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^arbart Collrgr l^tbrarg
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HORACE APPLETON HAVEN,
OF PORTSMOUTH, N, IL
(ClnH of 1»4«.l
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THE
THEOBY OF SOUND.
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THE
THEORY OF SOUND
BY
JOHN WILLIAM STRUTT, BARON RAYLEIGH, Sc.D., F.R.S.
HONORABT FELLOW OP TRINITY COLLEGE, CAMBRIDGE.
IN TWO VOLUMES
VOLUME I.
SECOND EDITION REVISED AND ENLARGED
MACMILLAN AND CO.
AND NEW YORK
1894
[All RighU reserved,]
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First Edition prinUd 1877.
Second Edition revised and enlarged 1894.
<7A]fBBU>aB: PBINTBD BY 0. J. CLAT, M.A. & SOHS,
AT THE UN1VBB61TT PBE88.
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PREFACE.
IN the work, of which the present volume is an instalment,
my endeavour has been to lay before the reader a connected
exposition of the theory of sound, which should include the
more important of the advances made in modem times by Mathe-
maticians and Physicists. The importance of the object which
I have had in view will not, I think, be disputed by those com-
petent to judge. At the present time many of the most valuable
contributions to science are to be found only in scattered
periodicals and transactions of societies, published in various
parts of the world and in several languages, and are often
practically inaccessible to those who do not happen to live in
the neighbourhood of large public libraries. In such a state of
things the mechanical impediments to study entail an amount
of unremunerative labour and consequent hindrance to the
advancement of science which it would be difficult to over-
estimate.
Since the well-known Article on Sound in the EncyclopoBdia
Metropolitana, by Sir John Herschel (1845), no complete work
has been published in which the subject is treated mathemati-
cally. By the premature death of Prof. Donkin the scientific
world was deprived of one whose mathematical attainments in
combination with a practical knowledge of music qualified him
in a special manner to write on Sound. The first part of his
Acoustics (1870), though little more than a firagment, is sufficient
to shew that my labours would have been unnecessary had Prof.
Donkin lived to complete his work.
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VI PREFACE.
In the choice of topics to be dealt with in a work on Sound,
I have for the most part followed the example of my predecessors.
To a great extent the theory of Sound, as commonly understood,
covers the same ground as the theory of Vibrations in general ;
but, unless some limitation were admitted, the consideration of
such subjects as the Tides, not to speak of Optics, would have
to be included. As a general rule we shall confine ourselves to
those classes of vibrations for which our ears afford a ready
made and wonderfully sensitive instrument of investigation.
Without ears we should hardly care much more about vibrations
than without eyes we should care about light.
The present volume includes chapters on the vibrations of
systems in general, in which, I hope, will be recognised some
novelty of treatment and results, followed by a more detailed
consideration of special systems, such as stretched strings, bars,
membranes, and plates. The second volume, of which a con-
siderable portion is already written, will commence with aerial
vibrations.
My best thanks are due to Mr H. M. Taylor of Trinity College,
Cambridge, who has been good enough to read the proofs. By
his kind assistance several errors and obscurities have been
eliminated, and the volume generally has been rendered less im-
perfect than it would otherwise have been.
Any corrections, or suggestions for improvements, with which
my readers may favour me will be highly appreciated.
TsBLiNa Place, Withax,
April, 1877.
IN this second edition all corrections of importfince are noted,
and new matter appears either as fresh sections, e.g. § 32 a,
or enclosed in square brackets [ ]. Two new chapters X a, X B
are interpolated, devoted to Curved Plates or Sheila, and to
Electrical Vibrations. Much of the additional matter relates to
the more difficult parts of the subject and will be passed over
by the reader on a first perusal.
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PREFACE. Vll
In the mathematical investigations I have usually employed
such methods as present themselves naturally to a physicist.
The pure mathematician will complain, and (it must be confessed)
sometimes with justice, of deficient rigour. But to this question
there are two sides. For, however important it may be to
maintain a uniformly high standard in pure mathematics, the
physicist may occasionally do well to rest content with argu-
ments which are fairly satisfectory and conclusive from his point
of view. To his mind, exercised in a different order of ideas,
the more severe procedure of the pure mathematician may appear
not more but less demonstmtive. And further, in many cases
of difficulty to insist upon the highest standard would mean
the exclusion of the subject altogether in view of the space
that would be required.
In the first edition much stress was laid upon the establish-
ment of general theorems by means of Lagi-ange's method, and
I am more than ever impressed with the advantages of this
procedure. It not unfrequently happens that a theorem can be
thus demonstrated in all its generality with less mathematical
apparatus than is required for dealing with particular cases by
special methods.
During the revision of the proof-sheets I have again had the
very great advantage of the cooperation of Mr H. M. Taylor,
until he was unfortunately compelled to desist. To him and
to several other friends my thanks are due for valuable sug-
gestions.
July, 1894.
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CONTENTS.
CHAPTER I.
PAGE
§§ 1 — 27 INTRODUCTION .... 1
Sound due to Vibrations. Finite velocity of Propagation. Velocity inde-
pendent of Pitch. Begnault's experiments. Sound propagated in water.
Wheatstone^s experiment. Enfeeblement of Sound by distance. Notes
and Noises. Musical notes due to periodic vibratioffitr Siren of Gagniard
de la Tour. Pitch dependent upon Period. Relationship between
musical notes. The same ratio of periods corresponds to the same
interval in all parts of the scale. Harmonic scales. Diatonic scales.
Absolute Pitch. Necessity of Temperament. Equal Temperament.
Table of Frequencies. Analysis of Notes. Notes and Tones. Quality
dependent upon harmonic overtones. Resolution of Notes by ear un-
certain. Simple tones correspond to simple pendulous vibrations.
CHAPTER II.
§§28 — 42 a HARMONIC MOTIONS . . . .19
Composition of harmonic motions of like period. Harmonic Curve. Com-
position of two vibrations of nearly equal period. Beats. Fourier's
Theorem. [Beats of approximate consonances.] Vibrations in perpen-
dicular directions/ Lissajous' Cylinder. Lissajous* Figures. Black-
burn's pendulum. Ealeidophone. Optical methods of composition
and analysis. The vibration microscope. Intermittent Illumination.
[Resultant of a large number of vibrations whose phases are accidentally
distributed.]
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X CONTENTS.
CHAPTER III.
PAOK
§§ 43—68 d SYSTEMS HAVING ONE DEGREE OF FREEDOM . 43
Independence of amplitude and period. Frictional force proportional to
velocity. Forced vibrations. Principle of Superposition. Beats due
to superposition of forced and free . vibrations. Various degrees of
damping. String with Load. Method of Dimensions. Ideal Tuning-
fork. Forks give nearly pure tones. Forks as standards of pitch.
[Dependence upon temperature. Slow ver$us quick heats.] Scheibler's
methods of tuning. Scheibler's Tonometers. Compound Pendulum.
Forks driven by electro-magnetism. [Phonic wheel.] Fork Interrupter.
Resonance. [Intermittent vibrations.] General solution for one degree
of freedom. [Instability.] Terms of the second order give rise to
derived tones. [Maintenance. Methods of determining absolute pitch.]
CHAPTER IV.
§§ 69 — 95 VIBRATING SYSTEMS IN GENERAL . .91
Generalized co-ordinates. Expression for potential energy. Statical theo-
rems. Initial motions. Expression for kinetic energy. Reciprocal
theorem. Thomson's [Kelvin's] theorem. Lagrange's equations. The
dissipation function. Coexistence of small motions. Free vibrations
without friction. Normal co-ordinates. The free periods fulfil a
stationary condition. An accession of inertia increases the free periods.
A relaxation of spring increases the free periods. The greatest free
period is an absolute maximum. Hypothetical types of vibration.
Example from string. Approximately simple systems. String of
. variable density. Normal functions. Conjugate property. [Introduc-
tion of one constraint. Several constraints.] Determination of con-
stants to suit arbitrary initial conditions. Stokes' theorem.
CHAPTER V.
§§96 — 117 VIBRATING SYSTEMS IN GFJJERAL. . . 130
Oases in which the three functions T, F, V are simultaneously reducible to
sums of squares. Generalization of Young's theorem on the nodal
points of strings. Equilibrium theory. Systems started from rest as
defiected by a force applied at one point. Systems started from the
equilibrium configuration by an impulse applied at one point. Systems
started from rest as deflected by a force uniformly distributed. Influ-
ence of rmall frictional forces on the vibrations of a system. Solution
of the general equations for free vibrations. [Routh's theorems. In-
stability.] Impressed Forces. Principle of the persistence of periods.
Inexorable motions. Reciprocal Theorem. Application to free vibrations.
Statement of reciprocal theorem for harmonic forces. Applications.
Extension to oases in which the constitution of the system is a function
of the period. [Reaction at driving point.] Equations for two degrees
of freedom. Roots of determinantal equation. Intermittent vibrations.
March of periods as inertia is gradually increased. Reaction of a
dependent system.
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CONTENTS. XI
CHAPTER VI.
PAGE
§§118 — 148 c TRANSVERSE VIBRATIONS OF STRINGS . . 170
Law of extension of a string. Transverse vibrations. Solution of the pro-
blem for a string whose mass is concentrated in equidistant points.
Derivation of solution for continuous string. Partial differential equa-
tion. Expressions for V and T. Most general form of simple harmonic
motion. Strings with fixed extremities. General motion of a string
periodic. Mersenne's Laws. Sonometer. Normal modes of vibration.
Determination of constants to suit arbitrary initial circumstances. Case
of plucked string. Expressions for T and F in terms of normal co-ordi-
nates. Normal equations of motion. String excited by plucking.
Young's theorem. String excited by an impulse. Problem of piano-
forte string. Friction proportional to velocity. Comparison with equi-
librium theory. Periodic force applied at one point. Modifications due
to yielding of the extremities. Proof of Fourier's theorem. Effects
of a finite load. Correction for rigidity. Problem of violin string.
Strings stretched on curved surfaces. Solution for the case of the
sphere. Correction for irregularities' of density. [Arbitrary displace-
ment of every period.] Theorems of Sturm and Liouville for a string
of variable density. [Density proportional to a;-'. Nodes of forced vibra-
tions.] Propagation of waves along an unlimited string. Positive and
negative waves. Stationary Vibrations. Reflection at a fixed point
Deduction of solution for finite string. Graphical method. Progressive
wave with friction. [Reflection at a junction of two strings. Gradual
transition. Effect of imperfect flexibility.]
CHAPTER VIL
§§149—159 LONGITUDINAL AND TORSIONAL VIBRATIONS
OF BARS 242
Classification of the vibrations of Bars. Differential equation for longi-
tudinal vibrations. Numerical values of the constants for steel. Solu-
tion for a bar free at both ends. Deduction of solution for a bar with
one end free, and one fixed. Both ends fixed. Influence of small load.
Solution of problem for bar with large load attached. [Reflection at a
junction.] Correction for lateral motion. Savart's '*son rauque."
Differential equation for torsional vibrations. Comparison of velocities
of longitudinal and torsional waves.
CHAPTER VIII.
§§160— 192 a LATERAL VIBRATIONS OF BARS . . . 255
Potential energy of bending. Expression for kinetic energy. Derivation
of differential equation. Terminal conditions. General solution for
a harmonic vibration. Conjugate property of the normal functions.
Values of integrated squares. Expression of F in terms of normal
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Xll CONTENTS.
PAGE
oo-ordinates. Normal equations of motion. Determination of constants
to suit initial conditions. Case of rod started by a blow. Bod started
from rest as deflected by a lateral force. In certain cases the series of
normal functions ceases to converge. Form of the normal functions for
a free-free bar. Laws of dependence of frequency on length and thick-
ness. [Numerical formuls for tuning-forks.] Case when both ends are
clamped. Normal functions for a clamped-free bar. Calculation of
periods. Comparisons of pitch. Discussion of the gravest mode of
vibration of a free-free bar. Three nodes. Four nodes. Gravest mode
for clamped-free bar. Position of nodes. Supported bar. Calculation
of period for clamped-free bar from hypothetical type. Solution of
problem for a bar with a loaded end. £ffect of additions to a bar.
Influence of irregularities of density. Correction for rotatory inertia.
Boots of functions derived linearly from normal functions. Formation
of equation of motion when there is permanent tension. Special ter-
minal conditions. Besultant of two trains of waves of nearly equal
period. Fourier^s solution of problem for infinite bar. [Circular Bing.]
CHAPTER IX.
§§193 — 213 a VIBRATIONS OF MEMBBANES . . 306
Tension of a membrane. Equation of motion. Fixed rectangular bound-
ary. Expression for V and T in terms of normal co-ordinates. Normal
equations of vibration* Examples of impressed forces. Frequency for
an elongated rectangle depends mainly on the shorter side. Cases in
which different modes of vibration have the same period. Derived
modes thence arising. Effect of slight irregularities. An irregularity
may remove indeterminateness of normal modes. Solutions applicable
to a triangle. Expression of the general differential equation by polar
co-ordinates. Of the two functions, which occur in the solution, one is
excluded by the condition at the pole. Expressions for Bessel's func-
tions. Formulie relating thereto. Table of the flrst two functions.
Fixed circular boundary. Conjugate property of the normal functions
without restriction of boundary. Values of integrated squares. Ex-
pressions for T and V in terms of normal functions. Normal equations
of vibration for circular membrane. Special case of free vibrations.
Vibrations due to a harmonic force uniformly distributed. [Force
applied locally at the centre.] Pitches of the various simple tones.
Table of the roots of Bessel's functions. Nodal Figures. Circular
membrane with one radius fixed. Bessers functions of fractional order.
Effect of small load. Vibrations of a membrane whose boundary is
approximately circular. In many cases the pitch of a membrane may
be calculated from the area alone. Of all membranes of equal area tbat
of circular form has the gravest pitch. Pitch of a membrane whose
boundary is an ellipse of small eccentricity. Method of obtaining limits
in cases that cannot be dealt with rigorously. Comparison of fre-
quencies in various cases of membranes of equal area. History of the
problem. Bourget's experimental investigations. [Kettle-drums. Nodal
curves of forced vibrations.]
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CONTENTS. XUl
CHAPTER X.
PAGE
§§ 214 — 235 a vibrations of plates . . 352
Potential Energy of Bending. Transformation of 8V. Superficial differ-
ential equation. Boundary conditions. Conjugate property of normal
functions. Transformation to polar co-ordinates. Form of general
solution continuous through pole. Equations determining the periods
for a free circular plate. EiichhofF's calctdations. Comparison with
observation. Badii of nodal circles. Oeneralization of solution. Ir-
regularities give rise to beats. [Oscillation of nodes.] Case of clamped,
or supported, edge. [Telephone plate.] Disturbance of Chladni's
figures. [Movements of sand.] History of problem. Mathieu's criti-
cisms. Rectangular plate with supported edge. Rectangular plate with
free edge. Boundary conditions. One special case (/li=0) is amenable
to mathematical treatment. Investigation of nodal figures. Wheat-
stone's application of the method of superposition. Comparison of
Wheatstone's figures with those really applicable to a plate in the case
/A=0. Gravest mode of a square plate. Calculation of period on hypo-
thetical type. Nodal figures inferred from considerations of symmetry.
Hexagon. Comparison between circle and square. Law connecting
pitch and thickness. In the case of a clamped edge any contraction of
the boundary raises the pitch. No gravest form for a free plate of
given area. In similar plates the period is as the linear dimension.
Wheatstone's experiments on wooden plates. Eoanig's experiments.
Vibrations of cylinder, or ring. Motion tangential as well as normal.
Relation between tangential and normal motions. Expressions for
kinetic and potential energies. Equations of vibration. Frequencies
of tones. Comparison with Chladni. [Fenkner's observations.] Tan-
gential friction excites tangential motion. Experimental verification.
Beats due to irregularities. [Examples of glass bells. Church bells.]
CHAPTER Xa.
§ 235 b — 235 h curved plates or shells . . 395
[Extensional Vibrations. Frequency independent of thickness. Inexten-
sional or flexural vibrations. Frequency proportional to thickness.
General conditions of inextension. Surface of second degree. Applica-
tion to sphere. Principal extensions of cylindrical surface. Potential
energy. Frequencies of extensional vibrations. Plane plate. Other
particular cases of cylinder. Potential energy of bending. Sphere.
Plane plate. Potential energy for cylindrical shell. Statical problems.
Frequency of flexural vibrations of cylindrical shell. Extensional
vibrations of spherical shell. Flexural vibrations of spherical shell.
Normal modes. Potential energy. Kinetic energy. Frequencies in case
of hemisphere. Saucer of 120°. References.]
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XIV CONTENTS.
CHAPTER Xb.
PAGE
§ 235 i — 235 y electrical vibrations . . . 433
[Calculation of periods. Forced vibrations. Insertion of a leyden equivalent
to a negative inductance. Initial currents in a secondary circuit. In-
versely as the number of windings. Reaction of secondary circuit.
Train of circuits. Initial currents alternately opposite in sign. Per-
sistences. Resistance and inductance of two conductors in parallel.
Extreme values of frequency. Contiguous wires. Several conductors in
parallel. Induction balance. Theory for simple harmonic currents. Two
conditions necessary for balance. Wheatstone's bridge. Generalized
resistance. Current in bridge. Approximate balance. Hughes' ar-
rangement. Interrupters. Inductometers. Symmetrical arrangement.
Electromagnetic screening. Cylindrical conducting core. Time-con-
stant of free currents. Induced electrical vibrations. Reaction upon
primary circuit. Induced currents in a wire. Maxwell's formulas.
Impedance. Kelvin's theory of cables. Heaviside's generalization.
Attenuation and distortion of signals. Bell's telephone. Push and
pull theory. Experiment upon bipolar telephone. Minimum current
audible. Microphone.]
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CHAPTEK I.
INTRODUCTION.
1. The sensation of sound is a thing aui generis, not com-
parable with any of our other sensations. No one can express
the relation between a sound and a colour or a smell. Directly
or indirectly, all questions connected with this subject must
come for decision to the ear, as the organ of hearing; and
from it there can be no appeal. But we are not therefore to
infer that all acoustical investigations are conducted with the
unassisted ear. When once we have discovered the physical
phenomena which constitute the foundation of sound, our ex-
plorations are in great measure transferred to another field lying
within the dominion of the principles of Mechanics. Important
laws are in this way arrived at, to which the sensations of the ear
cannot but conform.
2. Very cursory observation often suffices to shew that
sounding bodies are in a state of vibration, and that the phe-
nomena of sound and vibration are closely connected. When a
vibrating bell or string is touched by the finger, the sound ceases
at the same moment that the vibration is damped. But, in order
to affect the sense of hearing, it is not enough to have a vibrating
instrument ; there must also be an uninterrupted communication
between the instrument and the ear. A bell rung in vacuo, with
proper precautions to prevent the communication of motion,
remains inaudible. In the air of the atmosphere, however,
sounds have a universal vehicle, capable of conveying . them
without break from the most variously constituted sources to
the recesses of the ear.
3. The passage of sound is not instantaneous. When a gun
is fired at a distance, a very perceptible interval separates the
R. 1
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2 INTBODUCTION. [3.
report from the flash. This represents the time occupied by
sound in travelling from the gun to the observer, the retardation
of the flash due to the finite velocity of light being altogether
negligible. The first accurate experiments were made by some
members of the French Academy, in 1738. Cannons were fired,
and the retardation of the reports at different distances observed.
The principal precaution necessary is to reverse alternately the
direction along which the sound travels, in order to eliminate the
influence of the motion of the air in mass. Down the wind, for
instance, sound travels relatively to the earth faster than its
proper rate, for the velocity of the wind is added to that proper
to the propagation of sound in still air. For still dry air at a
temperature of O^C, the French observers found a velocity of 337
metres per second. Observations of the same character were
made by Arago and others in 1822 ; by the Dutch physicists Moll,
van Beek and Kuytenbrouwer at Amsterdam; by Bravais and
Martins between the top of the Faulhom and a station below;
and by others. The general result has been to give a somewhat
lower value for the velocity of sound — about 332 metres per
second. The effect of alteration of temperature and pressure on
the propagation of sound will be best considered in connection with
the mechanical theory.
4. It is a direct consequence of observation, that within wide
limits, the velocity of sound is independent, or at least very nearly
independent, of its intensity, and also of its pitch. Were this
otherwise, a quick piece of music would be heard at a little
distance hopelessly confused and discordant. But when the dis-
turbances are very violent and abrupt, so that the alteratious of
density concerned are comparable with the whole density of the
air, the simplicity of this law may be departed from.
6. An elaborate series of experiments on the propagation of
sound in long tubes (water-pipes) has been made by Regnault\
He adopted an automatic arrangement similar in principle to that
used for measuring the speed of projectiles. At the moment when
a pistol is fired at one end of the tube a wire convejdng an electric
current is ruptured by the shock. This causes the withdrawal of a
tracing point which was previously marking a line on a revolving
drum. At the further end of the pipe is a stretched membrane so
arranged that when on the arrival of the sound it yields to the
1 Mimmrti de VAcadimie de France, t. xzztil
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5.] VELOCITY OF SOUND. 3
impulse, the circuit, which was ruptured during the passage of the
sound, is recompleted. At the same moment the tracing point
falls back on the drum. The blank space left unmarked corre-
sponds to the time occupied by the sound in making the journey,
and, when the motion of the drum is known, gives the means of
detennining it. The length of the journey between the first wire
and the membrane is found by direct measurement. In these
experiments the velocity of sound appeared to be not quite inde-
pendent of the diameter of the pipe, which varied horn ©"'•lOS
to 1"**100. The discrepancy is perhaps due to friction, whose
influence would be greater in smaller pipes.
6. Although, in practice, air is usually the vehicle of sound,
other gases, liquids and solids are equally capable of conveying
it. In most cases, however, the means of making a direct measure-
ment of the velocity of sound are wanting, and we are not yet in
a position to consider the indirect methods. But in the case of
water the same difficulty does not occur. In the year 1826,
Colladon and Sturm investigated the propagation* of sound in the
Lake of Geneva. The striking of a bell at one station was
simultaneous with a flash of gunpowder. The observer at a
second station measured the interval between the flash and the
arrival of the sound, applying his ear to a tube carried beneath
the surface. At a temperature of 8®C., the velocity of sound in
water was thus found to be 1435 metres per second.
7. The conveyance of sound by solids may be illustrated by a
pretty experiment due to Wheatstone. One end of a metallic wire
is connected with the sound-board of a pianoforte, and the other
taken through the partitions or floors into another part of the
building, where naturally nothing would be audible. If a reso-
nance-boaixl (such as a violin) be now placed in contact with the
wire, a tune played on the piano is easily heard, and the sound
seems to emanate from the resonance-board. [Mechanical tele-
phones upon this principle have been introduced into practical
use for the conveyance of speech.]
8. In an open space the intensity of sound falls off with great
rapidity as the distance from the source increases. The same
amount of motion has to do duty over surfaces ever increasing as
the squares of the distance. Anything that confines the sound
will tend to diminish the falling off of intensity. Thus over the
flat surface of still water, a sound carries further than over broken
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4 INTRODUCTION. [8.
ground; the corner between a smooth pavement and a vertical
wall is still better ; but the most effective of all is a tube-like
enclosure, which prevents spreading altogether. The use of
speaking tubes to facilitate communication between the different
parts of a building is well known. If it were not for certain effects
(frictional and other) due to the sides of the tube, sound might
be thus conveyed with little loss to very great distances.
9. Before proceeding further we must consider a distinction,
which is of great importance, though not free from diflBculty.
Sounds may be classed as musical and unmusical; the former
for convenience may be called notes and the latter noises. The
extreme cases will raise no dispute; every one recognises the
difference between the note of a pianoforte and the creaking of a
shoe. But it is not so easy to draw the line of separation. In the
first place few notes are free from all unmusical accompaniment.
With organ pipes especially, the hissing of the wind as it escapes
at the mouth may be heard beside the proper note of the pipe.
And, secondly, many noises so far partake of a musical character
as to have a definite pitch. This is more easily recognised in a
sequence, giving, for example, the common chord, than by continued
attention to an individual instance. The experiment may be made
by drawing corks from bottles, previously tuned by pouring water
into them, or by throwing down on a table sticks of wood of suitable
dimensions. But, although noises are sonietimes not entirely
unmusical, and notes are usually not quite free from noise, there is
no difficulty in recognising which of the two is the simpler pheno-
menon. There is a certain smoothness and continuity about the
musical note. Moreover by sounding together a variety of notes —
for example, by striking simultaneously a number of consecutive
keys on a pianoforte — we obtain an approximation to a noise;
while no combination of noises could ever blend into a musical
note.
10. We are thus led to give our attention, in the first instance,
mainly to musical sounda These arrange themselves naturally
in a certain order according to pitch — a quality which all can
appreciate to some extent. Trained ears can recognise an enormous
number of gradations — more than a thousand, probably, within
the compass of the human voice. These gradations of pitch are
not, like the degrees of a thermometric scale, without special
mutual relations. Taking any given note as a starting point.
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10.] PITCH. 5
musicians can single out certain others, which hear a definite
relation to the first, and are known as its octave, fifth, &c. The
corresponding diflferences of pitch are called intervals, and are
spoken of as always the same for the same relationship. Thus,
wherever they may occur in the scale, a note and its octave are
separated by the interval of the octave. It will be our object later
to explain, so far as it can be done, the origin and nature of the
consonant intervals, but we must now turn to consider the physical
aspect of the question.
Since sounds are produced by vibrations, it is natural to
suppose that the simpler sounds, viz. musical notes, correspond to
periodic vibrations, that is to say, vibrations which after a certain
interval of time, called the period, repeat themselves with perfect
regularity. And this, with a limitation presently to be noticed,
is true.
11. Many contrivances may be proposed to illustrate the
generation of a musical note. One of the simplest is a revolving
wheel whose milled edge is pressed against a card. Each
projection as it strikes the card gives a slight tap, whose regular
recurrence, as the wheel turns, produces a note of definite pitch,
rising in the scale, cw the velocity of rotation increases. But the
most appropriate instrument for the fundamental experiments on
notes is undoubtedly the Siren, invented by Cagniard de la Tour.
It consists essentially of a stiflf disc, capable of revolving about its
centre, and pierced with one or more sets of holes, arranged at
equal intervals round the circumference of circles concentric with
the disc. A windpipe in connection with bellows is presented
perpendicularly to the disc, its open end being opposite to one of
the circles, which contains a set of holes. When the bellows are
worked, the stream of air escapes freely, if a hole is opposite to the
end of the pipe; but otherwise it is obstructed. As the disc turns,
a succession of puffs of air escape through it, until, when the
velocity is suflScient, they blend into a note, whose pitch rises
continually with the rapidity of the puffs. We shall have occasion
later to describe more elaborate forms of the Siren, but for our
immediate purpose the present simple arrangement will su£5ce.
12. One of the most important facts in the whole science is
exemplified by the Siren — namely, that the pitch of a note depends
upon the period of its vibration. The size and shape of the holes,
the force of the wind, and other elements of the problem may be
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6 INTRODUCTION. [12.
varied ; but if the number of puffs in a given time, such as one
second, remains unchanged, so also does the pitch. We may even
dispense with wind altogether, and produce a note by allowing
the comer of a card to tap against the edges of the holes, as they
revolve ; the pitch will still be the same. Observation of other
sources of sound, such as vibrating solids, leads to the same con-
clusion, though the difficulties are often such as to render
necessary rather refined experimental methods.
But in saying that pitch depends upon period, there
lurks an ambiguity, which deserves attentive consideration^
as it will lead us to a point of great importance. If a
variable quantity be periodic in any time t, it is also periodic
in the times 2t, 3t, &c. Conversely, a recurrence within a given
period T, does not exclude a more rapid recurrence within
periods which are the aliquot parts of t. It would appear
accordingly that a vibration really recurring in the time ^ (for
example) may be regarded as having the period t, and therefore
by the law just laid down as producing a note of the pitch defined
by T. The force of this consideration cannot be entirely evaded by
defining as the period the least time required to bring about a
repetition. In the first place, the necessity of such a restriction
is in itself almost sufficient to shew that we have not got to the
root of the matter ; for although a right to the period t may be
denied to a vibration repeating itself rigorously within a time ^r,
yet it must be allowed to a vibration that may differ indefinitely
little therefrom. In the Siren experiment, suppose that in one
of the circles of holes containing an even number, every alternate
hole is displaced along the arc of the circle by the same amount.
The displacement may be made so small that no change can be
detected in the resulting note; but the periodic time on which
the pitch depends has been doubled. And secondly it is evident
from the nature of periodicity, that the superposition on a vibra-
tion of period T, of others having periods Jt, Jt...&c., does not
disturb the period t, while yet it cannot be supposed that the
addition of the new elements has left the quality of the sound un-
changed. Moreover, since the pitch is not affected by their
presence, how do we know that elements of the shorter periods
were not there from the beginning?
13. These considerations lead us to expect remarkable rela-
tions between the notes whose periods are as the reciprocals of the
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13.] MUSICAL INTERVALS. 7
natural numbers. Nothing can be easier than to investigate the
question by means of the Siren. Imagine two circles of holes, the
inner containing any convenient number, and the outer twice as
many. Then at whatever speed the disc may turn, the period of
the vibration engendered by blowing the first set will necessarily
be the double of that belonging to the second. On making the
experiment the two notes are found to stand to each other in
the relation of octaves ; and we conclude that in passing from any
note to its octave, the frequency of vibration is doubled. A similar
method of experimenting shews, that to the ratio of periods 3 : 1
corresponds the interval known to musicians as the twelfth, made
up of an octave and a fifth; to the ratio of 4:1, the double
octave ; and to the ratio 5:1, the interval made up of two octaves
and a major third. In order to obtain the intervals of the fiffch
and third themselves, the ratios must be made 3 : 2 and 5 : 4
respectively.
14. From these expeiiments it appears that if two notes
stand to one another in a fixed relation, then, no matter at what
part of the scale they may be situated, their periods are in a
certain constant ratio characteristic of the relation. The same
may be said of their frequencies^, or the number of vibrations
which they execute in a given time. The ratio 2 : 1 is thus
characteristic of the octave interval. If we wish to .combine
two intervals, — for instance, starting from a given note, to take
a step of an octave and then another of a fifth in the same
direction, the coiTesponding ratios must be compounded :
2 3_3
1^2""r
The twelfth part of an octave is represented by the ratio v^2 : 1,
for this is the step which repeated twelve times leads to an
octave above the starting point. If we wish to have a measure
of intervals in the proper sense, we must take not the character-
istic ratio itself, but the logarithm of that ratio. Then, and then
only, will the measure of a compound interval be the sum of the
measures of the components.
1 A single word to denote the number of vibrations executed in the unit of time
is indispensable : I know no better than * frequency,* which was used in this sense
by Young. The same word is employed by Prof. Everett in his exceUent edition
of Desohaners Natural Philosophy,
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8 INTRODUCTION. [l5.
16. From the intervals of the octave, fifth, and third con-
sidered above, others known to musicians may be derived. The
difference of an octave and a fifth is called a fourth^ and has the
3 4
ratio 2-r^«K' This process of subtracting an interval from
the octave is called inverting it. By inverting the major third
we obtain the minor sixth. Again, by subtraction of a major
third from a fifth we obtain the minor third ; and from this by
inversion the major sixth. The following table exhibits side by
side the names of the intervals and the corresponding ratios of
frequencies :
Octave 2: 1
Fifth 3:2
Fourth 4:3
Major Third 5:4.
Minor Sixth 8:5
Minor Third 6 :5
Major Sixth 5:3
These are all the consonant intervals comprised within the
limits of the octave. It will be remarked that the corresponding
ratios are all expressed by means of snudl whole numbers, and
that this is more particularly the case for the more consonant
intervals.
The notes whose frequencies are multiples of that of a given
one, are called its harmonics, and the whole series constitutes
a harmonic scale. As is well known to violinists, they may all
be obtained from the same string by touching it lightly with the
finger at certain points, while the bow is drawn.
The establishment of the connection between musical intervals
and definite ratios of frequency — a fundamental point in Acoustics
— is due to Mersenne (1636). It was indeed known to the
Greeks in what ratios the lengths of strings must be changed
in order to obtain the octave and fifth; but Mersenne demon-
strated the law connecting the length of a string with the period
of its vibration, and made the first determination of the actual
rate of vibration of a known musical note.
16. On any note taken as a key-note, or tonic, a diatonic
scale may be founded, whose derivation we now proceed to ex-
plain. If the key-note, whatever may be its absolute pitch, be
called Do, the fifth above or dominant is Sol, and the fifth below
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16.] NOTATION. 9
or subdominant is Fa. The common chord on any note is pro-
duced by combining it with its major third, and fifth, giving the
ratios of frequency 1 • 7 • « or 4:5:6. Now if we take the
common chord on the tonic, on the dominant, and on the sub-
dominant, and transpose them when necessary into the octave
Ijring immediately above the tonic, we obtain notes whose fre-
quencies arranged in order of magnitude are :
Do Re Mi Fa Sol La Si Do
, 9 -5 4 3 5 15
8' 4' 3' 2' 3' 8 '
Here the common chord on Do is Do— Mi — Sol, with the
5 3
ratios 1 : -r : - ; the chord on Sol is Sol — Si — Re, with the ratios
2:-g-:2xg = l ^t^k; and the chord on Fa is Fa — La — Do,
still with the same ratios. The scale is completed by repeating
these notes above and below at intervals of octaves.
If we take as our Do, or key-note, the lower c of a tenor voice,
the diatonic scale will be
c d e f g a b c'.
Usage diflfers slightly as to the mode of distinguishing the
different octaves ; in what follows I adopt the notation of Helm-
holtz. The octave below the one just referred to is written with
capital letters — C, D, &c. ; the next below that with a suflSx —
C,, D„ &C. ; and the one beyond that with a double suffix — C,,, &c.
On the other side accents denote elevation by an octave — c', c",
&c. The notes of the four strings of a violin are written in this
notation, g — d' — a'— e". The middle c of the pianoforte is c'.
[In French notation c' is denoted by ut,.]
17. With respect to an absolute standard of pitch there has
been no uniform practice. At the Stuttgard conference in 1834,
c' = 264 complete vibrations per second was recommended. This
corresponds to a' = 440. The French pitch makes a' = 436. In
Handel's time the pitch was much lower. If c' were taken at 256
or 2*, all the c*s would have frequencies represented by powers
of 2. This pitch is usually adopted by physicists and acoustical
instrument makers, and has the advantage of simplicity.
The determination ab initio of the frequency of a given note is
an operation requiring some care. The simplest method in prin-
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10 INTRODUCTION. [l7.
ciple is by means of the Siren, which is driven at such a rate as to
give a note in unison with the given one. The number of turns
effected by the disc in one second is given by a counting apparatus,
which can be thrown in and out of gear at the beginning and end
of a measured interval of time. This multiplied by the number of
effective holes gives the required frequency. The consideration of
other methods admitting of greater accuracy must be deferred.
18. So long as we keep to the diatonic scale of c, the notes
above written are all that are required in a musical composition.
But it is frequently desired to change the key-note. Under these
cifcumstances a singer with a good natural ear, accustomed to
perform without accompaniment, takes an entirely fresh departure,
constructing a new diatonic scale on the new key-note. In this
way, after a few changes of key, the original scale will be quite
departed from, and an immense variety of notes be used. On an
instrument with fixed notes like the piano and organ such a
multiplication is iraprswsticable, and some compromise is necessary
in order to allow the same note to perform different functions.
This is not the place to discuss the question at any length; we
will therefore take as an illustration the simplest, as well as the
commonest case — modulation into the key of the dominant.
By definition, the diatonic scale of c consists of the common
chords founded on c, g and f. In like manner the scale of g con-
sists of the chords founded on g, d and c. The chords of c and g
are then common to the two scales ; but the third and fifth of d
introduce new notes. The third of d written f jt has a frequency
9 5 45
^ X J = ^ , and is far removed from any note in the scale of c.
9 3 27
But the fifth of d, with a frequency z^a^T^> differs but little
from a, whose frequency is ^ . In ordinary keyed instruments the
81
interval between the two; represented by ^ , and called a comma,
is neglected, and the two notes by a suitable compromise or '
temperament are identified.
19. Various systems of temperament have been used; the
simplest and that now most generally used, or at least aimed at,
is the equal temperament. On referring to the table of frequencies
for the diatonic scale, it will be seen that the intervals from Do to
Re, from Re to Mi, from Fa to Sol, from Sol to La, and from La
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19.]
EQUAL TEMPERAMENT.
11
9 10
to Si, are nearly the same, being represented by - or -q- ; while the
9
16
intervals firom Mi to Fa and from Si to Do, represented by j^ , are
about half as much. The equal temperament treats these ap-
proximate relations as exact, dividing the octave into twelve equal
parts called mean semitones. From these twelve notes the diatonic
scale belonging to any key may be selected according to the
following rule. Taking the key-note as the first, fill up the series
with the third, fifth, sixth, eighth, tenth, twelfth and thirteenth
notes, counting upwards. In this way all diflSculties of modulation
are avoided, as the twelve notes serve as well for one key as for
another. But this advantage is obtained at a sacrifice of true
intonation. The equal temperament third, being the third part of
an octave, is represented by the ratio v^ : 1, or approximately
1-2599, while the true third is 1'25. The tempered third is thus
higher than the true by the interval 126 : 125. The ratio of the
tempered fifth may be obtained from the consideration that seven
semitones make a fifth, while twelve go to an octave. The ratio is
T
therefore 2^"^ : 1, which = 1-4983. The tempered fifth is thus too
low in the ratio 1*4983 : 1*5, or approximately 881 : 882. This
error is insignificant ; and even the error of the third is not of
much consequence in quick music on instruments like the piano-
forte. But when the notes are held, as in the harmonium and
organ, the consonance of chords is materially impaired.
20. The following Table, giving the twelve notes of the chro-
matic scale according to the system of equal temperament, will be
convenient for reference'. The standard employed is a' = 440; in
c.
c.
C
c
c'
c"
c"'
c""
c
16-35
32-70
65-41
130-8
261-7
523-3
1046-6
2093-2
c«
17-32
34-65
69-30
138-6
277-2
544-4
1108-8
2217-7
D
18-35
36-71
73-42
146-8
293-7
587-4
1174-8
2349-6
^
19-44
38-89
77-79
155-6
311-2
622-3
1244-6
2489-3
r
20-60
41-20
82-41
164-8
329-7
659-3
1318-6
2637-3
F
21-82
43-65
87-31
174-6
349-2
698-5
1397-0
2794-0
n
23-12
46-25
92-50
185-0
3700
7400
1480-0
29601
G
24-50
49-00
98-00
I960
392-0
784-0
1568-0
3136-0
Q«
25-95
51-91
103-8
207-6
416-3
830-6
1661-2
3322-5 1
A
27-50
55-00
110-0
220-0
440-0
880-0
1760-0
3520-0 ;
AJ
29-13
58-27
116-5
233-1
466-2
932-3
1864-6
3729-2 1
B^
30-86
61-73
123-5
246-9
493-9
987-7
1975-5
3951-0 '
^ Zamminer, Die Musik und die mtuikaliichen Inttrumente, Giessen, 1855.
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12 INTRODUCTION. [20.
order to adapt the Table to any other absolute pitch, it is only
necessary to multiply throughout by the proper constant.
The ratios of the intervals of the equal temperament scale are
given below (Zamminer) : —
Note. Freqaency. | Note. Freqaency.
c =1-00000
Cjt 2^=1-05946
d 2^ = M2246
d$ 2^ = 1-18921
e 2^^=1-25992
A
fj{ 2^ = 1-41421
g 2^^=1-49831
gj 2^=1-58740
a 2^^7=1-68179
ajt 2^ = 1-78180
11
f 2^7=1-33484 I b 2^7=188775
c' = 2-000
21. Returning now for a moment to the physical aspect of the
question, we will assume, what we shall afterwards prove to be
true within wide limits, — that, when two or more sources of sound
agitate the air simultaneously, the resulting disturbance at any
point in the external air, or in the ear-passage, is the simple sum
(in the extended geometrical sense) of what would be caused by
each source acting separately. Let us consider the disturbance
due to a simultaneous sounding of a note and any or all of its
harmonics. By definition, the complex whole forms a note having
the same period (and therefore pitch) as its graveat element. We
have at present no criterion by which the two can be distinguished,
or the presence of the higher harmonics recognised. And yet— in
the case, at any rate, where the component sounds have an inde-
pendent origin — it is usually not difficult to detect them by the
ear, so as to effect an analysis of the mixture. This is as much as
to say that a strictly periodic vibration may give rise to a sensa-
tion which is not simple, but susceptible of further analysis. In
point of fisLct, it has long been known to musicians that under
certain circumstances the harmonics of a note may be heard along
with it, even when the note is due to a single source, such as a
vibrating string ; but the significance of the fact was not under-
stood. Since attention has been drawn to the subject, it has been
proved (mainly by the labours of Ohm and Helmholtz) that almost
all musical notes ai-e highly compound, consisting in fact of the
notes of a harmonic scale, from which in particular cases one or
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21.] NOTES AND TONES. 13
more members may be missing. The reason of the uncertainty
and difficulty of the analysis will be touched upon presently.
22. That kind of note which the ear cannot further resolve is
called by Helmholtz in German a ' ton* Tyndall and other recent
writei's on Acoustics have adopted ' tone' as an English equivalent,
— a practice which will be followed in the present work. The
thing is so important, that a convenient word is almost a matter
of necessity. Notes then are in general made up of tones, the
pitch of the note being that of the gravest tone which it contains.
23. In strictness the quality of pitch must be attached in the
first instance to simple tones only ; otherwise the difficulty of dis-
continuity before referred to presents itself. The slightest change
in the nature of a note may lower its pitch by a whole octave, as
was exemplified in the case of the Siren. We should now rather
say that the effect of the slight displacement of the alternate
holes in that experiment was to introduce a new feeble tone an
octave lower than any previously present. This is sufficient to
alter the period of the whole, but the great mass of the sound
remains very nearly as before.
In most musical notes, however, the fundamental or gravest
tone is present in sufficient intensity to impress its character on
the whole. The effect of the harmonic overtones is then to modify
the quality or character^ of the note, independently of pitch.
That such a distinction exists is well known The notes of a violin,
tuning fork, or of the human voice with its different vowel sounds,
&c., may all have the same pitch and yet differ independently of
loudness ; and though a part of this difference is due to accom-
panying noises, which are extraneous to their nature as notes, still
there is a part which is not thus to be accounted for. Musical
notes may thus be classified as variable in three ways: First, pitch.
This we have already sufficiently considered. Secondly, character,
depending on the proportions in which the harmonic overtones are
combined with the fundamental : and thirdly, loudness. This has
to be taken last, because the ear is not capable of comparing
(with any precision) the loudness of two notes which differ much
in pitch or character. We shall indeed in a future chapter give a
mechanical measure of the intensity of sound, including in one
system all gradations of pitch ; but this is nothing to the point.
^ German, 'Elangfarbe'— Fienoh, 'timbre.' The word 'character' is used in
this sense by Everett.
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14 INTRODUCTION. [23.
We are here concerned with the intensity of the sensation of
sound, not with a measure of its physical cause. The difference of
loudness is, however, at once recognised as one of more or less ; so
that we have hardly any choice but to regard it as dependent
cceteris paribue on the magnitude of the vibrations concerned.
24. We have seen that a musical note, as such, is due to a
vibration which is necessarily periodic; but the converse, it is
evident, cannot be true without limitation. A periodic repetition
of a noise at int^i-vals of a second — for instance, the ticking of a
clock — would not result in a musical note, be the repetition ever
so perfect. In such a case we may say that the fundamental tone
lies outside the limits of hearing, and although some of the
harmonic overtones would fall within them, these would not give
rise to a musical note or even to a chord, but to a noisy mass of
sound like that produced by striking simultaneously the twelve
notes of the chromatic scale. The experiment may be made with
the Siren by distributing the holes quite irregularly round the
circumference of a circle, and turning the disc with a moderate
velocity. By the construction of the instrument, everything
recurs after each complete revolution.
26. The principal remaining difficulty in the theory of notes
and tones, is to explain why notes are sometimes analysed by the
ear into tones, and sometimes not. If a note is really complex,
why is not the fact immediately and certainly perceived, and the
components disentangled ? The feebleness of the harmonic over-
tones is not the reason, for, as we shall see at a later stage of our
inquiry, they are often of surprising loudness, and play a prominent
part in music. On the other hand, if a note is sometimes perceived
as a whole, why does not this happen always ? These questions
have been carefully considered by Helmholtz\ with a tolerably
satisfactory result. The difficulty, such as it is, is not peculiar to
Acoustics, but may be paralleled in the cognate science of Physio-
logical Optics.
The knowledge of external things which we derive from the
indications of our senses, is for the most part the result of inference.
When an object is before us, certain nerves in our retinae are
excited, and certain sensations are produced, which we are
accustomed to associate with the object, and we forthwith infer its
presence. In the case of an unknown object the process is much
^ Tonempfindungeny 3rd edition, p. 9S.
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25.] ANALYSIS OF NOTES. 15
the same. We interpret the sensations to which we are subject so
as to form a pretty good idea of their exciting cause. From the
slightly different perspective views received by the two eyes we
infer, often by a highly elaborate process, the actual relief and
distance of the object, to which we might otherwise have had no
clue. These inferences are made with extreme rapidity and quite
unconsciously. The whole life of each one of us is a continued
lesson in interpreting the signs presented to us, and in drawing
conclusions as to the actualities outside. Only so far as we succeed
in doing this, are our sensations of any use to us in the ordinary
affairs of life. This being so, it is no wonder that the study of our
sensations themselves falls into the background, and that subjective
phenomena, as they are called, become exceedingly difficult of
observation. As an instance of thi^, it is sufficient to mention the
'blind spot' on the retina, which might a priori have been
expected to manifest itself as a conspicuous phenomenon, though
as a fact probably not one person in a hundred million would find
it out for themselves. The application of these remarks to the
question in hand is tolerably obvious. In the daily use of our ears
our object is to disentangle from the whole mass of sound that
may reach us, the parts coming from sources which may interest
us at the moment. When we listen to the conversation of a friend,
we fix our attention on the sound proceeding from him and
endeavour to grasp that as a whole, while we ignore, as far as
possible, any other sounds, regarding them as an interruption.
There are usually sufficient indications to assist us in making this
partial analysis. When a man speaks, the whole sound of his
voice rises and falls together, and we have no difficulty in recog-
nising its unity. It would be no advantage, but on the contrary
a great source of confusion, if we were to carry the analjrsis further,
and resolve the whole mass of sound present into its component
tones. Although, as regards sensation, a resolution into tones
might be expected, the necessities of our position and the practice
of our lives lead us to stop the analysis at the point, beyond
which it would cease to be of service in deciphering our sensa-
tions, considered as signs of external objects^
But it may sometimes happen that however much we may
wish to form a judgment, the materials for doing so are absolutely
1 Most probably the power of attending to the important and ignoring the
unimportant part of our sensations is to a great extent inherited — to how great an
extent we shall perhaps never know.
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16 INTRODUCTION. [25.
wanting. When a note and its octave are sounding close together
and with perfect uniformity, there is nothing in our sensations to
enable us to distinguish, whether the notes have a double or a
single origin. In the mixture stop of the organ, the pressing down
of each key admits the wind to a group of pipes, giving a note and
its first three or four harmonics. The pipes of each group always
sound together, and the result is usually perceived as a single
note, although it does not proceed from a single source.
26. The resolution of a note into its component tones is a
matter of very diflFerent difficulty with different individuals. A
considerable effort of attention is required, particularly at first ;
and, until a habit has been formed, some external aid in the shape
of a suggestion of what is to be listened for, is very desirable.
The difficulty is altogether very similar to that of learning to
draw. From the machinery of vision it might have been expected
that nothing would be easier than to make, on a plane surface, a
representation of surrounding solid objects ; but experience shews
that much practice is generally required.
We shall return to the question of the analysis of notes at a
later stage, after we have treated of the vibrations of strings, with
the aid of which it is best elucidated; but a very instructive
experiment, due originally to Ohm and improved by Helmholtz,
may be given here. Helmholtz^ took two bottles of the shape
represented in the figure, one about twice as large as the other.
These were blown by streams of air directed
across the mouth and issuing from gutta-percha
tubes, whose ends had been softened and pressed
flat, so as to reduce the bore to the form of a
narrow slit, the tubes being in connection with
the same bellows. By pouring in water when
the note is too low and by partially obstructing
the mouth when the note is too high, the bottles
may be made to give notes with the exact
interval of an octave, such as b and b'. The
larger bottle, blown alone, gives a somewhat muffled sound similar
in character to the vowel U ; but, when both bottles are blown,
the character of the resulting sound is sharper, resembling rather
the vowel O. For a short time after the notes had been heard
separately Helmholtz was able to distinguish them in the mixture ;.
^ Tonempfindungen, p. 109.
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26.] PENDULOUS VIBRATIONS. 17
but as the memory of their separate impressions fisMled, the higher
note seemed by degrees to amalgamate with the lower, which at
the same time became louder and acquired a sharper character.
This blending of the two notes may take place even when the high
note is the louder.
27. Seeing now that notes are usually compound, and that
only a particular sort called tones are incapable of further analysis,
we are led to inquire what is the physical characteristic of tones,
to which they owe their peculiarity? What sort of periodic vibra-
tion is it, which produces a simple tone? According to what
mathematical function of the time does the pressure vary in
the passage of the ear ? No question in Acoustics can be more
important.
The simplest periodic functions with which mathematicians
are acquainted are the circular functions, expressed by a sine or
cosine; indeed there are no others at all approaching them in
simplicity. They may be of any period, and admitting of no
other variation (except magnitude), seem well adapted to produce
simple tones. Moreover it has been proved by Fourier, that the
most general single-valued periodic function can be resolved into
a series of circular functions, having periods which are submultiples
of that of the given function. Again, it is a consequence of the
general theory of vibration that the particular type, now suggested
as corresponding to a simple tone, is the only one capable of
preserving its integrity among the vicissitudes which it may
have to undergo. Any other kind is liable to a sort of physical
analysis, one part being diflferently aflfected from another. If the
analysis within the ear proceeded on a different principle from that
effected according to the laws of dead matter outside the ear,
the consequence would be that a sound originally simple might
become compound on its way to the observer. There is no reason
to suppose that anything of this sort actually happens. When it
is added that according to all the ideas we can form on the subject,
the analysis within the ear must take place by means of a physical
machinery, subject to the same laws as prevail outside, it will be
seen that a strong case has been made out for regarding tones as
due to vibmtions expressed by circular functions. We are not
however left entirely to the guidance of general considerations like
these. In the chapter on the vibration of strings, we shall see
that in many cases theory informs us beforehand of the nature of
B. 2
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18 INTRODUCTION. [27-
the vibration executed by a string, and in particular whether any
specified simple vibration is a component or not Here we have
a decisive test. It is found by experiment that, whenever according
to theory any simple vibration is present, the corresponding tone
can be heard, but, whenever the simple vibration is absent, then
the tone cannot be heard. We are therefore justified in asserting
that simple tones and vibrations of a circular type are indissolubly
connected. This law was discovered by Ohm.
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CHAPTER II.
HARMONIC MOTIONS.
28. The vibrations expressed by a circular function of the
time and variously designated as simple, pendulous, or harmonic,
are so important in Acoustics that we cannot do better than devote
a chapter to their consideration, before entering on the dynamical
part of our subject. The quantity, whose variation constitutes
the ' vibration/ may be the displacement of a particle measured
in a given direction, the pressure at a fixed point in a fluid
medium, and so on. In any case denoting it by u, we have
w = acos
(v*-) (>^
in which a denotes the amplitude, or extreme value of • u ; r is
the periodic tims, or period, after the lapse of which the values
of u recur ; and e determines the phase of the vibration at the
moment from which t is measured.
Any number of harmonic vibrations of the sams period affect-
ing a variable quantity, compound into another of the same tj^e,
whose elements are determined as follows :
ti = Sa cos
(?-)
= cos — 2acos€ + sm — zasme
T T
= rco8(2^-6») (2),
if r={(2acoee)» + (2asine)«ji (3),
and tan ^ = Sa sin 6 -r Sa cos 6 (4).
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20 HARMONIC MOTIONS. [28.
For example, let there be two components,
t* = aco8( e\'^a coe( 6 I;
then r = {a« + a'«+2aa' cos (€-€)}* (5),
tantf = ; — ; -J (6).
acos6 + a cose
Particular cases may be noted. If the phases of the two com-
ponents agree,
. , ^ /27rt \
t^=s(a + ajcos( €j. y
If the phases differ by half a period,
w = (a — a ) cos ( € 1 ,
so that if a' "= a, t^ vanishes. In this case the vibrations are often
said to interfere, but the expression is rather misleading. Two
sounds may very properly be said to interfere, when they together
cause silence; but the mere superposition of two vibrations
(whether rest is the consequence, or not) cannot properly be so
called. At least if this be interference, it is difficult to say what
non-interference can be. It will appear in the course of this
work that when vibrations exceed a certain intensity they no
longer compound by mere addition; this mutual action might
more properly be called interference, but it is a phenomenon
of a totally different nature from that with which we are now
dealing.
Again, if the phases differ by a quarter or by three-quarters of
a period, cos (c — e') = 0, and
Harmonic vibrations of given period may be represented
by lines drawn from a pole, the lengths of the lines being pro-
portional to the amplitudes, and the inclinations to the phases
of the vibrations. The resultant of any number of harmonic
vibrations is then represented by the geometrical resultant of
the corresponding lines. For example, if they are disposed
symmeftrically round the pole, the resultant of the lines, or
vibrations, is zero.
29. If we measure off along an axis of x distances pro-
portional to the time, and take u for an ordinate, we obtain the
harmonic curve, or curve of sines.
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29.]
CX)MP08ITI0N.
21
u = aco8
(^-).
where X, called the wave-length, is written in place of t, both
quantities denoting the range of the independent variable corre-
sponding to a complete recurrence of the function. The harmonic
curve is thus the locus of a point subject at once to a uniform
motion, and to a harmonic vibration in a perpendicular direc-
tion. In the next chapter we shall see that the vibration of a
tuning fork is simple harmonic; so that if an excited tuning
fork be moved with uniform velocity parallel to the line of its
handle, a tracing point attached to the end of one of its prongs
describes a harmonic curve, which may be obtained in a permanent
form by allowing the tracing point to bear gently on a piece of
smoked paper. In Fig. 2 the continuous lines are two harmonic
curves of the same wave-length and amplitude, but of different
phases ; the dotted ^curve represents half their resultant, being
the locus of points midway between those in which the two
curves are met by any ordinate.
30. If two harmonic vibrations of different periods coexist^
/27rt \ , , /27rt A
I* = a cos ( 6 j -H a cos ( — -, — e j.
The resultant cannot here be represented as a simple harmonic
motion with other elements. If r and r be incommensurable, the
value of u never recurs ; but, if t and t' be in the ratio of two
whole numbers, u recurs after the lapse of a time equal to the
least common multiple of r and r'; but the vibration is not
simple harmonic. For example, when a note and its fifth are
sounding together, the vibration recurs after a time equal to
twice the period of the graver.
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22 HARMONIC MOTIONS [30.
One case of the composition of harmonic vibrations of different
periods is worth special discussion, namely, when the difference
of the periods is small. If we fix our attention on the course
of things diuring an interval of time including merely a few
periods, we see that the two vibrations are nearly the same as
if their periods were absolutely equal, in which case they would,
as we know, be equivalent to another simple harmonic vibration
of the same period. For a few periods then the resultant
motion is approximately simple harmonic, but the same har-
monic will not continue to represent it for long. The vibration
having the shorter period continually gains on its fellow, thereby
altering the difference of phase on which the elements of the
resultant depend. For simplicity of statement let us suppose
that the two components have equal amplitudes, frequencies
represented by m and n, where m — h is small, and that when
first observed their phases agree. At this moment their effects
conspire, and the resultant has an amplitude double of that of
the components. But after a time 1 -r 2 (m — n) the vibration
m will have gained half a period relatively to the other; and
the two, being now in complete disagreement, neutralize each
other. After a further interval of time equal to that above
named, m will have gained altogether a whole vibration, and
complete accordance is once more re-established. The resultant
motion is therefore approximately simple harmonic, with an
amplitude not constant, but varying from zero to twice that of
the components, the frequency of these alterations being m— n.
If two tuning forks with frequencies 500 and 501 be equally
excited, there is every second a rise and fall of sound corre-
sponding to the coincidence or opposition of their vibrations.
This phenomenon is called beats. We do not here fully discuss
the question how the ear behaves in the presence of vibrations
having nearly equal frequencies, but it is obvious that if the motion
in the neighbourhood of the ear almost cease for a considerable
fraction of a second, the sound must appear to fall. For reasons
that will afterwards appear, beats are best heard when the in-
terfering sounds are simple tonea Consecutive notes of the
stopped diapason of the organ shew the phenomenon very
well, at least in the lower parts of the scale. A permanent inter-
ference of two notes may be obtained by mounting two stopped
organ pipes of similar construction and identical pitch side by
side on the same wind chest. The vibrations of the two pipes
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30.] OF NEABLY EQUAL PERIOD. 23
adjust themselves to complete opposition, so that at a little
distance nothing can be heard, except the hissing of the wind.
If by a rigid wall between the two pipes one sound could be
cut off, the other would be instantly restored. Or the balance,
on which silence depends, may be upset by connecting the ear
with a tube, whose other end lies close to the mouth of one of the
pipes.
By means of beats two notes may be tuned to unison with
great exactness. The object is to make the beats as slow as
possible, since the number of beats in a second is equal to the
difference of the frequencies of the notes. Under favourable
circumstances. beats so slow as one in 30 seconds may be recog-
nised, and would indicate that the higher note gains only two
vibrations a minute on the lower. Or it might be desired merely
to ascertain the difference M the frequencies of two notes nearly
in unison, in which case nothing more is necessary than to count
the number of beats. It will be remembered that the difference
of frequencies does not determine the interval between the two
notes; that depends on the ratio of frequencies. Thus the
rapidity of the beats given by two notes nearly in unison is
doubled, when both are taken an exact octave higher.
Analytically
u = acos (2'rrmt — e) + a' cos (iimt — €'),
where m — n is small
Now cos (27171* — e') may be written
cos [2'rrmt — 27r (m — n) t — e'},
and we have
t* = rcos(27rm*— ^) (1),
where r" = a« + a''+ 2aa'cos{27r(m-n)* + 6' -6} (2),
^ asinc + a'sin {27r(m — n)* + 6'} .«v
tan u — ■ ; — ; rs — ; rr— — 7> \o).
a cos e + a cos [27r (m — n) f + e j
The resultant vibration may thus be considered as harmonic
with elements r and d, which are not constant but slowly varying
functions of the time, having the frequency m — n. The ampli-
tude r IB at its maximum when
cos {27r (m -7i) < + e' - €} = + 1,
and at its minimum when
COS {27r (m - n) * + 6' — 6} = — 1,
the corresponding values being a + a' and a — a respectively.
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24 HAKMONIC MOTIONS. [31.
31. Another case of great importance is the composition of
Vibrations corresponding to a tone and its harmonics. It is known
that the most general single-valued finite periodic function can
be expressed by a series of simple harmonics —
U = ao + l^i OnCOSl— 6nl (1),
a theorem usually quoted as Fourier's. Analytical proofs will be
found in Todhunter's Integral Calculus and Thomson and Taits
Natural Philosophy; and a line of argument almost if not quite
amounting to a demonstration will be given later in this work.
A few remarks are all that will be required here.
Fourier's theorem is not obvioua A vague notion is not un-
common that the infinitude of arbitrary constants in the series
of necessity endows it with the capacity of representing an arbi-
trary periodic function. That this is an error will be apparent,
when it is observed that the same argument would apply equally,
if one term of the series were omitted ; in which case the ex-
pansion would not in general be possibla
Another point worth notice is that simple harmonics are not
the only functions, in a series of which it is possible to expand
one arbitrarily given. Instead of the simple elementary term
we might take
formed by adding a similar one in the same phase of half the
amplitude and period. It is evident that these terms would
serve as well as the others; for
cos
f2mU \ ( /2'jmt \ 1 /^nt \]
1 ( /Svnt \ . 1 /Uimt \)
— ad infin.y
so that each term in Fourier's series, and therefore the sum of
the series, can be expressed by means of the double elementary
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31.] Fourier's theorem. 25
terms now suggested This is mentioned here, because students,
not being acquainted with other expansions, may imagine that
simple harmonic functions are by nature the only ones qualified
to be the elements in the development of a periodic function.
The reason of the preeminent importance of Fourier's series in
Acoustics is the mechanical one referred to in the preceding
chapter, and to be explained more fully hereafter, namely, that,
in general, simple harmonic vibrations are the only kind that are
propagated through a vibrating system without suffering decom-
position*
32. As in other cases of a similar character, e.g. Taylor's
theorem, if the possibility of the expansion be known, the co-
efficients may be determined by a comparatively simple process.
We may write (1) of § 31
W = ilo + 2^i ilnCOS-— -+2^.ifi„8in-— - (1).
T T
Multiplying by cos (inirt/r) or sin (2n7rt/T), and integrating
over a complete period fi:om ^ = to < = t, we find
. 2 r 2n7r< ,A
An = - I wcos at
T J T
T J T
An immediate integration gives
ilo = - Twcft (3),
T J
indicating that Aq is the mean value of v throughout the period.
The degree of convergency in the expansion of u depends in
general on the continuity of the function and its derivatives.
The series formed by successive differentiations of (1) converge
less and less rapidly, but still remain convergent, and arithmetical
representatives of the differential coefficients of u, so long as
these latter are everywhere finite. Thus (Thomson and Tait,
§ 77), if all the derivatives up to the m^ inclusive be free
from infinite values, the series for u is more convergent than
one with
(2).
for coefficients.
1 i 1 1 &c
* 2^' 3«' 4»»' '
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26 HARMONIC MOTIONS [32 a.
32 a. The general explanation of the b^ts heard when two
pure tones nearly in unison are sounded simultaneously has been
discussed in § 30. But the occurrence of beats is not confined to
the case of approximate unison, at least when we have to deal
with compound notes. Suppose for example that the interval
IB an octave. The graver note then usually includes a tone
coincident in pitch with the fundamental tone of the higher note.
If the interval be disturbed, the previously coincident tones
separate from one another, and give rise to beats of the same
firequency as if they existed alone. There is usually no difficulty
in observing these beats; but if one or both of the component
tones concerned be very faint, the aid of a resonator may be
invoked.
In general we may consider that each consonant interval is
characterized by the coincidence of certain component tones, and
if the interval be disturbed the previously coincident tones
give rise to beats. Of course it may happen in any particular
case that the tones which would coincide in pitch are absent from
one or other of the notes. The disturbance of the interval
would then, according to the above theory, not be attended
by beats. In practice faint beats are usually heard; but the
discussion of this phenomenon, as to which authorities are not
entirely agreed, must be postponed.
33. Another class of compounded vibrations, interesting from
the facility with which they lend themselves to optical observa-
tion, occur when two harmonic vibrations affecting the same par-
ticle are executed in perpendicular directions, more especially
when the periods are not only commensurable, but in the ratio
of two small whole numbers. The motion is then completely
periodic, with a period not many times greater than those of the
components, and the curve described is re-entrant. If u and v
be the co-ordinates, we may take
w = a cos (27m< — €), v = 6cos2im'^ (1).
First let us suppose that the periods are equal, so that n' ^n\
the elimination of t gives for the equation of the curve described,
representing in general an ellipse, whose position and dimensions
depend upon the amplitudes of the original vibrations and upon
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33.] IN PBRPENDICULAB DIRECTIONS. 27
the difference of their phases. If the phases differ by a quarter
period, cos € » 0, and the equation becomes
— 4--- = 1
In this case the axes of the ellipse coincide with those of
co-ordinates. If further the two components have equal ampli-
tudes, the locus degenerates into the circle
which is described with uniform velocity. This shews how a
uniform circular motion may be analysed into two rectilinear
harmonic motions, whose directions are perpendicular.
If the phases of the components agree, e » 0, and the ellipse
degenerates into the coincident straight lines
or if the difference of phase amount to half a period, into
ihtf'"-
When the unison of the two vibrations is exact, the elliptic
path remains perfectly steady, but in practice it will almost
always happen that there is a slight difference between the
periods. The consequence is that though a fixed ellipse represents
the curve described with sufficient accuracy for a few periods,
the ellipse itself gradually changes in correspondence with the
alteration in the magnitude of 6. It becomes therefore a matter
of interest to consider the system of ellipses represented by (2),
supposing a and b constants, but e variable.
Since the extreme values of u and v are ±a, ±b respectively,
the ellipse is in all cases inscribed in the rectangle whose sides
are 2a, 26. Starting with the phases in agreement, or e^O, we
have the ellipse coincident with the diagonal — 1~^' -^
€ increases from to ^v, the ellipse opens out until its equation
becomes
a»^6> ^•
From this point it closes up again, ultimately coinciding with
the other diagonal ~ + i == 0» corresponding to the increase of € ft^m
J^ to ^. After this, as € ranges from tt to 27r, the ellipse retraces
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28
HARMONIC MOTIONS,
[33.
its course until it again coincides with the first diagonal. The
sequence of changes is exhibited in Fig. 3.
F/G.3.
/
^;
\
7^
\
^^^\
C )
/"""
^ ^-^
1
The ellipse, having already four given tangents, is completely
determined by its point of contact P (Fig. 4) with the line v = 6.
A
F/ G. -4.
P A
/-^
"^ ^
'
^^
B'
*
\
In order to connect this with e, it is sufficient to observe that
when t; = &, cos27m^ = l; and therefore tt = acos€. Now if the
elliptic paths be the result of the superposition of two harmonic
vibrations of nearly coincident pitch, € varies uniformly with the
time, so that P itself executes a harmonic vibration along A A'
with a frequency equal to the difference of the two given fre-
quencies.
34. Lissajous^ has shewn that this sjrstem of ellipses may be
regarded as the different aspects of one and the same ellipse
described on the surfisice of a transparent cylinder. In Fig. 5
1 AwuOm de Chimie (8) lx. 147, 1857.
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34.]
LISSAJOUB CYLINDER.
29
A A' SB represents the cylinder, of which AS is a plane section.
Seen from an infinite distance in the direction of the common
tangent at il to the plane sections, the cylinder is projected into a
rectangle, and the ellipse into its diagonal. Suppose now that the
cylinder turns upon its axis, carrying the plane section with it.
Its own projection remains a constant rectangle in which the pro-
F/G. 6.
jection of the ellipse is inscribed. Fig. 6 represents the posi-
tion of the cylinder after a rotation through a right angle. It
appears therefore that by turning the cylinder round we obtain in
succession all the ellipses corresponding to the paths described by
a point subject to two harmonic vibrations of equal period and fixed
amplitudes. Moreover if the cylinder be turned continuously
with uniform velocity, which insures a harmonic motion for P,
we obtain a complete representation of the varjdng orbit de-
scribed by the point when the periods of the two components
differ slightly, each complete revolution answering to a gain or
loss of a single vibration^ The revolutions of the cylinder are
thus synchronous with the beats which would result from the
composition of the two vibrations, if they were to act in the same
direction.
36. Vibrations of the kind here considered are very easily
realized experimentally. A heavy pendulum-bob, hung fi^m a
fixed point by a long wire or string, describes ellipses under the
action of gravity, which may in particular cases, according to the
circumstances of projection, pass into straight lines or circles.
But in order to see the orbits to the best advantage, it is necessary
that they should be described so quickly that the impression
on the retina made by the moving point at any part of its course
has not time to fade materially, before the point comes round again
to renew its action. This condition is fulfilled by the vibration of
a silvered bead (giving by reflection a luminous point), which is
^ By a Tibrstion will always be meant in this work a complete oyole of changes.
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30
HARMONIC MOTIONS.
[35.
attached to a straight metallic wire (such as a knitting-needle)*
firmly clamped in a vice at the lower end. When the system is set
into vibration, the luminous point describes ellipses, which appear
as fine lines of light. These ellipses would gradually contract in
dimensions under the influence of friction until they subsided
into a stationary bright point, without undergoing any other
change^ were it not that in all probability, owing to some want
of symmetry, the wire has slightly differing periods according to
the plane in which the vibration is executed. Under these cir-
cumstances the orbit is seen to undergo the cycle of changes
already explained. #
36. So far we have supposed the periods of the component
vibrations to be equal, or nearly equal ; the next case in order of
simplicity is when one is the double of the other. We have
t^ = a cos (im-rrt — c), t; = 6 cos 2mrt.
The locus resulting from the elimination of t may be written
V
u
- = cose
a
i^i-^y^^-W^-t «■
which for all values of e represents a curve inscribed in the rect-
angle 2a, 26. If € « 0, or ir, we have
•^(•iS.
F/e.7
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36.]
CONSONANT INTERVALS.
31
representing parabolas. Fig. 7 shews the various curves for the
intervals of the octave, twelfth, and fifth.
To all these systems Lissajous' method of representation by
the transparent cylinder is applicable, and when the relative
phase is altered, whether from the different circumstances of
projection in different cases, or continuously owing to a slight
deviation from exactness in the ratio of the periods, the cylinder
will appear to turn, so as to present to the eye different aspects of
the same line traced on its surface.
37. There is no difficulty in arranging a vibrating system so
that the motion of a point shall consist of two harmonic vibrations
in' perpendicular planes, with their periods in any assigned ratio.
The simplest is that known as Blackburn's pendulum. A wire
ACB ia fastened at A and B, two fixed points at the same level.
The bob P is attached to its middle point by another wire CF.
For vibrations in the plane of the diagram, the point of suspension
is practically (7, provided that the wires are sufficiently stretched ;
but for a motion perpendicular to this plane, the bob turns about
D, carrying the wire ACB with it. The periods of vibration in
the principal planes are in the ratio of the square roots of CF and
DF. Thus if DO = 30P, the bob describes the figures of the
octave. To obtain the sequence of curves corresponding to
approximate unison, ACB must be so nearly tight, that CD is
relatively small.
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32 HARMONIC MOTIONS. [38.
38. Another contrivance called the kaleidophone was origin-
ally invented by Wheatstone. A straight thin bar of steel carrying-
a bead at its upper end is fastened in a vice, as explained in a
previous paragraph. If the section of the bar is square, or circular,
the period of vibration is independent of the plane in which it is
performed. But let us suppose that the section is a rectangle
with unequal sides. The stiffness of the bar — the force with
which it resists bending — is then greater in the plane of greater
thickness, and the vibrations in this plane have the shorter period.
By a suitable adjustment of the thicknesses, the two periods of
vibration may be brought into any required ratio, and the cor-
responding curve exhibited.
The defect in this ai*rangeraent is that the same bar will give
only one set of figures. In order to overcome this objection
the following modification has been devised. A slip of steel is
taken whose rectangular section is very elongated, so that as
regards bending in one plane the stiffness is so great as to amount
practically to rigidity. The bar is divided into two parts, and the
broken ends reunited, the two pieces being turned on one another
through a right angle, so that the plane, which contains the small
thickness of one, contains the great thickness of the other. When
the compound rod is clamped in a vice at a point below the junc-
tion, the period of the vibration in one direction, depending almost
entirely on the length of the upper piece, is nearly constant ; but
that in the second direction may be controlled by varying the
point at which the lower piece is clamped.
39. In this arrangement the luminous point itself executes
the vibrations which are to be observed ; but in Lissajous' form of
the experiment, the point of light remains really fixed, while its
image is thrown into apparent motion by means of successive
reflection from two vibrating mirrors. A small hole in an opaque
screen placed close to the flame of a lamp gives a point of light,
which is observed after reflection i)i the mirrors by means of a
small telescope. The mirrors, usually of polished steel, are attached
to the prongs of stout tuning forks, and the whole is so disposed
that wbeu the forks are thrown into vibration the luminous point
appears to describe harmonic motions in perpendicular directions,
owing to the angular motions of the reflecting surfaces. The
amplitudes and periods of these harmonic motions depend upon
those of the corresponding forks, and may be made such as to give
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39.] OPTICAL METHODS. 33
with enhanced brilliancy any of the figures possible with the
kaleidophone. By a similar arrangement it is possible to project
the figures on a screen. In either case they gradually contract as
the vibrations of the forks die away.
40. The principles of this chapter have received an important
application in the investigation of rectilinear periodic motions.
When a point, for instance a particle of a sounding string, is
vibrating with such a period as to give a note within the limits of
hearing, its motion is much too rapid to be followed by the eye ;
so that, if it be required to know the character of the vibration,
some indirect method must be adopted. The simplest, theo-
retically, is to compound the vibration under examination with a
uniform motion of translation in a perpendicular direction, as when
a tuning-fork draws a harmonic curve on smoked paper. Instead
of moving the vibrating body itself, we may make use of a revolv-
ing mirror, which provides us with an image in motion. In this
way we obtain a representation of the function characteristic of
the vibration, with the abscissa proportional to time.
But it often happens that the application of this method would
be difficult or inconvenient. In such cases we may substitute for
the uniform motion a harmonic vibration of suitable period in the
same direction. To fix our ideas, let us suppose that the point,
whose motion we wish to investigate, vibrates vertically with a
period r, and let us examine the result of combining with this a
horizontal harmonic motion, whose period is some multiple of t,
say, riT. Take a rectangiilar piece of paper, and with axes parallel
to its edges draw the curve representing the vertical motion (by
setting off abscissae proportional to the time) on such a scale that
the paper just contains n repetitions or waves, and then bend the
paper round so as to form a cylinder, with a re-entrant curve run-
ning round it. A point describing this curve in such a manner
that it revolves uniformly about the axis of the cylinder will
appear from a distance to combine the given vertical motion of
period T, with a horizontal harmonic motion of period nr. Con-
versely therefore, in order to obtain the representative curve of
the vertical vibrations, the cylinder containing the apparent path
must be imagined to be divided along a generating line, and
developed into a plane. There is less difficulty in conceiving the
cylinder and the situation of the curve upon it, when the adjust-
ment of the periods is not quite exact, for then the cylinder
B. 3
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34 HARMONIC MOTIONS. [40.
appears to turn, and the contrary motions serve to distinguish
those parts of the curve which lie on its nearer and further face.
41. The auxiliary harmonic motion is generally obtained
optically, by means of an instrument called a vibration-microscope
invented by Lissajous. One prong of a large tuning-fork carries
a lens, whose axis is perpendicular to the direction of vibration ;
and which may be used either by itself, or as the object-glass of
a compound microscope formed by the addition of an eye-piece
independently suppoiiied. In either case a stationary point i&
thrown into apparent harmonic motion along a line parallel to
that of the fork's vibration.
The vibration-microscope may be applied to test the rigour
and universality of the law connecting pitch and period. Thus
it will be found that any point of a vibrating body which gives
a pure miisical note will appear to describe a re-entrant curve,
when examined with a vibration-microscope whose note is in
strict unison with its own. By the same means the ratios of
frequencies characteristic of the consonant intervals may be
verified; though for this latter purpose a more thoroughly
acoustical method, to be described in a future chapter, may be
preferred.
42. Another method of examining the motion of a vibrating
body depends upon the use of intermittent illumination^ Suppose,
for example, that by means of suitable apparatus a series of
electric sparks are obtained at regular intervals t. A vibrating
body, whose period is also t, examined by the light of the sparks-
must appear at rest, because it can be seen only in one position.
If, however, the period of the vibration diflFer from t ever so
little, the illuminated position varies, and the body will appear
to vibrate slowly with a frequency which is the difference of that
of the spark and that of the body. The type of vibration can
then be observed with facility.
The series of sparks can be obtained from an induction-coil,
whose primary circuit is periodically broken by a vibrating fork,,
or by some other interrupter of sufficient regularity. But a better
result is afforded by sunlight rendered intermittent with the aid of
a fork, whose prongs carry two small plates of metal, parallel ta
the plane of vibration and close together. In each plate is a slit
^ Plateau, Bull, de VAead. roy. de Belgique, t. iii, p. 364, 1836.
