to construct the equivalent equation
from the roots

Xi = AI dz tn\j X 2 = A 2 db /#2)

X 4 + 2(Ai + A 2 )X 3 + (4AiA 2 + ni 2 + W 2 2 )X 2

+ 2(i 2 A 2 + # 2 2 Ai)X + n\ 2 n^ 2 = o.

We thus have, for the natural frequencies (neglecting the prod-
ucts bib2 and AiA 2 as relatively small quantities) by comparison
of coefficients,

o i 9 11 i ^22 o i o

#1 + #2 = 1 = #11 + #22 ,

Ci2 2

(73)
^! = n 2 n 2 - 4 -

and for the damping constants

An + Aaa,

(74)

Ai + A 2 = 1 = An + A 2 2,

ia\ 2^2

The equations are placed in 'this form to emphasize the redis-
tribution or repartition which takes place in the quantities A and
n 2 when the two unequal simple systems are coupled together:
a point which could not be made in the problem of ( 2i)because
of the symmetry there in nn 2 and An as compared with #2 2 2
and A 2 2-

To better express the new natural frequencies in terms of the
original natural frequencies (n 2 , w 2 2 2 ) we let n\ 2 + n% 2 = iA and

RESONATOR COUPLED TO A DIAPHRAGM 63

i 2 # 2 2 = 5 in (73), and eliminating one of these variables we
have the quadratic in ri 2

* 2 + B = o,
whence i 2 , 2 2 are the roots

2 = A

,90 i\ / N

= - =fc \J- - (11 2 22 2 - Wl2 4 ). (73*)

When #i 2 and 2 2 are found, the simultaneous equations (74),
are solved, giving for the new damping constants

Ai = Aii + A 22 A 2 .

After a tedious computation, according to (730) and (740) we
have the following values for the system of diaphragm and
coupled resonator:

/i = 1 120, Ai = 131.0; / 2 = 885, A 2 = 116.5;
the original values for diaphragm alone being

/ = looo, A = 200,

and for the resonator alone/ = 1000, A = 47.5. It may be noted
that by simply coupling the volume V* (without resonator prop-
erties, i. e., with the orifice closed) to the diaphragm, the stiff-
ness constant of the diaphragm would have been raised from
d ( = 1.97 X io 7 ) to en ( = 2.07 X io 7 ) i.e., only about 5 per cent,
thus increasing the natural frequency of the diaphragm with this

added constraint only to the value = 1022, or about 2 per

27T

cent. It appears then, that a very considerable change in nat-
ural frequencies, and a more equal distribution of damping in
the system has been brought about by coupling to the heavy
diaphragm a relatively light resonator system. The secret of

64 THEORY OF VIBRATING SYSTEMS AND SOUND

this effect is of course the kinetic energy residing in the mouth
of the resonator, where very rapid motion of a relatively small
mass takes place.

The study of the steady state behavior of the system is
one of the problems at the end of this chapter.

26. The Problem of the Loaded String; Filters

The problem of a vibrating string of beads, that is, a long
tense string loaded with equal masses, equally spaced, occupies
a key position in oscillation mechanics. In it, the general theory
reaches a climax, and from it have come important technical
applications, such as the periodic electrical structures or filters l
which are indispensable in telephony. The essentials of the
theory, neglecting dissipation, can be quickly given in view of
the preceding developments.

The length of the string is (m + i)/; its tension T; the m
masses, a\ each. The kinetic energy is then

T = ii(*i* + 2 2 + . . . {*). (75)

l The invention of the Electric Wave Filter is due to G. A. Campbell and is em-
bodied in his U. S. Patent No. 1,227,113, dated May 22, 1917. Lagrange gave in the
"Me"canique Analytique" the first general solution to the problem of the loaded
string. Routh ("Advanced Rigid Dynamics," 411) after discussing Lagrange's solu-
tion, points out that there may be a period of excitation of the string which is "so
short that ... no motion of the nature of a wave is transmitted along the string."
Rayleigh (I, 148*?) investigated the question of wave reflection, at the transition point
between two strings of equal tension, but of different loading, and derives an equation
for the ratio of reflected to incident amplitude, which is virtually the same as the equa-
tion for optical reflection, and identical with eq. (120), 31, which we shall encounter
in dealing with acoustic reflection. The analogy between the mechanical problem
and that of optical dispersion has been dealt with by various writers from the time of
Stokes. A paper by C. Godfrey (Phil. Mag. t 45, 1898, p. 356) deals with the propa-
gation of waves along the loaded string, and its optical analogy; and one by J. H.
Vincent (Phil. Mag., 46, 1898, p. 557) describes a dispersion model (a periodically
weighted spring) which has a definite frequency limit of Wave Transmission. Finally
there is a paper by H. Lamb (Mem. Manch. Lit. and PhiL Soc., 42, No. 3, 1898, p. i)
on waves in a medium having a periodic discontinuity of structure. In a one-dimen-
sional medium of this character. Lamb found that there were very definite selective
properties, for waves of certain frequencies, and in conclusion he points out that "a
dynamically equivalent problem is that of the propagation of sound waves along a tube
having a series of equidistant bulbous expansions ... ."

THE PROBLEM OF THE LOADED STRING

For small displacements, the forces on the first and last particles
are respectively (cf. 21),

the force on the rth particle being

and it is evident that these forces can be derived from a poten-
tial energy function (cf. eqs. 61, 6ia) y

V - in[& + (fe - {,) + (fo - fe) 2 + . . . + .]. (76)
For the natural oscillations we have the m equations of motion:

= o,

fe ~

- o) = o.

(77)

Now assuming motions of the type e* 9 (^ T = X 2 r), and letting

X 2
C = 2 + #1 , we have

Ci - & + o + . . =o

1 + C& 3 + - = o,

the determinantal equation of the coefficients being

C, - i, o
- i, C, - i

(78)

. . - i, C, - i
. . o, - i, C

(79)

m rows.

There are of course m roots in X 2 = n 2 , if the equation D m = o

66 THEORY OF VIBRATING SYSTEMS AND SOUND

is solved; the procedure is theoretically the same as in the prob-
lems of a few degrees of freedom we have already solved. But,
on account of algebraic difficulty a special device is applied.

By analogy from the problem of the string of length j/,
loaded with two particles, we infer that of the m modes, the
gravest will be with all the masses in phase, and as the fre-
quency of the mode rises, the string will show an increasing
number of nodes and loops until in the mode of highest fre-
quency each oscillating particle will be exactly out of phase
with its nearest neighbors. It can thus be assumed that the
spacing between the natural frequencies will be harmonic (or
nearly so) for the lower frequencies, while in the upper range,
up to the highest natural frequency, the spacing will become
closer and closer. A spectrum band is a fair analogy, with the
component lines becoming very close to each other as the head
of the band is reached.

To obtain the natural frequencies from the roots of D m (C) =o
we make use of a well-known trigonometric substitution which
is obtained as follows (Webster, "Dynamics," p. 166): D m (C)
is expanded in terms of its first minors, thus,

(80)

An = CAn-1 - An-2,

CD m ^ = D m + D m _ 2>

and this suggests the trigonometric relation

lA sin mB cos = A [sin (m + i)0 + sin (m i)0] (81)

provided the constant A is correctly determined; the transfor-
mation being from the variable C to 6. Now, if C = 2 cos 0, we
must have D m = A sin (m + T)0, by comparison of (81) with
(80); and since /)i(C) = C = 2 cos 0,

Di(C) = A sin (i + i)0 = iA sin cos 0,

or A = -. , and finally
sin J

D m (C) = Sm (m + I} *, in which C - 2 cos 0. (82)
sin

THE PROBLEM OF THE LOADED STRING

kir

For D m (C) to vanish, must equal . - - , (k being any inte-
ger), or

/Zi

rt | \ 2 f**

and since cos ix = i 2 sin 2 #,

--,

m + i

' (83)

! +"- 1 = x _ 2 sin 2

(84)

C\ .

kir

which gives for the natural frequencies (letting k = i, 2, . . . m)

/ i /a . i TT - i lc\ . m TT . .

JL = -\/ sm --- , ... / m = -\/- sin

7 7T ^^1 W + I 2' k/ 7T ^^1

+ I 2

Thus the natural frequencies are proportional to the abscissae
or ordinates in a quadrant divided into (m + i) equal parts,

1 TL

.fi f z

FIG. 8. SPACING OF RESONANT FREQUENCIES, LOADED STRING.

Fig. 8. If the number of particles, or sections of the string is
very large, we can write for the highest natural frequency

(85*)

68 THEORY OF VIBRATING SYSTEMS AND SOUND

The general solution for the natural oscillations is of the
type of equation (55^); thus neglecting dissipation

& = S B k A Tk cos (n k + <t> k ) y (86)

in which, it will be remembered, B k and 4> k are arbitrary, inde-
pendent of r, while A Tk must be consistent with the ratios

(~ r -j as given by the appropriate equations from the set
(78). Now in (78) we always have

r _i + C% T /+! = o,
or, for n ~n k from (86)

and if C k = 2 cos B kJ since

2 cos B k sin rO k = sin (r + i)0 A + sin (r i)0*,
we may take

i^ = sin rO k . (87)

= ~- -, as pr
m + r r

therefore

Now O k = , as previously determined (cf. eq. 83), and

We have finally, for the complete solution

& = S 5, sin -^- cos ( t / + 4 ),
fci w + i

with the 2m constants 5* and </>* subject to initial conditions.

In view of the connection between steady state and transi-
ent phenomena in structures of this sort, which we have previ-

THE PROBLEM OF THE LOADED STRING 69

ously emphasized, it should not be necessary to go further to
obtain an idea of the response of the system when steadily

driven by force of frequency ; but on account of the impor-

tance of the problem we desire to outline the behavior of the
system when w is varied, and so study its filtering properties.
Let a force *i = V e iut be applied to particle (i). Then we
have for the displacements of the particles nearest the begin-
ning and the end of the string (cf. eq. 60).

i = *X and fc. = m *i, (88)

and since the minors MI = <D w -i> and M m = i we have

SW I

(QQ^\

(88*)

Now we know, that if ra is large, D m _ l has a succession of
(m i) roots within the same region of frequency as those of
D m , that is, within the region of natural oscillations, and that

in this frequency range has appreciably large values, which

is to say that oscillations produced at one end of the structure
travel easily to the other end. The question is, what is the

value of the ratio when the driving force has a frequency

slightly in excess of the highest natural frequency of the system,
and this depends on the evaluation of D m _ l for co > 2-nf .

\ 2

In the expression C = 2 + we substitute w 2 for X 2 ,

and |- 2 for (cf. eq. 85^), thus

9 2

C = 2 cos 6 = 2 A-- or cos = i 2; (89)
n 2 n<? *

and it is seen that if w > 03 becomes imaginary; but we must

70 THEORY OF VIBRATING SYSTEMS AND SOUND

determine 6, to determine D m ^. Now if 6 = ix + IT, we may
write - cosh x = cos (ix + TT), and

in .which

sin m(ix -f
sin (ix +

' i) m * sinh
sinh x

C CO 2

= i 2 - = cosh x.
2

(90)

The general course of m _x(C) as a function of is indicated

7?o

D 9 =-106l

fj .__> *-P

-10.0
-6.0

**.

ots of D 10 (C

)=0

-G.O
-4.0

-2.0
-1.0

4-1.0

+2.0
410.0

' \

/'*

^ N

/ '

-' I;

L^M . /'-

^-^^"^*

^ ^ ^

s*^

^ r 1

0.4 O.6

FIG. 9. PROPERTIES OF D\o(C) AND D<t(C) (LOADED STRING). t

in Fig. 9, for the case m = 10. The roots of Z)io(C), (the 10
natural frequencies of the string) are also indicated by the verti-

FILTERS

cal lines. A study of this particular case, based on the behavior
of Dio(C) and D<j(C) shows clearly the filtering action of a
string equally loaded at only 10 points. At the point P, which

represents Dg for a value = 1.05, computed by means of (88)

the relatively small value of = is strikingly shown, as

u ^9

compared with values of in the resonant range. As is

^ I>9 5 Wo

further increased, DQ still rises rapidly and it is evident that
the string acts as a very efficient low-pass filter, cutting off all

vibrations of frequency greater than . If damping had been

2.7T

taken into account, the rise in DO outside the transmission range
would have been less rapid, but still sufficient to show the filter-
ing effect.

The general consideration of filters as such would take us
far afield. But we must not overlook the practical advantage
to be gained, if in any mechanical case we are able to substitute
the corresponding electrical network; for there is available a
complete theory of electrical structures of this kind, and their
properties are very well understood. 1 The electrical theory in
its simplest form relates to the steady state behavior of an

1 Care must be taken to preserve dimensional relations in comparing mechanical
systems to their electrical analogues. In most parts of the text it is convenient to take
the mechanical impedance as the ratio of maximum force to maximum velocity.
Strictly, the velocity is analogous to current density rather than to total current, and in
dealing with filters or similar structures this should be taken into account. It is evident
that we must consider the flux t== X as the equivalent of the electric current /.
For example, the kinetic energy in an orifice is:

2 K
This is analogous to the kinetic energy in an inductance, i/ 2 ; the inductance L is then

analogous to the quantity ~. The reader can easily make the necessary applica-
/C

tions of this idea, to the stiffness and resistance factors which also enter into the im-
pedance. To look at the matter in another way, the acoustic impedance per unit area,
divided by the area, is what corresponds most closely to the electrical impedance.

72 THEORY OF VIBRATING SYSTEMS AND SOUND

iterated structure of the ladder type, the longitudinal members
of which are known as series impedances (Z\) while the trans-
verse members (the rungs of the ladder) are known as shunt
impedances (Z 2 ). The first important property of such a network
is its iterative impedance, that is, its driving-point impedance
when the structure is infinitely long. It is easy to show that the
iterative impedance (at mid series) is

Zji f ,

(900)

Now, defining the propagation constant l P as the logarithm of
the ratio of the current in one section to that in the preceding
section, that is,

V>- = *', (90*)

-**
it follows that

P = 2log(j A J-^+Vi +~
= a + ift.

The factors a and ft are quite analogous to the attenuation and
phase factors which we shall encounter in Chap. Ill, in dealing
with wave propagation in acoustic media. Equation (90^)
may be written

IZi

P = 2

from which it follows that if the value of ~- falls within the

Z 2
y

range 4 < < o, P can then be expressed as a pure imaginary

Z 2

(P = z/3) and a. = o; that is to say, there is no attenuation within
the corresponding frequency range and the limiting frequencies

1 This term is a misnomer, but its usage is firmly fixed in the language. P is not a
constant; it is a function of the impedance, and hence of the frequency.

ACOUSTIC FILTERS 73

so defined are the boundaries of a frequency transmission band
of the filter. The conditions implied in this statement are real-
ized strictly only if we neglect dissipation; if dissipation is
taken into account, there will, of course, be attenuation in the

transmission band; but for moderate values of dissipation the

limiting frequencies determined from the condition 4 < ~ < o

/>2

by considering reactances only, will not be appreciably in error. 1
The elements of the simple theory of wave filters which we have
outlined will find application in the next section and in problems
17 and 1 8 at the end of the chapter.

27. Acoustic Filters

Following the development of electrical filters, G. W. Stew-
art devised in 1920 (Phys. Rev., March, 1921, p. 382) an acous-
tic low-pass filter, the theory of which is in all respects parallel
to that of the low-pass electrical filter, or of the string of beads
which we have just considered. The acoustic filters are merely
iterated resonators, and presumably for every electrical filter
known there is an acoustic analogue, though all are not equally
practicable.

In Figs. lOtf, iob y loc are shown respectively diagrams of
low-pass mechanical, acoustic and electrical filter structures,
the arrows indicating the displacement or velocity of motion
for the normal mode of highest frequency, that is, the limiting
frequency of transmission. In each case the filter is supposed
to have an infinite number of successive equal elements; the
oscillations are exactly out of phase from one element to the
next, in the mode considered. The simple formulae for limiting
frequency, assuming inertia and stiffness entirely concentrated
at a point, and taking no account of resistance, are shown
adjacent to the diagrams.

1 For a more extensive account of filter theory, the reader may refer to K. S. John-
son, "Circuits for Telephonic Communication," Chaps. XI and XVI. Many papers
by Bell System Engineers on Electrical Filters have appeared in the Bell System
Technical Journal e.g., O. J. Zobel, ibid., Jan., 1923, p. i.

74 THEORY OF VIBRATING SYSTEMS AND SOUND

If the conductivity k$ of the side orifices into the volume
is taken into account, the more accurate formula is

I +

(B')

according to a later paper of Stewart's which deals with the
general theory of Acoustic Wave Filters (Phys. Rev., Dec.,

One

Transmission. i w w i s
Conduit H . ! if ! /*t K,-f<

J V

1 \

' '/
} I/

- J-4 L

' ^

I x-

v i

*s I

1
1

Oense

Rare

, ,
(C J C,(Stiffness)= t

FIG. 10. LOW-PASS FILTER ANALOGUES.

1922, p. 528). This may be obtained by looking upon the con-
ductivity k\ as requiring a correction, due to the conductivities

eac h JL connected in series at either end of ki\ thus if k\ is the

corrected value we should have, adding "resistances,"

k<i

,4*iY

which accounts for the difference between the formulae in Fig.
(io) and equation (5 ; ). The latter equation "gives approxi-

ACOUSTIC FILTERS 75

mately the experimental values of/ and also explains satisfac-
torily the variation in/ with the conductivity of the orifices
leading from the transmission conduit to the volume in the
branch." (Stewart.) Note that in Stewart's formulae sub-
scripts (i) relate to properties of the "transmission conduit"
while subscripts (2) relate to branch elements which are in
shunt with the transmission conduit.

FIG. ii. ACOUSTIC HIGH-PASS FILTER (SCHEMATIC).

The high-pass filter is shown schematically in Fig. 11. The
approximate formula for / is shown adjacent to the figure.
The more accurate formula preferred by Stewart is

This is directly obtainable from the formula of Fig. 11, if we

consider that the conductivity iki = -j^ of a section of the trans-

mission conduit is disposed as a shunt with respect to the half
conductivity of each of the orifices which serve a given sec-

tion.

No ready explanation is forthcoming for the failure of the
published attenuation curves (Stewart, /oc. cit. y and later: Phys.
Rev., April, 1924, p. 520; also Peacock in the paper immediately
following) to compare favorably with those of the analogous
electrical filters in everyday use. (Note that Stewart's curves
show relative transmission to a linear scale, and not to a loga-

76 THEORY OF VIBRATING SYSTEMS AND SOUND

rithmic scale, in transmission units, 1 as is customary in the

telephone laboratory.) The damping --- characteristic of one

section of a good electrical filter can be made small, e.g., io or
less; but so indeed can the damping of the portion of the acous-
tic transmission conduit which serves as the neck of the cor-
responding resonator element. (For example, applying (69*),

A2 = r\ > substituting r = .75, which applies roughly to the

filter structures of Stewart, we have A2 = 24, taking- = .13 and

o> = 5 X io 3 .) Stewart has very properly made the main trans-
mission conduit a relatively wide tube; but the side conductiv-
ities (2) are likely to introduce complications particularly as
regards dissipation. The side conductivities in the low-pass
filters are due to a number of small holes; and we should expect
considerable dissipation therein. In the high-pass filter the side
conductivities will cause additional losses due to radiation. Al-
lowance for these complications and for the fact that the inertia
and stiffness elements are much more definitely "lumped" in
the electrical case may explain the relative shortcomings of the
acoustic filter.

A disadvantage due to multiple resonance 2 in the resonators
(which accounts for the failure of the low-pass filter to suppress
very high frequencies) is apparently inherent, due to the con-
tinuous structure of the acoustic medium within the resonator.
This disadvantage of course is lacking in discretely loaded
structures, such as the string of beads. It is an experimental

1 Two power levels, W\, W^ are said to differ by one transmission unit (TU)
when they are in the ratio = 1.25 (approximately). The general formula is

NTU = iolog w (TTT).

See K. S. Johnson, "Transmission Circuits," p. 12.

2 See the curves in Peacock's paper, referred to above; also a later paper by Stewart
(Phys. Rev. y 25, p. 90, Jan., 1925) in which he gives the necessary extension of the
theory.

FINITE STRING; NORMAL COORDINATES 77

fact that, by means of springs and weights very selective struc-
tures can be built, designing them for transverse, longitudinal
or torsional vibrations.

28. Finite String; Normal Coordinates; Normal Functions

We conclude with a discussion of the motion of a finite
stretched string of uniform density. There is a certain interest
in the problem as such, and in addition, we can use it to intro-
duce a further consideration of more general matters.

According to the straightforward method (cf. 10) take pdx
for the mass of an element of the string, the displacement being
(#) ; then since the net force in the direction of due to the ten-

9/3 A, , c . .

sion r is T--\ -lax, the equation of motion is,

a*\a*/ H

Without considering in detail the substitution = <t>(x)e xt it is
evident that one solution is of the type

= A cos (nt - 0) sin , (92)

A and 6 being arbitrary while n is to be related to the dimen-
sion / and the constant c of the system. For a string of length
/, choosing the origin at one end of the string, we must have
=o when x =o and x =/, whence

.- = fa (k any integer)

and the natural frequencies are

irC 2irC kirC

/ \

(93)
Thus, for any of the k normal modes

kirX

t = A K cos (n k t + B t ) sin > (94)

78 THEORY OF VIBRATING SYSTEMS AND SOUND

or, for any mode of arbitrary initial form, superposing in any
desired proportion a series of solutions of the type of (94),

{(#) = S A* cos (n k t + 6 k ) sin -y- (94*)

We may now compare the result as to w* with that which we
should obtain indirectly from the problem of the loaded string,
letting the spacing become so minute that the effect is of
"continuous" loading. Rewriting equation (85) and noting
that the spacing / of the former problem is the equivalent of
dx in the present case, we have, since

d = -j-, a\ = pdxy (m + i}dx = /,
\C\ . k TT IT. kdx TT

n k = 2\/ sin = 2\/- sin . ;

*a\ m + i 2 * pdx* L 2

or, since the argument of the sine function is infinitesimal,
and we can replace the sine by the argument itself,

__ kw IT __ kw

- TV; - y c -

Now, as to the factor sin -y, which must be obtained from
(87*), viz.:

r *

+ i

we note that for the rth element dx y as we proceed along the
string, rdx = x y and since (w + i)dx = /,

- . x dx, . , N

An = sin j- -j-kir = sm - (95)

To summarize, proceeding from the string loaded with discrete

FINITE STRING; NORMAL COORDINATES 79

particles, to the continuously loaded string, we have in the
limit

r = S E k A Tk cos (n k t + <*) = (#) = S 5* cos (w*/ + **) sin ~,

Jk-l t-1 /

(95*)

a result identical with (940). This was the method followed by
Lagrange, to whom we owe the solution of the beaded string
problem. The reader interested in the rapidity of convergence
for the fundamental mode will find an interesting table in Ray-
leigh, Vol. I, 120. It appears, for example, that for the case
of equal discrete loading at as many as six points, the frequency
is within about I per cent of that of the continuously loaded
string of the same total mass, length and tension. We shall not
carry the matter further.

One of the interesting questions which we wish to consider
is the relation of such dynamical developments as that of (940)
to the more purely mathematical notions of the properties of
Fourier's Series. To obtain (940), we have (consciously) used
[apart from the fact that sines and cosines are standard solu-
tions of (91)! only one mathematical idea, viz.: the principle of
superposition. This is that any finite sum of solutions of a
linear differential equation is also a solution. If this is granted,
and if the summation is extended to include a very large number
of separate solutions, then in solving the problem of a uniform,
continuous string we have virtually a dynamical proof of the
validity of the expansion of the arbitrary initial configuration
of the string into a sine or cosine series. From the practical
standpoint it is reassuring to note that "the physical induction
has been most fully corroborated by independent mathematical
proof." (Lamb, 39, which see by all means.) The reader will
enjoy discovering for himself, if he has not already done so, the
parallelism all along the way between the analytical theory of
Fourier's series and the purely physical ideas involved in the
application of these series to concrete problems. Instead of the
well known analytical device of integration, for example, to

8o THEORY OF VIBRATING SYSTEMS AND SOUND

determine the coefficients A* in (940), we can determine them
to as high a degree of approximation as we may desire by taking
a sufficient number of points on the string, writing for each
(for n k t + Bk = o),

(96)

>/ \ j - i j - i ,

(#1) = Ai sin -y- + Az sin --- h

yf \ j . j - r

(#2) = A i sin - - + As sin -- h

and solving the set of simultaneous equations. This is sound
enough from the physical standpoint, but generally more cum-
bersome than the integration method according to which

2 f* , .

/ J. * w

sn "

an easily proved mathematical relation. To follow the ideas
we have in mind, in making these brief suggestions, the chapter
on Fourier's Theorem in Lamb will be found very much to
the point.

Again consider the matter of normal coordinates. In dealing
with the continuously loaded string we have arrived, as it hap-
pens, at a solution in terms of normal coordinates; we began,
with the string loaded with discrete particles, using coordinates
that were not normal. In 10 (p. 25) we anticipated matters
to a certain degree by stating the fact that motions due to
arbitrary symmetrical distensions of the circular membrane
could be described by means of a certain series, each term of
which implies a normal coordinate. (The difference between

the factors sin and Jo(kjr) in the two cases is not material

at the moment.) It seems then, that for the string and the
membrane (in both of which the mass is uniformly distributed)
the use of normal coordinates (and the Principle of Superposi-
tion) in attacking the problem is a natural procedure; whereas
in the case of non-uniform distributions of mass, it would be

NORMAL FUNCTIONS 81

impracticable. In general, while we may not always be success-
ful in applying the method of normal coordinates to the vibra-
tions of a system with uniformly distributed constants, it is to
such systems that we look for the principal applications of the
method.

Finally, the success of the method of normal coordinates,
and the resulting gain of considering each natural oscillation as
independent of the others, depends on the availability of a set
of related normal functions in terms of which the shape of the
vibrating body can be stated at any moment of its history.
Mathematically a set of functions FI(X), F 2 (x) . . . F n (x) is
normal in the interval (a, b) if

A^FfWdx = i, (97)

in which \/A is known as a normalizing constant. The set of
functions, for example,

sin x sin 2x sin kx cos x cos ix cos kx

Vir ' VTT * " " A/V ' VTT ' A/TT ' VTT

is normal in the interval (o, 27r). A set of Bessel's functions can
likewise be normalized.

Now it is a fact that the normal functions which are useful in
physical problems are likewise orthogonali they satisfy the rela-
tion

Fj(x) -F b (x}dx = o for j * k y (98)

the interval (a y b} being taken as before. Thus we have
sinjx sin kxdx = o, etc., j 9* k\

xJn(<XjX}J n (ot k x)dx = O, /(;#) ^ Jn((X k a) = O

]**\.

to give only two examples. The latter set is of special interest

82 THEORY OF VIBRATING SYSTEMS AND SOUND

in connection with the problem of the circular membrane
(10, p. 25); we stated that any symmetrical initial distension
could be given by a series of the form (simplifying (29) by letting

0! = 2 = ),

(29*1)
and this is true provided the coefficients A k are given by

(99)

in which a is the radius of the membrane, and Jo(a t a) = o.

We state these results briefly, and without proof, in order
to give the reader an idea of the general situation in regard to
the use of normal functions in solving the problem of normal
modes of oscillation in certain systems. Fortunately, the ap-
propriate normal functions are known for the uniform string,
and the uniform membrane. (In the particular case of the
string, and similar problems for which the normal functions are
trigonometric functions > the expansion of a given function into
series of sines and cosines is known as the method of Fourier.)
In the case of vibrations of a rectangular plate ', the normal func-
tions are unknown; but in most problems with which we have
to deal, the properties of a certain set of known normal func-
tions are the key to the solution. Following the chapter on
Fourier's Theorem in Lamb, the more general treatment in
Whittaker and Watson's "Modern Analysis" will provide the
necessary background in the theory of normal functions.

PROBLEMS

11. Solve the problem of 21 [i.e., obtain equations (550)]
using two normal coordinates.

12. An electrical circuit (telephone receiver winding) of induc-
tance L and resistance R is arranged so that it exerts on a vibrating
member (diaphragm) a force Q\ (dynes) for each c.g.s. unit of current

PROBLEMS 83

flowing. The moving member induces in the circuit an electromotive
force #2 (c.g.s.) per unit of velocity. Taking the constants of the vi-
brating member as a y b y c (in the familiar notation), find the driving
point impedance of the electrical circuit when the vibrating member
is in resonance.

13. (a) A spherical resonator with thin walls con tains 400 cu. cm.,
and has a circular orifice of radius I cm. Find the natural frequency
and the damping coefficient, taking the velocity of sound in air at
ordinary temperatures as 3.4 X io 4 cm./sec.

(b) The orifice of the resonator is now fitted with a tube of diam-
eter 2 cm. and length 4 cm. Compute the natural frequency, and the
damping due to radiation under these new conditions; also give an
estimate of the damping due to friction in the neck.

14. In 25 a certain problem is solved for free oscillations. From
data there given, write the equations for the steady state velocities 1
and 2 when a force * cos w/ is applied at the center of the dia-

2
phragm, and compute the values of the ratio -.- at the resonant fre-

1
quencies of the system.

15. A string of 3 equally spaced beads is initially displaced so that

i = + i, 2=0 and 3= i. Taking = io 7 determine the constants

J9fc, ArK> Kk, and fa of equation (86) and (860) for zero initial velocity.

1 6. A flexible uniform string of length / is displaced and let go, the

/ / o/

displacements of points distant -, - and - - from one end being respect-

42 4

ively + i, o, i. Adjusting the fundamental of this string to agree
with that of problem (15), compare the motion of the three corre-
sponding points spaced along the 2 strings. (Use the first four terms
which do not vanish of the Fourier's series corresponding to (940).)

17. For the loaded string ( 26) construct the equivalent electrical
filter (that is, determine the mechanical reactances corresponding to
L and C), and verify this by finding the relation between the quantity

and the quantity C = 2 cos 0, (eq. 82) at the limiting frequencies

of transmission.

84 THEORY OF VIBRATING SYSTEMS AND SOUND

1 8. For the filter sketched in Fig. 12, sketch the electrical ana-
logue, and on this basis or by any other available method determine
the limiting frequencies of the filter.

FIG. 12. SEE PROBLEM 18.

19. Devise a low-pass torsional filter suitable for coupling between
two rotating shafts, and determine its upper limiting frequency.

20. A double resonator consists of two coupled resonators each
of volume f^ . The first resonator communicates to the air through
an orifice of area S and conductivity K\ the connection between the
two resonators is through a like opening. A plane wave of pressure
amplitude pie 1 ** is incident on the first orifice. Neglecting damping in
the inner orifice, find the natural frequencies of the coupled system,
and the overall velocity amplification of the system at these frequen-
cies. How do these coefficients compare with the velocity amplification
of the first resonator alone, at its resonant frequency ? How would you
diminish the coupling between the resonators, and what would be the
effect of this on the amplification at the selected frequency? (E. T.
Paris, Science Progress, XX, No. 77, 1925, p. 68.)

CHAPTER III

THE PROPAGATION OF SOUND

30. Properties of the Medium; Equation of Wave Motion

In Chapter I we dealt with the simplest type of vibrating
system. According to the point of view there adopted, all
parts of the system moved in phase, that is, any particular
state of vibration was manifested simultaneously in all parts
of the system. In Chapter II we have seen the result of coup-
ling together a number of equal simple systems in a long struc-
ture; we have observed that the vibrations of such a structure
may be regarded as wave phenomena, since there are periodic
changes of phase in passing from each element of the structure
to the next; and we have noted that in the limit, the wave-
transmitting properties of a long structure made up of equal
discrete particles, each equally coupled to the next, tend to
approximate those of a continuous medium. It is both logical
and expedient, at this point, to take up the subject of wave
propagation in acoustic media.

Sound waves are the inevitable result when vibratory
stresses are applied at the boundary, or in any part of a com-
pressible fluid. The physical properties of the medium must
therefore be stated; and we consider first the specific properties
in terms of the mean volume and density at a particular point
in the fluid. These are, for small increments of volume and
density,

(a) Dilatation

A = , i.e., v = (i + A); (101)

(b) Condensation,

, P = P O (I + s). (102)

86 THEORY OF VIBRATING SYSTEMS AND SOUND

(From (a) and (b) note that since pv = P O V O ,

s = - A and (i + s)(i + A) = i, (103)

neglecting the product $A as a small quantity of the second
order.)

For a fluid perfectly elastic as to small condensations we
have, experimentally,

-*L-L = c, c= T , (104)

v dp ' K v *'

in which C is the compressibility, and K the bulk modulus, or
coefficient of cubic elasticity. If the fluid is a gas undergoing
slow (i.e., isothermal) changes in volume, we also have/w = p v
whence

dp dv

= - =

and the coefficient of cubic elasticity is

\C J K == = PQ == "~ ,

8v s

~ * (105)

whence

For a perfect gas we have relations (101-103) as above, but,
for rapid changes in p and 0, we must use the adiabatic rela-
tion pv y p v y hence in this case

and the coefficient of cubic elasticity is

(*') *' = ypo = ^

whence ' (l 5 ^

The general relation which must hold for a given small
volume on any fluid, is the following:

PROPERTIES OF THE MEDIUM; WAVE MOTION 87

"The difference between the amounts of fluid which flow
in and out of a small closed surface during a small interval of
time $/ must be equal to the increase in the amount of fluid
during the same interval, which the surface contains/' This is
the principle of continuity \ we proceed to give it analytical
form, for motion in one dimension, namely along the x axis.
Let the volume considered be a lamellar element with oppo-
site faces of area A, normal to the x axis, the thickness of the
element being dx. Then the net flow (or flux) through the faces
of the element in time 8t is

while the increase in the amount of fluid contained is

a/'

hence the relation

IT/

which is the equation of continuity. This kinematical relation,
which can easily be generalized for motion in three dimensions,
is probably the most useful one in hydrodynamics. Replacing
p by P O (I + s) in (106) and neglecting s as compared with
, we have as more suitable for our purpose

1 . = o, (1060)

9/ QX

whence

9/9* Qx 2

Now if p is the excess pressure on one face of a lamellar element
A'dx, and p + dx- the pressure on the opposite face, then

the net force due to pressure is A'dx- y and taking into ac-

88 THEORY OF VIBRATING SYSTEMS AND SOUND

count the rate of change of momentum of the element we have
the equation of motion

= o. (107)

Writing for/), KS (105), and differentiating (107) with respect
to time, we have

2 , 2

+ P

3/ 2
and by comparison of (107*2) and (106^)

a/ 2

(I08 )

v y

This is the equation for the propagation of a sound wave y (com-

\~K .

pressional or longitudinal wa ve) in which c = \- is the velocity

of propagation. | is the "particle velocity" at any point in the
medium; the same equation would be obtained for as for |.
The same equation (as regards form) was obtained in studying
the problem of the vibrating string (cf. 91) but the wave veloc-

ity in that case (\/-) related to the propagation of transverse

disturbances along the string. In perfect fluids, only longi-
tudinal waves need be considered; in solids both longitudinal
and transverse waves are possible, the latter case depending
on the elasticity of the medium to shearing stresses. All the
steps leading to equation (108) apply equally well to the case
of an elastic solid rod, provided that fof * the longitudinal
elastic constant (Young's Modulus) is used. The physical con-

stants of various solid and fluid media, including c =*= \- and
_ ^ P

R == V/c- p, the two principal constants relating to sound trans-
mission, are given in Drysdale, "Mechanical Properties, etc.,"
pp. 288-292.

PROPERTIES OF PLANE WAVES OF SOUND 89

3 1 . Properties of Plane Waves of Sound

We propose to examine the more important properties of
plane (aerial) waves of sound, starting from the fact that each
of the quantities (, |, p, s, A, dp) which relate to the state of an
element of the medium must equally satisfy the equation:

f^ = ' 2 f2> ' = fc I, A, s, p or dp. (108)

From this equation we infer only the fact that solutions must
be of the type

= A-J(ct - x) + B-F(ct + *), (109)

that is to say, the general solution implies for any of the factors
0, two wave trains > each of arbitrary form, and proceeding in oppo-
site directions with velocity c. The values of A and B> and the
particular forms of f(ct x) andF(ct + x) must be determined
by the assigned boundary values in a particular problem. The
choice of which factor 6 (e.g., condensation, excess pressure,
particle velocity or what not) we wish to use in a given case
depends on the nature of the problem.

Thus in a closed tube, with recurrent reflections at either
end, we might, as in the string problem, wish to study the dis-
placement ; the solution would then be in the form

J . / , kirx\ . / kirx\l . kwx

= A\ sin I o>/ H T- ) sin ( o>/ ) = iA cos a?/ sin y>

(109*)

which clearly satisfies the differential equation and the bound-
ary conditions if

w/ . . . r% 2/ . , % / i\

T = c y that is if X* = -r y since / t -Xt.= c (ioo)
kw K

X being the wave length, and/* the frequency of one of the
natural oscillations of the tube. Suppose, however, we study the

90 THEORY OF VIBRATING SYSTEMS AND SOUND

propagation of a single pressure impulse whose original wave
form is given by the expression

*P\t -o =/W = J > for < * < *<> J ,
= o elsewhere; J

then, (neglecting dissipation due to friction) and considering
propagation in only the positive direction, we must have

dp = f(x cf) = i, for ct < x < ct + x ]

i u
= o elsewhere. J

These instances emphasize the purely kinematical information
which is available from (108).

Now consider a dynamical question, that of energy in a
plane wave. A piston of unit area is given an oscillating motion
cos w/, and this motion is communicated to the pear end of a
column of air of unit cross section^ which extends to infinity. At
any distance x along the column we must have, for the particle
displacement . ,

(*) = cos o> \t -Jj, (no)

consistently with (108), (109) and the special conditions we
have named.

Consider the potential energy first. The excess pressure dp
producing any given condensation $ is K'S. (Cf. io$a). Thus in
any lamellar element of volume dx we have

dx-V = dx( *'sds = Jrfx *'$*. (in)

The kinetic energy is

dx-T = \dx-p&. (112)

We can easily show that the kinetic energy is equal to the poten-

PROPERTIES OF PLANE WAVES OF SOUND 91

tial energy at any point in a plane wave train. If is the dis-
placement at point #, then at x + dx the displacement is

and we have

S EE - A = - -1

Now from (no)

| == = c- (114)

hence

cs = I and //v-T = \dx- P c 2 s 2 = -J</#Ys 2 , (ma)

a result 1 identical with (i 1 i). Thus the total energy dx(T + T 7 )
is half Kinetic and half Potential and "since has the same sign
as 5 an air particle moves forward (i.e., with the waves) as a
phase of condensation passes it, and backwards during a rare-
faction " (Lamb, 60). To obtain the average energy density E
in the medium we must integrate zT-dx or 2^-dx over an inte-
gral number of wave lengths and divide the result by the volume
considered, thus

r* fA r v+A / *\

E-\ = I iTdx K I s - cos 2 o>( / ]dx

J x J x \ c >

2 J x L \ cj }

1 The symbolic solution of the equation

=^1^^'' ( I0 8)

leads quickly to the result ^ = cs for the wave of velocity r. We have, treating the
dot and prime as operators,

whence

(*' + ^0 =o, or C = -
and

92 THEORY OF VIBRATING SYSTEMS AND SOUND

or noting that the integral of the periodic term vanishes (since
coX

c x = K's 2 X, = Kso 2 (115)

in which S is the maximum condensation.

The Intensity is defined as the rate of flow of energy past any
fixed plane normal to the column of air. It is clearly equal to the
product of the energy density and the velocity of flow, c. That is,

,.
2 K 2 pC

snce 5p = K'S.

This expression for Intensity in terms of Excess Pressure dp is a
most useful one. As will appear later, it is valid for divergent
as well as plane waves.

The rate of working of the piston in producing the outgoing

sound wave is dp. Now since dp = K'S and s = -

K 't
5p = and we have

r c

piston

(117)

also the simpler relation,

These important results define the quantity R = vVp which
is the radiation resistance l of the medium, per unit area. (Cf>
8, Ch. I). We note that in the form

V7; = /?= pc, (117*)

1 There is no consensus in naming this quantity. The engineer may prefer the term
characteristic impedance of the medium; Drysdale gives acoustic resistance a term due to
Brilli (Etude des Ondes Acoustiques, etc., le GMe Civi/ y 75, 1919: Aug. 23, p. 171
Aug. 30, p. 194, and Sept. 6, p. 218). The term radiation resistance was suggested by,
H. W. Nichols in Phys. Rev., 10, 1917, p. 193.

PROPERTIES OF PLANE WAVES OF SOUND 93

it appears in many formulae involving radiatipn; as for ex-
ample in the radiation damping of a resonator ( 24 and eq.

175). The tables for yVp in Drysdale have already been
mentioned.

One other property of plane waves, namely, normal reflection
at the boundary between two media, should be dealt with. Let
the plane x = o be the boundary between media (i) and (2).
Choosing a solution of (108) in the form of (109) we have for
waves in medium (i) to the left of the origin

. iw (-*-) , ., !'(+*) , n ,

= * Cl '' ^ Cl ' US

the two terms on the right representing respectively the particle
velocities of oncoming and reflected waves. Similarly for the
transmitted wave, in medium (2),

I. -to*'- (8)

In order to determine the ratios .-^ and .--~~ we require two

SOI SOI

equations of condition. These are available from the boundary
conditions which are that both velocity and excess pressure must
be continuous. (Some of the other quantities 6, (108) which
satisfy the differential equation must also be continuous; but
it is sufficient to deal with only two of them, the two chosen
here leading most directly to the end in view).

