,dvmtJ.
n
oJ^
A*
INTERNATIONAL SERIES IN PURE AND APPLIED PHYSICS
Cat. P. HARNWELL, Consulting Editor
Advisory Editorial Committee: E. U. Condon, Goorge R. Harrison,
Elmer HutcMasoo, K. EL Darrow
VIBRATION AND SOUND
The quality of Uie materials used in ike manufacture
of this book is governed hy continued postwar shortage*.
INTERNATIONAL SERIES IN
PURE AND APPLIED PHYSICS
G. P. HarkwbI/L, Consulting Editor
Rachhb and QoTOtMn— ATOMIC ENERGY STATES
Bktbb— INTRODUCTION TO FERROMAGNETISM
Brillouin— WAVE PROPAGATION IN PERIODIC STRUCTURES
Cauy— piezoelectricity
Clark— APPLIED XRAYS
Curtis ELECTEICAX MEASUREMENTS
DaveyCRYSTAL STRUCTURE AND ITS APPLICATIONS
Edwards— ANALYTIC AND VECTOR MECHANICS
Haujjv and PaaWH— TH3B PRINCIPLES OF OPTICS
IIauxwellELECTRICITY AND ELECTROMAGNETISM
HABNWBU. and Lmxcoon— EXPERIMENTAL ATOMIC PHYSICS
Houston— PRINCIPLES OF MATHEMATICAL PHYSICS
Hughes and DuBridgk— PHOTOELECTRIC PHENOMENA
HTOO> HIGHFREQUENCY MEASUREMENTS
PHENOMENA IN HIGHFREQUENCY SYSTEMS
KeublePRIXCIPLES OF QUANTUM MECHANICS
Keknabd— KINETIC THEORY OF GASES
Kollbb— THE PHYSICS OF ELECTRON TUBES
Morse— VIBRATION AND SOUND
Paulino and Goudsiot THE STRUCTURE OF LINE SPECTRA
Richtmyerand Kshhab©— INTBODtTCTlONTO MODERN PHYSICS
Ruark and Urey— ATOMS, MOLECULES AND QUANTA
Seitz— THE MODERN THEORY OF SOLIDS
Slatuib— INTRODUCTION TO CHEMICAL PHYSICS
MICROWAVE TRANSMISSION
Slater and Frank— ELECTROMAGNETISM
INTRODUCTION TO THEORETICAL PHYSICS
MECHANICS
SmytheSTATIC AND DYNAMIC ELECTRICITY
Stratton— ELECTROMAGNETIC THEORY
White— INTRODUCTION TO ATOMIC SPECTRA
Williams— MAGNETIC PHENOMENA
Dr. Lrso A. Du Bridge was consulting editor of the series from H139
to 1046.
VIBRATION
AND SOUND
By
PHILIP M, MORSE
Director, Broukhaven National Laboratory
SECOND EDITION
NEW YORK TORONTO LONDON
McGRAWHILL BOOK COMPANY, INC,
1948
VIBRATION AND SOUND
Copyright, 1936, 1048, by the
McCiuAWHiLL Book Company, Inc.
PRINTED IN THE UNITED STATES OF AMERICA
All riykts reserved. This book, or
parts thereof, may not be reproduced
in any form without per mission of
(he publishers,
THE MAPLE PRESS COMPANY, YORK, PA.
TO
PROFESSOR DAYTON C. MILLER
PREFACE TO THE SECOND EDITION
The recent war induced a considerable development in the science
of acoustics and in the mathematical techniques that are particularly
useful in theoretical acoustics. Studies in ultrasonics have quickened
interest in problems of radiation and scattering and in transient
phenomena; and the rapid development of microwave techniques,
which parallel acoustic techniques, has stimulated interest in the
general theory of wave motion.
Reflecting these developments, the present edition of this volume
includes more detail than the first edition on radiation problems and
introduces the important subject of transient phenomena and the
technique of the operational calculus. Both of these subjects arc
usually shunned because of their reputed difficulty. The writer is
convinced that they are not particularly difficult conceptually,
although they usually necessitate calculations of rather wearisome
extent. They are here discussed with the intent to bring out funda
mental ideas, rather than to ensure mathematical rigor of treatment.
In keeping with the plan of the first edition, the more difficult sub
jects are segregated at the terminations of each chapter, so that they
need not be assigned if the book is to be used for a beginning course.
The writer is pleased to acknowledge his indebtedness to a large
number of friends for valuable suggestions as to improvements in
exposition, grammar, arithmetic, and aJgebra. Some of the most
obvious errors in the first edition have been corrected.
Especial thanks are owing J. R. Pel lam, who has expended much
time and effort in checking the manuscript and mathematics, and to
Dr. Cyril Harris for his many helpful suggestions concerning subject
matter.
Philip M. Mohsk
Upton, n. Y.
January, 1948
rii
PREFACE TO THE FIRST EDITION
The following book on the theory of vibrations and sound is
intended primarily as a textbook for students of physics and of com
munications engineering. After teaching the introductory course in
this subject at the Massachusetts Institute of Technology for several
years, the author has become persuaded that there is need for a new
textbook in the field.
There are, of course, many other boobs on the theory of sound.
The author's excuse for adding another to the list is that in the past
ten years the rapid growth of atomic physics has induced a complete
reorganization of the science of acoustics. The vacuum tube and the
other applications of electronics have provided immensely powerful
tools for the measurement, recording, and reproduction of sound;
tools which have revolutionized acoustic technique. Another useful
tool, perhaps not so obvious, is the new mathematical technique which
has been developed for the working out of quantum mechanics, and
which is capable of throwing light on all problems of wave theory.
The last chapter of this book is an example of the utility of these
methods. In it the mathematical methods developed for the study of
the radiation of light from an atom are applied to the theory of the
acoustic properties of rooms.
During the recent rapid change in the science of sound, certain
parts of the subject have gained and other parts have lost importance.
The present book attempts to follow this change in emphasis and to
discuss the new development as well as those portions of the older
theory which are still important.
The book has been planned as a textbook with a twofold aim
in view. The first aim, of course, is to give the student a general
introduction to the theory of vibration and sound. An introduc
tory course in this subject must of necessity be more theoretical than
practical. In no other branch of physics are the fundamental measure
ments so hard to perform, and the theory relatively so simple; and in
few other branches are the .experimental methods so dependent on a
thorough knowledge of theory. Since this is so, the student must
first be given a physical picture of the fundamental theory of the
ix
x PREFACE TO THE FIRST EDITION
vibration of solid bodies and the propagation of sound waves before
he can appreciate the techniques used in the measurements of sound,
and before he can begin to design acoustical apparatus.
The second aim is to give the student a series of examples of the
method of theoretical physics; the way a theoretical physicist attacks
a problem and how he finds its solution. This subject is too often
neglected, especially in engineering courses. The student is usually
given a series of formulas to use in standardized cases, the formulas
sometimes introduced by a cursory derivation and sometimes with no
derivation at all. After such a course the student is capable of using
the formulas on standard problems, but he is unable to devise a new
formula to use in unusual cases.
In this book the author has tried to derive every formula from
the fundamental laws of physics {there arc a few exceptions to this
procedure) and to show in some detail tin: steps in these derivations
and their logical necessity. This docs not mean I. hut the mathematical
machinery is given in excessive detail, but that the steps in the physical
reasoning are brought out. Often generality and mathematical rigor
have been sacrificed to make the chain of logic more distinct. It has
been the author's experience that once the student can grasp the
physical picture behind a mathematical derivation, he can himself add
what extra generality and rigor he may need. Often, too, the author
has supplemented or replaced the rigid and esoteric technical vocab
ulary by more colloquial phrases, in order to make vivid a concept,
or to suggest a new point of view.
It is assumed that the student has a thorough knowledge of
calculus, and some acquaintance with the fundamental laws of
mechanics. A knowledge 6f differential equations is helpful but is
not necessary, for the solutions of the various differential equations
encountered are worked out in the text. Tables of the functions used
are given in the back of the book.
Although the book is designed primarily as a textbook, a certain
amount of material of an advanced nature has been introduced. In
this way, it is hoped, the volume will be useful as a fairly complete
reference work for those parts of the theory of sound which seem at.
present to be most important for the acoustical scientist. The
advanced material has been included in the form of extra sections
placed at the end of various chapters. The instructor may assign the
first few sections of these chapters for the introductory course, and the
student may refer to the other sections for further details when he
needs them.
PREFACE TO THE FIRST EDITION »
The author wishes to express his gratitude to Professor R. D.
Fay and to Dr. W. M. Hall, whose help in choosing subject matter
and methods of presentation has been invaluable. Ho is also indebted
to Dr. J. B. Fisk for his willing and painstaking aid in correcting proof,
and to many other colleagues in the Department of Physics at the
Massachusetts Institute of Technology, for their many helpful
suggestions.
Philip M. Morse
Cambridge, Mass.
August, 1936
CONTENTS
Preface to the Second Edition vii
Preface to the First Edition. . , ix
Chapter I
INTRODUCTORY
1. Definitions and Methods. 1
Units. Energy
2. A Little Mathematics 3
Tho Trigonometric Functions. Beasel Functions. The
Exponential, Conventions as to Sign. Other Solutions.
Contour Integrals. Infinite Integrals. Fourier Trans
forms
Problems . • 1?
Chapter II
THE SIMPLE OSCILLATOR
3. Free Oscillations 20
The General Solution. Initial Conditions. Energy of
Vibration
4. Damped Oscillations 23
The General Solution. Energy Relations
5. Forced Oscillations 27
The General Solution. Transient and Steady State.
Impedance and Phase Angle. Energy Relations. Electro
mechanical Driving Force. Motional Impedance. Piezo
electric Crystals
xiii
xiv CONTENTS
Section
6. Response to Transient Forces 42
Representation by Contour Integrals, Transients in a
Simple System. Complex Frequencies. Calculating the
Transients. Examples of the Method, The Unit Func
tion. General Transient, Some Generalizations. La
place Transforms
7. Coupled Oscillations . 52
The General Equation. Simple Harmonic Motion. Nor
mal Modes of Vibration, Energy Relations. The Case of
Small Coupling. The Case of Resonance. Transfer of
Energy. Forced Vibrations. Resonance and Normal
Modes. Transient Response
Problems . . 66
CfiAl'TEK III
THE FLEXIBLE STRING
8. Wares on a Siring 71
The Wave Velocity, The General Solution for Wave
Motion. Initial Conditions. Boundary Conditions. Re
flection at a Boundary. Strings of Finite Length
9. Simple Harmonic Oscillations 80
The Wave Equation. Standing Waves. Normal Modes.
Fourier Series. Initial Conditions. The Scries Coeffi
cients, Plucked String, Struck String. Energy of
Vibration
10. Forced Vibrations. . . , 91
Wave Impedance and Admittance. General Driving
Force. String of Finite Length. Driving Force Applied
Anywhere. Alternative Scries Form. Distributed Driv
ing Force, Transient Driving Force. The Piano String.
The Effect of Friction. Characteristic Impedances and
Admittances
11. Strings of Variable Density and Tension , . . , 107
General Equation of Motion. Orthogonality of Character
istic Functions. Driven Motion. Nonuniform Mass.
The Sequence of Characteristic Functions. The Allowed
CONTENTS XV
Section
Frequencies. Vibrations of a Whirling String, The Al
lowed Frequencies. The Shape of the String. Driven
Motion of the Whirling String
12. Perturbation Calculations. 122
The Equation of Motion. Firstorder Corrections. Ex
amples of the Method. Characteristic Impedances.
Forced Oscillation. Transient Motion
13. Effect of Motion of the End Supports. . . 133
Impedance of the Support. Reflection of Waves. Hyper
bolic Functions. String Driven from One End. Shape of
the String. Standing Wave and Position of Minima.
Characteristic Functions. Transient Response. Reca
pitulation
Problems M7
Chafer TV
THE V JURATION OF BARS
14. The Equation of Motion 151
Stresses in a Bar. Bending Moments and Shearing Forces.
Properties of the Motion of I he Bar. Wave Motion in an
Infinite Bai
ls. Simple Harmonic Motion 156
Bar Clamped at One End. The Allowed Frequencies.
The Characteristic Functions. Plucked and Struck Bar.
Clampedclamped and Freefree Bars. Energy of Vibra
tion. Nonuniform Bar. Forced Motion
16. Vibrations of a Stiff String 1G6
Wave Motion on a Wire. The Boundary Conditions.
The Allowed Frequencies
Problems 170
ClI.Vl'TKlt V
MEMBRANES AND PLATES
17. The Equation of Motion ', 172
Forces on a Membraue. The Laplacian Operator. Bound
xvi CONTENTS
Section
ary Conditions and Coordinate Systems. Reaction to a
Concentrated Applied Force
IS. Thz Rectangular Membrane. 177
Combinations of Parallel Waves. Separating the Wave
Equation. The Normal Modes. The Allowed Fre
quencies. The Degenerate Case. The Characteristic
Functions
19, The Circular Membrane . , 183
Wave Motion on an Infinite Membrane. Impcrmanence
of the Waves, Simple Harmonic Waves. Bessel Func
tions. The Allowed Frequencies. The Characteristic
Functions. Relation between Parallel and Circular
Waves. The Kettledrum. The Allowed Frequencies
2U. Forced Motion, The Condenser Microphone . 195
Neumann Functions. Unloaded Membrane, Any Force.
Localized Loading, Any Force. Uniform Loading, Uni
form Force. The Condenser Microphone. Electrical Con
nections. Transient Response of Microphone
21. The Vibration of Plates 208
The Equation of Motion. Simple Harmonic Vibrations.
The Normal Modes. Forced Motion
Problems , 213
Chapter VI
PLANE WAVES OF SOUND
22. The Equation of Motion. 217
Waves along a Tube. The Equation of Continuity.
Compressibility of the Gas, The Wave Equation. Energy
in a Plane Wave. Intensity. The Decibel Scale. Inten
sity and Pressure Level. Sound Power. Frequency Dis
tribution of Sounds. The Vowel Sounds
23. The Propagation of Sounds in Tubes . ........... 233
Analogous Circuit Elements. Constriction. Tank. Ex
amples. Characteristic Acoustic Resistance. Incident
and Reflected Waves, Specific Acoustic Impedance.
CONTENTS xvii
Snrnont
Standing Waves. Measurement of Acoustic Impedance,
Damped Waves. Closed Tube. Open Tube. Small
diameter Open Tube. Reed Instruments. Motion of the
Reed, Pressure and Velocity at the Reed. Even Har
monics. Other Wind Instruments. Tube as an Analo
gous Transmission Line. Open Tube, Any Diameter.
Cavity Resonance, Transient Effects, Flutter Echo
24. Propagation of Sound in Horns , . 265
Oneparameter Waves. An Approximate Wave Equation.
Possible Horn Shapes. The Conical Horn. Transmission
Coefficient A Horn Loudspeaker. The Exponential
Horn. The Catenoidal Horn, Reflection from the Open
End, Resonance. Woodwind Instruments. Transient
Effects
Problems 288
CnAI'TKlt VII
TBS RADIATION AND SCATTERING OF SOUND
25. The Wave Equation, . 294
The Equation for the Pressure Wave. Curvilinear Coordi
nates
26. Radiation from Cy folders , 297
The General Solution. Uniform Radiation. Radiation
from a Vibrating Wire. Radiation from an Element of a
Cylinder. Long and Shortwave Limits. Radiation from
a Cylindrical Source of General Type, Transmission inside
Cylinders. Wave Velocities and Characteristic Imped
ances. Generation of Wave by Piston
27. Radiation from Spheres 311
Uniform Radiation. The Simple Source. Spherical Waves
of General Form. Legendre Functions. Bessel Functions
for Spherical Coordinates, The Dipole Source. Radiation
from a General Spherical Source. Radiation from a Point
Source on a Sphere. Radiation from a Piston Set in a
Sphere
xviii CONTENTS
Section
28. Radiation from a Piston in a Plane Wall 326
Calculation of the Pressure Wave. Distribution of In
tensity. Effect of Piston Flexure on Directionality, Radi
ation Impedance, Rigid Piston. Distribution of Pressure
over the Piston, Nonuniform Motion of the Piston.
Radiation out of a Circular Tube. Transmission Coeffi
cient ■ for a Dynamic Speaker. Design Problems for
Dynamic Speakers. Behavior of the Loudspeaker,
Transient Radiation from a Piston
29. The Scattering of Sound 346
Scattering from a Cylinder. Short Wavelength Limit.
Total Scattered Power. The Force on the Cylinder.
Scattering from a Sphere. The Force on the Sphere.
Design of a Condenser Microphone. Behavior of the
Microphone
30. The Absorption of Sound at a Surface 360
Surface Impedance, Unsupported Panel. Supported
Panel. Porous Material. Equivalent Circuits for Thin
Structures. Formulas for Thick Panels. Reflection of
Plane Wave from Absorbing Wall
31. Sound Transmission through Ducts 308
Boundary Conditions. Approximate Solutions. Principal
Wave, Transient Waves. The Exact Solution. An Ex
ample
Problems , , . . 37(3
Chapter VIII
STANDING WAVES OF SOUND
32. Normal Modes of Vibration 381
Room Resonance, Statistical Analysis for High Fre
quencies. Limiting Case of Uniform Distribution. Ab
sorption Coerneinnt. Reverberation. Reverberation Time.
Absorption Coefficient and Acoustic Impedance. Standing
Waves in a Rectangular Room. Distribution in Frequency
of the Normal Modes. Axial, Tangential, and Oblique
Waves. Average Formulas for Numbers of Allowed Fre
quencies. Average Number of Frequencies in Band.
CONTENTS xix
SECTION
The Effect of Room Symmetry. Nonrect angular Rooms.
Frequency Distribution for Cylindrical Room
33. Damped Vibrations, Reverberation 401
Rcct tmgular Room, Approximate Solution. Wall Coeffi
cients and Wall Absorption. Reverberation Times for
Oblique, Tangential, and Axial Waves. Decay Curve for
Rectangular Room. Cylindrical Room. Second order
Approximation. Scattering Effect of Absorbing Patches
34. Forced Vibrations. 413
Simple Analysis for High Frequencies. Intensity and
Meansquare Pressure. Solution in Series of Characteristic
Functions. Steadystate Response of a Room, Rec
tangular Room. Transmission Response. The Limiting
Case of High Frequencies. Approximate Formula for
Response. Exact Solution. The Wall Coefficients,
Transient Calculations, Impulse Excitation. Exact Solu
tion for Reverberation
Problems 429
bibliography 433
Glossary of Symbols .,,,,.,.,,......... 435
Tables of Functions 438
I and II, Trigonometric and Hyperbolic Functions. Ill
and IV, Hyperbolic Tangent of Complex Quantity. V, VI
and YU, Bessel Functions. VTII, Impedance Functions
for Piston. IX, Legendre Functions. XII, General Im
pedance Functions for Piston. XIII, Absorption Coeffi
cients
Plates 453
I and II, Hyperbolic Tangent Transformation.. Ill,
Magnitude and Phase Angles of sinh and cosh. IV,
Standing Wave Ratio and Phase vs. Acoustic Impedance.
V, Hxact Solutions for Wave Modes in Rectangular Ducts
and Rooms. VI, Absorption Coefficient vs, Acoustic
Impedance
Index 459
VIBRATION AND SOUND
CHAPTER I
INTRODUCTORY
1. DEFINITIONS AND METHODS
The discussion of any problem in science or engineering has two
aspects: the physical side, the statement of the facts of the case in
everyday language and of the results in a manner that can be checked
by experiment; and the mathematical side, the working out of the
intermediate steps by means of the symbolized logic of calculus.
These two aspects are equally important and are used side by side in
every problem, one checking the other.
The solution of the problems that we shall meet in this book will,
in general, involve three steps: the posing of the problem, the inter
mediate symbolic calculations, and the statement of the answer.
The stating of the problem to be solved is not always the easiest part
of an investigation. One must decide which properties of the system
to be studied are important and which can be neglected, what facts
must be given in a quantitative manner and what others need only a
qualitative statement. When all these decisions are made for problems
of the sort discussed in this book, we can write down a statement some
what as follows: Such and such a system of bodies is acted on by such
and such a set of forces.
We next translate this statement in words into a set of equations
and solve the equations (if we can).
The mathematical solution must then be translated back into
the physical statement of the answer: If we do such and such to the
system in question, it will behave in such and such a manner. It is
important to realize that the mathematical solution of a set of equa
tions is not the answer to a physical problem; we must translate the
solution into physical statements before the problem is finished.
Units.— The physical concept that force causes a change in the
motion of a body has its mathematical counterpart in the equation
F = Jt (mv) (1.1)
2 INTRODUCTORY [LI
In order to link the two aspects of this fact, we must define the physical
quantities concerned in a quantitative manner; we must tell how each
physical quantity is to be measured and what standard units of
measure are to be used. The fundamental quantities, distance, mass,
and time, can be measured in any arbitrary units, but for convenience
we use those arbitrary units which most of the scientific world is using:
the centimeter, the gram, and the second. (The units of the few other
quantities needed, electrical, thermal, etc., will be given when we
encounter them.)
The units of measure of the other mechanical quantities are denned
in terms of these fundamental ones. The equation F = d(mv)/dt
is not only the mathematical statement of a physical law, it is also the
definition of the unit of measure of a force. It states that the amount
of force, measured in dynes, equals the rate of change of momentum in
gram centimeters per second per second. If force were measured in
other units than dynes, this equation would not be true; an extra
numerical factor would have to be placed on one side or the other of the
equality sign.
Energy. — Another physical concept which we shall often use is
that of work, or energy. The woundup clock spring can exert a force
on a gear train for an indefinite length of time if the gears do not move.
It is only by motion that the energy inherent in the spring can be
expended. The work done by a force on a body equals the distance
through which the body is moved by the force times the component
of the force in the direction of the motion; and if the force is in dynes
and the distance is in centimeters, then the work is given in ergs. The
mathematical statement of this is
W = fPds (1.2)
where both the force F and the element of distance traveled ds are
vectors and their scalar product is integrated.
If the force is used to increase the velocity of the body on which
it acts, then the work that it does is stored up in energy of motion
of the body and can be given up later when the body slows down
again. This energy of motion is called kinetic energy, and when
measured in ergs it is equal to mv 2 /2.
If the force is used to overcome the forces inherent in the system —
the "springiness" of a spring, the weight of a body, the pull between
two unlike charges, etc. — then the work can be done without increasing
the body's velocity. If we call the inherent force F (a vector), then
1.2] A LITTLE MATHEMATICS 3
the work done opposing it is W = — JF • ds. In general, this amount
depends not only on the initial and final position of the body but also
on the particular way in which the body travels between these posi
tions. For instance, owing to friction, it takes more work to run an
elevator from the basement to the third floor, back to the first, and
then to the fourth than it does to run it directly to the fourth floor.
In certain ideal cases where we can neglect friction, the integral W
depends only on the position of the end points of the motion, and in
this case it is called the potential energy V of the body at its final
position with respect to its starting point. If the body moves only in
one dimension, its position being given by the coordinate x, then
Fj>* or ,g)
(1.3)
In such a case we can utilize Newton's equation F = m(dv/dt) to
obtain a relation between the body's position and its velocity.
V = — I Fdx = — ra I (rrjdx = — ra I vdv = — %mv 2 + constant
or
imv 2 + V = constant (1.4)
This is a mathematical statement of the physical fact that when
friction can be neglected, the sum of the kinetic and potential energies
of an isolated system is a constant. This proof can be generalized
for motion in three dimensions.
2. A LITTLE MATHEMATICS
With these definitions given we can go back to our discussion of
the twofold aspect of a physical problem. Let us take an example.
Suppose we have a mass m on the end of a spring. We find that
to keep this mass displaced a distance x from its equilibrium position
requires a force of +Kx dynes. The farther we push it away from its
equilibrium position, the harder the spring tries to bring it back.
These are the physical facts about the system. The problem is to
discuss its motion.
We see physically that the motion must be of an oscillatory nature,
but to obtain any quantitative predictions about the motion we must
have recourse to the second aspect, the mathematical method. We
set up our equation of motion F — m(d 2 x/dt 2 ), using our physical
knowledge of the force of the spring F = — Kx.
4 INTRODUCTORY [1.2
^ + n 2 x = (2.1)
where n 2 stands for (K/m).
The physical statement corresponding to this equation is that
the body's acceleration is always opposite in sign and proportional
to its displacement, i.e., is always toward the equilibrium point
x = 0. As soon as the body goes past this point in one of its swings,
it begins to slow down; eventually it stops and then returns to the
origin again. It cannot stop at the origin, however, for it cannot
begin to slow down until it gets past the origin on each swing.
The Trigonometric Functions. — Equation (2.1) is a differential
equation. Its solution is wellknown, but since we shall meet more
difficult ones later, it is well to examine our method of obtaining the
solution, to see what we should do in more complicated cases. We
usually just state that the solution to (2.1) is
x = a cos (nt) + fli sin (nt) (2.2)
but if we had no table of cos or sin, this statement would be of very
little help. In fact, the statement that (2.2) is a solution of (2.1)
is really only a definition of the symbols cos and sin. We must have
more than just symbols in order to compute x for any value of t.
What we do — what is done in solving every differential equation —
is to guess the answer and then see if it checks. In most cases the
guess was made long ago, and the solution is familiar to us, but the
guess had nevertheless to be made.
So we shall guess the solution to (2.1). We shall make a pretty
general sort of guess, that re is a power series in t, and then see if some
choice of coefficients of the series will satisfy the equation:
x = a + ait + a 2 t 2 + a s t s + a^ A + • • •
For this to be a solution of (2.1), we must have d 2 x/dt 2 , which equals
2a 2 + Qa 3 t + I2a4 2 + 20a 5 t 3 + • ■ • plus n 2 x, which equals n 2 a +
n 2 a x t + n 2 arf, 2 + n 2 a z t z + • • • equal zero for every value of t. The
only way that this can be true is to have the sums of the coefficients
of each power of t each zero by themselves; i.e., a 2 = —n 2 a /2, a 3 =
— n 2 ai/6, a 4 = — ra 2 a 2 /12 = n 4 a /24:, etc. Therefore the series that
satisfies (2.1) is
( i _ VlIl a Vl!l _ Vl!l 4 ^
V 2 + 24 720 + ' ' ' /
V 6 + 120 ' ' /
X = Go
+ aAt
1.2] A LITTLE MATHEMATICS 5
By comparing this with (2.2) we see that what we actually mean by
the symbols cos, sin is
»2 «4 «*>
cos(*) = l 2  + 2l^+ t
sm(z) = 2 _._ + _ _ ^q +
(in the problem 2 = nt) and that when we wish to compute values of
cos or sin we use the series expansion to obtain them. For instance,
cos (0) = 1  + • • • = 1
cos(i) = 1  0.125 + 0.003 • • • = 0.878
cos (1) = 1  0.5 + 0.042  0.002 • • • = 0.540, etc.
We see that the mathematical solution involves certain arbitrary
constants «o and a\. These must be fixed by the physical conditions
of the particular experiment we have made and will be discussed
later (Sec. 3).
Of course, we usually mean by the symbols cos (z), sin (z) certain
ratios between the sides of a right triangle whose oblique angle is z.
To make our discussion complete, we must show that the trigonometric
definitions correspond to the series given above. In any book on ele
mentary calculus it is shown that the trigonometric functions obey
the following relations:
^ cos (z) = sin (z), £ sin (z) = cos (z)
cos(0) = 1, sin(0) =
By combining the first two of these relations, we see that both the
sine and cosine of trigonometry obey the equation d 2 y/dz 2 = — y,
which is equivalent to (2.1). Taking into account the third and
fourth relations, we see that the series solutions of this equation which
correspond to the two trigonometric functions must be the ones given
in (2.3).
Once this is known, we can utilize the trigonometric properties of
the sine and cosine to simplify our discussion of the solution (2.2)
[although, when it comes to computing values, the series (2.3) are
always used]. For instance, we can say that a; is a periodic function
of time, repeating its motion every time that nt increases by 2ir or
every time t increases by (2w/ri). The value of {2k /n) is called the
period of oscillation of the mass and is denoted by the symbol T.
The number of periods per second {n/2ir) is called the frequency of
6 INTRODUCTORY [1.2
oscillation and is denoted by v. Remembering the definition for n,
we see that
t =**4> (1)4
Bessel Functions. — More complicated differential equations can
be solved by means of the same sort of guess that we used above.
For instance, we can solve the equation
s+iS+0?)' ™
by again guessing that y = a + a x x + a 2 x 2 + a 3 x z + • • • and set
ting this guess in the equation. Thus:
a + dix + a 2 x 2 + a 3 x s + • • •
ai — a 3 x — aix 2 — a&x s — • • •
x
— + 2a 2 + 3a 3 x + 4a 4 x 2 + ba h x z + • • •
x
2a 2 + Qa 3 x + 12a 4 ;c 2 + 20a 5 £ 3 + ■ • ■
must equal zero. Equating coefficients of powers of x to zero as
before, we have
a = 0, o 2 = — 7^ = 0, a 3 = — g
rt a 3 ai ,
a 4 = 0, a 5 = ~ 24 = g . 24 * etc 
Therefore,
2/ =
1
minus ^ y =
X 2
, 1 rfw
plus  ~r =
x ax
1 d 2 y
P lus ^i =
/ z 3
^"2^4
2/ = a x [ x  ^ + 2 . 4 . 4 . 6  246468 +
We shall call this series a Bessel function, just as we call each of the
series in (2.3), a trigonometric function. To save having to write
the series out every time, we shall give it a symbol. We let
1/ x s
2\ X 24
Ji(x) = 2V X " 2^1 + 2446 ' ) (2 ' 5)
just as we represent series (2.3) by the symbols cos and sin.
Essentially the series (2.5) is no more complicated than series (2.3).
We can compute values of J\(x) in the same manner as we computed
values of cos (2).
1.2] A LITTLE MATHEMATICS 7
J x (0) =  • • • =
JiG) = 0.250  0.008 ■ • • = 0.242
Ji(l) = 0.5  0.063 + 0.003 • • • = 0.440, etc.
Once someone has obtained series (2.5), given us the symbol Ji(x)
and computed its values, we can use the symbol with as great freedom
as we use the symbol cos (z). We can say that the solution of (2.4)
is y = AJi(x), where the arbitrary constant A is to be determined
by the physical conditions of the problem, as we shall discuss later.
We notice that the solution of (2.4) involves only one function and
one constant A, whereas solution (2.2) involved two constants and
two different functions cos (nt) and sin (nt). Actually, there is another
solution of (2.4), designated by the symbol Ni(x) and called a Neumann
function; but Ni(x) becomes infinite at x = 0, and so we shall not be
able to use it for many of the problems discussed in this book. We
can say that the complete solution of (2.4) is y = AJi(x) + BNx(x)
but that, since a solution representing physical facts must not become
infinite, the constant B must be made zero in all cases where we let x
become zero. These functions will be dealt with more fully in Chap. V.
The Exponential. — Another very useful way of dealing with the
solution of (2.1) can be obtained by the following chain of reasoning.
We utilize the series method to show that a solution of the equation
(dy/dz) = y is the series called the exponential:
z 2 z 3 z 4 z 5
«,= !+* + _ + _ + _ + __+... (2.6)
By repeated differentiation we see that x = Ce at is a solution of the
equation (d 2 x/dt 2 ) — a 2 x = 0. If a 2 were to equal — n 2 , this equation
would be the same as (2.1). Therefore we can say that a solution of
(2.1) is Ce~ int , where i = \/—l (we could also use e +int ).
The function e~ iz is a complex number, with real and imaginary
parts, and can be represented in the usual manner by a point on the
"complex plane" whose abscissa is the real part of the function and
whose ordinate is the imaginary part. It is also represented by the
vector drawn from the origin to this point in the complex plane.
If we expand e~ iz in its series form
'*( 1 5 + H) < (*£ + i35')
we see immediately that the relation between the imaginary expo
nential and the trigonometric functions is
INTRODUCTORY
e i z _ cog (^ _ £ gm ^
cos (2) = i(e i!S + e~ iz )
sin (2) = — %i (e iz — e~ iz )
[L2
(2.7)
This shows that the number e~ iz can be represented on the complex
plane (see Fig. 1) by a vector of unit length inclined at an angle — z
Imag.
Axis
C=cio+icv
ContouiG.
Direction
of Rotation
Direction
of Rotation
Fig. 1. — Representation of complex numbers on the complex plane. Integration
around a contour G in the complex plane. Relation between i and j.
with respect to the real axis. It also shows th at the co mplex number
a + ib can be expressed as Ae'*, where A = Va 2 + b 2 and tan (<3?) =
(b/a). For Ae™ = A cos($) + iA sin(<i>); and since cos(3>) = (a/A)
1.2] A LITTLE MATHEMATICS 9
and sin ($) = (b/A), we have Ae i# = a + i&. The factor e** signifies
a rotation of the vector A through an angle $ in the complex plane.
Thus we see that Cer int is a solution of (2.1) and can be expressed
as a combination of cos (nt) and sin (nt), as is demanded by (2.2).
But this new solution is a complex number, and the results of physics
are, in general, real numbers, so the new solution would seem to have
little value to us. What can be done, however, is to write down the
solution as being Ce~ int and make the convention that we use only the
real part of this solution when we use it to check physical measure
ments. All through this book we shall be writing down complex
solutions of differential equations, with the convention that we use
only their real parts. It is possible to do this with the solution of
any linear differential equation (i.e., equations containing only the
first power of the unknown function and its derivatives) ; for if a com
plex function is a solution of a linear differential equation, then both
its real and imaginary parts by themselves are also solutions. (Why
is this not true for nonlinear differential equations?) We could, of
course, make a convention that we use only the imaginary part of
the function, for the imaginary part is also a solution of the equation,
but the usual convention is to 'take the real part.
We can therefore express solution (2.2) as the real part of
x = Ce~ int , C = a + iax (2.8)
This can be checked by the use of (2.7), for
Ce~ int = a cos (nt) + ia x cos (nt) — ia sin (nt) + ai sin (nt)
and the real part of this corresponds to (2.2). The advantages of
this method will become apparent as we use it. For instance, since
C = Ae { *, A 2 = al + a\, tan(f>) = (ai/o ), we see that the real part
of Ce~ int = 4 e *'c»**) is A cos (nt + <£>). Therefore we can express the
solution of (2.1) in any of four ways:
x = a cos (nt) + ai sin (nt) = A cos (nt — $) (2.9)
or
x = Ce~ int = Aert^v
where our convention requires that we take only the real part of the
last expressions. The constant A is called the amplitude of oscillation
of the mass, since it is the value of the maximum displacement of the
mass from equilibrium.
Our conventions as to the use of complex quantities must be used
with tact when we come to compute powers and energies, where squares
10 , INTRODUCTORY [1.2
Of quantities enter. For instance, in a given problem the power radi
ated may turn out to be Rv 2 , where v is the velocity of a diaphragm
and R is a real number. The velocity may be represented by the
expression v= Ce~ 2 * ivt , C = a + ia h just as x is in Eq. (2.9), But
to compute the average power radiated, we must take the real part
of the expression before squaring and averaging,
Rv 2 = RA 2 cos 2 (2irvt  $)
according to Eq. (2.9) . Since the average value of cos 2 is £, the average
power radiated will be
iRA 2 = iR(a 2 + a 2 ) = i#C 2 = %R\v\ 2
\v\ 2 = (real part of v) 2 + (imaginary part of v) 2
If z — x + iy is any complex number, the quantity \z\ is called the
magnitude of z. It is the distance from the origin to the point in the
complex plane corresponding to z, and it equals the square root of
the sum of the squares of the real and imaginary parts of z. There
fore another important rule concerning the use of complex quantities
in physical problems is the following: the average value of the square
of a quantity represented by the complex function Ce~ iat is equal to
onehalf of the square of the magnitude of C.
The angle <i> in Eq. (2.9) is called the phase angle of the complex
quantity C, since it measures the angle of lag of the quantity Ce~ int
behind the simple exponential e~ int . We notice that the phase angle
of i is 90 deg, that of (1) is 180 deg, and that of (i) is 270 deg or
90deglead. If a; = Ce' int , the velocity v = (dx/dt) =  inCer*" leads
the displacement x by 90 deg, as is indicated by the fact that v =
( — i)nx.
Conventions as to Sign. — In the present volume we use the letter i
to stand for V^l and the symbol e~ int to express simple harmonic
dependence on time. Many books on electrical engineering use the
symbol j instead of i and e jnt with the positive sign in the exponential.
Since we intend to take the real part of the result, the choice of symbol
and sign is but a convention; either convention is satisfactory if we
are only consistent about it.
As long as we are studying simple systems, with displacements a
function only of time, there is little to choose between the two con
ventions; as a matter of fact the positive exponential e int would be
slightly preferable. However, as soon as we come to study wave
motion, with displacement a function of position as well as of time,
it turns out that the form involving the negative exponential e~ int is
1.2] A LITTLE MATHEMATICS 11
rather more satisfactory than the positive. This will become appar
ent later in this book.
Since we are going to deal with problems of wave motion in this
book we shall consistently use the negative exponential e~ int . And
since this convention differs from the positive exponential used in
most electrical engineering books, we shall use the symbol i instead
of j, to emphasize the difference. It will, however, always be possible
to transform any of the formulas developed in this book into the usual
electrical engineering notation by replacing every iin the formulas by —j.
We might, if we wish, consider i and j as the two roots of ( — 1), so
that
i = j; (iy = (j)2 = _i
In this notation, the impedance of a circuit with resistance and induct
ance in series will be R — iuL.
Other Solutions. — Power series are not always the best guesses for
the solutions of differential equations. Now that we have denned the
exponential function, we can sometimes express solutions of other
equations in terms of exponentials. Consider the equation
^ + n 2 x = ae~ ipt (2.10)
where, to represent a physical problem, we must use our convention
on the righthand side of the equation. This could be solved by
expanding the exponential in a series and solving for the series for x
as before. But since an exponential is in the equation, and since we
know that Ce~ int is the solution of (2.10) when a is zero, it will be
simpler to guess that
x = Ce~ int + Be~ lpt
Setting this in the equation, we have that n 2 x = n 2 Ce~ int + n 2 Be~ ipt
plus d 2 x/dt 2 = n 2 Ce~ int  p 2 Be~ ipt must equal ae~ ipt . This means
that B = a/(n 2 — p 2 ). ,If we have used our convention on the term
ae i P t m (2.10), we must use it on the answer. The real part of
x = Ce~ int +
.i 2 — p 2
is
x = a cos (nt) + a x sin (nt) + ^ — ^ cos (pt) (2.11)
if a is a real quantity. If a is complex and equals De**, then we can
12 INTRODUCTORY [1.2
write
x = A cos (nt  4> ) + n2 _ 2 cos (p£  tfO
We notice again that in (2.11) our solution has two arbitrary con
stants a and a x . The arbitrariness corresponds to the fact that this
solution must represent all the possible sorts of motion which the
system can have when it is acted on by the forces implied by Eq.
(2.10). A mass on a spring can have different motions depending
on how it is started into motion at time t = 0. Therefore the par
ticular values of the arbitrary constants a and a x in (2.2) are deter
mined entirely by the physical statements as to how the system was
started into motion. These physical statements are called initial
conditions and are usually stated by giving the position and velocity
of the system at t = 0. More will be said on this point in the next
section.
Contour Integrals. 1 — In a number of cases discussed in this volume
it will be necessary to use integrals of complex quantities of the general
form $F(z) dz, where F = U + iV is a function of the complex varia
ble z = x + iy. Since z can vary over the x,?/plane, it will in general
be necessary to specify the path over which the integration is carried
out. A natural extension of our usual definition of integration indi
cates that the integral is the limit of a series of terms, a typical term
being the product of the value of F{z) for a value of z on the path, and
a quantity dz = \dz\e i& , with length \dz\ and with phase angle & deter
mined by the direction of the tangent to the path at the point z.
The integral is the limit of such a sum as [dz\ goes to zero. This is
shown in Fig. 1. Another definition is that jF dz = j[U + iV][dx
f i dy] where x and y are related in such a manner that the integral
follows the chosen path.
When the path chosen is a closed one, the integral is called a
contour integral (labeled f) and the chosen path is called a contour.
Contour integrals have certain remarkable properties, proved in
standard texts on theory of functions of a complex variable. We
have space here to outline only some of these properties. For sim
plicity it will be assumed that the function F{z) to be dealt with is a
smoothly varying, reasonable sort of function over most of the z
plane.
The value of a contour integral depends largely on whether the
integrand F(z) becomes zero or infinity for some value or values of z
1 This section need not be studied unless the later sections on transient phe
nomena are to be studied.
1.2] A LITTLE MATHEMATICS 1,3
inside the contour. Near such points F(z) would take on the form
(z — z ) n R(z) where R(z ) is neither zero or infinity and R(z) is not
discontinuous near z . When n is not an integer, positive or negative,
the point z is called a branch point and the problem becomes more
complicated than is necessary to discuss here. When n is a negative
integer, the point z is called a pole of F(z), the pole f or n = — 1 being
called a simple pole and that for n = —2,3 • • • being called a poZe
of second, third, etc., order. The statement at the beginning of this
paragraph can now be made more specific: the value of the contour
integral $F(z) dz depends on the behavior of F at the poles and
branch points that happen to be inside the contour.
When F has no branch points or poles inside the contour, the value
of the contour integral is zero. This can be verified by a tedious bit
of algebraic manipulation for those cases where F can be expressed
as a convergent power series in z.
When F has a simple pole at z = z , but has no other pole or branch
point inside the contour (see Fig. 1), then the contour integral can
be shown to be equal to that around a vanishingly small circle drawn
about z . Since z is specified as being a simple pole, F(z) has the
form R(z)/(z — z ) near z where R(z) is continuous and finite in
value near z . We can write the equation for the resulting circle as
z — z = ee iv and dz = ue iv d<p, where e is vanishingly small. The
contour integral then reduces to
£F(z) dz = R(z ) f ' l * e% \ d<P = iR(z ) I d<p = 2iriR(z )
Jo &* Jd
where R(z ) — lim [(z — z )F(z)) is called the residue of F{z) at its
z—*zo
simple pole zo.
When F(z) has N simple poles, at z , z x • • • zn, and no other poles
or branch points, then the contour integral j>F{z) dz equals (27ri)
times the sum of the residues of F at the poles that are inside the chosen
contour. We note that the direction of integration around the con
tour is counterclockwise; changing the direction will change the sign
of the result.
This is a remarkably simple result; in fact, it seems at first too
simple to be true, until we realize that the function F(z) we are dealing
with is a very specialized form of function. F is not just any complex
function of x and y; its dependence on these variables is severely
limited by the requirement that it be a function of z = x + iy. It
can have the form bz 2 + (C/z) or z sin(z), but not the form x sin(z) f
iy or \z\. For such a specialized function F(z), which is called an
14 INTRODUCTORY (1.2
analytic function, the positions of its branch points and poles, and its
behavior near these branch points and poles, completely determine
its behavior everywhere else on the complex plane. When this is under
stood, the result we have quoted for the value of a contour integral
does not seem quite so surprising.
As an example of the rule of residues, we can take the integral
<f sin (2) dz/(z 2 — a 2 ), where the contour is a circle of radius greater
than a, with center at the origin 2 = 0. The integrand has two
simple poles, at 2 = a and z = —a, with residues (l/2a) sin (a) and
— (l/2a) sin ( — a). The result is therefore
*^*OW>[*f]
A special case of the residue theorem can be stated in equation
form
§
z
dz = 2tiR(z ) (2.12)
where the contour of integration does not enclose a branch point or
pole of R(z). By differentiating once with respect to 2 on both sides
of the equation, we obtain the equation
§
, R(Z \ 2 dz = 2riR'(z ) (2.13)
\Z Zoj
where R'(z) = dR(z)/dz. This indicates how we can evaluate con
tour integrals around poles of second order. Cases for poles of higher
order may be dealt with by further differentiation of Eq. (2.13).
Infinite Integrals. — An oftencountered class of integral may be
evaluated in terms of contour integrals. Consider the integral
/". w
dz
where the integral is taken along the real axis. If F(z) has no branch
points or poles of higher order, if the simple poles of F are not on the
real axis, and if F(z) goes to zero as \z\ goes to infinity, then the integral
can be made into a contour integral by returning from + 00 to — ».."
along a semicircle of infinite radius. When t is positive (to make it
more general we should say when the real part of t is positive, but let
us assume that t is real), the exponential factor ensures that the integral
along this semicircle is zero if the semicircle encloses the lower half
of the 2 plane. For along this path, 2 = pe iv = p cos <p + ip sin <p,
1.2] A LITTLE MATHEMATICS 15
with p infinitely large and with ^ going from to — ny so that sm<p
is negative. Therefore along the semicircle
prizt ^ g— ipt cos <P + fit sin (P
As long as sin<p is negative this vanishes, owing to the extremely large
value of p. Therefore the addition of the integral around the semi
circle turns it into a contour integral but does not change its value.
The final result is that the integral equals (— 2xi) times the sum of
the residues of F(z)e~ izt at every one of its poles located below the real
axis. The negative sign is due to the fact that the contour in this
case is in the clockwise direction.
When t is negative, we complete the contour along a semicircle
enclosing the upper half of the zplane, which now has the vanishing
integral, and the resulting value is (27rc) times the sum of the residues
at each of the poles above the real axis. The limitations on F have
already been given.
As an example, we can write down the result
« 2 f " e ~** _  (a/2)e at (t < 0)
%r J . z 2 + a 2 dZ \ (a/2)e~ at (t > 0)
practically by inspection. Another result of interest later is
ia f °°
2^ J_
^ *{•?_ <!<2 (214)
z + ia [ ae~ at (t > 0)
In some cases of interest, F has poles on the real axis. These cases
can also be treated by making the convention that the integration is
not exactly along the real axis but is along a line an infinitesimal
amount above the real axis. With this convention, and for functions
F(z) that have only simple poles and that vanish at \z\ — * « , we have
the general formula
sum of residues of \
(2ti) I ^'^ at aU itS I it > 0)
\ poles on and below J
/
+ » „ , , . ± , I \the real axis /
F{z)e>»dz=( pi«~~» (2l5)
/sum of residues ot\
+ (2wi)( Fe ~ Ut at aU itS  (t < 0)
\ poles above the I
\real axis /
./^ f^^^eU
16 INTRODUCTORY [1.2
As an example, we can write down the very useful formulas
_^p e i, (t<0)
2tJ„z*  a*™ \ sin (<rf) (< > 0) C ^ 17;
_J_p^T 0>O)
Fourier Transforms. — When the quantities / and F are related by
the formula
/(0 = j^ a) F{z)e i " t dz (2.19)
then / is said to be the Fourier transform of F. A great deal of abstruse
mathematical reasoning, which we shall not go into here, is needed to
prove that if / is the Fourier transform of F, according to Eq. (2.19),
then
F{z) = ± J ^ f(f)e** dt (2.20)
so that, contrariwise, 2tF is the Fourier transform of /. Naturally
Eqs. (2.19) and (2.20) are valid only when F and / are reasonable sorts
of functions, which approach zero in a proper manner as \z\ approaches
infinity.
When f(t) is zero for t < 0, as it is in Eqs. (2.14), (2.16), (2.17),
and (2.18), this reciprocal integral may be written
2irJo
FHV) =2^J d f( l ) e ~ Pt dt (2.21)
where p = — iz. This integral converges satisfactorily as long as the
real part of p is positive, i.e., as long as the imaginary part of z is
positive.
Formula (2.21) provides a simple means of checking Eqs. (2.14)
to (2.18).
The usefulness of these integral formulas will become apparent
in the succeeding chapters, as we deal with problems of the response of
vibrating systems to transient {i.e., nonperiodic) forees. If, instead
of Eq. (2.10), we had put
d 2 x C "
^f + n 2 x = I F(p)e^ dp
Ml' A LITTLE MATHEMATICS 17
then the solution for x could have been expressed in terms of a con
tour integral. This problem will be considered in the next chapter.
Problems
1. A mass m. slides without friction on a horizontal table. It is attached to a
light string which runs, without friction, through a hole in the table. The other
end of the string is pulled downward by a constant force F. The mass (which is
too large to drop through the hole) is held at rest a distance D from the hole and
then let go Set up the equation of motion of the mass, and solve it. Is the
motion periodic? If so, what is its frequency, and how does the frequency depend
onZ)?
2. A bead of mass m slides without friction on a straight wire that is whirling
in a horizontal plane, about one end, with a constant angular velocity w radians
per sec. It is found that the centrifugal "force" on the bead is mw 2 r, where r is
the distance of the bead from the center of rotation. The direction of the force is
away from the center. Set up the equation of motion of the bead, and solve it.
Is the motion periodic, and, if so, what is the frequency?
3. Show that a solution of the equation
g+*g+»°
can be represented by the series
_ c A _ n 2 x 2 ntx* n 6 x" \
\ 2 2 ~*~ 2 2 • 4 2 2 2 4 2 6 2 ' " /
The series inside the bracket i3 called the Bessel function J (nx), where J (z)
= 1  (2 2 /2 2 ) + • • • . Compute values of /,(0), Jo(l), and J (l) to three
places of decimals.
4. Show that the solution of the equation
g + (1  kx 2 ) y =
is
y = a D e (k, x) + aiD„(k, x)
where D e and D are symbols for the following series:
nri.~\ . * s 7k . , Ilk + 15k 2 ,
D e (k,x) ooB* + i5 **3g 6 *« + ^j )Mr  **
n ti. \ • , * « 13* , , 17* + 63* 2 „
D (k,x)  sinx + 25 *•  2 52 q *' + ^720" *
Compute values of D e and D , to three places of decimals, for * = 1 and for x =» 0,
i, and 1.
6. Show that the solution of the equation
Tx [ (1 ~ x2) 2] + Xy = ° is y = a <>P e (\x) + ai P (\,x)
18 INTRODUCTORY 0L2
where P, and Pt, are symbols for the following series:
P.(X,x) = 1  ±x>  ^^ x«  M6X)(20A ) ;c< ,
P,(M)» + V' , + <a "" X) 5 ( ? 2 "" X) "'
 (2X)(12A)(30\) x , {
Compute values of P e and P , to three places of decimals, for X = 0, 1 and 2, for
x = 0, £, and 1.
6. Show that the solution of the equation
J^(* 2 0 + ( 1 §) y = O is »**<*>+«»*»<*>
where ji and ni are symbols for the following functions:
. , . sin a; cosx . N cosx sin a;
Ji(x) = — ; , n x (x) = 5
x % x ' x 2 x
What are the values of j t and rti at x = 0, £ and 1 ? What is the solution of the
equation
?s(£)+» M
7. What is the length of the line drawn from the origin to the point on the
complex plane represented by the quantity (a + ib)~*? What is the angle that
this line makes with the real axis? What are the amplitudes and phase angles of
the following quantities:
(a  to)*; (a + t&) 1 + (c + id) 1 ; (a + ib) n e iat
8. What are the real and imaginary parts, amplitudes, and phase angles of
Va + ib, log (a + ib), e  2Ti(  v+i ^ t , e~ ix (l + e^ 5 )" 1
9. The hyperbolic functions are defined in a manner analogous to the trigo
nometric functions [see Eq. (2.7)].
cosh (z) = §(e* + e  *); sinh (z) = §(e* — e~ g )
tanh (z) = , ,{ = — ,. . . (see Tables I and II)
cosh (z) coth (z) v '
Show that cosh (iy) = cos(y); cos (iy) =cosh(y); sinh (iy) =i sin(y); sin (iy)
= i sin (y). Find the real and imaginary parts of cosh (x + iy), cos (x + iy),
sinh (x + iy), sin (x + iy). What are the magnitudes and phase angles of these
quantities?
10. Find the real part R and imaginary part X of tanh (x + iy).
Show that
B. + ^ + ootCWj^
1.2] A LITTLE MATHEMATICS 19
Plot the curves on the complex plane for tanh (x + iy) when y is allowed to vary
but x = 0, (ir/10), oo ; when x is allowed to vary but y = 0, (tt/4), (x/2).
11. Where on the complex plane are the poles of the functions 1 /cosh (z) ; tan (z) ;
e «/ z 2( z 2 _ a 2). tan(«)/z(z 2 +a 2 )? Are they all simple poles? Calculate the
residues at all the simple poles of these functions.
12. Compute the values of the integrals
/» g—izt /• 00
, , , — 5T dz; I tan (z)e
 «. z(z 2 + a 2 ) J *
rfz
z
X2x
(1 — 2p cos + p 2 ) 1 d0, to a contour
integral over z, by setting e i9 = z. What is the shape of the contour? Where are
the poles of the integrand, when p is real and smaller than unity? Calculate the
value of the integral when < p < 1.
14. Calculate the value of the integral
16. Show that
_ il f°° ■
f
x 2 dx
» (1 + x 2 )(l  2x cos + x 2 )
l — (_l)n e i2(nir/u) I (^ < 0)
g2 _ m2 \dz = ^ sin {at) [0 < t < (nw/o>)]
( [(n7r/a>) < t]
CHAPTER II
THE SIMPLE OSCILLATOR
3. FREE OSCILLATIONS
Now that we have discussed, to some extent, the mathematical
methods that we shall need in our work, we shall come back to the
physics.
The whole study of sound is a study of vibrations. Some part
of a system has stiffness; when it is pulled away from its position of
equilibrium and then released, the system oscillates. We shall first
study the simplest possible sort of vibrations for the simplest sort of
system, a mass m fastened to some sort of spring, so that it can oscil
late back and forth in just one direction. A very large number of
vibrators with which we deal in physics and engineering are of this
type or are approximately like it. All pendulums (the "spring" here
is the force of gravity) are like this, and all watch balance wheels.
Loudspeaker diaphragms which are loaded so that their mass is con
centrated near their center are approximately like this (at least at
low frequencies), as are loaded tuning forks, etc. Even when an
oscillating system is more complex than the simple oscillator, many of
its properties are like it. Later, when we study these more compli
cated systems, we can simplify our discussion considerably by point
ing out first the properties wherein the system behaves like the simple
case and then showing where it differs.
The forces exerted by the various springs in the examples of the
simple oscillator mentioned above have one property in common:
The restoring force opposing the displacement of the mass from its
position of equilibrium is proportional to this displacement if it is
small enough (we remember the discussion of Hooke's law for a spring
and the discussion of the force on a single pendulum given in the ele
mentary physics course). In some problems it is not very accurate to
assume that the displacements are small enough, but in our work it is
a good assumption, for the vibrations with which we deal in sound have
displacements that are very small indeed. For instance, an air mole
cule needs to vibrate with an amplitude of motion of only about a
tenth of a millimeter to do its full part in transmitting away the racket
20
H.3] FREE OSCILLATIONS 21
generated at Times Square on New Year's Eve. Seldom does the amp
litude of oscillation of a loudspeaker diaphragm exceed a millimeter.
The General Solution. — We shall assume, then, that the "springi
ness" force acting on the simple oscillator can be expressed by the
equation
F = Kx (3.1)
where x is the displacement of the mass from equilibrium, K is called
the stiffness constant (its value depends on the sort of spring we use),
and the minus sign indicates that the force opposes the displacement.
This will be a very good assumption to make for most of the cases
dealt with in this book. The reciprocal of the stiffness constant,
C m = (1/K), is called the compliance of the spring.
To start with, we shall consider that no other forces act on the
oscillator. This is, of course, not a good assumption in many cases;
usually there are frictional forces acting, and sometimes external
forces come in. There are many cases, however, where the frictional
force is negligible compared with the springiness force, and we shall
treat these first, bringing in friction and external forces later.
This brings us to the equation of motion (2.1) discussed in Sec. 2:
^ = co 2 z, w 2 = (K/m) = (l/mC m ) (3.2)
We have already seen that the solution of this equation can be ex
pressed, in terms of our convention, as
x = Ce™ 1 , C = a + ia x
or as
x = a ccs {u)t) + «i sin (ut) (3.3)
We must now discuss the physical implications of this solution.
It has been shown in Sec. 2 that the moti on of the body is periodic,
having a frequency v = (o)/2ir) = (l/27r) s/K./m. This frequency is
larger for a stiffer spring (for a larger K) and is smaller for a heavier
mass.
Initial Conditions. — We have seen that the specific values of the
constants a and ai are determined by the way that the mass is started
into motion. Ordinarily, we start an oscillator into motion by giving
it a push or by pulling it aside and letting it go, i.e., by giving it, at
t = 0, some specified initial displacement and initial velocity. Once
we have fixed these two initial conditions, the motion is completely
determined from then on, unless we choose to interfere with it again.
It is not hard to see that a is the value of the initial displacement
22 THE SIMPLE OSCILLATOR [IL3
and ojai that of the initial velocity. Solution (3.3) can then be rewrit
ten in the following forms:
x = x cos (2rvot) + ( ~ J sin (2rvot)
= 4 cos (2™ * <*>); v = Q^^;} (3.4)
* * + &)'.■ tan * = fek) 
where ar is the initial displacement and v the initial velocity of the
mass.
These equations reemphasize the fact that only the initial value of
x and of (dx/dt) need be given to determine the subsequent motion of
an oscillator completely. Once x and v are specified, then, even if
we did not have a completely workedout solution of (3.2), we
could find the initial value of the body's acceleration by inserting x
on the righthand side of (3.2). The initial value of the third deriva
tive can be obtained by differentiating (3.2) with respect to t and
placing the value of v on the righthand side in place of (dx/dt);
and so on. If it is recalled that the behavior of a function is com
pletely specified by its Taylor'sseries expansion
(dj\ ,?(d?f\ ,t2(<pf\
W*o 2 W/«o "*" 6 \dt*/ t
m = /co) + 1 y; (=o +  2 ^ + ff ^ +  . .
(within a certain range of t whose limits are of interest in specific
problems, but which need not bother us here), we see that once x
and v are given, thus fixing the values of all the higher derivatives
at t — 0, the future motion is determined.
It is not hard to generalize this reasoning, to see that a body, acted
on by any sort of force that depends on x and on v, will have its motion
specified completely just by assigning definite values to its initial
position and velocity. The mathematical counterpart to this state
ment is the rule that the solution of any secondorder differential
equation (one having a second derivative term in it but no higher
derivative) has two arbitrary constants in it.
Another very important physical fact which can be deduced from
(3.4) is that the frequency of the oscillation depends only on K and
m and not at all on xo or v . This means that for a given mass and
a given spring, as long as the law of force of the spring is F = —Kx,
then the frequency of oscillation will be the same no matter how we
U.4] DAMPED OSCILLATIONS 23
start the system to oscillate, whether it oscillates with an amplitude
of motion of 1 cm or 0.001 mm. This is a very important fact in its
practical applications, for if the. law of force of actual springs were not
nearly F — —Kx, or if this property did not hold for solutions of
Eq. (3.2), then no musical instrument could be played in tune. Imag
ine trying to play a piano when the frequency of each note depended
on how hard one struck the keys! Compare this with Prob. 1, page
17.
We might have found this fact for a number of cases by a long
series of experimental observations, but our mathematical analysis
tells us immediately that every mass acted on by a force F = —Kx
has this property. Oscillations of this type are called simple harmonic
oscillations.
Energy of Vibration. — We shall need an expression for the energy
of a mass oscillating with simple harmonic motion of amplitude A and
frequency v . The energy is the sum of the potential and kinetic
energies
W = \mv 2 + JT* Kx dx = %mv 2 + iKx 2
= 2K 2 mv\A 2 sin 2 (2W  $) + \KA 2 cos 2 (2W  *)
But 4t 2 v% = (K/m) so that
W = i£A 2 [sin 2 (2W  $) + cos 2 (2W  $)]
or
W = \KA 2 = 2K 2 mv\A 2 = \mU 2 (3.5)
where U = 2ttv A is the velocity amplitude of the motion.
The total energy is thus equal to the potential energy at the body's
greatest displacement (iKA 2 ) or is equal to the kinetic energy at the
body's greatest speed (kmll 2 ). Expressed in terms of v and A, we
see that W depends on the square of these two quantities.
4. DAMPED OSCILLATIONS
So far, we have not considered the effects of friction on oscillating
systems. In general, friction does not play a very important role in
the problems that we shall consider in the first part of the book. If
we show its effects on the simplest system with which we deal, we can
deduce its effect, by analogy, on more complicated systems; therefore,
we shall mention friction only occasionally, later on in this book.
The friction that is most important in vibrational problems is the
resistance to motion which the air surrounding the body manifests.
24 THE SIMPLE OSCILLATOR [TLA
Energy in the form of sound waves is sent out into the air. From the
point of view of the vibrating system, this can be looked on as friction,
for the energy of the system diminishes, being drained away in the
form of sound. This resisting force depends on the velocity of the
vibrator, and unless the velocity is large (much larger than those with
which we shall deal), it is proportional to the velocity. It can be
expressed mathematically as —R m (dx/dt), where the constant R m is
called the resistance constant. The total force on a simple oscillator
acted on by both friction and springiness is therefore — R m (dx/dt) —
Kx, and the equation of motion becomes
dt 2 + dt + 47r VoX " °' 4tt 2 . 2 = (K/m) (4>1;
The value of v is the frequency that the oscillator would have if the
friction were removed (R m were zero) and is called the natural frequency
of the oscillator.
It is interesting to notice that the equation for the free oscillation
of charge in a circuit containing inductance, resistance, and capacitance
has the same form as (4.1). The inductance is analogous to the mass
m, the resistance to the resistance factor R m , and the inverse of the
capacitance to the stiffness constant K.
The General Solution. — To solve Eq. (4.1) we make use of the
exponential function again. We guess that the solution is Ce bt and
solve for b. Substitution in (4.1) shows that (6 2 + 2kb + 4r 2 v$)e bt
must be zero for all values of the time. Therefore
6 2 + 2kb + 4tt 2 ^ = 0, or b = k ± Vk 2  4tt 2 ^
In all the problems that we shall consider, the stiffness constant K
is much larger than the resistance constant R m {i.e., friction will not
be big enough to make the motion much different from that discussed
in the last section). Since 4r 2 vl is supposed to be larger than k 2 ,
then the square root in the expression for b is an imaginary quantity,
and we had better write
b = k ± %riv f , v f = vo Vl ~ (k/2rv ) 2 (4.2)
This means that, following our convention, we can write
x = Ce fc < 2 ™"< (4.3)
or
x = e ki [a cob (2rv ft) + «i sm(2irv f t)] = A e kt cos(2ttj7J  $)
The values of a and a x are again fixed by the initial conditions for
H.4] DAMPED OSCILLATIONS 25
the oscillator; a must equal the initial displacement x , and the initial
velocity in this case is equal to %cv } a x — ka , so that
eti
( t»o + kx<\
2rv f . J
The solution is not periodic, since the motion never repeats itself,
each swing being of somewhat smaller amplitude than the one before
it. However, if A is quite small compared with v f , we can say that
it is very nearly periodic. In any case, the frequency of the oscilla
tions is v f , which is very nearly equal to v if k is small. It again turns
out that the frequency is independent of the amplitude of the motion.
Of course, strictly speaking, we should not use the word frequency in
connection with nonperiodic motion. But when the damping is small,
the motion is almost periodic, and the word will have some meaning,
although a rather vague one.
There are several respects in which the motion of the damped oscil
lator differs from that of the simple oscillator. The most important
difference is that the amplitude of motion of the damped oscillator
decreases exponentially with time; it is A e~ kt , instead of being just A
(A is the initial value of the amplitude). The amplitude decreases
by a factor (1/e) in a time (1/k) sec (e = 2.718). This length of time
is a measure of how rapidly the motion is damped out by the friction
and is called the modulus of decay r of the oscillations. The fraction
of this decrease in amplitude which occurs in one cycle, i.e., the ratio
between the period of vibration and the modulus of decay, is called
the decrement 8 of the oscillations. Another method of expressing this
is in terms of the "Q of the system," where Q = (co m/# m ) is the num
ber of cycles required for the amplitude of motion to reduce to (1/e*)
of its original value. If these quantities are expressed in terms of
the constants of the system (the small difference between v f and v
being neglected), it turns out that
«fe)(?> **$ *£) r441
r=l = ^; 5 = l = l = J ( ( }
k irvn' Q v Q tvq J
The smaller R m is, the larger Q and r are, indicating that it takes
a longer time for the oscillations to damp out, and the smaller 5 is,
indicating that the reduction in amplitude per cycle is smaller. These
properties, of course, are independent of the way the oscillator is
started into motion.
Another difference between the damped and the undamped oscil
26 THE SIMPLE OSCILLATOR [II.4
lator is the difference in frequency. When (k/2rv ) is small, the
expression for v f can be expanded by means of the binomial theorem,
and all but the first two terms can be neglected.
Vf
=v^(sy = "(§^) + ' (4  5)
In most of the cases with which we shall be dealing, k and v have such
values that even the second term in the series is exceedingly small, so
that the change in frequency is usually too small to notice.
Energy Relations. — The subject of damped oscillations can be
considered from a quite different point of view — that of energy loss.
We must first develop an expression for the average energy of the
system at any instant. We cannot use the formulas (3.5) because
the amplitude of oscillation in the present case is not constant. The
sum of the kinetic and potential energies of a body of mass m acted
on by a springiness force Kx, whose displacement is given by the
formula A(t) cos{2irvt — <£), is
W(t) = \mv 2 + iKx 2
= 2K 2 mv 2 A 2  2mmv (^7 J A sm(2rvt  <3>) cos(2tt^  <S>)
+ im i^£\ cos 2 (2rvt  *)
When averaging this value over a single oscillation, the second term
on the right drops out. If A is very slowly varying, so that (dA/dt)
is small compared with vA, then the third term can be neglected,
and we have the approximate formula for the energy of motion and
position {i.e., the energy that can be recovered, that is not yet irre
vocably lost in heat)
W(t) ~%R[A(t)] 2 = 2T 2 mv 2 [A(t)Y = im[U(t)] 2 (4.6)
where the symbol ~ means "is approximately equal to." In the
case of the damped oscillator, this "free energy" is 2K 2 mv%A\e~ 2U \
which diminishes exponentially with time.
The rate of loss of energy due to friction is equal to the frictional
force opposing the motion R m v multiplied by the velocity v (since force
times distance is energy, rate of change of energy is force times
velocity). The rate of loss of energy is
P = R mV 2 = [Air 2 R m v f 2 sin 2 (2jrvft  4>)
+ 4;irR m Vfk sin (2irv f t — 3>) cos (2irv f t — 3>)
+ Rmk 2 cos 2 (2rv ft  $)]A§e~ 2fct
D.6] FORCED OSCILLATIONS 27
Using the same approximations as before, we have for the average loss
of energy per second •
&)
Wv\R m A\e™ = %R m [U(t)]* (4.7)
If we had started out without formula (4.3) for the details of the
motion but had simply said that the free energy at any instant was
given by (4.6) and that the energy loss was given by (4.7), where R m
is small, we could have found the dependence of W on the time by
means of (4.7). Eliminating U from Eqs. (4.6) and (4.7) results in
P ~ (R m /m)W = 2kW, so that we have
p = ~ (t)  2kW > or w  +2kW ( 4  8 )
The solution of this is
W~ W e~ 2kt = imU%e 2kt = %c i v\mA\e' lu
which checks with Eq. (4.6). We see from this that the damping out
of the motion is required by the fact that the energy is being lost by
friction. We might point out here that the fraction of free energy lost
per cycle is just (2k/v f ) ~ (2k/v ) = (2w/Q), where Q is given in
Eq. (4.4).
In nearly every more complicated case of vibrations, the effect of
friction is the same as in this simple case. The amplitude of vibration
slowly decreases, and the frequencies of natural oscillation are very
slightly diminished. Usually, the change of frequency is too small to
be of interest.
5. FORCED OSCILLATIONS
It often happens that a system is set into vibration because it is
linked in some way with another oscillating system (which we shall
call the driving system). For instance, the diaphragm of a microphone
vibrates because it is linked, by means of sound waves, to the vibra
tions of a violin string; and a loudspeaker diaphragm vibrates because
it is linked to the current oscillations in the output circuit of an ampli
fier. The system picks up energy from the driving system and oscil
lates. In the two instances mentioned, and in many others, the driven
system does not feed back any appreciable amount of energy to the
driving system, either because the linkage between the two is very
weak (as is the case with the violin and microphone) or else because
the driving system has so much reserve energy that the amount fed
28 THE SIMPLE OSCILLATOR [II.6
back is comparatively negligible (as is the case with the amplifier
and loudspeaker). In these cases the only property of the driving
system that we need to know is that it supplies a periodic force which
acts on the driven system. The more complicated case, where the
feedback of energy cannot be neglected, will be considered in the next
section.
The General Solution. — For the present, we ask what happens to
a simple oscillator when it is acted on by a periodic force F cos (2irvt),
or Fe~ 2lcivt , according to our convention. We wish to know what its
motion is just after the force has been applied and, more important,
what its motion is after the force has been acting for a long time. We
are also interested in how this behavior depends on the frequency v
of the driving force (which does not have to be the same as the natural
frequency v of the oscillator).
The total force on the oscillator is a combination of the springiness
and the frictional and the driving force R m v  Kx + Fe~ 2 * ivt . The
equation of motion is
^§ + 2k d 4i + 4ttV .t = ae 2 * M , a = (F/m), k = (R m /2m),
at 2 at
co = 2rv, 4tt 2 ^ = (K/m) (5.1)
We discussed a similar equation in Sec. 2, where we showed that a
choice of two exponentials, one corresponding to the free vibration
of the oscillator and the other to the forced motion, was a good guess
for the solution. Substituting x = Ce~ u '^ iv t t + Der 2 * M in (5.1), we
find that it is a solution if
n _ a = v(vl  v 2 ) + ivk = R itf
U ~ 4tT 2 (^  V 2 )  ^ivk ±TC\V% ~ V 2 ) 2 + \KV 2 k 2
where
B = (<*/4ir) _ (F/2tp)
\Ar 2 ("o ~ v 2 ) 2 + v 2 ¥
4
RI.+
o K
ZTV
and
tan# =
Rr
*{v\  v 2 ) (K/2kv)  2Tvvm
The solution can be thrown into a semblance of simplicity, and
the analogy with electric circuits can be made more apparent, if we
define a mechanical impedance for the system
II.5] FORCED OSCILLATIONS 29
Z m = R m  dam  — J = R m  iX m = \Z m \e~ i » (5.2)
co = 2wv t \Z m \ = ^Jr* + U m _ ±)
„ = *  90° = tan [ " m ~^ /c ° ]
We recall that the usual electricalengineering notation is obtained
by substituting —j for i, so we can see that the equation for Z m is
exactly analogous to the equation for the complex electrical impedance
of a series circuit, with the mechanical resistance R m analogous to the
electrical resistance, the mass m analogous to the electrical inductance
and the mechanical compliance C m = (l/K) analogous to the electrical
capacitance. The quantity X m = urn — (K/co) can be called the
mechanical reactance of the system. The units in which this mechanical
impedance are expressed are not ohms, for the quantity is a ratio
between force and velocity rather than between voltage and current.
The symbolic analogy is close enough, however, to warrant the use
of the same symbol Z with the subscript m to indicate "mechanical."
The units of mechanical impedance are dyneseconds per centimeter,
or grams per second.
The solution of Eq. (5.1) can therefore be written in either of the
two alternate forms
F
X = Q e hi1riv t t _ «"•'
lirivZn
or
\2Tv\Z m \J
x = e~ kt [a cos (2irv f t) + a t sin (2irv f t)] + I  — r T? — x ) cos (2irvt — &)
\ZTV\Z/ m \/
The constants a and ai are determined, as before, by the initial posi
tion and velocity of the mass.
Transient and Steady State. — When the force is first applied, the
motion is very complicated, being a combination of two harmonic
motions of (in general) different frequencies. But even if the friction
is small, the first term, representing the free, or "transient," vibra
tions, damps out soon, leaving only the second term, which represents
simple harmonic motion of frequency equal to that of the driving
force (see examples of this in Fig. 2).
30
TEE SIMPLE OSCILLATOR
IIL6
x *■ —
F
2irivZ r ,
e  *"', or x — »
i?
2tv\Z„
sin (%irvt — ■ <p)
where
y ~ * \ 7" ) e ~ iat ' 0r U — * \ l7~T ) C0S ^^ ~ ^
(com)  (X/co)
(5.3)
tan co = — cot# =
Rt!
) co = 2^^
This part of the motion is called the steady state. We see that it is
completely independent of the way in which the oscillator is started
V=" 2 A> f
V=Vf
\=2v*
Time*
Fig. 2. — Forced motion of a damped harmonic oscillator (A; = weff/10) Curve a
shows free oscillations, and curves b, c, and d show forced oscillations due to sudden
application of force at t = 0. Dotted curves give force as function of time; solid curves
give displacement. Effect of transient is apparent at the left side of the curves; at the
right side the steady state is nearly reached.
into motion, its amplitude, phase, and frequency depending only on
the constants F and v of the force and on the oscillator constants m,
R m , and K. No matter how we start the oscillator, its motion will
eventually settle down into that represented by (5.3).
II.5] FORCED OSCILLATIONS 31
Steadystate motion is motion of a system that has forgotten how
it started.
Impedance and Phase Angle. — The amplitude and velocity ampli
tude are proportional to the amplitude of the driving force and are
inversely proportional to the magnitude of the mechanical impedance
Z m . The analogy with electric circuits is therefore complete. The
velocity corresponds to the current, the mass to the inductance, the
frictional constant to the resistance, and the stiffness constant to
the reciprocal of the capacitance.
The impedance is large except when v = v , but at this frequency,
if the friction is. small, it has a sharply defined minimum. Therefore,
the amplitude of motion in the steady state is small except when
v = v , where it has a sharp maximum. The case where the frequency
v of the driving force equals the natural frequency v Q of the oscillator,
when the response is large, is called the condition of resonance. The
peak in the curve of amplitude against v is sharp if R m is small and is
broad and low if R m is large, as is shown in Fig. 3. This figure also
shows that the steadystate motion of the oscillator is not very sensi
tive to the value of the frictional constant except in the range of fre
quencies near resonance. The dotted curve for the amplitude of
motion is for a value of R m eleven times that for the solid curve, yet
the two are practically equal except in this frequency range.
The motion is not usually in phase with the force, the angle of lag
of the displacement behind the force being given by the angle &,
which is zero when v = 0, is (x/2) when v = v , and approaches x as
v approaches infinity (indicating that the displacement is opposite in
direction to the force). The angle of lag of the velocity behind the
force, <p = + (x/2) + #, is analogous to the phase angle in ac theory.
It is —k/2 when v = 0, zero when v = v , and +7r/2 when v is very
large.
In other words, when the frequency v of the driving force is much
smaller than the natural frequency v of the oscillator, then the ampli
tude is small, and the displacement is in phase with the force. As v
is increased, the amplitude increases and gets more and more out of
phase with the force, until at resonance the amplitude is very large
(if R m is small), and the velocity is in phase with the force. As v is
still further increased, the amplitude drops down and eventually
becomes very small. For these large values of v the displacement
is opposed to the force. Figures 2 and 3 illustrate this behavior
We use systems driven by periodic forces in two very different ways.
One type of use requires the system to respond strongly only to par
32
THE SIMPLE OSCILLATOR
[II.6
ticular frequencies (examples of this are the resonators below the bars
of a xylophone, the strings of a violin, and the human mouth when
shaped for a sung vowel). In this case we must make the friction
as small as possible, for then the only driving frequency that produces
a large response is that equal to the natural frequency of the driven
system. The other type of use requires the system to respond more
or less equally well to all frequencies .(examples are the diaphragms
of microphones and loudspeakers and the sounding board of a violin).
Fig. 3. — Amplitude and phase of forced motion, as functions of the frequency v
of the driving force. The frictional constant R m for the dotted curve is eleven times
that for the solid curve, all other constants being equal.
In some cases we wish the steadystate amplitude to be independent of
the frequency; in others we wish the amplitude of the velocity to be
constant; and in still other cases we would like the acceleration to
have a constant amplitude. One or another of these requirements can
be met within a certain range of frequencies by making the stiffness,
the friction," or the mass large enough so that its effects outweigh those
of the other two in the desired range of frequency.
These three limiting types of driven oscillators are called stiffness
controlled, resistancecontrolled, and masscontrolled oscillators, respec
(5.4)
II.6J FORCED OSCILLATIONS 33
tively. Their properties and useful ranges of frequency are
Stiffness controlled: K large ; Z m ~ i f  — h x ~ I =z \ e _iat
v considerably less than both ( ?r I » /— and ( n „ I
\27r/ \ m \2trR m /
Resistance controlled: R m large ; Z m ~ R m ; ( — J ~ ( — J e~ iut
v considerably less than I p^ \ larger than ( _ „ J
\2irm/ \2TR m /
Mass controlled: m large; Z m ~ — Qnrivm; ( r^ I ~ ( — ) e~ iwt
v considerably larger than both \wj */— and ( ^^ ]
It is to be noticed that every driven oscillator is mass controlled in
the frequency range well above its natural frequency v , is resistance
controlled near v (though this range may be very small), and is
stiffness controlled for frequencies much smaller than v . It simply
requires the proper choice of mechanical constants to place one or
another of these ranges in the desired place in the frequency scale.
It is also to be noted that there is always an upper limit to the fre
quency range over which an oscillator is stiffness controlled, a lower
limit to the range over which one is mass controlled, and both an
upper and a lower limit to the range over which an oscillator can be
resistance controlled. We can move these limits about by changing
the mechanical constants, but we never remove the limits entirely.
Energy Relations. — The average energy lost per second by the
oscillator due to friction, when in the steady state, is P = %R m U 2 =
iR m (F/\Z m \) 2 [see Eq. (4.7) of the last section]. The rate of supply of
energy from the driver to the driven oscillator is
F 2
vF cos (2irvt) = j=, cos (2irvt) cos (2irvt — <p)
\Z n
F 2
\Z m \
cos {2irvt) [cos <p cos (2Kvt) + sin <p sin (2Tcvt)]
The average value of this energy supplied per second by the driver
is HF 2 /\Z m \)cos<p = %R m (F/\Z m \) 2 (since cob* = RJ\Z m \), which
equals the loss of energy to friction P. One can say that the amplitude
and phase of the driven oscillator so arrange themselves that the energy
delivered by the driver just equals the energy lost by friction.
34 THE SIMPLE OSCILLATOR [II.5
Electromechanical Driving Force.— There is one particular form of
coupling between the driver and driven systems that is of particular
interest to us in the study of sound: the electromechanical coupling
between a vibrating diaphragm and an electrical driving circuit. In
practically all acoustical work at present, the sound vibrations dealt
with are transformed into electrical oscillations in order to amplify
them. After amplification they are then changed back into sound
waves, if need be, and an electromechanical coupling between the
amplifier and the sound generator is used. Such coupling can have a
variety of forms. For instance, the amplifier current can be sent
through a coil wound on a magnet, varying the magnetic field and thus
varying the force on a steel diaphragm (this is the coupling used in
the ordinary telephone receiver). Or the current can be sent through
a coil, attached to the diaphragm, which is placed in a magnetic field
whose direction is perpendicular to the coil winding (this coupling is
used in the socalled dynamic loudspeakers). Or the electric voltage
may be impressed across a piezoelectric crystal, causing it to change
size and shape (this coupling is used in crystal microphones and
loudspeakers).
In the magnetic forms of coupling the mechanical force exerted on
the driven oscillator is proportional to the instantaneous current I
through the coil, F = DI dynes, where the factor of proportionality D
has a value that depends on the particular sort of coupling system
that is used. For instance, if the coupling coil is fastened to the
diaphragm and consists of n turns, each of radius b cm, and if the coil
is placed in a radial magnetic field whose intensity is B gauss at the
position of the winding, then D = 2irnbB/10, if the current is measured
in amperes. In certain cases D may be a complex quantity, as, for
example, in an ordinary telephone receiver when hysteresis must be
taken into account.
If the current is an alternating one, represented by loe 2 "*', the
force on the oscillator is DIoe~^ ivt , and if the driven system behaves
like the simple oscillator discussed in this section, its steadystate dis
placement and velocity will be, according to Eq. (5.2),
x = ^° e 1 '**, v = §^° <r*"«, a = 2™ (5.5)
The motion of the driven system produces a back emf in the coil.
Owing to the reciprocal relation between the force on a current in a
magnetic field and the emf produced by motion in the same field, we
can immediately say that the back emf produced in the coil by moving
II.6] FORCED OSCILLATIONS 35
the driven system with a velocity v is vD/10 7 volts, where D is the
same constant that appeared in the expression for F. The back emf
produced in the coil due to its motion is therefore
Em = y~ he*"' = 151
where the constant r = D 2 /10 7 is called the electromagnetic coupling
constant. The effective electrical impedance in the amplifier circuit due
to the motion of the driven system is the ratio between Em and I. This
complex quantity Z M
r
Zm = Rm — iXM = 77 — TY m = T(G m — iB m )
<6m
where
G m —
( R m \ B _ _ (X m \ Y _ _1_ iv \ (5.6)
toj' Bm ~ \\z m \*)' Ym ~ \z m \ e (
\Z m \*=Kl + Xl; Z m = com^
Z M is called the motional impedance of the coil. It is a true electrical
impedance, measured in ohms, in contrast to the mechanical impedance
Z m . The quantity Y m = {\/Z m ) = G m — iB m is called the mechanical
admittance, with units seconds per gram or centimeters per dyne
second, having a real part G m , called the mechanical conductance, and
an imaginary part B m , called the mechanical susceptance. For the case
of electromagnetic coupling the electrical impedance due to the motion
of the coil is proportional to the mechanical admittance of the coil.
The constant of proportionality r has the units of ohmgrams per
second. We label all electrical impedances with a capitalletter sub
script to distinguish them from the mechanical impedances Z m , Z r , etc.
Equations (5.6) indicate that the larger the mechanical impedance
of the coil (i.e., the harder it is to move), the smaller the motional
impedance of the coil (the less the back emf due to its motion). A
little thought will show that this is a reasonable result.
Motional Impedance. — It is interesting to notice that an electrical
circuit with an inductance whose magnitude equals (T/K), a resistance
of magnitude (T/R m ) and a capacitance (m/T), all in parallel, will give
an electrical impedance of just the value given in (5.6). As far as the
electrical circuit is concerned, the loudspeaker mechanism is com
pletely equivalent to the circuit shown in Fig. 4. If the complex
impedance of the coil when the loudspeaker is rigidly clamped is
Re — 2irivLe, then its impedance when the loudspeaker is allowed to
36
THE SIMPLE OSCILLATOR
[H.5
move is Z E — Z c + Z M , where Z c = Rc — iwLc and Z M = T/Z m . Z c is
called the clamped or blocked impedance of the coil and Z E its total
electrical impedance.
When a current Z = Joe  ""* is sent through the coil and no external
force F e is applied, there will be a total voltage drop E = Z E I across
its terminals. This voltage can be considered to be the sum of the
drop across the blocked impedance Z C I and the motional emf E M —
Z m Im, which is the voltage drop across the part of the circuit enclosed
in a dotted line in Fig. 4. Once the equivalent electrical circuit is
determined, and the mechanical behavior of the coil, its velocity and
r
M echanical System _
Current /^Source F e /P
10
Im
<r/K)
^O'OOWO'OOWH
(r/Rm)
wwwvww
(m/r)
E m .
— ^OOOOO'OOW* — vwwwwv '
L c R c
Fig. 4. — Electrical circuit equivalent to the driving coil of a loudspeaker or
"dynamic" type of microphone. Effective electrical impedance of mechanical system
is equal to that of circuit enclosed in dashed line. Effect of external force F e applied
to coil is equivalent to constantcurrent source of strength (F e /D) applied as shown.
displacement, the power Po lost owing to heating the coil and the power
used in moving the coil P M can all be determined in terms of the voltage
Em across and the current Im through the mechanical part of the
circuit :
E M = ImZm volts
r
V
~f
X =
(= Z M IMF e = 0)
Em = VlO^T \~\ cm/sec
cm;
2iriv
Pm = %Rm(\Im\ 2 ) — ^rj?,
Rc
watts
(5.7)
\Z n
\v\
iR m tkv watts
In the last of these equations, we see the relation between the electrical
and mechanical units. On page 26 we showed that the expression
II.6] FORCED OSCILLATIONS 37
ii2 m v 2 is equal to the power dissipated by th'e mechanical resistance
R m . Mechanical power is measured in ergs per second, however, and
it takes 10 7 ergs per sec to equal 1 watt. The relationship between T
and D 2 provides just this factor, so that Pm, computed from the equiva
lent electrical circuit, comes out in watts.
There are several interesting corollaries from Eq. (5.7) which are
worth mentioning. The mechanical impedance discussed in the
example can be represented by a sum of impedances torn, R m , and
(K/—iw), as in a series circuit. The equations are more general than
this, however: if the mechanical load on the coil can be represented
by a mechanical admittance Y m = (1/Z TO ) which is any complex func
tion of co, then the motional impedance of the coil due to this load is
TY m . For instance, if the mechanical load consists of a spring and a
resistance in parallel, the mechanical admittance is (l/R m ) — (ioo/K)
and the motional impedance is Z M = (T/R m ) — (iwT/K) ohms, as
though an electrical resistance (T/R m ) and inductance (T/K) were in
series. Later in the book we shall consider impedances due to radia
tion loads. Here again, to find the motional impedance Z M) we multi
ply the total mechanical admittance of diaphragm plus air load by I\
If the coil is shortcircuited and is then moved by a mechanical
force, the mechanical load is greater than the mechanical impedance,
because the motion of the coil induces a current in the coil which pro
duces an additional reaction force. The additional mechanical imped
ance, due to current in the shortcircuited coil, is Z s = T/Z c g per sec
ana is the counterpart, for the mechanical system, of the motional
impedance Z M for the electrical circuit. The total mechanical imped
ance of the diaphragm and shorted coil is Z m + Z 8 . The quantity
Z a can be called the magnetomotive impedance.
This magnetomotive impedance can be utilized to obtain an alter
native formula for the velocity of the diaphragmcoil system when
driven by an emf E = Etfr™ % impressed across the coil
DIm DE DE
v =
Z m z m (Zc + Zm) Z m z c + r
= DE / Z c = Die
Zm ~T" Z s Z m \ Z s
where I c is the current that would flow through the coil if it were
clamped. This formula is of use in computing the change in velocity
of a diaphragm when the mechanical load is changed but the electrical
circuit is kept the same. In such a case D, I c , and Z„ would be
unchanged and only Z m would change.
38 THE SIMPLE OSCILLATOR [II.5
When an external force F e is applied to the coil, as occurs when the
system is used for a microphone, an additional term must be included
in the equations. It turns out that the corresponding addition to the
equivalent circuit is a constantcurrent generator, of strength (F e /D)
amp, applied across the "mechanical" part of the circuit, as shown in
Fig. 4.
For mechanical resonance the mechanical reactance arm — {K/w)
is zero, the magnitude \Z m \ of the mechanical impedance is a minimum,
and the magnitude Y/\Z m \ of the motional impedance is a maximum.
Other quantities being equal, at resonance \E M \ has its maximum value,
and consequently the velocity magnitude v is maximum. At still
higher frequencies there may be a series resonance between the induc
tance L c and the capacitance (jn/T).
So many kinds of impedance have been discussed in this section
that it might be well to list them together to contrast their meaning
and the units in which each are expressed:
Z E = total electrical impedance, in ohms
Z c = clamped electrical impedance, in ohms
Z M — motional impedance, electrical, in ohms
Z m = mechanical impedance, in g per sec [ w8)
Z r = radiation impedance, in g per sec
Z 8 = magnetomotive impedance, in g per sec
Throughout the book, whenever there is danger of confusion, we shall
use capitalletter subscripts for electrical impedances, ratios between
volts and amperes; and we shall use lowercase subscripts for mechanical
impedances, ratios between forces and velocities.
Piezoelectric Crystals. — Rochelle salt crystals are also electro
mechanical transducers, transforming electrical into mechanical energy,
and vice versa. For the "a>cut" type, for instance, the crystal is cut
in a rectangular form. A force is applied between two faces (between
face S and the rigid table T in the figure) by means of some mechanical
system (such as a diaphragm) with effective mechanical impedance
Z m . Voltage then develops between two other parallel faces, which
can be measured across the terminals A and B. Alternately, a voltage
applied across A and B will cause a force to be exerted on the mechanical
system attached to face S. The dimensions of the crystal a, b, and
d are shown in the figure.
In this case the displacement from equilibrium of face S, which
we. can label x, is due to the crystal expanding lengthwise. The
expansion per unit length ij = (x/q) is called the strain. The mechan
n.6]
FORCED OSCILLATIONS
39
ical force causing the expansion or contraction is the external stress X in
dynes per sq cm. This stress, in a piezoelectric crystal, causes an
electric polarization n, in_ this case perpendicular to the condenser
plates. The polarization in turn causes a charge to be formed On the
plates, in addition to that caused by the external applied field. For
ease in initial calculation we shall use electrostatic units for the first
steps. The relations between these internal quantities £, X, and n
External Force F»
echanical Load
I Y't'/'/W i
Source
J
Mechanical System
Fig. 5. — Typical arrangement of piezoelectric crystal, showing electrical connections
and direction of application of force F e  An equivalent electrical circuit is shown. For
other crystal cuts the metal plates would be on the top and bottom, rather than on a
pair of sides.
and the external quantities F e , x, the potential difference E between the
plates and the total charge Q on the plates are given by the set of
equations
F e = bdX + Z m v, v = — io)X = —iu£a
(5.9)
40 THE SIMPLE OSCILLATOR [II.6
where ki is the dielectric susceptibility of the unloaded crystal, su is
one of its elastic moduli, and du is the appropriate piezoelectric con
stant. The constants /c, s, and d vary markedly with temperature.
The first of the fwe equations (5.9) gives the relation between the
driving force F e , the internal stress, and the motion of the mechanical
load. The second is the usual relation between velocity and displace
ment for simple harmonic motion. The third is the usual electrical
relationship between surface charge density, applied field E/d, and
polarization II. The last two equations are peculiar to piezoelectric
materials. The equation for the polarization in the particular direc
tion we have chosen is related both to the applied field (which is
usual) and also to the mechanical stress. The strain in the vertical
direction depends in the usual manner on the stress; but there is an
additional dependence on the applied field. Our equations are given
for one particular cut of crystal, with force and field applied in a par
ticular way. Other arrangements with respect to the crystal axes
will change the last two of Eqs. (5.9), the modified equations involving
other constants k, d, and s labeled by different subscripts.
We have also assumed that the frequency of oscillation is smaller
than the lowest resonance frequency of the crystal; otherwise the
stress, strain, polarization, etc., will not be uniform throughout the
crystal. The more complicated case of high frequencies will be taken
up later.
After a large amount of algebra, we can obtain an expression for
the current I = —iuQ in terms of the applied voltage
t . n /o6\ /1 , ,, n ■ (bdg/su ZmjFe + ioi(abd\J sud)E
1 = " UaE \&Td) (1 + 4TKl) + ' 1  (l/ia,)(db/asuZ m )
Still further manipulation can be applied to show that this relation
between I and E is equal to that due to the equivalent circuit shown
in Fig. 5, where
^rff 4 io" radg { (510)
v d s 4 4 9 '
t = 300 (r^y 1 ) volts/dyne, a = 10 7 r £ ohmsec/g
and where the transition from electrostatic to practical units has been
made.
The constant a can be called the piezoelectric coupling constant,
II.6]
FORCED OSCILLATIONS
41
analogous to r in the electromagnetic case. It changes the mechanical
impedance of the load into electrical impedance in the equivalent cir
cuit. Values of these constants for Rochelle salt, for various tem
peratures in degrees centigrade, are given in Table 1.
Table 1.— Piezoelectric Constants for Rochelle Salt (See Fig. 5)
Tempera
Capacitance, nni
ture °C
t, juv/dyne
a, ohms/g
tor
T
(Cd/ab)
(Ccd/ab)
(C p d/ab)
br X 10 6
10
35.6
15.9
19.7
315
0.992
 5
25.6
13.3
12.3
370
1.369
19.4
10.5
8.9
425
1.806
5
16.7
9.2
7.5
462
2.134
10
16.7
8.8
7.9
454
2.061
15
22.2
10.9
11.3
396
1.568
20
42.2
16.7
25.5
291
0.847
23.7
Upper Curie poi
nt
24.7
132.3
24.4
107.9
209
0.437
25.7
71.5
19.7
51.8
240
0.576
28.2
38.1
15.4
22.7
303
0.918
31.0
23.4
12.2
11.2
382
1.459
From H. Mueller, Properties of Rochelle Salt, Phys. Rev., 57, 829, and 68,
565 (1940). These constants depend strongly on temperature, particularly near
the Curie points. The corresponding constants for quartz are much less dependent
on temperature.
We notice several important differences between this equivalent
circuit and the one in Fig. 4. In the first place, the part of the circuit
corresponding to the mechanical load is a series circuit, not a parallel
one; so that if m, R m , or K is very large, the equivalent electrical
impedance is very large, instead of very small, as is the case in Fig. 4.
A very large value of Z m corresponds to loading or clamping the crystal
so that its upper face cannot move. In this case the equivalent elec
tric circuit reduces to the condenser C c (the subscript c indicating
"clamped"). If the crystal is unloaded and the external force F e is
zero, the equivalent circuit consists of the two condensers in parallel,
having a combined capacitance of C farads.
In the present case the external applied force F e appears as a voltage
source, in series with the impedance equivalent to the mechanical
load, instead of as a current source parallel to this impedance, as in
Fig. 4.
The velocity of the top face of the crystal is proportional to the
current in the "mechanical" arm of the equivalent circuit. The equa
03
42 THE SIMPLE OSCILLATOR [H.6
tions giving this velocity and the power used in moving the connected
mechanical system are
v = 10 7 tI p , x =  10 7 ( ^ j I P
•*"• ' = 300 fe) i (511)
f\v\ 2 \
Pm = i(rR m \I P \ 2 = iR m I t^7 ) watts
We note that the equivalent capacitance C p corresponds to an addi
tional stiffness, which is the stiffness of the crystal. The effective
series capacitance of the mechanical arm of the circuit is Cm = [(1/C P )
+ Kt 2 ]~ x , and the corresponding total stiffness of the mechanical
system is
Mechanical resonance will occur when the equivalent reactance
t 2 [ojw — (K t /a))] is zero. At this frequency, the maximum current
will flow in the mechanical arm for a given applied voltage E. At a
higher frequency, parallel resonance will occur with the shunt capaci
tance C c .
6. RESPONSE TO TRANSIENT FORCES
In the previous section we dealt with the response of various simple
mechanical and electromechanical systems to simple harmonic driving
forces. In particular, we studied the steadystate response to such
forces after the transient effects, due to the force starting, have died
out. In this section 1 we wish to follow a little further the study of the
transient effects, particularly when the driving force itself is transient
and the system never does settle down into steady driven oscillation.
To carry this out we shall use the techniques of complex integration
discussed in Chap. I.
Representation by Contour Integrals. — The general method of
attack is as follows : we express our transient driving force as an integral
of the form of Eq. (2.19)
f(t) = f^° M F(w)er^ t da
such as the ones given in Eqs. (2.15) to (2.18), where F can be deter
mined by the use of Eq. (2.20). We next assume that the solution x
1 This section need not be studied unless the other sections in the book on
transient phenomena are to be studied.
II.6] RESPONSE TO TRANSIENT FORCES 43
is also of thisform : ^ v : 
x ® = f„ "X(fi>)e~ iut 'da> (6.1)
so that if X is known x can be determined; or, vice versa, if x is known,
X can be obtained by the related equation
2xJ_
X(<a) = y I x(t)e iu>t dt
By this mathematical trick we can change from a differential equation
relating x and / to an algebraic equation relating X and F. Finally,
Eq. (6.1) can be used to obtain x.
This method may seem rather mysterious and long winded.
Actually, it is a mathematical restatement of a rather simple principle.
Equation (2.19) says that a transient force can usually be expressed
as a limiting sum {i.e., an integral) of a whole series of component
simple harmonic forces of differing frequency (w/2tt) and amplitude
F((a) dca. The individual equations for each of the component forces
are solved to find the steadystate amplitudes X(a)) du>, and these are
then finally combined by the limiting sum of Eq. (6.1) to give the
required displacement x(t). To obtain the transient motion, we
analyze it into its component simple harmonic steadystate motions
and then recombine at the end.
In most problems of practical interest, we can assume that the
system is at rest at equilibrium until t = 0, when the transient is
suddenly applied. Therefore, we can assume (at least for the time
being) that the functions fit) and x(t) differ from zero only f or t > 0.
In this case we can use Eq. (2.21) to suggest a simpler form for X
X(ip) = 2^1 x(t)e*" dt (6.2)
where p = —i<a and the real part of p is positive. There is a similar
equation giving F(ip) in terms of f(t).
One general relation of great usefulness is the expression for the
Fourier transform of (dx/dt). This can be obtained from Eq. (6.2)
by integration by parts, or from Eq. (6.1) by differentiation inside
the integral sign, if this is permissible.
2xJ \dt)
e~ pt dt = pX '= — icoX
(6.3)
Additional terms must be added if x is not zero for t < 0,
44 THE SIMPLE OSCILLATOR [II6
Transients in a Simple System.— By the use of these formulas we
can transform the general differential equation
m w + Rm % + Kx = m * £» F(o})e ' iwt do}
into the algebraic equation
{K  ioiR m  co 2 m)X(co) = F(u)
by multiplying both sides of the differential equation by (l/2ir)e~ pt
and integrating over t. The differential equation, of course, corre
sponds to a simple mechanical system of mass m, resistance R m , and
stiffness K, acted on by force f(t).
The equation relating X and F is a familiar one, dealt with earlier
in our discussion of steadystate response to a simple harmonic force.
The equation
X(co) = F(w) = F{0,) (6.4)
K — iwRm — co 2 ra —ioiZ{u>)
where
Z(co) = R m — i (com J
is simply the expression for the amplitude of the component X(co),
having frequency co, in terms of the component i^(co) entering into
the integral for the total transient force. The final expression for the
displacement and velocity of the simple system acted on by force
/(0 is
x O = J TTn^ y w = ty x du (6.5)
J =<, — 2coZ(co) J «» Z(co)
where
F(p) = ^j Q fV) e  pt d *> V = t«
These equations are valid if the imaginary part of co is always positive
in the integration and if x, v; and / are all zero for negative values of
t [this incidentally means that F has no poles above the real axis;
see Eq. (2.15)]. In order that these integrals converge there must be
certain obvious restrictions on the way Z behaves as co goes to infinity
To solve the contour integrals for x and v we must investigate the
poles of the function F/(—iaZ). The poles of F(w) cannot be investi
gated until we choose a form for f(t), but we can discuss the factor
l/(— icaZ). For this factor to have poles, the function — z'coZ(co)
must have zeros for some values of u>, real or complex.
11.6}
RESPONSE TO TRANSIENT FORCES
45
Complex Frequencies. — We have thus arrived at a broadening of
our concept of impedance. Originally Z was defined as the complex
ratio between a simple harmonic force of frequency (co/2r) = v and
the corresponding steadystate velocity. We now are keeping to the
same definition, but are broadening the concept by allowing o> to be
a complex quantity, with real and imaginary parts. In particular,
we are searching for those complex values of co for which (— zcoZ) is
zero.
For a. simple mechanical system ( — ioZ) = — co 2 ra — iu>R m f K.
This can be factored to make the zeros apparent
(—iooZ) = — m(co  (0/ + ik)(ca + co/ + ik) (6.6)
where k = (R m /2m), u> f = co 2 , — k 2 , co§ = (K/m), in accordance with
Eqs. (4.1) and (4.2). The values of co for which this function goes
to zero therefore have for a real part plus or minus (2x) times the
frequency of free vibration of the mechanical system and have for an
imaginary part minus the damping constant of the system. At these
values of co the function F/(—io)Z) will have simple poles, unless by
extraordinary coincidence F has zeros or poles at these same values
of oj (which eventuality we shall not consider).
Calculating the Transients. — The poles of the function F/(—ioiZ)
that are due to Z are therefore closely related to the transient oscilla
tions of the system discussed in Sec. 5. The points on the complex
plane where these poles occur are symmetrically placed with respect
to the origin, are both a distance co = \/K/m from the origin, and
are both a distance k below the real axis. The residues of Fe' Mt /
(—iwZ) at these poles are
and
2mo)f
g — Utft
_ kt F(o> f i k)
2mco/
at co = — oif — ik
> at w = co/ — ik
As long as jP(co) has only simple poles, and as long as none of these
coincides with the poles of l/(— iuZ), we can utilize Eq. (2.15) to
write down the result for the displacement of the system
x(t) =
ir 1
x(t) = — — tf*'[e*»/«iP(— co/  ik) — er^'WCa/ 
(2iri) times the sum of the residues
oiFer^t/i—iwZ) at all simple poles
of F on and below the real axis of co
(for t < 0)
ik)]
(for t > 0)
(6.7)
46 THE SIMPLE OSCILLATOR [II.6
The first term is the transient motion of the system due to the sudden
onset of F(t) at t = 0. The terms due to the poles of F(w) are a sort
of generalized steadystate motion and depend on the specific form
<*/(*).
The corresponding expression for v{t) is
v{t) = (for t < 0)
v(t) = JL e kt [(a f + i^e^Fioif  ik)
+ (w/  itye^f'Ficof  ik)] ^ (6.8)
{2iri) times the sum of the residues of
F e iut/z at al i po i es f p on or b e i ow (for t > 0)
the real axis of o>
Further discussion requires choosing specific forms for / and F.
Examples of the Method. — We shall first take a case that we have
touched on earlier, the case where a simple harmonic force of frequency
v = (a/2r) is turned on at t = 0. The force function chosen is
f /0 (*<0)
J \F e iat (t > 0)
According to Eq. (2.18), the corresponding Fourier transform is
(F /2ti)
F(co) = 
co — a
To find the resulting displacement of the system we insert the
expression for F into Eq. (6.7) and turn the algebraic crank. After a
number of turns, the first part, due to the poles of l/(— icaZ), turns
out to be
F e kt ( e™' 1 e~ iaft \
2mo)f \co/ + a + ik o> f — a — ik)
while the term due to the pole of F is F e iat /[ — iaZ(a)]. Further sub
stitution of expressions for co , oi f , and k in terms of K, m, and R m
results in the final equation for a;
... F e iat F e~ kt , '... . ,
— iaZ(a) 2a) f [—iaZ(a)] '
+ («/ + a + ik)e™*\ (for t > 0)
which is to be compared with the results of page 29. The final
numerical results can be obtained by taking the real part of the expres
sions for f(t) and x (t).
II.6] RESPONSE TO TRANSIENT FORCES 47
No doubt this result could be obtained with less trouble by the
simpler methods outlined in Sec. 5. However, we shall find later that
the present method is more useful in discussing transient effects in
more complicated systems, where the simpler method breaks down.
The present examples are discussed so that we can become familiar
with the method.
The Unit Function. — Another force function which will be useful to
study is that of a force suddenly applied at t = and maintained at a
steady value thereafter. From Eq. (2.16) we see that the Fourier
transform of
*«««{? J > g » ™  " (as) (6.9)
The function u{t) is called the unit step function, often used in transient
problems. The resulting expression for the displacement of the same
simple system due to the application of the unit step force is therefore
(6.10)
1 1
x u (t) = I — — ~ e~ kt cos (<j) f t —
(*<0)
a)
(*>0)
tan a — —
<*f \/w§ — k 2
The first term is the "steadystate" displacement from equilibrium
caused by the unit force after t = 0, which comes from the residue at
the pole of F(«) (at o> = 0). The second term is the transient oscil
lation due to the sudden application of the force at t = 0, which comes
from the residues at the poles of l/( — icaZ).
One sometimes wishes to know what happens to a system that is
held aside from equilibrium for a long time and then is suddenly
released at t = 0. A force producing this behavior would be f(t) =
1 — u(t), a unit force applied steadily until t = and then released.
The response for such a force would be
*w = (A)  «.<« = i w* ? , „ , ('<«
\ K J I (wo/Ka> f )e kt cos (<a f t a) (t > 0)
If the force holding the system away from equilibrium is F instead of
unity, the corresponding motion is x = F x h (t).
If the force is an impulsive one, applied instantaneously at t = 0,
the integral of this force over time (the impulse of the force) must be
proportional to the unit step function. We can define a unit impulse
function 8(t) as one that is zero for every value of t except zero and
48 THE SIMPLE OSCILLATOR [H.6
that has there such a value that
jl ^ 5(r) dr = u(t) (6.11)
The function 8(t) is sometimes called the Dirac delta function. Look
ing back at Chap. I, we see that the function defined in Eq. (2.14)
approaches the function 8(t) as a approaches infinity.
It is therefore useful to compute the response of the simple mechan
ical system to a force of the type defined by Eq. (2.14). The corre
sponding Fourier transform is
n/ \ a 1
^(co) = 
2ri co + ia
and the expression for the displacement for t > is
ae~ at
a 2 m + aR m + K
aer kt r e %w f t e u, f t i
2mco/ Leo/ + i(k — a) to/ — i(k — a) J
The limiting case for a — > °° gives the response of the system to an
impulsive force 8(t)
(0 (*<0) (6.12)
XsKl) \ {l/mo> f )e kt sin (a f t) (t > 0)
The responses of the system to the two unit functions u{t) and 3(0
might be considered to be the basic transient responses. They are
closely related as Eq. (6.11) indicates. If it were not that u and 5
are discontinuous functions, we could say that 8(t) is equal to (du/dt).
From a formal point of view this is correct, since the limiting value of
the quantity
©
J Mfl  <t ~ A)]
as A goes to zero has the properties of the function 8{t). The relation
ship is also apparent between the two transient responses, for x$ =
dx u /dt, as a little close reasoning can prove.
General Transient. — The two basic transient responses can be
utilized to compute the response of the system to any more compli
cated force function. The delta function has the general property
that, for any function /,
f~J<?) W r)dr= f(t)
H.6] RESPONSE TO TRANSIENT FORCES 49
and, by integrating by parts, the step function has the property
/.
_ M ^r)^tfr=/(*)
The meaning of these integrals (if they have any!) is that a smooth
continuous function of time can be considered as being built up of an
infinite sequence of impulse functions, the one at time r having ampli
tude /(t), and so on. Or the function can be built up stepwise, the
increase at time t being proportional to df/dr. By utilizing the func
tions xs(t) and x u (t), the responses to the unit functions 8(t) and u(t),
we can write down the response of the system to any force f(t) as either
x(t) = j^ J{r)xi{t  r) dr
or ■ > (6.13)
x{f) = J ^ %j& Xu (t  r) dr
The second form should not be used with discontinuous force functions
(where df/dr is infinite) unless one adds terms proportional to x u (t — t„),
where r n is the location of the discontinuity in time and the propor
tionality factor equals the magnitude of the discontinuous step in /.
In a manner of speaking, Eqs. (6.13) form an alternative method to
Eq. (6.7) for obtaining the displacement function x{t) — although Eq.
(6.7) must be used to obtain the functions x s , x u used in Eqs. (6.13).
Actually, as might be suspected, these equations are two equivalent
ways of writing the same fundamental equation. This can be seen
by writing out the integral for x u in terms of Eq. (6.5)
^ ) = 2^J_ ^^ do)
C — Kilt
and, by differentiation,
o — lut
if" e
Substituting this into Eqs. (6.13) and changing the order of integra
tion yields
and, similarly, ^ (6.15)
50 THE 'SIMPLE OSCILLATOR [II.6
These expressions might be considered to be fundamental to both
Eqs. (6.7) and (6.13), the final form depending on whether we inte
grate first with respect to r or first with respect to w.
If we integrate initially with respect to w, we obtain the first of
Eqs. (6.13), and the second equation is obtained by integration by
parts. If we integrate initially with respect to t, Eq. (2.20) shows that
the factor in brackets is F(u>), and the result is just Eq. (6.5), from
which we have obtained Eq. (6.7). Which of the forms for calculation
of x(t) is to be used in a given problem depends entirely on the relative
ease of integration of the two forms.
Some Generalizations. — We can now review the method of com
plex integration (sometimes called the method of operational calculus),
so as to point the way to the application to the analysis of more com
plex systems. Suppose that we have a force f(t), which is zero for
t < 0, applied at some point to a mechanical system. To obtain the
response to the general force, we first obtain the steadystate response
of the system to a simple harmonic force of frequency v = (co/2ir).
All that is needed is the ratio between the force (applied at point 1)
and the steadystate velocity of some point of the system (either the
same point 1 or another point 2) which is called the impedance Zn(co)
or Zu(eo), a function of co. The ratio between the force and the corre,
sponding displacement is to2n(w) or —io)Z 12 (a).
Often it is more useful to obtain the reciprocal of the impedance,
the mechanical admittance Fn(«) or Y 12(a), the ratio between the
velocity and the force. The ratio between displacement and force
would then be Y (&>)/(—•&'«). This is a minor point, however, the
main objective being to obtain the relation between the response of the
system and the force for steadystate motion of frequency (co/27r).
A possible second step is to analyze the transient f(t) into its
simple harmonic components by obtaining the Fourier transform F(cj),
according to Eq. (2.21). The final response to force f(t) is then
obtained in terms of contour integrals of the form given in Eqs. (6.5),
where Z(«) may be either Zn or Z i2 , depending on whether the response
of the point of application or that of another point is required.
An alternative form of the second step is to find the response of the
system to an impulsive force 5(t), applied at point 1
where, again, Z may be Z n or Z 12 . The response to any force f(t)
II.6] RESPONSE TO TRANSIENT FORCES 51
applied to point 1 is then given by
x ® = Jl „ /( T )**(* ~ T ) dr, v(t) = f_ n f(T) Vi (t  t) dr (6.17)
These equations are of very general usefulness, as we shall see
later in this volume. The important steps in the process, for any
type system, are finding the impedance or admittance function, giving
the ratio between the simple harmonic force applied at some point
of the system and the corresponding steadystate response of the same
or another point, and computing the resulting contour integral, in
the form of Eqs. (6.5) or Eqs. (6.16).
It should be pointed out that the integrals of Eq. (6.16) do not
converge unless Z and coZ go to infinity faster than the first power
of co as co goes to infinity; otherwise Eqs. (6.15) must be used to find
the correct solution. For instance, if Z = R, a constant, Eqs. (6.16)
diverge. However, insertion of f(t) = 8(t) in Eq. (6.15) and utiliza
tion of the definitions of the Fourier transform would indicate that
v$(t) = (l/R)8(t) and, by integration, x&(t) = (\/R)u(t) in this case.
Sometimes a few tricks must be used to obtain results. For
instance, if Z were R + (iK/u>), we could utilize the obvious equations
1 = (co/fl) = 1_ _ (iK/R 2 )
Z co + i(K/R) R co + (iK/R)
The integral for x« then becomes (using the "definition" of u)
This integral is wellbehaved, and we finally obtain, for t > 0,
Xi(t) = 4 u(t)  4 + 4 etwt = ( ^ J ec*/*)«
R " w " R "*" R
The corresponding velocity can be obtained by differentiation, or by
using the same trick
vs{t) = G) m ~ {§*) e ~ {K/m > {t > o)
Later in the book we shall have further occasion to utilize this trick.
Laplace Transforms. — Another way of describing the procedure
utilizing the impulsive response is applicable to all systems we shall
encounter in this book. We first compute the steadystate behavior
of the system under consideration (simple oscillator, string, diaphragm,
52 THE SIMPLE OSCILLATOR [H.7
or room full of air) to a driving force (1/2tt) e~™ 1 applied at the point
(or area or volume) we wish to subject to transient force. After com
puting the behavior (displacement, velocity, shape of string, etc.)
we obtain the corresponding response to a unit impulsive force by
integrating the expression for this behavior over a> from — oo to + » .. .
The response for a general transient force can then be obtained by
using Eq. (6.17). This technique is of general utility, and we shall
refer to it from time to time throughout this book (keeping our eyes
open to be sure that the integrals converge, of course).
Equations (6.16) are of the general form
f(t) =7^J_ n Qiiah^da (6.18)
By Eq. (2.20) we see that $> can be expressed in terms of an integral
of /. Whenever f(t) is zero for t < 0, we obtain the following integral
for 3>(p), after substituting p for — iw.
*(P) = f ~ e~ pt f(t) dt (6.19)
which is related to Eq. (6.2). The function 3>(p) is said to be the
Laplace transform of f{t)
<*> = £(/)
In many cases where it is difficult to evaluate Eq. (6.18) directly in
terms of residues [when $ has other infinities besides simple poles, as
for instance a factor Vco + a, or where the convergence of (6.18) is
questionable] it turns out to be better to try to find an f(t) which,
when substituted in Eq. (6.19), gives the required form for $. In
order to aid in this procedure of working the problem backward,
several tables of Laplace transforms [tables of f(t) with their corre
sponding $(£>)] have been published; these can be used just as tables
of integrals are used. Such tables must be used if 3>(p) has terms with
p = —iu involving radicals. This will be discussed again in Chaps.