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42.] INTERMITTENT ILLUMINATION. 35
parallel to the prongs of the fork, and so placed as to afford a
free passage through the plates when the fork is at rest, or passing
through the middle point of its vibrations. On the openAig so
formed, a beam of sunlight is concentrated by means of a burning-
glass, and the object under examination is placed in the cone of
rays diverging on the further side*. When the fork is made to
vibrate by an electro-magnetic arrangement, the illumination is cut
oflF except when the fork is passing through its position of equi-
librium, or nearly so. The flashes of light obtained by this method
are not so instantaneous as electric sparks (especially when a
jar is connected with the secondary wire of the coil), but in my
experience the regularity is more perfect. Care should be taken
to cut off extraneous light as far as possible, and the effect is then
very striking.
A similar result may be arrived at by looking at the vibrating
body through a series of holes arranged in a circle on a revolving
disc. Several series of holes may be provided on the same
disc, but the observation is not satisfactory without some pro-
vision for securing uniform rotation.
Except with respect to the sharpness of definition, the result is
the same when the period of the light is any multiple of that of
the vibrating body. This point must be attended to when the
revolving wheel is used to determine an unknown frequency.
When the frequency of intermittence is an exact multiple of
that of the vibration, the object is seen without apparent motion,
but generally in more than one position. This condition of things
is sometimes advantageous.
Similar effects arise when the frequencies of the vibrations
and of the flashes are in the ratio of two small whole numbers.
If, for example, the number of vibrations in a given time be half
as great again as the number of flashes, the body will appear
stationary, and in general double.
42 a. We have seen (§ 28) that the resultant of two isoperiodic
vibrations of equal amplitude is wholly dependent upon their phase
relation, and it is of interest to inquire what we are to expect
from the composition of a large number (n) of equal vibmtions
of amplitude unity, of the same period, and of phases accidentally
determined. The intensity of the resultant, represented by the
square of the amplitude § 245, will of course depend upon the
1 Topler, Phil. Mag. Jan. 1867.
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36 HARMONIC MOTIONS. [42 a.
precise manner in which the phases are distributed, and may vary
from n^ to zero. But is there a definite intensity which becomes
more and more probable when n is increased without limit ?
The nature of the question here mised is well illustrated by
the special case in which the possible phases are restricted to two
opposite phases. We may then conveniently discard the idea of
phase, and regard the amplitudes as at random positive or negative.
If all the signs be the same, the intensity is n^ ; if, on the other
hand, there be as many positive as negative, the result is zero.
But although the intensity may range from to n', the smaller
values are more probable than the greater.
The simplest part of the problem relates to what is called in
the theory of probabilities the " expectation " of intensity, that
is, the mean intensity to be expected after a great number of
trials, in each of which the phases are taken at random. The
chance that all the vibrations are positive is (^)*^, and thus the
expectation of intensity corresponding to this contingency is
{\Y.n^. In like manner the expectation corresponding to the
number of positive vibrations being (/i— 1) is
(irn(n-2)«,
and so on. The whole expectation of intensity is thus
^ n(n-l)(n-2) ^^_g^^ I ^^^
Now the sum of the (n + 1) terms of this series is simply n, as
may be proved by comparison of coefficients of a;* in the equivalent
forms
(e« + e-*)** = 2« (l-h ia:» + . . .)*»
= e*** + n e <*»-*>* + '^ , "^ - e('»-^J*+ . . ..
X . z
The expectation of intensity is therefore n, and this whether n be
great or small.
The same conclusion holds good when the phases are unre-
stricted. From (3) § 28, if Oi = Oa = . . . = 1,
r* = (cos €i + cos es +...)" H- (sin €i + sin Cj + . . . )*
= 71 + 22 cos (€,-€,) (2),
where under the sign of summation are to be included the cosines
of the ^(?i — 1) diflFerences of phase. When the phases are
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42 a.] PHASES AT RANDOM, 37
accidental, the sum is as likely to be positive as negative, and
thus the mean value of r* is n.
The reader must be on his guard here against a fallacy which
has misled some eminent authors. We have not proved that when
n is large there is any tendency for a single combination to give
an intensity equal to n, but the quite different proposition that in
a large number of trials, in each of which the phases are dis-
tributed at random, the mean intensity will tend more and more
to the value n. It is true that even in a single combination there
is no reason why any of the cosines in (2) should be positive
rather than negative. From this we may infer that when n is
increased the sum of the terms tends to vanish in comparison with
the number of terms ; but, the number of the terms being of the
order n^ we can infer nothing as to the value of the sum of the
series in comparison with n.
So far there is no diiBculty; but a complete investigation of
this subject involves an estimate of the relative probabilities of
resultants lying within assigned limits of magnitude. For example,
we ought to be able to say what is the probability that the
intensity due to a large number (n) of equal components is less
than ^n. This problem may conveniently be considered here, though
it is naturally beyond the reach of elementary methods. We will
commence by taking it under the restriction that the phases are
of two opposite kinds only.
Adopting the statistical method of statement, let us suppose
that there are an immense number JV of independent combinations,
each consisting of n unit vibrations, positive or negative, and com-
bined accidentally. When N is sufficiently large, the statistics
become regular; and the number of combinations in which the
resultant amplitude is found equal to x may be denoted by
N ,f{n, x), where /is a definite function of n and a?. Now suppose
that each of the N combinations receives another random contri-
bution of ± 1, and inquire how many of them will subsequently
possess a resultant x. It is clear that those only can do so which
originally had amplitudes x — l, or a?+l. Half of the former,
and half of the latter number will acquire the amplitude x, so
that the number required is
iNf(n,x-^l) + iNfin,x + l).
But this must be identical with the number corresponding to
n + 1 and x, so that
/(n-hl,^)«i/(n,^-l)^i/(»,^-hl) (3).
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38 HARMONIC MOTIONS. [42 a.
This equation of differences holds good for all integral values
of X and for all positive integral values of n. If /(n, x) be given
for one value of n, the equation suflSces to determine / (n, x) for
all higher integral values of n. For the present purpose the
initial value of n is zero. In that case we know that /(a?) = for
all values of x other than zero, and that when x = 0,/(0, 0) = 1.
The problem proposed in the above form is perfectly definite ;
but for our immediate object it suffices to limit ourselves to the
supposition that n is great, regarding /(n, x) -sa a continuous
function of continuous variables n and x, much as in the analogfMiS
problem ofg 120, 121, 122.
Writing (3) in the form
f(n + 1, x) ^/(n, x) = i/(n, a^ - 1) + if(n, x+l) --/(n, x). . .(4),
we see that the left-hand member may then be identified with
dfldn, and the right-hand member with i^^fjda^, so that under
these circumstances the differential equation to which (3) reduces
is of the well-known form
dn^ 2dx^ ^^^•
The analogy with the conduction of heat is indeed very close ;
and the methods developed by Fourier for the solution of problems
in the latter subject are at once applicable. The special condition
here is that initially, that is when n = 0, / must vanish for all
values of x other than zero. As may be verified by differentiation,
the special solution of (5) is then
f{n,x) = ^e-^l^ <«>•
in which il is an arbitrary constant to be determined from the
consideration that the whole number of combinations is N. Thus,
if dx be large in comparison with unity, the number of combina-
tions which have amplitudes between x and x-^-dxia
er^f^dx\
AN [^->
while -T- / er^f^dx = N,
In J -00
80 that in virtue of the known equality
r+oo
e-'^dz^^Jir,
J —OP
A . V2^ =. 1.
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42 a.] PHASES AT RANDOM. 39
The final result for the number of combinations which have
amplitudes between x and x + dxis accordingly
-n^—\^'^^""^ W-
V(27m) ^
The mean intensity is expressed by
1 /■+*
V(27rn)J_«
as before.
We will now pass on to the more important problem in which
the phases of the n unit vibrations are distributed at random over
the entire period. In each combination the resultant amplitude
is denoted by r and the phase (referred to a given epoch) by 6 ;
and rectangular coordinates are taken so that
4? = r cos Q, y = ^ sin Q,
Thus any point (a?, y) in the plane of reference represents a
vibration of amplitude r and phase d, and the whole system of
N vibrations is represented by a distribution of points, whose
density it is our object to determine. Since no particular phase
can be singled out for distinction, we know beforehand that the
density of distribution will be independent of Q.
Of the infinite number N of points we suppose that
^/K ^, y) dxdy
are to be found within the infinitesimal area dxdy, and we will
inquire as before how this number would be changed by the
addition to the n component vibrations of one more unit vibration
of accidental phase. Any vibration which after the addition is
represented by the point a?, y must before have corresponded to
the point
fl?' = a? — cos <^, y'ssy — sin<^,
where ^ represents the phase of the additional unit vibration.
And, if for the moment ^ be regarded as given, to the area dxdy
corresponds an equal area dafdy. Again, all values of ^ being
equally probable, the factor necessary under this head is d<t>/2'n:
Accordingly the whole number to be found in dxdy after the
superposition of the additional unit is
Ndxdypfin, x\ yO #/27r ;
and this is to be equated to
Ndxdyf{n + 1, x, y) ;
/(n+1, x, rj)=.j^f{n, x\ y')d^\%ir (8).
so
that
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40 HARMONIC MOTIONS. [42 a.
The value of /(w, x, y) is obtained by introduction of the
values of x\ y' and expansion :
/(-'.y')=/(x.y)-fcos^-|8in<?+igco8'<?
80 that
Also, n being very great,
/(n 4- 1, X, y) -f{n, x, y) « dfjdn ;
and (8) reduces to
dn" ^Kdx'^ df) ^"^^^
the usual equation for the conduction of heat in two dimensions.
In addition to (9), /has to satisfy the special condition of
evanescence when n = for all points other than the origin. The
appropriate solution is necessarily symmetrical round the origin,
and takes the form
/(n, a?,y) = il7i-^e-'«'+i^/« (10),
as may be verified by differentiation. The constant ul is to be
determined by the condition that the whole number is N. Thus
N^NAn-'jje'^^'^y'^l''dxdy^NA2im-'re'^f''rdr^7rAN;
and the number of vibrations within the area dxdy becomes
^ e-.^^l^'dxdy (11).
If we wish to find the number of vibrations which have
amplitudes between r and r + dr, we must introduce polar
coordinates and integrate with respect to 0, The required number
is thus
2Nrr^e'-^l''rdr (12)^
The result may also be expressed by saying that the probability
of a resultant amplitude between r and r'\-dr when a large
number n of unit vibrations are compounded at random is
2rr^e^l''rdr (13).
1 PhiU Mag, Aug. 18S0.
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42 a.]
PHASES AT RANDOM.
41
The mean intensity is given by
Jo
as was to be expected.
The probability of a resultant amplitude less than r is
2n
-f.
e-^l^rdr = 1 - e-*"'" .
(14).
or, which is the same thing, the probability of a resultant ampli-
tude greater than r is
c-^'/n (15).
The following table gives the probabilities of intensities less
than the fractions of n named in the first column. For example,
the probability of intensity less than n is '6321.
■05
•0488
•80
•5506
■10
■0952
100
•6321
■20
■1813
r50
•7768
■40
•3296
2^00
•8647
60
■4512
' 300
•9502
It will be seen that, however great n may be, there is a
i-easonable chance of considerable relative fluctuations of intensity
in different combinations.
If the amplitude of each component be a, instead of unity, as
we have hitherto supposed for brevity, the probability of a resultant
amplitude between r and r + dr is
2
—-e-^/^Wdr
.(16).
The result is thus a function of n and a only through na", and
would be unchanged if for example the amplitude became ^a and
the number 4n. From this it follows that the law is not altered,
even if the components have different amplitudes, provided always
that the whole number of each kind is very great; so that if there
be n components of amplitude a, nf of amplitude 0, and so on, the
probability of a resultant between r and r + dr is
(in
"■|»««+n'/3«+..
wa' + n'^ + ,
rdr
That this is the case may perhaps be made more clear by the
* consideration of a particular case. Let us suppose in the first
place that n+4n^ unit vibrations are compounded at random.
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42 HARMONIC MOTIONS. [42 a.
The appropriate law is given at once by (13) on substitution of
n + 4tn' for n, that is
2(n + 4nO-'e-^''/<'»+*^Vdr (18).
Now the combination of n-|-4n' unit vibrations may be re-
garded as arrived at by combining a random combination of n
unit vibrations with a second random combination of 4n' units,
and the second random combination is the same as if due to a
random combination of n' vibrations each of amplitude 2. Thus
(18) applies equally well to a random combination of (n + ?i')
vibrations, n of which are of amplitude unity and nf of ampli-
tude 2.
Although the result has no application to the theory of vibra-
tions, it may be. worth notice that a similar method applies to the
composition in three dimensions of unit vectors, whose directions
are accidental. The equation analogous to (8) gives in place of
(9)
dn Q\d3? df"^ dsi*)'
The appropriate solution, analogous to (13), is
V(^)'
e-rVfn^d^ (18)^
expressing the probability of a resultant amplitude Ijdng between
r and r H- dr.
Here again the mean value of r*, to be expected in a great
number of independent combinations, is n.
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CHAPTER III.
SYSTEMS HAVING ONE DEGREE OP FREEDOM.
43. The material systems, with whose vibrations Acoustics is
concerned, are usually of considerable complication, and are sus-
ceptible of very various modes of vibration, any or all of which
may coexist at any particular moment. Indeed in some of the
most important musical instruments, as strings and organ-pipes,
the number of independent modes is theoretically infinite, and
the consideration of several of them is*essential to the most prac-
tical questions relating to the nature of the consonant chords.
Cases, however, often present themselves, in which one mode is
of paramount importance ; and even if this were not so, it would
still be proper to commence the consideration of the general
problem with the simplest case — that of one degree of freedom.
It need not be supposed that the mode treated of is the only one
possible, because so long as vibrations of other modes do not occur
their possibility under other circumstances is of no moment.
44. The condition of a system possessing one degree of free-
dom is defined by the value of a single co-ordinate w, whose origin
may be taken to correspond to the position of equilibrium. The
kinetic and potential energies of the system for any given position
are proportional respectively to ?i* and w* : —
T = \mu\ F = i/AU« (1).
where m and fi are in general functions of u. But if we limit
ourselves to the consideration of positions in the immedidte neigh'
hourhood of that corresponding to equilibrium, u is a small quantity,
and m and fi are sensibly constant. On this understanding we
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44 ONE DEGREE OF FREEDOM. [44.
now proceed. If there be no forces, either resulting from internal
friction or viscosity, or impressed on the system from without, the
whole energy remains constant. Thus
T+r= constant.
Substituting for T and V their values, and diflferentiating with
respect to the time, we obtain the equation of motion
wtt + zii^sO (2)
of which the complete integral is
. w = acos(nt — a) (3),
where ?i-=/A-*-m, representing a harmonic vibration. It will be
seen that the period alone is determined by the nature of the
system itself; the amplitude and phase depend on collateral cir-
cumstances. If the differential equation were exact, that is to
say, if T were strictly proportional to u\ and V to w", then, without
any restriction, the vibrations of the system about its configuration
of equilibrium would be accurately harmonic. But in the majority
of cases the proportionality is only approximate, depending on an
assumption that the displacement u is always small — ^how small
depends on the nature of the particular system and the degree of
approximation required ; and then of course we must be careful
not to push the application of the integral beyond its proper
limits.
But, although not to be stated without a limitation, the prin-
ciple that the vibrations of a system about a configuration of
equilibrium have a period depending on the structure of the
system and not on the particular circumstances of the vibration,
is of supreme importance, whether regarded from the theoretical
or the practical side. If the pitch and the loudness of the note
given by a musical instrument were not within wide limits in-
dependent, the art of the performer on many instruments, such
as the violin and pianoforte, would be revolutionized.
The periodic time _
. = 2^ = 2^^ (4).
so that an increase in m, or a decrease in /x, protracts the duration
of a vibration. By a generalization of the language employed in
ijie case of a material particle urged towards a position of equili-
brium by a spring, m may be called the inertia of the system, and
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44.] DISSIPAflVE FORCES. 45
ft the force of the equivalent spring. Thus an augmentation of
mass, or a relaxation of spring, increases the periodic time. By
means of this principle we may sometimes obtain limits for
the value of a period, which cannot, or cannot easily, be calculated
exactly.
45. The absence of all forces of a frictional character is an
ideal case, never realized but only approximated to in practice.
The original energy of a vibration is always dissipated sooner or
later by conversion into heat. But there is another source of loss,
which though not, properly speaking, dissipative, yet produces
results of much the same nature. Consider the case of a tuning-
fork vibrating in vacuo. The internal friction will in time stop
the motion, and the original energy will be transformed into
heat. But now suppose that the fork is transferred to an open
space. In strictness the fork and the air surrounding it consti-
tute a single system, whose parts cannot be treated separately.
In attempting, however, the exact solution of so complicated a
problem, we should generally be stopped by mathematical diffi-
culties, and in any case an approximate solution would be de-
sirable. The effect of the air during a few periods is quite insig-
nificant, and becomes important only by accumulation. We are
thus led to consider its effect as a disturbance of the motion which
would take place in vacuo. The disturbing force is periodic (to
the same approximation that the vibrations are so), and may be
divided into two parts, one proportional to the acceleration, and
the other to the velocity. The former produces the same effect as
an alteration in the mass of the fork, and we have nothing more
to do with it at present. The latter is a force arithmetically pro-
portional to the velocity, and always acts in opposition to the
motion, and therefore produces effects of the same character as
those due to friction. In many similar cases the loss of motion
by communication may be treated under the same head as that
due to dissipation proper, and is represented in the differential
equation with a degree of approximation sufficient for acoustical
purposes by a term proportional to the velocity. Thus
tt + /cti + n*u = (1)
is the equation of vibration for a system with one degree of
freedom subject to frictional forces. The solution is
u = ile-***cos{Vw^- J/c> . t-a] (2).
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46 ONE DEGREE OF FREEDOM. [45.
If the friction be so great that \/^ > n^ the solution changes its
form, and no longer corresponds to an oscillatory motion ; but in
all acoustical applications /e is a small quantity. Under these
circumstances (2) may be regarded as expressing a harmonic
vibration, whose amplitude is not constant, but diminishes in
geometrical progression, when considered after equal intervals of
time. The difference of the logarithms of successive extreme
excursions is nearly constant, and is called the Logarithmic Decre-
ment. It is expressed by \tcT, if t be the periodic time.
The frequency, depending on n' — \/c\ involves only the second
power o( k; so that to the first order of approximation the friction
has no effect on the peiHod, — a principle of very general application.
The vibration here considered is called the free vibration. It
is that executed by the system, when disturbed from equilibrium,
and then left to itself
46. We must now turn our attention to another problem, not
less important, — the behaviour of the system, when subjected to an
external force varying as a harmonic function of the time. In
order to save repetition, we may take at once the more general
case including friction. If there be no friction, we have only to
put in our results k = 0. The differential equation is
w+/«i + 7i*a= Ecospt (1).
Assume u = aco&(pt—€) (2),
and substitute :
a (n' — p') cos (p^ — e) — xpa sin (pt — e)
= ^cos € cos (p^ - e) - ^sin € sin {pt — e) ;
whence, on equating coefiicients of cos (pt — e), sin (pt — e),
a(n«-p») = ^cos€|
a.pK = Esm€} ^"^^^
so that the solution may be written
-ffsine / ^ V
w=~^<5os(p^-€) (4),
where tan€= - - — , (5).
71* — p* ^ ^
This is called sl forced vibration; it is the response of the system
to a force imposed upon it from without, and is maintained by the
continued operation of that force. The amplitude is proportional
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46.] FORCED VIBRATIONS. 47
to E — the magnitude of the force, and the period is the same
as that of the force.
Let us now suppose E given, and trace the effect on a given
system of a variation in the period of the force. The effects
produced in different cases are not strictly similar; because the
frequency of the vibrations produced is always the same as that of
the force, and therefore variable in the comparison which we are
about to institute. We may, however, compare the energy of the
system in different cases at the moment of passing through the
position of equilibrium. It is necessary thus to specify the moment
at which the energy is to be computed in each case, because the
total energy is not invariable throughout the vibration. During
one part of the period the system receives energy from the
impressed force, and during the remainder of the period yields it
back again.
From(4), ifu = 0,
energy oz u^ oc sin' e,
and is therefore a maximum, when sin e = 1, or, from (5),p = n. If
the maximum kinetic energy be denoted by T^, we have
2'=rosin»e (6).
The kinetic energy of the motion is therefore the greatest possible,
when the period of the force is that in which the system would
vibrate freely under the influence of its own elasticity (or other
internal forces), ttnthout friction. The vibration is then by (4)
and (5),
E
u = — sin nt ;
TIK
and, if k be small, its amplitude is very great. Its phase is a
quarter of a period behind that of the force.
The case, where p = n, may also be treated independently.
Since the period of the actual vibration is the same as that
natural to the system,
u + nhL = 0,
so that the differential equation (1) reduces to
KU^Ecospt,
whence by integration
u^ - \ co^pt dt== — smpt,
as before.
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48 ONE DEGREE OF FREEDOM. [46.
If p be less than n, the retardation of phase relatively to the
force lies between zero and a quarter period, and when p is greater
than n, between a quarter period and a half period.
In the case of a system devoid of friction, the solution is
U=-- ^C08 pt (7).
When p is smaller than n, the phase of the vibration agrees with
that of the force, but when p is the greater, the sign of the vibra-
tion is changed. The change of phase from complete agreement
to complete disagreement, which is gradual when friction acts,
here takes place abruptly as p passes through the value n. At the
same time the expression for the amplitude becomes infinite. Of
course this only means that, in the case of equal periods, friction
must be taken into account, however small it may be, and however
insignificant its result when p and n are not approximately equal.
The limitation as to the magnitude of the vibration, to which we
are all along subject, must also be borne in mind.
That the excursion should be at its maximum in one direction
while the generating force is at its maximum in the opposite
direction, as happens, for example, in the canal theory of the tides,
is sometimes considered a paradox. Any difficulty that may be
felt will be removed by considering the extreme case, in which the
** spring " vanishes, so that the natural period is infinitely long. In
fact we need only consider tha force acting on the bob of a com-
mon pendulum swinging freely, in which case the excursion on one
side is greatest when the action of gravity is at its maximum
in the opposite direction. When on the other hand the inertia of
the system is very small, we have the other extreme case in which
the so-called equilibrium theory becomes applicable, the force and
excursion being in the same phase.
When the period of the force is longer than the natural period,
the effect of an increasing friction is to introduce a retardation
in the phase of the displacement varying from zero up to a quarter
period. If, however, the period of the natural vibration be the
longer, the original retardation of half a period is diminished by
something short of a quarter period ; or the effect of friction is to
accelerate the phase of the displacement estimateii from that corre-
sponding to the absence of friction. In either case the influence
of friction is to cause an approximation to the state of things that
would prevail if friction were paramount.
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46.] PRINCIPLE OP SUPERPOSITION. 49
If a force of nearly equal period with the fi^e vibrations
vary slowly to a maximum and then slowly decrease, the dis-
placement does not reach its maximum until after the force has
begun to diminish. Under the operation of the force at its
maximum, the vibration continues to increase until a certain limit
is approached, and this increase continues for a time even although
the force, having passed its maximum, begins to diminish. On
this principle the retardation of spring tides behind the days of
new and full moon has been explained*.
47. From the linearity of the equations it follows that the
motion resulting from the simultaneous action of any number of
forces is the simple sum of the motions due to the forces taken
separately. Each force causes the vibration proper to itself,
without regard to the presence or absence of any others. The
peculiarities of a force are thus in a manner transmitted into the
motion of the system. For example, if the force be periodic in
time T, so will be the resulting vibration. Each harmonic element
of the force will call forth a corresponding harmonic vibration
in the system. But since the retardation of phase e, and the ratio
of amplitudes a: E,ia not the same for the different components,
the resulting vibration, though periodic in the same time, is dif-
ferent in character from the force. It may happen, for instance,
that one of the components is isochronous, or nearly so, with the
free vibration, in which case it will manifest itself in the motion
out of all proportion to its original importance. As another
example we may consider the case of a system acted on by two
forces of nearly equal period. The resulting vibration, being
compounded of two nearly in unison, is intermittent, according to
the principles explained in the last chapter.
To the motions, which are the immediate effects of the im-
pressed forces, must always be added the term expressing free
vibrations, if it be desired to obtain the most general solution.
Thus in the case of one impressed force,
u^^^co&(pt^€) + Ae-^(^{'^7^^J?.t-a] (1), '
where A and a are arbitrary.
48. The distinction between /orced and free vibrations is very
important, and must be clearly understood. The period of the
1 Airy'B Tide$ and Waves, Art. S2S.
K. 4
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50 ONE DEGREE OF FREEDOM. [48.
former is determined solely by the force which is supposed to act
on the system from without ; while that of the latter depends only
on the constitution of the system itself. Another point of differ-
ence is that 6o long as the external influence continues to operate,
a forced vibration is permanent, being represented strictly by a
harmonic function ; but a free vibration gradually dies away, be-
coming negligible after a time. Suppose, for example, that the
system is at rest when the force E cos pt begins to operate. Such
finite values must be given to the constants A and a in (1) of § 47,
that both u and u are initially zero. At first then there is a
free vibration not less important than its rival, but after a time
friction reduces it to insignificance, and the forced vibration is left
in complete possession of the field. This condition of things will
continue so long as the force operatea When the force is removed,
there is, of course, no discontinuity in the values of u or i, but
the forced vibration is at once converted into a firee vibration,
and the period of the force is exchanged for that natural to the
system.
During the coexistence of the two vibrations in the earlier part
of the motion, the curious phenomenon of beats may occur, in
case the two periods differ but slightly. For, n and p being nearly
equal, and k small, the initial conditions are approximately satis-
fied by
w = a cos (p^ — €) — ae~i*^ cos {Vn* — ^/c* . ^ — c}.
There is thus a rise and fall in the motion, so long as e"**^ remains
sensible. This intermittence is very conspicuous in the earlier
stages of the motion of forks driven by electro-magnetism (§ 63),
[and may be utilized when it is desired to adjust n and p to
equality. The initial beats are to be made slower and slower,
until they cease to be perceptible. The vibration then swells
continuously to a maximum.]
49. Vibrating systems of one degree of freedom may vary in
two ways according to the values of the constants n and /e. The
distinction of pitch is sufficiently intelligible ; but it is worth while
to examine more closely the consequences of a greater or less
degree of damping. The most obvious is the more or less rapid
extinction of a free vibration. The effect in this direction may be
measured by the number of vibrations which must elapse before
the amplitude is reduced in a given ratio. Initially the amplitude
may be taken as unity ; after a time t, let it be 6. Initially = e^K
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49.] VABIOUS DEGREES OF DAMPING, 51
2
If ^ = a?T, we have a? = log 0, In a system subject to only a
KT
moderate degree of damping, we may take approximately,
80 that a? = log^ (1).
KIT
This gives the number of vibrations which are performed, before
the amplitude falls to 0,
The influence of damping is also powerfully felt in a forced
vibration, when there is a near approach to isochronism. In the
case of an exact equality between p and n, it is the damping alone
which prevents the motion becoming infinite. We might easily
anticipate that when the damping is small, a comparatively slight
deviation from perfect isochronism would cause a large &lling off
in the magnitude of the vibration, but that with a larger damping
the same precision of adjustment would not be required. From
the equations
r=rosin»€, tan€ = -^,
w — jr
^eget __ir = ^__^ (2);
SO that if /c be small, p must be very nearly equal to n, in order to
produce a motion not greatly less than the maximum.
The two principal effects of damping may be compared by
eliminating /c between (1) and (2). The result is
-x-'"'\n p)\/t;^:t ^^>'
where the sign of the square root must be so chosen as to make
the rigl}t-hand side negative.
If, when a system vibrates freely, the amplitude be reduced in
the ratio after x vibrations ; then, when it is acted on by a force
{p\ the energy of the resulting motion will be less than in the
case of perfect isochronism in the ratio T i T^, It is a matter of
indifference whether the forced or the free vibration be the higher;
all depends on the interval.
In most cases of interest the interval is small; and then, putting
p s n + Sn, the formula may be written.
l9g^_29rSw / T
X n V T,'T ^*^-
4—2
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52
ONE DEGREE OF FREEDOM.
[49.
The following table calculated from these formulsB has been
given by HelmholtzM
T X 1 J* X J ^' I Number of Tibrations after which the
Interval oorresponding to a redaotioii ' *^ ***""«
of the resonance to one-tenth.
r:ro=i:io.
daced to one-tenth.
1^ tone.
38-00
J tone.
19-00
^ tone.
9-60
f tone.
6-33
Whole tone.
4-75
1^ tone.
3-80
f tone = minor third.
3-17
J tone.
2-71
Two whole tones = major third.
2-37
Formula (4) shews that, when Sn is small, it varies ccsteris
paribus as -.
60. From observations of forced vibrations due to known
forces, the natural period and damping of a system may be deter-
mined. The formulse are
Esmc
w =
pK
-cos(pf — e),
where
tan€ = -r^..
On the equilibrium theory we should have
U = -iCO&pt
The ratio of the actual amplitude to this is
^sin€ -£?_n*sin€
pK ' n^" pK '
If the equilibrium theory be known, the comparison of ampli-
tudes tells us the value of , say
px ^
n^sinc
= a,
Tonemj)/S7u2tmj7«n, 8rd edition, p. 221.
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50.] STRING WITH LOAD. 53
and € is also known, whence
, . /^ C08€\ J ©sine .-.
n* = i)»-5- 1 ), and k--^ (1).
'^ \ a J a — co8€ ^ ^
51. As has been already stated, the distinction of forced and
free vibrations is important ; but it may be remarked that most of
the forced vibrations which we shall have to consider as affecting
a system, take their origin ultimately in the motion of a second
system, which influences the first, and is influenced by it. A
vibration may thus have to be reckoned as forced in its relation
to a system whose limits are fixed arbitrarily, even when that
system has a share in determining the period of the force which
acts upon it. On a wider view of the matter embracing both the
systems, the vibration in question will be recognized as free. An
example may make this clearer. A tuning-fork vibrating in air
is part of a compound system including the air and itself, and
in respect of this compound system the vibration is free. But
although the fork is influenced by the reaction of the air, yet the
amount of such influence is small. For practical purposes it is
convenient to consider the motion of the fork as given, and that of
the air as forced. No error will be committed if the actual motion
of the fork (as influenced by its surroundings) be taken as the
basis of calculation. But the peculiar advantage of this mode of
conception is manifested in the case of an approximate solution
being required. It may then suffice to substitute for the actual
motion, what would be the motion of the fork in the absence of
air, and afterwards introduce a correction, if necessary.
52. Illustrations of the principles of this chapter may be
drawn from all parts of Acoustics. We will give here a few
applications which deserve an early place on account of their
simplicity or importance.
A string or wire ACB is stretched between two fixed points
A and £, and at its centre carries a mass M, which is supposed to
be so considerable as to render the mass of the string itself negli-
gible. When M is pulled aside firom its position of equilibrium,
and then let go, it executes along the line CM vibrations, which
are the subject of inquiry. AC^ CB = a. CM = x. The tension
of the string in the position of equilibrium depends on the amount
of the stretching to which it has been subjected. In any other
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54
ONE DEGREE OF FREEDOM.
[52.
position the tension is greater ; but we limit ourselves to the case
of vibrations so small that the additional stretching is a negligible
fraction of the whole. On this condition the tension may be
treated as constant. We denote it by T.
and
Thus, kinetic energy « J-Jfo",
^
potential energy = 2T { va' + ic* - a} = T -- approximately.
The equation of motion (which may be derived also inde-
pendently) is therefore
(1),
MS-h2T-'.
a
from which we infer that the mass M executes harmonic vibra-
tions, whose period
^ = 2--V^ (2).
The amplitude and phase depend of course on the initial cir-
cumstances, being arbitrary so £BLr as the differential equation is
concerned.
Equation (2) expresses the manner in which r varies with each
of the independent quantities T, M,a: results which may all be
obtained by consideration of the dimensions (in the technical sense)
of the quantities involved. The argument from dimensions is so
often of importance in Acoustics that it may be well to consider
this first instance at length.
In the first place we must assure ourselves that of all the
quantities on which t may depend, the only ones involving a
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52.] METHOD OF DIMENSIONS. 55
reference to the three fimdamental units— of length, time, and
mass — are a, M, and T. Let the solution of the problem be
written
T=f{fl,M,T) (3).
This equation must retain its form unchanged, whatever may
be the fundamental units by means of which the four quantities
are numerically expressed, as is evident, when it is considered
that in deriving it no assumptions would be made as to the mag-
nitudes of those* units. Now of all the quantities on which/'
depends, T is the only one involving time ; and since its dimen-
sions are (Mass) (Length) (Time)"*, it follows that when a and M
are constant, tx T"^; otherwise a change in the unit of time
would necessarily disturb the equation (3). This being admitted,
it is easy to see that in order that (3) may be independent of the
unit of lengtl)^ we must have t x T""* . a*, when M is constant ; and
finally, in order to secure independence of the unit of mass,
TxT-*.if*.a*.
To determine these indices we might proceed thus : — assume
Tx!r*.Jfy.a«;
then by considering the dimensions in time, space, and mass, we
obtain respectively
l = -2a?, = a? + ;8f, = a? + y,
whence as above ^ = — i, y = i, -s^ = i-
There must be no mistake as to what this argument does and
does not prove. We have asmmed that there is a definite
periodic time depending on no other quantities, having dimen-
sions in space, time, and mass, than those above mentioned. For
example, we have not proved that t is independent of the ampli-
tude of vibration. That, so far as it is true at all, is a consequence
of the linearity of the approximate differential equation.
From the necessity of a complete enumeration of all the
quantities on which the required result may depend, the method
of dimensions is somewhat dangerous ; but when used with proper
care it is unquestionably of great power and value.
53. The solution of the present problem might be made the
foundation of a method for the absolute measurement of pitch.
The principal impediment to accuracy would probably be the
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56
ONE DEGREE OF FREEDOM.
[53.
difficulty of making M sufficiently large in relation to the mass of
the wire, without at the same time lowering the note too much in
the musical scale.
<^
FiG.ia
oT
M
O
The wire may be stretched by a weight M' attached to its
further end beyond a bridge or pulley at B. The periodic time
would be calculated from
'^ir
4:
.(1).
The ratio of if' : Jf is given by the balance. If a be measured
in feet, and g ~ 32*2, the periodic time is expressed in seconds.
54 In an ordinary musical string the weight, instead of being
concentrated in the centre, is uniformly distributed over its length.
Nevertheless the present problem gives some idea of the nature of
the gravest vibration of .such a string. Let us compare the two
cases more closely, supposing the amplitudes of vibration the same
at the middle point.
no. II.
When the uniform string is straight, at the moment of passing
through the position of equilibrium, its different parts are ani-
mated with a variable velocity, increasing from either end towards
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54.] COMPARISON WITH UNIFORM STRING. 57
the centre. If we attribute to the whole mass the velocity of the
centre, it is evident that the kinetic energy will be considerably
over-estimated. Again, at the moment of maximum excursion,
the uniform string is more stretched than its substitute, which
follows the straight courses AM, MB, and accordingly the poten-
tial energy is diminished by the substitution. The concentration
of the mass at the middle point at once increases the kinetic
energy when a? = 0, and decreases the potential energy when i? = 0,
and therefore, according to the principle explained in § 44, prolongs
the periodic time. For a string then the period is less than that
calculated from the formula of the last section, on the supposition
that M denotes the mass of the string. It will afterwards appear
that in order to obtain a correct result we should have to take
instead of M only {ilir^)M. Of the factor 4/7r' by far the more
important part, viz. ^, is due to the difference of the kinetic
energies.
66. As another example of a system possessing practically but
one degree of freedom, let us consider the vibration of a spring, one
end of which is clamped in a vice or otherwise held fast, while the
other carries a heavy mass;
In strictness, this system like the last has
an infinite number of independent modes of vi- C j
bration; but, when the mass of the spring is ^'""^
relatively small, that vibration which is nearly
independent of its inertia becomes so much the FiOi2.
most important that the others may be ignored.
Pushing this idea to its limit, we may regard the
spring merely as the origin of a force urging the
attached mass towards the position of equilibrium,
and, if a certain point be not exceeded, in simple \^
proportion to the displacement. The result is a -^
harmonic vibration, with a period dependent on
the stiffiiess of the spring and the mass of the
load. ' :-
66. In consequence of the oscillation of the centre of inertia,
there is a constant tendency towards the communication of motion
to the supports, to resist which adequately the latter must be
very firm and massive. In order to obviate this inconvenience.
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58
ONE DEGREE OF FREEDOM.
[56.
O Q
two precisely similar springs and loads may be mounted on
the same framework in a symmetrical manner.
If the two loads perform vibrations of equal
amplitude in such a manner that the motions
are always opposite, or, as it may otherwise be
expressed, with a phase-difference of half a
period, the centre of inertia of the whole system
remains at rest, and there is no tendency to set
the framework into vibration. We shall see in a
future chapter that this peculiar relation of phases
will quickly establish itself, whatever may be the
original disturbance. In fact, any part of the
motion which does not conform to the condition
of leaving the centre of inertia unmoved is soon
extinguished by damping, unless indeed the
supports of the system are more than usually
firm.
vy
67. As in our first example we found a rough illustration of
the fundamental vibration of a musical string, so here with the
spring and attached load we may compare a uniform slip, or bar,
of elastic material, one end of which is securely fastened, such for
instcmce as the tongue of a reed instrument. It is true of course
that the mass is not concentrated at one end, but distributed
over the whole length ; yet on account of the smallness of the
motion near the point of support, the inertia of that part of
the bar is of but little account. We infer that the fundamental
vibration of a uniform rod cannot be very different in character
from that which we have been considering. Of course for pur-
poses requiring precise calculation, the two systems are suflSciently
distinct; but where the object is to form clear ideas, precision may
often be advantageously exchanged for simplicity.
In the same spirit mt^ may regard the combination of two
springs and loads shewn in Fig. 13 as a representation of a
tuning-fork. The instrument, which has been much improved
of late years, is indispensable to the acoustical investigator. On
a large scale and for rough purposes it may be made by welding
a cross piece on the middle of a bar of steel, so as to form a T, and
then bending the bar into the shape of a horse-shoe. On the
handle a screw should be cut. But for the better class of tuning-
forks it is preferable to shape the whole out of one piece of steel.
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57.] TUNING-FORKS. 59
A division running from one end down the middle of a bar is first
made, the two parts opened out to form the prongs of the fork,
and the whole worked by the hammer and file into the required
shape. The two prongs must be exactly symmetrical with respect
to a plane passing through the axis of the handle, in order that
during the vibration the centre of inertia may remain unmoved,
— unmoved, that is, in the direction in which the prongs
vibrate.
The tuning is effected thus. To make the note higher, the
equivalent inertia of the system must be reduced. This is done
by 'filing away the ends of the prongs, either diminishing their
thickness, or actually shortening them. On the other hand, to
lower the pitch, the substance of the prongs near the bend may
be reduced, the effect of which is to diminish the force of the
spring, leaving the inertia practically unchanged ; or the inertia
may be increased (a method which would be preferable for
temporary purposes) by loading the ends of the prongs with
wax, or other material. Large forks are sometimes provided with
moveable weights, which slide along the prongs, and can be fixed
in any position by screws. As these approach the ends (where the
velocity is greatest) the equivalent inertia of the system increases.
In this way a considerable range of pitch may be obtained from
one fork. The number of vibrations per second for any position
of the weights may be marked on the prongs.
The relation between the pitch and the size of tuning-forks is
remarkably simple. In a future chapter it will be proved that,
provided the material remains the same and the shape constant,
the period of vibration varies directly as the linear dimension.
Thus, if the linear dimensions of a tuning-fork be doubled, its
note falls an octave.
68. The note of a tuning-fork is a nearly pure tone. Imme-
diately after a fork is struck, high tones may indeed be heard,
corresponding to modes of vibration, whose nature will be subse-
quently considered ; but these rapidly die away, and even while
they exist, they do not blend with the proper tone of the fork,
partly on account of their very high pitch, and partly because
they do not belong to its harmonic scale. In the forks examined
by Helmholtz the first of these overtones had a frequency from 5*8
to 6*6 times that of the proper tone.
Tuning-forks are now generally supplied with resonance cases.
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60 ONE DEGREE OF FREEDOM. [58.
whose effect is greatly to augment the volume and purity of the
sound, according to principles to be hereafter developed Id
order to excite them, a violin or cello bow, well supplied with
rosin, is drawn across the prongs in the direction of vibration.
The sound so produced will last a minute or more.
69. As standards of pitch tuning-forks are invaluable. The
pitch of organ-pipes rapidly varies with the temperature and with
the pressure of the wind ; that of strings with the tension, which
can never be retained constant for long; but a tuning-fork kept
clean and not subjected to violent changes of temperature or
magnetization, preservers its pitch with great fidelity.
[But it must not be supposed that the vibrations of a fork are
altogether independent of temperature. According to the obser-
vations of McLeod and Clarke^ the frequency falls by *00011 of its
value for each degree Cent, of elevation.]
By means of beats a standard tuning-fork may be copied with
very great precision. The number of beats heard in a second is
the difference of the frequencies of the two tones which produce
them ; so that if the beats can be made so slow as to occupy half
a minute each, the frequencies differ by only l-30th of a vibra-
tion. Still greater precision might be obtained by Lissajous'
optical method.
Very slow beats being difficult of observation, in consequence
of the uncertainty whether a falling off in the sound is due to
interference or to the gradual dying away of the vibrations,
Scheibler adopted a somewhat modified plan. He took a fork
slightly different in pitch from the standard — whether higher or
lower is not material, but we will say, lower, — and counted the
number of beats, when they were sounded together. About four
beats a second is the most suitable, and these may be counted for
perhaps a minute. The fork to be adjusted is then made slightly
higher than the auxiliary fork, and tuned to give with it precisely
the same number of beats, as did the standard. In this way a
copy as exact as possible is secured. To facilitate the counting
of the beats Scheibler employed pendulums, whose periods of
vibration could be adjusted.
[The question between slow and quick beats depends upon the
circumstances of the case. It seems to be sometimes supposed
that quick beats have the advantage as admitting of greater
1 Phil Tram. 18S0, p. 12.
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59.] scheibler's tonometer. 61
relative accuracy of counting. But a little consideration shews
that in a comparison of frequencies we are concerned not with the
relative, but with the abaolvte accuracy of the counting. If we
miscount the beats in a minute by one, it makes just the same
error in the result, whether the whole number of beats be 60 or
240.
When the sounds are pure tones and are well maintained, it is
advisable to use beats much slower than four per second. By
choosing a suitable position it is often possible to make the
intensities at the ear equal; and then the phase of silence,
corresponding to antagonism of equal and opposite vibrations, is
extremely well marked. Taking advantage of this we may deter-
mine slow beats with very great accuracy by observing the time
which elapses between recuiTences of silence. In favourable cases
the whole number of beats in the period of observation may be
fixed to within one-tenth or one-twentieth of a single beat, a
degree of accuracy which is out of the question when the beats
are quick. In this way beats of periods exceeding 30 seconds may
be utilised with excellent effect ^]
60. The method of beats was also employed by Scheibler to
determine the absolute pitch of his standards. Two forks were
tuned to an octave, and a number of others prepared to bridge
over the interval by steps so small that each fork gave with its
immediate neighbours in the series a number of beats that could
be easily counted. The difference of frequency corresponding to
each step was observed with all possible accuracy. Their sum,
being the difference of frequencies for the interval of the octave,
was equal to the frequency of that fork which formed the starting
point at the bottom of the series. The pitch of the other forks
could be deduced.
If consecutive forks give four beats per second, 65 in all will
be required to bridge over the interval from c' (256) to c" (512).
On this account the method is laborious ; but it is probably the
most accurate for the original determination of pitch, as it is
liable to no errors but such as care and repetition will eliminate.
It may be observed that the essential thing is the measurement
of the difference of frequencies for two notes, whose ratio of
frequencies is independently known. If we could be sure of its
accuracy, the interval of the fifth, fourth, or even major third, might
^ Aooustical Observations, Phil. Mag, May, 1882, p. 342.
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62 ONE DEGREE OF FREEDOM. [60.
be substituted for the octave, with the advantage of reducing the
number of the necessary interpolations. It is probable that with
the aid of optical methods this course might be successfully
adopted, as the corresponding Lissajous' figures are easily recog-
nised, and their steadiness is a very severe test of the accuracy
with which the ratio is attained.
[It is essential to the success of this method that the pitch of
each of the numerous sounds employed should be definite, and in
particular that the vibrations of any fork should take place at the
same rate whether that fork be sounding in conjunction with its
neighbour above or with its neighbour below. There is no reason
to doubt that this condition is sufficiently satisfied in the case of
independent tuning-forks; but an attempt to replace forks by a
set of reeds, mounted side by side on a common wind-chest, has
led to error, owing to a disturbance of pitch by mutual inter-
action \]
The frequency of large tuning-forks may be determined by
allowing them to trace a harmonic curve on smoked paper, which
may conveniently be mounted on the circumference of a revolving
drum. The number of waves executed in a second of time gives
the frequency.
In many cases the use of intermittent illumination described
in § 42 gives a convenient method of determining an unknown
frequency.
61. A series of forks ranging at small intervals over an octave
is very useful for the determination of the frequency of any
musical note, and is called Scheibler's Tonometer. It may also
be used for tuning a note to any desired pitch. In either case
the frequency of the note is determined by the number of beats
which it gives with the forks, which lie nearest to it (on each
side) in pitch.
For tuning pianofortes or organs, a set of twelve forks may be
used giving the notes of the chromatic scale on the equal tempe-
rament, or any desired system. The corresponding notes are
adjusted to unison, and the others tuned by octaves. It is better,
however, to prepare the forks so as to give four vibrations per
second less than is above proposed. Each note is then tuned a
little higher than the corresponding fork, until they give when
sounded together exactly four beats in the second. It will be
1 Nature, xyii. pp. 12. 26 ; 1877.
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61. J scheibler's tonometer. 63
observed that the addition (or subtraction) of a constant number
to the frequencies is not the same thing as a mere displacement
of the scale in absolute pitch.
In the ordinary practice of tuners a! is taken from a fork, and
the other notes determined by estimation of fifths. It will be
remembered that twelve true fifths are slightly in excess of seven
octaves, so that on the equal temperament system each fifth is a
little flat. The tuner proceeds upwards from ol by successive
fifths, coming down an octave after about every alternate step, in
order to remain in nearly the same part of the scale. Twelve
fifths should bring him back to a. If this be not the case, the
work must be readjusted, until all the twelve fifths are too flat by,
as nearly as can be judged, the same small amount. The in-
evitable error is then impartially distributed, and rendered as little
sensible as possible. The octaves, of course, are all tuned true.
The following numbers indicate the order in which the notes may
be taken :
a% b c' c'jf d' d't e' fft g' g't a' a'i V c" c"jt d" d't e"
13 5 16 8 19 11 3 14 6 17 9 1 12 4 15 7 18 10 2
In practice the equal temperament is only approximately
attained; but this is perhaps not of much consequence, considering
that the system aimed at is itself by no means perfection.
Violins and other instruments of that class are tuned by true
fifths from a\
62. In illustration of forced vibration let us consider the case
of a pendulum whose point of support is subject to a small hori-
zontal harmonic motion. Q is the bob attached by a fine More to
a moveable point P. OP^x^. g_
PQ ss Z, and x is the horizontal
co-ordinate of Q. Since the
vibrations are supposed small,
the vertical motion may be
neglected, and the tension of
the wire equated to the weight
of Q. Hence for the horizontal
motion * + #«c + y (a? — a?o) = 0. rtci4.
Now iCp oc cos jp^ ; so that putting g-7-l^n\ our equation takes
the form already treated of, viz.
ai + ittP -h n"a? = E cos pt
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64
ONE DEQBEE OF FREEDOM.
[62.
I{ phe equal to n, the motion is limited only by the friction.
The assumed horizontal harmonic motion for P may be realized by
means of a second pendulum of massive construction, which carries
P with it in its motion. An efficient arrangement is shewn ia
the figure. A, B are iron rings screwed into a beam, or other firm
Fl Qf6.
support ; (7, B similar rings attached to a stout bar, which carries
equal heavy weights E, Fy attached near its ends, and is supported
in a horizontal position at right angles to the beam by a wire
passing through the four rings in the manner shewn. When the
pendulum is made to vibrate, a point in the rod midway between
C and D executes a harmonic motion in a direction parallel to
CD, and may be made the point of attachment of another pen-
dulum PQ. If the weights E and F be very great in relation
to Q, the upper pendulum swings very nearly in its own proper
period, and induces in Q a forced vibration of the same period.
When the length PQ is so adjusted that the natural periods of the
two pendulums are nearly the same, Q will be thrown into violent
motion, even though the vibration of P be of but inconsiderable
amplitude. In this case the difference of phase is about a quarter
of a period, by which amount the upper pendulum is in advance.
If the two periods be very different, the vibrations either agree
or are completely opposed in phase, according to equations (4)
and (5) of § 46.
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63.] RELATION OF AMPLITUDE AND PHASE. 65
63. A very good example of a forced vibration is afforded by
a fork under the influence of an intermittent electric cuirent,
Fl C. /&
c ^ '=r3
:5
n
2:^
whose period is nearly equal to its own. ACB is the fork ; E a
small electro-magnet, formed by winding insulated wire on an iron
core of the shape shewn in E (similar to that known as ' Siemens '
armature'), and supported between the prongs of the fork. When
an intermittent current is sent through the wire, a periodic force
acts upon the fork. This force is not expressible by a simple
circular function ; but may be expanded by Fourier's theorem in a
series of such functions, having periods t,\t^\ t, &c. If any of
these, of not too small amplitude, be nearly isochronous with the
fork, the latter will be caused to vibrate ; otherwise the effect is
insignificant In what follows we will suppose that it is the
complete period t which nearly agrees with that of the fork, and
consequently regard the series expressing the periodic force as
reduced to its first term.
In order to obtain the maximum vibration, the fork must be
carefully tuned by a small sliding piece or by wax, until its natural
period (without friction) is equal to that of the force. This is best
done by actual trial. When the desired equality is approached,
and the fork is allowed to start from rest, the force and com-
plementary free vibration are of nearly equal amplitudes and
frequencies, and therefore (§ 48) in the beginning of the motion
produce &eato, whose slowness is a measure of the accuracy of
the adjustment. It is not until after the free vibration has had
time to subside, that the motion assumes its permanent character.
The vibrations of a tuning-fork properly constructed and mounted
are subject to very little damping; consequently a very slight
deviation from perfect isochronism occasions a marked falling off
in the intensity of the resonance.
The amplitude of the forced vibration can be observed with
sufficient accuracy by the ear or eye; but the experimental
R. 5
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66 ONE DEGREE OF FREEDOM. [63.
verification of the relations pointed out by theory between its
phase and that of the force which causes it, requires a modified
arrangement
Two similar electro-magnets acting on similar forks, and in-
cluded in the same circuit are excited by the same intermittent
current. Under these circumstances it is clear that the systems
will be thrown into similar vibrations, because they are acted on
by equal forces. This similarity of vibrations refers both to phase
and amplitude. Let us suppose now that the vibrations are
effected in perpendicular directions, and by means of one of
Lissajous' methods are optically compounded. The resulting figure
is necessarily a straight line. Starting fi-om the case in which the
amplitudes are a maximum, viz. when the natural periods of both
forks are the same as that of the force, let one of them be put a
little out of tune. It must be remembered that whatever their
natural periods may be, the two forks vibrate in perfect unison
with the force, and therefore with one another. The principal
effect of the difference of the natural periods is to destroy the
synchronism of phase. The straight line, which previously re-
presented the compound vibration, becomes an ellipse, and this
remains perfectly steady, so long as the forks are not touched.
Originally the forks are both a quarter period behind the force.
When the pitch of one is slightly lowered, it falls still more behind
the force, and at the same time its amplitude diminishes. Let the
difference of phase between the two forks be e', and the ratio of
amplitudes of vibration a : a©. Then by (6) of § 46
a =« a© cos c'.
The following table shews the simultaneous values of a : a^
and €\
a : ao ' €'
1-0
■9
25» 50'
•8
36» 52'
•7
45«34'
•6
53» 7'
•5
60»
•4
66» 25'
•3
72»32'
•2
780 27'
•1
84» 15'>
Tonempfiudungen, 3rd edition, p. 190.
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63.] FORK INTERRUPTER. 67
It appears that a considerable alteration of phase in either
direction may be obtained without very materially reducing the
amplitude. When one fork is vibrating at its maximum, the
other may be made to differ from it on either side by as much as
60* in phase, without losing more than half its amplitude, or by
as much as 45^ without losing more than half its energy. By
allowing one fork to vibrate 45® in advance, and the other 45°
in arrear of the phase corresponding to the case of maximum
resonance, we obtain a phase difference of 90® in conjunction with
an equality of amplitudes. Lissajous' figure then becomes a circle.
[An intermittent electric current may also be applied to
regulate the speed of a revolving body. The phonic wheel, in-
vented independently by M. La Cour and by the author of this
work^ is of great service in acoustical investigations. It may take
various forms; but the essential feature is the approximate
closing of the magnetic circuit of an electro-magnet, fed with an
intermittent current, by one or more soft iron armatures carried
by the wheel and disposed symmetrically round the circumference.
If in the revolution of the wheel the closest passage of the
armature sjmchronises with the middle of the time of excitation,
the electro-magnetic forces operating upon the armature during
its advance and its retreat balance one another. If however the
wheel be a little in arrear, the forces promoting adv8mce gain an
advantage over those hindering the retreat of the armature, and
thus upon the whole the magnetic forces encourage the rotation.
In like manner if the phase of the wheel be in advance of that
first specified, forces are called into play which retard the motion.
By a self-acting adjustment the rotation settles down into such
a phase that the driving forces exactly balance the resistances.
When the wheel runs lightly, and the electric appliances are
moderately powerful, independent driving may not be needed. In
this case of course the phase of closest passage must follow that
which marks the middle of the time of magnetisation. If, as is
sometimes advisable, there be an independent driving power, the
phase of closest passage may either precede or follow that of
magnetisation.
In some cases the oscillations of the motion about the phase
into which it should settle down are very persistent and interfere
with the applications of the instrument. A remedy may be
found in a ring containing water or mercury, revolving concen-
1 Nature, May 23, 1878.
5—2
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68 ONE DEGREE OF FREEDOM. [63.
trically. When the rotation is uniform, the fluid revolves like a
solid body and then exercises no influence. But when from any
cause the speed changes, the fluid persists for a time in the former
motion, and thus brings into play forces tending to damp out
oscillations.]
64. The intermittent current is best obtained by a fork-
interrupter invented by Helmholtz. This may consist of a fork
and electro-magnet mounted as before. The wires of the magnet
are connected, one with one pole of the battery, and the other with
a mercury cup. The other pole of the battery is connected with
a second mercury cup. A U-shaped rider of insulated mre is
carried by the lower prong just over the cups, at such a height
that during the vibration the circuit is alternately made and
broken by the passage of one end into and out of the mercury.
The other end may be kept permanently immersed. By means
of the periodic force thus obtained, the eflTect of friction is com-
pensated, and the vibrations of the fork permanently maintained.
In order to set another fork into forced vibration, its associated
electro-magnet may be included, either in the same driving-circuit,
or in a second, whose periodic interruption is effected by another
rider dipping into mercury cups\
The modus operandi of this kind of self-acting instrument is
often imperfectly apprehended. If the force acting on the fork
depended only on its position — on whether the circuit were open
or closed — the work done in passing through any position would
be undone on the return, so that after a complete period there
would be nothing outstanding by which the effect of the frictional
forces could be compensated. Any explanation which does not
take account of the retardation of the current is wholly beside the
mark. The causes of retardation are two : irregular contact, and
self-induction. When the point of the rider first touches the
mercury, the electric contact is imperfect, probably on account of
^ I have arranged Beveral interrupters on the above plan, all the component
parts being of home manufacture. The forks were made by the village blacksmith.
The onps consisted of iron thimbles, soldered on one end of copper slips, the
further end being screwed down on the base board of the instrument. Some
means of adjusting the level of the mercury surface is necessary. In Helmholtz*
interrupter a horse-shoe electro-magnet embracing the fork is adopted, but I am
inclined to prefer the present arrangement, at any rate if the pitch be low. In
some cases a greater motive power is obtained by a horse-shoe magnet acting on a
soft iron armature carried horizontally by the upper prong and perpendicular to it.
I have usually found a single Smee cell sufficient battery power.
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64.] FORK INTERRUPTER. 69
adhering air. On the other hand, in leaving the mercury the
contact is prolonged by the adhesion of the liquid in the cup to
the amalgamated wire. On both accounts the current is retarded
behind what would correspond to the mere position of the fork.
But, even if the resistance of the circuit depended only on the
position of the fork, the current would still be retarded by its self-
induction. However perfect the contact may be, a finite current
cannot be generated until after the lapse of a finite time, any
more than in ordinary mechanics a finite velocity can be suddenly
impressed on an inert body. From whatever causes ari8ing\ the
effect of the retardation is that more work is gained by the fork
during the retreat of the rider from the mercury, than is lost
during its entrance, and thus a balance remains to be set off
against friction.
If the magnetic force depended only on the position of the fork,
the phase of its first harmonic component might be considered to
be 180* in advance of that of the fork's own vibration. The re-
tardation spoken of reduces this advance. If the phase-difference
be reduced to 90^, the force acts in the most favourable manner,
and the greatest possible vibration is produced.
It is important to notice that (except in the case just refeired
to) the actual pitch of the interrupter differs to some extent from
that natural to the fork according to the law expressed in (5) of
§ 46, € being in the present case a prescribed phase-difference
depending on the nature of the contacts and the magnitude of the
self-induction. If the intermittent current be employed to drive
a second fork, the maximum vibration is obtained, when the
frequency of the fork coincides, not with the natural, but with the
modified frequency of the iuterrupter.
The deviation of a tuning-fork interrupter from its natural
pitch is practically very small ; but the fact that such a deviation
is possible, is at first sight rather surprising. The explanation (in
the case of a small retardation of current) is, that during that half
of the motion in which the prongs are the most separated, the
electro-magnet acts in aid of the proper recovering power due to
rigidity, and so naturally raises the pitch. Whatever the relation
of phases may be, the force of the magnet may be divided into
1 Any desired retardation might be obtained, in default of other means, by
attaching the rider, not to the prong itself, bat to the farther end of a light
straight spring carried by the prong and set into forced vibration by the motion of
its point of attachment.
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70 ONE DEGREE OF FREEDOM. [64.
two parts respectively proportional to the velocity and displacement
(or acceleration). To the first exclusively is due the sustaining
power of the force, and to the second the alteration of pitch.
66. The general phenomenon of resonance, though it cannot
be exhaustively considered under the head of one degree of
freedom, is in the main referable to the same general principles.
When a forced vibration is excited in one part of a system, all
the other parts are also influenced, a vibration of the same period
being excited, whose amplitude depends on the constitution of the
system considered as a whole. But it not unfrequently happens
that interest centres on the vibration of an outlying part whose
connection with the rest of the system is but loose. In such a case
the part in question, provided a certain limit of amplitude be
not exceeded, is very much in the position of a system possessing
one degree of freedom and acted on by a force, which may be
regarded as given, independently of the natural period. The
vibration is accordingly governed by the laws we have already
investigated. In the case of approximate equality of periods to
which the name of resonance is generally restricted, the ampli-
tude may be very considerable, even though in other cases it
might be so small as to be of little account; and the precision
required in the adjustment of the periods in order to bring out
the effect, depends on the degree of damping to which the system
is subjected.
Among bodies which resound without an extreme precision of
tuning, may be mentioned stretched membranes, and strings asso-
ciated with sounding-boards, as in the pianoforte and the violin.
When the proper note is sounded in their neighbourhood, these
bodies are caused to vibrate in a very perceptible manner. The
experiment may be made by singing into a pianoforte the note
given by any of its strings, having first raised the corresponding
damper. Or if one of the strings belonging to any note be plucked
(like a harp string) with the finger, its fellows will be set into
vibration, as may immediately be proved by stopping the first.
The phenomenon of resonance is, however, most striking in
cases where a very accurate equality of periods is necessary in
order to elicit the full effect. Of this class tuning-forks, mounted
on resonance boxes, are a conspicuous example. When the unison
is perfect the vibration of one fork will be taken up by another
across the width of a room, but the slightest deviation of pitch
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65. J RESONANCK 71
is sufficient to render the phenomenon almost insensible. Forks
of 256 vibrations per second are commonly used for the purpose,
and it is found that a deviation from unison giving only one beat
in a second makes all the difference. When the forks ai*e well
tuned and close together, the vibration may be transferred back-
wards and forwards between them several times, by damping them
alternately, with a touch of the finger.
Illustrations of the powerful effects of isochronism must be
within the experience of every one. They are often of importance
in very different fields from any with which acoustics is concerned.
For example, few things are more dangerous to a ship than to lie
in the trough of the sea under the influence of waves whose period
is nearly that of its own natural rolling.
65 a. It has already (§ 30) been explained how the super-
position of two vibrations of equal amplitude and of nearly equal
frequency gives rise to a resultant in which the sound rises and
falls in beats. If we represent the two components by cos 27mit,
cos 27r7i^, the resultant is
2 008^(111— 71^)1, COS w(ni + n^)t (l)j
and it may be regarded as a vibration of frequency ^ (ui + n^), and
of amplitude 2 cos tt (rii - n,) t. In passing through zero the
amplitude changes sign, which is equivalent to a change of phase
of ISO"", if the amplitude be regarded as always positive. This
change of phase is readily detected by measurement in drawings
traced by machines for compounding vibrations, and it is a feature
of great importance. If a force of this character act upon a system
whose natural frequency is J (nj + n^), the effect produced is com-
paratively small. If the system start from rest, the successive
impulses cooperate at first, but after a time the later impulses
begin to destroy the effect of former ones. The greatest response
would be given to forces of frequency Ui and n,, and not to a force
of frequency i (w^ -f n,).
If, as in some experiments of Prof. A. M. Mayer \ an otherwise
steady sound is rendered intermittent by the periodic interposition
of an obstacle, a very different result is arrived at. In this case
the phase is resumed after each silence without reversal. If a
force of this character act upon an isochronous system, the effect
is indeed less than if there were no intermittence ; but as all the
* Phil. Mag. May, 1876.
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72 ONE DEGREE OF FREEDOM. [65 a.
impulses operate in the same sense without any antagonism, the
response is powerful. One kind of intermittent vibration or force
is represented by
2(l + cos2'7rmOcos27m^ (2),
in which n is the frequency of the vibration, and m the frequency
of intermittence ^ The amplitude is here always positive, and
varies between the values and 4. By ordinary trigonometrical
transformation (2) may be put in the form
2 cos 27rn^ + cos 27r (n + m) ^ + cos 27r(n — m) t (3):
which shews that the intermittent vibration in question is equiva-
lent to three simple vibrations of frequencies w, n + m, n — m.
This is the explanation of the secondary sounds observed by
Mayer.
The form (2) is of course only a particular case. Another in
which the intensity of the intermittent sound rises more suddenly
to its maximum is given by
^cos^irmtco&iimt (4),
which may be transformed into
f COB iimt + cos 27r (n + m) ^ + cos 27r (n — wi) t
+ icos27r(ri + 2m)^4-icos27r(n- 2m) t (5).
There are here four secondary sounds, the frequencies of the
two new ones differing twice as much as before fix)m that of the
primary sound.
The theory of intermittent vibrations is well illustrated by
electrically driven forks. A fork interrupter of frequency 128
gave a periodic current, by the passage of which through an
electro-magnet a second fork of like pitch could be excited. The
action of this current on the second fork could be rendered inter-
mittent by short-circuiting the electro-magnet. This was effected
by another interrupter of frequency 4, worked by an independent
current from a Smee cell. To excite the main cuirent a Grove
cell was employed. When the contact of the second interrupter
was permanently broken, so that the main current passed con-
tinuously through the electro-magnet, the fork was, of course,
most powerfully affected when tuned to 128. Scarcely any
response was observable when the pitch was changed to 124 or
132. But if the second interrupter were allowed to operate, so as
^ Cram Brown and Tait. Edin. Proe, June, 1S7S. Aooostioal Observations ii.
Phil Mag. April. 18S0.
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65 a.]
INTERMITTENT VIBRATIONS.
73
to render the periodic current through the electro-magnet inter-
mittent, then the fork would respond powerfully when tuned to
124 or 132 as well as when tuned to 128, but not when tuned to
intermediate pitches, such as 126 or 130.
The operation of the intermittence in producing a sensitive-
ness which would not otherwise exist, is easily understood. When
a fork of frequency 124 starts from rest under the influence of a
force of frequency 128, the impulses cooperate at first, but after J
of a second the new impulses begin to oppose the earlier ones.
After J of a second, another series of impulses begins whose effect
agrees with that of the first, and so on. Thus if all these impulses
are allowed to act, the resultant effect is trifling ; but if every
alternate series is stopped off, a large vibration accumulates.
Fig. 16 a.
The most general expression for a vibration of frequency n,
whose amplitude and phase are slowly variable with a frequency
m, is
::.:}
{Ao-^Ai cos 2'7rmt 4- -4 a cos 47rm^ + A^ cos GTnrU +
-f- Bi sin 27rmt -h J5, sin 47rwi^ -h B^ sin Qirmt
f Co 4- Ci cos 27rm^ + Cj cos 4'7rm^ 4- C^ cos 6frmt 4- . . .)
( -f- Asin 27rm^ 4 J5j sin 47rm^ +2), sin Qwrnt -f- . . . j
cos 2Tmt
sin27m^
•(6);
and this applies both to the case of beats (e.g. if Ai only be finite)
and to such intermittence as is produced by the interposition of
an obstacle. The vibration in question is accordingly in all cases
equivalent to a combination of simple vibrations of frequencies
n, n + m, n — m, n + 2m, n — 2m, &c.
It may be well here to emphasise that a simple vibration
implies infinite continuance, and does not admit of variations of
phase or amplitude. To suppose, as is sometimes done in optical
speculations, that a train of simple waves may begin at a given
epochs continue for a certain time involving it may be a large
number of periods, and ultimately cease, is a contradiction in terms.
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74 ONE DEGREE OF FREEDOM. [66.
66. The solution of the equation for free vibration, viz.
u4-#ci + n*a«0 (1).
may be put into another form by expressing the arbitrary con-
stants of integration A and a in terms of the initial values of u
and u, which we may denote by Mo and lio- We obtain at once
u = e"^^ juo ^^ -r- 4 Mo (cos n't + ^ sin n'n \ (2),
where n' = V/i^ - \k\
If there be no friction, /c = 0, and then
. sinn^ . . xQv
t^ssUn h^ocosn^ (o).
/I
These results may be employed to obtain the solution of the
complete equation
u4-/«i + n2w= U (4),
where CT is an explicit function of the time ; for from (2) we see
that the eflTect at time ^ of a velocity tu communicated at time
^'is
n
The effect of U is to generate in time dt' a velocity Udt\ whose
result at time t will therefore be
u= -, [7a«'e^«(«-«'»sinn'rt-n,
and thus the solution of (4) will be
M^^^fT^^^"^^ %vcLri {i^t')JJ di (5).
If there be no friction, we have simply
u = ^J*sinn(«-Of^d«' (6)*
U being the force at time H.
The lower limit of the integrals is so far arbitrary, but it will
generally be convenient to make it zero.
On this supposition u and u as given by (6) vanish, when
^ = 0, and the complete solution is
u = e-**^ jUo — 7— + Mo (cos nt + -^,smntj>
+ i, fe-i^tt-e') sin n' (t - 1') Udt' (7),
n Jo
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66.] INSTABILITY. 75
or if there be no friction
u = iio hiLoC08nt + - I smnit — tf) Udf (8).
When t is sufficiently great, the complementary terms tend to
vanish on account of the factor 6""**^ and may then be omitted.
66 a. In § 66 we have limited the discussion to the case of
greatest acoustical importance, that is, we have supposed that nf
is real, as happens when n^ is positive, and /e not too great. But
a more general treatment of the problem of free vibrations is not
without interest. Whatever may be the values of n^ and /c, the
solution of (1) § 66 may be expressed
w = ^e'*»* + J5e'*«* (1),
where /Ai, /i, are the roots of
/i«4-/c/i + n» = (2).
The case already discussed is that in which the values of fi are
imaginary. The motion is then oscillatory, with amplitude which
decreases if tc be positive, but increases if k be negative.
But if n\ though positive, be less than J/r*, or if n* be negative,
n' becomes imaginary, that is /i becomes real. The motion
expressed by (1) is then non-oscillatory, and it depends upon the
sign of fi whether it increases or diminishes with the time. From
the solution of (2), viz.
M = -i^±iV(«»~4n«) (3),
it is evident that if w* be positive (and less than J/c*) the two
values of fi are of the same sign, and that the sign is the opposite
of that of K. Hence if /c be positive, both terms in (1) diminish
with the time, so that the system, however disturbed, subsides
again into a state of rest. If, on the contrary, k be negative, the
motion increases without limit. ^
We have still to consider the case of n^ negative. The real
values of fi are then of opposite signs. It is possible so to start
the sjTStem from a displaced position that it shall approach asymp-
totically the condition of rest in the configuration of equilibrium ;
but unless a special relation between displacement and velocity is
satisfied, the motion tends to increase without limit. Under these
circumstances the equilibrium must be regarded as unstable. In
this sense stability requires that n* and k be both positive.
A word may not be out of place as to the eflTect of an im-
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76 ONE DEGREE OF FREEDOM. [66 a.
pressed force upon a statically unstable system. If in § 46 we
suppose a: = 0, the solution (7) does not change its form merely
because n^ becomes negative. The fact that a system is suscep-
tible of purely periodic motion under the operation of an external
periodic force is therefore no evidence of stability.
67. For most acoustical purposes it is sufiScient to consider
the vibrations of the systems, with which we may have to deal,
as infinitely small, or rather as similar to infinitely small vibra-
tiona This restriction is the foundation of the important laws
of isochronism for free vibrations, and of persistence of period
for forced vibrations. There are, however, phenomena of a sub-
ordinate but not insignificant character, which depend essentially
on the square and higher powers of the motion. We will therefore
devote the remainder of this chapter to the discussion of the
motion of a system of one degree of freedom, the motion not being
so small that the squares and higher powers can be altogether
neglected.
The approximate expressions for the kinetic and potential
energies will be of the form
2" = i (mo + miu) tt«, F= i (/io + f^u) u\
If the sum of T and V be difierentiated with respect to the
time, we find as the equation of motion
msiU + /i^u + m{(iu + \m{lk^ -h f /Ai^' = Impressed Force,
which may be treated by the method of successive approximation.
For the sake of simplicity we will take the case where mi = 0,
a supposition in no way aflfecting the essence of the question.
The inertia of the system is thus constant, while the force of
restitution is a composite function of the displacement, partly pro-
portional to the displacement itself and partly proportional to
its square — ^accordingly unsymmetrical with respect to the position
of equilibrium. Thus for free vibrations our equation is of the
form
tt-f-n*i* + aw^ = (1),
with the approximate solution
'2* = ^ cosn^ (2),
where A — ^the amplitude — ^is to be treated as a small quantity.
Substituting the value of u expressed by (2) in the last
term, we find
A^
u + n*M = — a -^ (1 H-cos 2nt),
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67.] TERMS OF THE SECOND ORDER. 11
whence for a second approximation to the value of u
M = ^cosni-2~ + g^^cos2/i^ (3);
shewing that the proper tone (n) of the system is accompanied
by its ocUive (2n), whose relative importance increases with the
amplitude of vibration. A trained ear can generally perceive the
octave in the sound of a tuning-fork caused to vibrate strongly by
means of a bow, and with the aid of appliances, to be explained
later, the existence of the octave may be made manifest to any
one. By following the same method the approximation can
be carried further; but we pass on now to the case of a system
in which the recovering power is symmetrical with respect to
the position of equilibrium. The equation of motion is then
approximately
u + M«w + /3a» = (4),
which may be understood to refer to the vibrations of a heavy
pendulum, or of a load carried at the end of a straight spring.
If we take as a first approximation t^ = J. cos ni, corresponding
to y8 = 0, and substitute in the term multiplied by y8, we get
u + tihi = — -r cos 3n^ i — cos rd.
4 4
Corresponding to the la*t term of this equation, we should
obtain in the solution a term of the form £sin7i^, becoming
greater without limit with L This, as in a parallel case in the
Lunar Theory, indicates that our assumed first approximation
is not really an approximation at all, or at least does not continue
to be such. If, however, we take as our starting point w = -4 cos mly
with a suitable value for m, we shall find that the solution may
be completed with the aid of periodic terms only. In fact it is
evident beforehand that all we are entitled to assume is that the
motion is approximately simple harmonic, with a period wp-
prosdmately the same, as if y3=sO. A very slight examination
is suflScient to shew that the term varying as it*, not only may,
but must affect the period. At the same time it is evident
that a solution, in which the period is assumed wrongly, no
matter by how little, must at length cease to represent the motion
with any approach to accuracy.
We take then for the approximate equation
.. . , 3/3^' , /3^' o . /^x
u-^-n^a^^ — —, — cos 771^ r-cos377it (o),
4 4 ^
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78 ONE DEGREE OF FREEDOM. [67.
of which the solution will be
u^Aoosmt^ -^ g- -, (6),
provided that m be taken so as to satisfy
or m» = n» + — r— (7).
4
The term in /) thus produces two effects. It alters the pitch
of the fundamental vibration, and it introduces the twelfth as
a necessary accompaniment. The alteration of pitch is in most
cases exceedingly small — depending on the square of the amplitude,
but it is not altogether insensible. Tuning-forks generally rise
a little, though very little, in pitch as the vibration dies away.
It may be remai'ked that the same slight dependence of pitch
on amplitude occurs when the force of restitution is of the
form w'wH-aw', as may be seen by continuing the approximation
to the solution of (1) one step further than (3). The result in th%t
case is
-'=-'-^' (8).
The difference m* — n' is of the same order in A in both cases ;
but in one respect there is a distinction worth noting, namely,
that in (8) m* is always less than n«, while in (7) it depends on
the sign of )3 whether its effect is to raise or lower the pitch.
However, in most cases of the unsymmetrical class the change
of pitch would depend partly on a term of the form ax(} and
partly on another of the form /8w*, and then
"^^^^ "6;i« -^-4- (^>-
[In all cases where the period depends upon amplitude, it is
necessarily an even function thereof, a change of sign in the ampli-
tude being merely equivalent to an alteration in phase of 180°.]
68. We now pass to the consideration of the vibrations
forced on an unsymmetrical system by two harmonic forces
Eao^pt, jPcos (j^ — €).
^ [A oorreetion is here introduced, the necessity for which was pointed oat to me
by Dr Burton.]
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68.J TERMS OF THE SECOND ORDER. 79
The equation of motion is
it + n*M = - aw» -h -Ecos|)f + -F' 008(9^ -€) (1).
To find a first approximation we neglect the term containing
a. Thus
i^ = ecosp^+/cos(g^--6) (2),
E F
where ^= i ^. /=-^ — , (3).
Substituting this in the term multiplied by a, we get
t6 4- n*w = J? cos p^ + jP cos (g^ — €)
- a I ^^ "*■ f ^^® 2pt^^Q0^ 2 {qt - 6)+ e/cos {(p -9) ^ + ej
+ e/cos{(^ + g)^-€} L
whence as a second approximation for u
u = e cos2)« +/C08 {qt - 6) ^^^^ - 2~(n» - V) "^"^ ^^^
-2(i&^^
-^-:p^.<^'>«Kp+9)<-*l (4)-
The additional terms represent vibrations having firequencies
which are severally the doubles and the sum and difference of
those of the primaries. Of the two latter the amplitudes are
proportional to the product of the original amplitudes, shewing
that the derived tones increase in relative importance with
the intensity of their parent tones.
68a. If an isolated vibrating system be subject to internal
dissipative influences, the vibrations cannot be permanent, since
they are dependent upon an initial store of energy which suffers
gradual exhaustion. In order that the motion may be maintained,
the vibrating body must be in connection with a source of energy.