We note that the velocities are vectors: the excess pressures
(of the nature of hydrostatic pressures) strictly speaking are
not. The phase difference (if any) between | 01 and ' ol is as yet
undetermined. To obtain the pressures we use the relation
Bp = zb Cp> paying particular attention to the sign before r,
which depends on the direction of propagation; this is neces-
sary because there can be no accumulation of uncompensated
pressure at the boundary.

94 THEORY OF VIBRATING SYSTEMS AND SOUND
We have therefore, at the boundary

fo. + *'o. = L> ( J

*A,i
i.e.,

VJO, - V,f'o, = f 2^0.>

Treating (119) and (i 190) as simultaneous equations, we obtain
on solution l

*' - 2/Zl and *'" - - * 2 " *'- (120)
- ^--Tf-jnT an a ---- #~TTb v I2 ;

foi *M ~r K% 01 ^-2 -r KI

Now if /?2 > /?i, i. e., if the second medium is more resistant
than the first, it is evident that in the reflected wave the particle
velocity has undergone a phase change of IT with respect to the
particle velocity of the incident wave; but there is no phase dif-
ference between 01 and | oa , the latter referring to the trans-
mitted wave. It is also evident that while the particle velocity
undergoes a phase change of ?ron reflection, the phase of excess
pressure is unchanged. In the transmitted wave the phases of
both the excess pressure and the particle velocity are unchanged
in any case.

Again, if R* < R\ the phase of the particle velocity is un-
changed on reflection, while the excess pressure suffers a phase
change of TT. And finally, if R 2 = Ri there is no reflected wave;
transmission to the second medium is unimpaired.

The first ratio in (120) may be called a transmission coeffi-
cient, ti2j the second ratio is the (amplitude) reflection coeffi-

cient^ r. Interchanging subscripts we have fci = /p \~W\ y
and the useful relation (Kl + * 2 '

(1200)

= i - r\
in which

2/2, /2 2 -

1 Equations (120) are identical with the Fresnel Equations for normal optical reflec-
tion, if the radiation resistance R is replaced by the refractive index.

SOUND TRANSMISSION IN TUBES 95

This is also obtainable by application of the energy principle to
the incident, reflected and transmitted waves.

The classical example of acoustic reflection is in air, from
a water surface; in this case [/?2 = 1.4 X io 5 , Ri = 40;
c< 2 = 1.4 X io 5 , ci = 3.3 X io 4 (Drysdale's table)], so that

(*'o,) 2 = -999(U 2 -

From water to steel, the transmitted wave contains about 13
per cent of the original energy in the incident wave.

One other consequence is of interest. If the boundary to
medium (i) is a (nearly) rigid wall, in which there is a small
orifice containing a diaphragm (or other sound-detecting ap-
paratus), we have, from (1190) and (120)

*A = 01 + /oi = 25 A>,> approximately. (121)

from which we can calculate the motion of the diaphragm if we
know its impedance.

32. Sound Transmission in Tubes

With the simple theory of the preceding sections we can
solve (though not very elegantly) some of the problems of
sound wave transmission in tubes. We shall have to make some
restrictions to avoid mathematical difficulty; we shall assume,
for example, that the walls of the tube are rigid, 1 and while we
shall make some allowance for friction to add interest to the
problem, we shall not specify the particular form of resistance

1 This restriction is important, for if the contained medium has inertia and stiffness
comparable with those of the wall material, the wall will yield appreciably to the excess
pressure within. There are two consequences of this; first, the effective stiffness of the
contained medium is diminished, hence a lowering of the wave velocity therein; and
second, owing to dissipation in the wall itself, and lateral radiation from the wall, the
wave in the contained medium suffers increased attenuation. This problem has been
treated by Lamb (62; also Mem. Manchester Phil, and Lit. Soc., 42, No. 9, 1898, p. i),
who found (for example) that if water were inclosed in a glass tube whose thickness was
one-tenth the radius, the phase velocity in the water would be diminished 24 per cent as
compared with the normal value, on the basis of a rigid containing wall. An interesting
experiment which the reader may try for himself will show the damping effect of thin-
walled rubber tubing on sound waves transmitted through air in the tube.

96 THEORY OF VIBRATING SYSTEMS AND SOUND

coefficient to be used. For this particular phase of the problem
the reader may consult Appendix A.

Choosing the tube of unit cross section, we derive the equa-
tion of wave motion anew, assuming that RI % is equal to that
component of the negative pressure gradient in phase with the
velocity. RI is analogous to the ohmic resistance per unit length
in an electrical transmission line: the whole problem is in fact
parallel to the electrical problem, if there is no leakage between
the pair of wires.

At any point x in the tube, let the excess pressure be

*'a , ...

p = K s = -- ; the net excess pressure acting on a lamina

of thickness dx is therefore dx- = x'dx -- - This is partly

2 r J

compensated by the rate of change of momentum of the lamina,
which is pdx- 1. The component of net excess pressure which is
expended in doing work against friction is dp f = R\$dx since

is the effective pressure gradient.

Adding all these forces and equating the sum to zero, we
have

pldx +Rikdx - *-dx = o, (122)

or, as we prefer to deal entirely with velocity,

This equation is identical, except for the friction term, with
(108). (The analogous expression in the electrical case is

, a 2 / , Qf i a 2 /

the constants L, R and C being per unit length of line as in
the acoustic problem.) (Fleming, " Propagation of Electric
Waves," Third Edition (1919), p. 125.)

SOUND TRANSMISSION IN TUBES 97

Assuming a solution = Ae fl we find on substitution

i P iRi

and if is a small quantity

i if iTJil
_ = - i -- LI

ci cl 2o>pJ

A complex value for the wave velocity appears as a natural re-
sult of dissipation in the transmission system, just as a complex
value for natural frequency appears when damping is present in
a vibrating system.

Placing the expression for in the assumed solution, we
have, for a wave traveling in the positive direction

t - w (i_? + !* 1 *)
c ""~" *!- ~~~* *l e

In equations of this sort, a is known as the attenuation factory
/3, the phase factor, and if we like, a + /|9 is ^^propagation con-
stant l , as remarked previously (26) in dealing with the iterated
electrical structure. (We shall often use k for the phase factor,
when a = o, i.e., when there is no attenuation due to friction).

In (124) the phase velocity is -; if a is small (as here), the phase

velocity is virtually the same as the unmodified wave velocity
in the case of no dissipation. If a is large, however (as in the
problems of 51, and Appendix A), the phase velocity will

be appreciably less than c = <\t for the free medium.

98 THEORY OF VIBRATING SYSTEMS AND SOUND

Now consider the phenomena in a finite tube of length /,
closed at either end by a vibrating piston. For definiteness, let
the piston at one end (taken as the origin) supply power of con-
stant frequency to the -system; at the other end the piston is
driven. by the excess pressure of the sound waves transmitted to
and reflected by its surface. The boundary conditions are sum-
marized thus:

# = x = /

Piston \

Impedance J ^ ''

Applied Force, = * <? l ' 6rf Zero

Velocity, = e lui ^e iut

Pressure, p e iwt p t e iut

Boundary 1 __ . __

Condition f ' " ^ Jl * 1 ^ l

Owing to the recurrent reflection in the tube, there will travel in
the positive direction from the origin a composite wave train,
each component of which differs from the others only in having a
different amplitude at the start, and a different phase constant
depending on the number of times it has traversed the double
length of the tube. These components form a series, of the type

* - f2ml <- *> - ' ( 2w? ~ + **) ***
+ = L //* -e -e

m = l

and it is clear that, since the amplitudes and phases of all the
components follow the same law as they progress down the
tube, they can all be summed up into one component of the
same type, thus

|+ = ^-< + tf>V

the constant A' being a carry-all for the series of. coefficients
each of which differs from its predecessor by constant attenu-
ation and phase factors. Dealing similarly with the reflected
wave system traveling in the negative direction from the point
x = /, we write

SOUND TRANSMISSION IN TUBES 99,

Consequently, for the velocity at any point in the tube

i( y \ = A . _L_ t ( /1 r p~ (** ~t~ *&)* D f *(<* + iP)*\ **ut fjor^

s\**/ S~T* l^ s VX^ * ** & y ^ \ AI *"J/

The excess pressure p is rp|, the plus sign being used for the
wave traveling in the positive direction, the minus sign for the
reflected wave, as in the preceding section. Thus

p = pc(A'c~ ( + *>* + 5 V + * x )^" 1 (i 26)

and considering maximum values only, we have

at x = o, { = A 9 - ', /> = prC//' +

at# = /5 ii A e a l U e a l i t 027)

Now in view of the boundary conditions as to pressure at the
piston faces, letting L = e~ (ot f /w we have

Z Q (A' - B') + pc(A' + B') =

/ B'\

Z(A'L ~jj-

E'
- = o.

The problem is thus reduced to that of a coupled system of two
degrees of freedom; solving equations (128) for the constants
A' and B' y and putting these values in (125) and (126) we have

.
and

in which J- (129)

Z, = f-<+^

and

Z) = (Z + pc)(Zi + pf) - L 2 (Z pc)(Zi pr)
= (Z Zi + p r )(i Z*)4

ioo THEORY OF VIBRATING SYSTEMS AND SOUND

From this point a number of applications are possible. If we
study the phenomena when the driving piston and the driven
are electrically connected by means of an amplifier so that the
motion of the piston at x I is enhanced and communicated to
the driving piston, we are dealing with the "howling" tele-
phone. This problem is interesting, but rather complicated, as it
involves detailed study of the roots of D = o in terms of Z ,
Zj and L in order to find the natural frequencies of the sys-
tem. A suggestion as to theory is given by H. W. Nichols 1
(Phys. Rev. X, 1917, p. 191); if the reader wishes experimental
details, these are available in Chap. XXIII of Kennelly's
"Vibration Instruments." A complete analysis of the Howling
Telephone Problem is given by H. Fletcher, in a paper in Bell
System Tech. Jour., V, Jan., 1926.

If we consider the steady state theory of wave transmission
down the tube, we are dealing with the basis of the Constan-
tinesco wave-system of power transmission, some of the properties
of which are interesting and relatively easy to visualize. Those
we shall note are of fundamental importance in any case.

Letting x = p in (129) we have for the driving-point im-
pedance 2

7 * ( Z

"" ...... "

"" ...... " (Z, + PC)- l?(Zt - PC)

(13)

1 The late H. W. Nichols (1886-1925) though known principally for his contributions
to Electrodynamics, was an inspiring colleague and helpful critic in Acoustics as well
because of his keen interest in and thorough familiarity with the classical theory, which
he often applied to practical problems. A biographical notice appeared in Nature,
Dec. 19, 1925, p. 909.

2 The form of (130), if dissipation is neglected, becomes

oo -5*

pc cos p/ H- tZi sin pi

as quoted in 56. At this point the reader should familiarize himself (if he has not
already done so) with Fleming's treatment of the parallel wire problem, in "The Propa-
gation of Electric Currents," Chap. III. For example, (288) is identical with Fleming's
equation (6 1), p. 99, if a = o. The reader will also be interested in comparing (288) with
the equation obtained for the driving-point impedance of the exponential horn, by taking
the ratio of (229) to (228), 46; see also problem 35, following Chap. IV. In problems
involving horns and tubes we have excellent examples of the value of impedance methods
in studying acoustic systems.

RESONANCE IN TUBES AND PIPES 103

Determining A and B, and substituting,

or |(*) = - s -- f sin /(/ - *) V"'. (135)

Thus "the amplitude becomes abnormally great, if sin = o,

k\ , f

or / = , & being integral " (Lamb, 62). The condition for

tube resonance is exactly the same as in the Constantinesco
scheme; but it must be recognized that the phenomena in the
Kundt's tube are somewhat different. In Kundt's experiment
there are standing waves whose kinematics are identical with
those of the stretched string of 28, p. 1 1 ; in the power trans-
mission scheme, not all the energy is reflected at the distant
end, owing to the yielding of the piston there. In this latter case
there is in the tube a composite of standing wave conditions and
wave transmission conditions if Z and Z/ are imperfectly re-
lated; the standing wave pattern tending to disappear when Z and
Z/ have no reactance components, or reactance components
which offset one another so that Z + Z/ is a pure resistance.

33. Approximate Theory oj Resonance in Tubes and Pipes

We may now deal with some of the properties of organ pipes,
which, ift the simple theory are tubes with ends either wide open
or rigidly closed. For closed ends, Z = Z/ = oo . In this case,
again, equation (130) is not adapted to the discussion; it merely
states, if we neglect pc as relatively small, the undoubted (but
useless) fact that Z 00 = Z , a very large quantity. Since the
tube with fixed ends cannot be driven from without, only the
transient solution is of interest, and this is left as a problem for
the reader, on the conclusion of this chapter.

If one or both ends of the tube are open we can obtain tran-
sient solutions on the basis of the steady state theory of equation
(129) by allowing the piston impedances to vanish at the open

102 THEORY OF VIBRATING SYSTEMS AND SOUND

we have

Zoo = (Z + Z,), if cos j8/ = i. (133*)

We conclude that the impedance of the piston at x = I is
transferred (as if bodily) to the driving point where it is added
to that of the directly driven piston. This is true with k either
even or odd, i.e., with pistons exactly in or out of phase. The
elastic medium furnishes in this ideal adjustment a massless,
frictionless coupling of infinite stiffness between the two pistons:
the ideal condition for wave power transmission.

The efficiency realized when the adjustment of frequency to
tube resonance has been made will depend on minimizing the

attenuation factor a. = For a given frequency, and medium

2pC

of given viscosity, this depends on using the shortest possible
tube, i.e., operating with tube one-half wave length long. In the
Constantinesco system, however, long tubes are frequently nec-
essary, with lateral connections at points along the tube from
which the alternating pressure in the tube is used to drive a
number of separate piston motors. It is obvious that in this
case the higher harmonics of the tube must be called into play
and that the lateral orifices must be disposed at points an inte-
gral number of half wave lengths apart. Those interested in the
Constantinesco system will find further references in Drysdale,
together with a practical discussion of the scheme.

In the familiar Kundt's tube experiment, Z, is infinite and
Z is unknown, the boundary condition at the origin being sim-
ply that a prescribed motion $ Q e iu * takes place, this being due to
the vibration at the end of a metal rod, itself in resonance. Equa-
tion (130) is therefore not adapted to the discussion. As in
(125), we take (neglecting dissipation in the air column)

t> (134)

and we know that

RESONANCE IN TUBES AND PIPES 103

Determining A and B, and substituting,

or |(*) = - s -- f sin /(/ - *) V"'. (135)

Thus "the amplitude becomes abnormally great, if sin = o,

k\ , f

or / = , & being integral " (Lamb, 62). The condition for

33. Approximate Theory oj Resonance in Tubes and Pipes

If one or both ends of the tube are open we can obtain tran-
sient solutions on the basis of the steady state theory of equation
(129) by allowing the piston impedances to vanish at the open

io 4 THEORY OF VIBRATING SYSTEMS AND SOUNO

ends. This is given as an interesting variation from the standard
textbook method. But as in the classical treatment we must
point out just what is involved in neglecting terminal impedance
at an open end.

The radiation resistance of a source of area S (cf. 24) has
been given as

2 . - c

in terms of frequency.

(This equation will appear as (175) later in this chapter.)

If we take S = i, b\ becomes for all moderate values of fre-
quency appreciably smaller than pr, the radiation resistance for
plane waves; thus no great harm is done in neglecting b\. But
this is not all; there is an inertia component of the impedance at '
an open end the effect of which is virtually to increase the length
of the tube by a small increment. This is a complicated matter
the discussion of which we shall defer to Chap. IV; in the pres-
ent situation we shall assume that pc is much greater than the
impedance of an open end, and obtain what information we can
on this basis.

The simplest case to discuss is that for which Z = o (i.e.,
Z < pc) and Z, is very great. Making this change in equation
(129), we have for the velocity, neglecting dissipation,

7 (*- i&* _ T2 p i9x\ty p^t

whence

>-^-

Thus the driving-point impedance is a pure reactance

( /V cot 07)

RESONANCE IN TUBES AND PIPES

105

and the natural frequencies are found by placing this quantity
equal to zero. Applying this condition we have

whence

that is,

or since

cos 07 = o,

fc-

(* = i, 3 5. )

/ -
4

The velocity is of course very great for these frequencies, for
which the tube is an odd number of quarter wave lengths long.
Since the denominator is not a function of #, we can take
the numerator of (1360) as indicating the distribution of veloc-
ity in the tube, under resonance conditions. We have then
e lftl = / and e~ tftl = / and we may write

= cos

= L

in which is the maximum value of for x = o. To obtain the
pressure distribution along the tube we cannot use the relation
dp = cp , because the waves in both positive and negative
directions have been combined to obtain |(#) as i n ( T 38). But
we always have

p = K'S = *' = 2 p (1050) and (113)

9# 9^*

whether the wave system is standing or moving. Thus, since
t

/CO

- _ 2 3_(L \i ~ - ' ' '

Z7 ~~* "" C p I . C. CJS jl5iV 1 6? ~ """" Z P**o ^*^* jOiV c \ 1 9/

and it appears that the pressure is zero at the point x = o, and

106 THEORY OF VIBRATING SYSTEMS AND SOUND

a maximum at #= /; the pressure distribution in the standing
wave system being complementary to the velocity distribution along
the tube. Moreover, the maxima of pressure are out of phase
with the maxima of velocity by one-quarter of a period; but
pressure and amplitude are in phase as far as time is concerned.
These relations are generally true in connection with standing
sound wave phenomena; the reader may note particularly the
contrast between this state of affairs and the progressive wave
phenomena emphasized in connection with the discussion of
energy, following equation (1120).

For the tube open at both ends (Z = Z t = o) the velocity
is, according to equation (129), (if dissipation is neglected)

., . P c(*-** + W)-V e i '* , .

(*> - -* (I40)

and *(/) = . (140*)

^ ' ^ * }

whence

cos Bl

*(o) = ~ PJ .

* v ' t P c sin pi ^ ' t pc sin pi

The driving-point impedance is now ipc tan ($/, and the natural
frequencies are given by

&X
0/ = for, (k = i, 2, 3, . . .) or / = ; (141)

a well-known result for the "open" pipe, usually obtained in a
more direct way. The velocity and pressure distributions in the
pipe may be obtained by the method applied to the tube closed
at one end, in which event equations similar to (138) and (139)
will be obtained; this is left to the reader. In view of the dis-
cussion that has been given it should not be necessary to deal
further with the mechanics of resonance in tubes. 1

1 The behavior of a cylindrical pipe in which lateral holes have been cut is treated
theoretically by W. Steinhausen, Ann. d. Phys.> 48, 1915, p. 693, and experimentally by
E. Ratz, Ann. d. Phys. y 77, 1925, p. 195. Incidentally Ratz determines the end correc-

PHYSICAL FACTORS AFFECTING TRANSMISSION 107

To give an exact account of the phenomena in tubes, dissipa-
tion would have to be included; but the pressure and velocity
relations in standing wave systems which we have obtained will
not be greatly affected thereby.

34. General Discussion of the Physical Factors Affecting Trans-
mission

Before going further with the general theory we must con-
solidate the position already gained (as was done in 8) by re-
viewing certain physical factors affecting transmission. These
phenomena can be treated only in barest outline; there is a
fairly complete mathematical theory of them, but it is too
lengthy and involved to include in an elementary course.

Consider absorption first: this may take two forms. If due
to resonance it will be selective, and the theory of the reaction of
resonant structures on sound-waves striking them is not what
we have in mind at present. There are many problems of this
sort, but they are more properly classed as radiation problems.
What we have in mind is inherent absorption due to friction in
the medium, and resulting in the extinction of a progressive
wave, after a certain distance of the medium has been traversed.
This is non-selective, except as "constants" of the medium vary
somewhat with frequency, and was illustrated to a marked de-
gree in the propagation of compressional waves in the thin
damping film of the air-damped transmitter. [ i r, eq. (37).]

We now wish to apply the theory of dissipation to sound
transmission in conduits of capillary dimensions. A solution of
this problem is given in Appendix A; for very narrow tubes it
appears that Poiseuille's law is applicable and that the inertia of
the fluid in the tube is negligible as compared with the viscosity,

tion for an (unflanged) pipe to be ot .y8r; this is between the figures given by Ray-
leigh for a flanged and an unflanged end. (See 43, Chap. IV.)

In Rayleigh (II), 318) a discussion is given of the absorption of energy from sound
waves transmitted through a tube, when they pass a lateral opening leading to a res-
onator. The reader interested in phenomena of this kind may refer to three recent papers
by G. W. Stewart on the Effect of Branches on Acoustic Transmission through a Conduit:
see Phys. Rev. y 25, 1925, p. 688; ibid., 27, 1926, p. 487 and p. 494.

io8 THEORY OF VIBRATING SYSTEMS AND SOUND

as far as the effect on transmission is concerned. The equation
of motion may be written

the solution of which shows that such waves as are transmitted
(or rather diffused) into a capillary tube are very highly damped.

It is natural to seek in these phenomena an explanation of
the absorption of sound by porous bodies. The pores or crev-
ices in the absorbing material are filled with air and therefore
present to an incident sound-wave an array of open ends of nar-
row conduits within which dissipation due to viscosity is very
high. The problem of producing a good absorbing material is
really twofold. We must have (to prevent reflection, 31) a
substance whose radiation resistance is as nearly as possible
equal to that of the adjacent medium from which the sound is to
be absorbed; and in addition we must provide in the material
as much friction as possible to extinguish the transmitted wave.
Considering felt (for example), if it is packed too loosely, while
its radiation resistance will nearly match that of the air (due to
its high percentage content of air), the air spaces in the material
will not be sufficiently constricted to bring viscosity forcibly into
play, and a gr^at thickness of the absorbing material is required
to produce extinction of the transmitted wave. If packed too
tightly, there is ample friction available, but the felt now has
too much inertia, its radiation resistance is too high as com-
pared with air, and consequently the sound waves are reflected
without penetrating the medium at all. The deduction as to
what constitutes a good absorbing material should be clear. We
shall consider the theory of absorbing materials, on the basis
sketched, in greater detail in Chap. V ( 51, 52).

While it is not our purpose to go into the properties of spe-
cific materials, mention should be made of a neat and effective
method used by E. C. Wente for testing the absorbing proper-
ties of various kinds of felt. A short tube (of sufficient diameter
to minimize internal friction) was driven by a telephone re-

PHYSICAL FACTORS AFFECTING TRANSMISSION 109

ceiver at one end; the other end was closed by a disc of the ab-
sorbing material backed by a rigid disc. The conditions of the
experiment are exactly those of the wave-transmission scheme
which we have considered at length ( 32); the standing wave
pattern (if the frequency is adjusted to tube resonance) tends to
disappear as more and more of the energy sent out by the tele-
phone is absorbed at the distant end of the tube. (The standing
wave pattern in the tube is determined by means of a small ex-
ploring tube connected to a sound measuring device, such as a
condenser transmitter). The experiment can be performed
in various ways; for a. fixed length of the tube in resonance, we
can determine the relation between the pressure maxima and
minima of the standing wave system, and the reflection coeffi-
cient of the absorbing surface, following the idea of H. O.
Taylor, P/iys. Rev., II, 1913, p. 270. This procedure would
be more or less along classical lines. (Another variation of the
experiment is illustrated in problem 41, p. 226.) But in prac-
tice Wente prefers to measure the driving-point impedance of
the tube, when its length is varied-, this will be a maximum when
the effective length of the tube is an integral number of half
wave lengths (cf. p. 102) and a minimum when this adjustment
of length is changed by a quarter wave length. From these
measured impedances the absorption coefficient of the layer
can be calculated.

We consider now the milder forms of dissipation, in an in-
finite medium, which tend to degrade sound waves. The de-
grading influences include everything which may change the
ordered oscillatory motion of the particles of the medium into
the disordered or statistical motions associated with the heat
content of the medium. These influences are particularly en-
hanced in large-scale phenomena, such as, for example, the
propagation of a pressure impulse originating in a distant ex-
plosion; while the local effect on the wave of departures from
ideal conditions in any one part of the medium may be moder-
ate, the cumulative effect during the passage of the wave from
beginning to end may be considerable. These effects, while not
of fundamental importance to those engaged in the study of

i io THEORY OF VIBRATING SYSTEMS AND SOUND

small-scale acoustic phenomena (as applied, for example, in
telephony) should at least be considered.

A typical degrading influence is viscosity, even if there are no
constricting boundaries. No substance can be expanded or con-
tracted, in one dimension, without dissipation; or to put the
matter another way, sound waves can traverse no medium, how-
ever light or elastic, without absorption to some degree. In the
case of a. gaseous medium it can be shown (Lamb, 64) that the
general equation for the propagation of a plane wave is:

in which /u is the viscosity coefficient. This equation is reminis-
cent of (122); the dimensions of the friction terms are exactly
the same in the two cases and the solution is obtained in a
similar way. Equations (142) and (122) are identical if we
write

2 2 ,v

(43)

i w c

as we may, because - ~ = -- -.,- for a progressive wave motion.
@x~ c**

Then taking as the solution of (142),

* ajr ia>U ) , r N , N

= Ae e < } (cf. 124) (144)

we have for the attenuation factor

Rl 2 M w 2 l^k' 2 , .

a = ---- = . = --- , (I44 tf )

2 P c 3 pc* 3 PC **

from (143). The coefficient - = v has a special name; it was

called the " kinematic" coefficient of viscosity by Maxwell. For air
it is about .132 cm. 2 /sec. at normal temperature and pressure.
The attenuation coefficient a. is small, judged by the standards
of small-scale phenomena in the laboratory; but it rises rapidly

PHYSICAL FACTORS AFFECTING TRANSMISSION in

with frequency. Thus, foghorn signals, when heard over great
distances, arrive with the fundamental tones relatively accen-
tuated at the expense of the higher frequency components.
(This effect incidentally is heightened by the "eddy structure"
of the atmosphere, another large-scale hazard that tends to de-
grade sound waves. See L. V. King, Phil. Trans. , 2i8A, p. 21 1,
1919, and G. I. Taylor, ibid., 215 A, p. i, 1915.)

Equation (142), which is due to Stokes, was derived for a
laterally unlimited isotropic medium. In the case of a gas,
K' = 7/>, that is, the effective bulk modulus for rapid vibrations.
In the case of longitudinal waves in a solid bar, K' must be re-
placed by Young's Modulus (Lamb, 43), and in the dissipa-
tive term the factor /* must be multiplied by (i +0-), a being
Poisson's ratio of lateral contraction to longitudinal extension.
A derivation of the equation in the form described is given by
S. L. Quimby, Phys. Rev., 25, 1925, p. 558.

An unpublished experiment of P. W. Bridgman and H. M.
Trueblood (1918) is of interest in this connection. They wished
to determine the phase velocity of compressional waves excited
in a column of rubber, when one end of the column was con-
nected normally to a vibrating telephone diaphragm. The at-
tenuation due to viscosity in the rubber is very rapid, and there
was no reflected wave from the distant end. The state of the
progressive wave was investigated by means of a light micro-
phone element attached at various points; and it appeared
that, from the phase of the vibration of the exploring micro-
phone, the wave length and hence the velocity of the sound in
the rubber (column) could be determined: this was of the order
of 80 metres/sec, for a certain sample. Theoretically it should
have been possible, by observing the diminution in intensity
with distance along the column, to determine the attenuation
constant, a, and from this to compute the coefficient of viscosity
of the rubber for the frequency of excitation of the system.
Unfortunately the experimental data did not permit this to be
done with any accuracy, but the data did show the variations
in viscosity from sample to sample, and a general effect of de-
creasing viscosity as the frequency was raised. This is quali-

ii2 THEORY OF VIBRATING SYSTEMS AND SOUND

tatively in agreement with a point discussed by Quimby (loc.
cit.) in dealing with the viscosity of glass and metals under the
rapidly alternating strains due to longitudinal waves. There is
every reason to expect that the effective viscosity of the ma-
terial will vary with frequency, and that the viscosity under
high-frequency excitation will be quite different from the static-
ally determined constant. Quimby found, for example, viscosity
coefficients of the order of io 3 for glass, aluminum, and copper
at frequencies of the order of 40,000, whereas the statically de-
termined constants are in the neighborhood of io 8 . The method
used by Quimby depends on the standing wave phenomena in
bars driven into longitudinal vibration, and is in itself an interest-
ing application of the theory of acoustic systems. 1

Heat conduction (i.e., failure of the laminar elements in the
medium to expand or contract adiabatically) is also a cause of
degradation. This was investigated by Kirchoff (cf. Lamb, 65)
who found that the attenuating effect was comparable to that
accounted for by viscosity.

When sound waves are of finite (i.e., large) amplitude, as
distinguished from the usually considered condition of "small
oscillations," degradation results to a marked degree. The wave
velocity depends on the amplitude-, the medium in effect does
not obey Hooke's Law. In the classical theory (Lamb, 63) it
is shown that the "crests" are propagated with slightly higher
velocity than the "troughs"; the former tend to overtake the
latter in transit and distortion of wave form is inevitable. This
case is of considerable importance to those working in the field
of large-scale acoustic transmission, and some recent experi-
mental work by M. D. Hart (Proc. Roy. Soc. IO5A, p. 80, Jan.
i, 1924) may interest the reader in this connection. It is evi-
dent that if the "crests" tend to pile up and become steeper,

1 The paper also gives an interesting discussion of the discrepancies between Stokes
theory and the observed data, arising when viscosity coefficients or displacements are
excessive. Again, the piezo-electric driving system is of special interest in connection
with other piezo-electric applications referred to in Appendix B.

Viscosity under alternating stresses is also discussed by R. W. Boyle, (Tram. Roy.
'Soc. Canada, Ser. Ill, 16, 1922, p. 293) " Compressional Waves in Metals."

SOUND WAVES IN THREE DIMENSIONS 113

a point will ultimately be reached at which disordered motion
takes the place of a certain part of the original acoustic energy.
To all the effects mentioned above there are added in any
practical case, meteorological hazards, such as refraction due to
varying temperature gradients in the atmosphere, or to varying
air currents. There are analogous disturbing factors in subma-
rine signalling, though not to so great a degree; and it may be
noted that the inherent losses in water (which depend on the
kinematic viscosity) are less than in air. It is indeed fortunate
that in telephony as ordinarily practiced no great use is made of
acoustic phenomena of the large-scale kind. 1

35. General Theory of Sound Waves in Three Dimensions

In most of the problems of sound radiation divergence plays a
dominant part; consequently we must extend the theory which
we have used for plane waves to include motion in three dimen-
sions. Instead of taking a thin lamina of the medium ( 30) we
consider a small element of volume dx-By- 5z. The net force in
the ^-direction due to the excess pressure on this element is

and since the force due to inertia is p\*dxdydz we have (cf. 107)

* + 1- - (H5)

for the equation of motion along the ,v-axis. If y and f are
respectively the displacements in the y and z directions, we
have the set of equations

*--! *--! *- <>

1 The reader specially interested in sound signalling in air, meteorological hazards,
etc., is referred to the monograph on "Principles of Sound Signalling," by M. D. Hart
and W. Whately Smith: London, 1925. (Reviewed in Nature^ 1 16, Dec. 12, 1925.)

1 1 4 THEORY OF VIBRATING SYSTEMS AND SOUND

In these equations p represents the excess pressure: and then

K '

fore/> = K'S (eq. 105^); also we may put c 2 = , so that the se

(145^) may be replaced by

- ' -t f -I

It is not an economy to deal with three separate equation
of motion in a case like this, any more than it would be in ar
ordinary oscillation problem to use %m equations for a systen
of m degrees of freedom (cf. 20). To unify the theory we seek j
single function from which all three of the quantities , 77, <
(or their time derivatives) can be derived. Such a functior
exists for all cases of fluid motion in which there is no circulation
(e.g., no eddies or vortices) and this is plainly the state of thing?
when the particles of the medium execute small oscillations in
transmitting sound waves. The function we are seeking is the
Velocity Potential > denoted by <|>, and is related l to the velocities
in any hydrodynamical problem as follows:

, a<t> a<t> 9* / AN

* = - a*' ' = - W f = " aT (I46)

For this fundamental idea we are again indebted to Lagrange.
(An alternative definition to (146) is

4>= ~ \ (kdx + My + fdz), (1460)

1 The reader will note that throughout we follow the convention of a negative
sign in such equations as

This usage is in accord with that of Lamb, but contrary to that of Rayleigh. It seems
to the writer only common sense to use the negative sign because the natural motions or
fluxes which take place in any field of force are from points of high pressure, or high
potential, to points of lower pressure or potential; in other words, the directional deriv-
ative or gradient must be negative with respect to the corresponding motion. If further
justification is required for this usage, the remarks of Lamb in the Preface to his
" Hydrodynamics " are to the point.

SOUND WAVES IN THREE DIMENSIONS 115

which is analogous to the line integral of the forces, used in
computing the work done, to obtain the Newtonian potential
function in mechanical problems; but (1460) is not as useful in
the present instance as (146). The function <|> is a scalar, just
as the mechanical potential function f\ s a scalar; but (like V)
its directional derivatives are vectors. The dimensions of <}> are
L-'T- 1 .)

To obtain <(> we first integrate (145^) obtaining

i = -c 2 ~ {

0*1/0

the constants | , ?} , f being the original component velocities,

when / = c. If we take for these constants = ----- -, etc.,

we have, on comparison of (147) and (146)

+ = r 2 f sdt + h, (148)

J Q

which is the velocity potential function sought. (148) yields at
once the relation

4> = c 2 s = -> ( I 49)

which is important in several ways as will appear.

One application of (149) is immediate. We have given the
equation of continuity (106, 1060) in one dimension, and by an
easy extension of the idea to three dimensions, we have

9 + 3| + 95 + 9f = . (I50)

3/ T 9* T 3y ^ 9z v i '

j. P.J.

Writing for s, 2 and for ^, -- and so on for the other two

C (jR

velocities, we have

ii6 THEORY OF VIBRATING SYSTEMS AND SOUND

or, using the accepted abbreviation V 2 <|> for the expression in
parentheses,

<j) c 2 V 2 $ = o, .(1510)

which is the general equation for the propagation of sound
waves in three dimensional space.

The general procedure in solving a three-dimensional prob-
lem is as follows: we first find a solution of (151^) which satis-
fies the boundary conditions; knowing <|> we can determine the
excess pressure p, the condensation j, or any other variable in
which we are interested from the relations (149) and (146).

36. Spherical Waves of Sound; the Point Source

Some of the simpler properties of spherical waves of sound
may now be investigated. In spherical coordinates we may
write

x = r sin 8 cos </>, y = r sin 6 sin 0, z = r cos 0,
and

a 2 , 2 a , i 3/ . . 3\ , i 3 2

Now if there is spherical symmetry about the origin (the case
of a small source of sound at x = o, y = o, z = o) <j> is a func-
tion of only r and / and (1510) becomes

+ 9*
r 3r

or, if a new function $ = r-<|> is chosen, we have instead of (153)

This equation is kinematically equivalent to (108) if we let
Q = (r<|>), consequently it must have a solution of the type

- + ^/ + r), (154)

SPHERICAL WAVES OF SOUND 117

the first term of which represents a spherical wave diverging
radially from the source, and the second a spherical wave in the
reverse direction just as we noted for equations (108) and (109)
for plane waves. A and B are arbitrary constants to which an
interpretation is to be given. It is evident at once that all ex-

A 2
pressions for the energy in the wave will contain <j> 2 or so

that the energy density at any point distant r from the origin
will vary as ; and the energy flow through any spherical sur-
face surrounding the origin must be a constant, independent of
r, consistently with the energy principle.

Fixing attention on only the divergent wave, the first rela-
tion we consider is that between the particle velocity (r) and

9<t>
the excess pressure. Since (r) = , and p(r) = p< (149)

we have

A* f , t ^ A'

(n = H f (ft r) H -

r r-

and

p(r) = j\ct r),

f(ct r) denoting the derivative of/ with respect to (ct r).
At appreciable distances from the origin the second term in the
first equation becomes small as compared with the first term,
so in any practical case

p = pc% = R% (r large). (156)

Also, since in any small element of volume dr-rdO-r sin 0-d<t> we
must have^> = K'S (p and 5 varying only with r)

cs = and p = {. (1560)

These are all relations which we have found for plane waves;
the only distinction between the two cases (if r is not too small)
is that we must consider (r) and (rp) for the divergent wave in

n8 THEORY OF VIBRATING SYSTEMS AND SOUND

place of | and p for the plane wave. This merely allows for the
decrease in energy density due to divergence.

It can also be shown that the maxima of kinetic and poten-
tial energy at any point are equal, as in the case of plane waves;
the same remarks as to energy and phase that were made follow-
ing (ilia) in 31 apply here.

Conditions at the source are especially interesting. Suppose
fluid is introduced periodically there so that

A A ' ft > ^ A ' (> *
* = f( c * ~ r ) ~ ~ cos "(/ -

then

A'u . /. r\ . A' / r

. ,

sin co (t - - + cos ut - , (155*)

and the total flux (or rate of flow of fluid) through a spherical
surface of radius r is

47rr 2 (r) = - ~ W r sin wU - -j + qxA' cos u(f - J. (157)

Now let r be the radius of the source, which we can take as a
very small sphere. Then if (r) = cos o>/, and if we take

(47ir 2 |o) cos ut = A cos o>/
we have in the limit, as r = r = o, from (157)

A COS / = 47ryf' COS a'/. ( J 59)

The quantity /f = 47rr 2 - is M^ maximum rate of emission of
fluid at the source and is called the strength of the source. We have
thus determined the constant A 9 of (1540) and we may write for
the velocity potential at any point in the medium due to a small
isolated source whose strength is A, and whose time periodic
factor is cos w/,

i A I r\ A , , N / s \

4> = cos col / -- 1 s cos (w/ kr). (160)
47rr \ cl 47ir

In making frequent use of this expression we shall be concerned
not so much with the exact shape of the source, as with its size.

THE POINT SOURCE 119

If it is irregular in shape, or flat (as, for example, one side of a
piston), it is important that it be small as compared with the
wave length, or (160) will not be applicable, due to the essential
lack of spherical divergence from the source. But it may be
noted that in many problems where we plainly do not have
spherical divergence from the source as a whole, we can apply an
expression similar to (160) to individual elements of the source
and by an integration obtain the velocity potential at any point
in the field, due to the source as a whole.

The rate of working of the system of sound waves from the
simple source must be the same at any spherical surface sur-
rounding the source. It is clearly obtainable as the product of
the flux and the pressure at any such surface. We have from
(160), for the velocity and pressure

| = - ^ = - ^sin (co/ - kr) + ~~~ cos (f - kr) (161)
and

p = p< = sin (co/ kr). (i6ia)

4?r

The rate of working is therefore

47rr 2 - /> = sin 2 (o>/ &r) -, sin (o>/ ^r) cos (utkr)

* 47T I r X J

>^u jf'' r T n

= p _ _ i cos i(utkr)j- sin 2(o>/ ytr) ; (162)
OTT I kr J

and we have for the average rate at which work is done by the
source

(163)

Ut (J II

in which the quantity

aw T

.___ ,

is evidently the sound intensity at the surface of a sphere of
radius r, due to the source of strength A at the origin. On the

iio THEORY OF VIBRATING SYSTEMS AND SOUND

principle of equation (116) the energy density in the medium
must be

i dW

E = ~c -7T =

This is obviously equivalent to

* /'max.

ax . = - f ~,->

^ff\

i 66)

as in the case of plane waves; and again we have for the inten-
sity the useful formula

1 p* m ^

2 pC

(l6 4 )

identical with (116) for plane waves.

37. The Pulsating Sphere as a Generator of Sound

The most interesting and instructive problem to which the
theory can be immediately applied is that of determining the
reactions on a pulsating sphere of finite radius, used as a sound
generator in the unlimited medium. Since we are not dealing
with a point source, we cannot neglect the second term of the ex-
pression of the particle velocity in (1550), and this as will appear
is an important element in the consideration of the problem.
It is desirable to make a new start, using complex quantities for
brevity in all operations.

The first boundary condition is that when r = r (the radius
of the sphere) the velocity shall be (the real part of)

l(ro) =

(167)

We must therefore take, for the solution of the differential equa-
tion (1530), for the divergent wave only, a function of the form

(l68)

PULSATING SPHERE AS GENERATOR OF SOUND 121

A second boundary condition, namely, that the velocity shall
vanish at infinity is obviously satisfied. From (168) we have,
for the velocity at the surface of the pulsating sphere

- <S1

. t

(I6 9 )

which must be identical with (167) ; thus comparing coefficients

If '

r li

' O

r 2 (i -

and the velocity potential is therefore

(170)

(170

From this we obtain at once the excess pressure at the surface of
the pulsating sphere:

p(r a ) = (4

(i + P

(172)

The radiation impedance of the device per unit area is evi-
dently, since = kc,

/>(*) (W + /*r.)

. .

(I73)

The first term in the numerator, a real quantity, is in phase
with the velocity and is therefore a resistance coefficient; the
imaginary quantity in the numerator is in phase with the accel-
eration (/w ) and is therefore an inertia coefficient.
These coefficients may be written

bi = pc-

(resistance),

ai = pr r+k^ (inerda) '

2^ = b\ + iwai (complex impedance).

(174)

122 THEORY OF VIBRATING SYSTEMS AND SOUND
The form given is to be preferred, but the reader may express

2?r

these in terms of wave length, by substituting for k. It will

be noted at once that if r becomes very small as compared
with the wave length (i.e., if the generator becomes the point
source of 36) the radiation resistance becomes

L2 o o

= pc-k 2 r 2 = - per unit area.