VI and VII./ /"   l )
7. COUPLED OSCILLATIONS
We must now treat the case that we avoided in the last section,
the behavior of two oscillators coupled tightly enough together so
that we cannot neglect the feedback of energy from the driven system
to the driver. ■ In this case both oscillators are on an equal footing;
we cannot call lone the driven and the other the driver, since each is
affected by the other. We shall expect that the results obtained
n.7i
COUPLED OSCILLATIONS
53
from our analysis will not be much different from those of Sec. 5 except
when the frequencies of the two oscillators are nearly equal, for in
any other case the amplitude of motion of one oscillator will be much
smaller than that of the other, and the feedback will not be large.
The General Equation. — The general case will be treated first,
however. We shall call one of the coupled oscillators No. 1, call its
mass mi and its displacement from equilibrium #i,.and call the other
oscillator No. 2, with mass m 2 and displacement x%. When x\ and x 2
are both zero, the system will be in equilibrium. We neglect friction
, Equilibrium ,
I Positions \
\ rC"" 1
f
fill ■ K 3\ ..;
'"' Masses ■' \
"'Stiffness Constants
m,
\\
K 3
k;
_ — r^TR^p — I
k;
Fio. 6. — Simple example of two coupled oscillators; two masses connected by
springs. The effective stiffness constant for mi is K\ = K\' + K%; that for m, is
Kt = Kt' + Ks. The analogous electrical circuit is also shown. A force applied to
mi is analogous to the voltage source F\, a force on mt to the source ^2.
for the time being, since it will only confuse the problem. It can
be considered later if need be.
Suppose it turns out that if we keep x% equal to zero, the force
on mass wii is equal to — K\X X (this will be true if Xi is small enough).
Then if we clamp oscillator 2 at x 2 — 0, oscillator 1 will vibrate with
a frequency v\ = (1/2jt) VKi/wi. Similarly, if we clamp mass mi
at xi = 0, the restoring for ce on m 2 is — K^x%, and its frequency of
vibration is v z = 0/2w) \jK%/m^. A displacement of mass mi,
however, produces a force on m 2 , for this is what we mean by coupling.
Suppose that this force is Kzpc\. Then owing to symmetry of the
system, the force on mi due to a displacement of m 2 is K^. The
54 THE SIMPLE OSCILLATOR [H.7
constant K s is called the coupling constant. It is usually very small
compared with Ki or K 2 . We shall not lose generality by assuming
that v\ is not less than v 2 .
The equations of motion of the two masses are
d 2 xi d 2 x 2 T . , _
m l ~^2 = —^1^1 + K SX2, m 2 J72 = K 2 X 2 + K3X1
If we change the scale of the displacements to allow for the difference
in mass of the two oscillators, letting x = xi V^i and y = x 2 Vmi,
then we can write these equations in a simpler form.
~ + Wv\x = 4ttVV, ^f + ^v\y = 4xV 2 x
4tt 2 ^ =
K, \ ( 7 D
If one of the oscillators is clamped down, then the other will
oscillate with simple harmonic motion; but if both oscillators are
allowed to move, the resulting motion will usually not even be periodic.
Curves c of Fig. 8 show a typical case. The curves for x and y as
functions of time are certainly not sinusoidal. This is annoying, for
we do not like to have to develop and define some new sort of function
to express the motion.
Simple Harmonic Motion. — So we first ask whether it is not possi
ble to start the two masses in some special way so that the motion
is simple harmonic, even though the motion in general is not. Of
course if the motion is to be simple harmonic, both oscillators must
be vibrating with the same frequency. We therefore try the solution
x = Ae 2 * ivt , y = Be 2 ™* in Eqs. (7.1) and see that it satisfies these
equations if the coefficients A and B are related in the following
manner: (v\  v 2 )A = ^B, (?f  v 2 )B = ^A.
From these two equations we can find the ratio of A to B and the
value of the frequency v. s Multiplying one equation by the other gets
rid of both A and B and leaves an equation for v, v i — (v\ + v \)v 2 +
v\v\ — n A = 0, having for a solution
v = [*(»? + 4) +i VW  v\y + 4 M 4 ]*
± i V(K 1 m 2  K 2 mxY +.4iqm 1 m s l (7.2)
H.7] COUPLED OSCILLATIONS 55
Thus there are two possible frequencies of simple harmonic oscil
lations for the combined system, and we shall see that (unlike the
simple oscillator) which of the frequencies it will vibrate with depends
on how we start the system into motion (i.e., what the values of A
and B are). Neither of the allowed frequencies is equal to either of
the natural frequencies v x or v 2 of the individual oscillators taken
separately. Suppose that we call the frequency involving the plus
sign in front of the radical v + and that involving the negative v_.
The value of v is smaller than either v\ or v 2 , that of v+ is larger than
both vx and v 2 . Coupling always spreads apart the natural frequencies.
Normal Modes of Vibration. — We can say that although the gen
eral motion of the system is not periodic, nevertheless if the masses
are started into motion in just the right way so that the amplitudes
of motion of m x and m 2 are related by B+ = A + (v\ — v\)/n 2 =
A + n 2 /{v\ — v\), then, and only then, will the system oscillate with
simple harmonic motion of frequency v+, these ratios between the
amplitudes of motion A+ and B + remaining the same throughout
the motion. Similarly, if the masses are started so that their
amplitudes are related by the equations #_ = A(v\ — v 2 _)/y. 2 =
A/i 2 /(i>% — vt), then, and only then, will the system oscillate with simple
harmonic motion of frequency V. If the motion is started in any other
way, there will be no permanent ratio between the displacements of the
two masses, and the motion will not be periodic. These two especially
simple ways of motion of the system are called its two normal modes
of vibration. By a little juggling of terms, these motions can be
represented as follows: If the frequency is v+,
x = C+ cos a 62""+', xi = [a+ cos (%rv + t)
Vmi
+ 6+ sin (2irv+t)] cos a
y = C+ sin a er 2 *****, x 2 = —= [a+ cos (2irv+t).
V m 2
+ b+ sin (2irv + t)] sin a
If the frequency is i>_, \ (7.3)
x = C_ sin a er 2ri '*, xi = — j= [a_ cos (2tvJ,)
VWi
+ b sin (2rvt)] sin a
y = C_ cos a er™'*, x 2 = —== [«_ cos (2ncvJL)
\/m 2
+ 6_ sin (2irv_t)] cos a
56
THE SIMPLE OSCILLATOR
[H.7
where the angle a has been introduced to unify and simplify the nota
tion. It is related to the v's, etc., as follows:
tan<
= {v\  vj) = _
("i  vi)
tan 2a =
2m 2
(v\  v\) (v\  v_)
n  "2
C+cosa = A+, C+sina = — B + , CSmet = A, CCosa = J5_
One may ask why we are so interested in these normal modes of
vibration when they are such a specialized way for the system to
oscillate. The answer is that as soon as these normal modes are
found the problem of determining the general motion of the system is
suddenly seen to be quite simple. For it turns out that the general
motion can always be represented as a combination of both the normal
modes of vibration. The general solution of Eqs. (6.1) is
(7.4)
\ y
\
\
\
\
\
a.
\
\
x = C+ cos a e*™* 1 + C sin a er M '*
y = — C+ sin a e 2wir+t + C_ cos a e' 2 ^" 1
■7
as can be verified by substitution in (011). The displacement of
each individual mass is a combination
of two oscillations of different frequen
ciesj which results in a nonperiodic mo
tion (except in the rare case where the
ratio of v+ to v_ equals the ratio of two
integers, when the motion will be peri
odic no matter how the system is
started. Why?).
We can represent the general mo
tion of the system in a still simpler
manner. Let X = C+er 2 ™* 1 and
Y = Ce 2 ™'. Then x = X cos a +
Y sin a, y = — X sin a + Y cos a. These
equations are just the ones used in
analytic geometry to represent the transformation of the coordinates
of a point in a plane with respect to the axes x and y to a
new set of axes X and Y inclined at an angle a to the first set,
as shown in Fig. 7. Suppose that we represent the position of the
system at any time by a point on a plane whose abscissa is x, the dis
placement (in the proper scale) of the first mass, and whose ordinate
is y, the displacement of the second. The motion of the system
\
\
Fig. 7. — Transformation to nor
mal coordinates for two coupled
oscillators.
n.7]
COUPLED OSCILLATIONS
57
corresponds to a motion of the point on the plane. What we have
said above is that as the point moves, corresponding to a general
sort of oscillation of the system, the projection of the point on the
rcaxis (the displacement of the first mass) moves back and forth in a
complicated nonperiodic way, and so does the projection of the point
on the #axis (the displacement of the second mass). However, the
(c) f— "
p IG g. — Motion of two coupled oscillators. Curves to the right show the dis
placements x and y as functions of time; those to the left show the path of the point
representing the system in the xy plane (configuration space). Cases (a) and (6)
show the two normal modes of vibration, when the system point travels along a normal
coordinate. Case (c) shows the general type of motion.
projection of the point on the Xaxis always moves back and forth
with simple harmonic motion of frequency v+ (since X = C+er 2 ™* 1 )
with an amplitude of motion C+, and the projection on the Faxis
moves with frequency ?_, with amplitude C_. Only when the system
is so started that its point moves along the Xaxis or along the Faxis
is its motion periodic (see Fig. 8 for examples). The plane in which
58 THE SIMPLE OSCILLATOR [H.7
the point moves is called the configuration plane of the system, and
the axes X and Y are called the normal coordinate axes for the system.
The general motion of the system can be written in the form given
in (7.4) or, less symbolically, as
si = —7=^ [a+ cos (2tv + {) + b+ sin (2wv + t)]
sin a , , n
+ —7= [« cos (frcvJL) + &_ sm (2jri»_i)]
VWi
[A+ cos a cos (27t»' + ^ — f> + )
(7.5)
+ yl_ sin a cos (2x^_/ — $_)]
a: 2 = _ ~Z. [o+ cos (Stov+O + 6 + sin (2irM)l
COS £tf
H 7= [0 cos (2x^0 + 6_ sin (2jrv.J)l
VW2
= — 7=[~~ A + sina cos (2x^+2 — $+)
+ A_ cos a cos(2rv_^ — $_)]
The general solution involves four arbitrary constants a+, b+, a_,
and 6_, or A+, <£> + , ii_, and $_, whose values are fixed by specifying
the initial displacements and velocities of the two masses.
Energy Relations.— Another example of the special simplicity
that is given to the equations of motion when normal coordinates
are used is the expression for the energy of the system. The kinetic
energy is, of course, frn^dx^dt)* + im 2 (dx 2 /dt)\ To find the
potential energy, we find the amount of work necessary to push the
system from equilibrium to the position where the displacements are
X! and x 2 . Since friction is negligible, we can do this process in any
order that we wish and get the same result. We choose to push m x
out first, pushing against a force —K x xi and so requiring an amount
of work JKiXt dx x = \K x x\. We next push m 2 out, pushing against
a force — K 2 x 2 + K 3 X! (by our definition of K z ) and doing an amount
of work \K 2 x\  K3X1X2. The total energy is therefore
mi \w) + m2 \w) + KlX * + K2X > ~ 2K ^^]
A3f) + (off + w ^ + 47r2 ^ 2 " 8*V*v]
an expression complicated by the term in xy.
2
= 1
2
H.7] COUPLED OSCILLATIONS 59
If we substitute the expressions for x and y in terms of X and Y,
expressing the functions of a in terms of functions of 2a, and use the
formula for tan (2a), we find a simple formula for W
w  "2@ 2 + 3©' + ***+** + *•,._ r (7.6)
without any term in XY. If X = A+ cos (2irv + t — $ + ) and Y =
A cos {2rcvJt — <£»_), as indicated in (6.5), then
W = 2ir 2 (ulA% + vlAl)
the sum of the energies of vibration along the two normal coordinates.
The Case of Small Coupling. — Now that we have obtained a
general solution for the motion of two coupled oscillators, it might be
well to see how the results correspond to the discussion of forced
oscillations given in Sec. 5. As we said at the beginning of this sec
tion, whenever the coupling is small and the frequencies of the two
oscillators are not equal, the amplitude of motion of one will be much
larger than that of the other. The oscillator showing the smaller
motion can then be considered as the driven oscillator, and its ampli
tude of vibration will be given by Eq. (5.3) of Sec. 5. For instance,
if m x is to be the "driver," its amplitude of motion should be larger
than that of m 2 . If the coupling is small, a is small, and the case
we should analyze is that for the frequency v + , where
(vW cos(a)
\ m 2 v\ — v\ \ m 2
Xl ~ I .. rzr ) cos(a)e 2a " 4 '''+'
and
#2
K z xi
£ 2  m 2 (2irv + ) 2
Since a is small, x 2 will be smaller than xi unless mi is very much
larger than m 2 .
From the point of view of m 2 , the quantity K3X1 is just the force
f(t) applied to m 2 through the coupling spring, due to the motion of
xi. The driver is oscillating with a frequency v+, so that/ = F^e~^ iv ^
— K3X1 where F = (i^ 3 C + /\/wi) cos a. The divisor is proportional
to the mechanical impedance of m 2 at the frequency v + , since
— 2nriv+Z 2 — K 2 — m i (2rv + ) 2
60 THE SIMPLE OSCILLATOR [H.7
Therefore the expression for the displacement of m 2 when it is being
"driven" by mi at a frequency v+ is
— Zrlv + Zi2
which is to be compared with Eq. (5.3).
The Case of Resonance. — However, our primary purpose in this
section is not to check the formulas given in Sec. 5 but to find out
what happens in the cases where the formulas of that section do not
hold, where the feedback of energy from m 2 to mi is appreciable. As
an example of this let us consider the case where the two oscillators
have the same natural frequency (Vi = j> 2 ) and where the friction is
negligible. If we try to use the formulas of Sec. 5 in this case, we
find that they predict that the amplitude of the driven oscillator will
be infinite. The infinity simply means that the amplitude of motion
of the driven mass will become large enough to absorb a large fraction
of the driver's energy, so that the formulas of Sec. 5 cannot be used.
To solve this problem, we must use the formulas of the present section.
According to Eq. (7.2), if v x — v i}
V+ = Vv{ + M 2 = vi + ^ +
,2
v = V4  M 2 = vi  2^ + * ' ' » a = 4
If ju is much smaller than v h only the first two terms in the series
expansions of the square roots need be considered. Therefore, if
two similar oscillators, each of natural frequency vx, are coupled
together, they can no longer oscillate with frequency vi but can
oscillate with a frequency either (^ 2 /2i'i) larger or this same amoun t
smaller than v\. If they are started so that x\ = —x% v / m 2 /mi,
then the system has only the higher frequency; but if they are started
so that Xi = Xi \/'mz/m\, then it has only the lower frequency. If
the system is started in any other way, the motion will be a combina
tion of both frequencies.
If the system is started by pulling mass 1 aside a distance x ,
while keeping aj 2 equal to zero, and letting both masses go at t = 0,
then the values of the constants to be used in (7.5) are 6+ = 6_ = 0,
a+cosa/VW = asma/s/mi = (x /2). The displacements of the
two masses will be represented by the equations
n.7]
COUPLED OSCILLATIONS
61
= X COS I IT — t) COS (2TVit)
(7.7)
= VS Xo ^Cy sin(w)
if the trigonometric equation for the sum or difference of two cosines
is used. The second form of the equations shows that, if the oscil
Fig. 9. — Motion of two coupled oscillators having the same natural frequency.
Solid curves show the displacements of the two oscillators as a function of time and
illustrate the alternation in amplitude.
lators are started in the manner described above and if the coupling
is weak, the motion is like an oscillation of frequency v\ whose ampli
tude of oscillation itself oscillates with the small frequency 0u 2 /2i>i).
This is illustrated in Fig. 9. Such motion is not simple harmonic
motion, since the amplitude changes with time. As the first forms
of Eqs. (7.7) show, it is a combination of two harmonic motions whose
frequencies differ by a small amount, so started that they at first
reinforce each other for Xi but after a while get out of phase and cancel
each other out, and so on. We notice that when the amplitude of
motion of mi is large, then that of w 2 is small, and vice versa.
Transfer of Energy. — We can compute the approximate value of
the free energy of oscillation of m x and m 2 by using formula (4.6),
W = 2w 2 mv 2 A 2 (t). Of course, it is not strictly correct to speak of
the energy of only a part of a system, and it is not right to use the
62 THE SIMPLE OSCILLATOR [II.7
formula for the energy of a simple oscillator to compute the energy
of a coupled oscillator; but if the coupling is weak, the formula has
some meaning. We find that the average "free" energy of mass mi
is 27r 2 mi^x§ cos 2 (tt/xV i 'i) an d * na * °f m 2 is 2r 2 miv\xl sm 2 (ir(i 2 t/vi).
It indicates that at t = all the energy of the system resides in mass
mi but that as time goes on the energy of the first mass diminishes
while that of the second increases until all the energy is 'transferred
to m 2 . Then the flow of energy reverses, that of mi building up again,
and so on. The total energy of the system is 2r 2 mivlxl, a constant,
as it should be.
This example illustrates how different from the usual forced oscil
lations can be the motion of two coupled oscillators if a large coupling
or an equality of frequencies of the two vibrators makes possible a
large transfer of energy from one to the other. It also illustrates an
obstacle encountered in building a resonating system that will respond
very strongly to only one frequency. In the last section it was pointed
out that such critical response required a system with very small
friction. We see now, however, that if we make a system with
negligible resistance tuned to a natural frequency v\ and try to drive
it by a system oscillating also with a natural frequency v\, the result
of the coupling will be that neither driver nor driven system can oscil
late with frequency vi but must oscillate with frequency v\ ± (n 2 /2vi).
The very fact that the response is so large at vi destroys the possibility
of vibration at that frequency.
Forced Vibrations. — We also wish to find the response of the sys
tem of two masses to an applied force. Here again the analysis goes
from the simple harmonic case to the more complicated transient
forces. We first find the response (i.e., the displacement and velocity
of mi and m 2 ) to a force F = i'V  "" applied to m x or m 2 . If it is
applied to m h Eqs. (7.1) become
^f + o*c  x 2 V = a*r"; ^ + o>\y  x 2 x = (7.8)
co? = (ifi/rai); cof = (i£ 2 /m 2 ), x 2 = \K z /\/m~m^\ a x = Fo/y/rni
Solving for the steadystate response consists in substituting x = xtf'™*
and y = y e~ iut in Eq. (7.8) and solving for x and y . The results,
when reduced back to expressions for the actual displacements, are
_ Fo (cof  to 2 )**" _ Kitr 1 ** Fp
•^l ... f O 9 \ / 9 \ 5 ^2
mi (co 2  co 2 .)^ 2  «i)' (co 2  co2.)(co 2  coi) mim 2
(7.9)
H.7] COUPLED OSCILLATIONS 63
where co + — 2ttv + , w_ = 2irv, with the v's given in Eq. (7.2).
«i = i(" 2 i + «?) + i V(col  col) 2 + 4 X 4
«L = *(«? + col)  i V(«J  cof) 2 + 4x 4
If we had applied the force to ra 2 , the corresponding response
would have been
*! ™ /..2 .* \/..?> «\> # 2 —
Wim 2 (to 2  oj2.)(co 2  »£.)' " m 2 (co 2  ^ + )(<o 2  «i)
(7.10)
The symmetry of the expressions is apparent.
Another way of writing the response is specifically in terms of
impedances corresponding to the circuit of Fig. 6. We define the
blocked impedance of Wi as the impedance when m 2 is held fixed at
equilibrium, Z± = —iwmx + i(K\/(a). The corresponding blocked
impedance of m 2 is Z 2 = — *com 2 + i(K 2 /(o). The coupling can be
expressed in terms of a mutual impedance M — i(K z /(S). By differ
entiating Eqs. (7.9) with respect to time, and by subsequent algebraic
juggling, we obtain expressions for the ratios between the simple
harmonic force applied to m\ and the corresponding velocities of m t
and m 2
„ F ZrZ %  M\ F ZrZ 2  M* ■
&\\ = — = ~ , ^12 = — = jj (.7.11)
V\ Z 2 Vi M
Z\\ is called the mechanical input impedance of the system at point 1;
Z 12 is called the mechanical transfer impedance between points 1 and 2.
If the force is applied to ra 2 , the corresponding impedances are
z 22
F ZrZi  M\
V2 Z X
5/21
_ F _
Vi "
Z X Z 2  M 2
M
Z12 (7.12)
This brings out the interesting fact that transfer impedances are
symmetrical. The response at point 2 due to a force at point 1 is the
same as the response at point 1 due to the same force applied at
point 2. Sometimes this result is referred to as the principle of
reciprocity.
Resonance and Normal Modes. — Each of the impedances given in
Eqs. (7.11) and (7.12) becomes zero at the two resonance frequencies
v+ and v. At these frequencies the displacements become infinite,
as is shown in Eqs. (7.9) and (7.10). The infinity is due to our neglect
of frictional forces; if a small amount of friction had been included,
the displacements would have been large but not infinite. The
ratio between y and x, when the driving frequency is v + , is x 2 /( w i — o>+)
64 THE SIMPLE OSCILLATOR [H.7
= l?/(A  "+), according to Eqs. (7.9). A glance at Eqs. (7.3)
shows that this is just the ratio maintained when the system is in
free vibration at frequency v + .
In other words, when the system is driven by a simple harmonic
force of frequency equal to one of the frequencies of free vibration of
the system, its response is large, and the relationship between the
motions of the parts (i.e., the configuration of the system) is the same
as if the system were in free vibration at that frequency. One of
the methods of finding the possible frequencies of free vibration of a
system and the configuration of the corresponding normal modes is to
drive the system by a simple harmonic force. The resonance fre
quencies are the frequencies of free vibration, and the configurations
at resonance are those of the normal mode.
In the present case, there are two natural frequencies of free
oscillation, so there are two resonance frequencies. For three masses
connected by springs there would be three normal modes and three
resonance frequencies, and so on.
When the present system is driven by a force of frequency consider
ably smaller than v, applied to m x , both masses move in phase with
the force, the ratio between x 2 and x\ being (K3/K2) = Ks/(K 3 + K'^),
so that X\, the displacement of the point of application of the force, is
greater than x 2 by a factor depending on the relative sti^ness of the
two springs connecting with m 2 .
According to Eqs. (7.9), as the driving frequency increases, the
amplitudes increase until the resonance at v_, which is a lower fre
quency than either vi or p 2 , the resonance frequencies of each mass
taken separately. For frequencies just above v, both displacements
are opposing the force (180 deg out .of phase).
At v 2f the displacement of mi, however, becomes zero (if friction is
not zero, xi is small, but not zero). At this driving frequency the
input impedance Zw of Eq. (7.11) becomes infinite, since the blocked
impedance Z 2 of m 2 is zero. What has happened is that at this fre
quency m 2 resonates when mi is held rigid. If we attempt to move
mi at this frequency, m 2 will immediately absorb all the available
energy with only an infinitesimal motion of mi. This corresponds to
a parallel resonance in the analogous electrical circuit shown in Fig. 6.
Above v 2 , X\ is in phase with the force and x 2 is opposed. At
v = v+, higher than vi or v 2 , the amplitudes are again infinite. For
higher frequencies, X\ opposes the force and x 2 is in phase with it.
Transient Response. — A few examples of the application of opera
tionalcalculus methods to the coupled system will be useful to discuss
H.7] COUPLED OSCILLATIONS 65
here, just to begin to show how the method works in more complicated
cases. Suppose that we compute the response to the impulsive
force 8(t) applied to mi. From Eqs. (7.9) we see that the quantities
(1/— icaZ) needed to compute the integral of Eq. (6.16) for the coupled
systems are
/ 1 \ = 1 i<4 ~ co 2 ) =
\ — iwZ u J mi (co 2 — co^Xco 2 r coi)
/ 1 \ = 1 X 2 = "2"
\ia)Zi 2 J y/mmz (« 2  co 2 h )(« 2  coi) ico
Since we are not considering friction, all the poles of these quantities
are along the real axis, being at ± co_ and + co+.
The residues of (l/2x) (e _iu '/ — icoZn) at w = ±« + are
— Vji
47rmico + \co 2 h — coi/
+ 6
The residues at the poles « = ± co_ are obtained by interchanging co +
and co_ in these expressions.
A great deal of algebra and utilization of the formulas
„ (co 2 .  CO 2 ) . , (cof  w 2 )
cos 2 a = f± 14; sin 2 a = ff =^
(C04. — col) (co^ — coi)
Y 2
£ sin (2a;) = sine* cosa = —^ — ^
u +
lead one finally to the expressions for the response of the coupled
system to an impulsive force applied to mi
t < 0, xu = x 2 & =
— icaZj
t > 0, xi S (t) = ^ J „ d»
1 [cos 2 ** . , . sin 2 a . , ..1
= ^T sm («+0 H sin (co_2)
Wi L co + T co_ J
Ar J» — tcoZ 12 I
mim 2 l 03 +
r 1
a COS a —
Leo
sin a cos a — sin (co+tf)
sin (<ai)
J
These formulas are to be compared with Eqs. (7.5). What the oper
ational calculus has done is to fix the constants a+, &+, etc., to corre
66 THE SIMPLE OSCILLATOR [II.7
spond to the initial conditions of m 2 at rest and m x being given a unit
impulse at t = 0.
Just after the impulse at t  0, x°& and va are both zero; Xi» is
zero but va starts at a value (1/wi), as it should for a mass ra x which has
just been given a unit impulse.
The unit impulsive response when the force is applied to ra 2 is
obtained by interchanging the subscripts 1 and 2 in Eqs. (7.13).
The response of the system to any force f(t), applied to mi is, from
Eqs. (6.17)
Xl (t) = J*^ f(r)x lS (t  t) dr) x*(t) = f'^ f(r)x»(t  t) dr (7.14)
This analysis of a simple coupled system has had no important
practical applications, but it has served to indicate the direction our
results will take for still more complicated systems. We should
expect to find complex systems (at least those coupled together by
springlike forces) to have a number of resonance frequencies. If the
system is started in just the right way, it will oscillate with simple
harmonic motion, being damped out if there is friction. There
should be several ways of starting the system to get simple harmonic
vibration, each different way corresponding to a different normal
mode and to a different natural frequency. In general, the system
will vibrate with nonperiodic motion which is a combination of several
natural frequencies.
If the system is driven at one point by a simple harmonic driving
force, the whole system will respond at that frequency. The steady
state response of any part will depend on the frequency and point of
application of the force, the ratio between force and response being
given in terms of the input transfer impedances of the system. From
these impedances, as functions of frequency, can be computed the
response of the system when subjected to a sudden unit impulse at
t = 0; and from the impulsive response can be obtained the response
to any transient force.
We shall work out these same results for a still more complicated
system, the simple string under tension, in the next chapter.
Problems
1. A vibrator consists of a 100g weight on the end of a spring. The spring's
restoring force is proportional to the weight's displacement from equilibrium;
if the weight is displaced 1 cm, this force is 10,000 dynes. The frictional force
opposing its motion is proportional to its velocity and is 100 dynes when its velocity
is 1 cm per sec. What is the modulus of decay of the oscillator? What is its
decrement? What "frequency" do the vibrations have? What frequency would
H.7] COUPLED OSCILLATIONS 67
they have if there were no friction? If the weight were originally at rest and then
were struck so that its initial velocity was 1 cm per sec, what would be its subse
quent motion? What would be its maximum displacement from equilibrium?
2. The diaphragm of a loudspeaker weighs 1 g, and the displacement of its
driving rod 1 mm from equilibrium requires a force of 1,000,000 dynes. The
frictional force opposing motion is proportional to the diaphragm's velocity and is
300 dynes when the velocity is 1 cm per sec. If it is assumed that the diaphragm
moves like a simple oscillator, what will be its natural frequency, and what its
modulus of decay? The driving rod is driven by a force of 100,000 cos (2irvt) dynes.
Plot a curve of the real and imaginary parts of the mechanical impedance of the
diaphragm as function of frequency, from v = to v — 1,000 cps.
3. The diaphragm of Prob. 2 is driven by a force of 100,000 cos (2irvt) dynes.
Plot a curve of the amplitude of motion of the diaphragm as function of frequency,
from v = to v — 1,000 cps. Over what frequency range is this loudspeaker mass
controlled?
4. What is the mechanical impedance of a mass m without spring or friction?
What is the impedance of a spring without mass? What will be the angle by
which the displacement of the mass lags behind an oscillating force? What will
be the angle of lag of the velocity of the mass behind the force? What are the
corresponding angles for the spring?
6. The diaphragm of Prob. 2 is coupled electromagnetically to an electric
circuit by means of a coil of negligible resistance and inductance, whose coupling
constant r = 10,000. An alternatingcurrent emf of 10 volts and variable fre
quency is applied across the coil. Plot the current through the coil and the ampli
tude of oscillation of the diaphragm as a function of frequency, from v = to
v ~ 1,000 cps. Plot the real and imaginary parts of the motional impedance of
the coil as function of frequency, from v — to 1,000 cps.
6. An ac emf of 10 volts and variable frequency is applied to the coil of
Prob. 5. Plot the current through the coil and the amplitude of oscillation of the
diaphragm as a function of frequency, from v = to v = 1,000 cps.
7. A mass m is attached to the lower end of a spring of stiffness constant K.
The upper end of the spring is moved up and down with an amplitude Be~ iu>t , and
the frictional force on the mass is proportional to the relative velocity of the mass
and the upper end of the spring (dr/dt), where r  x — Be~ iu>t . Show that the
equation of motion of the mass is
m (iO + Bm {§) + Kx = {K  *«««.)***"
show that the steadystate motion of the mass is
R m + i(K/co) _ . ,
x = — — ■ —  — — Be~ iwt
R m — i[wm — {K/o))]
and that the phase lag of x behind the displacement of the top of the spring is
taU_1 l * R m — ~ J + tan_1 ( K / wR ™) What is the amplitude of motion of xt
What is the phase and amplitude of x at very low frequencies? At very high
frequencies?
8. A loudspeaker diaphragm has mass and stiffness that can be neglected in
the useful frequency range. Its motion is opposed by a force equal to 400 v,
68 THE SIMPLE OSCILLATOR HI.?
where v is the diaphragm's velocity. The diaphragm is coupled to an electrical
circuit by means of a coil of resistance 25 ohms, inductance 10 mh, and coupling
constant r = 10,000. An ac emf of 10 volts, with variable frequency, is applied
to the coil. Assuming that all the energy lost by the diaphragm through friction
is transformed into sound waves, plot the sound energy radiated per second (in
watts) as a function of the frequency of the emf, from v = to v = 1,000.
9. A loudspeaker is coupled to the electrical circuit by means of a coil of
resistance 25 ohms, of negligible inductance, and of coupling constant r = 10,000.
It is found that the additional impedance due to the motion of the diaphragm is
equivalent to a resistance of 25 ohms, an inductance of 10 mh, and a capacitance
of 1 juf, all in parallel. What are the mechanical constants of the diaphragm? If
all the energy dissipated by the diaphragm goes into sound energy, plot the overall
efficiency of the loudspeakercoil system {i.e., the ratio of the power radiated as
sound to the total power dissipated by loudspeaker and coil) as a function of the
frequency, from v = to v = 1,000 cps.
10. The sharpness of resonance of a forced damped oscillator is given by the
"halfbreadth of the resonance peak," the difference between the two frequencies
for which the amplitude of oscillation is half that at the resonance frequency v .
Prove that if the natural period of oscillation is negligibly small compared with
2w times the modulus of decay {i.e., if (fc/47r) is small compared with v ), then this
halfbreadth equals (V3A) times the reciprocal of the modulus of decay of the
oscillator. What is the halfbreadth for the diaphragm of Prob. 2? What would
the frictional force have to be in order that the halfbreadth may be 20 cycles?
11. Two oscillators, each of mass m and natural frequency v {i.e., if one
oscillator is held at equilibrium, the other will oscillate with a frequency v ), are
coupled so that moving one mass 1 cm from equilibrium produces a force on the
other of C dynes. Show that if C is small compared with £x 2 mvl, and if one
oscillator is held 1 cm from equilibrium and the other at equilibrium and both are
released at t = 0, then the subsequent displacements of the masses will be
* = cos (i^bo) cos (w) ' Vt = sin G^To) sin (27r "°°
12. Suppose that each oscillator of Prob. 11 is acted on by a frictional force
equal to R times its velocity. Show that the modulus of decay of the oscillations
equals (2m /R).
13. Discuss the forced vibrations of the coupled oscillators described in Prob.
12.
14. Three masses, each of m g, are equally spaced along a string of length 4a.
The string is under a tension T dynes. Show that the three allowed frequencies
and the corresponding relations between the displacements for the normal modes
are
v = v y/l  V, if yi = ?/3 = 2/2/V2
v — vo, if y\ = —2/2 and y 2 =
v = vo \A + Vi, if 1/1 = 1/2= —I/2/V2
where (77a) = 2^v\m.
16. An "xcut" Rochelle salt crystal, mounted as shown in Fig. 5, has the
dimensions d = 1 cm, 6 = 3 cm, a = 2 cm. The external load is a diaphragm in
n.7] COUPLED OSCILLATIONS 69
contact with water, which has an equivalent mass load of 10 g, negligible stiffness,
and a resistance of 400,000 g per sec (radiation resistance). Plot the real and
imaginary parts of the electrical impedance of the crystal from v = to v — 10,000
cps for a temperature of 15°C.
16. What will be the opencircuit voltage across terminals A and B of the
crystal of Prob. 15 when an oscillating force of amplitude 10 dynes and frequency
1,000 cps is acting on its upper surface? What is the voltage if the frequency of
the driving force is 10,000 cps? Consider the temperature of the crystal to be
15°C.
17. A voltage 100 cos (2irvt) volts is impressed across the plates of the crystal of
Prob. 15. Assuming that all the power dissipated in the resistance R m = 400,000
is radiated into the water, plot the power radiated as a function of frequency v,
from v = to v = 10,000 cps. The crystal temperature is held at 15°C.
18. A simple mechanical system of impedance Z m = R m — t'[«m — (A /«)] has
applied to it a force
(0 « < 0)
F(t) = < sin (at) [0 < t < (mr/a)]
[ [(nr/o) < t]
Use the results of Prob. 15, Chap. I and of Eq. (6.7) to compute the displacement
of the system.
19. A dynamic speaker of the type discussed in Fig. 4 has negligible coil
resistance R c and mechanical stiffness K (the other quantities, R m , m and L are not
negligible). An impulsive voltage E = S(t) is impressed across the terminals of the
coil. Calculate the current through the coil, using the Fourier transform method
[see Eq. (6.14)]. Show that the motional emf is
M 2wLm J <e (w + wo + ik) (w — « + ik)  j e ~ kt sin (u t) (t > 0)
where k = (R m /2m) and o>l = (T/Lm) — (R m /2m) 2 . What is the velocity of the
diaphragm? What are the expressions if (R m /2m) 2 > (T/Lm)? If « = 0?
20. The speaker of Prob. 19 has constants m = 50, R m = 10,000, T  10,000,
L = 0.02. Plot the displacement of the diaphragm as a function of time for an
impressed emf E = 108(f) across the coil.
21. Use Eq. (6.13) to compute the velocity of the loudspeaker diaphragm of
Prob. 19 for an applied voltage.
!0 (t < 0)
E (0 < t < T)
(t > T)
Calculate the expressions for the three time ranges, t < 0, < t < T and t > T.
Plot the result for the case of Prob. 20, for E = 10 volts, T = 0.01 sec.
22. A crystal of Rochelle salt, mounted as shown in Fig. 5, has a mechanical
load which is pure resistive, Z m = R m . An impulsive force F e = 8(t) is applied to
the upper surface of the crystal. Using the formulas related to Eq. (6.14), com
pute the current through the equivalent circuit if terminals A, B are open, and
70 THE SIMPLE OSCILLATOR [II.7
show that the voltage across A, B is
ir /•" e iut dco . '
27rR m C c J» (<*+ i[(C c + C P ) aR m C c C P ] V ° ltS
(t < 0)
(l/10 7 ri2 m Cc) exp [(Cc + C P )t/SR m C c C P ] (t > 0)
= 1
23. The crystal of Prob. 22 has the following constants:
C c = C p = 10 10 , t = 10», a = 10, R m = 10«
The applied force is
!0 (t < 0)
10 dynes (0 < < < 0.001)
it < 0.001)
Plot the opencircuit voltage across A, B as function of time, from t — —0.001 to
* = 0.005.
24. The driving rod of the loudspeaker of Prob. 2 is driven by a force of
10,000[15 sin (200irf)  10 sin (600x<) + 3 sin (1000tt<)] dynes. Plot the displace
ment of the diaphragm during one cycle, and compare it with the curve for the
force.
25. The driving coil of Prob. 5 is actuated by an emf of [15 sin (200nJ)
— 10 sin (600ir£) + 3 sin (lOQOn2)] volts. Plot the displacement of the diaphragm
as a function of t for one cycle, and compare it with the curve for the emf.
26. The emf of Prob. 25 actuates the driving coil of Prob. 8. Plot the dis
placement of the diaphragm for one cycle, and compare it with the curve for the
emf.
CHAPTER III
THE FLEXIBLE STRING
8. WAVES ON A STRING
.... So far, we have been considering the vibration of a system whose
mass is all concentrated near one or two points, so that the motion
of the system is completely specified by giving the displacement
from equilibrium of the one or more masses as a function of time.
The vibrators we use are not usually of so simple a nature. Ordi
narily the mass is not concentrated at the end of a relatively weightless
spring but is spread along the spring. The vibrating string of a
violin cannot be considered as having all its mass concentrated at
the center of the string or even concentrated at a finite number of
points along the string; an essential property of the string is that its
mass is spread uniformly along its length. Similarly, a loudspeaker
diaphragm has a good portion of its mass spread out uniformly over
its extent. In these cases, each portion of the system will vibrate
with a somewhat different motion from that of any other portion.
The position of just a few parts of the system will not suffice to describe
its motion; the position of every point must be specified.
Presumably, we could attack the problem of the string, for instance,
by considering the motion of N equally spaced masses on a weightless
string and then letting N go to infinity. We should then have an
infinite number of equations of motion, whose solution would give the
position of every one of the infinity of points on the string as a function
of time. A solution of these equations can be obtained. The solu
tion shows, for one thing, that there are an infinite number of allowed
frequencies of oscillation of the string, as we could induce from the
discussions of the previous chapter. But this is a very awkward way
of solving a problem that is essentially simple. What is needed is a
new point of view, a new method of attack.
The new point of view can be summarized as follows: We must
not concern ourselves with the motion of each of the infinite number
of points of the string, considered as separate points, but we must
consider the shape of the string as a whole. At any instant the string
will have a definite shape, which can be expressed mathematically
by saying that y, the displacement from equilibrium of that part of
71
72
THE FLEXIBLE STRING
[m.8
the string a distance x cm from one end, is a function of x. The motion
of the string at any instant will depend on the shape of the string,
and the subsequent shape will depend on the motion; what we must
do is to find the relation between the shape and the motion. In other
words, the string's displacement y is a function of both x and t, and
we must discover the relation between y's dependence on x and its
dependence on t. The same point of view will be necessary in study
ing the vibrations of diaphragms and of air, as we shall see later.
The Wave Velocity. — Before becoming cluttered up with equa
tions, we shall utilize a simple, but rather clever, little trick to help us
in gaining a picture of how the shape of a string changes. Suppose
that we have a long flexible string of uniform mass e g per cm, unrolling
from one reel, threaded through
the glass tube shown in Fig. 10a,
and being wound up on another
reel. Suppose that the reels are
rotating so that the string is
traveling with a velocity v cm
per sec through the tube and so
that the string is under a tension
of T dynes. Suppose that we
pick out a small portion of the
tube of length As and ask what
force the string is exerting on
this part of the inside of the
tube (we neglect frictional forces). If a small enough length is picked,
the shape of the part of the tube (and the string in this part) will be
practically equivalent to the arc of a circle of radius R, where the value
of R will, of course, be different for different parts of the tube.
Now, if the tube is not bent too sharply anywhere {i.e., if R is every
where larger than the maximum deviation of the tube from the straight
dotted line in Fig. 10a), the tension on the string will everywhere
be T dynes. A study of Fig. 10b will show that the net force inward
on the tube due to the tension of the string is AT = <pT = T As/72.
In addition to this force is the centrifugal "force" due to the motion
of each successive portion of the string around the curved arc As.
This force is pointed outward, and, as we learned in elementary
physics, its value is the mass of the portion of the string eAs times
its velocity squared v 2 divided by the radius of curvature R. The
net inward force on the portion of the tube under consideration is
then (As/R)(T  ev 2 ).
Fig. 10. — Forces on a moving string.
m.8] WAVES ON A STRING 73
We notice that it is possible to run the string at a particular
velocity c ?= \/T/e such that there will be no net force on this portion
of the tube. And then we see that at this velocity c the string exerts
no net force on any part of the tube! For the force is zero regardless
of the size of R. If the string is run at the velocity c cm per sec, we
can carefully break away the tube from around the string and leave
the string moving with velocity c, still retaining the original form of
the tube, a wave form standing still in space.
Of course, running the string with a velocity c and having the
wave form stand still is the same as having the string stand still and
the wave form travel with a velocity c. What we have just proved is
that as long as the string has a uniform density e g per cm, and as long
as the displacement of the string
from equilibrium is not too great,
then a wave will travel along the
string with a velocity c = y/T/e
regardless of the form of the wave.
The last phrase is the important
part of the statement, for it means
that, subject to the qualifications „ ., w
' * ^ Fig. 11. — Wave motion on a
Stated, a wave travels along a String string. Curve (a) is the shape at
without any change of form as it * = 0; curve (6) the shape at ' = 1 
goes. These two properties of a string, or, rather, two aspects
of the same property, are peculiar to the uniform flexible string.
They are not possessed by a string of variable mass density, or a
string with stiffness, for example. This is just the property that
makes the string so useful in musical instruments, as we shall see.
The General Solution for Wave Motion. — The foregoing property
can be expressed mathematically by stating that the dependence
of the shape of the string on x and t when a wave is going in the posi
tive xdirection must be of the general form F(x — ct). Then at
t = the shape of the string is given by the function F(x), as is shown
in Fig. 11a. At t = 1 the wave form F is the same, but it has been
shifted bodily to the right a distance c cm. The general form for any
motion of the string must be
y=F(xd)+f(x + ct) (8.1)
representing a wave of the form F going to the right and one of the
form / going to the left. Every motion of the string can be considered
as a superposition of two waves, each having a speed c, traveling in
opposite directions.
74 THE FLEXIBLE STRING [HI.8
The symbol F(x — ct) simply means that, no matter what function
F is, the quantities x and t enter into it in the combination (x — ct).
Examples are A sm[k(x — ct)], B(x — ct) 3 er a < x  ct >% etc. With our
convention of taking real values, another example is
C exp [Hz (x — ct)] = Ce ikx e ikct .
A peculiarity of this sort of function of x and t, one that will be of use
to us in further discussion, is the following: F changes in exactly the
same way and by the same amount when t changes by an amount a as
when x changes by an amount — ca. A plot of the shape of the string
at a given instant of time is completely similar to a plot of the displace
ment of a given point on the string as a function of time. To find
the shape of F we can either run our eye over the whole extent of the
string at a given instant of time, or we can watch the motion of one
piece of the string as the wave passes by. This is, of course, still
another way of saying that the wave progresses along the string with
out change of shape. The same statement holds for f(x + ct), except
that the direction of x is reversed.
A mathematical way of stating the property of F and / discussed
above is
 F(x  ct) = c A F{x  ct), j t f(x + ct) = c A/(* + ct) (8.2)
where the symbol (d/dt) means the rate of change with respect to t
when x is held constant, and (d/dx) means the rate of change with
respect to x when t is held constant. A change in F due to a small
change in t is equal to (— c) times the change in F due to a change in
x of an equal size.
Initial Conditions. — The particular forms of the functions F and /
are determined by the initial conditions: the initial form and "velocity
form" of the string. If the shape of the string at t = is yo(x), and
its velocity at the point x is v (x), then F(x) + f(x) = y (x) and
d/dt [F(x — ct) + fix + ct)] = v (x) when t = 0, or, what is the same
thing,
«[?(*) +/(«)] (i)..(x)
using Eq. (8.2). Since v (x) must be integrated to solve the last
equation, we shall define the function
S(z) = r v (x) dx
in.8j
WAVES ON A STRING
75
where z = % ± ct. It is not difficult to see that the shape of the string
that corresponds to the specified initial conditions is represented by
the function
 \ \yo(x  ct) + y (x + ct)  \ S(x  ct) + \ S(x + ct) j
(8.3)
At t = 0, y = y (x), and (dy/dt) = dS(x)/dx = v (x), as is required.
The solution is built up of two "partial waves" going in opposite
directions, which combine to give the required behavior at t = and
spread apart thereafter. Two examples of this are given in Fig. 12,
^•f I
Fig. 12. — Motions of plucked and struck strings. The solid lines give the shapes
of the strings at successive times, and the dotted lines give the shapes of the two "partial
waves" traveling in opposite directions, whose sum is the actual shape of the string.
one for a string pulled aside and started with zero velocity, and one
for a string struck by a hammer so that it starts from equilibrium with
a specified velocity. The spreading apart of the partial waves is
apparent in the successive drawings.
Boundary Conditions. — So far, we have been treating the string
as though it had an infinite extent ; actually, it is fastened down some
where, and this fastening affects the motion of the string. The fact
that the string is fastened to a support is an example of a boundary
condition. It is a requirement on the string at a given point in space
which must be true for alL time, as opposed to initial conditions, which
fix the dependence of y and v on x at a given time. Boundary con
ditions are more important in determining the general behavior of the
string, its allowed frequencies, etc., than initial conditions are.
If the support is rigid, and the distance along the string is measured
76 THE FLEXIBLE STRING tDLS
from it, the boundary condition is that y must be zero when x = 0,
for all values of the time. If the support is springy, so that it is dis
placed sideward a distance CF cm for a sideward force F dynes, then
the boundary condition is that y must always be equal to C times the
component of the string's tension perpendicular to the equilibrium line:
y = CT(dy/dx) at x = 0. Many other sorts of boundary conditions
are possible.
Reflection at a Boundary. — Let us take the simple case of the rigid
support, requiring that y = when x = 0, and see what effect this
has on the motion of the string. The solutions y = F(x — ct) or
y = f(x + ct) cannot be used in this case, for they will not always
be zero when x = 0. However, the solution
y = F(x  ct) + F{x  ct)
will satisfy the boundary condition. At x = 0,
y = F(ct) + F(ct) = 0,
(a)
^ '*
(b)
^(c)
' Ky^
x\_.
Fig. 13. — Reflection of a wave from the end support of a string. The solid lines
show the shape of the string at successive instants of time; the dotted lines, the imagi
nary extension of the wave form beyond the end of the string.
for all values of t. To see what motion of the string this expression
corresponds to, let us suppose that F(z) is a function that is large only
when 2 is zero and drops off to zero on both sides of this maximum.
Then when t = —10, the function F(—x — ct) will have a peak at
x = 10c representing a single pulse traveling leftward along the string.
The function F(x — ct) at this time would have its peak at x = — 10c
if there were any string to the left of the support, but since no string
III.8]
WAVES ON A STRING
77
is there, the term —F(x — ct) is not apparent, in the shape of the
string at t = — 10. It has been represented in Fig. 13a by a (Jotted
line to the left of x = 0.
As t increases, the wave in the actual string moves to the left,
and the wave in the imagined extension of the string moves to the
right until they begin to coalesce at x = 0. During the coalescing
the displacement of the point at x = is always zero, for the effects
of the two waves just cancel each other here. A little later, the waves
have passed by each other, the wave that had been on the imaginary
part of the string now being on the actual string, and vice versa.
i^il_L
X^
Fig. 14. — Motions of plucked and struck strings fixed at one end. Dotted lines
show the traveling partial waves; their sum is the solid line, the actual shape of the
string.
The succession of events is pictured in Fig. 13. What has happened
is that the pulse which had originally been traveling leftward is
reflected at the point of support x = and comes back headed toward
the right, as a pulse of similar form but of opposite sign. The bound
ary condition at x = has required this reflection, and the particularly
simple sort of condition that we have imposed has required this very
symmetric sort of reflection. Most other boundary conditions would
require a greater difference between the original and the reflected
wave. When a wave strikes it, a rigid support must pull up or down
on the string by just the right amount to keep y zero; and in doing so
it "generates" a reflected wave.
78 THE FLEXIBLE STRING [III.8
The expression for the motion of a string satisfying the boundary
condition y = Oat a; = and the initial conditions y = y (x),v = v (x)
at t = is
y = ^ I Y(x  ct) + Y(x + ct)  c H(x ct) + ~H(x + c<) (S.4)
where
Y(z)
H{z)
= f y («) (z > 0)
\</o(z) (2 < 0)
= (S(z) (z > 0)
\S(z) (z < 0)
S(z) = f* v (z) dx
These definitions of Y and H are necessary because y and y are
defined only for positive values of x (where the string actually is),
whereas the form of the partial waves used to build up the subsequent
forms of the string must be given for all values of z = x + ct. The
particular forms of Y and H are chosen so that they automatically
satisfy the boundary conditions at x — for all values of t. Two
examples of the way in which the motion of the string can be built
up by the use of these partial waves are given in Fig. 14.
Strings of Finite Length. — Actual strings are fastened at both ends,
so that really two boundary conditions are imposed. For instance,
the string can be fastened to rigid supports a distance I cm apart, so
that y must always be zero both at x = and at x = I. The most
important effect of a second boundary condition of this sort is to
require that the motion of the string be periodic. A pulse started at
x = travels to the other support at x = I in a time (l/c), is reflected,
travels back to x = 0, and is" again reflected. If the supports are
rigid, the shape of the pulse after its second reflection is just the
same as that of the original pulse, and the motion is periodic with
a period equal to 21/ 'c. The motion in this case is not, in general, har
monic, as we shall see, but it is always periodic. This periodicity
of all motion of the string depends entirely on the fact that we have
imposed a particular sort of boundary condition; if other conditions
are imposed at a; = and x = I (i.e., if the supports are not perfectly
rigid), then it may not be true that every motion is periodic; in fact it
may never be periodic.
The quantitative manner of dealing with the two boundary con
ditions is by means of the partial waves F. When the string is only
IH.8] WAVES ON A STRINO 79
I cm long, we are free to give any shape to F(z) for z larger than I or
smaller than zero. "Free" is not the correct word, however, for we
must choose that form of F which satisfies both boundary conditions.
If we start out at t = with a pulse of the form F traveling to the
left, then, as before, we can satisfy the condition y = at x = by
setting y = F(x  ct) + F(x  ct). To have y = at x = I,
we must arrange the rest of the function F, beyond the limits of the
Ky=
V/^*~T
is^l 2s/
^\^r ^" N
Fig. 15.— Periodic motion of a string fixed at both ends. Solid lines give the shape
of the actual string at successive instants; dotted lines show the imaginary extension
of the wave form beyond the ends of the string. The motion is made up of two partial
waves going m opposite directions, each being periodic in x with period 21.
actual string, so that F (I  ct) = F(l  ct), or, setting z = I  ct,
so that F(z) = F(z + 21) for all values of z. This means that the
function F(z), which must be defined for all values of z, must be
periodic in z, repeating itself at intervals of 21 all along its length. An
illustration of how this sort of partial wave can be used to determine
the motion of the string is given in Fig. 15.
To satisfy the boundary conditions y = at x = and at x = I
and the initial conditions y = y (x), v = v (x) at t = 0, we build up
a combination similar to that given in Eq. (8.4):
V = \\Y{x  ct) + Y(x + ct)  ±H(x  ct) + \H{x + ct)] (8.5)
80 THE FLEXIBLE STRING [m.9
where
Y(z) =
yo(z)
(1 <z < 0)
2/0(2)
(0 < z < I)
Z/o(2Z 
z)
(I <z < 21)
yo(z  
21)
(21 < z < 31)
etc.
S(z)
(1 <z < 0)
S(z)
(0 < z < I)
S(2l  z)
(I <z < 21)
S(z  21)
(21 < z < SI)
etc.
H(z) =
S(z) = j" v (x) dx
Two examples of the motion of such strings are given in Fig. 16.
In Fig. 17 the displacement of a point on the string is plotted as a
^s S* ^v
k^' < i > < ^ :>< ^ t" — v — > — % 
e*^t
Fig. 16. — Motions of plucked and struck strings fixed at both ends. The solid
lines show the successive shapes of the string during one half cycle. Shapes for the
other half cycle are obtained by reversing the sign of the curves.
function of time, showing that the motion is periodic but not simple
harmonic.
9. SIMPLE HARMONIC OSCILLATIONS
It has been seen in the last section that the imposing of boundary
conditions limits the sorts of motion that a string can have and that
if the boundary conditions correspond to the fixing of both ends of
m.9]
SIMPLE HARMONIC OSCILLATIONS
81
the string to rigid supports the motion is limited to periodic motion.
This last result is an unusual one, for we found in the last chapter that
even as simple a system as a pair of coupled oscillators does not, in
general, move with periodic motion. It is not unusual for a system
to oscillate with simple harmonic motion (which is a special type of
periodic motion) when it is started off properly (we shall see that
practically every vibrating system can do this); what is unusual in
the string between rigid supports is that every motion is periodic, no
matter how it is started.
Our problem in this section is to find the possible simple harmonic
oscillations of the string (the normal modes of vibration) and to see
what the relation is between the frequencies of these vibrations that
makes the resulting combined motion always periodic. The problem
of determining the normal modes
of vibration of a system is not
just an academic exercise. For
systems more complicated than
that of the string between rigid
supports we have no method of
graphical analysis similar to that
of the last section, and the only
feasible method of discussing the
motion is to "take it apart" into
its constituent simple harmonic
components. There is also a
physiological reason for studying the problem, for the ear itself analyzes
a sound into its simple harmonic parts (if there are any). We dis
tinguish between a note from a violin and a note from a bell, for instance,
because of this analysis. If the frequencies present in a sound are all
integral multiples of a fundamental frequency, as they are in a violin,
the sound seems more musical than when the frequencies are not so
simply related, as in the note from a bell.
The Wave Equation. — To obtain the normal modes of vibration of
a string, we must take up the problem that we laid aside in the last
section, that of finding the equation relating the shape of a string and
its motion. This equation was almost derived in the previous section.
Looking back at Eqs. (8.2) given there, we find that, by repeating the
differentiation, both F(x — ct) and /(a; f ct) satisfy the equation
Fig. 17. — Displacements of the points
marked (c) on the strings shown in Fig. 16,
plotted as functions of the time.
d 2 y _ 1 d 2 y
Bx 2
at 2
c = \/T/e
(9.1)
82
THE FLEXIBLE STRING
[III.9
Since every motion of the string is a combination of the two waves,
every motion of the string must satisfy this equation. It is called
the wave equation. We shall meet it often in this book.
The wave equation corresponds to a number of statements con
cerning the motion of a string. We saw in the last section that it
implies that the wave motion travels with its shape unchanged, at a
velocity c, independent of this shape. Since the derivative (d 2 y/dx 2 ) is
proportional to the curvature of the shape of the string at a given
instant, Eq. (9.1) states that the acceleration of any portion of the
string is directly ■ proportional to the curvature of that portion. If the
curvature is downward, the acceleration is downward, and vice versa;
and the greater the curvature, the faster the velocity changes.
This second statement corresponds to the usual method of deriving
the wave equation. The net force
on a portion of string at any instant
is proportional to the curvature of
this portion at that instant. A
study of Fig. 18 shows us that the
net force perpendicular to the zaxis
on a piece of string of length ds is
T(sm<p 2 — sin<pi). If the displace
ment of the string from equilibrium
is not large (and we have already
had to assume this), the angles >i and <p 2 will be small, so that
sin^>i will be practically equal to tan<pi, and similarly for <p 2 . How
ever, tan <pi is equal to the slope of the string at the instant in question,
(dy/dx), at the point x. Similarly tan <p 2 is the slope at the point
x + dx. Now, from the definition of the symbol (d/dx), the value
of a function / of x at the point x + dx is equal to the value of / at
the point x, added to dx times the rate of change of / with respect to x,
* x+dx
Fig. 18. — Forces on an elementary
length of flexible string.
f(x + dx) = f(x) + dx
1
dx
(9.2)
so that the net vertical force on the element of string is
T \(dx\ _ (dy\ ] = Tdx ±(§y\ = Tdx ejy
l\dy/x+dx \dx/x] dx \dxj dx 2
The mass of the element of string is e ds; and if the angles <p are
small, this is practically equal to e dx. The equation of motion of the
element is therefore
HI.9] SIMPLE [HARMON IC OSCILLATIONS 83
edx^=Tdx^, or ^ € ^
edx dt2 1 dx dx2 , or ___ —
which is identical with Eq. (9.1).
Standing Waves. — But we are still looking for the possible ways
in which the string can execute simple harmonic vibrations. To exe
cute such motion, every portion of the string must oscillate with the
same frequency. The mathematical counterpart of this requirement
is that the shape of the string y(x, t) must be equal to the exponential
e 2*irt multiplied by a function of x alone: y = Y(x)e 2rirt . If this
expression is substituted in the wave Eq. (9.1), we find that Y must
satisfy the equation
d 2 Y 4tt 2 i/ 2
^+^"^ = 0, c*=(T/e) (9.3)
Solutions of this equation are the exponentials e 2vivx/c and e 2vivx/c .
All the simple harmonic motions of the string must therefore conform
to the expression
y = Q e (.2rir/c)(xct) _J_ Q _ e (2iciv/c){xct)
= A+ cos ^ (x  ct)  <S> + + A cos ^ (x + ct)  $_ (9.4)
representing two sinusoidal waves, of different amplitudes, traveling
in opposite directions. This result is another illustration of the
relation between the shape and the motion of the string; when we
require that the dependence of y on t be sinusoidal, the dependence
of y on x must also be sinusoidal.
The wavelength of these waves X = (c/v) is the distance between
the beginning and end of each cycle of the sinusoidal wave form.
In the special case when the amplitudes of the two waves are
equal, the trigonometric formula for the sum of two cosines can be
used, giving
A = 2A+ = 2A_
y = A cosf — x — Q J cos (2irvt — <f>), = £($_ + $ + ) (9.5)
$ = i($_  $ + )
Here the traveling waves combine to form standing waves. At points
on the string where cos ( — a — £2 J = 0, the two traveling waves
always just cancel each other, and the string never moves. These
84 THE FLEXIBLE STRING [HI.9
points are called the nodal points of the wave motion. In the case
that we are considering, where the density and tension are uniform,
the nodal points are equally spaced along the string a distance (c/2v)
apart, two for each wavelength. Halfway between each pair of nodal
points is the part of the string having the largest amplitude of motion,
where the two traveling waves always add their effects. This portion
of the wave is called a loop, or antinode.
Normal Modes. — So far, we have neglected boundary conditions.
If we require that y = when x = 0, the general form of (9.4) can
no longer be used; the number of possible harmonic motions is limited.
The expression for y that must be used is the standing wave form
(9.5) with the angle ft so chosen that a nodal point coincides with the
point of support x = 0:
y = A sin( x J cos (2irvt — $) (9.6)
This agrees with the discussion in the previous section. For the
simple boundary condition that we have used, the reflected wave has
the same amplitude as the incident wave; and when the incident one
is sinusoidal, the result is a set of standing waves. Any frequency
is allowed, however.
When the second boundary condition y = at x = I is added, the
number of possible simple harmonic motions is still more severely
limited. For now, of all the possible standing waves indicated in
(9.6), only those which have a nodal point at x = I can be used. Since
the distance between nodal points depends on the frequency, the
string fixed at both ends cannot vibrate with simple harmonic motion
of any frequency; only a discrete set of frequencies is allowed, the set
that makes sin ( ^ I ) zero. The distance between nodal points
must be I, or it must be (1/2), or (1/3) . . . etc. The allowed fre
quencies are therefore (c/2l), (2c/2l), (3c/2Z) . . . etc., and the differ
ent allowed simple harmonic motions are all given by the expression
y = A„sinfe) cos(™ t  ^ (n = 1, 2, 3, 4 • • •) (9.7)
The lowest allowed frequency v x = (c/2l) is called the fundamental
frequency of vibration of the string. It is the frequency of the general
IH.9] SIMPLE HARMONIC OSCILLATIONS 85
periodic motion of the string, as we showed in the last section. The
higher frequencies are called overtones, the first overtone being vz, the
second v%, and so on.
The equation for the allowed frequencies given in Eq. (9.7)
expresses an extremely important property of the uniform flexible
string stretched between rigid supports. It states that the frequencies
of all the overtones of such a string are integral multiples of the funda
mental frequency. Overtones bearing this simple relation to the
fundamental are called harmonics, the fundamental frequency being
called the first harmonic, the first overtone (twice the fundamental)
being the second harmonic, and so on.
Very few vibrating systems have harmonic overtones, but these,
few are the bases of nearly all musical instruments. For when the
overtones are harmonic, the sound seems particularly satisfying, or
musical, to the ear.
Fourier Series.— To recapitulate: The string has an infinite num
ber of possible frequencies of vibration; and if the supports are rigid,
these frequencies have a particularly simple interrelation. If such
a string is started in just the proper manner, it will vibrate with just
one of these frequencies, but its general motion will be a combination
of all of them:
y = Ai sin( y J cosf j t — 3>i J + A 2 sin( — p J cosf — j t — # 2 )
+ . . .
or, symbolically,
■  2<^x)[ B "°K ! T*) + c ""K s r)] (9 ' 8)
where the symbol 2 indicates the summation over the number n,
going from n = 1 to n = «> . The value of A n is called the amplitude
of the nth harmonic.
Equation (9.8) is just another way of writing Eq. (8.5). The
present form, however, shows clearly why all motion of the string
must be periodic in character. Since all the overtones are harmonic,
by the time the fundamental has finished one cycle, the second har
monic has finished just two cycles, the third harmonic just three
cycles, and so on, so that during the second cycle of the fundamental
86 THE FLEXIBLE STRING [HI.0
the motion is an exact repetition of the first cycle. This is, of course,
what we mean by periodic motion.
Equation (9.8) is in many ways more useful for writing the depend
ence of y on t and x than is Eq. (8.5). For it gives us a means of finding
the relative intensities of the different harmonics of the sound given
out by the string (corresponding to the analysis that the ear makes
of the sound) and thus gives us a method of correlating the motion
of the string with the tone quality of the resulting sound. We shall
have to wait until farther along in the book to discuss the quantitative
relations between the vibrations of bodies and the intensity of the
resulting sounds, but it is obvious that the intensity of the nth har
.monic in the sound depends on the value of the amplitude A n . Once
the values of all the A n 's are determined, the future motion of the
string and the quality of the sound which it will emit will both be
determined.
Initial Conditions. — The A n 's and $ n 's, or the B n 's and C n 's, are
an infinite number of arbitrary constants, whose values are fixed by
the initial conditions, corresponding to the infinite number of points
along the string whose positions and velocities must all be specified
at t = 0. Our analysis will not be complete until we devise a method
for determining their values when the initial shape and velocity shape
of the string are given.
The initial conditions must satisfy the equations, obtained from
(9.8) by setting t = 0,
y(x,0) = y (x) = ^ B n sm[~)
Series like the righthand sides of these equations are called Fourier
series.
Now, the initial shape and velocity shape of the string, the func
tions yo(x) and v (x), can be any sort of functions. that go to zero at
x = and x = I. It therefore must be possible to express any func
tion satisfying these boundary conditions in terms of a Fourier series
of the type S5„ sin (rnx/l). Subtle. mathematical reasoning must be
used to prove rigorously that this is true (in fact, a completely satis
factory proof has not yet been devised), but since we know as physicists
that the motion of the string is definitely specified by its initial shape
and velocity shape, we shall assume that it must be true.
m  9 ] SIMPLE HARMONIC OSCILLATIONS 87
The Series Coefficients.— The trick to obtain the values of the
B n 'a and C n 's consists in multiplying both sides of the preceding
equations by sin (irmx/l), where m is some integer, and integrating over
x from x = to x = I. The utility of the trick lies in the fact that
J o sin {irnx/l) sin (irmx/l) dx is zero unless n equals m, in which case
it equals (1/2), so that the infinite series of integrals on the righthand
side has only one term not zero, involving just one B m or C m . For
instance,
X sin (^) y ° (x) dx = % B « £ sin (^) sin (^) **
= \2j Bm
so that
„ 2 C . (irmx\ . . . \
m = T ) sin \T/ y °w )
and, similarly, ( (Q Q ,
Cwi = ^ Jo sin VT^o(x) rfz (m = 1, 2, 3, 4 ■ • • ) )
Equations (9.9) provide a means of determining the values of the
Bn& and C n 's in terms of the initial conditions.
Plucked String, Struck String.— A few examples will indicate how
the method works. For instance, if we pull the center of the string
out h cm and then let it go at t = 0, all the C m 's will be zero, and
8h . firmS
= ^ sin V27
{0 if m is an even integer
(_1)(— d/ 2 if w is an odd integer
Therefore,
y  ¥ r n Vi7 cos \tJ  9 sm \ir) cos {r) +'.■■■
• (9iP)
88
THE FLEXIBLE STRING
[m.9
Computing this series for y as a function of x and t gives the same
values for the shape of the string at successive instants as are shown
in the first sequence of Fig. 16. .Figure 19 shows how the correct
form is approached closer and closer the more terms of the series are
used.
At first sight, the foregoing series appears to be simply a more
awkward way of finding the shape of the string than the method used
in the previous section. However, the series can tell us more about
the string's motion than the results of the last section can. It tells
/\ First Te
First Four* Terms
Fig. 19. — Fourier series representations of the initial form of the string given in
Eq. (9.10) and the initial velocity form given in Eq. (9.11). Successive solid curves
show the effect of adding successive terms of the series; dotted curves show the actual
form, given by the entire series.
us, for instance, that the second, fourth, sixth, etc., harmonics will
be absent from the sound given out by the string, for they are not
present in the motion. It tells us that if, for example, the intensity
of the sound emitted is proportional to the square of the amplitude
of motion of the string, then the fundamental frequency will be 81 times
more intense than the third harmonic, 625 times more intense than
the fifth harmonic, etc.
The absent harmonics correspond to standing waves that have a
nodal point at the center, the point pulled aside. This is an example
of the general rule (which can be proved by computing the # required
integrals for B m ) that in the motion of any plucked string all those
harmonics are absent which have a node at the point pulled aside.
IH.9] SIMPLE HARMONIC OSCILLATIONS 89
If the string is struck, so that y = and
((*) ( 0< *<d
(o Q<»<»)
then all the B m 's are zero, and
iron ~ 2 f P z/4 A/xx\ . /xmx\ ,
so that
. 1 + V2 . /3tx\ . (Zicct\
+ — 27 sm VTJ sin V~ry
"m^—jsm^—J • • • J (9.11)
^ 7T°C
125
We note that the fourth, eighth, etc., harmonics, those having nodes
at x = (1/4:), are absent in this case.
Energy of Vibration. — There is a general analogy between the
amplitudes A n of the various harmonics and the amplitudes of motion
along the normal coordinates X and F discussed in Sec. 6. This
can be illustrated by computing the energy of vibration of the string.
The kinetic energy of the string is the integral of the kinetic energy
of each element of length: ie I i^Jdx. Its potential energy is
equal to the amount of work necessary to move the string into its
instantaneous form from the equilibrium form y = 0. Suppose that
its form at a given time tis y, a function of x. Then we can imagine
changing the string from equilibrium form to final form by making
its intermediate form be ky, where k changes from zero to unity. The
force on any element of string of the form ky is T f — 2 ky) dx, and the
90
THE FLEXIBLE STRINO
[III.9
force that we must use to oppose this is equal to this value and oppo
site in sign. As we displace the string from equilibrium by changing
k, the element of displacement is y dk, and so the work required to
bring this element of string into place is
J>(s)
^)kdxydk = Ty[^ 2
(5)*J>«>(S)
dx
The potential energy of the whole string, the work required to bring
the string from equilibrium into the form represented by the function
V, is
if we integrate by parts. Since y = at x = and x = I, the first
term is zero, and the potential energy is just (T/2)j(dy/dx) 2 dx. The
total energy of the string is therefore
When series (9.8) is substituted into (9.12), the expression for the
energy becomes
12
*(")'
I 4AM2 s i n (/
^ nA n cosf^y J cos(^ 3>„ J \ dx
A\ sin 2 (?f  <h) £ sin 2 (y) dx + • • •
$i ) sin Cr  * 2 ) Jo sin Vr) sin (nr) dx
+ • • •
+ A\ cos 2 fe*  * x ) Pcos 2 fe)^ + • ■ • }
All the integrals of the sort / sin (ttx/T) sin (2rx/l) dx are zero, whereas
integrals of the type of / sin 2 (rx/l) dx and / cos 2 (irx/l) dx are equal
to {1/2), This simplifies the expression enormously, and after adding
sin 2 to cos 2 , term by term, we can write
w = w.) 2 (™J A i = 2 2t ° (I) vlAl ' " ■■ (l) (9  13)
111.10] FORCED VIBRATIONS 91
The energy is therefore a series of terms, each term depending on
just one of the harmonics, an expression similar to that given in
Eq. (6.6). The different harmonics are the different normal modes
of vibration of the string, and the quantity A n is the amplitude of
vibration along the nth normal coordinate of the system. We can
say that the energy of a vibrating string is equal to the total energy
of an infinite number of equivalent harmonic oscillators, each having
a mass equal to half the total mass of the string {It/ 2), one having
frequency vi and amplitude A i, another having frequency v 2 and
amplitude A 2 , and so on.
10. FORCED VIBRATIONS
So far we have studied the particularly simple case of the free
vibrations of an idealized string. The analysis was useful, for it
gave us an insight into the general properties of the motion without
entangling us in the algebraic complications that crop up as soon as
some of the idealism is relaxed.
Before we study the effects of motion of the end supports, friction,
and nonuniformity on the behavior of the string, let us study the
response of the idealized string to a driving force. Here again we
find that a neglect of complications, at first, enables us to bring out
the salient behavior with the least amount of mathematical camouflage.
Wave Impedance and Admittance. — The simplest case is that of a
string of infinite length stretched at a tension T between supports
at x = and x = °o. The mass of the string per unit length is e,
so that t he v elocity of wave motion
is c = y/T/e. The support is rigid in
the xdirection, so as to sustain the
tension of the string; but it is hinged
so that it can move in the ^direction,
transverse to the string. A transverse
force applied to the support will move
the end of the string as well as the
support, so that the mechanical im
pedance offered to the force is the sum FlG  20  Forces on a flexible su PP° r *
of the transverse mechanical impedance of the support and the
mechanical impedance of the end of the string to transverse motion.
The impedance of the support is of the general type discussed in
Chap. II, and need not concern us here. What is now of interest is
the wave impedance of the string, the ratio of the transverse force
applied to the end of the string to the transverse velocity of the end
92 THE FLEXIBLE STRING [111.10
of the string, when the driving force is simple harmonic and the trans
verse impedance of the end support is neglected.
The displacement of the string at a distance x from the end and
at a time t is y(x,t). The angle that the end of the string makes with
the equilibrium line is 9 = tan _1 [(dy/dx)_o]. The longitudinal force
on the end of the string is — T cos 0. Since we are assuming, every
where in this chapter, that (dy/dx) is small compared with unity, we
can say that, to the first approximation, the longitudinal force is T,
the constant tension, and the transverse force exerted by the support
on the end of the string is
■  r (2)
rt».r.^/«> (io.i)
If the driving force is simple harmonic, / = Foe  *"', the shape of the
string must be sinusoidal, as is indicated in Eq. (9.3). In the present
case the wave must be going in the positive ^direction, since it is
being generated at x = 0. Therefore the space factor will be Ae+ ikx ,
where
— A oihx — iut
Incidentally, because we have chosen the negative exponential
e u>t f or our convention, it turns out that the sign of the exponential
e ikx indicates the direction of motion of the wave. A wave in the
negative ^direction would be e  i( ~ kx+a>t) , with a negative sign in the
a>part.
The value of A is determined from the expression for the driving
force in Eq. (10.1)
Foe™' =
so that
V =
\dx) x= o
1 ** ° e ik{xct)
— l(j) eC
V ~ dt ec 6
(10.2)
The input impedance of the string, the ratio between the driving force
and the transverse velocity of the string at x = 0, is
111.10] FORCED VIBRATIONS 93
ec = (±\ = y/Te (10.3)
This quantity, the input impedance of an infinite string, is sometimes
called the wave or characteristic impedance of the string. We see it
is real, being a pure resistance for a simple string. This implies that
energy is being continuously fed into the string, energy of wave
motion that never returns, since the string is infinite in extent. When
we come to consider strings of finite length, we shall see that the input
impedance differs from the characteristic impedance ec and is not
purely resistive.
The power input to the string is the average value of Fv at x =
p = i &!! = i €C  y p (io.4)
In many cases we wish to know the velocity of some other part
of the string when the force is applied at x = 0. This is obtained in
terms of the transfer impedance, already discussed in Sec. 7. It is
usually easier to deal with the reciprocal quantity, the transfer admit
tance, the ratio between the transverse velocity of the string at point x
and the simple harmonic driving force (in this case at x = 0). For
the simple case of the infinite string,
^=r„(0,*;«)= jgl^i;^ (10.5)
General Driving Force. — When the transverse force on the end of
the string is more complicated than a simple harmonic force, the
response of the string can be obtained by the methods of Sec. 6. We
find the Fourier transform of the force function
(W) = ^J
Then, in accordance with the analysis leading to Eqs. (6.15), we
obtain the equation for the velocity shape of the infinite string when
the end at re = is acted on by a transverse force f{t),
v(x,t) = f Y m {0,x;o>)e^ da> I ^ J ^ f(j)e™ dr J
.. /» 00 (* 00 .
= %T~ \ e (Wc)  i < tf ' du I /(r)e iwT dr
94 THE FLEXIBLE STRING [HI.10
Using the general formulas for Fourier transform given by Eqs. (2.19)
and (2.20), we find that the velocity shape of the string is
^sK'OM'?) (ia6)
This is a very interesting and simple result, one that we. should
have been able to obtain without recourse to the machinery of Fourier
transforms. It restates the simple fact that was discussed in Sec. 8,
that the shape of the string, as function of x for a constant time, is
simply related to the motion of one part of the string, as a function
of time. Since the input impedance of the infinite string is constant
and resistive, the velocity of the string at the point of application of
the force (x = 0) is proportional to the force no matter what form f(t)
has. Therefore for x = 0, v(0,t) = (l/ec)f(t), where ec is the input
resistance. And since the motion of the string causes waves in the
positive xdirection, the expression for v(x,t) must be that given in
Eq. (10.6), with the characteristic quantity (x — ct) appearing.
The corresponding expression for the displacement is obtained
by integrating /,
y(x,t) = \ c Q (t  fj; Q(Z) = j* f(t) dt (10.7)
We can now verify our calculations still further. From Eq. (10.1)
we have that
. *>*&L s[=«H)L
which checks, if we use Eq. (10.7) for Q and the relation c 2 = (T/e).
The reason that there is such a simple relationship between / and
y in this case is that the input impedance is a constant, independent
of «. There are no resonant frequencies, where the impedance is
zero and the admittance has a pole, for the infinite string; since there
is no reflection of waves from the far end, there is no periodicity of the
wave motion and, therefore, no natural frequencies of free vibration.
String of Finite Length. — When we come to consider the simple
string of finite length with a rigid support at x = I, we find that the
input impedance at x = is considerably more complicated. The
string now has resonance frequencies, so that the impedance goes to
zero for certain values of <a. The waves are reflected from the rigid
support at the far end (x ~ I), so that no energy is lost. Therefore
m.10] FORCED VIBRATIONS 95
if the support at x = I is rigid, and if there is no resistance to the
string's motion, then the input impedance is purely reactive, with
no resistive term.
We find the displacement of the string from Eq. (9.3). The two
simple harmonic waves of frequency (w/2tt) = v are combined to form
a standing wave of zero amplitude at the rigid support x = I,
y = A sin[k(l — xjje^, k = (w/c)
The amplitude o. the wave, A, is adjusted to fit the amplitude of the
force at x = 0,
, (dy\ . To, /<al\ . ,
1^1 — A — cos I — )e" at
\dx/ x= o c \ c )
Therefore the expressions for the displacement and velocity of a string
of length I driven from one end are
Foe* * =
El sin[(co/c)(S ~ x)] e _^ t
ecu cos (col/c)
.F 8w[(a/c)(l — x)]
ec cos (ool/c)
(10.8)
and the expression for the input admittance and the transfer admit
tance (reciprocals of the impedances) are
r(0,0; w )=(i)tan(^)
©MtM?)"©]
(10.9)
Y(0,x;co) =
Foe™ 1
This case, therefore, gives definite resonances; the amplitude of
motion becomes infinite (since we are neglecting friction) whenever
tan (o)l/c) becomes infinite. This occurs at the frequencies for which
M/c) = (tt/2), (&r/2), (&r/2) • • • ; or
Vn = (^) (2n + 1} (n = °' 1 ' 2 ' 3 ' ' ' >
io n = 2irv n = fe)(2n + 1)
For these frequencies the input admittance is infinite (the input imped
ance is zero).
For the frequencies v = (c/2l), (2c/2Z), (3c/ 21) • • • the input
admittance is zero, the input impedance is infinite (for the zero
friction case). This is analogous to the parallel resonance in electrical
circuits. At these antiresonance frequencies the motion of the end
\k*~
JJ^~
06 THE FLEXIBLE STRING [HI.10
point x = is infinitesimally small, though the rest of the string is in
motion.
For very low frequencies, the input impedance 1/F(0,0;«) has
the limiting value i(ec 2 /lu) = i(T/fa>), a " capacitative " reactance,
with an effective stiffness constant (T/l) and "capacitance" (l/T).
The next step in this analysis would be to compute the response
of this system to impulsive forces. This case will be left for a prob
lem, and we shall go on to a somewhat more useful example.
Driving Force Applied Anywhere. — Often the driving force is not
applied at one end of the string, but elsewhere along the string. Some
times, indeed, the force is distributed along the string, instead of being
concentrated at a point. A solution of this general case is best
obtained in terms of solutions for concentrated forces, however; so
the next problem we shall study is that of the string of length I, held
between two rigid supports, and acted on by a force F(£)e iwt applied
at the point x = £.
When we obtain the steadystate response for a simple harmonic
force applied at a point, we can use the operational calculus to obtain
the response to an impulsive force applied to a point. Once this is
obtained, the response of the string to a force that is any function
of time and that is distributed in any manner along the string is com
puted in terms of integrations over  (the point of application) and r
(the instant of impulse), analogous to the integrals of Eqs. (7.14).
If a simple harmonic force of frequency (w/27r) is applied to the
point x = £, the part of the string f or x < £ should be a part of a
standing wave that is zero at x — 0, such as a sin (cax/c). The part
for x > £ should be a standing wave that goes to zero at x = I, such
as b sin[(<o/c)(£ — x)]. There will be a sudden change of slope at
x = £, and — T times this change in slope must equal the applied
force. Calculations similar to those made in the previous example
show that the correct solution is
> s \* \ eco sin(«Z/c)
ecu sin {ool/c) \ c /
F(£)e™ 1 sin (w/c) .
{:>*>]
(10.10)
(x>
The transfer admittance (v/f) for this case is
i_ sm[(o)/c)(l  £)] sin (a>x/c) . < ^
_ i_ sin ((o^/c) sin[(co/c)(f  x)\ . > .
ec sin (wZ/c)
m.10] FORCED VIBRATIONS 97
The input admittance is obtained by setting x = £ in the expression
for Fax;co).
Alternative Series Form. — Equation (10.10) is a closed form for
the shape of the string driven by a simple harmonic foree applied to
x = £. Sometimes it is useful to express this function in terms of a
Fourier series of the type given in Eq. (9.8). As shown there, if
we set y = XB n sin (rnx/l) the integrals for B n can be worked out
?« = j I sin {jyJ v(x,t) dx
where y is given in Eq. (10.10). After a great deal of algebraic
manipulation, we obtain the final series
n = 1
which is equal to the expression given in Eq. (10.10). While it is an
infinite series, it is sometimes easier to use it for computation than is
the closed form of Eq. (10.10).
As a matter of fact we could have obtained this series in a much
more straightforward manner by going back to the equation giving
the force on each portion of the string. According to the discussion
following Eq. (9.2), the force on the element of string between x and
x + dx is T dx (d 2 y/dx 2 ), due to the tension. This is the only force
on most of the string in the present case; except for the point where the
driving force is applied, x = £. Here the applied force per unit length
of string is very large, so that the integral over x of the applied force
near x = £ is equal to .Foer**'. The applied force per unit length is
therefore given in terms of the delta function discussed on page 48.
The equation of motion of the string then becomes
If we are interested in the steadystate response, we consider that
the transient oscillations have died out, and all that is left is a shape
that oscillates with frequency («/2x). In brief, we set y{x,t) =
Y^e^K The resulting equation for Y is
d?Y
dx 2
® r [¥\«?o < io  is >
98 THE FLEXIBLE STRING [111.10
where c 2 =. (T/e). Now if we can find a Fourier series expansion for
8(x — £), we shall be able to find a Fourier series expansion for F.
A series expansion for the delta function can be found by the
methods of Eq. (9.9). The important property of the delta function
K x ~ £) i s that for any function f(x)
£f(x)8(x Z)dx=f(® (10.14)
Therefore if 8 can be expressed as a Fourier series SZ)„ sin {irnx/l), the
series coefficients are
D.f
X*e»(T)*?W ! f).
Formally, therefore, the series representing the delta function is
«(*  e  r 2 ""Or) sin (t 1 ) (10  15)
The only awkward part is that the series does not converge. This
should not be surprising, however, for one would expect a peculiar
sort of series to represent as peculiar a function as 8(x — £). It need
not disturb us particularly, however, for we do not intend to use series
(10.15) to compute 8(x — Q; we shall use it only as a means to com
pute the series for y. As long as we use the series for 5 only as a short
cut means of obtaining other series which do converge, we can perhaps
justify our use of a nonconvergent series. Naturally we must use the
short cut with care and check our results by other methods whenever
we can.
Going back to Eq. (10.13), we assume that Y = SA n sin (irnx/l)
and substitute this and the series for 8(x — ) into the equation. The
result is
Equating coefficients of sin (mx/l) on both sides of the equality sign
gives equations for A n , and the resulting series for y = Ye*™ 1 is just
that given in Eq. (10,12). This justifies the use of the series for
111.10] FORCED VIBRATIONS 99
b{x — £), since the series for y converges, and also indicates how much
shorter is the short cut than the earlier method.
The series of Eq. (10.12) shows very clearly the phenomenon of
resonance and the resonant properties of systems, which we began to
discuss in Sec. 7 on coupled systems. The steadystate shape of the
string is usually a combination of all the shapes of all the normal modes
of free vibration of the string, sin (irnx/l). When the driving frequency
(oj/2t) approaches one of the natural frequencies (nc/2Z), the contribu
tion of the corresponding mode to the shape of the string becomes
larger than all the others. In other words, as the driving frequency
approaches one of the natural frequencies, the amplitude of motion
increases without limit and the shape of the string approaches that of
the corresponding normal mode.
The series also indicates the effect of the point of application of the
force. If the force is applied at a node of one of the normal modes,
the corresponding factor sin (irn^/l) is zero and that normal mode and
resonance is absent from the motion.
Distributed Driving Force. — To find the response of the string to
a simple harmonic driving force that is distributed along the string,
we need only integrate the expressions given in Eqs. (10.10) and
(10.12) over the point of applications  of the force. If the previous
expressions give the response to a delta functio nTiorce at £, it is not
difficult to see that the response of the string to a force ^(De  ^ dynes
per cm length distributed along the string is given by the formulas
sin (tux/1)
(im/iy  (a>/c) 2 _
e**' (10.16)
One example of the use of this equation should probably be given.
When a plane wave of sound of frequency a)/2ir passes over a string,
it produces a force on it which is in phase over the whole length of the
string if the direction of the wave is perpendicular to the string, but
which varies in phase from point to point if the wave front is at an
angle. In general the force per unit length at the point £ cm from one
end can be given by the general expression
F(f)er*"« = /Toe*"*  *"', a = (2t/X s ) COS^
where X s is the wavelength of the sound wave in air and # is the angle
the direction of the wave makes with the string. The relation between
100 THE FLEXIBLE STRING [HI.10
F and the intensity of the sound will be discussed later in this book
[see Eq. (29.6)].
Taking the simple case of normal incidence first, where the force
Foe  "*' is uniform and in phase over all the string, we use the closed
form of Eq. (10.10) in Eq. (10.16) to obtain
■fc(!)]
J? lot ) C0S I
= b^_ < ,_„ N /■ _ ! i (10 17)
9 €0) 2 \ COS(cdZ/2c) )
This has resonances at every other harmonic (v = co/2tt = nc/2l;
n = 1,3,5,7 • • • ), the symmetry of the normal modes for the even
harmonics precluding their excitation. The shapes exhibited by the
string for different driving frequencies are shown in Fig. 21.
For the more general driving force Foe*"*  ™', it is easier to use Eq.,
(10.12) for y{£,x,t), and the final result can be expressed in the series
V °\ T J ^™* (xw) 2  (aiy
(10.18)
(ttw) 2  (a>Z/c) 5
This series is equal to the closed form of Eq. (10.17) for a = 0.
Transient Driving Force. — To calculate the response of the string
for a transient force, we use the operationalcalculus methods again.
We begin by computing the response of the string to an impulsive
force at t = applied at x = £; f(t) = 5(x — £)8(t). Referring to
Eq. (6.16), we see that the proper expression is given by the integral
Using Eq. (10.11), this becomes
, x . t) = _L f" *"*" s(»,z)
" ' 2irec J  «, o) sin (coZ/c)
where
(sin(^)sin[(f) (**)] (*>{)
The poles of the integrand are atu= (rnrc/l) where n is any integer,
positive or negative (these correspond to the natural frequencies).
Near one of the poles for an even n, the quantity sin (coZ/c) approaches
111.10]
FORCED VIBRATIONS
101
v = 4v, £
43
» s io»i
Fig. 21. — Shapes of steadystate motion of an undamped string between rigid sup
ports, driven by a uniform force of frequency v. The fundamental frequency of the
string is v\. Those parts of the string above the dashed equilibrium line are in phase
with the force; those below the line are 180 deg out of phase.
the value (l/c)[w — (ntrc/l)]. In other words, for <a very near (nirc/l)
(n even) the integrand becomes
1 Q—inrct/l
sin I
*  T ) sm V" )
2rel (rnrc/l) w — (nwc/l)
(*< Q
The residue of this expression, its limiting value when multiplied by
[w — (nirc/Vj\, as this factor approaches zero, turns out to be
1
2ir 2 ncc
o — invct/l
sin
r) sm KT)
(0 < x < I)
102 THE FLEXIBLE STRING [HI.10
For n odd, the factor sin (cd/c) approaches the quantity
fc\ ( rarc\
but the factor sin(nir — mr^/l) turns out to be — sm(nir£/l), so the
two minus signs cancel, and the result is the same as before. The
final value for the integral, — 2wi times the sum of the residues on or
below the real axis of <a, gives the value f or t >
/ (t < 0)
Vitefi = ]zl ^ I ^(^) ^J^A Pr ^,i {t > o)
l iwec ±^ n n \ I J \ I /
Utilizing the equation sinz = {l/2i)(e iz — e~ iz ), we have
/ (t < 0)
y s a,x;t) = ] _2_ ^ 1 S in (™*) sin (™) sin (^l) (t > 0)
1 n = 1
(10.19)
Incidentally, this equation could also have been obtained by substi
tuting series (10.12) in the contour integral.
The series for y& does not converge well, any more than the series
for the delta function does. However, as with the delta function, we
are not interested in computing y s , the behavior of the string after
being hit by an idealized impulsive force at a mathematical point on
the string. We only intend to use the series as an easy means of com
puting the behavior of the string when acted on by more realistic
forces, distributed along the length of the string and spread out in time.
For the most general type of force /(£,£), a function of time and of
position, the response of the string is
dr I d£ f(£,T)ys(£,x,t  t)
=3ijJ>[T (, ' ) ]*. (1 ° 20)
This series does converge for reasonable forms of /((•,<)•
The Piano String. — A reasonable approximation to the force of a
piano hammer on a string is
m.10] FORCED VIBRATIONS 103
/0 (icr > *> fr; . a*  if > $ > a* + fr)
*U0 = A cos [(^  * )] cos(^) (10.21)
\ (iff < * < i<r; z  if < £ < x + if)
The quantity a is the time duration of the application of force, and f
is the length of the portion of string acted on by the force. The force
starts acting at t = —io, rises to maximum at t = 0, and goes again
to zero at t = io. The distribution of the force along the string is also
like the positive half of the cosine curve, with the center of force, where
it is greatest, falling at £ = x .
Substituting this into Eq. (10.20) and carrying out the usual
accompaniment of trigonometry and algebra, we can work out the
expressions for the shape of the string. The integration over £ is
from x Q — if to x + if, but the integration over t is a bit more
elusive. When t is less than  i<r, y s (£,x; t  r) is zero for all values
of r for which F(£,r) differs from zero, so the integral is zero, as it
should be (since the string has not been hit yet). For t > io the
range of integration over t is from  io to + io; but f or  io < t < io
the only range over which Fy s is not zero is between r =  io, where
F goes to. zero, and t = t, where y»(£,x; t  t) goes to zero. Conse
quently, we have three expressions for the resulting shape y(x,t)
Fort< io, y(x,t) =
For  io < i < i<r,
For t > io,
. U ( X t) = *M. ^S i l" cog(™rr/2Q l ["cos (n7rco/2Z)l
x^ec ^ n Ll  (nf/0 2 J _1  (WO 2 J
D ;^ (TrnxA . (t71x\ . • (irnci\
8m \—) sm \r) sm \r)
This formula appears quite formidable, but it can be computed if need
be. Certainly it is not very difficult to obtain from it the relative
magnitude of the various harmonics in the free vibration after the
end of the blow (t > io).
Incidentally, this example is a good one to show the power of
104 THE FLEXIBLE STRING [111.10
the operationalcalculus methods. The algebraic gymnastics neces
sary to obtain Eq. (10.22) from the combination of Eqs. (10.20) and
(10.21) are not particularly easy for one mathematically musclebound.
Nevertheless, the calculations are only laborious, not subtle. On the
other hand, to obtain the final formula for y(x,t) by any other method
would involve still more labor, and more mathematical subtleties than
we care to include in this volume. Our applications of operational
calculus in Chap. II may have seemed rather like using a sledge ham
mer to drive a tack. We see now, however, that the problem of the
simple string already provides a spike worthy of the sledge.
To be honest, the resulting series for y(x,t) is not too good an
approximation for the actual motion of an actual piano string when it
is struck, partly because the actual piano string is not a perfect string
but has stiffness. We shall indicate how to correct for this in the
next chapter.
The Effect of Friction. — In the foregoing analysis we have neglected
friction, although it is present in every vibrating string. To complete
our discussion we should show, as with the simple oscillator, that the
effect of friction is to damp out the free vibrations and to change
slightly the allowed frequencies. To show that this is so is not diffi
cult by the use of operational calculus, although it would be difficult
by any other method.
The difficulty lies with the nature of the frictional term. The
resistive force per unit length opposing the string's motion is due to
the medium surrounding the string, the medium gaining the energy
that the string loses. Part of the energy goes into heating the medium,
the amount depending on the viscosity of the medium; and part goes
into outgoing sound waves in the medium, the amount depending on
the radiation resistance of the medium. The medium also adds an
effective mass per unit length, which may not be negligible if the
medium is a liquid. The important point, however, the one that is
responsible for our difficulties, is that the effective resistance due to
the medium (and also its added effective mass) depends on the fre
quency of the string's motion.
The equation Of motion for the string when friction is included is
d 2 y_ T d 2 y MsdV
e dF T dx* R{03) M
where R is the effective frictional resistance per unit length of string.
To find the "normal modes" of the string involves a sort of circular
process, since we cannot solve for the natural frequencies until we
IH.10] FORCED VIBRATIONS 105
know the value of R, and we cannot obtain the value of R unless we
know the frequency of motion. In this case it is actually easier to
start with steadystate driven motion, for then the frequency is known
and the value of R is definite.
The equation for the string acted on by a simple harmonic force
of frequency (a)/2ir) exerted on the point x = £ is
*4 + 2 *(co) f  c* g « m 8( x  fr** (10.23)
c 2 = (f/e), *(«) = R(<a)/2e
We are assuming that the added mass due to the reaction of the medium
is negligible compared with the weight of the string. The more
general case, where we must assume that e is also a function of <a,
will be discussed later.
As has been done previously, we shall assume that the steady
state motion of the string can be expressed in terms of a Fourier series
V = ^j °» sin ( nr) e_i "'
Substituting this in Eq. (10.23), multiplying both sides by sin (ttwz/Z)
and integrating over x from to I, gives an equation for a m , from which
one eventually obtains a series for y
le ~{ C" + ** ~~ «*»)(« + tk + tow)
where
w = [(T) 2 ^
We note that both w n and k are functions of w, the driving frequency.
Characteristic Impedances and Admittances. — An interesting
alternative method of writing this equation is in terms of transfer
admittances. The ratio between the string's velocity v = — iuy and
the driving force F{£)er iMt is
Y m (S,x;o>) = 2 \~^ ,
7T X \Zm (£,x;a},n)
 s \zr) J csc v— ) csc \— )
,106 THE FLEXIBLE STRING [111.10
The input admittance is obtained By setting x = f. Considering v
to be analogous to a current and F analogous to a voltage, the reaction
of the string (at x = £) is analogous to that of an electric circuit of
an infinite number of parallel branches, the nth branch consisting of an
inductance (le/2) csc 2 (rn^/l), a resistance of (IR/2) csc 2 (tw^/0> and a
capacitance of (2l/ir 2 n 2 T) sm 2 (irn^/l), all three in series.
The response of the string to a unit impulsive force concentrated
at x = £ is obtained by computing the contour integral
mU:X . t) .if { 2 < ?T* , \f* ( H!l ^ e ~n d »
y vs ' ' ' irk J „ ( ^J (w + ik — w n ){w + ik + w n ) )
One pole for the nth term occurs when a; + ik(o>) — w n (u>) is zero.
This may be difficult to solve algebraically if k(w) is a complicated
function of «. However, it can usually be solved graphically or by
successive approximations. We can write the solution symbolically as
<a = io n — ikn', k n = k(w n — ik n ) \ HO 26")
an = w n (<a n  ik n ) = [{nrnc/iy  A:*]* j
It turns out that the other pole of the nth term is at co = — u> n — ik n .
Taking residues at all the poles we finally obtain
/ (* < 0)
.J ^sin(^f)sin(^)sin M (t > 0)
y»(Z,x$) = 2\ e*»< ^ (Tn%\ ^ (irnx\ ^ / A (t ^ ft ^ (10.27)
which is to be compared with Eq. (10.19). This expression can be
used in Eq. (10.20) to obtain the response of the string, with friction,
to any transient force.
The expression for ys gives the free vibrations of a string when
started with an impulsive blow. It shows that the effect of friction
is to introduce a damping term e~ knt into each of the component vibra
tions. The frequencies of free vibration (co„/27r) do not greatly differ
from the harmonics for the undamped motion, (nc/2/), if the frictional
constant k n is small. In a good many cases k n increases as n increases,
so that the higher harmonics damp out more rapidly than the lower.
In such a case the sound emitted by the string will be harsh just after
the start of the motion, owing to the initial intensity of the higher
harmonics, becoming "smoother" as the motion damps out.
The amount of energy radiated by a string directly into the air is
quite small compared with that which can be radiated by a sounding
board attached to the string supports, as may be determined by com
in.il] STRINGS OF VARIABLE DENSITY AND TENSION 107
paring the intensity of sound radiated by a violin with and without
backboard. The effect of the transmission of energy from a string to
a sounding board, via the motion of the end support, will be discussed
in Sec. 13.
11. STRINGS OF VARIABLE DENSITY AND TENSION
The Fourierseries method discussed above is but one special case
of a method of dealing with vibrating bodies which we shall use in all
our subsequent work. We first find the shapes of the possible modes
of simple harmonic motion of the system, the modes that satisfy the
boundary conditions. In the case discussed above, these shapes were
given by the functions
A n sin
( irnx \
I )
but for other boundary conditions or for other systems {e.g., for a
string with nonuniform mass distribution or for a stiff bar) the func
tions of x will be different ones, which we can represent by the symbols
$n(x), A set of such functions, all satisfying the same equation and the
same boundary conditions, is called a set of characteristic functions,
and the corresponding allowed values of the frequency are called char
acteristic values.
General Equation of Motion.— The most general equation for the
string will involve a density e(x) and a tension T(x), both of which
vary with x. The net force on an element dx of string can be found
as before
K2)L[K2)1
The equation of motion of the string is finally
where f(x,t) is the applied force per unit length.
To study the normal modes of oscillation and to compute the char
acteristic functions and values, we consider the cases where f(x,t) = 0.
For simple harmonic motion we set y(x,t) = Y{x)e~^ ivt . The equation
dx
108 THE FLEXIBLE STRING [111.11
for Y is
T{x) ^1 + (2tv)Hx)Y = (11.2)
The solutions of this equation which satisfy the boundary conditions
at x = and x = I are the characteristic functions \f/ n , and the cor
responding allowed values of the frequency are the characteristic
values v n = (w«/2tt). Examples of these solutions will be given later
in this section.
The general free vibration of the system is then given by the series
y = % A ^(x)e^ (H.3)
71 = 1
where A n = B n + iC n = \A n \e^\ The determination of the con
stants B n and C n from the initial conditions is effected by the same
methods that obtained Eqs. (9.9). The characteristic functions turn
out to be orthogonal; i.e.,
J^(»)*.(*M«) «*«&. (*") (1L4)
where the constant M n is called the normalization constant, or the
effective mass, for the nth normal mode. We note that for the uniform
string of Sec. 9, M n = iZe.
Therefore if we are to represent the initial shape of the string, y ,
by the series
00
yo(x) = X B ^{x)
n = X
we obtain the values of B n by multiplying both sides by ^ m (x)e{x) and
integrating over x from to I. The result is
B m = Q) f yo(x)*~(x)e(x) dx (11.5)
which is analogous to Eqs. (9.9).
Orthogonality of Characteristic Functions.— The property
expressed by Eq. (11.4) is a general property of the characteristic
functions discussed here. This can be shown in the following maimer :
We multiply the equation for ^ n by \l/ m
*4( r t) »*■*
m.ll] STRINGS OF VARIABLE DENSITY AND TENSION 109
Next we multiply the equation for i// m by \(/ n and subtract the two
equations. The left side of the equation is
Both sides are now integrated over #, giving
[rw (*. f  *.*£)][ = (Bt  «a JJ *** *
In all the cases we are to consider, the quantity in brackets on the
left side of the equation is zero at both ends of the string, either because
the ^'s or their derivatives are zero, or because T is zero there. There
fore the righthand side of the equation is zero, and therefore the
integral must be zero unless o»2, = <a\, i.e., unless m = n. This is just
the orthogonality property given in Eq. (11.4).
Driven Motion. — If the string is driven by a simple harmonic force
of frequency (w/2r) which is concentrated at the point x = £, so that
f(x,t) = F(£)8(x — %)e iai , we find the steadystate motion by using
a series of characteristic functions, just as we did in the previous sec
tion. We expand the delta function for the force into the series
F(&S(x  = F(Q 2 (Jfc) <*)Mk)Mx)
as can be proved by multiplying both sides by f m (x) and integrating
over x. This series, of course, does not converge, but we are using it
as a shortcut means of getting y.
We also set y(t,x;o),t) = ^a n ^ n (x)e iat and insert both series into
Eq. (11.1), which gives
2 M*)(«i  «»)*•(*) = F(0 2 (^) <x)M&Mx)
Equating coefficients of $ n (x), we obtain an expression for a n , such that
the series for y is
^J M n (« — o>«)(ci> f «») v '
This is to be compared with Eq. (10.12) for the simple case.
The corresponding transfer admittance (l/F)(dy/dt) is given by
the series ■  ,
110 THE FLEXIBLE STRING [111.11
t j£1 M n (CO — «„)(« + <ti n )
n = l
It is purely imaginary, indicating that thelmpedance (1/F) is purely
reactive. At low frequencies (w < «i), y is in phase with the force,
and the impedance is stiffness controlled (see page 33).
The next stage is to use contour integration to obtain an expression
for the response of the string to an impulsive force at t = 0, localized
at x = £, symbolized by the expression S(t)8(x — £). The integral
for this is obtained from Eq. (6.16) and is
Each term in the series for Y m has two simple poles, atw = + o>„.
Adding the residues for each term, we finally obtain the impulsive
response
i0 (t < 0)
IS (V) M&Ux) sinW) (t > 0) (11.8)
This series does not always converge, any more than the series of Eq.
(10.19) did. The final result, however, will converge.
The desired final result is the shape of the string, y(x,t), when acted
on by a general force per unit length /(£,£), distributed along the string
in an arbitrary manner, and varying in an arbitrary manner with
time
y(x,t) = J dr J J^ y, (Z,x;t  t)/($,t) d$J
 % tM {£ . * ™ M  ^ ] [ J>*«> d "]} (lL9)
We notice that the motion may involve components with each of the
natural frequencies (co„/2tt). The relative magnitude of these com
ponents depends on the shape of /(£,£)•
This general series is the formal solution for the motion of the string
of variable mass and tension when acted upon by a transient force.
In the special case of the uniform string, it reduces to the series of
Eq. (10.20). For other cases, it will not be possible to go further
until we have worked out the specific form of the characteristic func
III.11] STRINGS OF VARIABLE DENSITY AND TENSION 111
tions \p n for the case we happen to be interested in. We shall work out
two cases in detail to show how these functions" may be computed
and then will indicate briefly how friction can be dealt with.
Nonuniform Mass. — Any variation in the distribution of mass
along the string also disturbs the harmonic relation between the over
> tones. As an example of this we shall consider a string of length I,
stretched between rigid supports, which is heavier at the center than
at the ends. Taking the point x = at the center of the string, we
specify that its mass per centimeter be e(x) = e [l — (x 2 /a 2 )], where a
must be larger than (1/2) . (Why ?) The wave equation then becomes
dx 2 \tJ\ a 2 ) dt 2
Waves do not travel with a constant speed along this sort of string,
nor is the shape of the wave unaltered during its travel. This means
that the graphical methods discussed in Sec. 8 cannot be used, and we
must analyze the motion into its simple harmonic modes of vibration.
If we set y = if/(x)e~ 2 * irt : , the wave equation reduces to
dfy . 471VeoA x 2 \ ,
We must find the solution of this equation and then pick out the
values of v that allow the solution to be zero at x = ±(l/2). Equa
tion (11.10) was considered in Prob. 4 of Chap. I. The solution was
shown to be
z = 2ttv Veo/Tx
and where
D.(k,z) = oos(2) + i s «  ggg*' + ■ • '
Bo(M) = sin(*)+±*.J 2 T+...
The function D e is even with respect to z = 0[i.e.,D e (k, —z) = D e (k,z)],
and Do is odd [i.e., D (k, —z) = —D (k, z)].
Since we have picked the origin at the Center of the string, we must
use either D e or D a as a possible solution to satisfy the boundary condi
tions. Any other combination will not have its nodal points placed
symmetrically about x = 0. The shape of the first mode of vibration,
which will vibrate with the fundamental frequency, will be given by
112 THE FLEXIBLE STRING [m.ll
D e for the value of v that makes the first nodal point of D e fall at x =
(1/2). Since D e is symmetric about x = 0, it will also have a node at
x — — (1/2) and will thus satisfy both boundary conditions. The shape
of the next mode, for the first overtone, will be given by D for the
value of v that makes the first nodal point of D (the first one, aside
from the one at x = 0) fall at x = (1/2). This function will also have »
a node at x = —(1/2). Going back to D e , if we increase v until the
second nodal point falls at x = (1/2), we obtain the third mode, or the
second overtone, and so on.
The Sequence of Characteristic Functions. — This illustrates a
general property of modes of vibration of strings, no matter what their
mass distribution or boundary conditions; if we arrange the charac
teristic functions giving the shapes of the different modes of vibration
in a sequence of increasing frequency of vibration, then each function
has one more nodal point between the points of support than the
preceding one, the curve for the fundamental having none, that for
the first overtone having one, that for the second overtone two, and
so on.
These functions can be labeled in order: fa, fa, fa ■ • • , a sequence
alternating between D e and D for successively larger values of v =
(u/2ir). The corresponding characteristic values or frequencies are
v\, v 2 . . . , or a>i, o>2 . •
The functions are orthogonal, as we have shown on page 108.
From them we can compute the motion of the string when acted on
by a driving force, by substituting in Eq. (11.9).
The general form of the string of variable mass which corresponds
to the initial conditions that y = y (x) and (dy/dt) = v (x) at t =
is given by the series
oo
V = % fa(x)[B n cos(2irv n t) + C n sm(2Trv n t)
n = l
where
B n = Ij^J J yofae(x) dx
Cn = ( o 1 \ T ) I v fae(x) dx, and N n = #€ dx
\27rv„iV w / Jo Jo
This can be shown by the method given in the first part of Sec. 10,
now that we have proved that the functions fa are mutually orthogonal.
The Allowed Frequencies. — The calculation of the allowed fre
quencies for the string corresponding to Eq. (11.10) is a somewhat
m.ll] STRINGS OF VARIABLE DENSITY AND TENSION 113
tedious task, but the results will be given here. In Table 2 are given
the values of the ratio between the frequency v n and the fundamental
frequency of a string of uniform mass e , for different values of l/2a
and of n. It should be noticed that the frequencies of the overtones
are not integral multiples of the fundamental, so that the general
motion of the string is not periodic.
n= I
n=2
n = 3
n = 4
Fig. 22.— Normal modes of a string of nonuniform density. Solid lines show the
shape of the characteristic functions for a string of variable density eo[l  (x/a) 2 ]
where (l/2a) = 0.8. Dotted lines show the corresponding sinusoidal functions for a
string of uniform density eo.
The shapes of the first four modes of vibration are given in Fig. 22
and are compared with the sinusoidal shape of the uniform string.
It is seen that the change in shape due to nonuniformity is not great.
It consists in a pushing out of those parts of the wave where the string
is heaviest (i.e., near the center) and a consequent pulling of the nodes
toward the center.
114
THE FLEXIBLE STRING
Table 2. — The String of Nonuniform Density
[in.li
0.2
0.5
0.8
1
1.003
1.016
1.042
2
2.012
2.071
2.181
3
3.019
3.117
3.299
4
4.026
4.161
4.411
5
5.033
5.203
5.521
6
6.039
6.246
6.630
7
7.046
7.289
7.738
8
8.053
8.331
8.846
Frequencies of the string of variable density e [l — (x 2 /a 2 )], stretched between
rigid supports a distance I apart. The table gives values of j3 n for different values
of n and of l/2a. The allowed frequency v n is equal to (1/2Z) y/T/eo multiplied
by/3„.
Vibrations of a Whirling String. — There are cases where gravi
tational, or "centrifugal/' forces act directly on the various portions
of the string, so that the tension varies from point to point along the
string. In these cases the wave equation for the motion of the string
takes on a different form. Returning to the discussion following
Eq. (9.2), we see that when the tension is a function of x the net
vertical force on an element of string is
\ dx/ x+ d x \ dx/ x ~ dx\ dx)
The resulting wave equation is
d_
dx
( r S)
d 2 y
dt 2
(11.12)
which reduces to the usual wave equation when T is independent of x.
As an example of such motion, let us consider the vibrations of a
string of length I pivoted at one end to a rigid support and whirling
about this support. We shall neglect gravity, so that the string's
motion will be in a plane. If we neglect the drag of the air, the position
of equilibrium will be a straight line rotating in a plane about the
central support with a uniform angular velocity co„ radians per sec.
The string can oscillate about this position of equilibrium if it is dis
turbed, as long as its total angular momentum remains constant.
In studying this oscillation we can neglect the uniform motion of the
equilibrium line and deal only with the string's displacement from this
IIL11] STRINGS OF VARIABLE DENSITY AND TENSION 115
line, #. The displacement is a function of t and of x, the distance
from the support. At first we take y perpendicular to the plane of
rotation.
To find the expressions for velocity and acceleration in terms of
these whirling coordinates, we study the behavior of a vector of vari
able length z, rotating with constant angular velocity co„ about its
end, as shown in Fig. 23. The velocity of the outer end of the vector
has two components: one along the vector, of magnitude (dz/dt), and
one perpendicular to it, of magnitude co^. There are two components
of acceleration parallel to z of magnitudes (d 2 z/dt 2 ) and  w *z and a
component perpendicular to z of magnitude <a a (dz/dt).
co a (az/at)
c) 2 z/dt*
Accelerations Displacements for the
Whirling String
Fig 23.— Velocity and acceleration components for a vector g rotating about one
SJ. Jit t^ f, , Veloci . t y "«•.,.!» the ca «e of the whirling string, vector x (pointing
along the line of dynamic equilibrium) rotates but does not change in length; vector y
the displacement from equilibrium, changes length and rotates.
In the case of the whirling string, we are to find the acceleration
of a point that is represented by the sum of two vectors, one of con
stant length x and the other, perpendicular to x, of variable length y
Both are rotating with angular velocity co a . If y is parallel to the axis
of rotation (perpendicular to the plane of rotation), then the accelera
tion of the point is calx along x (x is constant, so d 2 x/dt 2 = 0) and
d*y/dt* along y. But if y is in the plane of rotation, the acceleration
is <a a x  ca a (dy/dt) along x and (d 2 y/dt 2 )  a>ly along y.
Since dy/dt is assumed smaU compared with w a x, in both cases the
acceleration along x is ««*, which acceleration must be due to the
force of the tension in the string. The force on an element dx of the
string at a point x cm from the center will therefore be e dx x<S where
116 THE FLEXIBLE STRING [HI.11
e is the density of the string in grams per centimeter (which is supposed
to be uniform). The tension at the point x will be the sum of the
forces on all the elements of string beyond the point x out to the outer
tip of the string
T(x) = £ eulx dx = (?f\ (P 
The wave equation for vibrations perpendicular to the plane of rota
tion will then be
@*[<r *>$%
To study the normal modes of vibration of the string, we set y =
^/{x)e~ 2vivt ' } and to simplify the equation, we change the scale of length,
making x = h, so that z = is the center, and z = 1 is the outer end
of the string. The resulting equation for yp is
We can see without any further analysis that the frequencies of
vibration will depend oh the angular velocity co but will not depend
on the string's length or its density (as long as the density is uniform;.
For increasing the length or density increases the mass, which tends
to diminish the frequency; it also increases the tension, which tends
to increase the frequency; and the two tendencies just cancel. For
vibrations in the plane of rotation (coplanar) the additional term
— uly comes into the equation of motion, making the equation for $
become
The analysis is the same in both cases, since we are determining the
value of v in either case. If we determine the frequencies v p and the
modes of oscillation for the vibrations perpendicular to the plane of
rotation (transverse) the shapes of the modes will be the same for
the coplanar vibrations; but the allowed frequencies v a for the coplanar
case will be related to the transverse ones, v p , by the relation
4fey
m.ll] STRINGS OF VARIABLE DENSITY AND TENSION 117
We shall first find the frequencies for the transverse vibrations, per
pendicular to the plane of rotation.
The Allowed Frequencies. — To find the allowed frequencies of
oscillation, we must obtain the solutions of Eq. (11.13) which are
zero at z = 0. The easiest method of solving the equation is to assume
that \f/ is given by a powerseries expansion
\f/ = a\Z + a 2 z 2 + azz z + • • •
where we omit the constant term a in order to satisfy the boundary
condition at z = 0. If we set a = (Sir 2 v 2 / col), the equation becomes
I (1  z 2 ) ^j = 2a 2 + 2(3a 3  a x )z + 3(4a 4  2a 2 )z 2 +
dz _
= — oaf/ = ■— adiZ — aa 2 z 2
Equating powers of z, we obtain a 2 = a 4 = a 6 = • • • = 0; a 3 =
2 — a 12 — a
— g — ai; a 5 = — on — az ' ' ' • ^ ne solution of Eq. (11.13) which is
zero at z = is therefore the series
,4 + ii«, + p)ff.o ,
(2  q)(12  q)(30  «) _ , 1
" 1 " 5,040 z t • • J
Since this infinite series satisfies the one boundary condition no matter
what value a has, it would seem that all values of the frequency would
be allowed, since v is proportional to the square root of a. On the
other hand, when we pick some value of a and compute the values of
\f/ as a function of z, we find that the series becomes infinite at z = 1.