We have already considered cases of this kind under the head of
forced vibrations, where the system is subject to forces whose
amplitude and phase are prescribed, independently of the be-
haviour of the system. Such forces may have their origin in
revolving mechanism (such as electric alternators) governed so as
to move at a uniform speed. But more frequently the forces
under consideration depend upon the vibrations of other systems^
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80 ONE DEGREE OF FREEDOM. [68 a.
and then the question as to how the vibrations are to be main-
tained represents itself. A good example is afforded by the case
already discussed (§§ 63, 65) of a fork maintained in vibration
electrically by means of currents governed by a fork interrupter.
It has been pointed out that the performance of the latter
depends upon the magnetic forces operative upon it differing in
phase from the vibrations of the fork itself. With the interrupter
may be classed for the present purpose almost all acoustical and
musical instruments capable of providing a sustained sound. It
may suffice to mention vibrations maintained by wind (organ-
pipes, harmonium reeds, seolian harps, &c.), by heat (singing
flames, Rijke's tubes, &c.), by friction (violin strings, finger-
glasses), and the slower vibrations of clock pendulums and watch
balance-wheels.
In considering whether proposed forces are of the right kind
for the maintenance or encouragement of a vibration, it is often
convenient to regard them as reduced to impulses. Suppose, to
take a simple case, that a small horizontal positive impulse acts
upon the bob of a vibrating pendulum. The effect depends, of
course, upon the phase of the vibration at the instant of the
impulse. If the bob be moving positively at the instant in
question the vibration is encouraged, and this effect is a maximum
when the positive motion is greatest, that is, when the impulse
occurs at the moment of positive movement through the position
of equilibrium. This is the condition of things aimed at in
designing a clock escapement, for the effect of the force is then a
maximum in encouraging the vibration, and a minimum (zero to
the first order of approximation) in disturbing the period Of
course, if the impulse be half a period earlier or later than is
above supposed, the effect is to discourage the vibration, again
without altering the period. In like manner we see that if the
impulse occur at a moment of maximum elongation the effect is
concentrated upon the period, the vibration being neither en-
couraged nor discouraged.
In most cases the force acting upon a vibrating system in
virtue of its connection with a source of energy may be regarded
as harmonic. It may then be divided into two parts, one pro-
portional to the displacement u (or to the acceleration u), the
second proportional to the velocity ii. The inclusion of such
forces does not alter the form of the equation of vibration
u + KU + nhi^O (1).
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68 a.] MAINTENANCE OF VIBRATIONS. 81
By the first part (proportional to u) the pitch is modified, and by
the second the coeflScient of decay. If the altered k be still
positive, vibrations gradually die down ; but if the effect of the
included forces be to render k negative, vibrations tend on the
contrary to increase. The only case in which according to (I) a
steady vibration is possible, is when the complete value of k is
zero. If this condition be satisfied, a vibration of any amplitude
is permanently maintained.
When K is negative, so that small vibrations tend to increase,
a point is of course soon reached beyond which the approximate
equations cease to be applicable. We may form an idea of the
state of things which then arises by adding to equation (1) a
term proportional to a higher power of the velocity. Let us take
w+ictt+ic'u» + n»a = (2),
in which k and k are supposed to be small quantities. The
approximate solution of (2) is
u — ABmnt + —So- cos3n^ (3),
in which A is given by
A: + |ArV^« = (4).
From (4) we see that no steady vibration is possible unless k and
K have opposite signs. If k and k be both positive, the vibration
in all cases dies down ; while if tc and k be both negative, the
vibration (according to (2)) increases without limit. It k he
negative and tc' positive, the vibration becomes steady and
assumes the amplitude determined by (4). A smaller vibration
increases up to this point, and a larger vibration falls down to it.
If on the other hand tc be positive, while k is negative, the steady
vibration abstractedly possible is unstable, a departure in either
direction from the amplitude given by (4) tending always to
increase \
68 b. We will now consider briefly another and a very curious
kind of maintenance, of which the peculiarity is that the maintain-
ing influence operates with a frequency which is the double of
that of the vibration maintained. Probably the best known
example is that form of Melde's experiment, in which a fine string
is maintained in transverse vibration by connecting one of its
extremities with the vibrating prong of a massive tuning-fork,
^ On Maintained Vibrations, Phil, Mag., April, 18S8.
R. 6
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82 ONE DEGREE OF FREEDOM. [68 &.
the direction of motion of the point of attachment being parallel to
the lefigth of the string. The effect of the motion is to render
the tension of the string periodically variable ; and at first sight
there is nothing to cause the string to depart from its equilibrium
condition of straightness. It is known, however, that under these
circumstances the equilibrium may become unstable, and that the
string may settle down into a state of permanent and vigorous
vibration, whose period is the doMe of that of the fork.
As a simpler example, with but one degree of freedom, we
may take a pendulum, formed of a bar of sofb iron and vibrating
upon knife-edges. Underneath is placed symmetrically a vertical
bar electro-magnet, through which is caused to pass an electric
current rendered intermittent by an interrupter whose frequency
is twice that of the pendulum. The magnetic force does not tend
to displace the pendulum from its equilibrium position, but
produces the same sort of eflTect as if gravity were subject to a
periodic variation of intensity.
A similar result is obtained by causing the point of support
of the pendulum to vibrate in a vertical path. If we denote this
motion by ^ =/88in 2pt, the effect is as if gravity were variable by
the term 4p'/8 sin 2pt
Of the same nature are the crispations observed by Faraday*
and others upon the surface of water which oscillates vertically,
Faraday arrived experimentally at the conclusion that there were
two complete vibrations of the support for each complete vibra-
tion of the liquid.
In the following investigation*, relative to the case of one
degree of freedom, we shall start with the assumption that a
steady vibration is in progress, and inquire under what conditions
the assumed state of things is possible.
If the force of restitution, or " spring," of a body susceptible
of vibration be subject to an imposed periodic variation, the
differential equation takes the form
il + tcU'\-(n*-2asin2pt)u = (1),
in which k and a are supposed to be small. A similar equation
would apply approximately to the case of a periodic variation in
the effective mass of the body. The motion expressed by the
solution of (1) can be regular only when it keeps perfect time
I Phil, Traru. 1831, p. 299.
« Phil. Mag,, April, 1883.
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68 6.] MAINTENANCE OF VIBRATIONS. 83
with the imposed variations. It will appear that the necessary
conditions cannot be satisfied rigorously by any simple harmonic
vibration, but we may assume
u * ill sinpt + -Bi cosp^
•filjsin Spt -hBi cos 3pt+ As 8m bpt-^ (2),
in which it is not necessary to provide for sines and cosines of even
multiples of pt If the assumption be justifiable, the solution in
(2) must be convergent. Substituting in the differential equation,
and equating to zero the coefficients of sin pt, cos pt, &c we find
ill (n^ - j[)3) - #cpfi, - olBj 4- olB, = 0,
5i (n^-p") + KpAj^ - oili - ttils = ;
il , (n* - V) - 3^i>^J - (lBi + a2^8 = 0,
JS, (n»- V) + 3/«:pil3 + ttili - ail« = ;
A, {n^ - 2op^) - oicpB, " aB, + oBj = 0,
Bi (n* - 2op^) + OKpAji + aiia - ail, « ;
These equations shew that A^, B^ are of the order a relatively
to ill, Bi'j that ilj, Ba are of order a relatively to il,, J5„ and
so on. If we omit A^.B^ in the first pair of equations, we find
as a first approximation,
ili(n«-i>')-(/t/>H-a)A = 0,
4i(^i>-a) + (n>-p»)5i = 0;
whence ^ - —^ = ^""^/^ = V(«:i^p) /g.
Whence A,' 'cp + a n^ -^p^ ^/{a^ Kp) ^"^^^
and (n^-p'y^a^-Ky (4).
Thus, if a be given, the value of p necessary for a regular
motion is definite ; and p having this value, the regular motion is
u — Psin(pt + €),
in which €, being equal to tan""^ (BJAi), is also definite. On the
other hand, as is evident at once from the linearity of the original
equation, there is nothing to limit the amplitude of vibration.
These characteristics are preserved however far it may be
necessary to pursue the approximation. If ilam+i, ^wn+i may be
neglected, the first m pairs of equations determine the ratios of all
the coefficients, leaving the absolute magnitude open; and they
provide further an equation connecting p and a, by which the
pitch is determined.
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84 ONE DEGREE OF FKEEDOM. [68 b.
f
For the second approximation the second pair of equations
give
whence
aP
ti = Psin(p«4-€)+ _^^^cos(3/)^-H€) (5),
and from the first pair
tane:
1« — ll2 —
while /) is determined by
f'-i''-„._-9^)' = «'-*y (7)-
Returning to the first approximation, we see from (4) that the
solution is possible only under the condition that a be not less
than xp. If a = Kp, then p = n; that is, the imposed variation
in the "spring" must be exactly twice as quick as the natural
vibration of the body would be in the absence of friction. From
(3) it appears that in this case € = 0, which indicates that the
spring is a minimum one-eighth of a period after the body has
passed its position of equilibrium, and a maximum one-eighth of a
period before such passage. Under these circumstances the
greatest possible amount of energy is communicated to the
system ; and in the case contemplated it is just sufficient to
balance the loss by dissipation, the adjustment being evidently
independent of the amplitude.
I{a<Kp sufficient energy cannot pass to maintain the motion,
whatever may be the phase-relation ; but if o > ^q^, the balance
between energy supplied and energy dissipated may be attained
by such an alteration of phase as shall diminish the former
quantity to the required amount. The alteration of phase may
for this purpose be indifferently in either direction ; but if e be
positive, we must have
while if 6 be negative
If a be very much greater than /cp, € = ± Jtt, which indicates
that when the system passes through its position of equilibrium
the spring is at its maximum or at its minimum.
The inference from the equation that the adjustment of pitch
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68 6.] ABSOLUTE PITCH. 85
must be ?ibsolutely rigorous for steady vibration will be subject to
some modification in practice; otherwise the experiment could
not succeed. In most cases n* is to a certain extent a function of
amplitude; so that if n* have very nearly the required value,
complete coincidence is attainable by the assumption of an
amplitude of large and determinate amount without other
alterations in the conditions of the system.
The reader who wishes to pursue this subject is referred to a
paper by the Author " On the Maintenance of Vibrations by Forces
of Double Frequency, and on the Propagation of Waves through a
Medium endowed with a Periodic Structure,"^ in which the analysis
of Mr Hill" is applied to the present problem.
68 c. The determination of absolute pitch by means of the
siren has already been alluded to (§ 17). In all probability first-
rate results might be got by this method if proper provision, with
the aid of a phonic wheel for example, were made for uniform
speed. In recent years several experimenters have obtained excel-
lent results by various methods ; but a brief notice of these is all
that our limits will allow.
One of the most direct determinations is that of Koenig*, to
whom the scientific world has long been indebted for the construc-
tion of much excellent apparatus. This depends upon a special
instrument, consisting of a fork of 64 complete vibrations per
second, the motion being maintained by a clock movement acting
upon an escapement. A dial is provided marking ordinary time,
and serves to record the number of vibrations executed. The
performance of the fork is tested by a comparison between the
instrument and any chronometer known to be keeping good time.
The standard fork of 256 complete vibrations was compared with
that of the instrument by observing the Lissajous's figure appro-
priate to the double octave.
M. Koenig has also investigated the influence of resonators
upon the pitch of forks. Thus without a resonator a fork of 256
complete vibrations sounded in a satis&ctory manner for about 90
seconds. A resonator of adjustable pitch was then brought into
proximity, and the pitch, originally much graver than that of the
^ PhiL Mag., Angust, 1887.
^ On the Part of the Motion of the Lunar Perigree which is a Fnnction of the
Mean Motions of the Son and Moon, Acta Mathematica 8 ; 1, 1886. Mr Hill's
work was first published in 1877.
3 Wied. Ann. n. p. 394, 1880.
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86 ONE DEGREE OF FREEDOM. [68 C.
fork, was gradually raised. Even when the resonator w^ still a
minor third below the fork, there was observed a slight diminution
in the duration of the vibratory movement, and at the same time
an augmentation in the frequency of about '005. As the natural
note of the resonator approached nearer to that of the fork, this
diminution in the time and this increase in frequency became
more pronounced up to the immediate neighbourhood of unison ;
but at the moment when unison was established, the alteration of
pitch suddenly disappeared, and the frequency became exactly the
same as in the absence of the resonator. At the same time the
sound was powerfully reinforced; but this exaggerated intensity
fell off rapidly and the vibration died away after 8 or 10 seconds.
The pitch of the resonator being again raised a little, the sound of
the fork began to* change in the opposite direction, being now as
much too grave as before the unison was reached it had been too
acute. The displacement then fell away by degrees, as the pitch
of the resonator was fui-ther raised, and the duration of the
vibrations gradually recovered its original value of about 90
seconds. The maximum disturbance in the frequency observed
by Koenig was 035 complete vibrations. For the explanation
of these effects see § 117.
The temperature coefficient found by Koenig is '000112, so that
the pitch of a 256 fork falls '0286 for each degree Cent, by which
the temperature rises.
In determinations of absolute pitch' by the Author of this work
an electrically maintained interrupter fork, whose frequency may
for example be 32, was employed to drive a dependent fork of
pitch 128. When the apparatus is in good order, there is a fixed
relation between the two frequencies, the one being precisely
four times the other. The higher is of course readily compared
by beats, or by optical methods, with a standard of 128, whose
accuracy is to be tested. It remains to determine the frequency
of the interrupter fork itself.
For this purpose the intennipter is compared with the pendulum
of a standard clock whose rate is known. The comparison may be
direct, or the intervention of a phonic wheel (§ 63) may be invoked.
In either case the pendulum of the clock is provided with a silvered
bead upon which is concentrated the light from a lamp. Im-
mediately in front of the pendulum is placed a screen perforated
by a somewhat narrow vertical slit. The bright point of light
1 Nature, zvii. p. 12, 1S77 ; PhiL Tram. 1SS8, Part I. p. 316.
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68 C. j ABSOLUTE PITCH. 87
reflected by the bead is seen intermittently, either by looking over
the prong of the interrupter or through a hole in the disc of the
phonic wheel. In the first case there are 32 views per second, but
in the latter this number is reduced by the intervention of the
wheel. In the experiments referred to the wheel was so
arranged that one revolution corresponded to four complete vibra-
tions of the interrupter, and there were thus 8 views of the pen-
dulum per second, instead of 32, Any deviation of the period of
the pendulum from a precise multiple of the period of intermittence
shews itself as a cycle of changes in the appearance of the flash
of light, and an observation of the duration of this cycle gives the
data for a pi'ecise comparison of frequencies.
The calculation of the results is very simple. Supposing in
the first instance that the clock is correct, let a be the number of
cycles per second (perhaps ^) between the wheel and the clock.
Since the period of a cycle is the time required for the wheel to
gain, or lose, one revolution upon the clock, the frequency of revo-
lution is 8 + a. The frequency of the auxiliary fork is precisely 16
times as great, i.e. 128 ± 16a. If b be the number of beats per
second between the auxiliary fork and the standard, the frequency
of the latter is
128 ± 16a ± b.
An error in the mean rate of the clock is readily allowed for ;
but care is required to ascertain that the actual rate at the time
of observation does not diflfer appreciably from the mean rate.
To be quite safe it wouid be necessarj' to repeat the deter-
minations at intervals over the whole time required to rate the
clock by observation of the stars. In this case it would probably
be coDvenient to attach a counting apparatus to the phonic wheel.
In the method of M'Leod and Clarke^ time, given by a clock,
is recorded automatically upon the revolving drum of a chrono-
graph, which is maintained by a suitable governor in uniform
rotation. The circumference of the drum is marked with a grating
of equidistant lines parallel to the axis, and the comparison between
the drum and the standard fork is eflFected by observation of the
wavy pattern seen when the revolving* grating is looked at past
the edges of the vibrating prongs. These observers made a special
investigation as to the effect of bowing a fork upon previously
existing vibrations. Their conclusion is that in the case of un-
loaded forks no sensible change of phase occurs.
1 Phil. Trans, 1880, Part I. p. 1.
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88 ONE DEGREE OF FREEDOM. [68 C.
In the chronographic method of Prof. A. M. Mayer* the fork
under investigation is armed with a triangular fragment of thin
sheet metal, one milligram in weight, and actually traces its
vibrations as a curve of sines upon smoked paper. The time is
recorded by small electric discharges from an induction apparatus,
under the control of a clock, and delivered from the same tracing
point Although the disturbance due to the tracing point appears
to be very small, it is doubtful whether this method could compete
in respect of accuracy with those above described where the com-
parison with the standard is optical or acoustical On the other
hand, it has the advantage of not requiring a uniform rotation of
the drum, and the apparatus lends itself with facility to the deter-
mination of small intervals of time after the manner originally
proposed by T. Young*.
68d. The methods hitherto described for the determination of
absolute pitch, with the exception of that of Scheibler, may be
regarded as rather mechanical in their character, and they depend
for the most part upon somewhat special apparatus. It is possible,
however, to determine pitch with fair accuracy with no other
appliances than a common harmonium and a watch, and as the
process is instructive in respect of the theory of overtones, a short
account will here be given of it*.
The fimdamental principle is that the absolute frequencies of
two musical notes can be deduced from the interval between
them, i.e. the ratio of their frequencies and the number of beats
which they occasion in a given time when sounded together.
For example, if x and y denote the frequencies of two notes whose
interval is an equal temperament major third, we know that
y = 1*25992 x. At the same time the number of beats heard in a
second depending upon the deviation of the third from true
intonation, is 4^ — ox. In the case of the notes of a harmonium,
which are rich in overtones, these beats are readily counted, and
thus two equations are obtained from which the values of x and y
are at once found.
Of course in practice the truth of an equal temperament third
could not be taken for granted, but the difficulty thence arising
would be easily met by including in the counting all the three
^ National Academy of Sciences, Washington, 3/emoirf , Vol. iii. p. 43, 18S4.
« Lecturei, Vol. i. p. 191.
» Nature, Jan. 28, 1S79.
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68 d] ABSOLUTE PITCH. 89
major thirds which together make up an octave. Suppose, for
example, that the frequencies of c, e, ^rjf , c are respectively op, y, Zi
2a:, and that the beats per second between x and y are a, between
y and z are 6, and between z and 2a? are c. Then
4^ ~ 5^' 3s a, 4z — oy^ b, Sx — 5^ =» c,
from which a- = ^ (25a -f 206 H- 1 6c),
y = i (32a + 256 + 20c),
^«i(^W)a + 326 + 25c).
In the above statements the octave c — c' is for simplicity
supposed to be true. The actual error could readily be allowed
for if required ; but in practice it is not necessary to use c at all,
inasmuch as the third set of beats can be counted equally well
between ^rf and c.
The principal objection to the method in the above form is
that it presupposes the' absolute constancy of the notes, for
example, that y is the same whether it is being sounded in
conjunction with x or in conjunction with z. This condition is
very imperfectly satisfied by the notes of a harmonium.
In order to apply the fundamental principle with success, it is
necessary to be able to check the accuracy of the interval which is
supposed to be known, at the same time that the beats are being
counted. If the interval be a major tone (9 : 8), its exactness is
proved by the absence of beats between the ninth component of
the lower and the eighth of the higher note, and a counting
of the beats between the tenth component of the lower and the
ninth of the higher note completes the necessary data for de-
termining the absolute pitch.
The equal temperament whole tone (1*12246) is intermediate
between the minor tone (1*11111) and the major tone (1*12500),
but lies much nearer to the latter. Regarded as a disturbed
major tone, it gives slow beats, and regaixied as a disturbed
minor tone it gives quick ones. Both sets of beats can be heard
at the same time, and when counted (by two observers) give the
means of calculating the absolute pitch of both notes. If x and y
be the frequencies of the two notes, a and 6 the frequencies of the
slow and quick beats respectively,
, 9a?— .8y = a, 9y — 10a? = 6,
whence a? = 9a + 86, y = 10a + 96.
Theapplication of this method in no way assumes the truth of
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90 ONE DEGREE OF FREEDOM. [68 d.
the equal temperament whole tone, and in fact it is advantageous
to flatten the interval somewhat, so as to make it lie more nearly
midway between the major and the minor tone. In this way the
rapidity of the quicker beats is diminished, which facilitates the
counting.
The course of an experiment is then as follows. The notes C
and D are sounded together, and at a given signal the observers
begin counting the beats situated at about d!' and e" on the scale.
After the expiration of a measured interval of time a second signal
is given, and the number of both sets of beats is recorded.
For further details of the method reference must be made to
the original memoir, but one example of the results may be given
here. The period being 10 minutes, the number of beats recorded
were 2392 and 2341, giving x = 6709 as the pitch of C.
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CHAPTER IV.
VIBRATING SYSTEMS IN GENERAL.
69. We have now examined in some detail the oscillations
of a system possessed of one degree of freedom, and the results,
at which we have arrived, have a very wide application. But
material systems enjoy in general more than one degree of
freedom. In order to define their configuration at any moment
several independent variable quantities must be specified, which,
by a generalization of language originally employed for a point,
are called the co-ordinates of the system, the number of indepen*
dent co-ordinates being the index of freedom. Strictly speaking,
the displacements possible to a natural system are infinitely
various, and cannot be represented as made up of a finite number
of displacements of specified type. To the elementary parts of
a solid body any arbitrary displacements may be given, subject
to conditions of continuity. It is only by a process of abstraction
of the kind so constantly practised in Natural Philosophy, that
solids are treated as rigid, fluids as incompressible, and other sim-
plifications introduced so that the position of a system comes to
depend on a finite number of co-ordinates* It is not, however,
our intention to exclude the consideration of systems possessing
infinitely various freedom; on the contrary, some of the most
interesting applications of the results of this chapter will lie in
that direction. But such systems are most conveniently conceived
as limits of others, whose freedom is of a more restricted kind.
We shall accordingly commence with a system, whose position
is specified by a finite number of independent co-ordinates ^,,
^„ ^„ &c.
70. The main problem of Acoustics consists in the investi-
gation of the vibrations of a system about a position of stable
equilibrium, but it will be convenient to commence with the
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92
VIBRATING SYSTEMS IN GENERAL.
[70.
statical part of the subject. By the Principle of Virtual Velocities,
if we reckon the co-ordinates yp^u ^2, ifec. from the configuration
of equilibrium, the potential energy of any other configuratioD
will be a homogeneous quadratic function of the co-ordinates,
provided thart the displacement be sufficiently small. This quan-
tity is called F, and represents the work that may be gained in
passing from the actual to the equilibrium configuration. We may
write
F = iCn^i'+iCffl^2'+----^Ci,^iV^, + C^^,^,-h (1).
Since by supposition the equilibrium is thoroughly stable, the
quantities Cn, c^, Cii» &c. must be such that V is positive for all
real values of the co-ordinates.
71. If the system be displaced from the zero configuration
by the action of given forces, the new configuration may be
found from the Principle of Virtual Velocities. If the work done
by the given forces on the hypothetical displacement S-^i, S^a,
&c. be
^iSti + ^2% + (1),
this expression must be equivalent to SF, so that since Syjr^y S^,,
&c. are independent, the new position of equilibrium is deter-
mined by
'4i"^- ^|-*-«- <«.
or by (1) of §70,
. C-nti + C^ir^-^C^ylr^+ =^,l (3),
where there is no distinction in value between c,., and c,;..
From these equations the co-ordinates may be determined in
terms of the forces. If V be the determinant
Cii, Cij, Cis, ..
Ca> ^28* ^23> ••
C31, Cjftt, Cj3, ..
the solution of (3) may be written
dcii ^ dcia
w.
.(5).
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71.] RECIPROCAL RELATION. 93
These equations determine -^i, yft^, &c. uniquely, since V doea
not vanish, as appears from the consideration that the equations
dVjd^^ = 0, &c. could otherwise be satisfied by finite values of the
co-ordinates, provided only that the ratios were suitable, which is
contrary to the hypothesis that the system is thoroughly stable in
the zero configuration.
72. If Vrj, ... ^1, ... and ^/, ... ^/, ... be two sets of dis-
placements and corresponding forces, we have the following re-
ciprocal relation,
^i^/+.^«^/ + ...=^i>i + ^a't.-f (1),
as may be seen by substituting the values of the forces, when each
side of (1) takes the form,
Suppose in (1) that all the forces vanish except "9^ and '*'/;
then
^,V^/ = ^i'>^i (2).
If the forces ^j and '*'/ be of the same kind, we may suppose
them equal, and we then recognise that a force of any type acting
alone produces a displacement of a second type equal to the
displacement of the first type due to the action of an equal force
of the second type. For example, if A and B be two points
of a rod supported horizontally in any manner, the vertical de-
flection at A, when a weight W is attached at B, is the same as
the deflection at B, when W is applied at A *.
73. Since F is a homogeneous quadratic function of the co-
ordinates,
^•--iK+'+lf.*-^ (')•
or, if "^i, '^'a, &c. be the forces necessary to maintain the dis-
placement represented by V^ij-^sj&c.,
2F=^,^, + >Fa^.-h (2).
If ^^i + A-^i, '^2-|-A'^2, &0. represent another displacement,
for which the necessary forces are ^i-H A^i, %-|- A%, &c., the
1 On this sabjeot, see PhiU Mag,, Deo., 1S74, and Maroh, 1875.
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94 VIBRATING SYSTEMS IN GENERAL. [73.
corresponding potential energy is given by
+ A^i.^i + A>Fa.>/^a+...
+ A^i.A^i + A^a.A^,+ ...,
80 that we may write
2A7=2^.A>^ + 2A^.i|r + SA>F.A^ (3),
where AF is the difference of the potential energies in the two
cases, and we must particularly notice that by the reciprocal
relation, § 72 (1),
S^.A^r^SANl^.^r (4).
From (3) and (4) we may deduce two important theorems,
relating to the value of F for a system subjected t6 given dis-
placements, and to given forces respectively.
74. The first theorem is to the effect that, if given displace-
ments (not sufficient by themselves to determine the configuration)
be produced in a system by forces of corresponding types, the re-
sulting value of Ffor the system so displaced, and in equilibrium,
is as small as it can be under the given displacement conditions ;
and that the value of F for any other configuration exceeds this
by the potential energy of the configuration which is the difference
of the two. The only difficulty in the above statement consists
in understanding what is meant by * forces of corresponding types.*
Suppose, for example, that the system is a stretched string, of
which a given point P is to be subject to an obligatory displace-
ment; the force of corresponding type is here a force applied
at the point P itself. And generally, the forces, by which the
proposed displacement is to be made, must be such as would do
no work on the system, provided^nly that that displacement were
710^ made.
By a suitable choice of co-ordinates, the given displacement
conditions may be expressed by ascribing given values to the first
r co-ordinates ^i, '^a* ••• '^r, and the conditions as to the forces
will then be represented by making the forces of the i^emaining
types '^'r+n '*'r+si &c. vanish. If -^ -h A-^/r refer to any other con-
figuration of the system, and N?" -H A^ be the corresponding forces,
we are to suppose that A*^,, A*^], &c. as far as A^r c^U vanish.
Thus for the first r suffixes A-^ vanishes, and for the remaining
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74.] STATICAL THEOREMS. 95
suffixes ^ vanishes. Accordingly SNlT.A^ is zero, and therefore
SA'*'.'^ is also zero. Hence
2AF=2A^.AVr (1),
which proves that if the given displacements be made in any
other than the prescribed way, the potential energy is increased
by the energy of the diflference of the configurations.
By means of this theorem we may trace the effect on F of any
relaxation in the stiffness of a system, subject to given displacement
conditions. For, if after the alteration in stiffness the original equi-
librium configuration be considered, the value of V corresponding
thereto is by supposition less than before ; and, as we have just
seen, there will be a still further diminution in the value of V
when the system passes to equilibrium under the altered con-
ditions. Hence we conclude that a diminution in F as a function
of the co-ordinates entails also a diminution in the actual value of
V when a system is subjected to given displacements. It will
be understood that in particular cases the diminution spoken of
may vanish ^
For example, if a point P of a bar clamped at both ends be
displaced laterally to a given small amount by a force there ap-
plied, the potential energy of the deformation will be diminished
by any relaxation (however local) in the stifihess of the bar.
76. The second theorem relates to a system displaced by given
forces, and asserts that in this case the value of F in equilibrium
is greater than it would be in any other configuration in which
the system could be maintained at rest under the given forces, by
the operation of mere constraints. We will shew that the removal
of constraints increases the value of F.
The co-ordinates may be so chosen that the conditions of con-
straint are expressed by
^, = 0, V^, = 0, yJTr^O (1).
We have then to prove that when ^r+n ^r+a» &c. are given, the
value of F is least when the conditions (1) hold. The second
configuration being denoted as before by •^i + A'^i &c., we see
that for suffixes up to r inclusive -^ vanishes, and for higher
suffixes A'4^ vanishes. Hence
2V^.A^ = 2A^.^ = 0,
^ See a paper on Geneial Theorems relating to Equilibrium and Initial and
Steady Motions. Phil, Mag., March, 1875.
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96 VIBRATING SYSTEMS IN GENERAL. [75.
and therefore
2AF = 2A^.iX^ (2),
shewing that the increase in V due to the removal of the con-
straints is equal to the potential energy of the difference of the two
configurations.
76. We now pass to the investigation of the initial motion of
a system which starts from rest under the operation of given
impulses. The motion thus acquired is independent of any
potential energy which the system may possess when actually
displaced, since by the nature of impulses we have to do only
with the initial configuration itself. The initial motion is also
independent of any forces of a finite kind, whether impressed on
the system from without, or of the nature of viscosity.
If P, Q, R be the component impulses, parallel to the axes, on
a particle m whose rectangular co-ordinates are x, y, z, we have by
D'Alembert's Principle
Sm(^&r + ySy + iS-^) = 2(P&r + Q% + iJS^) (1),
where x, y, i denote the velocities acquired by the particle in virtue
of the impulses, and Bx, Sy, Sz correspond to any arbitrary dis-
placement of the system which does not violate the connection of
its parts. It is required to transform (1) into an equation expressed
by the independent generalized co-ordinates.
For the first side,
(dx dx d i \
, J , _ ( . dx , . dv , . dz\
+ dir,lm(.^+y^ + .^J^
-, . « / , dx . djf dz \
= «^j.i2m-^-(;r« + y« + i»)+
= st,|f + a^,jf+ (2).
where T, the kinetic energy of the system is supposed to be
expressed as a function of -^j, -^g, &c.
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76.] IMPULSES. 97
On the second side,
MP^ + Qh + RBz) = Sf.tm(p^+Q^ + R^) +
= f,S^i + f,S^.+ (3),
The transformed equation is therefore
(j|-f')«*'+(kl-f-)'*-+-=» <*>•
where S-^i.-S-^^a* &c. are now completely independent. Hence to
determine the motion we have
yr = fi» ,r = f9,&c (5),
where fi, f„ &c. may be considered as the generalized components
of impulse.
77. Since T is a homogeneous quadratic function of the gene-
ralized co-ordinates, we may take
T=-^(hiiri^-^ia^^^+ + 012^1^1^2 + aa'^2>^.+ (1),
whence
dyjr^
)
.(2),
where there is no distinction in value between Ors and a„^
Again, by the nature of T,
27=1^.1^+^,1^^+ = fttx + f,^.+ (3).
The theory of initial motion is closely analogous to that of the
displacement of a system from a configuration of stable equilibrium
by steadily applied forces. In the present theory the initial kinetic
energy T bears to the velocities and impulses the same relations
as in the former V bears to the displacements and forces respect-
ively. In one respect the theory of initial motions is the more
complete, inasmuch as T is exactly, while V is in general only
approximately, a homogeneous quadratic function of the variable&
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98 VIBRATING SYSTEMS IN GENERAL. [77.
If '^i, '^ai . . .i f i» f a » • • • denote one set of velocities and impulses
for a system started from rest, and -^Z, -^a', ..., f/, f,', ... a second
set, we may prove, as in § 72, the following reciprocal relation :
fi'^i+f.'^a+...=fi^/+fa^a'-f (4)^
This theorem admits of interesting application to fluid motion.
It is known, and will be proved later in the course of this work,
that the motion of a frictionless incompressible liquid, which
starts from rest, is of such a kind that its component velocities
at any point are the corresponding differential coefficients of a
certain function, called the velocity-potential. Let the fluid be
set in motion by a prescribed arbitrary deformation of the surface
S of a closed space described within it. The resulting motion is
determined by the normal velocities of the elements of S, which,
being shared by the fluid in contact with them, are denoted by
du/dn, if u be the velocity-potential, which interpreted physically
denotes the impulsive pressure. Hence by the theorem, if v be
the velocity-potential of a second motion, corresponding to
another set of arbitrary surface velocities dv/dn,
. \\^>-\\'> ■■ <«■
— ^an equation immediately following from Green's theorem, if
besides S there be only fixed solids immersed in the fluid. The
present method enables us to attribute to it a much higher gene-
rality. For example, the immersed solids, instead of being fixed>
may be free, altogether or in part, to take the motion imposed
upon them by the fluid pressures.
78. A particular case of the general theorem is worthy of
special notice. In the first motion let
^i=il, ^a = 0, f8 = f4 = f« =0;
and in the second.
Then f/ = f, (1).
In words, if, by means of a suitable impulse of the correspond-
ing type, a given arbitrary velocity of one co-ordinate be impressed
on a system, the impulse corresponding to a second co-ordinate
necessary in order to prevent it from changing, is the same as
would be required for the first co-ordinate, if the given velocity-
were impressed on the second.
^ Thomson and Tait, § 313 (/).
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78.] Kelvin's theorem. 99
As a simple example, take the case of two spheres A and B •-
immersed in a liquid, whose centres are free to move along certain
lines. If ^ be set in motion with a given velocity, B will
naturally begin to move also. The theorem asserts that the
impulse required to prevent the motion of B, is the same as if
the functions of A and B were exchanged : and this even though
there be other rigid bodies, C, D, &c., in the fluid, either fixed, or
free in whole or in part.
The case of electric currents mutually influencing each other by
induction is precisely similar. Let there be two circuits A and B,
in the neighbourhood of which there may be any number of other
wire circuits or solid conductors. If a unit current be suddenly
developed in the circuit A, the electromotive impulse induced in
B is the same as there would have been in A, had the current been
forcibly developed in B,
79. The motion of a system, on which given arbitrary velocities
are impressed by means of the necessary impulses of the corre-
sponding t)rpes, possesses a remarkable property discovered by
Thomson. The conditions are that -^i, yjr^, '^s, ...'^r are given,
while fr+i» fr+2, ••• vanish. Let yjti, -^a, •••?!, fa, &c. correspond to
the actual motion ; and
to another motion satisf}dng the same velocity conditions. For
each suffix either A*^ or f vanishes. Now for the kinetic energy
of the supposed motion,
2(r+Ar)=(f,+Af,)(^i + A^0 + ...
= 2T+f,A>^, + f,A^,+ ...
+ Afi.i^i + Afj.i^j+... + AfiAi|rj + Af,A>^a+....
But by the reciprocal relation (4) of § 77
faAi|ri-h...=Afa.'^i + ...,
of which the former by hypothesis is zero ; so that
2Ar=Af,Ai^i + Af,A^,+ (1),
shewing that the energy of the supposed motion exceeds that of
the actual motion by the energy of that motion which would have
to be compounded with the latter to produce the former. The
motion actually induced in the system has thus less energy than
any other satisfying the same velocity conditions. In a subsequent
chapter we shall make use of this property to find a superior limit
to the energy of a system set in motion with prescribed velocities.
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100 VIBRATING SYSTEMS IN GENERAL. [79.
If any diminution be made in the inertia of any of the parte
of a system, the motion corresponding to prescribed velocity
conditions will in general undergo a change. The value of T will
necessarily be less than before ; for there would be a decrease even
if the motion remained unchanged, and therefore a fortiori when
the motion is such as to make T an absolute minimum. Con-
versely any increase in the inertia increases the initial value of T,
This theorem is analogous to that of § 74. The analogue for
initial motions of the theorem of § 75, relating to the potential
energy of a system displaced by given forces, is that of Bertrand,
and may be thus stated : — If a system start from rest under the
operation of given impulses, the kinetic energy of the actual motion
exceeds that of any other motion which the system might have
been guided to take with the assistance of mere constraints, by the
kinetic energy of the difference of the motions^
[The theorems of Kelvin and Bertrand represent different
aspects of the same truth. Let us suppose that the prescribed
impulse is entirely of the first type fi. Then r = ifi^i, whether
the motion be free or be subjected to any constraint. Further,
under any given circumstances as to constraint, -^i is proportional
to fi, and the ratio fi : -^j may be regarded as the moment of
inertia ; so that
Kelvin's theorem asserts that the introduction of a consti^aint
can only increase the value of T when -^i is given. Hence whether
•^1 be given or not, the constraint can only increase the ratio of
27 to -^i' or of fi to -^i. Both theorems are included in the
statement that the moment of inertia is increased by the intro-
duction of a constraint.]
80. We will not dwell at any greater length on the mechanics
of a system subject to impulses, but pass on to investigate
Lagrange's equations for continuous motion. We shall suppose
that the connections binding together the parts of the system
are not explicit fiinctions of the time; such cases of forced
motion as we shall have to consider will be specially shewn to
be within the scope of the investigation.
By D'Alembert's Principle in combination with that of Virtual
Velocities,
2m Qthx + yhy + zhz) = 2 {Xhx + Yhy + Zhz) (1),
^ Thomson and Tait, § 811. Phil, Mag. March, 1875.
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80.] Lagrange's equations. 101
where Sx, Sy, Sz denote a displacement of the system of the most
general kind possible without violating the connections of its
parts. Since the displacements of the individual particles of
the system are mutually related, Sa;, ... are not independent. The
object now is to transform to other variables -^i, -^j,..., w^hich
shall be independent. We have
so that
xBx = -J- (xhx) — ^Sd^ ,
lm{xSx + ySy + zSz) = ^. 2m (xSx + y Sy + iS^) - ST,
But (§ 76) we have already found that
while ST^^B^, + ^^^,+ ...,
if r be expressed as a quadratic function of -^i, ^j, ..., whose
coeflScients are in general functions of V^u'^j,.... Also
d (dT ..\ d (dT\ ., dT ^..
dt\d;f.'^'f'')'dt[^^^
inasmuch as -^ S>^i ^B-rr-^i,
Accordingly
^SQ-f}^-* <^^
Thus, if the transformation of the second side of (1) be
2(X&c+y«y + ZS^) = ^iS^i + ^a8V^, + (3),
we have equations of motion of the form
if^).^^ = ^ (4).
dVdyjr' dy^ ^
Since '^'S^ denotes the work done on the system during a
displacement S^, "9 may be regarded as the generalized com-
ponent of force.
In the case of a conservative system it is convenient to
separate &om '^ those parts which depend only on the configura-
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102 VIBRATING SYSTEMS IN GENERAL. ' [80-
tion of the system. Thus, if V denote the potential energy, we
may write
dt\d'^) dit dyjr ^ ^'
where 'V is now limited to the forces acting on the system which
are not already taken account of in the term dV/d'^.
81. There is also another group of forces whose existence
it is often advantageous to recognize specially, namely those
arising from friction or viscosity. If we suppose that each
particle of the system is retarded by forces proportional to its
component velocities, the effect will be shewn in the fundamental
equation (1) § 80 by the addition to the left-hand member of
the terms
2 {kJcZx -f Kyi/hy H- Kg^ Bz\
where Kx, /Cy, Kg are coefficients independent of the velocities,
but possibly dependent on the configuration of the system. The
transformation to the independent co-ordinates -^i, y^^, &c. is
effected in a similar manner to that of
Sy/t {xSw + yBy + zSz)
considered above (§ 80), and gives
dF c , dF ^ , ,- .
^a^.-H^-^^sv^.-H (1).
where i" = ^2 (xg^ + Kyy^ + kJ;^)
= i6u^i' + i6«^2" + ... + 6i2^it« + b^yjr.yjt, + (2).
Fy it will be observed, is like T a homogeneous quadratic
function of the velocities, positive for all real values of the
variables. It represents half the rate at which energy is dissipated.
The above investigation refers to retarding forces proportional
to the absolute velocities ; but it is equally important to consider
such as depend on the relative velocities of the parts of the
system, and fortunately this can be done without any increase
of complication. For example, if a force act on the particle Xi
proportional to (d^ — ij), there will be at the same moment an
equal and opposite force acting on the particle x^. The additional
terms in the fundamental equation will be of the form
fC» (Xj — X2) BXi + Kx (Xt — Xi) &Cj,
which may be written
tCx (iJi - 4) B(Xi - ^j) = S^i -y- [^Kx (Xi - X^y] +
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81.]
THE DISSIPATION FUNCTION.
103
and 80 on for any number of pairs of mutually influencing
particles. The only eflfect is the addition of new terms to F,
which still appears in the form (2)\ We shall see presently that
the existence of the function F, which may be called the Dis-
sipation Function, implies certain relations among the coefficients
of the generalized equations of vibration, which carry with them
important consequences^
The equations of motion may now be written in the form
(3).
dtKd'f) df d^ df
82. We may now introduce the condition that the motion
takes place in the immediate neighbourhood of a configuration
of thoroughly stable equilibrium ; T and F are then homogeneous
quadratic functions of the velocities with coefficients which are
to be treated as constant, and F is a similar function of the
co-ordinates themselves, provided that (as we suppose to be
the case) the origin of each co-ordinate is taken to correspond
with the configuration of equilibrium. Moreover all three
functions are essentially positive. Since terms of the form dT/dyfr
are of the second order of small quantities, the equations of motion
become linear, assuming the form
dAdylrJ^d^jr^dyk'^ ^^^'
dt\d^J dyjt ' d'sjt
where under ^ are to be included all forces acting on the system
not already provided for by the differential coefficients of F and V.
The three quadratic functions will be expressed as follows : —
-P = i6nti»+i6a^,*+ ... +6i2^i^,-H ... ■ (2),
where the coefficients a, b, c are constants.
From equation (1) we may of course fall back on previous
results by supposing F and F, or F and T, to vanish.
A third set of theorems of interest in the application to Elec-
^ The differences referred to in the text may of coarse pass into differential
coefficients in the case of a body continuously deformed.
> The Dissipation Function appears for the first time, so far as I am aware, in
a paper on General Theorems relating to Vibrations, published in the Proeeedingi
of the Mathematical Society for June, 1873.
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104 VIBRATING SYSTEMS IN GENERAL. [82.
tricity may be obtained by omitting T €uid F, while F is retained,
but it is unnecessary to pursue the subject here.
If we substitute the values of T, F and F, and write D for djdt,
we obtain a system of equations which may be put into the form
^ii-^i + ^ij-^a + e^-^t + ... = ^1 '
.(3).
(6),
where en denotes the quadratic operator
e« = a«i)» + 6rJ> + Cr, (4).
It must be particularly remarked that since
art = ^try ^ri — ^trt <^r9 = ^tn
it follows that €„^etr (5).
[The theory of motional forces, i.e. forces proportional to the
velocities, has been further developed in the second edition of
Thomson and Tait's Natural Philosophy (1879). In the most
general case the equations may be written
where bn^btr, ^rf = Ar (7).
Of these the terms with the coefficients b can be derived from
the dissipation function
The terms in ^ on the other hand do not represent dissipation,
and are called the gyrostatic terms.
If we multiply the first of equations (6) by -^i, the second by
-^j, &c., and then add, we obtain
^^l^ + 2F^^,ir, + ^,yfr,-\- (8).
In this the first term represents the rate at which energy is
being stored in the system ; 2^ is the rate of dissipation ; and the
two together account for the work done upon the system by the
external forces.]
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83.] COEXISTENCE OF «MALL MOTIONS. 105
83. Before proceeding further, we may draw an important
inference from the linearity of our equations. If corresponding
respectively to the two sets of forces ^j, ^j,..., ^/, ^a', ... two
motions denoted by -^i, >^j, ..., i^/, i^,', ... be possible, then must
also be possible the motion '^i + >f^/, -^a + V^a'i • • • in conjunction
with the forces "5^1 + '*'i', '^'a + '*'«'» .••• Or, as a particular case,
when there are no impressed forces, the superposition of any two
natural vibrations constitutes also a natural vibration. This is the
celebrated principle of the Coexistence of Small Motions, first
clearly enunciated by Daniel Bernoulli. It will be understood
that its truth depends in general on the justice of the assumption
that the motion is so small that its square may be neglected.
[Again, if a system be under the influence of constant forces
"^1, &c., which displace it into a new position of equilibrium, the
vibrations which may occur about the new position are the same
as those which might before have occurred about the old position.]
84. To investigate the free vibrations, we must put '*'i, ^a, ...
equal to zero ; and we will commence with a system on which no
fiictional forces act, for which therefore the coefficients e^i* &c. are
even functions of the sjrmbol D, We have
eai-^i + ea^^sH- ....= \ (!)•
From these equations, of which there are as many (m) as the
system possesses degrees of liberty, let all but one of the valuables
be eliminated. The result, which is of the same form whichever
be the co-ordinate retained, may be written
Vt = (2),
where V denotes the determinant
•(3),
! ^u> ^ia> ^is> •••
^Ji> ^a2» ^as> •••
and is (if there be no friction) an even function of D of degree 2m.
I^t +X,, iXa, ..., ±\to be the roots of V=0 considered as an
equation in D, Then by the theory of differential equations the
most general value of '^ is
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106 VIBRATING SYSTEMS IN GENERAL. [84-
where the 2m quantities A, A\ B, B\ &c. are arbitrary constants.
This form holds good for each of the co-ordinates, but the constants
in the diflferent expressions are not independent. In fact if a
particular solution be
the ratios -4^ : -4, : -4, ... are completely determined by the
equations
6ii-4i + eMilaH-^i8^i + =0 \
en-4iH-Cail,-|-6a^8+ =0 > (5)i
where in each of the coefficients such as ^„» ^ is substituted for i).
Equations (5) are necessarily compatible, by the condition that Xi
is a root of V=0. The ratios A^ :Ai\At ... corresponding to
the root — Xj are the same as the ratios Ax'.A^iA^..., but for
the other pairs of roots Xj, - X^, &c. there are distinct systems of
ratios.
86. The nature of the system with which we are dealing
imposes an important restriction on the possible values of X, If Xi
were real, either \ or — Xi would be real and positive, and we
should obtain a particular solution for which the co-ordinates, and
with them the kinetic energy denoted by
increase without limit. Such a motion is obviously impossible for
a conservative system, whose whole energy can never differ from
the sum of the potential and kinetic energies with which it was
animated at starting. This conclusion is not evaded by taking Xi
negative ; because we are as much at liberty to trace the motion
backwards as forwards. It is as certain that the motion never was
infinite, as that it never will he. The same argument excludes the
possibility of a complex value of X.
We infer that all the values of X are purely imaginary, cor-
responding to real negative values of W Analytically, the fact
that the roots of V = 0, considered as an equation in i)*, are
all real and negative, must be a consequence of the relations
subsisting between the coefficients Ou, Oia, ..., Cu, Cu, ... in virtue of
the fact that for all real values of the variables T and V are
positive. The case of two degrees of liberty will be afterwards
worked out in full.
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86.] NORMAL CO-ORDINATES. 107
86. The form of the solution may now be advantageously
changed by writing irii for \i, &c. (where i= V — 1), and taking
new arbitrary constants. Thus
-1^1 = Ai cos (lilt - a) + -Bi cos {nj; - ^8) + Ci cos (n^t - 7) + ... '
-^2 = ila cos (ni^ — a) + jB, cos (wj^ — /8) + Ca cos (n,^ — 7) + ... ..,,(i)
-^/r, = ^, cos (nj; - a) + 5, cos (nj; — I3) + Ct cos (n^t - 7) + . .
where ni\ n,*, &c. are the m roots of the equation of m^ degree
in 71* found by writing — n^ for D* in V = 0. For each value of n
the ratios A^iAiiAt,.. are determinate and real.
This is the complete solution of the problem of the free
vibrations of a conservative system. We see that the whole
motion may be resolved into vi normal harmonic vibrations of
(in general) different periods, each of which is entirely indepen-
dent ef the others. If the motion, depending on the original
disturbance, be such as to reduce itself to one of these (rii)
we have
y^i^Ai cos {riit — a), y^^^^i cos {nj; - a), &c (2),
where the ratios -4^:^.2:^43 ...depend on the constitution of the
system, and only the absolute amplitude and phase are arbitrary.
The several co-ordinates are always in similar (or opposite) phases
of vibration, and the whole system is to be found in the configura-
tion of equilibrium at the same moment.
We perceive here the mechanical foundation of the supremacy
of harmonic vibrations. If the motion be sufficiently small, the
differential equations become linear with constant coefficients;
while circular (and exponential) functions are the only ones which
retain their type on differentiation.
87. The m periods of vibration, determined by the equation
V =0, are quantities intrinsic to the system, and must come out
the same whatever co-ordinates may be chosen to define the con-
figuration. But there is one system of co-ordinates, which is
especially suitable, that namely in which the normal types of
vibration are defined by the vanishing of all the co-ordinates but
one. In the first type the original co-ordinates '^u'^j, &c. have
given ratios; let the quantity fixing the absolute values be ^, so
that in this t3rpe each co-ordinate is a known multiple of 0,. So
in the second type each co-ordinate may be regarded as a known
multiple of a second quantity <^2i &nd so on. By a suitable deter-
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108 VIBRATING SYSTEMS IN GENERAL. [87.
mination of the m quantities ^, ^s> &c., any configuration of the
system may be represented as compounded of the m configurations
of these tjrpes, and thus the quantities ^ themselves may be looked
upon as co-ordinates defining the configuration of the system.
They are called the normal co-ordinates \
When expressed in terms of the normal co-ordinates, T and V
are reduced to sums of squares ; for it is easily seen that if the
products also appeared, the resulting equations of vibration would
not be satisfied by putting any m — 1 of the co-ordinates equal to
zero, while the remaining one was finite.
We might have commenced with this transformation, assuming-
from Algebra that any two homogeneous quadratic functions can
be reduced by linear transformations to sums of squares. Thus
where the coefficients (in which the double suffixes are no longer
required) are necessarily positive.
Lagrange's equations now become
ai^ + Ci<^i = 0, a,$, + Cj<^ = 0, &c (2),
of which the solution is
4>i = A cos(riit — a), 0, = £ cos (ti^ — /8), &c (3),
where A, 5..., a, /8... are arbitrary constants, and
ni»=Ci-rai, n^^c^-^a^, &c (4).
[The vibrations expressed by the various normal co-ordinates
are completely independent of one another, and the energy of the
whole motion is the simple sum of the parts corresponding to the
several normal vibrations taken separately. In fact by (1)
r+F=icx^« + iM,« + (3).
By the nature of the case the coefficients a are necessarily
positive. But if the equilibrium be unstable, some of the
coefficients c may be negative. Corresponding to any negative
c, n becomes imaginary and the circular functions of the time are
replaced by exponentials.
In any motion proportional to e^ the disturbance is equally
multiplied in equal times, and the degree of instability may be
considered to be measured by X. If there be more than one
^ Thomson and Tait's Natural Philoiophy, first edition 1867, § 337.
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87.] PERIODS OF FREE VIBRATIONS. 109
unstable mode, the relative importance is largely determined by
the corresponding values of X. Thus, if
in which Xj > Xj, then whatever may be the finite ratio o{ A : B,
the first term ultimately acquires the preponderance, inasmuch as
In general, unstable equilibrium when disturbed infinitesimally
will be departed from according to that mode which is most
unstable^ viz. for which X is greatest. In a later chapter we shall
meet with interesting applications of this principle.
The reduction to normal co-ordinates allows us readily to trace
what occurs when two of the values of n* become equal. It is
evident that there is no change of form. The spherical pendulum
may be referred to as a simple example of equal roots. It is
remarkable that both Lagrange and Laplace fell into the error of
supposing that equality among roots necessarily implies terms
containing ^ as a factor^ The analytical theory of the general
case (where the co-ordinates are not normal) has been discussed by
SomoP and by Routh'.]
88. The interpretation of the equations of motion leads to a
theorem of considerable importance, which may be thus stated*.
The period of a conservative system vibrating in a constrained type
about a position of stable equilibrium is stationarj- in value when
the type is normal. We might prove this from the original
equations of vibration, but it will be more convenient to employ
the normal co-ordinates. The constraint, which may be supposed
to be of such a character as to leave only one degree of freedom, is
represented by taking the quantities in gfven ratios.
If we put
i>,^A,0, <^, = il,^,&c (1),
d is a variable quantity, and Ai, A^, &c. are given for a given con-
straint.
The expressions for T and V become
r={K^i«+iMa»+ }^^
Vr={^c^Ai^ + ^cM+ }^,
1 Thomson and Tait, 2nd edition, § 343 7;i.
3 St Petenb, Acad, Sci, M6m. x. 1859.
' Stability of Motion (Adams Prize Essay for 1877). See also Bouth*8 Rigid
Dyfiamiet, 5th edition, 1892.
* Proceedingt of the Mathematical Society, June, 1873.
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110 VIBRATING SYSTEMS IN GENERAL. [88.
whence, if 6 varies as cos pt,
This gives the period of the vibration of the constrained type ;
and it is evident that the period is stationary, when all but one of
the coefficients -4i,ilj, ... vanish, that is to say, when the tj'pe
coincides with one of those natural to the system, and no constraint
is needed.
[In the foregoing statement the equilibrium is supposed to be
thoroughly stable, so that all the quantities c are positive. But
the theorem applies equally even though any or all of the c*s be
negative. Only if j^ itself be negative, the period becomes
imaginary. In this case the stationary character attaches to the
coefficients of ^ in the exponential terms, quantities which measure
the degree of instability.
Corresponding theorems, of importance in other branches of
science, may be stated for systems such that only T and -P, or only
Fand Fy are sensible ^
The stationary property of the roots of Lagrange's determinant
(3) § 84, suggests a general method of approximating to their
values. Beginning with assumed rough approximations to the
ratios A^-.A^iA^ we may calculate a first approximation to
jf from
^^iaii^i' + iaa^»+... + aiA^,+ ... ^^'
With this value of p* we may recalculate the ratios A^xA^.,. from
any (m — 1) of equations (5) § 84, then again by application of (3)
determine an improved value of p', and so on.]
By means of the same theorem we may prove that an increase
in the mass of any part of a vibrating system is attended by a
prolongation of all the natural periods, or at any rate that no
period can be diminished. Suppose the increment of mass to be
infinitesimal. After the alteration, the types of free vibration will
in general be changed ; but, by a suitable constraint, the system
may be made to retain any one of the former types. If this be
done, it is certain that any vibration which involves a motion of
the part whose mass has been increased will have its period
prolonged. Only as a particular case (as, for example, when a
load is placed at the node of a vibrating string) can the period
^ Brit. Asi. Rep. for 1S85, p. 911.
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88.] PERIODS OF FREE VIBRATIONS. Ill
remain unchanged. The theorem now allows us to sissert that
the removal of the constraint, and the consequent change of type,
can only affect the period by a quantity of the second order ; and
that therefore in the limit the &ee period cannot be less than
before the change. By integration we infer that a finite increase
of mass must prolong the period of every vibration which involves
a motion of the part affected, and that in no case can the period
be diminished ; but in order to see the correspondence of the two
sets of periods, it may be necessary to suppose the alterations
made by steps. Conversely, the effect of a removal of part of
the mass of a vibrating system must be to shorten the periods
of all the free vibrations.
In like manner we may prove that if the system undergo such
a change that the potential energy of a given configuration is
diminished, while the kinetic energy of a given motion is unaltered,
the periods of the free vibrations ai*e all increased, and conversely.
This proposition may sometimes be used for tracing the effects of
a constraint; for if we suppose that the potential energy of
any configuration violating the condition of constraint gradually
increases, we shall approach a state of things in which the
condition is observed with any desired degree of completeness.
During each step of the process every free vibration becomes
(in general) more rapid, and a number of the free periods (equal
to the degrees of liberty lost) become infinitely small. The
same practical result may be reached without altering the po*
tential energy by supposing the kinetic energy of any motion
violating the condition to increase without limit. In this case
one or more periods become infinitely large, but the finite
periods are ultimately the same as those arrived at when the
potential energy is increased, although in one case the periods
have been throughout increasing, and in the other diminishing.
This example shews the necessity of making the alterations by
steps; otherwise we should not understand the correspondence
of the two sets of periods. Further illustrations will be given
under the head of two degrees of freedom.
By means of the principle that the value of the free periods
is stationary, we may easily calculate corrections due to any
deviation in the system from theoretical simplicity. If we take
as a hypothetical type of vibration that proper to the simple
system, the period so found will differ fi'om the truth by quan-
tities depending on the squares of the irregularities. Several
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112 VIBRATING SYSTEMS IN GENERAL. [88.
examples of such calculations will be given in the course of
this work.
89. Another point of importance relating to the period of a
system vibrating in an arbitrary type remains to be noticed.
It appears from (2) § 88, that the period of the vibration cor-
responding to any hypothetical type is included between the
greatest and least of those natural to the system. In the case
of systems like strings and plates which are treated as capable
of continuous deformation, there is no least natural period;
but we may still assert that the period calculated from any hypo-
thetical type cannot exceed that belonging to the gravest normal
type. When therefore the object is to estimate the longest
proper period of a system by means of calculations founded
on an assumed t3rpe, we know a priori that the result will come
out too small.
In the choice of a hypothetical type judgment must be
used, the object being to approach the truth as nearly as can
be done without too great a sacrifice of simplicity. Thus the
type for a string heavily weighted at one point might suitably
be taken from the extreme case of an infinite load, when the
two parts of the string would be straight. As an example of
a calculation of this kind, of which the result is known, we
will take the case of a uniform string of length I, stretched
with tension Ti, and inquire what the period would be on
certain suppositions as to the type of vibration.
Taking the origin of a at the middle of the string, let the
curve of vibration on the positive side be
y = co8p<|l-(^)"| (1).
and on the negative side the image of this in the axis of y,
* n being not less than unity. This form satisfies the conditioa
that y vanishes when a? = ± JZ. We have now to form the ex-
pressions for T and V, and it will be sufficient to consider the
positive half of the string only. Thus, p being the longitudinal
density.
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89.]
PERIODS OF FREE VIBRATIONS.
113
Hence
i^=
2(n+l)(2n+l) 2\
2n-l 'pi*'
.(2).
If n = 1, the string vibrates as if the mass were concentrated
in its middle point, and
"-^'^
If n = 2, the form is parabolic, and
The true value of p^ for the gravest type is — j^ , so that
the assumption of a parabolic form gives a period which is too
small in the ratio ir : »J 10 or '9936 : 1. The minimum of p*,
as given by (2), occurs when n = J (V6 + 1) = 1*72474, and gives
©« = 9-8990^,.
The period is now too small in the ratio
TT : >/9^8990 = '99851 : 1.
It will be seen that there is considerable latitude in the
choice of a type, even the violent supposition that the string
vibrates as two straight pieces giving a period less than ten
per cent, in error. And whatever type we choose to take, the
period calculated from it cannot be greater than the truth.
[In the above applications it is assumed that there are no
unstable modes. When unstable modes exist, the statement is
that a constrained mode if stable possesses a frequency of vibra-
tion less than that of the highest normal mode, and if unstable
has a degree of instability less than that of the most unstable
normal mode]
90. The rigorous determination of the periods and types of
vibration of a given system is usually a matter of great diflSculty,
arising from the fact that the functions necessary to express the
modes of vibration of most continuous bodies are not as yet recog-
nised in analysis. It is therefore often necessary to fall back on
methods of approximation, refening the proposed system to some
other of a character more amenable to analysis, and calculating
corrections depending on the supposition that the difference be-
tween the two systems is small. The problem of approximately
R. 8
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114 VIBRATING SYSTEMS IN GENERAL. [90.
simple systems is thus one of great importance, more especially
as it is impossible in practice actually to realise the simple forms
about which we can most easily reason.
Let us suppose then that the vibrations of a simple system are
thoroughly known, and that it is required to investigate those
of a system derived from it by introducing small variations in
the mechanical functions. If ^, <^3, &c. be the normal co-ordi-
nates of the original system,
and for the varied system, referred to the same co-ordinates,
which are now only approximately normal,
F+SF=i (ci + Scn) <^»+ ... + Sc,0,<^+ .
in which Sa^, Soi,, Scu, Scia, &c. are to be regarded as small
quantities. In certain cases new co-ordinates may appear, but
if so their coefficients must be small. From (1) we obtain for the
Lagrangian equations of motion,
(ar+~Sa„ Z)« + Ci + Scn) (fh + (Ba^.2l> + Sc„) ^2 '
(&Zai)« + &M)<^ + (a, + Saa2)* + c, + Sc^)02 )- (2)-
In the original system the fundamental .types of vibration
are those which correspond to the variation of but a single co-
ordinate at a time. Let us fix our attention on one of them,
involving say a variation of <j>r, while all the remaining co-
ordinates vanish. The change in the system will in general
entail an alteration in the fundamental or normal types; but
under the circumstances contemplated the alteration is small.
The new normal type is expressed by the synchronous variation
of the other co-ordinates in addition to 0^ ; but the ratio of any
other <f)g to <f>r is small. When these ratios are known, the normal
mode of the altered system will be determined.
Since the whole motion is simple harmonic, we may suppose
that each co-ordinate varies as cosjOr^, and substitute in the
differential equations —pr^ for i>*. In the «'** equation 0^ occurs
with the finite coefficient
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90.] APPROXIMATELY SIMPLE SYSTEMS. 115
The coefficient of ^^ is
The other terms are to be neglected in a first approximation,
since both the co-ordinate (relatively to 0r) and its coefficient are
small quantities. Hence
*-*-- cr-^.«a. (^>-
Now - a,p,* + c, = 0,
and thus 0, : ^ ^ P^'f^""^ ^ (4)
the required result.
If the kinetic energy alone undergo variation,
*' = '^'"^7^'-^~ ^'^
The corrected value of the period is determined by the rth
equation of (2), not hitherto used. We may write it,
Substituting for ^, : (f>r from (4), we get
a ^ Cr + 8Crr y (^Crg -pr^'JarMy rr..
P' Or + SOrr a^(p,'^Pr') ^''^•
The first term gives the value of pr^ calculated without allow-
ance for the change of type, and is sufficient, as we have already
proved, when the square of the alteration in the system may
be neglected. The terms included under the symbol 2, in
which the summation extends to all values of 8 other than r,
give the correction due to the change of type and are of the
second order. Since a, and Or are positive, the sign of any term
depends upon that oi p^—pr^ If pg^>pr^, that is, if the mode
s be more acute than the mode r, the correction is negative,
and makes the calculated note graver than before; but if the
mode 8 be the graver, the correction raises the note. If r refer
to the gravest mode of the system, the whole correction is
negative ; and if r refer to the acutest mode, the whole correction
is positive, as we have already seen by another method.
91. As an example of the use of these formula, we may
take the case of a stretched string, whose longitudinal density p
is not quite constant. If x be measured from one end, and y
8—2
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116 VIBRATING SYSTEMS IN GENERAL. [91.
be the transverse displacement, the configuration at any time t
will be expressed by
y = <^8iny 4-<^8in-j-+<^3 8xn y + (1),
I being the length of the string. ^, 02,*.. are the normal
co-ordinates for /9 = constant, and though here p is not strictly
constant, the configuration of the system may still be expressed
by means of the same quantities. Since the potential energy
of any configuration is the same as if p = constant, SV==^0, For
the kinetic energy we have
T + Sr = ijV(*i8in^ + <^,sin?p' + ...yAr
a=4<^i" I psin"-=-da7 + i<^j* I psin'-v dx+...
Jo ^ .' ^
, 1 1 [^ . Tra? . 2irx , .
+ <pi9a I psin-T-sm— p cw? + ....
If p were constant, the products of the velocities would
disappear, since <^, ^, &c. are, on that supposition, the normal
co-ordinates. As it is, the integral coefficients, though not actually
evanescent, are small quantities. Let p = po-^- Bp] then in our
previous notation
«r = JVo> oarr= / op sui* — p cwj, oa„ = I Op sm -r- sm -y- rfd:.
Thus the type of vibration is expressed by
or, since pr' - P8^ = i^ i ^,
. . r* f^2Sp . rTra? . sirx , .^.
<>' •■ -^"^^ Jol^"'°-2 ^^°— '^ <2>-
Let us apply this result to calculate the displacement of the
nodal point of the second mode (r = 2), which would be in the
middle, if the string were uniform. In the neighbourhood of
this point, if x — ^l-hBx, the approximate value of y is
y^^^ism^ +<^,8m-2- + <^s8m-2-+...
7^ . TT 27r , 27r . )
TT ,
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91.] EXAMPLES. 117
Hence when y = 0,
^'=^J4>i-'h+ <!>.-- ] (3)
approximately, where
To shew the application of these formulae, we may suppose
the irregularity to consist in a small load of mass p^ situated
B,t x^H, though the result might be obtained much more easily
directly. We have
e. _ 2X f 2 2 2_ 2_ \
7rV2|P-4 8»-4 5«-4"^7«-4"^ J'
from which the value of Sx may be calculated by approximation.
The real value of Bx is, however, very simple. The series within
brackets may be written
which is equal to
-(■-i-^J4-A-H.
■/
ax.
ol + x*
The value of the definite integral is
TT -f- 4 sm J *,
J *!. t 2X 7rV2 X
as may also be readily proved by equating the periods of vibra-
tion of the two parts of the string, that of the loaded part being
calculated approximately on the assumption of unchanged type.
As an example of the formula (6) § 90 for the period, we
may take the case of a striug carrying a small load p^ at its
middle point. We have
ar = i/p», Sa^ = poXsin«-2-, &v, = poX sin -g- 81^-2"'
and thus, if P^ be the value corresponding to \ = 0, we get when
r is even, pr — Pr, and when r is odd,
1 Todhnnter'8 Int, Calc, § 255.
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118 VIBRATING SYSTEMS IN GENERAL. [91,
where the summation is to be extended to all the odd values
of 8 other than r. If r = 1,
Now 22--!— = 2 ^,-2 ^
V-1 8-1 8+V
in which the values of » are 3, 5, 7, 9.... Accordingly
v_L -1
and p,. = P..|i_^ + ^V I (6).
giving the pitch of the gravest tone accurately as far as the
square of the ratio X : l.
In the general case the value of p^, correct as far as the first
order in 8p, will be
,,.i>,|i-^}.i..|i-f0^.rJ!4..,r,
92. The theory of Vibrations throws great light on expansions
of arbitrary functions in series of other functions of specified
typea The best known example of such expansions is that
generally called after Fourier, in which an arbitrary periodic
function is resolved into a series of harmonics, whose periods
are submultiples of that of the given function. It is well known
that the difficulty of the question is confined to the proof of the
possibility of the expansion ; if this be assumed, the determination
of the coefficients is easy enough. What I wish now to draw
attention to is, that in this, and an immense variety of similar
cases, the possibility of the expansion may be inferred fix>m
physical considerations.
To fix our ideas, let us consider the small vibrations of a
uniform string stretched between fixed points. We know fix>m'
the general theory that the whole motion, whatever it may
be, can be analysed into a series of component motions, each
represented by a harmonic function of the time, and capable
of existing by itself. If we can discover these normal types,
we shall be in a position to represent the most general vibration
possible by combining them, assigning to each an arbitrary
amplitude and phase.
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92.]
NORMAL FUNCTIONS.
119
Assuming that a motion is harmonic with respect to time^
we get to determine the type an equation of the form
whence it appears that the normal functions are
. irx
y = sm-^,
y=sm--^,
y = 8in
^irx
T
, &c.
We infer that the most general position which the string can
assume is capable of representation by a series of the foim
. . irx . . 27rx . . Zirx
Ai sm -J- + Ai sm —j— + -4, sm -j — h
which is a particular case of Fourier's theorem. There would
be no diflSculty in proving the theorem in its most general form.
So far the string has been supposed uniform. But we have
only to introduce a variable density, or even a single load at
any point of the string, in order to alter completely the ex-
pansion whose possibility may be inferred from the dynamical
theory. It is unnecessary to dwell here on this subject, as
we shall have further examples in the chapters on the vibrations
of particuhu* systems, such as bars, membranes, and confined
masses of air.
92 a. In § 88 we have a formula for the frequency of vibration
applicable when by the imposition of given constraints the original
system is left with only one degree of freedom. It is of interest
to trace also the effect of less complete constraints, such as may
be expressed by linear relations among the normal co-ordinates of
number less by at least two than that of the (original) degrees of
freedom. Thus we may suppose that
9i<t>i+92<t>i-^g3<l>s + ... =
Ai^i + Aa<^a + ^8^8 + ... =
(!)•
If the number of equations (r) fall short of the number of the
degrees of freedom by unity, the ratios ^i:^s:^8... are fully
determined, and the case is that of but one outstanding degree of
freedom discussed in § 88.
This problem may be treated in more than one way, but the
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120 VIBRATING SYSTEMS IN GENERAL, [92 a.
most instructive procedure is to trace the effect of additions to T
and F. We will suppose that equations (1) § 87 are altered to
r=K<^* + ia,<^a» + ...+ia(/,<^4./.(^,+ ...)* (2),
7 = icA* + ic^.' + ... + i7(/A+/.^ + ...)" (3),
and that F, not previously existent, is now
F^1ifi{A<k+M.+ '")'-' (4).
The connection with the proposed problem will be understood
by supposing for instance that a = 0, )8 = 0, while 7 = oo . By (3)
the potential energy of any displacement violating the condition
/A+/A+...=0 (5)
is then infinite, and this is tantamount to the imposition of the
constraint represented by (5).
Lagrange's equations with X written for D now become
...(6).
If we multiply the first of these by/i/(aiX"+ c^), the second by
/^(OjX* -t- Ca), and so on, and add the results together, the factor
(fi<i>i-^fi<f>i+ '*-) will divide out, and the determinant takes the
form
f^ J. f^ 4. _i_ z = n\
(hX'-^C a^\^ + c^^ ^aX» + /3X + 7 ^ ^'
If any one of the quantities a, )8, 7 become infinite while the
others remain finite, the effect is equivalent to the imposition of the
constraint (5), and the result may be written
2/V(aX« + c) = (8V.
When multiplied out this equation is of degree (m— 1) in X*, one
degree of freedom having been lost.
If we put J3 = 0y (7) is an equation of the mth degree in X', and
the coefficients a, 7 enter in the same way as do Oi, Cj; a„ c,; &c.
In order to refer more directly to the case of vibrations about
stable equilibrium, we will write p* for — XI The values of p*
belonging to the unaltered system, viz. n^^ Wj',..., are given as
before by
Ci — air?^*=0, Cs — as722* = 0, &c., (9);
and we will also write
7-ai/« = (10),
1 Bouth'B Rigid Dynamics, 5th edition, 1892, § 67.
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92 a.] ONE CX)NSTRAINT. 121
where i/* relates to the supposed additions to T and V considered
as belonging to an independent vibrator. Let the order of magni-
tude of these quantities be
^*.W,«, nr\t^y7ir+i\ ^* (11).
We shall see that there is a root of (7) between each consecutive
pair of the quantities (11).
Our equation may be written
/i'(7-ap*)(c.-arf>*)(c8-a»pO
+/»"(7-ap")(ci-aj>«)(c-a,p«)
+
+ (Ci — OiP") (Ca — OaP') = (12).
When p^ coincides with any of the quantities (11), all but one
of the terms in (12) vanish, and the sign of the expression is the
same as that of the term which remains over. When p* < w,*, all
the terms are positive, so that there is no root less than ni\
When p^ = V, the expression (12) reduces to the positive quantity
/i« (7 - aV) (c - Ojni^) (Cs - ajTii')
When p^ rises to Wj', (12) becomes
/,«(7~ aw,«) (Ci- aiw,0(ca-a,72,«) ;
and this is negative, since the factor (ci — OiW,*) is now negative.
Hence there is a root of (12) between %" and n,*. When jp«= Wj«,
the expression is again positive, and thus there is a root between
n,* and n,* This argument may be continued, and it proves that
there is a root of (12) between any consecutive two of the (m + 1)
quantities (11). The m roots of (12) are now accounted for, and
there is none greater than rim^. If we compare the values of the
roots before and after the change, we see that the effect is to
cause a movement which is in every case towards 1^.^ Considered
absolutely the movement is in one direction for those roots that
are greater than v^ and in the opposite direction for those that
are less than 1^. Accordingly the interval from n^ to Wr+i', in
which v^ lies, contains after the change two roots, one on either
side of 1^.
If 1^ be less than any of the quantities n*, as happens when
7 = 0, one root lies between 1^ and Wj*, one between n^ and r^*, and
so on. Thus every root is depressed. On the other hand if
v" > n^, every root is increased. This happens if a = 0. (§ 88.)
^ It will be understood that in particular cases the movement may vanish.
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122 VIBRATING SYSTEMS IN GENERAL. [92 Ob.
The results now arrived at are of course independent of the
special machinery of normal co-ordinates used in the investigation*
If to any part of a system (wi', n^ ) be attached a vibrator
(v*) having a single degree of freedom, the eflFect is to displace all
the quantities nl^ ... in the direction of i^. Let us now suppose
that a second change is made in the vibrator whereby a becomes
a + a^ and 7 becomes 7 + 7'. Every root of the determinantal
equation moves towards l/'^ where 7' — aV = 0. If we suppose
that v^ = V*, the movements are in all cases in the same directions
as before. Going back now to the original system, and supposing
that a, 7 grow from zero to their actual values in such a manner
that 1^ remains constant, wet see that during this process the roots
move without regression in the direction of closer agreement
with I/*.
As a and 7 become infinite, one root of (12) moves to coinci-
dence with I/", while the remaining (m— 1) roots, corresponding to
the constrained system, are given by
2/V(c-a;)0 = (13),
and are independent of the value of i/*.
Particular cases are obtained by supposing either i/* = 0, or
j;» = 00 . Whether the constraint is eflFected by making infinite
the kinetic energy of any motion, or the potential energy of
any displacement, which violates it, makes no diflference to the
vibrations which remain. In the first case one vibration becomes
infinitely slow, and in the second case one becomes infinitely quick.
However the constraint be arrived at, the (m— 1) frequencies of
vibration of the constrained system sefparate^ the m frequencies
of the original system.
Any number of examples of this theorem may be invented
without diflBculty. Consider the case of a uniform stretched
string, held at both ends and vibrating transversely. This is the
original system. Now introduce a constraint by holding at rest a
point which divides the length in the proportion (say) of 3 : 2.
The two parts vibrate independently, and the frequencies for each
part form an arithmetical progression. If the frequencies proper
to the undivided string be 1, 2, 3, 4 ; those for the parts are
^ But in particular cases the " separation " may vanish. The theorem in the
text was proved for two degrees of freedom in the first edition of this work. In
its generality it appears to be due to Bouth.
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92 a.] ONE CONSTRAINT. 123
f (1, 2, 3, ...) and f (1, 2, 3, ...). The beginning of each series is
shewn in the accompanying scheme ;
and it will be seen that between any consecutive numbers in
the first row there is a number to be found either in the second
or in the third row. In the case of 5 and 10 we have an extreme
condition of things ; but the slightest displacement of the point
at which the constraint is applied will displace one of the fives,
tens &c. to the left And the other to the right.
The coincidences may be avoided by dividing the string
incommensurably. Thus, if x be an incommensurable number
less than unity, one of the series of quantities m/x, m/(l — a?), where
m is a whole number, can be found which shall lie between any
given consecutive integers, and but one such quantity can be found.
Again, let us suppose that a system is referred to co-ordinates
which are not normal (§ 84), and let the constraint represented by
-^1=8 be imposed. The determinant of the altered system is
formed from that of 'the original system by erasing the first row
and the first column. It may be called Vj, and from this again
may be formed in like manner a new determinant V,, and so on.
These determinants form a series of functions of p*, regularly
decreasing in degree; and we conclude that the roots of each
separate the roots of that immediately preceding^
It may be remarked that while for the sake of simplicity of
statement we have supposed that the equilibrium of the original
system was thoroughly stable, as also that of the vibration brought
into connection therewith, these restrictions may easily be
dispensed with. In any case the series of positive and negative
quantities, Wi*, n,", and i^, may be arranged in algebraic order,
and the effect of the vibrator is to cause a movement of every
value of jp* in the direction of i/*.
In order to extend the above theory we will now suppose that
the addition to 7 is
+ iaA(Ai<*>i + Ma+...)'+ (14)
^ Boath*B Rigid Dynamics, 5th edition, Part ii. § 58.