This result enables us to liquidate a long-standing obligation,
namely the determination of the radiation resistance of a small

source of area S y assuming spherical divergence of sound energy

therefrom. We have r 2 = and consequently, /0r the whole

, ^
= bi' = P c - = p ----- > (175)

4?r

area,

which is the relation previously quoted.

We conclude with a discussion of the physical significance of
the results obtained. Suppose first that the frequency of driving
of the generator is low, i.e., X is very great; we have already
noted the form of b\ for this case. The form of the inertia term a\
shows that, in effect, a mass of 'fluid r p per unit area of the gen-
erator surface must be pushed back and forth during the oscilla-
tion, much as if it were rigidly attached to the pulsating spheri-
cal surface of the generator. This is spoken of as the added mass
due to the medium; in the case a sound generator immersed in
water the added mass due to the water may be several times the
mass coefficient of the generator structure itself, with the result
that the natural frequency of the system is very much lowered.
The added mass for the whole surface of the sphere is 47rr 3 - p;
the total effect of added inertia due to the medium is thus equiv-
alent to a quantity of the fluid equal to that contained in a sphere
three times as large as the generator. This effect is analogous to
that obtained for linear (oscillatory or non-oscillatory) motion
of a sphere, at low velocities: in that case the added inertia is

PULSATING SPHERE AS GENERATOR OF SOUND 123

that of half the quantity of fluid displaced by the sphere
(Lamb, 77).

On account of their inherent interest curves have been com-
puted for the frequency variation of a\ and hi for a pulsating
sphere of unit radius, in fluid of unit density (see Fig. 13). (The
complementary variation of a\ and ^i is reminiscent of similar re-
lations observed previously, in pressure reactions as a function
of frequency it is interesting to compare the results shown in

rH.oxio 4

a 1

1,00
.80
.60
.40

.eo

Added Mass
/'per Unit Area

^ Resistance
per Unit Area

10.0 x'10 4

2.0 MO 4

10,000

20,000

30,000

40,000

50,000

Fpeouency

FIG. 13. REACTIONS ON PULSATING SPHERE OF UNIT RADIUS IMMERSED IN WATER.

Fig. 5.) For the pulsating sphere, the change in a\ and 1 as
higher frequencies are attained is striking; for very small values
of X, we have, since k is very great,

^ = pc = R y cf. (1170) (176)

the characteristic radiation resistance of the medium previously
obtained for plane waves. The inertia term (174) diminishes
very rapidly with X and is small if X is less than (say) ir in any
practical case. The generator now has to work against only a
radiation resistance (in addition to its own impedance) and its
sound-generating efficiency is a maximum.

i2 4 THEORY OF VIBRATING SYSTEMS AND SOUND

The results in this specific problem well illustrate the diffi-
culties involved in "driving" an acoustic medium, i.e., in radi-
ating sound without merely pushing the medium away at the
driving point. These results are general in their import, and
should be thoroughly understood by the reader before attacking
other radiation problems. The remarks of Stokes (Lamb, 80
or Rayleigh, II, 324) are to the point in this connection; for
his theory in full, reference may be had to his paper "On the
Communication of Vibration from a Vibrating Body to a Sur-
rounding Gas," in Phil. Trans. Roy. Soc., 158, June 18, 1868,
reprinted in his " Collected Papers," vol. IV, p. 299.

38. Reactions of the Surrounding Medium on a Vibratiyig String

The complete solution of this problem is a specialized and
difficult matter, but it may be discussed in this place because it
brings into one focus the important points relating to the inter-
action between vibrating system and medium that have been
previously developed in the text. These points are, the radia-
tion of energy from the system, the added mass, and finally the
dissipation due to viscosity in the medium. And in connection
with the last of these, there is an important application to the
string galvanometer.

In the net result, the radiation from a vibrating string is the
least important of the three effects, but it is necessary to deal
with it first in order to get it out of the way. Anticipating the
discussion of the next section ( 40) it may be observed that any
element of the string is virtually a double source, and therefore,
except at very high frequencies, a radiator of low efficiency.
Stokes, for example, in the classical memoir just cited pointed
out how feebly the vibrations of the string are communicated to
the surrounding gas. If lateral (i.e., non-radial) motion in the
neighborhood of the string could be prevented, the intensity of
the radiated sound would be enormously increased. "This
shows the [vital] importance of sounding boards in stringed in-
struments. Although the amplitude of vibration of the particles
of the sounding board is extremely small compared with that of

REACTIONS ON A VIBRATING STRING 125

the particles of the string, yet as it presents a broad surface to
the air it is able to excite loud sonorous vibrations, whereas were
the string supported in an absolutely rigid manner, the vibra-
tions which it could excite directly in the air would be so small
as to be almost or altogether inaudible." (Stokes, quoted by
Rayleigh, loc. cit.)

After a lengthy analysis, A. Kalahne (Ann. d. Phys., 45,
1914, p. 657) finds that the radiation from an infinite string de-
pends on the quantity

K - ^ 2 " (r) a * (I77)

2.7T

in which k = , for the medium (as is customary) and X' is
X

the wave length of the oscillations on the string. For steady
state conditions in the (infinite) surrounding medium, it is
found that for K real (that is, if the wave velocity on the string
is greater than the wave velocity in the medium) there is pro-
gressive wave motion, and hence radiation, from the string;
but if K is zero or imaginary, there are stationary waves through-
out the medium, and no radiation takes place. In both cases
the wave motion in the medium falls off as the distance from
the string is increased; and even in the first case, the radiation
is not very great. In view of these considerations and certain
analytical difficulties, Kalahne has advisedly neglected the
radiation resistance of the system in reckoning the added mass
due to the medium.

Kaliihne's work on the added mass (or lowering of the nat-
ural frequency of the system) is contained in two papers (Ann.
d. Phys.y 45, 1914, p. 321, and ibid., 46, 1915, p. i) which deal
with solid or hollow rods executing transverse vibrations in
liquids or gases. The added mass due to the medium is not great
unless the density of the medium is comparable to that of the
rod (or string). The resultant lowering of frequency, for a solid
string, is calculated to be

n n f L

---- =
n 2

126 THEORY OF VIBRATING SYSTEMS AND SOUND

in which p is the mean density of the medium, p the volume
density of the string, n the frequency in vacuum, n' the actual
frequency in the medium, and L a coefficient in terms of cylinder
functions which is approximately unity in all practical cases, but
which approaches zero for very high frequencies.

First we may remark that Kalahne's formula, at low fre-
quencies (for which L = i), is quite consistent with what we
should expect on the basis of "equilibrium theory." It is well
known that a cylinder moving sidewise in a fluid, at ordinary
speeds, is burdened with an added mass equivalent to the quan-
tity of fluid displaced by the cylinder. Consequently the added
mass is to the inherent mass of a solid cylinder as p : p . Since
the natural frequency of a stretched string is equal to a constant
divided by the square root of the density of its material ( 28),

i.e., n = ; , we must have, for any increase 5p = p in the

VPo

virtual density of the string, a negative increment drt of fre-

,. , . . np
quency which is equal to

To go deeper into the matter we may refer to Stokes' long
memoir "On the Effect of the Internal FYiction of Fluids on the
Motion of Pendulums," published in i856. 1 Section III and
the end of Section IV of this paper relate to a cylinder oscillating
in a fluid in a direction normal to its axis. Stokes considered the
viscosity of the fluid (which Kalahne did not) and was inter-
ested in the damping and the lengthening of the period of a cyl-
indrical pendulum due the reactions of the surrounding air.
Thus we should not expect his formulae to duplicate those of
Kalahne as far as frequency variations in the reactions are con-
cerned, but we should expect the low-frequency value of the
added mass to be the same, and should obtain in addition a
value of the damping due to friction.

Stokes* solution of the problem depends on a certain cylin-
der function F(ma) [equations 85, 87, 93, of the original paper]

l Camb. Philos. Soc. Trans.) IX, 1856, p. 8: reprinted in Stokes* "Papers," vol.
Ill, p. i.

'(i - -~Jr x )-/f = M'(k - ik')ink, [Stokes, 98]
\ Tn*ar'&\a)i

REACTIONS ON A VIBRATING STRING 127

a being the radius of the cylinder while m = \| . He ob-

tained for the reaction on the cylinder, in terms of the accelera-
tion inky the following expression:

in which M' = Tra 2 p-/, the mass of fluid displaced by a length /
of the cylinder. In this expression kM' is the added mass,
while nk'M' is the resistance coefficient; Stokes computed
tables for k and k' for a wide range of values of the argument

\\m\a.

For air, - = 7.0, approximately; consequently for a wire

of radius .05 cm., vibrating at 1000 cycles, we should have \rn\a
of magnitude about 10. In cases such as this we should expect
to apply Stokes' solution for large values of the argument,
namely,

k = i + 2 > k' = 2 -~ + - ' [Stokes, end of
\rn\a \m a \m 2 \a~ Section III]

From these we have, per unit length of cylinder, for the added

mass,

ai = Triple = *a 2 pi + - 2 \/ ); (178)

a * n p/

and for the resistance coefficient,

\_a \ a * inp/

It is evident that if M = o, the added mass is exactly that of the
equilibrium theory, and this value leads at once to the approx-
imate formula already given for the lowering of the natural fre-
quency. And considering only the effect of viscosity, this will be
greatest at low frequencies, and will vanish asymptotically as
the frequency is raised; but it must be noted that the formula
cannot be correct at extremely low frequencies, because the de-
velopment given depends on a value for \rn\a > i.

To find approximately the damping coefficient of the string
we take the ratio of b\ to twice the total (inherent and added)

128 THEORY OF VIBRATING SYSTEMS AND SOUND

mass of unit length of the string. The total mass is Tra 2 (p + P)>
to two terms. After a rough calculation we obtain

or, more simply still,

A = . (178.)

Potf

We may now consider some recent experimental work.
H. Martin (Ann. d. PJiys., 77, 1925, p. 627) made an experi-
mental test of Kalahne's formula for the lowering of the natural
frequency of steel wires when immersed in liquids. The fre-
quency n is taken as unchanged, for the string immersed in air,

on account of the smallness of the quantity ; for the case of

steel wires in water the formula is found to fit the facts very
well, that is, to within one or two per cent.

Martin's apparatus consisted of a stretched wire, immersed
in liquid and magnetically driven, the oscillations at various fre-
quencies being observed with a microscope. In this way he got
resonance curves, from which (if the vibrations were sufficiently
small, to preclude over-driving) both the resonant frequency
and the damping constant could be obtained. From a very care-
ful study he found for steel wires the following relation between
the damping constant of the system and the other quantities

^ -Po

which agrees closely with the formula we have just developed
from Stokes' calculations. 1 We have, therefore, for certain

1 Martin quotes a relation due to I. Klemencic (JVien. Ber. y 84, II Abt., 1881, p. 146),
namely

A = ^ pW/i 4- - M - ?l p 2 ,

a Po Pr/' 2 6/zp

for the damping constant of a cylinder immersed in a fluid, and oscillating about an axis
perpendicular to its own. It is evident that the first term of the formula based on
Stokes is in much better agreement with Martin's experiments than the first term of
Klemencic's formula; the second term is common to both expressions.

REACTIONS ON A VIBRATING STRING 129

cases, a fair theory of the damping of a vibrating string due to
the viscosity of the surrounding fluid. Incidentally we note the
close similarity between the damping constant as above given,
and the attenuation factor a (equation (/?), Appendix A) for
sound waves in a viscous medium surrounded by a tube; they
differ merely by the factor ic (twice the wave velocity) a
necessary matter of dimensions.

It would be interesting now if we could compute the added
mass and damping of a very fine wire immersed in air. One
would think that the " air damping " of a fine wire would be very
great, and indeed it is; the Einthoven String Galvanometer is
the outstanding application of this effect, as it depends for its
success more on the fineness of the string than on any other fac-
tor. Stokes* theory, carried to its logical conclusion for small
values of the argument \m a gives enormous values of added
mass and resistance due to the viscosity of the medium (equa-
tion 115, original paper), but Stokes distrusted the application
of the theory to this limiting case. Quoting from the end of Sec-
tion IV of his paper, " It would seem that when the radius of the
cylinder is very small, the motion of the cylinder would be un-
stable. This might well be the case with the fine wires used in
supporting the spheres employed in pendulum experiments. If
so, the quantity of fluid carried by the wire would be dimin-
ished, portions being continually left behind and forming eddies.
The resistance to the wire would on the whole be increased, and
would moreover approximate to a resistance which would be a
function of the velocity/' As one consequence of these specula-
tions Stokes thought that the damping of a fine wire would in
fact be greater than the value computed on the basis of the
theory; and in view of this uncertainty we find it necessary to
drop theoretical calculations at this point and deal with the
case of a fine wire for the most part on an experimental basis.

The most recent representative paper on the Einthoven Gal-
vanometer is that of H. B. Williams (Jour. Opt. Soc. Am.^ 9,
1924, p. 129); it also contains references to other literature, in-
cluding Einthoven's original contributions. Prof. Williams has
in preparation a second paper dealing with his more recent ex-

130 THEORY OF VIBRATING SYSTEMS AND SOUND

perimental study of the galvanometer; meanwhile he has kindly
made available this unpublished work, and his accumulated
experience with the instrument, for the purposes of the discus-
sion given here.

The substance of Williams' paper of 1 924 is the theory of the
motion of the current-carrying string, which is virtually a
"uniformly loaded member owing to the electrodynamic reac-
tion between all the elements of the string and the perpendicular
magnetic field in which it is placed. This theory of the action of
the string under a uniformly distributed alternating force is of
interest in itself, but that is not our present concern. It has been
noted as an experimental fact that if a natural oscillation of one
particular frequency is selected (harmonics being excluded) the
damping factor is constant over a considerable range of ampli-
tude (Williams, loc. cif. y p. 161). But in the theoretical studies
of the galvanometer per se y no allowance has apparently been
made for the variation of the damping constant with frequency;
and what interests us at the moment is the validity of this idea.

Einthoven's experiments are recorded in a two-part paper in
^nn. d. Phys. y 21, 1906, pp. 483-514 and pp. 665-700. He made
measurements on certain strings under widely varied adjust-
ment of the tension; of these we shall consider two typical
cases, namely, the high-tension or oscillatory case, and the
low-tension or over-damped case, for which n was less than
A. For the oscillatory case, Einthoven measured the loga-
rithmic decrement (d) and determined the mechanical resist-
ance r from the following relations:

d s

A = --, r = 2wA, m = ,

JL 7/o

in which T is the actual period of the oscillation, and the

27T

natural frequency. It is evident in this case that m is an
ideal or derived mass coefficient, which is valid for the case of
rapid oscillations, but which may not be applicable to the
slower motion of the string under reduced tension.

Now consider the over-damped case. Einthoven made

REACTIONS ON A VIBRATING STRING 131

a number of measurements of the mechanical resistance (r) of
the system which were based on the point of inflection of the
curve of the growth of the deflection when a constant force (cur-
rent) was applied. Now (as has been hinted in problem 2, Chap.
I), at the point of inflection, the acceleration is zero, and we
have

F .

Hence at this point r is given in terms of the stiffness constant
(j) and the graphically measured quantities q and |, but it must
be remembered that these relations depend on unvarying mass
and resistance constants, which Einthoven's analysis assumes.
It is safe to assume that the stiffness is constant, in a given
adjustment; but if mass and resistance vary, we should prefer
to write, for the relation between displacement and velocity
at the point of no acceleration,

2A*.

q = T&

in which the subscript / refers to conditions at the point of
inflection. This follows from the equation (obtained by put-
ting = o in the original differential equation),

which we should be compelled to use if the stiffness were taken
as constant in the experiments, and the variations in r, m and A
were unknown.

Now Einthoven apparently thought (loc. cit., p. 503) that r
was nearly constant, because when measured in the manner
described above, the value of r was nearly the same for a wide
variation in the tension of the string in one test the tension
varied over a range in the ratio of i : 324. But from the situa-
tion as we have examined it, it would seem more reasonable
to give the final results in terms of A rather than r, because of
the possibility that variations in r and m might escape notice,

132 THEORY OF VIBRATING SYSTEMS AND SOUND

as long as A remained a nearly constant quantity. It is
likely that both the added mass and the resistance of the string
vary; and we should expect them to vary in the way predicted
by Stokes' theory.

Indeed Einthoven (loc. a/., p. 670) has noted that under re-
duced tension the string behaves as if there were a variation in
mass his measurements being made on the assumption of a
constant resistance. (He found the mass to be greater, the
slower the motion which would be in harmony with the
theory.) We may now observe that it is entirely possible, within
a certain range of frequencies, for both mass and resistance to
vary in such a way that one compensates for the other, and a
more or less constant value of damping results. One of Wil-
liams' tests may be mentioned in this connection. The damping
and natural frequency (1200 cycles) of a galvanometer string
were first measured, for a certain tension; it was then calcu-
lated, on the assumption of constant damping, that the string
would be aperiodic at 400 cycles. The tension was then corre-
spondingly reduced, to give this natural frequency, and the
string was found to be aperiodic, to within one-third of one per
cent of the steady current deflection. It seems reasonable to
conclude that when the motion of the fine string is very slow,
there must be a certain compensating effect between the varia-
tions in added mass and in resistance. Thus from the experi-
mental standpoint, we have in the fine air-damped string a most
curious sequel to Stokes' theory, particularly when we note the
theoretical difficulties which he conceived, and his scruples in
dealing with them. Some rough computations on the basis of
Stokes* theory for a very fine wire give values of added mass and
resistance which are not incompatible with the experimental
phenomena we have described, and the reader who wishes to go
further into the matter should examine the behavior of Stokes'
functions k and k f for very small values of the argument (ma).

A word may finally be said on electrical damping. Any gal-
vanometer of low electrical resistance may be strongly damped
by suitably adjusting the external circuit. If the instrument in
itself has very variable mechanical characteristics, it is often of

PROBLEMS 133

advantage to stabilize it in this way. In the case of the Eintho-
ven Galvanometer the fineness of the string, to which the sen-
sitivity and the damping of the instrument are due, entails a
high electrical resistance, so that electrical damping is hard to
apply; but as we have seen, it is not required. There is, how-
ever, a very nice theoretical point in this connection; it will be
found in a paper by L. S. Ornstein (Kon. Akad. Amsterdam,
Proc., XVII, Nov. 28, 1914, p. 784) on the Theory of the String
Galvanometer. It is that electrical damping only occurs for
those frequencies which are odd multiples of the fundamental
frequency of the string and, since only the odd segment of the
string contributes damping, the most important cases are for
the vibrations of the lower frequencies. Ornstein's paper has
other interesting features, and it appears to be the most com-
pact treatment available of the theory of this instrument.

PROBLEMS

21. A plane sound wave in air has an intensity of i erg per second.
What is the force on 10 sq. cm. of an infinite wall (normal to the
wave), due to the impact of the wave?

22. Suppose in problem 21 the area considered in the wall is a pis-
ton whose constants are b 200 c.g.s., c io 7 c.g.s. and a = i gram.
What is the steady state equation of motion of the piston ?

23. What is the added resistance (if any) of the piston system due
to the fact that it is radiating into a semi-infinite medium?

24. A plane wave of sound in water strikes a plane water-air
boundary at an angle of 45. Investigate the properties of the re-
flected and transmitted waves. How are these changed if the rare and
dense media are interchanged with respect to the boundary?

25. In a wave power transmission line the diameter of the tube
is 3 cm. and the fluid used is water; the line is operated at 100 cycles.
Assuming Helmholtz's law of resistance in the tube [Appendix A,
eq. (/?)] what is your engineering opinion as to the maximum distance
for economic power transmission, and on what criteria are your con-
clusions based? How would your results be modified if oil of viscosity
n = i.o and density p = 0.9 were used in place of water?

134 THEORY OF VIBRATING SYSTEMS AND SOUND

26. Fluid of viscosity JJL is contained in a narrow crevice bounded
by parallel walls separated by distance d. Applying a method similar
to that of Appendix A, show that the mean resistance coefficient is

R = -r as given in equation (37), Chap. I.
a*

27. Pistons with elastic constraints, but free from resistance,
are disposed at either end of a tube of length /, containing air, as in
theory of 32. Neglecting dissipation in the tube, show that the
natural frequencies of the system are given by the equation

tan ftl =

pc(Z + Z/)

On the basis of this equation discuss the more interesting cases of
resonance of the system, for various values of /, and various relations
between Z , Z/, and pc.

28. Taking the velocity potential as

, / . cox <jox\

9 = 1 A cos --- \- B sin /

\ c c I

cos

find the natural frequencies of a tube of length /, rigidly closed at both
ends ( 33). Find also the space-distribution of amplitude and pres-
sure in the tube in the normal modes.

29. A pulsating sphere of radius 10 cm. is tuned to 1000 cycles and
has an inherent (mechanical) resistance of 20 c.g.s. per unit area at
that frequency. Assuming a force * <? ?wt per unit area applied to its
surface, find the ratio of radiated energy to total energy expended, at
1000 cycles, when immersed (i) in hydrogen, (2) in air, and (3) in
water. What do you find for the relative sound intensities at corre-
sponding points in the field, in the three media?

30. Two similar sirens are working, one in air, near the ground, in
flat country; the other just below the surface of the water, in a quiet
sea. What are their relative rates of working, in order that a listener
at a given distance may perceive sound of the same intensity, assum-
ing equally efficient listening devices in the two media and neglecting
all degrading factors? Now assuming dissipation due to viscosity in
the two media, how are your conclusions modified for components af
the siren tone of 1000 and 3000 cycles?

CHAPTER IV

RADIATION AND TRANSMISSION PROBLEMS

40. General Considerations; Single and Double Sources

In the preceding chapters the emphasis has been placed on
methods rather than results, and as a consequence it has been
possible to develop the general theory with some degree of
unity and coherence. In the present chapter departure to some
extent from this plan is unavoidable, for we shall have to con-
sider several applications in detail in order to give a fair idea
of the range of usefulness of the theory. In addition, there are
serious analytical difficulties, and it will be necessary at times
to depart from a strictly rigorous procedure in order to reach
practical conclusions without undue labor.

Before turning to typical radiation problems we may briefly
mention one of the ingenious conceptions of the classical theory,
namely the double source, or acoustic doublet. This is constituted
of two small simple sources, close together, and identical except
for the sign of the strength factor A\ the velocities of emission
of fluid at the two sources are 180 out of phase. The potential
theory of the acoustic doublet is analogous to that of the mag-
netic doublet, and of the electrostatic doublet; the idea origi-
nated from the necessity of determining the acoustic field due
to such devices as tuning forks, membranes with both sides
exposed to the medium, etc., which exert equal and opposite
driving forces on the medium at two nearby points.

Let L A 1 be the strength of either component of the double
source, and 5x be the distance between components, the axis of
the doublet being parallel to OX. It may then be shown that
the velocity potential at a point distant r from the doublet is

4 7T

136 THEORY OF VIBRATING SYSTEMS AND SOUND

in which A = ^/'S,v, a finite quantity ', and 8r = dx cos 0, being
the inclination of the radius vector r to the axis of the doublet.

Performing the differentiation we have, if k = = >

4, =

When r is small, i.e., for points near the doublet, the velocity
potential varies as ^cos 0; for large values of r, the correspond-

ik
ing factor in the coefficient is cos 6. Thus along the axis, at

great distances, conditions are not very different from those
prevailing in the field due to a single source. For = - (i.e.,

when r is normal to the axis) the radiation from the doublet is
zero; and it is evident that on the average (taking only energy
into account) the doublet is a poorer radiating device than the
simple source of equivalent strength.

According to present practice in acoustics we deal with
sound-generating and detecting devices in which only one side
of the vibrating element or diaphragm is exposed to the me-
dium; the notion of a double source is not of great use in these
problems and we shall not pursue it further. It is of greatest
advantage in the solution of such problems as the radiation
from a vibrating sphere (Lamb, 77; Rayleigh, II, 328) or the
scattering of sound waves by a spherical obstacle (Lamb, 81);
the reader interested in such matters may refer to the classical
treatises.

A major problem which we shall consider is the radiation
from a circular piston. To avoid analytical complications the
piston is made to reciprocate in a cylindrical hole in a plane
rigid wall; the radiation thus takes place into a semi-infinite
medium. The effect of the piston on the medium can be con-
sidered as the sum of the effects of elementary sources, each of
strength dS- 09 distributed uniformly over the area S of the

HIGH FREQUENCY RADIATION FROM A PISTON 137

piston. Since the divergence of the radiation from each ele-
mentary source is limited by the wall to a solid angle 2?r, in-
stead of the angle 4?r of eq. (160), we have, for the potential at
any point distant r from the elementary source

4> = ^Iv^-'*', (i 80)

and consequently the velocity potential due to the piston as a
whole is

t "** r r e ~' kr
= * ( ( dS.

^ J J* r

This useful formula (the existence of which was implied in the
discussion following eq. (160) has been obtained by simple rea-
soning on physical grounds; but it is a particular case of a gen-
eral formula which can be established rigorously. The general
formula (Rayleigh II, 278, 302) is

dS, (ISO*)

dn r

in which ----- is the normal velocity of the element dS of the
3 ;

g<j>

reciprocating surface; in (180) - = and, being a constant

with respect to dS, is removed from the integrand. Equation
(180^) or (i8o) can be taken as the starting point for the in-
vestigation of radiation into a semi-infinite medium from the

a<t> .

vibrating surface of any solid body, providing is known

for each point of the surface.

41. High-Frequency Radiation from a Piston; Diffraction

From the principles established in 37 and 40 we can
proceed to investigate the radiation from a flat piston into the
semi-infinite medium, the piston being driven at any frequency
whatever. Before dealing with this question generally, which
leads to some detailed applications, we had best dispose of the

138 THEORY OF VIBRATING SYSTEMS AND SOUND

particular case of radiation at high frequencies, with its accom-
panying diffraction phenomena.

Taking the piston of large diameter as compared with the
wave length, it may safely be assumed that the waves will be
nearly plane for a little distance from the piston, and conse-
quently there will be no added mass due to the medium. More-
over, the radiation resistance will be that for plane waves, i.e.,
#1 = pc per unit area of the piston. At some distance from the
piston, however, the radiation will begin to diverge; thereafter
the intensity must weaken and ultimately become inversely
proportional to the square of the distance from the source.
These conclusions are all analytically verified, in this and the
succeeding article.

FIG. 14.

It will suffice to investigate conditions on the axis of sym-
metry of the system, in which case the integration of equation
(1800) presents no difficulty. If the piston is given a motion
cos co/, the velocity potential is, in the notation of Fig. 14,

t tw ' s*v**R /"__tx7i . -A . \y a *R

.) - --j

(181)
in which it is understood that only the real component is to be

HIGH FREQUENCY RADIATION FROM A PISTON 139

retained. Suppose now that R = m\\ inserting the limiting
values of y y and reverting to real quantities,

<|> = - ? [ s i n (o* - kx*+ mW) - sin (f - *)], (182)

which is rigorously correct for all points on the axis. Since

27T

k = it is seen that for x = o, the maximum value of <|> at

the center of the piston face is zero or -,-, according as m = ,

or m = n + 2* n being an integer. It is also evident (on differ-
entiating with respect to x) that the velocity at x = o is
cos co/, as it should be.

To investigate conditions at points along the axis for which
x > 3#zX, we are justified in using the approximation

m 2 \ 2 \
+ ^ 2 j = x + a, (183)

and, since

. / , N A / n ka\ . ka

sin (6 ka) sin ^ = 2 cos ( -- ) sin >

we have

<}, = + M? sin ^ cos (/ - ** - y). (184)

The maximum value of the pressure is

. a . . TT , _ N

P = P<l>max. = , Sin = 2pff Sill - > (185)

K 2 2, X

as & = = - ; the intensity is therefore, by (1640)

A ^

From (186) we infer that there is an approximately parallel
beam of radiation out to the point x = w 2 X; that along the
axis, within the region o < x < m 2 \ there is a succession of
maxima and minima of intensity (analogous to the bright and

140 THEORY OF VIBRATING SYSTEMS AND SOUND

dark spots of optical diffraction phenomena) due to the inter-
ference of rays from the different portions of the piston; and
finally that for large values of x the intensity is, for points on
the axis,

It is evident that the radiation tends to diverge, ultimately in
the form of a cone, and that after the critical distance x = m 2 \
has been exceeded, this divergence becomes a dominating
factor. Equation (186*2) can be written in the form of (164)
and this leads to an interesting but somewhat paradoxical re-

suit. Noting that irR 2 = ir\ 2 m 2 = S y and X = , we have

from (i860)

dt S7T 2 X 2

It thus appears, on comparison with (164) that at great dis-
tances, using the axial radiation from a large piston source, a
fourfold gain in intensity at a given point can be obtained as
compared with that due to a small spherical source of equal
strength. It will be noted that aside from this numerical factor
there is no special factor in (186^) to differentiate it from (164);
that is, the particular angle subtended by the cone of rays has
no direct effect on the intensity along the axis of the cone.
This is not to say, however, that the solid angle containing the
cone of rays is indeterminate, or independent of the dimensions
of the particular piston used. A rough measure of this solid
angle is the angle subtended by the piston when viewed from
the point x = m 2 \\ this is

ir\ 2 m 2 TT / \

= (I87)

Evidently, for a given wave length, the divergence of the coni-
cal beam varies inversely as the area of the piston.

DIFFRACTION 141

For similar relations as between wave length and the size of
the source, the acoustic diffraction phenomena are exact ana-
logues of optical diffraction effects. The comparison is of inter-
est in the present case, if we regard the piston as analogous to
a circular aperture in a plane screen against which light waves
are normally incident. In the solution of this corresponding
optical problem, the Fraunhofer point of view may be adopted;
this assumes parallel rays in both the incident and diffracted
beam. The angle of diffraction for the first minimum of inten-
sity is .61--, if R is the radius of the aperture; this may be
K.

taken, without serious error, at moderate distances from R
(e.g., for x > 5/0 as the angle subtended by the limiting radius
of the first bright diffraction circle on a screen normal to the axis
when viewed from the aperture. The point we are leading up
to is that as long as first bright diffraction circle is smaller than
the size of the aperture, we have substantially a parallel beam,
and that, as the distance from the aperture is increased, the
first bright diffraction circle expands proportionately, while di-
vergence of the beam into nearly conical form takes place. The
solid angle of the cone thus asymptotically filled by this im-
portant component of the radiation is

-(6iX) _.,. (ig?a)

This may be taken as a more accurate expression for 12 than
that given in (187). To summarize, there is evidently some
point on the axis in the neighborhood of x = m 2 \ which roughly
marks the beginning of the transition from a parallel to a di-
verging beam of radiation. The hydrodynamical method of
studying diffraction problems is of great advantage in many
cases, and rigorous solutions can be obtained providing we avoid
such approximations as have been made in the present example.
The example we have treated finds some application in
high-frequency submarine signalling. Low-frequency submarine
signalling is well treated in Drysdale (Chap. IX); but there are

i 4 2 THEORY OF VIBRATING SYSTEMS AND SOUND

few references ! to high-frequency work in technical literature,
and a brief reference to the practical side of such operations
may therefore be permissible. The economy of high-frequency
signalling depends not only on the concentration of the radi-
ation (from the energy standpoint) but also on the associated
directive property. Obstacles, such as ships, icebergs, rocky
reefs, etc., can be located at distance of a mile or more by the
"echo" method. The best high-frequency sound generator
known is a sandwich-like tuned structure made of sections of
quartz crystals, operated piezo-electrically, which are firmly
mounted between two iron slabs each a quarter wave length
thick. One side of the vibrator is of course shielded from the
medium. When operated at (say) 50,000 cycles, by being
placed in connection with a source of high-frequency electrical
energy, the vibrator behaves as a condenser with a certain
amount of internal ohmic resistance; as much as one-third of
this may be due to its radiation resistance as a sound generator.
This radiating efficiency is considerable, being limited only by
such dielectric and elastic hysteresis effects in the structure as
are unavoidable with the substances used. For detection of
high-frequency sound waves a structure similar to that of the
vibrator may be used, or the vibrator itself as in echo work,
taking advantage of a quick change from sending to receiving
circuits through a multiple key. The area of the vibrator may
be of the order of 400 sq. cm.; it thus intercepts considerable
energy when used as a detector. Carbon microphones, and
piezo-electric crystals of rochelle salt may also be used as detec-
ors if suitably mounted in resonant structures. The electrical
circuits required for sending and receiving are similar to those
used in radio transmission.

1 The sketch following is of certain unpublished experiments made by the Columbia
University Group (Profs. M. I. Pupin, A. P. Wills, and J. H. Morecroft) in 1918. The
reader may compare a similar sketch of the Experiments of Langevin and Chilowsky
given in Nature, May 9, 1925, pp. 689-90, in the article "Echo Sounding"; see also
French Patent No. 505, 703 (1918) and British Patent No. 145,691 (1920) to Langevin,
for the Quartz Piezo-Electric vibrator. A more complete description of Langevin's
apparatus is given in "Ultra Sonic Waves for Echo Sounding," Hydrographic Review,
(Monaco) II, No. i, 1924, p. 57.

RADIATION FROM A PISTON 143

The frequency used in underwater high-frequency work
should be less than 100,000 cycles, owing to various practical
considerations. One of these is of course the kinematic viscosity
factor, which accounts for appreciable attenuation at very high
frequencies; other limiting factors are the difficulties encoun-
tered in the design of apparatus, and in utilizing available ma-
terials. Temperature and current conditions in the water are
also disturbing factors which are aggravated at high frequen-
cies; these are mentioned in 34.

42. Radiation from a Piston into a Semi-Infinite Medium

We return to the discussion of the general problem of radi-
ation from the vibrating piston at frequencies for which diverg-
ence near the piston must be taken into account. If we can de-
termine the reaction on the piston as a function of frequency,
we shall have material useful in the solution of several other
problems.

We start again from the equation

the geometrical relations being, for the moment, as in Fig. 14.
The pressure at any point y on the piston is therefore

in which n denotes a radius vector lying in the surface of the
piston and extending from the point y to the surface element
dS. Thus to compute the force on the piston we must perform
the integration in (188) and then integrate the resulting pres-
sure (a function of y) over the surface of the piston. The force
is then

144 THEORY OF VIBRATING SYSTEMS AND SOUND

in which dS' denotes a surface element at the point y. The
situation is as shown in Fig. 15: each element dS is to be
summed with respect to each element dS f over the surface of
the disc; the effect at dS due to dS' is the same as the effect at
dS' due to dS> so that in the double summation, the ultimate
result will be twice as great as if each pair of elements dS, dS f
were taken only once. The problem can thus be simplified if
we multiply the quantities to be integrated by two, and then

FIG. 15.

arrange the integration so that each pair of elements is taken
only once. This latter end is accomplished if for each value of
y (i.e., for each dS f ) we integrate with respect to dS only over
the inner circular portion of the disc; that is, the portion of
radius y, thus keeping dS f outside the region of the first inte-
gration.

We have first to evaluate the integral

The calculation is treated by Rayleigh (II, 302) but the fol-

RADIATION FROM A PISTON 145

lowing outline may be useful. We can take, instead of (190)
the integral

that is

rl r 2ycos9 2 C*

\ I I e- ikr *dnde = - I (*-**

*-^o /o IK. J o

= !L + ?' f V*' 2 * co

Expanding the exponential function, and integrating term by
term, we have

,
2- 2.4.41

since

. . -J,

f 2 7 1.3 ...( l) 7T .- .

I cos xdx = - -------- if w is even,

Jo 2.4 ... w 2

and

f* n J 2.4 . . . (tf l) . f . - ,

I cos n A:^ = - ------- if n is odd.

Jo 1.3 ... n

The first parenthesis in (190^) will be identified by the reader
as /o(2/fcy); the second parenthesis is a related but odd func-
tion which Rayleigh defines as

It must be noted that this is not a complete Bessel's function of
the second kind, since by itself it does not satisfy Bessel's equa-
tion.

Equation (190^) now becomes

(190*0

146 THEORY OF VIBRATING SYSTEMS AND SOUND
placing this in (1900) we have for the final result

r = -,[K(*ky) - i (i - /o(2*y))]- (i 9

V may be considered the Newtonian Potential at the edge of a
disc of radius y, the density of the disc being e~ lhri y if r\ is taken
with respect to the point on the edge.

In performing the second integration, namely I V-dS' we

note that dS f = ivydy\ consequently we shall need the follow-
ing formulae for the integration of the Bessel's functions:

and

jf-
jf

J (z)-zdz =

K(z)-zdz = Ki(z)

(192)

The first formula is that which follows eq. (47) of 12; the
second formula results from the definition of K(z) y if Ki(z) is
as given below in (195).

Using these relations, and taking twice the values obtained
from the integrations, we have, for the force on the piston

* = -~^- C

or, since o> = kc y

. .

, (193)

This expression is in the standard form for an impedance,
Z = S(bi + /ci>ai); in the first term (i.e., in ^i) we should expect

to van ^ s ^ w ^en ^/? is large (and it does), thus making
the radiation resistance of the piston irR 2 - pc at high frequen-

RADIATION FROM A PISTON 147

cies. 1 (This is consistent with the theory of 41). The gen-
eral behavior and the applicability of the functions Ji(lkR)
and K\(ikR) are fully dealt with by Rayleigh (loc. r/V.); these
are matters we shall take for granted. For our purposes it
suffices to note a few terms of the series

7 M - ?F _
Jl(Z) " 2L 1

and

In treatises on Bessel's functions it is shown that for targe values
of the argument

^ large

We are now prepared to discuss the reaction of the medium
on the piston at low frequencies, which is the more important
case, in general acoustics.

Since kR is small, we need take only the first term in the
lower series, equations (194); this gives for the resistance co-
efficient

k 2 R 2 pC'k 2 (trR 2 )' 2 pu 2 S 2
2 2?r lirC

It is interesting to compare this with what we should have ob-
tained for half the pulsating sphere of 37, assuming it to
radiate into a semi-infinite medium. In that case, taking
S = 27ir 2 we should have had, instead of (175)

k' 2 S 2

1 The resistance and reactance factors in the impedance of the piston are plotted as
X and Y (functions of kR) in Fig. 19 of 46.

148 THEORY OF VIBRATING SYSTEMS AND SOUND

Thus, as far as the radiation resistance is concerned the piston
is equivalent to the pulsating hemisphere of equal area, at low
frequencies. At high frequencies the resistance is pc per unit
area in either case.

However, at low frequencies the added mass factor due to
the medium is slightly greater for the piston of equal area, than
for the pulsating hemisphere. For the latter we have found, in

37,

S*
Sai = 27rr 2 -r p = ?;= (19?)

For the piston we have, at low frequencies, from (1930) and

(195)

luai = 2 3x

that is,

s ai = *&- P =

3?r 3

The numerical coefficients in (197) and (1980) are in the ratio
0.40 : 0.48, approximately; thus, at low frequencies, the added
mass due to the semi-infinite medium is 20 per cent greater for
the flat disc than for the hemisphere of equal area.

The added mass becomes vanishingly small at high fre-
quencies. Using the relation ( 195*2) and the imaginary term of
(1930) again, we have, for the piston

.03 pTT A.kR

= * 'to ' >

2/fe 3 7T

pX 2 IS

i j "ir

that is, since k = >

The corresponding factor for the pulsating hemisphere is, from
(174), since X is small,

~~ .

('74*)

END CORRECTIONS FOR A TUBE 149

The reader may easily convince himself that the inertia co-
efficients, in either case, are negligible at high frequencies, as
compared with the radiation resistance S&\ = pcS.

The utility of the results obtained for the piston will appear
presently. An immediate application is involved in determin-
ing the impedance of the semi-infinite medium at the end of a
tube of circular cross-section which is fitted with a plane flange
extending to infinity. Within the tube just before the end is
reached, the waves are plane; at the opening we can assume
(with the chance of a very small error) that they diverge into
the medium as if produced by a piston oscillating in the end of
the tube. Now abolishing the piston, it is evident that, if the
motion is to continue, a force must be applied at the circular
opening which is equal to the product of the velocity at the
opening and the impedance we have determined, namely
S(b\ + /o>0i), which represents the reaction of the medium on
the piston. The error involved in assuming that the distribu-
tion of velocity in the opening is the same as if the piston were
present is a small quantity of the second order.

43. End Corrections for a Tube; Impedance of a Circular Orifice

Before making the application suggested above, to "correct"
the theory of the finite open ended tube for the impedance of
the medium at the end, it may be well to inquire just what
such a correction implies. Damping (due to radiation) and
inertia (due to the divergence of the wave motion) are added
at the end; considering the tube as a simple vibrating system,
it is clear that a considerable increase in dissipation is required
to affect the frequency of the natural oscillations, while any
accession of inertia is directly effective in lowering the natural
frequency. Inasmuch as the natural frequencies of a finite tube
are inversely proportional to its length we should expect the
principal effect of the terminal impedance (i.e., the added
inertia) to result in a slight increase in the effective length of
the tube. The same conclusion is reached if the argument is
based on dimensional grounds.