Since we cannot allow an infinite displacement of the tip of the string,
this would seem to indicate that no value of the frequency is allowed!
The dilemma is resolved when we notice that if a equals 2, or 12, or
30, etc., the series is not an infinite one but breaks off after the first or
the second or the third, etc., term; in these cases ^ is not infinite at
2=1. The applying of the two boundary conditions, that ^ = at
z = and that rp be not infinite at z = 1, rules out all but a discrete
set of values of a and therefore allows only a discrete set of
frequencies, corresponding to the set of values a = 2, 12, 30 • • •
2n(2n — 1) • • • . If we always choose a\ so that if/ = 1 at z = 1, then
the allowed frequencies of vibration and the corresponding character
istic functions are
118 THE FLEXIBLE STRING [III.ll
v 2 = 2.4495 f^\ fc(z) = U5z 3  3z)
v z = 3.8730 (g^, * 8 (s) = (63z 5  702 3 + 15s) / (U ' 14)
v* = 5.2915 ( ( ^\, M*) = inr(429z 7  693s 6
+ 315s 3  35s
*fe)
Vw(2n  1)
The functions ^ n are equal to the Legendre functions P m (z) [denned
in (Eq. 27.6)] of odd order (m = 2n  1).
The fundamental vibration goes through one cycle every time the
equilibrium line rotates through 360 deg. The overtone frequencies,
however, are not integral multiples of the fundamental and therefore
bear no simple relation to the frequency of uniform rotation of the
equilibrium line. The forms of the string for the first four normal
Fig. 24. — Forms of the normal modes of vibration for the whirled string. The
displacements are from the dotted equilibrium line, which is supposed to be rotating
in a plane about its lefthand end with a constant angular velocity a> a .
modes are shown in Fig. 24. It is to be noticed that each function in
the sequence has one more node than the one preceding, also that these
nodes tend to concentrate near the outer end of the string, where the
tension is least.
The Shape of the String.— It can be verified that these functions
are mutually orthogonal; that
( (n ^ m)
i *•(«)*«(«) dz = { 1 (H.15)
{ (4n — 1) v '
ni.ll] STRINGS OF VARIABLE DENSITY AND TENSION 119
We can show, by the methods discussed earlier, that if the string is
started into oscillation by making its initial displacement from equilib
rium equal to y (x) and initial velocity (over and above the uniform
velocity of the equilibrium line) equal to v (x), the subsequent motion
will be given by the series
y(x
where
A n —
00
»0 = ^ #» (j) l A n cos(2jr* n + B n sin {2wvJ)] (11.16)
n = l
(4n  1)
Since the overtones are not harmonics, this motion will not be periodic
unless it is started in one of the normal modes, so that all the A's and
B's are zero except one pair. It should be noticed that the funda
mental mode of this transverse motion corresponds simply to a shifting
of the string's plane of rotation.
For the case of vibration in the plane of rotation, we mentioned
earlier that if a were made (8irV/coi[) + 2, instead of (8irV/«2), the
discussions on pages 117 and 118 would still be valid. This means
that the shape of the normal modes of vibration in the plane have
the same shape as the transverse modes, so that Fig. 24 is good for
both types of vibration. However, the allowed frequencies for the
two types differ; instead of those given in Eq. (11.14), the modes in
the plane have the frequencies
V! .= 0, V2 = 2.2361
fe) '■ = fe) V(«  D(2n + 1)
The equations for the ^'s are the same as given in Eq. (11.14). We
note that the lowest "frequency" is zero, indicating that the funda
mental mode is not a vibration but a steady motion, corresponding to
a change in angular velocity of the string.
Only the lowest modes for coplanar and transverse motions repre
sent a change in the string's total angular momentum, the coplanar
mode corresponding to a change in magnitude, and the transverse
mode corresponding to a change of direction. The higher modes
represent vibrations that do not affect this total angular momentum;
nor are these higher modes periodic with respect to the rotation of the
string about its support.
Driven Motion for the Whirling String.— The general formulas
worked out at the beginning of this section «an be applied to the prob
120 THE FLEXIBLE STRING pi.ll
lem of the transient or steadystate motion of the string about its
dynamic equilibrium. We shall see in Sec. 27 that the characteristic
functions that we have derived for the whirling string are proportional
to the Legendre functions $ n {z) = P^niix/l). Equations (27.7) give
some of the mathematical properties of the functions. One other
property is of use here
P (0)  ( 1V . 1 • 3 • 5 • • • (2n  1) J2n)^
Equation (11.6) indicates that, if a whirling string is subjected to a
simple harmonic force F(x)e~ i<ot dynes per cm distributed along the
length of the string, the steadystate displacement of the string from
its dynamical equilibrium will be
n = l L
where
*•(*) = P ^i(j\ M n = ^^rY «i = «J(»  D(2n + 1)
when the force applied is in the plane of rotation and F(x)e^ iat is the
component perpendicular to the equilibrium line of the whirling string.
A simple example of the use of this formula is the problem of the
effect of gravity on a string whirling in a vertical plane. The com
ponent of force per unit length perpendicular to the string, when the
string is at an angle 6 with the horizontal, is eg cos 0, uniform along the
string (g is the acceleration due to gravity). ■ Since the angular velocity
of the string is «„, 6 = co a t, so that the effective driving force is the
real part of ege^K The formula above can therefore be used, with
F(£) = eg. The driving force in this case has a frequency co equal to
the frequency co of rotation of the string; but since w does not equal
any of the natural frequencies w n of the whirling string, no resonance
occurs, so that the motion of the string away from equilibrium due to
the effect of gravity is not very great unless co is small.
To obtain numerical results we must compute the integral fegif/ n d£.
Using Eqs. (27.7) and the results given above,
f *.(*) # = I f P*ni(z) dz = ^zn l p ^~2(0)  P 2 „(0)]
_ , _°n*i Z 113 • • • (2n3)
" ^ L) l 2 • 4 • 6 • • • 2n
2n(2n2)l
~ y L ) l (2»n!) 2
STRINGS OF VARIABLE DENSITY AND TENSION
121
Putting this all together into Eq. (11.17), we finally obtain a series
for the displacement of the string from dynamical equilibrium y, or
better the dimensionless quantity y/l, analogous to z = (x/l),
(j)m^^ 1
13
(2n  3)
246
• 2n
4:71—1
2n 2  n  2
*©}
This gives the steadystate effect of gravity on a string whirling
in a vertical plane. We see that the relative magnitude of the effect
depends on (g/lafi), the ratio be
tween the acceleration of gravity
and the acceleration of the tip of the
string in its circular path (as should
have been expected).
Figure 25 shows the shape of the
string at the eight parts of the cycle
t = 0, (««/8), (« a /4), (3«„/8) • • •
(7««/8), for the ratio (g/la>l) = (i).
The dashed lines show the equilib
rium positions of the string if grav
ity were not acting. We note that
on the average the displacement is
out of phase with the force, being
above the equilibrium lines on both
sides, ahead of them when it is going
up, and behind them when it is going
down.
Physically, this means that the string goes slower at the top of its
swing than at the bottom, which corresponds to the wellknown rela
tion between kinetic and potential energy. Mathematically, this
comes about because the "driving frequency" (o) a /2ir) is above the
natural frequency v x = of the first normal node, so that this mode
(which is the largest term in the summation) is out of phase with the
force. The natural frequencies of all the other modes are higher than
(w /27t), so they are in phase with the force. The presence of the higher
modes accounts for the curved shape of the string. Physically, it is
due to the fact that there is a greater variation in the speeds of the
string, between the top and bottom of its swing, for the outer portion
of the string than for the portion near the axis.
Fig. 25. — Shapes of the whirling
string acted on by gravity, at eight
equally spaced parts of the cycle. Ratio
(ff/wa 2 Q has been chosen to be 0.20,
large enough to exaggerate the effect.
122 THE FLEXIBLE STRING [111.12
12. PERTURBATION CALCULATIONS
The motions of some strings, with simple distributions of mass and
tension, can be computed by the methods discussed in the previous
section. An exact solution of Eq. (11.2) can sometimes be obtained
without excessive labor, and the integrations of Eq. (11.9) can then
be worked out for the general motion. However, in many cases an
exact solution of Eq. (11.2) is not easy to obtain, so it will be useful
to work out an alternative, approximation method. This approxima
tion technique turns out to be most effective when the variations in
density and tension do not change greatly along the total length of
the string, so that the string does not differ much from a uniform
string. The deviations from uniformity are called perturbations,
and the technique of calculations is often called the perturbation
method.
The Equation of Motion. — As long as we are at it, let us make the
formulas as general as possible. Suppose that the mass of the string
per unit length at a distance x from one end is e [l + b(x)], where 6
is small compared with unity over the whole range of x from zero to
I. Suppose also that the tension is ^[l + h(x)] and that the trans
verse frictional resistance per unit length is V^Vo r (%), where both
h and r are everywhere small compared with unity. We assume,
however (until the next section), that the string supports are
rigid.
In one of the cases considered in the last section, b was equal to
(l/a) 2 [x — (1/2)] 2 and h and g were zero. When a is large com
pared with (1/2), b is everywhere small compared with unity. The
case of the whirling string, also considered in the previous section,
cannot be dealt with by perturbation methods, since the tension varies
from zero to a maximum value, and the corresponding h would not
be small compared with unity. The case of a string hanging vertically
between two rigid supports would be a workable example, however.
The tension at the bottom (x = 0) would be T Q , that at the top
(x = I) would be To + (e lg) (where g is the acceleration of gravity),
and the tension would vary linearly in between, so that h would equal
(e g/T )x. If the tension at the bottom, T , is larger than the weight
of the string, (e lg), then the perturbation method can be used. Many
other examples, some of practical utility, can be thought of.
The quantities b, h, and r are the perturbations. If t hey ar e all
zero, the string is a uniform one, with wave velocity c = y/To/eo, and
with normal modes and frequencies
IH.12] PERTURBATION CALCULATIONS 123
, » . (irnx\
yl(x) = sin I — \
(12.1)
If they are not zero, but are small, we can expect that the shape of the
nth. mode will be yl(x) plus a small correction and that the correspond
ing natural frequency will be vl plus a small correction. We wish to
find how to compute these corrections.
The equation of motion of the string, with no driving force applied,
is '
*U + «*)] + VT^, r(x)  = T ± J [1 + «,)] I j (12.2)
To find the correction terms for the free vibrations, we assume that
the corrected normal mode can be expressed in terms of a series
^ m (x)e ia  f = \y° m (x) + 2) a n yl{x)\ er*""*
c& = (2x^(1 + Vm ) = (x) 2 (5) (1 + Vm)
(12.3)
where the coefficients a n and the correction to the frequency, t] m , are
supposed to be small compared with unity. The summation indi
cated by the 2 sign is over all values of n except m (i.e., the symbol
5* means "not equal to").
We insert Eq. (12.3) into Eq. (12.2) and drop out "secondorder"
terms (i.e., bt], ah, etc.) and finally obtain
Firstorder Corrections. — The values of the coefficient a n can be
obtained by the device of multiplying all terms by sin (rsx/l) (where s
is some integer) and integrating over x from to I. The orthogonal
properties of this function make a large number of integrals vanish;
in fact all terms of the summation vanish except that for which n = s.
The resulting equation is
124 THE FLEXIBLE STRING [m.12
nm — (n/m)h nm + i(l/irm)r nm
a n <^
1 — (n/m)'
where
b nm = (j\ J o sin^j sin (^r) b W dx
^ = (?)^in(^)sin(^)Kx)^
where we have changed back from s to n after performing the integra
tions.
The value of the correction to the frequency is obtained by multi
plying Eq. (12.4) by sin (irmx/l) and integrating
r\ m ^ —b m m + h mm ~ 1 \~ZZ J Tmm J
a m = o) m  ik m ; ^ — ( 2) Tmm ( ( 12  6 )
C0 m CH ( — ; — 1 Vl + Jl mm — bmm J
Since b mm is proportional to the average value of the mass perturbation,
averaged over the wth mode, and since h mm is proportional to the simi
lar average of the tension perturbation, we can say that if the mass
perturbation tends to increase the mass the value of b mm will be posi
tive and the natural frequency («,»/2w) will be thereby decreased.
Conversely, if the tension perturbation is positive, the natural fre
quency will be correspondingly increased. (This is not surprising.)
The free vibrations are damped, due to the resistive term r, the expo
nential time factor for the wth mode being e***— «.
The coefficients a n representing the difference (to the first approxi
mation) between the shape of the perturbed mode of vibration and the
mode for the unperturbed (uniform) string are proportional to the
"transfer" integrals b nm , etc., which measure the amount by which
the perturbations " couple" the various/modes together. If the only
perturbation is a f rictional resistance r independent of x, then all
these transfer integrals are zero, the a's are zero, the perturbed modes
have the same shape as the unperturbed, and the sole effect of the
m.12] PERTURBATION CALCULATIONS 125
friction is to introduce damping into the amplitude of free vibration,
via the factor e~ kmt . Other examples are not as simple.
Examples of the Method. — We take up again the case of non
uniform density discussed in the previous section, in order to show
how easily (?) the calculations go. The perturbation is
and the other terms h and r are zero. The necessary integrals are
bmm = ~ \iw) l 1 ~ v^yj
b — mnl2 n + (—i) m + n ]
° nm ir 2 (m 2  n*)*a* L + { } J
and the corresponding natural frequencies and shapes of the normal
modes are, therefore,
, , N . Ux\ %l 2 . (3irx\
Mx) c* sm^J  ^^ sm^—j
(5ttx\
_ 5Z 2
34567r 2 a 2Sm
, . , . (2tx\ 2l 2 . /Wv V /io^
Mx) ^ sm \^)2^ sm \~r) ) (12  7)
_ SI 2 . (Qttx\ _
512x 2 a 2Sin \ I )
. , , . (3irx\ . 271 2 . (tx\
_ 135Z 2 . (5tx\ _
l,024x 2 a 2Sm \ I )
Since the perturbation is symmetric about x = (1/2), the series for the
normal modes contains either all even modes or all odd modes. (Why
is this?) We note again that the overtones are not strictly harmonic
in this case.
Incidentally, these results check fairly well with the values given
on page 114 and with the curves of Fig. 22. The amount of work
required for the present calculations is much less than for the exact
solution. The approximate formulas are no longer satisfactorily
126 THE FLEXIBLE STRING [111.12
accurate, however, when a becomes smaller than about (3Z/4), and
the exact solution is then the only satisfactory one.
The effect of other types of mass loading can be computed by
this method. A few examples will be given as problems.
The other perturbation considered here in detail will be that of
the string hanging vertically between rigid supports, where the tension
is (To + € gx), with x measured from the bottom end. The perturba
tion here is h(x) = (gx/c 2 ), where c 2 = (T /e ), and the required
integrals are
, _ feY h _ _ 2^o m 2 + n 2 _ ( _ 1)m+n]
n mm  \2ToJ' * 2 T Q (m 2  n 2 ) 2 L V J J
so that
_ 2lge mn(m 2 + n 2 ) , _ ,_.s m+n]
an ~ t 2 To (m 2  n 2 Y L V J; J
Therefore the natural frequencies and the first two modes are
n [To L , (lge \
(tx\ . (2lge \\l0 . faA , 68 . (4rx\
»(t s )+"]
sin
sin I
,290
+ 91261 Sm
etc.
Here the overtones are still harmonic (at least to this order of approxi
mation), and the even modes are modified by additional terms of odd
modes, and vice versa. The result is to move all the nodes toward
the bottom end, as will be seen if one plots out the i^'s.
Characteristic Impedances. — It is now time to expand on the
analogy, touched upon at the end of Sec. 10, which is suggested by
writing the reaction of each mode of the string as an impedance
or admittance. In Eq. (10.25) we denned a transfer impedance
Z m {Z,x)w)ri) for the nth mode, relating the motion at point x due to
force at point £. Now that we have seen that modes can be coupled
together by perturbations, we see that there must be a transfer imped
ance corresponding to this coupling, which can be labeled Z m (£,x;a>;n,m).
ni.12] PERTURBATION CALCULATIONS 127
This symbol may seem overmuch endowed with labels and subscripts,
but we shall see that its use "saves considerable time and space and
suggests several useful analogies. (We note here that the subscript
m denotes "mechanical" and the m in the brackets denotes an integer.
We shall omit the subscript in this section to reduce complexities.)
Before we come to the impedances, let us define an average mass,
resistance, and stiffness of the string, averaged over the nth normal
mode in a manner analogous to that of Eq. (10.25):
M{x,Z;n) = [yl(x)yl(k)] 1 j Q [^)J 2 e [l + b(u)}du
= (J (1 + U CSclyJcSclyj
R(x,!t;n) = faftaOriK*)] 1 J Q [yl(u)Y VnTo r(u) du
= (f Z ) r nn csc(^) csc(^) }> (12.9)
K(x,i;n) = te(x)M&\ 1 £ V°M £{w + *(«)]
iy°M
du
(12.10)
= y^r) (1 + hnn) csc \t) csg vi)
One can also write a corresponding "transfer" mass, etc., as follows
M(x,£;n,m) = I , f° ^J I Hu)yl(u)y° m (u) du
= (*Ab esc (—} esc (^)
D/ . /e cA (irnx\ /VraA
R{x,^;n,m) — I ^ \r nm cscl y ) cscl — j J
trr y \ /V 2 ranT \ , (irnx\ /VraA
K{x,^;n,m) = I — ^7 — J h ™ csc l ~T~ ) CSC \~T~ )
If this analogous symbolization is to have any meaning at all,
it should serve to express natural frequencies, etc., by formulas analo
gous to those for a simple oscillator. For example, from Eq. (4.1), we
see that the expression for the damping constant of the simple oscil
lator is k = (R/2m). The corresponding expression for the nth mode
of the perturbed string is, from Eqs. (12.9) and (12.6),
128 THE FLEXIBLE STRING [111.12
I" R(x,$;n) 1 /e\r r nn 1 ,M h
l2M{x,Z;n) J \2j _(1 + b nn ) J ~ W *** ~ "
if we neglect all but firstorder terms. Similarly, the expression for
the natural frequency of the simple oscillator is co = (2irv ) = y/K/m.
In the present case, we have
\ K (*,M "1* _ (*nc\ f (l + U 1*
Vl + Ann  K
again neglecting secondorder terms.
Therefore the analogy is a fruitful one. Each normal mode is
analogous to a simple oscillator, or analogous to a circuit of inductance
M(x,£;n), resistance R(x,£;ri), and capacitance l/K(x,l;ri) in series.
The characteristic impedance of the nth mode of vibration of the string
is ..
Z(x,Swn) = ia>M(x,Z;n) + R(x&n) + (m K(x,£;n) (12.11)
This impedance is a function of the frequency (co/2x), not only because
of the factors (— iw) and (i/<a), but also because the quantities 6, r,
and h which enter into M, R, and X may be functions of the frequency.
For instance, if the string radiates sound, then both r(x) and b(x)
will have radiation components that depend on frequency, as we shall
see in Chap. VII.
There is also a characteristic transfer impedance between the nth
and mth modes
Z(x,£;o>;n,m) = —i(j}M(x,£;n,m) + R(x,£yn,m)
+ (Mi£(z,S;n,m) (12.12)
which suggests that it. is a sort of mutual impedance, coupling the
nth and mth modes, as indeed it turns out to be. We notice that if
the string is truly uniform (6 = h = 0, r independent of x) then all
the transfer impedances are zero and there is no coupling between
modes, which is as it should be.
Finally, the expression for the perturbed normal mode has a par
ticularly simple form when written in terms of these impedances. For
instance, the transfer impedance between m and n at the frequency
<a m /2T of the mth mode [see (Eq. 12.6)] is, to the first order,
m.12] PERTURBATION CALCULATIONS 129
Z(x,£;a) m ;m,ri) m\—i ( ~yj [%) & ™ + \1T/ Tnm
Similarly, the impedance for the nth mode at the frequency of the
rath mode is, to the zeroth order,
A comparison of these formulas with Eq. (12.5) shows that an
alternative fashion of writing the series for the perturbed normal mode
is
This is interesting in that it shows that the amount by which the nth
mode modifies the mth mode is proportional to the transfer impedance
Z(x,£;o) m ;m,ri), which is in turn dependent on the perturbation integrals
Omnj l^mn) and. T mn .
Forced Oscillation. — The full usefulness of the characteristic
impedances defined in Eqs. (12.11) and (12.12) becomes apparent when
we take up the forced motion of the string. Suppose that we apply
a concentrated force at the point x — £. The equation of motion is
then
*(i + 6) g* + V^To r I  T 1 [(1 + h) I] = F{&B(x  0e
(12.14)
Setting
into this equation, multiplying it by sin (irmx/l), integrating over x
from zero to I, and finally dividing by sin (irm^/l), we obtain the funda
mental equations for the forced motion of the string
130 THE FLEXIBLE STRING [m.12
U m Z(x,Z;a>;m) + X U n Z(x,£wn,m) = F(Q  (12.15)
where
U m = —io)A m sin
( Tmx\
The quantity A n sin (irnx /I) is that component of the amplitude of
motion of the point x on the string which corresponds to the nth
mode, for a driving force F(£) applied at the point £. Similarly, the
quantities U n are the corresponding modal components of velocities.
The electrical analogue is therefore a network of infinitely many paths
in parallel, the selfimpedance of the rath path being Z(x,£;o);m), with
each path coupled to every other by mutual impedances Z(x,£;o);m,n).
The analogue to the applied force is a voltage F(g) applied across the
network ; and the U n 's correspond to the currents in the parallel paths.
The frequencies of free vibration correspond to the allowed frequencies
of vibration of the network when the generator is shortcircuited.
In order to calculate the values of the A' a or U's by successive
approximations, we first neglect the transfer impedances Z(x,£;a);m,ri)
which are supposed to be considerably smaller than the impedances
Z(x,%;o};m). (Why?) Therefore, to the "zeroth*' approximation,
A G in 6^— F ^
This can then be inserted back in the series in Eq. (12.15) and the
equation solved again to get the first approximation
a • f Tmx \ ~ to \  "5! ?&jwmo1
AmSin \ l ) ~~ iuZixfr^m) l l £d Zfofcwjn) J
— io)Z(x,£',ai;m) sin(xraa;/7)
if we utilize (and generalize) the definition of Eq. (12.13). Second and
higher approximations could be obtained by repeating this process;
but we shall let well enough alone.
Therefore the expression
y(x,^,t) c* 2i i.Z(x^;m) [' ~ Jj ~Z^^rJ (12 ' 16)
is an expression for the steadystate shape of the string when driven
by the force F(£)5(£ — x)e~ ia \ which is correct to the first order in the
small quantities [Z(x,%;a>;m,n)/Z(x,£;o>;ri)].
111.12]
PERTURBATION CALCULATIONS
131
When the string is uniform (6 = h = 0, r independent of x), then
the quantities Z(x,%;<ti)m,ri) are all zero, and the expression becomes
an exact one.
Fig. 26. — Sequences showing the successive shapes of a damped string, driven by
a force of frequency v, during a halfcycle of the steadystate motion. The force is
concentrated at the point marked by the circle, and the arrows show its successive
magnitudes and directions. The three sequences are for the driving frequency v, equal
to f , two and three times the fundamental frequency of the corresponding undamped
string.
Uniform String, Steady State.
y(x,£ya,t) = > }V m
<** —uaZ(x,S;u;n)
Z(x,$;a,;n) = [ ico (^) + (^f) } (12.17)
+ ©(^)M~M?)
132 THE FLEXIBLE STRING [111.12
which is identical with Eq. (10.25). Figure 26 shows the shapes of
such a string when driven by a force applied at £ = (1/3). The center
sequence shows a case of resonance, with large amplitude (not infinite
because the resistance is not zero). The lefthand sequence shows a
case that should be another resonance, were it not for the fact that the
force is applied at a node of the corresponding free vibration (in which
case sin (rn^/l) is zero and the corresponding Z is infinite) so only
nonresonant modes appear.
As noted in the discussion of Eq. (10.25), the analogy of the input
admittance to that of a parallel network of impedances Z(£,£;co;n) is
obvious. In the present case when the perturbations are not uniform,
the individual paths are coupled by the impedances Z(£,£;co;ra,?i.) =
Z(£,£;co;w,ra).
Transient Motion. — The calculation of the transient motion of the
perturbed string follows the same procedure as has been outlined in
Sec. 11. We set up the steadystate response of the string to a con
centrated force, which has been done in Eq. (12.16). We then express
the Z's in terms of co, b nm , r nm , etc., and perform the contour integra
tions necessary to obtain the response to a unit impulse at £
y*M;t) = ^ J_ m [f^] vOM;»,0 ^ ( 12  18 )
The details of computing the residues about all the poles of this
function are quite wearisome but involve nothing particularly different
from the cases discussed before. Each of the first terms in the series
for y(x,£;o),t) has two poles at co = ±co TO — ik m , with values that have
been given in Eq. (12.6). The second terms, which are summed over
both m and n, have four poles: two at co = ±co m — ik m , and two at
co = +co„ — ik n . When the residues have been computed and the
results expressed again in terms of the impedances, we have the
formula
t < 0, y»(x,$;0 =
t > 0, yb(x,%;t) ~ real part of
Op—kmt—iumt
{'2
^m(03m,^m(0} m ,X)
(xmeoc) Vl + h nn + b r ,
{ —  o — kmt — iifmt
^Mix^nOKix^m) (12 " 19)
TO
f ^ Z(x£',o) m ;m,n) _ ^ Z(x,%;o) m ;n,m) T\
L Si Z M'>">»ri Si Z(x^ m ;n) J/
in.13] EFFECT OF MOTION OF THE END SUPPORTS 133
which is correct to the first order of the small quantities b, h, and r
(but is not correct to the second order). Here we refer to Eq. (12.13)
for the formula for the perturbed characteristic functions, to Eqs.
(12.9) through (12.12) for definitions of the impedances, and we again
note that
k m c^
[ R(x,Z;m) 1 _ [ K(x,!;m) ~\
l2M(x,^m)]' " m lM(x,Z;m)]
Thus we have arrived again, by the route of the operational calculus,
to the result that ^ ro (w m ,x), as defined in Eq. (12.13), gives the shape
of the wth mode of free vibration of the perturbed string; and that
k m and w m , as defined in Eq. (12.6), are the damping constant and fre
quency of this vibration; all within the range of validity of the first
order approximation considered here. The formula of Eq. (12.19)
corresponds to that of Eq. (11.8). The integral for the general motion
due to a general force similarly corresponds to that of Eq. (11.9).
13. EFFECT OF MOTION OF THE END SUPPORTS
At the end of Sec. 10 we mentioned that strings are poor radiators
of energy by themselves. In fact the most effective method of radi
ating the energy of vibration of a string is to attach its supports to a
sounding board which will do the radiating. Before we can get very
far with a study of the effect of the sounding board, we must discuss
the effect on the motion of the string of the motion of the end supports.
Such an investigation will be doubly useful, for we shall find that the
methods developed here will also be necessary to the study of sound
waves in tubes and in rooms. We shall consider here only strings of
uniform density and tension: further complications of nonuniformity
can be treated, if need be, by the methods of the previous section.
Impedance of the Support. — Presumably the supports are capable
of bearing the string tension in the direction of the equilibrium line
of the string. Any force on the support at right angles to this line,
however, will cause some sidewise displacement of the support, small
or large depending on the transverse mechanical impedance of the sup
port. This impedance, the ratio between the sidewise force on the
support and the transverse velocity ( — iu times the transverse displace
ment) will be called Z for the support at x = and Zi for that at
x = I. The real part of Zq is a resistance R , and the imaginary part
is a reactance X (Z = Ro — iXo = \Z Q \e iv o) ) the phase angle being
<po = tan 1 (X /JSo).
134 THE FLEXIBLE STRING [111.13
When the string is in equilibrium, the supports are in equilibrium,
and their sideward displacements y(0) and y(l) are zero. When the
string is displaced, however, there is a transverse component of the
tension T, which will tend to pull the supports sideward, as shown in
Fig. 20 (on page 91). This transverse force on the support at x =
is T y = T sin 0, and when is small (as it usually is) it is almost
equal to T tan 0, which' is equal to T times the slope of the string,
(dy/dx), at x = 0. The new boundary condition is therefore that
T^dy/dx), at x = 0, is equal to the transverse impedance of the sup
port, Z , times the string velocity {—io>y), at x = 0. At x = I the
force is minus T times the slope at x = I. The two conditions may
therefore be summarized as follows, for strings oscillating with simple
harmonic motion of frequency (co/2tt) :
.,«*«> "ty®! (I8J)
<*» » + (£)(£))
We see that when the supports have infinite impedance the boundary
conditions reduce to the usual ones requiring y to be zero at x = and I.
Reflection of Waves. — A wave coming from the right and meeting
the support at x = would not undergo perfect reflection with change
of sign. Therefore we must first reexamine the reflection of simple
harmonic waves under these boundary conditions, in order to see
what the mathematical expression should be for the shape of the stand
ing wave. We consider first the support at x = 0, The incident
wave, coming from the right and traveling in the negative xdirection,
is AteW") (x+ct) , where c 2 ={T/e) and a = 2tv. The reflected
wave must also be simple harmonic, but its amplitude A r will differ
in magnitude and phase from Ai. To emphase this we can set A r =
A ie 2Ta +2Tip 0} w here the value of <x determines the reduction in ampli
tude of the reflected wave and the value of jS determines the phase
change on reflection.
The total displacement of the string is therefore
y — (Aie^ ivx/c) + A r e +(2Tivx/c) )e 2 * i ' ,t cm 2)
where X = (c/v) = (2tc/o)) is the wavelength of the incident (and
reflected) wave. The values of «o and /3 must be adjusted so that
the boundary condition at x = is satisfied. The slope of the string
is
m.i3]
EFFECT OF MOTION OF THE END SUPPORTS 135
(— ) = — [— ^^(eCaTisA) — ^2*(« i/j fa/X)) e &»*
The boundary condition of Eq. (13.1) for x = therefore requires that
*  a  *  (t)
5
e 2T(a i^ ) =
+ <z
i r
i + r
5 j0 jiK>d io9y = e
I Z 2 tr
4
(13.3)
321 1 1
a = Real Part of q.
Fig. 27. — Conformal transformation of the q = o — ib plane onto the f = — i\ plane,
and vice versa (when the drawing is turned upside down).
These equations display a very interesting reciprocal relationship
between q, the ratio between the amplitudes of the reflected and inci
dent wave, and f , the ratio between the transverse impedance of the
support causing the reflection and the wave impedance of the string
defined in Eq. (10.3). Relationships between complex quantities
136 THE FLEXIBLE STRING [111.13
expressable by equations relating them can be represented graphically
by what are called conformal transformations, from which approximate
values of the relationships can be read off. For instance, in this case
the line a = — i is a straight line on the gplane, parallel to the 6axis,
onehalf unit to the left of the origin; but on the f plane it is represented
by a circle of radius two units, with center at f = 2, as shown in Fig.
27. For the particular transformation symbolized by Eq. (13.3),
circles on the f plane go into circles on the <?plane, and vice versa.
A straight line, being a limiting form of a circle, also goes into a circle.
The coordinate lines of the gplane are drawn on the fplane in
Fig. 27 and constitute a representation of the transformation. From
it we can read off the values of and x corresponding to values of a
and 6. For instance, corresponding to q = — 0.5 + 0.'5iis{" = 1 — 2i,
and corresponding to q = — i is f = i, and so on. Since the reverse
transformation (f to q) has the same form, Fig. 27 need only be turned
upside down to have the reciprocal representation.
Several interesting things are apparent on studying Fig. 27. In
the first place, as long as is positive (and it must be in any actual
case, for the real part of an impedance must be positive) the magnitude
of q will be less than unity. This means that in any actual case the
reflected wave will never be larger than the incident wave (which
seems sensible). In the second place, when f = 1 (Z = ec) there is
no reflected wave at all (q = 0); when f = °o, q = —1; and when
f = o, q = 1. When f  = 1, the real part of q is zero, so that the
phase change on reflection is ± 90 deg. However, when the real part
of f is zero, \q\ = 1, i.e. , the reflected wave is as large as the incident one.
Hyperbolic Functions. — The relations displayed by Eq. (13.3) are
interesting enough, but a modified form of the transformation will
turn out to be much more useful in further calculations. The equa
tions will utilize hyperbolic functions, so that it will be well to review
their properties. The fundamental definitions are parallel to those
of Eq. (2.7), which defined the trigonometric functions in terms of the
exponential function,
cosh (x) = \{e x + e~ x ) ; e x = cosh (x) + sinh (x)
sinh (x) = \{e x — e~ x ) ; e* = cosh (x) — sinh (x)
j cosh (x) = sinh (x) ; ' r sinh (x) = cosh (x)
(13.4)
The relationship between these functions and the trigonometric
functions are given by the equations
in.13]
EFFECT OF MOTION OF THE END SUPPORTS
137
cosh (x + iy) = cosh (x) cos (y) + i sinh (x) sin (y)
sinh (x + ty) = sinh (x) cos (?/) + i cosh (x) sin (y)
cos ix + tV) = cos (x) cosh (y) — i sin (x) sinh (?/)
sin ( x + *2/) = sin (x) cosh (?/) + i cos (z) sinh (?/)
(13.5)
The amplitude and phase angle of cosh (a; + iy) and of sinh(x + iy) is
given in Plate I at the end of the book.
We also define a hyperbolic tangent and cotangent in an analogous
manner
tanh (x) =
sinh (x) _ 1 — e~
cosh (x) 1 + e~
coth (x)
tanh (x)
(13.6)
Tables of the hyperbolic sine and cosine are given on page 438.
More interesting to us at present are the values of the hyperbolic
functions of complex quantities. The values can be obtained by using
Eqs. (13.5), which result in the useful relations given in Eqs. (13.7).
cosh[ir(o! .— iff)] = cosh[7r(a — #)]er*o
cosh[x(a — i0)]\ = \/cosh 2 (7rQ:) — sin 2 (xj8);
tan £2 = tanh(ira) tan(x)3)
[*« if 0]
cosh[ir(a — ifi)] = i sinh  ir(a — ifi
= — cosh[7r(a — i(3 — i)]
tanh[ir(a  i&)\ = 6  i X = f = f \<T*
sinh(27ra) _ _ sin(2x/3)
e =
x =
cosh (2ira) + cos (2*0) ' A cosh (2™) + cos (2irj8)
2 + [x + cot(2x/3)] 2 = csc 2 (2tt8)
[6  coth(27m)] 2 + x 2 = csch 2 (2W)
tanh [ir(a — ip)] = tanh [ir(a — i@ + in)]
1
(13.7)
tanh [ir(a — ifi + in + i*')]
(n = 0, ±1, ±2
)
cosh (2x0:) — cos(27T8) _ _ T [* sin (2tt/8) T
cosh (2™) + cos (2x0) '^ ~ tan [sinn ( 27ra ) J
(1 + , )2 + x2]= ^ tanh _ i
4x L(l  0) 2 + X 2 .
l_sinh (2xa)
r 2 " i.
>£*""' [l^l?]/
From these equations we see that the line on the f = 6 — ix plane
which corresponds to ft = constant is a circle of radius esc (2x/3), with
138 THE FLEXIBLE STRING [111.13
center at f = i cot (2tt/3) ; and that the line corresponding to acon
stant is also a circle, of radius csch(2xa) and with center at f =
coth (2ra). This transformation is plotted in Plates I and II on pages
453 and 454. Plate III gives the magnitude and phase angle of the
hyperbolic sine and cosine. The transformation for the hyperbolic
tangent, shown in Plate I, is sometimes called the bipolar transforma
tion.  Tables III and IV, on pages 440 and 443, also give values for
the transformation, if greater accuracy is needed than can be obtained
from the plates.
( Both tables and plates are given only for values of /3 between
and 1, and only for positive values of a and 0. Only positive values
of and a are included, because for systems encountered in "real
life " the resistive term is never negative. The range of fi is restricted
because the whole transformation is a periodic function of 0, and
values for other ranges of /? can be obtained from the ones given.
Both and x can be considered as functions of a and /5; 0(a,/3), x(a,(i).
Values of and x for /3 increased (or decreased) by a whole integer are
equal to the original values:
0(a,/3 + n) = 0(«,/3); X (<*,P + n) = xK/3); n = 0,±l,+2 • • •
We can also see from Eqs. (13.7) that changing from fi to n —
changes the sign of x but leaves unchanged :
6(a,n  0) = 0(a,P); x(<*,n  0) = x(«,/3)
Therefore we see that 6(a,0. 2) = 0(a,1.2) = 0(a,27.2) = 0(a,6.8)
= 0(a,6.8) = 0(a,O.8) = 0(a,3.2), etc.; and x(«,0.2) = x(M2) =
x(«,27.2) .'= x(«,6.8) = x(«,6.8) = x(«,0.8) =  x ( a ,3.2), etc.
String Driven from One End. — We must now go back to our physi
cal problem to try to show why it is necessary to bring in such a lot
of new mathematical machinery. Suppose that the string is fastened,
at x = t, to a support having transverse mechanical impedance
Zi = ec£i and that a sideward force is impressed on_ the end x = 0.
We wish to compute the transverse impedance of the string at x =
and its shape and motion when acted on by the force. The impedance
of the support at x = Z will modify the reflected wave returning from
that end of the string and will, therefore, modify the whole reaction
of the string by an amount that we must now compute.
If the driving force is simple harmonic, with frequency (co/2x),
the motion of the string must be some combination of the two exponen
tials e {iax/6) and e~ (iax/c) , multiplied by the time term e~ iwt . Examination
of Eq; (13.4) shows that this combination can be expressed as
111.13] EFFECT OF MOTION OF THE END SUPPORTS 139
y(x,t) = A cosher a + j(y)  ifa fe™ 1
where X = (2irc/co) is the wavelength of waves on the string. The
constants a and j8 are to be determined by the value of the impedance
of the support at x = I, by use of Eq. (13.1)
A cosh[7r(aj — ifa)] = A (  — = ) sinh[7r(aj — ifa)]
\ia)cZiJ
or
(f)
tanh[x(ai  ifa)] = ( — ) = ft = 61  ix t (13.8)
where the constants for the end x = I are related to those for x —
by the equations
m = a ;
(f> ** + (?)
Therefore the calculation of the wave impedance of the string involves
a knowledge of the hyperbolic tangent of complex quantities and is
the excuse for the previous several pages of formulas.
The wave impedance of the string at x = is
Z(0,0;o>) = (J)(^) = ectanh[7r(«o  *j8 )]
= ec tanh<7raj — iir fa + ( y ) (
To go from the impedance of the support at x = I to the impedance of
string plus support, we find the values of a and P, on Plate I, corre
sponding to an equal value of a but a value of j3 increased by the
length of the string in half wavelengths, (21/X). We follow the circle
corresponding to the fixed value of a, from fa to fa f (2Z/X), and the
result is the transverse impedance of the string in units of (cc).
Shape of the String. — If the transverse driving force at x = is
Foe"^ 1 , the shape of the driven string is
^o cosh {ir[a + i(2x/\)  ifa ]} e _ iat
— t'coec sinh[7r(a — iPo)]
which corresponds to Eqs. (10.8) for the case of the rigid support.
By using some of Eqs. (13.7) we can express this in terms of amplitude
and phase angle:
140
THE FLEXIBLE STRING
[111.13
c
& m
ft .2
>
i
3 g
<3"
dTS
.C
H ft
"«
§ a
>
IS
5&
£ ft
O a]
^2
o
E
3
3
+3 ©
Ui
E
+3
03 ®
 c 
o
c
E
gS
a.
v .•
2
<
o* o
^_ Ets
o ^
O 3 3
u>E.^
ft 03
C x Q.
o E
o d
a.2<£
d ®
v 3
111.13] EFFECT OF MOTION OF THE END SUPPORTS 141
= El / cosh 2 Qr«o) ~ sin 2 (fcc  TftO V
y uec \ cosh 2 (xa ) — cos 2 (71ft) )
exp <t tan _1 [tanh(xo:o) tan (Ave — irfio)]
— t tan 1 [tanh (ira ) cot (flft)] — iwf + (  )>
where k = (2tt/X) = (w/c) and ft = ft + (2Z/X). Figure 28 shows
a time sequence of such wave motion for several different values of
ai and ft. The one for ai = is for no energy loss to the support at
x = I and is a true standing wave, with true nodal points spaced a
half wavelength apart. When ai is not zero, energy is lost to the
support, the reflected wave is not so large as the incident wave, and
at no point on the string is the amplitude exactly zero.
The amplitude of motion of the string, as a function of x, does
show maxima and minima, however, spaced a half wavelength apart.
The dependence of this amplitude on x is through the factor
\/cosh 2 (7rao) — sin 2 (fcc — xft)
= VcoshVaj)  sin 2 [(27r/X)(a;  I)  rft].
The maxima come when (2ir/\)(x — I) = 7rft + irn, (n = 0, ±1,±2
• • • ), and the maximum nearest the support at x — I is a distance
d = (I — x) = (X/2)(l — ft) from this support (we assume that the
value of ft lies between and 1, as it can always be made to do). The
minima come when (2ir/X)(x — I) = irfii + ir(n + i), and the mini
mum nearest the support comes at d = (I — x) = (X/2)(£ — ft) if
< ft < £, or (X/2)(f  ft) if 1 > ft > i. Therefore the value of
ft for the support at x = I can be determined by measuring what part
of a half wavelength lies between the support and the nearest point of
maximum amplitude and subtracting this fraction from 1.
At the maxima, the factor in the amplitude depending on x has
the value cosh(xaz) ; at the minima this factor has the value sinh(iraj).
Consequently, the ratio between the minimum amplitude of motion
of the string and the maximum is tanh(7raj), from which we can deter
mine ai. Therefore we can measure the transverse impedance of the
support at x = I by observing the driven motion of the string attached
to it. The ratio of minimum to maximum amplitude gives ai, the
distance of the maximum from the support gives ft, and the impedance
Zi can be obtained from Eq. (13.8) and the plates or tables at the back
of the book.
142 THE FLEXIBLE STRING [111.13
This same sort of analysis can be made for sound waves in a tube
closed at x = I by a diaphragm having mechanical impedance Z h as
will be shown in Chap. VI.
Standing Wave Ratio and Position of Minima. — An alternative
technique is to use Eqs. (13.3) and Fig. 27 to correlate measurements
of standing waves with the impedance of the driven support (the
load) or the impedance that the string exhibits to a transverse driving
force (the generator). The magnitude of q, \q\, the ratio between
the amplitudes of reflected and incident waves, is called the standing
wave ratio. At the points where the two waves are in phase, the
amplitude of motion Fmax is largest and is proportional to (1 + \q\);
at the points where the waves are out of phase, the amplitude 7^
is least and is proportional to (1 — g). Consequently, a measure of
the ratios between minimum and maximum amplitudes serves to
determine the standing wave ratio, or vice versa
P" = TT^ ! I?!  y I y* = «" (13.9)
t max L I j C^  l max r 1 min
The phase angle 2ir/3 of q = \q\e 2Ti ? gives the phase shift on reflec
tion. If it is zero, the amplitude at that point is maximum, and the
point of minimum amplitude is i wavelength away; if /? is ■£ (phase
angle 180 deg) the amplitude is there a minimum. Consequently,
the number of half wavelengths measured from the load to the nearest
minimum is equal to i  p lo&d ; or the number of half wavelengths
measured from the generator to the nearest minimum is equal to
i + iSgen (which is a restatement of the discussion on page 141).
This suggests using the chart of Plate IV, which is the portion of
Fig. 27 inside the circle = 0. The contours for the resistance and
reactance and % are plotted. The standing wave ratio \q\ is then
simply the radial distance from the center (0 = 1, % = 0) to the point
(0,x). The value of /3 is the angle of this radius vector.
If the value of the load impedance is known, the corresponding
point can be found on the chart and the standing wave ratio deter
mined by measuring the distance from the center to this point. The
impedance of the string for the generator can be found by traveling
along a circle with center (0 = 1, x = 0), starting from the point
(0ioad, Xioad), and traversing the circle through an arc in degrees equal
to 360 deg times the number of half wavelengths from load to gener
ator. We note that we must traverse the circle in a clockwise direc
tion in this case (increasing /3) .
111.13] EFFECT OF MOTION OF THE END SUPPORTS 143
If the value of the string impedance for the generator is known,
we find the load impedance by going from the point (0 gen , Xsen) along
a circle in the counterclockwise direction (decreasing /3) through an
arc equal to 360 deg times the number of half wavelengths from load
to generator.
If the ratio (Fmm/Fmax) is known and the position of one mini
mum (or maximum) is known, either the impedance of the load
or the string driving impedance can be computed by drawing a circle
of center (6 = 1, x = 0) and radius equal to \q\ = [1 — (Y^/Ym^)]/
[1 + Yrwn/Yma*)]. Going around this circle, from the point /? = i, in
the counterclockwise direction through an arc equal to 360 deg times
the number of half wavelengths from the minimum to the load, ends
at a point corresponding to the load impedance. Going around the
circle clockwise through an arc equal to 360 deg X (number of half
wavelengths between the minimum and the generator) reaches the
point corresponding to the drivingpoint impedance of the string.
Both types of charts (for a, j3 and for \q\, /3) are given at the back
of the book. One is more useful for some calculations, the other
more useful for others. It is well to get used to both.
Characteristic Functions. — To discuss the driven motion of the
string further we shall have to obtain the characteristic functions that
satisfy the boundary conditions at a given frequency. To make
the problem as simple as possible initially, without leaving out the
essential parts, we shall assume that the transverse impedance of the
supports at x = and x = I are large compared witTi ec, the wave
impedance of the string. In this case approximate methods can be
applied, and the technique is not too different from that employed
in Sec. 12. We consider that the string itself is uniform, with uniform
tension and zero distributed resistance. (These additional complica
tions can be brought in later; at present the only perturbations con
sidered are those due to the motion of the supports.)
We assume that the nth characteristic function is
^n(x,ca) = sin f — ) (x — a n )
(n = 1,2,3 • • • )
where w n and a n are to be determined by the boundary conditions.
Setting this in Eqs. (13.1) and assuming that (T/uZ) is small enough
to neglect quantities higher than the first order, we finally arrive at
the approximate formulas
144 THE FLEXIBLE STRING [111.13
a "^{ ! f)[ 1 + (dk) + (sass)]
where T is the tension (T = ec 2 ).
Therefore the characteristic function is
\l/n(x,w) ~ sin< (ira) 1  ( ^J (<r + tJto)  ( ^J (<ri + «i)
[(f) + (a) <•* + *•>]}
s^ ( — 7 J (o + Wo) (at a; = 0)
~ ( ~~ 1)n ~ l fe) (<Tl + ^ (at X = l)
where (ec/Z) = (1/r) = k  *V; k = ed?/(fl 2 + x 2 ) = 6/(6* + x 2 );
<r = ecX/(R* + x 2 ) = x/(6 2 + x 2 )
The quantity (Z/ec) is the specific impedance of the end support, the
impedance in units of the wave impedance of the string. The recip
rocal of this is the specific admittance of the support, here taken to be
a small quantity. The real part of this, k, can be called the specific
conductance and the imaginary part a, the specific susceptance of the
support. The* limiting values of the function yf/ n correspond to the
motions of the end supports produced by the motion of the string.
The characteristic function ip n is a complex quantity and is a function
of the driving frequency co, both explicitly and also through the depend
ence of the admittances k and a on co.
Following the procedure leading to Eq. (11.6), we see that the
shape of the string, when acted on by a simple harmonic driving force
concentrated at x = £, is
^J M n (co — w n
F(j) M MMx t u) ^
)(« + 0) n )
where
Mn ~ \2 ) l 1 \io>lZ ) [iuizj] — \2coJ
Transient Response. — The calculation of the response of the string
due to an impulsive force at t = involves the calculation of the resi
m.13] EFFECT OF MOTION OF THE END SUPPORTS 145
dues of y at its poles in the wplane. This is a little tedious, since co„
depends on w; but it can be carried out to the first order in the small
quantities k and <r. The most difficult point comes when we set the
approximate values w = + (rnc/l) into u n to calculate the first order
correction to the position of the poles. It is a general property of
impedances to change the sign of their imaginary parts (but not of
the real parts) when the real part of co changes sign. Consequently,
if
. «„(«) = y hrn + \^~1 ) ( KOn + Kln ~ t(r0n — *°"* n )
the two roots for « are, to the first order,
c , . V
J {irn — 0Qn — (Tin — 1*0n — Win)
t)
(—Tnc\ c , . . .v
— 7 — J ^ j {—irn + a 0n + <n n — *«0n — lain)
where we understand that the admittances in the nth term, <r» and
R n , are computed for the frequency (irnc/l). This behavior must also
be taken into account in evaluating $ n at the poles. The final result
is
!0 (t < 0)
Jj\imc) (13.10)
n = 1 N '
[ypn{%,  )^n(x, — )e i(  e/l) c™«>»«»)*
^(^,+)^„(a:,+)e i ^^^«»^] (t > 0)
where
&»(z,+) = sin<[7rn — (<r 0n + zVc 0n ) — («ri* + im*)]
[(i) + (£) ('+ *>]}
and
^ n (a?,) C~ Sinf ~) + COsf ^y J ( 1 — J J ((Ton — tKOn)
— J ((Tin — IKIn)
I
146 THE FLEXIBLE STRING [111.13
The series in (13.10) is correct only to the first order and, there
fore, does not give exactly the correct answer near t = 0; but the gen
eral behavior is a close enough approximation to see what is happening.
A string supported between nonrigid supports, started into motion
by any arbitrary force, vibrates with a complex of damped, harmonic
motions. The damping constant for the nth mode is (c/l) times the
sum of the specific mechanical conductances (/c 0w + ni n ) for transverse
motion of the two supports at the nth natural frequency. If the
supports are rigid or if their transverse impedances are purely reactive,
then these conductances will be zero and the string will not be damped
owing to support motion; though it may be damped owing to the
reaction of the air (which we have treated in Sec. 12, but are neglecting
here). But if the supports are attached to sounding boards that
radiate the energy of vibration, then the conductances will not be
zero, and the motion will be damped. We note that, because of the
factor (c/l), the longer the string is, the less will it be damped. It is
easy to see why this is so : for the same support conductance, a longer
string will feed a smaller fraction of its total energy to the supports in
any second.
We note also that the overtones of the string are not harmonic,
for the nth natural frequency is (cn/2l) — (c/2wl) (cr 0n + <xi n ), which
is only approximately equal to n times the fundamental frequency
(c/2l)  (c/2irl)(aoi + an).
If the supports are stiffness controlled their reactances are nega
tive, their susceptances <r = —ecX/(R 2 + X 2 ) are positive, and all
the natural frequencies are lowered. This is because the supports
will move in phase with the part of the string nearest them, and the
outer nodes, which are at the supports, if they are rigid, will be some
what outside the supports (virtual nodes, not on the actual string).
This means that the wavelength has been increased a bit, so the
frequency is lowered.
If the transverse impedance of the supports is mass controlled,
the susceptances are negative and all the natural frequencies are
raised. There is a node in the string a short distance in from each
support so that the support can move out of phase with the transverse
force from the vibrating string. If the conductances are not zero,
none of the nodes will be perfect; but the amplitude of motion of the
string will be a minimum at these points, and the phase lag of the
supports' displacement behind the string will be less than 180 deg.
This analysis will be taken up again when we study the behavior of
sound waves in tubes, and more accurate solutions will be derived.
111.13] EFFECT OF MOTION OF THE END SUPPORTS 147
Recapitulation. — We have gone into considerable detail in studying
the motions of the string, perhaps more detail than seems necessary.
This has been done because the string is the simplest case of a system
with an infinite number of allowed frequencies, and it is best to discuss
some of the properties common to all such systems for as simple a
system as we can find, lest the mathematical complications completely
obscure the physical ideas. The effect of friction, both on the system
and through the supports, and the phenomenon of multiple resonance
are both properties that are true of systems more complicated than
the string. The damping effect of the air's reaction will be more
important than the effect of the supports in systems more extended
than the string, but the general effect will be the same as that dis
cussed above.
We also have been developing methods of handling vibration prob
lems which will be exceedingly useful in our later work, developing
them on problems where the general method is not too much obscured
by details. In particular, we have been giving example after example
of the utility of the study of the normal modes of vibration of a sys
tem. Once the normal frequencies and corresponding characteristic
functions have been worked out for a system with a given set of
boundary conditions, we can determine its motion for any set of
initial conditions and for any sort of applied force. We can also
discuss, by methods similar to those developed in Sec. 12, the effect
on the motion of the system of slight changes in the system's properties
(such as its distribution of mass or its distribution of tension). And,
by expressing the applied force in terms of the characteristic functions,
we can work out the forced motion. We can show, for instance, that
when the driving frequency is equal to one of the allowed frequencies
of the system, then the system takes on the shape of the corresponding
characteristic function with an amplitude that is infinite if there is no
frictional damping force. (Compare this with the discussion in the
last paragraphs of Chap. II.)
All the methods discussed above will be used again later in more
complicated problems, either in the text or in the problems.
Problems
1. A string, clamped at one end, is struck at a point a distance D from the
clamp, by a hammer of width (D/4). The head of the hammer is shaped so that
the initial velocity given the string is maximum at the center of the head and is
zero at the edge, the initial "velocity shape" of this portion being like an inverted
V. Plot the shapes of the string at the times t = 0, (D/2c), (D/c)', (3D /2c),
148 THE FLEXIBLE STRING [111.13
(2D/c). Draw a curve showing the vertical component of the force on the clamp
as a function of time.
2. A harp string is plucked so that its initial velocity is zero and its initial
shape is
(i*r) x i° < x < 20J
KT)\2 X ) \& <x< 2o)
S20h\ , n an . . \
KW) {x ' l) V20 < ' < l )
Plot the successive shapes of the string during one cycle of the motion. Draw a
curve showing the vertical component of the force on one of the supports as a
function of time.
3. What are the total energies of the two strings shown in Fig. 16?
4. Show that if a string, having a wave velocity c, is plucked at a point whose
distance from one end is 1/3 the equation for its subsequent shape is
y = w L sm \t) cos vt) + 4 sin \rr) cos \tt)  tV sm vt) cos \rr)
, . (5ttX\ /5wct\ , . . /7rx\ /7irct\ . 1
 A sm {j j cos {— j + A ^ {—) cos ^ j + • • J
The string has a mass of 0.01 g per cm, is 25 cm long, and is under a tension of
1,000,000 dynes. Find the energy of vibration of the first four normal modes when
the string is pulled aside 1 cm (h = 1).
5. A uniform string with no friction is stretched between rigid supports a
distance I apart. It is driven by a force F e~ iut concentrated at its midpoint.
Show that the amplitude of motion of the midpoint is (F /2ecu) tan (ul/2c) . What
is the amplitude of motion of the point x = (1/4) ?
6. A uniform string of small electrical resistance is stretched between rigid
supports a distance I apart, in a uniform magnetic field of B gauss perpendicular
to the string. A current Iae~ i03t amp is sent through the string; what is the force
on the string per unit length? Show that the velocity shape of the string (assume
zero friction) is
BJoe*" {cos[(»/c)(a;  $Z)] ,) ,
v = ^. < / 7/0 ^ ~ 1 C cm /sec
lOzwe I cos (wZ/2c) )
Use the formula E = 10~ 8 j Bv dx volts to compute the motional emf induced
in the string by the motion and, therefore, the motional impedance.
7. A condenser is discharged through the string of Prob. 6, producing a current
S(t) amp. Compute the subsequent shape of the string.
8. The string of Prob. 5 is acted on by a force F(£) u(t) concentrated at the
point x — I [see Eq. (2.6) for a definition of u, and Eq. (6.10) for its use]. Show
that the shape of the string after t = is
HI.13] EFFECT OF MOTION OF THE END SUPPORTS 149
From this formula compute the shape of the string when it has been subjected to
the constant (independent of time) force F(£) up to t = and then released to
vibrate freely.
9. A string of length I and mass e g per cm is hung from one end, so that
gravitational forces are the only ones acting. Show that if the free end of the string
is taken as origin, the normal modes of vibration have the form
y  Jo^.^)
where v n = (8„/4) y/gjl) the function J (z) is given by the series (see Prob. 3 of
Chap. I)
z 2 z 4
■W^iT + ei
and the constants /3„ are the solutions of the equation Jo(irp n ) = [see Eq. (19.6)].
Is the motion periodic in general? What are the ratios of the lowest three allowed
frequencies to the frequency of oscillation that the string would have if all its mass
were concentrated at its lower end?
10. Utilizing the general formulas of Eqs. (11.6) and (11.8), obtain specific
formulas for the shape of the string of Prob. 9 when driven by a periodic force and
when struck by an impulsive force.
11. Choosing suitable values for the properties of the string of Prob. 9, plot the
shape of the string when it is vibrating at its lowest three allowed frequencies.
Plot its shape when driven at its free end at a frequency 1.5 times its fundamental.
12. The tip of the whirling string is struck an impulsive blow 5(0 in a direction
perpendicular to the equilibrium plane of motion. Calculate the series repre
senting the subsequent displacement. What is the amplitude of motion of the
lowest mode?
13. A string of infinite length is acted on by a force F(t) concentrated at the
point x = 0, where
(t < 0)
F(t) = { F (0 < t < to)
(t > U)
Plot the shape of the string at the times t = 0, (to/2), U, (32 /2), 2t . What is
the total energy given to the string?
14. A string of steel (density 7.7 g per cc) is stretched between rigid supports.
Its fundamental frequency of vibration is 500 cps. Sound of unit intensity and
frequency v falls on the string, normal to its axis. Compute the amplitude of
motion of the midpoint of the string for the frequencies v = 50, 100, 300, 450, 499,
550, 600, 1,000. Note. — The force per unit length on a string of crosssectional
radius a, due to an incident sound wave of intensity T and frequency v 2 , is 0.0074Ka 2
\/T e~ 2rin dynes per cm length.
15. Plot the shapes of the first three modes of the stretched hanging string,
described in Eqs. (12.8), for (lge /To) = 0.25. Compute the series for shape of the
string when driven by a periodic force concentrated at x = £.
150 THE FLEXIBLE STRING [111.13
16. What are the values of tanh[7r(a — i(5)] when a ■= 0, /S = 1.75; when
a = 0.2, j8 = 0.6; a = 0.1, /3 = 0.45; a = 0, '/3 = 0.5? What are the values of
cosh [ir (a — ip)] for these same values of a and /3?
17. A uniform string of length I and tension T is fastened at x = £ to a support
having transverse mechanical resistance R = ec tanh (ira — ^r) and zero reac
tance. The string is originally at equilibrium, and the end at x = is suddenly
acted on by a transverse impulsive force 8(t). Utilize Eqs. (13.10) and (6.14) to
obtain the formula for the shape of the string:
00
1 ^C^ p(ira/l)(.x+ct) I r^ "1
»<*» " k 2 (. + «■ + * I <" + §> sin [r <» + »<* + C,) J
+ a'cos ^ (n + %)(x + ct)\l
 s; 2 (»TwT 1 (w + j) 8in B ( " + t)(x  c<) ]
n =
+ « cos y (n + ) (x — cf) [ \ tanh (ira)
18. A string of density 0.1 g per cm is stretched with a tension of 10 s dynes
from a support at one end to a device for producing transverse periodic oscillations
at the other end. When the driving frequency has a given value, it is noted that
the points of minimum amplitude are 10 cm apart, that the amplitude of motion of
the minimum is 0.557 times the amplitude at the maximum, and that the nearest
maximum is 6 cm from the support. What is the driving frequency, what are the
values of on and ft, and what is the value of the transverse impedance of the
support?
19. A string is stretched between two supports having transverse mechanical
resistance R large compared with ec, and zero reactance. The string is driven by a
periodic force concentrated at a; = £ . What are the amplitudes of motion of the
end supports, and by how much do they lag behind the driving force?
20. The string of Prob. 19 is struck an impulsive blow at the point x =' £.
Compute the subsequent motion of the two end supports.
CHAPTER IV
THE VIBRATION OF BARS
14. THE EQUATION OF MOTION
It must have been rather obvious in the previous chapter that
we were analyzing the motions of a somewhat idealized string. In
the first place, we assumed that the string was perfectly flexible, that
the only restoring force was due to the tension. Secondly, we made
no mention of the possibility of longitudinal motion of alternate
compression and tension, which can be set up in any actual string as
well as in any other piece of solid material. This longitudinal wave
motion will be disregarded for a while longer; we shall spend the
whole of the last three chapters discussing it. However, we can no
longer put off studying the effect of stiffness on the string's motion.
And we shall begin the study by discussing the transverse vibrations
of bars.
There is no sharp distinction between what we mean by a bar
and what we mean by a string. In general, tension is more important,
as a restoring force, than stiffness for a string, and stiffness is more
important for a bar; but there is a complete sequence of intermediate
cases, from stiff strings to bars under tension. The perfectly flexible
string is one limiting case, where the restoring force due to stiffness
is negligible compared with that due to the tension. The rod or bar
under no tension is the other limiting case, the restoring force being
entirely due to stiffness. The first limiting case was studied in the
previous chapter. The second case, the bar under no tension, will be
studied in the first part of this chapter, and the intermediate cases
will be dealt with in a later part.
Stresses in a Bar.— To start with, we shall study the bending of a
straight bar, with uniform cross section, symmetrical about a central
plane. The motion of the bar is supposed to be perpendicular to this
plane, and we shall call the displacement from equilibrium of the
plane y.
When the bar is bent, its lower half is compressed and its upper
half stretched (or vice versa). This bending requires a moment M,
whose relation to the amount of bending we must find. To compress
151
152
THE VIBRATION OF BARS
[IV.14
a rod of crosssectional area £ and length I by an amount dl requires
a force QS(dl/l), where Q is a constant, called Young's modulus. The
values of this constant are given in Table 3, for some of the more
common materials. Now, imagine the bar
to be a bundle of fibers of crosssectional
area dS, all running parallel to the center
plane of the bar. If the bar is bent by an
angle $ in a length dx, then the fibers which
are a distance z down from the center sur
face (it is no longer a plane now that the
bar is bent) will be compressed by a length
2$, the force required to compress each
fiber will be Q dS(z$/dx), and the moment
of this force about the center line of the
bar's cross section will be (Q$/dx)z 2 dS. The total moment of these
forces required to compress and to stretch all the fibers in the bar will be
Fig. 29. — Moment acting on a
bent element of a bar.
M =
\dx) J
dS
(14.1)
where the integration is over the whole area of the cross section.
Table 3. — Elastic Constants of Materials
Material
Brass, cold rolled
Bronze, phosphor
Copper, hard drawn. .
German silver
Glass
Iron, cast
wrought
Ironcobalt (70 % Fe) .
Nickel
Nickeliron (5 % Ni) . .
Silver, hard drawn. . .
Steel, annealed
invar
Tungsten, drawn
Q
9 X 10 11
12 X 10"
10 X 10 11
11 X 10 u
6 X 10 11
9 X 10 11
19 X 10 11
21 X 10 11
21 X 10 11
21 X 10 11
8 X 10 11
19 X 10 u
14 X 10 11
35 X 10 11
8.6
8.8
8.9
8.4
2.6
7.1
7.6
8.0
8.7
.7.8
10.6
7.7
8.0
19.0
Values of Young's modulus Q in dynes per square centimeter, and of density p,
in grams per cubic centimeter, for various materials.
We define a constant k, such that k 2 = (l/S)jz 2 dS, where S is the
area of the cross section. This constant is called the radius of gyra
IV.14]
THE EQUATION OF MOTION
153
lion of the cross section, by analogy with the radius of gyration of
solids. Its values for some of the simpler crosssectional shapes are
as follows:
Rectangle, length parallel to center line 6, width perpendicular
to center line a:
Circle, of radius a:
VV12)
GO
Circular ring, outer radius a, inner radius b:
k = £ Va 2 + 6 2
Bending Moments and Shearing Forces. — Equation (14.1), giving
the moment required to bend a length dx of rod by an angle $, is then
M =
Q$Sk*
dx
(14.2)
If the rod does not bend much, we can say that $ is practically equal
to the difference between the slopes of the axial line of the rod at the
two ends of the element dx:
$
\dx/ x+dx \dx/
+ ( ^ = dx
Therefore the bending moment is
(14.3)
M+dM
F+dF
Fig. 30. — Bending moments and
shearing forces to balance.
This bending moment is not the same
for every part of the rod; it is a function
of x, the distance from one end of the rod.
In order to keep the element of bar in
equilibrium, we must have the difference
in the moments acting on the two ends of the element balanced by a
shearing force represented by F in Fig. 30 (moment and shear are,
of course, two different aspects of the single stress which is acting on
the bar). The moment of the shearing force is F dx and this must
equal dM for equihbrium, which means that
dx dx 3
(14.4)
154 THE VIBRATION OF BARS [IV.14
This equation is not exactly true when the bar is vibrating (since a
certain part of the moment must be used in getting the element of bar
to turn as it bends), but it is very nearly correct when the amplitude
of vibration is not large compared with the length of bar.
Properties of the Motion of the Bar. — The shearing force F is also a
function of x and may be different for different ends of the element of
bar. This leaves a net force dF = (dF/dx) dx acting on the element,
perpendicular to the bar's axis; and this force must equal the element's
acceleration times its mass pS dx where p is the density of the mate
rial of the rod. Therefore the equation of motion of the bar is
dx (dF/dx) = P S dx (d 2 y/dt 2 ), or
a^ _p_av (145)
dx* Qk 2 dt 2 K }
This equation differs from the wave equation in that it has a fourth
derivative with respect to x instead of a second derivative. The
general function F(x + ct) is not a solution, so that a bar, satisfying
Eq. (14.5), cannot have waves traveling along it with constant velocity
and unchanged shape. In fact, the term "wave velocity" has no
general meaning in this case, although it can be given certain special
meanings. For instance, a simple harmonic solution of Eq. (14.5) is
y = (7 e 2m(Ma:,« = ^ COS [2tt(^ ~ Vt) — $] (14.6)
where
"(ass?)' ' 2 " A >/?
This represents a sinusoidal wave traveling in the positive direction,
of just the sort one finds on strings. There is an important difference
between the two waves, however; for in the case of the bar the velocity
of the wave u = (v/n) = (4 7 r 2 Q/c 2 /p) i V~ v depends on the frequency of
oscillation of the wave, whereas in the case of the string it does not.
The velocity of propagation of a simple harmonic wave is called the
phase velocity and is one of the special kinds of "velocity" that have
meaning for a bar. In the case of the string the phase velocity is
independent of v and is equal to the velocity of all waves c. For the
bar the phase velocity depends on v, and there is no general velocity
for all waves. This is analogous to the case of the transmission of
light through glass, where light of different frequencies (i.e., of differ
ent colors) travels with different velocities and dispersion results. A
bar is sometimes said to be a dispersing medium for waves of bending.
IV.14] THE EQUATION OF MOTION 155
Wave Motion of an Infinite Bar.— Although the phase velocity is
not a constant, nevertheless we should be able to build up any sort of
wave out of a suitably chosen combination of sinusoidal waves of
different frequencies, in a manner analogous to the formation of a
Fourier series. At present, we are considering the bar to be infinite
in length, so that all frequencies are allowed, and the sum is an integral
y = .Co ^"W/O cos (7 VO + C(n) sin (7 VOJ dfi (14.7)
where y 2 = 4tt 2 k VQ/p, and where the functions B(n) and C(jx), corre
sponding to the coefficients B n and C n of a Fourier series, are deter
mined by the initial conditions. This integral is analogous to the
Fourier integral of Eq. (2.19). If the initial shape and velocity shape
of the bar are y (x) and t> (aO, a derivation similar to that given for
Eq. (9.9), for the Fourier series, shows that
B(m) = ( 4—3 J I y (x)e 2 ™» x dx \
) 1 \"f ( 14 8)
° (m) = V&W/ J_ m voWe 2 *^ dx J
as was shown in Eq. (2.20).
This may seem a rather roundabout way of obtaining solutions
of Eq. (14.5) which satisfy given initial conditions; but since the
useful functions F(x  ct) and f(x + ct) are not solutions of (14.5),
there is no other feasible method.
The case y = e * 2 / ia \ v Q = is one that can be integrated and
can be used to show the change in shape of the wave as it traverses
the bar. Solving Eqs. (14.8) and (14.7), we obtain
We can now see the utility of Eq. (14.7), for while we can show that
(14.9) is a solution of the equation of motion (14.5) by direct substi
tution, we can also see that it would be exceedingly difficult to obtain
(14.9) from (14.5) directly.
The shape of this function, at successive instants of time, is shown
by the solid fines in Fig. 31. For comparison, the sequence of dotted
lines shows the variation in shape of a flexible string with equal initial
conditions. In the case of the string, the two partial waves move out
ward with unchanged shape. The shape of each "partial wave" for
V
156
THE VIBRATION OF BARS
[IV.15
the bar, however, changes continuously as the wave travels outward.
In particular, notice the formation of subsidiary "ripples" ahead of
the principal "crest" of each wave. This is due to the fact, which
we have already noted, that the highfrequency, short wavelength parts
of the wave spread outward faster than the rest.
■p 1G . 31. — Comparison between the motion of waves on a bar (solid line) and on a
string (dotted line), both of infinite length. Each sequence shows the shapes at suc
cessive instants, after starting from rest in the form given at the top of the sequence.
The constant y was chosen to make the average "velocity" of the waves on the bar
approximately equal to c, the velocity of the waves on the string.
15. SIMPLE HARMONIC MOTION
We cannot pursue our study of the motions of the bar much further
without examining its normal modes of vibration. As with the
string, we must ask if there are any ways in which the bar can vibrate
with simple harmonic motion. We try setting y = Y{x)e 2vivt and
find that Y must satisfy the equation
d 4 Y
dx*
= 16.Vr, ,_£, 4^ (£) (15.1)
IV.16] SIMPLE HARMONIC MOTION 157
The general solution of this is
y = de 2 *"* + Cze 2 *** + Cge 2 ""* + Cie 2 ™"*
= a cosh (2irfix) + b sinh (2x,uz) + c cos (2x^r) + d sin (2tthx)
(15.2)
For a discussion of the functions cosh and sinh, see Eqs. (13.4), and
see Tables I and II at the back of the book.
This general solution satisfies Eq. (15.1) for any value of the fre
quency v. It is, of course, the boundary conditions that pick out the
set of allowed frequencies.
Bar Clamped at One End. — For example, if we have a bar of length I
clamped at one end x = 0, the boundary conditions at this end are
that both y and its slope (dy/dx) must be zero at x — 0. The par
ticular combination of the general solution (15.2) that satisfies these
two conditions is the one with c = —a and d = — b
Y = a[cosh (2ttmz)  cos (2w/ia;)] + 6[sinh (2t{jlx)  sin (2jr/is)] (15.3)
If the other end is free, y and its slope will not be zero, but the
bending moment M = QSn 2 (d 2 Y/dx 2 ) and the shearing force F =
—QSK 2 (d s Y/dx 3 ) must both be zero, since there is no bar beyond
x = I to cause a moment or a shearing stress. We see that two con
ditions must be specified for each end instead of just one, as in the
string. This is due to the fact that the equation for Y is a fourth
order differential equation, and its solution involves four arbitrary
constants whose relations must be fixed, instead of two for the string.
It corresponds to the physical fact that while the only internal stress
in the string is tension the bar has two: bending moment and shearing
force, each depending in a different way on the deformation of the bar.
The two boundary conditions at x  I can be rewritten as
(l/4ir 2 n 2 )(d 2 Y/dx 2 ) = and (l/8ir*n 3 )(d*Y/dx*) = at x = I. Sub
stituting expression (15.3) in these, we obtain two equations that fix
the relationship between a and 6 and between n and /:
a[cosh (%rnl) + cos (2r/*Z)] + 6[sinh (2^0 + sin (2thI)] =
afsinh (2jr/d)  sin (2ir/d)] + 6[cosh (2rtf) + cos (2r/*J)] =
or
b = a sin (2Tfxl)  sinh (2thI) = _ cos (2^1) + cosh (2t(iI)
cos (2ir(d) + cosh (2rfxl) a sin (2^0 + sinh (2jr/rf) ( ^
By dividing out a and multiplying across, we obtain an equation for /x:
[cosh(27TMZ) + cos(2ttmZ)] 2 = sinh 2 (2rAiO  sm 2 (2rnl)
158 THE VIBRATION OF BARS [IV.16
Utilizing some trigonometric relationships, this last equation can be
reduced to two simpler forms:
cosh (2ir/j.l) cos (2v/d) = — 1, or coth 2 (7ryuZ) = tan 2 (x/iZ) (15.5)
where coth(z) = cosh(20/sinh(z).
The Allowed Frequencies. — We shall label the solutions of this
equation in order of increasing value. They are (2irfj.il) = 1.8751,
(2*7*20 = 4.6941, (2*7*80 = 7.8548, etc. To simplify the notation,
we let (1/tt) times the numbers given above have the labels jS„, so that
(I)
(15.6)
where 0i = 0.597, /3 2 = 1.494, /3 3 = 2.500, etc.
It turns out that j8„ is practically equal to (n — i) when n is larger
than two.
By fixing /*, we fix the allowed values of the frequency. Using
Eq. (15.1), we have
(15.7)
or
0.55966
fl = l¥~
j> 2 = 6.267 vi
v z = 17.548vi
Vi = 34.387^1
etc.
Notice that the allowed frequencies depend on the inverse square
of the length of the bar, whereas the allowed frequencies of the string
depend on the inverse first power.
Equation (15.7) shows how far from harmonics are the overtones
for a vibrating bar. The first overtone has a higher frequency than
the sixth harmonic of a string of equal fundamental. If the bar
were struck so that its motion contained a number of overtones with
appreciable amplitude, it would give out a shrill and nonmusical sound.
But since these highfrequency overtones are damped out rapidly,
the harsh initial sound would quickly change to a pure tone, almost
entirely due to the fundamental. A tuning fork can be considered
to be two vibrating bars, both clamped at their lower ends. The fork
exhibits the preceding behavior, the initial metallic "ping" rapidly
dying out and leaving an almost pure tone.
IV.16] SIMPLE HARMONIC MOTION 159
The Characteristic Functions. — The characteristic function cor
responding to the allowed frequency v n is given by the equation
+ n = a n [cosh(^)  cos(^)]
6„[sinh(^)sin(^)
+
where
(15.8)
_k _ cosh (xpn) + cos (xj8„) _ sJnhQrftn) — sin(ir8 w )
n sinh (t0 u ) + sin (*r/3„) ~~ Un cosh (wj8„) + cos (w0«)
We shall choose the value of a n so that j ty\ dx = (Z/2), by analogy
with the sine functions for the string. The resulting values for
a n and b n are ai = 0.707, &i = 0.518, a 2 = 0.707, 6 2 = 0.721,
a 3 = 0.707, & 3 = —0.707, etc. For n larger than 2, both a n and b n
are practically equal to (I/a/2) Some of the properties of these
functions that will be of use are
J,, C (m ?± n)
o M*)*.(x) dx = j/j\ (m = n) * n 0) = (l)i V2
(^«(l)V5(^) and 0. ~ „*(«> 2) ^ (15  9)
it n ~: — = [ffrPnX/l __ (_Dnl e i/J B (ir0/fl
V2
V ^ 4/
+ sin(^ ^ (n> 2)
The shapes of the first five characteristic functions are shown in
Fig. 32. Note that for the higher overtones most of the length of
the bar has the sinusoidal shape of the corresponding normal mode
of the string, with the nodes displaced toward the free end. In terms
of the approximate form given above for \f/ n , the sine function is sym
metrical about the center of the bar; the first exponential alters the
sinusoidal shape near x = enough to make 4/ n have zero value and
slope at this point; and the second exponential adds enough near
x = I to make the second and third derivatives vanish. Note, also,
that the number of nodal points in yp n is equal to n — 1, as it is for
the string.
160
TH~E VIBRATION OF BARS
[IV.15
In accordance with the earlier discussion of series of characteristic
functions, we can show that a bar started with the initial conditions,
at t — 0, of y = y (x) and (dy/dt) =
v (x) will have a subsequent shape
n=( given by the series
n=2
n»3
y = 2) ^n{x)[B n co^{2irv n t)
+ C n sin (2rr n t)] (15.10)
n = l
where
B„ =
yo(x)$n(x) dx
Cm —
\irv n lj Jo
o(x)^„(x) dx
Fig. 32. — Shapes of the first five
characteristic functions for a vibrating
bar clamped at one end and free at the
other.
Plucked and Struck Bar. — Two
examples of such calculations will
be given. The first example is that
of a rod suddenly released from an
initial shape y$ = (hx/l), an undesir
able case in actual practice (for it
bends the bar quite severely at
x = 0) but one that can be easily
solved. The values of the coefficients B n turn out to be
£* = ^
JXTp n (x) dx
o
2h (
gp \a n [2 + ^(sinnx/^ — sinTrjSn) — cosh^n — cosx/3 w ]
+ &„[7rj3 n (cosh'7r/3 n + cos7Tj8„) — sinh7r/3„ — sinir/Sn]?
4h
On
when the ratios between a n and 6„, given in Eqs. (15.8), are used to
simplify the expression. The successive shapes taken on by the bar
when started in this manner are shown in the first sequence of Fig. 33.
Since the overtones are not harmonics, the motion is not periodic.
The other case to be considered is that of a rod struck at its free
end in such a manner that its initial velocity is zero everywhere except
at x = I, where it is very large, large enough so that jv dx = U.
This case corresponds to that of a tuning fork struck at one end. If
IV.16]
SIMPLE HARMONIC MOTION
161
the total impulse given to the end of the fork is P, then U = P/pS.
The coefficients B n are zero and
tM Jo 7ry„Z 7r 2 ^ \ Qk 2
B
H
Fig. 33.^ — Motions of a bar clamped at one end. Lefthand sequence shows the
successive shapes of a bar started from rest in the shape given in the topmost curve.
Eighthand sequence shows the motion of a bar struck so that its outer tip starts
upward at t =0. Since the overtones are not harmonics, the motion is not periodic.
when the value given for ip n (l) in Eqs. (15.9) is used. The shapes
of such a bar at successive instants of time are shown in the second
sequence of Fig. 33. This motion is also nonperiodic, as are all
motions of the bar that correspond to more than one normal mode of
vibration.
Clampedclamped and Freefree Bars. — Other boundary condi
tions will give rise to other characteristic functions and frequencies.
162 THE VIBRATION OF BARS [IV.16
The bar may be clamped at both ends, in which case expression (15.3)
must again be used for Y, but instead of Eq. (15.4) we must have
h — sin (2tt/xQ + sinh (2ir ixl) _ _ cos (2^) — cosh (2x^0
~ cos {2irid) — cosh (2irnl) si n (2rnl) — sinh (2rnl)
and instead of Eq. (15.5) Ave have
cosh(27TM0 cos {2tt id) = 1, or tanh 2 (717*0 = tan 2 (717*0
The allowed frequencies can be obtained from the formula
j8i = 1.5056
Vn==J W = w4^7^ ft = 2.4997 (15.11)
/3 n ~ n + i (n > 2)
Still another set of boundary conditions are the ones for a com
pletely free bar, where the second and third derivatives must be zero
at both ends. The characteristic functions (which we can call if/")
for this case can be obtained from the ones for the bar clamped at
both ends (which we can call \[/ n , to distinguish them from #,') by
simply differentiating twice with respect to x, $" = (d 2 if/ n /dx 2 ). By
using Eq. (15.1) we can show that (d 2 $"/dx 2 ) = (47r 2 ^p/Q/c 2 )^ n ,
so that if ip n and (dif/ n /dx) are both zero at x — and x = I, then
(dhl/'Jdx 2 ) and (dhp'^/dx 3 ) are both zero at the end points. The
functions ^" therefore satisfy the boundary conditions for a com
pletely free bar. It. is not difficult to see that $„ is a solution of the
equation of motion, corresponding to the same allowed frequency as
does the function ^„. The allowed values are given in Eq. (15.11).
We thus obtain the rather surprising result that the allowed fre
quencies for a free bar are the same as those for a similar bar clamped
at both ends and that the corresponding characteristic functions are
related by simple differentiation, although their shapes are quite
different.
Energy of Vibration. — It is sometimes useful to know the energy
of vibration of a bar. This energy can be computed by using the
expressions obtained in the previous section. We saw there [Eqs.
(14.2) and (14.3)1 that the moment required to bend an element dx
of the bar by an angle 3? is M = QSn 2 ($/dx). The amount of work
required to bend it from equilibrium to this angle is
J>— * ^^(3)'*
IV.16] SIMPLE HARMONIC MOTION 163
Therefore, the potential energy of the bar, the work required to bend
the bar into its final instantaneous shape y(x,t), is
The kinetic energy is, of course,
so that the total energy turns out to be
The energy of a bar, subject to one or another of the various
boundary conditions discussed above, is obtained by substituting
y = 2A W *„ cos(2W  fl B ) in Eq. (15.12). This results in a super
fluity of terms, each containing a product of two trigonometric
functions of time and an integral of a product of two characteristic
functions, or two second derivatives of these functions. The kinetic
energy terms, containing integrals of the products of two functions, can
be integrated by means of the first of Eqs. (15.9) and reduced to the
single sum
2T 2 r 2 nP lSAlsm 2 (2TvJ  O n )
The integrals in the potentialenergy terms can be integrated by
parts twice:
by using the equation of motion. The terms in the square brackets
are zero no matter which of the boundary conditions discussed
above are used. The resulting potentialenergy series is therefore
WvlplSAl cos 2 (2w  0.). The sin 2 of the kineticenergy terms
combines with the cos 2 in the potentialenergy terms and gives for the
total energy the simple series
W = %™(jf)<Al (15.13)
164 THE VIBRATION OF BARS [IV.15
Note the similarity with the corresponding expression for the energy
of the string, given in Eq. (9.13).
Nonuniform Bar. — Now that we know the properties of the normal
modes of vibration, we can solve a number of different sorts of prob
lems. For instance, we can find the change in the allowed frequencies
and characteristic functions when the bar is made somewhat non
uniform along its length. If the bar's density or its crosssectional
area or radius of gyration changes with x, a review of the derivation
of the equation of motion shows that the correct equation should be
iS(s)te»L w te'J Q dt2
which reduces to Eq. (14.5) when S, p, and k are all constant. To
study the normal modes of vibration, we set y — Y(x)e~ 2wivt and have
for the equation for Y,
1 d 2
S(x) dx 2
[«»*,) g]p=^a]r (1 5.H)
If the variation in p, S, and k is small, we can treat the problem by
a modification of the approximation method discussed in Sec. 12. If
we write
S = Soil + a(x)], k 2 = 4[l + °{x)], P = Poll + g(x)]
where So, k , and p are constants and a, <r, and g are small quantities,
then, by neglecting the products of small quantities, (15.14) can be
rewritten as
d * Y to ^ ■ n d(a + <r) d*Y d 2 {a + <y) d 2 Y
~dx~*~ {2Tfi) Y + l dx ~dx^ + dx 2 ~dx^
 (2x/x)%<r)F =
where ju 2 = (v/2v) VpoA&l If a > °"> and g were all equal to zero,
the solution of this equation would be one of the characteristic func
tions ^ n given in Eq. (15.8), and the allowed values of n would be
the values (j9 n /2Z) given in Eq. (15.6) or (15.11). Since a, <r, and g
are small, we assume that Y n = ZA^*,, where all the Akn's are small
except A nn , and that n n = (j8»/20 4 (l + Vn), where r\ n is small. Sub
stituting this in the equation, and again neglecting products of small
quantities, leaves
IV.15]
SIMPLE HARMONIC MOTION
165
d 2 {a + a) d 2 1
2 J &» =
+ 2 d(a + cr) d*_
dx dx 3
dx 2 dx 2
But (dtyk/dx*) = (Tfa/l)y k . Multiplying through by ^ n or ^ and
integrating gives the equations for the correction factors:
ij:4'*+k0[»^&
+
dx 3
d 2 (a + <r) c?V«
•<*&» —
2A nnJ 8
S)Jo**{
da; 2
da; 2
}
da;
2081  ft) Jo ^ ~ g) ^
+ (_ly r 2 <*<« + «o ^v»
W«/ L dx dx 3
(15.15)
+
d 2 (a + <r) dV
da; 5
da^J/^
As an example, consider the case of a tapered bar of rectangular
cross section having a constant thickness a in the direction of the
vibration and a varying width b (l  y Y j n this case, « and p
are constant, and S(x) = a b(l  y \ so that g = a = and
a = —(yx/l). The correction to m» is
Owing to the boundary conditions, the quantity in the brackets
reduces to  (i) (d^/dx)^ when the bar is clamped at one end and
free at the other. The corrected value of the frequency is therefore
"* V Po 4P V3po L + ir'flt V"da7A=J
7ra(n + i) 2
4l 2
} r 27 l
^ L 1 + T . (n + i )2 J (n > 1)
This equation states that tapering the bar so that its free end
is narrowest raises the natural frequencies over those for a bar whose
width is everywhere the same as at the clamped end. (This is hardly
166 THE VIBRATION OF BARS [IV.16
a surprising result.) If it is tapered so that the free end is three
quarters the width of the clamped end (7 = t), the fundamental
frequency is 1.077 times the fundamental of the nontapering bar
of the same width as that of the clamped end; the first overtone is
1.023 times the corresponding first overtone; the second, 1.008 times
the corresponding second; and so on.
Forced Motion. — Besides the case of the nonuniform bar, we can
discuss the forced motion of a uniform bar, now that we know the
characteristic functions. If the bar is driven by a force F(x)e 2 ' i " t
dynes per cm length, the equation for the shape of the bar during
the steady state is
&! **  4,rV7 = f{x), f = %&, y = Y(x)e**« (15.16)
3 expand Y and / in series of characteristic functions,
Y = Zhntnfr), f = ^Qntnix), Q n = \^jj J q F{x)^ n {x) dx
we obtain for the steadystate motion
y = &
^^tv^n(x)]e^ (15.17)
.n= 1 J
Resonance occurs whenever the driving frequency v equals one
of the natural frequencies v n , unless the corresponding g n is zero.
These results are analogous to those discussed at the beginning of
Sec. 11. The equation determining ^„ here differs from that used in
Eq. (11.1) ; but Eq. (15.17) has the same form as Eq. (11.6). In fact,
if we use the new characteristic functions we can utilize Eq. (11.9) to
determine the general transient behavior of the bar (provided we can
perform the integrations). The effects of friction can be handled as
they were handled for the string on page 104 and in Sec. 12. The
results are so similar that we do not need to go into detail about them
here. Some cases will be taken up in the problems.
16. VIBRATIONS OF A STIFF STRING
When a string is under a tension of T dynes, and also has stiffness,
its equation of motion is
1 dx 2 V dx* P dt 2 K
This equation can be obtained by combining the derivations in Sees.
IV ' 16 J VIBRATIONS OF A STIFF STRING 167
9 and 14. The constant 8 is the area of cross section, k its radius of
gyration, and p and Q are the density and modulus of elasticity of the
material.
Wave Motion on a Wire.— Sinusoidal waves can travel along such
a wire, for if we set y = Ce™^"> we obtain an equation relating
p and v:
or
TV + 47r 2 Q£/cV = P Su 2
n^a I r ?a enCy iS VeiT Sma11 (ix ' if v ' is ver y much small er than
I /16tt P Q£V), we expand the radical by the binomial theorem and
keep only the first two terms, and p 2 turns out to be approximately
equal to Q>Sv*/T). The phase velocity {v/p) is practically equal to
the constant value VT/p~S, which would be the velocity of every
wave if the wire had no stiffness (if QSk 2 were zero). The phase
velocity for the wire is not constant, however. It increases with
increasing v until for very high frequencies it is \/2lTv (Q* 2 / P )* the
phase velocity given in Sec. 14 for a stiff bar without tension.' In
other words, the wire acts like a flexible string for long wavelengths
and hke a stiff bar for short ones. This is not surprising.
The Boundary Conditions.— The usual boundary conditions cor
respond to clamping the wire at both ends, making y = and also
(dy/dx) = at x = and x = I. Setting y = Y(x)e^\ we have for Y
> (16.2)
7 2 = (v/2t) Vp/Qk> )
Setting Y = Ae 2 *»*, we obtain an equation for the allowed values
of p.: M 4  2/3 V  Y 4 = 0. This equation has two roots for p 2 and
therefore four roots for p:
P = ±p h p\ = V/3 4 + t 4 + 2 \
or /
P = ±ip2, p\ = V/3 4 + t 4  8 2 ( ^ 16 ' 3 ^
p\ = 2/3 2 + „l MlM2 = 7 2 /
The general solution of Eq. (1G.2) can then be written
tee 2 *" 1 * + be~ 2r > iix f cc 2ri " 2X f de~ 2Ti c* x }
Y = < A cosh (2ttp iX ) + Bsmh(2Tp 1 x) > (16.4)
' + C cos (2tp 2 x) + D sin (27r/i 2 a;) 1 .
168 THE VIBRATION OF BARS PV.16
If the boundary conditions are symmetrical, it will be useful to place
the point x = midway between the supports. The normal functions
will then be even functions, yp(x) = t(x)', « r the y w 111 be °^ ones >
$(x) = — iKz). In either case, if we fit the boundary conditions at
one end, x = (1/2), they will also fit at the other end, x = —(2/2).
The even functions are built up out of the combination
Y = A cosh (2irmx) f C cos (2wn$x)
and the odd functions from the combination
Y = B sinh (2ir/*i^) + D sin (2x/x 2 x)
The boundary conditions F = and (dY/dx) = at x = (2/2) corre
spond to the following equations for the even functions:
Acosh(xMiO = — Ccos(7r/x 2 0, mi^ sinh (tt/xiO = M2C sin (71W)
The Allowed Frequencies.— By the use of Eq. (16.3), these reduce
to
tan fcrW =  >/l + (^r) tanh (ri VmI + 2/8*). (16.5)
which can be solved for the allowed values of M2. The allowed values
of the frequency are obtained from the equation
v = 2X7 2 * /55 = 2^ 2 J(/*3 + 2/3 2 ) % 2
\ P \ \ P
These allowed values of v can be labeled v h v s , v 6 ... , in order of
increasing size.
The corresponding equation for /* 2 for the odd functions is
^l + (?^ tan (xZ M2 ) = tanh fcrf VmI + 2/3 2 ) (16.6)
The allowed values of v obtained from this equation can be labeled
vs, vi, v 6 . . . ; and then the whole sequence of allowed values will
be in order of increasing size, v x being the smallest (the fundamental),
v 2 the next (the fi rst overton e), and so on.
When 0(= VT/8t 2 QSk 2 ) goes to zero (i.e., when the tension is
zero) the equations reduce to those of a bar clamped at both ends,
with the following allowed values of v.
IV 16] VIBRATIONS OF A STIFF STRING 169
v% = 2.7565V1
= 3.5608 [Q? v 3 = 5.4039*! (16.7)
Vx P V p ' v 4 = 8.9330vi
etc.
obtained from Eq. (15.11) of the previous section. When is infinity
(i.e., when the stiffness is zero) the equations reduce to those of a
flexible string, and the allowed values of v are
v% = 2v\
05000 PF v z = 3vi (16.8)
I \ pS Vi = 4j>i
etc.
If is large but not infinite (i.e., if the wire is stiff, but the tension
is the more important restoring force), it is possible to obtain an
approxima te expres sion for the allowed frequencies. When is large
tanh (tI \/20 2 + n\) is very nearly unity, and Vl + (2/37/4) is a
large quantity for the lower overtones (i.e., as long as /*f is small com
pared with 0). Equation (16.5) for the even functions then states
that tan (trim) is a very large negative number, which means that (irl^)
is a small amount larger than (t/2) or a small amount larger than
(3x/2). In general, (tIh 2 ) = (m + iV + 5, where 5 is a small quan
tity and m is some integer. Expanding both sides of Eq. (16.5) and
retaining only the first terms, we obtain (1/5) = V20 2  — l or
8 = tt(2w + 1) VQSk 2 /T1 2 , where (2m + 1) is any odd integer.
Equation (16.6) for the odd functions states that tan (W)
must be very small, which means that for these cases (irlfi 2 ) is a small
amount larger than t or a small amount larger than 2w, or, in general,
(irl/i 2 ) = kir + 8, where 8 is small and k is some i nteger. The corre
sponding equation for 8 is 8 = ir(2k) VQSk 2 /T1 2 , where (2k) is any
even integer.
We have shown that for either even or odd functions we must
have (irlm) = (nir/2) + 8, where n is any integer and where 5 =
im VQSk 2 /T1 2 . The even solutions correspond to the odd values of
n (n = 2m + 1), and the odd functions correspond to the even values
of n (n = 2k). The a llowed values of /z 2 are given approximately by
\ ~^~ T \~T~j an( * ^ e anowec * values of the frequency are
n
21
" n ~ 21 \J7S K 1 ~*~ 1 \l :±: f L )' A more accurat e formula can be
" S( i+ ?^>
170 THE VIBRATION OF BARS [IV.16
obtained by retaining the next terms in the series expansions that
we have made:
? + (* + =£)$=]
• • ) (16.9)
This formula is valid as long as n 2 is smaller than (l 2 T /tt 2 QSk 2 ) .
When the "stiffness constant" {QSk 2 ) becomes negligible com
pared with l 2 T this formula reduces to (16.8), the equation for the
allowed frequencies of a flexible string. As the stiffness is increased
the allowed frequencies increase, the frequencies of the overtones
increasing relatively somewhat more rapidly than the fundamental,
so that they are no longer harmonics.
To obtain values for the higher overtones or for the cases where
(QSk 2 ) is the same size as or larger than (Z 2 T), we must solve Eqs.
(16.5) and (16.6) numerically.
Problems
1 Which of the materials given in Table 3 will give the highest fundamental
frequency for a bar of given size? Which will give the lowest frequency?
2. An annealed steel bar 20 cm long is clamped at one end. If its cross section
is a square 1 cm on a side, what will be the lowest four frequencies of vibration?
If the cross section is a circle 0.5 cm in radius, what will they be? If the cross
section is a rectangle of sides a and 2a, what must be the value of a to have the
bar's lowest frequency be 250 cps?
3 A bar of nickel iron of length 10 cm, whose cross section is a rectangle ol
sides 1 cm and 0.5 cm, is clamped at one end. It is struck at the midpoint of one
of its wider sides, so that v (x) = except at x = (1/2), and Jv dx = 100. What
is the shape of the bar a time (Ti/4) after the blow? (Z\ is the period of the bar s
fundamental.) What is its shape at a time (7V2)? What is the motion of the
end point of the bar? What is the amplitude of that part of the motion of the
end point corresponding to the fundamental? To the first overtone? To the
second overtone? . ,
4. Plot the shapes of the first three normal modes of vibration of a bar clamped
at both ends. Of a completely free bar. ■,,,•
5 What is the energy of vibration of the bar, struck at one end, which is
shown in Fig. 32? What is the ratio between the energy of the fundamental and
that of the first overtone? Of the second overtone?
6 A cylindrical bar of radius a, clamped at one end, is damped by the air
by a force equal to 1.25 X 10"* a*v* (dy/dt) dynes per cm length of bar where , is
the frequency of vibration and (dy/dt) the bar's velocity. Show that the modulus
of decay of the nth allowed frequency is 16 X 10* ( »p/o«,«), where ,„ is given
by Eq. (15.7).
IV.16] VIBRATION OF A STIFF STRING 171
7. A cylindrical bar of glass, 0.5 cm in radius and 10 cm long, is clamped at
one end. If it is struck at its free end, how long will it take for the amplitude of
the first overtone to diminish to onethousandth of the amplitude of the funda
mental? (Use the results of Prob. 6 to compute this.)
8. What are the first four allowed frequencies of the bar of Prob. 3? What
are the first four frequencies if the bar is loaded at its free end with a mass of 1 g?
The vibration is supposed to be perpendicular to the wider faces.
9. What are the shapes of the first two normal modes of vibration of the bar
of Prob. 3 when its free end is loaded with a mass of 1 g?
10. A bar of phosphor bronze, of length 10 cm and thickness 0.5 cm, is tapered
from a width' of 2 cm at its clamped end to a point at its free end. What are
approximate values for its first three allowed frequencies? Plot the shapes of
the normal modes corresponding to these frequencies.
11. An oscillating driving force of frequency v is applied to the free end of a
bar, clamped at the other end. Show that the mechanical impedance of the bar is
. npSl
\n = l
2^
12. Plot the mechanical impedance of the bar of Prob. 3 when driven at its
free end, as a function of v from v = to v = 2,000. The motion is perpendicular
to the wider faces.
13. The bar of Prob. 3 is driven electromagnetically at its free end by a coil of
coupling constant G = 1,000, resistance 10 ohms, and negligible inductance.
An alternating emf of 1 volt is impressed across the coil. Plot the amplitude of
motion of the end of the bar as a function of v, from v = to v = 2,000.
14. A bar is whirled about one end with an angular velocity «, in a plane per
pendicular to the axis of rotation. Show that the simple harmonic vibrations of
the bar from its position of steady rotation are given by y = F(z)e 2 *<>", where
Y is a solution of
£ + £[<'> £] + *ro
where «* = (Q K y P £«»Z«), z = (x/l), and M 2 = (&rV/«»). Discuss the solutions
when co is small; when w is large.
15. Calculate the transient motion of a clampedfree bar acted on by an impul
sive force S(t) concentrated at x = $. Obtain the equation corresponding to
(.lU.^&i )*
CHAPTER V
MEMBRANES AND PLATES
17. THE EQUATION OF MOTION
We must next consider the vibrations of systems extended in two
dimensions, whose equilibrium shape is a plane sheet. Surfaces whose
stiffness is negligible compared with the restoring force due to tension
are called membranes (examples are drumheads and the diaphragms
of condenser microphones). When the stiffness is the important
factor, the surfaces are called plates (examples are the diaphragms of
ordinary telephone transmitters and receivers).
The analysis of the motions of a membrane is more complicated
than that of the motions of the corresponding onedimensional sys
tem, the flexible string; for the membrane has much more freedom
in the way it vibrates than the string has. The form of the curve
giving the displacement of a given point on a string as a function of
time has a close connection with the form of the same string at some
instant, as we have seen in Sec. 8. The corresponding curve for the
motion of a point on a membrane can have no similar relationship to
the shape of the membrane at some instant, for the curve showing
the displacement of a point as a function of time is a onedimensional
line, whereas the shape of the membrane is a twodimensional surface.
It is possible for a membrane to behave like an assemblage of
parallel strings, having waves whose crests are in parallel lines, per
pendicular to their direction of propagation (like waves on the ocean).
The behavior of such waves is exactly like the behavior of waves on a
flexible string; the waves travel with unchanged shape (when friction
is neglected), and every such wave travels with equal speed. But
this is only the simplest form of wave motion that the membrane can
have. It can have circular waves, radiating out from a point or in
toward it; it can have elliptical ones, going out from or in to a line
segment; and so on. This more complicated sort of wave motion does
not conserve its form as it travels, nor does it have a speed independent
of its form. Much of it is so complicated in behavior that we are
unable to analyze it at all. In this book we shall content ourselves
with a treatment of parallel and circular waves.
172
V.17]
THE EQUATION OF MOTION
173
Tdy
Tdy
If the sheet of material is not perfectly flexible (i.e., is a plate),
then the motion is still more complicated than that for a membrane.'
We shall deal only with the simplest possible case of such motion, in
the last section of this chapter.
Forces on a Membrane.— Our first task is to set up the equation of
motion for the membrane. The procedure is similar to that for the
string. We shall find that any
part of the membrane having a
bulge facing away from the equilib
rium plane will be accelerated to
ward the plane, and vice versa.
In general, the acceleration of any
portion is proportional to its "bulg
iness" and opposite in direction.
We must find a quantitative mea
sure of this bulginess.
Suppose that the membrane
has a density of <r g per sq cm and
that it is pulled evenly around its
edge with a tension T dynes per
cm length of edge. If it is perfectly
flexible, this tension will be dis
tributed evenly throughout its
area, the material on opposite
sides of any line segment of length
dx being pulled apart with a force of T dx dynes. The displacement
of the membrane from its equilibrium position will be called 77. It is a
function of time and of the position on the membrane of the point in
question. If we use rectangular coordinates to locate the point, v will
be a function of x, y, and t.
Referring to the analogous argument on page 82 for the string
and to the first drawing of Fig. 34, we see that the net force on an
element dx dy of the membrane due to the pair of tensions T dy will be
T(r+dr)dy
Fig. 34. — Forces on an element of a
membrane in rectangular and polar
coordinates.
T
V [\dx) x+dx \dx)z.
T^dxdy
and that owing to the pair T dx will be T(d* v /dy 2 ) dx dy. The sum
of these is the net force on the element and must equal the element's
mass adxdy times its acceleration. The wave equation for the
membrane is therefore
174 MEMBRANES AND PLATES [V.17
The Laplacian Operator. — The lefthand side of this, equation is the
expression giving the measure of the bulginess (or, rather, the negative
bulginess) of the portion of the membrane under consideration.
We now encounter one of the extra difficulties of the twodimen
sional analysis; for we find that if we had picked polar instead of
rectangular coordinates, so that 77 was a function of r and <p, then the
resulting wave equation would have a different aspect. The net
force due to the tensions perpendicular to the radius (shown in the
second drawing of Fig. 34) is
_V d<p/ v+d<P \r d<P/<pJ r 2 dip 2
and that due to the tensions parallel to the radius is
^[('SL('S)J?K'S)'**
The resulting equation of motion in polar coordinates is
i±( r *i\ + L*i = L*! (172 )
r dr \ drj ^ r 2 dip 2 c 2 dt 2 K 
The lefthand side of this equation has a different form from the
lefthand side of (17.1). This does not mean that it represents a
different property of the membrane; for the value of the left side of
(17.2) for some point on a given membrane will have the same value
as the left side of (17.1) for the same point on the same membrane. It
simply means that when we wish to measure the bulginess of a portion
of a membrane by using polar coordinates, we must go at it differently
from the way we should have gone at it if we had used rectangular
coordinates.
We often emphasize the fact that the left side of both equations
represents the same property of the surface 17, by writing the wave
equation as
V*„4? (17.8)
where the symbol V 2 is called the Laplacian operator, or simply the
Laplacian. It stands for the operation of finding the bulginess of the
surface at some point. In different coordinates the operator takes
on different forms:
V.17] THE EQUATION OF MOTION 175
d 2 d 2
r~2 + ^~2 (rectangular coordinates)
^ 2 — \l d( d\ Id 2
r dr\ r d~r)~ i ~?dv* (p ° lar coordinates )
etc.
The fact that we have different forms for the Laplacian operator
corresponds to the fact that the membrane can have different sorts
of waves. The form for rectangular coordinates is the natural one
for parallel waves; that for polar coordinates is best for circular waves;
and so on. Although the Laplacian is a measure of the same property
of the membrane, no matter what coordinate system we use, never
theless there is a great variation in the facility with which we can
solve the wave equation in different coordinate systems. In fact,
the known methods for its solution are successful only in the case of a
few of the simpler coordinate systems.
Boundary Conditions and Coordinate Systems. — It should be
realized, however, that the difficulties with coordinate systems are not,
in one sense, a complete barrier to the discussion of waves of com
plicated form. For just as with the string, we can make up compli
cated wave motions by adding together simple ones. Circular waves
can be made up by adding together a large number of parallel ones,
each going in a different direction; parallel waves can be made up of a
suitable sum of elliptic waves; and so on. We shall see that it is
possible to study any sort of wave motion on a membrane of infinite
extent, by expressing the waves in terms of a suitable sum of parallel
waves and studying the properties of the sums (or rather, integrals,
since the sums usually turn out to be integrals). It often turns out
that the integrals are very difficult to evaluate, so that an immense
amount of numerical integration is required to find the wave form.
When the membrane is bounded, the difficulties encountered often
become insuperable. For the greater generality of the problem of the
membrane over that of the string is again apparent when we deal
with the effect of boundary conditions. In the case of the string we
needed only to specify boundary conditions at two points; but in the
case of the membrane we must specify conditions all along a boundary
line — and, in addition, must specify the shape of the line. A change
in shape of the boundary line will have as much effect on the motion
of the membrane as a change in the boundary conditions along the line.
It turns out that the only way we can deal with the effect of
boundary conditions is to solve the problem in coordinates suitable to
176
MEMBRANES AND PLATES
[V.17
the shape of the boundary: rectangular coordinates for a rectangular
boundary, polar coordinates for a circular boundary, and so on. And
if we cannot solve the wave equation in coordinates suitable to the
boundary chosen, we shall not be able to obtain the numerical results
that are necessary for our study of the vibrations.
Reaction to a Concentrated Applied Force. — Another point of
difference between the string and the membrane is in the reaction to
an applied force. A string of
length I pushed aside by a force
concentrated at the point x has a
form composed of the two line
segments shown in Fig. 35. The
shape is so arranged that the
vertical components of the ten
sion at the point of application
Th/x and Th/(l  x) add up to
equal the force F. The displace
ment h = Fx(l  x)/Tl of the
point of application is a finite
amount and is proportional to F.
The membrane, on the other
hand, cannot support a force
concentrated at a point, and the
displacement of the point of appli
cation is infinite, no matter how
small the force is. For instance, if the force is concentrated at the
center of a circular membrane of radius a, the displacement 17 of a
point a distance r from the center will be r\ — (2F/T) In (a/r), where
the symbol In means "natural logarithm of." This expression is a
solution of Eq. (17.2) for equilibrium conditions [i.e., when the right
hand side of (17.2) is zero]. It becomes zero at r = a and infinite
at r = 0.
This result means that the simplification of considering the force
to be applied at a point, which is allowable in the case of the string,
is too much of a simplification for the membrane. Actual forces are
not applied at a point but along a length (for the string) or over an
area (for the membrane), although the length or area can be small.
The point of the foregoing discussion is that, while small changes in
the length over which the force is applied make, very small changes in
the shape of the distorted string, changes in the size of the area of
application of a force to a membrane can make large changes in the
Fig. 35. — Equilibrium shapes of a string
and a circular membrane each acted on by
a constant force concentrated on a small
portion of the string and the membrane.
v  18 l THE RECTANGULAR MEMBRANE 177
value of the maximum displacement of the membrane, so that we must
take into account the size of this area.
We shall find it best, however, to solve problems of forced motion
where the force is concentrated in a small area by first solving the
problem with the force concentrated at a point. This will result in
a form for the membrane having a sharp peak of infinite height at the
point of application. Then we shall cut off the top of the peak to a
height such that the area of the top of the truncated peak is equal
to the actual area of application of the force. This is shown in
Fig. 35.
This discussion shows that it is not efficient to drive a membrane
by a force that is concentrated in a small area, for the amplitude of
oscillation of the rest of the membrane will be very much smaller than
that of the area of application.
If the sheet of material has a certain amount of stiffness, then a
force concentrated at a point will not produce an infinite displacement.
But unless the stiffness is considerable, the displacement of the point
of application will be considerably larger than the displacement of the
rest of the sheet.
18. THE RECTANGULAR MEMBRANE
The wave equation in rectangular coordinates is
w + w=7>dF' C = v^> (18.1)
A solution of this equation is v = F(x  ct), just as for the string.
It represents a wave moving with velocity c in the direction of the
positive a>axis, with its crests parallel to the yaxis. Its shape is
independent of the value of y, and the membrane behaves as if it were
made up of an infinite number of strings, all parallel to the zaxis.
Another solution is v = F(y  ct), having similar properties, except
that the direction of travel is parallel to the yasaa and the crests are
parallel to the zaxis. A still more general form is
V = F(x cos a + y sin a — ct) (18.2)
which represents a parallel wave traveling with velocity c in a direction
at an angle a to the zaxis.
Combinations of Parallel Waves.— In all these cases the mem
brane behaves like an assemblage of flexible strings, and the analysis
of the motion is the same as that given in Sec. 8. For instance, the
178
MEMBRANES AND PLATES
[V.18
reflection of such waves from a straight boundary along the zaxis is
given by the solution
7} = F(x cosa + y sina — ct) — F(x cosa — y sina — ct)
which is always zero at y = 0. This shows that when the angle of
incidence is a the angle of reflection is — a.
As soon as two or more waves going in different directions are
superposed, the membrane has to bend in more than one direction,
and the corresponding motion indicates that the membrane is more
than an assemblage of parallel strings. As we mentioned in the
previous section, every possible sort of wave can be built up out of a
suitable sum of simple waves. The mathematical form of this state
ment is that every possible motion of the membrane can be expressed
in terms of the integral
v = f 2 * F a (x cosa + y sina — ct) da (18.3)
where F a can have a different form for each different direction a.
The problem is to find the forms F a to correspond to a given motion
of the membrane. This will be done for a few cases, later in this
chapter.
The simple harmonic solutions of Eq. (18.1) can be built up out
of a sum of waves of the type
A a cos
'Ittv
(x cosa + y sina — ct)
over different values of a. This sum can be chosen so that it reduces
to a set of standing waves suitable for fitting to rectangular boundaries:
v = ^ i cos — (x cosa + y sina  ct)  $i
\2ttv
+ cos
+ COS
+ COS
(x cosa — y sina — ct) — *2
— (x cosa + y sina + ct) — $3
c
2irv
c
(x cosa  y sina + ct)  $2  *s + *i
= A cos fejx cos a  J cos (^ 2/ sin a  Q y J cos (frrvt 
where Q t = K* a + *»), fi * = ^ _ * z) ' * = ^ 3 ~ * l) '
(18.4)
V.18] THE RECTANGULAR MEMBRANE 179
Separating the Wave Equation. — Another, and perhaps more
straightforward, way of obtaining (18.4) is by separating the wave
equation (18. 1). If we wish to fit rectangular boundaries, we must use
a standing wave having a set of nodal lines (lines along which the dis
placement is zero) parallel to the x axis and another set parallel to
the y axis. Now, the only way that we can have a nodal line parallel
to the y axis {i.e., for i\ to be zero for a given value of x for all values
of y) is to have a factor of rj which is a function of x only and which
goes to zero at the value of x corresponding to the nodal line. The
nodal lines parallel to the x axis would require a factor depending on
y only. Finally, if the motion is simple harmonic, the time depend
ence must come in the factor e~ 2vivt . Therefore an expression for the
form of the membrane satisfying our requirements must be
r] = X(x)Y{y)e~ 2 * ivt
where substitution in (18.1) shows that
y^ = _±[V _ d*Y
dx 2 c 2 dy 2
or
1 d 2 X = _ i_
X dx 2 Y
/ 4*V d 2 Y\
\ c 2 dy 2 /
The lefthand side of this equation is a function of x only, whereas
the righthand side is a function of y only. Now, a function of y
cannot equal a function of x for all values of x and y if both functions
really vary with x and y, respectively; so that the only possible way
for the equation to be true is for both sides to be independent of both
x and y, i.e., to be a constant. Suppose that we call the constant
— (4x 2 f 2 /c 2 ). Then the equation reduces to two simpler equations:
dx 2 c 2 ' dy 2 c 2 K U
The solution of this pair of equations is
V = A cos( ~ x  Q x ) cosf — y  Qy\ cos (frtt VFT7 2  3>)
(18.5)
where we have set t = s/v 2 — f 2 . This solution is another way of
writing (18.4), with f instead of v cos a and r instead of v sin a.
The Normal Modes. — If the boundary conditions are that t\ must
be zero along the edges of a rectangle composed of the x and y axes,
180
MEMBRANES AND PLATES
[V.18
the line x = o, and the line y = b, it is not difficult to see that fi* 
8„ = tt/2, that (2f a/c) must equal an integer, and that (2t6/c) must
equal an integer (not necessarily the same integer). The character
istic functions, giving the possible shapes of the membrane as it
vibrates with simple harmonic motion, are
rj = A\f/
(x,y) cos (2Tv m J — 3>)
:\ • (nmy\
T XU \T)
. (irmx
(x,y) = sm^—
(18.6)
where
Vmn
=w!^/fe^w
(m = 1, 2, 3
(n = 1, 2, 3
m = l n = l
m=l n=2 m = 2 ^0=2
Fig. 36.— Shapes of the first four normal modes of a rectangular membrane. Arrows
point to the nodal lines.
The shapes of the first four normal modes are shown in Fig. 36. We
notice that the number of nodal lines parallel to the y axis is (m  1)
and the number parallel to the x axis (n  1).