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.(15).
....(le).
.)
.) + ... = 0...(17).
124 VIBRATING SYSTEMS IN GENERAL. [92 a.
and the addition to V
H (/i<th +M, + ...)» + H (S-.^ + <7W>. + •••)• +
If we set
a/V + 7/=-f", a,V+7„=<?',
and so on, Lagrange's equations become
(oiX.' + c.) ^ + F'f, if, in +/.^ + .
+ O'gi (g'i«^ + ^.<^ + -.) + S% (hi<fh + A^ +
(a,\» + c) ^ + F% {/^ +/,^ + . . .)
+ <?'<7.(5'i^ +5'*^+ ...) + H% (A.<^, + A,^ + ...) + ... = 0...(18),
and 80 on, the number of equations being equal to the number
(m) of co-ordinates ^, ^ .... The number of additions (r), corre-
sponding to the letters/, g,h,...,ia supposed to be less than m.
From the above m equations let r new ones be formed, as
follows. For the first multiply (17) by /,/(o,X' + c,), (18) by
/,/(a^' + c), and so on, and add the results together. For the
second proceed in the same manner, using the multipliers
giKih^* + <h), ffi/(<h^* + Ct), &c. In like manner for the third
equation use h instead of g, and so on. In this way we obtain r
equations which may be written
F' {fA +/A + • • •) IV-f" + F^' + ^.' + ^.' +• . .}
+ 0' ($i<f>i +g.<t>»+ ...) {F^Gr + F,G, + ...}
+ H'(lH<l>,-i-h,ip,+ ...){F,H^ + FJI,+ ...} + =0...(19),
F'(fr<k-^f^ + ...){G,F^ + GJ',+ ...}
4 G' ig,4>, -h <7,^ + . . .) {1/G' + G,» +(?,» + .. .}
+ fr'(A,^-t-A,^ + ...){(?,ff,-l-(?.ff. + ...}-l- =0...(20).
and so on, where for brevity
F,' = /.V(axX' -I- c,). Fi =UI{ais? + c,), Ac, «|
F,Gr=f^gJ{a,\' + c,). &c. J
The determinantal equation, of the rth order, is thus
l/F' + tF*, ^FG. tFH,.
1FG,1/G' + 2G', tGH,
1.FH, IGH, 1/H' + 1H\ .
= 0.
.(22).
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92 a.]
SEVERAL CONSTRAINTS.
125
If, for example, there be two additions to T and V of the kind
prescribed, the equation is
js^ + ^ + -^+2i*.2(?«-{2i'^}» = (23),
and herein
(i\» + j;'+ ...) (Gx» +(?,»+...)- (^i(?i + ^.<?. + ...)'
= -2,l.{F,G,-F,G,y ^24).
Equation (23) is in general of the mth degree in X*, and
determines the frequencies of vibration. In the extreme case
where F' and 0' are made infinite, the system is subject to the
two constraints
/.^+/,^+ ... = 0) ,
«7x<^ + «ir.*,+ "=0j ^^*''''
and the equation ' giving the (m — 2) outstanding roots is
(/ig«-/«yi)' (/ig»-/«gi)*
= 0.
.(26).
(a,X» + <h) (a,V + c) "^ (oiX' + c,) (a,X» + c.)
In general if the system be subject to the r constraints (1), the
determinantal equation is
IFF, IFO, -ZFH,...
1.F0, IGG, 20H....
IFH, I.OH. IHH,...
=
(27).
If r be less than m, this determinant can be resolved* into a
sum of squares of determinants of the same order (r). Thus if there
be three constraints, the first of these squares is
F^ F^ F^ »
ffi G, (?, (28),
Hi H^ Hi
and the others are to be found by including every combination of
the m suffixes taken three together. To fall back upon the original
notation we have merely in (28) to replace the capital letters
F, 0,... by/, g,..., and to introduce the denominator
(oiX* + Ci) (OaX' + c,) (a,X> + C3).
The determinantal equation for a system originally of m degrees
of freedom and subjected to r constraints is thus found. Its form
1 Thils result is due to Bouth, loe, cit, § 67.
* Salmon, Lessons on HigJier Algebra^ § 24.
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126
VIBRATING SYSTEMS IN GENERAL.
[92 a.
is largely determined by the consideration that it must remain un-
affected by. interchanges either of the letters or of the suffixes.
That it would become nugatory if two of the conditions of con-
straint coincided, could also have been foreseen. If r = m — 1,
the system is reduced to one degree of freedom, and the equation
is
g% g» 94'"
A, A, A4...
s
AAA-
9i 9* 9i"*
hi As A4...
.(o^' + c)-
.(29).
in agreement with § (88).
There are theories, parallel to the foregoing, for systems in
which T and jP, or V and P, are alone sensible. In these cases, if
the functions be intrinsically positive, the normal motions are
proportional to exponential functions of the time such as er*^\
The quantities Tj, t,,... are called the time-constants, or persis-
tences, of the motions, being the times occupied by the motions in
subsiding in the ratio of e : 1. The new persistences, after the
introduction of a constraint, will separate the original values. ^
The best illustrations of this theory are electrical, where the
motions are not restricted to be small. Suppose (to take an
electro-magnetic example) that in one branch of a net-work of
conductors there is introduced a coil of persistence (when closed
upon itself) equal to •/, the original persistences being Ti, Tj,....
Then the new persistences lie in all cases nearer to r, and they
separate the quantities t\ Ti, T3.... If t' be made infinite as by
increasing the self-induction of the additional coil without limit,
or be made to vanish as by breaking the contact in the branch,
the result is a constraint, and the new values of the persistences
separate the former onea
93. The determination of the coefficients to suit arbitrar}'
initial conditions may always be readily effected by the funda-
mental property of the normal functions, and it may be convenient
to sketch the process here for systems like strings, bars, mem-
branes, plates, &c. in which there is only one dependent variable
(I* to be considered. If 1^1,1^^... be the normal functions, and
■^, ^ ... the corresponding co-ordinates.
f=<^it*i + <^l«s-f <^,M,+ .
.(1).
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93.] INITIAL CONDITIONS.
The equations of free motion are
^ + wi'^ = 0, ^, + n,»<^, = 0,&c.
of which the solutions are
<l>i = Ai sin flit + B1CO8 flit
^ = 4a sin n,^ + -Bj cos nj;
127
.(2),
.(3).
The initial values of f and ^ are therefore
C = Biu, + 5jw, + 5,^, + ... I
(4).
and the problem is to determine Ai, A^,... Bi, ^, ... so as to
correspond with arbitrary values of (fo c^nd ^o-
It pdxhe the mass of the element dw, we have from (1)
= ^4>i^lptii*dx + ^^2^jpu^^dx + ... + ^^il
But the expression for T in terms of ^, <^j, &c. cannot contain
the products of the normal generalized velocities, and therefore
every integral of the form
/'
pUrU^^O.
.(5).
Hence to determine Br we have only to multiple the first
of equations (4) by pUr and integrate over the system. We thus
obtain
BrjpUr^dx^ jpUr^ifiUc (6).
Similarl}', nrArlpUr*dx=: jpii,t4^ 0)-
The process is just the same whether the element dx he bl line,
area, or volume.
The conjugate property, expressed by (5), depends upon the
fact that the functions u are normal. As soon as this is known
by the solution of a differential equation or otherwise, we may
infer the conjugate property without further proof, but the pro-
perty itself is most intimately connected with the fundamental
variational equation of motion § 94.
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I V
128 VIBRATING SYSTEMS IN GENERAL. [94.
94 If 7 be the potential energy of deformation, f the
displacement, and p the density of the (line, area, or volume)
element dx, the equation of virtual velocities gives immediately
SV+
jp'tS^dx^O (1),
In this equation 87 is a symmetrical function of ^ and B^,
as may be readily proved from the expression for V in terms
of generalized co-ordinates. In fact if
Suppose now that f refers to the motion corresponding to
a normal function t^, so that tJ + Wr*f=0, while Sf is identified
with another normal function u, ; then
SK=n^* iplLrU^,
Again, if we suppose, as we are equally entitled to do, that f
varies as a, and Sf as i/r, we get for the same quantity 87,
and therefore
(nr^-n,^)jpurv^ = (2),
from which the conjugate property follows, if the motions re-
presented respectively by t*r ai^d Ug have different periods.
A good example of the connection of the two methods of
treatment will be found in the chapter on the transverse vibrations
of bars.
96. Professor Stokes^ has drawn attention to a very general
law connecting those parts of the free motion which depend
on the initial displacements of a system not subject to frictional
forces, with those which depend on the initial velocities. If
a velocity of any type be communicated to a system at rest,
and then after a small interval of time the opposite velocity
be communicated, the effect in the limit will be to start the
system without velocity, but with a displacement of the corre-
sponding type. We may readily prove from this that in order
1 Dynamical Theory of Diffraction, Cambridge Trans, Vol. «. p. 1, 1866.
«
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95.] CONJUGATE PROPERTY. 129
to deduce the motion depending on initial displacements from
that depending on the initial velocities, it is only necessary to
diflFerentiate with respect to the time, and to replace the arbitrary
constants (or functions) which express the initial velocities by
those which express the corresponding initial displacements.
Thus, if <f> be any normal co-ordinate satisfying the equation
the solution in terms of the initial values of <f> and ^ is
^ ^ <f>ocos nt + - ^oBin nt (1),
n
of which the first term may be obtained from the second by
Stokes' rule.
R
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CHAPTER V.
VIBRATING SYSTEMS IN GENERAL
CONTINUED.
96. When dissipative forces act upon a system, the character
of the motion is in general more complicated. If two only of the
functions T, F, and V be finite, we may by a suitable linear trans-
formation rid ourselves of the products of the co-ordinates, and
obtain the normal types of motion. In the preceding chapter we
have considered the case of ^= 0. The same theory with obvious
modifications will apply when 7 = 0, or F = 0, but these cases
though of importance in other parts of Physics, such as Heat and
Electricity, scarcely belong to our present subject.
The presence of Motion will not interfere with the reduction of
T and V to sums of squares ; but the transformation proper for
them will not in general suit also the requirements of F, The
general equation can then only be reduced to the form
ai$i + 6u<^i + 6i2<^2 + ... +Ci<^i = <!>!, &c (1),
and not to the simpler form applicable to a system of one degree
of freedom, viz.
ai<^i + &i^ + c,<^ = *i. &c (2).
We may, however, choose which pair of functions we shall
reduce, though in Acoustics the choice would almost always fall
on T and V.
97. There is, however, a not unimportant class of cases in
which the reduction of all three functions may be effected; and
the theory then assumes an exceptional simplicity. Under this head
the most important are probably those when ^ is of the same form
as T or F. The first case occurs fi-equently, in books at any rate,
when the motion of each part of the sjrstem is resisted by a re-
tarding force, proportional both to the mass and velocity of the
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97.]
T, F, V SIMULTANEOUSLY BBDUCIBLE.
181
.(1).
part. The same exceptional reduction is possible when ^ is a
linear function of T and F, or when T is itself of the same form as
y. In any of these cases, the equations of motion are of the same
form as for a system of one degree of freedom, and the theory
possesses certain peculiarities which make it worthy of separate
consideration.
The equations of motion are obtained at once from T, F
and F:—
aifiH-6i<^i + Ci<^i = *i, )
o^ + h^^ + c,^, = <E>„ &c. j
in which the co-ordinates are separated.
For the free vibrations we have only to put O^ = 0, &c., and
the solution is of the form
^ = e-^ |<^, ^^* + 4>, (co8«'« + ^, sin n'«)| (2),
where K — hja, n*=^c/a, n' =^»J{n^-'\Kp),
and ^0 aiid ^q are the initial values of 4> and (^.
The whole motion may therefore be analysed into component
motions, each of which corresponds to the variation of but one
normal co-ordinate at a time. And the vibration in each of these
modes is altogether similar to that of a system with only one
degree of liberty. After a certain time, greater or less according
to the amount of dissipation, the free vibrations become insignifi-
cant, and the system returns sensibly to rest.
[If F be of the same form as T, all the values of k are equal,
viz. all vibrations die out at the same rate.]
Simultaneously with the free vibrations, but in perfect inde-
pendence of them, there may exist forced vibrations depending on
the quantities 4>. Precisely as in the case of one degree of free-
dom, the solution of
a^ + 6<^ + c<^ = <l> (3)
may be written
<^ = i /"%-**<'-«') sin n'(«-0<I>d«' (4X
where as above
K = 6/a, n* = cja, n' == tj(n^ — i /c").
To obtain the complete expression for <\> we must add to the
right-hand member of (4), which makes the initial values of <f>
and ^ vanish, the terms given in (2) which represent the residue
9—2
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132 VIBRATING SYSTEMS IN GENERAL. [97.
at time t of the initial values ^o and <^o* If there be no frictioo,
the value of <f> in (4) reduces to
<f> = ^j\inn(t-r)<S>df (5).
98. The complete independence of the normal co-ordinates
leads to an interesting theorem concerning the relation of the
subsequent motion to the initial disturbance. For if the forces
which act upon the sjrstem be of such a character that they do no
work on the displacement indicated by S<^, then <I>i = 0. No such
forces, however long continued, can produce any effect on the
motion ^. If it exist, they cannot destroy it ; if it do not exist,
they cannot generate it. The most important application of the
theorem is when the forces applied to the system act at a node of
the normal component <t>i, that is, at a point which the component
vibration in question does not tend to set in motion. Two extreme
cases of such forces may be specially noted, (1) when the force is
an impulse, starting the system from rest, (2) when it has acted so
long that the system is again at rest under its influence in a dis-
turbed position. So soon as the force ceases, natural vibrations
set in, and in the absence of friction would continue for an in-
definite time. We infer that whatever in other respects their
character may be, they contain no component of the type ^. This
conclusion is limited to cases where T, F, V admit of simultaneous
reduction, including of course the case of no friction.
99. The formulae quoted in § 97 are applicable to any kind of
force, but it will often happen that we have to deal only with the
effects of impressed forces of the harmonic type, and we may then
advantageously employ the more special formulae applicable to such
forces. In using normal co-ordinates, we have first to calculate the
forces ^1, *j, &c. corresponding to each period, and thence deduce
the values of the co-ordinates themselves. If among the natural
periods (calculated without allowance for friction) there be any
nearly agreeing in magnitude with the period of an impressed
force, the corresponding component vibrations will be abnormally
large, unless indeed the force itself be greatly attenuated in the
preliminary resolution. Suppose, for example, that a transverse
force of harmonic type and given period acts at a single point of
a stretched string. All the normal modes of vibration will, in
general, be excited, not however in their own proper periods, but
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99.]
EQQILIBRIUM THEORY.
133
in the period of the impressed force ; but any normal component,
which has a node at the point of application, will not be excited.
The magnitude of each component thus depends on two things :
(1) on the situation of its nodes with respect to the point at which
the force is applied, and (2) on the degree of agreement between
its own proper period and that of the force. It is important to
remember that in response to a simple harmonic force, the system
will vibrate in general in all its modes, although in particular
cases it may sometimes be sufficient to attend to only one of them
as being of paramount importance.
100. When the periods of the forces operating are very long
relatively to the free periods of the system, an equilibrium theory
is sometimes adequate, but in such a case the solution could
generally be found more easily without the use of the normal
co-ordinates. Bernoulli's theory of the Tides is of this class, and
proceeds on the assumption that the free periods of the masses of
water found on the globe are small relatively to the periods of the
operative forces, in which case the inertia of the water might be
left out of account. As a matter of fact this supposition is only
very roughly and partially applicable, and we are consequently
still in the dark on many important points relating to the tides.
The principal forces have a semi-diurnal period, which is not suffi-
ciently long in relation to the natural periods concerned, to allow
of the inertia of the water being neglected. But if the rotation of
the earth had been much slower, the equilibrium theory of the
tides might have been adequate.
A corrected equilibrium theory is sometimes useful, when the
period of the impressed force is sufficiently long in comparison
with most of the natural periods of a system, but not so in the
case of one or two of them. It will be sufficient to take the case
where there is no friction. In the equation
a^-hc(^ = 4>, or ^ + w'<^ = 4>/a,
suppose that the impressed force varies as cos pt Then
<^ = 4>-ra(n»-p») (1).
The equilibrium theory neglects jo» in comparison with n\
and takes
t^ = <l)~an« (2).
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134 VIBRATING SYSTEMS IN GENERAL. [100.
Suppose now that this course is justifiable, except in respect
of the single normal co-ordinate ^. We have then only to add
to the result of the equilibrium theory, the difference between
the true and the there assumed value of ^, viz.
A ==__*!_ *L=^ *^ n\
The other extreme case ought also to be noticed. If the
forced vibrations be extremely rapid, they may become nearly
independent of the potential energy of the s}rstem. Instead
of neglecting p" in comparison with n', we have then to neglect
w" in comparison with p^, which gives
^ = — <I>-4-ap* (4).
If there be one or two co-ordinates to which this treatment
is not applicable, we may supplement the result, calculated on
the hypothesis that V is altogether negligible, with corrections
for these particular co-ordinates.
101. Before passing on to the general theory of the vibrations
of systems subject to dissipation, it may be well to point out
some peculiarities of the free vibrations of continuous systems,
started by a force applied at a single point. On the suppositions
and notations of § 93, the configuration at any time is deter-
mined by
?=<^i^ + ^2Wa+<^^+ (1),
where the normal co-ordinates satisfy equations of the form
aJi>r-^Cr^r^^r (2).
Suppose now that the system is held at rest by a force applied
at the point Q. The value of <I>r is determined by the considera-
tion that ^f£<l>r represents the work done upon the system by the
impressed forces during a hypothetical displacement S^^B<f>r'Ur,
that is
thus ^r^lZUrdx^UriQ)jZdx;
80 that initially by (2)
c^^Vr{Q)fzdx (3).
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101.]
SPEflAL INITIAL CONDITIONS.
135
If the system be let go from this configuration at ^=^0, we
have at any subsequent time t,
Ur{Q)jZdx Ur{Q)jZdx
<^^=cosr^i =cosM v. (*),
*" n/ jp Ur^dx
and at the point P
Ur{P)ur{Q){zdx
f^ScosM . (5).
nr^ipVrdx
At particular points v>r(P) and %'{Q) vanish, but on the
whole
Ur(P)Ur(Q)^jpUr'dx
neither converges, nor diverges, with r. The series for f therefore
converges with 71^""'.
Again, suppose that the system is started by an impulse
from the configuration of equilibrium. In this case initially
a^r^j^fdt = Ur (Q)jz,da:,
whence at time t
This gives
<l>r=--—'Ur(Q)'\Zidx^ ^'IZ^dx (6),
Ur{P)Ur(Q)jZ,dx
f =2sinn,i
nr jpUr^dx
shewing that in this case the series converges with Wr~^ that
is more slowly than in the previous case.
In both cases it may be observed that the value of ^ is
symmetrical with respect to P and Q, proving that the displace-
ment at time t for the point P when the force or impulse is ap-
plied at Q, is the same as it would be at Q if the force or impulse
had been applied at P. This is an example of a very general
reciprocal theorem, which we shall consider at length presently.
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136 VIBRATING SYSTEMS IN GENERAL. [101.
As a third case we may suppose the body to start from rest
as deformed by a force uniformly distributed, over its length,
area, or volume. We readily find
Ur{P).Z.{u4ic
?=ScosM T-^ (8).
V IpUr^dx
The series for f will be more convergent than when the force
is concentrated in a single point.
In exactly the same way we may treat the case of a con-
tinuous body whose motion is subject to dissipation, provided
that the three functions T, F, V be simultaneously reducible,
but it is not necessary to write down the formulae.
102. If the three mechanical functions T, F and V of any
system be not simultaneously reducible, the natural vibrations
(as has already been observed) are more complicated in their
character. When, however, the dissipation is small, the method
of reduction is still useful ; and this class of cases besides being
of some importance in itself will form a good introduction to
the more general theory. We suppose then that T and V are
expressed as sums of squares
F = ic,^» + ic.0,« + ...J ^ ^'
while F still appears in the more general form
i^''=i6n<^i' + i6«<^,^ + ...+Mi*a+ (2).
The equations of motion are accordingly
Oi^i + bu^ 4- ftis^j + 6is<^3 + ... + Ci<l>i = I
a,^8 + iii<^i + i«<^2 + 6a<^s+-.. +Ca(^2 = > (3),
in which the coeflScients 6u, 6ia, &c. are to be treated as small.
If there were no friction, the above system of equations would
be satisfied by supposing one co-ordinate ^^ to vary suitably,
while the other co-ordinates vanish. In the actual case there
will be a corresponding solution in which the value of any other
co-ordinate (f>g will be small relatively to <^y.
Hence, if we omit terms of the second order, the r^ equation
becomes,
ar$r+6rr<^r+Cr<^r = (4),
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102.] SMALL DISSIPATIYE FORCES. 137
from which we infer that <f>r varies approximately as if there
-were no change due to friction in the type of vibration. If ^r
vary as e*^, we obtain to determine pr
arPr^-^brrPr + Cr^O (6).
The roots of this equation are complex, but the real part
is small in comparison with the imaginary part. [The character
of the motion represented by (5) has already been discussed
(§ 45). The rate at which the vibrations die down is proportional
to brr, and the period, if the term be still admitted, is approxi-
mately the same as if there were no dissipation.]
From the sf"^ equation, if we introduce the supposition that
all the co-ordinates vary as e^, we get
(Pr'a. + Ct) <l>g + brsPr^r = 0,
terms of the second order being omitted ; whence
<^. : ^, b^^pr_^_br^pr_ ,g.
This equation determines approximately the altered type
of vibration. Since the chief part of pr is imaginary, we see
that the co-ordinates <f>g are approximately in the same phase,
but that that phase differs by a quarter period from the phase
of ^r- Hence when the function F does not reduce to a sum
of squares, the character of the elementary modes of vibration
is less simple than otherwise, and the various parts of the system
are no longer simultaneously in the same phase.
We proved above that, when the friction is small, the value
of Pr may be calculated approximately without allowance for
the change of type ; but by means of (6) we may obtain a still
closer approximation, in which the squares of the small quantities
are retained. The r^^ equation (3) gives
<^Pr'+Cr+b„Pr + t-^;^^^^0 '..... (7).
The leading part of the terms included under 2 being real,
the correction has no effect on the real paiij of pr on which
the rate of decay c|iepends.
102 a. Following the electrical analogy we may conveniently
describe the forces expressed by jP as forces of resistance. In
§ 102 we have seen that if the resistances be small, the periods
are independent of them. We may therefore extend to this case
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138 VIBRATING SYSTEMS IN GENERAL. [l02 a.
the application of the theorems with regard to the effect upon
the periods of additions to T and F, which have been already
proved when there are no resistances.
By (5) § 102, if the forces of resistance be increased, the rates
of subsidence of all the normal motions are in general increased
with them; but in particular cases it may happen that there
is no change in a rate of subsidence.
It is natural to inquire whether this conclusion is limited to
BmaJl resistances, for at first sight it would appear likely to hold
good generally. An argument sufficient to decide this question
may be founded upon a particular case. Consider a system formed
by attaching two loads at any points of a stretched string vibrating
transversely. If the mass of the string itself be neglected, there
are two degrees of freedom and two periods of vibration corre-
sponding to two normal modes. In each of these modes both loads
in general vibrate. Now suppose that a force of resistance is
introduced retarding the motion of one of the loads, and that this
force gradually increases. At first the effect is to cause both kinds
of vibration to die out and that at an increasing rate, but after-
wards the law changes. For when the resistance becomes infinite,
it is equivalent to a constraint, holding at rest the load upon which
it acts. The remaining vibration is then unaffected by resistance,
and maintains itself indefinitely. Thus the rate of subsidence of
one of the normal modes has decreased to evanescence in spite of a
continual increase in the forces of resistance F, This case is of
course sufficient to disprove the suggested general theorem.
108. We now return to the consideration of the general
equations of § 84.
If '^i* '^a* &C. be the co-ordinates and ^i, ^a, &c. the forces,
we have
where e^ = ar,l> + KsD + c„ (2).
For the free vibrations ^i, &c. vanish. If V be the de-
terminant
^1 , ^2 > • • •
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103.]
GENERAL EQUATIONS.
139
the result of eliminating from (1) all the co-ordinates but one, is
V^ = (4).
Since V now contains odd powers of D, the 2m roots of the
equation V=0 no longer occur in equal positive and negative
pairs, but contain a real as well as an imaginary part. The
complete integral may however still be written
^ = -de^i« + ^>i'« + J5fl^« + 5'e^'« + (5),
where the pcdrs of conjugate roots are /*!, An' ; /^, /^' ; &c. Corre-
sponding to each root, there is a particular solution such as
in which the ratios A^ : A^ : -4,... are determined by the equa-
tions of motion, and only the absolute value remains arbitrary.
In the present case however (where V contains odd powers of D)
these ratios are not in general real, and therefore the variations
of the co-ordinates -^i, yjr^y &c. are not synchronous in phase. If
we put fh — ^i + i/^u /*i' = ai-*A, &c., we see that none of the
quantities a can be positive, since in that case the energy of
the motion would increase with the time, as we know it cannot
do.
103 a. The general argument (§§ 85, 103) from considerations
of energy as to the nature of the roots of the determinantal
equation (Thomson and Tait's Natural Philosophy, 1st edition 1867)
has been put into a more mathematical form by Routh\ His
investigation relates to the most general form of the equation in
which the relations § 82
Ort-a^y 6r»=6»r, C„ = C«. (1),
are not assumed. But for the sake of brevity and as sufficient
for almost all acoustical problems, these relations vdll here be
supposed to hold.
We shall have occasion to consider two solutions corresponding
to two roots fly V of the equation. ITor the first we have
^, = M,e^y ir,^M,e^', ^,^M,^\kc (2),
and for the second
^^^N.i^y ^,^N,^y ^, = i<r,e^,&c (3).
In either of these solutions, for example (2), the ratios
M,:M,:M,:
^ Rigid DynamieSf 5th edition, Ch. vii.
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140 VIBRATING SYSTEMS IN GENERAL. [103<L
are determinate when /a has been chosen. They are real when
fi is real ; and when fi is complex (a ± t^S), they take the fonD
P±iQ-
If now we substitute the values of '^ from (2) in the equations
of motion, we get
(OiiM" + 6iiA* + Cii) if, + (a,^* + 6wfA + Ci2)ifj+ =0
{a^ + 6ij/A 4- Cu) JJfi + (a«i/Lt» H- 6a/* + CM)ifj+ =0 r-'-W-
1
.=0)
.=0 p--(
The first result is' obtained by multiplying these equations in
order by ifj, ifj, &c, and adding. It may be written
^/A» + J5^+a = 0, (5),
where
-4=Jai,Jlf,« + Jaaif,» + auJfiif,+ (6),
J5 = \huM^^ + i6„ilf,» + 6«ilfiJlf,+ (7).
0=iCn^i' + iCaJf,» + Ci^iif,+ (8).
The functions -4, jB, (7, are, it will be seen, the same as we have
already denoted by T, F, and V respectively; but the varied
notation may be useful as reminding us that there is as yet no
limitation upon the nature of these quadratic functions.
The following inferences from (6) are drawn by Bouth : —
(a) If AyByC either be zero, or be one-signed functions of
the same sign, the fundamental determinant cannot have a real
positive root. For if fi were real, the coefficients M^, JIT,,
would be real. We should thus have the sum of three positive
quantities equal to zero.
(^) If there be no forces of resistance, i.e. if the term B be
absent, and if A and G be one-signed and have the same sign,
the fundamental determinant cannot have a real root, positive or
negative.
(7) If -4, J5, (7 be one-signed functions, but if the sign of
B be opposite to that of A and (7, the fundamental determinant
cannot have a real negative root.
The second equation is obtained as before from (4), except that
now the multipliers are iVj, N^,.,. appropriate to the root v. The
result may be written
A(ji,v)ii^-\'B{,jL,v)lx + G{^,v)^0 (9),
where
2A (fA, v) = OiiMiNi + aj\/fj^2 +
+ a,,(M,N,+ M^,)+ (10),
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103 a.] ROUTH's THEOREMS. 141
with similar suppositions for B(fi,v) and C(fi,v). A(fi,v) is
thus a symmetrical function of the Jtf s and N% so that
A<ji,v)=A(v,^) (11).
It will be observed that according to this notation A (fi, fi) is
the same as ^ in (6).
In like manner
A(ji,v)i^ + B(pL,v)v+C(jjL,p)^0 (12),
shewing that fi, v are both roots of the quadratic, whose co-
efficients are A (fi, v\ B (/Lt, v), C (fi, v). Accordingly
B(fi,v) P(M.y) n^^
A(fi,v) '^ A(/A,v)
We will now suppose that fA, v are two conjugate complex
roots, so that
where a, yS are real. Under these circumstances if ifj, Jfa> ••• be
A + tQi, Ps + iQa,..., then N^,N,,... will be Pi-iQj, P^-iQ,,
, the P's and 0*8 being real. Thus by (10)
2A(ji,v)^a,,{Pi^-{-Q,') + a^{P^+Q^) +
+ 2a,,(P,P, + Q,(2,)+
^2A{P) + 2A(Q) (14).
In (14) A(P), A(Q) are functions, such as (6), of real variables.
From (13) we now find
^—A-(pyfA(Q) <!«)'
"+'^-Z(P)TX(Q) <^^)-
From these Routh deduces the following conclusions : —
(S) If A and B be one-signed and have the same sign
(whether (7 be a one-signed function or not), then the real part a
of every imaginary root must be negative and not zero. But if B
be absent, then the real part of every imaginary root is zero.
(e) If A and C be one-signed and have opposite signs, then
whatever may be the character of B, there can be no imaginary
roots.
It may be remarked that if B do not occur, and if fi^ and j^
be different roots of the determinant, it follows from (9), (12) that
A(jjL,v)^C(ji,yp)^0 (17).
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142 VIBRATING SYSTEMS IN GENERAL. [103 <X.
When the number of degrees of freedom is finite, the ftinda-
mental determinant may be expanded in powers of fi, giving'
an equation / (/a) =0 of degree 2m. The condition of stability
is that all the real roots and the real parts of all the complex
roots should be negative. If, as usual, complex quantities a? + »y
be represented by points whose co-ordinates are x, y, the condition
is that all points representing roots should lie to the left of the
axis of y. The application of Cauchy's rule relative to the
number of roots within any contour, by taking as the contour the
infinite semi-circle on the positive side of the axis of y, is veiy
fully discussed by RouthS who has thrown the results into forms
convenient for practical application to particular casea
108 6. The theorems of § 103 a do not exhaust all that general
mechanical principles would lead us to expect as to the character
of the roots of the fundamental determinant, and it may be well
to pursue the question a little further. We will suppose through-
out that A is one-signed and positive.
If B and G be both one-signed and positive, we see that the
equilibrium is thoroughly stable ; for from (a) it follows that there
can be no positive root, and from (S) that no complex root can have
its real part positive.
In like manner the equations of § 103 a suffice for the case
where C is one-signed and positive, B one-signed and negative.
By (5) every real root is positive, and by (15) the real part
of every complex root. Hence the equilibrium is unstable in
every mode.
When C is one-signed and negative, all the roots are real (S) ;
but (5) does not tell us whether they are positive or negative.
When J? = 0, we know (§ 87) that the roots occur in pairs of equal
numerical value and of opposite sign. In this case therefore
there are m positive and m negative roots. We will prove that
this state of things cannot be disturbed by B, For if the determi-
nant be expanded, the coefficient of /x*^ is the discriminant of J,
and the coefficient of ij!^ is the discriminant of 0. By supposition
neither of these quantities is zero, and thus no root of the equation
can be other than finite. Hence as B increases from zero to its
actual magnitude as a function of the variables, no root of the
equation can change sign, and accordingly there remain m
1 Adams Prize Essay 1877 ; Rigid Dynamics § 290.
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103 6.] INSTABILITY. 143
positive and m negative roots. It should be noticed that in this
ai^ument there is no restriction upon the character of B.
In the case of a real root the values of Mi, ifj, ... are real, and
thus the motion is such as might take place under a constraint
reducing the system to one degree of freedom. But if this con-
straint were actually imposed, there would be two corresponding
values of fi, being the values given by (5). In general only one of
these is applicable to the question in hand. Othenvise it would
be possible to define m kinds of constraint, one or other of which
would be consistent with any of the 2m roots. But this could
only happen when the three functions -4, jB, C are simultaneously
reducible to sums of squares (§ 97).
When jB = 0, there are m modes of motion, and two roots for
each mode. In the present application to the case where C is
one-signed and negative, each of the m modes for J3 » gives
one positive and one negative root. The positive root denotes
instability, and although the negative root gives a motion which
diminishes without limit, the character of instability is considered
to attach to the mode as a whole, and all the m modes are said
to be unstable. But when B is finite, there are in general 2m
distinct modes with one root corresponding to each. Of the
2m modes m are unstable, but the remaining m modes must be
reckoned as stable. On the whole, however, the equilibrium is
unstable, so that the influence of B, even when positive, is in-
sufficient to obviate the instability due to the character of C,
We must not prolong much further our discussion of unstable
systems, but there is one theorem respecting real roots too
fundamental to be passed over. It may be regarded as an ex-
tension of that of § 88.
The value of fi corresponding to a given constraint ifi : if, : ...
is one of the roots of (5) : and it follows from (4) that the value of
fi is stationary when the imposed constraint coincides with one of
the modes of free motion. The effect of small changes in A, By C
may thus be calculated from (5) without allowance for the
accompan}ring change of type.
Let C, being negative for the mode under consideration,
undergo numerical increase, while A and B remain unchanged as
functions of the co-ordinates. The latter condition requires that
the roots of (5), one of which is positive and one negative, should
move either both towards zero or both away from zero ; and the
first condition excludes the former alternative. Whether it be
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144 VIBRATING SYSTEMS IN GENERAL. [103 6.
the positive or the negative root of (5) which is the root of the
determinant, we infer that the change in question causes the
latter to move away from zero.
In like manner if A increase, while B and C remain unchanged,
the movement of the root, whether positive or negative, is
necessarily towards zero.
Again, if A and C be given, while B increases algebraically
as a function of the variables, the movement of the root of the
determinant must be in the positive direction.
Ad algebraic increase in B thus increases the stability, or
decreases the instability, in every mode. A numerical increase
in C or decrease in A on the other hand promotes the stability
of the stable modes and the instability of the unstable modes.
We can do little more than allude to the theorem relating to
the effect of a single constraint upon a system for which G is
one-signed and negative. Whatever be the nature of B, the
(m— 1) positive roots of the determinant, appropriate to the
system after the constraint has been applied, will separate the m
positive roots of the original determinant, and a like proposition
will hold for the negative roots. Upon this we may found a
generalization of the foregoing conclusions analogous to that
of § 92 a. Consider an independent vibrator of one degree of
freedom for which C is positive, and let the roots of the frequency
equation be Vi, I'a, one negative and one positive. If we regard
this as forming part of the system, we have in all (2m + 2) roots.
The effect of a constraint by which the two parts of the system
are connected will be to reduce the (2m + 2) back to ^m. Of
these the m positive will separate the (m + 1) quantities formed
of the m positive roots of the original equation together with (the
positive) 1/2, and a similar proposition will hold for the negative
roots. The effect of the vibrator upon the original system is thus
to cause a movement of the positive roots towards v^y and a
movement of the negative roots towards Vj. This conclusion
covers all the previous statements as to the effect of changes in
Ay B, C upon the values of the roots.
Enough has now been said on the subject of the free vibra-
tions of a system in general. » Any further illustration that it
may require will be afforded by the discussion of the case of two
degrees of freedom, § 112, and by the vibrations of strings and other
special bodies with which we shall soon be occupied. We resume
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103 6.] EORCBD VIBRATIONS. 145
the equations (1) § 103, with the view of investigating further the
nature oi forced vSmxtions,
104. In order to eliminate irom the equations all the co-
ordinates but one ('^i), operate on them in succession with the
minor determinants
dV dV dV ^
and add the results together; and in like manner for the other
oo-ordinate& We thus obtain as the equivalent of the original
system of equations
(1).
in which the differentiations of V are to be made without re-
cognition of the equality subsisting between ert and e^.
The forces '^^i, ^9^, &c. are any whatever, subject, of course,
to the condition of not producing so great a displacement or
motion that the squares of the small quantities become sensible.
If, as is often the case, the forces operating be made up of two
parts, one constant with respect to time, and the other periodic,
it is convenient to separate in imagination the two classes of
effects produced. The effect due to the constant forces is exactly
the same as if they acted alone, and is found by the solution
of a statical problem. It will therefore generally be sufficient
to suppose the forces periodic, the effects of any constant forces,
such as gravity, being merely to alter the configuration about
which the vibrations proper are executed. We may thus without
any real loss of generality confine ourselves to periodic, and
therefore by Fourier's theorem to harmonic forces.
We might therefore assume as expressions for ^^^ &c. circular
functions of the time; but, as we shall have frequent occasion
to recognise in the course of this work, it is usually more con-
venient to employ an imaginary exponential function, such as
E^p\ where ^ is a constant which may be complex. When the
R. 10
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146 VIBRATING SYSTEMS IN GBNERAL. |^104.
corresponding symbolical solution is obtained, its real and
imaginary parts may be separated, and belong respectively to
the real and imaginary parts of the data. In this way the
analysis gains considerably in brevity, inasmuch as differentiations
and alterations of phase are expressed by merely modifying^ the
complex coefficient without changing the form of the function.
We therefore write
^1 = ^,^1^, "ir^^E^e^^ &c,
dV
The minor determinants of the type -p- are rational integral
functions of the symbol D, and operate on ^], &c. according to
f{D)^=f(ip)^'^ (2).
Our equations therefore assume the form
V^, = i4,en V^, = ^e**, &c (3),
where Ai, A^, &c. are certain complex constants. And the sym-
bolical solutions are
or by (2), ^, = ^,-^, &c (4),
where V (ip) denotes the result of substituting ip for D in V.
Consider first the case of a system exempt from friction.
V and its differential coefficients are then even functions of
D, so that V(ip) is real. Throwing away the imaginary part
of the solution, writing -Bie**» for Ai, &c., we have
p
^i = y^C0S(p« + tfi)> &C (5).
If we suppose that the forces "^^i, &c. (in the case of more
than one generalized component) have all the same phase, they
may be expressed by
EiCos(pt-\-a\ EiCosipt + a), &c. ;
and then, as is easily seen, the co-ordinates themselves agree
in phase with the forces:
t> = V^co8(l>« + a) (6).
The amplitudes of the vibrations depend among other things
on the magnitude of ^ (ip). Now, if the period of the forces
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104.] FORCED VIBRATIONS. 147
be the same as one of those belonging to the free vibrations,
V (%p) = 0, and the amplitude becomes infinite. This is, of
<x>urse, just the case in which it is essential to introduce the
consideration of friction, from which no natural system is really
exempt.
If there be friction, V (ip) is complex ; but it may be divided
into two pai-ts — one real and the other purely imaginary, of which
the latter depends entirely on the friction. Thus, if we put
V(ip) = Vi(i/)) + i>V,(ip) (7),
Vi, Vj are even functions of ip, and therefore real. If as before
Ai = i2i6^', our solution takes the form
or, on throwing away the imaginary part,
{V,(ip)|»+J)»V.7yS)l»}i ^ ^'
where tan7 = ~^,^^^ (9).
We have said that V, (ip) depends entirely on the friction ; but
it is not true, on the other hand, that V^ (ip) is exactly the same,
as if there had been no friction. However, this is approximately
the case, if the friction be small ; because any part of V (ip), which
depends on the first power of the coefficients of friction, is neces-
-sarily imaginary. Whenever there is a coincidence between the
period of the force and that of one of the free vibrations, V^ (ip)
vanishes, and we have tan 7 = — oo , and therefore
It,sm(pt + 0^)
indicating a vibration of large amplitude, only limited by the
friction.
On the hypothesis of small firiction, is in general small, and
so also is 7, except in case of approximate equality of periods.
With certain exceptions, therefore, the motion has nearly the
same (or opposite) phase with the force that excites it.
When a force expressed by a harmonic term acts on a system,
the resulting motion is everywhere harmonic, and retains the
original period, provided always that the squares of the displace-
10—2
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148 VIBRATING SYSTEMS IN GENERAL. [104.
meats and velocities may be neglected. This important principle
^as enunciated by Laplace and applied by him to the theory of
the tides. Its great generality was also recognised by Sir John
Herschel, to whom we owe a formal demonstration of its truths
If the force be not a harmonic function of the time, the types
of vibration in different parts of the system are in general different^
from each other and from that of the force. The harmonic
functions are thus the only ones which preserve their type un-
changed, which, as was remarked in the Introduction, is a strong
reason for anticipating that they correspond to simple tones.
106. We now turn to a somewhat different kind of forced
vibration, where, instead of given /orce« as hitherto, given inexora-
ble motions are prescribed.
If we suppose that the co-ordinates -^i, -^a, ••• V^r *re given
functions of the time, while the forces of the remaining types
•^y+i, '*',.+„ ... '*'»» vanish, the equations of motion divide them-
selves into two groups, viz.
and
er+1,1 ^1 + «r+l,i >/^2 + ... + ^r+l.m '^m = )
(2)-
emi '^1 + ^ms '^s+.-.+^mw '^m = J
In each of the m — r equations of the latter group, the first r
terms are known explicit functions of the time, and have the same
effect as known forces acting on the system. The equations of
this group are therefore sufficient to determine the unknown
quantities ; after which, if required, the forces necessary to main-
tain the prescribed motion may be determined from the liret
group. It is obvious that there is no essential difference between
the two classes of problems of forced vibrations.
106. The motion of a system devoid of friction and executing
simple harmonic vibrations in consequence of prescribed variations
of some of the co-ordinates, possesses a peculiarity parallel to those
considered in §§ 74, 79. Let
'^i = -4iCosp<, •^, = -4jCosp^, &c.,
^ Encyc. Metrop. art. 823. Also Outlinei of Astronomy , § 650.
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106.]
INEXORABLE MOTIONS.
149
in which the cjuantities Ai..,Ar are regarded as given, while the
remaining ones are arbitrary. We have from the expressions for
Tand 7, § 82,
+ {i(^i-p"^)-^i'+ ••• +(^-"l>'^«)-4i-4j+ •••} cos 2p<,
from which we see that the equations of motion express the con-
dition that E, the variable part of T + F, which is proportional to
i(^-l>"flbi)^i' + ...+(Ci2-l>*a»)ili^, + (1),
shall be stationary in value, for all variations of the quantities
Ar+i . . . Am' Let p'* be the value of p* natural to the system when
vibrating under the restraint defined by the ratios
Ai : A^.,,Ar : Ar+i l ...^my
then
8o that
J?=r(p'«-jp«){iauili« + ...+au4i^ + ...} (2).
From this we see that if p* be certainly less than p'* ; that is,
if the prescribed period be greater than any of those natural to
the system under the partial constraint represented by
Ai : ^a*" Arj
then E is necessarily positive, and the stationary value — ^there can
be but one — ^is an absolute minimum. For a similar reasoD, if the
prescribed period be less than any of those natural to the partially
constrained system, £ is an absolute maximum algebraically, but
arithmetically an absolute minimum. But when p* lies within the
range of possible values of p'*, E may be positive or negative, and
the actual value is not the greatest or least possible. Whenever
a natural vibration is consistent with the imposed conditions, that
will be the vibration assumed. The variable part of 74- F is then
zero.
For convenience of treatment we have considered apart the
two great classes of forced vibrations and free vibrations; but there
is, of course, nothing to prevent their coexistence. After the lapse
of a sufficient interval of time, the free vibrations always dis-
appear, however small the friction may be. The case of abso-
lutely no friction is purely ideal.
There is one caution, however, which may not be superfluous
in respect to the case where given motions are forced on the
system. Suppose, as before, that the co-ordinates -^i, '^a,...'^^ are
given. Then the free vibrations, whose existence or non-existence
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150 VIBRATING SYSTEMS IN GBNERAL. [106.
is a matter of indifference so far as the forced motion is concemed,
must be understood to be such as the system is capable of, when
the co-ordinates V^i.-.-^r <^'^^ w)f allowed to vary from zero. Ln
order to prevent their varjdng, forces of the corresponding types
must be introduced ; so that from one point of view the motion in
question may be regarded as forced. But the applied forces are
merely of the nature of a constraint ; and their effect is the same
as a limitation on the freedom of the motion.
106 a. The principles of the last sections shew that if
V^ii '^f-'^r be given harmonic functions of the time AiCoaptj
A2 cos pt,..,, the forces of the other types vanishing, then the
motion is determinate, unless p is so chosen as to coincide with
one of the values proper to the system when -^i, '^^...^r are
maintained at zero. As an example, consider the case of a
membrane capable of vibrating transversely. If the displacement
y^ at every point of the contour be given (proportional to cos pt),
then in general the value in the interior is determinate ; but an
exception occurs if p have one of the values proper to the
membrane when vibrating with the contour held at rest. This
problem is considered by M. Duhem^ on the basis of a special
analytical investigation by Schwartz. It will be seen that it may
be regarded as a particular case of a vastly more general theorem.
A like result may be stated for an elastic solid of which the
surface motion (proportional to cos pt) is given at every point. Of
course, the motion at the boundary need not be more than partially
given. Thus for a mass of air we may suppose given the motion
normal to a closed surface. The internal motion is then deter-
minate, unless the frequency chosen is one of those proper to the
mass, when the surface is made unyielding.
107. Very remarkable reciprocal relations exist between the
forces and motions of different types, which may be regarded as
extensions of the corresponding theorems for systems in which
only F or r has to be considered (§ 72 and §§ 77, 78). If we sup-
pose that all the component forces, except two — '^'i and '*', — are
zero, we obtain from § 104,
(1)-
1 Cours de Phyiique Math€matiqu€, Tome Second, p. 190, Paris, 1S91.
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107.] RECIPROCAL THEOREM. 151
We now consider two cases of motion for the same system ; first
when ^s vanishes, and secondly (with dashed letters) when '9i
vanishes. If ^, = 0,
^-^-'£^^ (^>-
Similarly, if ^,' = 0,
^''=^-'£^'' <^)-
In these equations V and its differential coefficients are rational
integral functions of the symbol D; and since in every case
^r« = «#r, ^ is a symmetrical determinant, and therefore
^^^^^ (4)
den deg,
zfft u^ir
Hence we see that if a force '*'i act on the system, the co-
ordinate '^^ is related to it in the same way as the co-ordinate '^Z
is related to the force '9^, when this latter force is supposed to act
alone.
In addition to the motion here contemplated, there may be
free vibrations dependent on a disturbance already existing at the
moment subsequent to which all new sources of disturbance are
included in '9 ; but these vibrations are themselves the effect of
forces which acted previously. However small the dissipation
may be, there must be an interval of time after which free vibra-
tions die out, and beyond which it is unnecessary to go in taking
account of the forces which have acted on a system. If therefore
we include under "V forces of sufficient remoteness, there are no
independent vibrations to be considered, and in this way the
theorem may be extended to cases which would not at first sight
appear to come within its scope. Suppose, for example, that the
system is at rest in its position of equilibrium, and then begins to
be acted on by a force of the first type, gradually increasing in
magnitude from zero to a finite value ^i, at which point it ceases
to increase. If now at a given epoch of time the force be sud-
denly destroyed and remain zero ever afterwards, free vibrations of
the system will set in, and continue until destroyed by friction.
At any time t subsequent to the given epoch, the co-ordinate yfr^
has a value dependent upon t proportional to '*'i. The theorem
allows us to assert that this value yfr^ beai*s the same relation to '9i
as yJTi would at the same moment have borne to '^/, if the original
cause of the vibrations had been a force of the second type in-
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152 VIBRATING SYSTEMS IN GENERAL. [107.
creasing gradually from zero to '^/, and then suddenly vaniBhing^
at the given epoch of time. We have already had an example of
this in § 101, and a like result obtains when the cause of the
original disturbance is an impulse, or, as in the problem of the
pianoforte-string, a variable force of finite though short duration^
In these applications of our theorem we obtain results relating to
free vibrations, considered as the residual effect of forces whose
actual operation may have been long before.
108. In an important class of cases the forces ^i and '9^ are
harmonic, and -of the same period. We may represent them by
Aie^^^, ilaV^*, where Ai and A^ may be assumed to be recdf if the
forces be in the same phase at the moments compared. The
results may then be written
^■.^,i}s^^l ™-
where ip is written for D. Thus,
^V« = ilit/.... (2). .
Since the ratio ^^i : il,' is by hypothesis real, the same is
true of the ratio -^Z : -^s; which signifies that the motions
represented by those symbols are in the same phase. Passing
to real quantities we may state the theorem thus: —
If a force ^i = AiC08pt, aoting on the system give rise to
the motion '^j = dAiCOs(pt — e); then will a force 'S^a' == A,' cos pt
produce the motion -^Z = dk^ cos (pt — c).
If there be no friction, e will be zero.
If ^1 = -4j', then >fr/ = -^j. But it must be remembered that
the forces "^i and 'V^ are not necessarily comparable, any more
than the co-ordinates of corresponding t)rpes, one of which for
example may represent a linear and another an angular dis-
placement.
The reciprocal theorem may be stated in several ways, but
before proceeding to these we will give another investigation,
not requiring a knowledge of determinants.
If ^1,^,,... -^tu V^,,... and ^/, ^/,... i|r/, -^Z,... be two sets
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108.] RECIPROCAL THEOREM, 153
of forces and corresponding displacements, the equations of
motion, § 103, give
+ ^/r/(e^i|ri + eM'^2 + ««V^.+ ...) + ••• •
Now, if all the forces vary as ^^^ the effect of a symbolic
operator such as en on any of the quantities -^ is merely to
multiply that quantity by the constant found by substituting
ip for D in €„. Supposing this substitution made, and having
regard to the relations «« = ««•, we may write
+ ^.(V^i>, + V^,>i) + (3).
Hence by the symmetry
which is the expression of the reciprocal relation.
109. In the applications that we are about to make it
will be supposed throughout that the forces of all tjrpes but
two (which we may as well take as the first and second) are
zero. Thus
^i^i' + >i^,l|r/«^/V^,+^,>, (1).
The consequences of this equation may be exhibited in three
different ways. In the first we suppose that
whence V^, : ^, = if-/ : ^/ (2),
shewing, as before, that the relation of '^^ to "9^ in the first
case when "9^^^ is the same as the relation of '^Z to ^,' in
the second case, when '*'i = 0, the identity of relationship ex-
tending to phase as well as amplitude.
A few examples may promote the comprehension of a law,
whose extreme generality is not unlikely to convey an impression
of vagueness.
If P and Q be two points of a horizontal bar supported in
any manner (e.g. with one end clamped and the other free), a
given harmonic transverse force applied at P will give at any
moment the same vertical deflection at Q as would have been
found at P, had the force acted at Q.
If we take angular instead of linear displacements, the
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154 VIBRATING SYSTEMS IN GENERAL. [109.
theorem will run: — A given harmonic couple at P will give the
same rotation at Q as the couple at Q would give at P.
Or if one displacement be linear and the other angular, the
result may be stated thus: Suppose for the first case that a
harmonic couple acts at P, and for the second that a vertical
force of the same period and phase acts at Q, then the linear
displacement at Q in the first case has at every moment the
same phase as the rotatory displacement at P in the second,
and the amplitudes of the tiro displacements are so related that
the maximum couple at P would do the same work in acting
over the maximum rotation at P due to the force at Q, as the
maximum force at Q would do in acting through the maximum
displacement at Q due to the couple at P. In this case the
statement is more complicated, as the forces, being of different
kinds, cannot be taken equal.
If we suppose the period of the forces to be excessively long,
the momentary position of the system tends to coincide with
that in which it would be maintained at rest by the then acting
forces, and the equilibrium theory becomes applicable. Our
theorem then reduces to the statical one proved in § 72.
As a second example, suppose that in a space occupied by
air, and either wholly, or partly, confined by solid boundaries,
there are two spheres A and B, whose centres have one degree
of freedom. Then a periodic force acting on A will produce
the same motion in B, as if the parts were interchanged; and
this, whatever membranes, strings, forks on resonance cases, or
other bodies capable of being set into vibration, may be present in
their neighbourhood.
Or, if A and B denote two points of a solid elastic body
of any shape, a force parallel to OX, acting at A, will produce
the same motion of the point B parallel to OF as an equal force
parallel to OY acting at B would produce in the point A,
parallel to OX.
Or again, let A and B be two points of a space occupied by
air, between which are situated obstacles of any kind. Then a
sound originating at A is perceived at B with the same intensity
as that with which an equal sound originating at B would be per-
ceived at ^.^ The obstacle, for instance, might consist of a rigid
^ Helmholtz, CreUe, Bd. lvii., 1859. The soands must be such as in the absenoe
of obstacles would diffuse themselves equally in aU directions.
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109.]
APPLICATIONS.
155
wall pierced with one or more holes. This example corresponds
to the optical law that if by any combination of reflecting or
refracting surfaces one point can be seen from a second, the second
can also be seen from the first. In Acoustics the sound shadows
are usually only partial in consequence of the not insignificant
value of the wave-length in comparison with the dimensions of
ordinary obstacles: and the reciprocal relation is of considerable
interest.
A frirther example may be taken from electricity. Let there
be two circuits of insulated wire A and B, and in their neigh-
bourhood any combination of wire-circuits or solid conductors
in communication with condensers. A periodic electro-motive
force in the circuit A will give rise to the same current in B
as would be excited in A if the electro-motive force operated
in B.
Our last example will be taken from the theory of conduction
and radiation of heat, Newton's law of cooling being assumed
as a basi& The temperature at any point A of sl conducting and
radiating system due to a steady (or harmonic) source of heat
at ^ is the same as the temperature at B due to an equal source
at A. Moreover, if at any time the source at B be removed, the
whole subsequent course of temperature at A will be the same as
it would be at fi if the parts of B and A were interchanged.
110. The second way of stating the reciprocal theorem is
arrived at by taking in (1) of § 109,
V^i = 0, i|r; = 0;
whence ^iV^/ = ^V^2 (1).
or ^, : ifr. = ^/ : ^/ (2),
shewing that the relation of ^j to yfr^ in the first case, when -^i = 0,
is the same as the relation of ^,' to yjti in the second case, when
v^;=o.
Thus in the example of the rod, if the point P be held at
rest while a given vibration is imposed upon Q (by a force there
applied), the reaction at P is the same both in amplitude and
phase as it would be at Q if that point were held at rest and
the given vibration were imposed upon P.
So if A and B be two electric circuits in the neighbourhood
of any number of others, C, D,,.. whether closed or terminating
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156 VIBRATING SYSTEMS IN GENERAL. [llO.
in condensers, and a given periodic current be excited in A by
the necessary electro-motive force, the induced electro-motive
force in B is the same as it would be in ^, if the parts of A
and B were interchanged.
The third form of statement is obtained by putting in (1)
of § 109,
%=o. v^;=o;
whence ^i>i + ^2>2 = (3),
or ^1 : V^, = -^/ :>/ (4),
proving that the ratio of -^i to yjr^ in the first case, when '9^ acts
alone, is the negative of the ratio of '9^' to '9i in the second case,
when the forces are so related as to keep -^Z equal to zero.
Thus if the point P of the rod be held at rest while a periodic
force acts at Q, the reaction at P bears the same numerical ratio
to the force at Q as the displacement at Q would bear to the
displacement at P, if the rod were caused to vibrate by a force
applied at P.
111. The reciprocal theorem has been proved for all systems
in which the frictional forces can be represented by the function F,
but it is susceptible of a further and an important generalization.
We have indeed proved the existence . of the function F for
a large class of cases where the motion is resisted by forces
proportional to the absolute or relative velocities, but there are
other sources of dissipation not to be brought under this head,
whose effects it is equally important to include ; for example, the
dissipation due to the conduction or radiation of heat. Now
although it be true that the forces in these cases are not for aU
possible motions in a constant ratio to the velocities or displace-
ments, yet in any actual case of periodic motion (t) they are
necesseuily periodic, and therefore, whatever their phase, ex-
pressible by a sura of two terms, oue proportional to the dis-
placement (absolute or relative) and the other proportional to the
velocity of the part of the system affected. If the coefficients
be the same, not necessarily for all motions whatever, hut for all
motions of the period r, the function F exists in the only sense
required for our present purpose. In fact since it is exclusively
with motions of period r that the theorem is concerned, it is
plainly a matter of indifference whether the functions T, F, V
are dependent upon r or not. Thus extended, the theorem is
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111.] RECIPROCAL THEOREM. 157
perhaps sufficiently general to cover the whole field of dissipative
forces.
It is important to remember that the Principle of Reciprocity-
is limited to systems which vibrate about a configuration of equi-
librium, and is therefore not to be applied without reservation to
such a problem as that presented by the transmission of sonorous
waves through the atmosphere when disturbed by wind. The
vibrations must also be of such a character that the square of the
motion can be neglected throughout; otherwise our demonstra-
tion would not hold good. Other apparent exceptions depend on
a misunderstanding of the principle itself. Care must he taken
to observe a proper correspondence between the forces and dis-
placements, the rule being that the action of the force over the
displacement is to represent work done. Thus couples correspond
to rotations, pressures to increments of volume, and so on.
Ill a. The substance of the preceding sections is taken £rom
a paper by the Author*, in which the action of dissipative forces
appears first to have been included. Reciprocal theorems of a
special character, and with exclusion of dissipation, had been
previously given by other writers. One, due to von Helmholtz,
has already been quoted. Reference may also be made to the
reciprocal theorem of Betti*, relating to a uniform isotropic elastic
solid, upon which bodily and surface forces act. Lamb* has shewn
that these results and more recent ones of von Helmholtz^ may
be deduced from a very general equation established by Lagrange
in the Mdcanique Analytique,
111 6. In many cases of practical interest the external force,
in response to which a system vibrates harmonically, is applied at a
single point. This may be called the driving-point, and it becomes
important to estimate the reaction of the sjrstem upon it. When
T and F only are sensible, or F and V only, certain general
conclusions may be stated, of which a specimen will here be given.
For further details reference must be made to a paper by the
Author •.
1 ** Some General Theorems relating to Vibrations," Froe, Math, Soc., 1878.
» II Nuovo Cimento, 1872.
» Proc. Math. Soc., Vol. xix., p. 144, Jan. 1888.
* CreUe, t. 100, pp. 137, 213. 1886.
B '*The Beaction upon the Driving-point of a System ezecnting Forced Harmonic
' Oscillations of various Periods." Phil. Mag., May, 1886.
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158 VIBRATING SYSTEMS IN GENERAL. [ill 6.
Consider a system, devoid of potential energy, in which the
co-ordinate yfri is made to vary by the operation of the harmonic
force '9i, proportional to e*^. The other co-ordinates may be chosen
arbitrarily, and it will be very convenient to choose them so that
no product of them enters into the expressions for Tand F, They
would be in fact the normal co-ordinates of the system on the
supposition that yfr^ is constrained (by a suitable force of its own
type) to remain zero. The expressions for T and F thus take the
following forms : —
+ <hlt^lit^ + (hz^l^Z + (h4^l4'4'^ ...(1).
The equations for a force '*'i, proportional to e*^, are accordingly
(tpOii -I- bn) ^1 + (tpOia + 6is) ^s + (tpo^ + ^w) ^s + . . . = ^i,
(ipoi^ + 6,a) ^^1 + (ipa„ -H b„) i/r, = 0,
(ipoi, + 6„) ^i -H (ipon + 63s) ^s = 0,
By means of the second and following equations -^j^ '^s ... are
expressed in terms of yjri. Introducing these values into the first
equation, we get
%l^^-ip<^r + K-^?^^-^^^^- (3).
The ratio '^^i/'^i is a complex quantity, of which the real part
corresponds to the work done by the force in a complete period
and dissipated in the system. By an extension of electrical
language we may call it the resistance of the system and denote it
by the letter R'. The other part of the ratio is imaginary. If we
denote it by ipL'^i^ or Z'-^i, L' will be the moment of inertia, or
self-induction of electrical theory. We write therefore
^, = (iJ'4-tpi0^i (4);
and the values of R and L are to be deduced by separation of the
real and the imaginary parts of the right-hand member of (3). In
this way we get
^-^--^h^^p^b^i^^Tp^) ^^>-
This is the value of the resistance as determined by the
constitution of the system, and by the frequency of the imposed
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1116.]
REACTION AT DRIVING-POINT.
159
vibration. Each component of the latter series (which alone
involves p) is of the form <iji^l{P + yp^)\ where a, /9, 7 are all positive,
and (as may be seen most easily by considering its reciprocal)
increases continually as jp* increases from zero to infinity. We
conclude that as the frequency of vibration increases, the value of
R increases continuously with it. At the lower limit the motion
is determined sensibly by the quantities h (the resistances) only, and
the corresponding resultant resistance R is an absolute minimum,
whose value is
&u-S(6„V6«) (6).
At the upper limit the motion is determined by the ineiiiia of
the component parts without regard to resistances, and the value
ofJZ'is
T V iha* . < (gi«&« - <hi>uy
h^d^
or
6„ + S(6«^|-26„^) (7).
When p is either very large or very small, all the co-ordinates
are in the same phase, and (6), (7) may be identified with
Also i'»a,-2^V2^^^^f4x (8).
In the latter series every term is positive, and continually
diminishes as p^ increases. Hence every increase of frequency is
attended by a diminution of the moment of inertia, which tends
ultimately to the minimum corresponding to the disappearance of
the dissipative terms.
If p be either very large or very small, (8) identifies itself
with 2TI^^\
As a simple example take the problem of the reaction upon
the primary circuit of the electric currents generated in a neigh-
bouring secondary circuit. In this case the co-ordinates (or rather
their rates of increase) ai-e naturally taken to be the currents
themselves, so that '^i is the primary, and -^3 the secondary
current. In usual electrical notation we represent the coefficients
of self-induction by i, N, and of mutual induction by M, so that
and the resistances by R and 8. Thus
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160 VIBRATING SYSTEMS IN QENBRAL. [ill 6.
and (5) and (8) become at once
^ -^-^¥+^* ^^^'
^-■^-S'-'+w ^ ^'
These formulae were given originally by Maxwell, who remarked
that the reaction of the currents in the secondary has the effect
of increasing the effective resistance and diminishing the effective
self-induction of the primaiy circuit.
If the rate of alternation be very slow, the secondary circuit is
without influence. If, on the other hand, the rate be very rapid,
iJ' = iZ -h M'SjN^ i' =Z - M'IN.
112. In Chapter ill. we considered the vibrations of a system
with one degree of freedom. The remainder of the present Chapter
will be devoted to some details of the case where the degrees of
freedom are two.
If w and y denote the two co-ordinates, the expressions for T
and V are of the form
2V==Aid' + 2Bxy + Cy*\ ^^'
so that, in the absence of friction, the equations of motion are
Lx-\-My-\-Ax-^By = X\ .
Mx-^Ny + Bx+Cy=Y] ^^^•
When there are no impressed forces, we have for the natural
vibrations
(LI> +A)x^{M]>-\-B)y^O\
(MD^ + B)x+{ND^-{'C)y = 0'
D being the symbol of differentiation with respect to time.
If a solution of (3) be x — le^, y = me**, X' is one of the
roots of
{L>}'\-A){N\^ + C)-{M\* + By = Q (4),
or
\'{LN^M^)^X^{LG'\-NA'-2MB) + AG^»^0 (5).
The constants L, M, N\ A, B, (7, are not entirely arbitrary.
Since T and V are essentially positive, the following inequalities
must be satisfied : —
LN>M\ AC>B' (6).
Moreover, L, N, A^ C must themselves be positive.
(3).
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112.] BOOTS OF DETERMINANTAL EQUATION. 161
We proceed to examine the effect of these restrictions on the
roots of (5).
In the first place the three coefficients in the equation are
positive. For the first and third, this is obvious £rom (6). The
,>!fei!identofV
ii^ which, as is seen from (6), JLNAG is necessarily greater than
MB, We conclude that the values of \*, if real, are both
negative.
It remains to prove that the roots are in fact real The
ition to be satisfied is that the following quantity be not
neglative : —
After reduction this may be brought into the form
4^{jLN,B-jAG.My
+ (^/I0 - jNly {{JLC - jNlr + 4 {JLNAG - MB)\
which shews that the condition is satisfied, since JLNAG --MB
is positive. This is the analytical proof that the values of X' are
both real and negative ; a fact that might have been anticipated
without any analysis from the physical constitution of the system,
whose vibrations they serve to express.
The two values of X' are different, unless both
JLN.B^JAC.M^O )
JLC'-JNA^O y
which require that
L : M : N^A : B : G (7),
The common spherical pendulum is an example of this case.
By means of a suitable force Y the co-ordinate y may be
prevented from varying. The system then loses one degree of
freedom, and the period corresponding to the remaining one is in
general different from either of those possible before the introduc-
tion of F. Suppose that the types of the motions obtained by
thus preventing in turn the variation of y and x are respectively
e^^, e^. Then /Lti', ftj* are the roots of the equation
(L\' + A)(N\' + G) = 0,
R. 11
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162 VIBRATING SYsWeMS^*^^ ^ ^^^^^^ [ll2.
being that obtained from (4) b)\ ' suppressing if and B. Hence
(4) may itself be put into the form^ ^p>J|
ZiV^(X«-/iiO(V-/^Y^*-^ = ^^^^ (^X
which shews at once that neither Vw^f the roots of V can be
intermediate in value between fj^^ aS^ id /*,* A Etle funiiS ^
examination will prove that one of the ro^Mal^ ^ grwiter tHi both,
the quantities /ii', /Xj', and the other less thaiSr^V^!.?o/if we pu^
/( V) = ZJV^ (V - /i,0 (V - Ata^) - (ilf V + 5)^
we see that when X* is very small, / is positive (AC-B^); whj
V decreases (algebraically) to fii\ f changes sign and becomes
negative. Between and /ii* there is therefore a root ; and
by similar reasoning between fta* and — x . We conclude thatSfche
tones obtained by subjecting the system to the two kinds of con-
straint in question are both intermediate in pitch between 4he
tones given by the natural vibrations of the system. In particu]
cases fii\ fi^ may be equal, and then
This proposition may be generalized Any kind of constraint
which leaves the system still in possession of one degree of free-
dom may be regarded as the imposition of a forced relation
between the co-ordinates, such as
aa?-|-^y = (10).
Now if cue + ^y, and any other homogeneous linear func-
tion of X and y, be taken as new variables, the same argument
proves that the single period possible to the system after the
introduction of the constraint, is intermediate in value between
those two in which the natural vibrations were previously per-
formed. Conversely, the two periods which become possible
when a constraint is removed, lie one on each side of the original
period.
If the values of X* be equal, which can only happen when
L : M : N=A : B : C,
the introduction of a constraint has no effect on the period ; for
instance, the limitation of a spherical pendulum to one vertical
plane.
113. As a simple example of a system with two degrees of
freedom, we may take a stretched string of length I, itself without
1
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113.]
FREE VIBRATIONS.
163
inertia, but carrying two equal masses m at distances a and b from
one end (Fig. 17). Tension = 2',.
Fig. 17.
K a and y denote the displacements,
2T = m(£* + y'),
2F=2\
a b — a I — b) '
Since T and V are not of the same form, it follows that the
two periods of vibration are in every case unequal.
If the loads be symmetrically attached, the character of the
two component vibrations is evident. In the first, which will have
the longer period, the two weights move together, so that x and y
remain equal throughout the vibration. In the second x and y are
numerically equal, but opposed in sign. The middle point of the
string then remains at rest, and the two masses are always to
be found on a straight line passing through it. In the first case
a; — y = 0, and in the second j;H-y = 0; so that a; — y and a? + y
are the new variables which must be assumed in order to reduce
the functions T and V simultaneously to a sum of squares.
For example, if the masses be so attached as to divide the
string into three equal parts.
m
from which we obtain as the complete solution,
x + y = Acosy^.t + ay
.(1).
.-y^Bcos{^.t + ^)^
.(2),
where, as usual, the constants A, a, B, jS are to be determined by
the initial circumstances.
114. When the two natural periods of a system are nearly
equal, the phenomenon of intermittent vibration sometimes pre-
sents itself in a very curious manner. In order to illustrate this,
11—2
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i
164 VIBRATING SYSTEMS IN GENERAL. [114.
we may recur to the string loaded, we will now suppose, with two
equal masses at distances from its ends equal to one-fourth of the
length. If the middle point of the string were absolutely fixed,
the two similar systems on either side of it would be completely
independent, or, if the whole be considered as one system, the two
periods of vibration would be equal. We now suppose that
instead of being absolutely fixed, the middle point is attached to
springs, or other machinery, destitute of inertia, so that it is
capable of yielding slightly. The reservation as to inertia is to
avoid the introduction of a third degree of freedom.
From the symmetry it is evident that the fundamental vibra-
tions of the system are those represented by a?H-y and x — y.
Their periods are slightly different, because, on account of the
yielding of the centre, the potential energy of a displacement
when X and y are equal, is less than that of a displacement
when X and y are opposite; whereas the kinetic energies are
the same for the two kinds of vibration. In the solution
a? + y = -4cos(ni« + a) |
we are therefore to regard n^ and n, as nearly, but not quite, equal.
Now let us suppose that initially x and x vanish. The conditions
are
-4 cos a -h -Bcos^ =
UiA sin a + n^B sin /8 =
which give approximately
^+-8 = 0, a = ^.
Thus x = A%m ^ ^ ^ t smf ^^"<H-aj
y = Acos -g— t cos f g ^ 4- a 1
The value of the co-ordinate x is here approximately ex-
pressed by a harmonic term, whose amplitude, being proportional
to sin J (n^ - Wi) *. is a slowly varying harmonic function of the
time. The vibrations of the co-ordinates are therefore intermittent,
and so adjusted that each amplitude vanishes at the moment that
the other is at its maximum.
This phenomenon may be prettily shewn by a tuning fork of
very low pitch, heavily weighted at the ends, and firmly held by
.(2).
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■^m
114.]
INTERMITTENT VIBRATIONS.
165
screwing the stalk into a massive support. When the fork vibrates
in the normal manner, the rigidity^ or want of rigidity, of the
stalk does not come into play ; bat if the displacements of the two
prongs be in the same direction, the slight yielding of the stalk
entails a small change of period. If the fork be excited by striking
one prong, the vibrations are intermittent, and appear to transfer
themselves bcu^kwards and forwards between the prongs. Unless,
however, the support be very firm, the abnormal vibration, which
involves a motion of the centre of inertia, is soon dissipated ; and
then, of course, the vibration appears to become steady. If the
fork be merely held in the hand, the phenomenon of intermittence
cannot be obtained at all.
116. The stretched string with two attached masses may be
used to illustrate some general principles. For example, the period
of the vibration which remains possible when one mass is held
at rest, is intermediate between the two free periods. Any in-
crease in either load depresses the pitch of both the natural
vibrations, and conversely. If the new load be situated at a point
of the string not coinciding with the places where the other loads
are attached, nor with the node of one of the two previously
possible free vibrations (the other has no node), the effect is still
to prolong both the periods already present. With regard to the
third finite period, which becomes possible for the first time after
the addition of the new load, it must be regarded as derived from
one of infinitely small magnitude, of which an indefinite number
may be supposed to form part of the system. It is instructive
to trace the effect of the introduction of a new load and its gradual
increase from zero to infinity, but for this purpose it will be
simpler to take the case where there is but one other. At the
commencement there is one finite period Ti, and another of in-
finitesimal magnitude r,. As the load increases r, becomes finite,
and both Ti and Tg continually increase. Let us now consider
what happens when the load becomes very great. One of the
periods is necessarily large and capable of growing beyond all
limit The other must approach a fixed finite limit. The first
belongs to a motion in which the larger mass vibrates nearly as
if the other were absent ; the second is the period of the vibration
of the smaller mass, taking place much as if the larger were fixed.
Now since Tj and r, can never be equal, Ti must be always the
greater ; and we infer, that as the load becomes continually larger,
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166 VIBRATING SYSTEMS IN GENERAL. [115.
it is Ti that increases indefinitely, and r, that approaches a finite
limit.
We now pass to the consideration of forced vibrations.
116. The general equations for a system of two degrees of
freedom including friction are
{MD'+l3D-hB)x + {ND' + yD-hC)y^Y] ^ ^•
In what follows we shall suppose that F = 0, and that X — ^.
The solution for y is
^ {A-p'L + iap)(G''P'N + iyp)^(B'-p'M + tfipy'"^^^'
If the connection between x and y be of a loose character, the
constants Jf, /8, B are small, so that the term (B — p^M + ifipY
in the denominator may in general be neglected. When this
is permissible, the co-ordinate y is the same as if a; had been pre-
vented firom varying, and a force Y had been introduced whose
magnitude is independent of N, 7, and C, But if, in consequence
of an approximate isochronism between the force and one of the
motions which become possible when ar or y is constrained to be
zero, either A—p^L-^tap or C—p^N'\-iyp be small^ then the
term in the denominator containing the coefficients of mutual
influence must be retained^ being no longer relatively unimportant ;
and the solution is accordingly of a more complicated character.
Symmetry shews that if we had assumed X = 0, F = 6*^, we
should have found the same value for x as now obtains for y. This
is the Reciprocal Theorem of § 108 applied to a system capable
of two independent motions. The string and two loads may again
be referred to as an example.
117. So far for an imposed force. We shall next suppose
that it is a motion of one co-ordinate (x = ^p*) that is prescribed,
while F=0; and for greater simplicity we shall confine ourselves
to the case where ^ = 0. The value of y is
y c-Np*+iyp ^^''•
Let as now inquire into the reaction of this motion on x.
We have
^-"^'^^'-'^m^ <^>
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117.] REACTION OF A DEPENDENT SYSTEM. 167
If the real and imaginary parts of the coefficient of e^ be re-
spectively A' and la'p, we may put
{MB'^B)y^A'x^tix (3),
^(G^Np^y+rfp^ ^ ^'
It appears that the effect of the reaction of y (over and above
what would be caused by holding y = 0) is represented by changing
A into A + A\ and a into a + a', where A' and a' have the above
values, and is therefore equivalent to the effect of an alteration in
the coefficients of spring and friction. These alterations, however,
are not constants, but functions of the period of the motion con-
templated, whose character we now proceed to consider.
Let n be the value otp corresponding to the natural frictionless
period of y {x being maintained at zero); so that (7 — n*iV = 0.
Then
A' = (B--MT^y A^(;>--nO ]
--iB^Mj^f ^^^^Jl^,^^^ ]
.(6).
In most cases with which we are practically concerned 7 is
small, and interest centres mainly on values of p not much differ-
ing from n. We shall accordingly leave out of account the
variations of the positive factor {B — Jfjp*)*, and in the small term
7^, substitute for p its approximate value n. When p is not
nearly equal to n, the term in question is of no importance.
As might be anticipated from the general principle of work,
a' is alwajrs positive. Its maximum value occurs when jp = n
nearly, and is then proportional to l/7n», which varies inversely
with 7. This might not have been expected on a superficial view
of the matter, for it seems rather a paradox that, the greater the
friction, the less should be its result. But it must be remembered
that 7 is only the coefficient of friction, and that when 7 is small
the maximum motion is so much increased that the whole work
spent against friction is greater than if 7 were more considerable.
But the point of most interest is the dependence of A' on p,
lip be less than n. A' is negative. As p passes through the value
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168 VIBRATINO SYSTEMS IN GENERAL, [117.
n. A' vanishes, and changes sign. When A' is negative, the in-
fluence of y is to diminish the recovering power of the vibration «,
and we see that this happens when the forced vibration is slower
than that natural to y. The tendency of the vibration y is thus
to retard the vibration x, if the latter be already the slower, but
to accelerate it, if it be already the more rapid, only vanishing in
the critical case of perfect isochronism. The attempt to make x
vibrate at the rate determined by w is beset with a peculiar
difficulty, analogous to that met with in balancing a heavy
body with the centre of gravity above the support. On which-
ever side a slight departure from precision of adjustment may
occur the influence of the dependent vibration is always to increase
the error. Examples of the instability of pitch accompanying a
strong resonance will come across us hereafter ; but undoubtedly
the most interesting application of the results of this section is to
the explanation of the anomalous refraction, by substances possess-
ing a very marked selective absorption, of the two kinds of light
situated (in a normal spectrum) immediately on either side of the
absorption band^ It was observed by Christiansen and Eundt,
the discoverers of this remarkable phenomenon, that media of the
kind in question (for example, ^t^oA^ne in alcoholic solution) refract
the ray immediately below the absorption-band abnormally in
excess, and that above it in defect If we suppose, as on other
grounds it would be natural to do, that the intense absorption is
the result of an agreement between the vibrations of the kind of
light affected, and some vibration proper to the molecules of the
absorbing agent, our theory would indicate that for light of some-
what greater period the effect must be the same as a relaxation of
the natural elasticity of the ether, manifesting itself by a slower
propagation and increased refraction. On the other side of the
absorption-band its influence must be in the opposite direc-
tion.