150 THEORY OF VIBRATING SYSTEMS AND SOUND

We know from the tube theory of 33 that the particle
velocity for a tube open at x o and closed at x = I is

{(*) = i./cos^)^ if cos = o, (cf. 137, 138),

\ C I C

being the velocity at x = o. Within any small element of
length 6/, measured inwardly from the opening, the variation
in velocity is negligible, consequently the kinetic energy is, in
this portion of the tube,

if S 7r/ 2 , the section of the tube. The kinetic energy in the
region just outside the opening is, from (198^) for low fre-
quencies

The total kinetic energy in the region considered, near the
opening is therefore

(O D\
U + ~l (200)

3 71 " /

Thus assuming a uniform velocity in the neighborhood of the
.opening, the effect of the added mass due to the medium is to

prolong the length of the tube by an amount a = ---- The

effective length of the pipe is therefore (/ + a) and the natural
frequencies are given by

(/+ a) . (/+ a) kir ., . , .

cos- v - = o, i.e., T = (A=i,3, 5. ) ( 201 )

C L 2*

which is similar to the simple result of eq. (137). It is evident,
in the approximate theory given, that the harmonic relations
of the simple theory between the overtones of lower frequency
will not be disturbed; but as the frequency rises, and the
radius of the tube becomes comparable to the wave length
(1980) is no longer valid. The result is that a becomes smaller

END CORRECTIONS FOR A TUBE 151

as the higher overtones are reached and the harmonic relation
no longer holds.

If the opening is unduly constricted, in comparison with
the main conduit, the theory becomes difficult, but the form of
the result can be inferred from general principles. There will be
increased kinetic energy in the neighborhood of the opening;
consequently the added mass will be enhanced, and there will
be a greater end correction; there will moreover be a large re-
flection coefficient for the waves in the tube as they arrive at
the constricted end.

The case of an unflanged pipe also involves theoretical diffi-
culties in determining the end correction due to inertia. From
experiments on flanged and unflanged pipes, Rayleigh estimates
(II, 314) that the effect of the flange is to raise the end cor-
rection by about .22/?; hence, subtracting this value from

87?
a = - = .8c/?, the end correction due to inertia for the un-

3"
flanged opening is about o.6R; this is supported by Blaikley's

determination (0.5767?), also quoted by Rayleigh.

In comparing the low-frequency radiation resistance of the
piston and the hemisphere [equations (196) and (1750), following]
we have found that the shape of the source was of little impor-
tance, as the same result was obtained in both cases, for radi-
ation into the semi-infinite medium. We are consequently
justified in assuming that for the unflanged end of a tube the
radiation resistance is, at low frequencies, substantially that of
an isolated point source, as in (175); that is, it is half as great
for complete spherical divergence as for divergence into the
semi-infinite medium. It is important to recognize that con-
centration of radiation in one direction is the essence of effi-
ciency in generating sound waves.

Another application of the inertia reaction on the piston,
though not giving exact results, is worthy of notice. In the
theory of resonators we have stated that the conductivity of a
circular opening in a thin wall is 2/?, the mass coefficient being

152 THEORY OF VIBRATING SYSTEMS AND SOUND

The exact derivation of this result is not a subject for simple
theory; but the reader may possibly have more confidence in it
if we can obtain, on simple grounds, a rough confirmation.
Imagine a massless plane piston in the opening, the effect of the
piston being to make the velocity uniform in all parts of the
opening. Now adding together the inertia reactions on both
sides of the piston, which are equal in all respects, we have
twice the coefficient of (198), namely

Sai = piR 3 . (2020)

The coefficients in (202) and (2020) are in the ratio 4.93 : 5.32,
or within 8 per cent of one another. The discrepancy is largely
due to the fact that in the actual case the velocity is not uniform
over the opening: the more refined theory takes this into ac-
count. As in 24, we again refer the reader to Rayleigh (II,
306; also ibid., Appendix A) for the exact theory. The re-
sult obtained on this basis for the correction for the open end
of a flanged tube is very accurately a = .82R. With this factor
determined, the mass factor for the circular aperture is within
5 per cent of twice the added mass at the flanged end of the tube
of equal sectional area.

44. Characteristics of Horns; Conical Horns

The theory has now reached the point where, if horns were
not available, it would be our plain duty to invent them, in
order to put to practical use the principles of sound radiation
which we have established. It must be clear to the reader who
has grasped the theory, that the function of the common flaring
horn is two-fold. In the narrow portion, which is coupled to the
loud-speaking receiver, or other form of sound generator, the
source is made to work at maximum efficiency, through the ex-
pedient of taking off the radiation in the form of plane waves,
for under these conditions the radiation resistance of a piston
source is a maximum. By flaring out to a large diameter at the
open end, the effect is to replace the source (which of itself has
very small area) by a large, nearly flat source of equal rate of

CHARACTERISTICS OF HORNS 153

working^ which is better adapted to radiate into the infinite
medium. By making the open end of the horn as large as is
practicable, without essentially destroying the planeness of the
waves within the horn we not only diminish the mass reactance
of the medium at the open end, but we also increase the radi-
ation resistance to a value comparable with that for plane
waves. Thus in a well-designed horn there is reflection from
the open end only at low frequencies \ transmission to the open
medium is accomplished without serious loss, and standing
waves or natural vibrations within the horn itself, which give
rise to unpleasant resonant effects, are reduced to a minimum.

Considering the horn to be a tube of varying section the
basis of the theory was stated by Rayleigh (II, 265) but it is
curious to note that horns have become an accomplished fact
largely through the application of empirical methods, rather
than as the logical result of the classical theories.

Taking S as the variable section of the tube (i.e., as a func-
tion of #, the distance along the axis) the equation of con-
tinuity (150) becomes

and since, by (149) and (146) we have

4> = c 2 s and = -- z ,
. 9*

the equation of propagation is

This may be written in the form

This equation is substantially the basis of A. G. Webster's
theory (Proc. Nat. Acad. Sci. 5, (1919) p. 275) and has been
used by several later writers. It is to Webster that we owe the
idea of "Acoustical Impedance " which he had made use of for

154 THEORY OF VIBRATING SYSTEMS AND DOUNS

some time prior to 1919; the paper just mentioned is impor-
tant as the starting point for horn theory generally.

Consider first a horn in the shape of a cone, the origin being
taken at the vertex. In this case S = ttx 2 if 12 is the solid angle
of the cone. Then

log S = log 12 + 2 log x,

and

(log S) = -,
so that (2040) becomes

9 2

9 = c 2 "T or

which is identical with (1530). On the principles used in the
solution of (1530), for periodic motions ($ = o> 2 <(>) the veloc-
ity potential is

A' t0--) E f i0 + *)
(j> = e c H e c . (206)

For a diverging wave due to a source of strength A situated at
the vertex we should have, analogously to (160)

<|> = <-**) (2060)

12* ' v '

since the radiation diverges to fill a solid angle 12 instead of 471-.
Equation (2060) is given by Rayleigh (II, 280) principally for
the purpose of showing how the energy from a source is con-
centrated by a megaphone.

It is clear that if a conical horn is to be of much use, in rais-
ing the efficiency of radiation from the source, the solid angle
12 will necessarily be small. Under these conditions any finite
conical horn will have little flare, and will therefore owing to
reflection at the open end possess resonance characteristics
which do not differ essentially from those of tubes of uniform
section. The effective length of the tube will be subject to an
end correction (dependent on the size of the opening and the

THE CONICAL HORN 155

frequency) which is similar to that obtained for the cylindrical
tube. Hence the utility of this form of horn is limited.

Important principles relating to the design of horns can,
however, be visualized if we consider a very long conical horn,
and have the choice of cutting off the horn at a variable dis-
tance Xi from the vertex, a piston source of constant strength
then being fitted to the opening at x = XL This system is equi-
valent to a conical element of the pulsating sphere system of
(37). If we let S\ = ttx\ 2 y the strength of the source is
A = i = &VI L> O ; and from (171) the velocity potential at
any point x > XL is, neglecting dissipation due to friction,

of which we shall retain only the real portion, that is, the por-
tion due to the motion A cos w/ at the source. The excess pres-
sure at a distant point x is the real portion of

that is,

__ Apu /kxi cos [co/ k(x #1)] sin [co/ k(x Xi)]\

p = COS fw/ k(x ~ X\ ) + 0] (lO%O)

ilxVk*Xi 2 + I

= tan- 1 -. );
and the intensity is, since S = 12* 2 ,

For a long conical horn, whose opening S is sufficiently great to
"match" the impedance of the infinite medium we are war-
ranted in the following conclusions:

156 THEORY OF VIBRATING SYSTEMS AND SOUND

1. For a source of given strength at a fixed position x = #1,
the intensity in the neighborhood of the mouth of the horn
varies inversely with the square of the angle of the cone.

2. At low frequencies (i.e., for kxi < i) the intensity rises
with the square of the frequency.

3. At high frequencies the intensity approaches asympto-
tically a value which is inversely proportional to the area (l2^i 2 )
of the section of the cone at the source.

4. The larger #1 can be made, the more nearly will the horn

Xj-20 cm

Frequency

.010

.008

.006

.004

.ooa

1,000

zpoo

3,000

4,000

FIG. 1 6. RELATIVE INTENSITIES AT THE MOUTH OF CONICAL HORN (NEGLECTING

REFLECTION).

afford distortionless transmission for all frequencies. The in-
terpretation of Fig. 1 6, which shows the frequency variation of
intensity for two values of #1 in the case of long horn is left to
the reader.

Two major factors enter into the design of the conical horn.
The length should be as great as is practicable, for thereby we
have at once the possibility a large area S at the mouth and a
small angle 2; the quantity S& is kept within reasonable
bounds. If S is large, resonance due to reflection is minimized,

THE CONICAL HORN 157

for reasons which we have previously given. In addition, if the
length is great the more serious natural vibrations will occur at
lower frequency where their effects are least harmful. Thus all
considerations of efficiency require a long horn, if it is in the
form of a cone.

The second important factor is the value of #1; this will be
determined by a compromise depending on how much we wish
to sacrifice efficiency at high frequencies for the sake of extend-
ing transmission to as low a frequency as possible and so obtain-
ing a more nearly uniform frequency characteristic.

In practice a very important question is that of properly
adapting the loud-speaking receiver element to the horn. This
is a specialized study, quite outside the limits of this text: but
one matter of importance may be mentioned. With any horn
that is designed to radiate properly over a wide range of fre-
quency from a source of uniform strength at all frequencies, it
can be safely assumed that it will behave well when driven by
such vibrating systems as are likely to be fitted to it. The
radiation damping provided by the horn will usually be suffi-
ciently great to smooth out any moderate amount of resonance
in the driving system. Indeed resonance to a certain degree in
the driving system may be used to advantage to improve the
transmission in the low frequency region. This principle has
been applied in the design of the piston loud speaker, which
has no horn; but it is equally applicable to sound generators ]
which are used in connection with horns. The poor transmis-
sion which is usually afforded at low frequencies is not neces-
sarily inherent in vibrators or horns, though it may be aggra-
vated if these are not properly designed; it is the fundamental
difficulty (which we first encountered in the problem of the
pulsating sphere) of driving the medium at low frequencies if
divergence of the radiation is permitted at the source.

The results obtained in this article are equivalent to those
given by E. W. Kellogg (Gen. Elec. Rev., XXVII, Aug., 1924,
p. 556). The theory of the conical horn of finite length is given

1 For references to loud-speaking apparatus of various types, see Appendix B.

158 THEORY OF VIBRATING SYSTEMS AND SOUND

by Webster, in the paper cited; it has been somewhat extended
by G. W. Stewart (Phys. Rev., XVI, Oct., 1920, p. 313).

45. Flaring Horns of Exponentially Varying Section

The flaring horn, of exponentially increasing section, is an
instrument of such great utility that it merits special attention.

Taking the origin at the small end, let S = Sie mx and we
have

(log S) = m,

so that, for periodic motions ($ = w 2 <|>) equation (2040) be-
comes

The solution of this equation, by the method of 3 (p. 9), is

(211)

in which

m , / /-.-- , .

MI = h - V *k 2 m 2 = a + /ft

2 2

(212)

2 2 4 '

analogously to equations (5). The velocity potential is there-
fore the real part of

<j> = e~ ax \A\e~ i&x + Bie^ iftJC \e iwt y (213)

that is, for wave motion in both directions,

cos / - 0x i cos

For a long horn, neglecting reflection at the open end (which is
presumably of large area) we should have for a divergent wave,

cos w/

This equation is similar to (124) 32 in its kinematical proper-
ties; the only difference being that the attenuation in (213^) is

LONG EXPONENTIAL HORN 159

due not to friction (which we have neglected) but to the lateral
release of pressure as the wave diverges to fill the horn. In both
(124) and (213^) the result of attenuation is not only to diminish
the amplitude, but to change slightly the velocity of propaga-
tion. In discussing (124) this latter effect was neglected as it
was not essential in the treatment of the tube problem; but it
cannot legitimately be neglected here. If there were no atten-

uation, the velocity of propagation would be -7- = c ; but under

the conditions which exist in the horn we must have, for sus-
tained wave motion.

c ~ ~ft ~ '

The velocity potential is thus slightly advanced m phase as com-
pared with that of a plane wave, assuming no dissipation. It is
apparent, on differentiating <|> with respect to #, and with
respect to /, that the velocity and the excess pressure are no
no longer in phase, as they are in the case of plane waves (31,
p. 91). Hence we must proceed with some care in using these
quantities to find the intensity of the radiation.

The immediate problem is to compare the frequency varia-
tion of the intensity at the mouth of a long "exponential" horn
with that for a conical horn for which the ratio of initial to final
section is the same. It is convenient in this problem to take
the velocity as the basis of phase, as we shall make the velocity
at the origin a prescribed quantity. As before, we have

fcc-o = 7T COS 0>/,
01

taking A as the strength of the source. It can then be shown
that the velocity potential due to this source is

* = ^ a cos ^ - P*) + P sin (<' - 0*)J> ( 2I 5)

160 THEORY OF VIBRATING SYSTEMS AND SOUND
and the velocity at any point x, is

* = - IT = A - e l~ cos ^ ~

3# Oi

since from (212), a 2 + 2 = P. This obviously satisfies the
condition imposed at the origin. The excess pressure is

p = p^ = [_ a s i n (/ - 0*) + cos (o>/ - |S*)]. (217)

01 AT

We shall not assume (though it would be legitimate, with a
simple reservation, as we shall find) that the intensity is given
by the formula ( 164*2) which we have repeatedly used for plane
and spherical waves. Returning to first principles, we have,
for the intensity

dt
Pa>

cos 2 (/ 0x) - a sin (/ - fix) cos (w/ -

or, taking mean values

42,>-'2<

since -7 = c.

We have already noted the relation or + & 2 = 2 ; a may
therefore be placed equal to k sin 0, and to k cos 0. Except
at very low frequencies a is a small quantity; it may be taken
as a measure of the phase difference between the velocity and

the excess pressure. The quantity cos = / may be looked

upon as a "power factor/'

We may now remark that equation (218) would have re-
sulted if we had written, instead of (217), the equivalent form

, . , N

p = =-r cos (co/ - fix + 0), (217*1)

LONG EXPONENTIAL HORN 161

and used for the intensity the relation similar to (164^)

dt 2 P c'

in which c f is taken, as in (214), equal to

^ n

cos B

The form of equation (218) must be changed slightly to
compare it with the corresponding equation (209) which was
obtained for the conical horn. We have

so that (218) becomes, on substitution
dW

The importance of the quantity in the theory of the exponen-
tial horn is clear. For a certain frequency (i.e., when 2co = me)
= o; for frequencies below this critical value, the horn trans-
mits nothing, or in other words behaves as a filter. As the fre-
quency rises above the critical frequency, /3 rises rapidly, cos 6
asymptotically approaching the value unity.

If we note that 12 = - y equation (209) for the cone may be

written

dW P cA* k 2 xi 2

" (2 9a)

which permits the immediate comparison of the two types of
horns, as A> S, and 1 are the same in both cases. If it is not
evident at once, that (except below the critical frequency) the
exponential horn is much superior, an example will make the
matter clear. In Fig. 17 are shown the two frequency char-
acteristics, according to (2180) and (2090); the length of the

horns being taken as 192 cm. and the ratio being (25)2 = 625.

1 62 THEORY OF VIBRATING SYSTEMS AND SOUND

From this ratio, the rate of taper for the exponential horn is
determined to be m = .033; the only other quantity required
in the calculations is #1 = 8 cm., the length measured from the
vertex of the cone to the point where the section is Si. The
figure should be self-explanatory.

The conclusion we have reached as to the superiority of the
exponential horn amply justifies its widespread application in
loud-speaking apparatus. The fundamental equations of the

For the Exp. Horn, 5 -

C-QC-033X

I Exponential so cm

c4 : =^LLi_

zoocm. H

as^

.80

.60

k 2 v 2

^-y- for Cone

Freauencij

1,000

2,000

3,000

4,000

FIG. 17. COMPARISON OF CONICAL AND EXPONENTIAL HORNS HAVING SAME INITIAL

AND FINAL OPENINGS.

exponential horn were given by A. G. Webster (/oc. cit.).
More recently the theory has been considered by Hanna and
Slepian (Trans. AJ.E.E.^ 43, 1924, p. 393), and by H. C.
Harrison (British latent No. 213,528, 1925). In the paper
by Goldsmith and Minton on Horns (Proc. Inst. Rad. Eng.,
Aug., 1924, p. 423) these writers are apparently in error (p. 454)
in finding that a given conical horn has better transmission
characteristics for certain frequencies above the critical fre-
quency than the corresponding exponential horn. We cannot
agree with their conclusion.

THE FINITE EXPONENTIAL HORN 163

Horns have been extensively studied by P. B. Flanders,
from whose most important memorandum the following quota-
tion is taken, for the purpose of summarizing this article. The
statement bears on the application of a horn to the short tube
leading from the loud-speaking receiver diaphragm.

"Neglecting reflection effects, the addition of a horn does
not effect an increase in the ultimate impedance at the end of a
receiver opening. It does, however, cause that impedance to
reach its ultimate value at a lower frequency; and the lower
that frequency, the better, of course is the horn. The imped-
ance "looking out" of a seven-tenths inch hole in an infinite
wall reaches 80 per cent of its ultimate value at 9300 c.p.s.; a
certain conical horn causes this 80 per cent value to be reached
at 4200 c.p.s.; while the corresponding exponential horn causes
this value to be reached at the relatively low frequency of
250 c.p.s. In a way these figures show why the exponential
horn is so much superior to the conical type."

46. The Finite Exponential Horn

The theory of horns, to be of practical use, must be applied
to horns of finite length; this problem we shall now consider,
taking for example the exponential horn because of its super-
iority. The method is straightforward, being based on that
already used in (32) for the tube problem; but the calcula-
tions are laborious, though some simplification is possible if we
make use of the impedance-methods which are well-known in
connection with the study of electrical networks. 1 We shall
first find the principal impedances which are characteristic of
finite and infinite horns, and then substitute for the given horn
an equivalent network based on these constants. It will then be
possible to find how the given horn behaves when placed be-
tween two given impedances, with a given driving force ap-

1 The method of applying the theory of electrical networks to the horn, as used
in this text, is due to P. B. Flanders and D. A. Quarles of the Bell Telephone Lab-
oratories. The advantages of the method will be evident, if a comparison is made
between the present text, and the treatment of the horn by A. G. Webster, cited
in 44.

164 THEORY OF VIBRATING SYSTEMS AND SOUND

plied at one end. Finally it will be necessary to make certain
approximations in dealing with the impedance of the medium
at the large (open) end; but this is not a matter of great diffi-
culty in view of the theory given in the earlier part of this
chapter.

From the general solutions already obtained [equations (216)
and (2170)] we may take for the velocity and the excess pressure

BS&]?*, (220)

and

p = pcc*[Ac-** + Be iftx ]e iut y (221)

in which, as before, 6 = tan* 1 -, the angle of lag of velocity

with respect to pressure, for a wave travelling in the positive
direction that is, the direction of increasing section of the
horn. Equations (220) and (221) thus provide for all possible
waves in both directions, d reversing its sign for propagation in
the negative direction.

We have at once for the impedance of an infinite horn, in
the positive direction

+ l), (222)

%
and similarly, in the negative direction (since then c is negative)

Z = pcSe~ ie = ^j-(0 - ia). (2220)

Now let the horn be terminated at x = o and x = / with
piston impedances Z and Zi respectively. The boundary con-
ditions are thus:

at x = o, Z f i + piSi = * ^, (223*)

at x = /, Z, 2 - p*S 2 = o; (223^)

in which * ? ta> ' is the applied force, and the subscripts i, 2,
refer to local values of velocity, pressure and section. Applying

THE FINITE EXPONENTIAL HORN 165

the condition (223^) to equations (220) and (221), we find the
relation

Eliminating 5 between (224) and (2230), we have

in which

20 = (Z z,

that is
D = (Z Zi + p 2 c 2 SiS 2 )i sin 0/

+ pr[ZjSi cos (0/ + 0) + Z S 2 cos (0/ - 0)]. (226)

We also have

Inserting in (220) and (221) the values for A and 5, we have
for the velocity and pressure distributions in the horn,

e""*
{ = - (Zi / sin 0(/ AT) + p^6*2 cos [/3(/ ^) ^])* <? , (228)

and

rt/ . p ax

/ sin /3(/ ^) + Zi cos []S(/ x) -

We can now construct the equivalent T-network for the
horn, considering it (in the positive direction) to have the pure
inertia coefficients Li, M and L* just as an ordinary trans-
former. These relations will be clear if we refer to Fig. 18. For
the transformer /wLi is the driving-point impedance at (i),
with the secondary circuit open; this corresponds to the driving

1 66 THEORY OF VIBRATING SYSTEMS AND SOUND

point impedance of the horn at (i ) when (2) is rigidly closed so
that 2 = o. The quantity iuL^ is determined similarly. The
mutual impedance M\% = / <*M is the ratio, for rigid closure of
the horn at *S* 2 , of the force (pzS<^) to the velocity f i. M is also
given by the simple formula

M = J

in which Z\ is the impedance at (i) with (2) electrically short
circuited, or in the acoustic case, with the impedance Zi = o.

L 2 -M

10 L,

(D |M

(a) (b) (c)

FIG. 1 8. FINITE HORN, AND NETWORK EQUIVALENT.

To determine L\, we put Z t = oo in (228) which becomes,

for x = o,

/ sin 8/'W e' wt

The driving-point impedance is therefore

* _ 7 * PC S I cos (# + ^)

. - - // H . -. . -

(l)max. l Sin &

Subtracting the piston impedance Z , we have for the imped-
ance of the horn itself,

= - k -(a - ft COt #), (2 3 2)

since cos 6 = . and sin = - To find L 2> we can avoid the
k k

labor of redetermining the constants A and B, for force applied
at point (2), which would involve a new formula in which (AT)
was taken as the positive direction; we merely note that in so

THE FINITE EXPONENTIAL HORN 167

changing our point of view, becomes 6 (pressure lagging
velocity) hence ( = k sin 0) becomes (a) while ft is un-
changed. We then have

Z 2 = i0 = " * S \ + ft cot pi). (233)

We proceed to determine M. Repeating (231), for Z = o,
/sin |3/- *<?'"'

The pressure at (2) is, for Z = o and Z L = oo , from (22)

<?- tt 'cos0-* <?' w ' / x

^--s^v^t+v (234)

and the force is

VP,, <?'"' . , s . .

c< , a ,->-rt> smce f = VTT- ( 2 35)

r k Sj COS (/3/ + 0) N i'2

The mutual impedance is, by definition,

,, . ,. s*p2 . pVSiS* , ,

A/ ls ,Af- ^ .- l ^- 8l --. (236)

The reader may verify this equation independently by deter-
mining Z\ and applying equation (230). This will incidentally
verify the assumed equivalent T-network shown in Fig. 18.

In the discussion above, both and ft have been considered
to be real quantities; that is, the frequency has been taken to
be within the transmission range. We shall not deal with phe-
nomena in the horn for frequencies below the critical frequency,
for which ft becomes imaginary, because in practice the critical
frequency is always very low. The point is mentioned, how-
ever, because in any complete theory of the exponential horn,
the whole range of frequencies must be considered.

To get a correct idea of the essential characteristics of the
finite horn, we must choose terminal impedances at the ends of
the network with some regard for actual operating conditions,
and we must assume a method of driving at the point (i) which

1 68 THEORY OF VIBRATING SYSTEMS AND SOUND

does not unduly distort the normal characteristics of the horn.
To avoid the complication of considering the characteristics of
the source of sound applied to the horn, we shall simply assume
that the fluid in the small end of the horn is driven with con-
stant periodic force, by means (say) of a piston whose imped-
ance is negligible. At the large end of the horn, we shall take
the impedance offered by the medium to be

Z l = pcS*(X + iY), (237)

in which X and Y are functions of frequency of the nature of
those discussed in 42 [eq. (193^)], when dealing with the re-
actions on a piston. The particular form of X + iY will be
determined later.

Making use of the equivalent network of the horn we have,
for the impedance at the driving point,

_ ^i^2 - ^J2_ T- ^l^Z , R v

^ ~ 7 \ 7 y V^J /

^2 + Zj

the velocity at the driving point being

*i = z > (23M

if ^ is the applied force, (W e lait ) independent of the frequency.
Using the relations (232), (233), and (236) it can be shown that
the expression, in (238),

If the horn were driven at constant periodic velocity, we
should use, to compute the velocity at the large end, the rela-
tion

l _ M "i.

but we shall see that this is not the best point of view from

THE FINITE EXPONENTIAL HORN 169

which to regard the horn. Substituting the values for Mi2, 2
and Zi [equations (233,) (236,) (237)] we have

/cos

S 2 (X sin ft/ + /[F sin ft/ - (sin 6 sin ft/ + cos cos ft/)])

(241)
When ft/ = TWTT, w being integral, this reduces to

k* = fi\Tr (conditions), (241/1)

and similarly, if/?/ = (2m + i) -, (condition <),

=f= / cos B . /A 1

It is now clear that, for constant driving velocity 1, no account
is taken [at least in condition (a)] of the output impedance
Zi = pcS 2 (X + iY) of the horn; in other words, in studying
2 as dependent on |i, no weight is given to the difficulty of
keeping 1 constant, due to the varying of the driving-point
impedance with frequency. In passing we note that condition
(a) corresponds to the case of an organ pipe open at both ends
[eq. (141), 33] and is a condition for maximum velocity at both
ends of the horn, due to resonance; for the case of a simple tube
it was shown in 32 that this was also the condition for maxi-
mum efficiency in power transmission, the driving-point im-
pedance being then the sum of that of the driving piston and
of the piston at the other end of the tube. This is of course
true here for the horn, as is evident also from equation (2440)
below.

If on the other hand we wish to drive the horn with constant
periodic force *, we use the relation l

1 In (242) and similar equations following, the reciprocal of the coefficient of ^
is of course the "transfer impedance" of the finite horn. (242) follows from (2384)
and (240).

1 70 THEORY OF VIBRATING SYSTEMS AND SOUND

Referring to (239), for the expression in parentheses, and noting
that

cot fl/ - a) - iX((* COt 0/ - a)], (243)

we have

. _ ipc cos

2 p 2 ^ 2 *?! 6*2 [sin ft! + (Y iX)(cos 6 cos ft/ sir

If ft/ mir (m an integer, as before)

I \n

t .

C'> ~~~ / '

and if ft/ = (im + i) >

T * cos

+ /T) sin - /]

(244^)

We may now deal with the terminal impedance (i.e., the
impedance of the medium at the open end) which is of the form

pcS*(X+iY). (237)

If this were the impedance offered to a piston in an infinite wall,
we should have (eq. 1930)

(245)

the added inertia due to the medium at zero frequency being

ipcS 2 Y 8 fa

= pS 2 - \J> (245^)

^ w = 37T ^ 7T

from (198). Now, inasmuch as we have no solution of the, prob-
lem of the reactions on a piston oscillating in the end of a non-
flanged tube, it is necessary to make some such assumption as
that of eq. (245), in order to obtain a manageable theoretical
solution of the problem of the finite horn. It seems reasonable
that the approximation involved will be sufficiently close for
the purpose, particularly since we have tacitly made other

THE FINITE EXPONENTIAL HORN 171

simplifying assumptions to obtain the theory in the simple form
given. (For example, no allowance has been made for dissipa-
tion; and no allowance has been made for the fact that the
wave front may have such a curvature that the conditions we
have assumed as to its expanding area as it proceeds may not
be quite fulfilled.) However, if it is tolerable to solve the prob-
lem as if the end of the horn were fitted with an infinite plane
flange, then it seems reasonable to go one step further, and
make the flange a very wide-angled cone. If this is done, we
can then consider the wave front, as it emerges from the open
end of the horn, to be equivalent to that produced by pulsating
spherical surface. The question then arises as to what the area
and curvature of the spherical surface should be to match prop-
erly; if this can be answered readily, we shall have available a
much simpler function for X + iY than that given in (245).

We have already noted [see eq. (175^) and remarks following]
that a piston and a pulsating hemisphere of equal area, both
flanged by an infinite plane, have the same radiation resistance
at high and at low frequencies, though for intermediate values,
the piston approaches the final maximum more rapidly. We
take therefore a spherical surface, whose curvature is to be de-
termined, but whose area is *V 2 , to be the approximate equiva-
lent of the piston in an infinite wall. To determine the curva-
ture (hence the solid angle of the conical flange) we adjust the
solid angle of the cone so that the maximum added inertia
(that at zero frequency) is the same in the two cases. We have
found [eq. (174)] that the added mass, for a spherical surface of
radius r () is pr per unit area at zero frequency. If we let

Sz fir () 2 , we have r = \/ > hence equating the mass pr S>2

for the spherical surface, to that for the piston of equal area
(245*0), we have

and a simple calculation gives 12 = i-397r, for the solid angle.
If R is the radius of the piston, then irR 2 *= i.^irr 2 and

172 THEORY OF VIBRATING SYSTEMS AND SOUND

r = -85/?. We then have, for the new functions X 1 and Y f .
from (174)

& r 2 l r

VI Jv / -Tjy K'O

X = Y -

27T

since o> = r, and = Finally, since the impedance

Zi = pc$2(X' + iY') we may write

Zi = pr 2 -cos 0V , </> = tan- 1 (248)

K.TQ

The method suggested for dealing with the impedance at the
end of a horn may have possible application to other acoustic
problems, hence it is interesting to compare the behavior of the
functions X and Y of (245) on the "piston" basis, with those of
(247) based on the "equivalent" spherical surface. This is
done in Fig. 19, both sets of functions being plotted in terms
of the argument kR = I.i8r ; it will be observed that the
greatest discrepancy between X and X' is about 20 per cent
(for kR = 2.5), while that between Y and Y' is about 40 per
cent, for kR = 1.3. The theoretical discrepancy is doubtless
greater than that which would be observed in any practical
case, because the wave front emerging from the large end of the
horn can hardly be strictly plane; it is much more likely to be
convex. And it may be noted that the smaller value for Y' as
compared with Y at low frequencies happens to be more nearly
in agreement with the smaller end correction for an unflanged
tube (as compared with a flanged tube) which was noted by
Rayleigh, and mentioned in 43, p. 151 above.

Placing the expression for Z/ (248) in (2440) we have, for
the values of 2 corresponding to the condition $1 = mw (drop-
ping the exponential factor)

For the other condition, rationalizing (244^) we have, since
y/2 = X '

THE FINITE EXPONENTIAL HORN 173

r i

, Y sin i

tan = vT~= >

Jf sin

when $1 = (2m + i) We can now state the general behavior
of the horn, within its transmission range, remembering that
for the lowest transmitted frequency sin = - = i, cos = o,

while as higher frequencies are reached sin = o and cos = i.
We may also refer again to the upper curve in Fig. 17 which is a
plot of cos for a certain exponential horn, this having been re-
quired to show the frequency variation of intensity at the
mouth, assuming no reflection there.
We note first that ( 45, p. 160)

sin = - = ; cos

for the phase angle characteristic of the horn. We then have
for the frequencies for which (250) is valid (that is for a condi-
tion of non-resonance in the horn),

m 2 r 2 mr

Y' cin2 /) /0 V' cin ft

A sin u ,;; - r~ x , * sin v /I0 . -~.

/ + o

It is evident that for these frequencies (since the quantities of
(252) are small as compared with i) the velocity f^U at the
mouth of the horn is approximately the same as if we were deal-
ing with an infinite horn: that is, proportional to cos 6. At the
higher frequencies, as cos approaches unity, <' approaches
zero.

For the frequencies for which 01 = mw, which are interca-
lated between the successive values of frequency dealt with
above, the finite horn is in resonance, the values of |{2| being

proportional to ; therefore (cf. 249) these points are a

174 THEORY OF VIBRATING SYSTEMS AND SOUND

series of maxima in the response of the horn, which decrease in
height with rising frequency and asymptotically approach the
level

which is the constant response characteristic of the horn at
high frequencies. When this range is reached, resonance is
eliminated since the impedance of the horn matches that of the
medium; the horn behaves as a simple transformer working at
maximum efficiency, and sends forth a nearly parallel beam of
sound radiation.

There are many interesting problems in connection with
horns, such as, for example, the directive effects in the radiation
after it emerges from the mouth; the effects of dissipation; the
effects of phase change, etc. Where these can be handled theo-
retically, they are left to the reader, on the basis of the prin-
ciples given in this chapter; they are too specialized to con-
sider here. It appears that some of the effects obtained with
horns require considerable experimental study in order to eluci-
date them; unfortunately no discussion of these can be given,
for lack of suitable data.

47. The Effect of Sound Waves on a Simple Vibrating System

When we dealt with the elementary theory of the resonator,
considered as a vibrating system ( 24), it was necessary to give
some account of the behavior of the system when acted on by
sound waves. Now, our discussion of typical radiation and
transmission problems would be incomplete if we did not con-
sider more fully the phenomena when a resonator or other
simple system is immersed in a field of sound waves, because
this problem is likely to arise whenever sound waves are to be
detected or measured. At the risk of repetition, we shall re-
open the question of selective absorption and amplification
(this time on an energy basis); in addition we shall give some
attention to the distortions of the original field, due to the
presence of the resonator; and finally we shall note the effect

EFFECT OF SOUND WAVES ON VIBRATING SYSTEM 175

of a resonator on the behavior of a nearby source of sound. All
these questions are difficult to answer rigorously, particularly
the second, because in addition to the purely resonant action of
the system, the mounting in which it is contained acts as a rigid
obstacle to further distort the field.

Consider again the mechanism of absorption, which is in-
evitably bound up with re-emission as in all general radiation
problems. Suppose (as in 24) that the resonator is tuned to
the frequency of the driving waves, which arise from a source of
strength A situated at a distance x from the mouth of the reso-
nator. The only impedance offered by the moving mass of air
in the resonator opening is its radiation resistance, which is

. - (.75)

if the dimensions of the resonator are small as compared with
the wave length. If we assume no resonator present, the veloc-
ity potential at the point x is

* = ~^/ (<at ~" X} (254)

and the excess pressure is

p = P<O = /wp<|> . (255)

The intensity is given by
dlV \i

IpC 1C

(256)

Now it is evident that when faced by the impedance (175),
(which is much less than the impedance pcS of an equivalent
area of the free medium) the excess pressure in the oncoming
wave will produce a much greater velocity at the point x than
if the resonator were absent. The energy must come from the
adjoining region of the medium, and the question is, how large
is the area from which this excess of energy is abstracted? A
rough answer is obtained if we compute the velocity from the

176 THEORY OF VIBRATING SYSTEMS AND SOUND

pressure according to (255) above, and the impedance (175).
We have then (using constants per unit area)

the resonator now behaving as a new source of strength

** = sin. = - ^ I4>,,| max . = - '- (258)

The maximum rate of working is ^a\p\ mM . that is,

/, _ CO _ 27T

~ ~

or, on the average

(259)

(259*)

by comparison with (256). From this it appears that the area
of wave-front from which energy is abstracted (and reradiated),

X 2
by the resonator is of the order of magnitude That is to say,

the energy density in the area S of the resonator mouth is

X 2
greater than that in the original field, in the ratio ^, this

TTu

energy-amplification coefficient being exactly that previously
obtained in 24 for the velocity amplification coefficient. This
is as it should be, because the driving force (Sp) is not in any
way amplified by the resonator; the rate of working (p%) is
merely increased linearly with the increase in velocity. The

X 2
area of the wave front from which energy is drawn, is by

hypothesis larger than the area S of the resonator; and we
infer that the resonator acts, as far as its effect on the primary
field is concerned, as a "sink" with a resultant convergence of
the stream lines from the primary field to the mouth of the

EFFECT OF SOUND WAVES ON VIBRATING SYSTEM 177

resonator. This notion of the mechanism of selective absorp-
tion is useful not only in acoustics, but has an application in
optics as well.

In addition to energy absorbed and reradiated, there is the
inevitable reflection or scattering of primary energy by the
resonator mounting, considered as a obstacle. In any practical
case, the impedance of the yielding member (e.g., the dia-
phragm) of a sound detector will be very great, as compared
with pcS y and the conditions we have pictured above for a
simple resonator would not obtain. We must consider the
whole sound detecting instrument as a fixed sphere or disc, and
to calculate the pressure on the diaphragm we must first de-
termine the effect of the obstacle (i.e., the detector) on the
sound field, from which investigation the velocity potential,
and hence the pressure at any point on the surface of the de-
tector can be found. The effect on the field of the yielding of
the diaphragm is practically ni/ y but even so the problem offers
great theoretical difficulty, and its solution cannot be attempted
here.

For low frequencies, in the case of a spherical obstacle of
radius a the total energy scattered in all directions is %(ka) 4
times the incident Energy. The corresponding figure for a thin
disc, is ^(kaYy if a is the radius. (Lamb, 81.) The propor-
tion of energy scattered is thus small for small obstacles, but it
is evident that it rises rapidly with the size of the obstacle, and
more rapidly with frequency than the radiation resistance b\ y
for example. As the frequency rises of course a greater propor-
tion of energy is reflected (i.e., scattered) from the face of the
obstacle toward the source until finally a condition of normal
reflection is reached, at that surface. At very high frequencies
the pressure on a stiff diaphragm facing the source is twice the
excess pressure of the incident waves. [Cf. eq. (121).] In experi-
mental work the only available recourse, to avoid the obstacle
phenomena and the allowance that must be made for them
when a very stiff detecting instrument is used, is to flange it
with an infinite rigid wall, in which case (121) is substantially
correct for all frequencies. If on the other hand it is possible to

I 7 8 THEORY OF VIBRATING SYSTEMS AND SOUND

use a light tuned detecting system, whose resistance (including
that due to radiation) is about equal to the radiation resistance
of the medium, good results will be obtained with little distor-
tion of the primary field.

Those who wish to pursue the relatively complicated calcu-
lations necessary to deal rigorously with obstacles, sound
shadows, and the scattering of radiation are referred to Lamb
( 79> 80, 81) and to Rayleigh (Chap. XVII).

We may finally consider how a resonator, when placed near
a source of sound, can be used to enhance sound vibrations.
According to (258) if kx is a small quantity the strength of the
source is effectively raised by virtue of the presence of the
resonator, in the ratio i : kx\ the intensity at some distant
point will therefore be greater in the ratio i : k' 2 x 2 . This seems
paradoxical, since if the resonator were distant from the source
it would only give out what energy it could abstract from a cer-
tain limited part of the wave system due to source (i). But it
must be recognized that if source (i) is, as we have assumed, a
point source, the principal component of the velocity is 90 out
of phase with the pressure there [cf. equations (161) and (161^),
36] so that the point source is not of itself a very efficient
radiator. The introduction of the resonator near the source pro-
vides a means whereby the excess pressure can produce a large
velocity with which it is in phase, with a consequent gain in
radiating power. That is to say, by decreasing the radiation
resistance /?, and at the same time compensating for the inertia
reactance of the source itself (through tuning the resonator)

p 2
we can make very large the quantity ^, which is the radiated

power.

48. Acoustic Radiation Pressure

In dealing with the impact of sound waves on a wall, it may
have occurred to the reader that here (as in the electrodynamic
case) there must be a positive pressure due to radiation. This
is indeed true, but it is a second order effect, which we omitted

ACOUSTIC RADIATION PRESSURE 179

for that reason in discussing plane waves in 31. The matter
has been considered by Rayleigh in two papers published since l
the second edition of the "Theory of Sound," and is not without
practical application in measurements of sound intensity. Ray-
leigh's method for finding the force required to confine the
vibrations at the end of a string is to be illustrated in Problem
39, which will give the reader an idea of the mechanism of radi-
ation pressure. This pressure depends primarily on the potential
energy residing in the medium, and is equal to the mean energy
density associated with the wave motion.

To give Rayleigh's proof for the acoustic case would require
the use of a general integral of the hydrodynamical equations
of motion which we have not dealt with; but we shall make use
of a simple and elegant treatment due to Larmor 2 for the gen-
eral case of plane waves of any sort, normally reflected from a
wall. The wall is free to move, normal to itself, and is pushed
with uniform velocity v to meet the advancing wave train, of
wave velocity r, and mean energy density E. If the wall were
stationary, the total energy density due to incident and re-
flected wave trains would be 2#. The length of the wave train
incident on the moving wall in unit time is c + v, and in trans-
mission, owing to the continuous encroachment of the wall,
this is compressed into a space of length c v. The energy
density in the reflected wave is thus increased in the ratio

E + A c + v .2v , , .

---,--;- I + r (26o)

since v is to be taken as a vanishingly small quantity. We there-

fore have A = ~- , and the total energy density is now
c

lE + AE. The increase in the total energy in the region of
length c before the moving wall is &E-c and this can be ac-
counted for only on the basis of the work done by the wall in

1 Phil. Mag. y III, 1902, p. 338, and X, 1905, p. 364; both reprinted in Vol. V of his
"Scientific Papers." A review of the matter is given by W. Weaver, Phys. Rev., 15,
1 920, p. 399.