The Allowed Frequencies.— The fundamental frequency is vu,
depending on T and a in a manner quite analogous to the case of the
string. Among the allowed frequencies are all the harmonics of the
fundamental, v 22 = 2vn, v 33 = 3vu, etc. But there are many more
allowed frequencies which are not harmonics. When a is nearly
equal to b, 2 extra overtones, v u and v 21 , come in between the first and
second harmonic; 6 extra, v 13 , v 3h v 23 , v S2 , v u , v A i, all come between the
second and third; 10 extra come between the third and fourth; and
so on. No matter what the ratio between a and b is, it is possible to
show that the average number of overtones between the nth and the
(n + l)st harmonics is hm(a 2 + b 2 )/ab, or that the average number
V.18]
THE RECTANGULAR MEMBRANE
181
of allowed frequencies in the frequency range between v and v + Av
is (2rvab/c 2 )Av.
In the case of the string the allowed frequencies are equally spaced
along the frequency scale, but in the case of the membrane the allowed
frequencies get closer and closer together the higher the pitch. The
higher the pitch, the more overtones there are in a range of frequency
of a given size. This property is true of all membranes, no matter
what shape their boundary has. It can be shown, for any membrane,
that the average number of allowed frequencies between v and v + A?
is (2tv/c 2 )Av times the area of the membrane. We shall show in
Sec. 32 how this can be proved.
X ^3Z^ y^TV
Fig. 37. — Various modes of simple harmonic vibration of a square membrane, for the
degenerate cases v\% = vi\ and v™ — vz\. Arrows point to the nodal lines.
The Degenerate Case. — An interesting phenomenon occurs when
the rectangular membrane is a square one {i.e., a = b), for then the
allowed frequencies become equal in pairs, v mn being equal to v nm .
There are fewer different allowed frequencies, but there are just as
many characteristic functions as there are when a is not equal to 6.
This is called a condition of degeneracy. There are two different
functions \l/ mn and \l/ nm , each corresponding to the same frequency
(except for the cases n = m, which are not degenerate). In such
cases the membrane can vibrate with simple harmonic motion of fre
quency v mn with any one of an infinite number of different shapes,
corresponding to the different values of y in the combination
rpmn COST + ypnm SU1 J
182
MEMBRANES AND PLATES
[V.18
Figure 37 shows the shapes of the normal modes of vibration of
the square membrane corresponding to v 12 and v 13 , for different values
of 7. The vibrations can be standing waves, corresponding to
V = {i>mn cos 7 + \f/ nm sin 7) cos (2rvmJ)
The nodal lines have a different shape for each different value of 7.
It is also possible to have traveling waves, corresponding to the
expression
V — bPmn cos {2irv mn t) + ^ nm sin (2irv m J)]
Fig. 38. — Successive shapes of a rectangular membrane struck at its center. Times
are given in terms of fractions of the fundamental period of vibration.
In this case the nodal lines go through the whole range of possible
shapes during each cycle. It is only in degenerate cases that it is
possible to have traveling waves of simple harmonic motion in a mem
brane of finite size.
The Characteristic Functions. — The characteristic functions for
the rectangular membrane have the following integral properties:
XT*
n&m'n' dx dy =
fO
{(ab/4)
(rn' = to
or n 7
and n'
n)
n)
(18.7)
V.19] THE CIRCULAR MEMBRANE 183
The behavior of a membrane having an initial shape y]o(x,y) and an
initial velocity v (x,y) is therefore
co oo
V = 2 X ^ n {x,y)[B m n aos{2icv mn t) + C mn sin (2irv mn t)] (18.8)
m= 1 n— 1
#m» = T I I l/O^nm dc <%, Cmn = ft& J I V<$ mn dx dy
This expression is the correct one for both the degenerate and the non
degenerate cases.
Figure 38 shows the motion of a membrane that has been struck
so that a small area around its center is started downward at t = 0.
In this case r?o = 0, and v is zero except near the point x = a/2,
y = 6/2, where it has a large enough value so that jjv dx dy = U.
The integrations for the coefficients C mn become simple, and the result
ing expression for 17
2U
v =
^ 7~~ tmnl g' 2) ^ mn ( X,V ^ Sln ( 2TV «» < )
xa6
The series does not converge if m and n both run to infinity, a corollary
of the fact that a concentrated force produces an infinite displacement
of the membrane. However, if the series does not continue for m
larger than some number M or for n larger than N, then there will be
only a finite number of terms to be added together, and the result
will never be infinite. Such a finite series corresponds approximately
to a starting area of dimensions (a/M) by (b/N).
The case shown in Fig. 38 is f or M = N = 10. The initial shape
and the shapes for successive eighths of the fundamental cycle are
shown. We see the initial pulse spread out and then reflect back to
pile up in the center in a "splash." Since not all the overtones are
harmonics, the motion is not periodic. We notice that the shape
of the pulse changes as it spreads out to the edge. More will be said
about this change of shape in the next section.
19. THE CIRCULAR MEMBRANE
Rectangular coordinates are useful to describe parallel waves and
normal modes for a rectangular boundary, but for the discussion of
circular waves and for the study of the normal modes for a circular
boundary we shall find it easiest to use polar coordinates. In this
study we shall encounter more forcibly than before the essential
184
MEMBRANES AND PLATES
[V.19
differences between wave motion on a string and that on a membrane.
To bring out these differences we start with a discussion of the general
vibrations of a membrane of infinite extent.
Wave Motion on an Infinite Membrane. — Let the initial shape and
velocity shape of the infinite membrane be vo(x,y) and v (x,y). By
letting a and b go to infinity in Eq. (18.8), we find that we can express
the subsequent shape of the membrane by the Fourier integral [see
Eq. (2.20)].
v(x,y,t) = j I I d$ dr e ( 2 ™/ c ><r*+^>
C J — 00 J — 00
[/» 00 /» 00
cos (2xVr 2 + t 2 I I da' dy' rj (x' ,y')e^ i/c ^^'+^
2xVr 2 + r 2 JJ
+
da;' dy 7 v (z',2/')e (2iri/c)(f *' +TI/ ' )
By a series of transformations too involved to discuss here, this
integral can be transformed into the
following simpler and much more in
teresting form :
v(x,y,t) = ^
RdRdd
y/cH*  R 2
* c l dt Jo Jo
J*ct flit
Vo(x',y')
o Jo
RdRdd
VcH*
ld& I
(19.1)
r
ct
~r
ct
Fig. 39. — Dependence of the wave
motion of a string and membrane on
initial conditions. The displacement
of the point (x,v) on the membrane at
the time t depends on the initial con
ditions for that part of the membrane
enclosed by a circle of radius ct with
center (x,y). The displacement of
point x on the string depends on the
initial conditions for the portion
within a distance ct on either side of x.
, Impermanence of the Waves,
corresponding expression for the
where R is the distance between the
point (x,y) and the point (x',y'), as
shown in Fig. 39.
Equation (19.1) shows that the
displacement of the membrane at the
point (x,y) at the time t depends on
the shape and velocity at t = of
all those parts of the membrane with
in a circle about (x,y) of radius ct.
The area that affects the displace
ment r](x,y,t) spreads out as t in
creases.
, — Let us compare this result with the
infinite string given in Eq. (8.3):
V.19I THE CIRCULAR MEMBRANE 185
y(x,t) = ^ \j t I Vo(x') dx' + I v (x') dx'
where we have expressed the equation in a form like (19.1). The
esse ntial differ ence between this expression and (19.1) is the factor
l/\/c 2 t 2 — R 2 in the integral for the membrane, which is missing in
Fig. 40. — Comparison between the behavior of a plucked string and a plucked
membrane. The first sketches show the initial shapes; the lower ones the shapes at
successive instants later. One quarter of the membrane has been cut away to show
the shape of the cross section.
the integral for the string. That the difference is important can be
seen by the following argument: Suppose that the initial disturbance
on the string is confined to a small range, from a; = XAtoa; =
X + A, y and v being zero elsewhere. Then, by the preceding
equation, the string at the point x (we assume that x is smaller than
X — A, to make the discussion less verbose) will be undisturbed as
long as ct is less than (X — x — A). The wave will reach the point x at
t = (l/c)(Z  x  A), and from this time to t = (l/c)(X  x + A)
186
MEMBRANES AND PLATES
[V.19
the displacement at x will be changing. However, from the time
t = (l/c)(X — x + A) on, the string at the point x will again be undis
turbed. This corresponds to the fact that the wave, as it travels, does
not change its shape; the pulse, as it passes the point x, leaves no
"wake" trailing behind it.
For comparison, let us suppose that the initial disturbance of the
membrane is confined to a small circular area of radius A, a distance
D away from the point (x, y). As with the string, no disturbance will
Fig. 41. — Comparison between the behavior of a struck string and struck membrane.
The initial displacements are zero, and the center portions of both string and membrane
are given an initial upward velocity.
occur at (x, y) as long as ct is less than (D — A). The wave will reach
(x, y) at t = (l/c)(D — A), and from this time until t = (l/c)(Z> + A)
the displacement at (x,y) will change. However, unlike the string,
the displacement at {x, y) will continue to change after t = (l/c)(Z> + A),
owing to the factor l/\^c 2 t 2 — R 2 in the integral expression. The
pulse, as it spreads out from its source, leaves a wake trailing behind
it. The wave therefore differs from the waves on a string and also
differs from waves on a bar, where ripples travel ahead of the "crest."
Figures 40 and 41 illustrate this property of the membrane and
V.19] THE CIRCULAR MEMBRANE 187
contrast it to that of the string. Figure 40 gives the shapes of the
membrane at successive instants, after it has been pulled out over a
small area and let go at t = 0. Figure 41 shows the resulting motion
when the central area in the first sketch is given an upward velocity
at t = 0. The corresponding motions of a string with an equal value
of c are also shown. In each case, the outermost part of the wave
on the membrane keeps its shape as it moves outward, being similar
to the shape of the corresponding part of the wave on the string. The
rest of the membrane, however, changes shape as it moves, more and
more of the crest being left behind. The initial conditions for these
two figures have been chosen to have very exaggerated forms, just
in order to show the above properties as clearly as possible.
Simple Harmonic Waves. — Now let us turn to the circular waves
that vibrate with simple harmonic motion. As in the rectangular
case, we separate the wave equation. In polar coordinates the equa
tion is
r dr\ dr/
r 2 d<p 2 c 2 dt 2
and by setting rj = R(r)${<p)e~ 2vivt we obtain the equations for R
and <£:
d 2 $
—— = — /z 2 <£>, <£ = cosO<p) or sin(jit^) (19.2)
a<p
W + r dF + V^" "" T 2 ) R ~ ° (19  3)
For these circular waves one "boundary condition" is required
even before the shape of the boundary line is decided upon. The
requirement is simply that the displacement rj be a single valued
function of position; for the coordinate <p is a periodic one, repeating
itself after an angle %r, and we must have rj(r, <p) equal to r\(r, <p + 2r).
This restricts the allowed values of /x to integers:
$ em = cos (m<p), $om = sin (m<p) (m = 0, 1, 2, 3 • • • )
This is not true for a membrane whose boundary is shaped like a
sector of a circle, where <p cannot go from zero clear around to 27r, but
such cases are of no great practical importance.
The foregoing requirement is the third different type of boundary
condition that we have encountered, the first type being the fixing
of the displacement or its slope (or both) at some point or along some
line (as with the string, the bar, and the rectangular membrane), the
188 MEMBRANES AND PLATES [V.19
second being simply that the displacement have no infinite values in
the range of interest (as with the whirled string). The third, the
condition of periodicity, will be used whenever any of the coordinates
are angles that repeat themselves.
Bessel Functions. — Equation (19.3) for the radial factor is Bessel's
equation. We have solved it for n = 1 in Sec. 2 and for n = in
Prob. 3 of Chap. I. The general solution J m (z), where z = (2wvr/c)
and where m is an integer, has the following properties :
Jm(z) = m\ \$) * (m+1)! W
+ I . (*) m+ * _ . . . \ (19.4)
^2!(m + 2)!\2/ / v
t ,\ 1 2" ( 2m + 1 \
i C 2w
Jm(z) = 7T~^ I e lzcoBW cos (mw) dw
2iri m Jo
Jml(z) + J m +l(z) = — J m (z)
z
T Jm(z) = i [J m \(Z)  J m +l(z)]
^[z m J m (z)] = z m J m ^(z), ^z m J m (z) = z m J m+1 (z)
I Ji(z) dz = —J (z), I zJ (z) dz = zJx(z)
I
s
I
Jl{z)zdz = Z ^[Jl(z)+J\(z)]
Ji(z)z dz = 2 [Jl(z)  J m i(z)J m+ i(z)]
2
J m (az)J m (pz)z dz = — — % [pJ m (az)J m _ 1 (fiz)
2 _ R2
 aJ m (pz)J m i(az)]
(19.5)
All these properties are proved in books on Bessel functions. Values
of Jo, J i, and J 2 are given in Table V at the back of this book.
The function J m {2irvr/c) is not the only solution of Eq. (19.3), for
it is a secondorder equation, and there must be a second solution.
This other solution, however, becomes infinite at r = and so is of
V.19] THE CIRCULAR MEMBRANE 189
no interest to us at present. It will be discussed in the next section, in
connection with forced vibrations.
The Allowed Frequencies. — Coming back to our problem, we can
now say that a simple harmonic solution of the wave equation which
is finite over the range from r = to r = <», which is single valued
over the range from <p = to <p = 2t, is
 cc)J m (^J
r] = cosm(<p — a)J m [ J cos(2n*Z — Q)
If the membrane is fastened along a boundary circle of radius a, the
allowed frequencies must be those that make J m (2Tva/c) = 0. For
each value of to there will be a whole sequence of solutions. We shall
label the allowed values of the frequency v mn , so that *oi, *02, *oi, etc.,
are the solutions of J (2Tva/c) = 0; *n, * X2 , *i3, etc., are the solutions
of J i(2t va/c) = 0; and so on. The values of * OTn are given by the
equations
Vmn =£0mn, 001 = 0.7655, 002 = 1.7571, 003 = 2.7546 • • •
2a
0n = 1.2197, 12 = 2.2330, 0i 3 = 3.2383
02i = 1.6347, 022 = 2.6793, 23 = 3.6987
0m« ^ n + 7T — t, if n is large
(19.6)
The frequency p i is the fundamental. Another way of writing these
results is
voi = 0.38274 ij^, vn = 1.5933poi, *2i = 2.1355*01 I ( 19>7 )
* 02 = 2.2954*oi, *3i = 2.6531*oi, v 12 = 2.9173* i • • • )
It is to be noticed that none of these overtones is harmonic.
The Characteristic Functions. — Corresponding to the frequency
*o« is the characteristic function
Wr, fp) = j(~f^j (19.8)
and corresponding to the frequency v mn (to > 0) are the two charac
teristic functions
*« = cos (m<p)J Jfe^J, tfw = sin (m<p) J m (^f\ (19.9)
190
MEMBRANES AND PLATES
[V.19
Except f or m = the normal modes are degenerate, there being two
characteristic functions for each frequency. The shapes of a few of
the normal modes are shown in Fig. 42. We notice that the (ra, n)th
characteristic function has m diametrical nodal lines and (n  1)
circular nodes.
where
Fig. 42. — Shapes of some of the normal modes of vibration of the circular membrane.
Arrows point to the nodal lines.
The integral properties of these functions are obtained from
Eqs. (19.5):
T ° f ^ temntem'nr dr d V = Jq Jq' ^Omnlfc>»V T dr d<p
( (m 7^ m' or n ?± n')
= \ra 2 A mn (m = m! and n = n') (19.10)
A „ = [J"i(ir/3 OTC )] 2 , A TO „ = i[J TO _i(x/3 m „)] 2 (m > 0)
The values of the constants A mn can be computed from the following
values of J m :
JfrM = +0.5191, Ji(tM = 0.3403,
Ji(7t/3o 3 ) = +0.2715, JiOtfoO = 0.2325
JoOnSii) = 0.4028, JoOnM = +0.3001,
JoWh) = 0.2497, JoOtfu) = +0.2184
/iCir/Sai) = 0.3397, J r i(x/3 22 ) = +0.2714,
J^&s) = 0.2324, J" 1 (7r i 824) = +0.2066
With the values of the constants A known, it is possible to com
pute the behavior of a circular membrane when started with the
V.19] THE CIRCULAR MEMBRANE 191
initial shape vo(r,(p) and the initial velocity v (r,<p). By methods
that we have used many times before, we can show that this behavior
is governed by the series
00 / 00
n= 1 v m =
where
[Bemn COS (2lTV mn t) + C emn Sm (2lTV mn t)\
+ J) tom n [B 0mn cos(2nrv mn t) + C 0m „ sin (2rv mn t)] > (19.11)
TO = 1 /
1 f ° f ^
■B emn = —T — I I r)Q\p emn r dr d<p
Td'Amn Jo Jo
i /»a /»2tt
C«mn = o~i — ^2l — I I V($ emn r dr d<p
4ir z v mn a 2 A mn Jo Jo
A similar set of equations holds for B Qmn and C Qmn .
Relation between Parallel and Circular Waves. — In Sec. 18 we
stated that a circular wave could be built up out of a series (or rather
an integral) of different parallel waves, and we set up a general expres
sion (18.3) for the form of the integral. We must now verify our
statements, by finding the form of F a which is to be used in (18.3) to
give circular waves. If the waves are to be simple harmonic, and if
we decide to make the dependence of F on a also periodic, Eq. (18.3)
becomes
it = I cos (ma) e (25riy/c) ( * °°* a+v Bin act) da
Since x — rcos<p and y = rsin^>, a; cos a + ysina becomes r(cosa
co8<p + sin a sin <p) = r cos (a — <p). Changing to the variable =
a — <p, the integral becomes
/»2*
V = e 2Tiyt I [cos (m<p) cos(m/3) — sin(m^) sin (m(3)]e 2irivr <*» v* dp
The term with sin (ra/3) integrates to zero, but the term with cos (m$)
can be transformed, by the aid of the last of Eqs. (19.4), into
v = e 2ri« C0S ( m(p )(2Tl m ) J J— J
Therefore the expression for a circular wave in terms of parallel waves
is
192 MEMBRANES AND PLATES [V.19
cos (m<p) Jrn\^~) e~ 2 * M
\2Tri m ) Jo
2ir
cos (ma)e (2ri ' /c) ( * C03 a+y sia a  ct) da. (19. 12)
There is an equivalent equation with sin (m<p) and sin (ma) in place
of the respective cosines.
Similarly, it is possible to express the simple harmonic parallel
wave e (2«»/c)o c «) _ e (2wiv/c) (r cos act) m terms of a series of circular
waves SA m cos (m<p)J m (2Trvr/c)e~ 2vi '' t . We use the series with the
cosine terms only, because the parallelwave function is symmetrical
in <p; a change in the sign of <p will not change the value of the expo
nential. Now, from the point of view of the dependence on <p, the
series of circular functions is a Fourier series, and by the Usual methods
of Sec. 9 we obtain
00
e (8»ir/e>(roo.reD — Q^ivi V C m {r) COS (nip)
m = Q
where
C °2*
i c 2 ' i r 2 "
— I e iMwr/e)m *da, C m =  I e (2 " v/c)ooa ^cos(m^) dtp
wr Jo t Jo
(m > 0)
From the last of Eqs. (19.4) we see that the first integral is just
Jo(2irvr/c) and the second 2i m J m {2irvr/c). Therefore, the series of
circular waves that build up into a parallel wave is
e (2 r i,/c)(xct) —  J  ~ n " r , j
+ ^2i m cos (m<p) J m r^ J e 2 ™' (19.13)
m = 1 \ / J
The real part of this series is
2p'(*e*)]
= W^p) 2coa(2<p)jjfe^\ + • • • cos(2rvt)
+ [ 2 cos (<p)Jr(^) ~ 2 cos (3*)J 3 (^)
+
sin (2tt^)
V.19] THE CIRCULAR MEMBRANE 193
The Kettledrum. — It sometimes happens that a circular membrane
is stretched over one end of a vessel that is airtight. This is the case
for the kettledrum and for some types of condenser microphones.
Here the tension is not the only restoring force, for the motion of the
membrane alternately compresses and expands the air in the vessel,
and this reacts back on the membrane, changing its natural frequencies
and its general behavior. If the diaphragm is very light and the
tension is extremely large, the speed of wave motion in the membrane
will be as large as the speed of sound in air, or larger; and the problem
would require for its solution all the techniques that will be developed
in Chap. VII. If the velocity of transverse waves in the membrane
is considerably less than the speed of sound, however, the problem
is much simpler, for then the compression and expansion of the air in
the vessel is more or less the same over the whole extent of the mem
brane and will depend on the average displacement of the membrane.
When the membrane is displaced from equilibrium to the shape
expressed by the function q, the volume of the vessel is diminished
by an amount I I rjr dr d<p. If the equilibrium volume inside
the vessel is TV and the equilibrium density of the air is p , then when
the alternations of pressure are rapid enough to be adiabatic changes,
the excess pressure inside the vessel will be
p = — I fj^r? J \ I rjr dr d<p dynes/sq cm
where c a is the velocity of sound waves in air at the equilibrium pres
sure and temperature in the vessel. At normal pressures and tem
peratures (temperature 20°C, pressure 760 mm of mercury) in air
the value of p cl is 1.44 X 10 6 ergs per cc. This expression for the
excess pressure will be proved in the next chapter. The pressure is
given with a negative sign because it is always in the direction opposite
to that of the average displacement of the membrane.
The equation of motion for the membrane is therefore
and the equation for simple harmonic vibrations is
V = Y(r,(p)e 2vivt
_2 7 ,2 / »\ l' a l* 2r
\Vot) Jo Jo
V*Y + ^Y=\$%)\ I Yrdrd<p (19.15)
194
MEMBRANES AND PLATES
[V.19
If Y = c ? 8 (m<p) J m (2rvr/c) (m > 0), the integral on the righthand
sin
side of this equation will be zero (owing to the integration over <p),
and the solution that satisfies the boundary conditions Y = at
r = a will be the characteristic functions given in Eqs. (19.9), with
the corresponding allowed frequencies. The presence of the airtight
Vessel, therefore, has no effect on the normal modes of vibration which
have one or more diametrical nodal lines {i.e., which have a factor
cosine or sine of (m<p), where m is not zero).
For the case where m = the integral on the right is not zero.
Since the solution of the equation without the integral is J (2rvr/c),
we try the function Y = J (2Tvr/c)  J (2Tvva/c), which satisfies
the boundary condition Y = at r = a. The integral then reduces to
where x = (irpocla 4 /VoT). Inserting this into Eq. (19.15) and utiliz
ing Eqs. (19.5) results in the equation
Jo(w) = — 2
■ vr
J (w) Ji(w)
w = (2irva/c)
)
4w
w 2
(19.16)
which determines the allowed values of the frequency for those normal
modes that are independent of <p (i.e., have no diametrical nodal lines).
Table 4. — Frequencies op the Kettledrum
X
Yoi
702
703
704
0.7655
1.7571
2.7546
3.7534
0.5
0.7880
1.7590
2.7550
3.7535
1
0.8097
1.7610
2.7555
3.7537
2
0.8510
1.7651
2.7566
3.7541
3
0.8899
1.7694
2.7576
3.7545
4
0.9265
1.7739
2.7587
3.7549
5
0.9604
1.7787
2.7598
3.7553
6
0.9914
1.7837
2.7609
3.7557
8
1.0445
1.7945
2.7632
3.7565
10
1.1101
1.8065
2.7657
3.7573
Frequencies of the symmetrical normal modes of vibration of a circular mem
brane closing an airtight vessel, as functions of x the effective restoring force of the
enclosed air. The nonsymmetrical modes (to > 0) are all independent of x The
frequencies in cps are given by the formula v = (y on c/2a).
V.20] FORCED MOTION. THE CONDENSER MICROPHONE 195
The constant x is a measure of the relative importance .of the air
confined in the vessel with respect to the tension, as a restoring force
on the membrane. It is small if the tension is large or if the volume
of the vessel is large. The limiting case x = is the one studied
earlier, and the allowed frequencies are given in Eqs. (19.6) and (19.7).
The allowed values v 0n of the frequency [or, rather, of y 0n = (2av 0n /c)]
for other values of x are given in Table 4. This table shows that the
presence of the vessel tends to raise the values of the allowed fre
quencies v 0n .
The Allowed Frequencies. — The allowed frequencies and corre
sponding characteristic functions for the membrane plus vessel are
therefore
^eOn = Joi T 2 J — Jo(iryOn) \
f emn = CO s(mv)J m (?^y ^ n = Bm{m^) jj^f^\ (19.17)
*(&)$ fe)^ (W>0) )
where the values of y 0n are given in Table 4 and the values of /3 m „
are given in Eq. (19.6). An approximate formula for y „, valid for
small values of x, is 7o« ^ jSo« + (2xAr 4 /3jjJ, and the corresponding
approximate formula for the allowed frequencies is
v 0n c* (vji.) [l + o_ 3T 71 a 2 ;..o ^4 I (19.18)
8T*Vo<rK4ny
where v^ n is the value of v 0n when x is zero, given in Eq. (19.7).
A series analogous to that of Eq. (19.11) can be set up to represent
the motion of the membrane started in any manner, by the use of
the integral properties of the characteristic functions given in Eq.
(19.10) and in the following equation:
j:
tf'eontfwr dr = < 2 _ (tt7o«) 2
{n f = n)
(n' t* n)
(19.19)
20. FORCED MOTION. THE CONDENSER MICROPHONE
So far, we have been dealing with circular solutions of the wave
equation which are finite everywhere. When we have to deal with
circular waves sent out by a simple harmonic driving force concen
196 MEMBRANES AND PLATES [V.20
trated at a point, we must expect to use solutions that become infinite
at the point of application of the force (which we set at r = 0). These
solutions are the other solutions of BessePs equation (19.3), for this
equation is a secondorder differential equation and must have two
independent solutions.
Neumann Functions. — The second solutions of Bessel's equation
are called Neumann functions and are given the symbol N m (z). They
have the following properties :
I d ( dN m \ , ( \ m 2 \
N (z) ^ () In (0.890536z) = (?) Qnz  0.11593) \ (20.1)
Ar , . (m  1)1 /2\ w , ^ m
N m (z) >  ^  (  1 (m > 0)
z»0 7T \Z /
o
N m i{z)J m (z)  N m (z)J m _!(z) = I —
The properties given in Eqs. (19.5) for the functions J m (z) are also
true for the corresponding functions N m (z).
The function representing a circular outgoing wave caused by
a force Fe~ 2vivi concentrated at r = is
'{$[<*?) + <*r).
(20.2)
From Eqs. (19.4) and (20.1) we can show that when r is very small
t? becomes very large, having the value — (2F/T)\nr er 2rirt , which
it must have in order to balance the force, as we showed on page
176. When r is very large, r\ will approach in value the function
{{ J / — e (2«i/c)(rc«)tv/4 j representing a circular simple harmonic
wave spreading outward with velocity c. The real part of this
function is
~ J ln(r) cos (2irvt)
r>0
^(l)V^ cos
*Z lr cO+l
V.20] FORCED MOTION. THE CONDENSER MICROPHONE 197
Although the motion of the whole wave with time is sinusoidal,
only the part of the wave at large distances from the driving force
approaches a sinusoidal dependence on r, and even at these distances
the amplitude of the waves diminishes with increasing r, owing to
the factor y/c/vr. Note, also, the phase lag (tt/4) of the outlying
wave behind the driving force, which is to be compared with the lag
of (ir/2) for the string discussed on page 92. These properties again
emphasize the difference between one and twodimensional waves.
Unloaded Membrane, Any Force. — The coupling between a mem
brane and the surrounding medium is generally much more effective
than is the corresponding coupling for a string. Here the vibrating
system is a surface, which must move a sizable portion of the medium
every time it moves, whereas strings are usually thin enough to avoid
disturbing the medium much by their motion. Consequently, a
calculation of the free and forced motions of a membrane in vacuum
is not so satisfactory an approximation to its behavior in air or water
as are the analogous formulas for a string.
The calculation of the effect of the medium on the motion of the
membrane is correspondingly more difficult than is the case for the
string, where the effects are always small and can be treated as per
turbations. Only when the membrane is heavy and the medium
light are the effects of the medium small enough to treat successfully
by the methods of Sees. 10 and 12. This case will be considered first,
however, since it is more straightforward.
As a matter of fact, we shall start with the simplest case, where
the effects of the medium are negligible, even though this is not often
applicable. By methods similar to those used in deriving Eq. (10.16),
we obtain an expression for the steadystate motion of a membrane
under the influence of a distributed force F(u ) v)e~ iat per unit area at
(u,v) (or r,<p) :
F(u,v)\f/ mn (u,v) du dv
«L — O) 2
ba ^J A mn IJ J
m,n
rc . , , , /0 (m! 9^ m or n' 9^ n)
II tmntmV du dv = < _ '
\*3**<mTl \ffl/ lib ctllLL lb — ft
(20.3)
where the integration is carried out over the surface of the membrane.
The functions \f/ mn are given by Eq. (18.6) for the rectangular mem
brane and by Eq. (19.9) for the circular membrane; A mn is given by
Eq. (18.7) (i.e., it equals J) or by Eq. (19.10); and w mn = %rv mn is
given by Eq. (18.6) or (19.6). S is the area of the membrane and
S<r its total mass.
198 MEMBRANES AND PLATES [V.20
When the force is concentrated near the point (u,v) and its total
value is Pe~ iat , this series becomes
SO ^J K m n 0>l n  CO 2
mn mn
When the frequency is zero, the sum gives the shape of the membrane
when pushed by a steady force P concentrated at the point (u,v).
m = ^Y(u,v;x,y); Y = ^ ^ n M^ n (x,y)
The function Y goes to zero at the boundary of the membrane and
approaches the function  (Sa/^T) \n[(x — u) 2 + {y — v) 2 ] when
(x,y) is close to (u,v). Using Y, we can reexpress the series for the
steadystate driven motion due to a concentrated force
" = s
[ Y(u,v, x,y) + ^ jr *%J? r~ ( 20 ' 4 >
L mn ■
The dependence of the shape on frequency is given by the second
sum, which converges much more rapidly than the sum written first.
Figure 43 shows the shapes of a square membrane driven by a
force concentrated near its center, for different values of the driving
frequency. The resonance frequencies are the fundamental ^n ajid
the oddnumbered overtones pi 3 = vu v5, vw = 3^n, vn = vu vl3,
etc. The evennumbered overtones do not appear, since the corre
sponding characteristic functions have a node at the midpoint of the
membrane and the terms in the sum vanish.
The nodal lines are shown in the figure by dotted lines. These
change their shape, spreading outward as v is increased, a new node
being introduced, near the center, after each resonance frequency has
been exceeded.
Localized Loading, Any Force. — When we begin to consider the
effects of the medium surrounding the membrane, we must first ask
whether the wave motion in the medium is faster or slower than trans
verse waves in the membrane. If the wave velocity in the medium
is much slower than that in the membrane, then the reaction of the
medium on any given portion of the membrane depends entirely on
the motion of that part of the membrane; for the different parts of
the medium are relatively slow in letting each other know what
motions they are undergoing.
V.20] FORCED MOTION. THE CONDENSER MICROPHONE 199
This case is seldom met in practice, but the calculations of driven
motion are not difficult. The load of the medium is expressible in
terms of a resistive term R per unit area and an additional reactive
load that can be added to the mass per unit area of the membrane to
give an effective mass <r e . Both R and <x e may vary with the driving
frequency; however, when the wave velocity of the medium is very
much less than that for the membrane, R is approximately constant,
equal to the product of the density of the medium with the wave
velocity in the medium, (pc a ), and the additional mass loading becomes
negligible.
Fig. 43. — Shapes of a square membrane driven by a force of frequency v concen
trated on a small area near the midpoint of the membrane. The resonance fre
quencies are the fundamental vu, P13 = 2.2361yn, v\z = 3vu, etc. At these frequencies
the amplitude will be infinite, since friction has been neglected. The nodal lines are
indicated by the dotted lines.
The equation of motion for a distributed driving force is
dt
dri
t =TVh,R^+ F{x,y)e
We substitute the usual series of characteristic functions, appropriate
to the boundary, for 17 and eventually obtain the familiar equation
for the steadystate motion:
lOiZn
(20.5)
where
fmn = ( or — ) I I F(x,y)\p mn dx dy over the membrane
200 MEMBRANES AND PLATES [V.20
and where
(2nrv mn ) 2 <r e
= R — t\ OXTe
(for localized loading)
is the effective impedance of the (m,n)th mode for transverse motion
of the membrane in the medium (at the frequency <a/2ir). As before,
the characteristic functions ^ m „ are given by Eq. (18.6) or (19.8).
The methods discussed in Sec. 10 can be used to compute the
transient motion of the membrane. No further" details need be dis
cussed here.
Uniform Loading, Uniform Force. — The other extreme, where the
wave velocity in the medium is much greater than that of the mem
brane, is more often encountered but is more difficult to solve. Here
the effect of the motion of one part of the surface is transmitted rapidly
through the medium to affect the other parts, so that in the limit the
reaction of the medium is uniform over the membrane, proportional
to its average displacement. This limiting case has already been dis
cussed in connection with the kettledrum behavior. Here we wish
to discuss the driven motion.
The case of greatest practical interest is that of the circular dia
phragm, which is related to the problem of the condenser microphone.
The constructional details of the microphone will be discussed some
what later. All that is necessary to know here is that the driving
force due to an incident sound wave is Fe~ i01t per unit area, approxi
mately uniform over the membrane, and that the reaction force per
unit area of the medium, on both sides of the membrane, is propor
tional to the average displacement of the membrane, the proportionality
factor being —iuz, where z is the effective specific acoustic impedance
of the medium (counting both sides of the membrane).
The real part of z, R, is composed in part of the radiation resistance
of the medium next to the outer part of the membrane. Formulas
for the dependence of this part of R on frequency and membrane size
are worked out in Chap. VII. The side facing the inside of the
microphone case may also have a resistive part in its reaction, par
ticularly if the case is pierced with small holes to equalize the pres
sure inside and out. Motion of the air through these holes produces
viscous friction. The reactive part of the impedance z due to the
outer air is masslike (i.e., X is positive), whereas the reaction of the
air inside the casing is usually stiffness controlled, as was the case
with the kettledrum.
V.20] FORCED MOTION. THE CONDENSER MICROPHONE 201
The equation of motion of the membrane is therefore
d 2 7}
dt 2
= TV 2 v + iazv + Fer* 1
(20.6)
where
' = («*) I dv i nTdr
By analogy with Eq. (10.17) we set down the following expression
'4 / {t)'{t)]'~ ^
for the steadystate motion of the membrane. This expression goes
to zero at the edge of the diaphragm (r = a). The average displace
ment rj can be found by using the integral formulas and the recursion
formulas given in Eq. (19.5) :
= AJ
(?)
Setting these expressions in Eq. (20.6) serves to determine the value
of the constant A
A =
(Fa 2 /T/JL 2 )
Join) + Q/itiJM
(x = (ua/c) = ir/3oi(v/i'oi) )
r = (za/ac) = (z/W^oi) > (20.8)
= 6 ix= (a/<rc)(R  iX))
The amplitude A is therefore proportional to the driving force F
and inversely proportional to the tension T. The dependence on
frequency is through the parameter /*. When the driving frequency
is equal to one of the resonance frequencies v 0n , of Eq. (19.6), the
quantity Jo(m) is zero, and if the air impedance parameter £* is small
compared to n, A will have its largest values at these frequencies.
In calculating the response of a condenser microphone, we must
compute the average amplitude of motion of the diaphragm. From
the results we have obtained, this is
(Fa 2 /Tti 2 )J»(n)e i0,t _ Fa 2
Join) + {$/iv)JM T
i/(ju)e i(a " Q)
(20.9)
where
and
H(n) =
_ JM J
M 2 I
+ "
©■*>r
tan £2 =
6JM
nJo(») — %JM
202 MEMBRANES AND PLATES [V.20
Figure 44 shows curves of the average amplitude function H and the
average phase lag £2 for x = 0, for different values of 0, plotted as"
functions of the frequency parameter n = (<aa/c) — irfio n (v/von)
These curves are interesting and important, for they are typical
of average response curves for membranes of any shape. At low
frequencies (m < 1) the average response is fairly independent of fre
quency. Using Eqs. (19.4) and neglecting all but the first power of
n, we have
* a * F er*" [a < (c/a)]
~ i ° 3Zm / v , x > (20.10)
This result is similar to that for a simple driven oscillator. The
effective driving force is the area of the diaphragm times the force
per unit area. The mechanical impedance is the area of the dia
phragm times the total specific acoustic impedance z of the medium
on both sides of the diaphragm, plus the equivalent mechanical
impedance of the diaphragm itself for low frequencies. This latter
corresponds to an effective mass of fourthirds of the total mass of the
membrane and an effective stiffness constant of 8t times T, the mem
brane tension in dynes per centimeter. The factors (i) and 8w come
in because all the membrane does not vibrate with the same amplitude
of motion, as does a mass on the end of a spring.
Formula (20.10) breaks down when the frequency comes near the
first resonance, and the exact formula (20.9) must be used. This
first resonance comes at n = 2.405, or v = v i [see Eq. (19.7)], and
the average amplitude is large unless R is large. As the frequency
is increased still further, the average amplitude decreases rapidly,
becoming zero at \x — 5.136, or c= vzi, no matter what value x and
6 have.
Just above the first resonance v i the membrane vibrates nearly
out of phase with the driving force (if 6 is small) ; and as the frequency
is increased still further a circular nodal line appears at the outer
edge and shrinks in toward the center, the part of the membrane
inside the node remaining out of phase and the part outside the node
being nearly in phase with the force. As the driving frequency
increases and the nodal circle shrinks, the motion of the outer part
cancels more and more of the motion of the inner part in the average
displacement, until at n = 5.136 the two parts completely cancel
each other out, and the average displacement is zero. At this point,
V.20] FORCED MOTION. THE CONDENSER MICROPHONE
203
«/ 2 (m) is zero and therefore the effect of the reaction of the medium
also goes to zero (as long as this reaction depends only on the average
displacement fj). For this reason the frequency of zero average dis
placement is not affected by the reaction of the air, represented by
the quantities x and 0.
180"
/I 90°
H 0.2
Fig. 44. — Response curves and curves of angle of lag of displacement for a damped
condenser microphone as functions of driving frequency, for different values of the
air resistance parameter 0. The quantity fj. is 2.405 times the ratio of the driving
frequency to the fundamental frequency of the diaphragm.
A little beyond this antiresonance frequency, when n = 5.520, is the
second resonance, and so on. Between each of the resonance frequen
cies v 0n and the next successive one is an antiresonance frequency v 2n .
The Condenser Microphone. — The condenser microphone corre
sponds approximately to the case just discussed. The diaphragm
204
MEMBRANES AND PLATES
[V.20
*l
P
J
N i,
B
Fig 45. — Simplified
cross section of a con
denser microphone.
of the microphone is metallic, and therefore it has stiffness; but the
diaphragm is often so thin and is under such great tension that the
effects of stiffness can be neglected (though we can take the effect
into account if necessary, as we shall see in the next section).
Figure 45 shows a simplified crosssectional view of a condenser
microphone. D is the diaphragm, which is usually thin enough and
under large enough tension to be considered as a membrane. This is
stretched over the end of a vessel B, which usually has vent holes
represented by H, so that it is not airtight.
Behind the diaphragm a short distance is a
plate P, sometimes pierced with holes so that
the air can penetrate easily, and insulated elec
trically from the diaphragm. This forms the
other plate of the condenser.
The driving force is the excess pressure, on
the outside of D, caused by a sound wave in
passing. Unless the wavelength of the sound
wave is smaller than the size of the microphone
(which happens only for frequencies higher than
5,000 cps, for most microphones), the excess
pressure can be considered to be uniform over the surface of the
diaphragm. The case for small wavelengths will be taken up in the
chapter on the scattering of sound.
We can also assume that the pressure varies sinusoidally with
time. If the sound is a combination of waves of many different
frequencies, the resulting motion of the diaphragm will be a com
bination of the motions due to each wave separately.
If the intensity of a simple harmonic wave of frequency v is T ergs
per sec per sq cm, then the excess pressure on the outside of the
diaphragm will be F == F e 2iriH , F = 9.2 y/T dynes per sq cm
(for air at standard conditions). This expression is correct only if
the presence of the microphone does not alter the motion of the sound
wave appreciably. In Chap. VII this question will be discussed in
detail. We have mentioned earlier that the intensity due to average
conversation from a person 3 ft away is about 1 erg per sec per sq cm.
Electrical Connections. — The output voltage of the condenser
microphone is not proportional to the amplitude A but is proportional
to the average displacement of the diaphragm. If the equilibrium
distance between D and P is A, then the equilibrium capacity of the
microphone is C = (wa 2 /4irA) in electrostatic units. If the displace
V.20] FORCED MOTION. THE CONDENSER MICROPHONE 205
ment from equilibrium of the diaphragm is ri(r,t), then the inverse
capacity of the microphone is
1 = i (A  „)„ = i(l  X J v*)
Using the expression for rj given above, we have
Figure 46 shows the usual circuit for connecting the microphone
to the amplifier. Resistance R is made
large enough so that the condenser can
not charge and discharge rapidly enough
to follow the alternations of capacitance
caused by the sound waves. The aver
age charge on the plates is E Q C , and the
average potential difference between the
plates is E . At any instant, however,
the potential difference will be the charge
E C (which will not change if R is large
enough) divided by the value of the microphone capacitance at that
instant:
To
amplifier
lililmi
Fig. 46. — Electrical circuit for the
condenser microphone.
tfi = (^°) = E
ii<
(¥)
The difference between E and this quantity is the emf impressed
on the amplifier:
E = B,  *,  (M\ j,(?22\ e —
The magnitude of the output voltage is therefore
E = 1.348
Eq
oA(i/ i) 2
Vt#0)
(20.12)
where E is the voltage impressed across the microphone, A the dis
tance between the diaphragm and backing plate, a the density of the
membrane in grams per square centimeter, v i its fundamental fre
quency of oscillation, and T the intensity of the sound in ergs per
second per square centimeter. The quantity H is the one given in
Eqs. (20.9), and the angle of lag of the output voltage behind the
206 MEMBRANES AND PLATES [V.20
sound pressure is the angle fl defined in the same equations. There
fore the curves of Fig. 44 are also curves of sensitivity and phase lag
of the microphone signal, provided that its construction corresponds
reasonably well to the assumptions as to air reaction we have made
in deriving Eqs. (20.9).
Transient Response of Microphone. — According to the discussion
following Eqs. (6.16) and (6.17), we can find the response of the
uniformly loaded diaphragm to a unit impulsive force spread uni
formly over the diaphragm, by integrating (l/27rF) times the 17 given in
Eqs. (20.9) over to from — ■ °o to + °° :
 (A  _^L f °° J"2(m) e^ /o)
This integral is extremely difficult to compute exactly, but it can be
evaluated approximately as long as f  is considerably smaller than
n(\z\ < < w, i.e., the load per unit area due to the medium on both
sides of the membrane is small compared with the mass reactance
of a unit area of the membrane itself). In this case the zeros of the
denominator of the integrand are very close to /z = ±(T/3 0n ). To
the first approximation, near \x = irl3on, «/o(m) — — (m — Trfion)Ji(jfio n ),
which can be obtained from Eqs. (19.5) and from the Taylor's expan
sion, J (n) = Jo(irp 0n ) + (m — *Po»)[dJo(n)/dfi] r p. Also, using Eqs.
(19.5), we can show that in this region J 2 (m) = (2/m)«A(m) — «/o(m) —
[/i  ir3on + (2/irj8on)]J"l(ir/3on) ^ (2/lT8o») Jl^Pon) for JU ~ Tl^On
Therefore .the contour integral breaks up into a sum of contour
integrals, each around the points m = ±7r/3o«. The one about \i =
+7T/3o« is approximately ( — 2iri) times the residue of
ca ( 1 Y M  T0on + (2/tt/3 *) e  (icilt/a)
( 1 V M  x,
%rT\Tp Q J n  Tpon + [2tf/0r/3 «) s
at its pole at n = (tj8o»)  [2#/(t0o») 2 ]. The contours around the
points fx = — 7t/3 „ can be similarly approximated, and the final result,
for the average displacement of the diaphragm in response to a uni
form impulsive force of unit impulse per unit area, is
rj & (t) ~ ^ ^ Q) 3 e^w^> sin6^) (20.13)
where R n is the resistive part of the medium's impedance per unit
area (z = R — iX) at the frequency v ». The response to a general
impulsive force F(«) per unit area is, according to Eq. (6.17),
V.20] FORCED MOTION. THE CONDENSER MICROPHONE 207
f, = §[ ^ F(T)rj s (t  t) dr
The corresponding series for the actual displacement t\ of the dia
phragm is obtained by multiplying the nth term in the series for m by
[7r/W2/i(ir/3on)]/o0r/3o«r/a). Figure 47 shows a time sequence of the
shapes of the diaphragm after being shock excited.
Unit Impulse at "t=0,spreaol uniformly over Membrane
I I I
t=Q • * * *
Fig. 47. — Crosssectional view of motion of a circular membrane with damping
proportional to average velocity (uniform damping), when subjected to a uniform
impulsive force. Compare with Fig. 38.
The motion corresponding to this formula is quite different from
the related motion of a string; for in this case the higher the overtone,
the less the damping. The damping factor here for the nth mode is
2R n /<T(irPo n ) 2 , which rapidly decreases as /3 « gets larger. The result is
due to the uniform nature of the coupling between the diaphragm and
the medium and is true only if the wave velocity in the medium is large
enough so that the reaction depends solely on the average displace
ment of the diaphragm, being uniform over the diaphragm. For
then, since the higher modes have a small average displacement com
208 MEMBRANES AND PLATES [V.21
pared with their maximum displacement, not much of their energy
of vibration gets carried away by the medium. By the same token,
the coupling between the higher modes and the uniform impulsive
force is small, so that the amplitude of the higher overtones is quite
small (it varies inversely as @\ n , as a matter of fact). Thus the series
converges rapidly.
The equation predicts that a kettledrum diaphragm, for instance,
when set into motion by a sound pulse, will have motion as a whole
(lowest mode) which is rapidly damped, but that smallamplitude
highfrequency short wavelength ripples in the diaphragm will persist
for a long time after the pulse. This is actually the way such a dia
phragm does behave, so our assumptions must not be far from the
correct ones for the drum. In the case of the condenser microphone
it is not quite such a good approximation, for the reaction of the air
inside the casing is not completely uniform, but is partly localized;
and the effect of localized damping is to damp out the higher modes
more rapidly than is the case when no localized reaction occurs.
The intermediate case, where the wave velocity in the membrane
is neither much larger nor much smaller than the speed of sound in
the medium, is too difficult to handle this early in the volume. We
shall return to it in Chap. VII.
21. THE VIBRATION OF PLATES
The study of the vibrations of plates bears the same relation to
the study of the membrane as the study of the vibrations of bars does
to the study of the flexible string. The effect of stiffness in both
cases increases the frequencies of the higher overtones more than it
does those of the lower overtones and so makes the fundamental fre
quency very much lower than all the overtones. However, the
motions of a plate are very much more complicated than those of a
bar, so much more complicated that we shall have to be satisfied with
the study of one example, that of the circular plate, clamped at its
edge and under no tension. The diaphragm of an ordinary telephone
receiver is a plate of this type, so the study will have some practical
applications.
The Equation of Motion. — The increased complications encoun
tered in the study of plates come partly from the increased complexity
of wave motions in two dimensions over those of one, but also come
about owing to the complex sort of stresses that are set up when a plate
is bent. The bending of a plate compresses the material on the inside
of the bend and stretches it on the outside. But when a material is
V.21] THE VIBRATION OF PLATES 209
compressed it tries to spread out in a direction perpendicular to the
compressional force, so that when a plate is bent downward in one
direction there will be a tendency for it to curl up in a direction at
right angles to the bend. The ratio of the sidewise spreading to the
compression is called Poisson's ratio and will be labeled by the letter s.
It has a value about equal to 0.3 for most materials. This complica
tion was not considered when we studied the vibration of bars, for we
tacitly assumed that the bar was thin enough compared with its
length so that the effects of a sidewise curl would be negligible.
The derivation of the wave equation for the plate involves more
discussion than is worth while here (it is given in books on theory
of elasticity). The equation is
y4„ + 3p(l ~ s 2 ) gg = (211)
v ^ Qh 2 dt 2 K }
where p is the density of the material, s its Poisson's ratio, Q its modulus
of elasticity, and h the halfthickness of the plate. Values of p and
Q are given for different materials in Table 3 of Chap. IV.
We shall not spend any time discussing the general behavior of
waves on a plate of infinite extent but shall simply remark that, like
the bar, the plate is a dispersive medium; waves of different wave
length travel with different velocities.
Simple Harmonic Vibrations. — To study the simple harmonic
motion of a plate, we insert the exponential dependence on time and
separate the factors depending on the individual coordinates. The
differential operator V 4 is difficult to separate in most coordinate
systems, but for polar coordinates the results turn out to be sufficiently
simple to justify our analyzing them in detail. Here, if we set r\ —
Y{r,(p)e~ 2Tivt , where Y's dependence on <p is by the factor cos or
sin (m<p), then the differential equation for Y can be written as
( V2_ 7 2 )(V 2 + t2 ) F = o, T 4 = 12TV ffi, " S2) '(21.2)
Therefore, Y can be a solution either of (V 2 F + y 2 F) = or of
(V 2 F  t 2 F) = 0.
Since V 2 and Y are to be expressed in polar coordinates, the solution
COS
of the first equation which is finite at r = is Y = . (nap) J m (yr),
^ sin
where m is an integer. This is the usual solution for the membrane,
with 7 instead of (2tv/c). The solution of the second equation is
obtained from the first by changing y into iy and necessitates a little
210 MEMBRANES AND PLATES [V.21
discussion of Bessel functions of imaginary values of the independent
variable. Let us call these hyperbolic Bessel functions and define
them by the equation I m (z) = r^J^iz). The properties of the
function I m (z) can be obtained from Eqs. (19.4) and (19.5) for J m (z).
The more useful formulas are
I m i(z)  I m+1 (z) = — I m (z), ^ I m (z) = i[Imi(z) + I m +i(z)]
fIo(z)z dz = zlriz), J7i(z) dz = I (z) (21.3)
Values of the functions 7o, Ji, and Z 2 are given in Table IV at the
back of the book.
The Normal Modes. — Possible solutions for the simple harmonic
oscillations of a plate are therefore given by the expressions
F ( r >*> = 1 S W [AJ m (jr) + BI m (yr)]
Sill
The boundary conditions corresponding to a circular plate of radius a,
clamped at its edges, are that Y(a,<p) = and (dY/dr) r=a = 0. The
first condition is satisfied by making
B = A
r jm(7«) i
_/ OT (7a)J
and the second condition is satisfied by requiring that y have those
values that make
d d
Im(ya) T J m (yr) — J m (ya) = I m (yr) =0 at r = a (21.4)
This equation fixes the allowed values of the frequency, for y depends
on v. We shall label the solutions of Eq. (21.4) by the symbols
y mn , where y mn = {ir/a)8 mn , and where
l8oi =■ 1.015, O2 = 2.007, O3 = 3.000)
0n = 1.468, 0i 2 = 2.483, 8 13 = 3.490 > (21.5)
21 = 1.879, 022 = 2.992, 23 = 4.000 )
m
Pmn, *■ n ~J~ ~~jr
The allowed values of the frequency are therefore
^ (8 Y
3p(1  s 2 )
irk /
Vmn 2a 2 '\ :
^ • 9342 &)^£? : S [ (2L6)
rn = 2.091^01, "21 = 3.426^01, "02 = 3.909P01
^12 = 5.983^01, etc.
V.21]
THE VIBRATION OF PLATES
211
The allowed frequencies are spread apart much farther than those
for the membrane, given in Eq. (19.7). The overtones are not
harmonic.
The characteristic functions are
L m \ a ) ImiirPmn) ™ \ « / J
fmn = COS(m
and a similar expression for ^ «n (for m > 0) where sin(m«?) is used
instead of cos (m<p). Some of these functions are shown in Fig. 48.
Fig. 48. — Shapes of a few of the normal modes of vibration of a circular plate clamped
at its edge.
The free vibrations of the plate corresponding to arbitrary initial
conditions can be expressed in terms of a series of these characteristic
functions.
Forced Motion. — As an example of the method of dealing with
forced motion, let us repeat the calculations for a fully damped con
denser microphone with a plate diaphragm instead of a membrane.
The equation of motion to be used instead of (20.6) is
dt 2
4x
F B
7 ^^ + ^° e " 2X "
R
//GO
r dr d<p (21.7)
■jra 2 hp
where t 4 = [12t 2 v 2 p(1 — s 2 )/Qh 2 ]. We now choose the function r?
to be
A p — 2irivt
V = 77 — r [Ii(ya)J (yr) + Ji(ya)I (yr)  Ii(ya)J (ya)
 Ji(ya)I Q (ya)]
which has zero value and slope at r = a and which has the following
average value:
212 MEMBRANES AND PLATES [V.21
TO" J J
vrdrdtp = 7 , . [Ii(ya)J 2 (ya)  J 1 {ya)l2{ya)]e Mvt
In order to compare the results of these computations with the
ones given in the previous section for the membrane diaphragm, we
shall use the same variables n = 2A05(v/v 01 ) &nd£ = 0.3828(Z/2hpi>oi)
that were used in Eq. (20.8); the only difference being that for v i,
instead of the value given in (19.7), we use the value given in Eq.
(21.6) for the fundamental frequency of the plate. In terms of these
variables, (aV) = 4.23 In or ay = 2.057 Vi*.
Substituting the expression for n into Eq. (21.7), and changing
to the new variables, we obtain an expression similar to Eq. (20.8)
(F/a, 2 h P ) 1
A = 7 1 (2.06 Vm) l "(m) + (TAm)^0*) (2L8)
L (m) = /i(2.06 VmVo(2.06 Vm) + /o(2.06 V^)J 1 (2.0Q Vm)
L 2 ( M ) = /i(2.06 Vm)«/ 2 (2.06 Vm>  / 2 (2.06 Vm)«/i(2.06 Vm)
The calculations for a condenser microphone with plate diaphragm
go through in a manner similar to that for the membrane, and the
expression for the output voltage is
E = 1.348 , , E \ 2A VTHMew™ (21.9)
flp{voi) A
r <0 .) = ^ {[lm Ilm] 2 + ($l>m]
tanfli =
\jiLo(ji)  xL 2 (n)]
Curves for Ht as function of n for different values of 6 are given in
Fig. 49. These may be compared directly with the curves in Fig. 44,
for the membrane diaphragm. The units have been so chosen that
when the two diaphragms have the same mass per unit area and the
same fundamental frequency then both horizontal and vertical scales
are the same. We see that below the first resonance frequency the
response curves are quite similar and that the chief difference above
this frequency is that the point of zero response is higher on the fre
quency scale for the plate than for the membrane — so much higher
that the second resonance peak for the plate is outside the range
plotted.
It is evident that the plate diaphragm has the advantage of having
a longer range of frequencies below the point of zero response. This
V.21]
THE VIBRATION OF PLATES
213
advantage, however, is almost neutralized by the disadvantage that
the average response of the plate is somewhat smaller than that of
the membrane, owing to the fact that the displacement must be small
near its edge. The chief disadvantage of the plate, however, is the
practical one that it is difficult to obtain and properly mount a plate
thin enough to give a small mass per unit area and yet stiff enough
to have a highfrequency fundamental (although plate diaphragms
have been successfully used on miniature condenser microphones).
0.3 r
Fig. 49. — Response curves for the damped condenser microphone with plate dia
phragm, plotted as a function of the frequency parameter fi for several different values
of the damping constant 0.
We shall not go into the details of the solution of the motions of a
plate under tension, the intermediate case between the membrane
and the plate without tension. It is obvious that the response curve
for a damped condenser microphone with such a diaphragm will be
intermediate between the curves of Figs. 44 and 49.
Problems
1. A membrane is made of material of density 0.1 g per sq cm and is under a
tension of 100,000 dynes per cm. It is wished to have the membrane respond best
to sound of frequency 250 cps. If the membrane is square, what will be the
length of one side? What will be the frequencies of the two lowest overtones?
2. A square membrane, 20 cm on a side, of mass 1 g per sq cm, is under a
tension of 10 8 dynes per cm. Its motion is opposed by a frictional force of 42(dq/dt)
dynes per sq cm. Find the modulus of decay of the oscillations. What are the
first four "frequencies" of the damped motion?
214 MEMBRANES AND PLATES [V.21
3. A square membrane, b cm on a side, of density <r and under a tension T, is
loaded at its center with a mass of M g. Show that the allowed frequencies are
approximately
(, T[m 2 , nn [, AM (b b\l)
as long as (AM/ab 2 ) is small. What will be the expression for the frequency if the
membrane is rectangular but not square?
4. Show that the energy of vibration of a rectangular membrane of sides
a and 6 is
^«jriT[(2) , +(g) , ^®>*
and that when the motion is given in terms of the series
«
t] = 2j A mn sin I J sin I =— J cos {2irv mn t + * mn )
m,n \ a / \ /
the series for the energy is
W = — (<rab) ^
2 ni,n
mn mn
5. A square membrane 20 cm on a side, with a = 1 and T = 10 6 , is started
from rest at t = with an initial shape r, = lCr b a;(20  x)y(20  y). What are
the energies of vibration corresponding to the fundamental and the lowest three
overtones, and what is the total energy of vibration of the membrane?
6. A rectangular membrane is pushed aside at the point (x ,yo) by a. steady
force F and is then suddenly released at t = 0. What is the expression for the
subsequent motion of the membrane? Neglect the reaction of the air.
7. A circular membrane of radius 10 cm, with a = 1, T = 36, is struck so
that a very small area A»S around its midpoint has an initial velocity (1/A/S) cm
per sec, all the rest of the membrane being at rest at t = 0. Plot the shape of a
cross section of the membrane (77 as a function of r) for t = 0, 1, 2, 3, 4, 5, 6 sec.
8. The tensile strength of aluminum is 2.5 X 10 9 dynes per sq cm, and its
density is 2.7 g per cc. What is the highest value of fundamental frequency
that can be attained with an aluminum membrane stretched over a circular
frame 3 cm in radius? If the aluminum is 0.005 cm thick, what will be the maxi
mum tension attainable?
9. A square membrane 4 cm on a side with a = 0.01 and T = 10 6 , which is
undamped, is driven by a uniform force of l,000e 2iriI " dynes per sq cm. Plot the
amplitude of vibration of the midpoint as a function of frequency from v =
to v = 2,000.
10. Suppose that the membrane of Prob. 9 is acted on by a damping force of
A2(dT)/dt) dynes per sq cm. Plot the amplitude of motion of the midpoint as a
function of v from v = to v = 2,000.
11. Find the effect on the allowed frequencies of a circular membrane due to a
small extra mass of M g attached to its midpoint.
V.21] THE VIBRATION OF PLATES 215
12. An undamped circular membrane of radius 2 cm, a = 0.1, and T — 631,700
is driven by a uniform force of 40,000e~ 2iriw dynes per sq cm. Plot the amplitude of
motion of the midpoint of the membrane as a function of v from v = to v = 1,200.
What are the lowest three frequencies of free vibration of the membrane? What is
the shape of the membrane (rj against r) when v = 600?
13. The circular membrane of a kettledrum has a radius of 50 cm, <r = 0.1,
and T = 10 8 . What is its fundamental frequency without the backing vessel?
The backing vessel raises the fundamental frequency to 1.45 times this value.
What is the volume of the vessel? What are the lowest five frequencies of the
membrane vessel system?
14. A condenser microphone diaphragm has a radius a = 5 cm, a = 0.005,
and T = 3 X 10 6 dynes per cm; A = 0.005 and E = 100 volts. What is the
lowest frequency of the undamped diaphragm? Suppose that the impedance of
the air per square centimeter of diaphragm is resistive, z = R = 100, the reaction
being uniform and proportional to y. Plot the response curve H for the micro
phone for the frequency range v = to v = 10,000.
16. A condenser microphone diaphragm is a membrane of radius a with local
ized air reaction, the resistive term being R(drj/dt) per sq cm, and the reactive
term being Krj (with no additional effective mass). Show that the effective
impedance to insert in Eqs. (20.5) is
= R
(£)[(*?*)+*]
This diaphragm is acted on by a distributed force 9.2 \/ye iat dynes per sq cm.
Show that the steadystate displacement of the diaphragm is
5.86 Vxr"' 2)
Jo(Tr@o n r/a)
_ fionJi (irl3on) ( — iuzon)
16. Compute the output voltage of the microphone of Prob. 15 as a function of
E , A, a, <r, T, R, K and driving frequency.
17. Calculate the transient response to a general impulsive force for the mem
brane of Eq. (20.5).
18. A membrane with uniform air reaction is acted on by a uniform impulsive
force 8(0 per sq cm, so that its motion is given by Eq. (20.13). Values of the
constants are (c/a) = 1, (4ca/T) = 1, (R n /<r) = 1. Plot value of ij(t) as function
of time t from t = to t = 10.
19. The diaphragm of Eq. (20.13) is acted on by a uniform force
(0 (* < 0)
F(t) = < A sin (ut) [3 < t < (»/«)]
(0 [t > (wtt/co)]
Compute the series for the average displacement of the membrane.
20. A plate diaphragm of steel Q = 19 X 10 11 , p = 7.6, s = 0.3 has dimen
sions a = 3 and h = 0.05. What is its fundamental frequency? It is driven by a
pressure of lOOe 2 *"*''' dynes per sq cm uniformly distributed over the diaphragm.
The mounting is so designed that the reaction due to the medium is negligible over
the useful range of v. Plot the displacement of the midpoint of the diaphragm
as a function of v from v = to v = 5,000.
216 MEMBRANES AND PLATES [V.21
21. Suppose the diaphragm of Prob. 20 is to be driven by a force of 100e _2iri,,t
dynes concentrated at its midpoint. Plot the amplitude of motion of the mid
point as a function of frequency from v = to v = 5000.
22. Plot the shape of the diaphragm of Prob. 21 (tj as a function of r) for v =
1,000, 3,000, 5,000.
23. A circular plate diaphragm is loaded at its midpoint by a mass M . Com
pute the approximate expressions for the allowed frequencies of the loaded plate,
valid when M is small.
24. Compute the transient response of a plate diaphragm to a uniform impul
sive force, and obtain a formula corresponding to Eq. (20.13).
CHAPTER VI
PLANE WAVES OF SOUND
22. THE EQUATION OF MOTION
We now come to the study of wave motion in air, the most impor
tant type of wave motion studied in the science of acoustics. Sound
waves differ from the waves that we have discussed heretofore in
several important respects. They are waves in three dimensions
and as such can be more complicated in behavior than waves in two
dimensions or in one. Sound waves also differ from waves on a string
or on a membrane by being longitudinal waves. So far, we have been
studying transverse waves, where the material transmitting the wave
moves in a direction perpendicular to the direction of propagation of
the wave. Each part of the string, for instance, moves in a direction
at right angles to the equilibrium shape of the string, whereas the
wave travels along the string. The molecules of air, however, move
in the direction of propagation of the wave, so there are no alternate
crests and troughs, as with waves on the surface of water, but alternate
compressions and rarefactions. The restoring force, responsible for
keeping the wave going, is simply the opposition that the gas exhibits
against being compressed.
Since there are so many points of difference between the waves
discussed earlier and the more complicated forms of sound waves,
it is well not to introduce all the complications at once. Accordingly,
we shall first study the motion of plane waves of sound, waves having
the same direction of propagation everywhere in space, whose "crests"
are in planes perpendicular to the direction of propagation. They
correspond to the parallel waves on a membrane. Waves traveling
along the inside of tubes of uniform cross section will usually be plane
waves. Waves that have traveled unimpeded a long distance from
their source will be, very nearly, plane waves.
Waves along a Tube. — Suppose that we consider the air in a tube
of uniform cross section of area S. When everything is at equilibrium
let us color red all the molecules in the plane, perpendicular to the
axis of the tube, at a distance xi along the tube from some origin;
217
218 PLANE WAVES OF SOUND [VI.22
color blue the molecules in the plane at z 2 ; and so on. When a sound
wave passes through the tube, these planes will be displaced from
their equilibrium positions back and forth along the tube. At some
instant the red plane will be at x x + £Oi), the blue plane at x 2 + £(2:2),
and so on. Each molecule of the gas originally a distance x from the
reference plane will be displaced in the xdirection a distance £. This
displacement depends on the time t and also on the particular mole
cule that we are watching {i.e., it depends on x).
Of course, this is a crude picture of what actually happens; the
real molecules are bouncing back and forth because of temperature
agitation, even when there is no sound. The quantity £(x,t) actually
measures the average displacement, due to the sound wave, of those
molecules whose average position was originally x. The average
velocity of this plane in the zdirection, (d£/dt), is called the particle
velocity u.
We must now find an equation giving the dependence of £ on x
and t. This equation will be obtained by combining three equations,
one of them Newton's equation of motion and the other two repre
senting two simple properties of a gas. One of these properties is
just a restatement of the law of the conservation of matter: The
amount of gas between the plane of red molecules and the plane of
blue molecules will remain the same as the planes move. The other
property is that relating the change in density of a perfect gas with
its change in pressure in the case when the gas is so rapidly compressed
that it cannot unload its gain in heat to the surrounding gas.
To put these properties into mathematical form we must make a
few definitions. We shall denote the equilibrium density of the gas
by p and its equilibrium pressure by P  The actual density at the
point x and time t will be denoted by p(x,t), and the relative change in
density will be denoted by 8(x,t), where
P (x,t) = P (l + 5), 8 = i p(x,t)  1 (22.1)
The difference between the actual pressure and the equilibrium pres
sure will be denoted by p{x,t). It is this excess pressure p that pro
duces the motion of a microphone diaphragm.
The Equation of Continuity. — Returning to the colored planes of
molecules, we see that unless the gas moves so much that turbulence
is set up the colored planes will remain planes as the gas moves.
Since the planes are made up of molecules of the gas, it is clear that,
unless there is turbulence, the gas ahead of one plane will always be
VI.22]
THE EQUATION OF MOTION
219
,>j(o) >(x,)/g(x 2 ) /j(x)
dx*
ahead of that plane, and the gas between two planes will always be
between those planes. Suppose that we consider the gas between two
planes as shown in the second sketch
of Fig. 50, which at equilibrium are
at the distances x and x + dx. At
equilibrium, the mass of the gas be
tween the planes will be its density
p times the volume cut off by the
planes and the surface of the tube,
S dx. By the argument above, the
mass of gas between the planes, when
they are displaced by the sound wave,
will still be pS dx. But when the
planes are displaced, the volume be
tween the planes may be altered, for
the displacement of one plane is (x)
and that of the other is £(x + dx) =
£(x) + dx(d%/dx). The volume en
closed is therefore S[dx + £(x + dx)
 £(*)] = Sdx + Sdx(d£/dx). The density of the gas between the
planes must be altered so that the total mass pS dx can remain unal
tered: p(x,t)[S dx + S dx(d£/dx)] = P S dx. With reference to Eq.
4(x)
(**>
N £(x+dx)
x x+dx
Fig. 50. — Longitudinal displace
ment in a sound wave. Gas particles
originally in the plane at x are dis
placed by an amount l(x).
(22.1), this gives pS dx = P (l + 8)Sdx
+ 2}
When the change
in density and displacement are small (and they are small in all but
the loudest sounds), we can neglect the product of the two small
quantities 8 and (d£/dx), so that
Therefore,
( 1 + g
(l + S)(l+^)^l + 5 +
1=1+5+
dx
or 5 = —
dx
dx
(22.2)
This is a form of the socalled equation of continuity. It states that
when the gas to the right of a given point is displaced more (to the
right) than the gas to the left of the point is displaced (i.e., if the two
colored planes are pushed apart by the displacement) then the density
of the gas is diminished at the point. We note for future reference
that if the crosssectional area of the tube S depends, on x, the correct
equation will be
220 PLANE WAVES OF SOUND [VI.22
* = s!< s » ■ (22  3)
Compressibility of the Gas.— The second property of the gas which
is used in deriving the equation of wave motion depends on some
thermodynamic properties of gases. All perfect gases obey the well
known relationship between the total pressure P, the volume occupied
V, and the temperature T which is PV = RT. Differentiating both
sides gives R dT = P dV + V dP, and on dividing by the undiffer
entiated equation we obtain
dT dP ,dV , 99 ^
T = ~P + T C
If the volume occupied by the gas is kept constant, and the pressure is
increased by heating, dV is zero and dP = (P/T) dT. The increase of
the heat energy of the gas during this process will be dQ = (dQ/dP) dP
= (dQ/dP)(P/T) dT, so that the rate of increase of heat energy
due to rise of temperature at constant volume is (dQ/dT)rconst =
(dQ/dP)(P/T). This rate of increase is called the specific heat of the
gas at constant volume and is denoted by CV. Therefore,
dP"P Cv
The temperature of the gas can also be raised while holding its
pressure constant, by increasing its volume by just the right amount
during the heating. In this case dP = 0, dV = (V/T) dT, and the
rate of increase of heating energy due to rise of temperature at con
stant pressure is (dQ/dT) Pcoast = (dQ/dV)(V/T). This rate of increase
is called the specific heat at constant pressure and is denoted by C P .
Therefore,
dV'V Cp
If, now, we allow the gas to change both volume and pressure, the
change in heat energy will be the rate of change of Q with respect to
V times dV, plus the rate of change with P times dP:
from the equations for CV and C P .
If the gas is expanded slowly enough to enable the temperature to
remain constant, then dT will be zero in (22.4), and the relation between
VI.22] THE EQUATION OF MOTION 221
the change in pressure and the change in volume is (dP/P) = — (dV/ V) .
However, the expansions and contractions in the sound wave are
entirely too rapid for the temperature of the gas to remain constant.
The alternations of pressure and density are so rapid that it is much
more nearly correct to say that no heat energy has time to flow away
from the compressed part of the gas before this part is no longer com
pressed. In this case, where the gas temperature changes but its
heat energy does not change, the compression is said to be adiabatic.
To find the relation between the change in pressure and the change in
volume we must use the equation for dQ, with dQ set equal to zero:
C P (dV/V)  =C v (dP/P). This is the equation that will be used to
find the elasticity of the gas.
To return to the study of the gas in the tube contained between
the planes originally at af and at x + dx, the volume of the gas is S dx,
and the change in volume is S dx(d£/dx) ; the pressure is P , and the
change in pressure is p. The preceding equation relating change of
pressure to change in volume in adiabatic compression takes on the
form
or, letting (C P /C V ) = y c ,
V = ~^ p ofj = ToPoS (22.5)
from Eq. (22.2). The constant y c , the ratio of the specific heat at
constant pressure to that at constant volume, has the value 1.40 for
air at normal conditions.
The Wave Equation. — From the requirement of conservation of
matter we have obtained Eq. (22.2) relating the change in density to
the rate of change of displacement; and from the thermodynamic gas
laws we' have obtained Eq. (22.5), relating the change in pressure to
the change in density. Each of the functions p, £, and 8 is a function
of x and t. By using one more equation, that relating the acceleration
of the gas between the planes to the net force on it, we shall have
enough equations to solve for all three quantities.
The force on the gas between the planes, due to the gas to the left,
is [P + p(x)]S; and that due to the gas to the right of the planes is
[Po + p(x + dx)]S = [P + p(x) + (dp/dx)dx]S. The difference be
tween these forces —Sdx(dp/dx) is the net force acting on the gas
between the planes and must therefore be equal to the mass of the
gas P S dx times its acceleration (d 2 £/dt 2 ). This gives us the needed
third equation:
PLANE
WAVES OF SOUND
a 2 i
9 at 2
du
dp
dx
222 PLANE WAVES OF SOUND [VI.22
(22.6)
By combining Eqs. (22.2), (22.4), (22.5), and (22.6) we obtain
formulas for £, p, and 5 and the change in temperature AT, in a plane
sound wave:
d 2 £
dx 2
1 d 2 £ d 2 p _ 1 d 2 p d 2 8 _ 1 d 2 8
~c~ 2 W dx 2 ~c 2 dt 2 ' dx 2 ~ ~c 2 W
c
V
iW pc.fi
AT
0^^ 1 )M'
(22.7)
The displacement, density, and pressure all obey the equation for
propagation of waves with a velocity c. For the simple case of plane
sound waves of not too large amplitude, these waves will be propa
gated without change of shape and with a velocity c independent of
shape. The three waves are not independent, being related by some
of the equations above; once the wave of particle displacement is
known (for instance), the waves of density and pressure are deter
mined. Where the pressure is greatest in magnitude, there the rate
of change of displacement with x is largest; and where the pressure
is zero, there the displacement is greatest (or least). Where pressure
is greatest, there density is greatest, and temperature is greatest, and
so on.
It is a very interesting and important result that plane waves of
sound, longitudinal waves, obey the same wave equation as do the
transverse waves on a string. All the results that we have worked
out for waves on a string can be taken over for plane sound waves,
except that the meaning of some of the terms — displacement, shape
of the wave, etc. — must be changed somewhat.
The velocity of sound in air at standard conditions (20°C, 760
mm of mercury) and the related constants are
c = 34,400 cm/sec
pc = 42 g/cm 2 sec
p = 0.00121 g/cm 3 , Tc = 140 ) (22.8)
pC 2 = Tc p = 1.42 X 10 6 ergs/cm 3
P = 1.013 X 10 6 dynes/cm 2
V1.22J
THE EQUATION OF MOTION
223
The constant pc is called the characteristic acoustic resistance of the
air, for reasons to be given in the next section.
Energy in a Plane Wave. — We can now obtain an expression for
the energy involved in a sound wave. The kinetic energy of any ele
ment of volume of the gas is ip(d£/d£) 2
dx dy dz. The potential energy of the
element is the amount of work required
to compress the element from its equilib
rium volume dx dy dz to its new volume
[1 + (d£/dx)] dxdydz = (1 — 8) dxdydz.
Figure 51 shows a graph of the volume
as a function of the pressure. The
work done in compressing the gas jPdV
turns out to equal the area of the four
sided figure (V, V , b, a). When p is
small the side (a, b) can be considered
to be ■ a straight line, and the area is
equal to (P + %p)(V Q — V). If the wave whose energy we are mea
suring is a simple harmonic one or one made up of a number of simple
harmonic components, the average value of the term Po(V — V)
will be zero. Since we are interested only in average values of the
energy, this term can be omitted. By the use of Eqs. (22.7) the re
maining term ip(V Q — V) becomes $pc 2 (d%/dx) 2 dx dy dz. The aver
age energy of a volume of gas due to the passage of a plane wave is
therefore
Average energy = \p I I (^)+ c2 (^r) dxdydz ergs (22.9)
where u = (di/di). For a simple harmonic wave of frequency v =
(w/27r), this can be expressed in terms of p alone:
v v v—
Fig. 51. — Relation between
pressure and volume in a gas.
The work done compressing the
gas from volume Vo to volume V is
equal to the area enclosed by the
figure (Vo, a, b, Vo).
Average energy =
1
2 P c'
///[©'©■
+ p 2
dx dy dz
These equations correspond to Eq. (9.12) for the average energy of a
string. The average energy per cubic centimeter will be denoted by
W.
Intensity. — The rate at which the energy is being transmitted
along the wave, per square centimeter of wave front, is called the
intensity of the sound wave and will be denoted by T. This will equal
224 PLANE WAVES OF SOUND [VI.22
the excess pressure p on the square centimeter, multiplied by the
velocity of the gas particles :
v (!) = " pc * JJ Tt er ^ seG ^ cm < 22  10 >
The average value of this is the intensity T.
If the wave is a simple harmonic one, with frequency v, the expres
sions for pressure, energy, and intensity are fairly simple. For a wave
going to the right, having a maximum pressure P+,
p+ = P +e *(*«), £ + = A + e ik ^~ ct \ f^tj = EZ+e*'*^')
P+ = pc(^) } (22.11)
»c
m
T + = ZirWpcA*. = () = ipcET*
To obtain the values of W and T given here it is necessary first to take
the real parts of the expressions for p and £ before substituting in
formulas (22.9) and (22.10) and averaging over time.
For a wave going to the left with maximum pressure P_,
rp_ — p_ e ik(x+ct) £__ _ A_ e ik(.x+ct) (_lr J = JJ_ e ik(x+ct)
A (dfe) u ~ = (^} p  = ~ pc (i)
T_ = 2w 2 v 2 pcAl =  (pA = ycUl
The intensity in this case is negative, since the wave is going in the
direction of the negative zaxis.
The expression used in Sees. 20 and 21 for the pressure on a con
denser microphone diaphragm is obtained from Eq. (22.11), for if the
wave is all going toward the diaphragm, the total intensity will equal
the square of the maximum pressure, divided by 2pc.  Since pc = 42,
(22.12)
VI.22] THE EQUATION OF MOTION 225
the maximum pressure will be V84T = 9.2 V T, which was the expres
sion used. This is not the correct expression if the microphone is large
enough to distort the wave as it passes, as we shall see in Chap. VIII.
The quantity (c/v) is called the wavelength X of the waves, and
the quantity (k/2w) = (v/c) = (1/X) is called the wave number. The
wavelength corresponding to y = 500 is 69 cm, or about 2 ft; that
corresponding top = 4,000 is 8.6 cm, or about 3 in.
The Decibel Scale. — While physical instruments can be designed
to measure intensity in ergs per sec per square centimeter or in micro
watts per square centimer (1 microwatt per sq cm = 10 ergs per sec
per sq cm = 10 6 watt per sq cm), the ear does not respond in a man
ner proportional to intensity or to amplitude. It is outside the scope
of this book to go into detail concerning physiological acoustics, but it
is necessary to include a brief outline of the subject in order to have an
idea of the ranges of values of intensity and frequency encountered
in acoustical work. 1
The human ear is a remarkably rugged yet sensitive organ. It
responds to a frequency range of about ten octaves, whereas the eye
responds to less than one. It responds to air vibrations whose ampli
tude is hardly more than molecular size; it. also responds without
damage to sounds of intensity 10 million million times greater (sound
much louder than this becomes painful) .
The response of the ear is not proportional to the intensity, how
ever; it is much more nearly proportional to the logarithm of the
intensity. If we increase the intensity of a sound in steps of what
seem to be equal increments of loudness, we find that the intensities
form a sequence of the sort 1, 2, 4, 8, 16 • • • or 1, 10, 100, 1000 • • •
and not of the sort 1, 2, 3, 4 • • • or 1, 10, 19, 28 • • • • Consequently,
a logarithmic scale is often chosen in which to express acoustic ener
gies and intensities. The unit is the decibel (abbreviated db).
A sound is said to be a decibel higher in level (or one decibel more)
than another sound if its intensity is 1.259 (= \/l0) times the inten
sity of the other. If the ratio of intensities is 10, the difference in
level is said to be 10 decibels (or one bel) ; if the ratio is 1,000, the differ
ence is 30 db, and so on. The difference in level of two sounds in
decibels is equal to ten times the logarithm to the base 10 of the ratio
between the intensities:
1 Many of the data quoted in this section have been obtained from H. Fletcher,
"Speech and Hearing," D. Van Nostrand Company, Inc., New York, 1929. The
xeader is referred to this book or to Stevens and Davis, "Hearing, Its Psychology
and Physiology," John Wiley & Sons, Inc., 1938, for further details.
226 PLANE WAVES OF SOUND [VL22
Level difference = 10 log 10 ( yM (22.13)
Intensity and Pressure Level. — At times it is convenient to deter
mine the absolute value of the intensity of some sound in the decibel
scale. This is done by giving its level above or below 10 10 microwatt
per sq cm (= 10 9 erg per sq cm per sec). Thus the intensity level
of the sound is
Intensity level = 10 log 10 (10 9 T) =90+10 logic (T) (22.14)
This reference level is chosen because it is approximately the minimum
audible intensity at 1,000 cps.
Unfortunately (or fortunately, perhaps) we seldom measure sound
intensity; what is measured is usually rootmeansquare pressure
amplitude p ims . If the sound is a sine wave, traveling in one direc
tion, then Eq. (22.11) shows the relation between p r m S and T; T =
(prms) 2 /42, for air at standard conditions, since (twO 2 = ip 2 . How
ever, if the wave is not a plane traveling wave, the relation between
T and p is not at all as simple. Indeed there are many cases when we
do not know enough about the sound field to be able to compute the
intensity once we have measured the pressure ; so that we cannot give
the value of the intensity level if we have measured only pressure.
To obviate this difficulty, we can define a level in terms of pressure
squared, the reference level being 0.000200 dyne per sq cm which cor
responds (within 2 per cent) to the pressure amplitude in a plane wave
of zero intensity level. The pressure level of a sound is
Pressure level = 20 logio L ^7oJ = 20 r °g»W + 74 < 22  15 )
When the sound is a plane wave going in a single direction, then its
pressure level is only 0.1 db higher than its intensity level; when the
sound has a more complex distribution, the pressure level, may differ
from the intensity level by many decibels ; but in any case if we change
the amplitude of a sound but not its distribution in space the decibel
difference between pressure and intensity level will remain constant,
the value of the difference being characteristic of the space distribution
chosen.
The response of the ear is not exactly proportional to the decibel
scale, though it corresponds much more closely than it would to the
intensity scale. Corresponding to the physical quantities intensity
and frequency are the physiological (or, rather, psychophysiological)
quantities loudness and pitch. The loudness of the sound depends on
VI.22]
THE EQUATION OF MOTION
227
both its intensity level and its frequency; its pitch depends chiefly
on frequency, but to seme extent on intensity. Contours of equal
loudness for the average person are plotted in Fig. 52 in terms of
intensity level and frequency.
The bottom curve, for zero loudness, is the threshold of hearing,
below which a sound of that frequency is inaudible to the average
person. The upper contour is the threshold of pain, above which
the sensation is more of pain than of sound (and the result is more or
150 ,.:,.,.W,
20
50 100
5000 10,000 20,000
200 500 1000 2000
Frequency, cycles per second
Fig. 52. — Contours of equal loudness, plotted against intensity level and frequency
for the average ear. Contours are numbered by correspondence with intensity leve
at 1000 cps. Unshaded portion, the auditory area, shows range over which sound can
be heard.
less damaging to the ear). About 5 db below this upper threshold
the sound begins to "feel uncomfortable"; at the threshold plotted
the sensation is perhaps best described as a "tickling sensation."
About 5 db above this, the sensation is "pure pain."
The area enclosed within the two thresholds represents the range
of audible sound in frequency and intensity. We see that the ear can
hear sounds of frequency as low as 20 cps and as high as 20,000,
although at these extreme limits the intensity range perceptible as
sound is very small. On the other hand, the range of perceptible
intensity at 1,000 cycles is as much as 125 db.
228 PLANE WAVES OF SOUND [VI.22
This area covers the range of frequency and intensity that sound
reproduction equipment must cover for perfect performance. The
ear is most sensitive at about 2,000 to 4,000 cps. If our abscissa
were pressure level at the ear diaphragm rather than intensity level
just outside the ear, the wavy appearance of the loudness contours
between 2,000 and 10,000 cps would not occur, for these fluctuations
are due to "cavity resonance" in the outer ear canal, increasing sensi
tivity at about 4,000 cps and decreasing it at about 8,000 cps. In
terms of pressure level at the ear diaphragm, the ear is most sensitive
at about 1,000 cps. Cavity resonance will be discussed at the end of
Sec. 23.
An interesting result of the peculiar response of the ear, evidenced
by the fact that loudnesslevel contours, as shown in Fig. 52, are
parallel neither to the intensitylevel contours nor to each other, is that
a complex sound changes its quality when its overall intensity is
changed. Suppose that one hears a sound consisting of many different
components, some at frequencies around 100 cps and some around
1,000 cps, first when most of the components have individual intensity
levels of about 100 db and then when their intensity levels are about
20 db (owing either to the fact that the source is farther away or that
the sound is recorded and played back at a lower level). In the first
case, when the components are at about 100 db, all components are
of approximately equal loudness (since the 100db loudness level
contour is roughly horizontal). In the second case, with the intensity
level of the components about 20 db, the 100cps components would
not be heard at all (for this level is below the threshold for 100 cps),
whereas the 1,000cps components would still be heard. Therefore
in order to change the loudness of a complex sound without changing
its quality, the different frequency components should be attenuated
by different amounts, according .to the loudnesslevel contours of
Fig. 52.
Sound Power. — The amount of power produced by sound gen
erators is very small compared with usual electrical powers. The
average power produced by a person talking in an ordinary conver
sational tone is about 10 5 watt, or 100 ergs per sec; although the
range of power that can be produced by the voice varies from about
1 erg per sec for very weak speech to about 10 4 ergs per sec for loud
speech. Greater powers can be produced by singing voices, the range
from pp to ff corresponding approximately to the variation from 10 3
to 3 X 10 6 ergs per sec.
VI.22] THE EQUATION OF MOTION 229
Some musical instruments can produce greater power than the
human voice. The violin cannot, but the trombone and cornet can
produce about 4 X 10 5 and the bass drum about 6 X 10 5 ergs per sec.
The ratio of maximum to minimum power produced by a band or
orchestra while playing may be as great as 10 5 . The important range
of frequency is from 30 to 8,000 cps.
If we assume that the acoustic power from any of these sources
radiates out equally in all directions (which it certainly does not do
in many cases), then the relation between intensity and power output
II would be T = Jl/^irr 2 . To this approximation, the intensity level
due to a source of power output II ergs per sec at a distance of 3 ft
(100 cm) would be 10 logio(n) + 39. The intensity level of a person
talking normally 3 ft away is therefore approximately +60, corre
sponding roughly to a loudness of 60, according to Fig. 52 (since most
components of spoken sounds are between 500 and 5,000 cps). The
relation between power output and soundintensity level farther from
the source, particularly in rooms, will be discussed in Chap. VIII.
Frequency Distribution of Sounds. — Since the ear does distinguish
between sounds of different frequencies, it is often important to
analyze the sound into its frequency components. This is, of course,
what we have been doing throughout the book, even when we have
discussed transient sounds. Referring to Eq. (2.19); we see that we
can express any dependence of pressure on time by the formula
p(t) = f "^ P{v)e^ ivt dv
where we can determine the function P(v) by the reciprocal relation
P(„) = J_ f p(*)e 2 ""d< (22.16)
The value of P (v) = \2P(v)\ or \2P{v)\ is the pressure ampli
tude of that part of the sound in a unit frequency band around v
(i.e., between v — \ and v + \ cps). If the sound is a plane wave,
the square of P , divided by 2pc, gives the intensity of this part of
the sound, T(v) = Pl(v)/2pc. The frequency distribution of a sound
is often given in a decibel scale, in terms of the socalled spectrum
level, which is the intensity level of that part of the sound in a unit
frequency band at frequency v,
Spectrum level = 10 logi [T(^)] + 90 (22.17)
~20 1ogio[Po(»')] + 71db
230 PLANE WAVES OF SOUND [VI.22
where the second line is valid only when the sound is approximately
a plane wave and when pc = 42.
It should be pointed out that a measurement of spectrum level
(which can be obtained from a sound analyzer) is not sufficient to
give the actual shape of the pressure fluctuation, for it gives values
of Po(v), which is proportional to the amplitude only of P{v) and of
P( — v). It follows that the phase angle of the frequency component
of pressure P(v) is not determined by the sound analyzer; there
fore Eq. (22.15) cannot be utilized to compute p(t), if we have only
soundanalyzer data. This usually does not matter, for the ear
seems to be less sensitive to the phase of P{v) than it is to its mag
nitude. This is, of course, not true of very short pulses, for here the
phases must be related in just the right manner to produce the pulse.
Consequently, specifying the spectrumlevel curve of a pulse does not
specify it uniquely.
A "pure" noise corresponds to a constant value of P over a large
range of frequency, whereas a note of one frequency has a P which
is zero everywhere except at just one frequency, where it is very large.
Musical sounds correspond to a series of sharp peaks in the curve for
P , equally spaced in frequency, the peak of lowest frequency corre
sponding to the fundamental, the next peak, at twice the frequency,
corresponding to the second harmonic, and so on. The area under
each peak corresponds to the amplitude of that harmonic.
The peaks for a musical sound are never in practice infinitesimally
narrow, corresponding to exactly one frequency, for there are always
slight variations in pitch and intensity during the production of the
sound to give it a "spread" in frequency. As a matter of fact, only
a sine wave that continues forever with undiminished amplitude and
unchanged frequency can have a peak of zero width. Any starting or
stopping of the vibration will spread out the peak. For instance,
we can utilize Eq. (22.16) to show that a sinusoidal oscillation of
finite length
(0 (t <  £A<)
p(t) = <P m cos (2rv t) ( $At < t < iM)
{ (t> iAt)
has a distribution in frequency corresponding approximately to
r> / \ f P A sinh^o  v)M]
VI.22]
THE EQUATION OF MOTION
231
(where we have neglected a term involving (v + v)~\ since it is
usually much smaller, at least when v is approximately equal to v ).
The approximation is valid if v is larger than (1/At), i.e., if the
pulse contains several cycles. The quantity P (v) has a peak at
v = vo, of height (P m At/2T), it falls to about half this value at
v = v ± (l/2At), and is very small for (v  v) large. Figure 53
Fig. 53.— Two pressure pulses, whose shape is plotted as a function of time, and their
corresponding distributioninfrequency for intensity.
shows curves of the distribution in frequency of the intensity [P§0)/2pc]
as functions of frequency. In order to build a pulse of finite duration
we must use waves of different frequency, so arranged that they all
cancel out except for the duration of the pulse. The shorter the pulse
the more different frequencies must be used to form it. A measure
of the "spread" of frequency needed for a pulse is the half width Av
of the peak in the distribution in frequency, the frequency difference
between the two points on the curve that have half the maximum
height. This half width is related to the length of the pulse b V the
relation
AvAt^l
(22.18)
This relationship is approximately true even if the pulse is not started
and stopped abruptly, as long as A* is taken as the half duration of the
232 PLANE WAVES OF SOUND [VI.22
pulse, the length of time between two parts of the pulse where the
amplitude is half the maximum amplitude.
Thus, the peaks in the curve of distribution in frequency of music
or speech have a width that is at least as wide as the reciprocal of the
length of time during which the note is maintained unchanged in
amplitude and frequency. In the case of music, the duration of the
tone is usually longer than ^ sec, and the half width is quite small,
the actual width of the peak being chiefly due to the vibrato effects
of the musician, who usually makes the frequency fluctuate slightly
during the production of the note. It may well be that this increased
spread in frequency is the reason that the vibrato note sounds better
than a pure tone; for we shall see in the last chapter that the wider the
spread in frequency a note has, the less marked are its resonance
effects in a room.
In the case of speech, however, the duration of any particular sound
is quite short, in the case of some consonants being as short as ^V. sec.
In this case the half width of the peaks is large, and the feeling that
the sound has a definite pitch is lost.
The Vowel Sounds. — The vowel and semivowel sounds are more or
less musical in nature, however; their distribution in frequency being
a sequence of many separate peaks, corresponding to the first 10 or
more harmonics of the natural frequency of the vocal cords, the width
of each peak corresponding to the duration of the sound. A bass
voice will have its fundamental at 100 cycles (or thereabouts) and will
have 10 peaks below 1,000 cps. A soprano voice with its fundamental
at 250 cycles will have only four peaks below 1,000 cycles. Ordinarily,
most of the peaks are small in height, only two or three of them being
large. The position of the large peaks in the frequency scale deter
mines the sort of vowel that is spoken. For instance, the vowel " ah "
has the peaks near 900 cycles enhanced, whether they correspond to
the eighth, ninth, and tenth harmonics for the bass voice with a funda
mental at 100 cycles or to the third and fourth harmonics for the
soprano voice of fundamental frequency 250 cycles. What seems to
be happening is that the oral cavity for all individuals shapes itself
for the vowel "ah" so that it resonates to a frequency of about 900,
enhancing whatever part of the sound, coming to it from the vocal
cords, is near this frequency.
The vocal cords differ from person to person, some sending to the
mouth cavity pulsations that can be analyzed into a Fourier series
with a fundamental frequency as low as 100 cycles, some with a funda
mental as high as 250 cycles, etc. The mouth, then, is shaped to pick
VI.23] PROPAGATION OF SOUND IN TUBES 233
from these frequencies the ones that correspond to the characteristic
frequencies of the vowel spoken; or, from another point of view, the
mouth is shaped so that the air in it is periodically set into vibration
at the characteristic frequencies of the mouth by the pulsations sent to
it from the vocal cords. The two points of view are essentially the
same, one corresponding to an analysis of the motion in time and the
other to an analysis in frequency.
The characteristic frequency of long "oo" is 400, that of long "o"
is 500, that of " ah " is 900. Some of the other vowels have two charac
teristic frequencies: Those of the "a" in "tan" are 750 and 1,600,
those of long "a" as in "tame" are 550 and 2,100, those of long "ee"
are 375 and 2,400, and so on.
23. PROPAGATION OF SOUND IN TUBES
A large number of soundgenerating devices are tubular in shape,
sound waves of large amplitude being set up inside the tube, and some
of this storedup energy being radiated out into the open. Organ
pipes, woodwind and brass musical instruments, and horn loud
speakers are sound generators of this type. Inside the tube the sound
waves are approximately plane waves and can be treated by the
methods of the preceding section. We shall discuss the behavior of
sound in tubes of uniform cross section in the present section and deal
with waves in tubes of changing cross section in the next section,
leaving the more complicated radiation in the open to be treated in
the next chapter.
In many cases the various parts of the tube (valves, constrictions,
etc.) are small in length compared with a wavelength of sound. When
this is the case the behavior of sound in the tube is analogous to the
behavior of electric current in a circuit with lumped circuit elements
of inductance, capacitance, etc. We shall take up this case first and
work out { he analogous electric circuits for a few examples. When the
tube elements are longer than a wavelength of sound, the electric
analogue is the transmission line. This more complicated case will
be taken up later in the section.
Analogous Circuit Elements. — A tube of variable cross section,
where the length of the elements are shorter than a half wavelength
of the transmitted sound, is analogous to an electric filter circuit.
The analogue of the pressure in a tube element is the voltage across
the corresponding part of the filter circuit, and the analogue of the
current at some point in the circuit is the total flow of air, the particle
234 PLANE WAVES OF SOUND [VI.23
velocity times the area of cross section 8 of the tube at its correspond
ing part.
The analogous impedance is thus not the specific acoustic imped
ance z (which is defined as pressure over velocity) but z divided by S.
It must be emphasized that the electric circuit is merely an analogue,
to help our analysis (since electric filters are more familiar to most of
us than acoustical filters). The analogous current comes out in cubic
centimeters per second, and the voltage is in dynes per square centi
meters. The capacitances and inductances are not in farads and
henrys but in the analogous acoustical units, and the power trans
mitted comes out, not in watts, but in ergs per second.
Tube elements are usually of two kinds: pieces of small cross
sectional area (opening at both ends into larger chambers), which we
shall call constrictions; and chambers of larger cross section (feeding
into constrictions at both ends), which we shall call tanks. The air
in the constrictions is mass controlled, and therefore the analogous
circuit element is an inductance; the air in a tank is stiffness controlled,
so the analogous element is a capacitance. The physical reasons
underlying these statements, and their proof, will have to be post
poned until Chap. VII, when we shall work out the necessary mathe
matics. In the meantime some very crude analysis will have to
suffice to compute the values of the analogous elements.
Constriction. — The air in a constriction has a total mass of pSl e g,
where S is the crosssectional area and l e the effective length of the
tube. We use an effective length because it turns out that some air
beyond the ends of the constriction moves along with the air in the
constriction, and this must be included to obtain the effective mass.
It will be shown in Sec. 28 that the effective length l e is related to the
actual length I by the formula
Z e ~ Z + 0.8 V£ (23.1)
Therefore even a hole in a thin plate has a nonvanishing effective
length, proportional to the square root of the area of the hole.
The force accelerating the mass pSl e is the difference in pressure
at the two ends of the constriction, multiplied by the area S. The
acceleration iwt* is therefore equal to S times the pressure drop,
divided by ( P Sl e ); and the analogous impedance, the pressure drop
divided by Su, is ioWS). A constriction is thus analogous to
an inductance of value
(f)g/cm 4
(23.2)
VI.23] PROPAGATION OF SOUND IN TUBES 235
where £ is the crosssectional area of the constriction, I its length, and
h is given by Eq. (23.1).
Tank.— If total flow of air Su is analogous to current, the analogue
of charge is S times the displacement £. If a volume of air S% flows
in through one opening of a tank, the increase in pressure in the tank
is obtained by using a variant of Eq. (22.5), p = pc\dV/V), where
V is the volume of the tank and dV is the volume of air introduced,
St The analogous capacitance is the ratio between analogous charge
(££) and analogous voltage p
Ca ~ (^) cm4 sec2 /g (23.3)
It will be shown later [see Eqs. (23.14) et seq.] that, when the output
end of a constriction opens into free space, the radiation load on the
opening is represented in the equivalent circuit by an analogous terminal
resistance
R T ^
fe)
(23.4)
which is shunted across the termination of the circuit, in series with
the analogous inductance L a of the output constriction. This resist
ance becomes very small for low frequencies, since a small opening
radiates very poorly at low frequencies. The power dissipated in this
resistance is the power radiated out of the acoustical net.
Examples.— By use of these formulas in the frequency range where
they are valid, we can build up various acoustic circuits and study their
properties, by studying the behavior of the analogous electric circuits
A few examples are shown in Fig. 54.
Case a is the simple Helmholtz resonator, analogous to the simple
LC series circuit. The analogous impedance at A is
and the resonance frequency is
v =
2tt VL a C a
cps (23.5)
Case b is a simple lowpass filter. Any standard text on filter
circuits will prove that the cutoff frequency, above which the sound
will not be transmitted, is at
236
PLANE WAVES OF SOUND
[VI.23
1 _1 lc*S
Case c is a simple highpass filter, where frequencies above this value
vo will be transmitted. Other, more complex, circuits may be built
up and analyzed in a similar manner.
l]~  ■*■! Area S
(«)
a "a
..J ^ AreoS
Area 5
(c)
Fig 54 —Acoustical circuits on left and analogous electrical circuits on right.
Constrictions of length I and crosssectional area 8 are analogous to inductances La ;
tanks of volume V are analogous to capacitances C«; as long as all dimensions oi indi
vidual elements are less than a halfwavelength of the transmitted sound. Analogous
impedance at A equals ratio of pressure to S times particle velocity at A.
The lumped circuit approximation is not so good a one for sound
as it is for electric currents, because of the considerably smaller speed
of sound and the corresponding shortness of wavelength for a given
frequency. JJsually the formulas given above cease to have any
validity for frequencies above 1,000 cps or so (for when an acoustical
system is longer than a halfwavelength it behaves like a transmission
line, and the wave formulas must be used).
We have now encountered three kinds of acoustic impedance, a
confusing redundancy only partly excused by the fact that the different
impedances are useful in different kinds of calculations: the analogous
impedance when we deal with lumpedcircuit elements at low fre
quencies, the specific acoustic impedance when we deal with transmis
VI.23] PROPAGATION OF SOUND IN TUBES 237
sionline calculations, and the usual mechanical radiation impedance
when we calculate the coupling between the waves in the tube and the
driving piston or the load at the output end. We list the three
impedances here for comparison; their ratio turns out to be the cross
sectional area S of the tube.
Analogous impedance Z a — (pressure/volume flow)
(Greek subscripts, see page 234). Used in analogous circuits.
Specific acoustic impedance z = (pressure/velocity) = SZ a
(Lower case z, see page 239.) Mechanical impedance per unit area.
Radiation impedance Z a = (force/velocity) = Sz = S 2 Z a
(Lower case subscripts, see page 238.) The part of the mechanical
impedance due to the sound field.
There is also the acoustic impedance ratio f = — ix = z/pc, the ratio
of the specific acoustic impedance to that of the medium, as indicated
on the next page.
In each of these cases we can draw equivalent electric circuits to
help us understand the behavior of the acoustical system. These
circuits are only analogous, however, for the impedances do not have
the dimensions of ohms; instead of wires and amperes and volts we
are dealing with tubes and air flow and pressure. Since we are more
familiar with electric circuits than with acoustic ones, the equivalent
circuits are often useful. The circuits equivalent to the analogous
impedances are usually most useful, for there is a direct relation
between currents and voltages in each part of the equivalent circuit
and the pressure and volume flow in the corresponding part of the
acoustical circuit. Another way of stating the same property is that
the analogous impedance does not change markedly with distances
along the tube of less than a half wavelength, even though the tube
cross section changes markedly. Equivalent circuits for z and Z a are
more difficult to interpret, but are sometimes helpful.
Characteristic Acoustic Resistance. — We are now in a position to
study the behavior of plane waves of sound in tubes of uniform cross
section, which are longer than a half wavelength. The treatment
will be quite analogous to that of Sec. 10 for wave motion on a string.
We first take the case of a tube of infinite length, with no reflected
wave, to get the simplest sort of air reaction. We suppose that we
have a tube of uniform crosssectional area S, starting at the origin
and extending to the right along the #axis for such a great distance
that no wave is reflected back to the origin from its far end. Suppose
that we fit the end at x = with a flattopped piston which can vibrate
and generate waves, so that its displacement, as a function of time,
238 PLANE WAVES OF SOUND (VI.23
is E(£) The wave produced in the tube must be such that the dis
placement of the air next to the piston (i.e., at x = 0) is equal to
S(0 Such a wave would be
05)
which travels to the right in the tube with velocity c.
The reaction F of the air back on the piston will be equal to the
area of cross section of the tube S multiplied by the excess pressure p
due to the wave. Using Eqs. (22.7) for the pressure, we have
F = Sp = Spc
(©"*[*(' 31
<, (dZ\ . ( . _ dZ(z)
(23.6)
The reaction on the piston is therefore proportional to the velocity
of the piston, a purely frictional reaction. The proportionality
constant (Spc) is called the radiation resistance of the air on the piston.
The resistance per unit area of the piston R = pc is called the charac
teristic acoustic resistance. Its value for air at normal conditions is
42, as we showed on page 222. This is the expression used on page
199 for the reaction on the diaphragm of a condenser microphone.
We have just seen that when the tube is so long that there is no
reflected wave the acoustic resistance on the piston is independent
of the type of motion of the piston. When the tube is not infinitely
long, however, there will be some wave motion reflected back from
the far end which will also affect the piston. When this happens, the
reaction on the piston will be different for different motions of the
piston. It will then be necessary to analyze this motion into its
simple harmonic components and to treat each component separately.
Incident and Reflected Waves. — When the sound wave is not all
radiated out of the far end of the tube, both incident and reflected
waves will be present. By referring to Eqs. (22.11) and (22.12), we
see that the pressure, particle velocity, energy density, and intensity
of a plane wave in a uniform tube with both waves present are
~y _ p gik(xct) _J_ p_g—ik(x+ct)
( zl \ — Jl (P e iHxct) _ p _ e ik{x+ct)y
J_ ,™ , ™s > (23.7)
2pc 2
J_
2pc
W^Pl+Pl)
t = ^r (P+  p )
VL23] PROPAGATION OF SOUND IN TUBES 239
To simplify the equations we shall express the ratio of the ampli
tudes of the waves in the two directions by means of an exponential:
(P/P+) = (A/A+) = (U/U+) = ew, where $ is, in general, a
complex number; in other words, $ = irao — iirfio. Twice the imagi
nary part of yf/, —2x^ , gives the phase angle between the two waves
at x = 0; and e~ 2Re ^ ) [where Re{<f) = ira is the real part of ^] is the
ratio between the amplitudes of the two waves, without regard to
phase. The ratio between the numerical magnitudes of the intensities
of the two component waves is e~* ReW = e~~ 4wa . The expressions for
the pressure and particle velocity, and their ratio, then become
= 2P + e+ 2 * i > t sinhU + ^llA
\Wai) =
<x) = (?P±) e *^« ..... L . *•■«* > (23 8 )
z ( x ) = \ a // aj ) = pc tanh
Specific Acoustic Impedance. — The quantity z(x) in Eq. (23.8),
the ratio between the pressure and particle velocity, is called the
specific acoustic impedance at the point x for the frequency v. It is a
most useful quantity, for, once it is known, the reaction of the air
on a vibrating system can be determined. As an example of its
utility, consider a uniform tube closed at one end by a cap that yields
somewhat to pressure and having a flattopped driving piston at the
other end. If we know the mechanical impedance of the cap, the
ratio of driving force to cap velocity, we can determine the specific
acoustic impedance of the air in the tube next to the cap's inner
surface. For the velocity of this air must equal the velocity of the
cap, and the pressure times the area of cross section of the tube must
equal the force on the cap. Knowing z at the cap, we can find ^; and
knowing \j/, we can find the value of z at the piston end of the tube.
If this value is known, we can find the ratio between the driving force
on the piston and its velocity, which is the effective mechanical
impedance Z m of the piston for the frequency v. If the piston is driven
by a force of magnitude i^o and frequency v, the velocity amplitude
of the piston will be F divided by the value of Z m for the frequency p.
Similarly, if the driving force is a combination of several components
of different frequencies, the piston velocity is a similar combination
of velocities of different frequencies, each with amplitude equal to
the ratio between the magnitude of the component force and the
impedance for the corresponding frequency. Thus a consideration
240 PLANE WAVES OF SOUND [VI.23
of the behavior of the specific acoustic impedance of a plane wave will
enable us to work out the details of its interaction with various mechan
ical systems.
In a great many cases it is most useful to express the acoustic
impedance in units of the characteristic impedance of the medium,
pc. The dimensionless quantity f = (z/pc) = 6 — ix is called the
acoustic impedance ratio, and its reciprocal 77 = (1/f) = k — ia the
acoustic admittance ratio. Its value indicates the amount of impedance
mismatch at any change of medium or change of cross section. Where
sound strikes the surface of a medium other than air (as, for instance,
the bounding surface of a tube or the wails of a room) then the value
of the admittance ratio 77 for the surface is a measure of the effect the
surface will have on the wave motion striking the surface, as will be
shown later.
Standing Waves. — Equations (23.7) and (23.8) are duplicates of
those discussed in Sec. 13 for waves in strings. The properties of the
hyperbolic functions were given in Eq. (13.7); and the relationship
between the behavior of the waves, which was discussed there, can
be applied to the present case, though the motion here is longitudinal
instead of transverse. The air displacement corresponds to the string
displacement and the pressure to the transverse force. We note that
the measurable quantity here is usually the pressure, whereas the
measurable quantity with the string is the displacement.
We shall need to distinguish between the real and imaginary parts
of the quantity in the brackets in Eqs. (22.14), as before, by setting
* + fe^*\ = x(a _ ij3)
a = (l/ir)ifc#); = (lA)/m(^)  (2x/X)
where Re(\p) is the real part and Im(4>) the imaginary part of rp, and
X = (c/v) is the wavelength of the sound wave. In terms of these
quantities the acoustic impedance ratio, the amplitude of the pressure
fluctuations, the amplitude of the air velocity, the energy density and
the sound intensity at the point x are, respectively,
I — J = f =  ix = tanh[x(«  #)]
pj = 2P+e* a Vcosh 2 ^)  cos 2 (7r/3)
\ u \ = (— ) e~* a Vcosh 2 (ra)  sin 2 (x/3) ^
VI.23J PROPAGATION OF SOUND IN TUBES 241
The minima of p are at the same places the maxima of string dis
placement were in the equations of page 141. This is because p corre
sponds to (dy/dx) for the string, whereas £ corresponds to y, so that
the quantities observed are opposite in behavior with respect to ft
Plates I, II, and III at the back of the book can be used in the present
case as well as for the string, however, as long as this difference is
kept in mind.
Suppose we consider the case where the "load" is at x = I and the
driving piston is at x = 0. The average impedance ratio of the load
is (Z m /pcS) = izi/pc) = 6i — ixi, which is related to the corre
sponding phase parameters for the load by the equation di — ixi =
tanh[7r(a!j — ipi)]. We are neglecting attenuation along the tube
so that a = ai, a constant. However, = /3o — (2x/\) = ft +
(2/X)(Z — x), so that /3 changes as we go along the tube, increasing
as we move from load to source, and decreasing as we go from source
to load. Therefore as we go from load to source the point correspond
ing to the impedance ratio moves from (aj,ft) on Plate I or II along
a curve of constant a, in the direction of increasing ft The point
moves in a clockwise direction by an amount of complete circuits equal
to the number of half wavelengths in the tube length. This motion
eventually brings us to the point (a ,ft>), where a = m, 0o = ft +
(2Z/X). From this we can compute the load on the driving piston due
to the air column plus load Z r = pcS tanh {ir[ai — ft — (2il/\)]}, by
means of Plates I or II.
The amplitude of the pressure fluctuation is a maximum where j3
is a half integer (•••—£,+ £,+ $•••). At these points the
amplitude of the particle velocity is a minimum. Where /3 is an
integer ( • • • — 1,0,1,2 ••• ) the pressure amplitude is a minimum
and the amplitude of the particle velocity is a maximum. Values
of pressure and velocity amplitudes and phase angles can be read
from Plate I in the back of the book.
The values of the real and imaginary parts of (z/pc), 0, and x,
corresponding to given values of a and ft are given in Plate II and
in Tables III and IV at the back of the book. Alternatively, the
magnitude of f and its phase angle can be obtained from Plate III
and Table III. Examples of the use of the figures for calculation were
given on page 138.
The usefulness of the foregoing analysis will become more apparent
later in the chapter. At present, we can see from Eqs. (23.8) that
in wave motion composed of two plane simple harmonic waves travel
ing in opposite directions, if the specific acoustic impedance at some
point x can be represented by the values R and X or by the con
242 PLANE WAVES OF SOUND [VI.23
stants a and /?o, then the impedance at some other point x is given
by the values of X and R corresponding to a = « and 18 = /3 —
(2/\)(x — x ). The different impedances for different points along
the wave correspond to the points on a circle of constant a in the
figures, with /3 equal to the distance along the wave in units of half
wavelength. Both R and X repeat themselves each half wavelength
increase of x.
The expression for the intensity in Eqs. (23.9) is obtained from
those for the maximum pressure and particle velocity by multiplying
the product of yw a nd (dg/da pmax by the cosine of the phase angle
between them, (R/\/R 2 + X 2 ). This gives the maximum intensity,
and the average intensity T is one half of this.
This intensity is independent of (3 and is therefore independent
of x. Although the pressure, the particle velocity, and the phase
angle between them all change with x, they change in such a manner
that the flow of energy along the wave is everywhere the same. This
must be, for otherwise the energy would pile up at certain points and
would continually diminish at others.
We have assumed heretofore that a is independent of x, as indeed
it is in a true plane wave, when there is no energy absorption during
the progress of the wave. In some cases in practice, however, energy
is absorbed by the walls of the tube (or in the air itself) as the wave
travels along, and a will also depend on x. This case will be discussed
later in this section.
Measurement of Acoustic Impedance. — We can use the results
of Eqs. (23.9) to devise a method for measuring specific acoustic
impedance. We fit on the end of the tube, at x = I, the material or
device whose impedance is to be measured; and a source of plane waves
is placed at the input end, x = 0. The mechanical impedance of the
device, Z m ( = force divided by velocity), divided by the crosssectional
area of the tube, S, is equal to the average specific acoustic impedance
z ( = pressure divided by velocity) at the end of the tube, x = I. This
terminal impedance is related to the quantities a and of Eq. (23.9)
by the usual formula
(s)
= 0i — ixi = tanh[7r(orz — ift)]
The dimensionless quantity f = (z/pc) is the acoustic impedance ratio
of the load, and and x are called the acoustic resistance and react
ance ratios respectively.
VI.23] PROPAGATION OF SOUND IN TUBES 243
The impedance, pressure, and particle velocity at any point of the
tube are then given by the equations
p = A \/cosh 2 (xa) — cos 2 (7rj8)
Id = (— I Vcosh 2 (7ro:) — sin 2 (x/?)
\pcj
m
where
a = oc i; j8 = ft + (2/X)(Z  a)
As we have mentioned in our discussion of Eqs. (23.9), the pressure
amplitude \p\ is minimum where is an integer, or (2/X)(Z — z) =
n — jSi, and the value of \p\ at these points is A sinh (rat). Conversely,
the pressure amplitude is a maximum where /3 is a halfinteger, or
(2/X)(£ — x) = n + i — ft, and its value there is A cosh (raj).