In order to trace the law of connection between A' and p, take,
for brevity, 7 n = a, Nijj^ — n") = x, so that
When the sign of a? is changed. A' is reversed with it, but pre-
serves its numerical value. When a; = 0, or ±00 , A^ vanishes.
1 Phil Mag,, May, 1872. Also Sellmeier, Pogg, Arm, t oxliii. p. 272, 1871.
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117.]
REACTION OF A DEPENDENT SYSTEM.
169
Hence the origin is on the representative curve (Fig. 18), and the
axis of a? is an asymptote. The maximum and minimum values of
A' occur when x is respectively equal to + a, or — a ; and then
X
af'+a:'
:.= ±
2a'
The corresponding values of p are given by
i^-'if
(7).
Hence, the smaller the value of a or 7, the greater will be the
maximum alteration of A, and the corresponding value of p wilt
approach nearer and nearer to n. It may be well to repeat, that in
the optical application a diminished 7 is attended by an increased
maximum absorption. When the adjustment of periods is such as
to fevour A' as much as possible, the corresponding value of a m
one half of its maximum.
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i
CHAPTER VI.
TRANSVERSE VIBRATIONS OF STRINGS.
118. Among vibrating bodies there are none that occupy a
more prominent position than Stretched Strings. From the
earliest times they have been employed for musical purposes,
and in the present day they still form the essential parts of such
important instruments as the pianoforte and the violin. To the
mathematician they must always possess a peculiar interest as the
battle-field on which were fought out the controversies of D'Alem-
bert, Euler, Bernoulli and Lagrange, relating to the nature of the
solutions of partial differential equations. To the student of
Acoustics they are doubly important. In consequence of the com-
parative simplicity of their theory, they are the ground on which
difficult or doubtful questions, such as those relating to the nature
of simple tones, can be most advantageously faced ; while in the
form of a Monochord or Sonometer, they aflFord the most generally
available means for the comparison of pitch.
The ' string ' of Acoustics is a perfectly uniform and flexible
filament of solid matter stretched between two fixed points — ^in
fact an ideal body, never actually realized in practice, though
closely approximated to by most of the strings employed in music.
We shall afterwards see how to take account of any small devia-
tions from complete flexibility and uniformity.
The vibrations of a string may be divided into two distinct
classes, which are practically independent of one another, if the
amplitudes do not exceed certain limits. In the first class the
displacements and motions of the particles are longittAdinal, so
that the string always retains its straightness. The potential
energy of a displacement depends, not on the whole tension, but
on the changes of tension which occur in the various parts of the
string, due to the increased or diminished extension. In order to
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118.]
TRANSVERSE VIBRATIONS OF STRINGS.
171
calculate it we must koow the relation between the extension of
a string and the stretching force. The approximate law (given by
Hooke) may be expressed by saying that the extension varies
as the tension, so that if I and r denote the natural and the
stretched lengths of a string, and T the tension,
^-■"-^ = ^ (1)
where ^ is a constant, depending on the material and the section,
which may be interpreted to mean the tension that would be
necessary to stretch the string to twice its natural length, if the
law applied to so great extensions, which, in general, it is far
firom doing.
119. The vibrations of the second kind are transverse ; that is
to say, the particles of the string move sensibly in planes perpen-
dicular to the line of the string. In this case the potential energy
of a displacement depends upon the general tension, and the
small variations of tension accompanying the additional stretching
due to the displacement may be left out of account. It is here
assumed that the stretching due to the motion may be neglected
in comparison with that to which the string is already subject in
its position of equilibrium. Once assured of the fulfilment of this
condition, we do not, in the investigation of transverse vibrations,
require to know anything further of the law of extension.
The most general vibration of the transverse, or lateral, kind
may be resolved, as we shall presently prove, into two sets of
component normal vibrations, executed in perpendicular planes.
Since it is only in the initial circumstances that there can be any
distinction, pertinent to the question, between one > plane and
another, it is sufficient for most purposes to regard the motion as
entirely confined to a single plane passing through the line of the
string.
In treating of the theory of strings it is usual to commence
with two particular solutions of the partial diflferential equation,
representing the transmission of waves in the positive and nega-
tive directions, and to combine these in such a manner as to suit
the case of a finite string, whose ends are maintained at rest;
neither of the solutions taken by itself being consistent with the
existence of nodes, or places of permanent rest. This aspect of the
question is very important, and we shall fully consider it; but it
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172 TRANSVERSE VIBRATIONS OF STRINGS. [119.
seems scarcely desirable to found the solution in the first instance
on a property so peculiar to a uniform string as the undisturbed
transmission of waves. We will proceed by the more general
method of assuming (in conformity with what was proved in the
last chapter) that the motion may be resolved into normal com-
ponents of the harmonic type, and determining their periods and
character by the special conditions of the system.
Towards carrying out this design the first step would naturally
be the investigation of the partial differential equation, to which
the motion of a continuous string is subject. But in order to
throw light on a point, which it is most important to understand
clearly, — the connection between finite and infinite freedom, and
the passage corresponding thereto between arbitrary constants
and arbitrary functions^ we will commence by following a some-
what different course.
120. In Chapter ill. it was pointed out that the fundamental
vibration of a string would not be entirely altered in character,
if the mass were concentrated at the middle point. Following
out this idea, we see that if the whole string were divided into a
number of small parts and the mass of each concentrated at its
centre, we might by sufficiently multiplying the number of parts
arrive at a system, still of finite freedom, but capable of represent-
ing the continuous string with any desired accuracy, so far at
least as the lower component vibrations are concerned. If the
analytical solution for any number of divisions can be obtained,
its limit will give the result corresponding to a uniform string.
This is the method followed by Lagrange.
Let I be the length, pi the whole mass of the string, so that
p denotes the mass per unit length, T^ the tension.
Fig. 19.
The length of the string is divided into m + 1 equal parts (a),
so that
(m + l)a = Z (1).
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120.]
MASS CONCENTRATED IN POINTS.
173
At the m points of division equal masses (jjl) are supposed con-
centrated, which are the representatives of the mass of the por-
tions (a) of the string, which they severally bisect. The mass of
each terminal portion of length ^a is supposed to be concentrated
at the final points. On this understanding, we have
(m+l)/A = /)i
(2).
We proceed to investigate the vibrations of a string, itself
devoid of inertia, but loaded at each of m points equidistant
(a) from themselves and from the ends, with a mass /a.
If i|^, ifr, V^nH-s denote the lateral displacements of the
loaded points, including the initial and final points, we have the
following expressions for T and F,
J'=4/^{^i' + ^s'H-...H-t'm+iH-t'm+4
(3)
with the conditions that '^i and y^m+n vanish. These give by
Lagrange's Method the m equations of motion.
Bf.+Ayfr, +Bylt, =0
Bfn. + Air^, + Bylt^+, =
where
A=fiD^ +
2^1
a
(5).
(6).
Supposing now that the vibration under consideration is one
of normal type, we assume that yjti, i^s, &c. are all proportional to
cos(ni — e), where n remains to be determined. A and B may
then be regarded as constants, with a substitution of — «' for D*.
If for the sake of brevity we put
(T).
C=A^B^-2+'^
the determinantal equation, which gives the values of n^, assumes
the form
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174
TEAN8VERSE VIBRATIONS OF STRINGS.
[120.
C, 1, 0, 0, 0.
1, C, 1, 0, 0.
0, 1. C, 1, 0.
0, 0. 1. C. 1.
0, 0, 0, 1, c.
m rows
= 0.
.(8).
From this equation the values of the roots might be fouud.
It may be proved that, if (7= 2 cos 6y the determinant is equivalent
to sin (774 H- 1) ^ -r sin ^ ; but we shall attain our object with greater
ease directly from (5) by acting on a hint derived from the known
results relating to a continuous string, and assuming for trial a
particular type of vibration. Thus let a solution be
yltr = P sin(r— l)/3 cos(n^ — e)
(9).
a form which secures that 1^1 = 0. In order that -^m+s may
vanish,
(mH-l)/3 = S7r (10),
where 8 is an integer. Substituting the assumed values of yp' in
the equations (5), we find that they are satisfied, provided that
2Bcosy3H-^=0 (11);
so that the value of n in term^ of /3 is
fia
A normal vibration is thus represented by
"=^«^f\/S <'')•
tr = P.8in^-^— ^co8(M-e,) (13).
m+1
where
V M
Sir
sin in — rr^
fia 2 (w + 1)
(14).
and Pgy €t denote arbitrary constants independent of the general
constitution of the system. The m admissible values of n are
found irom (14) by ascribing to 8 in succession the values 1, 2,
d...7n, and are all different. If we take « = m+l, yp'r vanishes,
so that this does not correspond to a possible vibration. Greater
values of 8 give only the same periods over again. If m + 1 be
even, one of the values of n — ^that, namely, corresponding to
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120.] MASS CONCENTRATED IN POINTS. 175
« = ^(m + l), — is the same as would be found in the case of only
a single load (m » 1). The interpretation is obvious. In the kind
of vibration considered every alternate particle remains at rest, so
that the intermediate ones really move as though they were
attached to the centres of strings of length 2a, fastened at
the ends.
The most general solution is found by putting together all the
possible particular solutions of normal type
tr = S*"" P. sin ^^^^^ COS (n.t-e.) (15).
and, by ascribing suitable values to the arbitrary constants, can
be identified with the vibration resulting from arbitrary initial
circumstances.
Let X denote the distance of the particle r from the end of the
string, so that (r — l)a = a?; then by substituting for fju and a
from (1) and (2), our solution may be written,
tree
'^{x) = P,sin8 J- co3(n,t — e,) (16),
2(m + l) /5\ . sir ,,,,
In order to pass to the case of a continuous string, we have
only to put m infinite. The first equation retains its form, and
specifies the displacement at any point x. The limiting form of
the second is simply
^ = TV7 (^^)'
whence for the periodic time,
"?-y^ ■ w
The periods of the component tones are thus aliquot parts of
that of the gravest of the series, found by putting s = h ' The
whole motion is in all cases periodic ; and the period is 21 \/(/>/7i).
This statement, however, must not be understood as excluding^
a shorter period; for in particular cases any number* of the
lower components may be absent. All that is asserted is that the
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176
TRANSVERSE VIBRATIONS OF STRINGS.
[120.
above-mentioned interval of time is sufficient to bring about a com-
plete recun'ence. We defer for the present any further discussion
of the important formula (19), but it is interesting to observe the
approach to a limit in (17), as m is made successively greater and
greater. For this purpose it will be sufficient to take the gravest
tone for which « = 1, and accordingly to trace the variation of
2(m + l)
TT
sm
TT
2(m4l)'
The following are a series of simultaneous values of the func-
tion and variable : —
m
1
2
3
4
9
19
39
2(m+l) . w
•9003
•9549
•9745
•9836
•9959
•9990
•9997
It will be seen that for very moderate values of m the limit is
closely approached. Since m is the number of (moveable) loads,
the case m = 1 corresponds to the problem investigated in Chapter
III., but in comparing the results we must remember that we there
supposed the whole mass of the string to be concentrated at the
centre. In the present case the load at the centre is only half as
great; the remainder being supposed concentrated at the ends,
where it is without effect.
From the fifict that our solution is general, it follows that any
initial form of the string can be represented by
-^ (^) = S (P cos €)g sin 8
*=i
TTX
T
(20).
And, since any form possible for the string at all may be
regarded as initial, we infer that any finite single valued function
of X, which vanishes at ^ = and x = l, can be expanded within
those limits in a series of sines of irx/l and its multiples, — which
is a case of Fourier's theorem. We shall presently shew how the
more general form can be deduced.
121. We might now determine the constants for a continuous
string by integration as in § 93, but it is instructive to solve the
problem first in the general case (m finite), and afterwards to
proceed to the limit. The initial conditions are
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121.] MASS CONCENTRATED IN POINTS. 177
• / \ A ' "^^ , A • a'''^ . A ' ^^
• / \ >! • 'Ttt , , . ^ Tra . . >« • ^"
where, for brevity, J., = P,co8€,, and '^(a), yjr{2a) ylt(ma)
are the initial displacements of the m particles.
To determine any constant Ag, multiply the first equation by
sin (87ra/l\ the second by sin (2«7ra//), &c., and add the results.
Then, by Trigonometry, the coefficients of all the constants, except
At, vanish, while that of -4, = ^ (m + 1)^ Hence
^'=^,^2^.^('-«)«"^^«r ^^>-
We need not stay here to write down the values of Bg (equal
to Pg sin €g) as depending on the initial velocities. When a becomes
infinitely small, ra under the sign of summation ranges by infi-
nitesimal steps from zero to L At the same time — — ^ = y ,
so that writing ra = iv, a==dx, we have ultimately
^ = f£v^Wsm(?^)d^ •• <2)'
expressing Ag in terms of the initial displacements.
122. We will now investigate independently the partial diflfer-
ential equation governing the transverse motion of a perfectly
flexible string, on the suppositions (1) that the magnitude of the
tension may be considered constant, (2) that the square of the
inclination of any part of the string to its initial direction may be
neglected. As before, p denotes the linear density at any point,
and Ti is the constant tension. Let rectangular co-ordinates be
taken parallel, and perpendicular to the string, so that x gives the
equilibrium and x, y, z the displaced position of any particle at
time t The forces acting on the element dx are the tensions at
. I Todhunter'fl ItU. Calc,, p. 267.
R. 12
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178 TRANSVERSE VIBRATIONS OF STRINGS. [122.
its two ends, and any impressed forces Yp dx, Zp dx. By D'Alem-
bert's Principle these form an equilibrating system with the
reactions against acceleration, ^pdhfjd^, —pd^zld^. At the
point X the components of tension are
rpdy rpdz
^'dx' ^'dx'
if the squares of dy/dx, dz/dx be neglected ; so that the forces
acting on the element dx arising out of the tension are
<{%h- '-iii)^-
Hence for the equations of motion.
' (1),
dt^ p da^
d^ p da?'^ )
from which it appears that the dependent variables y and z are
altogether independent of one another.
The student should compare these equations with the corre-
sponding equations of finite differences in § 120. The latter may
be written
Now in the limit, when a becomes infinitely small,
'^{x-a)'\-'>^{x'\-a)- 2i|r (a?) = i|r" (a?) a\
while fi = pa; and the equation assumes ultimately the form
agreeing with (1).
In like manner the limiting forms of (3) and (4) of § 120 are
^"i/"®'^ • «■
''-''•■/(i)"'^ «•
which may also be proved directly.
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122.] DIFFERENTIAL EQUATIONS. 179
The first is obvious from the definition of T, To prove the
second, it is sufficient to notice that the potential energy in any
con6guration is the work required to produce the necessary
stretching against the tension J\. Reckoning from the configura-
tion of equilibrium, we have
^-■''i{%-'>'
and, so far as the third power of -^ ,
dx * \dx)
123. In most of the applications that we shall have to make,
the density p is constant, there are no impressed forces, and the
motion may be supposed to take place in one plane. We may
then conveniently write
T
j-^' a).
and the differential equation is expressed by
d{aty dx" ^ ^*
If we now assume that y varies as cos ma^, our equation
becomes
|j+wi*y = (3),
of which the most general solution is
. y = {Asmmx-\'GQOH7nx)co&mat (4).
This, however, is not the most general harmonic motion of
the period in question. In order to obtain the latter, we must
assume
y = yi cos mat + y, sin mat (5),
where yi, y, are functions of x, not necessarily the same. On
substitution in (2) it appears that yi and y, are subject to equations
of the form (3), so that finally
y = {A sin ma + G cos mx) cos mat ] ,^v
+ (i?sin 7na?-hDcosm{r)sin7na^|
an expression containing four arbitrary constants. For any con-
tinuous length of string satisfying without interruption the differ-
12—2
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180 TRANSVERSE VIBRATIONS OF STRINGS. [123.
ential equation, this is the most general solution possible, under
the condition that the motion at every point shall be simple
harmonic. But whenever the string forms part of a system
vibrating freely and without dissipation, we know from former
chapters that all parts are simultaneously in the same phase,
which requires that
A : B^C : D (7);
and then the most general vibration of simple harmonic type is
y = {a sin WW? + )9 cos wwp} cos (ma^ — €) (8).
124. The most simple as well as the most important problem
connected with our present subject is the investigation of the free
vibrations of a finite string of length I held fast at both its ends.
If we take the origin of x at one end, the terminal conditions are
that when a? = 0, and when x^l, y vanishes for all values of t.
The first requires that in (6) of § 123
C=0, D = (1);
and the second that
sin mi = (2),
or that mZ = «7r, where 8 is an integer. We learn that the only
harmonic vibrations possible are such as make
^ = T (3).
and then
y = sm-^(ilcos-j- +5sm— =- 1 (4).
Now we know a priori that whatever the motion may be, it
can be represented as a sum of simple harmonic vibrations, and
we therefore conclude that the most general solution for a string,
fixed at and 2, is
y = 2^^ sm-j-|^^,cos-^ + i?,sm-pj (5).
The slowest vibration is that corresponding to «=1. Its
period (tj) is given by
^.4'-^Vf. w
The other components have periods which are aliquot parts
of Ti : —
T, = Ti^s (7);
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124.] FIXED EXTREMITIES. 181
so that, as has been already stated, the whole motion is under all
circumstances periodic in the time Ti. The sound emitted con-
stitutes in general a musical notey according to our definition of
that term, whose pitch is fixed by r,, the period of its gravest
component. It may happen, however, in special cases that the
gravest vibration is absent, and yet that the whole motion is not
periodic in any shorter time. This condition of things occurs, if
jli*-h-B,« vanish, while, for example, A^-\-B^ and A^-{-B^ are
finite. In such cases the sound could hardly be called a note;
but it usually happens in practice that, when the gravest tone is
absent, some other takes its place in the character of fundamental,
and the sound still constitutes a note in the ordinary sense,
though, of course, of elevated pitch. A simple case is when all
the odd components beginning with the first are missing. The
whole motion is then periodic in the time ^Tj, and if the second
component be present, the sound presents nothing unusual.
The pitch of the note yielded by a string (6), and the character
of the fundamental vibration, were first investigated on mechanical
principles by Brook Taylor in 1715 ; but it is to Daniel Bernoulli
(1765) that we owe the general solution contained in (5). He
obtained it, as we have done, by the synthesis of particular
solutions, permissible in accordance with his Principle of the
Coexistence of Small Motions. In his time the generality of the
result so arrived at was open to question ; in fact, it was the
opinion of Euler, and also, strangely enough, of Lagrange ^ that
the series of sines in (5) was not capable of representing an
arbitrary function; and Bernoulli's argument on the other side,
drawn from the infinite number of the disposable constants,
was certainly inadequate'.
Most of the laws embodied in Taylor's formula (6) had been
discovered experimentally long before (1636) by Mersenne. They
may be stated thus : —
^ See Riemann's PartieUe Differential Gleiehungenj § 78.
' Dr Toang, in his memoir of 1800, seems to have understood this matter quite
oorrectly. He says, ** At the same time, as M. Bernoulli has justly observed, since
every figure may be infinitely approximated, by considering its ordinates as
composed of the ordinates of an infinite number of trochoids of different magni-
tudes, it may be demonstrated that all these constituent curves would revert to
their initial state, in the same time that a similar chord bent into a trochoidal
curve would perform a single vibration ; and this is in some respects a convenient
and compendious method of considering the problem."
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182 TRANSVERSE VIBRATIONS OP STRINGS. [124.
( 1) For a given string and a given tension, the time varies as
the length.
This is fphe fundamental principle of the monochord, and
appears ta have been understood by the ancients^
(2) When the length of the string is given, the time varies
iDversely as the square root of the tension.
(S) Strings of the same length and tension vibrate in times,
which are proportional to the square roots of the linear density.
These important results may all be obtained by the method of
dimensions, if it be assumed that t depends only on I, />, and Tj.
For, if the units of length, time and mass be denoted re-
gpectively by [i], [T], [if], the dimensions of these symbols are
given by
i = [Z], p = [JlfX-a Z = [MLT^],
and thus (see § 52) the only combination of them capable of re-
presenting a time is Tr^.p^.l, The only thing left undetermined
is the numerical £eictor.
125. Mersenne's laws are exemplified in all stringed instru-
ments. In playing the violin diflferent notes are obtained from
the same string by shortening its efficient length. In tuning
the violin or the pianoforte, an adjustment of pitch is eflFected
with a constant length by varying the tension ; but it must be
remembered that p is not quite invariable.
To secure a prescribed pitch with a string of given material, it is
requisite that one relation only be satisfied between the length, the
thickness, and the tension; but in practice there is usually no great
latitude. The length is often limited by considerations of con-
venience, and its curtailment cannot always be compensated by
an increase of thicknea*^, because, if the tension be not increased
proportionally to the section, there is a loss of flexibility,
while if the tension be so increased, nothing is effected towards
lowering the pitch. The diflficulty is avoided in the lower strings
of the pianoforte and violin by the addition of a coil of fine wire,
whose effect is to impart inertia without too much impairing
flexibility.
^ Aristotle ** knew that a pipe or a chord of double length produced a sound of
which the yibratioDs occupied a double time; and that the properties of concords
depended on the proportions of the times occupied by the vibrations of the
aeparate aoundfl." — Young's Lectures on Natural Philosophy , VoL i. p. 404,
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125.] mersenne's laws. 183
For quantitative investigations into the laws of strings, the
sonometer is employed. By means of a weight hanging over a
pulley, a catgut, or a metallic wire, is stretched across two bridges
mounted on a resonance case. A moveable bridge, whose position
is estimated by a scale running parallel to the wire, gives the
means of shortening the efficient portion of the wire to any
desired extent. The vibrations may be excited by plucking, as
in the harp, or with a bow (well supplied with rosin), as in the
violin.
If the moveable bridge be placed half-way between the fixed
ones, the note is raised an octave ; when the string is reduced to
one-third, the note obtained is the twelfth.
By means of the law of lengths, Mersenne determined for the
first time the frequencies of known musical notes. He adjusted the
length of a string until its note was one of assured position in the
musical scale, and then prolonged it under the same tension until
the vibrations were slow enough to be counted.
For experimental purposes it is convenient to have two, or
more, strings mounted side by side, and to vary in turn their
lengths, their masses, and the tensions to which they are subjected.
Thus in order that two strings of equal length may yield the
interval of the octave, their tensions must be in the ratio of 1 : 4,
if the masses be the same ; or, if the tensions be the same, the
masses must be in the reciprocal ratio.
The sonometer is very useful for the numerical determination
of pitch. By varying the tension, the string is tuned to unison
with a fork, or other standard of known frequency, and then by
adjustment of the moveable bridge, the length of the string is
determined, which vibrates in unison with any note proposed for
measurement. The law of lengths then gives the means of
effecting the desired comparison of firequencies.
Another application by Scheibler to the determination of
absolute pitch is important. The principle is the same as that
explained in Chapter ill., and the method depends on deducing
the absolute pitch of two notes from a knowledge of both the
ratio and the difference of their frequencies. The lengths of the
sonometer string when in unison with a fork, and when giving with
it four beats per second, are carefully measured. The ratio of the
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184 TRANSVEBSE VIBRATIONS OF STRINGS. [l25.
lengths is the inverse ratio of the frequencies, and the diflTerence
of the frequencies is four. From these data the absolute pitch of
the fork can be calculated.
The pitch of a string may be calculated also by Taylor's
formula from the mechanical elements of the system, but
great precautions are necessary to secure accuracy. The tension
m produced by a weight, whose mass (expressed with the same
unit as p) may be called P; so that Ti=gP, where g = 32%
if the units of length and time be the foot and the second. In
order to secure that the whole tension acts on the vibrating
segment, no bridge must be interposed, a condition only to be
satisfied by suspending the string vertically. After the weight is
attached, a portion of the string is isolated by clamping it firmly
at two points, and the length is measured. The mass of the unit
of length p refers to the stretched state of the string, and may be
found indirectly by observing the elongation due to a tension
of the same order of magnitude as 7\, and calculating what
would be produced by T^ according to Hooke's law, and by
weighing a known length of the string in its normal state.
After the clamps have been secured great care is required to
avoid fluctuations of temperature, which would seriously influence
the tension. In this way Seebeck obtained very accurate results.
126. When a string vibrates in its gravest normal mode, the
excursion is at any moment proportional to sin (irx/ 1), increasing
numerically from either end towards the centre ; no intermediate
point of the string remains permanently at rest. But it is other-
wise in the case of the higher normal components. Thus, if the
vibration be of the mode expressed by
. STTX ( . sirat n ' 8wat\
y = Qm—^{AtCOB—j — hi>«sm— y- j ,
the excursion is proportional to sin (sttx/I), which vanishes at « — 1
points, dividing the string into s equal parts. These points of no
motion are called nodes, and may evidently be touched or held
fast without in any way disturbing the vibration. The produc-
tion of * harmonics ' by lightly touching the string at the points of
aliquot division is a well-known resource of the violinist. All
component modes are excluded which have not a node at the
point touched ; so that, as regards pitch, the efl^ect is the same as
if the string were securely fastened there.
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127.]
NORMAL MODES-
185
127. The constants, which occur in the general value of y,
§ 124, depend on the special circumstances of the vibration, and
may be expressed in terms of the initial values of y and y.
Putting ^ = 0, we find
yo = X,^i il,sin j-\ . yo= y 2,-i «^# sin -y- .
(1).
STTX
Multiplying by sin— r-, and integrating fix)m to Z, we obtain
. 2 r^ . 8irx J ^ 2 r' . . sirx , .^.
A.^jj^y,^-^dx', B.^—j^yoSm-^dx (2).
These results exemplify Stokes' law, § 95 ; for that part of y, which
depends on the initial velocities, is
'^t-* 2 . STTX . swat r' . . STTX ,
and from this the part depending on initial displacements may
be inferred, by differentiating with respect to the time, and
substituting y^ for y«.
When the condition of the string at some one moment is
thoroughly known, these formulae allow us to calculate the
motion for all subsequent time. For example, let the string be
initially at rest, and so displaced that it forms two sides of a
triangle. Then J?« = ; and
Fig. 20.
. 2y(f^x, airx , f^ l—x . airx , )
27^
. STrh
V«»6(i-6)"'^~r
(3).
on integration.
We see that -4, vanishes, if sin (sirb/l) = 0, that is, if there be
a node of the component in question situated at P, A more
comprehensive view of the subject will be afforded by another
mode of solution to be given presently.
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186 TRANSVERSE VIBRATIONS OF STRINGS. [128.
128. In the expression for y the coefficients of sin (sirx/l) are
the normal co-ordinates of Chapters iv. and V. We will denote
them therefore by ^«, so that the configuration and motion of the
t^ystem at any instant are defined by the values of <^< and ^«
according to the equations
y = <^iSin-=- + ^aSm-y- + ...+<^fSm-^ + ...
^ ^ \ ......(1).
y = <pi8m-y + ^,8m -^+ ... + 4>«sm-2- + ... I
We proceed to form the expressions for T and F, and thence
to deduce the normal equations of vibration.
For the kinetic energy,
T = ip/^' y'cfo = ip £ {s;:: ^. sin *7}'(ir
= i/o j 2,^1 <^,» sm» -J- dx,
the product of every pair of terms vanishing by the general
property of normal co-ordinates. Hence
T = ipl'Z',l'^.'' (2).
In like manner,
sirx
^-i<®'^'H{'^:-:*-T
■=°»-r
' dx
r\
=mx::^<t>'' (3).
These expressions do not presuppose any particular motion, either
natural, or otherwise; but we may apply them to calculate the
whole energy of a string vibrating naturally, as follows : — ^If M
be the whole mass of the string (p[), and its equivalent (a*/t>) be
substituted for Tu we find for the sum of the energies,
2'+F=iif.2;:r{<^.'-.^^ (4),
or, in terms of Ag and Bg of § 126,
T+v^-.'ifX::^^^ (5)-
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128.] young's theorem. 187
If the motion be not confined to the plane of xy^ we have
merely to add the energ}' of the vibrations in the perpendicular
plane.
Lagrange's method gives immediately the equation of motion
*-+(?)"*-l*- »■
which has been already considered in § 66. If <f>o and ^o be the
initial values of <^ and <^, the general solution is
. • sinn^ .
9 = 4*0 ^ 9o cos nt
^^{^^n{t-^1f)^dlf (7),
Ifm jo
where n is written for STrajL
By definition 4>« is such that <!>« Btfyg represents the work done
by the impressed forces on .the displacement B<f>g. Hence, if the
force acting at time ^ on an element of the string pdxhe p Ydx,
<I>, = fpFsin?^(ir (8).
I
In these equations ^t is a linear quantity, as we see from (1); and
4>, is therefore a force of the ordinary kind.
129. In the applications that we have to make, the only
impressed force will be supposed to act in the immediate neigh-
bourhood of one point a=^b, and may usually be reckoned as
a whole, so that
<!>, = sin ^jpFda: (1).
If the point of application of the force coincide with a node of
the mode (*), <1>, = 0, and we learn that the force is altogether
without influence on the component in question. This principle
is of great importance ; it shews, for example, that if a string be
at rest in its position of equilibrium, no force applied at its centre,
whether in the form of plucking, striking, or bowing, can generate
any of the even normal components ^ If after the operation of
the force, its point of application be damped, as by touching it
1 The observation that a harmonio is not generated, when one of its nodal
points is placked, is dae to Tonng.
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188 TRANSVERSE VIBRATIONS OF STRINGS. [l29.
with the finger, all motion must forthwith cease ; for those com-
ponents which have not a node at the point in question are
stopped by the damping, and those which have, are absent from
the beginning^ More generally, by damping any point of a
sounding string, we stop all the component vibrations which have
not, and leave entirely unaffected those which have a node at the
point touched.
The case of a string pulled aside at one point and afterwards
let go from rest may be regarded as included in the preceding
statements. The complete solution may be obtained thus. Let
the motion commence at the time ^ = 0; from which moment
^s = 0. The value of <^, at time t is
<^« = (^«)ocosne + -(<^,)o8inn< (2),
where {<Pt)o» {^»)o denote the initial values of the quantities
affected with the suffix s. Now in the problem in hand (^«)« = 0,
and {(t>g)o is determined by
„.(<^.)..2^.= ^rsin?^ (3).
if Y' denote the force with which the string is held aside at the
point b. Hence at time t
y = rp^ -S^ «in -^sin -J- -^ (o).
and by (1) of §128
Ip
where n = 87ra/L
The S3rmmetry of the expression (5) in x and 6 is an example
of the principle of § 107.
The problem of determining the subsequent motion of a string
set into vibration by an impulse acting at the point 6, may be
treated in a similar manner. Integrating (6) of § 128 over the
duration of the impulse, we find ultimately, with the same nota-
tion as before,
... 2 . sirb ^
(</>,), = ^sm-y-F„
^ A like resnlt ensnes when the point which is damped is at the same distance
from one end of the string as the point of excitation is from the other end.
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129.]
INITIAL CONDITIONS.
189
if IT'dt be denoted by F,. At the same time (^,)o = 0, so that by
<2) at time t
.. 2F, ^. . mrb . nrx sinnf .
Ip
n
The series of component vibrations is less convergent for a struck
than for a plucked string, as the preceding expressions shew.
The reason is that in the latter case the initial value of y is
continuous, and only dy/dx discontinuous, while in the former it
is y itself that makes a sudden spring. See §§ 32, 101.
The problem of a string set in motion by an impulse may also
be solved by the general formulae (7) and (8) of § 128. The force
finds the string at rest at ^ = 0, and acts for an infinitely short
time fix>m ^ = to t = T\ Thus (^t)o and (<^,)o vanish, and (7)
of § 128 reduces to
while by (8) of §128
r^.dt'-
J
Hence, as before,
"^'^Ifn^^"**
J
sm
snrb
T
i:
rdf' = 8iD?y F,.
, 2 17. . 8wb . -
(7).
Hitherto we have supposed the disturbing force to be concen-
trated at a single point. If it be distributed over a distance ^
on either side of 6, we have only to integrate the expressions (6)
and (7) with respect to 6, substituting, for example, in (7) in place
of Yism{87rb/l),
Yi sm -=- db.
b-fi ^
If F/ be constant between the limits, this reduces to
/:
^,21 . sir 8 . sirb
Yi — sm —Y- sm -7-
sir I I
(8).
The principal effect of the distribution of the force is to render
the series for y more convergent.
130. The problem which will next engage our attention is
that of the pianoforte wire. The cause of the vibration is here
the blow of a hammer, which is projected against the string, and
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190
TRANSVERSE VIBRATIONS OF STRINGS.
[130.
after the impact rebounds. But we should not be justified in
assuming, as in the last section, that the mutual action occupies
so short a time that its duration may be neglected. Measured by
the standards of ordinary life the duration of the contact is indeed
very small, but here the proper comparison is with the natural
periods of the string. Now the hammers used to strike the wires
of a pianoforte are covered with several layers of cloth for the
express purpose of making them more yielding, with the eflTect of
prolonging the contact. The rigorous treatment of the problem
would be diiBcult, and the solution, when obtained, probably too
complicated to be of use ; but by introducing a certain simplifica-
tion Helmholtz has obtained a solution representing all the
essential features of the case. He remarks that since the actual
yielding of the string must be slight in comparison with that of
the covering of the hammer, the law of the force called into play
during the contact must be nearly the same as if the string were
absolutely fixed, in which case the force would vary very nearly as
a circular function. We shall therefore suppose that at the time
^ = 0, when there are neither velocities nor displacements, a force
Fsiapt begjns to act on the string &tx = b, and continues through
half a period of the circular function, that is, until t = 7r/p, after
which the string is once more free. The magnitude of p will
depend on the mass and elasticity of the hammer, but not to any
great extent on the velocity with which it strikes the string.
The required solution is at once obtained by substituting for
<J>, in the general formula (7) of § 128 its value given by
4>, = jF sin^sinpt' (1),
the range of the integration being from to ir/p. We find
(t>7r/p)
w
6. = 1 — sm -y- I &inn(t — t) sin pt at
^ Inp I Jo
nir
'.-i — 7-- — ^. .jFsm-v- .sm
"('-i)
.(2),
and the final solution for y becomes, if we substitute for n and p
their values,
aiT^a . sirb
(*f\Q Sin '^■^~
^apl'F ^»~m 2pl ' I . 8irx . 8Tra( ir\ .^.
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130.]
PIANOFORTE STRING.
191
We see that all components vanish which have a node at the
point of excitement, but this conclusion does not depend on any
particular law of force. The interest of the present solution lies
in the information that may be elicited from it as to the depend-
ence of the resulting vibrations on the duration of contact. If
we denote the ratio of this quantity to the fundamental period of
the string by v, so that v^ira: 2pl, the expression for the ampli-
tude of the component 8 is
.(4).
V cos {STTV) . «7r6
and
We fall back on the case of an impulse by putting i/=0,
/»/p
FBlTLptdt^
2F
P '
When J/ is finite, those components disappear, whose periods
are |, |, f, ... of the duration of contact; and when 8 is very
great, the series converges with s~'. Some allowance must also
be made for the finite breadth of the hammer, the effect of which
will also be to favour the convergence of the series.
The laws of the vibration of strings may be verified, at least
in their main features, by optical methods of observation — either
with the vibration-microscope, or by a tracing point recording the
character of the vibration on a revolving drum. This character
depends on two things, — the mode of exciten^ent, and the point
whose motion is selected for observation. Those components do
not appear which have nodes either at the point of excitement, or
at the point of observation. The former are not generated, and
the latter do not manifest themselves. Thus the simplest motion
is obtained by plucking the string at the centre, and observing
one of the points of trisection, or vice versa. In this case the
first harmonic which contaminates the purity of the principal
vibration is the fifth component, whose intensity is usually in-
sufficient to produce much disturbance.
[The dynamical theory of the vibration of strings may be
employed to test the laws of hearing, and the necessary experi-
ments are easily carried out upon a grand pianoforte. Having
freed a string, say c, from its damper by pressing the digital, pluck
it at one-third of its length. According to Young's theorem the
third component vibration is not excited then, and in corre-
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192
TRANSVERSE VIBRATIONS OP STRINGS.
[130.
spondence with that fact the ear fails to detect the component ^.
A slight displacement of the point plucked brings ^ in again;
and if a resonator {g'^ be used to assist the ear, it is only with
difficulty that the point can be hit with such precision as entirely
to extinguish the tone. Experiments of this kind shew that the
ear analyses the sound of a stiing into precisely the same con-
stituents as are found by sympathetic resonance, that is, into
simple tones, according to Ohm's definition of this conception.
Such experiments are also well adapted to shew that it is not a
mere play of imagination when we hear overtones, as some people
believe it is on hearing them for the first time\
If, after the string has been sounded loudly by striking the
digital, it be touched with the finger at one of the points of
trisection, all components are stopped except the 3rd, 6th, &c., so
that these are left isolated. The inexperienced observer is usually
surprised by the loudness of the residual sound, and begins to
appreciate the large part played by overtones.]
131. The case of a periodic force is included in the general
solution of § 128, but we prefer to follow a somewhat different
method, in order to make an extension in another direction* We
have hitherto taken no account of dissipative forces, but we will
now suppose that the motion of each element of the string is
resisted by a force proportional to its velocity. The partial
differential equation becomes
s;-s-s-^-
■d).
by means of which the subject may be treated. But it is still
simpler to avail ourselves of the results of the last chapter,
remarking that in the present case the dissipation-function F is
of the same form as T. In fact
F=ip/cLI,^i4>,r
.(2),
where 0i, <f>2,... are the normal co-ordinates, by means of which
T and V are reduced to sums of squares. The equations of
motion are therefore simply
2
4>i + fC(l>, + n^<f>t=-^<Ps (3),
1 Helmholtz, Ch. it. ; Brandt, Pogg. Ann., Vol. czii. p. 324, 1861.
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131.] FRICTION PROPORTIONAL TO VELOCITY. 193
of the same form as obtains for systems with but one degree of
freedom. It is only necessary to add to what was said in
Chapter in., that since sc ib independent of s, the natural vibra-
tions subside in such a manner that the amplitudes maintain their
relative values.
If a periodic force Fcoapt act at a single point, we have
<I>, = ^sin-,- Gospt (4),
and §46 0,= IJ^^^^T ^^(^^"^) (^)'
where tan€= ^ — r (6).
If among the natural vibrations there be any one nearly
isochronous with cos jot, then a large vibration of that type will
be forced, unless indeed the point of excitement should happen to
fall near a node. In the case of exact coincidence, the component
vibration in question vanishes ; for no force applied at a node can
generate it, under the present law of friction, which however, it
may be remarked, is very special in character. If there be no
friction, ic = 0, and
lp4>, = -~--~ sm-^- cos pt (/),
which would make the vibration infinite, in the case of perfect
isochronism, unless sin (sTrb/l) = 0.
The value of y is here, as usual,
y = ^sm-^ + 0a8m y- + ^,sm -/-+ (8).
132. The preceding solution is an example of the use of
normal co-ordinates in a problem of forced vibrations. It is of
course to free vibrations that they are more especially applicable,
and they may generally be used with advantage throughout,
whenever the system after the operation of various forces is
ultimately left to itself. Of this application we have already had
Examples.
In the case of vibrations due to periodic forces, one advantage
of the use of normal co-ordinates is the facility of comparison with
the equUxbrium theory^ which it will be remembered is the theory
R. 13
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194
TRANSVERSE VIBRATIONS OF STRINGS,
[132.
of the motion on the supposition that the inertia of the system
may be left out of account. If the value of the normal co-ordinate
0, on the equilibrium theory be A g cos pt, then the actual value
will be given by the equation
n^As
0, = -
^ cos pt.
.(1),
so that, when the result of the equilibrium theory is known and
can readily be expressed in terms of the normal co-ordinates, the
true solution with the effects of inertia included can at once be
written down.
In the present instance, if a force F cos pt of very long period
act at the point b of the string, the result of the equilibrium
theoiy, in accordance with which the string would at any moment
consist of two straight portions, will be
- ^ 2F . sirb
lp<i>.^^«ai-j-co&pt.
.(2),
from which the actual result for all values of p is derived by simply
writing (w* — p^) in place of n\
The value of y in this and similar cases may however be
expressed in finite terms, and the difficulty of obtaining the
finite expression is usually no greater than that of finding the
form of the normal functions when the system is free. Thus in
the equation of motion
dt' "" d^^^'
suppose that Y varies as cos mat The forced vibration will then
satisfy
S+-!'-l.r W
If F= 0, the investigation of the normal functions requires the
solution of
and a subsequent determination of m to suit the boundar}'' con-
ditions. In the problem of forced vibrations m is given, and we
have only to supplement any particular solution of (3) with the
complementary function containing two arbitrary constants. This
function, apart from the value of m and the ratio of the constants^
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132,] COMPARISON WITH EQUILIBRIUM THEORY. 195
is of the same form as the normal functions ; and all that remains to
be effected is the determination of the two constants in accordance
with the prescribed boundary conditions which the complete
solution must satisfy. Similar considerations apply in the case
of any continuous system.
133. If a periodic force be applied at a single point, there are
two distinct problems to be considered; the first, when at the
point ^ = 6, a given periodic force acts ; the second, when it is the
actual motion of the point b that is obligatory. But it will be
convenient to treat them together.
The usual differential equation
dl?^" dt " da^ ^^^'
is satisfied over both the parts into which the string is divided at
h, but is violated in crossing from one to the other.
In order to allow for a change in the arbitrary constants, we
must therefore assume distinct expressions for y, and afterwards
introduce the two conditions which must be satisfied at the point
of junction. These are
(1) That there is no discontinuous change in the value of y ;
(2) That the resultant of the tensions acting at h balances the
impressed force.
Thus, if J?* cos pt be the force, the second condition gives
r,A(^)+^cosp^=:0 (2),
where A{dy/dx) denotes the alteration in the value of dy/dx
incurred in crossing the point x=b in the positive direction.
We shall, however, find it advantageous to replace cospt by
the complex exponential 6*^^ and finally discard the imaginary
part, when the symbolical solution is completed. On the assump-
tion that y varies as e^^, the differential equation becomes
S+'^'^=<^ <3>'
where \* is the complex constant,
'^'=^(p'-tl'«)-
(*).
13—2
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196 TRANSVERSE VIBRATIONS OP STRINGS. [l33.
The most general solution of (3) consists of two terms, pro-
portional respectively to sinX^, and cosXa;; but the condition to
be satisfied at a^sO shews that the second does not occur here.
Hence if 7 6*** be the value of y at a? = 6,
sinXd? .^ ,^.
y^'^i^^-'^ <^>'
is the solution applying to the first part of the string from x—0
to x^b. In like manner it is evident that for the second part we
shall have
sm\(l'-x) .^
y-'^si^w^b)'^ <«>
If 7 be given, these equations constitute the symbolical solution
of the .problem ; but if it be the force that is given, we require
further to know the relation between it and 7.
Differentiation of (5) and (6) and substitution in the equation
analogous to (2) gives
F sinXfc si n \ (? — 6)
'^'T, \~si^id ^^^•
Thus
_ F si nXj; 8inX(t — 6) ^^^
*^""Fi XsinAi
from x^O to x^b
^Fein \(l — x) sin \b ^
^""Ti XsinXZ ^
...(sy.
from x=ib to x^l
These equations exemplify the general law of reciprocity
proved in the last chapter ; for it appears that the motion at x
due to the force at b is the same as would have. been found at i,
had the force acted at x.
In discussing the solution we will take first the case in which
there is no friction. The coefficient sc is then zero ; while X is
real, and equal to p/a. The real part of the solution, correspond-
ing to the force F cob pt, is found by simply putting cospt for ^
in (8), but it seems scarcely necessary to write the equations again
for the sake of so small a change. The same remark applies to
the forced motion given in terms of 7.
It appears that the motion becomes infinite in case the force
^ Donkin'8 Acoutticif p. 121.
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133,]
PERIODIC K)RCE AT ONE POINT.
197
is isochronous with one of the natural vibrations of the entire
string, unless the point of application be a node ; but in practice
it is not easy to arrange that a string shall be subject to a force
of given magnitude. Perhaps the best method would be to attach
a small mass of iron, attracted periodically by an electro-magnet,
-whose coils are traversed by an intermittent current. But unless
some means of compensation were devised, the mass would have
to be veiy small in order to avoid its inertia introducing a new
complication.
A better approximation may be obtained to the imposition of
an obligatory motion. A massive fork of low pitch, excited by
a bow or sustained in permanent operation by electro-magnetism,
executes its vibrations in approximate independence of the re-
actions of any light bodies which may be connected with it. In
order therefore to subject any point of a string to an obligatory
transverse motion, it is only necessary to attach it to the extremity
of one prong of such a fork, whose plane of vibration is perpendicular
to the length of the string. This method of exhibiting the forced
vibrations of a string appears to have been first used by Melded
Another arrangement, better adapted for aural observation,
has been employed by Helmholtz. The end of the stalk of a
powerful tuning-fork, set into vibration with a bow, or otherwise,
is pressed against the string. It is advisable to file the surface,
which comes into contact with the string, into a suitable (saddle-
shaped) form, the better to prevent slipping and jarring.
Referring to (5) we see that, if sin 7d> vanished, the motion
(according to this equation) would become infinite, which may be
taken to prove that in the case contemplated, the motion would
really become great, — so great that corrections, previously insigni-
ficant, rise into importance. Now sin X6 vanishes, when the force
is isochronous with one of the natural vibrations of the first part
of the string, supposed to be held fixed at and b.
When a fork is placed on the string of a monochord, or other
instrument properly provided with a sound-board, it is easy to
find by trial the places of maximum resonance. A very slight
displacement on either side entails a considerable falling off in the
volume of the sound. The points thus determined divide the
string into a number of equal parts, of such length that the
natural note of any one of them (when fixed at both ends) is
1 Pogg. Ann. cix. p. 198, 1869.
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198
TRANSVEESE VIBRATIONS OF STRINGS.
[133.
the same aa the note of the fork, as may readily be verified. The
important applications of resonance which Helmholtz has made to
purify a simple tone from extraneous accompaniment will occupy
OUT attention later.
134. Returning now to the general case where \ is complex,
we have to extract the real parts from (5), (6), (8) of § 133. Far
this purpose the sines which occur as factors, must be reduced to
the form Re^. Thus let
sinXa? = i2a.6**«..* (1),
with a like notation for the others. From (3) § 133 we shall thus
obtain
R
.(2),
from a? 8= to a? = 6,
and from (6) § 133
R
y = 7 1^ cos (pt + e,-x - €i^\
from x — btox = l,
corresponding to the obligatory motion y = y cospt at b.
By a similar process from (8) § 133, if
X = a + i/3 (3),
we should obtain
from a? = to a7 = 6
y=r, V(«'+/3')'ig^ '''"(^*"^'"^'^'^"""^°"'^^^°^)
}-•(*).
from a? = 6 to a? = ?
corresponding to the impressed force F cos pt at 6. It remains to
obtain the forms of Rx, e^, &c.
The values of o and fi are determined by
«»-^ = g. 2«^ = -f (5).
and sin Xo; = sin our cos ifix + cos ax sin %/3x
= sin ax s 1- 1 cos ax ^ .
2
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134.J
FRICTION PROPOBTIONAL TO VELOCITY.
199
SO that
R,- = siu*ax {^'■^^--')\ci^*wc ( ^Y " )'••• (6)'
tancx
while
cot cu;
V(a»-h)8»)=:--y(p^ +;)>)«»)
(7).
(8).
This completes the solution.
If the friction be very small, the expressions may be simpli-
fied. For instance, in this case, to a sufficient approximation,
so that, corresponding to the obligatory motion at 6 y = 7 cos/>^, the
amplitude of the motion between ^ = and a; = 6 is, approximately
a 4a* c
.\pb . #c^6* .pb
n. iLn* n.
(9),
4a'
which becomes great, but not infinite, when sin {phia) = 0, or the
point of application is a node.
If the imposed force, or motion, be not expressed by a single
harmonic term, it must first be resolved into such. The preceding
solution may then be applied to each component separately, and
the results added together. The extension to the case of more than
one point of application of the impressed forces is also obvious.
To obtain the most general solution satisfying the conditions, the
expression for the natural vibrations must also be added; but
these become reduced to insignificance after the motion has been
in progress for a sufficient time.
The law of friction assumed in the preceding investigation is
the only one whose results can be easily followed deductively, and
it is sufficient to give a general idea of the effects of dissipative
forces on the motion of a string. But in other respects the con-
clusions drawn from it possess a fictitious simplicity, depending on
the fact that F — the dissipation-function — is similar in form to T,
which makes the normal co-ordinates independent of each other.
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200
TRANSVERSE VIBRATIONS OF STRINGS.
[134.
Id almofifc any other case (for example, when but a single point of
the string is retarded by friction) there are no normal co-ordinates
properly so called. There exist indeed elementary types of vibra-
tion into which the motion may be resolved, and which are
perfectly independent, but these are essentially different in cha-
racter from those with which we have been concerned hitherto, for
the various parts of the system (as affected by one elementary
vibration) are not simultaneously in the same phase. Special cases
excepted, no linear transformation of the co-ordinates (with real
coefficients) can reduce T, F, and V together to a sum of
squares.
If we suppose that the string has no inertia, so that 7^ = 0,
F and V may then be reduced to sums of squares. This problem
is of no acoustical importance, but it is interesting as being
mathematically analogous to that of the conduction and radiation
of heat in a bar whose ends are maintained at a constant tem-
perature.
136. Thus far we have supposed that at two fixed points,
^ = and x^l, the string is held at rest. Since absolute fixity
cannot be attained in practice, it is not without interest to inquire
in what manner the vibrations of a string are liable to be modified
by a yielding of the points of attachment; and the problem
will furnish occasion for one or two remarks of importance.
For the sake of simplicity we shall suppose that the system is
symmetrical with reference to the centre of the string, and that
each extremity is attached to a mass M (treated as unextended in
space), and is urged by a spring (/i) towards the position of equi-
librium. If no fiictional forces act, the motion is necessarily
resolvable into normal vibrations. Assume
y = {a sin wkc + i8 cos 7?w;} cos(ma^- e) (1).
The conditions at the ends are that
when a? = 0, Jlfy + /iy= 2\^]
7^ (2),
when a? = Z, My-\-fiy^-T^^]
which give
?. — ^ tan mZ — a _ M — Mdhrt} , .
/8~atanmZ + /S mT7~ ^
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135.] EXTREMITIES SUBJECT TO YIELDING. 201
two equations, sufiScient to detennine m, and the ratio of )8 to a.
Eliminating the latter ratio, we find
tan mi as y-— J (4),
if for brevity we write v for jp .
Equation (3) has an infinite number of roots, which may be
found by writing tan for v, so that tan ml == tan 20, and the result
of adding together all the corresponding particular solutions, each
with its two arbitrary constants a and e, is necessarily the most
general solution of which the problem is capable, and is therefore
adequate to represent the motion due to an arbitrary initial dis-
tribution of displacement and velocity. We infer that any function
of X may be expanded between x^Q and ^ = Z in a series of terms
^ (vi ainmi^ + cos rti^x) + 0, {v^ sin m^ + cos m^) -h (5),
mi, 772,, &c. being the roots of (3) and y^, v^, &c. the corresponding
values of v. The quantities ^i, ^, &c. are the normal co-ordinates
of the system.
From the symmetry of the sjrstem it follows that in each
normal vibration the value of y is numerically the same at points
equally distant from the micldle of the string, for example, at the
two ends, where a? = and x^L Hence v, sin mj, H- cos mJL = ± 1 ,
as may be proved also from (4). •
The kinetic energy T of the whole motion is made up of the
energy of the string, and that of the masses M, Thus
T^\p\ \%^(ymimx-\-Go&ma)Ydx
J
+ iJf {<^ -h </>,+ ...}» + iJf {<^i (i/i sin miZ -h cos miO + ... 1^
But by the characteristic property of normal co-ordinates, terms
containing their products cannot be really present in the expres-
sion for Ty so that
p I (Pr sin mfX + cos mrx) (i/, sin mgX + cos myx) dx
J
+ M + M(vrsmmfl -^ coswrl) (vgsijimgl + cosm^ = (6),
if r and 8 be different.
This theorem suggests how to determine the arbitrary con-
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202 TRANSVERSE VIBRATIONS OF STRINGS. [135.
\-
fitants, so that the series (5) may represent an arbitrary function
y. Take the- expression
pi y{vt sin m^ + cos m^)dx + My^ + Myi (v, sin m^ + cos m/). . .(7),
and substitute in it the series (5) expressing y. The result is a
series of terms of the type
pi ^r {vr BiT^ 7n^ + cos inja) {pg sin msX + cos m^)dx
J
+ M(l>r + Jf0r (j'r siu rWyZ + COS TTifl) (vg sin m,? + COS mj^),
all of which vanish by (6), except the one for which r=«. Hence
<f>g is equal to the expression (7) divided by
p I (v,sinm,a? + cosmja:)^da? + Jf+ Jf(i/,sin7?iaZ + cosm^)^..(8),
J
and thus the coeflScients of the series are determined. If Jf = 0,
even although fi be finite, the process is of course much simpler,
but the unrestricted problem is instructive. So much stress is
often laid on special proofs of Fourier's and Laplace's series, that
the student is apt to acquire too contracted a view of the nature
of those important results of analysis.
We shall now shew how Fourier's theorem in its general form
can be deduced from our present investigation. Let ilf = ; then
if /i = 00 , the ends of the string are fast, and the equation de-
termining m becomes tan ml = 0, or r)il = sw, as we know it must
be. In this case the series for y becomes
, . irx . . 27ra? . . Zirx ,^.
y = -dism-y- + ^jsm -y- +-d88m-y- + (9),
which must be general enough to represent any arbitrary functions
of X, vanishing at and I, between those limits. But now suppose
that /i is zero, M still vanishing. The ends of the string may be
supposed capable of sliding on two smooth rails perpendicular to
its length, and the terminal condition is the vanishing of dyjdx.
The equation in m is the same as before; and we learn that any
function y whose rates of variation vanish' at x=^Q and x^l, can
be expanded in a series
y =5iC08-y +-BjCos ,- +B3C0S-J- + (10).
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135.]
FOURIER S THEOREM.
203
This series remains unaffected when the sign of a; is changed,
and the first series merely changes sign without altering its
numerical magnitude. If therefore y^ be an even function of x,
(10) represents it firom —Ito + l. And in the same way, if y be
an odd function of x, (9) represents it between the same limita
Now, whatever function of x (j) (x) may be, it can be divided
into two parts, one of which is even, and the other odd, thus :
^ W 2 ■*■ 2 '
so that, if <^ (x) he such that (- i) = ^ (+ 1) and ^' (- 1) = i\> (+ 0,
it can be represented between the limits ± 2 by the mixed series
-d^sm-j- + J^i cos -J- + -4j sm -, +52C0s-t— + (11).
This series is periodic, with the period 21. If therefore <f> (x)
possess the same property, no matter what in other respects its
character may be, the series is its complete equivalent. This is
Fourier's theorem \
We now proceed to examine the effects of a slight yielding of
the supports, in the case of a string whose ends are approximately
fixed The quantity v may be great, either through fj, or through
M. We shall confine ourselves to the two principal cases, (1)
when /i is great and M vanishes, (2) when /i vanishes and M is
great.
In the first case
"^rlm'
and the equation in m is approximately
, 2 2T,m
tan m6 = — = .
Assume ml = 87r + x, where x is small ; then
22\.«ir
• tan a? = — -
^i
and
ml = sir
{^-'^
approximately,
.(12).
^ The befit * system * for proving Fourier's theorem from dynamical oonsidera-
tions is an endless chain stretched round a smooth cylinder (§ 189), or a thin
re-entrant column of air enclosed in a ring-shaped tube.
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204 TRANSVERSE VIBRATIONS OF STRINGS. [135.
To this order of approximation the tones do not cease to form
a harmonic scale, but the pitch of the whole is slightly lowered.
The effect of the yielding is in &ct the same as that of an increase
in the length of the string in the ratio 1 : IH ^ , as might
have been anticipated.
The result is otherwise if fi vanish, while M is great. Here
T,
2T
and tan mf = ^ ^ approximately.
Hence m? = ^tt + , , , ^ — (13).
Ma^ .air
The effect is thus equiva,lent to a decrease in I in the ratio
and consequently there is a rise in pitch, the rise being the
greater the lower the component tone. It might be thought
that any kind of yielding would depress the pitch of the string,
but the preceding investigation shews that this is not the case.
Whether the pitch will be raised or lowered, depends on the
sign of v^ and this again depends on whether the natural note of
the mass M urged by the spring fi is lower or higher than that of
the pomponent vibration in question.
136. The problem of an otherwise uniform string carrying
a finite load M B,t x — h can be solved by the formulae investigated
in § 133. For, if the force F cos pt be due to the reaction against
acceleration of the mass if,
F^yP'M. (1),
which combined with equation (7) of § 133 gives, to determine the
possible values of X (or p : a),
a^MXsin-Kb sin X (i - 6) = 2\sinXi (2).
The value of y for any normal vibration corresponding to X is
y = P sin Xa? sin X (Z — 6) cos (aXt — e) '
from ic = to a? = 6
y = P sin X (? — a;) sin X6 cos (aX^ — e)
from x^b to x = l
where P and € are arbitrary constants.
(3),
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136.] FINITE LOAD. 205
It does not require analysis to prove that any normal com-
ponents which have a node at the point of attachment are un-
afiFected by the presence of the load. For instance, if a string be
weighted at the centre, its component vibrations of even orders
remain unchanged, while all the odd components are depressed in
pitch. Advantage may sometimes be taken of this effect of a
load, when it is desired for any purpose to disturb the harmonic
relation of the component tones.
If Jlf be very great, the gravest component is widely sepa-
rated in pitch from all the others. We will take the case when
the load is at the centre, so that & = Z — &==^{. The equation in
X then becomes
. U (\Z \l pl\ ^ ...
^2"-l"2*^2-"5fp^ ^^)'
where pi : M, denoting the ratio of the masses of the string and
the load, is a small quantity which may be called o^. The first
root corresponding to the tone of lowest pitch occurs when ^X{ is
small, and such that
(i>i)«{l4i(i>i)*} = a» nearly,
whence JXZ = a (1 — J a'),
and the periodic time is given by
'ir
v^(><i) <'>■
The second term constitutes a correction to the rough value
obtained in a previous chapter (§ 52), by neglecting the inertia of
the string altogether. That it would be additive might have
been expected, and indeed the formula as it stands may be ob-
tained from the consideration that in the actual vibration the two
parts of the string are nearly straight, and may be assumed to be
exactly so in computing the kinetic and potential energies, with-
out entailing any appreciable error in the calculated period. On
this supposition the retention of the inertia of the string increases
the kinetic energy corresponding to a given velocity of the load in
the ratio of Jf : Jf+ J^i, which leads to the above result. This
method has indeed the advantage in one respect, as it might be
applied when p is not uniform, or nearly uniform. All that is
necessary is that the load M should be sufficiently predominant.
There is no other root of (4), until sin \\l = 0, which gives
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206 TRANSVERSE VIBRATIONS OF STRINGS, [136.
the second component of the string, — a vibration independent of
the load. The roots after the first occur in closely contiguous
pairs; for one set is given by ^Xl^sir, and the other approxi-
mately by jXZ = «7r+— ^, in which the second term is small
The two types of vibration for « = 1 are shewn in the figure.
Fig. 21.
^^
The general formula (2) may also be applied to find the efibct
of a small load on the pitch of the various components.
137. Actual strings and wires are not perfectly flexible.
They oppose a certain resistance to bending, which may be divided
into two parts, producing two distinct effects. The first is called
viscosity, and shews itself by damping the vibrations. This part
produces no sensible effect on the periods. The second is con-
servative in its character, and contributes to the potential eneray
of the system, with the effect of shortening the periods. A com-
plete investigation cannot conveniently be given here, but the
case which is most interesting in its application to musical
instruments, admits of a sufficiently simple treatment.
When rigidity is taken into account, something more must be
specified with respect to the terminal conditions than that v
vanishes. Two cases may be particularly noted : —
(i) When the ends are clamped, so that dy/dx = at the ends.
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137.]
CORRECTION FOR RIGIDITY.
207
(ii) When the termiDal directions are perfectly free, in which
case dh/jda? = 0.
It is the latter which we propose now to consider.
If there were no rigidity, the type of vibration would be
y X sin , - , satisfying the second condition.
The eflFect of the rigidity might be slightly to disturb the t)rpe;
but whether such a result occur or not, the period calculated
from the potential and kinetic energies on the supposition that
the type remains unaltered is necessarily correct as far as the first
order of small quantities (§ 88).
Now the potential energy due to the stiffness is expressed by
^'m'6.
8F
'^^Lwj
•(1).
where £ is a quantity depending on the nature of the material
and on the form of the section in a manner that we are not now
prepared to examine. The forrn of SF is evident, because the force
required to bend any element ds is proportional to ds, and to the
amount of bending already effected, that is to ds/p. The whole
work which must be done to produce a curvature l/p in da is
therefore proportional to ds/p^; while to the approximation to
which we work ds^dx, and l//) = d*y/(ic".
sirx
•(2).
Thus, if y = ^ sin -^
and the period of ^ is given by
T = T,^^1-22T p)
if To denote what the period would become if the string were
endowed with perfect flexibility. It appears that the effect of the
stiflhesa increases rapidly with the order of the component vibra-
tions, which cease to belong to a harmonic scale. However, in the
strings employed in music, the tension is usually sufiScient ta
reduce the influence of rigidity to insignificance.
The method of this section cannot be applied without modifi-
cation to the other case of terminal condition, namely, when the
ends are plamped. In their immediate neighbourhood the type of
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208 TRANSVBBSE VIBRATIONS OF STRINGS. [137.
vibration must differ from that assumed by a perfectly flexible
string by a quantity, which is no longer small, and whose square
therefore cannot be neglected. We shall return to this subject,
when treating of the transverse vibrations of rods.
138. There is one problem relating to the vibrations of strings
which we have not yet considered, but which is of some practical
interest, namely, the character of the motion of a violin (or cello)
string under the action of the bow. In this problem the tnodus
operandi of the bow is not sufficiently understood to allow us to
follow exclusively the a priori method : the indications of theoiy
must be supplemented by special observation. By a dexterous
combination of evidence drawn from both sources Helmholtz has
succeeded in determining the principal features of the case, but
some of the details are still obscure.
Since the note of a good instrument, well handled, is musical,
we infer that the vibrations are strictly periodic, or at least that
strict periodicity is the ideal. Moreover — and this is very import-
ant — the note elicited by the bow has nearly, or quite, the same
pitch as the natural note of the string. The vibrations, although
forced, are thus in some sense free. They are wholly dependent
for their maintenance on the energy drawn from the bow, and yet
the bow does not determine, or even sensibly modify, their periods.
We are reminded of the self-acting electrical interrupter, whose
motion is indeed forced in the technical sense, but has that kind
of freedom which consists in determining (wholly, or in part) under
what influences it shall come.
But it does not at once follow from the fact that the string
vibrates with its natural periods, that it conforms to its natural
tjrpes. If the coefficients of the Fourier expansion
y = <^ism -^ + <^,8m-j-+
be taken as the independent co-ordinates by which the configura-
tion of the system is at any moment defined, we know that when
there is no friction, or friction such that Fee T, the natural vibra-
tions are expressed by making each co-ordinate a simple harmonic
(or quasi-harmonic) function of the time ; while, for all that has
hitherto appeared to the contrary, each co-ordinate in the present
case might be any function of the time periodic in time r. But a
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138.] VIOLIN STRING. 209
little examination will shew that the vibrations must be sensibly
natural in their tjrpes as well as in their periods.
The force exercised by the bow at its point of application may
"be expressed by
F= %Ar COS ( Crj ]
so that the equation of motion for the co-ordinate ^« is
^.+ «^#+ — p- <^, = ^sm -^ . 2^^ cos ^-^ - €rj ,
h being the point of application. Each of the component parts of
4>, will give a corresponding term of its own period in the solu-
tion, but the one whose period is the same as the natural period
of ^t will rise enormously in relative importance. Practically then,
if the damping be small, we need only retain that part of ^,
which depends on J., cos ( e,j , that is to say, we may regard
the vibrations as natural in their typea
Another material fact, supported by evidence drawn both from
theory and aural observation, is this. All component vibrations
are absent which have a node at the point of excitation. "In
order, however, to extinguish these tones, it is necessary that the
coincidence of the point of application of the bow with the node
should be very exact. A very small deviation reproduces the
missing tones with considerable strengths"
The remainder of the evidence on which Helmholtz' theory
rests, was derived from direct observation with the vibration-
microscope. As explained in Chapter li., this instrument affords
a view of the curve representing the motion of the point under
observation, as it would be seen traced on the surface of a trans-
parent cylinder. In order to deduce the representative curve in
its ordinary form, the imaginary cylinder must be conceived to
be unrolled, or developed, into a plane.
The simplest results are obtained when the bow is applied at a
node of one of the higher components, and the point observed is
one of the other nodes of the same system. If the bow work
fairly so as to draw out the fundamental tone clearly and strongly,
the representative curve is that shewn in figure 22; where the
1 Donkin's Acovtvct^ p. 131.
14
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210 TRANSVERSE VIBRATIONS OF STRINGS. [l38.
abscissa} correspond to the time (AB being a complete period),
and the ordinates represent the displacement. The remarkable
Fig. 22.
fact is disclosed that the whole period t may be divided into two
parts To and t — To, during each of which the velocity of the
observed point is constant ; but the velocities to and fro are in
general unequal.
We have now to represent this curve by a series of harmonic
terms. If the origin of time correspond to the point A, and
AD^ FC = 7, Fourier's theorem gives
y=——/ .2 -sm— sm- It'-^] (1).
^ tt'to (t - To) •-! «* T T V 2/ ^ ^
With respect to the value of t©, we know that all those com-
ponents of y must vanish for which sin {sirx^jl) = {x^ being the
point of observation), because under the circumstances of the case
the bow cannot generate them. There is therefore i*eason to
suppose that Tq:t^x^:1\ and in fact observation proves that
AC : CB (in the figure) is equal to the ratio of the two parts into
which the string is divided by the point of observation.
Now the free vibrations of the string are represented in
general by
y = 2,-ism ^ j^,cos — +5,sm-^-k
and this at the point x^a-^ must agree with (1). For convenience
of comparison, we may write
2sTrt . n . 28Trt ri 257r/
. 2fi7r< , o . 2«7re ^, 257r/^ To\
-A, cos + jB,sm = C^cos — {^— o
T T T \ ^/
and it then appears that G^ = 0.
+ ABin^(<-t),
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138.] VIOLIN STRING. 211
We find also to determine D,
sin — y- . Dg - —1 — / 7 - sm -7- ,
whence
^•-^t.(t-t,)«' ^^^'
unless sin (sTrXo/l) = 0.
In the case reserved, the comparison leaves D, undetermined,
but we know on other grounds that Dg then vanishes. However,
for the sake of simplicity, we shall suppose for the present that
Dg is always given by (2). If the point of application of the bow
do not coincide with a node of any of the lower components, the
error committed will be of no great consequence.
On this understanding the complete solution of the problem is
The amplitudes of the components are therefore proportional to 5"^.
In the case of a plucked string we found for the corresponding
function 5^8in(s7r6/Z), which is somewhat similar. If the string
be plucked at the middle, the even components vanish, but the
odd ones follow the same law as obtains for a violin string. The
equation (3) indicates that the string is always in the form of two
straight lines meeting at an angle. In order more conveniently
to shew this, let us change the origin of the time, and the constant
multiplier, so that
y=^2-,sm-^sm~;^ (4)
will be the equation expressing the form of the string at any time.
Now we know (§ 127) that the equation of the pair of lines
proceeding from the fixed ends of the string, and meeting at a
point whose co-ordinates are o, )8, is
28fi ^ 1 . sva . sirx
Thus at the time t, (4) represents such a pair of lines, meeting at
the point whose co-ordinates are given by
a{l-
. STTO
Sin ^
= ±
. 2«7rf
sin
T
14—2
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212 TRANSVERSE VIBRATIONS OF STRINGS. [l 38.
These equations indicate that the projection on the axis of x
of the point of intersection moves uniformly backwards and
forwards between a? = and a? = Z, and that the point of inter-
section itself is situated on one or other of two parabolic arcs,
of which the equilibrium position of the string is a common
chord.
Since the motion of the string as thus defined by that of the
point of intersection of its two straight parts, has no especial
relation to x^ (the point of observation), it follows that, according
to these equations, the same kind of motion might be observed at
any other point. And this is approximately true. But the theo-
retical result, it will be remembered, was only obtained by as-
suming the presence in certain proportions of component vibrations
having nodes at x^, though in fact their absence is required by
mechanical laws. The presence or absence of these components is
a matter of indifference when a node is the point of observation,
but not in any other case. When the node is departed from, the
vibration curve shews a series of ripples, due to the absence of
the components in question. Some further Retails will be found
in Helmholtz and Donkin.
The sustaining power of the bow depends upon the fact that
solid friction is less at moderate than at small velocities, so that
when the part of the string acted upon is moving with the bow
(not improbably at the same velocity), the mutual action is greater
than when the string is moving in the opposite direction with
a greater relative velocity. The accelerating effect in the first
part of the motion is thus not entirely neutralised by the sub-
sequent retardation, and an outstanding acceleration remains
capable of maintaining the vibration in spite of other losses of
energy. A curious effect of the same peculiarity of solid friction
has been observed by W. Froude, who found that the vibrations
of a pendulum swinging from a shaft might be maintained or
even increased by causing the shaft to rotate.
[Another case in which the vibrations of a string are main-
tained is that of the Aeolian Harp. It has often been suggested
that the action of the wind is analogous to that of a bow ; but the
analogy is disproved by the observation^ that the vibrations are
executed in a plane iransfoerse to the direction of the wind. The
true explanation involves hydrodynamical theory not yet de-
veloped.]
1 VHL Mag., March, 1879, p. 161.
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139.] STRINGS STRETCHED ON CURVED SURFACES. 213
139. A string stretched on a smooth curved surface will in
equilibrium lie along a geodesic line, and, subject to certain con-
ditions of stability, will vibrate about this configuration, if dis-
placed. The simplest case that can be proposed is when the
surface is a cylinder of any form, and the equilibrium position
of the string is perpendicular to the generating lines. The student
will easily prove that the motion is independent of the curvature
of the cylinder, and that the vibrations are in all essential respects
the same as if the surface were developed into a plane. The case
of an endless string, forming a necklace round the cylinder, is
worthy of notice.
In order to illustrate the characteristic features of this class of
problems, we will take the comparatively simple example of a
string stretched on the surface of a smooth sphere, and lying,
when in equilibrium, along a great circle. The co-ordinates to
which it will be most convenient to refer the system are the
latitude measured from the great circle as equator, and the
longitude ^ measured along it. If the radius of the sphere be a,
we have
T^\\p{adYad4> = '^ld'di> (1).
The extension of the string is denoted by
su that
Now
d** = (odd)* + (o cos tf d^y ;
Thus
and
^ 1 \(^^W ./il* 1 l{d0\* 0* . .,
If the ends be fixed,
1 Cambridge Mathematical Tripos Examination, 1876.
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■^
214 TRANSVERSE VIBRATIONS OP STRINGS. [139.
and the equation of virtual velocities is
aV f'eSed4>-aT,fB0(j^^'\-e)d(l>=^O,
whence, since Bd is arbitrary,
•«'''^=^>(|^.+^) (^>
This is the equation of motion.
If we assume x cosjp^, we get
^+«-fv«-» <*).
of which the solution, subject to the condition that vani^faes
with <l>, is
tf = 4 8in|^^p»-f llV.cos;)^ (5).
The remaining condition to be satisfied is that vanishes when
o^ = Z, or ^ = a, if a = l/a.
This gives
where m is an integer.
The normal functions are thus of the same form as for a
straight string, viz.
tf = i4 sm — - co&pt (/),
but the series of periods is diflferent. The effect of the curvature
is to make each tone graver than the corresponding tone of a
straight string. If a > w, one at least of the values of jp' is nega-
tive, indicating that the corresponding modes are unstable. If
« = "W", jpi is zero, the string being of the same length in the dis-
placed position, as when ^ = 0.
A similar method might be applied to calculate the motion of
a string stretched round the equator of any surface of revolu-
tion^
140. The approximate solution of the problem for a vibrating
string of nearly but not quite uniform longitudinal density has been
fully considered in Chapter I v. § 91, as a convenient example of
> [For a more general treatment of this question see MioheU, Menenger of
Mathematics t vol. xiz. p. 87, 1890.]
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140.] VARIABLE DENSITY. 215
the general theory of approximately simple systems. It will be
sufficient here to repeat the result. If the density be p© + Sp» the
period t^ of the r^ component vibration is given by
If the irregularity take the form of a small load of mass m
at the point a; = 6, the formula may be written
"-ti>-ft-°-t! <^^
These values of r* are correct as far as the first power of the
small quantities hp and m, and give the means of calculating a
correction for such slight departures from uniformity as must
always occur in practice.
As might be expected, the effect of a small load vanishes at
nodes, and rises to a maximum at the points midway between
consecutive nodea When it is desired merely to make a rough
estimate of the effective density of a nearly uniform string, the
formula indicates that attention is to be given to the neighbour-
hood of loops rather than to that of nodes.
[The effect of a small variation of density upon the period is
the same whether it occur at a distance x from one end of the
string, or at an equal distance from the other end. The mea^}
variation at points equidistant from the centre is all that we need
regard, and thus no generality will be lost if we suppose that the
density remains symmetrically distributed with respect to the centre.
Thus we may write
T.» = ^^l + a,) (3)
where oir = jj — (l — cos— , \dx (4).
In this equation Sp may be expanded from to ^l in the series
-^ = ^0 + ^1 cos — ,- + ... -f -4rC0S— ^— -f (o),
Po ^ fr
where
A.-'jf'^d. (6),
. 4 r*' So 2Trrx , ,^^
Ar = j -^cos — j-dx (7).
IJo po I
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216 TRANSVERSE VIBRATIONS OP STRINGS. [l40.
Accordingly,
Or^Ao-iAr (8),
This equation, as it stands, gives the changes in period in
terms of the changes of density supposed to be known. And
it shews conversely that a variation of density may alwa3r8 be
found which will give prescribed arbitrary displacements to all
the periods. This is a point of some intei'est.
In order to secure a reasonable continuity in the density, it is
necessary to suppose that cti, a, . . . are so prescribed that Or assumes
ultimately a constant value when r is increased indefinitely. If
this condition be satisfied, we may take ilo ^ ««> a«nd then Ar tends
to zero as r increases.
As a simple example, suppose that it be required so to vary
the density of a string that, while the pitch of the fundamental
tone is displaced, all other tones shall remain unaltered. The
conditions give
Accordingly
and -4i = — 2ai.
Thus by (6)
Sp/pQ = — 2ai cos (iirx/l),]
141. The differential equation determining the motion of a
string, whose longitudinal density p is variable, is
^ dt^-^'d^ ^^>'
from which, if we assume y x cosjp^, we obtain to determine the
normal ftinctions
%^^f^ = ^ (2)>
where i^ is written for jpY^i. This equation is of the second
order and linear, but has not hitherto been solved in finite terma
Considered as defining the curve assumed by the string in the
normal mode under consideration, it determines the curvature at
any point, and accordingly embodies a rule by which the curve
can be constructed graphically. Thus in the application to a
stiing fixed at both ends, if we start fix)m either end at an arbitrary
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141.] VARIABLE DENSITY. 217
inclination, and with zero curvature, we are always directed by the
equation with what curvature to proceed, and in this way we
may trace out the entire curve.