2 Encyc. Brit. y Vol. 22, article on "Radiation."

i8o THEORY OF VIBRATING SYSTEMS AND SOUND

compressing the radiation. If * y<; is the mean radiation pressure,
the work done by the wall in unit time is MV#, hence

y R -v = &E-C = lE-v, (261)

that is, the mean radiation pressure is equal to the mean energy
density in the medium adjacent to the wall.

The intensity of the radiation is equal to the product of the
Energy Density and the wave velocity (31), hence radiation
pressure can be made the basis of sound intensity measurements
if a sufficiently sensitive radiometer is available. This has
actually been accomplished in experiments with high fre-
quency under-water waves. 1 The radiometer is a simple tor-
sion balance; a thin hollow metal box filled with air, on being
submerged serves as a very good totally reflecting vane. The
apparatus is simple and easily calibrated, and is an excellent
example of the technique of modern experimental acoustics. 2

PROBLEMS

31. In the sandwich-like high-frequency radiator described on
p. 142 (41) show that the two iron slabs must be one-quarter wave
length thick for resonance, assuming that the inside crystal layer
drives the adjacent boundary of the iron surface with a prescribed
velocity.

32. A telephone diaphragm (clamped circular plate) has a total
mass of 5 grams, a radius of 3 cm., a natural frequency of 1000, and a
damping coefficient of 200 in air. It is placed in water, with one side
only exposed to the fluid, and care is taken to prevent change in the
equilibrium position of the diaphragm due to hydrostatic pressure.
Find approximate values of the natural frequency and damping of the
system under these new conditions.

1 Such a radiometer was designed and used by A. P. Wills for measuring the radia-
tion from the high-frequency vibrator previously described (41). A similar instrument
was used by Langevin; see the article on "Echo Sounding*' previously cited.

2 The ponderomotive effect of one vibrating system on another in the same sound
field may be noted here. This has been investigated recently by E. Meyer (Ann. d.
Phys. y 71, 1923, p. 567), and is briefly discussed by E. Waetzmann (Phys. Zeit. y XXVI,

P- 746-747).

PROBLEMS 181

33. Assuming that the diameter of the end of a tube is small as
compared with the wave length, what is the reflection coefficient for
the energy when a wave within the tube reaches the open end?

34. A cylindrical tube of unit radius and length one metre is driven
at one end, the other end being open to the air. What is the driving
point impedance of the tube, for low frequencies?

35. Obtain the typical equations for the phenomena of transmis-
sion and the production of standing waves in cylindrical tubes, by a
suitable modification of the theory of the exponential horn.

36. For an exponential horn, find the phase relations between
pressure and velocity for frequencies below the critical frequency.

37. Derive a general expression for the behavior of a resonator in a
sound field, assuming that it is not tuned to the driving frequency, but
that the wave length is large as compared with the resonator.

38. In 43 we have seen that an inertia reactance at the end of a
tube is equivalent to an increase in length of the tube. It is likewise
true that the stiffness reactance due to a bulb at the end of a tube, is
equivalent to a decrease in the length of the tube. On these principles
calculate the natural frequencies of a tube of length /, and section S,
closed at one end by a rigid piston, and at the other end by a bulb of
capacity V. (Rayleigh, IT, 317.)

39. A long string of tension r passes through a heavy sleeve, free
to move along the string, at a point x = o near one end. A trans-
verse wave whose displacement is sin (a:/ + kx) travels along the
string to the sleeve; it is totally reflected there, as all transverse mo-
tion of the sleeve is prevented. Show that the mean (space) energy
density in the string is r& 2 2 ; and by a resolution of forces due to ten-
sion in the string at the point x o show that the mean (time) force
tending to displace the sleeve along the string is equal to r 2 2 , that
is, the mean energy density in the string.

40. A plane wave is normally incident on a rigid infinite wall, in
which there is a circular opening whose conductivity K is small as
compared with the wave length. Show that the rate of flow of energy
through the hole is to the intensity in the original wave as 2^ 2 :?r,
approximately (cf. Lamb, 82). Why are you justified in neglecting
the radiation resistance in calculating the velocity in the orifice?

CHAPTER V

THE ACOUSTICS OF CLOSED SPACES; ABSORPTION, REFLECTION
AND REVERBERATION

50. Architectural Acoustics

Most acoustical experiences take place in spaces bounded
by reflecting and absorbing walls which tend to confine the
sound energy and give rise to standing waves and other com-
plications which do not occur in an unlimited medium. On
account of their practical importance, considerable study has
been devoted to these effects and the theory of Architectural
Acoustics has been elaborated to deal with them. This is to
be the main concern in the present chapter.

Practically, Architectural Acoustics deals with the phenom-
ena which occur when speech or music are produced before an
audience in an assembly-room; consequently we must recognize
first of all what quality it is in sounds of this character which
lies at the bottom of the difficulty of correct transmission.
This quality depends on the fact that in speech and music the
component sounds, as originally produced, consist of wave-
trains of definite lengths, which overlap in a certain prescribed
way, and unless these are reproduced at the ear of the listener
with all the original sequence of phenomena (amplitude as a
function of time) he hears a distorted and often very unsatis-
factory version of the original composition. These distortions
are really two-fold, though both effects depend on multiple
reflection. In the first place it takes time for any source to
establish radiation of a given mean energy density in an enclo-
sure; and conversely, it takes time for the mean energy den-
sity to decay to zero, when the driving force is removed. This
of itself will introduce a time distortion of any individual tone-
component of limited duration. This effect throughout the

182

ARCHITECTURAL ACOUSTICS 183

whole enclosure, particularly the hang-over at the end of the
interval, is known as reverberation; it is analogous (but not
equivalent) to the persistence of the natural oscillations in a
system of several degrees of freedom, until all the energy is
drained away. Secondly, owing to the recurrent reflection,
there is produced in the enclosure a separate pattern or dis-
tribution of standing waves for each particular frequency of
tone; this is a kind of distortion in space, and at any point in
the enclosure, at a given instant, the resultant vibration due
to all these interference patterns is not likely to be a sound of
the same energy-frequency distribution as the original. To
distinguish this latter effect from reverberation we shall call it
local wave-distortion^ To summarize then, there is imposed on
the listener a confusion of two related effects, (a) reverbera-
tion, or the failure of the mean energy density to respond
faithfully to the power variation at the source, and (b) a local
effect of wave distortion, varying in both space and time, due
to the idiosyncrasies of the standing wave system.

It might be thought off-hand that the universal cure for
these effects would be to provide sufficient damping or absorb-
ing power at the boundaries of the enclosure to eliminate
standing waves entirely, or in other words, to simulate the
transmission conditions prevailing in open space. Band
music for example sounds mellow and agreeable in the open, if
there are no disturbing noises present. But here enter psycho-
logical or rather esthetic considerations: music and oratory
depend for some of their effects on sonority, and in these in-
stances reverberation is of advantage in prolonging or enhancing
certain tones in order to attain an artistic result. Reverber-
ation also increases the overlapping of one tone on another,
on the time scale, an effect which is sometimes desired. In
addition there is usually some limitation on the amount of
power which can or which should be expended at the source.
In the case of a brass band playing in the open, the character
of the music and the amount of available power usually suffice
to satisfy the sense of loudness of a large body of listeners, but
an equally agreeable effect would not be obtained with a string

i8 4 THEORY OF VIBRATING SYSTEMS AND SOUND

quartette playing chamber music, or with a vocal soloist, be-
cause of the too rapid dissipation of energy from the listening
area, in relation to a necessarily restricted power output at
the source. In these latter cases, it is necessary, in order to
fully appreciate the performance, to enclose the radiation and
make use of reverberation to increase the mean energy density
in the audience hall. All this implies, of course, that reverber-
ation must be under strict control, and we may remark that
in such purely physical investigations as sound analysis, etc.,
the distortions due to reverberation are not to be tolerated.

It is now clear that to solve any problem of Architectural
Acoustics a delicate adjustment is involved; first we must
know just how much reverberation is demanded by the per-
formance given (which is a matter of taste), and then we must
determine what physical measures are best adapted to bring
about this result in the given enclosure, which is a matter of
applied science. Here we have the whole basis l of Sabine's
work; he diligently collected data on which judgment could be
based as to the degree of reverberation desired in typical per-
formances, and he made a thoroughgoing investigation of the
physical factors which control reverberation. In what follows
we shall take for granted the judgment of qualified critics as
to what is optimum reverberation, and confine ourselves prin-
cipally to the purely physical aspects of the problem.

The subject of Architectural Acoutsics is sufficiently broad
to include in addition many special effects, such as echoes,
focal properties of reflecting surfaces, whispering galleries, etc.
Most of these, though possessing features of interest, are not
novel; many such cases have been previously treated, and
these we shall leave mostly to one side, with references to Ray-
leigh (II, Chap. XIV) and to Sabine, where interesting discus-
sions of these phenomena will be found. 2

Before proceeding to the theories of this chapter we must
mention the limitation which is imposed on listening con-

1 Wallace Clement Sabine, Proc. Am. Acad.> XLII, June 1906. Reprinted in "Col-
lected Papers on Acoustics," p. 69.

2 Sabine, "Papers," p. 255; "On Whispering Galleries."

ARCHITECTURAL ACOUSTICS 185

ditions whenever there are disturbing noises present. Let us
suppose that the energy of the sounds to which we are paying
particular attention fluctuates within a certain range; then if
the energy level J of the interfering noises encroaches on the
lower end of this range, clearly all the energy levels in the
range of the sounds for which we are listening must
be raised in order that they may stand out properly against
the background of noise. If this cannot be done (either for
mechanical or esthetic reasons) then the quality of the per-
formance is impaired. This problem of the preservation of
energy levels, both in themselves, and sufficiently above the
energy level of disturbing noises, arises in all cases of sound
reception; it is too technical for further discussion here, but the
reader must understand the reservation which is always im-
plied, regarding absence of disturbing noises, in the theories
which follow. For an example of work in this field, see V. O.
Knudsen, "Interfering Effect of Tones and Noise on Speech
Reception/' Phys. Rev., 26, 1925, p. 133. After giving due
attention to the mechanics of reverberation, in sections fol-
lowing, the reader should find of interest the series by P. E.
Sabine on "The Nature and Reduction of Office Noises," in
Am. Arch. y 121, 1922, p. 441, p. 487, and p. 527.

5 1 . Reflection and Absorption

Since reflection is the primary agent in confining radiation
within an enclosure it is important to gain an idea of the
mechanism of this effect, particularly at the surface of an ab-
sorbing wall. Wall surfaces vary in structure from hard
glazed tile and bricks which are nearly impervious to sound
waves, to coverings of fabrics which are light and porous and
which therefore transmit and absorb incident sound radiation
very well. In between these classifications are a variety of
materials; wood, plaster, paper, rough bricks, etc., all of which

1 The energy level is a somewhat loose but convenient term to indicate the energy
density or the power at the listening point, measured from any convenient zero.
The scale of Transmission Units (suggested on p. 76) is usually employed to des-
ignate differences in level.

1 86 THEORY OF VIBRATING SYSTEMS AND SOUND

are porous to various degrees. It is not necessary here to deal
with any particular material, but by a further development
of the ideas previously applied to account for sound absorption
( 34) we can arrive at a sufficiently good understanding of the
reflection and absorption phenomena at an average wall sur-
face.

In 34 we dealt with the classical conception (due to Ray-
leigh) of a wall surface of light material which was honeycombed
with sniall channels and crevices leading inward from the ex-
posed surface. The incident sound waves penetrate to a certain
degree into these narrow conduits where they are extinguished
by frictional resistance. (The theory of this effect is given in
Appendix A, the results of which should be familiar .to the reader
before proceeding with the following discussion.) In the pres-
ent instance we shall assume that the walls of the channels in
the absorbing material are hard and unyielding, also infinitely
thin; so that the channels are closely packed in a hexagonal
honeycomb structure, and each one can be regarded as a
small cylinder of circular section, without doing violence to
the argument. We can easily find the impedance presented
by the mouth of one of these conduits, and hence the driving
point impedance of the wall to sound waves; and by a com-
parison of this impedance with the radiation resistahce of the
free medium containing the incident and reflected sound waves
we can calculate how much of the incident radiation is reflected,
and how much is transmitted only to be absorbed.

In Appendix A it is shown that the particle velocity is
propagated in a very narrow tube according to the equation: .

$=i *-*V ( n>, (B' 1 )

in which

Rl = i. m

This implies for the excess pressure a similar equation

p = p *-W-M, (262)

REFLECTION AND ABSORPTION 187

from which we derive the presure gradient

g = - p(i + i)p. (263)

Making use of the familiar relation between the pressure
gradient, the velocity of flow, and the resistance coefficient /?i
appropriate to the conduit, we have:

in which

7 -
Z "

is the impedance of the conduit, per unit area of section.
From (K) (in which inertia effects are neglected) we have,

according to Poiseuille's law, /?i = ; hence, substituting also

7*0

for /?, the Jmpedance is: N

Z = - --~ =

M is a convenient facto\ containing the kinematic viscosity

:\ '

- ), the frequency, and the radius of the conduit. 1
p/. .

1 Inasmuch as we ha^e taken K = 7/> for the bulk modulus of the medium in the
conduit (see Appendix A) the constant c is of course the unmodified velocity of sound
in the free medium. In the discussion of this problem Rayleigh prefers for very small
tubes the isothermal bulk modulus, K /> > which of course leads to the Newtonian

velocity of sound, /~. The difference is* not material to the argument here; if we

VT
wished to use the Newtonian velocity, still retaining the convenient factor pc in

/' M

(2650), we might substitute for M the quantity M' , -. It must be particularly

/ VT

noted that neither the Newtonian nor the ordinary Laplacian value of the velocity is
the phase velocity of the waves which diffuse into a narrow tube; the latter velocity is
very much smaller, as shown in Appendix A. Incidentally we remark that for a con-
duit of diameter .02 cm. (as used in the example which is to follow) it is not certain
which is preferable for the bulk modulus the adiabatic value, yp Qy or the isothermal
value, /><>.

1 88 THEORY OF VIBRATING SYSTEMS AND SOUND

Inasmuch as Z differs from pr, the radiation resistance of the
adjacent free medium, there will be reflection from the face of
the wall. Rewriting eq. (120), 31, noting in that equation
that R<2 = Z and Ri = pc y we have for the particle velocity of
the reflected wave, in terms of that of a normally incident wave:

-i) -i

(266)

- The reflection coefficient for the incident energy, obtained
by squaring this ratio and rationalizing the result is:

i iM\ 2 iM 2 iM 4- i

l * 1 L) ^

(M + i - /M)* 2M 2

in which is the phase difference between </ and ^ , a quantity
with which we are not immediately concerned.

It can easily be shown that \R\ has a minimum value for

M = j-.* This means that for any given honeycomb struc-

ture made up of conduits of capillary dimensions there is a fre-
quency of minimum reflection, or maximum absorption. (The
bearing of this result will presently appear.) For example, sup-
pose that in the wall structure the conduits are 0.2 mm. in

diameter, or r = io~ 2 . Then, since = .13 (approximately,

for air) we find that for M = --,-, the corresponding frequency

is = 1600, roughly. For this frequency we compute, ac-

cording to (267), \R\ .17 (the minimum reflection coefficient),
and from this the fraction i \R\ = .83 of the incident energy
which passes on into the wall structure where it is absorbed.

Noting that M varies as -pi we arrive at the following series of

v w

values for \R\ and i |J?( which illustrate graphically the
theoretical behavior of this type of wall toward incident sound
waves in air:

REFLECTION AND ABSORPTION 189

THEORETICAL REFLECTION AND ABSORPTION COEFFICIENTS FOR A WALL OF
CLOSELY-PACKED HONEYCOMB STRUCTURE

Diameter of Conduits, .02 cm.

Frequency,

Impedance
Factor,

Reflection
Coefficient,

Absorption
Coefficient

1*1

i-|*l

200

.38

.62

400

.28

.72

800

.20

.80

l6oO

VI'

. 17 (min.)

. 83 (max.)

3200

.20

.80

6400

2\/2

.28

.72

Note first that since M varies as - -/-> these corresponding

r f> v w

values of M, \R\ and i \R\ will all be shifted to lower fre-
quencies if r is made greater than .01 cm., that is, if the wall is
made of coarser texture. If r is taken for example, equal to
.oi\/2, than the frequency corresponding to the quantities in
a given line of the table will be lowered by one octave.

If the absorption coefficient i \R\ is plotted against
frequency, a very good resonance curve is apparently obtained.
This resemblance is evidently accidental, as no resonance
phenomenon, or selective absorption (of the type described in
47) has been implied in the problem we are considering. But
it seems reasonable to infer from the results obtained on this
simplified theory of the mechanism of sound absorption 1 by

1 The essential soundness of the classical method of dealing with porosity, which we
have followed, is attested by an observation of P. E. Sabine (Phys. Rev. XVII, 1921,
p. 379). "Experiment indicated that the change in absorption with change in thickness
[for hair felt] follows the same law as the reduction of energy flow of an air stream
through the material. The dissipation of energy of flow through the material at con-
stant pressure approaches a limiting value as the thickness of the material is increased. "
In other words, the mechanism of dissipation in narrow conduits is essentially the same,
in both static and dynamic cases.

190 THEORY OF VIBRATING SYSTEMS AND SOUND

porous bodies, that we can never expect to find a homogeneous
material which will reflect and absorb sound waves according
to coefficients which are independent of frequency. The
reader is warned, however, not to confuse the selectivity due
to porosity alone, with selective absorption based on resonance
in layers of absorbing material. This latter effect will be dis-
cussed more fully in 52. The two effects are of course super-
imposed in the case of any yielding material such as felt.

The particular close-packed honeycomb structure we have
been studying has a high absorption coefficient, as all of its
area is effective for transmission purposes. But we can easily
imitate a harder or less permeable wall by filling up some of
the conduits with infinitely rigid material, taking care of course
to distribute the filled-up spaces so uniformly, that in any
reasonably small area, however chosen, the proportion of per-
forated to unperforated surface is always the same. The effect
produced is merely to increase the impedance of the wall. If
a is the part of unit area which is now of open conduits, and b
the unperforated part, then before, with both a and b open, the
impedence due to a and b in parallel was Z per unit area, or
adding admittances, we had:

1 a , b , L

, 7 -, 7 + -,,* since a + b = i .

/j /-j Z>

Now, with the impedance of the part b infinite, the impedance

^
per unit area of the wall is no longer Z, but Z' = ; and a can

be made any fraction less than unity. Putting Z' for Z in (266)
and (267) we have for the reflection coefficient of the modified
wall:

,_, 2A/ 2 - iMa + a 2

(An application of the equation of continuity to the flux in the
neighborhood of the openings (Rayleigh, II, 351) leads to the
same result.) From (2670) we observe that for a given value of
M (that is for constant impedance per conduit), the reflecting

REFLECTION AND ABSORPTION 191

power of the wall increases as a becomes smaller; \R\ finally
becomes equal to unity when a vanishes, that is, when all the
conduits are rigidly closed.

To this point we have dealt, in abstract terms, with reflec-
tion and absorption primarily for the purpose of understanding
them. In practice a more empirical method of dealing with
these properties is sufficient, and at the same time necessary,
because what interests the practitioner of Architectural Acous-
tics is the application of materials of known gross absorption
and reflection coefficients to control reverberation. One method
of measuring the absorbing power of a material has already
been described (cf. Wente's Experiment, 34), and older
methods are available, as for example the one due to Sabine,
which depends on the relation between the reverberation, the
dimensions of the enclosure, and the total absorbing power
of the bounding surface. Sabine very logically took as his
standard absorption coefficient, that of the free medium, or
more precisely, an open window of area one square meter. 1
The absorbing powers of everything else, e.g. walls (per square
meter), furniture (per piece), the audience (per person), etc.
were all fixed by his experiments, in terms of this standard.
(A great many absorption data are given in his "Collected
Papers/ 1 e.g., "Absorbing Power of an Audience, etc." pp.
52-60.) The utility of data of this sort will appear when we
come to consider reverberation.

1 P. E. Sabine (Phys. Rev., XIX, 1922, p. 402) has culled attention to the fact that,
due to diffraction, the shape as well as the area of an absorbing surface, or a window
opening, will affect the absorbing power of a given area. Some reflection from an open
window also takes place. It appears from P. K. Sabine's measurements that the smaller
and narrower the opening, the greater the transmission coefficient per unit area.

W. C. Sabine's earlier measurements were made by a substitution method, using
pew cushions as a standard of comparison, then comparing these with open window
space, assuming that the latter transmits perfectly, and in proportion to its area. In
his later work, however, he determined the absorption coefficient of a given material
by the introduction of a known area of material. The total absorbing power of the room
was determined by the "four organ pipe experiment," ("Collected Papers," p. 35).

Practically all the coefficients used are those obtained by the later method, through
Prof. Sabine does not note the change of method in any of his papers. (Comment of
P. E. Sabine, by letter, Jan. 30, 1926.)

192 THEORY OF VIBRATING SYSTEMS AND SOUND

52. Layers of Absorbing Material

In applying a layer of absorbing material (such as an inch
of felt) to a hard wall surface, if the sound waves are not extin-
guished in one transit back and forth through the material, it
becomes necessary to take into account the multiple reflections
in the layer, and the consequent emergence of sound waves
from within the layer which will increase its reflecting power.
Of course by actual measurement the gross absorption and
reflection coefficients of the structure can be found, and we
have observed that these data would be sufficient for engi-
neering purposes; but it may be desired to relate them to the
more fundamental constants which are based on the trans-
mission of sound waves in the absorbing material.

In general, there are two methods of attacking the problem;
these we may refer to as the impedance and the wave methods.
Both are empirical; to apply either method we assume that
sound waves are propagated in the material in accordance with
the familiar equation 1

= t Q e- ( *+ i *> t e i * y (268)

in which a and ft (the attenuation and phase factors) are to
be experimentally determined. The factor ft is established if

the phase velocity c' = is known. It is also necessary to

know either the density (p') or the radiation resistance
R 2 == pV; the quantity c' in this expression being approxi-

1 It has been established by P. K. Sabine (Phys. Rev. XVII, 1921, p. 378) for a po-
rous material (hair felt) that "the logarithm of the reduction in intensity of the trans-
mitted sound is proportional to the thickness of the material through which it passes"
which is in accordance with the assumption of eq. (268). The attenuation coefficients
as measured by P. E. Sabine vary with frequency, in much the same way as the typical
curves for felt obtained by W. C. Sabine, to be referred to as the end of 52. (See also
P. E. Sabine, "Transmission of Sound through Flexible Materials," Ant. Arch., Sept.
28, p. 215 and Oct. 12, 1921, p. 266.)

The work of P. K. Sabine on absorbing properties of materials, and other problems
in Architectural Acoustics was done, and is being continued at the Wallace Clement
Sabine Laboratory, Riverbank, Geneva, Illinois. The plan and the activities of this
Laboratory (which is a memorial to the late W. C. Sabine), are described in Am. Arch.>
116, 1919, p. iJJ-ijS.

LAYERS OF ABSORBING MATERIAL 193

mately equal to the phase velocity. (This is on the assumption
that a 2 is not comparable with 0-'; we are now dealing with a
yielding material whose stiffness, density and dissipation con-
stants differ from those of the pure air in a narrow conduit, for
which we found a = ft in the preceding section. In the present
case, on account of the inertia of the material, c* will be less
than the velocity of sound in the adjacent gaseous medium;
c' will be further lowered by a second order quantity dependent
on a, and a itself varies as the ratio of dissipation to density in
the material. As a result of all these effects, we should assume
that j8 > a except at very low frequencies. Strictly speaking
/? 2 SE p'c f is a complex quantity; but here we take c' as the
approximate phase velocity, which is a real quantity.) If we
know these fundamental constants we can estimate the sound
absorbing and reflecting characteristics of the given material,
in any configuration. The density can of course be checked by
weighing; but because of the composite structure of absorbing
materials, and the variation of their properties with the close-
ness of packing, it is usually not feasible to obtain precise results
for the other quantities.

If the properties of the material are known, the impedance
method is best adapted to determine the reflecting power of a
finite layer. To illustrate, if the layer is applied to a rigid wall
surface, the problem is similar to that of the tube closed at
one end, which was solved in 32 assuming that there was no
dissipation in the medium. There we found for the driving
point impedance of the tube, of length /,

Z = ipc cot $1. (136/0

This result can be applied to the problem of an absorbing layer
if we substitute for PC the radiation resistance /?2, character-
istic of the material, and for 0, the corresponding quantity
ia which takes account of dissipation, in the present prob-
lem. This equation may now be written:

i 9 4 THEORY OF VIBRATING SYSTEMS AND SOUND

and determines, at least theoretically, the impedance of the
layer. A comparison of this impedance with that of the adja-
cent free medium, making use of (120) as before, gives the
(amplitude) reflection coefficient for the layer, in both abso-
lute value and phase.

If on the other hand the constants of the material are not
known, and it is desired to measure them, what we shall call
the wave method has certain advantages in practice. The dif-
ference between the two methods is only in the point of view;
both lead to the same results, but it may be of interest to verify
this conclusion.

We shall suppose that by some scheme such as Wente's
( 34) we have measured the gross reflection coefficient of a
layer backed by a rigid wall, and that we wish to determine
the true reflection coefficient, that is the reflection coefficient
at the boundary of an infinitely thick layer. Let r equal this
latter quantity; then for unit amplitude of the incident wave,
the amplitude of the entering wave is, by eq. (1200), 31,

* = tl2 = = -

if /?2 is the radiation resistance of the absorbing material, and
R its relative radiation resistance with respect to the other
medium. On the same basis we have for the reflection coeffi-
cient

/?2 RI i R

to which we shall add the relations previously obtained
2/? 2 2/2

In passing through the layer to the rigid wall, the amplitude
fi2 becomes ti2e~ (a+iffil , or more concisely, tize"*. In reflection
from the wall, there is a phase change of I, and in transit
back through the layer a phase change of e~ x \ finally, in emerg-
ing from the layer at the boundary there is an amplitude reduc-

LAYERS OF ABSORBING MATERIAL 195

tion in the ratio fei : i. Combining all these effects, we have,
for the first internally reflected wave, on emergence from the
layer, the amplitude

fa = - ti2-t 2 i-e- 2 * = - (i - r 2 )*- 2 *,

which is added to the amplitude 1 = r of the externally re-
flected wave. Proceeding to the second internally reflected
wave, we note that it starts from the boundary with an ampli-
tude ti2e~ 2x -( r), and after undergoing transit through the
layer, reflection at the wall, and emergence, its amplitude is

fa = + <i2-fei-(- r)*- 4 * = - r(i - r 2 )*- 1 '.

We infer then that the gross amplitude of the wave reflected
from the layer is the sum

& = *i + fa + fa + - - = r - (i - r 2 )<r 2 * - r(i - r 2 )e~ 4 * . . .

(270)
that is,

(i - r 2 )*- 2 * r - e~ 2 *

summing the convergent geometric series which follows r. The
advantage of this formula is that if r (the ratio of reflected
to incident wave amplitude), is found for several different
thicknesses of the material, the quantities r and e~* = ^" (a+tftl
can be determined from the simultaneous equations con-
taining the experimental data. In solving the equations certain
approximations are necessary; these are best devised to fit the
necessities of each given set of data. Theoretically, if three
layers of different thicknesses are tested (as in Problem 41,
below), we have a method of separating r, , and 0, the three
principal constants of the absorbing medium. Knowing r is of
course equivalent to knowing /?2, since RI is known; equation
(2700) is easily obtained by substituting for R% in (120) the
quantity Z/ of (269).

To further illustrate the wave point of view, and a rough
determination of the phase velocity for an actual material, we

196 THEORY OF VIBRATING SYSTEMS AND SOUND

64 1Z8 256 512 1024 2048 4096

FreQuency

FIG. 20. ENERGY ABSORBING POWER OF ONE KIND OF FELT, IN LAYERS OF THICKNESS
INCREASING BY MULTIPLES OF i.i CM.

LAYERS OF ABSORBING MATERIAL 197

may consider some data taken by W. C. Sabine for a certain
kind of felt. This was nominally of wool, and very porous.
In Fig. 20 are shown the values of the (gross) absorption
coefficients of layers of various thicknesses of this material. 1
The first layer was 1.1 cm. thick, the second 2.2 cm. thick, and
so on in multiples of i.i cm. The increase of absorption with
thickness is marked, particularly at lower frequencies; there is
also noted a shift of the maximum absorption coefficient to
lower frequencies as the thickness is increased. This shows,
as pointed out by Sabine, a certain amount of resonance in the
material; this reaction may be viewed as due to standing waves
therein, if the thickness is sufficiently great, otherwise, to
simple compression. If we ignore the fact that due to porosity
alone, there will be a fixed frequency of maximum absorption,
we may suppose that a layer of the material will ordinarily
show resonance (and hence selective absorption) when it is
one-quarter of a wave-length thick. From this we can deter-
mine roughly the phase velocity in the material, since X/ = c f .
From the curves we take the frequencies of maximum absorp-
tion to be respectively 2048, 1024, 725, 610, 512, and 480, be-
ginning with the thinnest layer. Multiplying each of these
by four times the appropriate thickness, we get a series of values
for c 1 ranging from .91 X io 4 to 1.27 X io 4 , with rising thick-
ness, the mean being i.oi X jo 4 . This should be approximately
the value of the phase velocity in the material; the (apparent)
higher values of c r for the thick layers are what we should ex-
pect, as in these cases, the (apparent) frequency of maximum
selective absorption is doubtless raised owing to the presence of
a higher frequency of maximum absorption due to porosity
alone. The effect of increased attenuation at the higher fre-
quencies will also tend to lower the frequency of selective ab-
sorption in the thinner layers.

In Fig. 21 the same data are plotted as absorption coeffi-
cients in terms of thickness, each curve relating to a particular
frequency. "Thus plotted, the curves show the necessary

1 W. C. Sabine, Proc. Am. Acad., XLII, June, 1906; or "Collected Papers/' p. 99,
Figs. 12 and 13.

iy8 THEORY OF VIBRATING SYSTEMS AND SOUND

1 Z 3 4 5

ThicKness, cm.

FIG. 21. DATA OF FIG. 20 REPLOTTED TO SHOW, FOR DIFFERENT FREQUENCIES,
VARIATION OF ABSORBING POWER OF FELT WITH THICKNESS OF LAYER.

LAYERS OF ABSORBING MATERIAL 199

thickness for practically maximum ' absorption." It is evident
from the curves (with one exception) that they tend to pass
through the origin, that is, for zero thickness, the absorption
would be negligible. This Sabine interprets as showing that
there is no dissipative effect due to friction at the surface; we
have omitted any such speculation heretofore, but it is inter-
esting to note that this omission is justified. The main point
of this family of curves is that they show much greater atten-
uation in the material for waves of higher frequency, two thick-
nesses being sufficient to bring about virtually maximum ab-
sorption for / > 1024. Finally, we may observe that the
higher attentuation will enhance the reduction of the phase
velocity at the higher frequencies, the effect of which reduction
we have noted in the preceding paragraph.

In practice it may be necessary to deal with composite
absorbing structures built up of strata of various materials.
The methods we have illustrated are theoretically applicable in
the study of all such cases, but the relations obtainable between
the various quantities may be so cumbrous that they are of little
service. Here experiment must be the deciding factor; but the
procedure in many cases can be simplified if we use the theory
we have given as a guide to the study of the component mate-
rials.

The measurement of the leakage of sound energy from
enclosures (depending on transmission through windows, doors,
partitions, cracks in walls, etc.), while not at first sight germane
to this discussion of absorption, has its proper place in Archi-
tectural Acoustics, and should be mentioned. The point to be
made is that such leakages are to be taken into account as
additional dissipation from the enclosure, that is, as a virtual
increase in the absorbing power of the bounding surface. The
first experimental study of these effects was by W. C. Sabine 2 ;

1 Cf. P. E. Sabine, loc. cit. "The values of the absorption coefficient [for hair felt]
were found to increase with increasing thickness of absorbing material, but they
approach limiting values which are reached at smaller thicknesses for the higher
frequencies."

2 The Brickbuilder, XXIV, Feb. 1915; or "Papers," p. 237: "The Insulation of
Sound."

200 THEORY OF VIBRATING SYSTEMS AND SOUND

since that time much experimental testing has been done by P.
E. Sabine l and F. R. Watson ~ in order to establish the correct
transmission coefficients of doors, partitions, and other boun-
daries. Finally there is a very complete paper by E. Bucking-
ham 3 which not only considers the theory of the experiments
of sound transmission through walls, but gives a treatment of
reverberation which will interest the reader, in connection with
the theory of 54 following.

53. Reverberation in a Closed Tube

A natural approach to the general theory of reverberation
is through an analysis of the effects produced in a long tube of
unit section, closed with absorbing walls at either end, and
fitted at one end with a small source of sound. The source,
radiating sound at a constant rate, sends wave trains of any
desired length into the tube. The absorption of energy at the
ends of the tube is proportional to the energy density in the
tube, and since the energy density in the tube is constantly
added to by the radiation from the source, there comes a time
after starting the generator, when the radiated energy is in
equilibrium with that absorbed, and a steady state prevails.
The asymptotic growth of energy density toward this steady
value is the first characteristic effect to be noted; and con-
versely, on stopping the generator, the energy in the enclo-
sure decays exponentially to zero. Thus the time variation of
the energy density in the enclosure does not follow faithfully
that of the radiated energy, if the wave train is finite; and we
have the characteristic phenomena of reverberation. The
mean energy density is chosen as the best variable in terms of
which to state the theory, no account being taken of frequency
or phase.

1 P. E. Sabine, Am. Arch., July 28, 1920 "Transmission of Sound through Doors
and Windows"; see also ibid., Sept. 28 and Oct. 12, 1921, and July 4, 1923, the last
paper dealing with transmission through walls; see also Phys. Rev., 27, 1926, p. 116.

2 F. R. Watson, Univ. of Illinois Rng. Exp. Sta., Bulletin No. 127 (1922) "Sound
Proof Partitions'*; see also his "Acoustics of Buildings/' New York, Wiley, 1923,

3 U. S. Bureau of Standards, Sci. Paper No. 506, May 26, 1925.

REVERBERATION IN A CLOSED TUBE 201

It is assumed that the generator radiates a wave train of
energy density E\ into the tube; thus the rate of working of
the source is cE\ plus whatever power is lost locally due to the
adjacent absorbing surface at that end of the tube. The length
of the tube is /, and A = i R is the (energy) absorption
coefficient of the wall at either end, R being the (energy) reflec-
tion coefficient. The time required for n reflections of the origi-
nal wave at the ends of the tube is a trifle greater than ; the

number of reflections per second is m = . and the mean free

path between reflections is /. To find the energy density at any
moment after the generator has started working we must add
to the energy density E\ of the original wave, the energy density
of the once reflected wave (RE\), that of the twice reflected

wave (R 2 Ei) y and so on: hence just after the time / = the
energy density is

#1 / / x

-^- (i -/?"), (271)

taking the sum of the finite geometrical series. In terms of
time this may be written

A'ri

taking e A ' = R. The quantity A' is of the order of magnitude
of A\ since A' = log/? = log (i A), we have, on expan-
sion,

A 2 A*
A' = A + ^- + ?- + .... (272)

From (27 1 a) we note that the rate of growth of energy density
varies directly as -4', but the maximum energy density finally
attained in the steady state varies inversely as A. In other

202 THEORY OF VIBRATING SYSTEMS AND SOUND

words, a large absorption coefficient for the boundary surface
makes the enclosure more responsive to power variations at the
source, at the expense of a lessened maximum energy density
or loudness.

The growth of energy density, for constant rate of working
of the source, is analogous to the growth of charge density in a
leaky condenser, which is charged by a constant direct current.
This takes place according to the equation:

in which / is the constant charging current, C the capacity and
R the resistance of the leak between the plates of the condenser.

- , the charge density, is analogous to the energy density E.
The charging current 7 is analogous to 1 since both measure
rates. The conductivity - is analogous to the absorbing power

Ay and the capacity C corresponds to the volume / of the tube;
thus the analogy is complete.

Now suppose the sound generator is stopped, the energy

Ei
density in the tube having attained the value The number

of reflections in unit time is m = - and as the waves pass back

and forth between the absorbing surfaces an amount of energy
A-E is absorbed at each reflection. The rate of change of
energy density in the tube is therefore

(274)

and the solution of this gives, consistently with the boundary
condition,

_ Ei - c At / v

E = e l , (2740)

REVERBERATION IN A CLOSED TUBE 203

for the decay of energy density. The electrical analogue is the
same as before, the equation for the discharge of the leaky con-
denser being

| = IR*-**. (275)

A very real idea of what underlies the colloquial phrases
"filling a room with sound" and "deadening the room" (to
diminish reverberation) can be gained if we adopt another
analogy which happens to be virtually ready made. We have
found ( 48) that the energy density of radiation is equivalent
to its radiation pressure, a static effect. Hence for the immedi-
ate purpose we may substitute for the contained radiation a
compressible fluid whose density and pressure are proportional
to the density of the radiation. Fluid is introduced at constant
rate by a source; the pressure within increases until the rate of
dissipation of fluid through leaks or absorption is equal to the
rate of supply; and finally if the supply is shut off, the pressure
diminishes at a rate which is proportional to the product of the
pressure and the conductivity of the leaks, or the overall ab-
sorption coefficient of the boundary. These are the essential
ideas involved in the mechanism of reverberation, and may be
applied, with some modification of detail, to an enclosure of
any size, which has reasonably simple proportions.

If we define T, the reverberation time of the given enclosure
as the time required for the energy density to sink from one
prescribed level to another, then according to (2740), if any
two of the quantities T,Aorl are known, the other can be cal-
culated. Let subscripts I and 2 refer to the two energy levels;
then if E\. > 2, we have immediately

E'2 -jAT . I . El , s\

j=r = e > or AT log (276)

1 C 2

This equation states what might be called the^rj/ law of Archi-
tectural Acoustics; it will be sufficiently generalized in 54 to
cover all practical cases, in the form AT = K^ V being the
volume of the enclosure and K a constant. What we wish to

THEORY OF VIBRATING SYSTEMS AND SOUND

discuss here is the determination of A in terms of T, if / (or V}
is known. W. C. Sabine made a great many laborious experi-
ments using the threshold (i.e. minimum audible) energy den-
sity for E 2y and for E\ and energy level io 6 as great. By varying
the absorbing power of a given room, i.e. by adding or sub-
tracting absorbing surfaces on the floor or on the walls, he was
enabled, through the relation AiTi = A^T^ to measure the
absorbing powers of all the usual materials, objects, etc., in
terms of the open window of unit area for which he took A = i.
Since it was impracticable to line a room of any considerable
dimensions throughout with material of uniform absorbing
qualities, many differential experiments had to be made, in
order to accumulate a sufficient variety of absorption data for
working purposes; but when the data were finally accumulated,
it was found that the process could be reversed, and by calcu-
lating the total absorbing power of the boundary surfaces, and
the objects contained in the room, a sufficiently accurate
prediction of the reverberation time could be made.

The essential soundness of the reverberation method of
measuring absorption coefficients is not to be questioned, but
owing to the dependence on the ear for determining the time
at which the threshold intensity level was reached, repeated
observations had to be made in order to attain the required
degree of precision in the results. (Sabine customarily took io
or 20 observations in any given case.) A chronograph was
necessary to register time intervals; and there were other cum-
bersome features of his experiments, such as the use of the
constant pressure air reservoir (for actuating the organ pipes
used as sound sources) which entailed considerable manipula-
tion. The organ pipes had to be calibrated, in connection with
the air supply, in a physical laboratory. Taken altogether, the
work was difficult and tedious, as compared with present labo-
ratory practice, and if repeated today, would doubtless be
greatly facilitated by the use of better instruments which are
now available.

It seems, however, from the practical standpoint, that
small scale methods based on the mechanics of sound in tubes,

REVERBERATION IN THREE DIMENSIONS 205

and independent of reverberation, offer definite advantages
in the determination of the absorption and reflection coeffi-
cients at the surfaces of homogeneous materials. It is easy to
set up in the laboratory a compact arrangement similar to
that of Wente, to which reference has been made ( 34, 52),
in which electrical devices are employed for producing and
detecting sound waves, and to obtain, from a few readings of
an alternating current instrument, easily reducible data which
are equally as good as those previously obtained by the rever-
beration method. Here again we have a characteristic applica-
tion of the newer theory and technology of modern acoustics.

54. Reverberation in Three Dimensions

We proceed to apply the principles established in the pre-
ceding section, in order to obtain the formulae of Sabine for
reverberation in a closed three-dimensional space. Consider
first his factor a, the total absorbing power of the room. This
is not a pure number (that is, a ratio of energy values), but
has the dimensions of area; it is defined by the equation

a = AiSi + A 2 S 2 + . . . + A n S n , (277)

each term of which is the product of the area S } of one of the
component absorbing surfaces by its absorption coefficient,
relative to an equal area of open window space. Thus for six
square meters of carpet, of absorption coefficient Aj, we should
have a term 6Aj for the total absorbing power of this area. It
is clear that if we divide a by 2$j, we have the "effective" mean
absorbing power of the whole boundary surface, which must be
identical with the quantity A of 53. That is to say,

A - - ~i- (S = SA',), (278)

O ^Oj

in which it is understood that the summation is carried over
all the boundary surface, whatever its configuration, or local
absorbing power. This of course implies that all parts of the
boundary have equal opportunity to drain energy from the

206 THEORY OF VIBRATING SYSTEMS AND SOUND

enclosure, but this is not unreasonable on the static analogy of
radiation pressure which we have previously suggested. Expe-
rimentally the procedure is amply justified in the accuracy
which Sabine usually attained in his practical calculations.