Therefore we move a microphone (a small one to avoid distorting
the wave) along the tube, measuring the distances from the output
end of the tube (where the unknown impedance is placed) to the points
where the pressure amplitude is a maximum or minimum, and meas
uring the values of \p\ at these points. The distance between minima
(or between maxima) is a half wavelength (X/2). The distance
between the output end and the nearest minimum, divided by (X/2),
is 1 — ft. And the ratio between the value of p at its minimum to
its value at its maximum is tanh(7ra!z) Therefore both oti and 8 t
can be determined, and the value of the mechanical impedance of the
device, Spc(di — ixi), can be computed from Table III at the back of
the book. The method is particularly accurate for values of imped
ance corresponding to values of a larger than 0.02 and smaller than 0.5.
Damped Waves. — In actuality sound waves cannot transverse a
tube without losing at least a small part of their energy, either by
absorption at the walls of the tube or by the viscous friction impeding
the air nearest the wall, or else by absorption in the air throughout
the tube (or for all three reasons). Some of these effects will be
discussed in more detail in the next chapter; here we are interested
only in the result, which is that the amplitude of each wave is damped
out as it progresses along the tube. The incident wave has a factor
e Kx/c anc j tn e reflected wave a factor e KX/c , where k is the damping
constant for the wave. Therefore the correct expression for the pres
sure is
p(x) = 2P + e' a »+* T ^ iw ' sinh x I a — — — tjSo + i ^J
244 PLANE WAVES OF SOUND [VI.23
This means that in actual cases a, as well as /3, depends on x. The
dependence is not usually so pronounced as with /3, for (k/c) is usually
much smaller than 2w/\). However, when a depends on x we cannot
simply take the ratio of the minimum value of \p\ along the tube to
its maximum value to obtain the value of tanh (7ra) ; for each maximum
and each minimum will have a different value. What is necessary is
to plot the values of the maxima (against the corresponding values of
x where they occur) on semilogarithmic paper and join the points by
a smooth curve, also doing the same for the minima. The difference
between the two curves, extrapolated to x = I, will then give the value
of tanh^aj), which is used (via Table III) to obtain the acoustic
impedance at the termination.
The rest of the calculations of this section will be made tacitly
assuming that a is independent of x. In general, and for most pur
poses, this assumption is a good enough approximation so that the
additional calculations incident to the inclusion of damping are not
worth while. If damping must be taken into account, however, the
effects can usually be computed by considering that a varies linearly
with x, inserting a = ao — (kx/tc) into the formulas that are to be
developed.
Closed Tube. — The simplest case for calculation is that of a uniform
tube closed at x = I by a rigid plate. In this case the impedance at
x = I is infinite, corresponding to ai = 0; ft = g. The specific
acoustic impedance at x = is therefore
' , [ (l . 2l\~] . r/27rZ\ . tt] . /coZ\
pc tanh \ —tti I H + t"/ = —ipciem lyj + o =*pccot(— J
If a piston were placed at the input end of the tube (x = 0), it would
experience, because of the air in the closed tube, a mechanical imped
ance
Z r = iSpc cot (j) = iSpc cot (?~j (23. 10)
This is a pure reactance (although there would be a small resistive
term if we took damping into account).
At low frequencies (wl/c) becomes small and we can use the first
term in the series expansion for the cotangent
^6)W
[Z«(c/co)] (23.11)
VI.23] PROPAGATION OF SOUND IN TUBES 245
which is a stiffness reactance, as would be expected of air in a closed
tube. As noted earlier, the analogous impedance is equal to the corre
sponding mechanical impedance divided by S 2 , so that the analogous
impedance for a closed tube (for I smaller than \/%r) is
my
This justifies Eq. (23.3).
A closed organ pipe is a closed tube driven at the "open" end
x = by a jet of air blown across the opening. The jet is so adjusted
that it is very sensitive to flow of air into and out of the tube, so that
a small change in its direction makes a change in the pressure at the
open end. In addition, the jet of air is adjusted so that it strikes a
sharpedged "lip" at the opposite side of the opening, which tends to
set up a periodic motion perpendicular to the boundaries of the jet
(sometimes with periodic vortex formation), the period depending on
the speed of the air in the jet and on the distance between jet and
"lip." This motion (called an edge tone) produces a net pressure
fluctuation on the air in the closed tube. If the fluctuations are at
the frequencies for which the impedance near the mouth of the tube
is small, the tube response is large and, by reacting back on the jet,
the oscillations will settle down into a vibration that is nearly simple
harmonic.
The frequencies to which the closed tube most strongly responds
are those for which the Z r of Eq. (23.10) is smallest, i.e., for (wZ/c) =
(n 4 £V, or
Xn =~; v« = xi n (n = 1,3 » 5 ' ' ' ) (23>12)
If now the closed organ pipe is "blown" at just the right air speed,
the frequency of transverse vibration of the jet will equal the funda
mental frequency of the tube vi = (c/4Z), and the sound will consist
almost entirely of the fundamental; the odd overtones will be present
to some extent because the jet oscillations are not purely sinusoidal
even when aided by the tube resonance.
When the pipe is blown more strongly, the jet frequency is first
held near the fundamental free frequency of the tube by the strength
of coupling, but when the edge tone by itself would exhibit a frequency
close to the third harmonic of the pipe, the note suddenly changes to
this overtone and "locks in" at the new frequency. The pipe is then
said to be "overblown." As indicated in Eq. (23.12), only the odd
246 PLANE WAVES OF SOUND [V1.23
harmonics are present to any extent in the sound from a closed pipe;
the dependence of the amplitudes of the higher harmonics on the
dimensions of the driving jet of air and on the location and shape of
the "lip" regulating the edge tone is too complex to analyze in detail
in this book.
Open Tube. — The other case of particular interest is that of a
uniform tube (of circular cross section of radius a) which is open at
the end x = I. In Ghap. VII [Eq. (28.6)] we shall show that if the
open end is fitted with a flange that is wide compared with a wave
length (so that the open end is effectively a round hole in an infinite
plane wall) then the acoustic impedance at the open end, looking out
ward, is
Ji(w)]
zi ~ pc 1  2 ^Jp I  ipcM{w) = pc(0 o  *xo) (23.13)
where w = (2<oa/c) = (47ra/X), where a is the radius of the tube and
of the open end and J\ is the Bessel function given in Eq. (2.5).
Values of the acoustic resistance ratio O = 1 — [2Ji(w)/w] and of
the acoustic reactance ratio M(w), for an open end, are given in
Table VIII at the back of the book. Their behavior for limiting
values of w is
If
Xo = M(w)
9o = 1 _ 2 ZlW^ («V8) <»<0.6)
w [ 1 (w > 5)
(4w/3t) (w < 1)
(4/ttw) (w > 12)
(23.14)
The approximate formulas differ from the correct values by less than
10 per cent within the designated ranges of w.
Consequently, the limiting values of specific acoustic impedance
at the open end are
 (a  • \ I (p« 2 ^ 2 /2c)  io)(8 P a/Sir) (X > &ro)
zi  P c{d t X o) — \ pc _ (i/ w )(2pc 2 Aa) [X < fcra/3)]
At low frequencies the resistive term is quite small, so that very
little energy is lost from the open end. Open tubes having cross
sectional perimeter much smaller than the sound wavelength are
therefore nearly as good hoarders of energy as are closed tubes, for
only a small percentage of the stored energy can be radiated away in
any cycle. The reactive term is a mass load, equal to a mass of air
(8pa/37r)g per sq cm of opening. When there is no flange on the end
of the tube, or when the flange is less than a wavelength in size, the
VI.23] PROPAGATION OF SOUND IN TUBES 247
reactance is reduced somewhat in magnitude, changing from (8pa/3ir)
to approximately 0.6 pa. Equation (23.1) for the effective length of
a tube with two open ends, used an approximate correction inter
mediate between these limits.
For short wavelengths, the impedance is almost entirely resistive,
approaching in value the characteristic acoustic resistance pc. In
this case there is practically no reflected wave; the incident wave
radiates out of the open end with little subsequent spreading, as
though the tube were still guiding it. The sound wave, as it emerges
into the open, does not realize that it has left the tube and so sees no
need for sending back a reflected wave.
The values of a and corresponding to the impedance Zi for the
open end will be labeled a p and /3 P . Their values as functions of
w = (^irva/c) are given in Table VIII at the back of the book. For
very large or very small values of w they become
I 27r(a/X) 2 = (co 2 /8x) (X > &ra)
ap — \ (l/2ir) ln(artyX) = (l/2r) ln(«ti>/2) [X < (ra/3)] .
(16a/&rX) = (4w/3tt 2 ) (X > 8xa)
(i) [X < (W3)]
According to Eqs. (23.9), the specific acoustic impedance at the
driving end of the tube, x = 0, is
pc tanh LY a p  ip p  i — J (23.16)
. tan(27rZ p /X) +z27r 2 (a/X) 2 ,. ^ ,
~ lpC l^ 2 (a/X) 2 tan(2x^/X) (X > Swa)
pci^ e** il/ * [X < (W3)]
Twa
where l p = I + (8a/3x).
Smalldiameter Open Tube. — For wavelengths long compared with
the tube perimeter 2xa, the input impedance is almost purely reactive,
except at the resonances and antiresonances [where tan(27rZ p /X) is
zero, or where it is infinity]. When the frequency is low enough (so
that X > 2l p ) , then the input impedance becomes
z — —ipcy^ 2 ) = —iuplp = —iupll + ^)> X > 2l p ^> 4ra
z =
X
In this case the impedance is a mass load equal to a column of air of
length l p = I + (8a/37r) ; the correction for the effective length of the
"248 PLANE WAVES OF SOUND [VI.23
column being proportional to the tube radius a (if the flange is not
large, the correction is approximately 0.6a). If the tube were open
at x = also, the correction would be twice this: l e = I + (16a/37r).
The effective length of the inductive element given in Eq. (23.1) is
intermediate between 1.7a and 1.2a (where S is set equal to xa 2 ).
As* long as the wavelength is long compared with 4ira, resonance
occurs (zq minimum) when tan (2tI p /\) is zero, i.e., when
27 r
K = =g; v n = ^n (n = 1,2,3 • • • ) (23.17)
where l p = I + (8a/37r) for a large flange, ~ I + 0.6a for a small
flange. The impedance at the input end for these frequencies is
a pure resistance which is lowest for the fundamental frequency and
increases as n 2 for the higher harmonics.
An open organ pipe is driven by a jet of air similar to that for a
closed organ pipe. When it is blown at the right jet speed, most of
the energy is in the fundamental, but there are small amounts of all
the upper harmonics present. With increasing jet speed the second
or higher harmonics can be emphasized.
A flute is also an open tube, set into oscillation by a jet of air,
which the player blows across the end at x = 0. The various musical
notes are produced by varying the effective length of the tube (by
opening one or more holes in the side of the tube) and by varying the
strength of the jet, so as to emphasize one or the other of the har
monics. The inner bore of the actual flute is not a uniform tube, but
is conical in shape; the effect of the conical shape will be discussed
later in this chapter.
Reed Instruments. — Clarinets are tubes, fairly uniform in diameter,
open at the output end, which are driven by a reed set into vibration
by a blast of air into the tube. Coupling with the air column in the
tube sets the reed into periodic oscillation, the air blast through the
reed being modulated by the motion of the reed. The driving system
here is more analogous to a constantcurrent generator than to a
constantvoltage one; the greatest response is when the specific
acoustic admittance (1/zo) is smallest, or z largest. The natural
frequencies are when tan (2rrl P /\) is infinite, or when
X m = ^; v m = 4 m (m = 1,3,5 • • • ) (23.19)
in 4tj>
VI.23] PROPAGATION OF SOUND IN TUBES 249
In other words the natural frequencies of a "closed pipe" (i.e.,
open at only one end) excited at the "closed" end by a reed are the
same as those of a "closed pipe" excited at the "open" end by a jet
of air, as given in Eq. (23.12).
The impedance at the driving end of the tube at the antiresonance
frequencies given in Eq. (23.19) is
as long as X TO is considerably larger than 4ira.
The air jet through the reed or reeds is modulated by the reed
vibration, and the reed vibration is maintained by the pressure fluctu
ations caused by the modulation, so the complete analysis of the
motion would be quite intricate. A crude approximation to the
actual state of affairs for a clarinet can be obtained, however, by
assuming that the force driving the reed or reeds is an impulsive one,
with the fundamental frequency of the tube i
'''•SI 15 ^}
= gfo[ • • •  Kt + T) + 5(0  8(t  T) + 8(t  2T)  • • • ]
(where T = 2l p /c), corresponding to a positive impulse (tF /2) at the
beginning of each cycle and a negative one — (wF Q /2) a half period T
later. This may be caused by the production of vortices at the tip
of the reed; at any rate it is governed by the air vibration in the tube
and possibly by the player's tongue.
Motion of the Reed. — The reed has its own resonance frequency
(a)r/2r), which is usually several times larger than vi. The reed
impedance is therefore (m r / — io))(a)l — co 2 ) where m r is the equivalent
mass of the reed, and the displacement of the reed from equilibrium is
^l (F /m r ) r , . , _ ire
Ji =
(23.21)
The air flow through the reed is proportional to the difference
between the pressure P maintained in the mouth of the player and the
pressure p at the input end of the tube and to the width (y + y ) of
the opening at the reed, where y is the equilibrium width. The
particle velocity at the input end of the tube is therefore
250 PLANE WAVES OF SOUND [VL23
u^o = G(P  p)(y + y)
where G is the "conductivity constant "of the mouth piece. Jf u is
expressed in terms of a Fourier series in odd harmonics,
00
w*o = U + 2 #»e~~* ,,l(2n+1)< (23.22)
w =
then the pressure at x = is obtained by multiplying each term in the
series for the velocity by the corresponding specific acoustic resistance,
obtained from Eq. (23.20),
°°  ,
Po = fii 2 [ (2» + l).» J e_l ' wl(2n+1)< J «i = (8pcZ p 2 A 2 « 2 )
This is valid as long as [2l p /ira{2n + 1)] is larger than unity; otherwise
our approximate formula for the impedance of the open end [first
formula of Eq. (23.16)] is not valid. Ordinarily l p is approximately
50a, so that the series is valid for n smaller than 10 or 15 (2n + 1
smaller than 20 or 30) The resistance R ± is approximately equal to
10 5 , indicating that the pressure fluctuation at the mouthpiece is much
larger than the velocity fluctuation (which corresponds to the fact
that the mouthpiece is at a velocity node).
We next insert the series for y [Eq. (23.21)] and p in the expression
for u and solve to obtain the values of the constants U n . At first we
shall neglect the small term Gyp and set
u = Uo + 2 u » e ~™ l(2n+1)t = GP Vo + GPy ~ Gy p
^Li 1 — {a)i/ur) 2 {2n + 1)
e — iui(2n+l)«
— Gv a R\ ^? — e wi(2«+i)i
Equating coefficients of the different exponentials, we have
Uo = GPy )
v = B{2n + l) 2 } (23.23)
^ n [(2w + l) 2 + M 2 ][iV 2  (2» + l) 2 ] )
where 5 = (GPF /m»«!), M 2 = GRiy ; and where iV = (« r /«i) is the
ratio between the reed frequency and the fundamental frequency of
the tube (N ~ 10 for the clarinet).
Pressure and Velocity at the Reed. — The pressure in the tube, a
distance x from the mouthpiece is, therefore, to the first approximation,
VI.23]
n =
PROPAGATION OF SOUND IN TUBES
(_l)ngio> l (2n+l)t
[(2» + l) 2 + M*][N 2  (2n + l) 2 ] '
251
• sin \~ (2n + l)(l p x)+i g (2n + l) 2 ]
The expression for the fluctuations in the velocity can be obtained
from Eq. (22.6), so that
uc~ Uo +
RiB
pc
j(— \\n e iat(2n+l)t
» =
[(2» + l) 2 + M 2 ][N 2  (2ft + l) 2 ]
Ul
pc
cos I ~ (2ft + l)(l p  *)> i j£ (2n + l) 2
and the pressure and velocity at the effective output end of the tube
u One Period >
Time
Fig. 55. — Motion of reed, wave pressure and air flow at mouthpiece for a simplified
model of clarinet. Resonance frequency of reed is ten times the fundamental tube
frequency.
[x = l p = I } (8a/37r)] are
Pwmh „ pcB *y __._(»» + l)HXl)v*"<*+»«
U +
Ux—lv
[(2n + l) 2 + M 2 ][N 2  (2ft + l) 2 ]
RlB ^ t '(_l)ngi«i(2n+l)t
pc
[(2ft + l) 2 + M 2 ][N 2  (2ft + l) 2 ]
Curves for pressure and velocity at the mouthpiece are shown in Fig.
55. We confirm that at the beginning of each halfperiod there is a
252 PLANE WAVES OF SOUND [VI.23
peak of back pressure from the tube at the mouthpiece, and a corre
sponding reversal in the motion of the reed. During the rest of the
cycle the oscillations correspond closely to those for the free vibration
of the reed, which has a natural frequency ten times the tube funda
mental (N = 10) for the case shown. This justifies (approximately)
the assumption made to obtain Eq. (23.21) for the motion of the reed.
We notice several interesting facts concerning the air vibrations
in this approximate solution for a clarinet. Here, in distinction from
the flute and open organ pipe, the odd harmonics are excited and the
driving end is at a minimum of the ac component of particle velocity,
rather than a minimum of pressure. At the open end the ac com
ponent of velocity is large (because Ri is large) and the pressure is
small, as it must be at an open end of dimensions smaller than a
wavelength.
The frequencies in the sound which are emphasized are the funda
mental of the tube and those overtones which are near the resonance
frequency of the reed, which is near the ninth or eleventh harmonic
for tones in the middle register of the clarinet. The total output
power can be computed from the pressure and ac component of
velocity at either the mouthpiece or the output end, by multiplying
the ac velocity and pressure, term by term, averaging and adding
and finally multiplying by the tube area, S = ira 2 :
n ~ —
— 2
RlB2 2{[(2n + l) 2 + MW (2» + 1)»]} (23 ' 24)
where the expressions for the constants R lf B, M, and N are given on
the preceding pages. The individual terms in the series equal the
power output in the different harmonics, and the square root of each
term is proportional to the pressure component for that frequency
in the sound which is heard.
Even Harmonics. — In the actual sound from a clarinet there are
present a few components of even harmonics, which are not included
in the series given above. This is because the modulation of the air
flow by the reed is not purely linear : in other words we have neglected
the term Gyp in the expression for u. This is small, but it does bring
in even harmonics. The largest even terms are due, of course, to
the product of the largest terms in the series for y, which is for (2w + 1)
ca N — (a>r/coi), times the largest term in p, which is for n = 0.
Therefore the largest evenharmonic terms in the series for the velocity
at x = are (using the formula for the product of two cosines)
VI.23] PROPAGATION OF SOUND IN TUBES 253
{RyB 2 /2P)
(1 + M 2 )(N 2  1)[N 2  (2n r + l) 2 ]
[cos (2n r o)it) — cos[(2n r + 2)w^]}
where n r is the integer for which (2n r + 1) is closest to N (for the clari
net n r is 4 or 5). We have neg ected the small term (pc/Ri)(2n f l) 2 .
We see that the most important even harmonics are those just above
and just below the odd harmonic which is nearest the reed resonance.
The pressure at x = for these frequencies is obtained by using
Eq. (23.18) for the impedance for even harmonics. It is equal to
(8 P 2 c 2 B 2 /P)
(1 + M 2 )(N 2  1)[N 2  (2n r + l) 2 ] '
• {n 2 cos[2n r coid — (n r + l) 2 cos[(2w r + 2)wifl}
Other even harmonics can be computed by using the less important
terms in the series for x and p, but their amplitude is too small to
make it worth while including them. Other corrections would also
come in, owing to the fact that the actual tube may be slightly conical
rather than of uniform cross section. In the case of the oboe, the
conical shape is so pronounced as to modify considerably the empha
sized frequencies, as will be indicated later.
The behavior discussed in the example we have treated in such
detail is typical of most wind instruments. The air in a tube, open
or closed, is excited by the oscillations of a reed, a jet of air, the horn
player's lips, the vocal cords, or some similar mechanism. The
oscillations in the tube react back on the driving oscillator to modify
its motion. Often the coupling is sufficiently close for the driving
oscillator to move with the frequencies characteristic of the tube, as
is the case of the reed instruments (although the reed instruments can
also oscillate with the frequency of the reed when played by an
unskilled player). In such instruments the natural frequency of the
driving oscillator is usually higher than the tube fundamental.
Other Wind Instruments. — In other instruments the frequency of
the driving oscillator is adjusted to the fundamental or to some har
monic of the tube. This is the case of the flute, the organ pipe, and
the horns; in such cases the resulting note depends on the tuning of
the driver (which tube harmonic is picked out) as well as on the
effective length of the tube. In the case of the singing voice, the
fundamental frequency of the driver, the vocal cords, is lower than
that of the cavity, and the coupling is weaker than in the other
examples. Here the fundamental is determined primarily by the
driver, and the resonating "pipe," the throat and mouth, are (more
or less) adjusted to one of the harmonics of the driver.
254 PLANE WAVES OF SOUND [VI.23
In all these instruments the coupling between driver and tube is
nonlinear, which makes it possible to extract energy from a steady
blast of air, but which makes the analysis of the motion quite difficult.
Once the oscillations start, they build up at the frequencies that allow
the system to extract a maximum amount of energy from the blast.
For instance, in the case of the driving jet across the tube end (flute,
organ pipe) a vibration of the jet causes a pressure fluctuation, and
the frequencies emphasized are those with the smallest input imped
ance (to make the velocity large), so that the driving end is at a pres
sure node and at a velocity maximum. When a reed is the driver, a
vibration of the reed causes a velocity fluctuation; therefore the resonant
frequencies are those with the largest input impedance (to make the
pressure large) so that the driving end is at a velocity node and at a
pressure maximum.
In some cases, there are alternative choices of frequencies, either
choice giving large energy, as for instance the low and middle registers
in reed instruments. Once started in one or the other of these choices
the energy builds up, and the air blast must be nearly stopped to
change to the other type. • Part of the skill of the player comes in
avoiding undesirable oscillations of the instrument.
A great deal of experimental and theoretical work is needed before
we can say we understand thoroughly the behavior of any of the wind
instruments.
Tube as an Analogous Transmission Line. — From our discussion
at the beginning of this section we see that we can consider any tube
of uniform crosssectional area *S to be analogous to an electric trans
mission line, with distributed series inductance L T = (p/S) per unit
length and distributed shunt capacitance C T = (S/pc 2 ) per uni t leng th.
The analogous characteristic impedance of the line is Z a = \Z L T /C T =
(pc/S), and the analogous propagation constant is —ia> \/L T C r =
— ioi/c, as long as we neglect the energy loss of the wave as it travels
along the tube.
The analogous impedance of a round open end, of radius a, fitted
with a large flange is
Z„ — „
1 &
pc_
ira 2
L (2coa/c) J ira 2 \ c /
( (jk>*/2kc)  ^(8p/37r 2 a) [« « (c/2o)] m „_.
\ (pc/Tra 2 )  (;/a,)(2pc 2 Ar 2 a 3 ) (« » (c/2a)] V ' '
shunted across the end of the transmission line.
VI.23] PROPAGATION OF SOUND IN TUBES 255
A change of crosssectional area corresponds to a change of dis
tributed constants for the line; an increase of area corresponding to
an increase in the distributed capacitance and to a decrease in the
distributed inductance; and vice versa. Therefore a narrow portion
of the tube corresponds more nearly to a pure inductance and a wide
portion to a pure capacitance, as we assumed at the beginning of this
section.
A hole in the side of the tube will correspond to an inductance and
resistance shunted across the line at the proper point. According to
Eq. (23.25) the resistance should be approximately (p« 2 /27rc) and
the inductance should be approximately (0.5p/b) where b is the radius
of the open hole, and where we have taken into account the fact that
the effective mass of air in the hole extends somewhat inside the hole,
as well as outside, as was done in deriving Eq. (23.1).
After the analogous impedance at the input end has been com
puted, the specific acoustic impedance can be obtained by multiplying
by Si, and the mechanical impedance of the air column can be obtained
by multiplying the analogous impedance by Sf, where Si is the area
of the input end.
Open Tube, Any Diameter. — We now return to the general problem
of the piston and open tube and treat the intermediate case where the
diameter of the tube is neither large nor small compared with the
wave length. As we shall see in the next chapter, the acoustic imped
ance ratio at the open end of a cylindrical tube equipped with a baffle
flange is
?"»,
M  2 *^M _ iM(w)\ (23.26)
where w = (Anrva/c) = (4ra/X) and a is the radius of the tube. Values
of the functions 1 — [2Ji(w)/w] and M(w) are given in Table VIII
at the back of the book. Equation (23.26) is valid for all wavelengths;
the two limiting cases, corresponding to w — » and w — > °° , have been
discussed previously.
According to Eqs. (23.9), the direct and reflected waves inside the
tube must adjust themselves so that
tanh $ +
rv»)l
f%nP
is equal to the value of f (l) given above. To facilitate this calculation,
solutions of the equation
tanh [ir(a p  #„)] = 12 ^^  iM(w)
256 PLANE WAVES OF SOUND [VI.28
are also given in Table VIII. Once the values of a p and P are found
for a given value of w = (47ra/X), calculation of the properties of the
pistontube system is a simple matter. The ratio of reflected to
incident amplitude is e~ 2 * a . The specific acoustic impedance at the
piston is pc tanh Ur ( a p — i$ p — i — ) \> and the radiation impedance of
the piston itself is
%r = pc (q) tanh L f a p — i$ p — i— J = R r — iX r
R r = pee (fjf), x r = pcx (§)
Values of e and x can be obtained from Plates I and II, or from Table
III at the back of the book.
If the piston has a mechanical impedance Z p = R p — iX p , then
the total mechanical impedance of the piston is Z m = R m — iX m ,
where R m = R r + R P , X m — X r + X p . If the piston is driven by a
force of amplitude F , then its velocity u and amplitude of motion
So, the intensity T of the sound in the tube, and the power n radiated
out of the end of the tube are given by the formulas
Uq =
(23.27)
2(RI + XIY \SJ 2S{Rl + Xl)
As an example, the acoustic resistance ratio e and reactance ratio x
at the piston end are shown in Fig. 56 for a tube whose length I is ir
times its diameter 2a [so that w = (Ara/X) = 2Z/X]. The quantities
are plotted as functions of (2Z/X) = (2vl/c). For the low frequencies
the impedance at the open end is nearly all reactive, so that little of
the energy escapes, and large resonance peaks can occur. The points
of minimum impedance, where resonance occurs, are separated by
points of high impedance. As the frequency increases, more and more
of the energy reaching the open end is radiated out, less and less is
reflected back to help in the resonance, and the peaks and valleys in
the impedance curve get less and less pronounced. At very high
frequencies the reactance is zero, the resistance is pc, and no resonance
occurs.
The integral values of (2Z/X), shown by the dashed vertical lines,
correspond to the frequencies of resonance of a tube with an open end
VI.23]
PROPAGATION OF SOUND IN TUBES
257
which allows no energy to escape (i.e., a "perfect" organ pipe).. We
notice that the actual frequencies of resonance are slightly lower than
this for low frequencies and are considerably lower at high frequencies,
approaching the points midway between two lines.
The other two curves in Fig. 56 show the behavior of the acoustic
conductance and susceptance ratios, the real and imaginary parts of
the admittance (pc/z), at the end x = 0. If the driving piston at
x = has a constant driving force to apply to the air in the tube
[which would be the case in Eq. (23.27) if Z p <£ Z r ], then the power
3r
to
y
y
_J5
I I I
Zr
2
\U
1 
"I
n _• i
1 ,. ,1 1 1
4 6 8 Z 4
XllM (21 A)
Fig. 56. — Acoustic resistance, reactance, conductance, and susceptance ratios for
the driving end of a tube with other end open. The length of the tube is ir times its
diameter. If the driving piston is a "constant velocity generator," the power radiated
is proportional to the resistance; if the piston is a "constant pressure generator," the
power radiated is proportional to the conductance.
radiated would be proportional to the conductance ratio k —
0/(0 2 + x 2 ). The curve shows that if the piston were driven by a
".pure noise" force, having all frequencies present in equal magnitude,
those frequencies corresponding (approximately) to the first three or
four harmonics of the resonance fundamental (c/2l) will be strongly
reinforced, but the higher harmonics will not be reinforced very much.
If the piston impedance Z v is larger than Z r , the veloeity amplitude
of the piston will be nearly independent of v (if Z v is nearly constant)
and the power radiated will be proportional to 0, the acoustic resistance
258 PLANE WAVES OF SOUND [VI.2S
ratio. In this case, as the curves show, the first three or four of the
odd harmonics of the antiresonance frequency (c/4Z) will be reinforced,
and we have the analogue of the clarinet, as was discussed earlier in
this section.
In either case, however, the higher harmonics will not be reinforced
very much, so that a tube of this shape (I = 2ra), driven by a "pure
noise" force, will give out sound having only the first three or four
harmonics present in any appreciable intensity. A pipe that has a
larger diameter for its length than this will have even fewer high
resonance peaks and will give out a tone even poorer in harmonics.
One having a smaller diameter will have more peaks, and the tone
given out will be rich in harmonics.
Cavity Resonance. — Another problem that can be solved by the
hyperbolic tangent method is that of a plane wave falling on the
open end of a tube fitted with a flange and closed at the other end.
Suppose that the plane wave travels along the a>axis, which is the
axis of the tube and is normal to the flange, that the wave has fre
quency v and that far from the tube it has pressure amplitude P f and
velocity amplitude Uf = P//pc. When it strikes the open end of the
tube, it agitates the air in the open end, sending a wave down the tube
to the closed end where it is reflected. The reflected wave returns to
the open end, and part of it radiates out into the open again.
This method of analysis considers the wave reflected from the
tubeplusflange system to be made up of two parts : the wave reflected
from the flange, and the wave radiating from the open end. For wave
lengths long compared with the radius of the opening a, the wave
radiated from the mouth will spread out in all directions (as we shall
see in the next chapter), while the wave reflected from the baffle will
return along the #axis, if the baffle flange is large compared with
the wavelength. If the wavelength is small compared with a, both
waves will return along the negative zaxis without appreciable side
ward spreading, and there will be interference between the two waves.
For very short waves, the phase difference between the two reflected
waves is just (4xZ/X).
If the open end of the tube is provided with a large baffle flange,
the incoming plane wave will be reflected from it; and close to the
flange (which will be taken as the plane x = 0) the pressure due to the
plane wave has the value 2P f e~ 2 ' iyt , and the corresponding particle
velocity is zero. The baffle flange can sustain this pressure without
moving, but the air in the end of the tube cannot. To the degree of
approximation considered in this chapter, the air in the tube mouth
VI.23] PROPAGATION OF SOUND IN TUBES 259
will act as though it were a plane piston without mass or thickness,
driven by a pressure of amplitude Po = 2P f , sending waves down the
tube and also out into the open. The specific acoustic impedance
for each square centimeter of the piston is that due to a tube of length I
closed at the other end plus that due to radiation from a piston set in a
baffle. From Eqs. (23.10) and (23.13) the impedance ratio is
To = [l  2 1M]  i [M(w)  cot(^)]; w = 4WX
The velocity of the air in the mouth of the tube is therefore (d%/dt)
= (Po/pcf o^ 2 ™"', and the velocity and pressure inside the tube are
u = _ Po ,{ P< L, sin \^ax)\ e M *
f o sin(27rZ/X)
iPo
fo sin(27rZ/A)
sin y (Z  x)\
V = v ^/o_7/^ c os y (Z  x)
by Eq. (23.8). The pressure just outside the open end is the driving
pressure P e~ 2 * ivt given above, plus the pressure due to the outgoing
wave caused by the vibration of the air in the mouth.
The ratio of the pressure amplitude at the closed end to the pressure
amplitude at the open end is obtained from the foregoing equations:
®{[i»^r'(?)
+ [MMsin(^)cos(^
i
(23.28)
where w = (4ira/X). Tables of the functions involved are given at
the back of the book. The phase difference between the pressure at
x = I and what it would be at the same spot (x = I) if there had been
no tube or flange present turns out to be
q = cot" 1
cot(2TZ/X)  M(w)
1_2 Jl ^
w
(t)
When the wavelength is so long that w is very small, the pressure
p
at the far end of the tube becomes Pi * " 7/ . . Resonance
cos (2xi/X)
occurs whenever the frequency equals one of the natural frequencies
of vibration of the closed pipe v n = (nc/4Z), (n = 1, 3, 5 • • • )• What
happens is that some of the wave enters the tube and is trapped
260
PLANE WAVES OF SOUND
[VI.23
there, because of the low radiating efficiency of the open end for
long wavelengths. If the wavelength happens to be the proper size,
resonance occurs.
When the wavelength is so small that w is large, the pressure at
x — I becomes Pi — » P , independent of v. The efficiency of the open
end for radiating sound is so large that no wave can stay long enough
in the tube to produce resonance. The ratio Pi/Po is plotted in Fig.
57, for a tube whose length is t times its diameter (as in Fig. 56), as
H>
2l A
Fig. 57. — The ratio of pressure at the closed end to pressure at the open end,
for the tube shown in Fig. 56 when I = 2ira. The peaks are examples of cavity
resonance.
a function of (2Z/X) == (2lv/c). It shows the transition from strong
resonance at low frequencies to a nearly uniform response at very
high frequencies.
We have assumed above that the driving pressure at the open end
of the tube P is twice the pressure amplitude in free space P f . This
is true as long as the baffle flange is large compared with the wave
length of the sound; but when the wavelength is very large compared
with the size of the baffle, P becomes equal to P f . The curve for the
ratio of Pi to Pf will therefore be similar to the curve shown in Fig. 57,
over most of the range of (2Z/X), except that the vertical scale will be
doubled. If the flange is not very large, the curve for (Pi/Pf) will
be more like the dotted curve at small values of (2Z/X), reducing to
unit value (on the doubled scale) as a lower limit, owing to the reduc
tion of the value of (Po/Pf) from two to unity for long wavelengths.
This will be discussed in detail in the next chapter.
VI.23] PROPAGATION OF SOUND IN TUBES 261
Many condensermicrophone diaphragms are stretched by means
of a ring clamp set in front of the metal membrane, forming a short
tube with the diaphragm closing  one end. When the wavelength
of the sound is equal to four times the length of the tube, resonance
of the sort discussed above occurs and will be quite marked unless the
radius of the tube is much larger than its length. Such resonance in
condenser microphones is termed cavity resonance.
It is interesting to note that the dips in the contours of loudness
level, in Fig. 52, at 4,000 cps, are due to cavity resonance of the air
in the outer ear. At this frequency the pressure at the ear diaphragm
is several times the pressure just outside the ear; so that cavity
resonance acts to make the ear as a whole more sensitive at 4,000 cps
than at 1,000 cps or at 10,000 cps, although the inner ear is not more
sensitive at 4,000 than it is at 1,000 cps.
Transient Effects, Flutter Echo. — As an example of the application
of the contourintegral methods of calculation to waves in tubes, we
shall consider the case of a uniform tube of length I, fitted with a
piston at the end x = 0, and fitted at x = I with a termination having
a purely resistive mechanical impedance. The piston will be held
at rest until time t = 0, when it will be suddenly moved inward a
unit distance. This will cause a pulse to travel down the tube. The
pulse reflected from the resistive termination will not be so intense
as the incident pulse. By the time this reaches x = the piston is
at rest, so that the pulse will be reflected back down the tube without
loss of intensity. At every reflection from the end at x = I, how
ever, there will be a further reduction in amplitude.
One can guess the general form the reechoing pulse wave must take.
Presumably it can be expressed in terms of the following function:
yy [o<t<(i/ C )]
_ e  2 ™ 6 ^ _ 2 l^j [(l/c) < t < (2l/c)]
Flu(x,t) = \ ^" «V H IW C ) < « < <«/«>] (23.29)
f4±*)
[(4Z/c) < t < (52/c)]
etc.
262 PLANE WAVES OF SOUND [VI.23
This represents a pulse reflecting back and forth from one end
of the tube to the other, reducing in amplitude by an amount e~ 2va
every time it reflects from the end x = I. Such pulse echoes are
called flutter echoes. They are encountered in rooms with plane
parallel walls of high acoustic reflectivity. Any pulse wave generated
between the walls will bounce back and forth many times before
dying out. Proper auditorium design is aimed, in part, at the elimi
nation of flutter echoes, as will be explained in Chap. VIII.
Of course we have not yet shown that Eq. (23.29) is the correct
solution of the problem we outlined at the beginning of this section.
Before we use the contourintegration method to show that this
formula is the correct solution, it will be advisable to modify the form
of the expression for the function Flu(x,t), so we can recognize our
answer when it is obtained. A pleasant exercise for the reader would
be to juggle the delta functions around to arrive at
Flu(x,t) = <rW»<«<> ^ « 6 " 2m ' c + * )
jgjA.aLzJ^ (0<x<t)
m =
.g— Ora/0 (ct+z)
The two sums represent periodic waves, the first going in the positive
^direction, the second in the negative ^direction. Each sum is
modulated by an exponential damping term which in effect reduces
the amplitude of each succeeding pulse by a factor e~ 2 ™.
The sums, being periodic, can be expressed in terms of a Fourier
series. The first sum, for instance, can be expressed in terms of the
series
The coefficient A n can be obtained, as on page 87, by multiplying
both series by cos [(WO (^  x)] and integrating over [t  (x/c)]
from (l/c) to +(l/c). This results in the equations
1 = (^)a ; l = fVn (n = 1,2,3 • • • )
= (?); ev
Therefore the flutterecho function Flu(x,t) can be expressed in terms
of the series
VI.23] PROPAGATION OF SOUND IN TUBES 263
Ftu M = ^<« {£ + \ J cos [(=<) (*  f)]}
_ ^,(ira/I)(c<+x)
)2Z ' I _
provided, of course, that we waive questions of convergence (as we
mentioned on page 98). To complete our discussion we must show
that this series is, in fact, obtained by contour integration of the
transient function.
From Eqs. (23.8) and (23.9) we see that the air velocity in a simple
harmonic wave in a tube with terminal acoustic resistance R at x = lis
u = Ae'™ 1 cosh \ira — ~ (J — x)
f iia
I ira
= Ae~ iat sinh \ira (I — x)
(tic \
ira — ~ J = (pc/R) = k. If the piston
velocity is e™', the velocity in the tube is
_ _ faf sinhfra — (ia>/c)(l — x)] _ _ e'™ 1 sinhfra — (ua/c)(l  x) ]
~ sinh [ia — (icol/c)] i sin [((ol/c) + *Va:]
According to the discussion at the end of Sec. 6, if the piston
velocity is 8(t), the air velocity in the tube is
v ' Am J a, sin[(wt/c) + lira]
The poles of this integrand are at co = (c/l)(irn — iira), where n
is any integer, positive or negative (or zero). Close to the nth pole
the sine function is ( — l) n (l/c)[w — (irc/l){n — ia)] and the integral
around this pole, for the first exponential, becomes [since ( — 1)" = e™ n ]
_ _L_ (   (D e i«>(tx/c)+Ta(iul/c)+iirn ' dC0
 ia)\
4iri \lj J _ w ~ (irc/l)(n
— _£_ (,(Ta/l)(ctx)(iTn/l)(.ct—x)
There is another pole at the corresponding negative value of n, and
the two exponentials will combine to form the term (c/Z)e~ (Ta/Z)(c ' :r)
cos\{irn/l)(ct — x)]. Similar calculations for the second exponential
264 PLANE WAVES OF SOUND [VI.28
give similar terms; finally, by putting together all the terms, including
those for n = 0, we are able to show that
u s (t) = Flu(x,t) (23.31)
Returning to Eq. (23.29), we can use the relation p(du/dt) = — (dp/dx)
to obtain the corresponding pressure wave p&(t) = pcFlp(x,t), where
5 (t  fj [0< * < (l/c)]
e 2*a 8 L _ ^~j [(I/C) < t <(21/C)]
Flp(x,t) = < / 21 4 x\ (23.32)
e 4™ 5 h _ ^^) [(3Z/c) < i < (4Z/c)]
etc.
The pressure wave does not change sign on reflection, but it reduces
in amplitude on reflection from the end x = I by the same factor
e 2wa = (r _ P c)/(R + pc) as does the velocity wave (as, of course,
it must).
As in our earlier discussions of transient effects, the wave generated
by moving the piston in some arbitrary way,
{0 (t<0)
g(t) (0<t<A)
(t > A)
where g(t) is any arbitrary function of t, can be expressed in terms of
the pulse waves:
u{t,x) = f g(T)Flu(x,t — t) dr
p(t,x) = pc J[ A g(r)Flp(x,t  t) dr
If g(t) is a sensible sort of function of t, the resulting series for u and p
will converge satisfactorily, so that the question of the convergence of
the Fourier series for Flu is only academic, as was pointed out in
Sec. 10.
We note that if the duration A of the motion of the piston is less
than (21 /c), the reaction of the air during the motion is as though the
tube were infinite in length (in other words it exhibits its characteristic
resistance pc). We note also that, if the specific acoustic resistance R
of the termination were to depend on co, the reflected wave would not
in general be a simple pulse, but would be spread out more than the
VI.23] PROPAGATION OF SOUND IN TUBES 265
incident pulse. If the termination impedance is not pure resistive
but has a reactive part that also depends on w, the reflected wave would
also be "blurred out."
24. PROPAGATION OF SOUND IN HORNS
As we have noted before, and as we shall treat in detail in the next
chapter, a tube whose open end has a diameter smaller than the wave
length of the sound sent out is a very inefficient radiator of sound.
This inefficiency is a desideratum in organ pipes and other windinstru
ments, for in these cases resonance is important, and most of the wave
must be reflected back into the tube from the open end to have sharp
resonance.
The small radiating efficiency of a small opening (or a small dia
phragm) is a very great detriment in a loudspeaker, however, for a
loudspeaker should have no marked resonance frequencies. Two
general methods are in use for building loudspeakers so as to radiate
sound efficiently. One method is to make the vibrating diaphragm
large enough to radiate well (dynamic speakers are of this type).
This type of design has the advantage of compactness, but it requires
comparatively heavy moving parts with their consequent mechanical
inefficiencies. The other method is to use a small diaphragm and to
magnify its effective size by using a flaring tube, a horn. One purpose
of the loudspeaker horn is to spread the concentrated waves coming
from the diaphragm out over a large enough area so that they can
continue out from the mouth of the horn with very little reflection
back to the diaphragm. In this way a light diaphragm can be used,
and acoustic efficiency can be maintained at the same time. Another
occasionally important purpose of the horn is to concentrate the sound
into a directed beam, so that most of the radiated energy is sent out
in one direction.
The horn must not flare too rapidly, for the sound waves will then
not "cling" to the inner surface of the horn and spread out but will
act as though they were already out in free space, radiating from a
small diaphragm and exhibiting a correspondingly small efficiency.
Therefore in order that the mouth of the horn may be large in diam
eter, the horn must be long.
Oneparameter Waves. — The analysis of wave motion in a horn is a
very complicated matter, so complicated that it has been done in a
rigorous manner only for conical and hyperbolic horns. If the horn
does not flare too much, however, so that we can consider the wave
as spreading out uniformly over a cross section of the horn as it travels
266 PLANE WAVES OF SOUND [VI.24
outward, then we can use an approximate method of calculation and
obtain fairly satisfactory results. For in this case the displacement
of the gas molecules all over a surface perpendicular to the axis of the
horn will be the same, and the displacement, pressure, etc., will be
functions only of t and of the distance x along the horn.
The statements made in the last paragraph require further dis
cussion to bring out all their implications; although a detailed discus
sion would lead us into the far reaches of differential geometry and
would require a more complete knowledge of the properties of the
wave equation in three dimensions than we shall have room to cover
in the next chapter.' Nevertheless, a few paragraphs of general
discussion are needed here, to indicate the approximations and limita
tions of the assumption that a wave in a horn is a onedimensional
wave.
The shape of the inner surface of the horn of course affects the
shape of the wave, and since the horn cross section is not independent
of x, we should expect the wave to depend on the coordinates per
pendicular to x as well as on x, and thus not be a plane wave. What
we must discuss are the conditions under which the wave will approxi
mate the behavior of a plane wave, and what we should derive (but
will not because we have not in this book the requisite mathematical
tools) are the criteria showing when these conditions occur.
A plane wave, traveling in the positive ^direction, has a phase
which is the same everywhere over the surface of a plane perpendicular
to x. In the usual exponential notation,
p = Ae**™ 1 (24.1)
The quantity A is the amplitude of the wave and can be taken to be
real. The quantity 4>, also a real function, is the phase of the wave.
In the case of the plane wave <f> = {a/c)(x  x ), so that over a plane
perpendicular to the xaxis (x = a, for instance) <f> is constant. In a
plane wave the surfaces of constant phase are planes, perpendicular
to the xaxis, moving along x with a speed c.
Other waves can be set up which have the same general form as
Eq. (24.1), though their surfaces of constant phase are not planes,
and their amplitude is not constant. In the next chapter we shall
see that a spherical wave, radiating out from a point source, has the
form of Eq. (24.1). The amplitude A for the spherical wave is
inversely proportional to r, the distance from the source, and the
phase <t> = (w/c)(r  r ) is proportional to r. In this case, therefore,
the surfaces of constant phase are spheres, expanding outward from
VI.24]
PROPAGATION OF SOUND IN HORNS
267
the source. Just as with the plane wave, however, both amplitude
and phase are functions of only one coordinate.
In terms of these concepts we can restate our requirements for a
"good" horn. We first set up a system of mutually perpendicular
.H=I0
Fig. 58. — Coordinate surfaces corresponding to three types of horns. To the degree
of approximation of Eq. (24.2), surfaces of constant phase coincide with the jusurfaces
and the particle velocity is parallel to the lines = constant. See Eq. (24.3).
coordinates (/*,#) suitable for the horn, so that the inner surface of
the horn corresponds to # = # s (a constant), the axis of the horn (the
xaxis) corresponds to # = 0, and so that n = x when # = {i.e., the
scale factor for fi is the same as for x along the axis of the horn).
Samples of such coordinate systems are given in Fig. 58, for several
types of horn shapes.
268 PLANE WAVES OF SOUND [VI.24
We then set up the wave equation in these coordinates (this is, of
course, difficult and is part of the detailed discussion that must be
omitted here) and see whether we can obtain a solution of the form
given in Eq. (24.1), with amplitude and phase depending only on the
coordinate /*. If this can be done accurately, the wave is not a plane
wave, but it is a oneparameter wave, which can then be handled as
we have been handling planewave expressions. In this case the
problem is straightforward.
It turns out that very few coordinate systems give rise to a wave
equation simple enough to allow a solution that is a function of only
one coordinate (only the first of the three systems shown in Fig. 58
has this property). When the coordinates do not have this property,
the particle velocity will not be parallel to the /z coordinate lines, and
the wave will tend to reflect from the horn surface as it travels along,
rather than moving parallel to it. As we have seen in the previous
section, and shall see again in the next chapter, any reflection of the
wave during its' progress along a tube reduces the amount of energy
traveling out of the tube and tends to trap some of the energy inside,
causing resonance for some frequencies and poor transmission for
others. This condition is satisfactory for musical instruments, where
we desire strong resonances, but it is not desirable for loudspeaker
horns, where we desire a uniform transmission, independent of fre
quency. Consequently, we can say that one criterion for a good horn
is that it should be possible to set up oneparameter waves inside it;
that a possible solution of the Wave equation in coordinates suitable
for the horn should be a function of the coordinate n alone.
For some coordinate systems the correct solution is nearly a one
parameter function, and the properties of the corresponding horn,
deduced by using an approximate oneparameter solution, will be
close to the true behavior. It is always possible (though often diffi
cult) to check the degree of approximation involved for a given horn
by setting up the exact wave equation in the suitable coordinates,
substituting p = AGu)^ 00 * 61 ' in the equation and determining by
what amount this function does not satisfy the equation.
An Approximate Wave Equation. — If we find that a oneparameter
solution is approximately correct for some horn shape, we should be
able to set up a onedimensional wave equation that is approximately
correct, from which we can obtain the oneparameter approximate
solution. This approximate equation is not so difficult to obtain or
to work with as is the exact wave equation, and we are now in a
position to derive it.
VI.24] PROPAGATION OF SOUND IN HORNS 269
We consider a thin shell of air between two neighboring surfaces
of constant phase (between <t> and # + d<(>), rather than a plane sheet
between x and x + dx, as we did for the plane waves. If a one
parameter wave is a good approximation to the correct solution, this
is nearly equal to the air between the coordinate surfaces p and
/x. + dft. The total area of that part of the coordinate surface p
which is inside the horn is called S, which is a function of p only (if
the horn is a "good" one, S turns out to be proportional to the area
of the cross section of the horn which is tangential to the coordinate
surface p) . Therefore the volume of the gas in the shell at equilibrium
is S dp (or approximately S dx, since p = x along the horn axis).
During the passage of the wave, the surface of air originally at p is
displaced to p + £, and the new volume of the shell is
giving a fractional change in volume of (1//S) ^ (S£), which corre
sponds to Eq. (22.3) and to the equation p = ~("o")r (^)>
relating pressure and particle displacement. The net force on the
shell of air is everywhere perpendicular to the jusurf ace (if the approxi
mation is good), and therefore Eq. (22.6) still holds, since differenti
ation with respect to p is the same as differentiation with respect to
x along the horn axis. We next differentiate twice with respect to
time the equation for p given above, and substitute for (d 2 £/d£ 2 ) from
Eq. (22.6), obtaining, finally
l±(^dp\_L^p. §u _ldp (
Sdx\ dxj c 2 dt 2 ' dt P dx K }
where, as before, c 2 = (Po7c/p) and where u is the magnitude of the
particle' velocity (the direction is perpendicular to the jusurfaces,
parallel to the & coordinate). If the waves are simple harmonic, the
relation between u and p is
1 dp
tup ox
Possible Horn Shapes. — If approximate Eq. (24.2) is valid for a
given horn shape, then we should expect, to the same approximation,
that a wave traveling out of the horn would be represented by the
form given in Eq. (24.1). Moreover, we should expect that the
270
PLANE WAVES OF SOUND
[VI.24
amplitude A will (to the same accuracy) be inversely proportional to
the square root of S. For the intensity of sound in the wave should
be proportional to A 2 , and the total energy flow out of the horn would
therefore be A 2 S, which must be independent of x if energy is to be
conserved.
We set S(x) = ir[y{x)] 2 , where y is the effective radius of the
cross section of the horn at a distance x from its small end. Inserting
the expression p = {B/y)e i '^ iut into Eq. (24.2), we obtain
(ejy iMA_M 2
\dxj ^ y\dx 2 ) \c)
dx 2
The real and imaginary parts of this expression must be zero sepa
rately, for both y and <j> are real. The imaginary part indicates that
(d<f>/dx) must be a constant, which we can call (tco/c) for reasons that
will shortly be apparent.
Finally, in order that the approximations we have made in the
previous paragraph be all valid and selfconsistent, the real part of
the equation must also be zero, which means that the effective radius
of cross section of the horn must (at least approximately) satisfy the
equation
S(f)'o^«
A solution of this equation, properly adorned with constants, is
y = yo
S = So
T 2 = 1 ■
co 8 h() + rs inh()
co S h() + rsi nh()
\uhj \2rhJ
(24.3)
and the corresponding approximate solution for the plane wave travel
ing out of the horn is
V
Q>
1 pHa/c){rx—ct)
(24.4)
The shape of the horn represented by Eq. (24.3) is determined by
the relative values of the constants T and h. The constants y and
S = iryl fix the size of the throat of the horn at x = 0. The "scale
factor" h determines the rapidity of the "flare"; the smaller h is, the
greater is the curvature of the plot of y against x. The constant T
VI.24] PROPAGATION OF SOUND IN HORNS 271
is the "shape factor," determining the general properties of the horn
near the throat. When T = (h/x ) and h is allowed to go to infinity,
the horn is a conical horn, with angle tan" 1 (y /x ) between the axis
and one of the elements of the cone [the "angle of opening of the cone"
is 2tan 1 (Wzo)]. When T = 1, y = y e x/h , and the horn is an
exponential horn. When T = the shape of the generator of the
surface is a catenary, so the horn will be called a catenoidal horn. 1
This horn can join smoothly onto a uniform tube at x = 0, whereas
the conical and exponential horns have a discontinuity in slope of
y{x) at x = 0, making a worse impedance match with the uniform
tube. For large values of x, the catenoidal horn is indistinguishable
from the exponential horn. The shapes of all three types are shown
in Fig. 58. When the exact wave equation for the coordinates appro
priate to the horns of Eq. (24.3) is worked out, it can be shown that
Eq. (24.4) is a reasonably close approximation to an exact solution
as long as the radius y of the small end of the horn (the throat) is
much smaller than the scale factor h.
We shall next discuss in detail the acoustical properties of each
of the three main types of horns expressible in terms of Eq. (24.3),
for cases where the big end of the horn (the mouth) is large enough
so that no part of the wave coming out of the horn is reflected back
toward the throat. Finally we shall take up the case where the open
end is small compared with the wavelength (which is the case of some
wind instruments), where reflection from the mouth, and consequent
resonance, occurs.
The Conical Horn. — In a conical horn of angle # s (angle of opening
2#„ area of throat irxl sin 2 & a ) the areas of the phase surfaces are
S = So \l + yyl ; S = iryl y = 2*o sm(^J
where x is the distance back from the throat to where the apex of the
cone would be, if the cone were extended back to its apex.
Equation (24.2) turns out to be
1 d \, , , 2 dp] 1 d 2 p
(x + x ) 2 dxl K ' u/ dx] c 2 dt 2
which is the exact wave equation for a onedimensional spherical
1 Although it would also be appropriate to call it a "Salmon" horn, after the
person who first discussed the acoustical properties of the family of horns of Eq.
(24.3).
272 PLANE WAVES OF SOUND [VI.24
wave. The outgoingwave solution of this is
p
fp = gt(w/c) (x—ct)
X + X
The particle velocity at x is
pz L x + ZoJ ^
In this case the phase velocity of the wave inside the horn is the
same as it is in free space, c. Since the wave front spreads out as it
travels down the horn, the impedance looking out of the horn from
the small end is not the same as for a uniform tube, even if there is no
reflected wave:
(p\ _ pc _ pc
uj ~~ 1 + i(c/oox ) ~ 1 + i(\/2rx ) (24  5)
The analogous impedance of the throat of the conical horn, (z /S ),
is equivalent to a resistance (pc/S ) in parallel with an inductance
(pXo/So). Therefore any acoustical filter, or other circuit, that is
terminated by a conical horn has an equivalent electric circuit ter
minated by this shunt combination. The power lost in the resistive
arm represents the power radiated out of the open end. The shunt
inductance has very little effect on the behavior of the horn at high
frequencies (X < < 2inro), but at lower frequencies it shunts out the
resistor (pc/S ).
The specific acoustic impedance at the throat of the horn is there
fore
z = R — iX = \z\e^
R = pC Y = (pcX/27T3 )
1 + (\/2ttXo) 2 ' 1 +. (\/2irXo) 2
PS: _ pcxpw ^ (24.6)
\A + (X/27TX ) 2 VC 2 + (XOCO) 2
\z =
<p = tan
— i
\2irx J \XouJ
If the particle velocity at the throat of the horn, causing the out
going spherical wave, is Woe 2 ™"', then the power radiated out of the
horn is
IT  1 ,,2PO _ i pCMJ}ff _ ! „ , (2t;eop) 9
VI.24]
PROPAGATION OF SOUND IN HORNS
273
In this case the power radiated is small at low frequencies and rises
steadily as the frequency is increased, approaching asymptotically
the limiting value ?Sopcu$. The limiting value is the power that the
same piston would send into a uniform tube of infinite length.
Transmission Coefficient. — We are now in a position to define
what we mean by the term "radiating efficiency of a horn" which has
been so freely used above. A measure of this efficiency is the ratio
of the power radiated out of a given horn to the power radiated by the
same diaphragm, moving at the same velocity, into a cylindrical tube
of infinite length, having the same crosssectional area as the small end
of the horn. This ratio, for any type of horn, is defined as
2n
Sopcul
©
(24.7)
and will be called the transmission coefficient of the horn for sound.
This ratio is not, strictly speak
ing, an efficiency, for it is the
ratio of the actual power radiated
to the power radiated for a stand
ard case, not an ideal case. Con
sequently, t sometimes becomes
greater than unity. In general,
however, r varies between zero
and unity. When it is small the
diaphragm will have to vibrate
with large amplitude to radiate
much power. When the veloc
ity amplitude of the piston is
independent of frequency, the power radiated is proportional to the
transmission coefficient. In any case the power radiated is
n = iuiSopcr
The transmission coefficient for a long conical horn with wide
mouth is
1 (2tvx ) 2
0.2 0.4 0.6
Fig. 59. — Transmission coefficient for a
long conical horn with open end large
enough to eliminate resonance.
1 + (\/2tx q ) 2 c 2 + (2irvx y
(24.8)
It is plotted in Fig. 59 as a function of (x v/c) — (x /\) = lb/\(a — b),
where I is the length of the horn and a and 6 are the radii of its large
and small ends, respectively. This function has* the value  when
(xq/\) = \/2ir or when the frequency equals c(a — b)/2rlb. For fre
274 PLANE WAVES OF SOUND [VI.24
quencies above this the transmission coefficient approaches unity,
and the horn is relatively efficient as a radiator. For frequencies
below this the horn is quite inefficient. The smaller the flare of the
horn [i.e., the smaller the value of (a — b)/l\, the lower the frequency
at which the horn will be efficient.
A Horn Loudspeaker. — We can now put some of these formulas
to work by showing how they can be used in designing horn loud
speakers and in analyzing their behavior. The sketch at the top of
Fig. 60 shows the longitudinal cross section of a simplified version of a
conical horn loudspeaker. The piston P is considered to move as a
single mass of mechanical impedance (without the air load) Z p =
R P — i[wm p — (K p /a))] dynesec per cm. Its radiating surface has
area S p . The area of cross section of the throat of the horn, S , is
less than this, in order to improve the impedance match between the
piston and the air. The portion of tube between the piston and the
horn consists of a "tank" of volume V and a constriction of cross
sectional area So, of length I and of effective length l e ~ I f 0.5 \/$o
[this is a modification of Eq. (23.1) suitable for the present case].
Since the tube cross section changes, we shall find it easiest to
make our first analysis in terms of analogous impedances, Z a = (z/S),
for pressure and volume flow do not change discontinuously at changes
in tube cross section. The analogous impedance of the horn, looking
out of its throat, is equivalent to an inductance (px /So) and a resistance
(pc/So) in parallel. We use Eqs. (23.2) and (23.3) to complete the
rest of the equivalent circuit, the impedance of which is equal to the
ratio Z a between pressure and volume flow of air (uS p ) at the piston
face. This circuit is shown in the middle part of Fig. 60.
The power radiated out of the horn is represented in this circuit
by the power dissipated in the resistance (pc/So). This power is
diminished at the lower frequency end due to the shunting inductance
(pxo/So), which should therefore be made as large as possible (by
making xo large) to enable the horn to radiate well at low frequencies.
The sketch shows that for x to be large the conical angle # 8 must be
small; therefore the horn must be long in order that the mouth of the
horn be large enough to radiate away the power. This property of
conical horns has already been shown in Fig. 59. The power output
for the horn of Fig. 60 is also limited on the highfrequency side by the
shunting condenser (V/pc 2 ) and by the series inductance (pl e /S ).
Therefore the volume V in front of the piston must be as small as
possible and the constriction as short as possible in order that the
horn be efficient at high frequencies.
VI.24]
PROPAGATION OF SOUND IN HORNS
275
To include the effect of the impedance of the piston itself in the
analysis, we shall find it easier to turn from analogous impedance Z a =
(pressure) /(air flow) to mechanical impedance Z m = (force) /(velocity)
= SlZ a . We set up the equivalent circuit at the bottom of Fig. 60 to
aid in calculating this quantity. The equivalent voltage E equals
Longitudinal
Cross Section
of Conical Horn
— Loud Speaker
S =47TX 2 sin z (6 s /2)
Sp=Areoi Piston
J
P
°1
(ple/So)
,_V_
■pc«
Circuit for
Analogous
Impedance
Z a atthe
Piston P
(pleSp/S fl
m p R p
Sjpc* "
f/Kp
— 1— wvw
pXoSpi
pcSp;
Equivalent Circuit
giving Mechanical
Impedance of Piston
plus Radiation Load
"Impedance due to Air back of Piston
Fig. 60. — Idealized sketch of conical horn loudspeaker, with corresponding circuits for
analogous impedance of air load Z a , and for total mechanical impedance Z p + S p 2 Z a .
the force on the piston, and the current through m p equals the piston
velocity.
This circuit is not so convenient as the one above, for the currents
in the various arms do not everywhere correspond to air flow nor do
the voltages always correspond to pressures. The pressure at the
horn throat, for instance, turns out to be equal to the equivalent volt
age across the resistance (pcS*/So) divided by the area S (for in this
276
PLANE WAVES OF SOUND
[VI.24
circuit, voltage corresponds to force = pS). The air velocity in the
throat equals the equivalent current in the inductance (pl e Sl/S )
multiplied by (S p /S ). The power radiated is, of course, still equal
to the power dissipated in the resistance (pcSl/So)
We now can see why it is wcrth while to have the piston area S p
larger than the throat area So, even though this introduces the unde
sirable capacitance (F/£pc 2 ). The magnitude of the mechanical
impedance of the piston is usually much greater than pcS , so that there
r^JM^
v
pe>S 2 p
= px S 2 p
Rn"l
:pcS p
Equivalent circuit
giving Mechanical
Impedance of Mass
Controlled Piston
plus Radiation Load
Lc
SMAAM
Circuit giving
Electrical Impedance
z E =z c +(r/z m )
of Coil of Electro
dynamically driven
Piston
Motional Impedance
Fig. 61. — Equivalent circuit at top gives mechanical impedance of piston of Fig. 60
when piston is masscontrolled. If piston is driven by an electrodynamic coil, as dis
cussed on page 34, the electrical impedance of the coil whe n moving is equal to that of
lower cir cuit. Velocity of piston is equal to \Ao 7 /VE M . Velocity of air in throat
equals ■\/10 7 /T(S p /So)Er. We have here neglected the air load Z b on the back of the
piston.
is a poor impedance match between the piston and its air load unless
the effective air impedance is increased by the factor (S P /S Q ) 2 .
If the horn is well designed the stiffness and mechanical resistance
of the piston are small compared with the mass reactance over the
useful range of frequencies; in other words, the piston is mass con
trolled. In this case the equivalent circuit, representing the mechanical
impedance, is that shown at the top of Fig. 61. The inductance,
representing the piston mass, is in series with a parallel combination
representing the air load. We have omitted the inductance due to
VI.24] PROPAGATION OF SOUND IN HORNS 277
the constriction, because this can be made quite small compared with
m p .
The large mass of the piston plus driving mechanism is the chief
difficulty in providing for uniform radiating properties of the loud
speaker. By using a horn, we are able to make the piston smaller and
therefore considerably lighter than the diaphragmdriver unit of the
"dynamic" speaker. But it is extremely difficult, to reduce this
mass below 10 or 20 g. Consequently, the reactance due to m p at
1,000 cps will still be approximately 100,000 dynesec per cm. Since
pc is only 42 dynesec per cm 3 , we must make the factor (S%/S ) of the
order of 1,000 to have a reasonable impedance match. If, for instance,
So were 3 cm 2 , S p should be about 50 cm 2 (piston diameter ~ 3 in.).
But this large a piston is difficult to make stiff if it weighs as little as
10 g. All these practical limitations mean that it is quite difficult to
design a conical loudspeaker that is efficient over a wide range of
frequencies.
To illustrate these difficulties we shall calculate the horn behavior
for values of the constants which are difficult to attain in practice,
but which show clearly the general behavior
m p = 8 g; So = 3 cm 2 ; S p = 27 cm 2
Xo = 11 cm; # s = 5 deg; V = 8 cm 3
K v = 12.6 X 10 7 dynes/cm; m r = 3.2 g; R r = 10,000 dynesec/cm
The mechanical impedance of piston plus horn is plotted in the top
curve of Fig. 62. The mechanical resistance is maximum at 1,000 cps,
corresponding to the parallel resonance between K v and m r . The
horn would respond better to high frequencies if this resonance could
be moved to higher frequencies; but this requires either that #« or V
be made smaller, both of which are difficult. The reactance is large,
due to the large value of m p , which is almost impossible to reduce
much further.
Now suppose that the piston is driven by an electromagnetic
coupling system, such as that described on page 34 having coupling
constant T. The motional impedance of the coil, due to the motion of
piston and air in the horn, is equal to T times the mechanical admittance
of the pistonhorn system. This motional impedance is in ohms and
is to be added to the clamped impedance of the coil to give the total
electrical impedance of the driving coil. The reciprocal of the series
parallel circuit at the top in Fig. 61 is the parallelseries circuit inside
the dashed line in the lower sketch.
Consequently, this lower circuit has the same electrical behavior
278
PLANE WAVES OF SOUND
[VI.24
as the driving coil. We can show that the velocity of the piston in
centimeters per second is equal to \/10 7 /r times the voltage across
the conde nser ( m p /T) and that the air velocity at the throat of the
horn is Vl0 7 /r (S P /S ) times the voltage across the condenser (m r /T)
o ^
T5 —
^
I_Q
O
a. 
10 Volts Impressed
Across Driving Coil
10
200
400 600 1000
2000
4000 6000
32
V.
10 15
3.2 J
1.0 
cc
0.32 I
o
0.! Q 
10,000
Fig. 62. — Mechanical impedance of pistonplushorn of Fig. 60. Electrical imped
ance of driving coil if piston is driven by electromagnetic means (see page 34) . Power
radiated when coil is driven by 10 volt generator.
and the resistance (T/R r ). The power radiated out of the horn is,
of course, equal to the power dissipated by the resistance (T/R r ).
If we assume that r = 10 5 ohmscm per dynesec R c = 0.5 ohm,
and L c = 0.0008 henry (an improbably small value, chosen to show
the coil resonance separated from the piston resonance), then we can
use the equivalent circuit to calculate the total electrical impedance
VI.24] PROPAGATION OF SOUND IN HORNS 279
of the driving coil. This is shown in the center graph of Fig. 62.
The resistance is largest between 200 and 600 cycles; above 3,000 cycles
the mass load of the piston (the shunt capacitance m p /T) prevents
much radiation, and the resistance is that of the clamped coil. We
have adjusted the coil inductance L c so that it resonates with the con
densers at 2,000 cps, so as to strengthen the response above 1,000 cps.
Assuming that we drive the coil of the speaker with a constant
voltage generator of 10 volts emf but variable frequency, we should
obtain a plot of output power against frequency of the form given at
the bottom of Fig. 62. This output is fairly uniform over the fre
quency range from 200 to 3,000 cycles, but it drops very rapidly above
this range, due to the coil inductance L e and to the shunt capacitance
(Wp/r). In fact, the equivalent circuit corresponds to a rather
crudely devised lowpass filter, so it is not surprising to have the high
frequencies cut off. If the coil inductance L e were larger, the cutoff
would be at a still lower frequency.
It should be apparent by now that conical horn loudspeakers can
be made reasonably uniform in response over a range of frequency cor
responding to a factor of 10 (if they are carefully designed), but not
over a very much larger range. If the horn is to be reasonably effec
tive at 100 cps, it will probably not be very effective above 1,000 or
2,000 cps; if it is to be effective clear out to 10,000 cps (such high
frequency horns are called "tweeters"), it will not radiate well below
about 1,000 cycles. This general limitation on frequency range is
true of horn loudspeakers in general (as, in fact, it seems to be for
all loudspeakers). New developments in light stiff material (to
reduce m p ), or in "acoustic transformers'' to improve the impedance
match between piston and air load, are needed to produce a marked
improvement in behavior.
The Exponential Horn. — When the cross section of the horn has
the form S = S e 2x/h , the horn is called an exponential horn. The
length h is the distance in which the diameter of the horn crosssection
increases by a factor e = 2.718. We have already seen in Eq. (24.4)
that the pressure wave in such a horn can be expressed approximately
by the formula
^ _ Pq~ (x/k)+i(oi/c)(TX— ct) I
r = Vl  (XAo) 2 = Vl ~ {n/vY > (24.9)
Xo = 2rh; v = (c/2ttK) )
This wave travels out of the horn with everdiminishing amplitude,
with a velocity (c/t) larger than the speed of sound in the open.
280 PLANE WAVES OF SOUND [VI.24
This velocity is different for different frequencies, which makes the
exponential horn a dispersing medium for sound waves (see page 154).
When the frequency is equal to v (called the cutoff frequency) , the
velocity has become infinite; in other words, the air moves in phase
along the whole extent of the horn [this is, of course, only approxi
mately true, since Eq. (24.9) is an approximate solution]. Below this
cutoff frequency there is no true wave motion in the horn, and the
equation for the pressure is
V = P expj ^1  \^J  lj   iaA ( w < Wo ) (24.10)
where co = (c/h). Thus it turns out that the exponential horn is
very ineffective below a certain minimum frequency (the cutoff fre
quency) which is smaller the larger h, the scale factor, turns out to be.
Again a very long horn is needed to produce low frequencies effectively.
The air velocity in the horn for frequencies above the cutoff is
obtained by differentiating Eq. (24.9) :
Consequently, the specific acoustic impedance at the throat of the
exponential horn, for outgoing waves, is
z Q = R — iX = \z\er**; (« > « )
(24.11)
 / 0)ft\ I c\ I
\z\ = pc; <p = sin"
Below the cutoff frequency, no exponential horn can be considered long
compared with the wave length (inside the horn), so that the impedance
match at the mouth of the horn must be taken into account. This
will be considered later. Above the cutoff frequency the impedance
has a constant magnitude pc, its phase angle being 90 deg at the cutoff
frequency and approaching zero as the frequency is increased.
If the velocity amplitude of the driving piston at the throat of the
horn is Uo, the power radiated is
n = iulSopcyjl  (^\ [v >v = (c/2Th)]
Very little power is radiated for frequencies less than v Q .
VI.24]
PROPAGATION OF SOUND IN HORNS
281
The transmission coefficient for the long exponential horn, the
ratio of power actually radiated to the limiting value (ulSopc/2), is
'•  V 1  (t)' = V 1  (as)' <'>*>■ < 24  12 )
The curve for r e is shown in Fig. 63. Comparison with Fig. 59 for
the conical horn shows that for some purposes the exponential horn
is better than the conical horn. Although it radiates practically
nothing below the cutoff frequency, the transmission coefficient rises
0.6
0.8
0.4
(h/A)=(vh/c)
Fig. 63. — Transmission coefficient for a long exponential horn with open end wide
enough to eliminate resonance.
much more rapidly to unity above the cutoff. If h is made large
enough (i.e., if the horn does not flare too rapidly), r e will be constant
over practically the entire range of frequencies. Neither r c nor r e ever
become larger than unity, however. For this behavior we must turn
to the catenoidal horn.
The Catenoidal Horn. — When the cross section of the horn has the
form S = So cosh 2 (x/h), it is a catenoidal horn. The scale factor
h is again a measure of how slowly the horn flares out. At large dis
tances from the throat the catenoidal horn is indistinguishable from
an exponential horn, but at the throat the rate of change of ~S with x
is zero, in distinction to the exponential case.' The catenoidal horn
can therefore fit smoothly onto a uniform tube of cross section So,
whereas the conical and exponential types have a discontinuity in
rate of change of S, which tends to cause reflection back into the uni
form tube.
282 PLANE WAVES OF SOUND [VI.24
Referring to Eqs. (24.3) and (24.4) we see that an outgoing wave
is approximately represented by the onedimensional form
v = — p e iwo(Act) (24.13)
r cosh (x/h)
The wave velocity in this horn, (c/t), is larger than the velocity in
open space, as in the exponential horn, and there is again a cutoff
frequency v = (c/2irh), below which the horn is a poor radiator of
sound.
The corresponding particle velocity is
1
u = —
pc
+
£*■*©'
p [a > coo = (c/h)]
so that above the cutoff frequency the specific acoustic impedance at
the throat (x = 0) of the catenoidal horn is a pure resistance
' * " " VI W)' = "" VI  (X/frrfc)' (24  W)
In this case the specific acoustic resistance is greater than pc above
the cutoff frequency vo, rising to very large values as v — v diminishes
toward zero. For a constant velocity amplitude u at the throat, the
power radiated is
E = WoSopc l f ^ [v > vo = (c/2rfc)]
The transmission coefficient of the catenoidal horn is therefore
T  = ( l ) = /i \ /v {v > Vo) (24 ' 15)
A comparison between the transmission coefficients of conical, expo
nential, and catenoidal horns of the same overall dimensions is shown
in Fig. 64. The catenoidal horn has a cutoff frequency a little higher
than the exponential horn, but the transmission coefficient just above
the cutoff is considerably larger. Evidently, starting the horn with a
uniform cross section at the throat (dS/dx = at x = 0) can improve
the output of the horn just above the cutoff frequency. For instance,
for a catenoidal horn with h = 50 cm = 20 in. (cutoff frequency =
110 cps) the output at 150cps is nearly twice that of an exponential
horn of the same value of h and is very much larger than that of a
conical horn of the same size. As we have seen earlier, it is difficult to
VI.24]
PROPAGATION OF SOUND IN HORNS
283
obtain sizable power outputs at the low frequencies, so this property
of the catenoidal horn is often useful.
It is also apparent from Fig. 64 that the conical horn is the least
efficient of the series of shapes we have analyzed. Even though it
has no cutoff frequency its effectiveness at very low frequencies is
quite small and does not usually make up for the very slow rise of
transmission coefficient with frequency.
Frequency
Fig. 64. — Comparison between transmission coefficients of long, conical, exponen
tial, and catenoidal horns of the same overall dimensions. Length I is chosen to be
2.4 times the h for the exponential horn; I turns out to be 11 times the xo for the conical
horn; and 3.1 times the h for the catenoidal horn.
Reflection from the Open End, Resonance. — In many cases, horns
are used with open ends that are not large enough to radiate all the
sound coming to them from the throat. Many wind instruments, for
instance, are approximately exponential or catenoidal horns, with
open mouths designed to be small enough to , ensure resonance inside
the horn. Even horns designed as loudspeakers, to avoid resonance
at most frequencies, cannot avoid some resonance at the lowest fre
quencies. In this section we shall discuss the effect of such an imped
ance mismatch at the mouth of the horn.
We might as well carry out the calculations for the general form
given in Eq. (24.3), for any value of T. The formula for S can be
given in another form:
S =
So
cosh 2 (e)
cosh 2
G~>
T = tanh (e)
(24.16)
284
PLANE WAVES OF SOUND
[VI.24
When e = 0, the horn is catenoidal; when e = », the horn is expo
nential; and when e = (x /h) + i(ir/2), with h approaching infinity,
then the horn is conical. Intermediate values of e correspond to
horns of intermediate shape.
This can be substituted in Eq. (24.2) to give the approximate
wave equation and eventually to obtain a solution for the wave. In
this case we cannot neglect the wave reflected from the horn mouth,
so our solution has the form
V
Pe~
cosh
sinh^ + UaTX
.)
V
u = —
pc
z = pc
.,«h(, + =) + £—(! + .)
■(■
t coth( \j/ + l —\ + ^ tanhl t
•i
(HI
(24.17)
where t 2 = 1  (c/co/i) 2 . This is similar to Eqs. (23.8) for the uniform
tube. The differences in the expressions for the specific acoustic
impedance z lie in the factor t, corresponding to the fact that waves in
these horns travel with a speed different from waves in the open, and
in the term including tanh[(rr//i) + e], corresponding to the fact
that the surfaces of constant phase are curved surfaces, not planes as
in the uniform tube.
Our first task is to fit the wave at the mouth of the horn, at x = I.
Since the surfaces of constant phase are not planes, this fit is not so
simple as that for the uniform tube, which resulted in Eq. (23.13).
To a reasonable approximation, however, the joining of the wave
from the horn with the wave in the open effectively cancels the term
in tanh [(x/h) + e] and results in the equation for \J/
£ tanh(* + ^) = pc [l  2 ^]  ipcMiw) (24.18)
which is analogous to Eqs. (23.13). The quantity w is (2coa/c), where
a is the radius of the mouth of the horn. This joining equation holds
as well for frequencies below the cutoff frequency as for those above.
When (wh/c) < 1, we set
'^(t)'
and carry on with the calculations.
VI.24] PROPAGATION OF SOUND IN HORNS 285
We solve Eq. (24.18) for \f/ + (ioirl/c) = *(«& — ifa) and from this
compute the impedance at the throat of the horn:
z = pc <t coth ir(a h — tj8*) — ( ^— J H r tanh (e) >
For large values of w = (2«a/c), a h becomes large, and
*« » l / • / P ^ + u ^ (24.19)
w* oo t + {ic/a)h) tanh (e)
which corresponds to Eqs. (24.6), (24.11), and (24.14) for the special
cases discussed before. For these higher frequencies no reflected
wave is present in the horn.
For small values of w, the approximate expressions for ah and fih are
similar to those of Eqs. (23.15) :
ah
teX?)"' *©()
with a the radius of the mouth of the horn. The effective length of
the horn is l p = I + (8a /Sir), as was the case for the uniform tube.
In terms of this we can write the approximate expression for the
acoustic admittance ratio at the throat :
* =* \*™™' c) *^}f} + 4 tanh («) (24.20)
z (T/2)(aco/c) 2 — t tan (corZp/c) <ah '
(OW/C) « 1, T 2 = 1  (C/C0/1) 2
Woodwind Instruments. — If the tube of the flute or open organ
pipe has a shape corresponding to Eq. (24.16) for some value of c,
then the frequencies picked out are those for which z is real and as
small as possible. This corresponds to the requirement that (oyrl p /c)
be approximately equal to (nx), (n = 1,2,3 • • •). By using the
equation for t, we obtain the following equation for the resonance
frequencies :
In comparison with the uniform tube [see Eq. (23.17)] the overtones
here are not strictly harmonic, though if the tube flare is small (h large),
the discrepancy is small. The specific acoustic resistance at the throat
for the nth. "harmonic" is
l ~ i "l\'V) + WJ (24 ' 22)
which is to be compared with Eq. (23.18) for the uniform tube.
286 PLANE WAVES OF SOUND [VI.24
For a reed instrument with tube shaped according to Eq. (24.16),
the frequencies picked out will be those for which z is as large as
possible. We must choose a value of (cotZ p /c) so that (pc/z ) in Eq.
(24.20) is real and as small as possible. This is not obtained by setting
the tangent equal to infinity, for the term (c/oh) tanh e is usually
larger than (aco/c) 2 , and (pc/z ) would not be real or as small as pos
sible. Instead, we set (o)tI p /c) = ir(m + i) + 5, (m = 0,1,2 • • • )
where 5 is quite small. This corresponds to tan (o>tI p /c) ~ —(1/5),
which is quite large, but not infinite. We then adjust 5 so that, to
the first order in the small quant ties 5 and (aco/c) 2 , (pc/zo) is small
and real. This corresponds to making 5 = (c/toJi) tanhe, which is
small enough, since we are assuming that we are above the cutoff
frequency (co > coo = c/h). Setting this value for r in the equations
for co and for z , we have
Zanti
(iW m ; + © !+ P taiih '
C I p 1 I 1 ( p I p + Vi
Xiram) _ \irhmj ir 2 hm 2
(24.23)
where ra = 1, 3, 5 • • • . This is to be compared with Eqs. (23.19)
and (23.20) for the uniform tube. We note that for the catenoidal
case (e = 0) the term (8l p /Tr 2 h) tanh e is zero ; for the exponential horn
the term is (8l p /T 2 h) ; and for the conical horn this term is (SI p /t 2 x )
and the term (2l p /irh) 2 is zero.
The introduction of a nonuniform cross section thus makes the
overtones nonharmonic by a small percentage, makes the resistance
at resonance (z minimum) somewhat larger than for the uniform tube,
and makes the resistance at antiresonance (z maximum) somewhat
smaller than for the uniform tube. A certain small amount of flare
(h large but not infinite) does not alter v m and z very much, however,
and the larger mouth radiates more efficiently.
The case not included in this approximate solution is the one for a
conical horn when x is smaller than the wavelength X. This is the
case where the horn throat is extremely small compared with the open
end and is approximately the case for the oboe and the English horn.
For the conical horn, Eq. (24.20) is
pc ^ 1 — (t'/2)(aco/c) 2 tan(coZ p /c) , ic_
Zq (l/2)(aco/c) 2 — i tan(co? p /c) o)X
and when x is small enough, (c/W ) is large, instead of small, though
it is not usually so large as (c/aco) 2 . In this case, tan(coZ P /c) turns
VI.24] PROPAGATION OF SOUND IN HORNS 287
out to be approximately equal to — (wx /c), and the allowed frequencies
and the throat impedance at these frequencies turn out to be
)
Here the allowed frequencies include both the even and odd harmonics,
as is the case with the flute, rather than the clarinet. The specific
acoustic impedance at the throat is not particularly large, but it is
larger than any value of the resistance for any other frequency; so
the frequencies v n are the ones that are heard in the emitted note.
This lack of a pronounced resonance peak may partly explain the
difficulty of playing the oboe.
As a final item in this section, we note that the impedance at the
throat of a long horn, below the cutoff frequency, can be easily obtained
from Eq. (24.19). It is a pure mass reactance
z ~ io)(ph) Jl — I — J + tanhe (w < c/h)
It is approximately equal to the mass reactance of all the air in
the horn within a distance h of the throat.
Transient Effects. — Calculation of transients in horns of general
shape is difficult because of the complicated behavior of the impedance
function. We shall carry through one example here, however, to
show the general nature of the results. We consider the conical horn,
to keep the calculations as simple as possible. Suppose that we have
a piston in the throat of the horn, moving with specified velocity
C/(w)e _iw '. The pressure wave out of the horn, if it is long enough so
that reflections from the mouth are negligible, is
pCX e i(»/c)(xct)
x + x 1 + (ic/coxo)
We refer now to Eqs. (6.15) and (6.16) and the related discussion.
The expression for the pressure wave due to any velocity motion
u (t) of the piston is
OCX* f°° e ^>[t(x/e)] C °°
2ir(x + Xo) J «, 1 + (tc/oiXo) J«
288 PLANE WAVES OF SOUND [VI.24
To improve convergence, we shall utilize the trick mentioned in the
last part of Sec. 6:
1 = 1 — (* c /zo)
1 + {ic/wxo) (a f (ic/xo)
Therefore the integral becomes
= _£opc_ ( 1  e  iu[t  {x/c)] do} J u^ivr dT
x + x Q {2t J_„ J»
r °° r °° e MtT(x/c)]
 I w (t) <2t I i /• / \ dco
oJ» J» o) + (*c/a;o)
27TiCn .]
We now use Eq. (2.20) to calculate the first integral and use contour
integration to calculate part of the second:
Xopc
V = — r —
X + Xo
t\ y ^ T Mo ( T ) e (o/.o)(«r)+(^o) dr
c) (x + XojJ*
Therefore part of the wave has the same "shape" as the "velocity
shape" of the motion of the piston at the throat. Behind this wave,
however, is a "wake" represented by the second integral, which dies
out exponentially. This behavior is best seen if we cause the piston
to move suddenly forward at t = 0. Then u (t) = 8(t), and
( [t< (x/c)]
p,(0 = J _p^o_ 8 ( t _A_ • /*!_ ecc/^^/o] [t > {x/c)]
\X + Xz \ c) (x + Xo)
The original pulse moves out of the horn with velocity c and with
amplitude diminishing as (x + x ) _1 . After the sharp pulse comes a
negative pressure wave which dies out exponentially; the smaller
the constant x is, the more rapidly does this "wake" damp out but
the larger is its initial amplitude.
Problems
1. What is the speed of sound inside a gasolineengine cylinder just after com
bustion, when the pressure is two hundred times atmospheric and the temperature
is 1000°C; if the gas mixture has a value of y = 1.35 and would have a density of
0.0014 g per cc at 0°C and atmospheric pressure?
2. A plane wave in air has an intensity of 100 ergs per sec per sq cm. What
is the amplitude of the temperature fluctuation in the air due to the wave?
3. The specific acoustic impedance at a point in a tube is R = 42, X = —42
when v — 344. What is the specific acoustic impedance at points 25, 50, 75, and
100 cm farther along the tube?
4. A vibrating piston is placed at one end (x = 0) of a tube whose cross
section is 10 sq cm, and a second piston whose mechanical impedance is to be
VI.24]
PROPAGATION OF SOUND IN HORNS
289
measured is placed at the other end (x = 30). When the pressure due to the
sound wave is measured at different points along the tube, it is found that the
pressure amplitude is a maximum at the points x = 3, x = 15, x = 27> having an
amplitude of 10 at these points. The pressure amplitude is a minimum at x = 9,
x = 21, with a value of 6.57. From these data find the mechanical impedance
of the driven piston, the frequency of the sound used, and the amplitude of vibra
tion of the driving piston.
5. A vibrating piston is set in one end of a tube whose length is 86 cm and
whose cross section is 10 sq cm. Closing the other end of the tube is a diaphragm
whose overall mechanical impedance is to be measured. The measured radiation
impedance of the driving piston (not including the mechanical impedance of the
piston itself) has the following values:
V
100
200
300
400
500
600
700
800
R p
42
147
420
189
840
67
1,260
Xp
 71
1,260
365
126
840
76
1,680
What are the real and imaginary parts of the mechanical impedance of the
diaphragm at these frequencies? Plot a curve of the magnitude of the impedance.
6. A tube of length 86 cm and cross section 10 sq cm is closed at one end and
has a piston fitted in to the other end. The piston is driven electromagnetically
by a coil whose coupling constant is r = 1,000. The impedance of the coil is
measured at different driving frequencies, and the part of the impedance due to
the motion of the piston is found to be
V
100
150
200
250
300
350
400
450
500
Rm
1.92
1.17
1.68
1.47
0.54
1.75
0.67
Xm
0.37
0.98
0.75
0.88
0.89
0.64
0.95
Compute and plot the real and imaginary parts of the mechanical impedance of
the piston and of its radiation impedance.
7. Two similar pistons, each of mass 10 g, slide freely in opposite ends of a
uniform tube of length 34.4 cm and cross section 10 sq cm. One piston is driven
by a force 100e _2 ' r * > ' dynes. Plot the amplitude of motion of the other piston as
a function of v from v = to v = 500.
8. A piston of negligible mechanical impedance is fitted into the end of a
uniform tube of cross section 10 sq cm. The other end of the tube, 34.4 cm from
the piston, is closed by a rigid plate. The piston is driven by a force
: F v  42 sin (Stt*)  28 sin (24irt) + 8.4 sin (40ttO
Show that the velocity of the piston is
u P = ^ tan ( 2^5 J cos (8nrt) — ^ tan ( ^ J cos (24x0 + ^ tan (^q) cos ( 407rf )
Plot the values of F and of the piston displacement as a function of t from t =>
to I ■'* (I), What is the curve for piston displacement when the tube is 68.8 cm
long?
290 PLANE WAVES OF SOUND [VI.24
9. How long must an open organ pipe be to have its fundamental frequency
equal to 250 cps, if the pipe is a cylinder, with radius 2 cm?
10. When a gasoline exhaust valve first opens after the explosion cycle, the
volume occupied by the gas in the cylinder is 200 cc. The effective area of the
valve opening is 1 sq cm, and its effective length is 1 cm. Assuming the cylinder
valve system to act like a Helmholtz resonator, what will be its natural frequency
of oscillation if the air in the cylinder is at normal pressure and temperature?
What will be the frequency if the gas is at a temperature of 546°C and is at normal
pressure?
11. The exhaust valve in Prob. 10 opens into an exhaust pipe of 5 cm inside
diameter, 100 cm long. What will be the lowest natural frequency of the cylinder
valvepipe system for air at normal pressure and temperature?
12. A Helmholtz resonator has a cylindrical open neck 1 cm long and 1 cm
in diameter. If the resonating vessel is spherical, what must its radius be to
have the resonance frequency be equal to 400 cps?
13. A piston of mass 10 g, frictional resistance 1,000 dynes cm per sec, is at
one end of a tube 13.69 cm in diameter and 43 cm long, open at the other end.
Plot the real and imaginary parts of the total mechanical impedance of the piston
as functions of the frequency from v = to v = 2,000.
14. A simple "muffler" consists of a tank of volume V with input pipe at one
end and a cylindrical output pipe, of length I and radius a, at the other end.
Draw the equivalent circuit for the analogous impedance at the input end; include
the analogous radiation resistance R. If the volume flow into the tank at the
input end is simple harmonic, v e~ iat cm 3 per sec, what is the volume flow out
of the output pipe? What is the power radiated into the open? How does it
depend on frequency? What is the frequency above which the output radiation
falls off rapidly? Above what frequency will the equivalent circuit be invalid?
15. The back of a loudspeaker diaphragm looks into a "tank" of volume V,
which connects with free space by a constriction of negligible length, and area S
of opening. Set up the equivalent circuit for the analogous impedance at the back
of the diaphragm, and give the formula for the additional mechanical impedance
load on the diaphragm. Over what range of frequencies will the motion of air
in the constriction be in phase with the motion of the diaphragm with respect to the
outside (i.e., move out as the diaphragm moves out; be sure to express the phase
relations between input current. in the equivalent circuit and diaphragm motion
correctly) and be larger than the diaphragm motion? If the area of the constric
tion is 100 cm 2 , what volume must the "tank" have in order to have this reinforce
ment of the diaphragm motion come at and below about 100 cps? Above what
frequency will the equivalent circuit be invalid?
16. A tube 1 cm in radius and 68.8 cm long has a hole in its side, placed mid
way along its length, which is 1 cm in diameter and is provided with a short open
tube of the same diameter, 1 cm long. Plot the specific acoustic impedance at
one end of the long tube when its other end is closed. What are the lowest three
resonance frequencies of the system? Plot the specific acoustic impedance at one
end of the tube when its other end is open. What are the lowest three resonance
frequencies for this case?
17. An airconditioning system has a circulation fan that produces noise of
frequency chiefly above 200 cps. Design a lowpass acoustic filter, consisting of
two vessels and three narrow tubes in series, which will filter out the noise. The
VI.24]
PROPAGATION OF SOUND IN HORNS
291
narrow tubes cannot be less than 5 cm in diameter, and the vessels cannot have a
volume larger than 30,000 cc.
18. A condensermicrophone diaphragm is stretched across one end of a tube
2.5 cm in radius and 1.2 cm long, open at the other end. Compute and plot the
ratio between the pressure at the diaphragm and the pressure at the open end of
the tube as a function of v from v — to v = 10,000.
19. When the damping of wave motion along a tube must be taken into
account, show that the pressure amplitude at the point x is given by the formula
[ V \ = A \/cosh 2 Lvcci + (} (I  z)l  COS 2 \irf3i + (y) (I  X) J
where k is the damping constant. Plot the pressure amplitude as a function of
x, for a tube having
I = 100 cm, X = 50 cm, (£) = (j^A «* = 0.1, ft = 0.4
What is the acoustic impedance atz = Z? At a; = 0?
20. The pressure amplitude is measured in a tube of length 100 cm, as function
of the distance re along the tube from the driving end. The values of the maxima
and minima, and their positions along the tube are
Distance from driver x
12.5
25
37.5
50
62.5
75
87.5
Pressure p, max
1.899
1.478
1.204
Pressure p, min
1.934
1.335
0.869
0.489
From these data compute X, a*, ft, the damping constant k, and the specific acoustic
impedance at x = and x = 100. What is the pressure amplitude at x = 0?
At x i 100?
21. A tube of length I has a termination impedance such that ai and ft are
both independent of frequency. The damping constant k for waves in the tube
is likewise independent of frequency. The end at x = is fitted with a piston,
which is given a sudden push at t = 0; uo = 8(t). Show that the "flutter echo"
in the tube is represented by the expressions:
e KX/C g
e
2x(ajt/3i)
?
— 4x(ai— i/3i)
: iifld(f)(2J*) s f f _ 21  x \
[0 < t < (l/c)]
{{1/6) < t < (2l/c))
[{21/ c < t < {3l/c)}
[(3Z/c) < t < {4l/c}}
22. A conical horn has an angle of opening fi = 20 deg. The area of the neck
of the horn is 5 sq cm, and that of the mouth is large enough so that no sound is
292 PLANE WAVES OF SOUND [VI.24
reflected back into the horn. Plot the transmission coefficient of the horn as a
function of v from v = to v = 1,000. When the air in the throat is set into
vibration at a frequency of 500 cps, what will be the specific acoustic impedance
in the throat? If the velocity amplitude of the air in the throat is 1 cm per sec,
what is the intensity of sound in the horn 200 cm from the throat?
23. The conical horn of Prob. 22 is 200 cm long, and its open end has an area
of 4,000 sq cm. What value of x must be used in the formula for an exponential
horn that has the same area of throat and open end as the conical horn and is as
long? Plot the transmission coefficient of such an exponential horn as a function
of frequency from v = to v = 1,000. What is the specific acoustic impedance
in the throat of the horn at x = 500? If the air in the throat vibrates with a
velocity amplitude of 1 cm per sec, what is the intensity of the sound in the horn
200 cm from the throat?
24. The conical horn of Prob. 22 has its small end fitted to a driving piston of
area 25 sq cm and mechanical constants to = 1, R = 1,000, K = 1,000. Assum
ing that the horn is large enough so that Eqs. (24.6) are valid, compute and plot
the total mechanical impedance of the piston as a function of v from v = 100 to
v = 5,000. Is the radiation impedance an important part of this? A similar
piston is attached to the exponential horn of Prob. 23.. Plot its total mechanical
impedance as a function of v from v = 100 to v = 5,000.
25. The pistons of Prob. 24 are each driven by a force of 10,000e 2Xi,,< dynes.
Plot the power radiated by each of the horns as a function of v from v = 100 to
v = 5,000. In the light of these curves discuss the relative advantages of the
two types of horns.
26. What are the constants for a catenoidal horn having the same overall
dimensions as the horn of Prob. 23? Plot the transmission coefficient of such a
horn over the frequency range from to 1,000 cps. What is the specific acoustic
impedance in the throat at v = 500? What is the intensity at x = 200, if the
air in the throat vibrates at 500 cps with a velocity amplitude of 1 cm per sec?
27. The loudspeaker of Fig. 62 has its conical horn replaced by an exponential
horn of cutoff frequency v = 100 cps. What is the value of h? Recompute the
curves of Fig. '62 for frequencies from 100 to 4,000 cps for this horn. Is this a more
satisfactory loudspeaker? Can you suggest changes in values of the constants to
improve results?
28. Repeat the calculations of Prob. 27 for a catenoidal horn having a cutoff
frequency of 100 cps. Suggest further improvements.
29. One unit of a "tweeter" loudspeaker consists of a catenoidal horn of cutoff
frequency 1,000 cps. The driving piston, of effective mass 1 g, is set in the throat
of the horn, which has area So = 3 cm 2 . The piston is driven by electromagnetic
means, having a coil of inductance L c = 0.001 henry, resistance R c = 1 ohm, and
coupling constant r = 10 5 . Set up the equivalent circuit for the total electrical
impedance of the driving coil. Compute the power radiated for a 10volt driving
emf for the frequency range 1,000 to 10,000 cps. What changes in constants will
improve this behavior?
30. The "tweeter" unit of Prob. 29 has a conical horn with x& = 5 cm. Calcu
late the power radiated over the frequency range 1,000 to 10,000 cps.
31. The piston of Prob. 29 is driven by an xcut crystal of Rochelle salts, of
dimensions a = 2 cm, b = 2 cm, d = 1 cm (temp. = 15°C). Set up the circuit
for the total electrical impedance of the crystal. Suppose that the crystal is
VI.24] PROPAGATION OF SOUND IN HORNS 293
driven by an emf of 100 volts, compute the power radiated from v = 1,000 to
v = 10,000.
32. A catenoidal horn of shape formula S = So cosh 2 (x/h), of length I, is used
as a "speaking trumpet" for a dictating machine. Assume that the impedance
of the diaphragm at the throat is infinite and that a plane wave strikes the mouth
parallel to the axis of the horn. Modify Eqs. (24.18) and (23.28) to show that the
specific acoustic impedance presented to the plane wave at the mouth (x = I) is
*o = pc j [l  2 ^pH  i \m(w)  i cot (corZ/c)] 
where « = (4xa/X) = (2cm /c), t 2 = 1 — (c/coh) 2 , and a is the radius of the mouth
of the horn. Show that the pressure in the horn a distance x from the throat is
ipc po cosh (l/h) cos (utx/c)
t Zo cosh(x//i) sin (oxrl/c)
where po is the driving pressure at the mouth of the horn due to the incident plane
wave. Plot the ratio of the pressure at the throat (x = 0) to this driving pressure
gdh 1  2 ^]Mt) + [*<•>(?) — mv
for a horn of I = 34.4 cm, h = 17.2 cm, a = 3.44 cm as a function of « = 2wv from
v = to 500.
33. Use the relation
X'
/ : — pr 6 V P 1 + a 2
e* Jo(« Vt 2  b 2 ) dt = /
V p 2 + a 2
to help calculate transient sound in exponential or catenoidal horns. For a long
horn of shape given by Eq. (24.16), show that if a unit pressure impulse starts at
the throat at t = it will travel out of the horn according to the equation
1 cosh
V =
2tt cosh [(x/h) + e]
coshe
/ e <(?)^RIy^
M< : V4!)^^4V(iRly]}
cosh [(x/h) + e] } \ cj \ c) y/ t %  ( x / c )
where S and u are defined in Eqs. (6.9) and (6.11).
34. Use the equation quoted in Prob. 33 to show that the velocity wave out of a
long catenoidal horn due to an impulsive motion of a piston at its throat is
"*  o \tt7i\ f" T 1 +—■ } C/k) tanh(s/ft)~.
2x cosh (x/h) J « L *« VI  (c/coh) 2 J
. e< („*/c)Vi(^) 2 ^ do
, c sinh (x/h)
OCKV^F© 1 )
h cosh 2 (x/h)
Obtain an expression for the pressure pulse accompanying this velocity pulse.
CHAPTER VII
THE RADIATION AND SCATTERING OF SOUND
25. THE WAVE EQUATION
We must now consider the more complicated case of sound waves
that are not plane waves. When we studied the motions of a mem
brane we found that waves other than parallel waves are very difficult
to analyze; in the case of sound wave's we shall find an even greater
increase in complexity. Nonplane waves do not all travel with
constant speed or shape. Their motions must be expressed in terms
of coordinate systems which often are too complicated to handle. We
shall content ourselves, in this book, with a discussion of cylindrical
and spherical waves.
The Equation for the Pressure Wave. — The first task is to obtain a
wave equation for three dimensions. This can be obtained by a simple
generalization of the argument of Sec. 22. The air particle at the
point (x,y,z) can move in any direction, so we must find the three com
ponents of its displacement from equilibrium. This displacement is
a vector d with components £, 17, and f in the x, y, and 2directions,
respectively. The corresponding particle velocity u = (dd/dt) has
components u, v, and w. All these quantities, together with the
pressure p, are functions of x, y, z, and t. When displaced by the
sound wave, the elementary volume dx dy dz becomes a parallelepiped
of volume J dx dy dz, where
J =
dx
dx
dx
dj
dy
1+p
dy
it
dy
dj
dz
dr]
dz
dz
~ 1 + div(d)
where div(d) = — + — + ^ is called the divergence of the particle
displacement d.
Therefore the counterpart of Eq. (22.2) for the equation of con
tinuity is
294
VII.26] THE WAVE EQUATION 295
145 = 1; 5 ~ ^div(d)
Corresponding to Eq. (22.5) for the adiabatic compression of the gas
is the relation
1 + ■? = (1 + 8)t = ^ e ; V ^ VcPo div(d)
Finally, Newton's equation of motion is
= grad(p)
■©
where grad (p) is a vector, called the gradient of the pressure, with com
ponents (dp/dx), (dp/ By), and (dp/dz).
These equations can be combined as follows:
^ =  7c Po div(^ = ^ d iv[grad(p)] = Wp (25.1)
where c 2 = (7<JVp) and where
divferad( P )] = §* + $ + g = V*p
is called the Laplacian of the pressure. This is the wave equation
in three dimensions for the pressure. The particle velocity for simple
harmonic waves can be obtained from the pressure by using the
relation
\io>p/
grad(p) (25.2)
The potential energy of a unit volume of the gas at any instant
can be given in terms of the absolute temperature
pCvT ~ R (1 + 8) " ( 7 .  1) K + } (7c ~ IV 1
The equilibrium value of this, Po/(y c — 1), is the internal energy of
the gas at equilibrium and is not included in the expression for the
energy of the sound wave. When we expand the quantity (1/J 1 " 1 )
in a series of the small quantities (d£/dx), etc., we can leave out the
terms in the first powers of these quantities and terms involving cross
products, such as (d£/dy)(dn/dx), etc., because these terms have zero
average value. The only terms that are not zero on the average
296 THE RADIATION AND SCATTERING OF SOUND [VII.25
(unless we go to terms smaller than secondorder ones) will be those
involving the square of the quantities (drj/dy), etc.,
(7.  i) v** 1 /
~ r~TT)  ~ 2 l)y ° div 2 (d) + neglectable terms!
The total average energy of the sound wave is therefore the integral
W ~ ip f f f [Y^ + c 2 div 2 (d)l dx dy <fc
= JJJ Up" 2 + 2p7 2 P 2 J rf z dy dz
(25.3)
Curvilinear Coordinates. — The equations for pressure, particle
velocity, and energy have been given in terms of the rectangular
coordinates (x,y,z); they can also be given in terms of other coordi
nates. The differential operator V 2 in Eq. (25.1) can be written as
d 2 d 2 d 2
z—z +.^5 + t~i m rectangular coordinates
dx 2 By 2 dz 2
d 2 d 2 .
t— 5 + r:in cylindrical coords
V 2 = ( ■'■ °> \ or/ r d<i> 2 dz' ^ (25 4)
1 d
r dr\ drj r 2
1 d 2
+■ , • o « t  5 in spherical coords.
r 2 sm 2 #d99 2
The operator V 2 is called the Laplace operator, or the Laplacian: it
measures the concentration of a quantity (or, rather, the negative of
the concentration). The value of V 2 p at a point is proportional to
the difference between the average pressure near a point and the
pressure right at the point. When this is negative, there is a con
centration of pressure at the point, and when it is positive there is a
lack of concentration there. The wave equation simply states that, if
there is a concentration of pressure at some point, the pressure there
will tend to diminish. Compare this with the discussion on page 175
for the membrane.
In order to compute the particle velocity in curvilinear coordinates
we must have available expressions for the components of the vector
grad(p) along the coordinate axes. If, for instance, we label the
VTI.26] RADIATION FROM CYLINDERS 297
component of the gradient parallel to the cylindrical radius, grad r ( ),
that perpendicular to r and to z, grad^,( ), and so on, we have
grad,( ) = A; gradj/ ( ) = . ; g r ad 2 ( ) = A
in rectangular coordinates
grad r ( ) = A; grad,( ) = A; grad,( ) = i A
in cylindrical coordinates / v^^.o)
grad r ( ) =; grad^ . ) =  ;
1 8
grad»( ) = — : — r —  in spherical coordinates
rsm&d<p
These expressions can be substituted in Eq. (25.2) to compute the
components of particle velocity in these coordinates.
Finally, to calculate the sound energy in terms of particle displace
ment, we must give expressions for the divergence of a vector A :
dA x 6A y dA z . . . .. x
^ r r h t— m rectangular coordinates
in cylindrical coordinates
in spherical coordinates
With these formulas giving the different mathematical forms of the
physical quantities V 2 (p), grad(p), and div(d) when expressed in
these three coordinates, we are ready to study plane, cylindrical, and
spherical waves in free space.
26. RADIATION FROM CYLINDERS
We have already discussed circular waves on a membrane in
Sec. 19 and have developed there the set of functions needed to discuss
cylindrical waves. We showed there that circular waves have a more
complicated behavior than plane waves — for instance, that circular
waves change shape as they spread out, leaving a "wake" behind
them. Cylindrical waves, the threedimensional generalization of
circular waves, show the same behavior. We shall spend most of our
time here in discussing simple harmonic cylindrical waves.
298 THE RADIATION AND SCATTERING OF SOUND [VII.26
The General Solution. — The general solution of the wave equation
in cylindrical coordinates is a combination of functions of the type
p = cos (m<f>)[AJ m (kr) + BN^krW* 2 ™' (26.1)
Sill
where v = (c/2ir) s/k 2 + k 2 . The variable z measures distance along
the cylinder axis, r the perpendicular distance from the axis, and
is the angle that r makes with the reference plane. The functions
J m and N m have already been discussed; their properties are given in
Eqs. (19.4), (19.5), and (20.1), and the values of some of them are
given in Table III at the back of the book.
Uniform Radiation. — For waves spreading uniformly out from a
cylinder, we use the function for k z = and m = 0, which represents
outgoing waves [see Eq. (20.2)] :
>aJ% ^rcDiWA)^ k = (2tv/c) = (2tt/\) > (26.2)
— l (v) ln ^ e ~ 2 * irt
r— >0
Suppose that we have a long cylinder of radius a which is expanding
and contracting uniformly in such a manner that the velocity of the
surface of the cylinder is u = U er 2irivt . The constant A, to corre
spond to the radiated wave, must be chosen so that the velocity of the
air perpendicular to the cylinder surface u r = (l/2Trivp)(dp/dr) is
equal to u at r = a. If a is small compared with the wavelength,
this velocity is (A/ic*vpa)(r 2rivt } so that A must equal (T 2 v P aU ). The
pressure and velocity at large distances from the cylinder are then
\(*V
V > TrpaUo J— e*<~»«*/*>
Mr
yjcr
TaU *  e*<~»*<* /4 »
The product of the real part of each of these expressions gives the
flow of energy outward per second per square centimeter, and the
average value of this
T~U*pa 2 U 2 0?\ (26.3)
VH.2G] RADIATION FROM CYLINDERS 299
is the intensity of the sound at a distance r from the cylinder's axis.
The total energy radiated in ergs per second per centimeter length of
the cylinder is
n = T* P va*Ul
Radiation from a Vibrating Wire. — A somewhat more complicated
wave is generated by a cylinder of radius a vibrating back and forth
in a direction perpendicular to its axis, with a velocity U e~ 2Tipt . If
the plane of vibration is taken as the reference plane for <f>, the velocity
of the part of the cylinder's surface at an angle <£ from the plane of
vibration has a component U cos <j> e Mvt perpendicular to the sur
face. In this case we take the radiated wave to be
p = A cos0[J"i(At) + iNi(kry\er M '*, k = (2rv/c)
.Ac „ . ,
» — i—j— cosd>e~ 2rtrt
r _»o i r'rr y (26.4)
> A J~ ^(r~ct)i^/i) cog( x
r*co \TT l vr ^
If a is small, the radial velocity at r = a is
cos<f>e~ 2Tipt
\2ir 3 v 2 pa 2 J
which must equal the radial velocity of the surface, so that A =
(2ir z v 2 P a 2 U /c). The radial component of the particle velocity at
large distances from the cylinder is
Ur > I — J J^ e^(rc*)*(3x/4) cog ^
j— >«> \pc/ \7r J vr
There is a component of particle velocity perpendicular to the radius
r, but it diminishes as r* at large distances and so is negligible there
compared with u r . The intensity at large distances and the total
power radiated per centimeter length of vibrating wire are
t?^3co..* n = ^W3 (26 .5)
The amount of sound energy radiated by a vibrating string therefore
diminishes rapidly as the frequency of vibration of the string decreases
and diminishes very rapidly if the string's thickness is decreased. As
we have mentioned in Chap. Ill, a vibrating string is a very inefficient
radiator of sound.
300 THE RADIATION AND SCATTERING OF SOUND [VII.26
The reaction of the air back on the vibrating wire is obtained from
the expression for the pressure at r = a:
p ~ — iupaUo cos0e iw<
The net reaction force on the wire per unit length in the direction of
its motion is
•J".
2*
p cos <pd<t> = Foe™ 1 = —i(ap(ira 2 )UQer i » t
The ratio of this to the velocity of the wire is the mechanical impedance
per unit length of wire, due to sound radiation,
(jff) ~  to(*a 2 P ) (a « c/«)
which is equivalent to the reactance of a mass of air of volume equal
to that of the wire. The resistive part R of this impedance is too
small to be included in this approximation when a is small. We can
find the resistive part from Eq. (26.5) for the power radiated, for of
course II = %RUl,
Z tad c* i^a^p) + ( 7 ^) (a « oh)
This can be used to compute the effect of the presence of air on the
motion of a vibrating string, as discussed on page 105.
Radiation from an Element of a Cylinder.— To solve more com
plicated problems, where the velocity of the surface of the cylinder is a
less simple function of <p than the preceding examples, it is convenient
first to solve the problem where only a single line element on the
surface of the cylinder does the vibrating. Suppose that the radial
velocity of the surface r = a is
(da , da\
 — < 4> < + g J
( , da ^ , . da\
(^ + 2 < <t> < 2tt  T J
The Fourierseries expansion for this function of 4> is
\ / L OT= 1
U a =
2rivt
V1I.26] RADIATION FROM CYLINDERS 30 1
To fit this distribution of velocity at the surface we choose a general
sort of outgoing wave
00
V = 2 Am cos ( m ^ J ^ kr ) + iN m {kr)\e^\ k = (2rv/c)
m =
The corresponding radial particle velocity at r = a is
a = ( X 02) = \A^1 e'y + ^ AmCm e 1 ^ cos(m<f>)\ e~ 2 * irt
\2irivpo drj _ 2p c £J pc J
u a =
where
2 ^ [Jo(n) + iATo(M)] = iCoe**;
^ [/*(/*) + ii\T„Gz)] = iC«e^ (m > 0)
and where p. = ka. Therefore, using Eqs. (19.5),
Ji(ka) = %Co sin (70) ; Ni(ka) = — iC cos (70)
J m+ i(ka) — J m i(ka) = 2C m sin (y m ) (m > 0)
N m i(ka) — N m +i(ka) = 2C m cos (y m ) (m > 0)
Note the additional factor of two in C , to anticipate the factor ^ in the
sum for ra = 0. Limiting values of the amplitudes C m and phase
angles y m are given by the following approximate formulas:
When (ka) = (2ira /\) » m + £
Co c^ \/8/irka; y ~ka — (t/4)
C m ~ y/2/wka; y m ~ ka — hf(m + i) (m > 0)
When (ka) = (2ira/\) « m + i ) (26 6)
C ~(4Ma); 7o^7r(fca/2) 2
„ m!/2V +1 **» (ka\ m
Values of some of the C's and 7's are given in Table X at the back of
the book.
To fit the expression for u r at r = a to the expression for the velocity
of the cylinder u a , we must make
. pcU da .
A m =  — ~ — e tym
Since, at very large distances from the cylinder
J m (kr) + iN m (kr) ~ ^ e f [^i^+i)]
302 THE RADIATION AND SCATTERING OF SOUND [VII.26
we have the following expressions for the pressure, particle velocity,
and intensity of sound at the point r, <f> (when r is a large number of
wavelengths) and for the total power radiated by the element per unit
length of cylinder:
a U da */ e ik{r  ct) \{/(<p) ; p ~ pcu
\w 3 ka ^u (J m
r
00
_ / .N r Tm + £ (2m+1)
T
pc\Uda) 2 *S? cos (m<f>) cos (n0). } (26.7)
2r*pr ^J n C m C n
•cos [y m — y n + ^7r(m — n)]
n = pc 2 (£/^) 2 [_2_ , ^ J. 1
Wv LCI £{ ell
To find the intensity we have, of course, multiplied the real parts of
u r and p together and averaged over time. The total power II per
Long and Shortwave Limits. — When the wavelength is quite
long compared with 2ira, we can use the second part of Eqs. (26.6) to
compute the radiation. The largest values of (l/C m ) are for m = 0,
so that to the first approximation
^(0)c A /pe** /4 (26.8)
T ~ (*£') (IT *»)•; TL^(^)(Uday
At these low frequencies the sound radiates out with equal intensity
in all directions, and the amount radiated is small. The expression
for intensity is the same as that given in Eq. (26.3) for a uniformly
expanding cylinder, if we substitute for U in the earlier expression
the average velocity (U da/2n) of the surface.