If the assumed value of i/* be right, the curve will cross
the axis of x at the required distance, and the law of vibration
will be completely determined. If i/* be not known, different
values may be tried until the curve ends rightly; a sufficient
approximation to the value of v^ may usually be arrived at by a
calculation founded on an assumed type (^ 88, 90).
Whether the longitudinal density be uniform or not, the
periodic time of any simple vibration varies costeris paribus as the
square root of the density and inversely as the square root of the
tension under which the motion takes place.
The converse problem of determining the density, when the
period and the type of vibration are given, is always soluble. For
this purpose it is only necessary to substitute the given value of y,
and of its second differential coefficient in equation (2). Unless
the density be infinite, the extremities of a string are points of
zero curvature.
When a given string is shortened, every component tone is
raised in pitch. For the new state of things may be regarded as
derived fi:x)m the old by introduction, at the proposed point of
fixture, of a spring (without inertia), whose stiffiiess is gradually
increased without limit. At each step of the process the potential
energy of a given deformation is augmented, and therefore (§ 88)
the pitch of every tone is raised. In like manner an addition to
the length of a string depresses the pitch, even though the added
part be destitute of inertia.
142. Although a general integration of equation (2) of § 141
is beyond our powers, we may apply to the problem some of the
many interesting properties of the solution of the linear equation
of the second order, which have been demonstrated by MM. Sturm
and Liouville^ It is impossible in this work to give anjrthing
like a complete account of their investigations ; but a sketch, in
which the leading features are included, may be found interesting,
and will throw light on some points connected with the general
^ The memoirs referred to in the text are contained in the first volame of
liioaville's Journal (1836).
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218 TRANSVERSE VIBRATIONS OF STRINGS. [142.
theory of the vibrations of continuous bodies. I have not thought
it necessary to adhere very closely to the methods adopted in the
original memoirs.
At no point of the curve satisfjdng the equation
S + '^''2/ = o (1).
can both y and dyjdx vanish together. By successive differen-
tiations of (1) it is easy to prove that, if y and dyjdx vanish
simultaneously, all the higher differential coeflScients dh/lda?y
d^y/dafy &c. must also vanish at the same point, and therefore
by Taylor's theorem the curve must coincide with the axis of x.
Whatever value be ascribed to i^, the curve satisfying (1) is
sinuous, being concave throughout towards the axis of x, since
p is everywhere positive. If at the origin y vanish, and dy/dx
be positive, the ordinate will remain positive for all values of x
below a certain limit dependent on the value ascribed to v*.
If i;* be very small, the curvature is slight, and the curve will
remain on the positive side of the axis for a great distance. We
have now to prove that as i/* increases, all the values of x which
satisfy the equation y = gradually diminish in magnituda
Let y' be the ordinate of a second curve satisfying the equa-
tion
% + ^'W-O ...(2).
as well as the condition that y' vanishes at the origin, and let us
suppose that i/'* is somewhat greater than i/*. Multipljring (2) by
y, and (1) by y\ subtracting, and integrating with respect to x
between the limits and x, we obtain, since y and y' both vanish
with X,
y'f.-y%=<''"-''^l?yy''^ <^>-
If we further suppose that x corresponds to a point at which
y vanishes, and that the difference between p'^ and i^ is very small,
we get ultimately
y't-^^'iy^'^ w-
The right-hand member of (4) being essentially positive, we
learn that y and dy/dx are of the same sign, and therefore that,
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142.] Sturm's theorem. 219
whether dyjdx be positive or negative, y' is already of the same
sign as that to which y is changing, or in other words, the value
of X for which y vanishes is less than that for which y vanishes.
If we fix our attention on the portion of the curve lying
between d? = and a?=Z, the ordinate continues positive through-
out as the value of i^ increases, until a certain value is attained,
which we will call v^. The function y is now identical in form
with the first normal function t^ of a string of density p fixed
at and I, and has no root except at those points. As i^ again
increases, the first root moves inwards from x=^l until, when a
second special value v^ is attained, the curve again crosses the
axis at the point a; = Z, and then represents the second normal
function u^. This function has thus one internal root, and one
only. In like manner corresponding to a higher value v^ we
obtain the third normal function u^ ^with two internal roots, and
so on. The n^^ function u^ has thus exactly n— 1 internal roots, and
since its first differential coefficient never vanishes simultaneously
with the frmction, it changes sign each time a root is passed.
From equation (3) it appears that if Ur and u, be two diflferent
normal functions,
pUrUgdx=0 (.5).
/:
A beautiful theorem has been discovered by Sturm relating
to the number of the roots of a function derived by addition
from a finite number of normal functions. If Um be the component
of lowest order, and v^ the component of highest order, the function
f{x)=^<f)mUm + <f>m+ill-m+i't -\r <l>nUn (6),
where <f>m, (f>m+i, &c. are arbitrary coefficients, has at least m — 1
internal roots, and at most n — l internal roots. The extremities
B,t x=^0 and a,t x = l correspond of course to roots in all cases.
The following demonstration bears some resemblance to that given
by Liouville, but is considerably simpler, and, I believe, not less
rigorous.
If we suppose that f{x) has exactly /a internal roots (any
number of which may be equal), the derived function f (x) cannot
have less than /a + 1 internal roots, since there must be at least
one root of/* (x) between each pair of consecutive roots o{f(x), and
the whole number of roots of /(a?) concerned is /i.+ 2. In like
manner, we see that there must be at least /a roots of f" (x),
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220 TRANSVERSE VIBRATIONS OF STRINGS. [142.
besides the extremities, which themselves necessarily correspond
to roots; so that in passing from /(a?) to f (x) it is impossible
that any roots can be lost. Now
= - P {Vm^ 4>m Um + V^m+i 4>m+i t^tn+i + + I'n' ^n ^n)- • (7),
as we see by (1); and therefore, since p is always positive, we
infer that
has at least fj, roots.
Again, since (8) is an expression of the same form as f(x),
similar reasoning proves that
has at least /a internal roots ; and the process may be continued
to any extent. In this way we obtain a series of functions, all
with fi internal roots at least, which diflfer from the original
function /(a?) by the continually increasing relative importance of
the components of the higher orders. When the process has been
carried sufficiently far, we shall arrive at a function, whose form
differs as little as we please from that of the normal frinction of
highest order, viz. u^, and which has therefore n — 1 internal roots.
It follows that, since no roots can be lost in passing down the
series of functions, the number of internal roots of /(a?) cannot
exceed n — 1.
The other half of the theorem is proved in a similar manner
by continuing the series of functions backwards fit)m /(a?). In
this way we obtain
^t4,„+ <f>m+iUm+i-\r + ^»^
Vnr^<l>mUm-\rV'^fn+i<l>m+iUn^i-\- +I'n"'<^nWn
arriving at last at a function sensibly coincident in form with the
normal function of lowest order, viz. iim, and having therefore
m — 1 internal roots. Since no roots can be lost in passing up the
series from this function to f{x), it follows that f(x) cannot have
fewer internal roots than m — 1 ; but it must be understood that
any number of the m — 1 roots may be equal.
We will now prove that f(x) cannot be identically zero, unless
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142.] EXPANSION IN SERIES OP NORMAL FUNCTIONS. 221
all the coefficients ^ vanish. Suppose that <f>r is not zero.
Multiply (6) by pUr, and integrate with respect to x between the
limits and L Then by (5)
I purf{x)dx^il>r\ pur^dx (9);
Jo Jo
from which, since the integral on the right-hand side is finite, we
see thsLt/{x) cannot vanish for all values of x included within the
range of integration.
Liouville has made use of Sturm's theorem to shew how a
series of normal functions may be compounded so as to have an
arbitrary sign at all points lying between a? = and x=:l. His
method is somewhat as follows.
The values of x for which the ftinction is to change sign being
a, 6, c, ..., quantities which without loss of generality we may
suppose to be all diflferent, let us consider the series of determi-
nants,
Wi(a), tii{x)
1*1 (a), Wi(6), Ui(x) I
t*8(a), Wa(6), tk(x) I
u^(a)y M,(6), u^(x) l,&c.
The first is a linear function of v^ (x) and ti, (x), and by Sturm's
theorem has therefore one internal root at most, which root is
evidently a. Moreover the determinant is not identically zero,
since the coefficient of v^ (x), viz. t*i (a), does not vanish, whatever
be the value of a. We have thus obtained a function, which
changes sign at an arbitrary point a, and there only internally.
The second determinant vanishes when x = a, and when x = b,
and, since it cannot have more than two internal roots, it changes
sign, when x passes through these values, and there only. The
coefficient of w,(ic) is the value assumed by the first determinant
when x = b, and is therefore finite. Hence the second determinant
is not identically zero.
Similarly the third determinant in the series vanishes and
changes sign when a? = a, when a? = 6, and when a? = c, and at these
internal points only. The coefficient of U4(x) is finite, being the
value of the second determinant when x = c.
It is evident that by continuing this process we can form
functions compounded of the normal functions, which shall vanish
and change sign for any arbitrary values of x, and not elsewhere
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222 TRANSVERSE VIBRATIONS OF STRINGS. [142.
internally ; or, in other words, we can form a function whose sign
is arbitrary over the whole range from x = to x = L
On this theorem liouville founds his demonstration of the
possibility of representing an arbitrary function between a; = and
a? = i by a series of normal functions. If we assume the possibility
of the expansion and take
/(^) = </>i^(^) + <^*t^,(a?) + <^,w,(a?) + (10),
the necessary values of <f>i, (fh, &c. are determined by (9), and we
find
/{x) = ^\vr{x)j pUr(x)f{x)dx^j pUr"" (x) cLcY (11).
If the series on the right be denoted by F(x\ it remains to
establish the identity of /(a?) and F(xy
If the right-hand member of (11) be multiplied by pUr(x) and
integrated with respect to x from a? = to a? = Z, we see that
I p tlr (x) F{x)dx—\ pUr {x)f{x) dx,
Jo Jo
or, as we may also write it,
^ {F(x)-f(x)}pur(x)dx==0 (12),
Jo
where iv(a?) is any normal function. From (12) it follows that
/:
{F(x) -f(x)] [A,u, (x) + A^u^ (x) + i4,w,(a?)+ •. .} pdx^O.. .(13),
where the coeflBcients Ai, A^^ &c. are arbitrary.
Now if F{x) -f(x) be not identically zero, it will be possible
so to choose the constants Aj, -4j, &c. that A^Ui (x) + A^v^(x)-{' ...
has throughout the same sign as F (x) —f(x)y in which case every
element of the integi-al would be positive, and equation (13) could
not be true. It follows that F(x)—f(x) cannot differ from zero,
or that the series of normal functions forming the right-hand
member of (11) is identical with /(a?) for all values of a? from x =
to a? = i.
The arguments and results of this section are of course ap-
plicable to the particular case of a uniform string for which the
normal functions are circular.
[As a particular case of variable density the supposition that
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142.] VARIABLE DENSITY. 223
p = ax^ is worthy of notice, § 148 6. In the notation there
adopted
m«+i = n«=pVyi (14),
and the general solution is
y = .4ic*+*"*+-Bi»*-»*"* (15).
If the string be fixed at two points, whose abscissae x^, x^ are
as r to 1, the frequency equation is r^^ = 1, or
"'^i + ofgC). «
where s denotes an integer. The proper frequencies thus depend
only upon the ratio of the terminal abscissae. By supposing r
nearly equal to unity we may fall back upon the usual formula
(§ 124) applicable to a uniform string.
The general form of the normal function is
142 a. The points where the string remains at rest, or nodes,
are of course determined by the root.s of the normal functions,
when the vibrations are free. In this case the frequency is limited
to certain definite values ; but when the vibrations are forced, they
may be of any frequency, and it becomes possible to trace the
motion of the nodal points as the frequency increases continuously.
For example, suppose that the imposed force acts at a single
point P of a string AB, whose density may be variable. So long
as the frequency is less than that of either of the two parts AP,
PB (supposed to be held at rest at both extremities) into which
the string is divided, there can be no (interior) node (Q). Other-
wise, that part of the string AQ between the node Q and one
extremity {A\ which does not include P, would be vibrating
freely, and more slowly than is possible for the longer length AP,
included between the point P and the same extremity. When the
frequency is raised, so as to coincide with the smaller of those
proper to AP, PB, say -4P, a node enters at P and then advances
towards A. At each coincidence of the frequency with one of
those proper to the whole string AB, the vibration identifies itself
with the corresponding free vibration, and at each coincidence with
a frequency proper to AP, or BP, a new node appears at P, and
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224 TRANSVERSE VIBRATIONS OF STRINGS. [142 a.
advances in the first case towards A and in the second towards B.
And throughout the whole sequence of events all the nodes move
outwards from P towards A or B.
Thus, if the string be uniform and be bisected at P, there is
no node until the pitch rises to the octave {&) of the note (c) of the
string. At this stage two nodes enter at P, and move outwards
sjnmmetrically. When g' is reached, the mode of vibration is that
of the free vibration of the same pitch, and the nodes are at the
two points of trisection. At c" these nodes have moved outwards
so far as to bisect AP^ BP, and two new nodes enter at P.
143. When the vibrations of a string are not confined to one
plane, it is usually most convenient to resolve them into two sets
executed in perpendicular planes, which may be treated inde-
pendently. There is, however, one case of this description worth
a passing notice, in which the motion is most easily conceived and
treated without resolution.
Suppose that
. SfTX 287rt
y = sm -,- cos
^ I T
. STTX . 28'7rt
z = sin — f- sm
I T
Then
.(1).
STTX
r = V(y» + ^«) = sin-p (2),
and z : y = tan (2s7r^/T) (3),
shewing that the whole string is at any moment in one plane,
which revolves uniformly, and that each particle describes a circle
with radius sin {sirxjl). In fact, the whole system turns without
relative displacement about its position of equilibrium, completing
each revolution in the time rja. The mechanics of this case is
quite as simple as when the motion is confined to one plane, the
resultant of the tensions acting at the extremities of any small
portion of the string's length being balanced by the centrifugal
force.
144. The general differential equation for a uniform string,
viz.
iy-^a?^ (V)
d1? " da? ^^^'
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144.] UNLIMITED STRING. 225
may be transformed by a change of variables into
^1 = <^ (2),
where w = a? — erf, v = a; + a^. The general solution of (2) is
y^f{u)^-F{v)^f{x^at) + F{x^'aA) (3)S
/, F being two arbitrary functions.
Let us consider first the case in which F vanishes. When
t has any particular value, the equation
y^fix-ai) (4).
expressing the relation between x and y, represents the form of the
string. A change in the value of ^ is merely equivalent to an
alteration in the origin of Xy so that (4) indicates that a certain
form is propagated along the string with uniform velocity a in the
positive direction. Whatever the value of y may be at the point
X and at the time t, the same value of y will obtain at the point
x + a^tB,\, the time ^ + A^.
The form thus perpetuated may be any whatever, so long as it
does not violate the restrictions on which (1) depends.
When the motion consists of the propagation of a wave in the
positive direction, a certain relation subsists between the inclina-
tion and the velocity at any point. Differentiating (4) we find
%-'% »
Initially, dy/eft and dyjdx may both be given arbitrarily, but if
the above relation be not satisfied, the motion cannot be repre-
sented by (4).
In a similar manner the equation
y:=^F{x^-at) (6)
denotes the propagation of a wave m the negative direction, and
the relation between dyjdt and dyjdx corresponding to (5) is
l-«S ('^
In the general case the motion consists of the simultaneous
propagation of two waves with velocity a, the one in the positive,
1 [Equations (1) and (3) are due to D'Alembert (1750).]
R. 15
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226 TRANSVERSE VIBRATIONS OF STRINGS. [144.
and the other in the negative direction; and these waves are
entirely independent of one another. In the first dyjdt = — a dyjdx,
and in the second dy/dt = a dy/dx. The initial values of dy/dt
and dy/dx must be conceived to be divided into two parts, which
satisfy respectively the relations (5) and (7). The first constitutes
the wave which will advance in the positive direction without
change of form ; the second, the negative wave. Thus, initially.
whence
/»=i(l-^S)
.(8).
equations which determine the functions /' and F' for all values
of the argument from a? = — xtoa?=x,if the initial values of
dy/da smd dy/dt be known.
If the disturbance be originally confined to a finite portion of
the string, the positive and negative waves separate after the
interval of time required for each to traverse half the disturbed
portion.
Fig. 28.
Suppose, for example, that AB ia the part initially disturbed.
A point P on the positive side remains at rest until the positive
wave has travelled from A to P, is disturbed during the passage
of the wave, and ever afber remains at rest. The negative wave
never affects P at all. Similar statements apply, mutatis mutandis,
to a point Q on the negative side of AB. If the character of the
original disturbance be such that a dyjdx — dyjdt vanishes initially,
there is no positive wave, and the point P is never disturbed at
all ; and if a dyjdx + dyjdt vanish initially, there is no negative
wave. If dyjdt vanish initially, the positive and the negative
waves are similar and equal, and then neither can vanish. In
cases where either wave vanishes, its evanescence may be con-
sidered to be due to the mutual destruction of two component
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144.] POSITIVE AND NEGATIVE WAVES. 227
waves, one depending on the initial displacements, and the other
on the initial velocities. On the one side these two waves con-
spire, and on the other they destroy one another. This explains
the apparent paradox, that P can fail to be affected sooner or later
after AB has been disturbed.
The sabsequent motion of a string that is initially displaced
mthout velocity, may be readily traced by graphical methods.
Since the positive 8md the negative waves are equal, it is only
necessary to divide the original disturbance into two equal parts,
to displace these, one to the right, 8md the' other to the left,
through a space equal to at, and then to recompound them. We
shall presently apply this method to the case of a plucked string
of finite length.
146. Vibrations are called stationary, when the motion of each
particle of the system is proportional to some function of the time,
the same for all the particles. If we endeavour to satisfy
dt-"'' da^ • ^^^'
by assuming y = XT, where X denotes a function of x only, and
T a function of t only, we find
TdWy^Xd^"^"^ (a constant),
fio that
7= -4 cos mat + 5 sin rnat h .^.
X = cosmx +Dsmnix J ^ **
proving that the vibrations must be simple harmonic, though of
arbitrary period. The value of y may be written
y = P cos (mat — e) cos {mx — a)
= i P cos (mat + 7M« — € — a) -h J P cos (wo^ — ma — € + a).,.(3),
shewing that the most general kind of stationary vibration may
be regarded as due to the superposition of equal progressive vibra-
tions, whose directions of propagation are opposed. Conversely,
two stationary vibrations may combine into a progressive one.
The solution y^f{x — at)'^F{x + at) applies in the first
instance to an infinite string, but may be interpreted so as to
give the solution of the problem . for a finite string in certain
15—2
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228
TRANSVERSE VIBRATIONS OF STRINGS.
[145.
casea Let us suppose, for example, that the string terminates
at x==0, and is held fast there, while it extends to infinity in
the positive direction only. Now so long as the point ^ =:
actually remains at rest, it is a matter of indiflference whether
the string be prolonged on the negative side or not. We are
thus led to regard the given string as forming part of one doubly
infinite, and to seek whether and how the initial displacements
and velocities on the negative side can be taken, so that on
the whole there shall be no displacement at a; =: throughout the
subsequent motion. The initial values of y and y on the positive
side determine the corresponding parts of the positive and negative
waves, into which we know that the whole motion can be resolved.
The former has no influence at the point x — 0. On the negative
side the positive and the negative waves are initially at our dis-
posal, but with the latter we are not concerned. The problem is
to determine the positive wave on the negative side, so that in
conjunction with the given negative wave on the positive side
of the origin, it shall leave that point undisturbed.
Let OPQRS... be the line (of any form) representing the
wave in OX, which advances in the negative direction. It is
Fig. 24.
evident that the requirements of the case are met by taking on
the other side of what may be called the contrary wave, so that
is the geometrical centre, bisecting every chord (such as PP^
which passes through it. Analytically, if y =/(aj) is the equation
of 0PQR8 , -y=^f(-co) is the equation of OFQ'RS
When after a time t the curves are shifted to the left and to
the right respectively through a distance at, the co-ordinates
corresponding to a; = are necessarily equal and opposite, and
therefore when compounded give zero resultant displacement
The effect of the constraint at may therefore be represented
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145.]
REFLECTION AT A FIXED POINT.
229
by supposing that the negative wave moves through undisturbed,
but that a positive wave at the same time emerges from 0. This
reflected wave may at any time be found from its parent by the
following rule :
Let APQRS... be the position of the parent wave. Then the
reflected wave is the position which this would assume, if it were
Fig. 26.
turned through two right angles, first about OX as an axis of
rotation, and then through the same angle about OT. In other
words, the return wave is the image of APQRS formed by
successive optical reflection in OX and OY, regarded as plane
mirrors.
The same result may also be obtained by a more analytical
process. Id the general solution
the functions /(^), F(z) are determined by the initial circumstances
for all positive values of z. The condition at a? = inquires that
f(-at) + F(at)^0
for all positive values of t, or
/(-z) = -Fiz)
for positive values of z. The functions / and F are thus de-
termined for all positive values of x and t
There is now no difficulty in tracing the course of events when
iuH) points of the string A and B are held fast. The initial dis-
turbance in AB divides itself into positive and negative waves,
which are reflected backwards and forwards between the fixed
points, changing their character from positive to negative, and
vice versd, at each reflection. After an even number of reflec-
tions in each case the original form and motion is completely
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TRANSVERSE VIBRATIONS OF STRINGS.
[145-
recovered. The process is most easily followed in imagination
when the initial disturbance is confined to a small part of the
string, more particularly when its character is such as to give rise
to a wave propagated in one direction only. The pulse travels with
uniform velocity (a) to and fro along the length of the string, and
after it has returned a second time to its starting point the
original condition of things is exactly restored. The period of
the motion is thus the time required for the pulse to traverse
the length of the string twice, or
T = 2l/a (1).
The same law evidently holds good whatever may be the character
of the original disturbance, only in the general case it may
happen that the shortest period of recurrence is some aliquot part
of T.
146. The method of the last few sections may be advantage-
ously applied to the case of a plucked string. Since the initial
velocity vanishes, half of the displacement belongs to the positive
and half to the negative wave. The manner in which the wave
must be completed so as to produce the same effect as the con-
straint, is shewn in the figure, where the upper curve represents
Fig. 26.
the positive, and the lower the negative wave in their initial
positions. In order to find the configuration of the string at any
future time, the two curves must be superposed, after the upper
has been shifted to the right and the lower to the left through a
space equal to at.
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146. J
GRAPHICAL METHOD.
231
The resultant curve, like its components, is made up of straight
pieces. A succession of six at intervals of a twelfth of the period,
Kg. 27.
shewing the course of the vibration, is given in the figure (Fig. 27),
taken from Helmholtz. From the string goes back again to A
through the same stages \
It will be observed that the inclination of the string at the
points of support alternates between two constant values.
147. If a small disturbance be made at the time t at the
point X of an infinite stretched string, the effect will not be felt
at until after the lapse of the time xja, and will be in all
respects the same as if a like disturbance had been made at
the point a; + A^ at time t — ^w/a. Suppose that similar dis-
turbances are communicated to the string at intervals of time
T at points whose distcmces from increase each time by a Br,
then it is evident that the result at will be the same as if the
disturbances were all made at the same point, provided that the
time-intervals be increaised from t to t + St. This remark con-
1 This method of treating the vibration of a plucked string is due to Toung.
Phil. Trant., 1800. The student is recommended to make himself &miliar with it
by actually constructing the forms of Fig. 27.
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232 TRANSVEBSE VIBRATIONS OF STRINGS. [l47.
tains the theory of the alteration of pitch due to motion of the
source of disturbance ; a subject which will come under our notice
again in connection with aerial vibrations.
1498. When one point of an infinite string is subject to a forced
vibration, trains of waves proceed from it in both directions ac-
cording to laws, which are readily investigated. We shall suppose
that the origin is the point of excitation, the string being there
subject to the forced motion y^A^\ and it will be sufficient to
consider the positive side. If the motion of each element (b be
resisted by the frictional force Kpyda, the differential equation is
d&^^dt''' da? ••••^^^'
or since y oc c^,
%-&-^y-^'y •<«)•
if for brevity we write V for the coefficient of y.
The general solution is
y={C«r*» + D6+*»}6<'^ (3).
Now since y is supposed to vanish at an infinite distance, D
must vanish, if the real part of X be taken positive. Let
where a is positive.
Then the solution is
y=^6-<»+*>*+<^ (4),
or, on throwing away the imaginary part,
y^Aer^ co&(pt-/3w) (5),
corresponding to the forced motion at the origin
y = -4 coapt (6).
An arbitrary constant may, of course, be added to t
To determine a and )3, we have
«.-/3. — g; 2«/3 = f (7X
If we suppose that k is small,
j3=p/a, a = K/2a nearly,
and y=:Ae-'^^ cos (pt—^xj (8).
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148.] DAMPING OP PROGRESSIVE WAVES. 233
Thi8 solution shews that there is propagated along the string
a wave, whose amplitude slowly diminishes on account of the
exponential factor. If k^O, this factor disappears, and we have
simply
y = 4 cos (pt-^) (9).
This result stands in contradiction to the general law that,
when there is no friction, the forced vibrations of a system (due
to a single simple harmonic force) must be synchronous in phase
throughout. According to (9), on the contrary, the phase varies
continuously in passing from one point to another along the string.
The &ct is, that we are not at liberty to suppose /c^O in (8),
inasmuch as that equation was obtained on the assumption that
the real part of X. in (3) is positive, and not zero. However long
a finite string may be, the coe£Scient of friction may be taken so
small that the vibrations are not damped before reaching the
further end. After this point of smallness, reflected waves begin
to complicate the result, and when the friction is diminished
indefinitely, an infinite series of such must be taken into account,
and would give a resultant motion of the same phase throughout.
This problem may be solved for a string whose mass is supposed
to be concentrated at equidistant points, by the method of § 120.
The co-ordinate -^i may be supposed to be given {—H^^\ and
it will be found that the system of equations (5) of § 120 may all
be satisfied by taking
^r = «^^^i (10),
where is a complex constant determined by a quadratic equa-
tion. The result for a continuous string may be afterwards
deduced
[In the notation of § 120 the quadratic equation is
5^ + il^ + 5 = (11),
where ^=-/*p» + =^', B^-^ (12).
The roots of (11) are
«.-^±^'(^:=«■) („),
and are imaginary if 4^ > il^ that is, if
^<*i <")•
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234 TRANSVEESE VIBRATIONS OF STRINGS. [148.
a condition always satisfied in passing to the limit where a and /jl
are infinitely small. In any case when (14) is satisfied the
modulus of ^ is unity, so that (10) represents wave propagation.
If, however, (14) be not satisfied, the values of are real In
this case all the motions are in the same phase, and no wave
is propagated. The vibration impressed upon y^i ib imitated upon
a reduced scale by i^j, i^, , with amplitudes which form a
geometrical progression. In the first case the motion is pro-
pagated to an infinite distance, but in the second it is practically
confined to a limited region round the source.]
148 a. So long as the conditions of § 144 are satisfied, a
positive, or a negative, wave is propagated undisturbed. If
however there be any want of uniformity, such (for example) as
that caused by a load attached at a particular point, reflection
will ensue when that point is reached. The most interesting
problem under this head is that of two strings of different
longitudinal densities, attached to one another, and vibrating
transversely under the common tension Tj. Or, if we regard the
string as single, the density may be supposed to vary dis-
continuously from one uniform value (pi) to another (pj). If
Oi, Os denote the corresponding velocities of propagation,
a,«=rV/>i, a,' = r,//>, (1),
and M = ai/aa = VW/)i) (2).
The conditions to be satisfied at the junction of the two parts
are (i) the continuity of the displacement y, and (ii) the continuity
of dy/da. If the two parts met at a finite angle, an infinitely
small element at the junction would be subject to a finite force.
Let us suppose that a positive wave of harmonic type, travelling
in the first part (pj), impinges upon the second (/)j). In the latter
the motion will be adequately represented by a positive wave,
but in the former 'We must provide for a negative reflected wave.
Thus we may take for the two parts respectively
y = fi6*» <•»*-«» -fiTe*** <*»*+*> (3),
y=Z;6**«w-«) (4),
where ki = 27r/Xi , k^ = 2'7r/X5 ,
so that kjOa — k^ (5).
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148 a.] BBFLECTION AT A JUNCTION. 235
The conditions at the junction {x = 0) give
H+K^L (6),
hH-hK^kJL '. (7);
whence 5-^^^' = -^ W-
Since the ratio KjH is real, we may suppose that both
quantities are real; and if we throw away the imaginary parts
from (3) and (4) we get as the solution in terms of real quantities
y = J?cosA:i(ai^-a?)+ircosii(ai^ + a?) (9);
y«(fl' + Z)cosA?,(a,«-a?) (10).
The ratio of amplitudes of the reflected and the incident
waves expressed by (8) is that first obtained by T. Young for
the corresponding problem in Optics.
148 6. The expression for the intensity of reflection established
in § 148 a depends upon the assumption that the transition from
the one density to the other is sudden, that is occupies a distance
which is small in comparison with a wave length. If the
transition be gradual, the reflection may be expected to tsX\ off,
and in the limit to disappear altogether.
The problem of gradual transition includes, of course, that of
a variable medium, and would in general be encumbered with
great difficulties. There is, however, one case for which the
solution may be readily expressed, and this it is proposed to
consider in the present section. The longitudinal density is
supposed to vary as the inverse square of the abscissa. If y,
denoting the transverse displacement be proportional to e**, the
equation which it must satisfy as a function of a?, is (§ 141),
g + n'a;-^ = (1),
where v? is some positive constant, of the nature of an abstract
number.
The solution of (1) is y = ila^+^n ^. ^^im (2),
where m^^v?'-^ (3).
If m be real, that is, if w > \, we may obtain, by supposing
A ss 0» as a final solution in real quantities,
y as (7^ cos (p< — wloga-fe) (4),
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236 TRANSVERSE VIBRATIONS OP STRINGS. [148 6.
which represents a positive progressive wave, in many respects
similar to those propagated in uniform media.
Let us now suppose that, to the left of the point x = Xi, the
variable medium is replaced by one of uniform constitution, suck
that there is no discontinuity of density at the point of transition ;
and let us inquire what reflection a positive progressive wave in
the uniform medium will undergo on arrival at the variable
medium. It will be sufficient to consider the case where m is
I'eal, that is, where the change of density is but moderately rapid.
By supposition, there is no negative wave in the variable
medium, so that ^ = in (2). Thus
Jiud, when x — Xi, -y='- (5).
ydx Xi ^ ^
The general solution for the uniform medium, satisfying the
equation d^jdnf + v?x{'^ = 0, may be written
y^He *» -{-Ke *» (6),
from which, when oo^Xi,
dy in H-K .,..
ydx^ x,H + K ^^
In equation (6), H represents the amplitude of the incident
positive wave, and K the amplitude of the reflected negative
wave. The condition to be satisfied at a; 3=0^ is expressed by
equating the values of — ^ given by (5) and (7). Thus
jy"i(n+m)-i ^ ^'
which gives, in symbolical form, the ratio of the reflected to the
incident vibration.
Having regard to (3), we may write (8) in the form
^- -:ii- (9)-
so that the amplitude of the reflected wave is ^(n + m)"^ of
that of the incident. Thus, as was to be expected, when n and m
are great, %.e., when the density changes slowly in the variable
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I48 6.] GRADUAL TRANSITION. 237
medium, there is but little reflection. As regards phase, the
result embodied in (9) may be represented by supposing that the
reflection occurs at ^ = a^, and involves a change of phase amount-
ing to a quarter period.
Passing on now to the more important problem, we will
suppose that the variable medium extends only so far as the point
x=^x^, beyond which the density retains uniformly its value at
that point. A positive wave travelling at first in a uniform
medium of density proportional to Wi"^, passes at the point x = Xi
into a variable medium of density proportional to ar^, and again, at
the point x^x^, into a uniform medium of density proportional to
x^. The velocities of propagation are inversely proportional to
the square roots of the densities, so that, if /li be the refractive
index between the extreme media,
f^'l (lO)-
The thickness (d) of the layer of transition is
d-x^ — Xi (11).
The wave-lengths in the two media are given by
_ 27rXi _ 27rdg, ^
n n
«othat „ = ___=.^__^ (12).
For the first medium we take, as before,
y^He '^ -hiTe *. (6),
giving, when x^x^,
dy inH — K in0^ ....
ydi*"""^2r+ir~ "^ ^ ^'
if, for brevity, we write 6 for „ r-^ •
For the variable medium,
y = ila:*+*"«+5a:*-*'« (2),
giving, when a? = a?i,
^y - ^ -1 (i + im)Ax,^ + {\ - im) BxT'-
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238 TRANSVERSE VIBRATIONS OF STRINGS, [i48 6.
Hence the condition to be satisfied at a? = d^ gives
whence ^ = ^-^«^^--^^-|^* (14).
The condition to be satisfied at a? = aj, may be deduced firom (14),
by substituting x^tor a^, putting at the same time ^ = 1 in virtue
of the supposition that in the second medium there is no negative
wave. Hence, equating the two values ot A\B, we get
^ t^ + tn^ + i""^* im + tn + i ^^^^'
as the equation from which the reflected wave in the first medium
is to be found. Having regard to (3), we get
^ _ ^r-iT ^ m + n + ii + /A»~ (rn - n - ii)
" H-{-K m + n-ii + /[i»^« (m- n + it) '
so that rr=o/ \ . o aim/ \ (16)-
-ff 2 (m + n) + 2/Lt*** (m — n) ^
This is the symbolical solution. To interpret it in real quantities,
we must distinguish the cases of m real and m imaginary. If the
transition be not too sudden, m is real, and (16) may be written
K ^ % — 1 -f cos (2m log ^) + 1 sin (2m log /a)
fl" *" 2 m + n + (m — n) cos (2m log fi) +i{m — n) sin (2m log ft) "
Thus the expression for the ratio of the intensities of the reflected
and the incident waves is, after reduction.
1
sin'
4m»
in'(mlog /i) .^y.
+ sin' (m log /*) ^
If m be imaginary, we may write im = m'] (16) then gives for
the ratio of intensities,
or, if we introduce the notation of hyperbolic trigonometry § 170,
sinh' (m' log fi)
sinh' (mf log/x) + 4m''
For the'critical value m = 0, we get, from (17) or (19),
.(19).
4 + (log^)' ^^"^-
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1 48 6.J GRADUAL TRANSITION. 239
These expressions allow us to trace the effect of a more or
less gradual transition between media of given indices. If the
transition be absolutely abrupt, n = 0, by (12); so that rn!=^\.
In this case, (18) gives us (§ 148 a) Young's well-known formula
U + lJ
.(21).
Since increases continually from a: = 0, the ratio (19)
X
increases continually bom. m=0 to m's=J, i.e,, diminishes
continually from the case of sudden transition m' = ^, when its
value is (21), to the critical case m = 0, when its value is (20),
after which this form no longer holds good. When m' = 0, w = J,
and, by (12), d = (X, - \)l Anr.
When n>\, (17) is the appropriate form. We see from it
that with increasing n the reflection diminishes, until it vanishes,
when mlog/A = 7r, %,e. when
"■=i^<i^y •• • «
With a still more gradual transition the reflection revives, reaches
a maximum, again vanishes when m log /bb «= 27r, and so on^
148 a In the problem of connected strings, vibrating under
the influence of tension alone, the velocity in each uniform part is
independent of wave length, and there is nothing corresponding to
optical dispersion. This state of things will be departed from if
we introduce the consideration of stiffness, and it may be of interest
to examine in a simple case how far the problem of reflection is
thereby modified. As in § 148 a, we will suppose that at a? = ()
the density changes discontinuously from pi to pa, but that now
the vibrations of the second part occur under the influence of
sensible stiffness. The differential equation applicable in this
case is, § 188,
or, if y vary as e*»*,
-'^S+«'*S+"*y=<^ <!>'
so that, if y vary as e***,
y8»fe* + a,«A:»-n* = (2).
^ Proe, Math. Soc., vol. zi. February, 1S80 ; where will also be found a numeri-
cal example illustrative of optical conditions.
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240 TRANSVERSE VIBRATIONS OF STRINGS. [148 C.
In consequence of the stiffiiess represented by ^ the velocity
of propagation deviates from a^y and must be found from (2). The
two values of Ar* given by this equation are real, one being positive
and the other negative. The four admissible values of k may thus
be written + k^, ± iK* so that the complete solution of (1) will be
y = -4e*^ + jBe-*««+(7e-*«* + i)6*^ (3),
Aa, k^ being real and positive. The Velocity of propagation is njk^
In the application which we have to make the disturbance of
the imperfectly flexible second part is due to a positive wave
entering it from the first part. When x is great and positive, (3)
must reduce to its second term. Thus
^=0, i) = 0;
and we are left with
y = Ber*^ + Ce-^ (4).
This holds when x is positive. When x is negative, corresponding
to the perfectly flexible first part, we have
y = ire-^t«+ire«t* (5),
in which ki-n/oi (6).
The " refi^iCtive index " is given by
fi^kjk, (7).
The conditions at the junction are first the continuity of y and
dyldx. Further, d^y/dx^ in (4) must vanish at this place, inasmuch
as curvature implies a couple (§ 162), and this could not be
transmitted by the first part. Hence
H + K=B-hC (8),
k,{H'-K) = k^^ih,C (9),
-A8»5 + VC=0 (10).
From these we deduce
H_+K_kj(h^±ik^
H-K' kji, ^^^^'
K_h^ (k^ - Aj) + ik^ . . ^
H' h {k\ + k,) + ikA ^^^^'
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148 c.] IMPERFECT FLEXIBILITY. 241
and thence for the intensity of reflection, equal to Mod^ (K/H),
If the second part, as well as the first, be perfectly flexible,
/8 = 0, Aj = 00 , and we fall back on YouDg's formula In general,
the intensity of reflection is not accurately given by this formula,
even though we employ therein the value of the refractive index
appropriate to the waves actually under propagation.
16
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CHAPTER VII.
LONGITUDINAL AND TORSIONAL VIBRATIONS OF BARS.
149. The next system to the string in order of simplicity
is the bar, by which term is usually understood in Acoustics a
mass of matter of uniform substance and elongated cylindrical
form. At the ends the cylinder is cut off by planes perpendicular
to the generating lines. The centres of inertia of the transverse
sections lie on a straight line which is called the axis.
The vibrations of a bar are of three kinds — longitudinal,
torsional, and lateral. Of these the last are the most important,
but at the same time the most difficult in theory. They are
considered by themselves in the next chapter, and will only be
referred to here so far as is necessary for comparison and contrast
with the other two kinds of vibrations.
Longitudinal vibrations are those in which the axis remains
unmoved, while the transverse sections vibrate to and fro in the
direction perpendicular to their planes. The moving power is
the resistance offered by the rod to extension or compression.
One peculiarity of this class of vibrations is at once evident.
Since the force necessary to produce a given extension in a bar
is proportional to the area of the section, while the mass to be
moved is also in the same proportion, it follows that for a bar of
given length and material the periodic times and the modes of
vibration are independent of the area and of the form of the
transverse section. A similar law obtains, as we shall presently
see, in the case of torsional vibrations.
It is otherwise when the vibrations are lateral. The periodic
times are indeed independent of the thickness of the bar in the
direction perpendicular to the plane of flexure, but the motive power
\
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149.] CLASSIFICATION OF VIBRATIONS. 243
in this case, viz. the resistance to bending, increases more rapidly
than the thickness in that plane, and therefore an increase in
thickness is accompanied by a rise of pitch.
In the case of longitudinal and lateral vibrations, the mechan-
ical constants concerned are the density of the material and the
value of Young's modulus. For small extensions (or compressions)
Hooke's law, according to which the tension varies as the extension,
- -J J xi. .1 ^ • • actual length — natural length
holds good. If the extension, viz. ^ ^-^i -r — ,
° natural length
be called e, we have T=qe, where q is Young's modulus, and T
is the tension per unit area necessary to produce the extension €.
Young's modulus may therefore be defined as the force which would
have to be applied to a bar of unit section, in order to double its
length, if Hooke's law continued to hold good for so great exten-
sions ; its dimensions are accordingly those of a force divided by an
area
The torsional vibrations depend also on a second elastic con-
stant fjL, whose interpretation will be considered in the proper
place.
Although in theory the three classes of vibrations, depending
respectively on resistance to extension, to torsion, and to flexure
are quite distinct, and independent of one another so long as the
squai-es of the strains may be neglected, yet in actual experiments
with bars which are neither uniform in material nor accurately
cylindrical in figure it is often found impossible to excite longi-
tudinal or torsional vibrations without the accompaniment of some
measure of lateral motion. In bars of ordinaiy dimensions the
gravest lateral motion is far graver than the gravest longitudinal
or torsional motion, and consequently it will generally happen that
the principal tone of either of the latter kinds agrees more or less
perfectly in pitch with some overtone of the former kind. Under
such circumstances the regular modes of vibrations become
unstable, and a small irregularity may produce a great eflfect. The
difficulty of exciting purely longitudinal vibrations in a bar is
similar to that of getting a string to vibrate in one plane.
With this explanation we may proceed to consider the three
classes of vibrations independently, commencing with longitudinal
vibrations, which will in fact raise no mathematical questions
beyond those already disposed of in the previous chapters.
16—2
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244
LONGITUDINAL VIBRATIONS OF BARS.
[150.
160. When a rod is stretched by a force parallel to its length,
the stretching is in general accompanied by lateral contraction in
such a manner that the augmentation of volume is less than if
the displacement of every particle were parallel to the axis. In the
case of a short rod and of a particle situated near the cylindrical
boundary, this lateral motion would be comparable in magnitude
with the longitudinal motion, and could not be overlooked without
risk of considerable error. But where a rod, whose length is great
in proportion to the linear dimensions of its section, is subject
to a stretching of one sign throughout, the longitudinal motion
accumulates, and thus in the case of ordinary rods vibrating
longitudinally in the graver modes, the inertia of the lateral
motion may be neglected. Moreover we shall see later how a
correction may be introduced, if necessary.
Let X be the distance of the layer of particles composing any
section from the equilibrium position of one end, when the rod
is unstretched, either by permanent tension or as the result of
vibrations, and let f be the displacement, so that the actual
position is given by d? + f. The equilibrium and actual position
of a neighbouring layer being x+ Bx, a? + &c+f+-^&p re-
spectively, the elongation is d^/dx, and thus, if T be the tension
per unit area acting across the section,
T = q
^1.
dx'
.(1).
Consider now the forces acting on the slice bounded by x
and X + Bx, If the area of the section be o), the tension at a; is
by (1) qcod^/dx, acting in the negative du-ection, and at ar + Sa?
the tension is
(d^ d'i
-(S-S^).
acting in the positive direction ; and thus the force on the slice
due to the action of the adjoining parts is on the whole
q<o
dx"
Sx,
The mass of the element is pa) &r, if p be the original density,
and therefore if X be the accelerating force acting on it, the
equation of equilibrium is
^^fj-o W
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150.] GENERAL DIFFERENTIAL EQUATION. 245
In what follows we shall not require to consider the operation
of an impressed force. To find the equation of motion we have
only to replace X by the reaction against acceleration — ^, and
thus if g : p = a^ we have
t^-a^tl (ON
df " da? ^^>-
This equation is of the same form as that applicable to the
transverse displacements of a stretched string, and indicates the
undisturbed propagation of waves of any type in the positive and
negative directions. The velocity a is relative to the unstretched
condition of the bar ; the apparent velocity with which a disturb-
ance is propagated in space will be greater in the ratio of the
stretched and unstretched lengths of any portion of the bar. The
distinction is material only in the case of permanent tension.
161. For the actual magnitude of the velocity of propagation,
we have
a^ = q : p = qa} : pcD,
which is the ratio of the whole tension necessary (according to
Hooke's law) to double the length of the bar and the longitudinal
density. If the same bar were stretched with total tension T,
and were flexible, the velocity of propagation of waves along it
would be aJ{T : pay). In order then that the velocity might be
the same in the two cases, T must be qay, or, in other words, the
tension would have to be that theoretically necessary in order to
double the length. The tones of longitudinally vibrating rods
are thus very high in comparison with those obtainable from
strings of comparable length.
In the case of steel the value of q is about 22 x 10* grammes
weight per square centimetre. To express this in absolute units
of force on the c. G.s.* system, we must multiply by 980. In
the same system the density of steel (identical with its specific
gravity referred to water) is 78. Hence for steel
a = /y/
^^^-^^f^^ = 530,000
approximately, which shews that the velocity of sound in steel is
about 530,000 centimetres per second, or about 16 times greater
^ Centimetre, Gramme, SecoDd. This system is recommended by a Committee
of the British Association. Brit, Ass. Report, 1873.
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246
LONGITUDINAL VIBRATIONS OF BARS.
[151.
than the velocity of sound in air. In glass the velocity is about
the same as in steel.
It ought to be mentioned that in strictness the value of q deter-
mined by statical experiments is not that which ought to be used
here. As in the case of gases, which will be treated in a subsequent
chapter, the rapid alterations of state concerned in the propaga-
tion of sound are attended with thermal effects, one result of
which is to increase the effective value of q beyond that obtained
from observations on extension conducted at a constant tempera-
ture. But the data are not precise enough to make ibis correction
of any consequence in the case of solids.
162. The solution of the general equation for the longitudinal
vibrations of an unlimited bar, namely
f =/(a7 - aO + ^(^ + aO»
being the same as that applicable to a string, need not be further
considered here.
When both ends of a bar are free, there is of course no perma-
nent tension, and at the ends themselves there is no temporarj'
tension. The condition for a free end is therefore
dx
=
.(1).
To determine the normal modes of vibration, we must assume
that f varies as a harmonic function of the time— cos nat. Then
as a function of ^, ^ must satisfy
did'
+ n>f = 0.
.(2).
of which the complete integral is
f = J. cosrw7+ Bsmnx (3),
where A and B are independent of x.
Now since d^jdx vanishes always when x = 0, we get £ = 0; and
again since d^jdx vanishes when x = I — the natural length of the
bar, sin TiZ = 0, which shews that n is of the form
n =
ITT
I
.(4),
i being integral.
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152.]
BOTH EXTREMITIES FREE.
247
Accordingly, the normal modes are given by equations of the
form
-. J. XTTX iirat ,^v
f = -4 cos-,- cos— ^ (o),
in which of course an arbitrary constant may be added to f, if
desired.
The complete solution for a bar with both ends free is there-
fore expressed by
XTTCU )
f = 2
.-.^« I
{..
iirat rt •
cos — ; — I- Bi sin
I
.(6),
vrhere -4^ and Bi are arbitrary constants, which may be determined
in the usual manner, when the initial values of f and ^ are
given.
A zero value of i is admissible ; it gives a term representing a
displacement f constant with respect both to space and time,
and amounting in fact only to an alteration of the origin.
The period of the gravest component in (6) corresponding to
1 = 1, is 2l/a, which is the time occupied by a disturbance in
travelling twice the length of the rod. The other tones found
by ascribing integral values to i form a complete harmonic scale ;
so that according to this theory the note given by a rod in
longitudinal vibration would be in all cases musical.
In the gravest mode the centre of the rod, where x = ^Z, is a
place of no motion, or node ; but the periodic elongation or com-
pression d^/dx is there a maximum.
163. The case of a bar with one end free and the other fixed
may be deduced from the general solution for a bar with both
ends free, and of twice the length. For whatever may be the
initial state of the bar free at a? = and fixed at x = l, such dis-
placements and velocities may always be ascribed to the sections
of a bar extending from to 21 and free at both ends as shall
make the motions of the parts from to i identical in the two
cases. It is only necessary to suppose that from I to 21 the dis-
placements and velocities are initially equal and opposite to those
found in the portion from to Z at an equal distance from the
centre x=^L Under these circumstances the centre must by
the symmetry remain at rest throughout the motion, and then the
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248
LONGITUDINAL VIBRATIONS OF BARS.
[153.
portion from to i satisfies all the required conditions. We con-
clude that the vibrations of a bar free at one end and fixed at the
other are identical with those of one half of a bar of twice the
length of which both ends are free, the latter vibrating only in the
uneven modes, obtained by making i in succession all odd integers.
The tones of the bar still belong to a harmonic scale, but the
even tones (octave, &c. of the fundamental) are wanting.
The period of the gravest tone is the time occupied by a pulse
in travelling /owr times the length of the bar.
164. When both ends of a bar are fixed, the conditions to
be satisfied at the ends are that the value of f is to be invariable.
At a? = 0, we may suppose that f = 0. At a; = Z, f is a small
constant a, which is zero if there be no permanent tension. In-
dependently of the vibrations we have evidently f = a: a -r Z, and
we should obtain our result most simply by assuming this term
at once. But it may be instructive to proceed by the general
method.
Assuming that as a function of the time f varies as
A cos nat + B sin nat,
we see that as a function of x it must satisfy
of which the general solution is
^ = (7 cos Tio! •}- D sin nx (1).
But since | vanishes with x for all values of ^, (7 = 0, and thus
we may write
f = 2 sin 7WJ \A cos nat -f B sin nat].
The condition at a? = Z now gives
2 sin TiZ {A cos nat + B sin nat] = a,
from which it follows that for every finite admissible value of n
sin nZ = 0, • or n =
But for the zero value of w, we get
Aq sin nl = a.
ITT
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154.] BOTH EXTREMITIES FIXED. 249
and the corresponding term in f is
^ . . sinrw? x
f = -do sin 7WJ = a -i — i = a r .
* sinm ^
The complete value of f is accordingly
f = a y + 2^.1 sm y- V^i cos —v- + 5f sm — p k..(2).
The series of tones form a complete harmonic scale (from
which however any of the members may be missing in any
actual case of vibration), and the period of the gravest com-
ponent is the time taken by a pulse to travel twice the length
of the rod, the same therefore as if both ends were free. It
must be observed that we have here to do with the unstretched
length of the rod, and that the period for a given natural length
is independent of the permanent tension.
The solution of the problem of the doubly fixed bar in the
case of no permanent tension might also be derived from that
of a doubly free bar by mere differentiation with respect to x.
For in the latter problem d^jdx satisfies the necessary differential
equation, viz.
dt' \dx) da^ {dx) '
inasmuch as { satisfies
dt''^^ dx^'
and at both ends d^jdx vanishes. Accordingly dUdx in this
problem satisfies all the conditions prescribed for f in the case
when both ends are fixed. The two series of tones are thus
identical.
165. The effect of a small load M attached to any point of
the rod is readily calculated approximately, as it is sufficient
to assume the type of vibration to be unaltered (§ 88). We
will take the case of a rod fixed at a: = 0, and free at x = I, The
kinetic energy is proportional to
*/.
pay sm^ -^j- dx + J^Jlf sm« -^ ,
p(ol /- 2M . . i7rx\
«^*^ 4 (l + po,i«^°'l^J•
/GoogIe
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250 LONGITUDINAL VIBRATIONS OF BARS. [155.
Since the potential energy is unaltered, we see by the prin-
ciples of Chapter iv., that the eflfect of the small load 3f at a
distance x from the fixed end is to increase the period of the
component tones in the ratio
The small quantity M : p<d is the ratio of the load to the
whole mass of the rod.
If the load be attached at the free end, sinn*Va?/2Z) = l, and
the effect is to depress the pitch of every tone by the same small
interval. It will be remembered that i is here an uneven integer.
If the point of attachment of Jlf be a node of any component,
the pitch of that component remains unaltered by the addition.
166. Another problem worth notice occurs when the load at
the free end is great in comparison with the mass of the rod.
In this case we may assume as the type of vibration, a condition
of uniform extension along the length of the rod.
If f be the displacement of the load M, the kinetic energy is
r=iJif|'+i|»fV«^Ar = ilniJf+i/>a,o (1).
The tension corresponding to the displacement f is qca^/l,
and thus the potential energy of the displacement is
F^?^ (2)
The equation of motion is
and if f X cos pt
P' = ^^{M + ip<ol) (3).
The correction due to the inertia of the rod is thus equivalent
to the addition to M of one-third of the mass of the rod.
166 a. So long as a rod or a wire is uniform, waves of longi-
tudinal vibration are propagated along it without change of type,
but any interruption, or alteration of mechanical properties, will
in general give rise to reflection. If two uniform wires be joined,
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156 a. J CORRECTION FOR LATERAL MOTION. 251
the problem of determining the reflection at the junction may be
conducted as in § 148 a. The conditions to be satisfied at the
junction are (i) the continuity of f, and (ii) the continuity of
qcod^/dx, measuring the tension. If pi, ps, <Oi, o),, Oy, Oa denote
the volume densities, the sections, and the velocities in the two
wires, the ratio of the reflected to the incident amplitude is
given by
H pi (Oiai + piCo^a^
The reflection vanishes, or the incident wave is propagated
through the junction without loss, if
PiCOitti = p.,a}.,ai (2).
This result illustrates the difficulty which is met with in obtaining
effective transmission of sound from air to metal, or from metal to
air, in the mechanical telephone. Thus the value of pa is about
100,000 times greater in the case of steel than in the case of air.
157. Our mathematical discussion of longitudinal vibrations
may close with an estimate of the error involved in neglecting
the inertia of the lateral motion of the parts of the rod not
situated on the axis. If the ratio of lateral contraction to longi-
tudinal extension be denoted by /a, the lateral displacement of a
particle distant r from the axis will be /jlvc in the case of equili-
brium, where e is the extension. Although in strictness this
relation will be modified by the inertia of the lateral motion, yet
for the present purpose it may be supposed to hold good, § 88.
The constant /a is a numerical quantity, lying between and ^.
If fjL were negative, a longitudinal tension would produce a lateral
swelling, and if p, were greater than i, the lateral contraction
would be great enough to overbalance the elongation, and cause
a diminution of volume on the whole. The latter state of things
would be inconsistent with stability, and the former can scarcely
be possible in ordinary solids. At one time it was supposed
that fi was necessarily equal to ^, so that there was. only one
independent elastic constant, but experiments have since shewn
that fi is variable. For glass and brass Wertheim found experi-
mentally /i = ^.
If fj denote the lateral displacement of the particle distant r
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LONGITUDINAL VIBRATIONS OF BARS.
[157.
from the axis, and if the section be circular, the kinetic energy
due to the lateral motion is
Thus the whole kinetic energy is
In the case of a bar free at both ends, we have
iTTX df ITT . iirx
f X cos ~, , ^ X — r sm -=- ,
* I ax I I
and thus
748^:7=1 + ^^^.
The effect of the inertia of the lateral motion is therefore to
increase the period in the ratio
1:1 +
4 ¥'
This correction will be nearly insensible for the graver modes of
bars of ordinary proportions of length to thickness.
[A more complete solution of the problem of the present
section has been given by Pochhammer^ who applies the general
equations for an elastic solid to the case of an infinitely extended
cylinder of circular section. 'The result for longitudinal vibrations,
so far as the term in r*/P, is in agreement with that above deter-
mined. A similar investigation has also been published by Chree^
who has also treated the more general question* in which the
cylindrical section is not restricted to be circular.]
168. Experiments on longitudinal vibrations may be made
with rods of deal or of glass. The vibrations are excited by
friction § 138, with a wet cloth in the case of glass ; but for metal
or wooden rods it is necessary to use leather charged with powdered
i-osin. " The longitudinal vibrations of a pianoforte string may be
excited by gently rubbing it longitudinally with a piece of india
rubber, and those of a violin string by placing the bow obliquely
across the string, and moving it along the string longitudinally,
keeping the same point of the bow upon the string. The note is
unpleasantly shrill in both cases."
» Crelle, Bd. 81, 1876.
» Ihid, Vol. 23, p. 317, 1889.
a Quart, Math. Joum., Vol. 21, p." 287, 1886.
\
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158.]
TORSIONAL VIBRATIONS.
253
" If the peg of the violin be turned so as to alter the pitch of
the lateral vibrations very considerably, it will be found that the
pitch of the longitudinal vibrations has altered very slightly. The
reason of this is that in the case of the lateral vibrations the
change of velocity of wave-transmission depends chiefly on the
change of tension, which is considerable. But in the case of the
longitudinal vibrations, the change of velocity of wave-transmis-
sion depends upon the change of extension, which is comparatively
slight \'*
In Savart's experiments on longitudinal vibrations, a peculiar
sound, called by him a "son rauque," was occasionally observed,
whose pitch was an octave below that of the longitudinal vibra-
tion. According to Terquem * the cause of this sound is a trans-
verse vibration, whose appearance is due to an approximate
agreement between its own period and that of the sub-octave of
the longitudinal vibration § 68 6. If this view be correct, the
phenomenon would be one of the second order, probably referable
to the fact that longitudinal compression of a bar tends to produce
curvature.
169. The second class of vibrations, called torsional, which
depend on the resistance opposed to twisting, is of very small
importance. A solid or hollow cylindrical rod of circular section
may be twisted by suitable forces, applied at the ends, in such a
manner that each transverse section remains in its own plane.
But if the section be not circulai*, the effect of a twist is of a
more complicated character, the twist being necessarily attended
by a warping of the layers of matter originally composing the
normal sections. Although the effects of the warping might pro-
bably be determined in any particular case if it were worth
while, we shall confine ourselves here to the case of a circular
section, when there is no motion parallel to the axis of the rod.
The force with which twisting is resisted depends upon an
elastic constant different from g, called the rigidity. If we de-
note it by n, the relation between q, n, and fi may be written
.(1)',
n ?_
2(^-hl)
^ Donkin's Acoustics, p. 154.
« Atm. de ChinUe, Lvn. 129—190.
' Thomson and Tait, § 688. This, it shonld be remarked, applies to isotropic
material only.
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254
LONGITUDINAL VIBRATIONS OF BARS.
[159.
shewing that n lies between Jg and ^q. In the case of /A = i,
n = fg.
Let us now suppose that we have to do with a rod in the form
of a thin tube of radius r and thickness dr, and let denote the
angular displacement of any section, distant x from the origin.
The rate of twist at x is represented by dOjdx, and the shear of the
material composing the pipe by rddjdx. The opposing force per
unit of area is nr dOjdx ; and since the area is 27rr dr, the moment
round the axis is
d0
^mrr^ dr
dx'
Thus the force of restitution acting on the slice dx has the
moment
d^e
2mn^ dr dx
d^'
Now the moment of inertia of the slice under consideration
is 2irrdr.dx,p.r^, and therefore the equation of motion assumes
the form
d^e d'0
Plt^^^'d^'
.(2).
Since this is independent of r, the same equation applies to a
cylinder of finite thickness or to one solid throughout.
The velocity of wave propagation is *J(n/p), and the whole
theory is precisely similar to that of longitudinal vibrations, the
condition for a free end being dO/dx = 0, and for a fixed end ^ = 0,
or, if a permanent twist be contemplated, = constant.
The velocity of longitudinal vibrations is to that of torsional
vibrations in the ratio V? - V*^ or \/(2 + 2/a) : 1. The same ratio
applies to the frequencies of vibration for bars of equal length
vibrating in corresponding modes under corresponding terminal
conditions. If /i = ^, the ratio of frequencies would be
Vg: Vw = V8 : V3 = l-63,
corresponding to an interval rather greater than a fifth.
In any case the ratio of frequencies must lie between
V2 : 1 = 1-414, and V3 : 1 = 1732.
Longitudinal and torsional vibrations were first investigated by
Chladni
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CHAPTER VIII.
LATERAL VIBRATIONS OF BARS.
160. In the present chapter we shall consider the lateral
vibrations of thin elastic rods, which in their natural condition are
straight. Next to those of strings, this class of vibrations is per-
haps the most amenable to theoretical and experimental treatment.
There is diflSculty sufficient to bring into prominence some im-
portant points connected with the general theory, which the fami-
liarity of the reader with circular functions may lead him to pass
over too lightly in the application to strings ; while at the same
time the difficulties of analysis are not such as to engross attention
which should be devoted to general mathematical and physical
principles.
Daniel Bernoulli ^ seems to have been the first who attacked
the problem. Euler, Riccati, Poisson, Cauchy, and more recently
Strehlke ', Lissajous ', and A. Seebeck ^ are foremost among those
who have advanced our knowledge of it.
161. The problem divides itself into two parts, according to
the presence, or absence, of a permanent longitudinal tension.
The consideration of permanent tension entails additional compli-
cation, and is of interest only in its application to stretched
strings, whose stiffiiess, though small, cannot be neglected al-
together. Our attention will therefore be given principally to the
two extreme cases, (1) when there is no permanent tension,
(2) when the tension is the chief agent in the vibration.
^ Comment. Acad, Petrop. t. xiii. > Pogg. Ann. Bd. xxvii. p. 505, ISSS.
» Ann. d. CUmie (3), xxx. 386, 1860.
« AbhandUmgen d. Math. Phys. Classe d. K. Sdchs. Gesellschaft d, Witten-
tehaften. Leipzig, 1862.
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256
LATERAL VIBRATIONS OF BARS.
[161.
With respect to the section of the rod, we shall suppose that
one principal axis lies in the plane of vibration, so that the bending
at every part takes place in a direction of maximum or minimum
(or stationary) flexural rigidity. For example, the surface of the
rod may be one of revolution, each section being circular, though
not necessarily of constant radius. Under these circumstances the
potential energy of the bending for each element of length is pro-
portional to the square of the curvature multiplied by a quantity
depending on the material of the rod, and on the moment of
inertia of the transverse section about an axis through its centre of
inertia perpendicular to the plane of bending. If od be the area
of the section, k^o) its moment of inertia, q Young's modulus, ds the
element of length, and dV the corresponding potential energy for
a curvature 1 -s- ii of the axis of the rod,
dV = iqK^<o^
.(1).
This result is readily obtained by considering the extension of
the various filaments of which the bar may be supposed to be
made up. Let rj be the distance from the axis of the projection
on the plane of bending of a filament of section da). Then the
length of the filament is altered by the bending in the ratio
li being the radius of curvature. Thus on the side of the axis for
which 7f is positive, viz. on the outward side, a filament is extended,
while on the other side of the axis there is compression. The
force necessary to produce the extension rjlR is q rf/R . da> by the
definition of Young's modulus; and thus the whole couple by
which the bending is resisted amounts to
/■
^i
. d(li) = -^ K^CO,
if CD be the area of the section and k its radius of gyration about
a line through the axis, and perpendicular to the plane of bending.
The angle of bending corresponding to a length of axis ds is ds-i-It,
and thus the work required to bend ds to curvature 1 -r- iZ is
1 a ^
since the mean is half the final value of the couple.
[For a more complete discussion of the legitimacy of the
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161.] POTENTIAL ENERGY OF BENDING. 257
foregoing method of calculation the reader must be referred to
iiirorks upon the Theory of Elasticity. The question of lateral
vibrations has been specially treated by Pochhammer^ on the
basis of the general equations.]
For a circular section /c is one-half the radius.
That the potential energy of the bending would be proportional,
ccBteris paribus, to the square of the curvature, is evident before-
hand. If we call the coeflScient -B, we may take
or, in view of the approximate straightness,
^'^K^^ «•
in which y is the lateral displacement of that point on the axis of
the rod whose abscissa, measured parallel to the undisturbed posi-
tion, is X, In the case of a rod whose sections are similar and
similarly situated £ is a constant, and may be removed from under
the integral sign.
The kinetic energy of the moving rod is derived partly from
the motion of translation, parallel to y, of the elements composing
it, and partly from the rotation of the same elements about axes
through their centres of inertia perpendicular to the plane of vibra-
tion. The former part is expressed by
i I /t>ft> y^dx ,
■(3),
if p denote the volume-density. To express the latter part, we have
only to observe that the angular displacement of the element dx is
dy/dx, and therefore its angular velocity d^y/dt dx. The square of
this quantity must be multiplied by half the moment of inertia of
the element, that is, by ^/v'/xo dx. We thus obtain
2'=i//,a.y'(te + i/«»p«(Hy«ir (4).
1 CreUe, Bd. 81, 1876.
R. 17
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258 LATERAL VIBRATIONS OF BARS. [l62.
162. In order to form the equation of motion we may avail
ourselves of the principle of virtual velocities. If for simplicity we
confine ourselves to the case of uniform section, we have
'"St-^^-'i^^ «•
where the terms free from the integral sign are to be taken between
the limits. This expression includes only the internal forces due
to the bending. In what follows we shall suppose that there are
no forces acting from without, or rather none that do work upon
the system. A force of constraint, such as that necessary to hold
any point of the bar at rest, need not be regarded, as it does no
work and therefore cannot appear in the equation of virtual velo-
cities.
The virtual moment of the accelerations is
Thus the variational equation of motion is
in which the terms free from the integral sign are to be taken
between the limits. From this we derive as the equation to be
satisfied at all points of the length of the bar
while at each end
or, if we introduce the value of B, viz. y/c*©, and write q/p = 6^
^'y + 6V»'^^-*»-^ = (4)
l\-
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162.] TERMINAL CONDITIONS. 259
and for each end
In these equations b expresses the velocity of transmission of
longitudinal waves.
The condition (5) to be satisfied at the ends assumes different
forms according to the circumstances of the case. It is possible to
conceive a constraint of such a nature that the ratio S (dy/dx) : Sy
has a prescribed finite value. The second boundary condition is
then obtained fi:om (5) by introduction of this ratio. But in all
the cases that we shall have to consider, there is either no constraint
or the constraint is such that either 8 {dyldx) or 8y vanishes, and
then the boundary conditions take the form
das'
<t)-'- {^-"U]^'" <«^
We must now distinguish the special cases that may arise. If
an end be free, Sy and B{dy/dx) are both arbitrary, and the
conditions become
a^ ' dt^dx daf
the first of which may be regarded as expressing that no couple
acts at the free end, and the second that no force acta
If the direction at the end be free, but the end itself be con-
strained to remain at rest by the action of an applied force of the
necessary magnitude, in which case for want of a better word the
rod is said to be supported, the conditions are
S=0' «y=o (8).
by which (5) is satisfied.
A third case arises when an extremity is constrained to main-
tain its direction by an applied couple of the necessary magnitude,
but is free to take any position. We have then
^i)-"'
17—2
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260 LATERAL VIBRATIONS OF BARS. [162.
Fourthly, the extremity may be constrained both as to
position and direction, in which case the rod is said to be clamped.
The conditions are plainly
^i)-"'
Sy = o (10).
Of these four cases the first and the last are the more
important; the third we shall omit to consider, as there are
no experimental means by which the contemplated constraint
could be realized. Even with this simplification a considerable
variety of problems remain for discussion, as either end of the
bar may be free, clamped or supported, but the complication
thence arising is not so great as might have been expected.
We shall find that diflferent cases may be treated together,
and that the solution for one case may sometimes be derived
immediately from that of another.
In experimenting on the vibrations of bars, the condition
for a clamped end may be realized with the aid of a vice of
massive construction. In the case of a free end there is of course
DO difficulty so far as the end itself is concerned ; but, when both
ends are free, a question arises as to how the weight of the bar
is to be supported. In order to interfere with the vibration
as little as possible, the supports must be confined to the neigh-
bourhood of the nodal points. It is sometimes sufficient merely
to lay the bar on bridges, or to pass a loop of string round the bar
and draw it tight by screws attached to its ends. For more exact
purposes it would perhaps be preferable to cany the weight of
the bar on a pin traversing a hole drilled through the middle of
the thickness in the plane of vibration.
When an end is to be 'supported,' it may be pressed into
contact with a fixed plate whose plane is perpendicular to the
length of the bar.
163. Before proceeding further we shall introduce a sup-
position, which will greatly simplify the analysis, without seriously
interfering with the value of the solution. We shall assume that
the terms depending on the angular motion of the sections of
the bar may be neglected, which amounts to supposing the
inertia of each section concentrated at its centre. We shall
afterwards (§ 186) investigate a correction for the rotatory in-
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163.] HARMONIC VIBRATIONS. 261
ertia, and shall prove that under ordinary circumstances it is
small. The equation of motion now becomes
S^-'-S-o «•
xmd the boundary conditions for a free end
g-». s-» *^^
The next step in conformity with the general plan will be
the assumption of the harmonic form of y. We may conveniently
take
y^u cos
(ym'f) (3),
where I is the length of the bar, and 7n is an abstract number,
whose value has to be determined. Substituting in (1), we
obtain
d^^T"" ^*>-
If t/ = eP"^' be a solution, we see that p is one of the fourth
roots of unity, viz. +1, —1, +i, — i; so that the complete
solution is
t* = 4cosm|+58iiim| + (7e«*" + jDe-^/' (4a),
containing four arbitrary constants.
[The simplest case occurs when the motion is strictly periodic
with respect to x, C and D vanishing. If \ be the wave-length
and T the period of the vibration, we have
2ir m 27r ,m*
«^*^** ^=2^6 ^*^>-l
In the case of a finite rod we have still to satisfy the four
boundary cx)nditions, — two for each end. These determine the
ratios A : B : C : D, and furnish besides an equation which m
must satisfy. Thus a series of particular values of m are alone
admissible, and for each m the corresponding u is determined in
everything except a constant multiplier. We shall distinguish the
different fiinctions u belonging to the same system by suffixes.
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262 LATERAL VIBRATIONS OF BARS. [l63.
The value of y at any time may be expanded in a series of
the functions u (§§ 92, 93). If <^, ^, &c. be the normal co-
ordinates, we liave
j^ = <^Wi + <^ti,+ (5),
and r = i/>« |(^t^i + ^atij-f ...ydx
= i/>»Ui*[wi»da7 + 4>,«A/,»dar+...l (6),
We are fully justified in asserting at this stage that each
integrated product of the functions vanishes, and therefore the
process of the following section need not be regarded as more
than a verification. It is however required in order to determine
the value of the integrated squares.
164. Let Umy Unc denote two of the normal functions cor-
responding respectively to m and m\ Then
or, if dashes indicate differentiation with respect to {mxjl),
tim"" = t^, t^'''" = U^' (2).
If we subtract equations (1) after multiplying them by m,« ,
M,rt respectively, and then integrate over the length of the bar,
we have
•"''"*'d^ "'''"' da^^ dx dx' '^ dx dx' ^•^^'
the integrated terms being taken between the limits.
Now whether the end in question be clamped, supported, or
f^ee^ each term vanishes on account of one or other of its
^ The reader sbonld observe that the oases here specified are partioolar, and
that the right-hand member of (S) yanishes, provided that
Bnd ^^ : ^"«»=^"«>' ; ^^',
dx ' dx' dx ' dx*
These conditions indade, for instance, the case of a rod whose end is urged
towards its position of eqnilibrinm by a force proportional to the displacement, as
hy a spring without inertia.
>
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164.] CONJUGATE PROPERTY. 263
fetctors. We may therefore conclude that, if Um, Um' refer to two
modes of vibration (corresponding of course to the same terminal
conditions) of which a rod is capable, then
/«
jUmthn'dx^O (4),
provided m and m' be different.
The attentive reader will perceive that in the process just
followed, we have in fact retraced the steps by which the funda-
mental differential equation was itself proved in § 162. It is the
original variaHonal equation that has the most immediate con-
nection with the conjugate property. If we denote y by w and 8y
ajid the equation in question is
^/SS*'+^'"/**'*'=^ <^>-
Suppose now that u relates to a normal component vibration,
so that H + v?u = 0, where n is some constant ; then
n'p<^juvdw^BJ^^dx.
By similar reasoning, if v be a normal function, and u represent
any displacement possible to the system,
n'*p<.fuvdx = BJ^^d<v.
We conclude that if u and v be both normal functions, which
have different periods,
\yvdx = (6);
/•
and this proof is evidently as direct and general as could be
desired.
The reader may investigate the formula corresponding to (6),
when the term representing the rotatory inertia is retained.
By means of (6) we may verify that the admissible values of n^
are real. For if w* were complex, and u = a + ifi were a normal
function, then a -1^9, the conjugate of u, would be a normal
function also, corresponding to the conjugate of n', and then the
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^
264 LATERAL VIBRATIONS OF BARS. [l64.
product of the two functions, being a sum of squares, would not
vanish, when integrated^
If in (3) 771 and m be the same, the equation becomes iden-
tically true, and we cannot at once infer the value of jv^dir.
We must take m' equal to m + Sm, and trace the limiting form of
the equation as hm tends to vanish. [It should be observed that
the function tt^+sm is not a normal function of the system ; it is
supposed to be derived from v^ by variation of m in (4fa) § 163,
the coeflScients A, B,C, D being retained constant.] In this way
we find
4m^ r 2^ « _^ ^ _ ^ ^ 4. ^ A. ^^ _du d d?u
b J dmda^ dmda? da^dmdx dxdmdx^'
the right-hand side being taken between the limits.
*T du m , o du X , a
Now dS = T"''^-' d;» = l"'"^'
and thus
in which v!'" = w, so that
-y I Wm^Cto = 3mm " + -p t^« p U m'" - w'll" + -j- (it ')'. - .(7),
between the limits.
Now whether an end be clamped, supported, or free,
and thus, if we take the origin of x at one end of the rod,
[\t«(ia;= ft (tt»- 2w'ir + u"«)y
= ii(M«-2M'u'" + w"Vf .....(8).
The form of our integral is independent of the terminal con-
dition at a? = 0. If the end ir = Z be free, u" and u'" vanish, and
accordingly
[ u^dx = \l u^{iy (9).
^ This method is, I belieye, doe to Poisson.
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164.] VALU£8 OF INTEGBATED SQUARES. 265
that is to say, for a rod with one end free the mean value of u^ is
one-fourth of the terminal value, and that whether the other end
be clamped, supported, or free.
Again, if we suppose that the rod is clamped at a; = Z, u and u'
vanish, and (8) gives
Since this must hold good whatever be the terminal condition at
the other end, we see that for a rod, one end of which is fixed and
the other free,
( u^dx = ^lu^ (free end) = \W^ (fixed end),
shewing that in this case u^ at the free end is the same as w"" at
the clamped end.
The annexed table gives the values of four times the mean of u'
in the different cases.
clamped, free
free, free
w' (free end), or w"' (clamped end)
w* (free end)
m"' (clamped end)
- 2u'u"' (supported end) = 2u"
M* (free end), or - 2u'u'" (supported end)
w"* (clamped end), or - 2m' m'" (supported end)
clamped, clamped ...
supported, supported
supported, free
supported, clamped
By the introduction of these values the expression for T
assumes a simpler form. In the case, for example, of a clamped-
free or a free-free rod,
2'=^{^*V(0+^.''«,'(0 + ...} (10),
where the end x=^l is supposed to be free.
166. A similar method may be applied to investigate the
values of ju'^dx, and jv!'^dx. In the derivation of equation (7) of
the preceding section nothing was assumed beyond the truth of
the equation v!'" ^u, and since this equation is equally true of any
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266
LATERAL VIBRATIONS OF BARS.
[165.
of the derived functions, we are at liberty to replace u by u' or u\
Thus
taken between the limits, since the term u vl' vanishes in all three
cases.
For a free-free rod
~j u'^dx = 3 (uu')i - 3 {uu\ + m (u'O/
= 6(Mi£'), + m(tt'«), (1),
for, as we shall see, the values of uu' must be equal and opposite
at the two ends. Whether u be positive or negative at a? = /,
u v! is positive.
For a rod which is clamped at a; = and free at a; = /
*PJu'^dx = 3 {uuy + mui'^ + {u''u'"\.
[We have already seen that Uo'=± ui; and it may be proved
from the formulee of § 173 that
Wo'" _ w/ _ cos m + cosh m
u{ " vi sin m sinh m *
so
Thus
. 1 . (m"w'")o (cos m 4- cosh my ,
tnat -7-7 — r- = — r-T r— rr = — l.J
{uu)i sm'msmh'm
yju'*da!^2(uu)i + mur' (2),
4m r'
lo'
a result that we shall have occasion to use later.
find
By applying the same equation to the evaluation of 1 u"*dx, we
=rm(w"«-2w'a'" + t*'')/,
since w'u" and wu'" vanish.
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165.] NORMAL EQUATIONS. 267
Comparing this with (8) § 164, we see that
ju''^dx=ju*dx (3),
whatever the terminal conditions may be.
The same result may be arrived at more directly by integrating
by parts the equation
m* . d*u
166. We may now form the expression for V in terms of the
normal co-ordinates.