To include the volume fin the analysis we must deal with a
factory, which Sabine called the "mean free path between re-
flections." In 53 the mean free path was clearly the length
of the tube, (/) just as was stated. By a tedious course of experi-
ment, which need not be described here, Sabine established the
relation p = .62 v V\ that is to say, the mean free path be-
tween reflections (which is to be regarded as an ideal acoustic
dimension, independent of the length, width, or shape of an
irregular enclosure) is somewhat less than the cube root of the
volume. We can now obtain the general equation for reverber-
ation in a volume V by substituting in (276) the quantity

p = .62 v^ for /, and the ratio -- for A y thus:

(279)

tance -- Thus we have the fundamental relation

To eliminate the quantity S y we note that for any enclosure
of reasonably compact proportions we should have approxi-
mately, S\/y = 6^, since the volume of any elementary cone
based on a surface element dS of the boundary of the room is
(very nearly) one third of the product of the base by the dis-

(.80)

the importance of which has already been pointed out. As
deduced here, the constant K depends on the determination
of the mean free path in terms of the \/V\ Sabine followed the
opposite course, first establishing the relation aT =

REVERBERATION IN THREE DIMENSIONS 207

which led him to the relation p = .62 y/V. For the standard
ratio -=r = io 6 (as used by Sabine) we compute, taking c = 340
meters per second,

K = .15 sec./meter. (2800)

The meter units of Sabine are retained as best suited to the
large spaces dealt with in actual practice. Sabine gives a value
K = .164 as the best determination 1 of this constant; the
discrepancy noted is due to the very approximate character of
the calculation we have made, and (probably to some extent)
to a relatively poorer determination of the quantity/). 2

At this point we might well consider the problem of calcu-
lating the mean free path/), or what is the same thing, deducing
the law of architectual acoustics (eq. 279) a priori. Such a cal-
culation was given first by W. S. Franklin (Phys. Rev. y XVI,
1903, p. 372); a similar calculation is also given by K. Bucking-
ham 3 and made use of by E. A. Eckhardt in an interesting
paper on reverberations. 4

According to this theory, the number of energy units coming
from all directions reaching any bounding surface dS in unit

time is > in which it is assumed that all the trains of inci-

4
dent waves are of energy density E. The amount of energy

, , , . ... 0-1 AS- EC aEc .
absorbed in unit time by an area o is then = > in

which a is Sabine's area-absorption coefficient. If now E is
the rate of emission of the source, then the differential equation
for the energy density in a closed space of volume V is

^ + \caE = E, (281)

1 American Architect, 1900, "Reverberations"; also Collected Papers," p. 50.

2 W. C. Sabine, "Papers," p. 40.

3 U. S. Bureau of Standards , Sci. Paper No. 506, May 26, 1

4 Jour. Fr. Inst., 195, 1923, p. 799.

208 THEORY OF VIBRATING SYSTEMS AND SOUND

The solution of this equation, for the particular case of decay
from an energy density level E\ is

E - <

~ = e "'; (282)

while for the case of growth of energy density to a maximum,
the solution is

since for / = oo, the rate of absorption of energy by the walls
of the room must be equal to the generated power.

Equation (282) is identical with (279) if we replace the con-
stant therein (.62 X 6 = 3.72) by 4; or what is the same thing,
if we take the mean free path as/) = .67 *$/V. Using this theo-
retical value of />, we have instead of (2800), K = .161, which
is in good agreement with Sabine's experimental value (K =
.165). This in a way justifies the calculation, which is very
rough, from the nature of things. Eckhardt's paper contains
a number of graphs illustrating the growth and decay of energy
density in rooms of different absorbing power. Before going on
we may observe that (282) and (283) make possible a second
definition of reverberation time, namely as the time of decay
from the maximum energy density level due to a source of constant
-power placed within the room. This conception (due to P. E.
Sabine) is one of the newer developments, to which we shall
return presently, after a further consideration of the applica-
tions which W. C. Sabine made of his ideas.

A characteristic application of W. C. Sabine's theory of
Architectural Acoustics was made when he was consulted on
the design and interior treatment of Symphony Hall (Boston)
in advance of its construction. 1 The plan was virtually to con-
struct a hall whose acoustic quality (reverberation) should be
the same as that of the Gewandhaus (Leipzig), but which was to

1 W. C. Sabine, "Papers," p. 60.

REVERBERATION IN THREE DIMENSIONS 209

a 70 per cent larger audience; when the new hall was com-
pleted, its volume was about 60 per cent greater than that of
the Gewandhaus. The reverberation time of the Gewand-
haus was T = 2.30 sec., as calculated by Sabine, due allowance
being made for the presence of the audience, in addition to the
sum of the absorbing powers of all objects and the boundary
surfaces. In collaboration with the architects a design was
made for Symphony Hall such that its reverberation time
should be 2.31 seconds. This design, together with the prior
researches on which it was based, may justly be considered a
classic of Applied Acoustics. Sabine makes the interesting point
that it would not have done (in the absence of a correct theory)
to merely enlarge proportionately the Gewandhaus design; for
the reverberation time in that event would have been increased
to T = 3.02 sec., corresponding to the necessary increase of
the volume of the Gewandhaus from 11,200 cu. m. to 25,300
cu. m. to obtain the required seating capacity. This would
have differed from the chosen result by an amount that would
have been very noticeable. How serious such an error would
have been will appear in due course.

To investigate the accuracy of musical taste in judging
optimum reverberation time, Sabine carried on a series of
tests, cooperating with several competent musicians, leading
to the best adjustment of each of five rooms for listening to
piano music. 1 When the critics were satisfied with the final
adjustments, the reverberation time for each room was meas-
ured; a series of values T = .95, 1.10, i.io, 1.09 and 1.16 sec.
for the five rooms was obtained. This indicates pretty definitely
that the mean reverberation time T = 1.08 sec. is characteristic
of a room of moderate size acoustically good for this purpose;
this result is quite consistent with the results of later work of
a similar kind.

P. E. Sabine in a recent paper 2 gives the following rule:
"The time of reverberation for an auditorium with its maxi-
mum audience as computed by equation [2870 below] should

1 W. C. Sabine, Proc. Am. Acad., XLII, 1906; "Collected Papers," p. 71.

2 P. E. Sabine, "Acoustics in Auditorium Design," Am. Architect, June 18, 1924.

no THEORY OF VIBRATING SYSTEMS AND SOUND

lie between one and two seconds. For speech and light music
it should fall in the lower half of this range, while for music of
the larger sort, it may lie nearer the upper limit." To fully
understand this we must discuss his newer idea of reverber-
ation time, to which reference has already been made. If a
generator of constant power output E is placed in a room of
volume V and area-absorption coefficient a the energy density
will build up finally to the value

Ei = ^- (2830)

If decay now takes place from this level, we have, substituting
this value of Ei in (282),

E = -e'y (285)

ac ^ *'

whence

(286)
ca

(It is clear that if the ratio of initial to final energy density is

4?-,oe

caE I0 '

we return at once to W. C. Sabine's first law, namely
aT = .1

as given by (2800), using the accurate value of K.) But the
purpose of the present calculations is to get away from the
concept of a fixed initial energy density and base the definition
of reverberation time on a fixed power output at the source.
This is because, in practice, we are more likely to deal with
sources of constant power than with cases of constant maximum
energy density in rooms of different characteristics. And it is

REVERBERATION IN THREE DIMENSIONS 211

an empirical fact that, with the reverberation time defined on
the power basis, there is a better correlation between the
reverberation time and the apparent goodness of the room,
judged by the artistic test.

Following P. E. Sabine, let us take the power of the source,
E y as io 10 cubic meters of sound of threshold density (E) ', and
so obtain a more practical form for (286), which gives the rever-
beration time TI on the new basis as

This is equivalent to 2

aT\ = .0271^8.07 Iogi a]

which may be compared to the earlier definition (of W. C.
Sabine) given in equations (280, 2800); it is equivalent if
8.07 logio a = 6. Broadly speaking, we have in (2870) a
relation which virtually "corrects" W. C. Sabine's formula
aT = KV y for unduly large values of a, and gives a new concept
of reverberation time more nearly in accordance with the actual
conditions under which concert rooms are used.

The following table gives a few comparisons for typical
large halls, of T\ as computed by P. E. Sabine, and T as deter-
mined by the older method. These halls are all supposed to be
acoustically satisfactory, though not perfect.

1 "This particular value of E has been chosen, since it approximates the acoustic
power of the organ pipes used by Professor Sabine in his investigations. Further, a
source of sound of this power, will give in empty audience rooms of the usual propor-
tions, with seating capacity of 500 to 1000 persons and with upholstered seats, an in-
tensity of io 8 times the threshold level. Finally, sound chamber measurements of the
power of musical instruments and the human voice indicate that this is a fair approxi-
mation to the power of the sounds with which we are dealing in auditorium acoustics."
(From an unpublished paper of P. E. Sabine.)

2 Equation (2870) is of course in metric units. If a is given in square feet, and V
in cubic feet, we have

aTi = .0083 ^(9.1 - logiofl), (287*)

as given by P. E. Sabine, loc* cit.

2i2 THEORY OF VIBRATING SYSTEMS AND SOUND

Hall

Volume
(cu. m.)

Seats

(P. F, Sabine)

Authority
forr

i. Masonic Temple, Madison.. . .

8,580

1500

i-37

i. 60

P. E. Sabine

2. Unions Hall, Moscow. . . .

12 .CQO

1600

I A. A.

I 7C

Lifshitz

/ J

3. Symphony Hall, Boston

I8.4.OO

2600

I . Q2

2 11

W.C Sabine

4. Eastman Theatre, Rochester.

22,400

3340

1.6 5

2 08

Watson

5. Auditorium Theatre, Chicago

26,200

3640

I. 4 8

1.90

P. E. Sabine

This table contains only a few of the available data, but
they are intended to be representative. It will be noted that
there is less of a trend toward higher reverberation time with
greater volume, for TI than for T, even if we exclude No. i and
No. 5 which represent halls of the theater type, having rela-
tively large seating capacity for a given volume, that is, a
larger ratio a\V. If data on more halls were given, they would
show markedly less variation of T\ than of T. A general mean
of Ti for large halls of good acoustics has been figured by P. E.
Sabine to be 1.47 seconds, and he concludes that this is a very
satisfactory standard; it will be noted that this value lies
nicely within the range from one to two seconds given in his
statement first quoted. And finally we note that the orchestra
director Nikish considered the Unions Hall, Moscow, the best
hall in Europe; while the Chicago Auditorium is unanimously
considered satisfactory for both theater and concert purposes.
Considering the diversities of opinion which may be encoun-
tered among musical people on the optimum degree of reverber-
ation it seems reasonable, for all practical purposes, to follow
the general standard which P. E. Sabine has proposed.

A study of reverberation from the artistic standpoint has
been made by S. Lifshitz 1 using the variation in the number of
the audience present to vary the absorbing power of the room.
For a small room (e.g., for V < 350 cu. m.) it was found that
the optimum reverberation times for baritone, violin and
violoncello music were nearly the same, and centered about the
value T = 1.03 sec. For a certain orator in a room of volume

Optimum Reverberation for an Auditorium (Phys. Rev., 25, 1925, p. 391). Un-
fortunately this paper seems to be based on a rather small amount of data.

REVERBERATION IN THREE DIMENSIONS 213

126 cu. m. the optimum time was T = 1.06. These critical
observations were apparently made with considerable assur-
ance. For three large halls of volumes 12,500, 13,800 and 17,000
cu. m., the reverberation times were taken to be, respectively,
1.75, 2.00 and 1.55 sec. with all the audience present. The
acoustic properties of the first room (the Unions Hall, Moscow)
are notably good; the second is too reverberatory; the last is
too dead, unless some of the seating capacity is vacant. As a
result of experiments of this sort, and certain theoretical con-
siderations l Lifshitz proposed a formula for optimum rever-
beration time as an increasing function of volume. According
to this formula the times for the three rooms described should
have been, respectively, 1.75, 1.79, and 1.85 sec. On the same
basis, Symphony Hall (Boston) would be improved if its rever-
beration time were lowered to about 1.9 sec. While this may
seem somewhat of a refinement over the effect actually striven
for in Symphony Hall (i.e., T = 2.30), it is fortunate indeed
that as a result of acoustic design, the more serious error was
avoided of making its reverberation time any greater. The
opinion may be hazarded from the preceding discussion that if
there is any tendency toward changing our present standards
of reverberation, it is likely to be toward lower values.

Recently great advances 2 have been made in recording
and reproducing music on the phonograph, with the result that
from the standpoint of frequency distortion (that is, resonance,
or the undue suppression of high or low frequency sounds), the
quality of the reproduction is nearly ideal. In addition, the
correct reproduction of energy-level relations (that is, the
elimination of distortion due to widely varying amplitudes),

1 S. Lifshitz, "Architectural Acoustics," Moscow, 1923. In a second paper in the
Phys. Rev. (27,1926, p, 61 8), Lifshitz has clarified his argument and given another
empirical equation for optimum reverberation time. He also quotes from Watson's

( Acoustics of Buildings " a number of data on American halls. In comparing Lif-
shitz's conclusions with those of P. E. Sabine, above given, the reader must note that
Lifshitz uses T as defined by W. C. Sabine, while P. E. Sabine uses Ti for his latest
data.

2 Some of these improvements are described by J. P. Maxfield and H. C. Harrison, in
a paper on High Quality Recording and Reproducing of Music and Speech, J.A.LE.E.)
March, 1926, p. 243.

2i 4 THEORY OF VIBRATING SYSTEMS AND SOUND

has been accomplished to a notable degree. These improve-
ments bring forcibly to the attention certain requirements
from the standpoint of Architectural Acoustics in the making
and use of such records, in order to obtain the illusion of exact
reproduction. Clearly the best result is obtained if, when the
record is reproduced, the inherent quality in the record and the
accompanying effect due to the acoustics of the listening room
exactly duplicate the effect that would have been obtained if
the original sound had been produced in the regular way in a
room of optimum reverberation time. The most straight-
forward way of insuring this result is to record the original
sound with no accompanying reverberation whatsoever, and
then to play the record (which is by hypothesis a faithful copy
of the original sound) in a room of exactly the acoustic
properties demanded for best appreciation of the original
sound itself. This is a counsel of perfection, and of course
assumes that the user of the phonograph is prepared to take
some care in adjusting listening conditions. If this is not prac-
ticable, the alternative is for the record maker, at the cost of
considerable experimental work, to make less perfect records
which are, by a compromise of some kind, better adapted to
universal use under average listening conditions. The point
we wish to make is that the application of acoustic principles
both to recording and to reproduction is now appreciated by
the phonograph makers; every effort is made to control record-
ing conditions, and insure a good copy of the original sound;
but, granting all this, the best results will not be obtained in
final reproduction, without intelligent cooperation on the part
of the user of the phonograph, in adjusting the loudness of the
sound and the reverberation time of the listening room.

55. Standing Wave Systems; Focal Properties of an Enclosure;
Acoustic Difficulties

The most unsatisfactory feature of the acoustics of a closed
space (equally from the standpoint of theory and of practice)
is the inevitable system of standing waves. If the walls have an
absorption coefficient of unity, the problem becomes that of an

STANDING WAVE SYSTEMS 215

open space; but reflection is inherent in Architectural Acous-
tics: in practice the various wall surfaces and wall coverings
which are available have absorption coefficients which on the
average are far from unity. The absorption coefficients of
brick and plaster walls, are for example, 2 or 3 per cent (W.
C. Sabine, loc. cit. in 51); a carpeted floor may have a coeffi-
cient of 20 per cent; a good grade of acoustic felt, when laid
next to the wall, 70 per cent at 1000 cycles (Sabine, "Collected
Papers," p. 158); the most extreme value is probably that for
the audience, which is given by Sabine as 0.96 in square

meter units. Thus the over-all absorption coefficient ( A = - )

is not likely to be large in any practical case. To take an
example, suppose a cubical room of volume 1000 cu. m. has a
reverberation time of 1.6 seconds, which would probably fit it
for musical purposes; according to equation (280) we should
have aT = .16^ which gives a = 100. Since S = 600 for such

a room we have A = - - = . Hence, on the average, it

would require four reflections to diminish the amplitude of any
given wave, as generated, to 49 per cent of its original value.
We infer from this that in any enclosure whose "acoustics are
good" there will be a sufficient number of multiple reflections
to produce standing waves.

Standing wave systems, while easily analyzed in the one-
dimensional tube problem which we have frequently studied,
are so difficult of analysis in any practical three-dimensional
case that they are virtually incalculable. In order to appre-
ciate the complexity of the phenomena encountered in prob-
lems of this kind, we shall consider a rather academic example,
namely, the steady state wave system corresponding to a small
source of sound at one of the foci of an ellipsoidal enclosure
having a perfectly reflecting boundary. The problem is sev-
eral degrees removed from those occurring in practice; rooms do
not often have clearly defined sound foci, and if they do, the
source of sound is not always located there; rooms never have
perfectly reflecting boundaries, nor unbroken walls of regular

216 THEORY OF VIBRATING SYSTEMS AND SOUND

shape; consequently such conclusions as we shall draw will be
rather inductive. But this example is one of the few which
permits of analysis, and is not without a certain interest; there
is at least one large audience hall in existence which has nearly
this shape.

A section through the foci of the ellipsoid is shown in Fig.
22. To simplify matters the major axis (la) and the distance
between foci (id) are each taken equal to an integral number

FIG. 22. GEOMETRY OF THE ELLIPSE.

of wave lengths, e.g., la = 8X, and id = 5\. No great sac-
rifice in generality is involved in the particular dimensions
chosen. The quantities a and d are connected by the relation

in which ib is the minor axis. From the geometry of the
ellipse, the excess pressure at a given point P = (x y y) in the
interior will be the sum of the excess pressures inherent in two
sets of converging and diverging waves, originating from a
source placed at F\. One set of waves may be said to radiate
from the focus Fi along the lines marked A to P; in this set
FiP is the radius of a divergent wave, and F^A^P the radius

STANDING WAVE SYSTEMS 217

of a convergent wave, if (as in the figure) F?AiP > F\P. The
length la of the path F\A\F\ traversed by the wave through
p2 before entering on the path F^A^P need not be taken into
account, as this length is 8\ and involves no phase change; the
result at P is as if there were two equal sources, in exact phase,
located at F\ and F 2 . In other words F* is the (positive) image
of Fiy to borrow the familiar analogy from electrostatics, or
from the theory of mirrors. Similar conditions hold regarding
the paths of the other set of waves, which are marked B.

Now in either set of waves meeting at any point P y the
radius of curvature of -the convergent and divergent compo-
nents is the same, hence the maximum values of the amplitudes
are identical, and only periodic factors (phases) need be con-
sidered. Constructive interference (i.e. maxima of excess pres-
sure) in set A will take place whenever F^A^P = F\P + k\ y
k being an integer. The loci of all such points are circles about

P\ as a center, whose radii are , if m is integral. These

circles are one set of those represented by the lighter lines in
Fig. 23. In addition to this set of circles of maximum excess
pressure, there is superimposed another set of such circles,
centered at F\ and similarly derived from the paths B y which
also are traversed by the sound waves arriving at P. The
crosses which indicate the intersections between the circles of
the two sets therefore mark points where the excess pressure
is greater than at other points on these circles; for here the
pressure maxima in the two systems are additive, since they
are in phase. (These points will describe circles normal to the
major axis, when the section shown is rotated to generate the
ellipsoid.) The regions bounded by circular arcs between the
crosses, and containing dashed lines, represent areas of pressure
minima, and will develop into rings when the section is rotated
about the major axis.

Other interesting properties relate to the general pattern
of the standing wave system, and the relative values of pres-
sure and velocity maxima at different points. From one stand-
point the crosses are, from their construction, located on con-

218 THEORY OF VIBRATING SYSTEMS AND SOUND

focal ellipses, as shown by the dashed lines; or equally, they are
located on the corresponding system of confocal hyperbolas,
some of which are drawn in. Thus the complete system of
regions of maximum pressure may be defined as the circles
(and the points on the major axis) which are the intersections
of the system of confocal ellipsoids and hyperboloids so spaced
that there is a change of phase of a half wave length in passing

FIG. 23. SECTION OF THE STANDING WAVE-SYSTEM IN AN ELLIPSOID FOR WHICH

2</=5X, 2 = 8X.

from one surface to the next. These maxima of pressure are
not all of equal intensity or energy density; if we exclude the
foci from the discussion, the two crosses marked M in the
diagram are the points of most enhanced excess pressure in the
whole standing wave system. (This can easily be shown by
the reader, and is suggested as a problem at the end of the
chapter.) The regions of maximum particle velocity fall in the
center of the diamond-shaped and lens-shaped elements de-

FOCAL PROPERTIES 219

fined by the intersecting circles. In those diamond-shaped
elements through which the hyperbolas are drawn, the result-
ant velocity at the center of the element is parallel to the hyper-
bola, that is, normal to the corresponding ellipsoidal surface;
while in the remaining elements the resultant velocities are
parallel to the ellipsoidal surfaces; thus there is no normal
component of velocity at the boundary. We may observe,
however, that in any practical case it is usually sufficient to
determine the distribution of pressure maxima, since pressure
driven instruments, such as diaphragms or the ear, are most
likely to be used for the purpose of detecting the sound. Finally
we note that if in any enclosure a point F* can be found which
is the image of another point F\ the enclosure is a true whis-
pering gallery l that is, all the reflections conspire to make the
energy density greatest at Fz if a source of sound is placed at
FI. It is unnecessary to emphasize further the application of
the method of images to the study of sound reflection from
curved surfaces which are so regular that they have focal
properties.

It is evident that altering the size or the proportions of the
ellipsoid will make no essential change in the general distri-
bution of the standing wave system assuming that the rela-
tions between the different parameters are such that a stand-
ing wave system is possible at the given frequency. But any
change that is made to simulate practical conditions, such as

1 For a concrete illustration we may refer to the focal properties of the Great Mor-
mon Tabernacle at Salt Lake City. In general, the interior is shaped like a short ver-
tical cylinder of nearly elliptical cross-section, surmounted by a domed roof which is
a little too shallow to be half of the corresponding ellipsoid of revolution. In spite of
these deviations from an exactly ellipsoidal shape, the enclosure possesses two conju-
gate foci, and at one of these the speaker's desk is placed. The listening conditions are
very unequal in different parts of the enclosure, unless a large audience is present; the
best listening point is known to be at the edge of a balcony, in the end of the building
opposite the speaker's desk. This is the conjugate focus; here faint sounds (such as
the drop of a pin) originating at the speaker's desk can be heard with ease: without
reflections from the walls this would be impossible, owing to the great distance between
the foci.

A description and photographs of the building are given in W. C. Sabine's paper on
Whispering Galleries, previously cited.

220 THEORY OF VIBRATING SYSTEMS AND SOUND

for example, making the source of finite size, placing the
source out of the focus, or introducing irregularities in the
boundary surface will tend to complicate matters so that
theory becomes ineffective; it must be emphasized that we
have introduced every conceivable geometrical simplification,
in order to obtain a theoretical solution. (Even the problem
of determining the effect of a uniform lining of absorbing mate-
rial at the boundary has its difficulties, though it can be solved.)
Since a priori calculations are impracticable in dealing with
most standing wave systems, we must as a rule be reconciled
to a series of experimental surveys of the given enclosure, one
for each frequency for which the information as to distribution
is desired. The experimental procedure is not particularly
difficult, and can easily be imagined by the reader; the labor
involved is another matter. Astonishing complexities will
often be revealed by such an investigation; see for example
the result of a survey of a certain room made by W. C. Sabine,
shown in Fig. 12, p. 152 of the "Collected Papers." 1 For the bene-
fit of the reader who does not have access to this diagram, it
need only be stated that it is made up of contour lines, showing
the loci of points of equal energy density, which have every
conceivable size, shape and orientation. The only evidence of
regularity in the diagram is a certain symmetry about the
longitudinal and transverse axes of the room; this effect is
natural enough since the source of sound was placed at the
center of the room.

In listening to sounds in an enclosure it must be realized
that there will be, depending on the frequency, many possible
distributions of standing waves. If the sound is at all complex,
a number of these patterns are superimposed, and since the
maxima and minima for sounds of different frequencies will
never exactly coincide, the result will be what we have called
local wave distortion ( 50). In listening to a sustained sound
no matter how well damped the enclosure may seem to be (by

J The Correction of Acoustical Difficulties, Arch. Quarterly of Harvard Univ.,
March 1912. The diagram also appears as Fig. 8 in the paper Architectural Acoustics,
Jour. Fr. Inst. t Jan. 1915, which summarizes much of Sabine's work.

ACOUSTIC DIFFICULTIES 221

a reverberation test), it is easy to get a different impression
of the sound by merely shifting the head a short distance. If
the sound is a pure tone, the maxima and minima of the stand-
ing waves will be in evidence; if the sound is complex, a consid-
erable difference in quality will be noted. If the character of
the sound changes as in music or speech, then the standing
wave patterns, during the course of the reverberations, are in
a state of flux, thus introducing further complications. In lis-
tening to concerts in a given hall, it may be found by experience
that certain positions are to be avoided, because of excessive
local wave distortion, obstructions, or what not; about the
only generalization that can be given is that, if the hall is
crowded, one should not be too far from the source of sound.
We may point out, however, that if the two ears could be
located on the surface of one of the reflecting walls of the
auditorium, the listener would stand a better chance of correct
audition than in the open space of the enclosure, because a re-
flecting surface is a unique locus of maxima of pressure (veloc-
ity nodes) for all frequencies. Not only should we expect a more
representative frequency distribution of excess pressure at the
average point on such a surface, but we should expect less varia-
tion in the frequency distribution, from point to point on the
wall surface, than from point to point within the enclosure.

The most notorious cases of acoustical difficulty, which
experts have been called on to correct, have involved both
excessive reverberation and aggravated local wave distortion,
focal properties, or echoes. These latter effects of course result
in a very unequal distribution of energy density in the enclo-
sure. The first course of treatment is obviously to add absorb-
ing material; to be most effective this should be located near
the regions of greatest energy density. If these regions are
near regularly curved wall surfaces, a twofold advantage is
often gained by placing the absorbing material on such sur-
faces. If the result is still unsatisfactory after adding all the
damping the room will stand (without too great a decrease
in reverberation), the regularity of the offending reflecting sur-
faces should next be modified so as to substitute wave scattering

222 THEORY OF VIBRATING SYSTEMS AND SOUND

for regular reflections. This may be done by coffering (as in a
domed ceiling) or by the introduction of obstacles (such as
large chandeliers) suitably disposed; with the aid of such expe-
dients many originally poor auditoriums have been made
serviceable.

For the complete elimination of standing waves, in acoustic
experiments, it is possible to employ devices which would not
be practicable in an auditorium. The most radical, and prob-
ably the most effective device for breaking up the standing
wave system, and so insuring a uniform energy density in an
enclosure is that used in the W. C. Sabine Laboratory, River-
bank, Geneva, Illinois. This consists of a large steel baffle, or
reflecting surface which is slowly and silently rotated about a
vertical axis in the room. 1 On the other hand, if it is desired
merely to measure the mean energy density produced in a
room by a given source under steady state conditions, another
expedient may be resorted to; the detector may be placed on
a rotating arm, so that the mean reading of the detector will
measure the mean energy density around the path traversed.

56. The Reaction of an Enclosure on a Source of Sound

It remains to consider the reaction of the enclosure on a
sound generating apparatus working in it. The key to this
reaction is the impedance of the enclosure, at the point where
it is driven; this depends on the state of the standing wave
system there. In the mean energy density calculations of 53
and 54 it was not necessary to consider the phase of the wave
motion, because the rate of working of the source was fixed; but
in general to fix the rate of radiation of the source, which is
important in physical measurements, the driving point impe-
dance, and its variation from point to point in the enclosure
must be determined.

Sabine describes an interesting experiment 2 in which an
apparently paradoxical result was obtained. The source of

1 For a detailed description, see Set. Am.> Sept. 1923, pp. 154-155.

2 W. C. Sabine, "Collected Papers," Appendix, p. 278.

REACTION ON THE SOURCE 223

sound was a diaphragm vibrating with prescribed amplitude
at the base of a resonating chamber. The emitted sound was
measured when the source was set up in a certain position in
the room. The room was then strongly damped by placing
felt on the floor, and the emitted sound again measured; it was
found to be eight times louder than the original value, for the
same amplitude of motion of the diaphragm. The explanation
is of course that the introduction of the damping material
changed the distribution of the standing wave system, and that
whereas, in the first experiment, the diaphragm was located at
a point of minimum pressure (or loop of particle velocity), in
the second experiment, it was located at a point of maximum
excess pressure, and was able to do more work, for a given
amplitude of motion, on the surrounding medium.

This explanation can be made more precise if we again apply
the familiar theory of the simplest acoustic system: a tube of
length / closed at the distant end with a piston of arbitrary
impedance. In the discussion following equation (1310) (32)
we have shown that the condition for maximum power trans-
mission from the driving piston is that in which the tube, by
virtue of its length, contributes no reactance to the transfer
impedance, and that in this adjustment, the impedance of the
distant piston is transferred to the driving point. From (130)
the driving point impedance is

Zj cos 01 + ipc sin 01 ( .

Z 00 = PC T, ~7 -.-. - f (2GX)

pc cos ftl + iZi sin 01

and this becomes Z 00 = Z, L if sin $1 = o, that is, if cos /?/ = db i,
or / = , k being integral. With this virtual transfer of the

distant piston to the driving point, if we let it represent an
absorbing boundary, for example, the conditions are those of
maximum dissipation at the boundary for a given motion at the
source. Again, if the distant piston is to simulate a rigid wall,
and we are interested only in increasing the energy density in
the tube, using a source of prescribed velocity, the mechanism
is that of the Kundt experiment, and the phenomena are as

224 THEORY OF VIBRATING SYSTEMS AND SOUND

stated in eq. (135). We now have a maximum energy density

k\
in the tube, for the same condition as before, namely / = ---;

that is, the energy density is a maximum if the driving piston
is located at a maximum pressure point, since the pressure is a
maximum at the distant end.

In certain experiments made in the past, it has been assumed
that if the source worked at a prescribed amplitude it was cer-
tain to produce, irrespective of its location, a definite energy
density in a given space. But if, in the two cases we have con-
sidered, the sources were located at pressure minima, instead
of pressure maxima, the energy density, and the absorption of
power at the boundary, would have been very much reduced;
hence the fallacy of ignoring the relative position of the source.
Moreover we observe that for any chance disposition of the
source, an output more independent of location is likely to
result if the source, instead of being driven at constant ampli-
tude, is driven with constant force on the moving element. In
this case the amplitude of motion at the source will be restricted
partly by its internal impedance, and partly by the driving
point impedance of the enclosure, with the result that there
is less variation in radiating power as its position is shifted.
(These statements can best be illustrated by a numerical ex-
ample, such as that suggested in problem 49 below.) But we
can never eliminate the point to point variation in the impe-
dance against which the source is made to work, and due allow-
ance must be made for this in all acoustical measurements in
enclosures.

It is obviously impossible to arrange the members of a choir
or orchestra so that each individual source of sound is correctly
placed, with respect to the concert hall, for every frequency of
tone emitted. On the average, we should expect one compact
arrangement of independent sources to be as good as another,
assuming that the acoustics of the auditorium are tolerable
and that any exaggerated directive effects are eliminated. The
conventional arrangement of the orchestra, with the more
powerful instruments in the rear, seems to be justified princi-

REACTION ON THE SOURCE 225

pally from the standpoint of the conductor, whose judgment
and skill in controlling the various parts are aided by having
the basic nucleus of the orchestra (the strings) nearest him. In
addition, this arrangement gives a good perspective view of
the ensemble. But both of these are artistic rather than phys-
ical considerations; there is no basic acoustic reason for the
accepted arrangement. For one detail of orientation some
excuse may be found, on scientific grounds; the horns are sup-
posed to sound better (i.e., to emit tones which sound more
mellow) if their bells are not pointed directly at the audience.
The sounds they emit are rich in shrill, high-frequency compo-
nents, so that the source (that is the bell of the horn) is some-
what more directive for these higher harmonics. The effect
may become pronounced if a number of horns are played to-
gether with the bells in line; the directive effect normal to such
a line of sources increases with the length of the line. But in
general, in a large room, in which reverberation is taking placCj
directiveness counts for little: subject always to the consider-
ation that as reverberation becomes diminished by added damp-
ing, and conditions approximate more closely to those of open
space, any directiveness (orits opposite, divergence) inherent in a
particular aggregation of sources will become more and more
noticeable. What is best regarding the disposition of an indi-
vidual instrument can be found by trial; the result is to be
judged according to the canons of musical taste. In any event
the points mentioned here are secondary, from the standpoint
of the listener, to the matter of choosing a good listening point
in the auditorium.

Here we may well conclude the argument. Throughout,
the emphasis has been placed on the physical principles in-
volved; all the problems considered have been chosen with
this end in view. It is not to be expected that the reader whose
interest lies in Applied Acoustics will be satisfied with the lim-
ited applications of the theory which have been made; nor on
the other hand, will the mathematically inclined reader be dis-

226 THEORY OF VIBRATING SYSTEMS AND SOUND

posed to accept as complete, in every case, the calculations
given. To the critical reader of either class, we may observe
that the solution of his favorite problem is best left to his own
particular devices; and these, of course, must be based on the
well-worn and enduring principles of the classical Theory of
Sound.

PROBLEMS

41. In Taylor's experiment (p. 109) a layer of sound absorbing
material 2 cm. thick reduces the amplitude of the standing waves by
50 per cent (as compared with the amplitudes obtained when the
material is removed from the end of the tube) and a layer 4 cm. thick
reduces the amplitude by 60 per cent. A further increase in the thick-
ness of the felt produces virtually no effect. What thickness of this
material would be required to reduce a transmitted wave to .01 of its
original intensity? Certain approximations may be necessary in
the solution; note that the reflecting power of the felt is a negative
quantity.

42. From Figs. 20 and 21 you can derive the fundamental con-
stants for a certain kind of felt at any frequency within a certain
range. A layer of this material 2 cm. thick, when placed on a hard
wall, absorbs 50 per cent of the incident energy, at 512 cycles. What
is the absorption coefficient of the layer at the same frequency when
it is separated from the wall by 5 cm. of air space?

43. A lecture room whose walls and ceiling are of wood (pine) is
12 meters long, 5 meters high, and 8 meters wide. There are four
windows on each side, each of area 2 square meters. The room con-
tains 100 chairs for the audience, the floor is covered with a carpet.
What is the reverberation time T for the room, with half the audience
present, the windows being closed? How is this changed when all
the audience is present, and the windows are halfway open, i.e., with
all the sash raised? Use the following coefficients in computation:

ABSORBING POWER DATA. (W. C. SABINE)
(In Square Meter Units)

Pine wood surface, per sq. m., .06 Chairs, each 01

Carpet on Floor, per sq. m., .20 Audience, per person 44

Window Glass, per sq. m., .03

PROBLEMS 227

44. Two adjoining rooms of volumes V\^ V^ energy density EI
2, and area-absorption coefficients ai, 02, are separated by a partition
of area ?, and of energy transmission coefficient k per unit area. A
source whose energy rate is K is working in V\. What are the maxi-
mum steady state values of E\ and 2? Show that, under certain
conditions, the constant k can be determined from the relation

if Ti y T2 are the respective reverberation times of V\^ V%. If there is
no reverberation in /^ what is the intensity of the sound on emerg-
ing from the partition? (A. H. Davis, Phil. Mag., 50, July, 1925,

P- 75-)

45. Derive an algebraic expression for the reverberation time of
a spherical enclosure lined with absorbing material, assuming that
the original distribution of energy density is that of the steady state
produced by a constant small source placed at the center.

46. A certain auditorium contains 20,000 cu. m., with seating
capacity of 3,000. With all the seats occupied, such a hall might
possibly have a total absorption coefficient (W. C. Sabine) of 2 X io 3
sq. m. units. Is the room acoustically good?

A quarter note is sounded in this auditorium by an instrument
working at the rate of io 10 threshold units of energy per second, and
the note is maintained for 0.5 second by the player. W 7 hat is the
average energy density at the end of the note, in terms of the thresh-
old level? What is the time of decay (Ti) to the threshold level?

47. In the ellipsoid of 55 show that the energy density is greatest
at the points of maximum pressure nearest the foci.

48. In Wente's experiment (p. 109) find the relation between the
maximum and minimum driving-point impedances of the tube, and
the reflection and absorption coefficients of the layer at the end of
the tube.

49. A piston whose impedance is a R (a pure resistance) is used
to drive one end of a tube of air, of unit section, R being the radia-
tion resistance of air. The other end of the tube is closed with a layer
of absorbing material whose impedance is also a R. Two adjustments

of tube length are made: (i) / = > and (2), / = Show

228 THEORY OF VIBRATING SYSTEMS AND SOUND

that, if the piston is driven at a prescribed velocity in both adjust-
ments, the relative rates of radiation are as a 2 : i. Show also that if
the piston is driven with a prescribed force in both cases, the relative
rates of radiation are as

4 +i
NOTE. In all practical cases a is likely to be greater than unity.

50. What is the driving point impedance of a spherical enclosure,
if a small spherical source is placed at the center?

APPENDIX A

RESISTANCE COEFFICIENTS FOR CYLINDRICAL CONDUITS

The viscosity of a substance is measured by the tan-
gential force on unit area of either of two horizontal
planes of indefinite extent at unit distance apart, one of
which is fixed, while the other moves with unit velocity,
the space between being filled with the viscous substance ."

MAXWELL'S DEFINITION.

A fuller consideration may be worth while of the phenom-
ena of resistance to fluid motion in tubes of circular section,
which effects have been rather casually treated in the text.
Two cases must be distinguished, according as the section of
the tube is effectively small or of moderate size, the classifica-
tion depending not only on the relation of the constants of the
fluid to the actual width of the tube, but also on the frequency
at which the fluid is driven. As will appear, the discriminant
between the two cases is the quantity

/ /pco

r\ >

* II

in which r is the radius of the tube. If | kr \ is not greater than
unity, the tube is effectively a " narrow" tube; the reaction due
to the inertia of the fluid is much less than the frictional resist-
ance, and ratio of the driving force to the mean velocity over

the section is Poiseuille's Coefficient ( R = 8 - ), neglecting

the inertia component. In this case there is lamellar motion
throughout the section, the velocity varying from zero at the
wall of the tube to a maximum at the center: this condition is

229

230 THEORY OF VIBRATING SYSTEMS AND SOUND

illustrated roughly in Fig. 240. Theoretically the situation is
analogous to that in a vibrating system when both mass and
damping are adding to the system; but the damping or choking
effect in the tube is so great that compressional waves are trans-
mitted only with great difficulty, and the resemblance is to an
aperiodic system.

In 32 the problem of wave transmission was treated on the
general assumption that some resistance coefficient RI was
applicable which was small compared with pco. For the steady

(a)

FIG. 24. LAMELLAR MOTION OF FLUID IN SMALL AND MODERATELY LARGE TUBES.

flow of liquids in pipes, it is doubtless sound enough to apply
Poiseuille's Coefficient; but for pipes of such sizes as are likely
to be used for sound transmission, as for example in the Con-
stantinesco scheme, it would seem that the inertia of the fluid
plays such a dominant role that Poiseuille's law no longer
applies.

It must be admitted that here, as in certain other hydro-
dynamical problems, the reconciliation of theory and practice
is by no means complete and it may well be that empirically
determined coefficients are necessary in order to deal accurately

RESISTANCE COEFFICIENTS FOR CONDUITS 231

with this situation. It would be well, for example, if the upper
frequency limit for which Poiseuille's coefficient is valid could
be established experimentally; the ease of computation accord-
ing to a simple formula of this sort would be of advantage in
many cases. Unfortunately reliable data of this kind are not
available.

It happens that in most problems involving sound trans-
mission in tubes, the factor \kr\ is large as compared with unity;
in other words the tube is effectively " large" and does not
choke or damp the oscillations to a very great degree. The
situation is then analogous to a mildly-damped vibrating
system, the effect of the damping being to decrease the velocity
of propagation by a small quantity of the second order. This
defect in velocity was computed by Helmholtz, whose solution
of the problem we shall follow. There have been numerous
experimental tests, which we shall refer to later, of the Helm-
holtz formula; unfortunately these are not concordant, nor do
they fully support the simple theory. The purpose of the fol-
lowing discussion is therefore limited to placing before the
reader the best available idea of the mechanism of the losses
due to viscosity, and the essential difference between the
effects in wide and in narrow tubes.

The motion in the case of the wider tube is roughly as shown
in Fig. 24^. Owing to the oscillation of the fluid along the axis
of the tube, viscosity waves are diffused radially from within
the fluid (where the velocity is greatest) toward the walls of
the tube, where the velocity is nil. It is a peculiar property of
diffusion waves that they are virtually extinguished after tra-
versing a distance of one wave length. One result of this is
that the effect of viscosity, for sound waves in tubes wider than
(say) double this wave length, is confined to a shell of thickness
approximately one wave length next to the wall of the tube.
There remains a region in the center of the tube in which the
dragging effect is practically absent, and in which the axial
velocity at any point does not vary greatly with its distance
from the center. The conception is then of a cylindrical core
of air, oscillating as a unit in the center of the tube, and

THEORY OF VIBRATING SYSTEMS AND SOUND

impeded in its motion by the reactions which take place in a
thin layer of fluid between its cylindrical boundary and the
inner wall of the tube. These reactions involve both added
inertia and resistance as can be seen from the following con-
siderations.