Values of T and II are plotted in Fig. 65. Polar curves of the
intensity are shown for different values of n = ka = (2ira/X) and a
curve is given for II as a function of /z. We notice that at long wave
lengths very little power is radiated, and the intensity has very little
directionality. As the wavelength is decreased more power is radi
ated, and the intensity has more directionality; the cylinder begins
VII.26]
RADIATION FROM CYLINDERS
303
to cast a "shadow," and a smaller proportion of the energy is sent
out on the side of the cylinder opposite the radiating element. For
very short waves the intensity is large and uniform from <£ = — {ir/2)
to d> = +(x/2) and is zero from <t> = (ir/2) to <t> = (3tt/2), in the
shadow. In the intermediate range of n, where the wavelength is
about the same size as a, interference effects are noticeable. The polar
curve for ju — 3 shows that a fairly intense beam is sent out from the
cylinder in a direction diametrically opposite to the position of the
line source (<£ = 180 deg).
The general properties illustrated by this set of curves are a charac
Cylinder
2k<*
teristic of all wave motion when
it strikes an obstacle. When the
wavelength is large compared with
the size of the obstacle, the wave
pays hardly any attention to its
presence. The first polar curve
in Fig. 65 shows that for long
waves the intensity is distributed
in approximately the same manner
as it would be if the line source
were not in the side of a cylinder
but were all by itself, radiating
into free space. On the other
hand, when the wavelength is
very small compared with the size
of the obstacle, the motion resem
bles the motion of particles, the
waves traveling in straight lines,
and the obstacles casting sharp
edged shadows. Light waves
have this raylike property in most
cases; geometrical optics is a valid
approximation because the light waves are very much shorter than the
size of most of the obstacles that they encounter.
When the wavelength is about the same size as the obstacle, com
plicated interference effects can sometimes occur, and the analysis of
the behavior of the waves becomes quite complicated.
Radiation from a Cylindrical Source of General Type. — If the line
source is not at <f> = on the surface of the cylinder but is at <f> = a,
the pressure and velocity at large distances are, by Eqs. (26.7),
Fig. 65. — Power radiated and distri
butioninangle of intensity from a vibrat
ingline source set in a rigid cylinder, for
different values of n = 2ira/X.
304 THE RADIATION AND SCATTERING OF SOUND [VH.26
p — > pc(U da) .J \}/(<j> — a)e ik<  r  ct \
U r +(Uo da) J 4>(4>  a )e mr  c »
the axis of the polar diagram being turned through an angle a. If
several sources are distributed over the surface of the cylinder, each
for a different value of a, the resulting radiated wave will be the sum
of all the waves for the individual sources taken separately. This fact
can be used to express the radiation from a cylinder whose surface
vibrates with any arbitrary distribution of velocity amplitude. If the
distribution is such that the surface at <f> = a has the radial velocity
TJ{a)e~' iTivt , then the wave may be considered to be the result of an
infinite sequence of line sources, the one at <f> = a having the velocity
amplitude U(a), etc. The pressure and radial velocity at large
distances are then obtained by integrating the expressions given
above for a single line source:
p~ p cj® e ik ( r  c » J iK0  a) U(a) da (26.9)
fe)
For instance, if a section of the cylinder between a = —a and
a = \a is vibrating, so that U(a) = U for — a Q < a < a , and
is zero for the rest of the values of a, then the pressure wave at large
distances from the cylinder is
V = pcUo J^ e ik (*°» J ° iK0  «) da
= 2pcU /7 tt(fWrf) >h sin (map) cos (m<ft) g i[\m+f(2m + i)]
t 2 \j/r ^J mC m
(26.10)
where we use the convention that  —  — — = a when m = 0.
m
The intensity and total power radiated are
_ 2pc 2 Ul >ri sin (map) sin (na ) cos {m<j>) cos (n<f>)
T*vr ^Li mnC m Cn
• cos 7m — 7n + (m — n) I
Vn.26] RADIATION FROM CYLINDERS 305
_ 2pc 2 U 2 [2a ^h sm 2 (m<x ) ]
11 ~ tt 3 ? C 2 "^ 2j m*Cl J
When the frequency is very small, so that X is much larger than a,
the intensity and power radiated are
T > P^H n>7rpm 2 t/^
to the first approximation. The expression for T is quite similar to
Eq. (26.3) for a uniformly expanding and contracting cylinder. Even
though the velocity of the surface of the cylinder, in the present case,
is not symmetrical about the axis, nevertheless the radiation at very
long wavelengths is symmetrical, behaving as though it came from a
uniform cylindrical source whose velocity amplitude is (ZJo<xo/t) (the
average velocity amplitude of the actual surface). This is another
example of the fact, mentioned above, that wave motion is insensitive
to details smaller in size than the wavelength.
Transmission inside Cylinders. — Now that we have discussed the
behavior of cylindrical waves of general type, it is appropriate to
return to the subject discussed in Chap. VI, the transmission of sound
inside tubes, and indicate what happens when the sound is not a one
dimensional wave. As an example, we take a tube of circular cross
section of radius a, driven at the end x = 0. The three coordinates
are r and 4>, giving position in the crosssectional plane, and x, the
distance along the axis (we use x instead of z to correspond to the
usage of the last chapter). The solutions for the pressure waves
inside the tube will be some combination of the characteristic waves
™(»*V«(Av)« l( *^" l> ; *? + *2= , (fY
The functions N m cannot be used here, for they become infinite at
r = 0. The values of k r are determined by the boundary conditions
at the inner surface of the tube, r = a.
Ordinarily the tube is fairly solid, but it does yield slightly to the
pressure of the waves on the inside, and in many cases this yielding
has important effects. At any rate we cannot always neglect it
entirely. We shall assume that each portion of the tube moves out
ward with a velocity proportional to the pressure of the wave at that
point:
u r = —) at r = a
Zt
306 THE RADIATION AND SCATTERING OF SOUND [VII.26
The factor of proportionality z t is called the specific impedance of the
inner tube surface and the ratio of this to pc is the impedance ratio
of the wall, ft. It may be due to the elasticity of the tube walls (in
which case z t is mostly reactive), or it may be due to the fact that the
inner surface of the tube is covered with porous material which allows
a small amount of air to penetrate the pores (in which case z t is often
mostly resistive). Of course z t will depend on frequency. The
reciprocal of the impedance ratio, the admittance ratio, will be written
(pc/z t ) = (1/ft) = (i<t — i<r t ), where a t is the conductance ratio of the
tube and c t is its susceptance ratio.
We can compute the radial velocity from the pressure wave by
using Eqs. (25.2) and (25.5). Confining ourselves here to the radially
symmetric waves, the boundary condition is
p = AJoiKrfeW**^; u r = i l — J AJi(k r r)e i(kxX  iut ^
Jo(k r a) = i(^f)ji(k r a); k = (£), ft = (ji\
(26.11)
If the specific impedance z t is large compared with the characteristic
impedance of air (as it usually is), then ft is large compared with unity,
and the first approximation to the solution for k r is that —Ji(kra) =
jfj^x [Jo(kra)] is equal to zero. Solutions for this can be obtained
from tables of Bessel functions:
fc r a~xa „; aoo = 0; a i = 1.220; a 02 = 2.233 • • • (26.12)
(For further values of the a's, see page 399.) The case a o = is the
plane wave case we discussed in the previous chapter. We note
that a 0n equals the quantity /3i„ given in Eq. (19.6) for the circular
membrane.
The next approximation is obtained by expanding the functions J
and J i about k r a = wa 0n and taking the first terms:
+
{k r a)Ji(k r a) = (xaon)«/i(7ra n) + (k r a — ira 0n ) j zJ x (z)
Therefore Eq. (26.11) becomes, to the next approximation (assuming
that ft is large),
7 ■ I ka . \ . / oioa ,\
kra ~ ira n — 1 1 U ] ~ ira 0n — * [ U I
\ira 0n / yraion /
(26.13)
k = ( w /c); ft = (z t /pc)
VII.26] RADIATION FROM CYLINDERS 307
except in the case n = 0. In this special case, Eq. (26.11) reduces to
1 = li(k*a$i/k), or
*'~V!s ; M+ fe) (n=0) ■
The general expression for axially symmetric waves in the tube is
(26.14)
+ 2^.[(^)<^)
. g— (iKi/o)+i[ (&)T 0n /c) + (<rj/a) Jx— iu «
where (1/ft) = x t — *Vt = (pc/z t ).
Wave Velocities and Characteristic Impedances. — These waves all
travel with different velocity, even in the case of a rigid tube (f t = oo ).
This is due to the factor ro n entering into the term (r „oja;/c) and is
fundamentally due to the fact that all the waves except that for n =
(a o = 0) have components of motion perpendicular to the tube axis.
The lowest mode {n = 0) represents the plane wave solution we
discussed in the previous chapter, for the case of ft = oo . When the
tube wall impedance is riot infinite, even this wave has a small com
ponent of motion in the radial direction, for p depends to some extent
on r. To the second approximation the pressure wave for this lowest
mode is (from the first two terms in the expansion for J )
V
( r 2 \1
\ _1_ £ w I 1 g— (xKt/a)+i[(a/c)+(<rt/a)]x— iwt
\2acf t /J
Therefore, when the tube wall impedance is taken into account, the
pressure wave is not quite uniform over a plane perpendicular to the
zaxis; there is a part of the wave, out of phase with the usual plane
part, which is zero at the axis r = and reaches a maximum at r = a.
If the tube admittance (l/z t ) = (1/pcft) = (l/pc)(/c, — ia t ) has a
real part {n t /pc = conductance) not equal to zero, the wave dies out
exponentially as it travels down the tube. If the tube susceptance
(<r t /pc) is not zero, the wave velocity differs from that in free air, even
for this "planewave" mode. To the first approximation, the wave
velocity is c[l  <r t (\/2ira)], as long as f< is very large. A stiffness
reactance corresponds to a positive susceptance (f ~ ipc{K/u) ;
(1/f) — —if = ~i((a/Kpc), so that a tube with "stiff" walls has a
308 THE RADIATION AND SCATTERING OF SOUND [VII.26
wave velocity somewhat less than c, whereas a tube with walls pre
senting masslike reactance has a wave velocity somewhat larger than
c. The departure from velocity c is greatest at long wavelengths
(unless the variation of <x t with co cancels this effect).
The higher modes travel with a wave velocity which is larger than
that in free air. Even when the tube impedance is infinite the veloc
ities are (c/ron) = c/s/l — (Xa n/2a) 2 . Below a certain frequency
v 0n = (ca 0n /2a), called the nth cuioff frequency, the nth mode cannot
be transmitted down the tube as wave motion. The quantity t »
becomes imaginary, and the pressure dies out exponentially for any
lower frequencies. In other words, for a given frequency v only the
first n modes are propagated down the tube, where n is such that
a 0n is just smaller than (2av/c). For a tube of 1 ft diameter only the
lowest, "planewave," symmetrical mode is transmitted for fre
quencies below about 1,300 cps, and for a smaller tube the higher
modes are damped out for even higher frequencies. At the cutoff
frequency the wave motion for the "cutoff" mode is purely transverse.
Only at higher frequencies does this mode have any motion "to spare"
for travel along the tube.
The nonsymmetrical modes, involving cosine and sine of (ra<£),
also have cutoff frequencies v mn = (ca mn /2a), where
aio = 0.586; an = 1.697; a 12 = 2.717
a 20 = 0.972 •
a 3 o = 1.337 } (26.15)
TYh 1
a mn — w + 9 + 7 ( n large) (see also page 399)
Only the "planewave" mode (0,0) is transmitted at all frequencies.
We note that these a's are not equal to any of the /3's in Eq. (19.6)
for the circular membrane.
The ratio of pressure to air velocity in the direction of wave
motion for a wave traveling in one direction is defined as the char
acteristic impedance for the mode in question. For free air we have
found that it is pc. In a tube with walls of impedance pc{ t the char
acteristic impedance for the "planewave" mode is approximately
pc[l — (\/2Ta)((Tt + iKt)]. Therefore if the tube impedance is purely
reactive, the characteristic impedance is purely resistive; if the tube
impedance has a real component, the characteristic impedance has a
reactive term. The difference between the characteristic impedance
for this mode and that of free air is proportional to the tube specific
admittance (pc/£ t ) as long as f t is large.
VH.26] RADIATION FROM CYLINDERS 309
The nth symmetrical mode has a c haracteristic im pedance (pc/T 0n )
•[1 — i(X/2iraT 0n ^t)], where r 0n = \A — (Xaon/2a) 2 . Therefore the
characteristic impedance for the higher modes differs from pc even
when the tube impedance is infinite. At frequencies well below the
cutoff for the mode, the impedance is reactive, being approximately
equal to that of a mass of (pa/ira n) g per sq cm of piston surface.
If a reflected wave is also present, due to mismatch at the far end,
the impedance for each mode has a hyperbolic tangent factor in it,
analogous to the cases worked out in the previous chapter.
Generation of Wave by Piston. — The sort of wave that travels
along the tube is determined by the nature of the driving piston at
x = 0. To illustrate this part of the calculation, we consider a rigid
tube and confine ourselves to symmetric oscillations of the piston;
hence only the symmetric modes will be present. The extension to
nonsymmetric modes and to nonrigid tubes will then be evident.
Suppose that the velocity of the driving piston is given by u x =
Ua{r)e~ iat at x = 0, and suppose that the tube is long enough so we
can neglect reflected waves. The functions Jo(ira 0n r/a) are orthogonal
characteristic functions, analogous to those discussed in connection
with Eq. (19.10). Their integral properties are
jJ*J^)j (^) rdr = K 2 (26.16)
o \ a / \ a / )x JKiraon) (to = n)
Therefore the function u (r) can be expanded into the series
U n Jo(Tra 0n r/a)
n =
where
Uo(r) = > =) f
^4 Joiiraon)
n =
^» ~ TTl "T I u (r)J ( 7 ^IL_ \ r dr
la 2 Jo{ira Qn )_\ Jo \ a /
The coefficient Uo = (2/a 2 ) f u (r)r dr is the average velocity of
the piston.
We next fit the series of Eq. (26.14) (for K t = a t = 1/ft = 0) to
this series as a boundary condition at x = 0, by first computing u x and
then equating term by term. We find that A n = {pc/tq^} U n /J (Traon),
so that the pressure wave traveling away from the piston is
310 THE RADIATION AND SCATTERING OF SOUND [VII.26
(pc/r 0n ) U n Jo(Tao n r/a)e i(  a/c) fro.**)
2
n =
Jo(Traon)
•Jo (~~) e^Wo**^ (26.17)
TOn = Vl — (TTCQIOn/wa) 2
If the motion of the piston is uniform over its surface, then only the
first term (n = 0) remains, and we are back to the planewave case
discussed in the previous chapter. If the piston is not perfectly stiff,
the motion will not be uniform and some of the higher modes will
come in.
As an example, suppose that the driver is a membrane, driven so
that its motion is u (r) = UJ (Tt3 ir/a), where O1 is adjusted so that
Jo(irpoi) = [see Eq. (19.6)]. Then, from formulas for the integration
of two Bessel functions, it turns out that
TT _ (2C7/3oiA) t ( r \ 0.432E/
Un ~ ~Q2 2 «/ lW30lj = "H /i or . a Ton
POI — «0n U — (1.306o!0n) 2 ]
The average velocity of the piston is U Q = 0.432 U.
The average pressure back on the diaphragm is the coefficient of
the first term in the series, pc times the average velocity. The pres
sure at a point on the diaphragm a point r from its center is
oo
Voir) = pcUo > pj n Qnft — '' . r — (26.18)
^Jll — (1.306o:oJ 2 l/n(7ra;n„.)Tn» '
n =
which is not equal to the average pressure. In fact this pressure is
no t exactly in pha se with the velocity because of the factors t „ =
\/l — (caon/2av) 2 , which become imaginary for the larger values of n.
At this point we can refer back to our discussion of the air reaction
on a membrane, given on page 193. There we said that if the velocity
of waves on the membrane was very much smaller than the velocity
of sound waves in air, then the pressure was fairly uniform over the
membrane and was proportional to the average velocity of the mem
brane. In the present case, if the membrane is in free vibration at
its lowest frequency, the driving fr equency will be v = (c TO /3 i/2a), so
that the factors t » will equal \A — (ca 0n /c m ^oi) 2 (where c is the
sound velocity and c m the velocity of waves in the membrane). If c is
VII.27] RADIATION FROM SPHERES 311
very much larger than c m , then all the t's, except t o = 1 (for apo = 0),
will be large and imaginary. Consequently, the most important term
in the series (26.18) for the pressure will be the one for n = 0, which
is just pc times the average velocity of the membrane, the higher terms
being out of phase and smaller the larger the ratio (c/c m ) is.
On the other hand, if (c/c m ) is quite small, a large number of t's
will be approximately equal to unity; therefore series (26.17) will
become p c^pcwo(r)e i(o,/c)<:af ~ c<) , except for small terms for higher n.
In this case, therefore, the reaction at a point on the membrane is
just pc times the velocity at that point. This is the other limiting case
mentioned on page 198. The intermediate case is, as we see, quite
complicated.
The same sort of calculation can be made for waves radiating into
the open (we shall discuss this in the next section) and the same
general conclusions concerning wavevelocity ratios and reaction on
the membrane can be reached.
The transmission of sound through ducts will be taken up again
in this chapter, after we have discussed the nature of wall impedance
more in detail.
27. RADIATION FROM SPHERES
More important for radiation problems than cylindrical coordi
nates are the spherical coordinates r, &, and <p; r being the distance
from some center, # being the angle between r and a polar axis, and
<p the angle between the plane through r and the polar axis and a
reference plane. Few radiators of sound are so much longer than
they are wide that they behave like long cylinders, but many radiators
behave like spherical sources, especially when their dimensions are
small compared with the wavelength of the sound emitted.
Uniform Radiation. — The simplest sort of outgoing spherical wave
is one that is radiated by a sphere uniformly expanding and contract
ing, so that the wave does not depend on # or <p. The wave equation
in this case is
d 2 p
dt 2
A general solution of this equation which is finite everywhere except
at r = is
r 2 dr\ dr ) c 2
p =±F(rcQ+±f(r + cQ
consisting of a wave of arbitrary form going outward from the center
and another wave focusing in on the center,
312 THE RADIATION AND SCATTERING OF SOUND [VII.27
This solution, except for the factor (1/r), is similar to Eq. (8.1)
for waves on a string and to the equation for plane waves of sound
given on page 238. This means that spherical waves are more like
plane waves than they are like cylindrical waves. Plane waves do
not change shape or size as they travel ; spherical waves do not change
shape as they spread out, but they do diminish in amplitude owing
to the factor (1/r) ; whereas cylindrical waves change both shape and
size as they go outward, leaving a wake behind. Figures 40 and 41
show that, if a cylinder sends out a pulse of sound, the wave as it
spreads out has a sharp beginning but no ending; the pressure at a
point r from the axis is zero until a time t = (r/c) after the start of the
pulse, but the pressure does not settle back to its equilibrium value
after the crest has gone by. With both plane and spherical waves
the wave for a pulse has a sharp beginning and ending, the pressure
settling back to equilibrium value after the pulse has gone past. This
behavior is another example of the general law (proved in books on
the mathematics of wave motion) that waves in an odd number of
dimensions (one, three, five, etc.) leave no wake behind them, whereas
waves in an even number of dimensions (two, four, etc.) do leave
wakes.
Spherical waves do resemble circular waves on a membrane, how
ever, in that they have infinite values at r = 0. As we have seen on
page 176, this simply means that the size of the source must be taken
into account; every actual source of sound has a finite size, so that the
wave motion never extends in to r = where it would be infinite.
Suppose that a sphere of average radius a is expanding and con
tracting so that the radial velocity of its surface is everywhere the
same function of time U{t). The rate of flow of air away from the
surface of the sphere, in every direction, is Anra % U{t) = Q(t). To
obtain an expression for the pressure wave radiated from the sphere,
we write Newton's equation as p(du r /dt) = — (dp/dr). If p is chosen
to be an arbitrary outgoing wave p = P(r — ct)/r, the requirement at
the surface of the sphere,  ^ 5 = — r 1 —  9 —> at r = a serves to
7 r or r z 4xcr at
determine the shape of the wave P.
The Simple Source. — If the vibrating sphere is very small (more
specifically, if a is small compared with the wavelength of the sound
radiated), the sphere is called a simple source of sound. In this case
(P/r) is much larger than (dP/dr) at r = a, and P ~ (p/4nr)(dQ/dt)
at r = a. The pressure wave at a distance r from the center of the
simple source is therefore
VH.27] RADIATION FROM SPHERES 313
(27.1)
&«■('§)
where Q'(z) = r Q(z). The function Q gives the instantaneous value
of the total flow of air away from the center of the source. The pres
sure at the distance r is proportional to the rate of change of this flow
a time (r/c) earlier.
In some cases the source of sound is small enough or is so placed
with respect to boundaries that source plus boundaries (if they are
present) can be replaced by a distribution of simple sources, such that
the element of volume dx' dy' dz' at (x',y',z') has an equivalent outflow
of air q(x' ,y' ,z' ;t) dx' dy' dz' cm 3 per sec. The function q is called the
source function or density of sources. This method of calculation will
be particularly appropriate for the calculation of the radiation from a
piston and for the calculation of sound radiation in a room.
It is not difficult to see that a logical extension of Eq. (27.1) gives
for the pressure wave at the point (x,y,z)
< a d C C C dx' dy' dz' ( , , , R\ /nfr „ N
V^y,z;t) = p  j J J —JL a [x',y',z';t  ) (27.2)
where R is the distance between (x,y,z) and (x',y',z'). The differential
equation for the pressure wave in the presence of a source distribution
of density q is
An important special case is when the simple source has a simple
harmonic flow of air, so that Q(t) = Qoe™*, where Q , the magnitude
of the volume flow of air at the surface of the source, is called the
strength of the simple source. In this case, at large distances from the
source (i.e., many wavelengths away) the pressure wave is
p ~ iw U~\ Qoe^o"), k = (u/c) = (27I/A) (27.4)
The corresponding particle velocity at great distances is (p/pc), and
the intensity and total power radiated from a simple source are
8cr 2 2c
When the wavelength of sound is much longer than the overall
dimensions of the radiator, the radiation will be much the same no
314 THE RADIATION AND SCATTERING OF SOUND [VII.27
matter what shape the radiator has, as long as the motion of all parts
Of the radiator is in phase. In such limiting cases the formula (27.4)
for the simple source can be used. For instance, the open end of an
organ pipe, or of any woodwind instrument, is usually small enough
to be considered as a simple source. If the average velocity of the
air in the mouth of the tube is U e 2rivt and the cross section of the
tube has area S, then the strength of the equivalent simple source is
UoS, and the power radiated is (wp/S V £7g/2c) .
Spherical Waves of General Form.— To discuss any more com
plicated waves radiating from a sphere, we must consider the solutions
of the wave equation that depend on # as well as on r and t. In this
book we shall limit the discussion to waves that depend on # but are
symmetric about the polar axis and therefore are independent of <p.
To consider the waves that depend also on <p would add more compli
cation than is needed for the problems to be treated.
The wave equation to be solved is
r 2 dr V dr J ^ r 2 sin tf d& \ * d&) c 2 dt 2
If p = R(r)P(&)e 2wirt , the equation becomes
The left side of the equation is a function of r only, and the right
hand side is a function of # only. Since they are equal for all values
of r and #, they must both have the same constant value, which we
can label C.
Legendre Functions. — We shall first solve the equation for P
1 * ( shi »§) + CP =° < 27  5 )
sin # d&
or, letting cos# =
x,
d [d " **) ^] + CP =
dx\ dx
We solve this equation by setting P equal to an arbitrary series
p = a + aix + a 2 x 2 + • • • and substituting this in the second
equation. We find that the coefficients a must be such that
VII.27] RADIATION FROM SPHERES 315
P = a
1 _ £ X 2 _ C(6 ~ O x < _ C(6  0(20  O r6
2! 4! 6!
+ ax he H g— x s + i ^ i a; 5 +
The equation and solution have been treated in Prob. 5 (Chap. I) and
on page 118. We have found that unless C has certain specific values
the function P will become infinite at x = ±1. The only cases where
P stays finite are for the following sequence of values of C and a or
a x : C = 0, a x = 0; C = 2, a = 0; C = 6, a x = 0; C = 12, a = 0; etc.
The allowed values of C are therefore
C = m(m +1) (to = 0,1,2,3 • • • )
The solution of Eq. (27.5) which is finite over the range of x from —1
to +1, corresponding to C = m(m + 1), is labeled P m (x). It can
be obtained by substituting the proper value for C in the series given
above, making one a equal to zero and giving the other a the value
that makes P m (l) = 1. The resulting solutions are
m = o, C = 0, P (x) = 1
Po(cos#) = 1
w = 1, C = 2, Px(x) = X
Pi(cos#) = cos#
to = 2, C = 6, P 2 (x) = i(3z 2  1)
P 2 (cos#) = K3 cos 2* + 1)
to = 3, C = 12, P,(x) = H5* 3  Sx)
P 3 (cos#) = (5 cos 3^ + 3 cos)*
(27.6)
The function P TO is called a Legendre function of order to. It can
be shown to have the following properties:
d 2 P dP
{x2 ~ l) ^ + 2x ^~ w(m + l)Pm = (* = c °s*)
1 d m
Pm{x) = ¥^.dx^ {x2 ~ l)m
(2m + l)a;P m (aO = (m + l)P m+1 (z) + mP m ^{x)
(2to + l)P m (s) = * [P m+1 (x)  P m i(x))
(n J* 5 m)
(n = to)
(27.7)
316 THE RADIATION AND SCATTERING OF SOUND [VII.27
Values of some of these functions are given in Table IX at the back
of the book.
The last equation of (27.7) shows that the functions P m (x) constitute
a set of orthogonal characteristic functions. Any function of x in the
range from a; = 1 to a; = — 1 can be expanded in terms of a series of
these functions:
CO
F(x) = 2) B„P m (x), B m = (m +i) f 1 F(x)P m (x) dx (27.8)
m =
The expression for the coefficients B m can be obtained by the method
discussed on page 108.
Bessel Functions for Spherical Coordinates. — We must now solve
the equation for the radial function R:
Only the allowed values of C can be used; otherwise the related func
tion of # will not be finite everywhere. Changing scale, to get rid of
numerical factors, we obtain
g+si+O^)**^* (2 , 9 )
This equation looks very much like the Bessel equation (19.3), and
in fact solutions of the equation are R = (l/\/z)J m+ i(z). The solu
tions can be expressed in a simpler way, however, in terms of trigono
metric functions. Direct substitution will show that two solutions of
Eq. (27.9) are the spherical Bessel and Neumann functions j m (z)
and n m (z), where
3
»(*)
=
smz
z
n (z) = —
cosz
z
3
iGO
=
sin z
COS2
z
n\(z) =
sinz
z
cosz
3<
.(*)
ft
 J smz
2/
3
k cosz
z 2
ni{z) =
3 .
5 sinz
z 2
_(3.
\z 3
(27.10)
 I cosz
(27.11)
VH.27] RADIATION FROM SPHERES 317
These functions have the following properties:
3m(z) = yJ^Jm+itz), n m {z) = J^N m+i (z)
jm(  z) T^l'35 ••• (2m + 1)
„.(,) _  1185 u;( 2ml)
... 1 / mf 1 \
3m(z) ~^~ cosl 2 2"— T )
. . 1 . / m+ 1 \
n m (z) > sinlz £ — t )
Z— » oo Z \ if
fft(z)z* dz = ~ [jl(z) +' no(*)iiOO]
2 3
/n§(2)2 2 dg =  [ng(«)  io(2)n!(2)]
n m _i(2) < 7* OT (2)  n m (2)j m _i(2) = f^J
and have the following properties, for either j m or n m :
3mAZ) + Jm+l(2) = J m (z)
J J
di* 7 '" 1 ^ = 2m + 1 W»i(*) ~ ( m + l)i+i(2)]
g^VOO] = ^ + %i(2), ^[2^(2)]  ~zj m+1 (z)) ( 27 12)
Jj'lCa) dz = jo(z), Jjo(z)2 2 cfe = 2 2 Ji(2)
2 3
Jim(2)z 2 dz = g \.Jl(z)  i»i(2)y*+i(2)] (m > 0)
In all the equations (27.12) the function n can be substituted for j.
Values of some of the functions j and n are given in Table VII at the
back of the book.
Returning to the problem of wave motion, we can now see that
the general solution of the wave equation which is symmetrical about
the polar axis and is finite everywhere except at r = is a combination
of the functions
V = P.(cos^) [Aj m (?^) + Bn m (^] e w (27<13)
This corresponds to Eq. (26.1) for cylindrical waves.
318 THE RADIATION AND SCATTERING OF SOUND [VII.27
The Dipole Source. — The first example of the use of the functions
discussed above will be that of the radiation from a sphere of radius a
whose center vibrates along the polar axis with a velocity Uoe~ 2 * ivt .
The radial velocity of the surface of the sphere is Uo cost? er 2Tivt }
where # is the angle from the polar axis; and this must equal the radial
particle velocity of the air at r = a. Therefore the pressure wave
must be
p = AP!(cos#)[ji(/cr) + in x {kr)]e^ ivt , k = (2*/X) (27.14)
> — A ( ^— ) cost? e ik(  T ~ ct)
r—>«> \lirvr/
The radial particle velocity is
 * dP
r 2wivp dr
= Acos * [j (kr) + in (kr)  2j 2 (kr)  2in 2 (kr)]e** i *
oipc
cost? e~ 2 * ivl
»o \ pc ) \kr)
If the radius of the sphere is small compared with the wavelength of
the sound radiated, the limiting expression for u r can be equated to
Uo cos t? e~ 2wivt at r = a to determine A :
A ~
/4fVW \
A small vibrating sphere of this sort is called a simple dipole source.
The intensity and the total energy radiated are
2irV 4 a 6 ^oCOS 2 <?
(27.15)
J»2ir Pit
d<p
o Jo
Tr 2 smt? dt? = — ~x — 
6c 3
We note that the dipole source is less efficient than the simple source
[given in Eq. (27.4)] for radiating at low frequencies, for II diminishes
as v 4 instead of as v 2 .
To find the reaction back on the source due to its motion, we first
obtain the pressure at the sphere's surface. To the first approxima
VII.27] RADIATION FROM SPHERES 319
tion, this is p r==a ^ —rivpaUo cos# e~ 2irivt , for a small. The net' force
on the sphere is in the direction of vibration and is
F = ° 2 Jo' d<p Jo ( pr=a cos ^ sin ^ d * — ~ (^'"Xl^pa 3 ) Uoe 2 ™ 1
This force is entirely reactive (since it is 90 deg out of phase with the
velocity) and is equivalent to the reaction of a mass of air having
volume equal to half the volume of the sphere. The resistive reaction,
the part in phase with the velocity, is not given by this approximate
expression, which simply means that the resistive part of the reaction
for the dipole is so much smaller than the reactive part that it is not
included in the first approximation. To find it we must compute the
second approximation for A and p r=0 . However, since we know that
the total energy radiated must equal (E/o/2) times the real part of F,
we can find this real part from Eqs. (27.15). The radiation impedance
of the simple dipole is therefore
7 FWw*pv*a? , .wi ax  '"■'
r = U 3c 5 (2Tiv)(hrp a*)
Radiation from a General Spherical Source. — We take up next the
general case of a sphere, not necessarily small, whose surface vibrates
with a velocity U{d)e 2Tivt , where U is any sort of function of #. We
first express the velocity amplitude £/(#) in terms of a series of Legendre
functions
m =
U m = (m + i) £ U(&)P m (cos&) sin# d& (27.16)
To correspond to this we also express the radiated pressure wave in a
series
00
p = 2) AmPmicoB^lj^kr) + in m (kr)]e 2 " rt
m =
where the values of the coefficients A m must be determined in terms
of the known coefficients U m . The radial velocity of the air at the
surface Of the sphere is
=  ^ A TO Z> m P m (cos#)e* 8 »' i
pc
n» =
320 THE RADIATION AND SCATTERING OF SOUND [VII.27
where
 j [jm(ij) + in m {n)] = iD m e^ m , and n = (ka) = I ^ J
Therefore, using Eqs. (27.11) and (27.12),
mn m i(ka) — (m + l)n m+ i(ka) = (2m + l)D m cos 8 m
(m + l)jn^i(ka) — mj m i(ka) = (2m + \)D m sin 8 m
When ka » m + g
D m ~lj); 8 m ~ ka — £n(m + 1)
When k«m + 
^ 1  3  5 ';f; + r 1)(m+1) («>o)
5 ™ — 12 . 32 . 52 . . . (2m  l) 2 (2m + l)(m + 1)
Values of the amplitudes D m and of the phase angles 8 m are given in
Table XI at the back of the book.
The radial velocity of the air at r = a must equal that of the sur
face of the sphere, and equating coefficients of the two series, term by
term, we obtain equations for the coefficients A m in terms of U m :
(27.17)
Am —
( pcU m \
\D m )
The pressure and radial velocity at large distances {i.e., many
wave lengths) from the sphere can then be expressed
©
w r ~ C/ [  J e ik(  r  ct) ^(&) ; p ~ p cu r
\ka)^i
*(*) = ( h ) ^j^P^cost^^^+u
where we have multiplied and divided by Uo, the average velocity of
the surface of the sphere. The air velocity near the sphere is, of
course, not in phase with the pressure, nor is it entirely radial; but
far from the sphere the velocity is radial and in phase with the pressure.
When kr is very large, the intensity at the point (r,#) and the
total power radiated are
Vn.27] RADIATION FROM SPHERES 321
F 'W = (jzrr) 1 2 ^#^(cos^)P n (cost?) • . /
\ kaU °/ TO ^o ^"^ > (27.18)
• cos [5 TO — 8 n + ir(m — n)]
n= fd* r r 2 Tsint? ^ = pg!_/^ ffl
Jo Jo 2™ 2 £J (2m + l)D 2 m
The function /?,(#) is called the angledistribution function for radiation
from a sphere.
When ka = (2ira/\) is quite small, all but the first terms in these
series can be neglected, and, using Eq. (27.17), we have for the long
wavelength limit
p ca  ia ( ~ J (4ra 2 Uo)e ik(  r  c ^ (27. 19)
which is identical with Eq. (27.4) for the simple source, if Q , the
strength of the source, is set equal to (4wa 2 U )
Radiation from a Point Source on a Sphere. — To show graphically
how the radiation changes from symmetrical to directional as the
frequency of the radiated sound increases, we shall work out the
details of the radiation problem for two cases.
The first case is that of a point source at the point # = on the
surface; i.e., the surface velocity of the sphere is zero except for a
small circular area of radius A around & = 0. The definition of
U(&) is
77<W = i Uo (0 ^ * < A/a >
K) \0 (A/a<#<C7r)
and the coefficients U m are
U m = (m + i)u f^ a P m (coad) sm&dfr — >$(m + i)u (A/a) 2
J0 A+0
since P m (l) = 1. The intensity and power radiated can be obtained
fromEqs. (27.18):
• cos[3 w — S„ f j7r(m — n)]
322. THE RADIATION AND SCATTERING OF SOUND [VII. 27
, / A 4 \ 4tt ^O 2m + 1 /27ra\
At very low frequencies, only the first terms in these series are
important, and the pressure wave again has the familiar form for the
simple source
V
'^(^)
(7rA 2 Wo)e ifc(rc0
with a strength equal to the velocity w times (xA 2 ), the area of the
radiating element.
' Figure 66 gives curves ;f or the distribution in angle of the intensity
radiated from a point source on a sphere for different ratios of wave
"0 90° 180°
Angle from Spherical Axis ,9
Fig. 66. — Distribution in angle of intensity radiated from. a point source set in the
surface of a sphere of radius a, for different values of /x = (2tto/X). Curves also give
meansquared pressure at point (a, #) on surface of sphere due to incident plane wave
traveling in negative x direction;
VH.27] RADIATION FROM SPHERES 323
length to sphere circumference. We see again the gradual change
from radiation in all directions to a sharply directional pattern as the
frequency is increased. These curves are of particular interest
because of their dual role, as required by the principle of reciprocity,
which will be discussed in Sec. 29. As computed, the curves give
intensity or pressure amplitude squared at a point (r,#) , a considerable
distance r from the sphere (when kr » 1, \p\ = \/2pcT) at an angle
# with respect to the line from the center of the sphere through the
radiating element of area at point (a,0). But the principle of reci
procity says that the pressure at a point Q due to a unit simple source
at point P is equal to the pressure at point P due to a unit source at
point Q. Consequently, the curves of Fig. 66 also represent the
square of the pressure amplitude at a point (a,0) on the surface of
the sphere due to a point source of strength (7rA 2 w ) at the point
(r,#). Therefore the curves are useful as an indication of the direc
tional properties of the ear plus head, or of a microphone in a roughly
spherical housing.
Radiation from a Pistion Set in a Sphere. — The other example to be
worked out is that of a piston of radius a sin (# ) set in the side of a
rigid sphere. As long as # is not too large, this corresponds fairly
closely to the following distribution of velocity on the surface of the
sphere :
The general formulas in Eq. (27.18) can be used, with
U m = (m f i)w f P m (x) dx = %u [P m i (cos#o) — P m +i (costfo)]
Jcos&o
where, for the case m = 0, we consider Pi(x) = 1.
These expressions for U m can be substituted in Eq. (27.18) to
give series for intensity and power. Curves for T as function of #
are given in Fig. 67. When the wavelength is long compared with
(2xa), the pressure and intensity are those for a simple source of
strength 47ra 2 Wo sin 2 (t? /2).
The radiation impedance for the piston set in a sphere can be
computed by integrating the expression for the pressure at r = a over
the surface of the piston. Alter quite a little algebraic juggling, we.
find
324
THE RADIATION AND SCATTERING OF SOUND [VII.27
Z p — E p — iXp = pdira 2 sin 2 f ~ J (0 P — ix P )
d = i^ [Pmi(cost?o) P TO+ i(costV)] 2
X P
= *2
TO=0
M 2 (2m + l)Dl
[P m l(C0S ff )  P OT+ i(cos flp)] 2
(2m + l)D m
(27.20)
• [jm(ir) sin (5 m ) — n m (n) cos (5 m )]
where fi = (2tt va/c) = (%ca/\).
The quantities 8 P and Xp are the average acoustic resistance and
reactance ratios at the diaphragm. These are plotted, in Fig. 68, as
2 ju 4 6
Fig. 67. — Distribution in angle of intensity and total power radiated from a piston set
in a sphere.
functions of the ratio of equivalent diaphragm circumference to
wavelength, (2ira p /\) = {wa p /c) [a p = 2a sin (#p/2)], for several differ
ent values of #o = 2 sin 1 (a p /2a). We notice that the resistive terms
all are small at low frequencies, increasing as o 2 ; rising to a value of
approximately unity at wavelengths a little smaller than one third
VH.27]
RADIATION FROM SPHERES
325
times the equivalent diaphragm circumference; and then, for very
high frequencies, approaching the usual limit of 1 (z = pc). The
reactance is always positive, representing a mass load. It first
increases linearly with frequency, as the reactance due to a constant
2 4 6
Ratio Piston Circumference to. Wavelength, (Zircip/A)
Fia. 68. — Values of radiation, resistance and reactance ratios (R r , X r /Tra p 2 pc),
where a p = 2a sin (#<,/2), as function of (27ra p /X) = (wa p /c) for different values of t? ,
for radiation of sound from a piston set in a sphere of radius, a.
mass would, but then it reaches a maximum at X~ 2xa, and for
higher frequencies it diminishes more or less rapidly. The value of
the initial slope is responsible for the factor in the end correction
given in Eq. (23.1) and on page 247.
These curves indicate the sort of radiation load one would expect
on the diaphragm of a dynamic loudspeaker, set in a spherical case.
326 THE RADIATION AND SCATTERING OF SOUND [VII.28
The diaphragm of tho dynamic speaker is not usually the surface
of a sphere but often has the shape cf an inverted cone in order to
increase its mechanical strength. The radiation from such a cone
would naturally differ from that from a section of a sphere; but it
turns out that the average radiation impedance on a piston is approxi
mately the same, no matter what its shape, as long as its circum
ference is not changed, and as long as the volume of the mounting
case is not changed. Therefore the radiation reaction on the dia
phragm of a dynamic speaker of outer circumference 2ira p , set in a
cabinet of volume V P = 4x« 3 /3, is approximately given by {tca\pc)
times the curves of Fig. 68 for # = 2 sin _1 [apOr /6 V ',)*], with the fre
quency scale equal to (2Trva p /c).
Not only is the radiation impedance for a diaphragm, set in a
cabinet of roughly spherical shape, given by the curves of Fig. 68;
the average specific acoustic impedance at the open end of a tube or
horn, of mouth circumference (2nra f ), is approximately equal to (pc)
times the curves of Fig. 68 for # ^ 2 sin 1 (Tra p /2a f ) . This is the
impedance referred to in Sees. 23 and 24.
A plane piston, or an open end of a tube, of radius a p , set in an
infinite plane rigid wall, corresponds to the case of a — » °o*, &o — > 0.
Fairly simple expressions for the distribution in angle of the radiation
and for the average specific acoustic impedance at the opening, can
be obtained for this case by the use of Eq. (27.2). This will be dis
cussed in the next section.
28. RADIATION FROM A PISTON IN A PLANE WALL
In this section we shall consider a useful example of the way we
can build up the radiation from an extended source by considering it
to be an assemblage of simple sources. This method has been used
for cylindrical waves to obtain Eqs. (26.10); in the present discussion
we shall build up the radiation out of spherical waves.
Suppose that a circular flattopped piston of radius a is set flush
in an infinite plane wall and vibrates with a velocity u e 2iri,, \ radiating
sound out into the space in front of the wall. It does not matter, for
the purposes of our analysis, whether the piston is an actual one or is
simply the vibrating layer of air at the open end of a tube or horn;
the effect on the air outside is the same. Actually, the air in the
open end does not all vibrate with the same velocity, as the top of a
piston does, but the velocity is nearly uniform in many cases, and the
results of the following analysis will be nearly correct when used for
VH.28] RADIATION FROM A PISTON IN A PLANE WALL
327
the open ends of pipes. Later in this section we shall indicate how
the approximation can be improved.
Calculation of the Pressure Wave.— The method that we shall
choose to solve our problem is to consider each element of area of the
piston as a simple source of sound and to add all the waves from all
the elements together to obtain the resulting wave. The strength of
the simple source corresponding to the element dS is Q = 2u dS; the
factor 2 is used because the amount u dS can radiate only into the
space to the right of the wall. The effect of the wall can be replaced
by an image elementary source u dS radiating to the left of the plane
of the wall. The actual element and its image together form a simple
source of strength 2w dS.
The radiation pressure at the point P in Fig. 69, due to the element
dS, is obtained from Eq. (27.3) :
dp =
. fpvuo dS
■tyJT^)*
( h ct )} k = ( 2 *Vc)
The particle velocity due to this wavelet is entirely radial, so that
the velocity along the surface of the
wall is everywhere parallel to the
surface, as it must be.
The quantity h is the distance
from the element dS to the point P.
If the point P is far from the piston,
so that a is much smaller than r,
then (l/h) is practically equal to
(1/V), and h is approximately equal to
r — y sin & cos yj/, where y is the dis
tance of dS from the piston center, #
is the angle between r and the per
pendicular from the center cf the
piston, and \p is the angle between the
plane defined by this perpendicular and r and the plane defined by the
perpendicular and y. Therefore, when r is much larger than a, the
expression for the pressure wave at P due to dS is, approximately,
Fig. 69. — Radiation from a piston
set in a plane wall. The wave at P
is the resultant of all the wavelets
radiated from all the elements of area
of the piston.
dp ~
<(*=*)
Qik{r— cV)q— iky&ir>dcos$y £y ^r
where dS = y dy d\p.
The total pressure at P due to all the simple sources making up
the piston is the integral
328 THE RADIATION AND SCATTERING OF SOUND [VII.28
p ~ _ ih]^l J e * ( rc«) J ydy\ e 11 *™*™* d$
= %riv(^Je ik(r ^ f J (kysm&)ydy
= ™ P u a 2 ^[ ^^ J < (28.1)
using Eqs. (19.4) and (19.5). The radial velocity at this large dis
tance is (p/pc).
Distribution of Intensity. — Using the expression developed above
for p, the radiated intensity at large distances turns out to be
• 2Jx(MBintf) j' (2jrtl/x) (28 . 2)
T = Ipculu 2
Values of [2J x {x)/x~\ can be obtained from Table V or VIII. It is
unity when x is zero, remains nearly unity until x is about (x/2), goes
to zero at x about 1.27T, falls to about —0.13 at a; about l.lir, goes to
zero again at x about 2.27T, and so on, having a sequence of maxima
and minima which diminish in size.
When the wavelength X of the sound radiated is longer than the
circumference 2wa of the piston, then the value of (n sin &) is less than
(t/2) even for # = 90 deg (i.e., even along the wall), and the term in
brackets is practically unity for all values of #. Therefore, long
wavelength sound spreads out uniformly in all directions from the
piston, with an intensity four times that due to a simple source of
strength ira 2 u . If the wall were not present and the "piston" were
the open end of a pipe, this end would act like a simple source of
strength ira 2 u for long wavelengths; so that the wall, or baffle plate,
produces a fourfold increase in intensity. The sound reflected from
the baffle reinforces the sound radiated outward, thereby doubling
the amplitude of the wave and thus quadrupling the intensity, which
depends on the square of the amplitude. Of course, to have the
baffle give this considerable increase in amplitude, it must be con
siderably larger than the wavelength of the sound radiated, so that
it will act as though it were infinite in extent.
If X is smaller than 2xa, then the reflected sound still reinforces
the sound radiated straight ahead, and the intensity has its maximum
value pv 2 (ira 2 u ) 2 /2cr 2 at # = 0. At points off the axis, however,
the reflected sound interferes with that radiated directly, and T
diminishes in value as & increases, falling to zero when sin # is about
equal to 0.6(X/a), then rises to a secondary maximum (where there
VH.28] RADIATION FROM A PISTON IN A PLANE WALL
329
is a little reinforcement) of about 0.02 times the intensity for # = 0,
then goes again to zero, and so on. Therefore, highfrequency sound
is chiefly sent out perpendicular to the wall, with little spread of the
beam. Most of the intensity is inside a cone whose axis is along the
axis of the piston and whose angle is about sin 1 (X/2a). This main
beam is surrounded by diffraction rings, secondary intensity maxima,
whose magnitude diminishes rapidly as we go from one maximum to
the next away from the main beam.
Values of the intensity as a function of # are given in polar diagrams
in Fig. 70, for different values of ju = (2tt va/c). The increased direc
tionality with increase in frequency is apparent. It is this direc
\5a
^
A. 2a
Fig. 70. — Polar diagrams of distribution in angle of radiated intensity from a piston
set in a plane wall, for different ratios between the wave length X and the radius of the
piston a. In the curve for X = o, the small loops at the side are diffraction rings.
Compare this with Fig. 67.
tionality that makes it difficult to use a single loudspeaker in outdoor
publicaddress systems: although the lowfrequency sound is spread
out in all directions, only the people standing directly in front of the
loudspeaker will hear the highfrequency sound. In rooms of moder
ate dimensions the sound is scattered so much by the walls that the
directionality does not matter particularly (unless it is very marked),
and one loudspeaker is usually adequate.
Effect of Piston Flexure on Directionality. — The equations derived
on the last few pages have been for a plane rigid piston, which moves
as a unit, every part of the surface having the same velocity. It is of
interest to see what effect variations in motion from point to point
over the piston surface will have on the sound radiation. This will
be particularly useful when the "piston" in the plane wall turns out
to be the air in the open end of a tube or horn; for it is then not a very
330 THE RADIATION AND SCATTERING OF SOUND [VTI.28
good approximation to assume that the velocity is uniform over the
open end.
In a previous section we discussed the transmission of various
modes down a tube of circular cross section. There we used (for
the case of rigid walls) the following set of characteristic functions:
J a{irao n y / a)
*n(y)
Jo(iraon)
^cmn{y,4>) _ COS / ,\ \/ , 2ira mn J m (Tra mn y/a)
Vsmniyrf) Sin J m (Ta mn ) VWmn) 2 ~ W 2
dJ m (ra mn ) = ^ Eqg (26 12) and (26 15)]
f * dty J ^f 2 y dy = ira 2
to describe the distribution of the waves across the tube. These
functions will be useful for expressing the motion of any sort of piston,
with any sort of distribution of velocity u (y,i/)e~ iat over its face:
00 00
U0\y } Y) = ?\ U n¥n ~T~ ^, \U cmn™cmn \ U smnXsmn)
n = m,n — \
U " = J ^\ W \ My,t)*n(y)y dy > ( 28  3 )
ira 2 Jo Jo [
Ucmn hi r^ rMy,+)Z cmn (y,<t>)ydy
™ Jo Jo
U sm „ wa 2 Jo Jo ' ^
smn
For instance, we can set this series into Eq. (28.1) and utilize the
properties of the Bessel functions given in Eqs. (19.4) and (19.5) to
obtain a series for the radiated pressure wave at considerable distances
(many wavelengths) from the piston, at the point (r,&,<p) in spherical
coordinates centered at the center of the piston:
p ik(r—ct) I <r>
p ~ irvpa* —^ \ >► ^«*».W (28.4)
'n =
00 .
+ ^ i m [U cmn cos (nap) + U smn sin (imp)] 2 **<n — ^(^ I
S 2 — (Ta m n) 2 J
m,n= 1
, 2sJi(s) . , A . _ fJmlis)  J m+1
where s = ka sin # = (2ira/\) sin #.
The term for Uo gives the radiation due to the average motion of
the piston and turns out, of course, to be equal to the final formula
VII.28] RADIATION FROM A PISTON IN A PLANE WALL
331
of Eq. (28.1), since a o = 0. The other terms give the modification
of the angular distribution due to nonuniform distribution of velocity
over the surface of the piston. The more important terms are those
for m = 0, for the velocity distribution is usually radially symmetric
(independent of \f/). Curves for some of these are shown in Fig. 71,
as functions of s. We notice that the functions <£„ for n > add
very little to the central part of the main beam (# ~ 0) but tend to
modify the shape of the diffraction bands at the edge of the main
na
\
0.6
\
V$o
02

A^
A
i ^
,*6
n

02
04
V'
*h
$5'"
C
t
> L
\ c
> i
s i
3 1
2 1'
\ 1
3 1
3 20
Angle Parameter s = (2ira/Msin0
Fig. 71. — Angle distribution factor <£>„ for radiation from a piston with velocity
distribution proportional to the characteristic function Jo(.irao n r/a). <I>o gives dis
tribution for uniform velocity (rigid piston).
beam. The zeros of each function $ n (s) coincide with those of the
others, except that each function has one zero missing, the nth one;
$o is not zero at s = 0, $i is not zero at s = 7ra i, and so on. All the
radially symmetric waves are in phase, unless the U n 's are not in
phase. The intensity at {r,&,<p) is, of course, (l/2pc)p 2 , as long as
r » X.
By juggling with the various $„'s it is possible to adjust the shape
of the radiated beam to any desired distribution (within certain
limits) and then to work back from the corresponding U n 's to deter
mine the distribution of velocity of the piston which will give this
332 THE RADIATION AND SCATTERING OF SOUND [VII.28
beam. For instance, the first "side lobe" in the diffraction pattern
may be suppressed, or the first three or so can be drastically reduced
in value (which is important if one needs a sharply defined beam).
The limits of possible variation of the beam distribution are strin
gent, however, and can soon be learned by attempting to "tailor
make" a beam. The most important restriction is that the main
lobe (0 < sin# < a iX/2a) can never be made narrower by juggling the
velocity distribution of the piston. The beam can be made broader
by such juggling, but it can be made narrower (for a given wave
length) only by increasing the size of the piston. The minimum
angular size of the main lobe is therefore given by the inequality
(valid for X < a/2)
Angular width of main lobe = (& for first zero)
is larger than (35X/a) = (1, 200,000/ va)
where the angle is in degrees, X and a are in centimeters and v is in
cycles per second.
Radiation Impedance, Rigid Piston. — Often it is more important
to compute the reaction of the air back on the piston than it is to com
pute the distribution in angle of the radiated sound. The radiation
impedance of the piston is necessary in calculating the mechanical
and electrical properties of a loudspeaker, for instance. We have
already given curves in Fig. 68 for the impedance ratios for a piston
of effective radius a p = la sin (#o/2) set in a sphere of radius a. The
limiting case of #o * corresponds to a plane rigid piston of radius
o p set in a sphere of infinite radius {i.e., in a plane wall). This case
can be computed in closed form directly from the analysis developed
in this section, as will be shown.
To find the reaction of the air on the piston we must find the
pressure at the point (r,<t>) on the piston, due to the motion of the
element of area of the piston at (y,i). From Eq. (28.1) we see that
this is
dp = IPV fy J e mhct) y d y fy
where h = \A 2 + y 2 — 2ry cos ty — 4). The total pressure at (r,<£)
is
p = —ipvu<>e iu>t
J>"J>*(f) (28  5)
VII.28] RADIATION FROM A PISTON IN A PLANE WALL 333
where we are now using a, instead of a p , to denote the radius of the
piston. To obtain the total force on the piston this must be inte
grated over the whole surface of the piston:
F = —ipvuoe™'
i % *r rf *r^f e Kf)
(28.6)
The calculation of this integral involves some special mathe
matical tricks which will not be elaborated on here. The results give
the total radiation impedance of a rigid piston of radius a set in a
plane wall:
■p
Z* = — =uTt = Ta 2 pc(0o — ixo) = tci 2 (Ro — iXo)
d = 1  (1/1(10); w = 2ka = (4ra/\) = (2<oa/c)
Xx/2
sin (to cos a) sin 2 a da
U(ka) 2 (ka*0) f(8kd/ZTr) (ka > 0)
"* (1 (ka * oo ) ; Xo ~~* ((2/ik) (ka ~> oo)
The properties of. these functions have already been discussed in the
previous chapter, and their limiting values have been given in Eq.
(23.14). These are the functions we have used to compute approxi
mate values of the average impedance of the open end of a tube.
Tables of values for different values of w are given at the back of the
book. Curves of d and xo as 'functions of (w/2) = (27ra/X) are given
in Fig. 68, labeled # = 0.
Comparison of the curves in Fig. 68, and of the curves of Figs. 67
and 70, show that a change from plane to spherical baffle makes a
large change in distribution in angle of the radiated intensity, but
makes little change in the average radiation impedance load on the
piston. Sound is sent out in different directions, but its reaction
back on the piston depends more on the ratio of piston circumference
%ca (2wa p in Fig. 68) to wavelength X than it does on baffle shape.
Distribution of Pressure over the Piston. — In many cases (the cases
discussed in Chap. VI, for instance) it is sufficient to know the average
reaction of the air on the rigid piston, which is given by the functions
do and xo. To go further into the details of the problem, or to study
the reaction of the air on a nonrigid piston, we must know the depend
ence on r of the reaction pressure p given in Eq. (28.5). The integral
of Eq. (28.5) is a complicated function of r, which can most easily be
expanded in a series of characteristic functions appropriate to a cir
334 THE RADIATION AND SCATTERING OF SOUND [VH.28
cular piston of radius a. These functions have already been given in
Eq. (26.17). In terms of them we can set
n = L
Jo(ir<XOn) .
where the coefficients in the expansion for the case of the rigid piston
are
= i I d$ I # \ r , ' r dr rydy (28.7)
Jo Jo Jo xXoVoCiraon) Jo fr
The coefficient (do — ixo) has already been discussed. It gives the
average acoustic impedance ratio over the surface of the piston.
The coefficients f„ = (0„ — *x?0 for n > represent corrections
to the distribution of radiation load over the rigid piston. Their
average effect is zero, since the average value of each of the higher
characteristic functions
, Jo (iraonr/a) , m
Jo{Trao n )
over the surface of the piston is zero. Expressions for these imped
ances can be given in terms of simple integrals:
1 J\(u)u du
On =
% = 2 M P 
Jo [u
2 — (iraon) 2 ] V/* 2  u 2
( M 4 /37r 2 a 2 J 0»>O)
(7a n/2/i 2 ) (m>°°)
J\(u)u du
{,
(28.8)
Xn = 2M J„ '[u 2  (Traon) 2 ] V« 2 " M 2
_^(9nH 0**0)
\(2A/i) (m*«)
M = (2ira/X) = fca,
0i = 0.092, 02 = 0.0356, g 3 = 0.0194, £ 4 = 0.0116 • • •
Some of these functions are plotted in Fig. 72, and values are given
in Table XII at the back of the book.
We see that for very small and for very large values of n = ka =
(<aa/c) both d n and x« (f° r n > 0) vanish, being important only for
values of ka near ira 0n (i.e., for frequencies near ca 0n /2a). Therefore
VH.28] RADIATION FROM A PISTON IN A PLANE WALL 335
at very high and very low frequencies the reaction pressure of the air
on the front of a uniformly vibrating piston is uniform over the face
of the piston; only for frequencies such that the wavelength is the
same general size as the piston dimensions is the pressure markedly
nonuniform over the piston.
10
4 6 8, 10 "0 2 4 6
Frequency Parameter p.=ka=.(2iraA)
Fig. 72. — Coupling resistance and reactance ratios between radiation and circular
piston of radius a in infinite plane wall. Ratio f „ = 6 n  i'x» gives coupling between
zero (uniform) velocity mode and the nth pressure mode.
Nonuniform Motion of the Piston. — If the piston is not rigid, the
analysis of the radiation reaction must be carried still further. We
can express the velocity of the piston surface in terms of a series of
characteristic functions of the sort we have been using, ^„(r) =
Jo(Trao n r/a)/Jo(jra n) '
u (r) = ^UnVnir); u n = (~\ I u(r)* n (r)r dr
n =
as in Eq. (26.17). By analogy with Eq. (28.7) we can expand the
reaction pressure of the air at a point on the piston a distance r from
its center:
oo oo
p(r) = pC ^ ^ Untnm^m(r)
71 = m =
336 THE RADIATION AND SCATTERING OF SOUND [VII.28
where the impedance ratio f„ ro = (d nm  i Xnm ) = f«« are obvious
generalizations of the impedance ratios £« (now written £ n) given in
Eq. (28.7):
Un = ^ J d * Jo ** Jo * m(r> dT Jo * n{y) T V dy
The average pressure over the piston is the coefficient of ^ (r) = 1
in the series
Pav = pC 2) U n £ n0 , (f n0 = fn)
» =
At low frequencies the quantity fro == f o is larger than any of the
other f's. Therefore at frequencies small compared with (c/2tto) the
pressure on the piston is approximately uniform, nearly equal to
pcfo = pc(6o — ixo) times the average velocity u of the piston. This
approximation was used in Chap. VI for the calculation of radiation
out of the open end of a tube.
For frequencies high compared with (c/2wa) the coefficients f mm are
approximately equal to unity and the terms £ nm (n ^ m) are quite
small. Therefore at high frequencies the pressure at r is
00
P(r) ~ pc ^ MmSmmtmW ~ pcu{r) (since f mm ~ 1)
l m =
which is pc times the piston local velocity at r. This is the result
mentioned on pages 198 and 311.
Radiation out of a Circular Tube. — We can now indicate how the
radiation of sound out of the open end of a circular tube (fitted with
a wide flange) can be calculated with greater accuracy than was pos
sible in Chap. VI. As an example we take the case where the tube
radius a is smaller than (ca i/2v), so that the velocities (c/T 0n ) [see
Eq. (26.17)] of all the higher normal modes are imaginary and only
the plane wave (n = 0) mode is transmitted without attenuation.
Therefore by the time the wave has gone from the piston to the open
end, these higher modes have become small, and to the first approxima
tion only the plane wave mode S^o survives.
In this case, to the first approximation, the velocity amplitude of
the air at the open end is uniform and in a direction parallel to the
tube axis (perpendicular to the plane of the open end). Suppose
that this velocity has amplitude u . According to Eq. (28.7) the
VH.28] RADIATION FROM A PISTON IN A PLANE WALL 337
pressure in the plane of the open end, a distance r from the tube
axis, due to the radiation of the plane wave mode into the open, is
«
n =
Vn(r) = JoiTaonr/^/Joiiraon)
The average pressure due to this radiation is of course pcu (6 — ixo),
which we have used in the previous chapter for calculating the acoustic
impedance at the other end of the tube. But we now see that there
are other, smaller, nonuniform terms in the reaction pressure, which
cause local irregularities in the sound field near the open end. These
corrections are very small for low and for high frequencies, so that it
is usually not worth taking them into account. They are discussed
here to indicate how these and similar details may be calculated if
need be.
The additional terms in the pressure modify the velocity distri
bution over the open end, by sending back down the tube waves
corresponding to the higher modes V n . Since, by our assumption,
these waves attenuate rapidly, they will not extend back into the
tube any appreciable distance, but they will cause a modification near
the open end. We have seen on page 309 that the nth mode of trans
mission in the tube has a characteristic impedance
pc/ron = iaiap/iraonTn), where T n = y/l  (v/p 0n ) 2 , v^ = (ca 0n /2a)
Therefore, corresponding to the higher terms in the series for the
pressure, we have a correction to the velocity distribution at the open
end. Insteafl of being u it is, to the first approximation in the small
quantities 0„ and x«,
1*3=0 ~ Uoe™ 1
~ 2 fe) ^k + **)*.(r)
where the sign in front of the summation symbol is minus because
these higher modes are being sent back into the tube (negative zaxis).
In turn, each correction term for the velocity gives rise to a second
order correction to the pressure. The average value of this correction
n = l ^ '
338 THE RADIATION AND SCATTERING OF SOUND [VII.28
and therefore the average acoustic impedance of the open end for
the plane wave mode is, for the case we are considering,
pc
{e  ixo + 2 (S) V 1 " {wj [26nXn + i{el " X " )] J
to the second order of approximation in the small quantities n , Xn
Usually the summation can be neglected in comparison with O and xo
We can also utilize Eq. (28.4), together with the equation for
w z =o given above, to obtain a somewhat more accurate expression for
the distribution in angle of the wave radiated out of an opening in a
plane wall.
Transmission Coefficient for a Dynamic Speaker. — We have shown
in the preceding pages that a dynamic speaker of radius a set in a
plane wall, moving with velocity Uoe~ io>t , radiates the total energy
fra 2 pcU?,d = bra 2 pcUf } ( 2toa / c ) I
into the open. The quantity do is therefore the transmission coeffi
cient for a piston in a baffle, and its curve should be compared with
those in Fig. 64 for various horns. As with the earlier cases, the
coefficient is small at low frequencies, which means that the velocity
amplitude must be increased at the low frequencies to obtain uniform
response. This can be done by making the piston masscontrolled
(which is not difficult) for then the velocity amplitude is inversely
proportional to v over the frequency range of mass control.
Since do is also the transmission coefficient for the open end of a
horn, Fig. 72 shows that the open end of a horn must have its circum
ference larger than the longest wavelength that it is wished to trans
mit; otherwise, the open end will reflect an appreciable part of the
sound back into the horn, causing strong resonance.
Most dynamic speakers are designed so that over the useful fre
quency range the inertial reactance of the piston itself is the largest
part of the total mechanical impedance (i.e., the speaker is mass con
trolled). Therefore, above some minimum frequency the velocity
amplitude of the piston driven by a force of amplitude F is u —
(F /2t vrrip). Above this minimum frequency the energy radiated
by the speaker is
n ~ —
cm
\n*Q>)[i¥**\ (289)
VH.28] RADIATION FROM A PISTON IN A PLANE WALL
339
The values of f — 5 ) * <A(w) are plotted in Fig. 73 as a function
of w = (47ra/X). The curve shows that the response of this type of
loudspeaker is good for lowfrequency sound (as long as the fre
quency is above the lower limit of the mass control) but that as soon
as the wavelength becomes smaller than the circumference of the
speaker the power radiated diminishes.
If the speaker is resistance controlled and is driven by a force
whose amplitude is independent of v, the power radiated will be pro
portional to (do), shown in Fig. 72. In this case the higher frequencies
in the useful range will be radiated best.
1.0
8R0
pew 2
0.5
flower limit of masscon trot
2 4 6 8
W
Fig. 73. — Power radiated by a "dynamic" speaker which is mass controlled and which
is driven by a force with amplitude independent of the frequency.
Design Problems for Dynamic Speakers. — To review some of the
material we have developed in this section, and to show how it can
be applied in the solution of practical problems, we shall discuss the
design of a dynamic speaker. We cannot go into very many of the
tricks of design used to improve the behavior of presentday dynamic
speakers; for it is not our aim to discuss engineering practice. 1 We
shall discuss some of the major problems encountered, however, to
bring out the methods of analyzing the problems and the way in
which the analytic techniques we have developed can help us in the
design of acoustical equipment.
We of course wish the dynamic speaker to reproduce passably
well sound of frequency between 80 and 4,000 cps, a range of about six
1 For a discussion of the engineering aspects of this problem, see Olsen and
Massa, "Applied Acoustics," The Blakiston Company, Philadelphia, 1934, for
instance; or M'Lachlan, "Loud Speaker Design," Oxford University Press,
Oxford, England, 1934.
340 THE RADIATION AND SCATTERING OF SOUND [VH.28
octaves; and we wish it to produce 10 6 ergs of sound energy per second
if need be. (A person speaking with an average conversational tone
produces about 100 ergs per sec and when shouting produces about
10,000 ergs per sec.) We shall assume that we can obtain a magnetic
field as large as 10,000 gauss at the driving coil.
We must first decide on the size of the loudspeaker cone, or piston.
Since the piston is to be mass controlled (see the comments on page
338), we shall expect the output of the speaker to be more or less
constant as long as the radiation resistance is increasing with the
square of the frequency and to fall off when the radiation resistance
becomes constant. Figure 73 shows that this output is greater than
half its maximum value as long as w is less than 4 or as long as (va) is
less than 10,000. For this limit to be at v = 4,000 we should have
a = 2.5, or about 1 in. However, a piston of 1 in. radius would have
to vibrate with an amplitude of more than a centimeter at v = 200 to
radiate 10 6 ergs per sec from its small surface, so that we must sacrifice
some of the intensity at the higher frequencies to obtain a piston
large enough to radiate efficiently. We shall choose a to be 10 cm; in
this case the intensity at c = 4,000 is onesixteenth of the intensity
at v = 1,000, a considerable diminution but perhaps not too great.
The next problem is the driving coil. Increasing the number of
turns on the coil increases the electromagnetic coupling constant G,
but it also increases the mass of* the moving parts and the resistance
of the coil. We choose the coil to be 1 cm in radius and to have 160
turns of wire of such a size as to make the resistance of the coil 5 ohms.
In such a case it may be possible to make the mass m p of the coil
piston system as small as 30 g. If the magnetic field on the coil is
10,000 gauss, the electromagnetic coupling constants are D — 10 6 ,
G = 10 5 . It may be possible to keep the selfinductance of the coil as
low as 0.2 mh.
The next question is what to do about the space in back of the
piston. We can, of course, let the back radiate into another room,
but this is not usually done. If we enclose the back of the speaker,
the enclosure will form a resonator that will add stiffness to the piston.
If the inside of the enclosure is made of hard material, standing waves
can be set up, and the enclosure can have a large number of resonance
frequencies; so we shall line the interior with absorbing material to
destroy most of the resonance due to standing waves. This will not
appreciably alter the resonance of the Helmholtz type (see page 235)
however, so we must make the volume of the enclosure large enough
so that this resonance frequency is below the range that we wish to
VTI.28] RADIATION FROM A PISTON IN A PLANE WALL 341
reproduce. The stiffness constant of the piston due to an enclosure
of volume V is given in Eq. (23.3) : K p = (tci 2 ) 2 (pc 2 /V). If we choose
V to be 20,000 cc, th e natural frequency of the resonatorpiston
system (l/27r) y/K p /m p will be 76 cps, which is just below the lower
limit that we have set. This resonance will increase somewhat the
response at lower frequencies. It will be assumed that the frictional
forces on the piston, other than radiation resistance, are negligible
(although in actual pistons the frictional resistance is usually larger
than the radiation resistance).
The constants assumed for the loudspeaker are therefore
m p = 30, K p = 6.9 X 10 6 , a = 10, V = 2 X 10 4 \
D = 10 6 , G = 10 6 , Re = 5, Lc = 2 X 10~ 4 /
/a \ / \ ( (28.10)
™v = i.3x 10 <, «> = (¥) = (m) I
These are not recommended values of the constants, nor are all of them
typical values. They have been chosen to show, as clearly as possible,
how each property of the loudspeaker enters into the radiation process.
Behavior of the Loudspeaker. — To compute the behavior of the
speaker we first obtain its mechanical impedance:
\Z m \ = s/Rl + XI
R m = Tra^o = 1.3 X lO^Tl  2^1
*^a_, + 1rtlfXo > (28.11)
11 y 106
' A + 190 v + 1.3 X 10W (w)
X m = jr — ? + 2rvm p + ira 2 X
Zttv
where R , X , and M are given in Eq. (28.5). Curves of R m and Z m
are given in Fig. 74. We notice that over most of the range of fre
quency (for v larger than 200) Z m is practically equal to the reactance
due to the mass of the piston Z m ~ 190j>. This is, of course, what is
meant by saying that the loudspeaker is mass controlled. The
sudden drop in Z m on the lowfrequency side is due to the resonance
of the vessel enclosing the back of the piston.
The electrical impedance of the coil is
\Z,\ = VfiJ + XI Rb ~ 5
342
THE RADIATION AND SCATTERING OF SOUND
[VII.28
Curves of these three quantities are shown in Fig. 75. Since the
resistance of the coil is fairly large, Z E stays fairly constant over most
of the specified range of frequency. The motional reactance due to
the piston just cancels that due to the impedance of the coil at about
600 cps. If the coil resistance were lower, this would cause a pro
nounced resonance peak in the response. The impedance rises on
10,000
Fig. 74. — Total mechanical resistance and impedance of a "dynamic" loudspeaker
as a function of the driving frequency. The constants of the loudspeaker are given
inEqs. (28.10).
the highfrequency side owing to the inductance of the coil and on
the lowfrequency side because of the mechanical resonance of the
loudspeaker plus enclosure. This rise in electrical impedance due to
mechanical resonance is more than enough to cancel out the corre
sponding peak in the piston response (see the discussion on page 38).
We shall assume that an emf of \0e Mvt volts is applied across the ;
coil. In actuality, the voltage amplitude will also vary with fre
quency, but we must cease adding complications somewhere and so
VH.28] RADIATION FROM A PISTON IN A PLANE WALL
343
shall assume that the voltage amplitude is constant. The current
through the coil is then (10/Z*), the force on the piston is(10D/\Z*$,
and the velocity amplitude of the piston is u p = (10D/\Z E \\Z m \): The
power radiated is \u\R m . Curves of u p and of n, the power radiated,
are given in Fig. 76. They show that II is reasonably constant over the
range 100 < v < 1,000 and that u p is inversely proportional to v over
most of the frequency range. Below the mechanical resonance at low
frequencies the curves for u p and II drop off. The falling off of u p \
below the straight line of the (1/v) curve, for high frequencies, is due
to the inductance of the coil. The drop in II for high frequencies' is
5000
Fig. 75. — Total electrical resistance, reactance, and impedance of the driving coil of the
loudspeaker whose constants are given in Eqs. (28.10).
due partly to the inductance but is also due to the fact that the radi
ation resistance is not rising any more, so that it cannot make up for
the decrease of u p . In actual loudspeakers the drop is not so rapid,
because at high frequencies only the central portion of the piston
vibrates, and the effective mass is reduced. Over the useful range
of frequencies the overall efficiency of the loudspeaker is about
1 per cent.
At v = 100 the amplitude of motion of the piston (u p /2rv) is
about 0.2 cm, so that the mechanism should be designed for an ampli
tude of about 0.4 Cm if it is to radiate 0.1 watt without rattling.
The efficiency of the loudspeaker could be increased by decreasing
the coil resistance, by increasing the magnetic field, by increasing the
344
THE RADIATION AND SCATTERING OF SOUND JVIL28
number of turns on the coil, or by decreasing the mass of the moving
system. If we decrease the coil resistance much, the electrical imped
ance will not be constant over the useful range, and the resonance
between piston mass and coil inductance will become prominent. If
we increase the number of turns on the coil, we increase the resistance
and also the mass, which is not desirable. It would be quite difficult
to increase the magnetic field to any extent, though if this could be
done the efficiency could be increased without any concomitant ill
effects. If we decrease the mass of the system much, it will no longer
be purely mass controlled, and the response will not be so uniform.
io 5 n
 100
1000
10,000
Fig. 76.
Velocity amplitude u p and power radiated II when the loudspeaker of Eqs.
(28.10) is driven by an emf of 10 volts amplitude.
The curves of Fig. 76 are for the loudspeaker out in the open with
no obstructions near by. The effect of the resonance of a room on
the response curves will be discussed in the next chapter.
Transient Radiation from a Piston. — The general equation (27.1)
for radiation from an elementary source can be used to compute the
transient pressure wave from a piston in a plane wall. Suppose that
the velocity of the piston is U(f) and its acceleration is A(t) = dU/dt.
Study of Fig. 77 shows that a part of the pressure wave arriving at
the point (r,#) at the time t is that due to the acceleration of the strip
of length d = 2 \/a 2 — y 2 and width dy a time (l/c)(r — y sin#)
VII.28] RADIATION FROM A PISTON IN A PLANE WALL
345
earlier (if r is much larger than a). The amplitude of this component
of the pressure pulse is, from Eq. (27.1),
dp ~^r a/o ' ~ y 2 A\t  (r  ysm&)\dy
The total pressure is obtained by integrating this over the surface
.d1
Fig. 77. — Radiat ion of a transient from a piston. Sound reaching point P came
from strip d = 2 \/a 2  V 2 a time (l/c)(r  2/ sin #) earlier. Lower curve shows
pressure fluctuation at P due to a velocity pulse U = d(t) of the piston.
of the piston, from a to +a. Change of integration variable and
integration by parts yields the general formulas
pc
p = H
2irr sin 2 # _
,2 Cf
_ pc l
J Vo 2 sin 2 ??  (ct + r  ct) 2 A(r) dr
(r + ct — ct) U{r) dr
2irr sin 2 # J e Va 2 sin 2 ??  (r + ct — ct) 2
(28.12)
where the limits of integration are e = t  (l/c)(r + a sin/?) and
f = t  (l/c)(r  asintf).
This is a very interesting formula, for it shows, perhaps more
clearly than Eq. (28.1), the dependence of the pressure wave on the
346 THE RADIATION AND SCATTERING OF SOUND [VII.29
angle #. Directly ahead (# = 0) the pressure wave reproduces the
piston acceleration exactly:
»=te>H> ( *= o)
The force on a microphone diaphragm directly ahead of the piston is
therefore proportional to the piston acceleration; and if the diaphragm
is mass controlled, its acceleration is proportional to the force, so
that the accelerations, velocities, and displacements of piston and
diaphragm are proportional. As & is increased, however, the integral
covers a larger and larger interval of time t, so that more and more
of the piston motion gets blurred together in the pressure wave
arriving at P.
A simple example of this is for the case when the piston suddenly
moves outward a distance A. In this case the displacement of the
piston is Au(t), where u{t) is the step function defined in Eq. (6.9);
the velocity is A 8(0, proportional to the impulse function; and the
acceleration A8'(t) is formally proportional to the derivative of the
impulse function, a "pathological" function going first to plus infinity
and then to minus infinity in an infinitesimal period of time. Its inte
gral properties are V ^ h'{r  a)j{j) dr = ~[(df(a)/da)u(t  a)]. The
resulting pressure at (r,&) is
!0 (ct < r — a sin#)
pc2A r  ct , _ asiiit? <ct<r + asm #)
27rrsm 2 # v 7 a 2 sm 2# _ ( r _ ct y
(c/ > r '+ a sin #)
This pulse, shown in Fig. 77, is a stretched out version of the 8' func
tion; the greater the angle &, the greater the stretch. Only directly
ahead of the piston, at # = 0, is the pressure pulse as instantaneous
as the piston pulse.
29. THE SCATTERING OF SOUND
f When a sound wave encounters an obstacle, some of the wave is
deflected from its original course. It is usual to define the difference
between the actual wave and the undisturbed wave, which would be
present if the obstacle were not there, as the scattered wave. When a
plane wave, for instance, strikes a body in its path, in addition to the
undisturbed plane wave there is a scattered wave, spreading out from
the obstacle in all directions, distorting and interfering with the plane
VH.29] THE SCATTERING OF SOUND 347
wave. If the obstacle is very large compared with the wavelength
(as it usually is for light waves and very seldom is for sound), half of
this scattered wave spreads out more or less uniformly in all directions
from the scatterer, and the other half is concentrated behind the
obstacle in such a manner as to interfere destructively with the
unchanged plane wave behind the obstacle, creating a sharpedged
shadow there. This is the case of geometrical optics; in this case the
half of the scattered wave spreading out uniformly is called the
reflected wave, and the half responsible for the shadow is called the inter
fering wave. If the obstacle is very small compared with the wave
length (as it often is for sound waves), then all the scattered wave
is sent out uniformly in all directions, and there is no sharpedged
shadow. In the intermediate cases, where the obstacle is about the
same size as the wavelength, a variety of curious interference phe
nomena can occur.
In the present chapter, since we are studying sound waves, we
shall be interested in the second and third cases, where the wavelength
is longer or at least the same' size as the obstacle. We shall not
encounter or discuss sharply defined shadows. So much of the
scattered wave will travel in a different direction from the plane wave
that destructive interference will be unimportant, and we shall, be,
able to separate all the scattered wave from the undisturbed plane
wave. We shall be interested in the total amount of the wave that
is scattered, in the distribution in angle of this wave, and in the effect
of this scattered wave on the pressure at various points on the surface
of the obstacle.
Scattering from a Cylinder. — Let us first compute the scattering,
by a cylinder of radius a, of a plane wave traveling in a direction per
pendicular to the cylinder's axis. If the plane wave has intensity
To, the pressure wave, if the cylinder were not present, would be
Vv = P e ik ^~^ = p„e*(« «»*««>, p = V2pcT^ k = — ;
A
where the direction of the plane wave has been taken along the positive
zaxis.
In Eq. (19.13) we expressed this plane wave in terms of cylindrical
waves:
Vv = Poe*< ro "* rf > = P [J (kr) + 2 j? i™ cos (m0)J«(Jiy)]r w '» (29.1)
m = \
348 THE RADIATION AND SCATTERING OF SOUND [VTL29
The radial velocity corresponding to this wave is
w«r — "
(?j\ iiJ^kr) + 2 i m+1 [Jm+i(kr)  Jailer)] cos(mtf>)l
, — 2irivt
When the cylinder is present with its axis at r = 0, the wave
cannot have the form given by the above series, for the cylinder dis
torts the wave. There is present, in addition to the plane wave, a
scattered outgoing wave of such a size and shape as to make the radial
velocity of the combination zero at r = a the surface of the cylinder.
We shall choose the form of this outgoing wave to be the general series
CO
p, = 2^ cos (m4>)[J m (M + iN m (kr)]er™"
m =
u„ = (L\ iiAo[Ji(Jcr) + iNxikr)] + 1 ^ A ™ cos (m<j>)[J m +i(hr)
 J m i(kr) + iN m+1 (kr)  iN m !(kr)]\ e 2 ""'
The combination J + iN has been chosen because it ensures that all
the scattered wave is outgoing.
Our first task is to find the values of the coefficients A which
make the combination u pr + u sr equal zero at r = a. Equating
u„ to — Upr at r = a term by term, we obtain
A m = e^Po^+^^sinCTm); Po = V^pcjo )
Ji(fco) . _ Jmx(ka)  J m +x(ka) > (29.2)
tanTo = "" mM)'' tanTm ~ N m+1 (ka)  N m ^(ka) )
where eo = 1 and e m = 2 for all values of m larger than unity. These
phase angles y m have already been defined in Eq. (26.6), in connection
with the radiation of sound from a cylinder. Values of some of them
are given in Table X at the back of the book. The behavior of these
phase angles completely determines the behavior of the scattered wave.
It is interesting to notice the close connection between the waves
scattered by a cylinder and the waves radiated by the same cylinder
when it is vibrating. The quantities needed to compute one are also
needed to compute the other.
The pressure and radial velocity of the scattered wave at large
distances from the cylinder are
vn.29]
THE SCATTERING OF SOUND
349
4pcT a , , N .,/ « Pa
H — — $ s (4>)e tk(r  c » ; u s ~ —
TV pC
^.(0) = —7= 2 Cm sin (Tm)^ 1 ' 7 " 1 cos (m^)
TO=0
The intensity of the scattered part, at the point (r,<f>)(kr^> 1), is,
therefore,
\U<f>)\'<
(29.3)
A=ttoi
00
^« 2 = j— ^j e m e„ sin7 TO sin7„ cos (y m — y n ) cos (m^) cos (n<f>)
m,n =
where e = 1, e m = 2 (m > 0). This intensity is plotted as a function
of ^ on a polar plot in Fig. 78, for
different values of p = (2rva/c) =
(2iroA).
It is interesting to notice the
change in directionality of the
scattered wave as the wavelength
is changed. For very long wave
lengths (n small) but little is scat
tered, and this is scattered almost
uniformly in all the backward
directions. As the frequency is
increased, the distribution in
angle becomes more and more
complicated, diffraction peaks
appearing and moving forward,
until for very short wavelengths
(much shorter than those shown
in Fig. 78) onehalf of the scat
tered wave is concentrated
straight forward (the interfering
beam), and the other half is spread
more or less uniformly over all the
other directions, giving a polar
plot which is a cardioid, inter
rupted by a sharp very high peak
Fig. 78. — The scattering of sound
waves from a rigid cylinder of radius a.
Polar diagrams show the distribution in
angle of the intensity of the scattered
wave, and the lower graph shows the
dependence of the total scattered intensity
on fi = 2ira/X.
in the forward direction, as will be shown in Eq. (29.4).
For very long wavelengths only the two cylindrical waves corre
350 THE RADIATION AND SCATTERING OF SOUND [VTI.29
sponding to m = and m = 1 are important in the scattered wave.
As shown in Eq. (26.6),
(^)
To ^ — 7i ~ I ~~ 4~ J ^ ~ 72 ' —, V3 ' ' ' > wnen ^« — >
The firstorder approximation for the scattered intensity at long wave
lengths is, therefore,
~ \ 8c 3 r /
T (l  2cos<f>) 2 (coa «c)
Short Wavelength Limit. — For wavelengths very small compared
with the circumference of the cylinder, the relatively simple approxi
mation of "geometrical acoustics" is valid, with the "scattered"
wave dividing into two parts, the "reflected" and the "shadow
forming" waves. However, the process of demonstrating that the
series of Eqs. (29.3) really does behave in a simple manner involves
mathematical manipulation of considerable intricacy.
In the first place the series of Eqs. (29.3), which is in a form useful
for calculating scattering at longer wavelengths, is not particularly
suitable for showing in detail the interference between part of the
"scattered" wave and the primary wave to form the shadow. In
optics, one differentiates between "Fraunhofer diffraction," where
intensities are measured at distances so large that the angle subtended
by the diffracting object is small compared with the ratio (X/2ira) =
(1/ka), and "Fresnel diffraction," for distances large compared with
the wavelength but not extremely large compared with {%ra). The
Fresneldiffraction formulas show the shadow with its related diffrac
tion bands, but at the great distances involved in the Fraunhofer
diffraction formulas, the shadow has become blurred out again.
Series (29.3) is for distances corresponding to Fraunhofer diffraction,
so that what can be demonstrated is the separation of the "scattered
wave" into a "reflected wave" and a "shadowforming wave" but
not the details of the interference with the incident wave, which are
characteristic of the Fresnel formulas.
In the second place, the simplicity of the formulas for very short
wavelengths only appears as an average; the scattered intensity varies
rapidly with angle in a complicated sort of way, and only the average
intensity per degree (or per minute) varies smoothly. This rapid
fluctuation is seldom measured, however, for any small change of
frequency or of position of the cylinder will blur it out, leaving only the
average intensity. Consequently, our calculations should separate
VII.29] THE SCATTERING OF SOUND 351
the rapid fluctuations of intensity from the average behavior of the
"reflected wave."
When all the necessary manipulations are made, the expression
for the scattered intensity at short wavelengths is
T ' ~ Tr sin (l) + SR COt! (l) Sin2 {k " Sin *>
+ (rapidly fluctuating terms), kr^>ka^>l (29.4)
The first term of this expression constitutes the "reflected"
intensity, which, for a cylinder, reflects more in the backward direction
(# ~ 7r) than in a forward direction (<^ ~ 0). The second term is the
"shadowforming" beam, concentrated in the forward direction within
an angle {j/ka) = (X/2o) which is smaller the smaller X is compared
with a. The third term contains rapidly fluctuating quantities that
average to zero, and so may be neglected.
Total Scattered Power. — The total power scattered by the cylinder
per unit length is obtained by multiplying T by r and integrating over
<t> from to 2ir. The cross terms in the sum of Eq. (29.3) disappear,
owing to the integral properties of the characteristic functions cos (ra0),
leaving the result
x / m =
/ (67r 5 a 4 A 3 ) To (X » 2xa)
l4aT (X«27ra)
(29.5)
The limiting value for total scattered power for very short wave
lengths is the power contained in a beam twice as wide as the cylinder,
(4a). This is due to the fact, discussed above, that the scattered
wave includes both the reflected and the shadowforming waves, the
first and second terms of Eq. (29.4). The integral of r times the first
term is the total reflected power, which is just 2a T ; and the integral
of r times the second term is also 2a T , showing that the shadow
forming wave has enough power just to cancel the primary wave
behind the cylinder.
The quantity (n s /4aT ) is plotted in Fig. 78 as a function of
ka = (2ra/\). Notice that in spite of the various peculiarities of
the distribution in angle of the intensity, the total scattered intensity
turns out to be a fairly smooth function of ka. We also should note
that II S is not usually measured experimentally, because of the diffi
culty of separating primary from scattered waves at small angles of
scattering. What is usually measured is more nearly the quantity
352 THE RADIATION AND SCATTERING OF SOUND [VII.29
n ezp ~ 2 jT' r ~ A % r d<f>
where A is a small angle, if the experimental conditions are good, but
it is never zero. It turns out that n exp is very nearly equal to II, for
the longer wavelengths; but as the wavelength is made smaller, the
shadowforming beam (the second term in Eq. 29.4) is less and less
included in the integral; until for very short wavelengths n exp is equal
to in 8 . The transition from n s to iU s comes at values of ka near
(ir/2A), at wave lengths X near 4aA.
The Force on the Cylinder. — Returning now to the expression
for the total pressure due to both the undisturbed plane wave and
the scattered wave, we find, after some involved juggling of terms
[the use, for instance, of the last of Eqs. (20.1)], that the total pressure
at the surface of the cylinder at an angle from the xaxis is
P. = (p, + *)„ = g£) «« 2 2^ «' [ " T " +?] (29.6)
m =
where the quantities C m are defined in Eq. (26.6). This expression
is proportional to the expression in Eq. (26.7), giving the pressure at
some distance from the cylinder due to a vibrating line element on the
cylinder, when we make the necessary change from <j> to tt — <f> (since
now = is in the direction opposite to the source). This is an
example of the principle of reciprocity. The pressure at a point A
due to a source at point B is equal to the pressure at B due to a source
at A, everything else being equal. Therefore the polar curves of
Fig. 65 show the distribution of intensity about a cylinder having a
line source, and they also show the distribution of the square of the
pressure on the surface of the cylinder due to a line source at very
large distances away from the cylinder (distances so large that the
wave has become a plane wave by the time that the wave strikes the
cylinder).
When n is very small (X » a) the expression for the pressure at
r = a reduces to
p — ► P (l + 2ika cos 4>)e~ 2Tivt
M>0
which approaches in value the pressure P e 2Tivt of the plane wave
alone as n goes to zero.
The net force on the cylinder per unit length is in the direction
of the plane wave and is
Vn.28]
THE SCATTERING OF SOUND
353
f
(29.7)
f = a p cos <f>d(f> = 4aP ( j—jr ) e^*  *™***/ 2
' zco(47T 2 a 2 /c)Poe iat [« « (c/o)]
/ 4aX P e i( » /C > («f')+ix/4 ( X « 27ra)
This force lags behind the pressure, which the plane wave would have
at r = if the cylinder were not there, by an angle — 71 + (ir/2).
The limiting* formula, for co small, has been used on page 149 in dis
cussing the forced motion of a string (see Prob. 14).
The quantity F/P is the net force on the cylinder per unit length
per unit pressure of the plane wave. This quantity, divided by 2ra,
is plotted in Fig. 79 as a function of ka = ix. We note that for small
0.8 r
Fig. 79. — Amplitude of sideward force per unit length F on a, cylinder of radius o
due to the passage of a plane wave of pressure amplitude P„, plotted as a function of
ft = 2ira/\.
frequencies the force is proportional to the frequency (i.e., to ft) but
that when /* becomes larger than unity (i.e., when the wavelength
becomes smaller than the circumference of the cylinder) the linear
relation breaks down and the force diminishes with increasing fre
quency thenceforth. This result is of interest in connection with the
socalled velocityribbon microphone, which consists of a light metal
strip more or less open to the atmosphere, pushed to and fro by the
sound wave. The ribbon is in a transverse magnetic field, so that the
motion induces an emf along the ribbon, which actuates an amplifier.
The net force on the strip is, of course, not exactly the same function
of ft as that given in Eq. (29.7) for the cylinder, but the behavior will
be the same in general. The force on the strip will be proportional
354 THE RADIATION AND SCATTERING OF SOUND [VII.29
to the frequency for small frequencies, but this linear dependence will
break down when the wavelength becomes smaller than twice the
width of the strip.
There is an approximate method of finding the net force on the
cylinder which gives the correct result for wavelengths longer than
the circumference of the cylinder. If the pressure in the plane wave
is P e ik(Xr  ct) , the pressure at the surface of the cylinder due* to the plane
wave is P^^^^ ivt . If ^ is small, this can be expanded into p p ~
P (l + i/x cos <}>)e 2irivt . This is the pressure due to the plane wave;
there is also a scattered wave, enough of a wave to make the radial
velocity at the surface come out to be zero. This scattered wave con
tributes a term ifiPo cos <$> e~ 2 * ivt to this approximation to the pressure,
so that the net pressure is that given in Eq. (29.6), and the net force
is the limiting value given in (29.7). We thus see that even for very
long waves the distortion of the plane wave due to the presence of the
cylinder contributes a factor 2 to the net force on the cylinder.
Scattering from a Sphere. — The analysis of the scattering of waves
from a spherical obstacle follows exactly the same lines as that for the
cylinder. The expression for a plane wave traveling to the right
along the polar axis is
oo
p p = p oe «(rcostf c t) = p ^ (2m + \)i m P m {cosd)j m {kr)e~^ ivt (29.8)
m =
where P = \/2p c T , and where the factors P m and j m are defined
in Eqs. (27.6) and (27.10), respectively. The expression for the wave
scattered from a sphere of radius a whose center is the polar origin is
oo
Vt = p ^ (2w + l)i m+1 e iS » sin 5 TO P TO (cost?).
m =
• [j m (kr) + m OT (/cr)]e 2 ™< (29.9)
where the angles 8 m have been defined in Eqs. (27.17) in connection
with the radiation from a sphere. The values of some of them are
given in Table XI at the back of the book.
The intensity of the scattered wave and the total power scattered
are
a 2 To
00
1_ ^ (2m+l)(2n + l)
r 2 k 2 c
m,n =
• sin 8 m sin 8 n cos (8 m  8 n )P m (cos#)P n (cos#) ( (29.10)
( (16ir 4 i' 4 a 6 To/9c 4 r 2 )(l  3 cos#) 2 (ka « 1)
\ (a 2 /4r 2 ) + (a 2 /4r 2 ) cot 2 (&/2)J\(ka sin &) {ka » 1)
VII.29]
THE SCATTERING OF SOUND
355
(29.11)
n, = 27ro 2 To(2/fc 2 a 2 ) J; (2m + 1) sin 2 5 m
m =
( (2567r 5 a 6 /9X 4 ) T • (X » %ra)
(27ra 2 To (X « 2wa)
The discussion concerning the short wavelength limit for the
scattering from a sphere is similar to that preceding and following
Eq. (29.4) for the cylindrical case. The total power scattered is that
X=2jtci
n/^2
12 3 4
Fig. 80. — Distribution in angle of intensity scattered from a sphere of radius a and
total power scattered II, per unit incident intensity.
contained in an area of primary beam equal to twice the cross section
ira 2 of the sphere. Half of this is reflected equally in all directions
from the sphere (the first term in the last expression of Eq. 29.10);
and the other half is concentrated into a narrow beam which tends to
interfere with the primary beam and cause the shadow (the second
term in the last expression). If the experimentally measured "total
scattered power" includes everything from # = A to & = ir — A,
n exp will equal II 8 for wavelengths longer than (4a/ A) and will approach
ill s for wavelengths much shorter than (4a/ A).
Figure 80 shows polar curves of the scattered intensity as a function
356
THE RADIATION AND SCATTERING OF SOUND [VH.29
of the angle of scattering #, for different values of n = ka, and a
curve n s as function of ix. As with the cylinder, the directionality
of the scattered wave increases as the frequency increases.
The Force on the Sphere. — The total pressure at a point on the
sphere an angle # from the polar axis (note that the point # = is the
point farthest away from the source of the sound) turns out to be
oo
p a = iVr*"«(l/A;a) 2 ^5 ??Ltl p m (cos #)**<«■*»«»>
m =
~ (1 + $ika cosifyPoer** (ka « 1) (29.12)
As with the cylinder, this expression is proportional to that for the
pressure, at large distances, due to a simple source set in the sphere.
The curves of Fig. 66 therefore show the dependence of pi on &.
v % 1
8
Fig. 81. — Ratio of pressure amplitude at a point on a sphere to pressure amplitude
of the plane wave striking the sphere, plotted as a function of ix = (2ira/\). Solid line
is the pressure at a point facing the incident wave; dotted line is the average pressure
for a circular area around this point with an angular radius of 30 deg.
The amplitude of the pressure at the point nearest the source of
sound (# = ir), for a plane wave of unit pressure amplitude, is plotted
in Fig. 81 as a function of /*. We notice that for wavelengths long
compared with the circumference of the sphere the pressure at # = t
equals the pressure of the plane wave but that for shorter wavelengths
the distortion of the wave due to the presence of the sphere makes p a
differ from P . This general fact will be true for obstacles of other than
spherical shape, even though the dependence of p a on n will be some
what different from that given in Eq. (29.12). Therefore, a micro
phone measures the pressure of the wave striking it only as long as its
circumference is smaller than the wavelength of the wave. For
VII.29]
THE SCATTERING OF SOUND
357
smaller wavelengths a correction must be made for the distortion due
to the presence of the microphone.
By using Eqs. (27.7) we can find the average value of the pressure
on that portion of the surface of the sphere contained between the
angles # = t and # = k — # :
Pav =
j r*
cos# J*~» V '
sinddd
= Poe^ 1
>b i m e i5m pmi (cosflp)  P m +i (cos^o) l
^3 k 2 a 2 D m I P_ x (cos# )  Pi (cos #„)
iPoe^l + £ifca(l + cos do)] (ha « V
(29.13)
where P_i(cos# ) = 1. The dotted line in Fig. 81 gives the values
of (pav/Poe  ^) as function of n = ka for the case # = 30 deg. This
curve will be useful in discussing the behavior of microphones later
in this chapter.
Design of a Condenser Microphone. — As an example of the use of
the scattering formulas derived above, let us discuss the behavior of a
condenser microphone in a spherical
housing. We shall have to make a
number of simplifications that do not
correspond to actuality in order to
avoid confusing complications. We
shall try to include, however, enough
of the complications encountered in
actual practice to show the difficulties
involved and the relative effect of the
various complicating factors.
The first simplification that we
shall make concerns the container
for the microphone. We shall
assume it to be spherical in shape,
because we have analyzed the effect of a sphere in distorting a plane
wave, and we have not so analyzed the effect of a cube, for instance.
The effect of the sphere will be enough like the effect of any other
object of about the same size so that the behavior that we compute
will be typical.
The microphone that we shall use will be a resonatorcoupled
condenser microphone, arranged as shown in Fig. 82. The radius
of the sphere will be 5 cm and that of the diaphragm will be 2.5 cm,
so that the angle of the circular opening in the sphere is 30 deg = # .
Fig. 82. — Simplified cross section
of the condenser microphone whose
constants are given in Eqs. (29.14).
358 THE RADIATION AND SCATTERING OF SOUND [VH.29
The diaphragm is set back 1.2 cm from the opening; any larger dis
tance would make the first cavity resonance come at too low a fre
quency. The spacing between the condenser plates will be 0.02 cm.
We wish the microphone to respond up to 8,000 cps, and, referring
to Fig. 44, we see that the lowest resonance frequency of the diaphragm
cannot be less than 4,000. If the diaphragm is aluminum, 0.0015
cm thick, the membrane density <r is 0.005 g per sq cm, and the tension
must be 3 X 10 6 dynes per cm for the frequency v \ to be 4,000 [see
Eq. (20.9) and the paragraph preceding].
If we design the air space in back of the diaphragm properly, we
can make the damping constant 6d [see Eq. (20.8) come out to be 3,
by using damping material behind the plates. The constants of the
system are therefore
a = 5; b = 2.5; # = 30°; 6 d = 3; I = 1.2
a = 0.005; A = 0.02; v i = 3,800
ha = (j^) ~ (V1,000); n = (~\ ~ OMa \ (29.14)
©0.06^ ■ »(£*) *.
where c a is the velocity of sound in air and Ca that of transverse waves
in the membrane.
Behavior of the Microphone. — Our first task is to determine the
average pressure at the surface of the opening in the sphere. This
has been done in Fig. 81, for the case when the opening is pointed
straight at the oncoming plane wave. There will be, of course, a
different curve if the microphone is pointed in some other direction.
Next, we must determine the pressure at the diaphragm, which
is set at the back of' a tube of length I and radius 6. This quantity
is given by Eq. (23.28) for cavity response and is shown in Fig. 57.
Equation (23.28) must be modified so as to use expression (27.20)
for the impedance of a hole in a sphere, instead of expression (28.6)
for the impedance of a hole in a plane. The modification is not large.
The ratio of the average pressure at the diaphragm to the pressure
of the undistorted plane wave is shown in Fig. 83 as a function of
frequency. The slow rise of the curve is due to the increasingly
important distorting effect of the sphere on the wave as v is increased;
the large peak at about 4,000 cps is due to cavity resonance.
Finally, having obtained the pressure at the diaphragm, we can
use Eq. (20.12) to determine the electrical response of the microphone.
This response can be expressed as follows: If the intensity of the sound
VH.29J
THE SCATTERING OF SOUND
359
in the plane wave is T , and if the microphone plate is charged to a
potential of E volts, then the amplitude of the emf delivered to the
10,000
Fig. 83. — Ratio of pressure amplitude at the diaphragm of the microphone shown in
Fig. 82 to the pressure amplitude of the incident plane wave.
2.0r
500 1000
Frequency, v
Fig. 84. — Response curve for the microphone shown in Fig. 82.
is given in terms of Hiy) by Eq. (29.15).
amplifier is
E = 0.93 X 10 3 VTo EoH(v)
5000 10,000
The output voltage
(29.15)
where H(v) is plotted against v in Fig. 84. A resonance peak for the
diaphragmresonator system has merged with the one for cavity
resonance at the upper part of the useful range. The sudden drop
in response above this is due to the fact that the waves on the dia
360 THE RADIATION AND SCATTERING OF SOUND [VII.30
phragm become short enough so that the motion of one portion cancels
that of another, as was discussed on page 202. Compare this with
the second curve of Fig. 44.
The response could be made larger by decreasing the resonance
frequency of the diaphragm, by increasing E , by decreasing the
spacing A, or by reducing the diaphragm density a. The diaphragm
has already been made as thin as it can reasonably be, so that <x cannot
be much reduced. Presumably, E is made as large as it can be for
the spacing A. Decreasing the resonance frequency voi would mean
reducing the useful range of frequencies of the microphone, which is
undesirable.
The diaphragm cannot be set any farther back in the sphere, for
then the cavity resonance frequency would intrude on the useful
range. The sphere should not be made any larger, for it would then
distort the plane wave to a greater extent. Perhaps a slight advan
tage would be gained by reducing the values of a and I below those
chosen.
30. THE ABSORPTION OF SOUND AT A SURFACE
In the previous discussions of scattering we have assumed that the
scattering surfaces were rigid, an approximation that is not always
good enough. It is now time for us to take up the task of discussing
just what a surface does when it is acted on by a sound wave and of
computing what effect this behavior has on the sound wave.
Surface Impedance. — Of course different sorts of materials react
differently to sound, depending on their structure. Nonporous mate
rials yield slightly, since they are never perfectly rigid, and porous
materials also allow some air to penetrate below the surface, producing
an additional effective motion of the surface. In any of these cases
we can express the reaction of the surface in terms of a specific acoustic
impedance, a ratio between pressure at the surface and the normal
velocity of the surface. This impedance in general depends on the
nature of the surface material, on the frequency of the wave, and on
its angle of incidence (we at first consider only plane simple harmonic
waves; later we shall build up more complex waves out of these).
Just as occurred with the problem of the reaction of the air on a
membrane, discussed on page 335, we find that the amount of depend
ence of the impedance on the angle of incidence of the wave depends
on how well wave motion can travel, in the surface material, parallel to
the surface. If such wave motion is rapidly attenuated or is con
siderably slower than that of sound in air, then the impedance of the
VII.30] THE ABSORPTION OF SOUND AT A SURFACE 361
surface is nearly independent of angle of incidence, for in this case
one part of the surface is not aware of the motion of another part, and
the reaction of one part of the surface is proportional to the local
pressure at that point. Such surfaces can be called locally reacting
surfaces. On the other hand if the wave motion in the surface is
not attenuated, and is as fast as or faster than that in the air, then
the reaction of one part of the surface will depend on the motion of
other parts of the surface and the surface impedance will depend on
the angle of incidence of the wave. Such surfaces can be called
surfaces of extended reaction.
We shall consider a few typical cases in order to illustrate these
general statements.
Unsupported Panel.— One of the simplest sorts of surface is an
impervious plane panel of dimensions that are large compared with
the wavelength in air. We assume that its outer surface at equilibrium
is in the (?/,2)plane and that the panel supports are far enough apart
so that they do not play a part in the reaction between panel and air.
According to Eq. (21.1) the equation of transverse motion of the
panel is
«.£<">, + 9, m ;:^t /3(1 _ s2)] CO.!)
where p is the pressure of the air, Q and p the modulus of elasticity
and density of the material, 2h its thickness, and v its displacement
away from the air (in the negative ^direction). The constant g is
called the flexural rigidity of the panel, and m s is its mass per unit
area. The velocity of simple harmonic transverse waves in the panel
is c a = Vco (g/m.)*.
Now suppose that the wave in air has an angle of incidence $, so
that the pressure has the form
V = 2Poe*<«"> <»■•»**> cos [7— ) cos* + *]
The air velocity perpendicular to the surface is
u x = i —^ c *(./e)c»«^ef) sin \h£\ cos* + *1
The value of the phase angle ^ is determined by solving Eq. (30.1)
for n and then setting  (dv/dt) equal to u x at x = 0.
The solution for — (dij/dt) is
362 THE RADIATION AND SCATTERING OF SOUND [VII.30
_dy _ / 2Po cos * \ c i(a/c) (^^^
dt \ z* )
z s = ^ = ia>m s + ig (^)sin 4 $
and the solution for the phase angle is
tan 1 ^ = il — J = *'(«• ~ **») (30.2)
where z s is the specific acoustic impedance of the panel for the angle
of incidence <S>, and k 3 and a s are its acoustic conductance and sus
ceptance ratios.
The equation giving z s in terms of a, m„ g, and * shows that this
impedance is purely reactive and that at high frequencies it depends
very strongly on the angle of incidence. At frequencies low enough
so that c s , the speed of transverse waves in the panel, is considerably
smaller than c, the speed of sound in air, the specific acoustic imped
ance of the panel becomes
z s ~ i(am s (« « c 2 y/m,/g)
a pure mass reactance, due to the mass of the panel. This illustrates
the general statement made at the beginning, that if the speed of wave
motion in the surface is much slower than that of sound in air the
surface has local reaction and the impedance is independent of angle
of incidence. When the impervious panel is quite thin (as with the
fabric lining of an airplane cabin, for instance), the flexural rigidity
is extremely small and z s c^ i<am 8 over the whole of the useful fre
quency range.
For some laminated panels the flexural rigidity factor is complex,
g = g s — ir s , so that the impedance has a real part
= wrn, + (ig. + r.) I ^jsin 4 *
and energy is lost by motion of the surface. The amount of energy
lost is strongly dependent on frequency and on angle of incidence,
however. For such a panel, with very little support, the impedance
becomes very small at very low frequencies.
Supported Panel.— This first example, which has just been dis
cussed, is one seldom encountered in practice; it is more usual to have
the surface material supported by a heavier structure, with distances
VII.30] THE ABSORPTION OF SOUND AT A SURFACE 363
between the points of support of the same size or smaller than a wave
length. In this case, transverse waves cannot be propagated far
along the surface without being stopped by the supporting structure,
so that the panel is, on the average, a locally reacting surface. Each
portion of panel between supports is a plate with effective specific
admittance (l/s P ) equal to the ratio of average velocity u x , averaged
over the portion of panel, to the driving pressure. At the higher
frequencies this impedance will show a mass reactance z s ~ t'ww B , but
at low frequencies the stiffness due to the support structures becomes
important and z s ~ iK s /a>. There is usually a resistive term.
Consequently, the specific acoustic impedance of a supported,
impervious panel can be represented by an equivalent circuit of
inductance ra s , resistance R s , and capacitance C s = 1/K S in series.
Porous Material.— A case often encountered is that of a surface
which is porous enough so that the air motion normal to the surface
is due more to air motion into and out of the pores than to motion of
the panel as a whole. Here we must take into account the wave
motion of the air in the pores. The properties of the pore system can
be expressed in terms of three quantities: flow resistivity r p , porosity
P p , and effective air density pm p .
Under steadystate conditions a pressure drop (Ap) across a thick
ness d of the material will force a flow of air through the pores of
u x = r p d(Ap) cc per sec per sq cm of surface of the material. This
defines the flow resistivity r p .
If the inner side of the material is made impervious, then forcing
air into the material will raise the pressure of the air in the pores.
If the air were compressed adiabatically, as would happen in the
open, the rise in pressure would be (pc*/P p d)% [see (Eq. 23.3)] where £
is the displacement of the air into the outer surface (in cubic centi
meters per square centimeter of surface) andP p is the fraction of the
volume of the material that is available for the air to flow into (this
defines the porosity P p ). In many cases the pores are so small that
the air loses its heat to the pore walls, so that the expansion is iso
thermal instead of adiabatic. Then the rise in pressure would be
(pc 2 /ycPpd)%, where y c = 1.4 for air.
Finally, the air in the pores may have an effective density greater
than that in free space because some filaments of the pore material
may move with the air, so that m p may be greater than unity.
The equation of motion of the air in the pores is therefore equiva
lent to that for electric current in a transmission line, with line resist
ance r p per unit length, line inductance P m p per unit length, and shunt
364 THE RADIATION AND SCATTERING OF SOUND [VII.30
capacitance (ycP p /pc 2 ) per unit length. The equations of motion of
the air in the pores, for simple harmonic waves, are
—iupnipU + r p u = — grad p 
„ , .ycP pP o>, . , n > (30.3)
V 2 p + i  — ~ (r p — ia)m p p)p = I
pc )
and the characteristic impedance of the material is
= P Cyj
m p + i(r p /po)
7cP P
which is a complex quantity, indicating attenuation of the wave. .
Equivalent Circuits for Thin Structures. — If the flow resistance
r p is fairly large, wave motion will attenuate rapidly in the material
and the pores will exhibit local reaction to the sound waves. This is
the case usually encountered in practice, although for some materials
(such as hair felt) extended reaction is not negligible. When local
reaction predominates, so that the impedance is independent of angle
of incidence, and when the thickness of the porous material is small
compared with the wavelength, then the specific acoustic impedance
of the porous material can be expressed in terms of equivalent circuits
for different structures.
Figure 85 shows some equivalent circuits for a few structures. In
the first case the porous material is mounted on a relatively rigid and
impervious wall, so that the air motion in the pores is responsible for
the acoustic impedance. The important part of the impedance is
the capacitative reactance of the pore volume. The effective resist
ance and inertia are reduced by a factor i, because the air motion in
the pores nearest the backing wall is constrained to move less than
the air in the pores near the outside.
In the second example shown in Fig. 85 the panel is held away
from the impervious backing wall by a framework, which prevents
flexural waves from traveling along the panel, and which also dis
courages lateral wave motion in the air space behind the panel. Here
the panel can move by flexure, as well as allowing air to move through
the pores, so that the equivalent circuit shows three parallel paths,
one for flexure, one for transmission through the pores, and the third
due to the stiffness of the air in the pores.
The third example shows that laminated structures are equivalent
to filter networks and that the equivalent circuit can be used to calcu
VH.30] THE ABSORPTION OF SOUND AT A SURFACE
365
late transmission of sound through wall structures as well as absorp
tion of sound by the structure.
Porous Rigid
material { backing
o°<y
°oo° °
° '
O oo 0<
d
V//;
J
AWVAAr
YcPpd
pc*
mppd
^S
S^ o
Impervious
septum a
mass ma
■I
i
Impervious
septum b
mass m/,
r p d m p pd
jL,
pc 2 '
^AAAAArJpTinT 1 ' B
m b
"pc 2
r
Fig. 85. — Equivalent electrical circuits giving specific acoustical impedance of
various wall structures. Equivalence is valid as long as wavelength is long compared
to t, I or d.
Formulas for Thick Panels.— When the thickness of porous mate
rial is greater than a wavelength of sound in the material, the analysis
becomes more difficult. The wave equation for motion of air in the
pores can be written in the form
te . + <» + *>'(!) , i»o
d 2 p
(n + i q y = y J> p [m p + i(^]
366 TEE RADIATION AND SCATTERING OF SOUND [VII.30
and the solution, for the case of material of thickness d f backed by a
wall (or air space) of characteristic impedance z b (at x = — d)
p = A cosh I J {q  in)(x + d) + ^
  feL  * n ^f **[© ( *  in)d + *] (m4)
^ = tanh^To^^/pc^ + n)\
When the frequency is high enough, or the thickness of the material
is great enough, the wave effectively dies out before it is reflected
from the back of the material, the hyperbolic tangent is approxi
mately equal to unity, and the wall impedance becomes the character
istic impedance of the material, z pc . For lower frequencies, Plates I
and II can be used to compute the impedance.
The quantity n can be called the index of refraction of the material
for sound waves, and q can be called its attenuation index. When n
is larger than unity, the wave travels more slowly in the medium
than in air; and when q is large, the wave attenuates rapidly. In
either case the material would be (more or less) locally reactive, and
the wall impedance would be (approximately) independent of angle
of incidence.
Reflection of Plane Wave from Absorbing Wall.— We must next
investigate the behavior of sound waves that strike absorbing surfaces.
We have seen from the preceding discussion that in a great many
cases the surface reacts locally, so that its specific acoustic impedance
is practically independent of angle of incidence. We shall make our
calculations for this case, because the results will be valid for most
types of surface material, and also because the calculations are easier.
In the rest of this chapter and in the next chapter, therefore, we
assume that the acoustic properties of a surface are given by its specific
acoustic impedance, the ratio between the pressure at a point on the
surface and the normal velocity, into the wall, of the air at the surface,
pet = pcir!***
The ratio of the impedance z to the characteristic impedance of air
pc will be called the acoustic impedance ratio of the surface, and its
reciprocal will be called the acoustic admittance ratio of the surface.
fe)' — * (?)
v = K — ia (30.5)
VII.30]
THE ABSORPTION OF SOUND AT A SURFACE
367
where 0, x, *, and o" are the acoustic resistance, reactance, conductance,
and susceptance ratios of the material, respectively. A thin mem
brane has a specific reactance coM and susceptance — (1/coilf), where
M is the mass of the membrane per unit area; a stiff supported panel
of small mass has specific reactance —(K/u>) and susceptance {<u/K),
and so on.