"'•^/i^s-*-^^-}'*
"2"
If the functions u be those proper to a rod free at a; = /, this expres-
sion reduces to
F=^|^|7n,*K(0?*,« + 7i^*K(0?*,«+...} (2).
In any case the equations of motion are of the form
p(DWdx 4>i'h—ijr^nH*Wdx ^ = *i (3),
and, since ^iS^ is by definition the work done by the impressed
forces during the displacement S<^,
^1 = j YtLipcadx (4).
if YptDdx be the lateral force acting on the element of mass ptodx.
If there be no impressed forces, the equation reduces to
*> + ^t^V = (5).
as we know it ought to do.
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268
LATERAL VIBRATIONS OF BARS.
[167.
167. The significance of the reduction of the integrals
Ju^dnc to dependence on the terminal values of the function and
its derivatives may be placed in a clearer light by the following
line of argument. To fix the ideas, consider the case of a
rod clamped at a? = 0, and free at fl5 = Z, vibrating in the normal
mode expressed by u. If a small addition Al be made to the
rod at the free end, the form of u (considered as a function of
x) is changed, but, in accordance with the general principle
established in Chapter iv. (§ 88), we may calculate the period
under, the altered circumstances without allowance for the change
of type, if we are content to neglect the square of the change.
In consequence of the straightness of the rod at the place where
the addition is made, there is no alteration in the potential
energy, and therefore the alteration of period depends entirely
on the variation of T, This quantity is increased in the ratio
I u^dx : I V
Jo Jo
v?dx^
or
Ij^^l^^
Jlu^dx'
which is also the ratio in which the square of the period is
augmented. Now, as we shall see presently, the actual period
varies as P, and therefore the change in the square of the period
is in the ratio
1:1 + 4Ai/i.
A comparison of the two ratios shews that
ui"
: ju^dx=^
4 : L
The above reasoning is not insisted upon as a demonstration,
but it serves at least to explain the reduction of which the in-
tegral is susceptible. Other cases in which such integrals occur
may be treated in a similar manner, but it would often require
care to predict with certainty what amount of discontinuity in the
varied type might be admitted without passing out of the range
of the principle on which the argument depends. The reader
may, if he pleases, examine the case of a string in the middle
of which a small piece is interpolated.
168. In treating problems relating to vibrations the usual
course has been to determine in the first place the forms of the
normal functions, viz. the functions representing the normal
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168.] INITIAL CONDITIONS. 269
types, and afterwards to investigate the integral formulse by
means of which the particular solutions may be combined to
suit arbitrary initial circumstances. I have preferred to follow
a different order, the better to bring out the generality of the
method, which does not depend upon a knowledge of the normal
fiOictione. In pursuance of the same plan, I shall now investigate
the connection of the arbitrary constants with the initial circum-
stances, and solve one or two problems analogous to those treated
under the head of Strings.
The general value of y may be written
y = f -4i cos y niiH + Bi sin ^- miHj Ui
/ tcb Kb \
+ ( -4j cos -^ rn^H + B^ sin y m^Hj u^
+ (1),
so that initially
yo = ^iWi + -4,t4,+ (2),
Kb
yo = -|i^{^"Ai*i + ^'^j^ + ...} (3).
If we multiply (2) by t^ and integrate over the length of the
rod, we get
[y^xirdx — Ar lur^dx (4),
and similarly from (3)
■l^jyoUrdx=-nir''Brjur^dx (5),
formulae which determine the arbitrary constants Art Br*
It must be observed that we do not need to prove analytically
the possibility of the expansion expressed by (1). If all the
particular solutions are included, (1) necessarily represents the
most general vibration possible, and may therefore be adapted
to represent any admissible initial state.
Let us now suppose that the rod is originally at rest, in its
position of equilibrium, and is set in motion by a blow which
imparts velocity to a small portion of it. Initially, that is, at
the moment when the rod becomes free, y© = 0, and y© differs from
zero only in the neighbourhood of one point {x = c).
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270 LATERAL VIBRATIONS OF BARS. [16,
From (4) it appears that the coefBcients A vanish, and frcn
(5) that
rrir^Br I Ur^dx ^-j^u^(c) I jff^dx.
Calling fyopwdx, the whole momentum of the blow, Y, we
have
J. _ 1*7 t/,(c)
^'~Kb(m mr'fur'dx ^^^'
and for the final solution
l^y {Me)uy(a!).('cb \
-^5s^(f-'0- } (^)-
In adapting this result to the case of a rod free at ^ = 2, we
may replace
fur^dx by \l[ur(l)]\
/«
If the blow be applied at a node of one of the normal com-
ponents, that component is missing in the resulting motion. The
present calculation is but a particular case of the investigation
of § 101.
169. As another example we may take the case of a bar,
which is initially at rest but deflected from its natural position
by a lateral force acting at x = c. Under these circumstances
the coeflScients B vanish, and the others are given by (4), § 168.
Now
and on integrating by parts
j^y^l^^^y'd^^ di d^
in which the terms free from the integral sign are to be taken
between the limits ; by the nature of the case j/q satisfies the
same terminal conditions as does tir, and thus all these terms
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169.] SPECIAL CASES. 271
vanish at both limits. If the external force initially applied
to the element dx be Ydx, the equation of equilibrium of the
bar gives
,«^6«g«=F (1).
and accordingly
If we now suppose that the initial displacement is due to
a force applied in the immediate neighbourhood of the point
a; = c, we have
and for the complete value of y at time t,
In deriving the above expression we have not hitherto made
any special assumptions as to the conditions at the ends, but
if we now confine ourselves to the case of a bar which is clamped
at d; » and free at a? = i, we may replace
Lr^dx by ii[Wr(OP-
If we suppose further that the force to which the initial deflection
is due acts at the end, so that c = /, we get
When ^ = 0, this equation must represent the initial displace-
ment. In cases of this kind a difficulty may present itself as
to how it is possible for the series, every term of which satisfies
the condition j/" ^0^ to represent an initial displacement in
which this condition is violated. The fact is, that after triple
differentiation with respect to x^ the series no longer converges
for a? = Z, and accordingly the value of j/" is not to be arrived
at by making the differentiations first and summing the terms
afterwards. The truth of this statement will be apparent if
we consider a point distant dl from the end, and replace
M"\l^dl) by v:"{l)-u^{l)dl,
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272 LATERAL VIBRATIONS OF BARS. [l69.
in which v}"^ (I) is equal to
For the solution of the present problem by normal co-ordinates
the reader is referred to § 101.
170. The forms of the normal functions in the various par-
ticular cases are to be obtained by determining the ratios of the
four constants in the general solution of
If for the sake of brevity w be written for (mx/l), the solution
may be put into the form
w = -4 (cos X 4- cosh x) + B (cos sc' - cosh a?')
+ G(smx +sinh a?') +i5(sina?' — sinha?') (1),
where cosh x and sinh x are the hyperbolic cosine and sine of x,
defined by the equations
coshir = J(6* + e-*), sinha? = J(e*-c-*) (2).
I have followed the usual notation, though the introduction of
a special symbol might very well be dispensed with, since
cosh X = cos ix, sinh a? = — i sin ta? (3),
where % = V(— 1) J ^^^ ^^^^ ^^^ connection between the formulae of
circular and hyperbolic trigonometry would be more apparent. The
rules for differentiation are expressed in the equations
-7- cosh X = sinh x, -y- sinh x = cosh x
ax ax
-j-^ cosh X = cosh X, -7-^ sinh x = sinh x.
In differentiating (1) any number of times, the same four com-
pound functions as there occur are continually reproduced. The
only one of them which does not vanish with x is cos x + cosh x\
whose value is then 2.
Let us take first the case in which both ends are free. Since
d^u/dx!^ and d^ujda? vanish with a?, it follows that jB = 0, i) = 0, so
that
11 = A (cos x' + cosh x') + C (sin x' + sinh x) (4).
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170. J NOBMAL FUNCTIONS FOR FRBE-FREB BAR. 273
We have still to satisfy the necessary conditions when a? = Z, or
X = 7». These give
-4 (—cos mH- cosh m)+ 0(- sinm + sinhm) — )
A{^ sinw + sinhm)+ C(-co8mH-coshm) = j ^ ''
equations whose compatibility requires that
(cosh m — cos mf = sinh" m — sin" m,
or in virtue of the relation
cosh'm — sinh'w = l (6),
cosm cosh w = l (7).
This is the equation whose roots are the admissible values of m.
If (7) be satisfied, the two ratios oi A \ G given in (6) are equal,
and either of them may be substituted in (4). The constant multi-
plier being omitted, we have for the normal function
u = (sin m — sinh m) jcos -j- + cosh -j- 1
— (cos m — cosh w) -jsin ,-H-sinh-y|- (8),
or, if we prefer it,
u = (cos m — cosh m) jcos -j- + cosh -p[
+ (sin mH- sinh 7w) -jsin -^ + sinh j-y (9);
and the simple harmonic component of this type is expressed by
tb
y=:Pttcos(~m«^+e) (10).
Kb
171. The frequency of the vibration is o~ii^'» ^^ which b is
a velocity depending only on the material of which the bar is
formed, and m is an abstract number. Hence for a given material
and mode of vibration the frequency varies directly as k — ^the
radius of gyration of the section about an axis perpendicular to the
plane of bending — and inversely as the sqimre of the length. These
results might have been anticipated by the argument from dimen-
sions, if it were considered that the frequency is necessarily
determined by the value of Z, together with that of xb — the
only quantity depending on space, time and mass, which occurs in
R. 18
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274
LATERAL TIBBATION8 OF BAB8.
[171.
the differential eqaation. If eyerything ooncenuDg a bar be given,
except its absolute magnitude, the frequency yaries inversely as
the linear dimension.
These laws find an important application in the case of tuning-
forks, whose prongs vibrate as rods, fixed at the ends where they
join the stalk, and free at the other ends. Thus the period of vibra-
tion of forks of the same material and shape varies as the linear
dimension. The period will be approximately independent of the
thickness perpendicular to the plane of bending, but will vary
inversely with the thickness in the plane of bending. When the
thickness is given, the period is as the square of the length.
Jn order to lower the pitch of a fork we may, for temporary
purposes, load the ends of the prongs with soft wax, or file away
the metal near the base, thereby weakening the spring. To raise
the pitch, the ends of the prongs, which act by inertia, may be
filed.
The value of b attains its maximum in the case of steel, for
which it amounts to about 5287 metres per second. For brass
the velocity would be less in about the ratio 1*5 : 1, so that a
tuning-fork made of brass would be about a fifth lower in pitch
than if the material were steel.
[For the design of steel vibrators and for rough determinations
of frequency, especially when below the limit of hearing, the
theoretical formula is often convenient. If the section of the bar
be rectangular and of thickness t in the plane of vibration, k* « i^;
and then with the above value of 6, and the values of m given
later, we get as applicable to the gravest mode
(clamped-free) frequency = 84590 f/P,
(free-free) frequency = 538400 t/P,
I and t being expressed in centimetrea
The first of these may be used to calculate the pitch of steel
tuning-forks.
The lateral vibrations of a bar may be excited by a blow, as
when a tuning-fork is struck against a pad. This method is also
employed for the barmonicon, in which strips of metal or glass are
supported at the nodes, in such a manner that the fi*ee vibrations
are but little impeded. A frictional maintenance may be obtained
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171.] NORMAL FUNCTIONS FOR CLA.MPED-CLAMPED BAR. 275
with a bow, or by the action of the wetted fingers upon a slender
rod of glass suitably attached. The electro-magnetic maintenance
of forks has been already considered, § 64. It may be applied with
equal facility to the case of metal bars, or even to that of
wooden planks carrying iron armatures, free at both ends and
supported at the nodes. The maintenance by a stream of wind
of the vibrations of harmonium and organ reeds may also be
referred to.
The sound of a bar vibrating laterally may be reinforced by a
suitably tuned resonator, which may be plcu^ed under the middle
portion or under one end. On this principle dinner gongs have
been constructed, embracing one octave or more of the diatonic
scale.]
172. The solution for the case when both ends are clamped
may be immediately derived from the preceding by a double dif-
ferentiation. Since y satisfies at both ends the terminal con-
ditions
^y^o ^y=o
it is clear that y" satisfies
■-»• f-».
which are the conditions for a clamped end. Moreover the general
differential equation is also satisfied by y". Thus we may take,
omitting a constant multiplier, as before,
u = (sin m — sinh m) {cos a?' — cosh x'}
— (cos m — cosh m) {sin a?' — sinh a?'} (1),
while m ia given by the same equation as before, nam ely,
cosm coshm=l (2).
We conclude that the component tones have the same pitch in the
two cases.
In each case there are four systems of points determined by
the evanescence of y and its derivatives. Where y vanishes, there
is a node ; where y' vanishes; a loop, or place of maximum displace-
ment; where y" vanishes, a point of inflection; and where y"'
vanishes, a place of maximum curvature. Where there are in the first
case (free-firee) points of inflection and of maximum curvature, there
18—2
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276 LATERAL VIBRATIONS OF BARS. [172.
are in the second (clamped-clamped) nodes and loops respectively;
and vice versd, points of inflection and of maxiraum curvature for
a doubly-clamped rod correspond to nodes and loops of a rod whose
ends are free.
173. We will now consider the vibrations of a rod clamped at
a? = 0, and free at a? = Z. Reverting to the general integral (1)
§ 170, we see that A and C vanish in virtue of the conditions at
^ s 0, so that
t^ = JB(cosa?' — coshir') + D(sina;'-8inha?') (1).
The remaining conditions at x = l *give
JB( cosm + cosh m) + i)(sin m -|-sinhm) = )
B (- sin m + sinh m) + D (cos m + cosh w) = j *
whence, omitting the constant multiplier,
u = (sin m + sinh m) jcos , cosh -j-y
— (cos m + cosh m) jsin -, - sinh -,- 1 (2),
or
u = (cos m + cosh m) jcos -j — cosn -j-Y
+ (smm — smhm) jsm —, — smh-y^ > (3),
where m must be a root of
cosm cosh m + 1=0 .(4).
The periods of the component tones in the present problem are
thus different from, though, as we shall see presently, nearly re-
lated to, those of a rod both whose ends are clamped, or free.
If the value of w in (2) or (3) be differentiated twice, the re-
sult {u") satisfies of course the fundamental differential equation.
At a? = 0, d?u"lda^, d^u^jda^ vanish, but at a? = i u" and du"ldx
vanish. The function u" is therefore applicable to a rod clamped
at I and free at 0, proving that the points of inflection and of
maximum curvature in the original curve are at the same distances
from the clamped end, as the nodes and loops respectively are
from the free end.
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174.]
CALCULATION OF PERIODS.
277
174 In default of tables of the hyperbolic cosine or its loga-
rithm, the admissible values of m may be calculated as follows.
Taking first the equation
cosm coshm=sl (1),
we see that m, when large, must approximate in value to
^(2i + 1) TT, 1 being an integer. If we assume
m = i(2t + l)^-(-l)»/8 (2),
P will be positive and comparatively small in magnitude.
Substituting in (1), we find
cot i)9 = 6"* = e*<*<+^>' e-<-')^ ;
or, if c*^'"'"''' be called a,
atanii8=re(-W (3),
an equation which may be solved by successive approximation after
expanding tan^/8 and ^~'^'^ in ascending powers of the small
quantity /3. The result is
^ 2 , ^,,4 34 , ,wll2
(4)S
which is sufficiently accurate, even when % = 1.
By calculation
A « 0179666 - 0003228 + 0000082 - 0000002 = 0176518.
/Ss) fit* 0*f fit are found still more easily. After ^o the first term of
the series gives fi correctly as far as six significant figures. The
table contains the value of fi, the angle whose circular measure is
/3, and the value of sin ^/3, which will be required further on.
Free-Free Bar.
P-
P expiesaed in degrees,
minntes, and seoondg.
-.f.
1 1 10-' X -176518
2 1 10-» X -777010
3 10-« X -335506
4 10-» X -144989
6 ' 10-' X -626556
1
1* 0' 40"-94
2' 40''-2699
6"-92029
-299062
-0129237
10-* X -88258
10-' X -38850
10-* X -16775
10-» X -72494
10-' X -31328
^ This process is somewhat similar to that adopted by Strehlke.
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278 LATERAL VIBRATIONS OF BARS. [l74.
The values of r?i which satisfy (1) are
Wi= 4-7123890 + i8,= 4-7300408
r?i,= 7-8539816-^,= 7*8532046
f/is = 10-9955743 + /S, = 10*9956078
m, « 14-1371669 - A ^ 141371655
m, = 17-2787596 + /^^ = l7-2787o9«
after which m = ^(2i+ l)7r to seven decimal places.
We will now consider the roots of the equation
cosm co8hm = — 1 (5)^
[Assuming
w, = i(2i-l)7r-(-iyai (6),
we have e"^ = cot ^Oi = e*^*'"*^' . e^^^^^^t ,
or a tan^Oi+i = e~<~^^*«+i (7),
a having the value previously defined.
Thus, as in (4),
«"-'-(-v^,^.-(-»'"J+ (»)■
a,>, being approximately equal to /Si.
The values calculated from (8) are
a, = 10-^ X -182979, a, = 10"^ x 335527,
a, - lO-» X -775804, a, = 10-» x 144989,
after which the diflerence between «»•+! and fit does not appear.]
The value of Oi may be obtained by trial and error from the
equation
logio cot i ai - -6821882 - -43429448 a, = 0,
and will be found to be
a, = -3043077.
Another method by which mi may be obtained directly will be
given presently.
The values of m, which satisfy (5), are
Tfh^ 1-5707963 + ai = 1875104
m,= 4-7123890 -0^= 4-694098
m,= 7-8539816 + a, = 7-854757
m, = 10-9955743 - 04 = 10-995541
VI, = 14-1371669 + a« = 14137168
m. =: 17-2787596 - a. = 17-278759 ,
1 The caloolation of the roots of (5) giyen in the first edition was affected by an
enor, ^hich has been pointed oat by Greenhill {Math, Me$s»^ Deo. 1SS6).
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174.] CALCULATION OF PERIODS. 279
after which m = ^ (2t — 1) v sensibly. The frequencies are propor-
tional to m', and are therefore for the higher tones nearly in the
ratio of the squares of the odd numbera However, in the case of
overtones of very high order, the pitch may be slightly disturbed
by the rotatory inertia, whose effect is here neglected.
175. Since the component vibrations of a system, not subject
to dissipation^ are necessarily of the harmonic type, all the values
of 711*, which satisfy
cosm coshm= ± 1 (1),
must be real. We see further that, if m be a root, so are also
— m, m»J{— 1), — m^/('~ 1). Hence, taking first the lower sign, we
have
^(c<.smcoshm+l) = l-j2 + j^2j^-
-i^-m-w)'- «■
If we take the logarithms of both sides, expand, and equate co*
efficients, we get
2— ; = Tij; 2— :=St^.s^; &c (3).
This is for a clamped-free rod.
From the known value of 2m~^, the value of m^ may be derived
with the aid of approximate values of ms, m, We find
2w-^ = 006547621,
and mf^ = -000004242
mf^ = -000000069
mr" = 000000005,
whence m{-^ = 006543305
giving rrii = "1875104, as before.
In like manner, if both ends of the bar be clamped or free,
•-igs^--(>-5)('-S)*- <*'■
whence 2 — r = tctoc ^^-y where of course the summation is exclu-
nr IZ.O0
sive of the zero value of m.
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280
LATERAL VIBRATIONS OF BARS.
[176.
nv
176. The frequencies of the series of tones are proportional to
The interval between any tone and the gravest of the series
may conveniently be expressed in octaves and fractions of an
octave. This is effected by dividing the difference of the logarithms
of m* by the logarithm of 2. The results are as follows :
1-4629
2-6478
2-4358
4-1332
3-1390
51036
3-7382, &c.
6-8288, &c.
where the first column relates to the tones of a rod both whose
ends are clamped, or free ; and the second column to the case of a
rod clamped at one end but free at the other. Thus from the
second column we find that the first overtone is 2*6478 octaves
higher than the gravest tone. The fractional part may be reduced
to mean semitones by multiplication by 12. The interval is then
two octaves + 7*7736 mean semitones. It will be seen that the
rise of pitch is much more rapid than in the case of strings.
If a rod be clamped at one end and free at the other, the pitch
of the gravest tone is 2 (log 4-7300 - log 1-8751) -r- log 2 or 2-6698
octaves lower than if both ends were clamped, or both fr^e.
177. In order to examine more closely the curve in which the
rod vibrates, we will transform the expression for u into a form
more convenient for numerical calculation, taking first the case
when both ends are free. Since w = ^(2i + l)7r — (— 1)* /8,
cosm = 8in/8, sinm = cosiV x cos)8; and therefore, m being a
root of cos m cosh m = 1, cosh m = cosec /8.
Also
sinh' m = cosh' m — 1 = tan' m = cot^ /8,
or, since cot /8 is positive,
sinh m = cot /8.
Thus
sin m — sinh m 1 — cos tV sin /3
cos m — cosh m
cos/8
(cos ^ff — cos tV sin ^/8)'
(cos J/3 — cos tV sin J/8)(cos i/8 + cos iir sin ^fi)
_ cos ^/8 COS ITT — sin ^ff
"" cos i/8 COB tTT + sin i^/S '
\
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177.] GRAVEST MODE FOR FREE-FREE BAR. 281
We may therefore take, omitting the constant multiplier,
u = (cos i/8 cos ITT + sm ijS) jsm -y- + smh -j- >
cos -J- + cosh -y- r
= V2cosi7rsin|^-^ + (-iy||
+ 8ini)8e««''-cosi7r cos^/Se-^*" (1).
if we further throw out the factor V2, and put Z « 1, we
may take
W^J^i + i^ + i's,
where
-Fj^cosiTTsinfma? — J7r + i( — l)*i8} \
logJ^,= Twa? log 6 + log sin ^^ — log V2 \ (*)»
log±-F, = - Twojloge + logcosi/S — logV2 J
from which u may be calculated for different values of i and x.
At the centre of the bar, a? = ^, and -P,, F^ are numerically
equal in virtue of ^ == cot ^ /3. When i is even, these terms cancel.
For ^1, we have ^i = (— 1)* sin \ tV, which is equal to zero when
% is even, and to ± 1 when t is odd. When i is even, therefore,
the sum of the three terms vanishes, and there is accordingly a
node in the middle.
When a? = 0, w reduces to - 2 (- 1)< sin (J tt - i (- 1)* /8}, which
(since fi is always small) shews that for no value of t is there a
node at the end. If a long bar of steel (held, for example, at the
centre) be gently tapped with a hammer while varying points of
its length are damped with the fingers, an unusual deadness in
the sound will be noticed, as the end is closely approached.
178. We will now take some particular cases.
Vibration witk two nodes. i=l.
If i = 1, the vibration is the gravest of which the rod is capable.
Our formulae become
/\ « - sin [x{nO^ + V 0' 40" -94) - 45« - 30' 20" -47}
log F^ = 2-054231 x + 3-7952391
log jP, = - 2-054231 X + 1-8494681,
from which is calculated the following table, giving the values of
u for X equal to *00, 05, -10, &c.
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282
LATERAL VIBRATIONS OF BARS.
[178.
The values o( u:u (-5) for the intermediate values of x (in the
last column) were found by interpolation formulae. If o, p, q, r, e, t
be six consecutive terms, that intermediate between q and r is
X
Fi
F,
F.
u
u : u{-o)
1
•000
+ •7133200
+ -0062408
+•7070793
+ V4266401
+ 1-645219
•026
• • •
.. .
...
1454176
•050
•5292548
•0079059
•5581572
10953179
1-263134
•075
. . .
...
...
...
1072162
•100
•3157243
•0100153
•4406005
•7663401
•8837528
•125
, ,,
...
...
•6969004
•150
+ -0846166
•0126874
•3478031
•4451071
•5133028
•175
...
. . .
...
...
•3341625
•200
- 1512020
•0160726
•2745503
+ •I 394209
+ 1607819
•225
< ••
...
...
- 0054711
•250
•3786027
•0203609
•2167256
- ^1415162
•1631982
•275
< • •
. ..
...
...
•3109982
•300
•5849255
•0257934
•1710798
•3880523
•4475066
•325
...
...
...
...
•5714137
•350
•7586838
•0326753
•1350477
•5909608
•6815032
•375
. • •
...
•7766629
•400
•8902038
•0413934
•1066045
•7422059
•8559210
•425
...
...
...
...
•9184491
•450
•9721635
•0524376
•0841519
•8355740
•9635940
•475
•••
. . .
...
•9908730
•500
-I 000000
+ ^0664285
•0664282
- -8671433
- 1^0000000
Since the vibration curve is symmetrical with respect to the
middle of the rod, it is unnecessary to continue ihe table beyond
X = '5. The curve itself is shewn in fig. 28.
Fig. 28.
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178.]
FKEE-FREE BAR WITH THREE NODES.
283
To find the position of the node, we have by interpolation
^ = -2<^ + SrO-<>2^ = -22418.
which is the firaction of the whole length by which the node is
distant Jrom the nearer end.
Vibration with three nodes, i » 2.
J*! = sin { (450» - 2' 40" -27) a; - 45» + 1' 20" -135}
logFt= S-410604;e+ 4-438881 6
log (- i",) = - 3-410604 X + 1-8494850.
SB
« : - M (0)
X
I
«:-u(0)
•000
-1^0000
•260
+ •5847
•025
•8040
•276
•6374
•050
•6079
•300
•6620
•075
•4147
•325
•6569
•100
•2274
•350
•6246
•125
- 0487
[ 376
•5652
•150
+ 1175
•400
•4830
•175
•2672
1 ^425
•3805
•200
•3972
•450
•2627
•225
•6037
•475
•1340
•600
•0000
In this table, as in the preceding, the values of u were calcu-
lated directly for x = '000, "050, '100 &c , and interpolated for the
intermediate values. For the position of the node the table gives
by ordinary interpolation a? = '132. Calculating from the above
formulae, we find
i^(1321) = --000076.
1^ (-1322)=: + -000881,
whence a? = •132108, agreeing with the result obtained by Strehlke.
The place of maximum excursion may be found from the derived
function. We get
u (-3083) = + 0006077, u' (3084) = - -0002227,
whence u' (-308373) = 0.
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284
LATERAL VIBRATIONS OF BARS.
[178.
Hence ti is a maximum, when a; = '308373; it then attains
the value '6636, which, it should be observed, is much less than
the excursion at the end.
The curve is shewn in fig. 29.
Fig. 29.
Vibration with four nodes, i ^ 3.
^1 = - sin j (630^ + 6"'92) « - 45« - 3'H6},
log F, =: 4-775332 x + 5-0741527,
log F, « - 4-775332 x + 1'8494850.
From this ^(0) = 1'41424, w(i) = 1-00579. The positions of
the nodes are readily found by trial and error. Thus
u (3558) = - -000037 u ('3559) = + '001047,
whence u ('355803) = 0. The value of x for the node near the end
is -0944, (Seebeck).
The position of the loop is best found from the derived
function. It appears that u' = 0, when x = -2200, and then
u=s — -9349. There is also a loop at the centre, where however
the excursion is not so great as at the two others.
Fig. 30.
We saw that at the centre of the bar F^ and F^ are numerical!;
equal. In the neighbourhood of the middle, F^ is evidently veiy
small, if i be moderately great, and thus the equation for the nodes
reduces approximately to
l 4^^ ^^ 2'
tnir,
n being an integer. If we transform the origin to the centre of
the rod, and replace m by its approximate value ^(2t + l)9r, we
find
05 _ ± 2n — i
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178.] GRAVEST MODE FOR A CLAMPED-FREE BAR. 285
shewing that near the middle of the bar the nodes are uniformly
spaced, the interval between consecutive nodes being 22-f-(2t + 1).
This theoretical result has been verified by the measurements of
Strehlke and Lissajoua
For methods of approximation applicable to the nodes near
the ends, when i is greater than 3, the reader is referred to the
memoir by Seebeck already mentioned § 160, and to Donkin's
Acoustics (p. 194).
179. The calculations are very similar for the case of a bar
clamped at one end and free at the other. If uoc F, and
F^Fi+Fi + Fi, we have in general
jPi = cos {mx + i 7r + i (- 1 )»a},
C - IV 1
= ^-^-sinia^«*; ^, = ---^cosia6-^.
If t = 1, we obtain for the calculation of the gravest vibration-
curve
log (- Ft) = mx\oge + T-0300909.
log (- F,)='-mx log e + 1-8444383.
These give on calculation
/'(0) = -000000.
if(-2) = -102974,
-F (-4) = -370625,
i'C -6)= -743452,
F( -8) = 1-169632.
^(10) = 1-612224,
from which fig. 31 was constructed.
Fig. 81.
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286
LATERAL VIBBATI0N8 OF BARS.
[179.
The distances of the nodes from the free end in the case of a
rod clamped at the other end are given by Seebeck and by Donkin.
2"^ tone -2261.
3"" tone -1321, 4999.
4'" tone -0944, -3558, -6439.
-i, 1-3222 4-9820 9-0007 4/- 3 4t- 10-9993 4»- 7-0175
^"^4i-2' 4i-2' 4i-2'4i-2' 4t-2 ' 4i-2 '
"The last row in this table must be understood as meaning
that
4;-3
, . Q may be taken as the distance of the j^ node from the
free end, except for the first three and the last two nodes."
When both ends are free, the distances of the nodes from the
nearer end are
1* tone -2242.
2"* tone 1321
-5.
3"* tone -0944
•3558.
^ , 1-3222
intone -^.^2
4-9820
4i + 2
9-0007
41 + 2
4;- 3
4i + 2
The points of inflection for a free-free rod (corresponding to
the nodes of a clamped-clamped rod) axe also given by Seebeck ;—
1"' point
2°<1 point
■rth
point
1»' tone
2°d tone
3^ tone
t'^ tone
•5000
•3593
50175
4t+2
No inflection point.
8-9993
4t + 2
4ic+l
4t+2
rs
Except in the case of the extreme nodes (which have no
corresponding inflection-point), the nodes and inflection-points
always occur in close proximity.
180. The case where one end of a rod is fr^ee and the other
supported does not need an independent investigation, as it maybe
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180.] POSITION OF NODES. 287
referred to that of a rod with both ends free vibrating in an even
mode, that is, with a node in the middle. For at the central node
y and y" vanish, which are precisely the conditions for a supported
end. In like manner the vibrations of a clamped-supported rod
are the same as those of one-half of a rod both whose ends are
clamped, vibrating with a central node.
181. The last of the six combinations of terminal conditions
occurs when both ends are supported. Referring to (1) § 170, we
see that the conditions at a? — 0, give -4. = 0, fi = ; so that
M = (0 + i)) sin a?' + (C - i)) sinh a:'.
Since u and u" vanish when a:' = m, C — D = 0, and sin m « 0.
Hence the solution is
. iirx i^it^Kh ^ ,_.
y=8m ^- cos— p— e (1),
where i is an integer. An arbitrary constant multiplier may of
course be prefixed, and a constant may be added to t
It appears that the normal curves are the same as in the case
of a string stretched between two fixed points, but the sequence of
tone is altogether different, the frequency varying as the square
of 1. The nodes and inflection-points coincide, and the loops
(which are also the points of maximum curvature) bisect the
distances between the nodes.
182. The theory of a vibrating rod may be applied to illustrate
the general principle that the natural periods of a system fulfil the
maximum-minimum condition, and that the greatest of the natural
periods exceeds any that can be obtained by a variation of
type. Suppose that the vibration curve of a clamped-free rod is
that in which the rod would dispose itself if deflected by a force
applied at its free extremity. The equation of the curve may be
taken to be
y = -3Za:» + ^,
which satisfies d^yjdx^ = throughout, and makes y and y' vanish
at 0, and y" at I, Thus, if the configuration of the rod at time t be
y^i-Zla^-^-a*) Qospt (1),
the potential energy is by (1) § 161, % qi^ <ol* cob^^ pty while the
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288 LATERAL VIBRATIONS OF BARS. [l82.
kinetic energy is ^ptoP p* sm* pt ; and thus jj* = — . .
Now pi (the true value of p for the gravest tone) is equal to
^x(l-8751)»;
SO that
2>,:p = (l-8751)«y^ = -98556,
shewing that the real pitch of the gravest tone is rather (but
comparatively little) lower than that calculated from the hypo-
thetical tjrpe. It is to be observed that the hypothetical tyi>e in
question violates the terminal condition y^''= 0. This circumstance,
however, does not interfere with the application of the principle,
for the assumed type may be any which would be admissible as an
initial configuration ; but it tends to prevent a very close agree-
ment of periods.
We may expect a better approximation, if we found our calcu-
lation on the curve in which the rod would be deflected by a force
acting at some little distance from the free end, between which
and the point of action of the force {x^c) the rod would be
straight, and therefore without potential energy. Thus
potential energy = 6 g^oic* cos* 2>^.
The kinetic energy can be readily found by integration from
the value of y.
From to c y = — 3ca?' 4- si^ ;
and from ciol y^c^{C'- Zx),
as may be seen from the consideration that y and y' must not
suddenly change at a? = c. The result is
whence
kinetic energy = p<» p^ sin* l^d yg c' + i c* (Z - c) (c* + ZV) ,
The maximum value of l//>* will occur when the point of
application of the force is in the neighbourhood of the node of the
second normal component vibration. If we take c = f Z, we obtain
a result which is too high in the musical scale by the interval
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182.] LOADED END. 289
expressed by the ratio 1 : '9977, and is accordingly extremely near
the truth. This example may give an idea how nearly the period
of a vibrating system may be calculated by simple means without
the solution of diflFerential or transcendental equations.
The type of vibration just considered would be that actually
assumed by a bar which is itself devoid of inertia, but carries a
load M at its free end, provided that the rotatory inertia of M could
be neglected. We should have, in fact,
F= 6gr/c*a>Z' cos» pt, T = iMV'p^ sin* j)t,
BO that P'--M¥ (3>-
Even if the inertia of the bar be not altogether negligible in
comparison with if, we may still take the same type as the basis
of an approximate calculation :
F=65'#c*a>Z"cos22>^,
33
whence
r= {2MI' + 1? pwp"^ p» sin«pe,
that is, M is to be increased by about one quarter of the mass of
the rod. Since this result is accurate when M is infinite, and does
not differ much from the truth, even when if =0, it may be re-
garded as generally applicable as an approximation. The error
will always be on the side of estimating the pitch too high.
183. But the neglect of the rotatory inertia of M could not
be justified under the ordinary conditions of experiment. It is as
easy to imagine, though not to construct, a case in which the inertia
of translation should be negligible in comparison with the inertia of
rotation, as the opposite extreme which has just been considered.
If both kinds of inertia in the mass M be included, even though
that of the bar be neglected altogether, the system possesses two
distinct and independent periods of vibration.
Let z and denote the values of y and dyjdx at a; = Z. Then
the equation of the curve of the bar is
R. 19
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290
and
LATEBAL VIBBATI0N8 OF BABS.
v^^s^
[183.
(1);
^^{9^-3110+ I'd*]
while for the kinetic energy
T=iM£*+\MK'^^- (2),
if k' be the radius of gyration of M about an axis perpendicular to
the plane of vibration.
The equations of motion are therefore
Mk'*'e + ^-p^i-3lz + 21*6) =
whence, if z and 6 vary as cos pt, we find
corresponding to the two periods, which are always different.
If we neglect the rotatory inertia by putting tc' = 0, we fiJl
back on our previous result
(3);
P' =
_3qifa>
Ml*
The other value of p" is then infinite.
1{ k' '.I be merely small, so that its higher powers may be
neglected,
^_4iqi(^(of, . 9«'^"\
P' =
P Ml* \ 4 I*))
(5\
If on the other hand k^ be very great, so that rotation is
prevented.
^ _ 12 qfc^(o
or
MU'^
(6),
the latter of which is very small. It appears that when rotation
is prevented, the pitch is an octave higher than if there were no
rotatory inertia at all. These conclusions might also be derived
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183.] EFFECT OP ADDITIONS. 291
directly from the diflFerential equations ; for if ic' = oo , tf := 0, and
then
but if «' = 0, = Szf2l, by the second of equations (3), aud in
that case
Mi + --^ — z = 0.
184. If any addition to a bar be made at the end, the period
of vibration is prolonged. If the end in question be free, suppose
first that the piece added is without inertia. Since there would be
no alteration in either the potential or kiuetic energies, the pitch
would be unchanged; but in proportion as the additional part
acquires inertia, the pitch falls (§ 88).
In the same way a small continuation of a bar beyond a
clamped end would be without eflFect, as it would acquire no
motion. No change will ensue if the new end be also clamped ;
but as the first clamping is relaxed, the pitch falls, in consequence
of the diminution in the potential energy of a given deformation.
The case of a ' supported ' end is not quite so simple. Let the
original end of the rod be A, and let the added piece which is at
first supposed to have no inertia, be AB. Initially the end A is
fixed, or held, if we like so to regard it, by a spring of infinite stiflF-
ness. Suppose that this spring, which has no inertia, is gradually
relaxed. During this process the motion of the new end B
diminishes, and at a certain point of relaxation, B comes to rest.
During this process the pitch falls. J5, being now at rest, may be
supposed to become fixed, and the abolition of the spring at A
entails another fall of pitch, to be further increased 2isAB acquires
inertia.
186. The case of a rod which is not quite uniform may be
treated by the general method of § 90. We have in the notation
there adopted
ar = I pcj^Ur^dx, Sar = I BpcoUr^dx,
19—2
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292
LATERAL VIBRATIONS OF BARS.
[185.
whence, Pr being the uncorrected value ofpr,
Pr -rr Y+ r /(Pu^V^ "TTI
.(1).
[If the motion be strictly periodic with respect to x, Ur" is
proportional to w^, and both quantities vanish at a node. Ac-
cordingly an irregularity situated at a node of this kind of motion
has no eflFect upon the period. A similar conclusion will hold good
approximately for the interior nodes of a bar vibrating with
numerous subdivisions, even though, as when the terminals are
clamped or free, the mode of motion be not strictly periodic with
respect to a?.]
If the rod be clamped at and free at I,
^"^ p<oA l^fhBr lufJopay,
The same formula applies to a doubly free bar.
The effect of a small load dM is thus given by
, 5om* f, , u^dM]
p^ = 1:= — i 1 — 4 — ~iyr
where M denotes the mass of the whole bar. If the load be at
the end, its effect is the same as a lengthening of the bar in the
ratio M : M + dM. (Compare § 167.)
[In (2) dM is supposed to act by inertia only ; but a similar
formula may conveniently be employed when an irregularity of
mass dM depends upon a variation of section, without a change
of mechanical properties. Since B = gr/c'©,
S-B/£o = S(/c«a>)/(iC«a>)o;
so that the effect of a local excrescence is given by
(2),
pVi^ = i +
dx
lufj 'i
dx .
.(3).
If the thickness in the plane of bending be constant, S/c* = 0,
and S (f^o>)/(K*(o)Q = Sa)/a)Q,
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185.] COERECTION FOR BOTATORY INERTIA. 293
TT. _xi_ [S(odx dM
and thus ^/i>. = i+4^'i^' (4).
If, however, the thickness in the plane perpendicular to that of
bending be constant, and in the plane of bending variable (£7),
then 8 (tc'w)/(K*<o)o = Brf/yo* = 3 87/70 = 38€d/©o ;
and in place of (4)
p^lP^^i^^^—^^ :-(5).
If a tuning-fork be filed {dM negative) near the stalk (clamped
end), the pitch is lowered ; and if it be filed near the free end, the
pitch is raised. Since u^'^^ufy the effects of a given stroke of
the file are equal and opposite in the circumstances of (4), but in
the circumstances of (5) the effect at the stalk is three times as
great as at the firee end.]
186. The same principle may be applied to estimate the
correction due to the rotatory inertia of a uniform rod. We have
only to find what addition to make to the kinetic energy, supposing
that the bar vibrates according to the same law as would obtain,
were there no rotatory inertia.
Let us take, for example, the case of a bar clamped at and
firee at I, and assume that the vibration is of the type,
y^ucospt,
where u is one of the functions investigated in § 179. The kinetic
energy of the rotation is
= ^^^^^8in»;>e (2uu' + mu%,
by (2) §165.
To this must be added
^ 'f sin*/>^ I u^ dx, or ^ p* ain^|)< uf ;
so that the kinetic energy is increased in the ratio
1 : l+TT 2-4-m— .
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294
LATERAL VIBRATIONS OF BARS.
[186.
The altered frequency bears to that calculated without allow-
ance for rotatory inertia a ratio which is the square root of the
reciprocal of the preceding. Thus
p:P=l-i-7r- 2 — + m — J ,
.(I).
By use of the relations cosh m = ~ sec m, sinh m = cos tTr.tan m,
we may express u' : u when a? = Z in the form
^ —
u ""
— slnm
cosa
cos^7^+cos??i~ 1
— cosiVsina'
if we
substitute for
m from
m = i(2i-
.l)7r-
-(-lya.
In the case of the gravest tone, a = *3043, or, in degrees and
minutes, a = 17*^ 26', whence
Thus
- = •73413,
u
2~+m^ = 2-4789.
le
;>:P = 1-2-3241 J
.(2),
which gives the correction for rotatory inertia in the case of the
gravest tone.
When the order of the tone is moderate, a is very small,
and then
u' \u = \ sensibly.
and
p..P = l-{x^-)— (3),
shewing that the correction increases in importance with the
order of the component.
In all ordinary bars « : Z is very small, and the term depending
on its square may be neglected without sensible error.
187. When the rigidity and density of a bar are variable
from point to point along it, the normal functions cannot in
general be expressed analytically, but their nature may be investi-
gated by the methods of Sturm and Liouville explained in § 142,
If, as in § 162, B denote the variable flexural rigidity at any
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187.] ROOTS OF COMPOUND FUNCTIONS. 295
point of the bar, and pw dx the mass of the element, whose length
is (2a?, we find as the general differential equation
^(*£)--i'-« «■
the effects of rotatory inertia being omitted. If we assume that
y (X cos pt, we obtain as the equation to determine the form of the
normal functions
U^S)-'^y «-
in which i^ is limited by the terminal conditions to be one of an
infinite series of definite quantities Vi^, i/,', y,*
Let us suppose, for example, that the bar is clamped at both
ends, so that the terminal values of y and dy/da vanish. The first
normal function, for which i/* has its lowest value i/i*, has no
internal root, so that the vibration-curve lies entirely on one side
of the equilibrium-position. The second normal function has one
internal root, the third function has two internal roots, and,
generally, the r^^ function has r — 1 internal roota
Any two different normal functions are conjugate, that is to
say, their product will vanish when multiplied by p<odx, and
integrated over the length of the bar.
Let us examine the number of roots of a function f(x) of
the form
/(«) = <^mt^m(a?) + <^m+iWm+i(a?)+...+<^nt«n(^) (3),
compounded of a finite number of normal functions, of which the
function of lowest order is Umi^) and that of highest order is
Ujt, (x). If the number of internal roots oif{x) be /i, so that there
are /li -I- 4 roots in all, the derived function f (x) cannot have less
than fjL-\-l internal roots besides two roots at the extremities, and
the second derived function cannot have less than fi + 2 roots.
No roots can be lost when the latter function is multiplied by B,
and another double differentiation with respect to x will leave at
least /i internal roots. Hence by (2) and (3) we conclude that
has at least as many roots as f{x). Since (4) is a function of the
same form as f{x)y the same argument may be repeated, and a
series of functions obtained, every member of which has at least
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296
LATERAL VIBRATIONS OF BARS.
[187.
as many roots as f{x) has. When the operatibn by which (4) was
derived from (3) has been repeated sufficiently often, a function is
arrived at whose form differs as little as we please from that of the
component normal function of highest order Un{x)\ and we con-
clude that f{x) cannot have more than n — 1 internal roots. In
like manner we may prove that f{x) cannot have less than m — 1
internal roots.
The application of this theorem to demonstrate the possibility
of expanding an arbitrary function in an infinite series of normal
functions would proceed exactly as in § 142.
[An analytical investigation of certain cases where the section
of a rod is supposed to be variable, will be found in a memoir by
Kirchhoff^].
188. When the bar, whose lateral vibrations are to be con-
sidered, is subject to longitudinal tension, the potential energj' of
any configuration is composed of two parts, the first depending on
the stiffness by which the bending is directly opposed, and the
second on the reaction against the extension, which is a necessary
accompaniment of the bending, when the ends axe nodes. The
second part is similar to the potential energy of a deflected string ;
the first is of the same nature as that with which we have been
occupied hitherto in this Chapter, though it is not entirely
independent of the permanent tension.
Consider the extension of a filament of the bar of section dm,
whose distance from the axis projected on the plane of vibration
is 17. Since the sections, which were normal to the axis originally,
remain normal during the bending, the length of the filament
bears to the corresponding element of the axis the ratio -R + 17 : iJ,
jR being the radius of curvature. Now the axis itself is extended
in the ratio j : ? + T^ reckoning from the unstretched state, if
Tto denote the whole tension to which the bar is subjected.
Hence the actual tension on the filament is {r+i7(r+gr)/jB}c?o),
from which we find for the moment of the couple acting across the
section
1 Berlin Monataber.y 1879 ; Collected Works, p. 339. See also Todhanter and
Pearson's HUtory of the Theory of EUutieity, Vol. 11., Part ii., § 1802.
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188.] PERMANENT TENSION. 297
and for the whole potential energy due to stiffness
i(j+T)«»a,/(gy(ir (1),
an expression differing from that previously used (§ 162) by the
substitution of (q + T) for q.
Since q is the tension required to stretch a bar of unit area to
twice its natural length, it is evident that in most practical cases
T woald be negligible in comparison with q.
The expression (1) denotes the work that would be gained
during the straightening of the bar, if the length of each element
of the axis were preserved constant during the process. But
when a stretched bar or string is allowed to pass from a displaced
to the natural position, the length of the axis is decreased. The
amount of the decrease is iJidy/dxydx, and the corresponding
gain of work is
Thus
F=i(?+r)«-«/(g)'d.H-i2'«/(|)'i^ (2).
The variation of the first part due to a hypothetical displace-
ment is given in § 162. For the second part, we have
i«/(l)''^-/lf^-llM-/S*^ <'>•
In all the cases that we have to consider, ^ vanishes at the
limits. The general differential equation is accordingly
or, if we put g + 7= 6*p, T = a'p,
*V dor* dx^dt'l "^ dx'^ dt'^ ^*^"
For a more detailed investigation of this equation the reader is
referred to the writings of Clebsch^ and Donkin.
189. If the ends of the rod, or wire, be clamped, dyjdx = 0, and
the terminal conditions are satisfied. If the nature of the support
be such that, while the extremity is constrained to be a node, there
^ Theorie der EloMticitHt fe$ter Kdrper, Leipzig, 1862.
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J
298 LATERAL VIBRATIONS OF BARS. [189.
is DO couple acting on the bar, dhfjda? must vanish, that ia to say,
the end must be straight. This supposition is usually taken to
represent the case of a string stretched over bridges, as in manj
musical instruments ; but it is evident that the part beyond the
bridge must partake of the vibration, and that therefore its length
cannot be altogether a matter of indifference.
If in the general differential equation we take y proportional
to cos nt, we get
'•('■S--S)-S-"V-« ox
which is evidently satisfied by
y = 8ini , cosn^ (2),
if n be suitably determined. The same solution also makes
y and /' vanish at the extremities. By substitution we obtain
for 71,
'^ - P P + i>7r»ic^ ^^^
which determines the frequency.
If we suppose the wire infinitely thin, n* = i*7r*a'-4- P, the same
as was found in Chapter VL, by starting from the supposition of
perfect flexibility. If we treat /e : Z as a very small quantity, the
approximate value of n is
lira
For a wire of circular section of radius r, «• = J r*, and if we
replace b and o by their values in terms of q, T, and p.
7{>-rpf} <*)■
which gives the correction for rigidity ^ Since the expression
within brackets involves i, it appears that the harmonic relation
of the component tones is disturbed by the stiffness.
190. The investigation of the correction for stiffness when the
ends of the wire are clamped is not so simple, in consequence of
the change of tjrpe which occurs near the enda In order to pass
from the case of the preceding section to that now under con-
^ Donkin's Acoustics, Art. 184.
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190.]
PERMANENT TENSION.
299
sideration an additional constraint must be introduced, with the
effect of still further raising the pitch. The following is, in the
main, the investigation of Seebeck and Donkin.
If the rotatory inertia be neglected, the differential equation
becomes
2)*-
a
' ^-&)^=»-
where D stands for -5-
dx
In the equation
.(1).
one of the values of L* must be positive, and the other negative.
We may therefore take
-0*-^^-iS = (^-«')(i>' + /3')
•(2).
and for the complete integral of (1)
y = A cosh ax + 3 sinh ouc + (7 cos ^a? + i) sin jSx (3),
where a and fi are functions of n determined by (2).
The solution must now be made to satisfy the four boundary
conditions, which, as there are only three disposable ratios, lead
to an equation connecting a, /3, L This may be put into the form
sinhoZ sin/32
2afi
= 0.
1 - cosh al cosfil 0^ — ^
The value of -r-^^ , determined by (2), is — r— , so that
.(4).
sinh al smfil 2nbK _ ^
1 — cosh al cos fil
From (2) we find also that
a"
.(5).
a*
n^b^K"
s^-}
.(6).
Thus far our equations are rigorous, or rather as rigorous as
the differential equation on which they are founded ; but we shall
now introduce the supposition that the vibration considered is but
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300 LATERAL VIBRATIONS OF BARS. fl90.
slightly affected by the existence of rigidity. This being the case,
the approximate expression for y is
. iirx fi'rr ^\
y = sm -T- cos I y a^ J ,
and therefore
/B^tTr/l, n^iirajl (7).
nearly.
The introduction of these values into the second of equations
(6) proves that n*6'/r*/a* or h^t^ja^J} is a small quantity under the
circumstances contemplated, and therefore that aH^ is a large
quantity. Since cosh aZ, sinhof are both large, equation (5) re-
duces to
, 2n6/e
tani8Z= ^, ,
or, on substitution of the approximate value for fi derived fiom
(6).
nl ^ nbK
tan — = 2
a 0/
2 '
The approximate value of nlja is iir. If we take idja^vir-^B,
we get
tan(iV-htf) = tantf=tf=.2^f = 2tV^p
so that n = t
. "wa /, c^b ic^
I
{'<t) <'^
According to this equation the component tones are all raised
in pitch by the same small interval, and therefore the harmonic
relation is not disturbed by the rigidity. It would probably be
otherwise if terms involving i^ : Z* were retained ; it does not there-
fore follow that the harmonic relation is better preserved in spite
of rigidity when the ends are clamped than when they are free,
but only that there is no additional disturbance in the former
case, though the absolute alteration of pitch is much greater. It
should be remarked that 6 : a or V(? + T) : ^T, is a large quantity,
and that, if our result is to be correct, k : I must be small enou^
to bear multiplication by 6 : a and yet remain small.
The theoretical result embodied in (8) has been compared with
experiment by Seebeck, who found a satisfactory agreement. The
constant of stifihess was deduced from observations of the rapiditj
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^^4>0.] RESULTANT OF TWO TRAINS OF WAVES. 301
of the vibratioDS of a small piece of the wire, when one end was
clamped in a vice.
[As the result of a second approximation Seebeck gives
(foe. cit.)
„. = ^.|l+4*^ + (12 + ».^)^j (9)].
191. It has been shewn in this chapter that the theory of bars,
even when simplified to the utmost by the omission of unimportant
quantities, is decidedly more complicated than that of perfectly
flexible strings. The reason of the extreme simplicity of the
vibrations of strings is to be found in the fact that waves of the
harmonic type are propagated with a velocity independent of the
wave length, so that an arbitrary wave is allowed to travel without
decomposition. But when we pass from strings to bars, the con-
stant in the differential equation, viz. cPy/rft* + ie*6'd*y/(ic* = 0, is
no longer expressible as a velocity, and therefore the velocity of
transmission of a train of harmonic waves cannot depend on the
differential equation alone, but must vary with the wave length.
Indeed, if it be admitted that the train of harmonic waves can
be propagated at all, this consideration is sufficient by itself to
prove that the velocity must vary inversely as the wave length.
The same thing may be seen from the solution applicable to
waves propagated in one direction, viz. y=^co8---(Vt — x), which
satisfies the differential equation if
v=^^ (1).
A.
Let us suppose that there are two trains of waves of equal
amplitudes, but of different wave lengths, travelling in the same
direction. Thus
= 2co8 7r
!,-eo,2.(^-!)+oo.2^(i-Q
..(2).
If t' — T, X' — X be small, we have a train of waves, whose
amplitude slowly varies from one point to another between the
values and 2, forming a series of groups separated from one
another by regions comparatively free fix)m disturbance. In the
case of a string or of a column of air, X varies as r, and then the
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302
LATERAL VIBRATIONS OF BARS.
[191.
groups move forward with the same velocity as the component trains
and there is no change of type. It is otherwise when, as in the cas€
of a bar vibrating transversely, the velocity of propagation is a
function of the wave length. The position at time t of the middle
of the group which was initially at the origin is given by
which shews that the velocity of the group is
If we suppose that the velocity F of a train of waves varies as
X**, we find
d(l/X) d(l/X) ^ ^
.(3).
In the present case n = — 1, and accordingly the velocity of the
groups is tvnce that of the component waves^
192. On account of the dependence of the velocity of propaga-
tion on the wave length, the condition of an infinite bar at any
time subsequent to an initial disturbance confined to a limited
portion., will have none of the simplicity which characterises the
corresponding problem for a string; but nevertheless Fourier's
investigation of this problem may properly find a place here.
It is required to determine a function of a? and <, so as to
satisfy
d*V d*y ^ „v
and make initially y = (f> {x\ y = '^ {x).
A solution of (1) is
y=^QOHqH cosg(ir — o) (2),
where q and a are constants, from which we conclude that
y = j da F(a) I dq cos g^t cos q(x — a)
^ In the oorresponding problem for waves on the Burfftce of deep water, the
Telocity of propagation varies directly as the square root of the wave length, bo
that n s } . The velocity of a group of such waves is therefore one half of that of
the component trains. [See note on Progressive Waves, appended to this volume.]
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192.] Fourier's solution. 303
is also a solution, where F(a) is an arbitrary function of a. If
now we put ^ = 0,
r+oo r+flo
yo = I daF(a) I dq cos j (a? — a),
which shews that ^(a) must be taken to be 5- ^ (a), for then by
Fourier's double integral theorem yo = ^(^). Moreover, y = 0;
hence
I r+flo r+ao
y = ^ daif>(a)j dqcoaqH cosg(a?-a) (3)
satisfies the differential equation, and makes initially
y = <l>(x), y = 0.
By Stokes' theorem (§ 95), or independently, we may now
supply the remaining part of the solution, which has to satisfy the
differential equation while it makes initially y = 0, y = -i/r (a?) ; it is
I r+ao /•+» I
y = g;-\ daylr(a)j dq-ain^t 0035(0? — a) (4).
ZirJ __oo ,' _ao q^
The final result is obtained by adding the right-hand members
of (3) and (4).
In (3) the integration with respect to q may be effected by
means of the formula
j^^dqcosqHcosqz^A^^ sin^J + |^) (5),
which may be proved as follows. If in the well-known integral
formula
we put a? -h 6 for x, we get
/:
a
Now suppose that a* = i = e^', where i = V (- 1), and retain
only the real part of the equation. Thus
r+oo
I COS (x' -f 26a?) dx = */7r sin (6» + i tt),
.' —00
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n
304 LATERAL VIBRATIONS OF BARS. [l92.
whence
/+• _
cos aj» cos 26a: dx^Jir sin (6* + \ir\
—00
from which (5) follows by a simple change of variable. Thus
equation (3) may be written
.^ a — a?
y^j^j *ti/i'(co8/i« + 8in/i«)^(« + 2/*V0 (6)-
192 a. If the axis of the rod be curved instead of straight,
we obtain problems which may be regarded as extensions of.
those of the present and of the last chapters. The most impor-
tant case under this head is that of a circular ring, whose section
we will regard as also circular, and of radius (c) small in
comparison with the radius (a) of the circular axis.
The investigation of the flexural modes of vibration, executed
in the plane of the ring, is analogous to the case of a cylinder
(see § 233), and was first effected by Hopped If « be the number
of periods in the circumference, the coeflScient p of the time in
the expression for the vibrations is given by
^4 !-!-«> pa' ^^^'
where q is Young's modulus and p the density of the material.
This may be compared with equation (9) § 233. To fisill back
upon the case of a straight axis we have only to suppose
8 and a to be infinite in such a manner that 27ra/« is equal to the
proposed linear period. The vibrations in question are then purely
transverse.
In the class of vibrations considered above the circular axis
remains unextended, and (§ 232) the periods are comparatively
long. For the other class of vibrations in the plane of the ring,
Hoppe found
p'-(^^^^l^* (2>
1 CrdU, Bd. 68, p. IfiS, 1871.
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192 a.] CIRCULAR RINGS. 305
The frequencies are here independent of c, and the vibrations
are analogous to the longitudinal vibrations of straight rods.
If « = in (2), we have the solution for vibrations which are
purely radial.
For flezural vibrations perpendicular to the plane of the
riag, the result^ corresponding to (1) is
i ^(^-iy gg (3)
the difference consisting only in the occurrence of Poisson's ratio
(jjt) in the denominator.
Our limits will not allow of our dwelling further upon the
problem of this section. A complete investigation will be found
in Love's Treatise on Elasticity, Chapter XYiii. The effect of
a small curvature upon the lateral vibrations of a limited bar
has been especially considered by Lamb*.
1 Miohell, Menenger of Mathenuitiei, zix., 1889.
* Proe. Lond. McUh. Soe., xn., p. 365, 1888.
20
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CHAPTER IX.
VIBRATIONS OF MEMBRANES.
193. The theoretical membrane is a perfectly flexible and
infinitely thin lamina of solid matter, of uniform material and
thickness, which is stretched in all directions by a tension so great
as to remain sensibly unaltered during the vibrations and displace-
ments contemplated. If an imaginary line be drawn across the
membrane in any direction, the mutual action between the two
portions separated by an element of the line is proportional to the
length of the element and perpendicular to its direction. If the
force in question be T^ ds, Ti may be called the tension of the memr
hrune; it is a quantity of one dimension in mass and — 2 in time.
The principal problem in connection with this subject is the
investigation of the transverse vibrations of membranes of different
shapes, whose boundaries are fixed. Other questions indeed may
be proposed, but they are of comparatively little interest ; and,
moreover, the methods proper for solving them will be suflBciently
illustrated in other parts of this work. We may therefore proceed
at once to the consideration of a membrane stretched over the
area included within a fixed, closed, plane boundary.
194. Taking the plane of the boundary as that of xy^ let fo
denote the small displacement therefrom of any point P of the
membrane. Round P take a small area 8^ and consider the forces
acting upon it parallel to z. The resolved part of the tension is
expressed by
^'it^
where da denotes an element of the boundary of £•, and dn an
element of the normal to the curve drawn outwards. This is
balanced by the reaction against acceleration measured by pSw,
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194.] EQUATION OF MOTION. 307
p being a symbol of one dimension in mass and —2 in length
denoting the superficial density. Now by Green's theorem, if
j^ds^jjV'wdS^VHu.S ultimately,
and thus the equation of motion is
dt^^ p\di^^ dy'J ^^^•
The condition to be satisfied at the boundary is of course w = 0.
The differential equation may also be investigated from the
expression for the potential energy, which is found by multiplying
the tension by the superficial stretching. The altered area is
and thus
''-i^-//((sr-(l)'}«' <^^
firom which SF is easily found by an integration by parts.
If we write Ti -r /> = c*, then c is of the nature of a velocity, and
the differential equation is
dt^-'^Vdx^^d^^) ^^^•
196. We shall now suppose that the boundary of the mem-
brane is the rectangle formed by the coordinate axes and the lines
ar = a, y = 6. For every point within the area (3) § 194 is satisfied,
and for every point on the boundary w = 0.
A particular integral is evidently
w = sin sm— T^cosp^ (1),
where p. = c»^g + »;) (2),
and m and n are integers; and from this the general solution may
be derived. Thus
w=^z S sm-^— sm -r^ {^tiin cosp^ -h £,»n smpq (3).
20—2
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308 VIBEATIONS OF MEMBRANES. [195.
That this result is really general may be proved a posteriori,
by shewing that it may be adapted to express arbitrary initial
circumstances.
Whatever function of the co-ordinates w may be, it can be ex-
pressed for all values of x between the limits and a by the series
y^sm — -h Fjsm- — + ,
a a
where the coefficients Fi, F,, &c. are independent of x. Again
whatever function of y any one of the coefficients F may be, it can
be expanded between and h in the series
C,sin7+C.sin^-^ + ,
where G^ &c. are constants. From this we conclude that any
function of x and y can be expressed within the limits of the rect-
angle by the double series
m-flo n-oo ^^^ ^„
m-l n-1 d
(*)■
and therefore that the expression for w in (3) can be adapted to
arbitrary initial values of w and w. In fEtct
The character of the normal functions of a given rectangle,
sm sm— r^ ,
a b
as depending on m and n, is easily understood. If m and n be both
unity, w retains the same sign over the whole of the rectangle,
vanishing at the edge only; but in any other case there are
nodal lines running parallel to the axes of coordinates. The
number of the nodal lines parallel to a; ifi n — 1, their equations
being
h 26 (n-1) 6
y^:^^ 1^^ — -
n n n
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195.] RECTANGULAB BOUNDARY. 809
In the same way the equations of the nodal lines parallel to y
are
_a 2a (m — l)a
X—— y , f
mm m
being m — 1 in number. The nodal system divides the rectangle
into mn equal parts, in each of which the numerical value of tc; is
repeated
196. The expression for w in terms of the normal functions
is
t(; = 22<^^n8in^^sin^^^- (1),
where if>nm &c. are the normal coordinates. We proceed to form
the expression for V in terms of ^«n. We have
In integrating these expressions over the area of the rectangle
the products of the normal coordinates disappear, and we find
■-nmhm^^y
-f't'^e'^S*-' <2)'
the summation being extended to all integral values of m and n.
The expression for the kinetic energy is proved in the same
way to be
r=|^42^» (3).
firbm which we deduce as the normal equation of motion
In this equation ^
^mn = j ] Z sm -^ am -^dxdy (d),
ii Zdxdy denote the transverse force acting on the element dxdy.
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310 VIBRATIONS OF MEMBRANES. [196.
Let us suppose that the initial condition is one of rest under
the operation of a constant force Z^ such as may be supposed to
arise firom gaseous pressure. At the time ^ = 0, the impressed
force is removed, and the membnCM left to itself Initially the
equation of equilibrium is
<^'^(S+3<^>'=^*^»» ^^
■whence (<^mn)o is to be found. The position of the system at time \
is then given by
</>mn = (<Atnn)oCOsfy' — H-^.Clrtj (7),
in conjunction with (1).
In order to express *m»> we have merely to substitute for Z its
value in (5), or in this case simply to remove Z fix)m under the
integral sign. Thus
, -r*f* . mirx . ntry J ,
= Z r (1 — COS imf) (1 — COS nir\
We conclude that *„»» vanishes, unless m and n are hoih odd, and
that then
Accordingly, m and n being both odd,
16Z COSJO^ ,J^^
^^^^ ii-p^n^ ^^^'
where p.= c»7r«(^ + g) (9).
This is an example of (8), § 101.
If the membrane, previously at rest in its position of equili-
brium, be set in motion by a blow applied at the point (a, /3), the
solution is
^mn = ^ sm-^ 8m--^jJM;oCtody.smpe...(10>
[As an example of forced vibrations, suppose that a harmonic
foree acts at the centre. Unless m and n are both odd, <I>iiiji-0>
aud in the case reserved
^mn =±^1 cos 3« (11),
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«t»m»= .Z.
196.] KECTANGULAR BOCJNDARY. 311
where Z^ is the whole force acting at time t, and ± represents
sin^mTT sin JnTT. From (4) and (9) we have
and w is then given by (1).
In the case of a square membrane, j> is a symmetrical function
of m and n. When m and n are unequal, the terms occur in pairs,
such as
±iZiCORqt f. m'rrx . rwrv . ri'rrx . miry] ,,-.v
~r7 r-75\ ^^^ ^ sm - ^ + sm sm — ^k..(13),
-&' combination symmetrical as between x and y. The vibration is
of course similarly related as well to the four sides as to the four
corners of the square.
In the neighbourhood of the centre, where the force is applied,
the series loses its convergency, and the displacement w tends to
become (logarithmically) infinite.]
197. The frequency of the natural vibrations is found by
ascribing different integral values to m and n in the expression
P.
27r
c Im^ n^ .^.
For a given mode of vibration the pitch falls when either
side of the rectangle is increased. In the case of the gravest
mode, when m = l, n = l, additions to the shorter side are the
more effective; and when the form is very elongated, additions
to the longer side are almost without effect.
When a? and 6* are incommensurable, no two pairs of values
of m and n can give the same frequency, and each fundamental
mode of vibration has its own characteristic period. But when
a' and b' are commensurable, two or more fundamental modes
may have the same periodic time, and may then coexist in any
proportions, while the motion still retains its simple harmonic
character. In such cases the specification of the period does
not completely determine the type. The full consideration of
the problem now presenting itself requires the aid of the theory
of numbers; but it will be suflScient for the purposes of this
work to consider a few of the simpler cases, which arise when
the membrane is square. The reader will find fuller information
in Riemann's lectures on partial differential equations.
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312
YIBBATIONS OF MEMBRANES.
Ifa = 6.
VI /• ,
^ - /«i»-l-««
2ir-2a'^"' +"
[197.
(2X
The lowest tone is found by patting m and n equal to unity,
which gives only one fundamental mode : —
w — sin — sm -^cosp^ (3).
Next suppose that one of the numbers m,ni8 equal to 2, and
the other to unity. In this way two distinct types of vibration
are obtained, whose periods are the same. If the two vibrations
be synchronous in phase, the whole motion is expressed by
w-
(^ . 2irx . 'try ^ J. . trx . 27ry) ^ ,^.
\ a a a a ] ^ ^ '^
so that, although every part vibrates Sjmchronously with a
harmonic motion, the type of vibration is to some extent arbitrary.
Four particular cases may be especially noted First, if D = 0,
sm sm-^cosp^ (5),
which indicates a vibration with one node along the line x = \(k
Similarly if (7 "= 0, we have a node parallel to the other pair of
edges. Next, however, suppose that C and D are finite and
equal Then w is proportional to
. ^irx .Try . tto? . 27ry
sm sm -^ 4- sm — sm — - ,
a a a a
which may be put into the form
g. . TTX . iry / irx ^ iry\
2 sm — sm -^ cos h cos -^ .
a a \ a a]
This expression vanishes, when
sin irx\a = 0, or sin iry\a =
or again, when
cos irx\a + cos Try/a = 0.
The first two equations give the edges, which were originally
assumed to be nodal ; while the third gives y + a; = a, representing
one diagonal of the square.
In the fourth case, when (7 = — D, we obtain for the nodal
lines, the edges of the square together with the diagonal y^^
The figures represent the four cases.
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197.] CASES OF EQUAL PERIODS. 313
Fig. 82.
2) = 0. C = 0. C~D«0. a + i) = o.
[Frequency (referred to gravest) = 1'68.]
For other relative values of C and D the interior nodal line
is curved, but is always analytically expressed by
Ccos— +Dcos^ = (6),
a a
and may be easily constructed with the help of a table of logarith-
mic cosines.
The next case in order of pitch occurs when m = 2, w « 2.
The values of m and n being equal, no alteration is caused by
their interchange, while no other pair of values gives the same
frequency of vibration. The only type to be considered is
accordingly
. 2irx . 27rv
w = sm sm — - cos 1)^,
whose nodes, determined by the equation
Fig. 88.
. ttiju . Try TTu, tru -.
sm — sm -^ cos — cos -^ = 0,
a a a a
are (in addition to the edges) the straight lines
Fig. (33)
x — ^a y=ia.
[Frequency = 2*00.]
The next case which we shall consider is obtained by ascribing
to m, » the values 3, 1, and 1, 3 successively. We have
(^ . ^irx . Try ^ r\ - itx , 37ry)
w = -^1/ sm sin — ^ + i/sm — sm — - \ cos »^.
\ a a a a } ^
The nodes are given by
sin^sin ^{c(4co3«^- l) + i)(4coe»^- l)Uo,
or, if we reject the fint two ftrCtors, which correspond to the edges.
(7(4co8»^'^-l) + i)(4co8»^-l) = (7).
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314
VIBRATIONS OF MEMBRANES.
[197.
If (7 = 0, we have . y^^ia, y=»|a.
If2) = 0, a? = ia, a? = Ja.
irx try
cos — == ± cos — ^ ,
a a
If (7 = -D,
whence, y = a?, y^a-^x,
which represent the two diagonals.
Lastly, i{ C=D, the equation of the node is
^trx , ,7ry ,
cos* h cos* -^ = i,
a a "
or
., . 27ra: . 27ry
1 + cos h cos — ^ s
a a
Fig. 84.
Z) = 0. 0+D = 0.
(8),
[Frequency = 2-24.]
In case (4) when a: = ^a, y = Ja, or fa; and similarly when
y = ^a, a? = Ja?, or fa. Thus one half of each of the lines joining
the middle points of opposite edges is intercepted by the curve.
[The diameters of the nodal curve parallel to the sides of the
square are thus equal to ^a. Those measured along the diagonab
are sensibly smaller, equal to ^\/2 . a, or '471 a.]
It should be noticed that in whatever ratio to one another
G and D may be taken, the nodal curve always passes through
the four points of intersection of the nodal lines of the first two
cases, (7=0, 2) = 0. If the vibrations of these
cases be compounded with corresponding phases,
it is evident that in the shaded compartments of
Fig. (35) the directions of displacement are the
same, and that therefore no part of the nodal curve
is to be found there ; whatever the ratio of ampli-
tudes, the curve must be drawn through the un-
shaded portions. When on the other hand the phases are opposed,
the nodal curve will pass exclusively through the shaded portions.
Fig. 95.
H
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197. J EFFECT OF SLIGHT IRREGULARITIES. 315
When m = 8, n = 3, the nodes are the straight lines parallel
to the edges shewn in Fig. (36).
The last case [Frequency = 2'56] which we
shall consider is obtained by putting
m==S, n=:2, or m = 2, n = 3.
The nodal system is
[Frequency = 3*00.]
Cam sm — ^-^JJsm — -sin - =0,
a a n a
or, if the fiu;tors corresponding to the edges be rejected,
c(4cos«^~l)cos^ + Dcos^(4co8«^-l) = (9).
If C or i) vanish, we fall back on the nodal systems of the
component vibrations, consisting of straight lines parallel to the
edges. If C = J5, our equation may be written
fcos^ + cos^)f4cos^cos^~l) = (10),
of which the first factor represents the diagonal y + ^ = a, and
the second a hyperbolic curve.
If (7= — i), we obtain the same figure relatively to the other
diagonals
198. The pitch of the natural modes of a square membrane,
which is nearly, but not quite uniform, may be investigated by
the general method of § 90.
We will suppose in the first place that m and n are equal.
In this case, when the pitch of a uniform membrane is given,
the mode of its vibration is completely determined. If we now
conceive a variation of density to ensue, the natural type of
vibration is in general modified, but the period may be calculated
approximately without allowance for the change of t}rpe.
We have
= i ^m» |p. \ + jjSp sin' ^ 8in» ^ dxdy^ ,
I lami, Letont iwr ViUutieitf, p. 129.
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r
316 VIBRATIONS OF MEMBRANES. [198.
of which the second term is the increment of T due to ip. Hence
if w X Qospty and P denote the value of p previously to variation,
we have
i>™„':P^»=l-y"r^8in.^siB«^yd.dy (1).
where Pmm^^ — ^ , and c»= Tj -^ po.
For example, if there be a small load M attached to the middle of
the square,
4tAf nr
lW':P«.m'=l-* sin'mj (2), ;
CL pQ ^
in which sin* ^tt vanishes, if m be even, and is equal to unity, if
m be odd. In the former case the centre is on the nodal line of
the unloaded membrane, and thus the addition of the load produces
no result.
When, however, m and n are unequal, the problem, though re-
maining subject to the same general principles, presents a pecu-
liarity diflFerent from anything we have hitherto met with. The
natural type for the unloaded membrane corresponding to a speci-
fied period is now to some exteut arbitrary ; but the introduction
of the load will in general remove the indeterminate element. In
attempting to calculate the period on the assumption of the undis-
turbed tj^, the question will arise how the selection of the undis-
turbed tj^ is to be made, seeing that there are an indefinite
number, which in the uniform condition of the membrane give
identical periods. The answer is that those types must be chosen
which differ infinitely little from the actual types assumed under
the operation of the load, and such a type will be known by the
criterion of its making the period calculated from it a maximum
or minimum.
As a simple example, let us suppose that a small load M is
attached to the membrane at a point lying on the line x^^^a, and
that we wish to know what periods are to be substituted for the
two equal periods of the unloaded membrane, found by making
m = 2, n.= l, or m = l, n = 2.
It is clear that the normal types to be chosen, are those whose
nodes are represented in the first two cases of Fig. (32). In the
first case the increase in the period due to the load is zero, which
is the least that it can be; and in the second case the increase
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198.] SOLUTIONS APPLICABLE TO A TRIANGLE. 317
is the greatest possible. If /3 be the ordinate of if, the kinetic
energy is altered in the ratio
2 4^f4 + 2«"^'^'
and thus p^^ : P^«= l~^f sin*--^ (3)
while l>ii' = Pii' = Pi,».
The ratio characteristic of the interval between the two natural
tones of the loaded membrane is thus approximately
1+ - sm»-- (4).
a^p a
I( /3 = ^a, neither period is affected by the load.
As another example, the case where the values of m and n
are 3 and 1, considered in § 197, may be referred to. With a load
in the middle, the two normal types to be selected are those
corresponding to the last two cases of Fig. (34), in the former
of which the load has no effect on the period.
The problem of determining the vibration of a square mem-
brane which carries a relatively heavy load is more difficult, and
we shall not attempt its solution. But it may be worth while to
recall to memory the £su;t that the actual period is greater than
any that can be calculated from a hypothetical type, which differs
from the actual one.
199. The preceding theory of square membranes includes a
good deal more than was at first intended. Whenever in a vibrat-
ing system certain parts remain at rest, they may be supposed to
be absolutely fixed, and we thus obtain solutions of other questions
than those originally proposed. For example, in the present case,
wherever a diagonal of the square is nodal, we obtain a solution
applicable to a membrane whose fixed boundary is an isosceles
right-angled triangle. Moreover, any mode of vibration possible to
the triangle corresponds to some natural mode of the square, as
may be seen by supposing two triangles put together, the vibra-
tions being equal and opposite at points which are the images of
each other in the common hypothenuse. Under these circum-
stances it is evident that the hjrpothenuse would remain at rest
without constraint, and therefore the vibration in question is in-
cluded among those of which a complete square is capable.
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I
318 VIBRATIONS OF MEMBRANES. [l99.
The frequency of the gravest tone of the triangle is found by
putting m s: 1, n =s 2 in the formula
£ = Av(^.+„.) (1^
and is therefore equal to C'Jojia.
The next tone occurs, when m = 3, n = 1. In this case
JL^^lJ^ (2)
as might also be seen by noticing that the triangle divides itself
into two, Fig. (37), whose sides are less
than those of the whole triangle in the ^' ^'^'
ratio V2 : 1.
For the theory of the vibrations of
a membrane whose boundary is in the
form of an equilateral triangle, the
reader is referred to Lamp's Lefons
8ur r^ldsticitd. It is proved that the frequency of the gravest
J L tone is c-r-h, where h is the height of the triangle, which is the
same as the frequency of the gravest tone of a square whose
diagonal is h.
200. When the fixed boundary of the membrane is circular,
the first step towards a solution of the problem is the expression
of the general differential equation in polar co-ordinates. This
may be effected analytically ; but it is simpler to form the polar
equation de novo by considering the forces which act on the polar
element of area rdO dr. As in § 194 the force of restitution acting
on a small area of the membrane is
r
!'w _ (d^w Idw 1 d^w \ .
i¥~^\dr^~'^rd^'^f^dS^] ^^^•
and thus, if TJp = c* as before, the equation of motion is
d^w . {d^w
The subsidiary condition to be satisfied at the boundary is that
t(; = 0, when r=^a.