Let an infinite plane wall oscillate in its own plane in con-
tact with the fluid, with the result that viscosity waves are
diffused in the X direction, normal to the oscillating plane.
The force due to the inertia of an element of fluid of unit area
and thickness dx is p\dx. The net force on the element due to

shearing stress is (by definition of //) -- ( /* )dx t the

"Qx \ fix/

negative velocity gradient being used because of the falling
away in velocity with increasing x. The equation of motion is
therefore

and if x = <* ~ **> we have

ft = v ; since V / =

V~2

The solution is therefore, for waves in the positive direction,

and X = = 27r\/ This is the distance to which the

transverse vibrations are of any consequence: fore"^ = ~ 2ir ,
a very small quantity. The reaction R% on the driving wall is,
per unit area, the product of the viscosity and the velocity
gradient, that is, dropping the exponential factor,

a*,.,

which gives a resistance coefficient R = nft (i + /'). We note
that the imaginary part of this expression is in phase with the

RESISTANCE COEFFICIENTS FOR CONDUITS 233

acceleration of the driving surface, and is therefore of the nature
of a mass reactance; it is evident, in applying this to the tube
problem that there will be a slowing down of the propagation
velocity of the sound waves in the tube, because of the mass
reactance on the oscillating core of the medium in the tube, the
effect of which is virtually to increase its density. This corre-
sponds to the lowering of the natural frequency of a vibrating
system by loading it with added mass, and is a larger effect
than that due to the added damping.

The total force on unit length of the surface of the core of
moving gas in the tube is ijrr.R; so that, per unit sectional
area of the tube the resistance coefficient is

Rl = .R = (, + 0,

In applying (C), obtained for plane waves, to the surface of a
cylinder, and so deriving equation (D), the curvature of the
cylindrical surface has been neglected. This is legitimate, for
the thickness X of the region between the oscillating core, and
the wall of the tube is relatively small as compared to the radius
of the tube in which these effects are supposed to take place.
(For example X = .6 mm. for a frequency of 500 in air.)

The results obtained above are confirmed by a more general
treatment, which takes into account the cylindrical structure
of the tube, and the whole range of driving frequencies; and in
addition an insight is gained into the meaning of the criterion
first given, which depends on the magnitude \kr\ for differen-
tiating between the two types of resistance phenomena.

We proceed as in 10, except that the circular section (ar,, 2 )
of fluid is substituted for the circular membrane, the axial
driving force now being *&-dx per unit area; * is of course the
negative pressure gradient parallel to the axis of the tube.
The total driving force on an annular ring of fluid of vol-
ume iirrdr-dx is Vdx-iirrdr; this is opposed by a reactance
x due to inertia. The opposing force due to friction

on the inner surface of the ring is 2irrdx-fj. y using the

234 THEORY OF VIBRATING SYSTEMS AND SOUND

negative velocity gradient as before, because of decrease of
with increasing r. The net force on the annulus due to friction
is therefore

3/ . 3\,

I 27ir ax u }dr*
9r\ M Sr/ '

and hence the equation of motion

h -$)]<- <

in which only \ is a function of r. This may be written

[+^+4~* (*--?) <*>

the solution being

k = - + 4J (kr\ (F)

for finite velocity when r o. [Cf. eq. (32.)] Exactly as before
(for the membrane) the velocity must vanish at the boundary,
r = r ; determining A we have

Integrating \(r) over the section, we have for the mean velocity

that is

according to the method of 12, eq. (47). The equation is now

in the form = and it only remains to discuss the values

taken by R as a function of frequency.

RESISTANCE COEFFICIENTS FOR CONDUITS 235
If \kr\ is not greater than unity we may take

and

v 2 x 4
Jo(kr ) = /(*) = i - + --,

4 t>4

Ji(kr ) = /!(*) =

(cf. 25)

192

using only 3 terms of these well known series expansions. We
then have,

*/.(*)

X 2 X*

X 2

T __

Thus (G) becomes

and

since P =

The pure resistance term in this equation is Poiseuillis Co-
efficient, R\ = ; the reactance -/o>p represents a increase

. r . ^

in effective mass or density due to the diffusion effects in the

narrow tube, as compared with the reactance ipw in the un-
limited medium. The inertia component is of little importance,
i.e., if \kr \ = i> the inertia reactance is only of the resistance
factor: the whole behavior of the tube as a wave-transmitting
system is profoundly modified, and in fact approximates to an

236 THEORY OF VIBRATING SYSTEMS AND SOUND

aperiodic system, as has been previously stated. The equation
of wave motion, in terms of mean velocity, is

which becomes, if we neglect the reactance
^ = <*?*. <* = -',

This is in the same form as eq. (A), and solving in the same
manner we have

$= { *-<V<- (')

in which

I /co/i co / 4^1 co 2

~~ c^ 2p ~ c ^pr a " f |r | 2 "

The phase velocity is now

\kr I 2
,' = f 1 *^-, (L)

in which c~ is as usual, the unmodified velocity of sound. It is
evident at once that the phase velocity is very greatly reduced,
this effect being inevitable as the sequel to the high damping
and attenuation in the narrow tube.

To evaluate R for values of \kr \ in the range T < \kr \ < 10
it is necessary to make use of the relation

/<>(# V /') = ber x + i bei #,

in much the same way as was done in 1 1 to solve the air
damping problem. We shall not enter into these rather prolix
calculations because in practice the values of R in this fre-
quency range are not of much concern. If r is i cm. and
co > 20, \kr \ > 12 (for example) in air; thus for all acoustic
frequencies we can rigorously apply certain simple formulae to
evaluate the Bessel's functions, whose arguments are then com-
plex variables of large absolute value.

RESISTANCE COEFFICIENTS FOR CONDUITS 237
From (G) we have

1 = _ _L [ T _ JL Zi
R rf*L kr J

R
to which we apply the relation (cf. Jahnke and Emde, p. 101),

Ji(xV-i) . ,/--- ....

-, / -- = - *, xV-t = r ----- , (M)
J (xV- t)

which is very nearly true for x > 10. We then have
i i

D /o [ ^'1 [

# = - )U^ 2 I - T- = /PCO I

L ArJ L

This reduces to

R = h + (i + i), ^

the second term of which confirms the result previously ob-
tained for the large tube [cf. eq. (D)]. The inertia reactance ipw
is the normal inertia effect present in any case, if there are no
viscosity effects. The principal interest in this solution of the

resistance problem lies in the additional reactance due to

r r

the combined viscosity and inertia effects in the tube of mod-
erate width; this gives rise to a second order effect of dimin-
ished propagation velocity.

The equation of wave motion is now, since / = ~>

or, say,

238 THEORY OF VIBRATING SYSTEMS AND SOUND

Solving this equation exactly as before ( 32, p. 96-97), we
have the following relations:

(P)

in which it will be noted that

, /7 v7 / M\

C = \/ = ..= r^ = H I )j

x p / / 2ufl\ \ wpr/
i +

approximately. Remembering that = ^ [eq. (B)] we may
summarize the results in terms of the simpler constants thus:

2 / /i

P V = pc( i + - vrr: )> K < = = - r v 2 ^^,

= - -7-7 = \/ > approximately.

As an example of the magnitude of the attenuation effect,
according to the Helmholtz formula, let the radius of the tube
be 5 mm., and w = 6000, in air. The change in velocity is
c c' = .007 r; and a' = 1.3 X io~ 3 . It is evident therefore
that in many problems these effects are so small that we are
justified in neglecting them.

These calculations of course ignore the loss due to heat con-
duction to the wall of the tube, which is a more difficult effect
to deal with, though the calculations are not dissimilar. The
diffusivity of a gas is a coefficient of the same order of magnitude
as the kinematic viscosity, hence the effect of losses due to heat
conduction will be equivalent to a sensible increase HI the
kinematic viscosity, in the general case. In the special case of a

RESISTANCE COEFFICIENTS FOR CONDUITS 239

narrow metal tube, if there were no viscosity to contend with,
the volume changes in the gas would take place isothermally,
just as we have pointed out in 1 1 when calculating the pres-
sure changes in a thin film of air bounded by metal surfaces.
In such a case the velocity of propagation would become in the

limit, that of the Newtonian formula, c 2 = But we have

already seen [eq. (L)] that in an arrow tube, on account of vis-
cosity alone, the phase velocity is very greatly reduced. In
this particular case the effect of viscosity predominates.

According to the extended treatment of Kirchoff's con-
tribution to the theory (Rayleigh, II, 348), which takes
account of the effects of heat conduction, we should add to the

factor v = - a factor (a-, say) which represents the "diffusivity"
P

or " temperature conductivity" of the gas. If we write
( 7 ')2 v -|_ ^ then we have for the modified velocity of sound
in the tube of moderate width

on which basis most investigators have reduced their results.
For air at normal temperature and pressure we should have
v == .13, 0- = .17^ and hence 7' = .54, in c.g.s. units.

We may finally discuss some of the experimental tests that
have been made of the Helmholtz-Kirchoff formula; the older
work is well reviewed in Barton's " Textbook of Sound,"
Chap. X (516-529). Following the work of Kundt (who
first observed the defect in velocity, in a tube), Schneebeli
(1869) and A. Seebeck (1870) experimented with tubes of
various diameters, using sound waves of somewhat limited
frequency range. From these experiments it appeared that

the defect in velocity varied as --> in accordance with the for-

mula, .but it varied with frequency as -p not as ,--. Kaiser,

w v co

2 4 o THEORY OF VIBRATING SYSTEMS AND SOUND

later, was enabled to reconcile his tests with the formula pro-
vided p, were given a large value: a proposition not inconsis-
tent with increasing /* to take account of losses due to both heat
conduction and viscosity. Blaikley (1884), after very careful
tests at a single frequency (256), using smooth trass tubes of
diameters from i to 9 cm., came to the conclusion that the
Helmholtz formula was correct as to variation of the effect in
accordance with the inverse diameter of the tube; the values
obtained also favored the view that the effect was inversely
proportional to V w. But in 1903 J. Muller concluded that the
Helmholtz formula was not valid, and that the speed of sound
waves in a tube depended among other things on the material
of the tube. (This is doubtless true, to a certain extent, as we
have pointed out in 32; but we should surely expect the
yielding of a thick metal wall to be a very small factor, for
sound waves in a gas within.) F. A. Schulze (1904) experi-
mented with very narrow tubes (ca. I mm. diameter), but we
should not expect these results to bear on the Helmholtz
formula, for reasons we have given.

In the work of E. H. Stevens, (Ann. d. PAys. y 7, 1902, p.
285) since he was interested in the velocity of sounder se, the
correction factor 7' was eliminated, by comparing measure-
ments made with tubes of various diameters a common
practice in work of this kind. He gives a discussion of the pre-
vious experimental studies of the Helmholtz-Kirchoff effect,
and obtained from his own measurements a value for 7' (air)
somewhat larger than that predicted by theory. But the impor-
tant point is that for tubes from 2 to 4 cm. in diameter, he found
7' constant, and hence adopted the Helmholtz-Kirchoff formula.
E. Griineisen and E. Merkel (Ann. d. Phys. y 66, 1921, p. 344)
in a similar study of the velocity of sound, obtained for 7' the
value 0.49 c.g.s. (air) as compared with the theoretical value
of 0.54 c.g.s.

From all these experiments it seems reasonable to conclude
that in some of the earlier work inconclusive results were obtained
through failure to control conditions so that a straightforward
test of the Helmholtz-Kirchoff Formula was possible. There

RESISTANCE COEFFICIENTS FOR CONDUITS 241

seems to be an opportunity for an extended series of measure-
ments on the transmission of sound waves of a wide range l of
frequency, in good smooth tubes. And until comprehensive
transmission data of this sort are available, it seems well to
abide by the outline of theory we have considered.

1 As bearing on the frequency range to be covered, we note an experimental study
of the transmission of very low frequency air waves in a long rubber tube, made by
Simmons and Johansen, Phil. Mag., 50, 1925, p. 553. The results show that the at-
tenuation in the tube is greater than that given by the theory; this may be due to the
yielding of the wall ( 32, note). The wave velocity for the low frequencies used (30
to 60 per. per min.) was somewhat less than the Newtonian value (that is, less than 278
meters per second). The observations quoted relate to a pipe 9.5 mm. in diameter;
the authors found that when the diameter was reduced, a very considerable reduction
in wave velocity took place.

APPENDIX B
RECENT DEVELOPMENTS IN APPLIED ACOUSTICS

To the reader interested in acoustic research or technology
the following notes 1 may supply such references as are needed to
supplement those given in the text, in order to get quickly into
touch with current practice. No specific references are added
to those already given in the discussion of (i) Acoustic Filters
( 27); (2) Tubes and Pipes (33, 43; Appendix A); (3) High
Frequency Underwater Sound Devices (41); (4) Horns (45,
46, 47) and (5) the Acoustic Radiometer (48). With the
exception of tubes and pipes, on which a great deal of older
acoustic work has been done, there is little further dependable
information to offer on these devices. Some supplementary
data on the condenser transmitter (cf. the air-damped vibrating
system, n) and on resonators (cf. 24) may be of interest, in
addition to the necessary introductory notes on other experi-
mental apparatus.

For a useful general bibliography of some comparatively
recent work, reference is first made to a Bulletin of the National
Research Council 2 which was prepared by a committee of
representative American Physicists in 1922. For economy of
space, most of the references to be given will concern work
published since that time.

Piezo-Electric Resonators

A brief discussion of a typical experiment will serve to
illustrate the contrast between the older technique and the

1 These have been prepared with the able assistance of Miss H. M. Craig, Research
Librarian of the Technical Library, Bell Telephone Laboratories. In addition to sup-
plying many of the references, she has taken every possible care to verify the entire
list.

2 Vol. 4, Part 5, Nov., 1922, " Certain Problems in Acoustics."

242

PIEZOELECTRIC RESONATORS 243

new; we select for the purpose Kundt's experiment, the theory
of which has been given in 32. For an account of the original
experiment we may turn to Barton's J " Text Book of Sound,"
(London, 1914), (p. 530), which contains a very comprehensive
Chapter (X) on Acoustic Determinations. It is of interest to
note that in 1868 Kundt, with his hand-operated rod and tube,
measured the relative velocities of sound in various fluids, and
determined the effect on the velocity of narrowing the tube,
which (as we have seen in Appendix A) decreases the velocity
more or less according to theHelmholtz formula. According to
most recent practice the resonant metal rod is equipped at the
pressure maximum (the node, or point of support, half way
along the length) with piezo-electric quartz crystals, and the
electro-mechanical system thus constituted is placed as a con-
denser in an electrical oscillator circuit. With this apparatus
W. G. Cady (Phys. Rev.,2i, 1923, p. 371 ;#/</., 23, 1924,^.558),
using electrically driven rods whose natural frequencies are
very accurately known, has provided a convenient and precise
method for determining the velocity of sound in tubes.

The theory of longitudinal vibrations of a viscous rod is
given by Cady in Phys. Rev., 19, 1922, p. i, and the theory
of the vibrating piezo-electric crystal follows in Proc. I.R.E.,
10, 1922, p. 83; the determination of the equivalent electric
network of the system is further treated by K. S. Van Dyke,
Phys. Rev. y 25, 1925, p. 895. There is also a paper by M. v. Laue
(Zeit.f. Phys., 34, 1925, p. 347) on piezo-electric oscillations in
quartz rods. The piezo-electric resonator is inherently stable
and permanent, and has been developed in various forms by
Cady as a frequency standard, with particular application to
radio frequency measurements; see Proc. I.R.E., 12, 1924, p.

1 The late E. H. Barton (1859-1925) a Fellow of the Royal Society, was a consistent
contributor to the literature of Acoustics. In collaboration with his students he wrote
many papers dealing with the performances of musical instruments, a subject which
he was specially qualified to discuss. These papers have appeared (interspersed with
his papers on resonance, and on coupled systems) for some years in the Philosophical
Magazine; the latest one will be found in vol. 50 (1925) p. 957, and deals characteris-
tically with the tones of the trumpet and the cornet. A biographical notice of Prof.
Barton appeared in Nature^ Nov. 7, 1925, p. 685.

244 THEORY OF VIBRATING SYSTEMS AND SOUND

805; also Jour. Opt. Soc. Am.> io, 1925, p. 475; also F. E. Nan-
carrow, P.O.E.E.J., J8, 1925, p. 168; also G. W. Pierce, Proc.
Am. Acad.y 59, No. 4, 1923, p. 81. The use of electrically con-
trolled tuning forks for standardizing frequency is described by
Horton, Ricker and Marrison, Trans. A.I.E.E.^ 42, 1923, p. 730.

Piezo-electric crystal oscillators have been applied by G.
W. Pierce (Proc. Am. Acad.> 60, No. 5, 1925, p. 271) to the
precision measurement of the velocity of sound over a range of
high frequencies (4 X io 4 to io 6 cycles) in air and in carbon
dioxide. The method depends on the reaction, at the driving
point, of a standing wave system produced in the medium be-
tween the driving crystal and a nearby reflecting wall. (The
action of the piezo-electric system is similar to that of other
electro mechanical devices in that the velocity of its moving
member is translated into an electrical " motional" impedance,
which, added to the inherent electrical impedance of the appa-
ratus, determines the net electrical impedance between termi-
nals. Thus from one standpoint, in an experiment of this kind,
we have a problem in impedances quite similar to those we
have met in the preceding text; or if we prefer, we may by
analogy call Pierce's apparatus an Acoustic Interferometer.)
Among the experimental results obtained by Pierce are, an
apparent frequency variation in the velocity of sound in both
gases, and a large sound absorption effect by carbon dioxide
at high frequencies. Low-frequency determinations based on
resonance in a 4 cm. tube were also made, using a simple tele-
phone receiver as a source. The factor for the defect in velocity
due to thd tube was eliminated in reducing the observations.
The paper also gives references to a number of modern deter-
minations of the velocity of sound in air.

These references, with those below on the telephone re-
ceiver, are cited to emphasize the important field of work in
which acoustics and alternating currents have become very
closely associated. This development has not only greatly
increased the resources of the acoustic laboratory, but has
also resulted in better working ideas in the theory of vibrating
systems generally.

THE TELEPHONE RECEIVER 245

The Telephone Receiver

This instrument, invented by Alexander Graham Bell in
1875, is not only one of the oldest, but has become one of the
most indispensable devices of the laboratory. References have
already been given (7) to the work of A. E. Kennelly and his
associates on the motion of the diaphragm, and other matters
which are treated in 4< Electrical Vibration Instruments/' A
good introductory paper on the telephone is that of L. V. King,
Jour. Fr. Inst., 187, 1919, p. 611. The most general treatment
of the receiver is that of R. L. Wegel (J. A.I.E.E 40, 1921, p.
791). Two papers by Hahnemann and Hecht are referred to in
the footnote (7) p. 19. Their work has been continued in two
papers on the Receiver in Ann. d. Phys. y 60, 1919 p. 454; 63,
1920, p. 57; and 64, 1921, p. 671. Two additional articles by
these authors, translated in Engineering, 106, 1918, p. 756 and
107, 1919, p. 224, deal with sound radiation and sound gene-
rators, in a different style 1 from that adopted in the present
text. A recent paper by Mallet and Button, Jour. I.E.E., 63,
1925, p. 502 (discussion later, ibid., p. 715), describes some
interesting acoustic experiments with receivers, and gives ref-
erences to earlier work by these writers.

The Standard Phone, and the Phonometer of A. G. Web-
ster, used for sound-intensity measurements, are described in
Proc. Nat. /lead. 6V/., 5, 1919, p. 173 and 275.

The best paper on the behavior of a diaphragm immersed
in a sound field is that of D. A. Goldhammer (Ann. d. Phys., 33,
1910, p. 192). A paper by E. Meyer (Ann. d. Phys., 71, 1923, p.

1 See also a paper by W. Schottky (Zeif.f. Phys., 36, 1926, p. 689) on the Law of
Low [Frequency] Reception in Acoustics. This is a restatement of the Principle
of Reciprocity; the principle dates from Helmholtz, and, as applied to sound fields,
is given in Rayleigh, II, p. 145. Rayleigh applied it to the conical horn (II, p. 146);
also to elucidate a certain experiment of Tyndall's. (See "The Application of the
Principle of Reciprocity to Acoustics," Proc. Roy. *SV., 25, 1876, p. 1 18, or "Scientific
Papers," I, p. 305.) There are certain restrictions on its applicability, and indeed, in
Tyndall's experiment it was found to be not applicable. Schottky's application is to
the comparison between the receiving-efficiency and the radiating-efficiency of the
" Band-Sprecher," a device noted in the next section.

246 THEORY OF VIBRATING SYSTEMS AND SOUND

567) deals with the force due to the impact of sound waves on
resonant membranes. The vibrations of a disc clamped at the
center are treated by R. V. Southwell, Proc. Roy. Soc., AIOI,
1922, p. 133.

Loud Speaking Telephones

The typical loud speaker with a horn is an outgrowth of the
telephone, and as might be expected, there are many current
models of the device. The Western Electric model, which is
representative, is described in Electrician, 84, Mar. 12, 1920,
p. 300; Te/eg. and Tel. Age, 40, July 16, 1922, p. 321; also in
various bulletins, supplied by the Western Electric Company.
Circuits for use with the apparatus on a large scale are described
by Green and Maxfield, Trans. A.I.E.E., 42, 1923, p. 64; with
discussion later, p. 83; see also the paper by Martin and Clark,
Trans. A.I.E.E., 42, 1923, p. 75. The Marconi Loud Speaker
and circuits for operating it are described by H. J. Round in
Wireless World, XV, Dec. 17, 1924, p. 365.

To get away from horns, many devices have been proposed;
most of these employ some sort of a light diaphragm, closely
coupled to an electromagnetic driving element. Typical is
the Western Electric model 540 AW, popularly known as the
cone type, as the diaphragm is a large paper cone. (See Sci.
Am., Dec. 1924, p. 390; Q.S.T., 8, Dec. 1924, p. 27; also Bul-
letin No. T-75O supplied by the Western Electric Company.)

The device of C. W. Hewlett, in which a large conducting
diaphragm, placed in a strong magnetic field, is driven by
the induced eddy currents, is described in Jour. Opt. Soc.
Am., 6, 1922, p. 1059, and its calibration is discussed in Phys.
Rev., 23, 1924, p. 310; the theory of the device is given in Radio
Broadcast, 7, Aug. 1925, p. 508. The Gaumont loud speaker,
in which thin driving coil is mounted on the diaphragm, the
diaphragm being placed in a strong magnetic field, is described
by Bonneau in Bull. Soc. Franc, des Rlectriciens, 4, 1924, p. 157;
see also le Genie Civil, 84, 1924, p. 526. The German u Band-
Sprecher " is described by E. Gerlach (Phys. Zeit., 25, 1924, p.
675; Zeit.f. Tech. Phys., 5, 1924, p. 576) and W. Schottky (in

LOUD SPEAKING TELEPHONES 247

Zeitf. Tech. Phys., 5, 1924, p. 574 or Phys. Zcit., 25, 1924, p. 672),
also in Electr. Nachr. Tech., 2, 1925, p. 157, and the publications
of the Siemens- Konzern, who manufacture the device. Interest-
ing, but so far non-commercial devices are the Frenophone of
S. G. Brown (J.LE.E. 62, 1924, p. 283) and the Johnsen-
Rahbek device (J.I.E.E. 61, 1923, p. 713; see also K. Rott-
gardt, Zeit. fur Tech. Phys., 2, 1921, p. 315; and Jahrb. der
Draht. Teleg.j 19, 1922, p. 299). Some general information on
loud speakers is given in a paper by Rice and Kellogg, J.A.I.
E.E. y 44, 1925, p. 982 (discussion p. 1015 following) entitled
"Notes on the Development of a New Ty? e f Hornless Loud
Speaker/' with a description of an instrument of the piston
type they have recently developed. Interesting discussions
of various phases of the loud speaker problem are given in
J.I.E.E.y 62, 1924, p. 265, by a number of members of the
Institution of Electric Engineers and the London Physical
Society; also reported in Proc. London Phys. Soc., 36, 1924 p.
114 and p. 211. An excellent paper on the general theory of
loud speakers is that of H. Riegger, Wissensch. Vervjjent. aus
d. Siemens-Konzern, III, No. 2, 1924, p. 67; a paper by
Trendelenburg, ibid^ IV, No. 2, 1925, p. 200, describes a New
Method for testing these instruments. A recent paper by C. R.
Hanna (Proc. LR.E. y 13, 1925, p. 437) deals with the design of
the balanced-type element for driving the loud speaker.

The loud speaker in various forms is in a state of rapid
development, and it is likely that instruments both with and
without horns have possibilities that have not yet been fully
exploited. Many designers of loud speaking apparatus all but
ignore acoustical principles, in operating and testing their
devices, but with the revival of interest in theoretical acous-
tics, and the dissemination of more reliable information as to
the duty expected of the loud speaker, the more faulty appa-
ratus is in process of elimination.

The Thermophone

This device, while inefficient as compared with a telephone
receiver, has the advantage of being free from resonance. The

248 THEORY OF VIBRATING SYSTEMS AND SOUND

theory is given by Arnold and Crandall (Phys. Rev., X, 1917,
p. 22); also in a later paper by E. C. Wente (ibid., XIX,
1922, p. 333) which precises the calibration of the instrument.
The thermophone has also been applied by Wente to the cali-
bration of the condenser transmitter.

Resonators; Hot Wire Microphones

Interesting recent work on resonators has been done by
Hahnemann and Hecht (Phys. Zeit., 21, 1920, p. 187; ibid. y 22,
1921, p. 353); also by A. T. Jones (Phys. Rev., 25, 1925, p. 696
and p. 705). In this connection, the reader may be interested in
the action of the Tucker Microphone, which consists of a hot
wire placed in the mouth of a resonator. This is described by
Tucker and Paris (Phil. Trans. Roy. Soc., vol. A22I, 1921, p.
389) followed by further work of A. T. Paris (Proc. Roy Soc.
AIOI, 1922, p. 391; Phil. Mag., 48, 1924, p. 769) on double
resonators; there is a good paper by Paris on the magnification
of acoustic vibrations by resonators in Science Progress, XX,
No. 77, 1925, p. 68. The pin-hole resonator of C. Barus (Proc.
Nat. Acad. Sci., 8, 1922, p. 163) has been studied by P. E. Sa-
bine, Phys. Rev., 23, 1924, p. 116.

The hot wire microphone in itself is considered by A. V.
Hippel, Ann. d. Phys., 76, No. 6, 1925, p. 590; a thermo-couple
instrument for Sound (Intensity) Measurements is described
by W. Spaeth, Zeit.f. Tech. Phys., 6, 1925, p. 372.

The Rayleigh Disc

It is of interest to note that Mallet and Dutton (J.I.E.E.
63, 1925, p. 502) make use of the Rayleigh Disc to measure the
sound output of the telephone; thus Lord Rayleigh's device of
1882 (Phil. Mag.,\lV. p. 1 86, 1882, or " Papers," Vol. II, p. 132;
see also " Sound," II, p. 44) is still in current use. It has the ad-
vantage of giving a steady deflection, proportional to the square
of the particle velocity in the undisturbed field; the radiometer
(48) is the only other purely acoustic measuring instrument
of which this can be said. The sensitiveness of the disc can be

THE CONDENSER TRANSMITTER 249

increased by placing it at a velocity loop in a resonant chamber;
under these conditions the device is most serviceable for meas-
urements at a single frequency. Care must be taken not to
make the disc so large that its size is comparable to the wave
length of the sound; its anomalous behavior under these cir-
cumstances is treated by C. H. Skinner, Phys. Rev. y 27, 1926,
p. 346.

The Phonodeik

This instrument, used by D. C. Miller in his analyses of
speech and music, is described fully in his " The Science of
Musical Sounds" to which reference has already been made
( i). A recent paper by S. H. Anderson (J. Opt. Soc. Am. y n,
1925, p. 31) deals with design and calibration, and gives some
further useful references.

The Condenser Transmitter

With microphones as such, the acoustic laboratory must
proceed with caution, due to the fact that the sensitiveness of
the microphonic substance is inherently hard to control. For
precision work in sound detection over a range of frequen-
cies electromagnetic or electrostatic devices which have been
highly damped are much more reliable. Representative of the
first class is the Marconi-Sykes Magnetophone, described by
H. J. Round in the Wireless World, XV, Nov. 26, 1924, p. 260.
The condenser transmitter of E. C. Wente is representative of
instruments of the latter class; its more recent features, and the
calibration of the device, are described by him in Phys. Rev. y
XIX, 1922, p. 498. Information on recent models can be had
from the Western Electric Co., Inc.

The Kondensator-Mikrophon of H. Riegger (described by
F. Trendelenburg, Wiss. Verojjent. d. Siemens-Konzern y III,
No. 2, 1924, p. 46) is quite different from Wente's instrument,
both in its construction, and in the much less straightforward
way in which it is used. It seems to have more complicated
mechanical characteristics, and it is used in a high-frequency

250 THEORY OF VIBRATING SYSTEMS AND SOUND

circuit so that incident speech waves modulate high-frequency
oscillations; a detailed argument, according to which the in-
strument functions as a faithful recorder of complex sound
waves, is contained in the reference cited.

The condenser transmitter of Wente has been applied in
many studies by the staff of the Bell Telephone Laboratories
which required distortionless transmission of speech sounds.
See for example, the paper by H. Fletcher on " The Nature of
Speech and its Interpretation" (Jour. Fr. Inst. y 193, 1922, p.
729); the paper of Crandall and MacKenzie on " Energy Dis-
tribution in Speech" (Phys. Rev., XIX, 1922, p. 221); also
Crandall and Sacia " Dynamical Study of the Vowel Sounds,"
(Bell System Tech. Jour., Ill, 1924, p. 232). In work of this
kind a distortion-free amplifier circuit is necessary, and if
records of sounds are to be made with the electrical circuit, the
oscillograph used mustalso be distortionless. The oscillograph
vibrator, whose general construction is explained by Kennelly
(" Electrical Vibration Instruments," Ch. XV) must be highly
damped and carefully calibrated; since it is a relatively simple
vibrating system this is entirely practicable. A description of
the amplifier used by Crandall and Sacia for making accurate
records of vowel and consonant sounds appears in the paper
by Crandall noted below. The important requirements for any
amplifier used in connection with precision apparatus are to
make the frequency response as nearly constant as possible over
the working range, and to eliminate distortion due to over-
loading, for the maximum output furnished by the amplifier.

Speech and Hearing

The voice and the ear are, broadly speaking, the most
important acoustical apparatus in use; and though many
acoustical experiments may be made independently of speech
and hearing, no worker in acoustics is likely to ignore these
interesting phenomena, whatever his special interests may be.
The references given here to recent work on speech and hearing
do not discriminate between the fundamental study of these
phenomena in themselves, and the more restricted work, which

SPEECH AND HEARING 251

relates to their bearing on other fields of acoustics, and to their
utility in the laboratory.

Hearing has been extensively studied by members of the
staff of the Bell Telephone Laboratories; a paper by H. Fletcher
in Bell System Tech. Jour., II, Oct. 1923, p. 145 (also in Jour.
Fr. Inst. y 196, 1923, p. 289) deals with Physical Measurements
of Audition and gives a very complete bibliography of the sub-
ject. The recent letter of R. L. Wegel in Nature (116, 1925,
p. 393) gives a good statement of the present status of the reso-
nance theory of hearing. 1 See also Bell System Tech. Jour., IV,
July, 1925, p. 375, for a compilation of the best available data
on the Constants of Speech and Hearing. In this paper will be
found references to the data of Fletcher and Steinberg, Fletcher
and Wegel, and Wegel and Lane; also references to other recent
work on various phases of hearing all important if the ear is
to be used as a sound-detecting instrument. The Audiometer,
an instrument for measuring the sensitiveness of the ear, is
described by Fletcher in the Trans. Coll. of Physicians of Phila.,
45, Series 3, 1923, p. 489; also in several bulletins furnished by
the Western Electric Company. Audiometric Methods and
their Applications are .described by E. P. Fowler and R. L.
Wegel in Trans. Am. Laryng. Rhinol. and Otol. Soc. y 1922, p. 98.

The experiments of F. W. Kranz of the Riverbank Labora-
tories on the sensitivity of the ear (Phys. Rev., 21, 1923, p. 573,
and ibid., 22, 1923, p. 66) are being continued. The relative
sensitivity at different levels of loudness is treated by D. Mac-
Kenzie in Phys. Rev., 20, 1922, p. 331. V. O. Knudsen has a
paper on the sensibility of the ear in Phys. Rev., 21, 1923, p. 84;
also a paper on the effect of tones and noise on speech recep-
tion by the ear, in Phys. Rev. y 26, 1925, p. 133, The relation
between the loudness of a sound and its physical stimulus is
treated by J. C. Steinberg, Phys. Rev. y 26, 1925, p. 507.

1 The monograph by George Wilkinson arid Albert A. Gray entitled " The Mechan-
ism of the Cochlea" (London, 1924) is an interesting and well written " Restatement
of the Resonance Theory of Hearing." Many of the questions which were still open
when this volume was written have been answered by the more recent work noted, but
the monograph is a distinct contribution to the literature of hearing.

252 THEORY OF VIBRATING SYSTEMS AND SOUND

A general study of important German contributions to the
theory of hearing (and some of the American contributions
already mentioned) is given by E. Waetzmann, Phys. Z^//.,
XXVI, 1925, p. 740. Waetzmann also has a book on the " Reso-
nance Theory of .Hearing," Braunschweig, 1912.

In using the ear for detecting purposes it may be pointed
out that in general, many more observations are required for
a given degree of precision than if a mechanical or electrical
instrument is used. (Cf. 53, remarks on Sabine's technique.)
The ear is at its best if two sounds of the same wave form and
the same intensity are to be compared; the accuracy obtainable,
of the order of five per cent for a single observation, is compar-
able to that obtainable with the eye in photometrical measure-
ments.

The principal technical application of the Binaural Effect
is to direction finding, as in submarine detection and airplane
location. These subjects will be dealt with below.

The most original work on the speech sounds in recent
years has been in synthesizing them, from the transient vibra-
tions of electrical or mechanical respnators. (Most of the
vowels, as is evident from their energy-frequency spectra, can
be imitated by suitably exciting a double-resonator system;
in such a system the external orifice corresponds to the mouth,
the chamber behind it to the buccal cavity, and the inner
chamber to the pharynx. The coupling between the two
chambers may be varied by changing the size of the orifice
between them; and since the volumes of the chambers them-
selves may be varied, a,number of arrangements of loose or
close coupling can be found for producing, with sufficient
exactitude, a given doubly-resonant effect.) The analogous elec-
trical method was used by J. Q. Stewart, reported in Nature, 1 10,
Sept. 2, 1922, p. 311; the mechanical method by Sir Richard
Paget (Proc. Roy. Soc., Aio2, 1923, p. 752) for Vowel Sounds;
and again by Paget for Consonant Sounds, ibid., Aio6, 1924,
p. 150; see also Nature > ] 1 1, Jan. 6, 1923; Proc. Land. Phys. Soc. y
36, 1924, p. 213; and J.I.E.E.y 62, 1924, p. 963 (lecture deliv-

SPEECH AND HEARING 253

ered Mar. 20, 1924). In this connection the Artificial Larynx
of Fletcher and Lane (a device used in connection with the
mouth cavities, to produce speech when the natural larynx has
been removed) is of interest. This is described in Western
Electric News, Jan. 1925; Philadelphia Record, Jan. 18,1925,
and other news sources of that time; also in literature supplied
by the Western Electric Company.

The structure of vowel sounds and the frequencies of certain
consonant sounds are treated by C. Stumpf, Ber. d. Preuss.
Akad., Berlin, 1918, p. 333; also ibid., 1921, p. 636; a later
paper by Stumpf appears in Beitr. z. Anat., PhysioL, Pathol.,
des Ohres, etc., 17, 1921, p. 15 r ; see also ibid., p. 143, and p. 182.
Another German investigation, by F. Trendelenburg, is given
in Wissensch. f^eroffenf. dem Siemens-Konzern, III, Heft 2,

1924, p. 43, and ibid., IV, Heft T, 1925, p. i ; see also E.T.Z., 46,

1925, p. 915; the first of these papers contains a bibliography
of previous German work. A recent paper by Riegger and
Trendelenburg, Zeit. /. Tech. Phys., 7, 1926, p. 187, describes
further studies in speech, with applications to the loud speaker
problem.

Some recent work on speech at the Bell Laboratories has
already been noted. The general problem of High Quality Re-
production of Speech is treated by Martin and Fletcher in Trans.
A.J.E.E., 43, 1924, p. 384. The work of Crandall and Sacia on
Speech is being continued; a paper by Sacia on " Speech Power
and Energy" and one by Crandall on (the wave forms of) "The
Sounds of Speech " appear in Bell System Tech. Jour., IV, Oct.,
1925. The results of all the investigations on speech point to def-
inite characteristic frequencies and energy distributions, both in
time and frequency, for the individual sounds. (The semi-vowel
sounds, for example, have frequency spectra which suggest
systems of four degrees of freedom, as might be expected, with
the naso-pharynx and nasal cavities brought into play.) For
discussion of the mechanism of the various sounds the reader
had best refer to the original sources cited, and the collateral
references there to be found; and other papers on this subject
may be expected in due course from the Bell Laboratories.

254 THEORY OF VIBRATING SYSTEMS AND SOUND

Sound Analyzing Devices

The analysis of wave forms by computation, or by graphical
methods belongs to the mathematical laboratory, but sound-
analyzing devices making use of tuned circuits or other physical
means may be mentioned here. The apparatus of Wegel and
Moore (Bell System Tech. Jour., Ill, 1924, p. 299) for electrical
analyses, and the photomechanical method of C. F. Sacia (J.
Opt. Soc. of Am.) 9, 1924, p. 487) are representative of purely
physical devices for analysing complex wave forms.

The hot-wire microphone (noted above) has been applied by
A. Fage (Proc. Roy. Soc. y Aioy, 1925, p. 451) to analyze the
vibrations of air screws.

Submarine Signalling Apparatus; Underwater Sound Detection ,
and "Depth Finding

As a result of the war, interest has been focussed on sound-
detecting apparatus generally and many applications have
been made to Submarine Signalling, Acoustic Depth-finding,
Submarine and Airplane Detection, Sound-Ranging, and Geo-
phones.

Reference has already been made to two general sources
of information on submarine signalling. The older art is repre-
sented by several papers noted in section II of " Certain Prob-
lems in Acoustics " (Bull. Nat. Res. Council, loc. cit.) ; more recent
work described in C. V. Drysdale's Chapter (IX) of the collec-
tive work " Mechanical Properties of Fluids," and in his Elev-
enth Kelvin Lecture, J.I.E.E., 58, 1920, p. 572. Another
source of general information is the " Unterwasserschalltech-
nik" of F. Aigner, Berlin (Krayn) 1922.

The theory of receiving sound in water is dealt with by H.
A. Wilson (Phys. Rev., XV, 1920, p. 178). The reaction of the
medium (water) on elastic plates is the subject of a paper by
H. Lamb (Proc. Roy. Soc. y Ag8, 1920, p. 205). The mechanics
of diaphragms immersed in liquids is also treated by Powell
and Roberts in Proc. London Phys. Soc., 35, 1923, p. 170, and by
J. H. Powell, ibid., 37, 1925, p. 84; these writers confirm Lamb's

SUBMARINE SIGNALLING APPARATUS 255

theoretical work. Diaphragms capable of continuous tuning are
described by L. V. King in Proc. Roy. Soc., Ayg, 1921, p. 163.
A paper by Barlow and Keene (Phil. Trans. A222, 1922, p. 131)
deals with the analysis of sounds in water. Two papers by H.
Hecht on underwater acoustics will be found in Zeit. fur Tech.
Phys., 2, 1921, p. 265 and p. 337. A paper by H. Lichte (Phys.
Zeit., 20, 1919, p. 385) discusses the effect of a temperature
gradient in water on the signalling range.

Explosions under water are treated by H. Lamb (Phil. Mag.
45, 1923, p. 257) and by Ramsauer (Ann. d. Phys. y 72, 1923, p.
265). There is a paper on Cavitation in the Propagation of
Sound, by R. W. Boyle, Proc. and Trans, of Roy. Soc. Canada,
16, Series III, 1922, p. 157.

The publications of H. C. Hayes summarize many impor-
tant features of American practice in underwater sound detec-
tion. There are two articles in Vol. LIX, Proc. Am. Phil. Soc.,
1920; No. i, p. i, and No. 5, p. 371. The first of these articles
contains, for example, descriptions of the Compensator and
several arrangements of multiple unit acoustic receivers used
with it, which were described by M. Mason in an unpublished
paper before the Am. Phys. Soc., at the Washington meeting,
April, 1919; see also British Patent No. 146,192, Dec. 28,
1921, and U. S. Patent No. 1,422,876, July 18, 1922. The
electrical compensator of G. W. Pierce is described in 287 of
his book on "Electric Oscillations and Electric Waves;" see
also British Patent No. 146,163, Oct. 25, 1921, and Proc. Am.
Acad., 57, No. 8, May 1922. The Western Electric Seaphone (the
prototype of the rubber-diaphragm hydrophone, with inertia
microphone attached) is covered by U. S. Patent No. 1,581,334.