Now suppose again that the surface is at the (?/,z)plane, with air
on the positive x side. An incident wave comes in from the right at
an angle of incidence * and is reflected from the wall. The equation
for the pressure wave is
sq = p . (>i(fi>/c) (y sin*— x cos*— ct) p g— i(w/c)(y s in*+a; cos *— ct)
where the ratio between the reflected pressure and the incident pres
sure is e 2*(«*»(p r = p t . e 2 1 r(ai^)) an( j th e ra ^ between the reflected
and incident intensity is e 4 ™. The pressure and air velocity can be
expressed in terms of hyperbolic functions :
u x —
p = 2Poe~ ,r(a_ ^ ) e i(a,/c)(j ' 8in *~ c0 sinh<ir (?) cos*  a + \>
2 e ir(a*0) e »(<o/c) („ sin * ct ) cog $ g^ J ^ U ^ J CQ g $ _ a __ ^ K
The values of a and are determined by the impedance of the
surface
P — P
tanh[7r(a  iff)] = * r = f cos* = (0  i%) cos* (30.6)
±i ~~\~ ± r
so that, if the specific acoustic resistance and reactance of the surface
are known, the values of a and /3 for a given angle of incidence * can
be computed from Plates I or II at the back of the book.
The ratio of reflected to incident energy is
— o — iva =
1 — f COS*
1 + £ cos*
2 = (1 — 0cos*) 2 + x 2 cos 2 * f „
(1 + cos*) 2 + x 2 cos 2 * ^ '
Curves giving values of this ratio, the fraction of incident intensity
which is reflected, as function of f cos*, for different phase angles <p
of the wall impedance, are shown in Fig. 86. We note that the amount
reflected has a minimum value (and therefore the fraction of incident
energy absorbed has a maximum) when f cos* = 1. The larger
the power factor cos <p for the absorbing surface, the greater the frac
tion of energy absorbed, or the smaller the fraction reflected. When
the power factor is zero (<p = ±90 deg) , no energy is absorbed.
368
THE RADIATION AND SCATTERING OF SOUND
[VH.30
The curves also show that the fraction of energy reflected
approaches unity (fraction absorbed approaches zero) as f cos<I>
approaches zero, i.e., as the angle of incidence approaches 90 deg.
In fact Eq. (30.7) indicates that the surface would not absorb any
energy from a wave traveling parallel to the surface, no matter what
the value of z. This seems to be a contradiction of terms, for the
pressure fluctuations in a wave parallel to the surface would cause
motion of the surface in a direction perpendicular to the assumed
direction of the wave. The fact of the matter is that a plane wave
5
ISkos$
Fig. 86. — Reflection coefficient for plane waves incident at angle of incidence $ on a
plane surface of acoustic impedance 2 = pcf \e — iip.
cannot travel parallel to an infinite plane surface of noninfinite acoustic
impedance. This will be shown in more detail' in the next section.
It can also be seen from the equations for p and u x . We notice that
both of these go to zero when $ = 0, so that a solution in terms of
plane waves alone is not possible.
31. SOUND TRANSMISSION THROUGH DUCTS
Before we discuss the behavior of sound waves in rooms, we should
return once more to the problem of the transmission of sound through
ducts, in order to investigate the effect of the impedance of the duct
wall on the sound wave. This investigation was begun in Sec. 26,
where Eq. (26.14) gave an approximate expression for the transmis
sion of sound in a cylindrical duct. Now we wish to carry the analysis
further. In recent years the problem of sound transmission in venti
lation ducts has become important, so it will be useful to show how to
compute the relation between sound attenuation and duct wall imped
ance, in order to be able to design duct linings to absorb undesirable
VII.31J SOUND TRANSMISSION THROUGH DUCTS 369
sound. Since most ventilating ducts are rectangular in cross section,
we shall confine ourselves to this simple type in the present section.'
Boundary Conditions.— We assume that the duct cross section is
rectangular, of width l v in the ^direction and I, in the zdirection.
We also assume that the impedance of the duct walls is uniform, the
acoustic impedance ratio being f for each wall. The pressure, for a
wave traveling to the right along the duct, is given by the equation
cosh /2^\ cosh/2*^ x _ ct)
smh \ ly / smh \ h /
where r 2 = 1 + (27rc/a>) 2 [(&A)2 + fo./Z,) 2 ], and where the ^'s are
determined by the boundary conditions at y = ± (l v /2) and z = ± (l z /2)
(we are taking the central axis of the duct to be the origin of the lateral
coordinates y and z).
The air velocity into the duct wall at y = +(l y /2) must be equal
to the specific acoustic admittance (77/pc) = (1/pcf) times the pres
sure at the wall.
,, _ (2*yv\ a sinft / \ COSh/^TIYkzX ., . ,,
U " ~ WJ A c 0S h <■*> sink (,177 6
(r}\ . cosh, cosh (%cqj\ .. . .,
This produces the equation for determining g y in terms of r\ :
Toth ^> =4„)ft) ; X = (WU) (3L1 '
where the hyperbolic tangent is used for the waves that are symmetric
about the center plane y = . (p proportional to cosh(2jr^ tf y/^», and
the tangent is used for the antisymmetric modes (p proportional to
the sinh).
This equation also ensures that the boundary condition at y =
— (lv/2) is satisfied, for everything is symmetrical (or antisymmetrical)
about the center plane.
The boundary condition for the surfaces z = ± (l z /z) is a similar
equation
tanh
coth
to*) = *(^(k); x = (2Wco)
The exact solution of these equations is a complicated process, so
we shall first obtain an approximate solution, valid for values of wall
admittance ratio 17 small compared with (\/l) (large wall impedance).
370 THE RADIATION AND SCATTERING OF SOUND [VII.31
Approximate Solution. — When i?(^/X) is small compared with
unity, Eq. (31.1) can be solved approximately. There are a large
number of solutions, each with a characteristic value of g. They can
be put in order of increasing size and labeled with an index number
n v = 0,1,2 • • • . For reasons that will be apparent shortly, the
solutions using the hyperbolic tangent will be labeled with the even
values of %, 0,2,4 . . . , and the solutions using the cotangent will be
labeled with the odd values, 1,3,5 .... The properties of the
functions tanh and coth which are used to obtain the approximate
solutions are
tanh (w + iirn) ~ w; coth (w + iirn + iwr) c^ w, w « 1
Using these expressions, we find that the approximate solutions
for the characteristic values of g„ and g e are
***+(^)*^ +0 ' < ' ) fe) ( "" 1A8 ' ■ '
with the even values of the n's going with the cosh factor in the expres
sion for the pressure and the odd values going with the sinh factor.
The solution for n y = n z = is the "plane wave" or principal wave,
which usually has the smallest attenuation and carries the greatest
part of the power.
The equation for the transmission coefiicient t gives, to the first
approximation in the small quantity rj,
t(0,0) c* 1 + ^g (<r + in), n v = n e =
n* =
(31.2)
r(n y ,0) cr (n y ,0) + ^fcfi^th + * 1 ^ + «)■
r(0,n.) ^ r (0,n 2 ) + ^y ^ (h + 2l y ){a + «), n y = ^ ^^
r{n v ,n z ) ^ro(n y ,n g ) + ^^ ~ (<r + «)
(ny > 0, n* > 0)
ro(n,,n.) = [l  Qg) ~ (^) J
VII.31] SOUND TRANSMISSION THROUGH DUVTS 371
where S = l y l z is the crosssectional area and L = 2(l y + l z ) is. the
perimeter of the duct cross section. We note that when n u = the
duct walls perpendicular to the yaxis (l z ) are half as effective in chang
ing r as is the case for n y > 0, and similarly for n z and l y . Since the
case n y = corresponds to waves traveling "parallel" to the sides l a
and the cases n y > correspond to waves being reflected back and
forth between these walls, we can reword our remark to say that the
walls have (to this approximation) half as much effect on waves travel
ing "parallel" to them as they do on waves that are obliquely incident.
The reason we have put quotation marks around the word parallel
is that the case n y = does not correspond to waves exactly parallel
to the I, walls. In Eq. (30.7) we showed that, if a plane wave could
move parallel to a plane wall, the wall impedance would have no effect
on the wave. We also showed that this was impossible, and the
present results reinforce this, for we find that when the wave is as
"parallel" to the wall as we can make it (n„ = 0) the wall impedance
still makes itself felt; only half as much as for oblique waves, but not
zero times as much.
Principal Wave. — This property of waves and nonrigid walls can
be clarified to some extent by studying the behavior of the principal
wave (n y = n z = 0). Remembering that rj is small and that the
expansion of cosh (u) for u small is 1 + %u 2 , we have
V ^ A [l + ijL (a + iK)y^ \l +q(? + i*> 2
. e <V2S)(t<r«)* e i( M /e)(*rf) > (jl y = Uz = Q) (31.4)
In the first place, we see that the wave is not a perfect plane wave,
for the pressure amplitude depends on y and z and therefore there is
some air motion perpendicular to the #axis (just enough to take into
account the effect due to the sideward motion of the duct walls). In
addition the wave is damped, as indicated by the factor e(^««/ss) j by
an amount proportional to the acoustic conductance ratio k of the
walls, proportional to the circumference of the duct, and inversely
proportional to its crosssectional area S.
The attenuation of the principal wave in decibels per centimeter is
therefore approximately (4.34Lk/£) (if L is measured in feet and S in
square feet, the attenuation will be 4.34Lk/£ db per ft). We note that
the greater the wall area is, compared with the duct volume (L is,
compared with S), the greater the attenuation.
The attenuation of the higher modes is larger than this, being
approximately (SMLK/Sroiny,^)) for n y > 0, n z > 0; and being
372 THE RADIATION AND SCATTERING OF SOUND [VTI31
(4.34/c//STo(n„,0)) (2l y + l z ) for n g = 0, etc., as long as n y , n z are small
enough so that to is real. For all modes above certain values of
n y , n z , to is imaginary for a given frequency, and no true wave motion
can occur; the attenuation for all higher modes being very much
greater than for the ones below the cutoff (see page 308).
The characteristic acoustic impedance of the principal wave is
u x i(a)/c) — (Lrj/2s)
(i _ _£_r
\pc 2icoSzp/
where z v is the specific acoustic impedance of the duct walls. There
fore the characteristic wave impedance of the principal wave is anal
ogous to a circuit with resistance pc in parallel with an impedance
—ica(2Sz p /Lc). For instance, if the duct walls are covered with
porous material of thickness d, flow resistivity r p , porosity P p , and
negligible density m p , the equivalent circuit giving the characteristic
impedance of the principal wave has one parallel arm a pure resistance
pc and the other parallel arm a resistance pc(2S/jcPpLd) in series with
an inductance (2Sr p d/3Lc).
Transient Waves. — We can use the results of the preceding discus
sion to obtain a first approximation to the transient behavior of sound
in a long duct. Suppose that the average air velocity at the input end
is the arbitrary function u x0 (t). Then the pressure in the principal
wave, a distance x along the tube (assuming no reflection from the
far end), is
*»H£j>( i+ 5' ,+ *' , X i &r
t* 00
. e (L V /2s)x+i(w/c) X iwt j w a . (r)e iuT dr
In several cases, where the wall admittance 77 has a simple behavior,
this integral can be evaluated.
For instance, if the side walls are stiffness controlled, 17 = — iwe
where e is the specific elastance of the walls, we use the equations,
derived from Eqs. (2.19) and (2.20),
.. /» * /» 00
f(t — a) = = I e iaa ~ iat du I f(T)e iaT dT
2ir J — 00 J — *
1 f°° f °° d 2
J I ^ w 2 e* dco j_J(r)e^dr =  ^f{t)
VH.31] SOUND TRANSMISSION THROUGH DUCTS 373
(to be used with due regard for convergence;) to obtain
P(t) ~ 1 + (Lce/2S) f'° [' ~ X (~c + M)] '
;8+d£['6 + S)]}
It is not very difficult to show that if the wall impedance is a pure
resistance, independent of 00(77 = k, a constant), then
p(t) ~pcj t <~ I ewwuxo (t ~)dw
K£ + d^"*M)} (31  5)
The proof of this formula will be left to the reader.
The transient waves in the higher modes can be computed in a
similar manner, though the calculations are more difficult.
The Exact Solution. — In a number of cases, the specific acoustic
impedance ratio of the wall, f , is small enough so that the approxi
mations used in Eqs. (31.2) to (31.4) are not valid. Since these cases
are also of practical importance, we must set about finding an exact
solution of the equations
O*)**; ,«+>; k& ^ (31 . 6 )
where z p = Izpler** = (pc/rj) is the specific acoustic impedance of the
duct wall. In the approximate solutions, n turned out to be approxi
mately half an integer and  was small; we cannot expect this to be
true for the general solution. We shall find a series of solutions,
however, with a series of allowed values for n, which can be arranged in
order of increasing size, and labeled /i , in, m • • • . Corresponding
to each ju„ is a solution for £„, and therefore a g(n) = £„ + ifi n , which
is the nth characteristic value. The corresponding characteristic
posh
function is $ n (y) = ^^ l^rg y (n y )y/l v ] where the cosh function is used
for the even values of n y [which are obtained by solving Eq. (31.6) with
tanh] and where the sinh is used for odd values of n y [for which coth
is used in Eq. (31.6)].
The easiest way to solve Eqs. (31.6) is by graphical methods.
The solutions are displayed on Plate V at the back of the book, in
terms of contours for constant values of h and <p drawn on the (£,/x)
plane. We see that the solutions approach the integer and half
374 THE RADIATION AND SCATTERING OF SOUND [VH.31
integer limits on the imaginary axis, as h gets small, as demanded by
the approximate solution. The heavy dashed lines separate the modes
one from the other. For each pair of values of h and <p there
is a point within each modal region. The point in the lowermost
region in the "tanh" chart corresponds to go (cosh is used in the
expression for p); the point in the lowest region in the "coth" chart
corresponds to gi (sinh is used for p) ; the point between the first and
second dashed lines (second sheet) of the coth chart corresponds to g 2
(cosh is used for p) ; and so on.
The dependence on x is by means of the exponential e i(  WTX/c) , where
= 1 +
WW] +m)
(v)"0
= 1 _ ( ^ j I My(^y)  kl(n y ) n\(n,)  £(n y )
2j M»(Wy) €»(%) _ 2l  /*«(»«) €«(»».
']
The real and imaginary parts of tn (N stands for the pair of numbers
n Vf n e ) correspond to the "index of refraction" and the damping
index for each wave in the sequence. This equation also cannot be
solved by approximate means when r is large. The final solutions
= cosh\ 2Tg v (n y )y ~\ cosh f 2irflr«(n,)g l g(Wc)(7VC _ rf) (31 7)
Y sinh L l v J sinh \_ U \
are characteristic functions that can be used to fit initial conditions
and to compute the effect of perturbations, as was discussed in Sees.
11 and 12.
It is obvious that the behavior of the sound waves, even of the
principal wave, is quite complicated when the admittance ratio of the
walls is not small compared with unity. If one adds to this the fact
that many porous acoustic materials have impedances that vary
considerably with change in frequency, it becomes apparent that very
few sweeping generalities can be made concerning the behavior of
sound in ducts with highly absorbent walls. An examination of
Plate V shows that for negative phase angles (stiffness reactance) and
for large values of h (small values of impedance and/or high fre
quencies) the value of £ can become quite large, and therefore even
the principal wave can be highly damped. Further examination
shows that in these cases the principal wave is far from a uniform
plane wave, the negative reactance of the walls having in some way
pulled most of the energy of the wave away from the center of the
VH.31]
SOUND TRANSMISSION THROUGH DUCTS
375
duct to the periphery, where it is more quickly absorbed as it travels
along. A positive phase angle (mass reactance) has the opposite
effect. The change of acoustic behavior with frequency is greatest
Frequency Parameter, (1/a)
Frequency Para meter, (l/a)
Fig. 87. — Transmission of sound through square duct of width I covered with
acoustic material of acoustic impedance pc(6 — i%), for principal (0,0) and a higher
(1,1) model wave. Quantities proportional to attenuation per unit length and phase
velocity are plotted against frequency. Sudden rise of quantities for (1,1) mode illus
trates fact that higher modes cannot be transmitted at low frequencies.
when h and # are close to one of the "branch points" of Eq. (31.6),
shown as circles on the dashed lines separating the modes.
An Example. — As an example, we show in Fig. 87 transmission data
for a case of a porous wall material with specific acoustic impedance
376 THE RADIATION AND SCATTERING OF SOUND [VII.31
equivalent to a resistance and capacitance in series. To simplify the
problem we set l y = l z = I and choose for a frequency parameter
(l/\) = (cdZ/27rc). We choose the constants of our acoustic material
so that f = 1.5 f 1.5i(\/l), which is stiffness controlled at very low
frequencies, but is fairly ""soft" at (X/Z) = 1. The first curves in
Fig. 87 show the variation of acoustic impedance and admittance
ratios with frequency. They are fairly typical curves. The next
curves show the variation of the characteristic values /io, m, £o, £i for
the principal wave and the first higher mode. Finally we show curves
for the attenuation of two actual waves [the principal wave (0,0)
and the (1,1) wave] in decibels per centimeter and the ratio between
their phase velocity and that of sound in the open, l/(real part of t).
We note that the attenuation of the principal mode is less, than the
higher mode for all frequencies shown (this occurs in nearly every
case encountered in practice). We note also the very sudden and
large increase in attenuation of the (1,1) mode as the frequency is
lowered below its cutoff frequency. Few other generalizations can
be drawn from these curves. Slight changes in impedance and in
size of duct can produce very considerable changes in the shape of the
curves of £ and n and of all the quantities derived from them.
Problems
1. A flexible wire 50 cm long is stretched at a tension of 10 9 dynes between
rigid supports. The wire is cylindrical, having a radius of 0.1 cm, and weighs
0.4 g per cm length. The midpoint of the wire is pulled aside 0.5 cm and then
let go at t = 0. What is the expression for the total power radiated by the string
in the form of sound? What is its value at t = and at t = 1? What is the ratio
between the intensity of the fundamental and that of the first harmonic at t =
and at t = 1?
2. A long cylindrical tube of radius 10 cm has a long slit 1 mm wide in its
wall parallel to the tube axis. Air is forced back and forth through the slit at a
velocity of 10 e' 2 ™. Plot the total power radiated by the slit cylinder per centi
meter of its length, as a function of v from v = to v = 1,000. Plot on a polar
diagram the distribution in angle of the radiated intensity for v = 547.5, v = 876.
3. The portion of the surface of a long cylinder of radius a, which is between
—4>a and +<£o, vibrates with velocity normal to the surface, while the rest of the
cylinder is rigid. Obtain a series analogous to Eqs. (27.20) for the average acoustic
impedance over the vibrating surface.
4. Compute the distribution in angle of the pressure ]^(<£)j radiated by a line
source on a cylinder of radius a, for a wavelength X = ira [see Eqs. (26.7)].
5. A hollow cylindrical tube of inner radius 54 cm is lined with acoustic
material of specific acoustic impedance pc[10 + (2,000/o>)i]. Compute the cutoff
frequencies and plot the characteristic impedances as function of w for « from to
10,000, for the first three symmetric modes of wave transmission along the tube,
VU.31] SOUND TRANSMISSION THROUGH DUCTS 377
6. A hollow cylindrical tube of inner radius a, lined with material of specific
acoustic impedance z, is closed at x = I with a rigid plane plate. The other end
of the tube, x = 0, is driven by a plane piston. Derive the formulas for the pres
sure amplitude at any point x along the tube and the formula for the radiation
impedance on the driving piston, including the effects of the impedance z to the
first order of approximation in (pc/z). Show how it is possible to measure the
wall resistance and reactance by measuring the standing wave. What measure
ments must be made, and what formulas must be used?
7. The surface of a sphere of radius a vibrates in such a manner that the
radial component of velocity at the surface is
u a = iC/" (3cos2i> + l) e * ri '*
Show that when X is large compared with a the radiated intensity and power
radiated are
T = Hpo ^ t/*[P 2 (cos#)p, n = iUpo—Jr Ul
Such a source of sound is called a quadrupole source. Plot the distribution in
angle of the intensity on a polar plot.
8. A piston of radius 10 cm is set in the surface of a sphere 20 cm in radius.
It is vibrating with a velocity \Qe~ 2irivt . Plot the total power radiated as a func
tion of v from v = to v = 1,000. Plot on a polar diagram the distribution in
angle of the radiated sound intensity at v = 164.2 and v = 438.
9. The mechanical constants of the piston in Prob. 8 are to = 10, R  1,000,
K = 1,000. Plot the total mechanical impedance of the piston as a function of
v from v = 100 to v = 1,000.
10. The piston of Prob. 9 is driven by a force of lOjOOOe 211 "^' dynes. Plot
the total power radiated by the pistonsphere system as a function of v from
v = 100 to v = 1,000.
11. A dynamic loudspeaker of radius 27.4 cm is set flush in a large flat wall.
Plot the transmission coefficient of the speaker as a function of driving frequency
from v = to v = 1,000. When the speaker is oscillating with a certain velocity
amplitude at a frequency of 400 cps, the sound intensity at a point 500 cm away
from the wall, straight out from the center of the piston, is 100 ergs per sec per
sq cm. What is the intensity at a point close to the wall, 500 cm from the center
of the piston? If the diaphragm vibrates with the same velocity amplitude as
before but at a frequency of 200 cps, what will be the intensity of the sound at
these two points? At 800 cps?
12. A cylindrical tube of radius 5.47 cm and length 34.4 cm has its open end
set flush in a large plane wall (the tube is inside the wall with its axis perpendicular
to the surface) . The other end of the tube is provided with a piston that oscillates
with a velocity of lOOe^"'. Plot the total energy radiated out of the tube as a
function of v from v = to v = 5,000, and plot the distribution in angle of the
radiated intensity at v = 500, 1,000, 2,000.
13. A piston of radius 5.47 cm, set flush in a large plane wall, is vibrating with
a velocity lOOe 2 ™'"'. What are the total power radiated and the distribution in
angle of the intensity at v = 200, 500, 1,000, 2,000? If the same piston is set in
the small end of an exponential horn of constant x„ = 34.4 cm, whose open end
378 THE RADIATION AND SCATTERING OF SOUND [VII.31
has a radius of 54.7 cm and is set flush in the wall, what are the total power radi
ated and the distribution in angle of the intensity at v = 200, 500, 1,000, 2,000?
Assume that the open end is large enough so that all the energy is radiated out
of it. What do these results indicate about the directional properties of horns?
14. Suppose that the pressure at the mouthpiece end of a wind instrument is
given approximately by the Fourier series
po = Po 2* cos [2(2n + l)ir Vl t)
n =
where the frequency of the driving mechanism has adjusted itself so that the
impedance at the fundamental vi is maximum [see Eq. (23.17)]. Compute the
air velocity u e at the open end of the instrument. Assuming that this open end
acts as a simple source of sound, compute the sound pressure p r a large distance r
from the open end. Assuming reasonable values for the constants involved, plot
u e and p r as functions of time.
15. A velocityribbon microphone consists of a conducting ribbon of mass m
per unit length and of width 2a, suspended so that it moves freely at right angles
to a magnetic field H and at right angles to its surface {i.e., the mechanical imped
ance of the ribbon per unit length is approximately 2wivm over the useful range
of v). The motion of the ribbon induces an emf in it of magnitude Hlu X 10—
volt. Assuming that the force on the ribbon due to a plane wave of sound falling
vertically on it is equal to the corresponding force given in Eq. (29.7) for a cylinder,
show that the emf induced in the ribbon is
4a%m p. x 10— > 2 ^*!p x io
mcy?C\ ft*o mc
where P is the pressure amplitude of the incident plane wave.
16. Using the results of Prob. 15, design a velocityribbon microphone and
plot its response curve. The pole pieces of the magnet are 5 cm long and 2 cm
apart, and a field of 1,000 gauss can be maintained between them. An aluminum
(density 2.7) ribbon can be used, of thickness 0.001 cm. What width ribbon
should be used to have a response constant to within 10 per cent for frequencies
below 5,000 cps. Plot this response for the range < v < 10,000.
17. A circular membrane of radius a is set under tension, in a plane wall that
is otherwise rigid. It is set into vibration at its fundamental frequency v a i. Plot
the distribution in angle of the radiated pressure, and compute the total power
radiated for (2iravoi/c) = 10 and for suitable values of the other constants (c
here is the speed of sound in air, not the speed of waves on the membrane). Use
the curves in Fig. 71 to compute the angle distribution.
18. The distribution in angle of radiated pressure given by the following
equation:
(*)
e iHrct) [$„ _ 0.45*1  0.1*2]
[where the functions *„ are defined in Eq. (28.4) and plotted in Fig. 71] has very
small "side lobes" for radiation at frequencies such that ka = 10. Plot the dis
VH.S1] SOUND TRANSMISSION THROUGH DUCTS 379
tribution in angle of the radiated intensity as function of &, and plot the velocity
amplitude of the piston, as function of y, which will produce this radiation.
19. Use the curves of Fig. 72 and the formula of Eq. (28.8) to compute the
pressure on a flat piston of radius a, set in a rigid plane, vibrating with frequency
v. Plot this pressure as function of r for (2wav/c) = ka = 0.2, 1, 4.
20. Suppose that the condenser microphone whose constants are given in
Eqs. (29.14) were set in a sphere of twice the radius, a = 10. Plot the altered
response curve of the microphone from v = to v = 10,000, and compare the
curve with that of Fig. 84.
21. A ribbon of width 0.5 cm and length 5 cm is set in the side of a cylinder
2.19 cm in diameter, with its length parallel to the axis of the tube. The interior
of the tube is so designed that the effective mechanical constants of the ribbon
per unit length are w = 0.01, R = 20, K = 50. The ribbon is in a magnetic
field of 1,000 gauss, perpendicular to the axis of the cylinder and parallel to the
face of the ribbon. Plot the response curve of this "pressureribbon microphone"
for the frequency range v = to v — 10,000.
22. Discuss the directional properties of the condenser microphone whose
constants are given in Eqs. (29.14). Plot the response curve for the microphone
for the axis of the diaphragm pointing at right angles to the direction of incident
sound and the curve for the axis pointing away from the source of sound. Com
pare these curves with that of Fig. 82.
23. A plane wave of sound of the form
[15 sin (z)  10 sin (3z) + 3 sin (5z)]
where z = l,000n It 1» strikes the microphone whose constants are given in
Eqs. (29.14). Plot the emf output of the microphone for one cycle, and compare
it with the pressure wave in free space.
24. The pressure variation from a "warbletone" generator corresponds to
the expression
p s yg— 2t»j> «— t(Ai'/j'„) cos (2xj>„0
By using the last of Eqs. (19.4) show that this expression is equal to the Fourier
series
p  ie 2T<v o' V J n l — ) g2Tt>„»i<(»v*)
X '•(£)
Compute the values of the amplitudes for the component frequencies, for A = 10,
v = 250, Av = 50, v„ = 10. Plot the curve for the real part of p for the time
range t — to t = 0.1.
25. Obtain the formula for the ratio of reflected to incident sound energy as
function of angle of incidence, for a plane wave falling on a plexiglass window of
density m 8 g per sq cm and of flexural rigidity g s  ir„. Discuss the difference
between these results and those shown in Fig. 86.
26. The flow resistivity of a certain acoustic material is r p = 50 cc per dynesec
cm. Its porosity is P p = 0.7, its effective mass is m p = 5, and its thickness is
2 cm. Plot the specific acoustic impedance of the material when its back surface
380 THE RADIATION AND SCATTERING OF SOUND [VII.31
is glued to a rigid wall and also when the panel is spaced out 4 cm from the rigid
wall (assume that the effect of panel flexure is negligible). Plot resistance and
reactance for both cases as a function of frequency v from v = to v = 2,000 cps.
Assume that the equivalent circuit is valid over this range.
27. A plane wave falls normally on the acoustic material of Prob. 26. Plot,
as a function of p.(0 < v < 2,000), the fraction of incident power absorbed by the
material for the two cases mentioned.
28. A ventilating duct 100 cm square in cross section is to be lined with mate
rial to attenuate sound of 1,000 cps most effectively. The acoustic material
described in Prob. 26 is to be used with the material fastened directly to the duct
walls (which are assumed rigid) and with the thickness d of material to be chosen
for optimum results. How thick should the material be to give the greatest
attenuation per foot for 1,000 cps sound?
29. The outlet for a propellor test stand is a honeycomb of ducts, each of 6
in square cross section, lined with material of resistivity 84, porosity 0.7, effective
mass m p = 2, and thickness 5 cm. Compute the attenuation of the principal
wave for 200 and 1,000 cps by the approximate formula, and also by use of Plate
V. How long must the duct be to attenuate the 200cps sound by 60 db? How
much will this length attenuate 1,000cps sound?
30. Derive Eq. (31.5). Plot curves illustrating the properties of the equation.
31. A duct has a square cross section 34.4 cm on a side. Use Plate V to
calculate the optimum wall impedance to give maximum attenuation per length
of duct at 400 cps for the least attenuated mode. Repeat the calculations for
1,000 cps.
32. A dynamic loudspeaker has a conical diaphragm with outer radius a,
set in a large plane baffle. When the diaphragm is vibrating with frequency
(u/2ir) and velocity amplitude V , the air in the plane of the baffle has a normal
velocity approximately equal to
F [*o(rO +*( f f)*i( r )] (r<a)
where 6 is a constant related to the "height" of the cone and where the functions
¥ are given in Eq. (28.3). Show that the distribution in angle of the radiated
intensity at great distances from the diaphragm is
T ^  pcuVln 2 '
* +
(?H
where n = (2xo/X) and where the functions * are given in Eq. (28.4). Plot this
distribution as a function of # f or n = 10 and (2irb/X) = 2.
CHAPTER VIII
STANDING WAVES OF SOUND
32. NORMAL MODES OF VIBRATION
In the preceding chapter we studied sound that is radiated into
open space, tacitly assuming that there were no obstacles opposing its
free flow outward. In most cases, however, sound generators are in
rooms of size small enough so that the waves produced are reflected
back and forth many times a second. When this occurs we cannot
say that the waves generated are all radiating outward from the
source; rather we must say that the source sets into motion one or
more of the normal modes of vibration of the air in the room.
This consideration makes it necessary to alter considerably our
picture of the distribution of intensity about a source of sound. We
cannot expect, for instance, that the intensity will vary inversely as the
square of the distance from the source; in some rooms the intensity
at some point far from the source may be considerably greater
than at intermediate points. Nor can we expect that the intensity of
sound in a room is simply related to the power radiated by the sound
generator. The mechanical behavior of a loudspeaker, its mechanical
and electrical impedance and total power radiated, will be practically
unaltered by the properties of the room, but the intensity of the sound
produced, and the distribution of this intensity, will be greatly altered.
Room Resonance. — Our point of view with respect to the problem
of sound in a room can be stated as follows: We look on the air in the
room as an assemblage of resonators, standing waves that can be set
into vibration by a source and that will die out exponentially when
the source is stopped. When the source is started there will be set
up a steadystate vibration, having the frequency of the source, and a
transient free vibration, having the frequencies of the normal modes,
which will die out. The steadystate vibration may be considered to
be made up of a large number of the standing waves (just as the
forced motion of a string can be built up out of a Fourier series) whose
amplitudes depend on the frequency of the source, the "impedance"
of the standing wave in question, and the position of the source in the
room. The transient vibration will have the form necessary to satisfy
381
382 STANDING WAVES OF SOUND' [VHI.32
the initial conditions in the room when the source is started and will
therefore also be made up of many standing waves; but each normal
mode of the transient vibration will vibrate with its own natural fre
quency. We shall study these frequencies in the present section.
After the transient has died out, the steadystate vibration remain
ing will have only the frequency of the source. We shall study the
shape of this vibration, how it differs from the free radiation from a
source in the open, and how it can be built up out of the standing
waves, in Sec. 34. When the source is turned off, these standing
waves remain, only now they have their own natural frequencies,
damping out exponentially according to their free vibration properties,
and perhaps interfering with each other (making beat notes) as they
do so. The damping of these free vibrations, which is called the
reverberation, will be discussed in Sec. 33.
From another point of view the room is a transmitter of sound from
the speaker to the listener; a generalized horn, so to speak. As such,
it ought to have more the qualities of the horn than of a musical
instrument. In other words, it ought to transmit all frequencies
equally well, and its transient characteristics must not distort the
sound wave noticeably. This of course could be done by arranging
to have the walls almost perfect absorbers of sound, so that they
would not reflect sound back into the room. If this is done, however,
we lose the effect of the walls in enhancing the sound level by reflection
and require considerably greater output power in order to be heard
throughout the room. In the open, sound intensity from a simple
source diminishes as the square of the distance. In a welldesigned
room, however, the sound level 100 ft away from the speaker may be
only 5 db below the level 10 ft away, in contrast to the 20db drop
which would occur in the open. The engineering problem in room
acoustics is to design the shape and acoustic impedance of the walls
so that the room is as uniform a sound transmission system as possible,
without losing completely the reinforcing effect of the wall reflections.
Statistical Analysis for High Frequencies. — We shall see later in
this section that at low frequencies there are few resonances and at
high frequencies there are many resonances in any given band width.
Consequently, a sound of average wavelength about the same size as
the room will excite only a few standing waves in the room; whereas
a highfrequency sound, with average wavelength small compared
with the room dimensions, will excite hundreds of standing waves.
Obviously, the methods of calculation that are easy to use for the
lowfrequency case will be difficult for the highfrequency case, and
Vin.32] NORMAL MODES OF VIBRATION 383
vice versa. The situation is analogous to the difference between the
methods of statistical mechanics, which deal with the average behavior
of a large number of bodies, and those of ordinary mechanics, which
deal with the detailed motions of one or two bodies.
The statistical case, appropriate for high frequencies, is the simplest
to analyze and is the one usually most useful in practical problems.
Here hundreds of normal modes of the room are excited by the source;
the sound is usually fairly uniformly distributed throughout the room
and is traveling in all directions. When the steady state is reached,
the sound at any place in the room can be represented as an assem
blage of a large number of plane waves, each with the frequency of
the driver but going in all different directions.
p(x,y,z) = f 2ir d<p fj Afot^e***"' sin# d& (32.1)
where k is a vector of magnitude k = (co/c) pointing in the direction
of the wave, given by the spherical angles (&,<p); vector r connects
the origin of coordinates with the point (x,y,z) where the pressure is
measured; and kr is their scalar product (k times r times the cosine
of the angle & between them).
The quantity A gives the amplitude and phase of the component
plane pressure wave in the <p,#direction at the point (x,y,z). In
general this is a function of the coordinates (x,y,z) and of the direction
(<p,&) of the component wave, corresponding to the fact that the
sound is not usually completely uniformly distributed in position or
direction. The average sound energy density at (x,y,z) is propor
tional to the square of A, averaged over all directions of k [for the
energy density in a plane wave is (A 2 /2pc 2 )] so that
w{x,y,z) = — 2 J dip J  A (*>,#)  2 sin #d#
The sound intensity in this case is denned as the net flow of energy
per second into a square centimeter of area (it is to be noted that this
is not identical with our definition of intensity for plane waves).
Suppose that 3> is the angle of incidence of the plane wave Ae^ T on
the square centimeter in question. The flow of energy into a square
centimeter perpendicular to k would be (l/2pc)  A I 2 , and that flowing on
the square centimeter with angle of incidence <S> is (l/2pc)A 2 cos3>,
so that the net intensity is
1{x,y,z) = j c I dtp \ ^(^#) 2 cos<S>sin<I><f<S>
384 STANDING WAVES OF SOUND [VHI.32
which is, in general, dependent on the orientation of the square centi
meter in question (since A is a function of direction).
The criterion for a room that is satisfactory for hearing is usually
that A be as independent of <p, &, x, y, and z as possible. A non
uniform A produces annoying irregularities in sound intensity, both
as a function of frequency and also of position in the room. It turns
out that irregularity of wall shape tends to make A more uniform.
This is because irregularities tend to scatter sound waves in all direc
tions, particularly if they are about the size of a wavelength. Smooth
concave walls produce nonuniform sound distributions, on the other
hand, for they tend to focus sound and give rise to localized regions
of high energy density and marked directionality. Even smooth flat
walls are undesirable, as we shall see later in the chapter.
Limiting Case of Uniform Distribution. — The simplest possible
case to analyze is that where the sound is uniform in density and
intensity throughout the room. This is a case seldom attained in
practice but one which it is desirable to approximate, as we have indi
cated above. We assume that the room is large enough so that the
sound from the speaker excites many natural modes of oscillation
of the room, that the wall shape is irregular enough so that sound is
scattered in every direction throughout the room, and that there is
enough absorbing material on the walls so that the sound does not take
an unreasonable length of time to die out after the power source is
shut off (how long a time is "unreasonable" will be discussed later).
In this case the amplitude factor A of the component plane waves
has a magnitude that is independent of (x,y,z) and of (<p,&), so that the
energy density is
w = — Li ergs/cm 3 (32.2)
which is assumed to be independent of (x,y,z) even in the transient
state. Likewise, the intensity is
T = M^J 2 = fe?\ ergs/cm* sec (32.3)
independent of direction.
We now assume that the energy density and intensity stay uni
form even during the transient stage. This is very rarely true in
actual rooms, but it is a condition that is approached in acoustically
satisfactory rooms, and it allows a great simplification to be made
VIH.32] NORMAL MODES OF VIBRATION 385
in calculating the intensity of sound in the room. For it means that
we do not need to calculate the behavior of the amplitudes A(<p,&), but
can concentrate our attention on the average quantities T and w.
Our assumption corresponds to the requirement that w always equals
(4T/c), even when both are functions of time.
To determine how w and T depend on time and on the power
output of any speaker in the room in this simplified case, we set up
the equation for energy balance in the room. The power input into
the room is, of course, the power output II (t) of the speaker, which
may vary with time. Energy is lost by conversion of sound into heat
in the air and at the walls. At high frequencies (above about 6,000
cps) the air can absorb a fairly large amount of energy, particularly
if the humidity of the air is high. But below about 2,500 cps the great
majority of the energy lost is lost at the walls, and we can neglect the
absorption in the air. Each portion of the wall will absorb a certain
fraction of the energy incident upon it ; and since we have assumed that
the intensity is uniform, we can conclude that the power lost to the
walls is proportional to the area of wall surface and to the instantaneous
intensity T(t).
Absorption Coefficient. — The fraction of incident energy that is
absorbed by a portion of the wall depends on the physical character
istics of the wall (e.g., its acoustic impedance) and on the distribution
of sound in the room (e.g., the dependence of A on <p and #). When
A is independent of <p and #, as we have assumed here, the fraction
of power lost to the wall depends only on the wall and is called the
absorption coefficient a of the material. The relation between a and
the specific acoustic impedance of the material will be discussed
shortly, after we have set up the energybalance equation. Values
of a for different materials and different frequencies are given in Table
XIII at the back of this book.
The sum of the products of the absorption coefficients a 8 of each
material composing the walls, floor, and ceiling of the room times the
exposed areas A 8 of each is called the absorption a of the room
a = £ <x a A a (32.4)
s
It is not hard to see that the total power lost to the walls (when. the
sound is uniformly distributed) is Ta. The total energy in the room
at any instant is equal to the volume of the room V times the energy
density (w = (4T/c)). Consequently, the energybalance equation (for
uniformly distributed sound below 5,000 cps) is
386 STANDING WAVES OF SOUND [Vin.32
(32.5)
Reverberation. — The solution of this equation is
T = (^p. J e~ act/iV I e a °" iV H(t) dr (32.6)
indicating that the intensity at a given instant depends on the power
output n for the previous (4F/ac) sec, but depends very little on the
power output before that time (due to the exponential inside the
integral). If the power n fluctuates slowly, changing markedly in a
time long compared with (4V/ac), then the intensity T will be roughly
proportional to II, and Eq. (32.6) reduces to
Intensity level ~ 10 log ( — ) f 90 db
if II is in ergs per second and a in square centimeters. If n is in watts
and a in square feet, the equation is
©
Intensity level ~ 10 log I J +. 130 db
This result is easily seen from Eq. (32.5), for if (dn/dt) is small then
d(4:VT/c)/dt can be neglected and oT~H. The intensity is thus
inversely proportional to the room absorption a, so that for steady
state intensity to be large, a should be small.
On the other hand, if II varies widely in a time short compared with
(4F/ac), then the intensity will not follow the fluctuations of II and the
resulting sound will be "blurred." If the sound is shut off suddenly
at t. = 0, for instance, the subsequent intensity will be
T = %e actMV (32.8)
Intensity level = 10 log T + 90  4.34 ( ^ J db
The "blurring" of rapid fluctuations of speaker power is known as
reverberation. It is related to the fact that the intensity level in the
room does not immediately drop to zero when the power is shut off,
but drops off linearly, with a slope — 4.34(ac/4F) db per sec. This
linear dependence of intensity level on time is typical of rooms with
Vm.32] NORMAL MODES OF VIBRATION 387
uniform sound distribution. We shall discuss cases, later in the
chapter, that have more complex behavior.
Reverberation Time. — The slope of the decay curve (the intensity
level plotted against time after the power is shut off) indicates the
degree of fidelity with which the room follows transient fluctuations
in speaker output. The length of time for the level to drop 60 db
is used as a measure of this slope and is called the reverberation time T.
If lengths are measured in centimeters, this time is
= 60 \§i ac )
sec
When lengths are measured in feet, and for air at normal conditions
of pressure and temperature, the reverberation time is
T 0.049 g) = <^ sec (32 . 6)
When the speaker output changes slowly compared with T, then the
intensity follows the output; but when the speaker output changes
markedly in a time less than onetenth of the reverberation time, then
the fluctuations will not be followed.
Therefore in order that the room transmit transient sound faith
fully, the reverberation time should not be large. For this require
ment a should be large, in contradiction to the requirement that a
be kept small to keep the steadystate intensity large. A compromise
must be worked out between these opposing requirements, a com
promise that varies with the size of the room. For a small room
(V ~ 10,000 cu ft) T can be as small as 1 sec and the average intensity
will still be satisfactorily high; but for a large room (V c^ 1,000,000
cu ft) T may need to be as large as 2 sec for the intensity to be high
enough throughout the room. If the room is used primarily for speech,
which fluctuates rapidly, the reverberation time should be about two
thirds of this, for if the hall is large the intensity can be increased by a
publicaddress system. If the room is used chiefly for music, we can
allow more reverberation without detriment (in fact the music does
not sound "natural" unless there is a certain amount of reverberation).
Thus an analysis of an extremely simplified example of sound in
a room indicates the sort of compromise between reinforcement and
absorption that must be reached for any sort of room, even if
the sound is not uniformly distributed throughout its extent. The
analysis has also indicated that a useful criterion to indicate the
degree of uniformity of the sound distribution is the shape of the decay
388 STANDING WAVES OF SOUND [VIII.32
curve for the sound after the source is shut off. If this is a straight line
(on a decibel scale), then the chances are that the sound is fairly evenly
spread throughout the room; but if it is a curve, then it is certain that
the sound is not uniformly distributed, either in space or in direction of
propagation or both.
Absorption Coefficient and Acoustic Impedance. — Before we finish
our discussion of the idealized case of uniform distribution of sound,
we must compute the relationship between the specific acoustic imped
ance of the wall material and the absorption coefficient a.
As stated above, this quantity is the average fraction of power
absorbed by the wall when sound is falling on it equally from all
directions.
To obtain this average, we go back to our discussion of the ampli
tudes A(<p,&). Suppose that we choose <p and t? so that the polar axis
is perpendicular to the wall (assumed plane) and so that t? is the
angle of incidence of the wave of amplitude A(<p,&). In the case we
are at present considering, A is independent of <p and t?, so that the
power falling on a unit area of wall is
T = ^ \ d<p A 2 cost? sin t?dt? = ^ A 2
2pc Jo Jo pc ' '
But from Eq. (30.7) we can show that the fraction of power lost by a
wave of angle of incidence t?, on reflection from a plane surface of
specific acoustic impedance z (acoustic admittance ratio rj = % ~ *<0, is
a(t?) = 1
77 — cost?) 2 4k cost?
y] + cost?) (k + cost?) 2 + <r 2
Therefore the average value of a, which is to be used in the case
of uniform sound distribution, is given in terms of the acoustic con
ductance and susceptance ratios of the wall by the formula
a = T I d<p \ «WI^I 2 cost? sin t?rft?
(32.10)
Values of this quantity, in terms of values of acoustic impedance ratios
f = (1/r?) = 6 — ix = \^\e~ i<p = (z/pc), can be obtained from the
contour plot in Plate VI at the back of the book. This plot shows
that the maximum value of absorption coefficient (a = 0.96) comes
Vin.32] NORMAL MODES OF VIBRATION 389
when the specific impedance is a pure resistance, a little bit larger than
pc{$ ^ 1.25). As the impedance is increased or decreased from this
value, the absorption coefficient diminishes, and at very large values
of f, a is approximately equal to 8k = 86/(6* + x 2 ) There is no
set of values of k and <r for which a is unity. A plane wave can be
completely absorbed by material of proper impedance; a mixture of
plane waves can never be completely absorbed.
Therefore when the room is so designed (irregular in shape, high
frequencies, sufficient absorption) that the sound in it is fairly uni
formly distributed throughout its volume, then the acoustical charac
teristics of the room are given by Eqs. (32.6) to (32.8) and the slowness
of response to transient sounds is measured by the reverberation time;
the absorption coefficient entering into these equations is given in
terms of the physical properties of the wall material by Eq. (32.10).
If, however, the sound is not uniformly distributed, then Eqs.
(32.6) to (32.8) will not be valid, and Eq. (32.10) for the absorption
coefficient will have no application; in fact the term "absorption
coefficient" will have no application. To analyze this less idealized
(and often encountered) case, we must return to our study of the
individual standing waves in a room.
Standing Waves in a Rectangular Room. — To commence the study
we consider a limiting case of another sort, a room with perfectly
smooth, rigid walls. Here we are sure that the sound will not be
uniformly distributed. We choose a rectangular room with sides
l x , ly, hy for simplicity, and note that the boundary condition for a
rigid wall is that the air velocity perpendicular to the wall is zero at
the wall.
The wave equation in rectangular coordinates is
d 2 p d 2 p , d 2 p _ 1 d 2 p
dx 2 + 7Jy~ 2 + ~dz 2 ~ ~c~dt?
If we choose the origin at the midpoint of the room, the standing
waves must have a symmetry about the origin. The solution of the
wave equation is
cos/ x'
p= . Itfa;
sm\ c, ^ x „ , ^ x „, ^ (32U)
v = 2^ \/o)l + u>l + <a\
where either the cosine or the sine can be used. In either case the
pressure wave behaves in the same manner at x = l x /2 as at # = — l x /2,
etc.
°) eos L«) cos L*)
:/ sm\ c/ sm\ c/
390 STANDING WAVES OF SOUND [VIIL32
The velocity in the xdirection is
am/ *W jAcmAA
cos\ c/ sm\ "c/sm\ c/
This must be zero at a; = ± Z x /2. For the sine function, corresponding
to the cosine function in the pressure wave, to be zero at x = +l x /2,
03 x must have the following values:
03 X = j n x T (n x = 0, 2, 4, 6 • • • )
In this case u will also be zero at x = —l x /2. For the cosine function,
corresponding to the sine function in the pressure wave, to be zero
at x = ± l x /2,
<*x = f n x ir {n x = 1, 3, 5 • • • )
i x
Therefore, the characteristic functions for the rectangular room are
the functions given in Eqs. (32.11) with the following characteristic
values of the co's and of the frequency:
(32.12)
When n x is an even number, cos (co^/c) is used in the expression for
p; when n x is odd, sin (co*x/c) is used; and similarly for n y and n 2 .
The normal mode corresponding to any particular set of values of
n x , n y , and n 2 can be produced by starting a plane wave in the direction
given by the direction cosines u x /u, «„/«, and co 2 /co and letting it be
reflected from the various walls until it becomes a standing wave. If
the values of w x , <a y , co z , and co = 2ttv are related to n x , n y , and n z in the
manner specified by Eqs. (32.12), then the reflected parts of the wave
will combine in such a manner that the resulting motion oscillates with
simple harmonic motion. If we start any other sort of wave, its
reflected parts will interfere with each other, and the motion will not
be periodic, so it will not correspond to a normal mode of vibration.
Distribution in Frequency of the Normal Modes. — The last expres
sion in Eqs. (32.11) suggests that v be considered as a vector with
components (« x /2jt), (ay/tor), and (»,/2jr). The direction of the
vector gives the direction of the wave producing the standing wave,
and the length of the vector the frequency. A normal mode of oscil
co x
_ OK
"X
CTT
<t) y — y tl y ,
h
0) z
C7T
= j n z
"Z
(n x ,
n y , n z = 0, 1,
2,3
• • • )
V
CO
m+(
'nA 2
<m
VIII.32]
NORMAL MODES OF VIBRATION
391
lation can therefore be considered as a point in "frequency space,"
whose x component is an integral number of unit lengths (c/2l x ), whose
y component is an integer times (c/2l y ), etc. The length of the line
joining this point and the origin is the frequency of the normal mode,
and the direction of this line is the direction of the wave that can be
used to generate the standing wave. Some of these "characteristic
points" are shown in Fig. 88, and it can be seen that they correspond to
the intersections of a rectangular lattice with x, y, and z spacings
equal to (c/2l x ), (c/2l y ), (c/2l z ), respectively. It can also be seen that
all the normal modes are included among the points in the octant of
space between the positive v x , v y ,
and v z axes, for any of the waves
of frequency v with directions
corresponding to
(cn x /2l x , cn y /2ly,cn z /2lz),
( — cn x /2l x , cn y /2l y , cn z /2l z ),
(cn x /2l x , —cn y /2l y , cn z /2l z ), etc.,
will generate, by reflection, the
same standing wave.
This picture of a lattice of
characteristic points in frequency
space is extremely useful in dis
cussing the number and type of
normal modes having frequencies
within a given frequency range.
For instance, since there are
(8F/c 3 ) lattice cells per unit vol
ume of frequency space (V = ld y l z ), there will be, on the average,
(8V/c s )(irv 3 /6) normal modes having frequency equal to or less than
v (the factor irv 3 /6 being the volume of an eighth of a sphere of radius
v). The actual number of modes having frequency less than v varies
in an irregular manner as v increases, being zero until v equals the
smallest of the three quantities (c/2l x ), (c/2l y ), (c/2l z ), when it suddenly
jumps to unity, and so on.
Axial, Tangential, and Oblique Waves. — Referring to Sec. 31, and
anticipating the results of the next section a little, we note that waves
traveling "parallel" to a wall are affected by the wall (are absorbed
by it, for instance) to a lesser extent than waves having oblique inci
dence. Therefore we separate our standing waves into three cate
gories and seven classes:
Fig. 88. — Distribution of allowed fre
quencies in "frequency space" for a rec
tangular room of sides l x , l v , and l z . The
length of the vector from the origin to one
of the lattice points is an allowed frequency,
and the direction of the vector gives the
direction cosines of the corresponding
standing wave in the room.
392 STANDING WAVES OF SOUND [VUL32
Axial waves (for which two n's are zero)
xaxial waves, parallel to the xaxis (n y , n z = 0)
?/axial waves, parallel to the yaxis (n x , n z = 0)
2axial waves, parallel to the zaxis (n x , n y = 0)
Tangential waves (for which one n is zero)
y, ztangential waves, parallel to the y, 2plane (n x = 0)
x, 2tangential waves, parallel to the x, 2plane (n y = 0)
x, ^tangential waves, parallel to the x, yplane (n z = 0)
Oblique waves (for which no n is zero)
It will turn out that, even in the first approximation, waves of
different classes have different reverberation times and, to the first
30,
2
"o
•c
E
es
CO
.£
1 20
10
L=220, A=l80O,V=450O
Bond Width 10 cps
o jrrprf
150
200
100
Frequency, v
Fig. 89. — Number of standing waves with frequencies between v — 5 and v + 5
in a room 10 by 15 by 30 ft. Irregular solid line gives exact values; dashed smooth
curve is plot of Eq. (32.14), giving approximate values of dN.
approximation, waves of the same class (with v's approximately
equal) have the same reverberation time.
Consequently, it will be quite important to count the number of
standing waves of a given class having frequency less than v. The
representation in a lattice system is again useful here; for the axial
Vm.32] NORMAL MODES OF VIBRATION 393
waves have their lattice points on the corresponding axis in "frequency
space" and the tangential waves have their points in the corresponding
coordinate planes. Again the number of lattice points can be counted,
or a "smoothedout" average number can be computed.
To take an example of practical interest: suppose that the source
sends out a pulse of sound of frequency v and duration At. According
to Eq. (22.18), if this pulse is to be transmitted in the room without
serious distortion of shape there must be a sufficient number (it turns
out that a "sufficient number" is more than 10) of standing waves
with frequencies within a frequency band between v — (Aj//2) and
pq + (Av/2), where Av = (l/At), in order to "carry" the sound. If,
for instance, we should wish to have the room transmit adequately
a pulse of length tV sec, we would be interested in counting the number
of resonance frequencies of the room between *» — 5 and v + 5. If
this number is less than 10 for a certain value of v , then a pulse of
frequency v and of duration xtr sec would not be transmitted with
fidelity in the room. If the number is larger than about 10, and if
in addition the reverberation time of each the standing waves involved
is less than about a second, then the pulse will be transmitted with
reasonable fidelity. Figure 89 shows a curve (solid irregular line)
representing a count of this sort for a room 10 by 15 by 30 ft. It
indicates that a pulse of to sec duration would not be reproduced
adequately unless its frequency were larger than about 150 cps.
Average Formulas for Numbers of Allowed Frequencies. — It is
quite tedious to count the individual allowed frequencies less than a
given frequency or in a given frequency band, so that it is useful to
obtain "smoothedout" formulas for average values of the counts.
This can be done by considering that each lattice point "occupies"
a rectangular block of dimensions (c/2l x ), (c/2l y ), (c/2l g ) in frequency
space, with the actual lattice point at the center of the block. Then
the average number of points can be obtained by dividing the volume
of frequency space considered by the volume (c 3 /87)(7 = l x l y h) of
each block.
As an example we can count up the numbers of different classes
of waves having frequencies less than v. The average number of
xaxial waves is just v divided by the lattice spacing in the v x direc
tion, (2vl x /c) [i.e., it is the number of blocks in a rod of cross section
(c 2 /4l y l z ) and length v] and the average number of all axial waves with
frequencies less than v is
ZV ax
©
394 STANDING WAVES OF SOUND [VIII.32
where L = 4(^ f l v + h) is the sum of the lengths of all the edges
of the room.
The average number of y, ztangential waves is the number of
blocks in a quarter of a disk, of thickness (c/2l x ) and of radius v,
minus a correction to allow for the axial waves, which have been
counted separately. This correction in volume is one half the space
"occupied" by the y and the z axial lattice points, viz.,
{v C y$>v)(i y + u)
The factor onehalf comes in because only onehalf the volume "occu
pied" by the axial lattice points is inside the angular sector formed
between the y and zaxes, which bounds the quarter disk. Therefore
the average number of y, ztangential waves having frequencies less
than v is
and the average number of all tangential waves with frequencies less
than v is
N (cw)'(€)
where A = 2(1 J y + l x l z + l y l z ) is the total wall area. We are neglect
ing the corrections for the overlapping regions at the origin, v = 0, for
they are independent of v and are small in magnitude.
The volume "occupied" by the lattice points for the oblique waves
of frequency less than v is the volume of oneeighth sphere minus the
volume already counted for the other classes of wave :
where V = l x l y L is the volume of air in the room. Therefore the
total number of standing waves of all classes which have frequencies
less than v is
The correct value for JV fluctuates above and below this average
value but is seldom more than one or two units away, unless the room
is too symmetrical; this will be discussed later.
VIII.32] NORMAL MODES OF VIBRATION 395
Average Number of Frequencies in Band. — The number of stand
ing waves with frequencies in a band of width dv is obtained by differ
entiating the formulas given above:
dN ax ,x — (2^/c) dv, etc.
dN ax ~ (L/2c) dv
dN ta , yz ~ [(2tv/c 2 )I v I z  (l/c)(ly + I,)] dv, etc.
dN ia ~ [(ttM/c 2 )  (L/2c)] dv } (32.14)
dN ob ~ [(4™ 2 7/c 3 ) ■ (7rM/2c 2 ) + (L/8c)] dv
dN ~ [(4rv 2 V/c») + (7r^/2c 2 ) + (L/8c)] dv
£ = 4(Z a + ^ + Z,); A = 2(l x l u + ZJ, + l y l z ); V = l x l y l z
The value of dN, obtained from this formula, for dv = 10 and for
appropriate values of the other constants is shown as the dashed line
in Fig. 89. It is seen that this curve is a good " smoothedout "
approximation to the correct step curve. At very high frequencies
just the term proportional to v 2 is important.
We notice that the average number of allowed frequencies in a band
increases with the square of the frequency at the higher frequencies.
If we assume that the average intensity of sound in a room (for a con
stant output source) is proportional to the number of standing waves
that carry the sound (i.e., the number with frequency inside the band
characteristic of the driver), then the intensity in the room increases
as the square of the frequency, for high frequencies, according to
Eqs. (32.14). This is very interesting, because the power output
into free space from a simple source is proportional to v 2 , according
to Eq. (27.4). Therefore the power transmitted from source to
receiver in a room varies, on the average, with frequency as it does in
the open; but superimposed on the smooth rise are fluctuations (as
shown in Fig. 89) due to the fluctuations of the number of standing
waves in the frequency band of the driver. These irregularities of
response are more pronounced, the more symmetric is the shape of the
room, or the narrower is the frequency band of the sound source.
When dN, as given by Eqs. (32.14), becomes two or less, then the
fluctuations become so large that they appreciably reduce the fidelity
of transmission. For the room referred to in Fig. 89 this lower limit,
for a band width of 10 cps, is about 50 cps; for a band width of 5 cps,
it is about 100 cps; etc.
The Effect of Room Symmetry. — We have mentioned several times
in the preceding pages that the response curve of a room, as evidenced
by the exact curves for dN, is more irregular when the room is more
symmetrical. This is due to the increase in the number of degenerate
396
STANDING WAVES OF SOUND
[VIII.32
modes, standing waves with different n's that have the same frequency.
As an example of the effect of degeneracies we can consider the sequence
of rooms of dimensions l x , l y = ql x , l z = L/q, so chosen that the volume
Table 4. — Characteristic Frequency Parameters (2lv/c) and Corresponding
Quantum Numbers for Standing Waves in a Cubical Room of Side I
and in a Room of Dimensions I, I \/2> Z/a/2
Cubical
room
Non cubical room
(2lv/c)
n x
n y
n,
(2lv/c)
n x
n y
71*
fl
0.707
1
1.000 .
°
1
1.000
1
k
1
1.225
1
1
1.414
(i
1
1
1.414
{o
2
1
lo
1 '
1
1.581
1
1
1.732
1
1
1
1.732
{\
1
f 2
2
2.000
2
1.871
1
1
1
lo
2
lo
/2
1
2.000
2
1
ll
2
{I
1
1
2.236
) 2
)l
1
2
2.121
2.236
3
2
1
1°
2
1
2.345
1
3
\0
1
2
{;.
1
f 2
1
1
2.449
2
2.449
1
2
1
{o
1
1
u
1
2
2.550
3
1
f 2
2
2.739
i
3
1
2.828
2
2
ii
2
lo
2
2
2.828
4
/3
\i
1
2
lo
3
2.915
3
3.000
Jo
)2
2
3
1
3.000
P
2
I 2
1
2
li
4
\1
2
2
V = (l x ) z remains the same but the relative dimensions change as we
change q. The natural frequencies are
&)Mf)
+ q 2 n 2 t
VUI.32] NORMAL MODES OF VIBRATION 397
which change in spacing as we change q. We shall write down the
lowest allowed frequencies for two rooms of this sequence: one for
q 2 = 1, a cubical room, the most symmetric; and one for q 2 = 2, which
is not symmetric.
Table 4 gives the allowed frequencies and the combinations of
integers (n x ,n y ,n z ) which label the corresponding characteristic func
tions for these two cases. We notice immediately the tendency of all
the characteristic frequencies to "clump together" in the cubical
room case. Threefold and even sixfold degeneracies are common
even at these low frequencies (for instance 2.236 and 3.000). These
result in large ranges of frequency within which there is no character
istic value, so that the response is very irregular. In contrast, the
case of q 2 = 2 never gives more than twofold degeneracies in the
frequency range considered, and the allowed frequencies are therefore
more evenly spaced along the scale. We note that, because the room
volumes are equal, there are approximately the same number of
frequencies equal to (3c/2Z) or less (28 in one case, 27 in the other)
but the particular values of (n x ,n y ,n z ) included, and their order on the
frequency scale differs. If we had picked an incommensurate value
for q 2 (the cube root of 5 for instance), we would have had no degener
acies at all, and the allowed frequencies would have been still more
evenly spaced along the scale. We can never get absolutely uniform
spacing with a rectangular room, of course, because the lattice in
frequency space is always rectangular. A room with irregular walls
would correspond to a more random arrangement of lattice points in
frequency space and, perhaps, to a more uniform response.
Nonrectangular Rooms. — Our analysis of standing waves has
depended to some extent on the fact that we have chosen to study
rectangular rooms. This is not a serious limitation, for most rooms
approximate a rectangular form. Nevertheless, it would be more
satisfactory if it could be shown that Eqs. (32.13) and (32.14) hold
for all room shapes. This cannot be done for several reasons. In
the first place, although it is not difficult to generalize the quantities V
(room volume) and A (area of walls) to rooms of other shape, the
quantity L (total length of edge) becomes a problem. (For example,
if L for a cylindrical room is just 4irR, what is L for a room of octagonal
floor plan, or with a floor plan that is a polygon of a large number of
sides — approaching a circular form — and what is L for a spherical
enclosure?)
In the second place it becomes progressively more difficult to
define axial and tangential waves as the room shape is made more com
398 STANDING WAVES OF SOUND [VHI.32
plex (this of course is another aspect of the reciprocal relationship
between uniformity of room shape and uniformity of wave behaviors
all waves are oblique waves if the room is irregular enough).
As an example of how these questions can be answered in one
other case, we shall outline the solution for a cylindrical room, with
circular floor plan of radius a and of height I. The solution of the
wave equation in cylindrical coordinates is
cos
<t)'(?)
V = gin (m<p) cos^j J»yf) <r M «, " = ^ Va>i + «? (32.15)
where z = is at one of the end "walls," the floor. The properties
of the Bessel functions J m have been given in Eqs. (19.4) and (19.5).
In order to have the z component of the particle velocity zero at
z = and z = I, we must have the derivative of p with respect to z
zero at these points. The derivative is zero at z = 0, because we
have already chosen the cosine function. For it to be zero at z = I,
we must have (wj/c) = n z ir, (n z = 0,1,2,3 • • • )• I n order to have
the radial particle velocity zero at the cylinder walls, we must have
(dJ m /dr) = at r — a. For this to be true, we must have ((o r a/c) =
Tra mn where a mn is a solution of the equation [dJ ' m {ira) / 'da] = 0. The
characteristic functions for the cylindrical room are therefore those
given in Eq. (32.15), and the characteristic values are
j J (n, = 0,1,2 • • • )
co r = I — ^_ J (where a mn is given in Table 5)
c
" = 2
V(t)' + ( s S b )' ™
The quantities <x mn have already been mentioned in Eqs. (26.12) and
(26.15), in connection with the cylindrical pipe.
If I is not longer than 1.71a, the mode with the lowest frequency
is a transverse one, with one diametric mode, the air "sloshing" back
and forth across the cylinder.
The next problem is to label the "axial" and "tangential" waves.
The 2axial waves are obvious ones; they correspond to those for which
m = n = 0, where the motion is parallel to the zaxis. Similarly,
the waves for which n z = are obviously parallel to the floor and
ceiling and should, probably, be called <p, rtangential waves, by
analogy with the rectangular case. The waves for which motion is
VIIL32]
NORMAL MODES OF VIBRATION
399
entirely radial (parallel to r) for which n z = m — are those which
focus the sound along the cylinder axis. They can perhaps be called
raxial waves, though they are not parallel to any wall. The waves
for which n z = n = are those which travel close to the curved walls
and have little motion near the cylindrical axis. These can be called
<paxial waves, though the air motion is not entirely in the ^direction
(perpendicular to r and to z) . We shall see in the next section that the
raxial waves are least absorbed by acoustic material on the curved
walls and the ^>axial waves are very strongly absorbed.
Table 5. — Characteristic Values a mn for the Cylindrical Room Solutions
OP dJ m (ira) /da =
1
2
3
4
0.0000
1.2197
2.2331
3.2383
4.2411
1
0.5861
1.6970
2.7140
3.7261
4.7312
2
0.9722
2.1346
3.1734
4.1923
5.2036
3
1.3373
2.5513
3.6115
4.6428
5.6624
4
1.6926
2.9547
4.0368
5.0815
6.1103
5
2.0421
3.3486
4.4523
5.5108
6.5494
6
2.3877
3.7353
4.8600
5.9325
6.9811
7
2.7304
4.1165
5.2615
6.3477
7.4065
8
3.0709
4.4931
5.6576
6.7574
7.8264
«m0 — (w/tt)
OCmn ^ n + \m + \
(m » 1)
{n ^> 1, n > m)
Except for m = 0, each mode is doubly degenerate, corresponding
cos
to the duality . (m<p) in the characteristic function.
J sin v
This can be
represented formally by allowing m to take on negative as well as
positive values, with negative m corresponding to the cos(m^>) case
and positive m to the sin(m<p) case (where, of course, a^ n = a mn ).
Frequency Distribution for Cylindrical Room. — A certain amount of
arbitrariness comes in when we set up the lattice in frequency space
for computing allowed frequencies. Here the standing wave is made
up of waves with components in all directions perpendicular to the
zaxis, so that there is no obvious direction that corresponds to the
v m , n ,n z vector, as there was for the rectangular room.
To be strictly accurate, we should represent each wave by a portion
of a circle in frequency space, corresponding to the fact that a cylin
drical wave is made up of a distribution of plane waves in a variety
of directions. Formally, however, we can set up arbitrarily a sym
400
STANDING WAVES OF SOUND
[VIII.32
metrical grid of points on the (v x ,v y ) plane, each the proper distance
from the origin, which can represent all the allowed frequencies.
A possible representation is given in Fig. 90, where the raxial
waves (w = 0) lie along the main diagonal and the doubly degenerate
^axial waves (n = 0) lie along
the v x and i^axes. This type
of representation is useful for,
as we shall see later, the ^>axial
waves depart most radically from
the average reverberation behav
ior, being much more rapidly
absorbed than the rest. In fact
it is not unreasonable to expect
waves tangential to any concave
curved wall to be absorbed more
rapidly than oblique waves, and
waves tangential to a flat or a
convex curved wall to be ab
sorbed less rapidly than the rest.
The third dimension of the
lattice (the ^direction) is ob
tained by adding similar two
dimensional arrays, one above
the other, spaced a distance (c/2l) apart. The process of counting
the number of standing waves with frequencies less than v is now
carried out as before. The "smoothedout" formula is somewhat
more difficult to calculate, for the asymptotic formulas for the a's for
large m and n must be utilized. The equations analogous to Eqs.
(32.14) are
(c/a)
(Zc/a)
Fig. 90. — Formal representation of
allowed frequencies in a cylindrical room.
Allowed frequencies given by radial dis
tance from origin to one of the small circles.
Lattice is made up of a parallel sequence of
similar planes, spaced (c/2h) apart.
dN aXyZ ~ (22/c) dv
dNax, v ^ (2xa/c) dv
dN ta ,r<p^[2irv/c 2 )(ira 2 )  (7ra/c)]dv
dN ta , z<p ~[(2rv/c 2 )(2T<il)  (2va/c)  (2l/c)]dv
dN ~ [(±tv 2 V/c s ) + (tvA/2c 2 ) + (L/8c)] dv
V = iraH; A = 27ra 2 + 2wal; L = lira + U
(32.17)
This corresponds to our definitions of volume V and area A for the
earlier case; but L has a form that could not easily have been predicted.
It appears likely that the equation for dN is a general one as far
as the terms in v 2 and v go, but that the constant term (if it is needed)
must be worked out for each case. It is seldom that this constant
Vm.33) DAMPED VIBRATIONS, REVERBERATION 401
term in the series is needed, however. Indeed, in many cases only the
first term (4.irv 2 V/c z ) dvis sufficiently accurate. It can be proved that
this term has the correct form for a room of volume V of any shape
whatsoever.
The curve for dN from Eq. (32.17) is roughly similar to Fig. 89, but
the allowed frequencies in this case are not so evenly spaced as with the
rectangular room, so that the actual value of dN does not approach the
average curve for so low a frequency as in the rectangular case. This
is due to the fact that at the higher frequencies there are many modes
having nearly the same frequency, the asymptotic form of a showing
that a m i t n+\ ^ <w, etc. This "bunching together" of the allowed
frequencies is essentially due to the symmetry of the enclosure about
the cylindrical axis, so that a number of standing waves can be set up,
having different directions but having about the same frequency.
The distribution in frequency of the standing waves in a spherical
enclosure is even less regular than that in a cylinder, and still higher
frequencies must be u3ed before the actual curve for dN is smoothed
out and approaches the average curve. Such an enclosure would not
be satisfactory for use as a room, because of the fluctuation in its
resonating characteristics as the frequency is changed. In fact, we
can state as a general rule that the more symmetrical an enclosure is,
the larger will be the range of frequency over which the resonance
properties fluctuate, and the less desirable will it be for use as an
auditorium. The curve for dN for a room of the same volume as
that used for Fig. 89, but having very irregular walls, will approach the
smooth average curve still more rapidly than the curve shown does.
Irregular walls also serve to spread out the sound energy more
or less uniformly over the room. Most of the highfrequency standing
waves in a rectangular enclosure have an average amplitude that is
nearly the same everywhere in the room, but many of the standing
waves in a spherical room have larger amplitudes near the center than
near the wall. In rooms having smooth concave surfaces, focal points
of considerable excess intensity may occur to render the room undesir
able as an auditorium.
33 DAMPED VIBRATIONS, REVERBERATION
The standing waves discussed in the last section do not continue
in the enclosure forever; they lose energy and damp out. Over the
range of frequencies useful in acoustics, most of this energy is lost at
the walls, some of it being transmitted through to the outside, and
some going to heat the wall. At very high frequencies an appreciable
402 STANDING WAVES OF SOUND [VIII.33
portion of the energy is lost in heating the gas itself, but for the fre
quencies in which we are interested we can consider that this loss is
negligible compared with that lost at the walls.
We have already computed [see Eq. (32.6)] the reverberation in a
room with sufficiently irregular walls so that the sound is uniformly
distributed throughout its volume. In this section, we shall discuss
the reverberation of individual standing waves in a room. Pursuing
the analogy mentioned at the beginning of Sec. 32, the analysis of this
section is more analogous to the detailed analysis of ordinary mechanics
than to the calculations of statistical mechanics.
Rectangular Room, Approximate Solution. — We start first with the
rectangular room of sides l x , l v , l z , with walls (and floor and ceiling)
that are fairly hard (specific acoustic impedance large compared with
pc). In this case the standing waves are not much different in form
from those given in Eq. (32.11) for a rectangular room with rigid walls:
_ cos /im x x\ cos f Kn y y \ cos / irn z z \
* N ~" sin\ l x ) sin\ l y ) sin\ l t )
— «vfc) ,+ fey +&)■*"•'
(33.1)
where the origin of coordinates is at the center of the room. Even
values of n correspond to the use of the cosine in \f/ N , odd values of n
correspond to the use of the sine. If the wall impedance has a resis
tive term, energy is lost and the free vibration of each standing wave
must contain an exponential factor e~ kt , corresponding to the loss of
energy. In Eq. (4.8) we related the damping factor k for a simple
oscillator to the fractional loss in energy of free vibration per second.
This relationship holds for any oscillation and can be used here to
obtain a first approximation to the damping constant for a standing
wave of sound.
The total energy of the standing wave corresponding to the trio of
numbers N = n x , n„, n z is [using Eqs. (33.1) and (25.3)]
^' = 5 J J/ k + ^ "*]*""*! (33 . 2)
VAx * _ /i / \ \
= K — 2» An ~~ W € »* e »» e »J j
where e = 1 but d = e 2 = e 3 = • • • =2, and where V = U y l z .
We notice that the axial waves have more energy content than the
VIII.33] DAMPED VIBRATIONS, REVERBERATION 403
tangential, and the tangential have more than the oblique. This is
because each cosine or sine factor has meansquared value %, whereas a
constant factor (n = 0) has average value 1. If there are nodal planes
in all three directions (oblique waves), the average value will be least.
The energy lost per second at a square centimeter of wall surface is
the average value of the pressure p times the velocity normal to the
wall; this equals (/cp 2 /2pc), where k is the acoustic conductance ratio
of the wall material. The rate of total power loss at the walls is
therefore
= WcSf K
(S)p 2 (S) dS
where the integration is over every element of all area for which k is not
zero.
Referring to Eq. (4.8), we see that the damping constant k is
equal to i times the ratio between power loss and total energy of
vibration. In the present case, we can write the equation for k as
follows:
k ^ Ac [average value of tap* over all walls]
N 2V [average value of ^* over volume of room] ^ ^
This expression is correct to the first order in k for standing waves in
rooms of any shape. A is the wall area and V is the room volume.
The "Q of the room" for the iVth wave [see Eq. (4.4)] is (o> N /2k N ).
Wall Coefficients and Wall Absorption.— The exponential decay
factor for energy or intensity of the standing wave will therefore have a
term e~ 2k ^ (because energy or intensity is proportional to the square of
p). Comparison of this with Eq. (32.8) for the statistical case of
uniformly distributed sound shows that the quantity that takes the
place of the average absorption coefficient a in the case of a single
standing wave is the average wall coefficient a N , where
5 ^ average value of 4k^ over all walls
average value of ^ over room volume (33.4)
to the first approximation in the (assumed small) quantity k. In the
statistical case, a was independent of the sound; in the present case,
a N depends on the standing wave as well as on the wall materials.
These formulas show that absorbing material (material with a
nonzero value of k) should be placed at those parts of the wall where
the pressure is largest if it is wished to absorb the standing wave most
404 STANDING WAVES OF SOUND [VTII.33
rapidly. Material placed at the corners of the room is twice as, effec
tive, on the average, as if it were placed at other locations, because all
standing waves have pressure maxima there.
One might be tempted to place strips of absorbing material in a
regular pattern on the wall, corresponding to the maxima of a particu
lar \f/ N , in order to damp out the wave having a particular trio of values
of n x , n y , n z . This would *absorb that particular standing wave
rapidly, but other waves, with different values of the w's, would not
have their maxima coincide with the strips of absorbing material, and
so would be much less rapidly damped. Whenever there is a wide
variation in the reverberation times of different standing waves, the
acoustical conditions in the room turn out to be unsatisfactorily
nonuniform; so any regular pattern of patches of absorbing material
will usually lead to unsatisfactory acoustics in one or more frequency
ranges. In order to maintain as much acoustic uniformity as possible,
it is best to cover several walls completely with material, or else to
distribute patches of material in a nonuniform manner over all the
walls. A random arrangement of patches has the additional advan
tage of scattering sound and tending to distribute it more uniformly
throughout the room. This scattering effect is not calculable by first
order approximations, but will be discussed briefly later in this section.
If each wall has a uniform acoustic impedance, or if its impedance
varies in a completely random manner, then the maxima and minima
of each standing wave fall with equal likelihood on the most absorbing
parts of the wall. In this case the average value of (4k^) is equal to
the product of the average value of (4k) times the average value of
(^), each averaged separately over the wall in question. In this
case Eqs. (33.3) and (33.4) can be further simplified.
Suppose that we multiply the acoustic conductance ratio of each
wall material by eight times its exposed area and sum over all the area
of the two walls perpendicular to the zaxis, calling this quantity
a*~ Y (8k s )A* (33.5)
x walls
the absorption (firstorder approximation) of the x walls. Comparison
with Eq. (32.4) shows that, to the first order in k, the quantities (8/c)
for the rectangular room correspond to the absorption coefficients a
in irregularly shaped rooms with uniformly distributed sound. The
quantities a y and a z can similarly be defined.
Finally, utilizing the properties of the squares of sine and cosine to
obtain average values for \[/%, we obtain approximate formulas for the
Vra.33] DAMPED VIBRATIONS, REVERBERATION 405
damping constant kn and corresponding reverberation time for the
standing wave specified by N = (n x , n y , n e ):
0.0497 ^ (33,6)
T N
\^n x dx + $*n Jly + Tt^nflz)
where Tn is given for dimensions in feet and for air at standard condi
tions. These formulas are good only to the first order in the (sup
posedly) small quantities k; more accurate formulas will be discussed
later in the section.
The factors ^e„ (= ^ for n = 0; = 1 for n > 0) constitute the
difference between this formula and that of Eq. (32.9) for the room with
uniform distribution of sound. When n x (for instance) is zero, the x
walls contribute onehalf as much (to this approximation) to the
absorption of the wave as they do for waves with n x > 0. This is
due to the properties of the functions \f/ N ; the ratio between the average
energy in the room and the average intensity on the x walls for n x =
(waves parallel to the x walls) is onehalf what it is for the waves that
reflect from the x walls (n x > 0), so that for the parallel waves the
fraction of total energy absorbed by the x walls is onehalf the usual
fraction.
Reverberation Times for Oblique, Tangential, and Axial Waves. —
These formulas illustrate some of the effects of room regularity on
sound absorption. A rectangular room has walls smooth enough so
that some standing waves move parallel to two or more walls. For
these standing waves the walls parallel to the motion have less effect
on the damping (onehalf as much, to the first approximation) as they
do for oblique waves. This behavior is, in general, true for smooth
plane walls and smooth convex walls, in rooms of other than rectangu
lar shape (by "smooth" we mean that the radius of curvature of any
part of the Wall is considerably larger than a wavelength). For
smooth concave walls it turns out that the waves "parallel" to its
surface are more strongly absorbed than are oblique waves. In either
case, however, the waves moving tangential to one or more walls are
absorbed at a different rate from the rest.
As an example, it can be mentioned that the standing waves in a
room with a floor plan that is an isosceles right triangle also has three
classes of tangential waves: those tangential to the planeparallel
floor and ceiling (for which the floor and ceiling absorptions are
multiplied by onehalf), those "tangential" to the two walls at right
406 STANDING WAVES OF SOUND [VHI.33
angles (for which the absorption for these walls is multiplied by
threequarters), and those tangential to the diagonal wall (the factor
for this wall is also threequarters) . Another example is the cylindrical
room, which will be worked out later.
In a rectangular room (or indeed in any room with "smooth"
walls), if a sound source exciting a number of standing waves is sud
denly turned off, the various standing waves will take different lengths
of time to die out. The oblique waves will damp out most quickly,
then the tangential waves (if the walls are not concave), and the axial
wave parallel to the most absorbent walls will linger nearly twice as
long as the oblique waves. Therefore, the distribution of sound in
the room will change as it dies out, being at first fairly random, but
eventually being mostly parallel to the absorbent walls. Experience
shows that this behavior, if the differences of reverberation time are
pronounced, is acoustically objectionable.
Therefore, if a rectangular room cannot be changed in shape by
introduction of irregularities, and if it is to be used for purposes of
speech or music, requiring good acoustics, the absorbing materials
should be placed about equally on all walls, so that a x , a y , and a z are
nearly equal. This will produce the least difference between rever
beration times for a given value of total absorption. Irregularities of
wall shape, to scatter the axial and tangential waves, would improve
the acoustical conditions considerably, however, and these should be
introduced whenever possible.
Decay Curve for Rectangular Room. — If the source in the room
emits sound in a frequency band dv (between v — \dv and v + \dv),
there will be excited, on the average, dN " & oblique waves, dN ta tan
gential waves, and dN ax axial waves, where the formulas for the
quantities dN are given in Eqs. (32.14). If each of these wave groups
is excited and contributes to the resulting average intensity by an
amount proportional to the dN's, we can write down a formula for
the average attenuation of sound in a rectangular room
T ~ ? D x (t)D v (t)D.(t)
where a x , a y , a z are defined in Eq. (33.5). To this approximation,
therefore, the sounddecay curve (the intensity level vs. time) for a
rectangular room can be expressed in terms of a sum of individual
terms, 10 log(D x ), etc. Each of these terms starts at t = as a straight
line with negative slope proportional to a x . A time (8V/ca x ) In (c/2vl x )
Vm.33] DAMPED VIBRATIONS, REVERBERATION 407
later the curve has a break, ending up, beyond this, as a straight line
with slope proportional to \a x , half the initial slope. Since each of
the additive terms has it's "break" at a different time (unless the room
is completely symmetrical), the resulting decay curve is quite far from
being the straight line that was indicated in Eq. (32.8) for a uniform
distribution of sound in an irregular room..
At high enough frequencies, the "breaks" come late enough so
that the first 20 or 30 db of the curve is nearly straight, with a slope
and indicated "reverberation time" corresponding to the oblique
waves. If, by chance, this result were assumed to correspond to that
given in Eq. (32.8) for an irregular room with diffuse sound, then the
quantities (8k) (where k is the wallconductivity ratio) would be
presumed to equal the absorption coefficient a. We have seen, how
ever, that this is an inaccurate correlation, which may work fairly
well for very stiff walls [k very small; see the comments on Eq. (32.10)]
but which fails for more absorptive walls, when the break in slope of
the decay curve is more pronounced.
If most of the absorbing material is concentrated on the two
opposite walls of a room, then those standing waves that do not
reflect from the absorbing walls will take about twice as long to die
out as do all other waves. When a sound with a "spread" of fre
quency is used to excite a number of standing waves at the same time,
the dying out of these waves after the source is shut off is a rather
complicated phenomenon. When only two or three standing waves
are excited, these waves as they die out may alternately reinforce and
interfere with each other, owing to the fact that they are of slightly
different frequency. The intensity will then fluctuate instead of
decreasing uniformly, the sort of fluctuations obtained depending on
the position of source and microphone in the room and on the manner of
starting the sound.
If more than three standing waves have been excited, these
fluctuations will be more or less averaged out, and the resulting
intensity will first diminish uniformly at a rate dependent on the
reverberation time of the standing waves which are reflected from all
six walls. After these waves have died out, the rest of the sound, due
to waves not striking the most absorbent walls, remains and dies out
more slowly. The intensity level as a function of time approximates a
broken line, the steeper initial part corresponding to the most of the
standing waves, and the less steep later part due to the waves that do
not strike some walls.
In such cases the term "reverberation time " has a specific meaning
408
STANDING WAVES OF SOUND
[VHI.33
only in connection with the damping out of single normal modes, i.e.,
in connection with the slopes of the two portions of the broken line of
intensity level against time. The actual length of time that it takes
for the intensity level to drop 60 db will depend on the relative amounts
of energy possessed by the rapidly and the slowly damped standing
waves.
Figure 91 illustrates these points; it shows curves of intensity level
in a room 10 by 15 by 30 ft as a function of the length of time after
the sound is shut off. The two smallest walls are supposed to be much
more absorbent than the other four. The solid line is the curve for
Time
Fig. 91. — The decay of sound in a room with two opposite walls more absorbent
than the rest. Solid curve shows the average decay of a large number of normal
modes; dotted line shows the interference effects possible when only two normal modes
have been excited.
intensity when the room has been excited by a tone with wide enough
frequency "spread" to excite 10 standing waves about equally, so
that a smooth decay results, and the "break" in the curve is apparent.
The dotted line is the curve when only two standing waves are excited,
the resulting fluctuation, due to interference, masking the exponential
decay.
These interference oscillations and breaks in the curve for decay of
intensity level are present even when all the walls are about equally
absorbent, but they are less pronounced.
Cylindrical Room. — We can use the characteristic functions given
in Eqs. (32.15) and (32.16) to compute the damping constants for the
different standing waves in a cylindrical room of radius a and height I.
If the absorbing material is distributed either at random or com
pletely uniformly over each wall, the quantity k can be averaged
Vm.33] DAMPED VIBRATIONS, REVERBERATION 409
separately as before. The wall behavior is then expressed in terms of
the absorptions.
a z = X(Sk s )A s (over flat floor and ceiling)
a r = 2(8k s )A s (over cylindrical surface)
The wave properties are evidenced through the averaged values of
p 2 over the walls and throughout the volume. The meansquare
value of the Bessel function is
Therefore the damping constant for the iVth wave (N = m,n,n z ) in
the cylindrical room is
*'^[***+ l ( i/„^, ] (337)
which is to be compared with Eq. (33.6). Values of a mn are given on
page 399.
The r, ^tangential waves (n z = 0) are absorbed half as much by
the flat floor and ceiling as are the waves for n z > 0, as is the case for
the flat walls in a rectangular room. The curved cylindrical walls
absorb most waves less effectively than do flat walls; for the factor
l/[2 — 2(m/7raw) 2 ] is smaller than unity unless n is small. Most
waves in a cylindrical room are focused toward the center, away from
the curved wall, and so are not affected as much by this wall as they
would if it were flat.
In contrast, the standing waves for which n = (the ^,2tangential
waves) are much more strongly absorbed than if the wall were flat;
for these waves l/[2  2(m/Tra mn ) 2 ] is much larger than unity. These
(p, ztangential waves move "parallel" to the curved wall and have
large amplitude only near this wall, so it is natural that it will absorb
them strongly.
When a sound source in a cylindrical room is shut off, the <p,z
tangential waves damp out most rapidly and the raxial waves least.
Therefore the sound nearest the curved walls vanishes first, eventually
leaving the radial wave motion, which focuses the sound along the
axis and which attenuates most slowly. This effect was mentioned on
page 399, where we discussed the classes and numbers of standing
waves in a cylindrical room.
Secondorder Approximation.— In the foregoing calculations, we
took into approximate account the effect of the flow of air into and out
410 STANDING WAVES OF SOUND [VIII.33
of the wall surface on the power absorption at the surface, assuming
that the standing wave shape is not changed by the yielding of the
walls. Actually, of course, the fact that the wall impedance is not
infinite does affect the shape of the standing waves, and the second
approximation to the solution must take this into account.
How this correction is to be calculated can be seen most easily
by assuming that we gradually increase the admittance of the walls,
starting from zero and ending at the actual distribution of admittance
which is under consideration. When the walls are all rigid, the stand
ing waves are given by the characteristic functions ^n given in Eq.
(33.1) or (32.15), no energy is absorbed by the walls, and the natural
frequencies are (o)n/2tt), with the characteristic values ca N given in
Eqs. (32.12) or (32.16). As the wall admittance is increased slightly
the shape of the characteristic functions is not changed very much at
first; there is a slight air velocity into the wall, of an amount equal to
the specific acoustic admittance of the wall (still small) times the
pressure at the wall surface. This motion of air produces radiation
out into the room; in other words the original standing wave is modified
by the scattered waves produced by the reaction of the wall to the
standing wave.
Suppose that we start with the standing wave \f/N(P)e~ iaNt [where
P stands for the point (x,y,z)]. When the wall admittance is not
zero, the air velocity out from the wall at the point Q = (X, Y,Z) on
the wall surface is — (l/pc)^w(Q)i?(Q), where rj = (pc/z) is the acoustic
admittance ratio (small) of the wall at the point Q. The element of
wall area dS Q at Q therefore acts like a simple source of sound, modify
ing the pressure at P by an amount [ivN^ N (Q)ii(Q) dS Q /ch]e i < u « /c '><* r . et \
where h is the distance between P and Q [see the discussion of (Eq.
28.1)]. Consequently, to the first order in the small quantity rj, the
modified characteristic function for the pressure at P = (x,y,z) is
walls
¥ W (P) ~ MP) + {£%) J J [ Y J ««•"•>»"> dS Q (33.8)
walls
This modified standing wave, vibrating within nonrigid walls,
does not vibrate with quite the same frequency as does the wave ip N
inside rigid walls. To find the new natural frequency we set the
expression of Eq. (33.8) into the wave equation. After a great deal
of involved integration, which does not need to be gone into here, we
find that the square of the corrected characteristic value is given
to the second order in the small quantity 17 by the expression
Vm.33] DAMPED VIBRATIONS, REVERBERATION 411
2^ 2 L ■ _ (ic\ average of [r)(P)ty N (P)\f/ N (P)] over all walls \
N \ \u>nJ average of [^(P)] over room volume )
(33.9)
where Vn(P) is the function given in Eq. (33.8).
To put these formulas into a form that can be calculated with
reasonable ease, we assume that the characteristic functions yj/ N have
the usual properties of orthogonality
fff+»(P)MP) dV p = { ° [* I $ ohuo)
room s '
where V is the volume of the room, N and M are trios of numbers
labeling different standing waves, and Aw is some dimensionless con
stant depending on the shape of the function $ N . In terms of these
functions, the outgoing wave in Eq. (33.8) is given in terms of a series
[a simplified form of Eq. (34.4)]
where h is the distance between the point P (which can be anywhere
in the room) and the point Q (which is somewhere on the wall). The
function ^ N (P) can also be expanded into a series of characteristic
functions. As long as 97 is everywhere small, the largest term in the
series will be i[/ N , the rest being correction terms:
where the functions
Y MN = Gmn — iBmn, 7] = K — l(T
Gmn ' k(Q)
= / / ^ (Q) *"( Q) ds<1
Bmn walls <r(Q)
are the transfer admittances of the walls, coupling the Nth and Mth
standing waves. They have the dimensions of an area, as do the
absorption constants a x , given in Eq. (33.5). The prime on the sum
mation symbol 2 indicates that the sum does not include the term
M = N, for this term has been written down separately as the first
and largest term.
412 STANDING WAVES OF SOUND [Vm.33
Inserting Eq. (33.11) into Eq. (33.9) gives us the natural fre
quency of the iVth wave. The time dependence of the free vibration
of the Nth wave is given, to the second order of approximation, by
the exponential
exp — icon —
cY NN . c 2 Y* NN
2FAtf SV 2 usA%
"^AS#^jJ f (33 ' 13)
The real part of the exponent gives the damping constant kx, to
the second order in the small quantities Y.
, cGnN , C 2 COjV GnnBnN , ^0' AGmnBmN /„q .is
The first term in this, of course, corresponds to the firstorder
expression given in Eq. (33.3). The second term is the correction due
to the distortion of the standing wave by the wall admittance.
The imaginary part of the exponent also carries a correction to the
unperturbed angular frequency un. The firstorder correction is
— (cBnn/2VAn). While the effect of the wall conductance (to the
first approximation) is to attenuate the free vibrations, the effect of
the wall susceptance is to change the frequency of free vibration. The
secondorder correction is of the same general form as that in Eq.
(33.14) for k N , except that the quantities i((? 2 — J5 2 ) appear instead
oiQB.
Scattering Effect of Absorbing Patches. — We can now begin to see
the effect of wall irregularity on the distribution of each standing wave.
As long as the admittances Ymn in Eq. (33.12) are small, the standing
wave ^n differs only slightly from the symmetrical wave ^r for smooth
rigid walls. But as the walls are made softer some of the F's increase
in size; and if the absorbing material is properly placed so that many
of the F's get large, each standing wave becomes a more or less random
mixture of a large number of ^'s, corresponding to wave motion in
many directions. If enough F's are large enough, each standing wave
is a random mixture of waves, all waves of about the same frequency
have about the same value of damping constant, and we arrive at
the uniform sounddistribution case discussed at the beginning of this
chapter.
Thus a placement of absorbing material so that as many Ymn's
as possible are as large as possible will cause the acoustic conditions to
Vin.34] FORCED VIBRATIONS 413
approach the uniform acousticdistribution case as closely as possible.
Uniform distribution of absorbing material over any wall will not pro
duce this effect, for if t)(Q) is constant over any wall (the yz wall for
instance) a part of Ymn will equal r\  I ^m(Q)^n(Q) dy dz, which is
x =
zero unless m y = n y and m z = n z , so that in this case many of the F's
would be zero.
A study of the integrals defining Ymn shows that, for as many of
them as possible to differ from zero, the absorbing material should be
placed on the walls in patches, of dimensions neither large nor small
compared with the wavelength, the patches being placed in a random
manner over all walls. In this way, the absorbing material will scatter
the waves as effectively as possible.
A study of the scattering effect of small modifications of shape of
the walls (which will not be computed here) shows that shape irregu
larities, if they are about the size of a wavelength and are irregularly
placed, are even more effective in scattering than are irregularities of
admittance (though, of course, they do not cause absorption). A
combination of "bumps" and patches of absorbing material, irregu
larly placed, produces the best acoustic effect.
34. FORCED VIBRATIONS
We are now in a position to calculate the excitation of the standing
waves in a room by a sound generator and thus obtain an idea of the
effect of the acoustic properties of the room on the quality of the sound
produced by the generator. We shall see that in a moderately absorb
ing room the shape of the room will have very little effect on the output
of the generated sound at high frequencies. The intensity will be
greater and will be more uniformly distributed over the volume than
is the case when the source is in free space; but the dependence of this
intensity on frequency will be the same, its value being simply a con
stant times the power output of the source for free space. This is due
to the fact that at high frequencies the energy is "carried" by a large
number of standing waves, all of nearly the same natural frequency,
so that the resulting effect is fairly uniform.
At low frequencies, however, the majority of the energy will be
carried by one or two standing waves, and large fluctuations in the
amplitude of these waves can occur as the frequency is changed. The
variation of output with frequency will depend more on the character
istics of the room than on those of the generator at low frequencies.
414 STANDING WAVES OF SOUND [VIII.34
Simple Analysis for High Frequencies.— The intensity of high
frequency sound due to a source of output power n can be obtained
in a very simple manner. If it is possible to assume that the intensity
of the sound T is uniform throughout the room, then the formulas
derived at the beginning of this chapter can be used. In particular, if
the source output II changes little during a reverberation time, Eq.
(32.7) can be used. Close to the source most of the sound is radiating
outward from the source, and the intensity varies with the inverse
square of the distance, as from a simple source. At larger distances
the outward radiation is lost beneath the randomly scattered waves,
which have more or less uniform intensity everywhere in the room.
If II is the power output of the source in watts, r the distance from the
source in feet, a the room absorption in square feet [see Eq. (32.4)],
then the intensity close to the source in watts per square centimeter is
(II/4,0007rr 2 ), where the factor 1,000 is approximately the number of
square centimeters in a square foot. For large values of r, the inten
sity is n/l,000a).
Intensity and Meansquare Pressure. — At this point we must call
a halt, to point out the difference between our definition of intensity
and the way sound "intensity" is usually measured. We have
defined intensity as sound power falling on one side of a square centi
meter of area. This can conceivably be measured, but in many cases
the result may depend on the orientation of the area. Close to the
source the intensity is all flowing outward, so that we must arrange
that the square centimeter be placed perpendicular to the radius r, if
our measurement is to equal (n/4,000jrr 2 ) ; if it were placed parallel tor
the intensity measured would not be at all as large. On the other
hand, throughout the rest of the room, according to our assumption,
the intensity flows equally in all directions, and the intensitymeasur
ing device need not have any special orientation to measure the
predicted amount.
In actual practice, sound intensity is rarely measured directly;
what is measured is meansquare pressure, as was mentioned in con
nection with Eq. (22.15) (p 2 ms = ipi 2 ) This quantity is simply
related to the average energy density w by the relation p 2 ^ = pc 2 w,
but it is not simply related to the sound intensity. If the intensity is
flowing in only one direction the relation is piL = pcT, but if it is
flowing equally in all directions the relation is p 2 ^ = 4pcT, as was
shown in Eq. (32.3). Consequently, the quantity to compute, which
can be checked directly with measurement, is the meansquare pres
sure, rather than the intensity, or else the pressure level (whieh is
20 log (prms) + 74 = 10 log (w) + 136) rather than the intensity level.
VHI.S4] FORCED VIBRATIONS 415
In terms of these quantities the statements made above become
C 10 log (n)  20 log (r) + 49 db (r 2 < a/50)
Pressure level c^ < (34.1)
( 10 log (n)  10 log (a) + 66 db (r 2 > a/50)
for the statistical case, where II is source power in watts, r the distance
from the source in feet, and a the room absorption in square feet.
The criterion for range of validity (r 2 vs. a/50) is obtained by equating
the two formulas. If the power II is measured in ergs per second, r in
centimeters, and a in square centimeters, the formulas for meansquare
pressure are
2 f (pcll/4irr 2 ) (r 2 < a/50)
Prms — ( (4 pc n/a) (r 2 > a/50)
a = ^a s A s
s
Close to the source, the first expression is larger and is used; far from
the source the second term, representing the random sound, is larger
and is valid.
If the sound generator is a simple source of strength Q Q , then Eq.
(27.4) shows that the power generated will be (po; 2 Q§/8xc). Conse
quently, over most of the volume of the room the meansquare pres
sure due to a simple source is
(pWQ$\
\ 2ira J
A   DsfV (34  2)
This equation is obtained here, for the case of the room with uniform
distribution of sound, so as to compare it with the expression we shall
obtain for a rectangular room, where the assumption of uniform dis
tribution is not valid.
Solution in Series of Characteristic Functions. — The preceding
equations are satisfactory for high frequencies, where the resonance
frequencies are close enough together for the response to be fairly uni
form, and for rooms of sufficiently irregular shape for the sound to be
uniformly spread over the room. It certainly will not be valid at low
frequencies, where the response is far from uniform. To determine
the range over which Eqs. (34.1) and (34.2) are valid, we must analyze
the coupling of the source with the individual standing waves.
For a given room, the characteristic functions, being the solutions
of the wave equation which satisfy the proper boundary conditions,
can be represented by the sequence yp N , where N represents a trio of
416 STANDING WAVES OF SOUND [VHI.34
integers and the ^'s satisfy the orthogonality condition given in Eq.
(33.10). The corresponding characteristic values can be written as
(o) N — ik N ). If the walls are rigid k N = 0, and if the walls have some
simple shape (rectangular, cylindrical, etc.) the values of ca N can be
computed, as has been done in Sec. 32. If the walls are not rigid, the
values of w N and kn can be obtained to the first approximation by Eqs.
(33.1), (33.3), (33.6), or (33.7), to the second approximation by Eqs.
(33.13) and (33.14). In any case the functions \f/ N are solutions of the
equation
2 Aw + A\vY ,
VVjv = I ) yp N
Any source of sound may be considered to be an assemblage of
simple sources, as was indicated in Eqs. (27.2) and (27.3) and as
was done in Eq. (28.1) in computing the radiation from a piston.
A source distribution qixa^e™' can be expressed as a series of
characteristic functions :
Qm = (?b) / // q{x',y',z')+ M (x',y',z') dx' dy' dz' ) (34 ' 3)
room
The steadystate distribution of sound in the room can likewise be
represented as a series XA m ^m, with the series satisfying Eq. (27.3),
which becomes
V A n [(i0)N + &JV) 2 + i0 2 ]\pM — lO)pC 2 ^ Qm$M
Solving this for the coefficients A, we finally obtain a series solution for
the steadystate pressure
p — pce ^ 2 ^ An + .^ _ w2) Mx,y 4 z) t<54.4)
where we have assumed that ## is small enough so that we can neglect
k% compared with a>%. This corresponds to the series of Eqs. (10.16) or
(11,6) for the string.
The equation shows that the steadystate pressure wave at a point
(x,y,z) is the sum of the waves corresponding to the different normal
modes of the room, each with amplitude proportional to the values of
Vm.34] FORCED VIBRATIONS 417
the standing wave at the source and at (x,y,z) and inversely propor
tional to the "impedance" of the standing wave:
(2o) N kif/o>) — i[u — (co^/co)].
Steadystate Response of a Room. — The energy density at any
point in the room is (p 2 /2pc 2 ), and the intensity, by Eq. (32.3), is
(p 2 /Spc) in those parts of the room where the energy is fairly uniformly
distributed (i.e., not too near the source). When k is small the only
functions yp n that have large coefficients in expression (34.7) are those
having allowed values of oo n nearly equal to 2rv (i.e., those standing
waves whose allowed frequency is almost equal to the frequency of
the source). Even these coefficients may be small if Jjjqrf/ n dv is small
(i.e., if the source of sound is located near & node of the nth standing
wave).
The only important terms in the expression for the intensity at a
point (x,y,z) are those for waves whose natural frequency is close to the
frequency of the source and which are so distributed in space that
neither the source nor the point (x,y,z) is at a node of the wave. At
low frequencies the allowed frequencies are spaced so widely apart that
for some frequencies no coefficients A M will be large, and for some
others only one may be large ; and this will be large only when the source
is put in particular places in the room. Only for large frequencies,
when several standing waves have frequencies near enough to that of
the source to be excited strongly, will the intensity be distributed more
or less uniformly over the room and will the intensity at any point
be uniform when the frequency of the source is changed. The
intensity close to the source will always be larger than that some
distance away, as a calculation by Eq. (34.4) will show; .but at high
frequencies the intensity throughout the rest of the room will be fairly
uniform.
As mentioned earlier the measurable quantity is not energy density
or intensity but meansquare pressure. This is obtained by squaring
the series for p, integrating over the room volume, and dividing by 2V.
In virtue of the integral properties of \J/ N , given in Eq. (33.10), this
becomes
Pma ~ 2V 2 ZA (2u> N k N /u) 2 + \(col/o)  u,) 2 ld4  D;
N
where Ajv is the average value of *//% over the room. If the largest
dimension of the sound source is less than a half wavelength, the inte
gral Jjjq^NdV becomes Qo^n(S), where Qo is the equivalent source
418 STANDING WAVES OF SOUND [VIIL34
strength (amplitude of total air outflow from source) and where
iPn(S) is the value of ^n at the position S of the source.
Rectangular Room. — If the room under consideration is a rec
tangular one of sides l x , l y , U, the characteristic functions are those
given in Eq. (32.11). To the first approximation the allowed values
of un are given in Eq. (33.1) and those of the damping constant k N are
given in Eq. (33.3) or (33.6). The pressure at the point P = (x,y,z)
due to a simple source Qoe'™* at S = (x ,yo,Zo) is then
_ PC 2 Qo iut ^ 6n,e w ,6n.^y(^)^(P) ,„, fi ,
P— V e 2j (2coaW") + *'[K/co)  «1
JV
where eo = 1, e 2 = e 3 = • • • =2. The meansquare pressure
throughout the room is then
Prms— 2 y 2 ^ (2u, N k N /u) 2 + [ K/co)  0>] 2 ^ ' '
N
The source factor E N (S) = e^e^e^i/^^) is unity if the source is
placed at random in the room (or is moved about in the room to average
out the effect of source position), but is e nx e ny e ni if the source is placed
at a corner of the room.
Transmission Response. — Figure 92 shows response curves for a
room 10 by 15 by 30 ft (the one used for Fig. 89), with a damping
constant for oblique waves of 1c n = 10 (assumed independent of
frequency) corresponding to a reverberation time of about 1 sec. The
source is located at one corner of the room, thereby making \[/n(S)
unity for all values of N and making the variation in energy depend
only on the impedance functions for the normal modes and on the
position of the observer. Curves for the energy density at the center
of the room, at the center of an opposite wall, and for the average
density are plotted as functions of v. We see that at low frequencies
the response is very nonuniform, both as to position and as to fre
quency, but that it becomes more uniform at higher frequencies.
These curves should be compared with Fig. 89, which gives the
number of allowed frequencies between v — 5 and v + 5 for the same
room. In Sec. 32 we did not include the damping of the standing
waves, and we had to assume that the frequency distribution of the
sound source was "spread out" in order to obtain a uniform response
even at high frequencies. For a spread of Av we found that the
response would be fairly uniform for frequencies larger than
\/c 3 /£tVAv [according to Eq. (32.14)]. In the present section we
see that absorption at the walls spreads out the response of each
VHI.34]
FORCED VIBRATIONS
419
standing wave so that, instead of responding "infinitely well" to just
one frequency and only moderately well to others, there will be a range
of frequencies over which the response will be about equally large.
We can show, from Eq. (34.7), that the range of frequency, over which
the response of a given standing wave is larger than onehalf its
maximum response, is between v n — \/3(A/27r) and v n + V 3(A/2r),
50 100 50
y v
Fig. 92. — Response curves for a room 15 by 30 by 10 ft with reverberation time
1 sec. Dotted lines correspond to approximate formula (34.12), which the actual
curves approach asymptotically.
so that the response has a "spread" of Sv = \/3(k/ir) c~ (k/2). This
means that if the frequency of the source is high enough so that the
average difference between the allowed frequencies of the standing
waves that resonate is less than (k/2), then several standing waves will
have large amplitude, and the response of the room will be uniform
even if the source sends out a single frequency.
Utilizing the fact that k ~ (7/T), where T is the reverberation
time, and using Eq. (32.14), we see that the general rule for the lower
420 STANDING WAVES OF SOUND [VHI.34
limit of uniform response is the following: If a source is sending out
energy over a frequency range from v to v + Av in a room of volume V
and reverberation time T, the response of the room will be fairly
uniform in frequency and in distribution over the room for all fre
quencies greater than v„n a , where
1 £f__V
\4a7 [A»+ (4/T)]/
iovvT (34  8)
for V in cubic feet
VAv + (4/7 1 )
For a pure tone (Av = 0), and for a reverberation time between 1 and
3 sec, this JWn is approximately equal to 10 4 / \/F. For an auditorium
as large as 50 by 100 by 200 ft this lower limit is 10 cps, so that no large
resonance fluctuations will occur in the useful frequency range (30 to
10,000 cycles). For a room as small as 10 by 12 by 20 ft, however, the
response will fluctuate considerably for frequencies below 200 cycles,
unless the spread in frequency of the source is as large as 100 cycles or
larger.
The preceding analysis, of course, does not take into account the
possible focusing effects of the walls. If some of the walls are smooth
and curved, a considerable concentration of the energy may occur in
some parts of the room, with detrimental effects on the uniformity of
response and the reverberation time of the room.
The Limiting Case of High Frequencies.— As a last example of
the method of standingwave analysis of sound in a room, we shall
show how the standingwave formula (34.7) for a simple source goes
over into the simple formula (34.2) when the frequency is high enough
for uniform response. In such a case the allowed frequencies are close
enough together so the quantity {(wwA^/tt^) 2 + [(<a%/%rv) — Zwv] 2 } 1 ,
when considered as a function of a) N , does not change much as u N
changes from one of its allowed values to the next. When this is true
we can change the summation of Eq. (34.7) into an integration over the
variable u = (a N . We shall have to separate the axial from the tan
gential from the oblique waves in performing the integration, since they
have different values of ku, even to the first order. The number of
such waves having values of a N between u and u + du can be obtained
from Eqs. (32.14), by letting u = 2rv. All these terms are obtained by
modifying the series of Eq. (34.7) from a series over N to a number of
integrals over dN, where the dN's for different types of waves are
given in Eqs. (32.14). The largest term of all comes from the chief
term (AttvW/c 3 ) dv = (u 2 V/2t 2 c s ) du in the expression for dN Q b and is
VIII.34]
FORCED VIBRATIONS
421
P 2 cQW
4tt 2
E (S)
!.
u 2 du
(2uko) 2 + (u 2  co 2 ) 2
(CJ >>> ^Vaan)
where 2£ (£) is unity if the source is not near a wall; is 2 if it is on a wall,
not near an edge; is 4 if it is on an edge, not near a corner; and is 8 if it
is near a corner ("near" is less than a wavelength). The quantity k Q
is given to the first order, according to Eq. (33.6), by the expression
^° ~ \w)' a ° = a * + ay + a * ~ 2 ^ k ^ a "
^ ' all
walls
where a is the room absorption for the oblique waves at the frequency
v (k may depend on v) and is the expression that comes closest to the
absorption a given in Eq. (32.4) for a room with uniform distribution
of sound.
The integrand written above is large only when u is nearly equal
to co (since k is usually much smaller than co). Therefore no great
error is made if we extend the lower limit of integration from to minus
infinity. If this is done, the integral can be readily performed.
/:
u 2 du
if.
dx
(2uk ) 2 + (w 2  a> 2 ) 2 — 4 J_ . k 2 + x 2
Therefore this largest term in the expression for w becomes
P 2 co 2 Q 2
7T
Wo
T^VE Eo{s)
2tOo
Eo(s)
which is to be compared with the expression for p 2 ^ given in Eqs. (34.2)
for the statistical case of uniform distribution of sound.
Approximate Formula for Response. — The other terms in the
integral, for the oblique, tangential, and axial waves, can be computed
in a similar manner. The result is, to the second order in (v mia /v),
J P f «H28
Pi.
B x =
2rrao
B X B V B*
^2L
Ox "f* dy 4" 0>z
■%0>x j CLy 4" Q>z
dx + CLy + a z
2 + —
2i x \j£a x + Oy + a z
etc.
(source not
near an xwall)
(source close
to an xwall)
(34.9)
— 2 (8K a )A B ; a = a x + a y + a z ; X = (2irc/co)
xwalls
etc.
422 STANDING WAVES OF SOUND [VIII.34
When the room dimensions are large compared with a wavelength,
this reduces to Eq. (34.2) which was derived for the case of uniform
sound distribution.
To recapitulate: at low frequencies, below jw ^ 10 4 s/T/kV, a
room is an irregular transmitter of sound, having alternate bands of
good and bad transmission, more or less irregularly placed depending
on the values of the resonance frequencies of the various standing
waves (which are widely spread on the average at low frequencies).
If any of the room dimensions is equal (l x = l y , etc.) or is a simple frac
tion of each other (l, = %l y , etc.), then multiple degeneracy will occur
and the irregularity at low frequency will be still more pronounced.
At frequencies above jw the individual resonance peaks merge
together over most of the frequency range and most of the irregularity
disappears. At high enough frequencies the meansquare pressure is
practically uniform over the room and equal to the value computed
statistically, given in Eq. (34.2).
The same sort of analysis 1 can be carried through for the pressure
wave itself, given by Eq. (34.6). The analysis is somewhat more
difficult than that for p^, owing to the fact that when (x,y,z) is
near the source the phase difference. between *[/ N (S) and \f/ N (x,y,z) must
be taken into account. The results are that when v is larger than v min
the amplitude of the pressure wave at a distance r fro m a sou rce in a
rectangular room is (j>vQ /2r) when r is less than V^o/loV and is
ap proxima tely constant, equal to 2pvQ y/v/a, when r is larger than
■x/ao/l&ir. (This holds when neither the source nor the point (x,y,z) is
very close to a wall.) Close to the source the wave behaves as if the
source were not in a room, being radiated outward in a form corre
sponding to that given by Eq. (27.4) for the pressure wave from a
simple source in free space. Far from the source the waves are uni
formly distributed and are traveling in all directions, though there is
everywhere a slight preponderance of waves coming from the source
and a slight diminution in amplitude as r is increased. The more
highly damped the room is, the greater will be the portion of the room
filled with radially outgoing waves, and the smaller will be the portion
having uniform density, as was indicated in Eq. (34.1). Of course, if'
the room has focusing walls, these results will be considerably altered.
When either source or observer is close to a wall, interference
effects will occur at certain wavelengths, owing to the reinforcing
1 See E. Fermi, Quantum Theory of Radiation, Reviews of Modern Physics,
4, pp. lOOjf. (1932), for a discussion of the methods used, details of the calculation
of the integrals involved, etc.
vm.34] FORCED VIBRATIONS 423
effect of the reflected wave. When the source is sending out sound of
frequency v (larger than v min ), the meansquare pressure a small
distance 8 from a wall will be proportional to cos 2 (2rv8/c), unless the
walls are very highly absorbent. The phenomenon is the analogue of
the optical effect utilized in the Lippmann process of color photography.
Exact Solution. — All the calculations we have made so far have
been by the use of approximate solutions of the boundary condi
tions. In some cases these first (or second) order approximations are
sufficiently accurate ; but if the walladmittance ratios get large (magni
tude larger than about 0.2) the approximate solutions lose their
validity, and the formulas given in this and the previous section may
be several hundred per cent in error. In most cases, when this is
true, the problem cannot be computed, but in a few cases, where the
wall admittance is constant over each wall, an exact solution can be
obtained.
As an example of such a case, and of the exact solution that can
be obtained, we consider a rectangular room of dimensions l x , l y , l z
(origin of coordinates at corner of room) with the wall at x = l x having
an acoustic impedance [acousticadmittance ratio (pc/z) = t\ = k. — ia]
constant over the whole wall and with the other five walls perfectly
rigid. This is a somewhat idealized version of a small sound room set
up to. measure the absorbing properties of acoustic material. The
material is sometimes placed on one wall, the other walls being bare
concrete.
The general method of solution has already been discussed in Sec.
31. We set
(34.10)
N =? n,n y ,n z
The boundary condition at x = l x is that p = zu x , or
tanh(7T0„) = iiP^b X = (2rc/co) (34.11)
The equation is similar to that of Eq. (31.1) or (31.6) and can be
solved graphically by means of one part of Plate V, at the back of
the book. Note, however, that in this case (five walls rigid, one wall
soft) h = (2i?Z/X), an extra factor of 2. The solution is complex,
g = £ + i\i We note that the whole set of functions ^n depends on
424 STANDING WAVES OF SOUND [VIII.34
the driving frequency {w/%r) through the complex quantity g n . Con
sequently, these characteristic functions are useful primarily for expres
sing the steadystate driven motion of air in the room in terms of a
series expansion. From the steadystate response we compute the
transient behavior by the operationalcalculus methods.