Hayes* second article deals with the hydrophone as an aid
to navigation. An article by Hayes in Jour. Fr. Insf., 197, 1924,
p. 323, deals with measuring ocean depths by Acoustic Methods;
see also his article on Sonic Depth Finding, Proc. Am. Phil. Soc.,
LXIII, 1924, p. 134. Other papers by H. C. Hayes will be
found in the Marine Review, 51, 1921, p. 404; p. 466; p. 493.
The work of L. V. King on Sounding is given in Nature, 114,
1924, p. 122; see also ibid., 113, 1924, p. 463.

256 THEORY OF VIBRATING SYSTEMS AND SOUND

The work of H. C. Hayes on Sounding is also described in
the Hydrographic Review (Monaco) 2, No. i, 1924, p. 93. (This
issue has also been referred to in connection with the work of
Langevin and Chilowsky on High-Frequency Signalling (41).)
Other French work is described by F. Collin in le Genie Civil,
86,1925, p. 38, and p. 64. A method of echo sounding developed
by Behm at Kiel is described in Bull. Technique du Bureau
Veritas, 6, 1924, p. 161.

The methods used by the Submarine Signal Co. (Boston)
are described by H. V. Hayes in the Engineer, 129, 1920, p. 491,
and in literature available from that concern.

British practice in sound detection centers about the work
of Sir William Bragg and his associates; see Engineering, 107,
1919, p. 776, or Nature, 103, 1919, p. 467. Wood and Young
treat the directional phase of sound detection under water in
two papers: Proc. Royal Soc., Aioo, 1921, p. 252 and p. 261.
Other papers of interest are by J. C. McLennan, Trans. North-
East Coast Inst. of Engrs. and Shipbuilders, 35, 1919, p. 386;
and bydu Bois-Reymond, Zeit.f. Tech. Phys., 2, 1921, p. 234;
see also the London Times Engineering Supplement, July, 1919,
p. 220, and Nature, 104, 1919, p. 28 (Inaugural Address of Sir
Charles A. Parsons).

Important work done by the French on Sound Detection is
described by A. Troller, La Nature, 49, 1921, p. 4; see also le
Genie Civil, 79, 1921, p. 375, p. 393 and p. 417; also A. Marce-
lin, Bulletin de Recherches et Inventions, Nos. 10, 11, 12, 1920,
p. 513, p. 577, and p. 641 ; also H. Brillie, le Genie Civil, 80, 1922,
p. 378, p. 397, and p. 427. Brillie's earlier papers (1919) in le
Genie Civil have already been noted in 31.

The work of the Signal Gesellschaft Laboratories at Kiel is
described in " Submarine Acoustic Signalling Apparatus," by
W. Hahnemann, Proc. Inst. Rad. Eng., n, 1923, p. 9. A good
description of the large sound generator is given by du Bois-
Reymond, Hahnemann and Hecht in Zeit.f. Tech. Phys., II,
1921, p. i and 33; see also H. Gerdien, Phys. Z.eit., 22,
1921, p. 679. Hahnemann and Lichte describe the develop-
ment of submarine detectors in Germany during the war, in

RADIO-ACOUSTIC SIGNALLING 257

Die Naturwissenschaftcn, 8, rgio, p. 871. The natural oscilla-
tions of microphones for underwater detection are treated by
P. Ludewig, Phys. Zeit., 21, 1920, p. 305. Barkhausen and
Lichte (Ann. d. Phys., 62, 1920, p. 485) give the results of a
comprehensive study of underwater transmission conditions;
Aigner has a paper in Zeitf. Phys., I, 1920, p. i6r on the most
economical operating frequency. Many more references to
German practice and to the earlier art, are available in the
book of Aigner, cited above; and finally a resume of signalling
devices is given by W. Wolf, Zeit.f. Fernmeldetechnik, 2, 1921,
p. 81.

A very good bibliography, particularly as to patents, on
subaqueous signalling of all kinds is given in one of the abstracts
of miscellaneous methods of communication published by The
Radio Review, II, Sept., 1921, p. 487. (A number of significant
patents in this field were granted by the British Patent Office
at about that time.) Here will be found references to the war-
time inventions of P. Langevin, M. I. Pupin, H. C. Hayes,
R. L. Williams, R. A. Fessenden, M. Mason, G. W. Pierce, and
the members of the staff of the Signalgesellschaft at Kiel, some
of which have already been noted. Another good bibliography
is given in a very complete article by M. Tenani on the general
problem of sound detection, in Rivisfa Marittima, 57, 1924, p.
319. The field under review is wide, and embraces a good deal
besides acoustics.

Radio-Acoustic Signalling

A combination of radio with acoustic signalling has been
devised for the purpose of position finding at sea. In one typical
arrangement, the acoustic signals are sinusoidal vibrations; an
embodiment of this scheme is seen in British Patent No. 146,125,
1920, to R. L. Williams of the Submarine Signal Company.
Many others might also be mentioned; the field is not new,
and has been developed with increased impetus recently, due
to advances in the radio art, and the acoustic developments
resulting from the war. In another arrangement, a bomb is
used to produce the acoustic signal; representative of this class

258 THEORY OF VIBRATING SYSTEMS AND SOUND

is the joint work of the U. S. Bureau of Standards and the U. S.
Coast and Geodetic Survey, described by S. R. Winters in
RadiOy 6, July, 1924, p. ]o; see also Special Publication, No.
107, 1924, of the U. S. Coast and Geodetic Survey: " Radio-
Acoustic Method of Position Finding" by Heck, Eckhardt
and Keiser.

Sound Ranging Apparatus

Reference has been made above to the Tucker microphone,
which was used as a detector for sound ranging, that is, locating
the direction and distance of cannon and projectiles. Other
references to British practice will be found in Drysdale's
Kelvin Lecture, cited above, and C. A. Parsons' address,
Nature, 104, 1919, p. 28; see also p. 313 of " Mechanical Prop-
erties of Fluids," and Nature 104, 1919, p. 278. The American
practice in reducing observations is described by Col. A. Trow-
bridge, Jour. Fr. Inst., 189, 1920, p. 133; the work of the Army
Engineers in collaboration with the Western Electric Company
on apparatus and methods is described by E. B. Stephenson
in U. S. Engineer School, Occasional Papers, No. 63, 1920, on
Sound Ranging; see also J. B. Cress, Military Engr., 12, 1920,
p. 275. This work was based on British practice (the Bull-
Tucker System). Certain theoretical problems are discussed
by E. Eschlangon, Rev. Scientifique, 59, 1921, p. 164, also by
H. W. Hodgkins in Jour. U. S. Artillery, 52, 1920, p. 41. Sub-
aqueous sound ranging is treated by F. E. Smith in the Engineer
138, 1924, p. 534; also by H. C. Allen, Coast Artillery Jour.,
59, 1923, p. 35; see also J. U.S. Artillery, 54, 1921, p. 69; also,
the article by R. B. Webb, Coast Artillery Jour., 59, 1923,
p. 17.

Direction Finding; Airplane Location

The subject of binaural hearing has necessarily entered
the discussion given in several papers cited under Sound Rang-
ing and Submarine Detection; but its most outstanding appli-
cation is to the problem of locating the direction of a sound

GEOPHONES 259

source in air. Outstanding in the field of binaural hearing are
the contributions of R. V. L. Hartley (Phys. Rev. y XIII, 1919, p.
373), G. W. Stewart (Phys. Rev., XV, 1920, p. 425 and 432), and
Hartley and Fry (Phys. Rev., XVIII, 1921, p. 431); see also
the bibliography (already cited) by Fletcher in Jour. Frank.
Inst. y Sept., 1923. Some further references are available in a
paper by E. Meyer (E.T.Z., 46, 1925, p, 805) on " Stereo-
acoustic Hearing." A recent paper by C. E. Lane (Phys. Rev.,
26, 1925, p. 401) on Binaural Beats is also of interest in con-
nection with this subject.

The effect of irregularities in the air on sound transmission
is considered by G. W. Stewart, Phys. Rev., XIV, 1919, p. 376;
his work on Aircraft Location is described in the same volume,
p. 1 66. There is a paper on Aircraft Location by E. Waetz-
mann, in Zeit.f. Techn. Phys. 2, 1921, p. 191. In vol. 4, 1923,
p. 99, ibid., the paper by E. Liibcke describes water-tight appa-
ratus for submarine listening to aircraft sounds. The analysis
of aircraft sounds is treated by E. Waetzmann, ibid., 2, 1921,
p. 1 66, and by L. Prandtl, ibid., 2, 1921, p. 244. There is a
paper by W. S. Tucker on Sound Reception as Applied to Air
Defence in Jour. Roy Aeronaut. Soc., 28, 1924, p. 504.

Geophones

The geophone was developed originally in France for lis-
tening to the sounds sent through the ground, as the result of
military mining operations. Since the war, mining engineers
have been interested in the device, but few references are
available to original work. An excellent general account of
Geophones is given by Alan Leighton, U. S. Bureau of Mines,
Technical Paper No. 277, 1922, A brief note by Ackley and
Ralph in Jour. Fr. Inst., 198, 1924, p. 71 1 or p. 834, on Improve-
ments in Geophones by the use of Electrical Sound Amplifiers is
followed by a more detailed account in a publication of the
C7, S. Bureau of Mines, Report of Investigation No. 2639. There
is an excellent paper by H. S. Ball in Jour. Inst. of Min. and
Met., 28, 1919, p. 189, which contains material on the Geo-
phone and other devices.

260 THEORY OF VIBRATING SYSTEMS AND SOUND

Addenda: Measurement of Absorbing Properties of Materials

There has recently come to hand a paper by Eckhardt and
Chrisler (U. S. Bureau of Standards, Sci. Paper No. 526,
April 28, 1926) on Transmission and Absorption of Sound;
this is a continuation of the work at the Bureau (Sci. Paper
No. 506) noted in Chap V. In the more recent paper a number
of data are given on transmission and absorption by panels of
various materials; and of particular interest is the method of
determining absorption coefficients, which is based on that of
H. O. Taylor to which reference has already been made (p. 109).
In Eckhardt's method, the length of the tube infixed (for reso-
nance at a given frequency) and the reflecting power of the
layer is given in terms of the pressure maxima and minima in
the standing wave system, which are determined by an explor-
ing tube. A correction is also made for the attenuation in the
tube (Helmholtz effect) if this is required.

Some further notes on Wente's experiment may be of inter-
est here, following the brief outline given on p. 109. The
velocity of the driving piston is kept virtually constant, and
the pressure near the piston (as determined by the exploring
tube) thus measures the driving-point impedance of the
apparatus. If p\ represents the maximum driving point
pressure (corresponding to an effective length of tube of an
integral number of half wave lengths) and p2 the minimum
driving point pressure (the length of tube now differing by a
quarter wave length from that of the first adjustment) the
amplitude reflection coefficient of the layer at the end of the
tube is

r =

V/>l//> 2 + I

from which the absorption coefficient of the layer can be
calculated (cf. problem 48, p. 227).

A more complete account of Wente's analysis and of* his
determination of the absorbing properties of materials, by this
method, is in preparation.

INDEX OF NAMES

[ The numbers refer to pages]

ACKLEY, W. T., 259
AlGNER, F., 254, 257

ALLEN, H. C., 258
ANDERSON, S. H., 249
ARNOLD, H. D., 248

BALL, H. S., 259
BARKHAUSEN, H., 257
BARLOW, G., 255
BARTON, E. H., 3, 239, 243
BARUS, C., 248
BEHM, 256
BELL, A. G., 245
BESS EL, F. W. (see subject index)
BLAIKLEY, D. J., 151, 240
BONNEAU, P. E., 246
BOYLE > R. W., 112, 255
BRAGG, W. H., 3, 256
BRIDGMAN, P. W., 1 1 1
BRILLIE, H., 92, 256
BROWN, S. G., 247
BUCKINGHAM, E., 200, 207

CADY, W. G., 243
CAMPBELL, G. A., 64
CARSON, J. R., 49
CHILOWSKY, C., 142, 256
CHRISLER, V. L., 260
CLARK, A. B., 246
COLLIN, F., 256

CONSTANTINESCO, G., 1O2, 103, 125

CRANDALL, I. B., 248, 250, 253
CRESS, J. B., 258

DAVIS, A. H., 227

DRYSDALE, C. V., 19, 31, 88, 92, 93, 95,

102, 141,254,258
DU BOIS-REYMOND, A., 256
BUTTON, G. F., 245, 248

ECKHARDT, E. A., 207, 208, 258, 260
ElNTHOVEN, W., 129, 130, 131, 132, 133

EMDE, F., 24, 237

ESCHLANGON, E., 258

FACE, A., 254

FESSENDEN, R. A., 257

FLANDERS, P. B., 163

FLEMING, J. A., 95, 100

FLETCHER, H., 10x0,250, 251, 253,25(^,314

FOURIER, J. B. J. (see subject index)

FOWLER, E. P., 251

FRANKLIN, W. S., 207

FRAUNHOFER, J., 141

FRESNEL, A. J. (see subject index)

FRY, T. C., 50, 259

GERDIEN, H., 256

GERLACH, E., 246
GODFREY, C., 64

GOLDHAMMER, D. A., 245

GOLDSMITH, A., 162
GRAY, A., 22
GRAY, A. A., 251
GREEN, I. W., 246
GRUNEISEN, E., 240

HAHNEMANN, W., 15, 245, 248, 256
HANNA, C. R., 162, 247
HARRISON, H. C., 162, 213
HART, M. D., 112, 113
HARTLEY, R. V. L., 259
HAYES, H. C., 255, 257, 312
HAYES, H. V., 256
HEAVISIDE, O., 49
HECHT, H., 15, 245, 248, 255, 256
HECK, N. H., 258
HELMHOLTZ, H. L. F., 245
(see also subject index)

16 1

262

INDEX OK NAMES

HEWLETT, C. W., 246
HIPPEL, A. V., 248
HODGKINS, H. W., 258
HOOKE, R. (see subject index)
HORTON, J. W., 244

JAHNKE, E., 24, 237

JOHANSEN, F. C., 24!
JOHNSEN, A., 247

JOHNSON, K. S., 73, 76
JONES, A. T., 248

KAISER, 239

KALAHNE, A., 3, 125, 126, 128

KEENE, H. B., 255

KEISER, M., 258

KELLOGG, E. W., 157, 247

KENNELLEY, A. E., 15, 17, 37, 38, 100,

245, 250
KING, L. V., 111,245,255

KlRCHOFF, G. R., 112, 239, 240

KLEMENCIC, I., 128
KNUDSEN, V. O., 185, 251
KRANZ, F. W., 251

KtlNDT, A., 102, 103, 223, 239, 243

LAGRANGE, J. L., (see subject index)
LAMB, H.

"Sound," 2, 21, 31, 39, 45, 54, 79, 80,
82, 91, 103, no, in, 112, 123, 124,
136, 177, 178, 181

"Hydrodynamics," 31, 114

Papers by, 64, 95, 254, 255
LANE, C. E., 251, 253, 259
LANGEVIN, P., 142, 180, 256, 257
LARMOR, J., 179
LAUE, M. v., 243
LEIGHTON, A., 259
LICHTE, H., 255, 256, 257

LlFSHITZ, S., 212, 213

LOVE, A. E. H., 38

LiJBCKE, E., 259
LUDEWIG, P., 256

MACKENZIE, D., 250, 251

MACLENNAN, J. C., 256

MARCELIN, A., 256

MALLET, E., 245, 248
MARRISON, W. A., 244
MARTIN, H., 128
MARTIN, W. H., 246, 253
MASON, MAX, 255, 257
MATHEWS, G. B., 22
MAXFIELD, J. P., 213, 246
MAXWELL, J. C, 6, no, 229
MERKEL, E., 240
MEYER, E., 180, 245, 259
MILLER, D. C., 3, 249, 253

MlNTON, J. P., l62

MOORE, C. R., 254

MORECROFT, J. H., 142
MlJLLER, J., 240

NANCARROW, F. E., 244
NICHOLS, 11. W., 92, 100

OHM, G. S. (see subject index)
ORNSTEIN, L. S., 133

PAGET, R. A. S., 252

PARIS, E. T., 84, 248

PARSONS, C. A.j 256, 258

PEACOCK, H. B., 75, 76

PIERCE, G. W., 15, 16, 244, 255, 257

POISEUILLE, J. L. M. (see subject index)

POISSON, S. D. (see subject index)

POWELL, J. H., 254

POYNTING, J. H., 3

PRANDTL, L., 259

PUPIN, M. L, 142, 257

QUARLES, D. A., 163

QUIMBY, S. L., Ill, 112

RAHBEK, K., 247

RALPH, C. M., 259

RAMSAUER, C., 255

RATZ, E., 1 06

RAYLEIGH, LORD,

"Sound," 2, 6, 19, 22, 37, 51, 52, 53, 54,

64, 79, l 7> "4 I2 4> i*5> *3 6 > X 37

H5> '47> l $ l > X 5 2 > *53 J 54, i7 2 >

178, 181, 184, 186, 187, 190, 239, 245

V Papers," 179,245,248

INDEX OF NAMES

263

RICE, C. W., 247
RICKER, N. H., 244

RlEGGER, H., 2 4 7, 249, 253

ROBERTS, J. H. T., 254
ROUND, H. J., 246, 249

ROTTGARDT, K., 247

ROUTH, E. J., 64
RUSSELL, A., 32

SABINE, P. E., 185, 189, 191, 192, 199*

200, 208, 209, 211, 212, 213, 248
SABINE, W. C, 3, 184, 191, 192, 197, 199,

2O4, 205, 2O6, 207, 208, 209, 210, 211,
212, 213, 215, 219, 220, 222, 227, 252

SACIA, C. F., 250, 253, 254
SEEBECK, A., 239

SCHNEEBELI, 239
SCHOTTKV, W., 245, 246

SCHULZE, F. A., 240

SIMMONS, L. F. G., 241

SKINNER, C. H., 249

SLEPIAN, J., 162

SMITH, F. E., 258

SMITH, W. WHATELY, 113

SOUTHWELL, R. V., 246

SPAETH, W., 248

STEINBERG, J. C., 251

STEINHAUSEN, W., 106

STEPHENSON, E. B., 258

STEVENS, E. H., 240

STEWART, G. W., 73, 74, 75 7 6 IO 7 J 5 8

259

STEWART, J. Q., 252
STOKES, G. G., 64, in, 112, 124, 125, 126,

127, 128, 129, 132
STUMPF, C., 253

TENANI, M., 257
THOMSON, J. J., 3
TRENDELENBURG, F., 247, 249, 253
TROLLER, A., 256
TROWBRIDGE, A., 258
TRUEBLOOD, H. M., in
TUCKER, W. S., 248, 258, 259
TYNDALL, J., 245 (note)

VAN DYKE, K. S., 243
VINCENT, J. H., 64

WAETZMANN, E., 19, 180, 252, 259

WATSON, F. R., 200, 212, 213

WATSON, G. N., 28, 82

WEAVER, W., 179

WEBB, R. B., 258

WEBSTER, A. G., 66, 153, 158, 162, 163,

*45

WEGEL, R. L., 15, 26, 245, 251, 254
WENTE, E. C., 36, 108, 191, 194, 205, 227,

248, 249, 250, 260
WHITTAKER, E. T., 28, 52, 82
WILKINSON, G., 251
WILLIAMS, H. B., 129, 130, 132
WILLIAMS, R. L., 257
WILLS, A. P., 142, 1 80
WILSON, H. A., 254
WINTERS, S. R., 258
WOLF, W., 257
WOOD, A. B., 256

YOUNG, F. B., 256

YOUNG, T. (see subject index)

TAYLOR, G. I., HI

TAYLOR, H. O., 15, 109, 226, 260

ZOBEL, O. J., 73

INDEX OF SUBJECTS

[The numbers refer to pages]

Absorption, 1 07-111
by walls, 186-190
coefficients, felt, 196-199
selective, 175-176, 188, 197

Absorbing

materials, 108, 186-190, 196-199
power, (Sabrne) 191, 197, 199, 204

Acoustic
doublet, 135
filters (see Filters)
impedance 71 (note), 153
interferometer, 244
radiometer, 180

Acoustics,

classical theory of, 2
development and scope of, 2
experimental, (see Experiment)
recent work noted, 242-259
(see also Architectural acoustics)

Added mass, (see Mass)

Adiabatic compression, 86

Air-dampsd system, 29
general solution, 33
resistance at low frequencies, 35

Air-damping of vibrating string, 129-132
Airplane detection, recent work noted, 259

Amplification of sound, by resonators, 57,
58, 84, 175-176

Amplitude, - r}

prescribed, 30, 102, 120, 143, 155, 224
finite, (see Finite, also Non-Linear vi-
brations, also Over driving)

Analysis of sound, noted, 2, 254

Architectural acoustics, 182-185, 205-214
correction of difficulties, 221, 222
first law of, 203, 207, 210

Architectural acoustics, cont.,

of particular buildings, 208, 209, 212,
213

Attenuation factor, 72, 97
due to dissipation in medium, no
due to high viscosity in solid, in
in absorbing material, 193
in exponential horn, 158
of filter, 72, 75

Audiometer, Western Electric, 251
Audition, (see Hearing)

Ber and Bei functions, 32

Bessel's functions, 21-27, 31, 37
applied to cylindrical tube, 234-237
applied to vibrating piston, 145-147
applied to vibrating string, 126, 127
as normal functions, 81-82
integration of, 38, 146
of zero order, 22, 37

Binaural beats, recent work noted, 259

Binaural hearing, recent work noted, 258-
259

Biquadratic equation, approximate solu-
tion of, 62

Capillaries, dissipation in, 108, 187, 229,

Circle diagram of velocity, 14, 15

Circular membrane,
arbitrary distension, 25, 82
equilibrium theory, 20
general theory, 24
mass coefficient, 27
natural frequencies, 23
steady state theory, 25-26
264

INDEX OF SUBJECTS

265

Circular plate, clamped at edge,

mass constant of, 38, 41

motion of, 37

natural frequencies of, 37
Compensator, of Max Mason, noted, 255
Compensator ', electrical, of G. W. Pierce,

noted, 255

Complex quantities, 9
Condensation, 85
Condenser transmitter, 29

design of, 36

use of, 109, 249, 250
Conduction of heat, \ 1 2, 239
Conductivities, combination of, 55, 74
Conductivity of an orifice^ 54, 151-152
Cone, (see Horns)

Conservation of energy, (see Energy Prin-
ciple)

Consonant sauna's, recent work noted,

252, 253

Continuity, equation of, 87, 115, 153

Constantinesco System, 100-102

Coupled systems, (two degrees of freedom),

43, 46, 59~ 6 3
Critical or limiting frequencies

of filters, 72, 74, 75, 84

of horn, 161

Critical Tables, International, 3
Cylinder functions, (see Bessel 's fuuctions)

Cylinder, moving sidewise in fluid, 126,
127

d'Alemberis Principle, noted, 42
"Damping coefficient, 7

for general system, 46

for resonators, 55, 56

for vibrating string, 128, 130-132

in tubes, 229-241

measurement of, 1 5

Damping, electrical, 132, 133

Damping film, (see Air-damped system)

Degradation of sound, 109, 143

Detection of sound, 174, 248, 249, 250, 254-

259
in water, recent work noted, 142, 254-

258

Detection,

of aircraft, noted, 259

of cannon and projectiles, 258

of submarines, noted, 254-257
Determinantal equations for natural fre-
quencies, 44, 46, 49, 72, 65, 100
Determinant "D,"

as function of frequency, 69-71

role of, 49

trigonometric equivalent, 66
Diaphragms, 20-28, 36-39

recent work on, noted, 245, 246

(see also Circular plate)
Diffraction, 137, 140-141, 178
Diffusion

equation, 232

into small conduits, 10$, 187, 231
Dilatation, 85

Direction finding, devices noted, 254-259
Directive radiation, 139, 174

possible from orchestra, 225
Disc, (see Rayleigh disc)
Dissipation of energy, 1 8

effect of, on wave velocity, 97, 193, 231,
236, 238-240

from a resonator, 55

function, 51, 52

in narrow conduit, 107-108, 186, 230

2 ? 5 .
in unlimited viscous medium, 109-112

in water, 143

(see also Air-damping)
Distortion,

by reverberation, 182-184

in phonograph recording, 213-214

of frequency spectrum, 213

(see also Wave distortion)
Divergence, 113

Divergent waves, 113, 116-120
(see also Horns)

Double Source, 135

266

INDEX OF SUBJECTS

Echoes, 184, 122

(see also Reverberation)
Einthoven siring galvanometer, theory of,

129-133

Elastic hysteresis, effects of, 19
Elastic modulus

of fluid, 86

of solid, 88, in
Electrical analogies,

addition of resistances, 18

filter problems, 73-76, 83, 84

for finite tube, 100 (note), 202-203

for inertia in an orifice, 71

of finite horn, 165-169, 174

of loaded string, 64, 73-74, 83

of resistance of conduit (Ohm's Law),
31, 232-238

of reverberation, 202-203

of simple system, 6

of sound waves in tube, 96

"power factor" in horn, 160
Electrical damping, 132, 133

Ellipsoid, wave motion and energy dis-
tribution in, 215-219

Enclosure, reaction of, on generator, 222-
225

End correction to tube, (see Mass)
Energy density, 91, 117, 120, 179, 224

in ellipsoid, 218, 227

in reverberation, 185, 201, 203, 207

irregular distributions of, 220

measurement of mean, 222

on string, 181

Energy levels, 76, 185, 203, 208, 213

Energy Principle, 4
limitations of, 42

Equations of motion
of air-damped system, 31
of circular membrane, 22, 24
of divergent waves, 113-116
of long loaded string, 65
of plane sound waves, 88, 89

symbolic solution, 91 (note)
of plane waves, with dissipation, no
of resonator-diaphragm system, 61

Equations of motion, cont.,
of simple system, 5
of sound in tube, 96

in narrow tube, 108, 233-238
of string loaded at two points, 44
of waves in horns, 153, 154, 158, 164

Equilibrium theory
of circular membrane, 20
of circular plate, 38

Equivalent piston, 29, 36

application to clamped cicrular plate,

38,60

use of the idea, 36

Experiment

of Bridgeman and Trueblood, 1 1 1

of Columbia University Group, 142, 180

of Eckhardt and Chrisler, 260

of Kundt, 102, 243

of Langevin, 142

of W. G. Cady, 243

of G. W. Pierce, 244

of Quimby, 112

of P. E. Sabine, 189, 191, 192, 199, 200,

211, 222

of W. C. Sabine, 184, 197, 206, 209, 222
of H. O. Taylor, 109, 226, 260
of E. C. Wente, 108-109, 191, 194, 205,

227, 260

of H. B. Williams, 132
of A. P. Wills, 1 80

Felt, absorption by, 108, 196-199, 226

Filter, wave

acoustic (Stewart) 73-76
development of idea, 64 (note)
electric (Campbell) 64 (note)

general properties, 72
high pass, 75

loaded string as, 48, 69-71
low pass, 69-71, 73-74
torsional, 84

Finite amplitude, distortion of waves due
to, 112

Focal properties of walls, 184, 219
Forced vibrations, (see Steady state theory)

INDEX OF SUBJECTS

267

Fourier s series, 3, 4, 25, 82, 83
determination of coefficients, 80
methods suggested, 10
physical basis of, 79-81

Fresnel Equations, noted, 94

Frequency standards, recent work on,
noted, 243-244

Friction (see Viscosity)

Frictional force, 5, 19, 56

Generalized

coordinates, 43, 52

forces, 43, 52
Generator, (see Sound Generator, also

Source)
Geophones, recent work on, noted, 259

Hearing,

binaural, noted, 258, 259

ear as detector, 252

recent work noted, 250-252, 259
Helmholtz formula, for dissipation in tube,

Helmholtz resonator, 53, 57, 58
Honeycomb structure, absorption and re-

flection by, 189
Hooke's Law, noted, 1 12
Horns,

adjustment of impedance at mouth, 171
comparison of conical and exponential,

161-162
conical, 154-157

design of, 155, 156
critical or limiting frequency of, 161
effect of, on impedance of an orifice,

'63

exponential, 158-161

exponential, finite, 163-170, 172-174

function of, 152

horn as a transformer, 165-166, 174

wave motion in, 153-154, 158

Impedance*

acoustic, 71 (note), 153

at end of a tube, 151, 153, 163, 170, 174

at mouth of narrow conduit, 187

Impedance, cont.,

at surface of pulsating sphere, 121-123,

172
at surface of vibrating piston, 146, 170

174

motional, of receivers, 1 5 (note), 244
of absorbing layer, 193-195
of enclosure, 222-223
of finite exponential horn, 165-167
of infinite exponential horn, 164
of iterated structure, 72
of medium, (see Resistance, radiation)
of perforated wall, 187, 190
of simple system, 9
of tubes and pipes, 101-102, 104, 106
of tuned resonator, 57, 175

Impedance, driving point,
of finite horn, 168
of loaded string, 48, 72
of receiver, 1 5, 82, 83
of tube, 100, 273

Impedance, transfer,
of horn, 169
of loaded string, 48
of tube, 101

Inertia, (see Mass)
Interferometer, acoustic, 244

Intensity of sound,

at mouth of horn, 155, 161
defined, for plane waves, 92
on axis of vibrating piston, 140
for spherical waves, 119, 120

Interference, (see Standing waves; sec
Noise)

Isothermal compression, 30, 86, 187

Kinetic energy, 4, 50, 52
example of, for membrane, 20
example of resonator-diaphragm sys-
tem, 6 1, 64
function, 50, 52
of divergent wave, 120
of plane wave, 90-91

Kirchoff formula, for dissipation in tube,
(see Helmholtz formula)

268

INDEX OF SUBJECTS

Lagrange,

equations of, 51

method of, 43, 50-^52

solution of, for string, 64, 79

velocity potential due to, 114
Larynx, artificial, 253
Layers, absorbing, 192-199
Leakage of sound, through cracks, etc., 199
Least Action, principle of, noted, 42
Length correction to tube, (see Mass)
Linear differential equation, 6

general solution, 10
Listening, in an auditorium, 183, 220, 22 1>

223
Loud speaker, (see Telephone)

Magnetophone, of F. J. Round, 249
Mass, added,

at end of tube, 150-151

due to medium, 121, 122, 126, 127, 148
Mass coefficients, 4, 17

due to medium,

on pulsating sphere, 121-123
at end of tube, 150-151
on piston, 146, 148
on vibrating string, 125-129

for circular orifice, 54, 71 (note), 151-
152

for resonator, 54

illustration of, 17

of circular plate, 38, 41

of membrane, 20, 27

of telephone diaphragm in water, 180
Megaphone, 1 54, 245 (note)
Meterological hazards, 113, 259
Microphone, carbon, 142, 249, 255
Microphone, hot wire, 57, 248
Microphone, of W. S. Tucker, 248

underwater, noted, 256
Mode of vibration

of ellipsoidal enclosure, 217-219

of loaded string, 45, 66

of membrane, 21-23

of tubes, 105, 106, 169

of uniform string, 78

Music,

orchestral, 225

quality of, 182, 184, 209, 213, 221, 243,

249
Musical instruments, Barton's work noted,

*43
Musical taste, and reverberation, 1 84, 209,

212, 213, 214
in using phonograph, 214

Natural Frequency

of finite tube, closed by pistons, 100,134

of circular plate, 37

of finite horn, 173

of loaded string, 44, 67

of membrane, 21

of resonator, 54

of stretched uniform string, 78

of tubes, 105, 106, 134
Natural oscillations,

audible, 12

general solution for, 45-46

of loaded string, 68

of uniform string, 77

theory of, for simple system, 6-8

theory of, two degrees of freedom, 43,

45
Newtonian velocity of sound, noted or used,

88, 239, 241

Noise, effect of, on sound reception, 185, 251
Non-linear vibrations, 19, 112,213
Normal

coordinates, 52, 80, 82

functions, 4, 81
Normalizing constant, 81

Obstacles,

detection of, 142

scattering by, 177, 178, 222
Ohm's Law, 31
One degree of freedom, 4-17

starting from rest, 1 1
Optical analogies,

diffraction, 137, 140-141

interferometer, 244

INDEX OF SUBJECTS

269

Optical analogies, cont.,

reflection, 64 (note), 94 (note)
selective absorption, 177
sound foci, 184, 219

Orchestra, conventional arrangement of, 225

Orthogonal functions, 81

Over driving 19, 38, 213

Phase factor, 72, 97

in absorbing material, 193

of filter, 72

of horn, 159-160
Phase velocity, 97

in absorbing material, 193

in exponential horn, 159

in narrow tube, 187, 236

in tube of moderate width, 97, 231,

238-240

Phonodeik, of D. C. Miller, noted, 249
Phonograph recording, dependence of, on

correct acoustics, 213, 214
Phonometer and standard phone, of A. G.

Webster, noted, 245
Piezo-electric apparatus, ill, 142

recent contributions noted, 242-244
Piston, radiation from, 143-149

at high frequency, 137-142

(see also Equivalent piston)
Piston, reactions on vibrating, 146-148
Piston loud speaker, i 57, 247
Plane waves, 89-94

due to viscosity, 232

energy in, 91, 92

in tubes, 95-103

reflection of, 93-95

theory of dissipation for, no
Point source, 116-119
Poiseuille's Law, 31, 107, 187, 229, 230,

231, 235

Poisson's Ratio, used, 37, in
Porous materials,

absorption by, 108-109, 186-190, 196-
199

reflection from, 108-109, 186-190

Potential energy, 4, 50, 52
function, 50, 52
of divergent wave, 120
of membrane, 21
of plane wave, 90-91
of resonator-diaphragm system, 61

Potential, of disc, 146

(see also Velocity potential)

Prescribed motion,
of generator, 223-224
of piston, 30, 102, 143, 227
of sphere, 120

Propagation constant,
acoustic, 97
of filter, 72

Radiation pressure, 1 79

Radiation resistance,
of piston, 138, 147
of resonator, 55

of vibrating string, negligible, 124-125
(see also Resistance)

Radiator, (see Sound generator)

Radiometer, acoustic, 180, 248

Radio acoustic signalling, noted, 257-258

Rayleigh disc, 248

Reactance, (see Mass, also Stiffness)

Reciprocity, Principle of, noted, 245

Reflection,

at end of tube, 151, 153, 181

from absorbing material, 109, 186-190,
194-195

from rigid wall, 95, 221

in an ellipsoid, 216-219

in finite horn, 169, 174

in tube, 98, 174, 201

of plane waves, 93-95

or scattering by obstacle, 177, 178

recurrent, in layers, 194-195
Repartition, of natural frequencies and
damping, on coupling, 62

Resistance coefficient,

due to air damping film, 33-35

for fluid between parallel walls, 31, 134

270

INDEX OF SUBJECTS

Resistance coefficient, cont.,
for plane waves in absorbing medium,

for resonator neck, 56
for tube, 95-96
for viscosity waves, 232

Resistance constant, 5, 19

a function of displacement, 19

due to eddies, 19

due to viscosity, 18-19

of bending plate, 18

of string in fluid, 127, 130-132

Resistance, radiation, 18
at mouth of horn, 170-172
increased by a horn, 163
of absorbing material, 194
of hemisphere, 147
of Langevin generator, 142
of medium, 92, 123
of piston, 138, 147-148
of pulsating sphere, 121-123
of resonator, 55, 175
of vibrating string, 124, 125

Resonance,

distortion effects of, 213

in finite horn, 153, 169, 174

in general, 48, 49

in Kundt's tube, 103

in layers of absorbing material, 197

of simple system, 14

Resonator-diaphragm system, 59-63

Resonators, 53-58, 84
absorption by, 175-177
amplification by, 57, 58, 84, 176, 178
multiple resonance in, 59, 76, 248
natural frequency of, 54
piezo-electric, recent work noted, 242-

244

radiation from, 55, 175-177
recent work noted, 248

Reverberation, 182-184

and phonograph technique, 214
inherent, in architectural acoustics, 183,

215

method for absorbing power, 204
optimum, 212-213

Reverberation, cont.,

theory of, in tube, 200-203

in general, 205-214
time, 203, 206, 211-213
(see also Architectural acoustics)

Sabine Memorial Laboratory, 192, (note)

222

Sabine' s (original) Reverberation Constant,

203, 206
P. . Sa bine's modification, 208, 211

Scattering, by obstacle, 177, 178, 222
Seaphone, 255

Signalling, radio-acoustic, noted, 257-258
(see also Submarine signalling)

Sink, resonator as, 176

Sound,

analysis of, 2, 243, 250, 253, 254

detection of, 254-259

foci, 219

generator, (see also Source)
effect of horn on, 1 53, 1 56
Langevin's 142
piezo-electric, 142, 243, 244
of Signal-Gesellschaft, noted, 256
reaction of enclosure on, 222-224
vibrating piston, 139-142, 143-148

ranging, recent work on, noted, 258

Sounding, acoustic, 255-256
by echo method, 142, 256

Source,

double, 135-136

effect of resonator on, 178

of finite area, 137, 143

orchestra as heterogeneous, 224

point, 116

strength of, 36, 118

(see Sound generator, also Submarine

signalling)
Speech,

apparatus for recording, 250

power and energy, recent work noted,

^53

quality of, 182, 250, 252-253
recent work noted, 250, 252-253
sounds, synthetic, noted, 252

INDEX OF SUBJECTS

271

Sphere,

pulsating, 120-123
reactions on, 121

vibrating, 122, 136
Spherical enclosure, 227, 228
Spherical waves, 1 1 6- 1 20
Standing waves, 103, 104, 106, 109, 112,
182,183,214-221,223

elimination of, 183, 222

in ellipsoid, 215-219

in horn, 169, 174

(see also Wave distortion, local)
Steady state theory,

general outline of, 46-50

of conical horn, 1 54-1 55

of energy density, with reverberation,
201-203

of exponential horn, 158-161

of finite horn, 163-173

of finite tube, 99-107, 223

of loaded string, 69-71

of pulsating sphere, 120-123

of simple system, 8-9

as function of frequency, 12-17

of vibrating piston, 138-139, 144-147

of vibrating string, noted, 125, 128

of viscosity waves, 232

of waves in ellipsoidal enclosure, 2 1 6-2 1 9

two degrees of freedom, 47

(see also Wave motion)
Stiffness coefficient, 4

a function of displacement, 19

for resonator, 54

of membrane, 21
Stokes 1 equation, 1 10, 1 1 1
Strength of source, 36, 118

(see also Source)
String galvanometer, 1 29-1 33
String, loaded,

general theory of, 64-71

of two degrees of freedom, 43-45
forced oscillations of, 47-48
normal modes of, 45
String, uniform stretched, 77-78

as limit of discrete loading, 78

steady state theory of, noted, 130

Submarine signalling,

recent work noted, 254-257

high frequency*, 142, 256
Submarine sound detection, recent work

noted, 254-257

Superposition, Principle of, 3, 25, 79
Symphony Hall, (Boston), 208-209, 213,
Systems (see Vibrating systems)

Telephone, howling, noted, 100
Telephone, loud speaking, 157
recent work noted, 246-247
Telephone receiver, 1 5, 28
recent work noted, 245
Thermophone, recent work noted, 247
Transformer, horn as, 165-166, 174
Transformer network, applied to horn,

165-166
Transients (see Natural oscillations, also

Reverberation)

Transmission, (see Wave motion, also Ab-
sorbing materials)
Transmission through partitions, 199-200,

227, 260
Transmission coefficient of waves across

boundary, 94, 194
Transmission in air, disturbing effects

noted, 111-113,259
Transmission in tubes, 95-103, 229-241

(see Horns, also Tubes)
Transmission Units, defined, 76, (note)
Tubes and pipes, (see also Horns)
driven by piezo-crystals, 243
finite tube, 98-107, 223
impedance of, at unflanged end, 151
impedance of, loo, 101, 102, 104, 106,

223

Kundt's experiment, 102-103, 2 43
with open and closed ends, 104-106
wave propagation in, 96-97, 229-241
with flange, impedance at end, 150
with lateral holes, 106 (note).
Tuning forks, electrically controlled, 244

Ultra sonic waves, (see Wave motion)

272

INDEX OF SUBJECTS

Velocity ', Illustration of mean, for a system,

37-38
Velocity of sound, i, 88, 97, 159, 193, 231,

236, 238-241, 243, 244
effect of dissipation on, 97, 193, 230-

231, 236, 238-241
Velocity potential, 114
due to piston, 138-139, 143
in conical horn, 154-155
in exponential horn, 158
of double source, 135
of point source, 118
Vibrating systems, (see also One degree

of freedom; also Coupled systems)
effect of horn on, 157, 163
general conclusions regarding natural

oscillations, 46
general conclusions regarding steady

state, 48-49

general solution outlined, 49
general theory, 43-52
Vibrator, (see Sound generator)
Viscosity, 19, 55-56, 107, 112, 229, 232,

239
damping due to, 19

in resonator neck, 56

(see also Alr-damplng)
effect of, on waves in tubes, 229-241
kinematic, 1 10, 239

Viscosity, cont.,

Maxwell's definition, 229

of solids, iu-112, 243

waves, 232

Vowel sounds, recent work noted, 250, 252-
253

Wave-distortion, local, 183, 215,220,221,
227

surveys of, 220
Wave motion, I

in absorbing material, 108, 193, 197

in ellipsoid, 216-219

in horns, 153-165

in medium, 88-92, no, 113-120

in narrow conduits, 31, 108, 187, 229-

2 3 W^

in solids, 88, 111-112, 243

in space, 113-116

in tubes, 95-107, 223, 229-241

of finite amplitude, 112

on string, 77

scattering of, by obstacle, 177-178

ultrasonic, 142, 256
Wave power transmission, 100-102, 133

optimum condition for, 102, 169, 223
Whispering gallery, 184, 219

"Youngs Modulus, used, 37, 88, in

ERRATA

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