The equation for the forced motion due to a simple source at
S = £o,2/o,Zo is, from Eq. (27.3),
where P = x,y,z and An is the average value of \[>% over the room
volume. The calculation of the series expansion for p is carried
through in the same manner as for Eq. (34.6) . The result is
V " V ^J A N (co 2  un)
As we have mentioned, the constant un is now a complex quantity.
The real part co% corresponds to the resonance frequency, and the
imaginary part k% corresponds to the damping constant. If we can
neglect (k° N ) 2 compared with (<4) 2 , as we usually can, the quantity
wj, can be written as «) 2  2ua%K But from the definition of w 2 .
given in Eq. (34.10), using the notation g N = (fw + iiur (where £ and n
can be determined from Plate V), we can write
co  a**  w [(?y + fey + fey  <*^)]
or
7TC
V I /._ I \ I... I \ i,„ I I
(34.12)
/^■V2„„t„\ v
n w
vey+ey+ey
o ~ , / V 2 c v,vgA
The resulting series for p and for the meansquare pressure (when the
source is placed at random away from the walls) is
PW ~ V 2± An o) 2  « ~ **&) 5 ,
* , \ (34.13)
Pmu, — 2F 2 ^ (2<^) 2 + [«) 2  0> 2 ] 2
We note that since $n and /in, solutions of Eq. (34.11), are functions
Vin.34] FORCED VIBRATIONS 425
of w, all the functions rpx and all the quantities &$ and k% are functions
of the driving frequency (w/27r).
The expression for pL, is quite similar to that of Eq. (34.7), and
the discussion of its behavior would follow the same lines as that given
earlier. The difference is in the value of k%, which is now
~w~)\w)\~irr
where \ N = (2irc/w° N ) and A a = l y l z , the area of the wall that is covered
with acoustic material. Equation (33.6) would give k^ as
(»)
(4e w /c s J,
to the first order in the conductance ratio k s , for the case under consid
eration. Detailed examination of the solutions of Eq. (34.11) shows
that the quantity
a „ sW = (4*w^. n = (e^ (34 . 14)
is approximately equal to (ie„/c s ) when k is small (0.1 or less), but is a
much more complicated function of k and of <o than this when k is not
small.
The Wall Coefficients.— The quantity <x% N is the closest analogue
to the absorption coefficient for the acoustic material that one can have
in a rectangular room with at least one "soft" wall. It has been
called the wall coefficient [see Eq. (33.4)]. We notice that it is not only
dependent on the acoustic impedance of the material (as is the a for a
room with uniform sound distribution), but also depends on the size
of the room and on the particular standing wave that is being damped.
For hard walls (k small) the variation with room size becomes negli
gible, and the only dependence on the type of wave is in the difference
between tangential and oblique classes. For larger values of k the
quantity a N (u>) may change appreciably from one value of N to
the next (and it can be shown to depend also on the impedance of the
opposite wall). In other words, each standing wave has a different
reverberation time; and the decay curve for a sound involving several
standing waves may be far from a straight line.
At present our analysis is for the steadystate, driven motion of
the room, so that we cannot yet say that k% is the damping constant
for the reverberation. As we shall see shortly, the damping constant
426 STANDING WAVES OF SOUND [VIII.34
(and the related wall coefficient a) for the free vibration of a standing
wave is obtained by setting w equal to <  ik° N in solving for £ y and
hk. Since < also depends on co, this means a series of successive
approximation calculations to find the correct values of k and a to use
for the transient decay rate.
Transient Calculations, Impulse Excitation. — To show how the
free vibrations of the room can be computed by contour integral
evaluation from Eqs. (34.13) we shall take as a first example the
response to a sudden explosion of unit strength at the point S = x ,yo,z.
This corresponds to a simple source at S of strength 8(t), where 5 is
the impulse function defined in Eq. (6.11). By an obvious generaliza
tion of Eqs. (6.16), the resulting pressure distribution in the room is
Pi(0
(« < 0)
where p(co) is the series given in Eq. (34.13).
Finding the locations of the poles of each term in the series for p(u)
is rather complicated. Both < and k N vary as co is varied (since £
and n depend on co) ; in fact, k% changes sign when co becomes negative
(since £ changes sign). However, in principle, a value of «°' and k%
can be obtained for every value of co, even for complex values. The
position of a pole is found by varying co over the complex plane until
it equals <a%  ik% for that same value of co. The value of co at that
point can be called <  ik r N , the corresponding values of $ and v can
be called & and » r N , and that of a° xN can be labeled just a xN .
The other pole turns out to be at  a r N  ik r N , as would be expected
from the symmetry of the equations. The corresponding values of
£, fi, and a are just — &, n N , and <x x n. # _
The characteristic functions $», which are complex quantities, are
also functions of co. Their values at the poles can be represented by
the following symbols:
For a, = o> N  iky, mp) = r K (P)*"™; A * = W**
For co = <  ik N ; MP) = MP)e^ p) ; A* = A^r^
where ¥ N is a real function of x, y, z, the magnitude of the pressure
wave; and <p N , the phase lag of the JVth wave, also depends on x, y, z.
The average value of $\ also has its magnitude and phase.
We can now compute the residues about the poles of the function
p(co). The resulting expression for the "impulse wave" is
VHI.84] FORCED VIBRATIONS 427
^.j^&cop^,. (<<0) (34 , 5)
\ • cos [tf„t  <p' N (P)  ^(Q) + ^] ( t > 0)
The individual standing waves damp out exponentially, each with the
damping constant k r N , given in terms of the wall coefficients <x xN by
Eq. (34.14), where the parameters n r N , & are computed for the reso
nance frequency « = «£  iV N , as mentioned above. This series
does not converge, corresponding to the fact that the excitation cor
responds to an impulse function. But for a physically possible fluctua
tion of source strength Q(t), the corresponding pressure wave
P = jl „ Q(j)p»(t  r) dr (34.16)
analogous to Eq. (6.17) is given by a series that does converge. This
integral is the counterpart, for the exact solution, of Eq. (32.6) for
the statistical solution.
Exact Solution for Reverberation.— As a last example we shall
compute the response of the rectangular room, with the wall at x = l x
covered with material of admittance ratio v = K  ia and the other
five walls rigid, for a simple source of strength <? sin (2™*) from
t oo to t = 0, and of strength zero from t = to t = oo (j e
the source, of frequency v, is shut off at t = 0). This can be computed
by the use of Eq. (34.16), or it can be computed by obtaining the
Laplace transform for the source function and calculating another
contour integral for the pressure. We choose the latter method to
provide one further example of its use. '
We first compute the steadystate pressure wave in the room if the
source is not shut off at t = 0. We calculate the parameters * and M
for the frequency v and from them compute the steadystate functions
t or positive values of %rv : For negative values of 2™ :
<  iK = u f N e ia »° < + ik° N = «/e"fc
MP)  f»(?)^(« MP) = r N (P)e i ^°^
A* = A» e*** An = A >i**°
Then, using the series for p(„) given in Eq. (34.13), we have that the
steadystate wave due to a source Q sin (2wvt) is
428 STANDING WAVES OF SOUND [Vm.U
V 2irV 2i A%G%{2nrv)'
N
p.(2r,0 = ^ real part of ^w ^ _ (2 ^ )2 _ (< _ ^ )t j
cos[2*v*  <p%{P)  <p%(S) + ^ + r^  (34.17)
where
[G° N (2irv)f = (ZkvY + K) 4  2(2ttv) 2 (co/) 2 cos (2Q&)
and
«> = [s^)] sinW)
Next we compute the pressure oscillations in the room for a source
that is zero for t < and Qo sin (2irv0 for t> 0. To do this we need
the Laplace transform [see Eq. (6.19)] for this source function
X27TV
sin (2rrf)«r* dt = ^^ + g2
Then the resulting pressure is, using Eq. (34.13) again,
p u (t) = ^ I *(iw)p(w) dco
oc 2 Qo r  7 _• * "^ * w $n(PWn(Q)
= ""V" ^ ,2l JW[co 2  (2™) 2 ][co 2  (co»  t/#)*J
J * AT
The residues about the poles a> = ±2irv result in the function
p s (2rv) of Eq. (34.17); whereas the residues about the poles
w = ± W £  tA* [by a calculation similar to that for Eq. 34.15)]
give — p r (t), where, for t > 0,
m _ P c2 £° >? M p )^( S
Vrit) ~~ ir 2a A^(2x.)
pc 2 Qo ^ + r N ( P)MS) p  kN rt.
• cosK*  <P r N (P)  <&(&) '+ *Sr + r r J (34.18)
where
[G* N (%tv)Y = (2rv)» + K  ik r N \ 4 ~ 2(2rv)*K ~ **&!' cos (2^)
and
and
VHI.34] FORCED VIBRATIONS 429
Finally, the effect due to a source that has been of strength
Qo sin (2rrvt) from t = — <*> to t = 0, and is turned off at t = 0, must
be the difference between the steadystate response ps(2irv) and the
expression p s (?Trv) — Pr(t) which represents the effect if the source is
zero before t = and is turned on at t = 0. Consequently, the pres
sure wave in the room due to a source that has been on for a long time
and is turned off at t = is
/ p.(2rv) (t < 0) [see Eq. (34.17)]
V \ Pr(t) (t > 0) [see Eq. (34.18)]
The steadystate response p 8 and the reverberation p r are similar
series. There are differences in the values of the amplitudes of the
terms, represented by the superscripts and r, owing to the fact that
the driving frequency (2rv) is used in p s for the calculation of £# and
hn (and thus w%, k%, \f/%, etc.), whereas the "natural frequency"
(<a N — ik r N ) is used in p r for the calculation of £ and p. The change in
amplitude is not large for those standing waves having resonances near
the driving frequency (a> N nearly equal to 2irv).
The major difference between the series is in the exponential e~ klfH
and in the terms 2irvt or w N t in the cosine terms. The reverberation
series p r damps out in time, whereas the steadystate pressure p 8 has
a constant amplitude (as, of course, it must). In addition, the indi
vidual standing waves in the reverberation each oscillate with their
own natural frequency co^, whereas every term in the steadystate
solution oscillates with the driving frequency (2rv). Since there are
many natural frequencies u r N with nearly the same value, complicated
"beatnote" effects often occur in the reverberation (see Fig. 91).
It is a far cry from auditorium acoustics to second quantization;
yet the methods used above to analyze the behavior of sound in rooms
are quite similar to the methods Dirac used to predict the existence of
the positron three years before its experimental discovery. This
similarity is an interesting example of the unifying influence of theo
retical physics.
Problems
1. A rectangular office room is 15 ft high, 20 ft wide, and 30 ft long. The
walls are of plaster, wood, and glass, with absorption coefficient 0.03. The floor is
covered with a carpet (a = 0.2) and the ceiling with acoustic material (5 = 0.4).
Ordinarily, six persons are present in the room. What is the reverberation time?
If four typewriters are going in the room, each producing 1 erg of sound energy per
second, what will be the intensity level in the room?
430 STANDING WAVES OF SOUND [VIII.34
2. An auditorium is 30 ft high, 50 ft wide, and 100 ft long and contains 500
wooden seats (a = 0.15 apiece). The walls, ceiling, and floor have an absorption
coefficient of 0.03. What is the reverberation time when the auditorium is empty?
When it is full? How much acoustic material (a = 0.4) must be placed on the
ceiling and walls in order that the reverberation time may be 2 sec when the audi
torium is empty? What is then the reverberation time when the room is filled?
What must then be the power output of a publicaddress system in order to have
the average intensity level in the filled auditorium 90 db? What will be the
intensity when the auditorium is empty?
3. Calculate and plot the absorption coefficient, as a function of frequency of
the material given in Prob. 23, Chap. VII, for both values of spacing from the wall,
for < v < 2,000 cps.
4. A room with cylindrical walls, of radius 5 m, has a flat floor and ceiling,
4 m apart. Plot the number of allowed frequencies in the room between v and
v + 5 as a function of v from v = to v = 50. Above what frequency will this
curve become fairly uniform?
5. A rectangular corridor is 2 m wide, 3 m high, and 10 m long. Plot the
number of allowed frequencies in the enclosure between v and v + 5 as a function
of v from v = to v — 100. Above what frequency will this curve be fairly
uniform?
6. A cubical room 5 m on a side has an average absorption coefficient for floor
and ceiling of 0.2; for the walls, a value of 0.04. What is the reverberation time
for those waves which strike floor and ceiling? For those waves which do not
strike floor and ceiling? List all the allowed frequencies between zero and 100 cps
and give the position of the nodal planes and the reverberation times of each
corresponding standing wave.
7. List frequencies, nodes, and reverberation times of the normal modes
between v = and v = 100 for the room of Prob. 6 when walls, floor, and ceiling
all have an average absorption coefficient of 0.1.
8. The air in the room of Prob. 6 is started into vibration so that all the
normal modes between v = 98 and v = 102 are set into motion with equal initial
amplitudes. What normal modes are excited? Plot the decay curve of intensity
level against time after the source is shut off at the midpoint of the room ; at the
midpoint of one wall; at a point 167 cm out from two walls and 250 cm up from
the floor.
9. A room has constants l v = l z = 500, l x = 505, a x = a y = 0.02, a z = 0.2.
All the allowed frequencies between v = 81 and v = 83 are excited with equal
amplitude. Plot the decay of intensity level at the midpoint of the room, at the
midpoint of the floor, at a point in one corner of the room halfway between floor
and ceiling (assume that at this point the pressure due to the different standing
waves are in phase when the sound shuts off). Will the decay curve be the same
for the corresponding points in the other three corners?
10. The allowed frequencies of the room, discussed in Prob. 9, between v — 66
and v = 68 are excited with equal amplitude and are in phase at the midpoint of
the room at the instant when the source is shut off. Plot the decay curves for the
midpoint of the room; for the midpoint of one of the walls perpendicular to the
zaxis; for a point in one corner between walls, halfway between floor and ceiling.
Will the decay curve be the same for the corresponding points in the other three
corners?
Vin.34] FORCED VIBRATIONS 431
11. The room of Prob. 6 has a point source of strength Qo ■» 10 located at its
midpoint. Plot the average energy density in the room as a function of v from
v ■= to v = 100. The source is stopped at the instant that its source function
q is a maximum, when its frequency is 100 cps. Plot the decay curve of sound
at the midpo'nt of one wall.
12 The point source of Prob. 11 is relocated at the midpoint of one of the
walls. Plot the energy density as a function of v from v = to v = 100 at the
midpoint of the room; at the midpoint of the wall opposite the source.
13. A point source of source function
q = 15 sin(40rf)  10 sin (120x0 + 3 sin(200ir*)
is placed at the midpoint of one of the walls of the room of Prob. 6. Plot the
curves of pressure against time for one cycle of the sound for the midpoint of the
room; for the midpoint of the opposite wall; for one of the corners of the room
farthest from the source.
14. Discuss the reverberating qualities of a cubical room 5 m on a side, having
two opposite walls with an absorption coefficient 0.8, the other walls with a coeffi
cient 0.05. Find the "average reverberation time" from Eq. (33.6), and find the
reverberation time for the lowest five standing waves by use of Eqs. (34.12).
15. Calculate the exact values (by Plate V) of the damping constants and the
corresponding reverberation times for the lowest five standing waves in a room
8 by 12 by 16 ft with acoustic material on one of the 8 by 12 walls and with the
other five walls rigid. The specific acoustic impedance of the "soft" wall is
3pce i(ir/6) (z = 3pc, <p =» —30°) for the frequencies of these five standing waves.
BIBLIOGRAPHY
This is not intended as an exhaustive classification of all the books on vibrations
and sound; it is simply a list of the books that the writer has found particularly
useful as collateral reading or as reference works giving particular problems in
greater detail than can be given in the present volume.
A good book on the philosophy of the scientific method, with many examples
in mechanics and theory of vibrations, is "Foundations of Physics," by Lindsay
and Margenau. A good general text on theoretical physics is Slater and Frank
"Introduction to Theoretical Physics." Reference works on theoretical physics
are Byerly, "Fourier Series and Spherical Harmonics"; Bateman, "Partial Dif
ferential Equations of Mathematical Physics"; and Webster "Partial Differential
Equations of Mathematical Physics." Reference works on the type of mathe
matics used in this book are Whittaker and Watson, "Modern Analysis"; Courant
and Hilbert, " Mathematische Physik"; and Watson, "Theory of Bessel Func
tions." A good set of tables of the functions used, more complete than those given
below, is in JahnkeEmde, "Tables of Functions."
In the field of vibrations and sound the standard reference works are Rayleigh,
"Theory of Sound"; and Helmholtz, "Sensations of Tone." Useful books for
collateral reading are Lamb, "Dynamical Theory of Sound"; Richardson, "Sound";
Watson, "Sound"; Miller, "Science of Musical Sound"; Crandall "Vibrating
Systems and Sound"; Trendelenberg, "Klange und Gerausche"; Stewart and
Lindsay, "Acoustics"; and Bergmann and Hatfield, "Ultrasonics."
Especially useful are Olsen and Massa, "Applied Acoustics," for its discussion
of recent practical applications; Fletcher, "Speech and Hearing"; and Stevens and
Davis, "Hearing, Its Psychology and Physiology," for their discussion of the
physicophysiological aspect of acoustics.
In the field of architectural acoustics the classical reference is the "Collected
Papers" of W. C. Sabine. Books for collateral reading are Knudsen, "Archi
tectural Acoustics"; Knudsen's article in the Reviews of Modern Physics, 6, 1 (1934) ;
Watson, "Acoustics of Buildings"; and P. E. Sabine, "Acoustics and Architecture."'
Of course, the issues of the Journal of the Acoustical Society of America should
be consulted. Two articles in the Reviews of Modern Physics, one by V. O. Knud
sen on page 1, Vol. 6 (1933) and one by Morse and Bolt on page 69, Vol. 16 (1944),
have material of interest in architectural acoustics.
433
34
320
GLOSSARY OF SYMBOLS
The more commonly used symbols are listed below, together with their usual
meanings and a reference to the page where the symbol is denned (if a definition is
needed) . Only those symbols used in several sections are listed.
SymbGl , Meaning p
a 3 Radius of cylinder or circular diaphragm, in cm 298
I Absorption constant of a room, in sq ft or sq cm 3g 5
A 5 Amplitude of oscillation, in cm 9
} Area of room in sq ft 38 _
B \ Mechanical susceptance in sec per g on
} Magnetic induction, in gauss 34
c Wave velocity, in cm per sec 73 222
C Capacitance in farads '
C m Radiation amplitude for a cylinder 301
du Piezoelectric constant
D Ratio mechanical force to current, in dynes per amp
D m Radiation amplitude for a sphere
1 J~ 2  71828 * * • , base of natural logarithms
A Emf, in volts
< General function
} Fourier transform of force
F Force, in dynes
G Mechanical conductance, in sec per g ,.
h Half thickness of a plate Jj?
H Specific response function for a diaphragm
* " V ?F I \ . To corre spond to the usual electricalengineering nota
tion, the x in this book is equal to minus j ■, ,
/ Current, in amps ^
/« Hyperbolic Bessel function .rjj
3 = ~i See note on i
jm Spherical Bessel function J.Z
J m Cylindrical Bessel function ^Jl
k (Damping constant (R/2m) „_}!!
I Wave number („/c) = (2x/X) = (2™/c) ' "*
A Stiffness constant of spring <T
I Length of string or tube, in cm ?*
ln(x) Natural logarithm of x
log (a;) Logarithm to base 10 of x
L Inductance, in henrys
«C Symbol representing Laplace transform ?
Mass, in g bZ
201
M Reactance function for piston in plane wall J
435
436 GLOSSARY OF SYMBOLS
Symbol Meaning Page
m,n Integers labeling normal modes of vibration 84, 390
n m Spherical Neumann function 317
N m Cylindrical Neumann function 196
p Excess pressure in sound wave, in dynes per sq cm 218
j Power, in ergs per sec 26
I Pressure amplitude, in dynes per sq cm 224
P n Legendre function 315
q Source density, in sec 1 313
!= (o} m/R) (The "Q" of a circuit or other system) 25, 403
Modulus of elasticity, in dynes per sq cm 152
Strength of simple source, in cc per sec 313
r Radial distance, in cm 311
S Electrical resistance, in ohms 36
I Mechanical resistance, in g per sec 24
{Elastic modulus of crystal 40
Poisson's ratio 209
S Area of cross section, in sq cm 217
t Time, in sec 2
I Tension, in dynes m 72
( Reverberation time, in sec ' 387
u Unit step function , 47
u,v,w Components of velocity, in cm per sec
U Velocity amplitude, in cm per sec 23
( Potential energy, in ergs 3
I Volume, in cc or cu ft 394
w Energy density in sound wave, in ergs per cc 383
W Energy, in ergs 2
x,y,z Rectangular coordinates
( Stress in crystal, in dynes per sq cm 39
] Mechanical reactance, in g per sec 29
Displacement of string 71
Y Mechanical admittance, in sec per g 35
z Specific acoustic impedance (p/u) in g per sq cm per sec 237
( Electrical impedance, in ohms 36, 38
I Mechanical impedance, in g per sec 29, 38
a Absorption coefficient 385, 403
a,/3 Acoustic impedance parameters for plane wave 139, 240
a^ n ,(3„ Characteristic numbers for Bessel functions 189,210,399
ey\ Wavelength constant for bar and plate 155, 209
& e Ratio of specific heats = 1.4 for air 221
y m Phase shift for scattered cylindrical wave 301
r Electromechanical coupling constant 35
5 Unit impulse function 47
/ Change in density of air due to sound wave 218
S m Phase shift for scattered spherical wave 320
A Separation of plates in condenser microphone 204
V 2 Laplace operator, Laplacian 174, 296
s Mass per unit length of string, in g per cm 72
X
y
v
GLOSSARY OF SYMBOLS 437
Symbol Meaning
«o =* 1, «i = 62 •= es =» • • • = 2 402
f Acoustic impedance ratio = (z/pc) = — ix 144, 240
t\ (Displacement of membrane, in cm 173
\ Acoustic admittance ratio = (pc/z) = k — ia 144, 240
( Phase angle 31
I Angle in spherical coordinates 311
B Acoustic resistance ratio 135, 240
( Dielectric susceptibility 40
} Acoustic conductance ratio 144, 240
X Wavelength, in cm 83
m = (wa/c) = ka 320
v Frequency, in cps 6
j Strain in crystal, in cm per cm length 38
I Displacement of air in sound wave, in cm 218
7t = 3.14159 • • •
(Electric polarization 39
( Power radiated, in ergs per sec or in watts 229
\ Density 154
p / Density of air at standard conditions 218
pc Characteristic impedance of air for sound = 42 g/cm ! sec 222
! Piezoelectric coupling constant, in ohms per g 40
Mass per unit area, in g per sq cm 173
Acoustic susceptance ratio 144, 240
2 Summation symbol 85
\ Ratio propagation velocity to sound velocity c 307
I Transmission coefficient 273
T Intensity of sound, in ergs per sec per sq cm 223
(Phase angle 31
(Angle in spherical or polar coordinates 296, 311
4> Angle in cylindrical coordinates 296
\ Laplace transform of / 52
( Angle of incidence 367
X Acoustic reactance ratio 135, 240
^ Complex reflection phase for plane wave = ir(a — i0) 239
SP Characteristic function
<*> = 2irv, angular velocity 21
Q Mechanical phase angle 201
Cii Approximately equal to
oo Infinity
+. Approaches as a limit
> Is greater than
< Is less than
J> Contour integral 12
Z Magnitude of Z = IP + X 2 10
Capital subscripts for Z,R,X, etc., denote electric impedance in ohms, lower
case subscripts denote mechanical impedance in grams per second, Greek sub
scripts denote analogous impedances (see pages 38 and 237).
438
TABLES OF FUNCTIONS
Table I. — Trigonometric and Hyperbolic Functions
(See pages 4 and 136)
X
sin (x)
COS (x)
tan (a;)
sinh (x)
cosh (x)
tanh {x)
e*
e~ x
0.0
0.0000
1.0000
0.0000
0.0000
1.0000
0.0000
1.0000
1.0000
0.2
0.1987
0.9801
0.2127
0.2013
1.0201
0.1974
1.2214
0.8187
0.4
0.3894
0.9211
0.4228
0.4018
1.0811
0.3799
1.4918
0.6703
0.6
0.5646
0.8253
0.6841
0.6367
1.1855
05370
1.8221
0.5488
0.8
0.7174
0.6967
1 . 0296
0.8881
1.3374
0.6640
2.2255
0.4493
1.0
0.8415
. 5403
1 . 5574
1.1752
1.5431
0.7616
2.7183
0.3679
1.2
0.9320
0.3624
2 . 5722
1 . 5095
1.8106
. 8337
3.3201
0.3012
1.4
0.9854
+0.1700
+ 5.7979
1.9043
2.1509
0.8854
4.0552
0.2466
1.6
0.9996
0.0292
34.233
2.3756
2.5775
0.9217
4.9530
0.2019
1.8
0.9738
0.2272
4.2863
2.9422
3.1075
. 9468
6.0496
0.1553
2.0
0.9093
0.4161
2.1850
3.6269
3.7622
0.9640
7.3891
0.1353
2.2
0.8085
0.5885
1.3738
4.4571
4.5679
0.9757
9.0250
0.1108
2.4
0.6755
0.7374
0.9160
5.4662
5.5569
0.9837
11.023
0.0907
2.6
0.5155
0.8569
0.6016
6.6947
6.7690
0.9890
13.464
0.0742
2.8
0.3350
0.9422
0.3555
8.1919
8.2527
0.9926
16.445
0.0608
3.0
+0.1411
0.9900
0.1425
10.018
10.068
0.9951
20.086
0.0498
3.2
0.0584
0.9983
+ 0.0585
12 . 246
12 . 287
0.9967
24.533
0.0407
3.4
0.2555
0.9668
0.2643
14.965
14.999
0.9978
29.964
0.0333
3.6
0.4425
0.8968
0.4935
18.285 .
18.313
0.9985
36 . 598
0.0273
3.8
0.6119
0.7910
0.7736
22.339
22.362
0.9990
44.701
0.0223
4.0
0.7568
0.6536
1.1578
27.290
27.308
0.9993
54 . 598
0.0183
4.2
0.8716
0.4903
1.7778
33.335
33.351
. 9996
66.686
0.0150
4.4
0.9516
0.3073
3.0963
40.719
40.732
0.9997
81.451
0.0123
4.6
0.9937
0.1122
+8.8602
49.737
49.747
0.9998
99.484
0.0100
4.8
0.9962
+0.0875
11.385
60.751
60.759
0.9999
121.51
0.0082
5.0
0.9589
0.2837
3.3805
74 . 203
74.210
0.9999
148.41
0.0067
5.2
0.8835
0.4685
 1 . 8856
90.633
90.639
0.9999
181.27
0.0055
5.4
0.7728
0.6347
1.2175
110.70
110.71
1 . 0000
221.41
0.0045
5.6
0.6313
0.7756
0.8139
135.21
135.22
1.0000
270.43
0.0037
5.8
0.4646
0.8855
0.5247
165.15
165.15
1 . 0000
330.30
0.0030
6.0
0.2794
0.9602
0.2910
201.71
201.71
1 . 0000
403.43
0.0025
6.2
0.0831
. 9965
0.0834
246.37
246.37
1 . 0000
492.75
0.0020
6.4
+0.1165
0.9932
+0.1173
300.92
300.92
1.0000
601 . 85
0.0016
6.6
0.3115
0.9502
0.3279
367.55
367 . 55
1.0000
735.10
0.0013
6.8
0.4941
0.8694
0.5683
448.92
448.92
1 . 0000
897 . 85
0.0011
7.0
0.6570
0.7539
0.8714
548.32
548.32
1 . 0000
1096.6
0.0009
7.2
0.7937
0.6084
1 . 3046
662 . 72
662 . 72
1.0000
1339.4
0.0007
7.4
0.8987
0.4385
2.0493
817.99
817.99
1 . 0000 "
1636.0
0.0006
76
0.9679
0.2513
3 . 8523
999.10
999.10
1 . 0000
1998.2
0.0005
7". 8
0.9985
+0.0540
+ 18.507
1220.3
1220.3
1.0000
2440.6
0.0004
8.0
0.9894
0.1455
6.7997
1490.5
1490.5
1.0000
2981.0
0.0003
TABLES OF FUNCTIONS
439:
Table II. — Trigonometric and Hyperbolic Functions
(See pages 4 and 136)
X
sin (xas)
cos (kx)
tan (trx)
sinh (irx)
cosh (irx)
tanh (rx)
e xx
e rx
0.00
0.0000
1.0000
0.0000
0.0000
1.0000
0.0000
1.0000
1.0000
0.05
0.1564
0.9877
0.1584
0.1577
1.0124
0.1558
1.1701
0.8546
0.10
0.3090
0.9511
0.3249
0.3194
1.0498
0.3042
1.3691
0.7304
0.15
0.4540
0.8910
0.5095
0.4889
1.1131
0.4392
1.6019
0.6242
0.20
0.5878
0.8090
0.7265
0.6705
1.2040
0.5569
1.8745
0.5335
0.25
0.7071
0.7071
1.0000
0.8687
1.3246
0.6558
2.1933
0.4559
0.30
0.8090
0.5878
1.3764
1.0883
1.4780
0.7363
2 . 5663
0.3897
0.35
0.8910
0.4540
1.9626
1.3349
1.6679
0.8003
3.0028
0.3330
0.40
0.9511
0.3090
3.0777
1.6145
1.8991
0.8502
3.5136
0.2846
0.45
0.9877
+0.1564
6.3137
1.9340
2.1772
0.8883
4.1111
0.2432
0.50
1.0000
0.0000
00
2.3013
2.5092
0.9171
4.8105
0.2079
0.55
0.9877
0.1564
6.3137
2.7255
2.9032
0.9388
5.6287
0.1777
0.60
0.9511
0.3090
3.0777
3.2171
3.3689
0.9549
6.5861
0.1518
0.65
0.8910
0.4540
1.9626
3.7883
3.9180
0.9669
7.7062
0.1298
0.70
0.8090
0.5878
1.3764
 4.4531
4.5640
0.9757
9.0170
0.1109
0.75
0.7071
0.7071
1.0000
5.2280
5.3228
0.9822
10.551
0.09478
0.80
0.5878
0.8090
0.7265
6.1321
6.2131
0.9870
12.345
0.08100
0.85
0.4540
0.8910
0.5095
7.1879
7.2572
0.9905
14.437
0.06922
0.90
0.3090
0.9511
0.3249
8.4214
8.4806
0.9930
16.902
0.05916
0.95
+0.1564
0.9877
0.1584
9.8632
9.8137
0.9949
19.777
0.05056
1.00
0.0000
1.0000
0.0000
11.549
11.592
0.9962
23.141
0.04321
1.05
0.1564
0.9877
0.1584
13.520
13.557
0.9973
27.077
0.03693
1.10
0.3090
0.9511
0.3249
15.825
15.857
0.9980
31.682
0.03156
1.15
0.4540
0.8910
0.5095
18.522
18.549
0.9985
37.070
0.02697
1.20
0.5878
0.8090
0.7265
21.677
21.700
0.9989
43.376
0.02305
1.25
0.7071
0.7071
1.0000
25.367
25.387
0.9992
50.753
0.01970
1.30
0.8090
0.5878
1.3764
29 . 685
29 . 702
0.9994
59.387
0.01683
1.35
0.8910
0.4540
1.9626
34.737
34.751
0.9996
69.484
0.01438
1.40
0.9511
0.3090
3.0777
40.647
40.660
0.9997
81.307
0.01230
1.45
0.9877
0.1564
6.3137
47.563
47.573
0.9998
95.137
0.01051
1.50
1.0000
0.0000
00
55.654
55.663
0.9998
111.32
0.00898
1.55
0.9877
+0.1564
6.3137
65.122
65.130
0.9999
130.25
0.00767
1.60
0.9511
0.3090
3.0777
76 . 200
76.206
0.9999
152.41
0.00656
1.65
0.8910
0.4540
1.9626
89.161
89.167
0.9999
178.33
0.00561
1.70
0.8090
0.5878
1.3764
104.32
104.33
1.0000
208.66
0.00479
1.75
0.7071
0.7071
1.0000
122.07
122 . 08
1.0000
244.15
0.00409
1.80
0.5878
0.8090
0.7265
142.84
142.84
1.0000
285.68
0.00350
1.85
0.4540
0.8910
0.5095
167.13
167.13
1.0000
334.27
0.00299
1.90
0.3090
0.9511
0.3249
195.56
195.56
1.0000
391.12
0.00256
1.95
0.1564
0.9877
0.1584
228.82
228.82
1.0000
457 . 65
0.00219
2.00
0.0000
1 . 0000
0.0000
267.75
267.75
1.0000
535.49
0.00187
MQ
TABLES OF FUNCTIONS
Table III. — Hyperbolic Tangent of Complex Quantity
tanh [x(o  i0)\  6  i x  f e~**
tanh (ia)
e
X
in
<P
X
in
<f>
a
=
D.00
P =
3.05
0.0000
0.00
0.0000
0.0000
0.0000
090°
0.0000
0.1584
0.1584
90.00°
0.0159
0.05
0.0500
0.0000
0.0500
0.00
0.0512
0.1580
0.1660
72.03
0.0319
0.10
0.1000
0.0000
0.1000
0.00
0.1025
0.1567
0.1872
56.82
0.0481
0.15
0.1500
0.0000
0.1500
0.00
0.1537
0.1547
0.2180
45.20
0.0645
0.20
0.2000
0.0000
0.2000
0.00
0.2048
0.1519
0.2549
36.56
0.0813
0.25
0.2500
0.0000
0.2500
0.00
0.2558
0.1482
0.2956
30.09
0.0985
0.30
0.3000
0.0000
0.3000
0.00
0.3068
0.1438
0.3388
25.12
0.1163
0.35
0.3500
0.0000
0.3500
0.00
0.3577
0.1386
0.3836
21.18
0.1349
0.40
0.4000
0.0000
0.4000
0.00
0.4084
0.1325
0.4293
17.98
0.1543
0.45
0.4500
0.0000
0.4500
0.00
0.4589
0.1256
0.4758
15.32
0.1748
0.50
0.5000
0.0000
0.5000
0.00
0.5093
0.1181
0.5228
13.05
0.1968
0.55
0.5500
0.0000
. 5500
0.00
0.5596
0.1096
0.5702
11.08
0.2207
0.60
0.6000
0.0000
0.6000
0.00
0.6095
0.1005
0.6177
9.36
0.2468
0.65
0.6500
0.0000
0.6500
0.00
0.6593
0.0905
. 6654
7.82
0.2761
0.70
0.7000
0.0000
0.7000
0.00
0.7088
0.0798
0.7133
6.43
0.3097
0.75
0.7500
0.0000
0.7500
0.00
0.7581
0.0683
0.7612
5.15
0.3497
0.80
. 8000
0.0000
0.8000
0.00
0.8070
0.0561
0.8090
3.97
0.3999
0.85
. 8500
0.0000
0.8500
0.00
0.8558
0.0432
0.8569
2.88
0.4686
0.90
0.9000
0.0000
0.9000
0.00
0.9041
0.0295
0.9047
1.87
0.5831
0.95
0.9500
0.0000
0.9500
0.00
0.9523
0.0151
0.9524
0.91
=
0.10
=
0.15
0.0000
0.00
0.0000
0.3249
0.3249
90.00°
0.0000
0.5095
0.5095
90.00°
0.0159
0.05
0.0553
0.3240
0.3286
80.32
0.0629
. 5079
0.5118
82.93
0.0319
0.10
0.1104
0.3213
0.3398
71.03
0.1256
0.5031
0.5186
75.98
0.0481
0.15
0.1655
0.3169
0.3575
62.43
0.1878
0.4951
0.5296
69.22
0.0645
0.20
0.2202
0.3106
0.3808
54.67
0.2493
0.4841
0.5445
62.75
0.0813
0.25
0.2746
0.3027
0.4087
47.78
0.3099
0.4700
0.5629
56.61
0.0985
0.30
0.3286
0.2929
0.4402
41.72
0.3692
0.4531
. 5845
50.82
0.11,63
0.35
0.3820
0.2815
0.4745
36.39
0.4273
0.4333
0.6085
45.40
0.1349
0.40
0.4349
0.2684
0.5110
31.68
0.4838
0.4110
0.6347
40.18
0.1533
0.45
0.4871
0.2537
0.5492
27.51
0.5385
0.3860
. 6626
35.63
0.1748
0.50
0.5386
0.2374
0.5886
23.79
0.5914
0.3589
0.6917
31.25
. 1968
0.55
0.5893
0.2196
0.6289
20.44
0.6423
0.3295
0.7219
27.16
0.2207
0.60
0.6390
0.2003
0.6697
17.41
0.6911
0.2982
0.7527
23.34
. 2468
0.65
0.6880
0.1796
0.7110
14.63
0.7378
0.2652
0.7840
19.76
0.2761
0.70
0.7358
0.1576
0.7525
12.08
0.7822
0.2305
0.8155
16.43
0.3097
0.75
0.7827
0.1342
0.7941
9.73
0.8243
0.1945
. 8469
13.28
0.3497
0.80
0.8285
0.1096
0.8375
7.54
0.8642
0.1573
. 8783
10.32
0.3999
0.85
0.8731
0.0838
0.8771
5.48
0.9015
0.1191
0.9094
7.52
. 4686
0.90
0.9166
0.0569
0.9184
3.56
0.9367
0.0800
0.9401
4.88
0.5831
0.95
0.9589
0.0289
0.9594
1.73
0.9695
0.0403
. 9704
2.38
TABLES OF FUNCTIONS 441
Table III. — Hyperbolic Tangent op Complex Quantity.— (Continued)
tanh (ira)
e
X
in
<P
e
X
in
V
a
/3 =
0.20
13 = 0.25
0.0000
0.00
0.0000
0.7265
0.7265
90.00°
0.0000
0.0000
1.0000
90.00°
0.0159
0.05
0.0763
0.7238
0.7278
83.98
0.0998
0.9950
1.0000
84.28
0.0319
0.10
0. 1520
0.7145
0.7315
78.01
0.1980
0.9802
1.0000
78.58
0.0481
0.15
0.2265
0.7019
0.7375
72.12
0.2934
0.9560
1.0000
72.93
0.0645
0.20
0.2993
0.6831
0.7458
66.34
0.3846
0.9230
1.0000
(67.38
0.0813
0.25
0.3698
0.6593
0.7560
60.72
0.4706
0.8824
1.0000
61 .93
0.0985
0.30
0.4376
0.6312
0.7680
55.27
0.5504
0.8348
1.0000
56.60
0.1163
0.35
0.5023
0.5989
0.7816
50.01
0.6236
0.7818
1.0000
51.42
0.1349
0.40
0.5635
0.5627
0.7964
44.96
0.6896
0.7241
1.0000
46.40
0.1543
0.45
0.6212
0.5235
0.8123
40.12
0.7484
0.6632
1.0000
41.55
0.1748
0.50
0.6749
0.4814
0.8290
35.50
0.8000
0.6000
1.0000
36.87
0.1968
0.55
0.7247
0.4370
0.8462
31.09
0.8446
0.5355
1.0000
32.38
0.2207
0.60
0.7703
0.3907
0.8637
26.89
0.8824
0.4706
1.0000
28.07
0.2468
0.65
0.8120
0.3430
0.8815
22.91
0.9139
0.4060
1 . 0000
23.95
0.2761
0.70
0.8497
0.2943
0.8992
19.11
0.9395
0.3423
1.0000
20.02
0.3097
0.75
0.8835
0.2451
0.9169
15.51
0.9600
0.2800
1.0000
16.27
0.3497
0.80
0.9136
0.1955
0.9343
12.08
0.9757
0.2195
1.0000
12.68
0.3999
0.85
0.9401
0.1459
0.9514
8.82
0.9869
0.1611
1.0000
9.27
0.4686
0.90
0.9632
0.0967
0.9681
5.73
0.9945
. 1050
1.0000
6.03
0.5831
0.95
0.9831
0.0480
0.9843
2.79
0.9986
0.0512
1.0000
.2.93

0.30
/} 
0.35
0.0000
0.00
0.0000
1.3764
1.3764
90.00°
0.0000
1.9626
1.9626
90 . 00°
0.0159
0.05
0.1440
1 . 3664
1.3740
83.98
0.2403
1.9391
1.9539
82.93
0.0319
0.10
0.2841
1 . 3373
1.3671
78.01
0.4670
1.8708
1.9283
75.98
0.0481
0.15
0.4164
1 . 2904
1.3450
72.12
0.6699
1.7653
1.8882
69.22
0.0645
0.20
0.5382
1 . 2282
1.3408
66.34
0.8408
1 . 6326
1.8365
62.75
0.0813
0.25
0.6455
1.1537
1.3228
60.72
0.9776
1.4832
1.7762
56.61
0.0985
0.30
0.7419
1.0701
1.3021
55.27
1.0809
1.3261
1.7109
50.82
0.1163
0.35
0.8223
0.9803
1.2794
50.01
1 . 1540
1.1701
1.6432
45.40
0.1349
0.40
0.8885
0.8873
1 . 2556
44.96
1.2007
1.0140
1.5755
40.18
0.1543
0.45
0.9413
0.7933
1.2311
40.12
0.2267
0.8793
1 . 5092
35.63
0.1748
0.50
0.9820
0.7005
1.2063
35.50
1.2359
0.7500
1.4457
31.25
0.1968
0.55
1.0120
0.6102
1.1819
31.09
1.2324
0.6323
1.3852
27.16
0.2207
0.60
1.0326
0.5336
1.1578
26.89
1.2198
0.5263
1.3284
23.34
0.2468
0.65
1.0449
0.4415
1 . 1344
22.91
1.2002
0.4313
1.2755
19.76
0.2761
0.70
1.0507
0.3640
1.1121
19.11
1.1762
0.3467
1.2262
16.43
0.3097
0.75
1.0509
0.2916
1.0906
15.51
1 . 1493
0.2713
1 . 1808
13.28
0.3497
0.80
1.0465
0.2240
1.0703
12.08
1 . 1202
0.2039
1.1386
10.32
0.3999
0.85
1.0387
0.1612
1.0511
8.82
1.0902
0.1440
1.0996
7.52
0.4686
0.90
1.0278
0.1032
1.0330
5.73
1.0599
0.0905
1.0637
4.88
0.5831
0.95
1.0148
0.0494
1.0160
2.79
1 . 0296
0.0428
1.0305
2.38
442 TABLES OF FUNCTIONS
Table III. — Hyperbolic Tangent of Complex Quantity. — {Continued)
tanh (ia)
e
X
m
V
8
X
in
<p
a
= 0.40
= 0.45
0.0000
0.00
0.0000
3.0777
3.0777
90.00°
0.0000
6.3138
6.3138
9p.00°
0.0159
0.05
0.5115
2.9990
3.0423
80.32
1.8580
5.7272
6.0211
72.03
0.0319
0.10
0.9565
2.7833
2.9431
71.03
2.9217
4.4691
5.3394
56.82
0.0481
0.15
1.2948
2.4799
2.7976
62.43
3.2313
3.2535
4.5855
45.20
0.0645
0.20
1.5189
2 . 1427
2.6265
54.67
3.1500
2 . 3362
3.9220
36.56
0.0813
0.25
1.6444
1 . 8124
2.4473
47.78
2.9260
1.6953
3.3820
30.09
0.0985
0.30
1 . 6960
1.5119
2.2720
41.72
2.6722
1.2524
2.9511
25.12
0.1163
0.35
1.6966
1.2501
2 . 1074
36.39
2.4310
0.9417
2.6070
21.18
0.1349
0.40
1.6652
1.0277
1.9569
31.68
2.2154
0.7188
2.3291
17.98
0.1543
0.45
1.6144
0.8408
1 . 8208
27.51
2.0269
0.5550
2.1015
15.32
0.1748
0.50
1 . 5547
0.6853
1 . 6989
23.79
1.8633
0.4319
1.9126
13.05
0.1968
0.55
1.4901
0.5414
1.5901
20.44
1.7210
0.3371
1.7538
11.08
0.2207
0.60
1 . 4247
0.4466
1.4932
17.41
1 . 5972
0.2632
1.6187
9.36
0.2468
0.65
1.3609
0.3553
1.4065
14.63
1.4887
0.2044
1.5026
7.82
0.2761
0.70
1.2994
0.2781
1.3289
12.08
1.3931
0.1568
1.4019
6.43
0.3097
0.75
1.2412
0.2129
1.2593
9.73
1.3083
0.1180
1.3137
5.15
0.3497
0.80
1 . 1862
0.1569
1.1966
7.54
1.2331
0.0857
1.2361
3.97
0.3999
0.85
1 . 1349
0.1090
1 . 1401
5.48
1.1655
0.0587
1.1670
2.88
0.4686
0.90
1.0867
0.0674
1.0888
3.56
1 . 1047
0.0360
1 . 1053
1.87
0.5831
0.95
1.0418
0.0314
1.0423
1.73
1 . 0499
0.0167
1 . 0500
0.91
00
475
= 0.50
0.0000
0.00
0.0000
12.706
12.706
90.00°
00
0.0000
00
090°
0.0159
0.05
5.780
9.030
10.725
57.35
20.000
0.0000
20 . 000
0.00
0.0319
0.10
6.213
4.811
7.859
37.75
10.000
0.0000
10.000
0.00
0.0481
0.15
5.260
2.682
5.905
27.01
6 . 6667
0.0000
6 . 6667
0.00
0.0645
0.20
4.356
1.636
4.653
20.58
5.0000
0.0000
5.0000
0.00
0.0813
0.25
3.662
1.075
3.817
16.35
4.0000
0.0000
4.0000
0.00
0.0985
0.30
3.138
0.7445
3.225
13.35
3 . 3333
0.0000
3.3333
0.00
0.1163
0.35
2.736
0.5367
2.789
11.10
2.8571
0.0000
2.8571
0.00
0.1349
0.40
2.422
0.3979
2.454
9.33
2 . 5000
0.0000
2 . 5000
0.00
0.1543
0.45
2.170
0.3008
2.190
7.90
2.2222
0.0000
2.2222
0.00
0.1748
0.50
1.9638
0.2304
1.9773
6.70
2.0000
0.0000
2.0000
0.00
0.1968
0.55
1.7927
0.1778
1.8014
5.67
1.8182
0.0000
1.8182
0.00
0.2207
0.60
1.6486
0.1375
1.6543
4.76
1.6667
0.0000
1.6667
0.00
0.2468
0.65
1.5238
0.1060
1 . 5293
3.97
1 . 5385
0.0000
1 . 5385
0.00
0.2761
0.70
1.4194
0.0809
1.4217
3.27
1.4286
0.0000
1.4286
0.00
0.3097
0.75
1.3269
0.0605
1.3284
2.61
1.3333
0.0000
1.3333
0.00
0.3497
0.80
1.2458
0.0438
1.2465
2.02
1.2500
0.0000
1.2500
0.00
0.3999
0.85
1 . 1737
0.0300
1.1741
1.46
1.1765
0.0000
1 . 1765
0.00
0.4686
0.90
1 . 1095
0.0183
1 . 1097
0.94
1.1111
0.0000
1.1111
0.00
0.5831
0.95
1.0518
0.0085
1.0519
0.46
1 . 0526
0.0000
1 . 0526
0.00
TABLES OF FUNCTIONS
443
Table IV. — Inverse Hyperbolic Tangent of Complex Quantity
7r(a  ifi) = tanh 1 (0  t'x)
(See page 137)
a
a
P
«
a
/3
a
6
X =
X = 0.2
x = 0.4
X = 0.6
x = 0.8
0.0
0.0000
0.0000
0.0000
0.0628
0.0000
0.1211
.0.0000
0.1720
0.0000
0.2148
0.2
0.0645
0.0000
0.0619
0.0664
0.0552
0.1250
0.0468
0.1762
. 0386
0.2186
0.4
0.1349
0.0000
0. 1281
0.0796
0.1118
0.1379
0.0931
0.1894
0.0760
0.2302
0.6
0.2206
0.0000
0.2041
0.0936
0.1703
0.1640
0.1373
0.2135
0.1103
0.2500
0.8
0.3497
0.0000
0.2955
0.1426
0.2255
0.2110
0.1749
0.2500
0.1386
0.2776
1.0
00
00.5
0.3672
0.2659
0.2593
0.2814
0.1985
0.2964
0.1576
0.3106
1.2
0.3816
0.5000
0.3271
0.3894
0.2562
0.3524
0.2041
0.3436
0.1661
0.3445
1.4
0.2852
0.5000
0.2681
0.4394
0.2322
0.4013
0.1962
0.3826
0.1655
0.3750
1.6
0.2334
0.5000
0.2255
0.4610
0.2060
0.4307
0.1823
0.4111
0.1593
0.3999
1.8
0.1994
0.5000
0.1950
0.4723
0.1832
0.4488
0.1678
0.4312
0.1505
0.4198
2.0
0.1748
0.5000
0.1721
0.4792
0.1644
0.4605
0.1536
0.4454
0.1409
0.4341
2.2
0.1561
0.5000
0.1542
0.4841
0.1490
0.4686
0.1411
0.4557
0.1317
0.4454
2.4
0.1412
0.5000
0.1399
0.4868
0.1361
0.4743
0.1302
0.4634
0.1230
0.4541
2.6
0.1291
0.5000
0.1281
0.4890
0.1252
0.4786
0.1208
0.4692
0.1151
0.4610
2.8
0.1188
0.5000
0.1181
0.4908
0.1159
0.4819
0.1124
0.4737
0.1079
0.4665
3.0
0.1103
0.5000
0.1097
0.4921
0.1080
0.4845
0.1052
0.4773
0.1016
0.4709
3.2
0.1029
0.5000
0.1025
0.4931
0.1010
0.4865
0.0988
0.4802
0.0959
0.4744
3.4
0.0965
0.5000
0.0961
0.4940
0.0950
0.4875
0.0931
0.4826
0.0907
0.4774
3.6
0.0908
0.5000
0.0905
0.4947
0.0895
0.4892
0.0880
0.4845
0.0860
0.4799
3.8
0.0858
0.5000
0.0855
0.4953
0.0847
0.4906
0.0834
0.4862
0.0817
0.4820
4.0
0.0813
0.5000
0.0812
0.4958
0.0804
0.4916
0.0793
0.4876
0.0778
0.4838
X = 1.0
X = 12
X = 14
X = 16
X = 2.0
0.0
0.0000
0.2500
0.0000
0.2789
0.0000
0.3026
0.0000
0.3222
0.0000
0.3524
0.2
0.0316
0.2532
0.0259
0.2814
0.0213
0.3046
0.0178
0.3238
0.0127
0.3534
0.4
0.0619
0.2686
0.0506
0.2890
0.0417
0.3106
0.0348
0.3285
0.0249
0.3564
0.6
0.0892
0.2786
0.0729
0.3012
0.0602
0.3185
0.0503
0.3360
0.0362
0.3612
0.8
0.1118
0.2993
0.0916
0.3173
0.0760
0.3326
0.0639
0.3459
0.0464
0.3675
1.0
0.1281
0.3238
0.1058
0.3360
0.0885
0.3472
0.0749
0.3574
0.0552
0.3570
1.2
0.1373
0.3493
0.1150
0.3558
0.0974
0.3628
0.0832
0.3699
0.0623
0.3833
1.4
0.1403
0.3734
0.1197
0.3750
0.1028
0.3783
0.0890
0.3826
0.0679
0.3920
1.6
0.1383
0.3944
0.1207
0.3926
0.1054
0.3930
0.0924
0.3949
0.0719
0.4008
1.8
0.1341
0.4120
0.1190
0.4080
0.1056
0.4064
0.0938
0.4064
0.0745
0.4093
2.0
0.1281
0.4262
0.1157
0.4211
0.1042
0.4182
0.0937
0.4169
0.0760
0.4174
2.2
0.1216
0.4376
0.1114
0.4321
0.1017
0.4284
0.0926
0.4262
0.0766
0.4249
2.4
0.1150
0.4468
0.1067
0.4412
0.0985
0.4372
0.0906
0.4344
0.0764
0.4318
2.6 •
0.1087
0.4542
0.1019
0.4488
0.0950
0.4446
0.0883
0.4375
0.0756
0.4381
2.8
0.1027
0.4602
0.0973
0.4551
0.0913
0.4510
0.0855
0.4478
0.0743
0.4437
3.0
0.0974
0.4652
0.0927
0.4604
0.0878
0.4564
0.0828
0.4531
0.0729
0.4488
3.2
0.0924
0.4693
0.0884
0.4648
0.0843
0.4610
0.0799
0.4579
0.0712
0.4533
3.4
0.0877
0.4727
0.0844
0.4686
0.0808
0.4633
0.0771
0.4619
0.0695
0.4573
3.6
0.0835
0.4756
0.0807
0.4718
0.0776
0.4684
0.0743
0.4655
0.0676
0.4609
3.8
0. 0796
0.4781
0.0773
0.4746
0.0745
0.4714
0.0717
0.4686
0.0657
0.4641
4.0
.0.0760
0.4802
0.0739
0.4789
0.0716
0.4740
0.0691
0.4713
0.0639
0.4668
444
TABLES OF FUNCTIONS
Table V. — Bessel Functions for Cylindrical Coordinates
Jnix) and N n (x) (see pages 188 and 196)
X
J»{x)
No(x)
Ji(x)
iVi(x)
Jz(x)
N 2 (x)
0.0
1.0000
GO
0.0000
— w
0.0000
CO
0.1
0.9975
1.5342
0.0499
6.4590
0.0012
127.64
0.2
0.9900
1.0811
0.0995
3.3238
0.0050
32.157
0.4
0.9604
0.6060
. 1960
1.7809
0.0197
8.2983
0.6
0.9120
0.3085
0.2867
1.2604
0.0437
3.8928
0.8
0.8463
0.0868
0.3688
0.9781
0.0758
2.3586
1.0
0.7652
+0.0883
0.4401
0.7812
0.1149
1.6507
1.2
0.6711
0.2281
0.4983
0.6211
0.1593
1.2633
1.4
0. 5669
0.3379
0.5419
0.4791
0.2074
1.0224
1.6
0.4554
0.4204
0.5699
0.3476
0.2570
0.8549
1.8
0.3400
0.4774
0.5815
0.2237
0.3061
0.7259
2.0
0.2239
0.5104
0.5767
0.1070
0.3528
0.6174
2.2
0.1104
0.5208
0.5560
+0.0015
0.3951
0.5194
2.4
+0.0025
0.5104
0.5202
0.1005
0.4310
0.4267
2.6
0.0968
0.4813
0.4708
0.1884
0.4590
0.3364
2.8
0.1850
0.4359
0.4097
0.2635
0.4777
0.2477
3.0
0.2601
0.3768
0.3391
0.3247
0.4861
0.1604
3.2
0.3202
0.3071
0.2613
0.3707
0.4835
0.0754
3.4
0.3643
0.2296
0.1792
0.4010
0.4697
+0.0063
3.6
0.3918
0.1477
0.0955
0.4154
0.4448
0.0831
3.8
0.4026
+0.0645
+0.0128
0.4141
0.4093
0.1535
4.0
0.3971
0.0169
0.0660
0.3979
0.3641
0.2159
4.2
0.3766
0.0938
0.1386
. 3680
0.3105
0.2690
4.4
0.3423
0.1633
0.2028
0.3260
0.2501
0.3115
4.6
0.2961
0.2235
0.2566
0.2737
0.1846
0.3425
4.8
0.2404
0.2723
0.2985
0.2136
0.1161
0.3613
5.0
0.1776
0.3085
0.3276
0.1479
+0.0466
0.3677
5.2
0.1103
0.3312
0.3432
0.0792
0.0217
0.3617
5.4
0.0412
0.3402
0.3453
+0.0101
0.0867
0.3429
5.6
+0.0270
0.3354
0.3343
0.0568
0.1464
0.3152
5.8
0.0917
0.3177
0.3110
0.1192
0.1989
0.2766
6.0
0.1507
0.2882
0.2767
0.1750
0.2429
0.2299
6.2
0.2017
0.2483
0.2329
0.2223
0.2769
0.1766
6.4
0.2433
0.2000
0.1816
0.2596
0.3001
0.1188
6.6
0.2740
0.1452
0.1250
0.2858
0.3119
+0.0586
6.8
0.2931
0.0864
0.0652
0.3002
0.3123
0.0019
7.0
0.3001
0.0259
0.0047
0.3027
0.3014
, 0.0605
7.2
0.2951
+0.0339
+0.0543
0.2934
0.2800
' 0.1154
7.4
0.2786
0.0907
0.1096
0.2731
0.2487
0.1652
7.6
0.2516
0.1424
0.1592
0.2428
0.2097
0.2063
7.8
0.2154
0.1872
0.2014
0.2039
0.1638
0.2395
8.0
0.1716
0.2235
0.2346
0.1581
0.1130
0.2630
TABLES OF FUNCTIONS
; *45
Table VI. — Hyperbolic Bessel Functions
I m (z) = i~ m J m {iz) (see page 210)
z
Tc(z)
Ii«
ItW
0.0
1.0000
0.0000
0.0000
0.1
1.0025
0.0501
0.0012
0.2
1.0100
0.1005
0.0050
0.4
1.0404
0.2040
0.0203
0.6
1.0921
0.3137
0.0464
0.8
1 . 1665
0.4329
0.0843
1.0
1.2661
0.5652
0.1358
1.2
1 . 3937
0.7147
0.2026
1.4
1.5534
0.8861
0.2876
1.6
1.7500
1.0848
0.3940
1.8
1.9895
1.3172
0.5260
2.0
2.2796
1.5906
0.6890
2.2
2.6292
1.9141
0.8891
2.4
3.0492
2.2981
1.1111
2.6
3.5532
2.7554
1.4338
2.8
4.1574
3.3011
1.7994
3.0
4.8808
3.9534
2.2452
3.2
5.7472
4.7343
2.7884
3.4
6.7848
5.6701
3.4495
3.6
8.0278
6.7926
4.2538
3.8
9.5169
8.1405
5.2323
4.0
11.302
9.7594
6.4224
4.2
13.443
11.705
7.8683
4.4
16.010
14.046
9.6259
4.6
19.097
16.863
11.761
4.8
22.794
20.253
14.355
5.0
27.240
24.335
17.505
5.2
32.584
29.254
21.332
5.4
39.010
35.181
25.980
5.6
46.738
42.327
31.621
5.8
56.039
50.945
38.472
6.0
67.235
61.341
46.788
6.2
80.717
73.888
56.882
6.4
96.963
89.025
69.143
6.6
116.54
107.31
84.021
6.8
140.14
129.38
102.08
7.0
168.59
156.04
124.01
7.2
202.92
188.25
150.63
7.4
244.34
227.17
182.94
7.6
294.33
274.22
222.17
7.8
354.68
331.10
269.79
8.0
427.57
399.87
327.60
446 TABLES OF FUNCTIONS
Table VII. — Bessel Functions fob Sphekical, Cooedinates
j«(x) = y/tc/2x J n+ \(x), Unix) = y/ir /2x N n+i (x)
(see page 317)
X
io(x)
no(x)
h(x)
ni(x)
h(.x)
ws(«)
0.0
1.0000
— X
0.0000
_»
0.0000
— 00
0.1
0.9983
9.9500
0.0333
 100 . 50
0.0007
3005.0
0.2
0.9933
4.9003
0.0664
25.495
0.0027
377.52
0.4
0.9735
2.3027
0.1312
6.7302
0.0105
48.174
0.6
0.9411
1.3756
0.1929
3.2337
0.0234
14.793
0.8
0.8967
0.8709
0.2500
1.9853
0.0408
6.5740
1.0
0.8415
0.5403
0.3012
1.3818
0.0620
3.6050
1.2
0.7767
0.3020
0.3453
 1 . 0283
0.0865
2.2689
1.4
0.7039
0.1214
0.3814
0.7906
0.1133
1.5728
1.6
0.6247
+0.0183
0.4087
0.6133
0.1416
 1 . 1682
1.8
0.5410
0.1262
0.4268
0.4709
0.1703
0.9111
2.0
0.4546
0.2081
0.4354
0.3506
0.1985
0.7340
2.2
0.3675
0.2675
0.4346
0.2459
0.2251
0.6028
2.4
0.2814
0.3072
0.4245
0.1534
0.2492
0.4990
2.6
0.1983
0.3296
0.4058
0.0715
0.2700
0.4121
2.8
0.1196
0.3365
0.3792
+0.0005
0.2867
0.3359
3.0
+0.0470
0.3300
0.3457
0.0630
0.2986
0.2670
3.2
0.0182
0.3120
0.3063
0.1157
0.3084
0.2035
3.4
0.0752
0.2844
0.2623
0.1588
0.3066
0.1442
3.6
0.1229
0.2491
0.2150
0.1921
0.3021
0.0890
3.8
0.1610
0.2082
0.1658
0.2158
0.2919
0.0378
4.0
0.1892
0.1634
0.1161
0.2300
0.2763
+0.0091
4.2
0.2075
0.1167
0.0673
0.2353
0.2556
0.0514
4.4
0.2163
0.0699
+0.0207
0.2321
0.2304
0.0884
4.6
0.2160
+0.0244
0.0226
0.2213
0.2013
0.1200
4.8
0.2075
0.0182
0.0615
0.2037
0.1691
0.1456
5.0
0.1918
0.0567
0.0951
0.1804
0.1347
0.1650
5.2
0.1699
0.0901
0.1228
0.1526
0.0991
0.1871
5.4
0.1431
0.1175
0.1440
0.1213
0.0631
0.1850
5.6
0.1127
0.1385
0.1586
0.0880
+0.0278
0.1856
5.8
0.0801
0.1527
0.1665
0.0538
0.0060
0.1805
6.0
0.0466
0.1600
0.1678
+0.0199
0.0373
0.1700
6.2
0.0134
0.1607
0.1629
0.0124
0.0654
0.1547
6.4
+0.0182
0.1552
0.1523
0.0425
0.0896
0.1353
6.6
0.0472
0.1440
0.1368
0.0690
0.1094
0.1126
6.8
0.0727
0.1278
0.1172
0.0915
0.1243
0.0875
7.0
0.0939
0.1077
0.0943
0.1029
0.1343
0.0609
7.2
0.1102
0.0845
0.0692
0.1220
0.1391
0.0337
7.4
0.1215
0.0593
0.0429
0.1294
0.1388
+0.0068
7.6
0.1274
0.0331
0.0163
0.1317
0.1338
0.0189
7.8
0.1280
0.0069
+0.0095
0.1289
0.1244
0.0427
8.0
0.1237
+0.0182
0.0336
0.1214
0.1111
0.0637
TABLES OF FUNCTIONS
447
Table VIII. — Impedance Functions for Piston in Infinite Plane Wall
0o  ixo = 1  (2/w)Ji(w) — iM(w) = tanh[x(« p  ip P )]; w  (4ira/X)
(see page 333)
to
So
xo
a p
fip
0.0
0.0000
0.0000
0.0000
0.0000
0.5
0.0309
0.2087
0.0094
0.0655
1.0
0.1199
0.3969
0.0330
0.1216
1.5
0.2561
0.5471
0.0628
0.1663
2.0
0.4233
0.6468
0.0939
0.2020
2.5
0.6023
0.6905
0.1247
0.2316
3.0
0.7740
0.6801
0.1552
0.2572
3.5
0.9215
0.6238
0.1858
0.2800
4.0
1.0330
0.5349
0.2175
0.3008
4.5
1 . 1027
0.4293
0.2517
0.3194
5.0
1.1310
0.3231
0.2899
0.3353
5.5
1 . 1242
0.2300
0.3344
0.3460
6.0
1.0922
0.1594
0.3868
0.3456
6.5
1.0473
0.1159
0.4450
0.3207
7.0
1.0013
0.0989
0.4788
0.2600
7.5
0.9639
0.1036
0.4594
0.2050
8.0
0.9413
0.1220
0.4241
0.1887
8.5
0.9357
0.1456
0.3980
0.1958
9.0
0.9454
0.1663
0.3839
0.2132
9.5
0.9661
0.1782
0.3799
0.2344
10.0
0.9913
0.1784
0.3845
0.2565
10.5
1.0150
0.1668
0.3964
0.2774
11.0
1.0321
0.1464
0.4153
0.2958
11.5
1 . 0397
0.1216
0.4410
0.3097
12.0
1.0372
0.0973
0.4734
0.3158
12.5
1.0265
0.0779
0.5101
0.3083
13.0
1.0108
0.0662
0.5421
0.2810
13.5
0.9944
0.0631
0.5490
0.2409
14.0
0.9809
0.0676
0.5316
0.2117
14.5
0.9733
0.0770
0.5073
0.2032
15.0
0.9727
0.0881
0.4877
0.2092
15.5
0.9784
0.0973
0.4758
0.2231
16.0
0.9887
0.1021
0.4718
0.2406
16.5
1.0007
0.1013
0.4750
0.2591
17.0
1.0115
0.0948
0.4852
0.2767
17.5
1.0187
t).0843
0.5017
0.2914
18.0
1.0209
0.0719
0.5247
0.3007
18.5
1.0180
0.0602
0.5522
0.3010
19.0
1.0111
0.0515
0.5798
0.2879
19.5
1.0021
0.0470
0.5968
0.2610
20.0
0.9933
0.0473
0.5940
0.2314
448
TABLES OF FUNCTIONS
Table IX. — Legendre Functions foe Spherical Coordinates
(See page 315)
&
Pi  Pe
Pi(cost?)
P2(COS t?)
Pt(costf)
P4(cos#)
0°
1.0000
1.0000
1.0000
1.0000
1.0000
5
1 . 0000
0.9962
0.9886
0.9773
0.9623
10
1 . 0000
0.9848
0.9548
0.9106
0.8352
15
1 . 0000
0.9659
0.8995
0.8042
0.6847
20
1 . 0000
0.9397
0.8245
0.6649
0.4750
25
1 . 0000
0.9063
0.7321
0.5016
0.2465
30
1.0000
0.8660
0.6250
0.3248
0.0234
35
1.0000
0.8192
0.5065
0.1454
0.1714
40
1.0000
0.7660
0.3802
0.0252
0.3190
45
1.0000
0.7071
0.2500
0.1768
0.4063
50
1 . 0000
0.6428
0.1198
0.3002
0.4275
55
1.0000
0.5736
0.0065
0.3886
0.3852
60
1 . 0000
0.5000
0.1250
0.4375
0.2891
65
1.0000
0.4226
0.2321
0.4452
0.1552
70
1 . 0000
0.3420
0.3245
0.4130
0.0038
75
1 . 0000
0.2588
0.3995
0.3449
+ 0.1434
80
1 . 0000
0.1736
0.4548
0.2474
0.2659
85
1.0000
0.0872
0.4886
0.1291
0.3468
90
1.0000
0.0000
0.5000
0.0000
0.3750
t»
P 5 (cost?)
Ps(cos t?)
P7(cos#)
P8(cos t5)
P»(cost»)
0°
1 . 0000
1 . 0000
1.0000
1.0000
1.0000
5
0.9437
0.9216
0.8962
0.8675
0.8358
10
0.7840
0.7045
0.6164
0.5218
0.4228
15
0.5471
0.3983
0.2455
0.0962
0.0428
20
0.2715
0.0719
0.1072
0.2518
0.3517
25
0.0009
0.2040
0.3441
0.4062
0.3896
30
0.2233
0.3740
0.4102
0.3388
0.1896
35
0.3691
0.4114
0.3096
0.1154
+0.0965
40
0.4197
0.3236
0.1006
+0.1386
0.2900
45
0.3757
0.1484
+0.1271
0.2983
0.2855
50
0.2545
+0.0564
0.2854
0.2947
0.1041
55
0.0868
0.2297
0.3191
0.1422
0.1296
60
+0.0898
0.3232
0.2231
0.0763
0.2679
65
0.2381
0.3138
0.0422
0.2411
0.2300
70
0.3281
0.2089
0.1485
0.2780
0.0476
75
0.3427
0.0431
0.2731
0.1702
+0.1595
80
0.2810
0.1321
0.2835
+0.0233
0.2596
85
0.1577
0.2638
0.1778
0.2017
0.1913
90
0.0000
0.3125
0.0000
0.2734
0.0000
TABLES OF FUNCTIONS
449
Table X. — Phase Angles and Amplitudes for Radiation and Scattering
from a Cylinder
M = ka = (27ra/\) = (wa/c) (see page 301)
[Dashes indicate values that can be computed by Eq. (26.6)]
ka
Co
7C
Ci
71
C 2
72
Cz
73
Ct
74
0.0
00
0.00°
CO
0.00°
00
0.00°
00
0.00°
00
0.00°
0.1
12.92
.44
63.06
.45
2546
0.00
—
0.00
—
0.00
0.2
6.651
1.71
15.55
1.82
318.2
0.01
9565
0.00
—
0.00
0.4
3.583
6.28
3.875
6.97
39.71
0.14
600.7
0.00
—
0.00
0.6
2.585
12.82
1.844
13.62
11.72
0.69
119.6
0.01
1595
0.00
0.8
2.091
20.66
1.199
18.73
4.922
2.09
38.20
0.06
382.9
0.00
1.0
1.793
29.39
0.9283
20.50
2.529
4.77
15.81
0.20
127.3
0.00
1.2
1.593
38.74
0.7884
18.94
1.503
8.91
7.712
0.57
52.03
0.02
1.4
1.447
48.52
0.7035
14.80
1.012
14.06
4.212
1.35
24.54
0.06
1.6
1.335
58.62
0.6453
8.84
0.7627
19.03
2.504
2.77
12.85
0.16
1.8
1.246
68.96
0.6019
1.61
0.6309
22.49
1.596
5.08
7.290
0.37
2.0
1.173
79.49
0.5676
+6.52
0.5573
23.69
1.086
8.44
4.405
0.79
2.2
1.112
90.15
0.5392
15.31
0.5130
22.56
0.7898
12.71
2.801
1.55
2.4
1.060
100.93
0.5152
24.57
0.4836
19.45
0.6158
17.32
1.861*
2.80
2.6
1.014
111.81
0.4944
34.20
0.4624
14.75
0.5136
21.41
1.287
4.73
2.8
0.9743
122.75
0.4760
44.11
0.4457
 8.84
0.4535
24.15
0.9265
7.46
3.0
0.9389
133.76
0.4597
54.24
0.4319
 1.99
0.4175
25.09
0.6965
11.01
3.2
0.9071
144.82
0.4450
64.55
0.4198
+ 5.59
0.3952
24.19
0.5496
15.13
3.4
0.8785
155.92
0.4317
75.01
0.4090
13.73
0.3804
21.64
0.4566
19.29
3.6
0.8524
167.06
0.4195
85.58
0.3992
2.33
0.3698
17.71
0.3987
22.81
3.8
0.8286
178.23
0.4084
96.25
0.3901
31.29
0.3617
12.66
0.3631
25.11
4.0
C.8067
189.42
0.3980
107.01
0.3816
40.55
0.3549
6.72
0.3412
25.90
4.2
0.7865
200.64
0.3885
117.83
0.3737
50.06
0.3489
0.04
0.3275
25.14
4.4
0.7678
211.88
0.3796
128.72
0.3662
59.77
. 3434
+7.22
0.3187
22.95
4.6
0.7503
223.14
0.3713
139.65
0.3592
69.66
0.3383
14.97
0.3126
 19 .54
4.8
0.7341
234.42
0.3635
150.64
0.3525
79.70
0.3334
23.13
0.3081
15.10
5.0
0.7188
245.71
0.3562
161.66
0.3462
89.87
0.3287
31.62
0.3044
 9.81
ka
Ci
75
Ce
76
Ci
77
C 8
78
C 9
79
2.0
22.07
0.04°
130.8
0.00°
903.5
0.00°
7144
0.00
0.00
2.2
12.82
0.10
68.99
0.00
432.1
0.00
3099
0.00
—
0.00
2.4
7.834
0.22
38.65
0.01
221.4
0.00
1452
0.00
—
0.00
2.6
4.999
0.45
22.78
0.03
120.2
0.00
725.6
0.00
4941
0.00
2.8
3.309
0.86
14.03
0.06
68.58
0.00
383.4
0.00
2418
o.oo
3.0
2.261
1.53
8.967
0.13
40.86
0.01
212.6
0.00
1248
0.00
3.2
1.590
2.59
5.922
0.25
25.27
0.02
122.9
0.00
674.6
0.00
3.4
1.149
4.18
4.025
0.47
16.16
0.04
73.80
0.00
380.0
0.00
3.6
0.8534
6.41
2.805
0.83
10.65
0.07
45.80
0.00
222.1
0.00
3.8
0.6539
9.35
2.000
1.41
7.200
0.14
29.29
0.01
134.1
0.00
4.0
0.5190
12.92
1.456
 2.30
4.985
0.26
19.24
0.02
83.43
0.00
4.2
0.4287
16.83
1.082
 3.60
3.526
0.46
12.95
0.04
53.32
0.00
4.4
0.3693
20.62
0.8211
 5.42
2.542
0.77
8.907
0.08
34.92
0.01
4.6
0.3312
23.74
0.6374
 7.85
1.865
1.26
6.252
0.14
23.40
0.01
4.8
0.3071
25.76
0.5081
10.88
1.391
2.00
4.471
0.25
16.00
0.02
5.0
0.2921
26.44
0.4177
14.40
1.054
3.06
3.251
0.42
11.15
0.04
450
TABLES OF FUNCTIONS
Table XI. — Phase Angles and Amplitudes fob Radiation and Scattering
from a Sphere
n = ka = (2tto/X) = (ua/c) (see page 320)
[Dashes indicate values that can be computed by Eq. (27.17)]
ka
Do
So
Di
Si
Di
Si
D,
Si
Da.
St
0.0
00
0.00°
00
0.00°
00
0.00°
00
0.00°
00
0.00°
0.1
100.5
0.02
2000
0.01
—
0.00
—
0.00
—
0.00
0.2
25.50
0.15
250.1
0.08
5637
0.00
—
0.00
—
0.00
0.4
6.731
1.12
31.35
0.58
354.6
0.01
5906
0.00
—
0.00
0.6
3.239
3.41
9.408
1.82
70.73
0.06
785.5
0.00
—
0.00
0.8
2.001
7.18
4.101
3.80
22.67
0.25
188.9
 0.01
2058
0.00
1.0
1.414
12.30
2.236
6.14
9.434
0.70
62.97
 0.02
547.8
0.00
1.2
1.085
18.56
1.426
8.11
4.646
1.59
25.82
 0.08
186.9
0.00
1.4
0.8778
25.75
1.021
8.97
2.583
3.07
12.22
 0.22
75.74
0.01
1.6
0.7370
33.68
0.7931
8.25
1.584
5.19
6.426
 0.51
34.84
0.02
1.8
0.6355
42.19
0.6529
5.87
1.057
7.77
3.667
 1.05
17.66
0.06
2.0
0.5590
51.16
0.5590
1.97
0.7629
10.40
2.236
 1.97
9.669
0.15
2.2
0.4993
60.49
0.4918
+ 3.21
0.5901
12.49
1.444
 3.38
5.635
0.32
2.4
0.4514
70.13
0.4411
9.44
0.4837
13.51
0.9823
 5.34
3.459
0.64
2.6
0.4121
80.01
0.4011
16.50
0.4148
13.14
0.7036
 7.80
2.220
1.18
2.8
0.3792
90.08
0.3686
24.23
0.3676
11.31
0.5308
10.54
1.481
2.02
3.0
0.3514
100.32
0.3415
32.49
0.3333
8.11
0.4214
13.16
1.024
3.27
3.2
0.3274
110.70
0.3184
41.18
0.3071
3.73
. 3508
15.17
0.7334
5.00
3.4
0.3066
121.20
0.2985
50.23
0.2862
+ 1.65
0.3042
16.17
0.5443
7.23
3.6
0.2883
131.79
0.2811
59.57
0.2688
7.86
0.2723
15.94
0.4195
9.83
3.8
0.2721
142.47
0.2657
69.15
0.2540
14.75
0.2496
14.41
0.3364
12.58
4.0
0.2577
153.22
0.2519
78.92
0.2411
22.20
. 2326
11.67
0.2807
15.10
4.2
0.2448
164.03
0.2396
88.88
0.2296
30.12
0.2193
 7.84
0.2432
17.00
4.4
0.2331
174.91
0.2285
98.97
0.2194
38.44
0.2084
 3.08
0.2174
17.95
4.6
0.2225
185.83
0.2184
109.20
0.2101
47.08
0.1992
+ 2.47
0.1994
17.79
4.8
0.2128
196.79
0.2091
119.55
0.2016
56.00
0.1912
8.70
0.1863
16.47
5.0
0.2040
207.79
0.2006
129.98
0.1939
65.16
0.1840
15.48
. 1764
14.04
ka
Dt,
Ss
De
5<s
Di
57
Z>8
5s
Dt
59
2.0
51.31
 0.01°
323.7
0.00°
2370
0.00°
—
0.00°
—
0.00°
2.2
27.14
 0.02
155.2
0.00
1030
0.00
7790
0.00
—
0.00
2.4
15.25
 0.04
79.69
0.00
483.5
0.00
3343
0.00
—
0.00
2.6
9.021
 0.10
43.38
0.01
242.2
0.00
1541
0.00
—
0.00
2.8
5.573
 0.20
24.83
0.01
128.2
0.00
755.6
0.00
5002
0.00
3.0
3.576
 0.37
14.83
0.03
71.28
0.00
390.7
0.00
2407
0.00
3.2
2.371
 0.68
9.206
0.06
41.34
0.00
211.7
0.00
1219
0.00
3.4
1.620
1.16
5.907
0.11
24.88
0.01
119.5
0.00
645.8
0.00
3.6
1.137
 1.91
3.904
0.21
15.49
0.02
70.01
0.00
356.1
0.00
3.8
0.8183
 2.99
2.649
0.38
9.933
0.03
42.39
0.00
203.6
0.00
4.0
0.6043
 4.48
1.841
0.66
6.545
0.06
26.44
0.00
120.2
0.00
4.2
0.4583
 6.43
1.308
1.09
4.418
0.12
16.94
0.01
73.09
0.00
4.4
0.3577
 8.79
0.9486
1.73
3.050
0.22
11.13
0.02
45.66
0.00
4.6
0.2881
11.45
0.7015
2.65
2.148
0.37
7.479
0.04
29.24
0.00
4.8
0.2399
14.15
0.5290
3.92
1.542
0.61
5, 130
0.07
19.16
0.00
5.0
0.2065
16.56
0.4072
5.58
1.126
0.98
3.587
0.12
12.82
0.01
TABLES OF FUNCTIONS
451
Table XII. — Impedance Functions for Piston in Infinite Plane Wall
Jo(irao n r/a)
l^n — ^Xn)
n =
p(r) = pcuoe** V (e n  »x.) /o(7r "°" r/a) ; M = (2*»/A) (see page 334)
M
0o
xo
0i
XI
02
*2
03
X»
04
X4
0.0
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.5
0.1199
0.3969
0.0014
0.0480
0.0004
0.0180
0.0002
0.0096
0.0001
0.0062
1.0
0.4233
0.6468
0.0200
0.1128
0.0058
0.0404
0.0028
0.0214
0.0016
0.0131
1.5
0.7740
0.6801
0.0656
0.1806
0.0228
0.0616
0.0106
0.0324
0.0062
0.0204
2.0
1.0330
0.5349
0.1966
0.2138
0.0496
0.0750
0.0228
0.0387
0.0132
0.0236
2.5
1.1310
0.3231
0.3496
0.1664
0.0788
0.0653
0.0356
0.0392
0.0204
0.0238
3.0
1.0922
0.1594
0.4404
0.0340
0.0868
0.0486
0.0386
0.0303
0.0220
0.0210
3.5
1.0013
0.0989
0.4458
+0.1502
0.0776
0.0408
0.0344
0.0286
0.0196
0.0188
4.0
0.9413
0.1220
0.3462
0.3078
0.0680
0.0604
0.0302
0.0374
0.0172
0.0272
4.5
0.9454
0.1663
0.1830
0.3740
0.0852
0.1016
0.0360
0.0532
0.0202
0.0355
5.0
0.9913
0.1784
0.0260
0.3420
0.1480
0.1348
0.0554
0.0644
0.0302
0.0390
5.5
1.0321
0.1464
+0.0568
0.2394
0.2404
0.1202
0.0784
0.0605
0.0414
0.0398
6.0
1.0372
0.0973
0.0646
0.1389
0.3268
0.0409
0.0920
0.0448
0.0470
0.0325
3.5
1.0108
0.0662
0.0188
0.0820
0.3516
+0.0882
0.0868
0.0286
0.0446
0.0251
7.0
0.9809
0.0676
0.0260
0.0811
0.3006
0.2170
0.0738
0.0362
0.0382
0.0301
7.5
0.9727
0.0881
0.0366
0.1072
0.1850
0.2901
0.0756
0.0646
0.0386
0.0423
8.0
0.9887
0.1021
0.0140
0.1246
0.0540
0.2836
0.1140
0.0970
0.0516
0.0544
8.5
1.0115
0.0948
+0.0143
0.1121
+0.0296
0.2272
0.1832
0.0989
0.0716
0.0552
9.0
1.0209
0.0719
0.0264
0.0828
0.0538
0.1370
0.2602
0.0768
0.0870
0.0428
9.5
1.0111
0.0515
0.0130
0.0568
0.0264
0.0795
0.2962
+0.0487
0.0864
0.0270
10.0
0.9933
. 0473
0.0072
0.0514
0.0116
0.0666
0.2710
0.1609
0.0748
0.0254
452
TABLES OF FUNCTIONS
Table XIII. — Absorption Coefficients fob Wall Material*
(See page 385)
Based on Bulletin of Acoustic Materials Association, VII, 1940
Material
Values of absorption coefficient a
Frequency, cps
128
256
512
1,024
2,048
0.27
0.50
0.88
0.80
0.26
0.79
0.88
0.76
0.40
0.50
0.80
0.55
0.03
0.03
0.04
0.05
0.25
0.37
0.34
0.27
0.27
0.76
0.88
0.60
0.08
0.11
0.25
0.30
0.27
0.50
0.80
0.82
0.01
0.02
0.02
0.02
0.04
0.03
0.03
0.03
0.04
0.03
0.03
0.03
0.04
0.03
0.03
0.02
0.01
0.01
0.02
0.02
0.34
0.74
0.76
0.75
0.02
0.03
0.04
0.04
0.40
0.54
0.52
0.50
0.50
0.40
0.35
0.30
0.07
0.06
0.06
0.06
4,096
Acoustex 40R*
Acoustone D*
Acoustic plaster
Brick wall, unpainted . .
Carpet, lined
CelotexC3*
Curtains, light
heavy drapes
Floor, concrete
wood
linoleum
Glass
Marble or glazed tile. . .
Permacoustic*
Plaster, smooth on lath
Temcoustic F2*
Ventilator grill
Wood paneling
Object
0.16
0.13
0.30
0.02
0.11
0.25
0.04
0.10
0.01
0.05
0.04
0.04
0.01
0.19
0.02
0.33
0.50
0.08
Values of absorption a = aA in sq ft
Frequency, cps
128
Adult person, seated in audience
Chairs, metal or wood
Theater chair, wood veneer
Leather upholstered
Plush upholstered
1.5
0.15
0.18
1.0
1.5
256
512
1,024
2,048
2.5
3.8
5.0
4.8
0.16
0.17
0.19
0.20
0.20
0.22
0.25
0.25
1.2
1.4
1.6
1.6
2.0
2.5
2.8
3.0
0.70
0.74
0.50
0.05
0.24
0.25
0.30
0.75
0.02
0.03
0.03
0.02
0.01
0.74
0.04
0.42
0.25
0.06
4.5
0.19
0.23
1.5
3.0
* Measurements for acoustic material cemented to plaster or concrete. Other means of mount
ing give other values for a.
Acoustical Constants for Air and Water
Air at 760 mm mercury, 20°C, cgs units:
P = 0.00121, c = 34,400, P c = 42, P c 2 = 1.42 X 10 6
y e = 1.40, Po = 1.013 X 10 3
Water at 20°C, cgs units:
P = 1.0, c = 146,000, P c = 1.5 X 10 s , pc* = 2.1 X 10 10
PLATES
453
o
\
\
/
/
A
2
7 b
^
i
\
y
v.
./
'
s
\<
^2;^
<jX,
a^oA.
A
\
y
r
g
N
/
\
c'
■
.^v
/P7
>
s
X
/
\
7T^i
•
k
?v
•?
r
^
<
/^^
\
\
i
\
— ^
PK
Jl
rf
pas
—
kio\
*PjPk
p 5
r
)y s
V
fi®Ks\
J"
V£
>
\
/
Vl\
/
y
.1
\k$
\f£°
' lb \
^es
\
^
<
\
Itfr
T '
<3\
"Op
y
jc
^Ct
^
, o^X
/
^Pl
y^
\^
/
— —
T
v
•2
\
s?>
\.
— —
V
^'
— — 
A;
s*
N?
h
\
>
At
^
\
\
■7,
\>X\
\
\
.
1 2 3
Resistance Ratio = (R/pc)
Plate 1. — Bipolar plot of wavereflection parameters a and /3 against impedance
ratio f = (z/pc) = — i% = tanh [ir(a — t'/3)j. Standing wave ratio = e~ z * a . See
pages 137 and 240.
454
PLATES
•*» 3 =
1.5
0.8
0.6
0.4
0.3
0.2
0.15
=>
'"^i
/
\
/£s
5^ *X
^ /
II /
^
^A
M^
***• /
$\
fe./
\0\
^Oeo>
SfOAU
f=o.arh^
\^
^n£035
0=0.70
p075
\ 1 1
0=0.30
Kri
III
*3
0=0.25
p0.80
. g.
"0=0.20
p=085V^
J^aTT
B^OBO^
j^Jy
r^r£—
— Vo^
\o.o>
^C TV
1^1
V?
n .q6X.
/ Q
^> \
J
k #
— ^
"C
V
90°
60 €
30° 0° 30°
Phase Angle ,ij?
60'
90°
Plate II. — Plot of wavereflection parameters a and /3 against magnitude and phase of
.; impedance ratio (z/pc) = f = If \e~ i,p = tan h [ir(a  t/3)]. See pages 137 and 240.
PLATES
455
.06
«09 O oe o * o0£+ o09+
sinh[tr(aip)]=pe■ , '' cp
120°
Phase angle of sinh
Plate III. — Magnitude and phase angle of sinh [*(«  «£)] and cosh frr(a  *£)].
See pages 137 and 241.
456
PLATES
Maximum
Pressure!
Minimum
Displacemei
Maximum
Displacement
Scale for Standing Wave Ratio  q. I = I A r /A;
0.5 i.O
Ui.
I i ' i i ' i '  ' i '
1.0 05
Corresponding Values of (A m in/Am<w)
Plate IV.— Ratio g of reflected to incident plane wave, in phase and amplitude
(\q\ = standing wave ratio), for different valves of resistance and reactance ratios B
and X . where (z/oc) = r = 6  i X . Standing wave ratio is distance from center of
plot to point 6, x Angular measure for in units of halfwavelength along tube or
fttring. See pages 137 and 241.
PLATES
457
sN
\
pni
3\
V
°4\
UJ
Q
ZO
UJ2
/ a.
o
UJ
£i
z
V
7

\
y y
$>
^
l
«:
\^
H>
1 1
1
u
Ou
ii
%2
'
*7
'v^'
**
M
\$\
PN
^
' A
"A
L ^
^fr
A
sJ&Sf
T^
/$S
b— 1
c
c
II £ § II
^ o 
s *<
ST3
••a 5
3 r ^5
©
5 o
m s
(0
£ >>
a s i» ^
■73 73
>5 £
$ 2" *
ftC.£S
03 s Si.
RT3
— a
o a ll ^3
!«d
.9 S^a
a o
3«Sfi
03 J? <S rf3
•3 S ft ""
vj. ft oj §
II 2 T3
r^ X ® ft
» H (j a
S °5 ^
b • **■* ©
d o t>
go ©
£ °^
.sis 83
T3 T) CJ "O
5 § « c so
glft2^
ISIg'S
o *■> a ajR
©
a 5*
» 5
■^03 <,.
00 03 ©
H« ? ? .
"3 ^^
■ I 3 _
© So"
P ftoS
458
PLATES
90°
60°
30'
0° 30°
Phase Angle cp
60°
90°
Plate VI. — Contours for absorption coefficient a corresponding to various values
of magnitude and phase angle of acoustic impedance ratio (z/pc) = fe  **' for wall
material. Only to be used for rooms and frequencies such that sound is uniformly dis
tributed in room. See page 388.
INDEX
Absorption, of sound, 360, 385
wall, 403405
Absorption coefficient (see Coefficient)
Absorption constant, 385, 452
Acoustic admittance ratio, 240
of horn, 285
of surface, 306, 366, 369, 410
Acoustic conductance ratio, 240, 306,
367, 371, 403, 411
Acoustic constants of air and water,
222, 452
Acoustic filters, 235, 290
Acoustic impedance, 237
analogous, 234, 237
of constriction, 234
of horn, 272, 285
of opening, 235
of tank, 235
of catenoidal horn, 282
characteristic, of air, 223, 238, 307,
452
of conical horn, 272
at diaphragm, 200, 333
of exponential horn, 281
measurement of, 242, 456
of piston in sphere, 324
in wall, 332, 447, 451
radiation, 237
of surface, 360, 361, 365, 368, 388
ratio, 240
specific, 234, 237, 239, 287
of surface, 366, 368, 388
Acoustic power, sources of, 228
of voice, 228
Acoustic reactance ratio, 240, 367
Acoustic resistance (see Acoustic imped
ance)
Acoustic resistance ratio, 240, 367
Acoustic susceptance ratio, 240, 306,
367, 411
Adiabatic compression, 221, 363
Admittance, acoustic (see Acoustic
admittance)
mechanical, of oscillator, 35, 37, 50
of string support, 144
transfer, 93, 95, 105
wave, 91
Air, acoustic properties of, 222, 452
load of, on membrane, 310, 335
localized, 198, 361
uniform, 193, 200, 361
Allowed frequencies (see Frequencies,
natural)
Amplitude, average, of diaphragm, 201
of harmonic, 87, 91
of oscillation, 9, 87, 190, 201
radiation, 301, 320
velocity, 23, 224
of wave, 266
Analytic function, 14
Analogous circuits, 233, 365
Analogous impedance (see Acoustic
impedance)
Angle, of lag (see Phase angle)
Angledistribution function, 321
Angular distribution, of radiated sound,
from cylinder, 304
from dipole, 318
from piston, 323, 328, 331
from point on sphere, 321
of scattered sound, from cylinder,
349
from sphere, 354
Angular momentum, of whirled string,
119
Antinode, 84
Approximate calculation (see Perturba
tion calculations)
459
460
VIBRATION AND SOUND
Attenuation index, for porous material,
366
Attenuation, of sound, in ducts, 307,
371
Axial waves, 391, 405
B
Bar, 151170
clampedclamped, 161, 170
clampedfree, 157, 170
freefree, 162, 170
nonuniform, 164, 171
plucked and struck, 160
whirled, 171
Bel, 225
Bending, of plate, 209
Bending moments, in bar, 153
Bessel functions, 6, 17, 188, 298, 444
spherical, 316, 446
Boundary conditions, 75, 117, 175, 187
for air in duct, 306, 369
for bar, 157, 162
for flexible string, 75, 84, 135
for membrane, 175, 194
at plane surface, 360, 367, 380, 388
for plate, 210
for stiff string, 167
at wall of room, 388, 423
Branch point, 13
Capacitance, of crystal, 41, 69
Catenoidal horn, 267, 271, 281283, 292
Cavity resonance, 228, 258, 359
Characteristic frequencies, of vowel
sounds, 233
Characteristic functions, 107—109
for bar, 159, 162, 171
for membrane, 180183, 189191,
195
for plate, 211
for sound in room, 389, 402, 415, 423
in tube, 309, 373
for string, 107, 112, 118, 143, 149
Characteristic impedance, of air, 223,
238, 364, 452
in tube, 308, 372
Characteristic impedance, for string, 93,
126
of water, 452
Characteristic values, 107
for bar, 158, 162
for Bessel functions, 189, 399
for membrane, 180, 189, 195
for plate, 210
for sound in room, 390401, 411
for string, 107, 113, 117, 149
for wire, 168
Circular membrane, 183195, 214, 297,
306
Circular waves, 184187, 297311
Clamped bar, 157, 161
Clamped capacitance, 41
Clamped impedance, 36, 38
Clarinet, 248, 285
Closed tube, 244, 249, 258261, 291
Coefficient, absorption, 385, 388, 452
wall, 403405, 425
Coefficients, of Fourie: series, 87
of power series, 4, 6, 7
of series of characteristic functions,
108, 160, 309, 428
of series for scattered wave, 348, 354
Coil, driving, electromagnetic, 34, 67,
277, 289, 292
Complex exponential, 7
Complex numbers, 8
Complex plane, 8
integrals in, 1219, 4244
Compliance, 21, 29
Compressibility, of air, 220, 363
Condenser microphone, 195208, 211
215, 357360, 379 ..
Conditions, boundary (see Boundary
conditions)
initial (see Initial conditions)
Conductance, acoustic (see Acoustic
conductance)
mechanical, 35, 144
Configuration, plane of, 58
of system, 57, 64
Conformal transformation, 136, 373,
453457
Conical horn, 271279, 291292
Constriction, in tube, 234, 247
Continuity, equation of, 218, 294
INDEX
461
Contour integrals, 1216, 19, 4246,
100, 132, 144, 206, 263, 288, 428
Convention, for complex numbers, 9
Coordinate systems, 174, 296
Coordinates, normal, 56, 58, 9091
Cosine, 4, 438439
Coupled oscillators, 5266, 68
Coupling, small, 59
of source, with standing waves, 415—
418
Coupling constant, 54
electromagnetic, 35
piezoelectric, 40
Crystal, piezoelectric {see Piezoelec
tric force)
Cutoff frequency, 235, 280, 282, 308
Cylinder, force on, 352, 378
radiation from, 297305, 376
scattering from, 347—352
transmission inside, 305311, 377
Cylindrical coordinates, 296
Cylindrical room, 398401, 408409
D
Damped vibrations, 2327
of air, in room, 386, 401409, 427429
in tube, 262264
of bar, 170
of membrane, 206208
of string, 106, 132, 145146
Damped waves, 243, 291, 307, 371,
375, 380
Damping constant, 24, 133, 243, 386,
402, 416, 419, 426429
Decay, modulus of, 25
Decay curve, for sound, in room, 406
Decibel scale, 225
Decrement, 25
Degeneracy, 181, 190, 395
Delta function, 48, 97
Density of air, 218, 363, 452
of solids, 152
Design, of condenser microphone, 357
360
of dynamic speaker, 339344
of horn loudspeaker, 274279
Dielectric susceptibility, 40
Diffraction, of sound, 329, 350
Dipole source, 318
Dirac, 429
Dirac delta function (see Delta func
tion)
Directionality, of sound from piston, 329
(See also Angular distribution)
Dispersion, of waves, 154, 209, 307
Displacement, of air in wave, 218
Distribution, of natural frequencies,
390401
of sound, in room, 383, 401
Divergence, of vector, 294
Driver, 27, 59, 62
Ducts, sound transmission through,
368376
Dynamic loudspeaker, 34, 6769, 323
326, 338344, 377379
Dyne, 2
E
Ear, response of, 226228
Echo, flutter, 261264, 291
Effective length, of tube, 235, 247
Elastic modulus, of bar, 152
of crystal, 40
of plate, 209
Electrical analogue, 233, 275, 365
Electromagnetic drive, for bar, 171
for diaphragm, 341344, 377
for horn, 276279, 289, 292
for oscillator, 34—38
for string, 148
Electromechanical driving force, 34
Element, of cylinder, radiation from,
300303
of sphere, radiation from, 321323
Energy, 2
kinetic, 2, 89, 163
in normal coordinates, 59, 91
potential, 2, 90, 163
of vibration, 23
of bar, 162
of coupled oscillators, 58
of driven oscillator, 33
of oscillator, 23
of sound wave, 223, 224226, 402,
414
of string, 89
462
VIBRATION AND SOUND
Energy density of sound, 223, 240, 383,
414
Energy loss, 26, 33, 300, 319, 367, 403
Energy transfer, 61
Erg, 2
Exponential function, 7, 438439
Exponential horn, 279281, 292
Extended reaction of surface to sound,
200, 361
F
Fermi, Enrico, 422
Filters, acoustic, 235, 290 ,
Flange, on tube, 246, 258, 326, 336
338
Flexible string (see String)
Flow resistivity, 363
Fluctuations of sound in room, 408
Flute, 248
Flutter echo, 261264, 291
Force, 2
on cylinder, due to wave, 352
reaction of membrane to, 176
on sphere, due to wave, 356
transient, 16
Forced vibration, of air, in tube, 242
258, 305311, 368372
in horn, 271287
of bar, 166
of coupled oscillators, 62
of membrane, 195206
of oscillator, 27
of plate, 211213
of sound, in room, 413429
of string, 91100, 119121, 129132
138144
Fourier series, 85, 97
Fourier transform, 16, 43, 51, 94, 229,
288,372
Fraunhofer diffraction, 350
Frequencies, complex, 45, 424
natural, 22, 24, 107, 133
of bar, 158, 162
of coupled oscillators, 55, 68
of kettledrum, 194
of membrane, 180, 189, 195
of plate, 210
of resonating horn, 286
Frequencies, natural, of sound, in room,
390401, 411
of string, 84, 107, 113, 117, 149
of wire, 168
Frequency, 5
antiresonance, 203, 212
cutoff, 235, 280, 282, 308
fundamental, 84
natural (see Frequencies, natural)
resonance, 31, 95, 198, 418420
Frequency distribution, of normal
modes, in room, 390401
of sounds, 229
Frequency space, 391400
Frequency spread, of sounds, 229, 419
420
Fresnel diffraction, 350
Friction, effect of, 23, 27
on air in pores, 363—366
on bar, 166
on membrane, 198203, 207
on string, 104106, 130
Functions, analytic, 14
Bessel, 6, 17, 188, 298, 316, 444446
characteristic (see Characteristic
functions)
delta, 48, 97
hyperbolic, 18, 136, 438439
Legendre, 18, 118, 314, 448
Neumann, 7, 196, 444446
step, 47
trigonometric, 4, 438—439
Fundamental frequency, 84
(See also Frequencies, natural)
G
Gradient, 295
Gyration, radius of, 152
H
Halfbreadth, of resonance peak, 68
Half width of pulse, 231
Hanging string, 149
Harmonics, 85
even and odd, 88
in clarinet, 252
in closed tube, 245
in open tube, 248
INDEX
403
Helmholtz resonator (see Resonator)
Horn, 265271
catenoidal, 267, 281283
conical, 271279
exponential, 267, 279281
resonance in, 263—287
shape of, 269271
transients in, 287288
transmission coefficient of, 273, 281,
282
waves in, 265269
Hyperbolic Bessel functions, 210, 445
Hyperbolic functions, 18, 136, 438439
Hyperbolic tangent, 136, 239243, 255,
284, 369, 423, 438443
Imaginary units i and j, 10
Impedance, acoustic (see Acoustic
impedance)
blocked, of coupled oscillator, 63
clamped, of coil, 36, 38
input, of coupled oscillators, 63
magnetomotive, 37, 38
mechanical (see Mechanical imped
ance)
motional, 38
of crystal, 41
mutual, of coupled oscillator, 63
radiation (see Radiation load)
of string support, 133, 144
of surface, 360
transfer, of coupled oscillators, 63
for string, 93, 95, 105, 128
of wave, in room, 417
wave, for string, 91, 139
Impulse excitation, 426
Impulse function, 47
Incident and reflected waves, 76, 134,
238, 366, 388
Index of refraction, for porous material,
366
Initial conditions, 21
for bar, 160
for membrane, 183
for oscillator, 21
for string, 74, 86
Integral, contour, 1216, 19, 4246,
100, 132, 144, 206
infinite, 14^16
Intensity, 223, 226, 414
of radiation, from cylinder, 297305,
376
from piston, 328332, 377379
from simple source, 312—314, 386,
414^15
from sphere, 311324, 377
of reflected sound, 367368, 388
of scattered sound, from cylinder,
350351
from sphere, 354—356
of sound, in room, 383, 386, 414
in plane wave, 223, 226
in standing wave, 240
Intensity level, 226
of sound, in room, 386, 415
Interference of waves, 258, 303, 328,
349351, 355
Interference fluctuations, of sound, 408
Isothermal expansion, 221, 363
K
Kettledrum, 193195
L
Laplace transform, 51, 293, 428
Laplacian operator, 174, 295
Legendre functions, 18, 118, 314, 448
Level, intensity, 226, 386, 414
pressure, 226, 415
spectrum, 229
Lippman color photographs, 423
Loaded bar, 171
Loaded membrane, 214
Loaded plate, 216
Local reaction of surface to sound, 198,
336, 361
Loop, 84
Loudness, 226228
Loudspeaker, dynamic, 34, 6769,
323326, 338344, 377379
Loudspeaker horn, 265293
464
VIBRATION AND SOUND
M
Magnetomotive impedance, 37
Magnitude, of complex number, 10
Mass, 2
effective, for air load, 234, 247, 300,
319
for membrane, 202
for string, 108, 124, 127
nonuniform, of bar, 164
of string, 111114
Masscontrolled vibrations, 33
Mechanical admittance, 50
Mechanical impedance, 29, 38
of coupled oscillators, 63
of oscillator, 29, 31, 50
Mechanical input and transfer imped
ance, 63
Mechanical reactance, 29
Mechanical resistance, 24, 29
Membrane, 172208
circular, 183208
forced motion of, 195208
forces on, 173, 176
plucked, 185
rectangular, 177183
struck, 183, 186
transient motion of, 206208
waves on, 172, 184187
Microphone, condenser, 195—208, 211
215, 357360, 379
ribbon, 353, 378
Modulus, of decay, 25
of elasticity, 40, 152, 209
Moment, bending, 153
Motional impedance, 35, 38
N
Natural frequency {see Frequencies,
natural)
Neumann functions, 7, 196, 444446
Nodal line, 180182, 190, 199, 202,
211
Nodal point, 84, 140, 241
Noise, analysis of, 230
Nonuniform bar, 164
Nonuniform string, 107121, 123130
Normal coordinates* 56, 58
Normal modes of oscillation, 84, 107
109
of bar, 156160, 171
of coupled oscillators, 55
of membrane, 179183, 189191, 195
of plate, 210211
for room, 390, 402, 415, 423
for sound in tube, 309, 373
of strings, 84, 91, 107, 112, 118, 143,
149
effective impedance for, 128
Normalization constant, 108
O
Oblique waves, 391, 405
Oboe, 286
Open tube, 246247, 255258
smalldiameter, 247253
Operational calculus, 50, 104
Organ pipe, 245, 248
Orthogonality, of characteristic func
tions, 108
Oscillations (see Vibrations)
Oscillator, coupled, 5266
clamped, 2327
forced, 2742
simple, 2023
energy of, 23, 33
Overtones, 84, 158
Panel, reaction of, to sound, 361
Parallel waves, 172
Particle velocity, in sound wave, 222,
224
Period of vibration, 5
Periodic motion, of string, 78, 86
Perturbation calculations, 122—133,
164165, 402412, 415, 422
Phase, constant, surfaces of, 266269
Phase angle, 10
for driven oscillator, 31
Phase shift, scattering, for cylinder,
301, 449
for sphere, 320, 450
Piano string, 102104
Piezoelectric constant, 40
Piezoelectric force, 38
INDEX
465
Piston, in duct, 309311
in sphere, 323326
in plane wall, 326336, 344346,
447, 451
Pitch, 226
Plate, vibrations of, 208213
Point source, 313
Poisson's ratio, 209
Polar coordinates, 174, 187, 398, 409
Polarization, electrical, 39
Pole, of complex function, 13, 45
Porosity, 363
Porous surface, 363366
Power, absorbed by wall, 367368, 385,
388
lost to friction, 33, 36
radiated, from cylinder, 299, 302, 305
from horn, 272, 278, 280, 282
from opening, 235
from piston, in plane, 338, 344
from simple source, 313
from sphere, 313, 318, 321, 322
from various sources, 228
scattered, by cylinder, 351
by sphere, 355
Power series, 4
Pressure, of air, 218, 452
mean square of, 414
in tube, 240
maxima and minima of, 241
Pressure level, 226, 415
Principal wave, in duct, 308, 371
Pulse, of sound, frequency distribution
of, 230
in room, 126
in tube, 261264, 291
Q
Q of system, 25, 403
Quadrupole source, 377
Quality, tone, of sound, from bar, 158
dependence on intensity, 228
from string, 86
R
Radiation amplitudes, for cylinder, 301,
449
for sphere, 320, 450
Radiation of sound, from cylinder,
298305
from dipole, 318
from piston, in plane, 326336
in sphere, 323326
in tube, 309311
from distributed source, 313
from dynamic speaker, 338—344
from open tube, 246247, 336338
from simple source, 313
from sphere, 311—326
from vibrating wire, 299300
Radiation load, due to medium, 38, 104
on diaphragm, 193, 198, 200
on dipole source, 319
on piston in sphere, 324
in wall, 332, 451
on wire, 300
(See also Acoustic impedance)
Radius of gyration, 152
Reactance, acoustic (see Acoustic react
ance)
mechanical, 29
Reaction, of surface, to sound, 310,
333
extended and local, 361
Reciprocity, principle of, 63, 352
Rectangular membrane, 177—183, 199
Reed instruments, 248—253
Reed motion, 249
Reflection, of waves, from absorbing
surface, 366
from obstacles, 347, 355
on string, 76, 134
Residue of function at pole, 13
Resistance, acoustic (see Acoustic resist
ance)
mechanical, 24, 29
Resistance constant, 24
Resistancecontrolled vibrations, 33
Resistivity, flow, 363
Resonance, 31
cavity, 228, 258261
of coupled oscillators, 64
of horn, 283, 286287
of membrane, 202
of oscillator, 31
of plate, 212
of room, 381, 418420
466
VIBRATION AND SOUND
Resonance, of string, 95, 99
of tank, 235
of tube, 245, 248, 252, 253, 256
Resonance peak, halfbreadth of, 68,
419
Resonator, Helmholtz, 235
Response, of microphone, 205, 212, 358
of room, 395, 419, 421
transient (see Transient vibrations)
Reverberation, 382, 386, 401
Reverberation time, 387
Ribbon microphone, 353, 378
Rigidity, of panel, 361
Room, cubical, 396
cylindrical, 398401, 409
nonrectangular, 397401
rectangular, 389394, 402408, 418
429
response of, 395, 419, 421
reverberation in, 386388
S
Salmon, Vincent, 271
Scattered and reflected sound, 347, 350
Scattering, of sound, 346347
from absorbing patches, 410, 412
from cylinder, 347352
from sphere, 354357
Scattering shift, of phase, for cylinder,
348, 449
for sphere, 354, 450
Separation of variables, 179
Series, of characteristic functions, 108,
183, 415
Fourier, 85, 97
power, 4
Series coefficients, 87, 108
Shadow, acoustic, 350
Shearing force, in bar, 153
Simple harmonic vibrations, 23, 80,
156
Simple source (see Source)
Sine, 4, 438439
Sound power of various sources, 228
Source, dipole, 318319
simple, 312313, 415, 418421
on sphere, 321322
Source function, 313
Specific acoustic impedance (see Acous
tic impedance)
Specific heats of gas, 220, 452
Specific impedance of surface, 306
Spectrum level, 229
Sphere, radiation from, 311326
scattering from 354—357
Spherical Bessel functions, 316; 446
Spherical coordinates, 296, 311
Spherical room, 401
Spring supports, for string, 146
Spring, 3
mass on, 3, 2033
Standing waves, in air in tube, 240
in room, 381, 389
on string, 83, 140
Standing wave ratio, 142, 456
Statistical analysis, of sound in room,
382385
Steady state, 29, 31
Step function, 47
Stiff string, 166170
Stiffness constant, 21
Stiffnesscontrolled vibrations, 33
Strain in crystal, 39
Strength of simple source, 313
Stress, in crystal, 39
String, damped, 104106
energy of, 8990
forced motion of, 91104, 129132,
138141
free vibration of, 8490
hanging, 149
impedance of, 91, 95, 126129
characteristic, 93
nonuniform, 111113
plucked and struck, 80, 87
stiff, 166
waves in, 7280, 134136
weights on, 68
whirling, 114121
String support, effect of, 76, 134
impedance of, 133
Surface, acoustic impedance of, 362
porous, impedance of, 363366
sound absorption at, 360368, 388
Susceptance, acoustic (see Acoustic
susceptance)
mechanical, of oscillator, 35
INDEX
467
Susceptaace, mechanical, of string sup
port, 144
Susceptibility, dielectric, 40
Symmetry of room, effect of, 395
Tangential waves, 391, 405
Tank, 235
Temperature change due to sound, 222,
288
Tension, on membrane, 173
on string, 72
variable, 116
Thermodynamic relations for gas, 220
221
Threshold, of hearing, 227
of pain, 227
Transducer, electromagnetic, 3438
electromechanical, 3842
Transfer admittance and impedance,
of coupled oscillator, 63
for string, 93, 95, 105
"for walls of room, 411
Transform, Fourier, 16, 43, 93
Laplace, 51, 293, 428
Transformation, conformal, 136
Transient force, 16, 42, 93, 110
Transient vibrations, 4252
of air, in horn, 287288, 293
in room, 386, 426129
in tube, 261264, 372
of coupled oscillators, 64
of diaphragm, 206208
of oscillator, 29, 4452
of radiation, from piston, 344346
of string, 93, 100106, 110, 132, 145,
147148
Transmission, of sound, through ducts,
368376
inside cylinders, 305311
inside horns, 265283
through porous material, 365366
in room, 418425
inside tubes, 224, 237240, 254
258
Transmission coefficient, 273
of horn, 273, 281, 282
of piston, 338
Transmission line, analogous, 254, 366
Trigonometric functions, 4, 438
Tube, analogy of, with transmission line,
254
cavity resonance in, 258—261
closed, 244245
constriction in, 234
effective length of, 234
flaring (see Horn)
open, 246253, 255258
waves in, 222225, 238244, 305311,
368376
Tuning fork, 158
U
Unit impulse function, (see Delta
function)
Unit step function, (see Step function)
Units, physical, 2
Velocity, particle (see Particle velocity)
wave (see Wave velocity)
Velocity amplitude, 23
Vibrations, damped (3ee Damped vibra
tions)
free, 2023
of air, in room, 390401
of bar, 156160, 168
of coupled oscillators, 5459
energy of, 23, 58, 89, 214, 223, 296
of membrane, 180183, 187191
of plate, 208
simple harmonic, 2023
of string, 8491, 112118
forced (see Forced vibrations)
Vibrator (see Oscillator)
Vowel sounds, 232
W
Wall coefficient (see Coefficient)
Warble tone, 379
Water, acoustic constants of, 452
Wave, 7274
axial, tangential and oblique, 391
400, 405, 409
468
VIBRATION AND SOUND
Wave, in bar, 155156
circular, 184, 191
damped, 243, 307, 371
longitudinal and transverse, 217
in membrane, 172, 184186, 191
plane, 217, 266268
in porous material, 365366
principal, in duct, 308, 371
standing, 83, 140, 240, 381, 389
in string, 7180, 9293
in wire, 167
Wave admittance, 91
Wave equation, 81
for air, 221, 294
for horn, 269
for membrane, 174
for porous material, 364
separation of, 179, 268
for string, 81
Wave impedance, of principal wave, in
duct, 372
for string, 91, 139
(See also Acoustic impedance)
Wave number, 225
Wave velocity, 72
Wave velocity, in air, 222, 452
in bar, 154
in membrane, 177, 187
in* string, 72
in tube, 307, 371374
in water, 452
Wavelength, 83, 134, 225
Weighted string, 68, 71
Width, of pulse, 231
of resonance peak, 68
Whirled bar, 171
Whirled string, 114
Wind instruments, 253, 285
Wire, radiation from, 299
vibrations of, 166169
Work, 2
Young's modulus of elasticity, 152
Z
Zero response of diaphragm, 202203,
212213
hrt
Ll
^