In order to investigate the normal component vibrations we
have now to assume that w \& b, harmonic function of the tima
A
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200.] POLAR CO-ORDINATES. 319
Thua, if w oc oob (2>^ — c), and for the sake of brevity we write
pjc ss k, the differential equation appears in the form
cPw Idw 1 d^w . , - ^ ,-,v
dr- + rd7 + r«rf^ + *'«'-^ <2).
in which k is the reciprocal of a linear quantity.
Now whatever may be the nature of to as a function of r and 0,
it can be expanded in Fourier's series
w = W9 + WiCO8{0-hai) + w, cos 2 (tf + Oj) + (3),
in which Wo, «^, &c. are functions of r, but not of 0. The result
of substituting from (3) in (2) may be written
the sunmiation extending to all integral values of n. If we
multiply this equation by cos n (^ + On), and integrate with respect
to between the limits and iir, we see that each term must
vanish separately, and we thus obtain to determine Wn SiS sl
function of r
d^Wn . Idwn
■*lt-H''-%h.-o W.
dr» r dr \
in which it is a matter of indifference whether the factor
cos h (^ + On) be supposed to be included in Wn or not.
The solution of (4) involves two distinct functions of r,
each multiplied by an arbitrary constant. But one of these
functions becomes infinite when r vanishes, and the corresponding
particular solution must be excluded as not satisfying the pre-
scribed conditions at the origin of co-ordinate& This point may
be illustrated by a reference to the simpler equation derived from
(4) by making k and n vanish, when the solution in question
reduces to w — logr, which, however, does not at the origin
satisfy V*w =s 0, as may be seen from the value of f(dw/dn) ds, inte-
grated round a small circle with the origin for centre. In like
manner the complete integral of (4) is too general for our
present purpose, since it covers the case in which the centre of
the membrane is subjected to an external force.
The other function of r, which satisfies (4), is the Bessel's
function of the rfi^ order, denoted by J^ (kr), and may be expressed
in several ways. The ascending series (obtained immediately
from the differential equation) is
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320 VIBRATIONS OF MEMBRANES. [20C
!1
i
2 . 4 . 2n + 2 . 2n + 4
2 . 4.6 . 2n + 2 . 2n + 4 . 2w + 6 "*■
«^»(*) = 2»r(n + l)r~2.2n + 2''';
..} (5X
from which the following relations between functions of conseca
tive orders may readily be deduced :
/,'(*) J,{z) (6),
2/„'(*) = J^,(«)-J„+.(z) (7).
2n
— /„(*) = J"^, (*) + /„+,(«) (8).
z
When n is an integer, J^ {z) may be expressed by the definiU
integral
Jn{z) = - I cos (-e: sin fi) — no)) do) (9),
WJo
which is Bessel's original form. From this expression it is evidenl
that «7n <md its differential coefficients with respect to z are always
less than unity,
n M V The ascending series (5), though infinite, is convergent for all
values of n and z\ but, when z is great, the convergence does not
begin for a long time, and then the series becomes useless as a
basis for numerical calculation. In such cases another series
proceeding by descending powers of z may be substituted with
advantage. This series is
-^-(") = V^r- 1.2(8.)> + }^('-4-^2J
/ 2 [ l«-4n' (l'-4n') (3«-4n') (5«-4n«) )
'^y irz\ 1.8^ 1.2.3.(8^)» "^ |
^^(^"i"^l) .....(10);
it terminates, if 2n be equal to an odd integer, but otherwise, it
runs on to infinity, and becomes ultimately divergent. Nevertheless
when z is great, the convergent part may be employed in calcula-
tion ; for it can be proved that the sum of any number of terms
differs from the true value of the function by less than the last
term included. We shall have occasion later, in connection with
another problem, to consider the derivation of this descending seriea
As Bessel's functions are of considerable importance in theo-
retical acoustics, I have thought it advisable to give a table foi
the functions Jq and Ji, extracted from Lommers^ work, and due
1 Loxnmel, Studien Hher die Be$$eV$chen Funetionen. Leipzig, 1868.
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200.]
BESSEL S FUNCTIONS,
321
originally to Hanseu. The functions J© and Jj are connected by
the relation Jo' = — «A •
1 »
•^o(')
z
45
.^.(«)
JM
z
J,{?)
•2453
0-0
10000
00000 1
•3205
•2311
9^0
•0903
1 01
•9975
•0499 ,
4-6
•2961
•2566
91
•1142
•2324
' 0-2
•9900
•0995 ,
4^7
•2693
•2791
9^2
•1367
•2174
0-3
•9776
•1483 '
48
•2404
•2985
93
•1577
•2004
0-4
•9604
•1960
49
•2097
•3147
9^4
•1768
•1816
OS
•9385
•2423
50
•1776
•3276
9-5
•1939
•1613
0-6
•9120
•2867
5^1
•1443
•3371
96
•2090
•1395
0-7
•8812
•3290
52
•1103
•3432
97
•2218
•1166
0-8
•8463
•3688
53
•0758
■3460
9-8
•2323
•0928
0-9
•8075
•4060
5 4
•0412
t3453
9^9
•2403
•0684
1-0
•7652
•4401
5-5
-•0068
•3414
10^0
•2459
■0435
M
•7196
•4709
5-6
+ ^0270
•3343
101
•2490
+ •0184
1-2
•6711
•4983
5^7
•0599
•3241
10-2
•2496
-•0066
1-3
•6201
•5220
5-8
•0917
•3110
103
•2477
•0313
1-4
•5669
•5419 1
6-9
•1220
•2951
10^4
•2434
•0555
1-5
•5118
•5579
6^0
•1506
■2767 '
10^5
•2366
•0789
1-6
•4554
•5699 :
61
•1773
•2559
106
•2276
•1012
1-7
•3980
•5778
62
•2017
•2329 ;
10-7
•2164
•1224
1-8
•3400
•5815
63
•2238
•2081 ,
108
•2032
•1422
1-9
•2818
•5812 1
6-4
•2433
•1816
10-9
•1881
•1604
2-0
••2239
•5767 1
6-5
•2601
•1538
ll^O
•1712
•1768
2-1
•1666
•5683
6-6
•2740
•1250
HI
•1528
•1913
2-2
•1104
•5560 '
67
•2851
•0953
112
•1330
•2039
2-3
•0555
•5399
6^8
•2931
•0652
113
•1121
•2143
2-4
+ •00-25
•5202 '
69
•2981
•0349
114
•0902
•2225
2-5
-•0484
■4971
7^0
•3001
-0047
11-5
•0677 •
•2284
2-6
•0968
•4708 1
7^1
•2991
+ •0252
11-6
•0446
•2320
2-7
•1424
•4416
7-2
•2951
•0543
117
-•0213
•2333
2-8
•1850
•4097
7^3
•2882
•0826 I
ir8
+ •0020
•2323
2-9
•2243
•3754
7^4
•2786
•1096
119
•0250
•2290
3-0
•2601
•3391 ,
75
•2663
•1352
12^0
•0477
•2234
31
•2921
•3009
76
•2516
•1592 ,
12-1
•0697
•2157
3-2
•3202
•2613
7^7
•2346
•1813
12^2
•0908
•2060
3-3
•3443
•2207
7-8
■2154
•2014 '
123
•1108
•1943
3-4
•3643
•1792
7-9
•1944
•2192
12^4
•1296
•1807
3-5
•3801
•1374
8^0
•1717
•2346
12-5
■1469
•1655
3-6
•3918
•0955
8-1
•1475
•2476 ,
12^6
■1626
•1487
3-7
•3992
•0538
8^2
•1222
•2580 ,
12^7
•1766
•1307
3-8
•4026
+ -0128
8-3
•0960
•2657
128
•1887
•1114
3-9
•4018
-•0272
8^4
•0692
•2708
12^9
•1988
•0912
4-0
•3972
•0660
8-5
•0419
•2731
13-0
•2069
•0703
4-1
•3887
•1033
8-6
+ •0146
•2728 1
131
•2129
•0489
4-2
•3766
•1386
8^7
- 0125
•2697 I
132
•2167
•0271
4-3
•3610
•1719
8-8
•0392
•2641 '
133
•2183
-0052
44
•3423
•2028
8^9
•0653
•2559
13-4
•2177
+ •0166
u.
21
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322 VIBRATIONS OF MEMBRANES. [201.
201. In accordaace with the notation for Bessers functions
the expression for a normal oompouent vibration may therefore be
written
w^PJn(kr) co8n(tf + a) co&(pt + €) (1);
and the boundary condition requires that
Jn{ka)^0 (2),
an equation whose roots give the admissible values of k, and
therefore o{ p.
The complete expression for w is obtained by combining the
particular solutions embodied in (1) with all admissible values of
k and n, and is necessarily geneml enough to cover any initial
circumstances that may be imagined. We conclude that any
function of r and may be expanded within the limits of the
circle r=a in the series
w = 22 /n(*r){^ cos ntf + ^ sin n^} (3).
For every integral value of n there are a series of values of k,
given by (2); and for each of these the constants ^ and '^ are
arbitrary.
The determination of the constants is effected in the usual
way. Since the energy of the motion is equal to
J p r rw^rdddr (4),
and when expressed by means of the normal co-ordinates can only
involve their squares, it follows that the product of any two of the
terms in (3) vanishes, when integrated over the area of the circle.
Thus, if we multiply (3) by Jn{kr) cos nO, and integrate, we
find
I / w Jn{kr) cos nOrdrdO
^(l>jj[Jn(kr)Ycos^n0rdrd0 = <l>.irr[Jn{h')']^rdr (5).
by which <f> is determined. The corresponding formula for ^Jr is
obtained by 'writing sin n6 for cos nd, A method of evaluating
the integral on the right will be given presently. Since tfe and -^
each contain two terms, one varying as cos^^ and the other as
sin pt, it is now evident how the solution may be adapted so as to
agree with arbitrary initial values of w and w.
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202.] CIRCULAR BOUNDARY, 323
202. Let us now examine more particularly the character of
the fundamental vibrations. If n == 0, i«r is a function of r only,
that is to say, the motion is symmetrical with respect to the centre
of the membrane. The nodes, if any, are the concentric circles,
whose equation is
/o(A?r) = (1).
When n has an integral value different from zero, wias, func-
tion of d as well as of r, and the equation of the nodal system
takes the form
Jn{kr) cosn(tf--o) = (2).
The nodal system is thus divisible into two parts, the first con-
sisting of the concentric circles represented by
Jn(kr)^0 (3),
and the second of the diameters
^ = a + (2m + l)7r/2w (4),
where m is an integer. These diameters are n in number, and
are ranged uniformly round the centre; in other respects their
position is arbitrary. The radii of the circular nodes will be
investigated further on.
203. The important integral formula
Vn(ir) J„(A:V)rdr = (1),
Jo
/o
where k and k' are different roots of
Jn{ka)^0 (2),
may be verified analytically by means of the differential equations
satisfied by J»(fcr), t/«(A:V); but it is both simpler and more
instructive to begin with the more general problem, where the
boundary of the membrane is not restricted to be circular.
The variational equation of motion is
where
and therefore
SV+pjjiuBwdxdy = (8)
^-i''!l{&<t)h'' <*>■
'^-'M'^'f.wi'"' <'>•
•21—2
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324 VIBRATIONS OF MEMBRANES. [203
III these equations w refers to the actual motion, and St(; to a hypo
thetical displacement consistent with the conditions to which th(
system is subjected. Let us now suppose that the system is exe
cuting one of its normal component vibrations, so that w=^u, and
u-hp^u^^O (6),
while Sw is proportional to another normal function v.
Since k^p/c, we get from (3)
i-//«t;da.dy=//jg| + ||}dxdy (7>
The integral on the right is symmetrical with respect to u and v
and thus
{hf^-lc'){\uvdxdy = (8),
where h'^ bears the same relation to v that 1<^ bears to u.
Accordingly, if the normal vibrations represented by u and v
have different periods,
jjuvdxdy = (9).
In obtaining this result, we have made no assumption as to the
boundary conditions beyond what is implied in the absence of re-
actions against acceleration, which, if they existed, would appear
in the fundamental equation (3).
If in (8) we suppose k' = k% the equation is satisfied identically,
and we cannot infer the value of jju^dxdi/. In order to evaluate
this integral we must follow a rather diflferent course.
If u and V be functions satisfying within a certain contour the
equations V^w + k*u = 0, V^v + k'^v = 0, we have
(k'-' - k^) jjuvdxdy^^l Hv V^u - u V^v^dxdy
by Green s theorem. Let us now suppose that v is derived from
n by slightly varying k, so that
V = If + -IT SA-, kf =^k-\-ik)
substituting in (10), we find
St//..^.y./(S£-.^;U (U);
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203.] VALUES OF INTEGRATED SQUARES. 325
or, if u vanish on the boundary,
a//..,l.iy-/||* (12).
For the application to a circular area of radius r, we have
u ^ C0& nd Jn(kr)) . ^.
v = cosneJn(kV)] ^^"^^^
and thus from (10) on substitution of polar co-ordinates and integra-
tion with respect to 0,
(A:'* - k^) rj^ (At) J^ (AV) rdr
Jo
^rJ^{k'r)^~Jn{kr)-rJn(kr)^^^ (^*)-
Accordingly, if
^^Jnik'r) : J, (*'r) = ^^Jn{M : J„(ir),
and k and k' be different,
fjn(kr)J^(k'r)rdr^O (15).
an equation first proved by Fourier for the case when
Jnikr) = Mk'r)='0.
Again from (11)
dJdJ , d^r
1 UK iir iiV dk
dashes denoting differentiation with respect to kr. Now
and thus
2fjn'(kr)rdr = i^J^Hkr) + 7- (l - ^f^^j Jn\kr) (16).
This result is general ; but if, as in the application to membranes
with fixed boundaries, /„ (kr) = 0,
then 2['jn^{kr)rdr^r^J^^(kr) (17).
Jo
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326 VIBRATIONS OF MEMBRANES. [204.
201. We may use the result just arrived at to simplify the
expressions for T and F. From
W = 22 {^«»/n iJCwnr) cos ud + ^^^nJn i^^mnT) siu Tltf j (1),
we find
r«ip7ra«22J'„'»(*m«a){<^mn' + ^mn'; (2),
F » i p7ra« Upmn^Jn'^ {Kna) {^«n' + ^mn'\ (3) ;
whence is derived the normal equation of motion
^+i»«„'^„-— .jr-/,'^,^-^) (4).
and a similar equation for '^mn* The value of 4>mn is to be found
from the consideration that ^wn^^mn denotes the work done by the
impressed forces during a hypothetical displacement S^mn ; so that
if Z be the impressed force, reckoned per unit of area,
^mn^uZJn{krnnr)co^n0rdrde (5).
These expressions and equations do not apply to the case n = 0,
when ^ and '^ are amalgamated. We then have
r^^pira'p„,o'Jo'{kmoa)il>mo'\ ^ ^'
*,
As an example, let us suppose that the initial velocities are zero,
and the initial configuration that assumed under the influence of a
constant pressured; thus
^„,o = Z.2tr\ Jo {k„,or) rdr,
Jo
Now by the differential equation,
rJ, (kr) - - {r Jo'' (kr) + 1 Jo' (^t)}.
and thus
Vo(*r)rdr = -^Jo'(Jta) (8);
A/
80 that 4>^ ^« ZJ,' (k^a).
Substituting this in (7), we see that the initial value of ^^o is
— 2Z
I
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204] SPECIAL PROBLEMS. 327
For values of n other than zero, <t> and the initial value of ^^n
vanisL The state of the system at time t is expressed by
<A»n« = (^mo)e-o.COSp^e (10),
W = 2^,«,Jo(*»nor) (11),
the summation extending to all the admissible values o( tCm^'
As an example o{ forced vibrations, we may suppose that Z, still
constaut with respect to space, varies as a harmonic function of the
time. This may be taken to represent roughly the circumstances
of a small membrane set in vibration by a train of aerial waves.
If Z=cos qt, we find, nearly as before,
w^ ^cosg^Z , ,,'^^^^''L, , (12).
The forced vibration is of course independent of 0. It will be seen
that, while none of the symmetrical normal components are missing,
their relative importance may vary greatly, especially if there be a
near approach in value between q and one of the series of quanti-
ties j[)^. If the approach be very close, the eflfect of dissipative
forces must be included.
[Again, suppose that the force is applied locally at the centre.
By (5)
4>,^ = ZiCosg^ (13),
if Zi COS qt denote the whole force at time t From (7)
^ ^i_??l?i nA\
'''"^-p7ra'(l>«o»-3«)^o'MUa) ^'*^'
and w is then given by (11). The series is convergent, unless
r = 0.
But this problem would be more naturally attacked by including
in the solutions of (4) § 200 the second BesseFs function § 341.
In this method k—qjc\ and the ratio of constants by which the
two functions of r are multiplied is determined by the boundary
condition. When q coincides with one of the values of j», the
second function disappears from the solution.]
205. The pitches of the various simple tones and the radii of
the nodal circles depend on the roots of the equation
Jn(*a)-c/n(^) = 0.
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328
VIBRATIONS OF MEMBRANES.
[20!
If these (exclusive of zero) taken in order of magnitude b
called Zn'^^^ Zn^^\ Znf^^ Zn}"^ , then the admissible values of
are to be found by multiplying the quantities Zn^*^ by c/a. Tb
particular solution may then be written
W = Jn [z^^*^ -\ {^n<'> cos 1X0 + B^^*^ siu nO] cos l^^r,,^'^^ -€n <"[...(!).
The lowest tone of the group n corresponds to ^,/** ; and since i
this case Jn (-^n"^ rja) does not vanish for any value of r less than c
there is no interior nodal circle. If we put « = 2, «/„ will vanisl
when
Ob
that is, when
r^a
zJ^'
which is the radius of the one interior nodal circle. Similarlj
if we take the root Zn^''\ we obtain a vibration with « — 1 noda
circles (exclusive of the boundary) whose radii are
a
All the roots of the equation J^ {ka) = are real. For, \\
possible, let Aa = X + ifi be a root ; then A;'a = X — i^i is also a root
and thus by (14) § 203,
4i\/A I Jn {kr) Jn {k'r) rdr = 0.
Now Jn (kr\ Jn (k'r) are conjugate complex quantities, whos(
product is necessarily positive ; so that the above equation requires
that either \ or fi vanish. That \ cannot vanish appears from
the consideration that if ka were a pure imaginary, each term ol
the ascending series for Jn would be positive, and therefore the
sum of the series incapable of vanishing. We conclude thai
/Lt = 0, or that k is reaP. The same result might be arrived at
from the consideration that only circular functions of the time
can enter into the analytical expression for a normal component
vibration.
The equation Jn (z) = has no equal roots (except zero). Froii
equations (7) and (8) § 200 we get
J'--J -7
Z
^ RiemaDD, PartielU DiffereniialgUichungen, Brauoschweig, 1869, p. 2C0.
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205.] ROOTS OF bessel's functions. 329
whence we see that if «/«, Jn vanished for the same value of z, Jn+i
would also vanish for that value. But in virtue of (8) § 200
this would requii-e that all the functions /„ vanish for the value
of z in question \
206. The actual values of z^ may be found by interpolation
from Hansen's tables so far as these extend ; or formulae may be
calculated from the descending series by the method of successive
approximation, expressing the roots directly. For the important
case of the symmetrical vibrations (n = 0), the values of z^ may be
found from the following, given by Stokes':
z^^*^ ^. '050661 '053041 2 62051 .-.
For n = 1, the formula is
z^_ -151982 015399 _ '245270
It -f+^^ 4«+r"^(4«-My (45+l)» ^ ^*
The latter series is convergent enough, even for the first root,
corresponding to « = 1. The series (1) will suffice for values of b
greater than unity; but the first root must be calculated
independently. The accompanying table (A) is taken from
Stokes' paper, with a slight diflFerence of notation.
It will be seen either from the formulae, or the table, that the
diflference of successive roots of high order is approximately tt.
This is true for all values of «, as is evident from the descending
series (10) § 200.
[The general formula, analogous to (1) and (2), for the roots of
Jn C-^) has been investigated by Prof. M*^Mahon. If m = 4w*, and
a = j7r(2n-l+4«) (3),
32(»t-l)(83w''- 982ot + 3779)
15 (8a)»
.(4).
^ Bourget, " M($moire sur le mouyement vibratoire des membranes circulaires,"
Ann, de Vecole nortnale, t. iii., 1866. In one passage M. Boarget implies that he
has proved that no two Bessel's functions of integral order can have the same root,
bat I cannot find that he has done so. The theorem, however, is probably true ;
in the case of functions, whose orders differ by 1 or 2, it may be easily proved from
the formuls of § 200.
^ Camb, Phil, Trans. Vol. ix. ** On the numerical calculation of a class of defi-
nite Integrals and infinite series." [In accordance with the calculation of Prof.
M*^Mahon the numerator of the last term in (2) has been altered from '245835
to -246270.]
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330
VIBRATIONS OF MEliBBANES.
[206.
This formula may be applied not only to integral values of n as Id
(1) and (2), but also when n is fractional. The cases of n = 4, and
71 == } are considered in § 207.]
M. Bourget has given in his memoir very elaborate tables of
the frequencies of the different simple tones and of the radii of
the nodal circles. Table B includes the values of z, which satis^*
*/„(-?), for w = 0,1,... 5, « = 1, 2, ... 9.
Table A.
«
-for /,(«) = 0.
1
•7655
2
1-7571
3
2-7546
4
3-7534
5
4-7527
6
5-7522
7
6-7519
8
7-7516
9
8-7514
10
9-7513
11
10-7512
12
11-7511
Diff.
-fory,(«) = 0.
Diff.
•9916
•9975
•9988
•9993
•9995
•9997
•9997
•9998
•9999
•9999
•9999
1^2197
2-2330
3-2383
4-2411
5-2428
6-2439
7-2448
8^2454
9-2459
10-2463
11-2466
12 2469
1^0133
1^0053
1-0028
1-0017
1^001 1
1-0009
1-0006
1-0005
1-0004
1-0003
1-0003
When n is considerable the calculation of the earlier roots
becomes troublesome. For very high values of n, -?»<'7^ approxi-
mates to a ratio of equality, as may be seen from the consideration
that the pitch of the gravest tone of a very acute sector must tend
to coincide with that of a long parallel strip, whose width i3 equal
to the greatest width of the sector.
Table B.
— \
«
n =
n=l
n = 2
n = 3
n = 4
n = 5 1
1
2-404
3-832
5-135
6-379
7-586
8-780
2
5-520
7-016
8-417
9-760
11-064
12-339
3
8-654
10-173
11-620
13-017
14-373
15-700
4
11-792
13-323
14-796
16-224
17-616
18-982
5
14-931
16-470
17-960
19-410
20-827
22-220
6
18-071
19-616
21-117
22-583
24-018
25-431
7
21-212
22-760
24-270
25-749
27-200
28-6-28
8
24-353
25-903
27-421
28-909
30-371
31-813 ,
9
27-494
29-047
30-571
32-060
33-512
34-983 1
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206.]
NODAL FIGURES.
331
£.000
2.206
.456
9A56
3.0 52
1.594
%.f4«
The figures represent tlie more important normal modes of
vibration, and the numbers affixed give the frequency referred to
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332
VIBRATIONS OF MEMBRANES.
[20f
the gravest as unity, together with the radii of the circular node
expressed as fractions of the radius of the membrane. In the cas
of six nodal diameters the frequency stated is the result of a rougl
calculation by myself.
The tones corresponding to the various fundamental modes o
the circular membrane do not belong to a harmonic scale, bu
there are one or two approximately harmonic relations which ma;
be worth notice. Thus
^ X 1-594 = 2125 = 2-136 nearly,
f X 1-594. = 2-657 = 2-653 nearly,
2 X 1-594 = 3188 = 3*156 nearly;
so that the four gravest modes with nodal diameters only woul(
give a consonant chord.
The area of the membrane is divided into segments by thi
nodal system in such a manner that the sign of the vibratioi
chunges whenever a node is crossed. In those modes of vibratioi
which have nodal diameters there is evidently no displacement o
the centre of inertia of the membrane. In the case of symmetri
cal vibrations the displacement of the centre of inertia is proper
tional to
|Vo (h-) rdr = - j^ U" (kr) + ji J^' (kr) I I'dr = - | ^o' (*«),
iiD expression which does not vanish for any of the admissible
values of k, since Jo (z) and J^ (z) cannot vanish simultaneously
111 all the symmetrical modes there is therefore a displacement o
the centre of inertia of the membrane.
207. Hitherto we have supposed the circular area of th(
membrane to be complete, and the circumference only to b(
fixed; but it is evident that our theory virtually includes th(
solution of other problems, for example — some cases of a mem
brane bounded by two conceutric circles. The complete theorj
for a membrane in the form of a ring requires the second Bessel't
function.
The problem of the membrane in the form of a serai-circle
may be regarded as already solved, since any mode of vibration
of which the semi-circle is capable must be applicable to the
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207.] FIXED BADIUS. 333
complete circle also. In order to see this, it is only necessary
to attribute to any point in the complementary semi-circle the
opposite motion to that which obtains at its optical image in
the bounding diameter. This line will then require no constraint
to keep it nodal. Similar considerations apply to any sector
whose angle is an aliquot part of two right angles.
When the opening of the sector is arbitrary, the problem
may be solved in terms of Bessels functions of fractional order.
If the fixed radii are d = 0, d = )8, the particular solution is
virO
w — PJ„fp(kr) sin ^ cos(2>^ — c) (1),
where i/ is an integer. We see that if fi be an aliquot part of tt,.
mrfff is integral, and the solution is included among those ab eady
used for the complete circle.
An interesting case is when fi = 2ir, which corresponds to the
problem of a complete circle, of which the
radius = is constrained to be nodal. ^'^8' ^®'
We have
w = PJ^y (kr) sin ^v0 cos (pt — e).
When V is even, this gives, as might be
expected, modes of vibration possible without
the constraint; but, when v is odd, new
modes make their appearance. In fact, in
the latter case the descending series for /
terminates, so that the solution is expressible in finite terms.
Thus, when i/ = 1,
sin fc7*
^"^V(F)^'''*^ cos(p^-6) (2).
The values of k are given by
sin ka = 0, or ka = stt.
Thus the circular nodes divide the fixed radius into equal
parts, and the series of tones form a bar- j.j 39
monic scale. In the case of the gravest
mode, the whole of the membrane is at any
moment deflected on the same side of its
equilibrium position. It is remarkable that
the application of the constraint to the
radius 0^0 makes the problem easier than
before.
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[2(
(3).
(4>
334 VIBBATIONS OF MKMBRANES.
If we take v = 3, the solution is
In this case the nodal radii are Fig. (39)
and the possible tones are given by the equation
tan ka=:ka
To calculate the roots of tan a? = a; we may assume
a: = (* + i)7r-y = Z-y,
where y is a positive quantity, which is small when x is large.
Substituting this, we find cot y = X - y,
whence
* zv -a: Z' v 3 15 315 •••
This equation is to be solved by successive approximatio
It will readily be found that
y=z-.+?jr-.+!|x-+J|x-+...,
60 that the roots of tan x^^x are given by
• (5).
r^.
where X = (« 4- i) tt.
In the first quadrant there is no root after zero since tan x >
and in the second quadrant there is none because the signs
X and tana; are opposite. The first root after zero is thus
the third quadra it, corresponding to « = 1. Even in this ca
the series converges sufficiently to give the value of the ro
with considerable accuracy, while for higher values of a it
all that could be desired. The actual values of x/ir are 1-430
2-4590, 3-4709, 4-4747, 5-4818, 6-4844, &c.
208. The effect on the periods of a slight inequality in tl
density of the circular membrane may be investigated by tl
general method § 90, of which several examples have alread
been given. It will be sufficient here to consider the case of
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208.] EFFECT OF SMALL LOAD. 335
small load M attached to the membrane at a point whose radius
vector is /.
We will take first the symmetrical types (n = 0), which may
still be supposed to apply notwithstanding the presence of M. The
kinetic energy T is (6) § 204 altered from
and therefore
/>«. .i»»-i p^„.j-;,(^^„) w.
where P^ denotes the value of je>„w'i when there is no load.
The uns3anmetrical normal types are not fully determinate for
the unloaded membrane ; but for the present purpose they must
be taken so as to make the resulting periods a maximum or
minimum, that is to say, so that the effect of the load is the
greatest' and least possible. Now, since a load can never raise
the pitch, it is clear that the influence of the load is the least
possible, viz. zero, when the type is such that a nodal diameter (it
is indifferent which) passes through the point at which the load is
attached. The unloaded membrane must be supposed to have two
coincident periods, of which one is unaltered by the addition of the
load. The other type is to be chosen, so that the alteration of
period is as great as possible, which will evidently be the case
when the radius vector / bisects the angle between two adjacent
nodal diameters. Thus, if r' correspond to d = 0, we are to take
V) = ^nn Jn {Kinr) COS Yid \
SO that (2) § 204
T= i pira? 4>^n' Jn' (kfnna) + i if ^mn' Jn' {KnV).
The altered p^n' is therefore given by
Pmn^r^n p^a« /,/« (i^„a) ^^^'
Of course, if r' be such that the load lies on one of the nodal
circles, neither period is affected.
For example, let M be at the centre of th^ membrane. J^ (0)
vanishes, except when n = 0; aud Jo(0) = l. It is only the
symmetrical vibrations whose pitch is influenced by a central load,
and for them by (1)
M
P-^"''^^-"^^''j-^(k,^a)~^a^ (3).
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1
)8.
336 VIBRATIONS OF MEMBRANES. [208.
By (6) §200 Jo(z)^-Ji{zl
so that the application of the formula requires only a knowledge of
the values of «7i (z), when J^ (z) vanishes, § 200, For the gravest
mode the value of Jo (Jcm<^) is '51903*. When Ar^o^ is consider-
able,
t/i' (^mott) =2-5- irkmod
approximately; so that for the higher components the influence of
the load in altering the pitch increases.
The influence of a small irregularity in disturbing the nodal
system may be calculated from the formulsB of § 90. The most
obvious effect is the breaking up of nodal diameters into curves
of hj^rbolic form due to the introduction of subsidiary sym-
metrical vibrations. In many cases the disturbance is favoured
by close agreement between some of the natural perioda
209. We will next investigate how the natural vibrations of
a uniform membrane are affected by a slight departure from the
exact circular form.
Whatever may be the nature of the boundary, w satisfies the
equation
<Pw I dw ^ 1 d^w , ,. ^ .^^
d> + ;: di^+i^ d^+*'«' = o (!)•
where ifc is a constant to be determined. By Fourier's theorem w
may be expanded in the series
w = Wo + Wi COS {0 + ai) •{• w^cos 2 (0 + OLt) '\-
+Wncosn(0+an) + ,
where w©, Wi, &c. are functions of r only. Substituting in (1), we
see that Wn must satisfy
d^Wn 1 dwn
-lt<^->'0-
dr^
of which the solution is
Wn^Jnif^r);
for, as in § 200, the other function of r cannot appear.
The general expression for w may thus be written
w = AoJo{kr) + Ji{kr)(AiCO8 + Bisin0)
+ ... + Jn(kr)(AnCO8n0 -h Bnsmn0) + (2).
For all points on the boundary w is to vanish.
1 The succeeding Talues are approximately -341, *271, '282, *206, -187, Ac,
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209.] NEABLY CIRCULAR BOUNDARY. 337
In the case of a nearly circular membrane the radius vector is
nearly constant. We may take r — a + Br, 8r being a small
function of 0. Hence the boundary condition is
• O^A,[Jo(ka) + kBrJ,'(ka)] +
4- [Jn (lea) -{-kSr Jn (ka)] [An cos n0 + Bn sin nO]
+ (8),
which is to hold good for all values of 0.
Let us coQsider first those modes of vibration which are nearly
symmetrical, for which therefore approximately
w = AoJo(kr).
All the remaining coefficients are small relatively to Aq, since
the type of vibration can only differ a little from what it would
be, were the boundary an exact circle. Hence if the squares of
the small quantities be omitted, (3) becomes
A^ [Jo {ka) + ASr Jo (ka)] 4- Ji (ka) [A^ cos ^ + jBi sin 0]
+ . . . + t/n (ka) [AnCO8n0 + BnSmn0]+ ...=0 (4).
If we integrate this equation with respect to between the
limits and 27r, we obtain
27r Jo (ka) + Jo (ka) r'kBrdd = 0,
Jo
ka + kfjSr ^1=0 (5),
or Jo-
which shews that the pitch of the vibration is approximately the
same as if the radius vector had uniformly its mean value.
This result allows us to form a rough estimate of the pitch of
any membrane whose boundary is not extravagantly elongated.
If a denote the area, so that pa is the mass of the whole mem-
brane, the frequency of the gravest tone is approximately
(27r)-^ X 2-404 x y/^> (6)^
In order to investigate the altered type of vibration, we may
* [A numerical error is here corrected.]
22
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338 VIBRATIONS OF MEMBRANES. [209.
multiply (4) by cos nO, or sinn^, and then integrate as before.
Thus
/•2ir \
Ao Jo(ka) I k Br cos n0 dd-^irAn Jn (ka) =
Jo
'0
f'
..(7),
AoJo(ka)r'kBrsmn0d0 + irBnJn(ka)^O
which determine the ratios An : Aq and Bn : Ao.
If Sr = Sro + 5riH- ... + Srn+...
be Fourier's expansion, the final expression for w may be writt'en,
w iAq^Jq (kr)
When the vibration is not approximately symmetrical, the
question becomes more complicated. The normal modes for the
truly circular membrane are to some extent indeterminate, but the
irregularity in the boundary will, in general, remove the indeter-
minateness. The position of the nodal diameters must be taken,
so that the resulting periods may have maximum or minimum
values. Let us, however, suppose that the approximate type is
w^A,J^{kr)Q(^ve (9),
and afterwards investigate how the initial line must be taken in
order that this form may hold good.
All the remaining coefficients being treated as small in com-
parison with ^^, we get from (4)
Aq Jq (ka) + ... + Ay[J^ (ka) + kirJJ (4a)] cos vd
'\-B^J,(ka)8iav0 •{•
'\-Jn{ka)[AnCiysn0'\'BnCosnd] + .,.^0 (10).
Multiplying by cos v0 and integrating,
/•ftr
IT J, (ka) + k J/ (ka) Br cos* v0d0 = 0,
or
J^ pa + Ar I Sr cos* i/^ — I = 0,
which shews that the effective radius of the membrane is
a4-/>cos.^f (IIX
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209.] NEARLY CIRCULAR BOUNDARY. 339
The ratios of An and Bn to Ay may be found as before by in-
tegrating equation (10) after multiplication by cos nO, sin nd.
But the point of greatest interest is the pitch. The initial line
is to be so taken as to make the expression (11) a maximum or
minimum. If we refer to a line fixed in space by putting d — a
instead of 0, we have to consider the dependence on a of the
quantity
' 5rcos« !/(<?-- a) dd,
r
'0
which may also be written
cos* va I Sr cos^ i0d0 + 2 cos i/a sin va j Br cos i^O sin v0d0
-hsin^vaj Srsm^v0d0 (12),
and is of the form
A cos* va 4- 25 cos va sin va + C sin* i/a,
A, B, C being independent of a. There are accoi-dingly two
admissible positions for the nodal diameters, one of which makes
the period a maximum, and the other a minimum. The diameters
of one set bisect the angles between the diameters of the other
set.
There are, however, cases where the normal modes remain inde-
terminate, which happens when the expression (12) is independent
of a. This is the case when Br is constant, or when Br is propor-
tional to cos v0. For example, if Br were proportional to cos 20,
or in other words the boundary were slightly elliptical, the nodal
system corresponding to n = 2 (that consisting of a pair of per-
pendicular diameters) would be arbitrary in position, at least to
this order of approximation. But the single diameter, correspond-
ing to n = l, must coincide with one of the principal axes of
the ellipse, and the periods will be different for the two axes.
210. We have seen that the gravest tone of a membrane,
whose boundary is approximately circular, is nearly the same as
that of a mechanically similar membrane in the form of a circle of
the same mean radius or area. If the area of a membrane be
given, there must evidently be some form of boundary for which
the pitch (of the principal tone) is the gravest possible, and this
22—2
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w
r^
340 VIBRATIONS OF MEMBRANES. [21(
form can be no other than the circle. In the ca^e of approximat
circularity an analytical demonstration may be given, of which th
following is an outline.
The general value of w being
u; = ilot/o(^)4- ... + Jn(AT) (-4nC08 7i^4-5sinn6?) + (1),
in which for the present purpose the coeflScients A^B^,... are smal
relatively to -4©, we find from the condition that w vanishe
when r = a-\-tr,
A,J,{ka) + kA,J,'{ka)Zr + \l^A,J^\ka),{h^)''^
+ 2 [[Jn{ka) + 1cJn{ka) 8r + ...}{ilHCOsn^ + JBnsin n^}] = 0...(2).
Hence, if
8r = a, cos ^ + /8i sin d + . . . + a„ cos n^ + )8„ sin wd + (3),
we obtain on integration with respect to from to 27r,
2A, Jo + i k^A, Jo" 2^]* «•• + Pn^)
+ A:2^^J(a,,4n + /3nfin)Jn'] = (4).
from which we see, as before, that if the squares of the smal
quantities be neglected, Jo (ka) = 0, or that to this order of ap
proximation the mean radius is also the eflfective radius. Ii
order to obtain a closer approximation we first determine A^ : A
and Bn : -4© by multiplying (2) by cos nO, sin nd, and then in
tegrating between the limits and 27r. Thus
AfiJn = "■ kanA Jo , Bf^Jn = " kpf^ AqJo (5).
Substituting these values in (4), we get
J, (ka) = i *» t^ ^(a„^ + /3„') 1*^' - i J,' jl (6).
Since Jo satisfies the fundamental equation
Jo" + j^Jo' + Jo = (7),
and in the present case Jo = approximately, we may replac
Jo" '^y ~ IT *^o'- Equation (6) then becomes
/,(A:a) = U-^/„'2;:;[(«..' + /3,.'){^' + 2j-}] (8).
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210.] FORM OP MAXIMUM PERIOD. 341
Let us now suppose that a + cia is the equivalent radius of the
membrane, so that
Jo [* (a + da)] = Jo (lea) + Jo (ka) kda = 0.
Then by (8) we find
'^=-**ch'-»-^-'){^'-*-2y] <^>-
Again, if a + da' be the radius of the truly circular membrane
of equal area,
A»' = ^X^"(a„» + /9„') (10);
SO that
The question is now as to the sign of the right-hand member.
If n = 1, and z be written for ka,
vanishes approximately by (7), since in general J^^ — J^\ and
in the present case Jo {z) = nearly. Thus da' — da = 0, as should
evidently be the case, since the term in question represents merely
a displacement of the circle without an alteration in the form of
the boundary. When n = 2, (8) § 200,
t/2 — — t/i — t/oi
Z
from which and (7) we find that, when Jo = 0,
j;^^'-4
J, 2z
whence
(12),
da'-da = ^(a,«+A»)g'-l) (13),
which is positive, since -» — 2*404.
We have still to prove that
Jn{z)
is positive for integral values of n greater than 2, when z » 2*404.
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342
VIBRATIONS OF MEMBRANES.
[211
I
For this purpose we may avail ourselves of a theorem given i
Riemann's Partielle Differentialgleichungen, to the effect thi
neither «/» i^or J^ has a root (other than zero) less than n. Tt
differential equation for Jn may be put into the form
d(\ogzy
+ (^-n»)/„(^)=t);
while initially Jn and Jn (as well as dJn/d log z) are positive. A(
cordingly dJn/d log z begins by increasing and does not cease to d
so before z = n, from which it is clear that within the range z =
to ^ = n, neither Jn nor Jn can vanish. And since Jn and /«' ai
both positive until z^n/it follows that, when n is an integer greats
than 2*404, da' — da is positive. We conclude that, unless o,, )8
a„ ... all vanish, da' is greater than da, which shews that in tb
case of any membrane of approximately circular outline, the circl
of equal area exceeds the circle of equal pitch.
We have seen that a good estimate of the pitch of an approx
mately circular membrane may be obtained from its area alon<
but by means of equation (9) a still closer approximation may b
effected. We will apply this method to the case of an elh*ps(
whose semi-axis major is R and eccentricity e.
The polar equation of the boundary is
r^R {l-Je>-^e* + +Je«cos2d + } (14);
so that in the notation of this section
a = /i(l-ie»-^e*), Oa^ie^iJ.
Accordingly by (9)
*^=-w-*^-{:^V) + 2.}'
or by (12), since A:iJ = -a: = 2*404,
o4
Thus the radius of the circle of equal pitch is
^, i>fi 1^ 9-779 e*)
in which the term containing e* should be correct.
,(15).
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210.] ELUPTICAL BOUNDARY. 343
The result may also be expressed in terms of e and the area <r.
We have
and thus
from which we see how small is the influence of a moderate eccen-
tricity, when the area is given.
211. When the fixed boundary of a membrane is neither
straight nor circular, the problem of determining its vibrations
presents difficulties which in general could not be overcome
without the introduction of functions not hitherto discussed or
tabulated. A partial exception must be made in favour of an
elliptic boundary ; but for the purposes of this treatise the im-
portance of the problem is scarcely sufficient to warrant the
introduction of complicated analysis. The reader is therefore
referred to the original investigation of M. Mathieu^
[The method depends upon the use of conjugate functions. If
a? + iy = 6Cos(f + ii;) (1),
then the curves tf = const, are confocal ellipses, and f = const, are
confocal hjrperbolas. In terms of f , i; the fundamental equation
(V* + A*)tt = becomes
^ + g+A;''(co8h«i,-co8«f)tt = (2).
where k^ =« ke.
The solution of (2) may be found in the form
u.H(f).H(i,) (3),
in which H is a function of f only, and H a function of tf only,
provided
^-(i'»cos»f-.a)H = (4),
J" + (Jb'»cosh>i;-a)H = (5),
a being an arbitrary constants
^ LiouTille, xm., 1S6S; Cour$ de physique mathimatique^ 1878, p. 122.
' Pockels, Oher diepartielle DifferentiaXgleichung ^u-k-l?u^Of p. 114.
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344
VIBRATIONS OF MEMBRANES.
[21
Michell* has shewn that the elliptic transformation (1) is tl
only one which yields an equation capable of satisfiewtion in tl
form (3).]
Soluble cases may be invented by means of the gener
solution
w = AoJo(kr)'\- .,.+{AnCosn0 + BnSinn6)Jn(kr)+
For example we might take
w = Jo (kr) — \ J"i (At) cos 0,
and attaching different values to \, trace the various forms
boundary to which the solution will then apply.
Useful information may sometimes be obtained from tl
theorem of § 88, which allows us to prove that any contraction <
the fixed boundary of a vibrating membrane must cause an elevj
tion of pitch, because the new state of things may be conceived 1
differ from the old merely by the introduction of an addition
constraint. Springs, without inertia, are supposed to urge tl
line of the proposed boundary towards its equilibrium positio
and gradually to become stiffer. At each step the vibratioi
become more rapid, until they approach a limit, corresponding
infinite stiffness of the springs and absolute fixity of their poin
of application. It is not necessary that the part cut off shou
have the same density as the rest, or even any density at all.
For instance, the pitch of a regular polygon is intermedia
between those of the inscribed and circumscribed circles. Clas
limits would however be obtained by substituting for the circur
scribed circle that of equal area according to the result of § 21
In the case of the hexagon, the ratio of the radius of the circle
equal area to that of the circle inscribed is I'OoO, so that the m«
of the two limits cannot differ from the truth by so much as 2^ p
cent. In the same way we might conclude that the sector of
circle of 60° is a graver form than the equilateral triangle obtaini
by substituting the chord for the arc of the circle.
The following table giving the relative frequency in certa
calculable cases for the gravest tone of membranes under simil
mechanical conditions and of eqiial area (or), shews the effect of
greater or less departure from the circular form.
* Messenger of Mathematics, vol. xix. p. 86, 1890.
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211.] MEMBBANES OF EQUAL AREA. 345
Circle 2404 . a/tt « 4-261.
Square ^2 . tt = 4*443.
Qaadrant of a circle -^— . \/w = 4'551.
Sector of a circle 6(y> 6-379 a/| = 4*616.
Rectangle 3x2 a/^ . tt = 4624.
Equilateral triangle 27r . Vtan 30* = 4-774.
Semicircle 3-832 W ^ = 4-803.
1 = 4-967.
Rectangle 2 x 1
Right-angled isosceles triangle
Rectangle 3 X 1 w- A/-n- = 5-736.
For instance, if a square and a circle have the same area, the
former is the more acute in the ratio 4*443 : 4*261, or 1*043 : 1.
For the circle the absolute frequency is
(2^)-^x2-404c>y/^, where c=-^/T^^^p,
In the case of similar forms the frequency is inversely as the
linear dimension.
[From the principle that an extension of boundary is always
accompanied by a fall of pitch, we may infer that the gravest
mode of a membrane of any shape, and of any variable density, is
devoid of internal nodal lines.]
212. The theory of the free vibrations of a membrane was
first successfully considered by Poisson^ His theory in the
case of the rectangle left little to be desired, but his treatment
of the circular membrane was restricted to the sjonmetrical
vibrations. Kirchhoff's solution of the similar, but much more
difficult, problem of the circular plate was published in 1850,
and Clebsch's Theory of EUtstidty (1862) gives the general theory
of the circular membrane including the effects of stiffness and
1 M6m. de VAcadSmie, t. vin. 1829.
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346 VIBRATIONS OF MEMBRANES. [212.
of rotatory inertia^ It will therefore be seen that there was not
much left to be done in 1866 ; nevertheless the memoir of Bourget
already referred to contains a useful discussion of the problem
accompanied by very complete numerical results, the whole of
which however were not new.
213. In his experimental investigations M. Bourget made use
of various materials, of which paper proved to be as good as any.
The paper is immersed in water, and after removal of the superfluous
moisture by blotting-paper is placed upon a frame of wood whose
edges have been previously coated with glue. The contraction of the
paper in dr3ring produces the necessary tension, but many fidlures
may be met with before a satisfactory result is obtained. Even
a well stretched membrane requires considerable precautions in
use, being liable to great variations in pitch in consequence of the
varying moisture of the atmosphere. The vibrations are excited
by organ-pipes, of which it is necessary to have a series proceeding
by small intervals of pitch, and they are made evident to the eye
by means of a little sand scattered on the membrane. If the
vibration be sufficiently vigorous, the sand accumulates on the
nodal lines, whose form is thus defined with more or less precision.
Any inequality in the tension shews itself by the circles becoming
elliptic.
The principal results of experiment are the following : —
A circular membrane cannot vibrate in unison with every sound.
It can only place itself in unison with sounds more acute than
that heard when the membrane is gently tapped.
As theory indicates, these possible sounds are separated by less
and less intervals, the higher they become.
The nodal lines are only formed distinctly in response to
certain definite sounds. A little above or below confusion ensues,
and when the pitch of the pipe is decidedly altered, the membrane
remains unmoved. There is not, as Savart supposed, a continuous
transition from one system of nodal lines to another.
The nodal lines are circles or diameters or combinations of
circles and diameters, as theory requires. However, when the
^ [The reader who wishes to pursue the subject from a mathematical poiDt of
view is referred to an excellent discussion by Pockels (Leipzig, 1891) of Uh
di£Ferential equation \i^u-\-kHmOJ]
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213.] OBSERVATIONS OF M. BOURGET. 347
number of diameters exceeds two, the sand tends to heap itself
confusedly towards the middle of the membrane, and the nodes
are not well defined.
The same general laws were verified by MM. Bernard and
Bourget in the case of square membranes^; and these authors
consider that the results of theory are decisively established in
opposition to the views of Savart, who held that a membrane
was capable of responding to any sound, no matter what its pitch
might ba £ut I must here remark that the distinction between
forced and free vibrations does not seem to haTe been sufficiently
borne in mind. When a membrane is set in motion by aerial
waves having their origin in an organ-pipe, the vibration is
properly speaking forced. Theory asserts, not that the membrane
is only capable of vibrating with certain defined frequencies, but
that it is only capable of so vibrating freely. When however the
period of the force is not approximately equal to one of the
natural periods, the resulting vibration may be insensible.
In Savart's experiments the sound of the pipe was two or three
octaves higher than the gravest tone of the membrane, and was
accordingly never far from unison with one of the series of over-
tones. MM. Bourget and Bernard made the experiment under
more &vourable conditions. When they sounded a pipe somewhat
lower in pitch than the gravest tone of the membrane, the sand
remained at rest, but was thrown into vehement vibration as unison
was approached. So soon as the pipe was decidedly higher than the
membrane, the sand returned again to rest. A modification of the
experiment was made by first tuning a pipe about a third higher
than the membrane when in its natural condition. The membrane
was then heated until its tension had increased sufficiently to
bring the pitch above that of the pipe. During the process of
cooling the pitch gradually fell, and the point of coincidence
manifested itself by the violent motion of the sand, which at the
beginning and end of the experiment was sensibly at rest.
M. Bourget found a good agreement between theory and obser-
vation with respect to the radii of the circular nodes, though the
test was not very precise, in consequence of the sensible width of
the bands of sand ; but the relative pitch of the various simple
tones deviated considerably from the theoretical estimates. The
^ Am. de Ghim. lx. 449—479. 1860.
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348
VIBRATIONS OF MEMBRANES.
[21
committee of the French Academy appointed to report
M. Bourget's memoir suggest as the explanation the want
perfect fixity of the boundary. It should also be remembered tl
the theory proceeds on the supposition of perfect flexibility-
condition of things not at all closely approached by an ordina
membrane stretched with a comparatively small force. B
perhaps the most important disturbing cause is the resistance
the air, which acts with much greater force on a membrane th
on a string or bar in consequence of the large surface expose
The gravest mode of vibration, during which the displacement
at all points in the same direction, might be affected ve
differently from the higher modes, which would not require
great a transference of air from one side to the other.
[In the case of kettle-drums the matter is further complicat
by the action of the shell, which limits the motion of the air up
one side of the membrane. From the fact that kettle-drums i
struck, not in the centre, but at a point about midway betwe
the centre and edge, we may infer that the vibrations which it
desired to excite are not of the symmetrical class. The sound
indeed but little affected when the central point is touched wi
the finger. Under these circumstances the principal vibration {I]
that with one nodal diameter and no nodal circle, and to tl
corresponds the greater part of the sound obtained in the nom
use of the instrument. Other tones, however, are audible, whi
correspond with vibrations characterized (2) by two nodal diamet
and no nodal circle, (3) by three nodal diameters and no no<
circles, (4) by one nodal diameter and one nodal circle. "
observation with resonators upon a large kettle-drum of 25 incl
diameter the pitch of (2) was found to be about a fifth above (
that of (3) about a major seventh above (1), and that of (4) a lit
higher again, forming an imperfect octave with the principal to
For the corresponding modes of a uniform perfectly flexible me
brane vibrating in vdcuo, the theoretical intervals are th«
represented by the ratios 1-34, 1'66, 1*83 respectively \
The vibrations of soap films have been investigated by Meld
The frequencies for surfaces of equal area in the form of the cin
the square and the equilateral triangle, were found to be
if —
1 Phil Mag., vol. vii., p. 160, 1879. ^
a Pogg. Ann,, 169, p. 275, 1876. Akustik, p. 131, 1883.
A
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2 13. J FORCED VIBRATIONS. 349
1*000 : 1'049 : 1'175. In membranes of this kind the tension is
due to capillarity, and is independent of the thickness of the film.]
213 a. The forced vibrations of square and circular membranes
have been further experimentally studied by Elsas^ who has
confirmed the conclusions of Savart as to the responsiveness of a
membrane to sounds of arbitrary pitch. In these experiments the
vibrations of a fork were communicated to the membrane by means
of a light thread, attached normally at the centre ; and the position
of the nodal curves and of the maxima of disturbance was traced
in the usual manner by sand and lycopodium. A series of figures
accompanies the memoir, shewing the effect of sounds of pro-
gressively rising pitch.
In many instances the curves found do not exhibit the
symmetries demanded by the supposed conditions. Thus in
the case of the square membrane all the curves should be similarly
related to the four comers, and in the case of the circular mem-
brane all the curves should be circles. The explanation is probably
to be sought in the difficulty of attaining equality of tension. If
there be any irregularity, the effect will be to introduce modes of
vibration which should not appear, as having nodes at the point of
excitation, and this especially when there is a near agreement of
periods. Or again, an irregularity may operate to disturb the
balance between two modes of theoretically identical pitch, which
should be excited to the same degree. Indeed the passage through
such a- point of isochronism may be expected to be highly unstable
in the absence of moderate dissipative forces.
The theoretical solution of these questions has already (§§ 196,
204) been given, but would need much further development for
an accurate determination of the nodal curves relating to periods
not included among the natural periods. But the general course
of the phenomenon can be traced without difficulty.
If the imposed frequency be less than the lowest natural
frequency, the vibration is devoid of (internal) nodes. For a nodal
line, if it existed, being of necessity either endless or terminated
at the boundaiy*, would divide the membrane into two parts. Of
^ Nova Acta der K$l. Leap. Carol. Deutschen Akademie^ Bd. xlv. Kr. 1. Halle,
1882.
' Otherwise the extremity would have to remain at rest under the action of
component tensions from the surrounding parts which are all in one di^ction.
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1
350 VIBRATIONS OF MEMBRANES. [21 J
these one part would be vibrating freely with a frequency
than the lowest natural to the whole membrane, an imposs
condition of things (§ 211). The absence of nodal curves under
above-mentioned conditions is one of the conclusions drawn
Elsas from his observations.
As the frequency of the imposed vibration rises through
lowest natural frequency, a nodal curve manifests itself round
point of excitation, and gradually extends. The course of thi
is most easily followed in the case of the circular membn
excited at the centre. The nodal curves are then of necessity i
circles, and it is evident that the first appearance of a nodal ci]
can take place only at the centre. Otherwise there would b
circular annulus of finite internal diameter, vibrating freely wit
frequency only infinitesimally higher than that of the entire cir
At first sight indeed it might appear that even an infinitely sn
nodal circle would entail a finite elevation of pitch, but a c
sideration of the solution (§ 204) as expressed by a combinatioi
Bessel's functions of the first and second kinds, shews that thi
not the case. At the point of isochronism the second funct
disappears, and immediately afterwards re-enters with an infinit
small coefficient. But inasmuch as this function is itself infio
when r = 0, a nodal circle of vanishing radius is possible. Acco
ingly the fixation of the centre of a vibrating circular membrs
does not alter the pitch, a conclusion which may be extended
the fixation of any number of detached points of a membrane
any shape.
The eflfect of gradually increasing frequency upon the na
system of a circular membrane may be thus summarized. Bel
the first proper tone there is no internal node. As this point
reached, the mode of vibration identifies itself with the cor
sponding free mode, and then an infinitely small nodal cir
manifests itself. As the frequency further increases, this cir
expands, until when the second proper tone is reached, it coinci(
with the nodal circle of the free vibration of this frequen
Another infinitely small circle now appears, and it, as well as 1
first, continually expands, until they coincide with the nodal spt
of a free vibration in the third proper tone. This process c<
tinues as the pitch rises, every circle moving continually outwar
At each coincidence with a natural frequency the nodal syst<
identifies itself with that of the free vibration, and a new cir
begins to form itself at the centre.
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213 a.]
NODAL CURVES.
351
The behaviour of a square membrane is of course more difficult
to follow in detail. The transition fjx)m Fig. (34) case (4), corre-
sponding to m = 3, n = 1, and m = 1, n = 3, to Fig. (36) where m = 3,
n = 3, can be traced in Elsas's curves through such forms as
Fig. 39 a.
o
CTXIPX^
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k
CHAPTER X.
VIBRATIONS OF PLATES\
214. In order to form according to Green's method the equ
tions of equilibrium and motion for a thin solid plate of unifor
isotropic material and constant thickness, we require the expressi(
for the potential energy of bending. It is easy to see that for ea
unit of area the potential energy F is a positive homogeneo
symmetrical quadratic function of the two principal curvaturi
Thus, if pu Pihe the principal radii of curvature, the expressii
for V will be
Aa^\^^) (1).
where A and /i are constants, of which A must be positive, ai
fM must be numerically less than unity. Moreover if the materi
be of such a character that it undergoes no lateral contracti*
when a bar is pulled out, the constant /i must vanish. Tl
amount of information is almost all that is required for o
purpose, and we may therefore content oiu*selves with a me
statement of the relations of the constants in (1) with those 1
means of which the elastic properties of bodies are usually d
fined.
From Thomson and Tait s Natural Philosophy, §§ 639, 64
720, it appears that, if 2A be the thickness, q Youngs moduli
^ [This Chapter deals only with flexural vibrations. The extensional vibratio
of an infinite plane plate are briefly considered in Chapter X.a, as a particoi
case of those of an infinite cylindrical shell. They are not of much acoustic
importance.]
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214.] POTENTIAL ENERGY OF BENDING. 353
aod fi the ratio of lateral contraction to longitudinal elongation
when a bar is pulled out, the expression for V is
gA' ffi , ly 2(1-/*) ) .2V
[Equation (2) gives the interpretation of the constants of (1)
in its application to a homogeneous plate of isotropic material ;
.but the expression (1) itself is of far wider scope. The material
composing the plate may vary from layer to layer, and the elastic
chiEuracter of any layer need not be isotropic, but only symmetrical
with respect to the normal. As a particular case, the middle
layer, or indeed any other layer, may be supposed to be physically
inextensible.
Similar remarks apply to the investigations of the following
chapter relating to curved shells.]
If w be the small displacement perpendicular to the plane
of the plate at the point whose rectangular coordinates in the
plane of the plate are x, y,
11-^ 1 d^wd^w fd^wy
Pi pi ' Pipi dx" df \dxdy) '
and thus for a unit of area, we have
>-3(^,[<'-»)'-^<'-'')i£f#-(l|)"}]«.
which quantity has to be integrated over the surfa<^ {S) of the
plate.
^ The foUowiDg comparison of the notations used by the principal writers may
save trouble to those who wish to consult the original memoirs.
Rigidity =n (Thomson) =^ (Lam6).
Young's modulus =£ (Clebsch) =3f (Thomson) = -^ - (Thomson)
O/C + H
^ n(8TO~n) (Thomson) =g (Kirchhoflf and Donkin)=2JS:^^ (Kirchhoff).
Ratio of lateral contraction to longitudinal elongation =/x (Clebsoh and Donkin)
=ir (Thomson) =!^'* (Thomson) = ^-J^^ (Kirchhoff) r:^-^^^ (Lam6).
Poiseon assumed this ratio to be }, and Wertheim |.
R- 23
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354 VIBRATIONS OF PLATES. [21
216. We proceed to find the variation of V, but it Bhould
previously noticed that the second term in F, namely (I-
represents the total curvature of the plate, and is therefore d
pendent only on the state of things at the edge.
80 that we have to consider the two variations
jjV^w.V^iw.dS and jjS(pipt)-'dS.
Now by Green's theorem
jh^w . V^Sw . dS = jh*w ,&w,dS
-J-d^r-^-^+r'^^*' <2>
in which ds denotes an element of the boundary, and d/dn denoi
differentiation with respect to the normal of the boundary dra?
outwards.
The transformation of the second part is more difficult. ^
have
■11^.-111
d^ivd^Sw d}wd}ho ^^ d}w cPStt^l ,„
(ic* dy^ dy* da^ dxdy dxdy]
The quantity under the sigu of iutegratiou may be put ii
the form
, d^ /dSw d^w _ d8w d^w \ d_ /dSw d^w _ d£w d^w \
dy \ dy da? dx dxdy) dx\dx dy* dy dxdy) '
Now, if -f be any function of x and y,
//^^dy = /^sin^d.|
||^(irdy = jVcos^cfej
where 6 is the angle between x and the normal drawn outwaK
and the integration on the right-hand side extends round t
boundary. Using these, we find
^ rr^- f^ • g{d£wdhv ^d^ d?w)
J J PiRi ~ i \dy da? dx (}xdy)
.,.(3)
+ Id^cos^l
diw d?w ^ dSw d^w )
{da? dy^ dy dxdy)'
\
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215.]
EXPRESSION FOR 8F.
355
If we substitute for dSw/dx, dBw/dy their values in terms
dSw/dfiy d^/ds, from the equations (see Fig. 40)
dSw dSw ^ dhv . ^
- = _— cos ^ — J— sin d
dx an as
dSw dSw . >, . dSw ^
dy dn as
Fig. 40.
.(4),
we obtain
^ffdS [jdBwi.^^d^w^ 8/)^'^ o- D n<^'^]
+ |i.^jco8^sin^(^-^)+(3m»^-cos'^)^-f...(o).
The second integral by a partial integration with respect to
8 may be put into the form
Ck>llecting and rearranging our results, we find
:'^-.a-,)^(c..sin.(0-^^)
+ (cos..-sin«.)||)}
•3(1
-^{1." +
.f^^{.V...(l-.)(co..^-^sin..^;
^ ^ . ^ d^w \ ) '
+ 2cos^sm^ 1— ^j[
..(6).
23—2
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356
VIBRATIONS OF PLATES.
[5
There will now be no difficulty in forming the equation
motion. If p be the volume density, and Z,p,7Ji,dS the transv
force acting on the element dS,
BV " jf 2ZphBwdS+ if 2phivBwd&=0 (7
is the general variational equation, which must be true whati
function (consistent with the constitution of the system) Sw i
be supposed to be. Hence by the principles of the Calculu
Variations
-V^w-Z+iv^O.
(8
3p(l-/i«)
at every point of the plate.
If the edges of the plate be free, there is no restriction on
hypothetical boundary values of Sw and dSw/dn, and therefore
coefficients of these quantities in the expression for S Fmust vao
The conditions to be satisfied at a free edge ai'e thus
.^+(I-.M)^|cos5sm^(-^^-^)
<df
+ (cos«^.sin«^)|^}=0
.(9
;xV^ + (l-M){cos«5^ + sin«<?*^
+ 2co8^sind:^^l=0
dxdy)
If the whole circumference of the plate be clamped, iw-
dSw/dn = 0, and the satis&ction of the boundary conditioDi
already secured. If the edge be ' supported'*, Sw == 0, but diw
is arbitrary. The second of the equations (9) must in this case
satisfied by w,
216. The boundary equations may be simplified by gett
rid of the extrinsic element involved in the use of Cartes
coordinates. Taking the axis of x parallel to the normal of
bounding curve, we see that we may write
d^w
Also
COS* d j-r + sm* ^ 3-^ + 2 COS ^ sm ^ ^—5- -
dx^ d'if dxdy
— - cPw . d^w
dn" dc^
.(1)
* The rotatoiy inertia is here neglected.
* Compare § 163.
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216.] CONDITIONS FOR A FRKK EDGE. 357
where <risA fixed axis coinciding with the tangent at the point
under consideration. In general d^w/da* differs fix)m d^wjdsK
To obtain the relation between them, we may proceed thus.
Expand w by Maclaurin's theorem in ascending powers of the
small quantities n and <r, and substitute for n and a their values
in terms of «, the arc of the curve.
Thus in general
dw dw , ^w , d^w , d^w ,
(Wio da-o ^ dn^ dfiodco ^da^^
while on the curve o- = « + cubes, n = — ^ «*//> -»- . ., , where p is
the radius of curvature. Accordingly for points on the curve,
, dw ^ dw . d?w . . , J.
t(; = Wo — ij + j— * + i J— ¥ ^ + cubes of *,
and therefore j-. * j~v 3- (2);
cw" At* P dn '^ '
whence from (1)
— , d^w \dw . cPw ,„.
^**""d;^+p5;^ + d«» <«)•
We conclude that the second boundary condition in (9) § 215
may be put into the form
d^w , (IdAV ^d?w\ ^ ,..
d^+^ipdi^-^SF)'^ <*)•
In the same way by putting ^ == 0, we see that
^ . >, fd^w d^w\ . .^ . _ ^. d^w
18 equivalent to d^w/dndc, where it is to be understood that
the axes of n and <r are fixed. The first boundary condition now
becomes
l'--<>-''>5(il)- <«>
If we apply these equations to the rectangle whose sides are
parallel to the coordinate axes, we obtain as the conditions to be
satisfied along the edges parallel to y,
^{S"-*-")^"}-") ,,,
d^w d^w ^
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■^
358 VIBRATIONS OF PLATES. [216.
In this case the distinction between o- and 8 disappears, and p, the
radius of curvature, is infinitely great The conditioos for the
other pair of edges are found by interchanging x and y. These
results may be obtained equally well from (9) § 215 directly, with-
out the preliminary transformation.
217. If we suppose Z = 0, and write
3p(l-/.»r^ ^^^'
the general equation becomes
ib + c'V^w^O (2),
or, if w oc cos(p^ — e),
V*u; = A*t(; (3X
where k^^p'/c^ (4).
Any two values of w, u and v, corresponding to the same
boundary conditions, are conjugate, that is to say
/f'
uvdS^O (5),
provided that the periods be different. In order to prove this
from the ordinary differential equation (3), we should have to
retrace the steps by which (3) was obtained. This is the method
adopted by Kirchhoff for the circular disc, but it is much simpler
and more direct to use the variational equation
SV'\'2phjfw&ivdS=-0 (6X
in which w refers to the actual motion, and £u; to an arbitrazy
displacement consistent with the nature of the system. SFis a
symmetrical function of w and Sw, as may be seen fix)m § 215, or
from the general character of V (§ 94).
If we now suppose in the first place that w = u, Sw^v, we
have
SV^iphp'jjuvdS;
and in like manner if we put w=^v,Sw — u, which we are equally
entitled to do,
SV^2php''jjuvdS,
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217.]
whence
CONJUGATE PROPERTY.
359
(^-/»)//l
uvdS=0
.(7).
This demonstration is valid whatever may be the form of the
boundary, and whether the edge be clamped, supported, or free, in
whole or in part
As for the case of membranes in the last Chapter, equation
(7) may be employed to prove that the admissible values of p^ are
real ; but this is evident frx)m physical considerations.
218. For the application to a circular disc, it is necessary to
express the equations by means of polar coordinates. Taking the
centre of the disc as pole, we have for the general equation to be
satisfied at all points of the area
(V*-/fc*)w = (1),
1 _d 1 d«
rdr r" 5^ '
where (§ 200)
V*= 7, + -^ + ^
To express the boundary condition (§ 216) for a free edge
(r = a), we have
dn dr
§LV^ ^/d*w^\ d d^(dw\ d^w _ d^w
dr ^' d8\dnd^)~'~adddr\rd0)' d^^oH^'
p ss radius of curvature = a ; and thus
dr\dr^ r dr) dd^ \ a* dr a*
'd}w 1 dmA ^ / 2-^
\d7^ rdr) de\ a«
dr^'^^\id^'^ a^de^)
(2).
After the diflferentiations are performed, r is to be made equal
to a.
If w be expanded in Fourier's series .
w = Wo + Wi + ...H-t(;„ + ...,
each, term separately must satisfy (2), and thus, since
Wn X cos {nO — a),
d_ [d*Wn 1 dWfa\
dr \ dr^ r dr
dr^
■(«>
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A I
1
360 VIBRATIONS OF PLATES. [218.
The superficial differential equation may be written
(V» + i*)(V«-i*)w = 0,
which becomes for the general term of the Fourier expansion
/# Id n*^\fd^ Id w' ^\ _^
shewing that the complete value of Wn will be obtained by adding
together, with arbitrary constants prefixed, the general solations of
(£-?^-"*'-)»-<' <*^
The equation with the upper sign is the same as that which
obtains in the case of the vibrations of circular membranes, and
as in the last Chapter we conclude that the solution applicable
to the problem in hand is WnOcJn (At), the second function of r
being here inadmissible.
In the same way the solution of the equation with the lower
sign iawnocjn (tAr), where « s ^ (. 1) as usual. [See § 221 a]
The simple vibration is thus
Wn = cos ntf {aJn (At) + fiJn (ikr)] + sin n0 {yJn (At) + SJn (ikr)}.
The two boundary equations will determine the admissible
values of k and the values which must be given to the ratios
a : fi and 7 : S. From the form of these equations it is evident
that we must have ol : fi^y : S,
and thus Wn may be expressed in the form
Wn-PcosinO'-a) {Jn(kr)'\'\Jn(ikr)} cos (pt-e) (0).
As in the case of a membrane the nodal system is composed of
the n diameters symmetrically distributed rotind the centre, but
otherwise arbitrary, denoted by
cos(n^-a) = (6),
together with the concentric circles, whose equation is
Jn(AT) + XJn(tAT) = (7).
219. In order to determine \ and A; we must introduce the
boundary conditions. When the edge is free, we obtain from
(3) § 218
nV-1) likaJn\ika)-'Jni%ka)} + %lifa*Jn'i%ka)
_^_ ifi-l){kaJn'(ka)^n*Jn{ka)}-'k'a*Jn{ka)
(/* - 1) {ikaJn' (ika) - w« /« (ika)} + A"a« /« {ika) )
..(1).
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219.] POLAK CO-ORDINATES. 361
in which use has been made of the differential equations satisfied
by Jn(kr), Jn(ikr). In each of the fractions on the right the
denominator may be derived from the numerator by writing ik in
place of k. By elimination of X the equation is obtained whose
roots give the admissible values of k.
When n » 0, the result assumes a simple form, viz.
2(i-.)t«.i^.*.J^!-o., <.).
This, of course, could have been more easily obtained by neglecting
n from the beginning.
The calculation of the lowest root for each value of n is trouble-
some, and in the absence of appropriate tables must be effected
by means of the ascending series for the frmctions Jn{kr), Jn(ikr).
In the case of the higher roots recourse may be had to the semi-
convergent descending series for the same functions. Kirchhoff
finds
tan(A:a-in7r) ^-^^^^^--^-^—^^ (3).
where
-B==7(l-4n«)-8,
(7=7(1 -4n«) (9 -4»«) + 48(l + 4n«),
D = - 7 J {(1 - 4n«) (9 - 4m') (13 - 4n«)} + 8 (9 + 136n« + SOn').
When ka is great,
tan (ka - ^ nw) = approx. ;
whence
ia = i'7r(n + 2A) (4),
where h is an integer.
It appears by a numerical comparison that h is identical with
the number of circular nodes, and (4). expresses a law discovered
by Chladni, that the frequencies corresponding to figures with a
given number of nodal diameters are, with the exception of the
lowest, approximately proportional to the squares of consecutive
even or uneven numbers, according as the number of the diameters
is itself even or odd. Within the limits of application of (4), we
see also that the pitch is approximately unaltered, when any
number is subtracted from h, provided twice that number be
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362
VIBRATIONS OF PLATES.
[219.
added to n. This law, of which traces appear in the following table,
may be expressed by saying that towards raising the pitch nodal
circles have twice the effect of nodal diameters. It is probable,
however, that, strictly speaking, no two normal components have
exactly the same pitch.
*!
»» =
»=1
Ch.
P.
W.
Ch.
P.
W.
...
• • •
...
...
...
* • •
1 Ois
QiB +
A +
b
h-
0-
2 gui' +
1
V-
b'+
e" +
r+
fi»" +
h\
n=2
n = 3
, Ch.
P.
W.
Ch.
P.
W.
1 C
C
C
d
dis-
dis-
1 g'
gi8' +
a'-
d".di8"
di8" +
a"-
The table, extracted from Eirchhoff's memoir, gives the pitch
of the more important overtones of a free circular plate, the gravest
being assumed to be C^ The three columns under the heads
Ch, P, W refer respectively to the results as observed by Chladni
and as calculated from theory with Poisson's and Wertheim's
values of ffr. A pltLS sign denotes that the actual pitch is a little
higher, and a mimts sign that it is a little lower, than that written.
The discrepancies between theory and observation are considerable,
but perhaps not greater than may be attributed to irregularity in
the plate.
220. The radii of the nodal circles in the symmetrical case
(n = 0) were calculated by Foisson, and compared by him with
results obtained experimentally by Savart. The following numbers
are taken from a paper by Strehlke^ who made some careful
measurements. The radius of the disc is taken as unity.
ObseiTation. Galcalation.
. 0-67816 0-68062.
fO-39133 0-39151.
|0-84!l49 0-84200.
[o-25631 0-25679.
Three circles |o-59107 0-59147.
io-89360 0-89381.
^ Gifl correspondB to G^ of the English notation, and hiob nataral.
« Pogg. Ann. xcv. p. 677. 1866.
One circle .
Two circles.
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220.]
KIRCHHOFFS THEORY.
363
The calculated results appear to refer to Foisson's value of /i, but
would vary very little if Wertheim's value were substituted.
The following table gives a comparison of Eirchhofifs theory
(n not zero) with measurements by Strehlke made on less accurate
discs.
Radii of Circular Nodes.
Observation.
« = !, A=l
n = 2, A=l
» = 3, A=l
n=l, h^2
0-78136
082194
0-84523
0-49774
0-87057
0-78088
0-82274
0-84681
0-49715
0-87015
The most general motion of the uniform circular plate is
expressed by the superposition, with arbitrary amplitudes and
phases, of the normal components already investigated. The
determination of the amplitude and phase to correspond to
arbitraiy initial displacements and velocities is effected precisely
as in the corresponding problem for the membrane by the aid of
the characteristic property of the normal functions proved in § 217.
221. When the plate is truly symmetrical, whether uniform
or not, theory indicates, and experiment verifies, that the position
of the nodal diameters is arbitrary, or rather dependent only on
the manner in which the plate is supported, and excited. By
vaiying the place of support, any desired diameter may be made
nodal It is. generally otherwise when there is any sensible
departure fix)m exact symmetry. The two modes of vibration,
which originally, in consequence of the equality of periods, could
be combined in any proportion without ceasing to be simple
harmonic, are now separated and affected with different periods.
At the same time the position of the nodal diameters becomes
determinate, or rather limited to two alternatives. The one set is
derived from the other by rotation through half the angle included
between two adjacent diameters of the same set. This supposes
that the deviation from uniformity is small ; otherwise the nodal
system will no longer be composed of approximate circles and
diameters at all The cause of the deviation may be an irregu-
larity either in the material or in the thickness or in the form of
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364
VIBRATIONS OF PLATES,
[22
the boundary. The effect of a small load at any point may 1
investigated as in the parallel problem of the membrane § 20
If the place at which the load is attached does not lid on a nod
circle, the normal types are made determinate. The diameti
system corresponding to one of the types passes through the f\m
in question, and for this type the period is unaltered. The peri(
of the other type is increased.
[The divergence of free periods, which is due to slight i
equalities, would seem to afford an explanation of some curioi
observations by Savart^ When a circular plate, vibrating wil
nodal diameters, is under the influence of the bow applied at ai
part of the circumference, the nodal diameters indicated by sand a
so situated that the bow lies in the middle of a vibrating segmei
If, however, the bow be suddenly withdrawn, the nodal syste
oscillates, or even revolves, during the subsidence of the motio
It is evident that no such displacement could be expecte
were the plate absolutely symmetrical. The same would be tn
even in the cas6 of asymmetry, if the bow were so applied as
excite one only of the two determinate vibrations then possib
But in general the effect of the bow must be to excite both kin
of vibrations, and then the matter is more complicated. It won
seem that so long as the constraining action of the bow lasts, bo
vibrations are forced to keep the same time, and the effect
much the same as in the case of symmetry. But on withdraw
of the bow the free vibrations which then ensue take place each
its proper frequency, and a phase difference soon arises by whi
the effects are modified.
Let us suppose that the origin of ^ is so chosen in relati
to the irregularities that the types of vibration are represent
by cos 710, sin n0. Then in general the free vibrations, result!
from the action of the bow at an arbitrary point of the circu
ference, may be taken to be
cosna 8mn0 COS pt -- sin na cosnd coa(pt + e) (1),
where e is the difference of phase which has accumulated sii
the commencement of the fi^e vibrations. In the case
symmetry e « 0, and (1) becomes
sin n(0 — a) cos pt (2),
1 Ann. Chim,f toI. 86, p. 257, 1827.
"^
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221.] OSCILLATION OF NODES. 365
which represents a fixed nodal system
0^a + m{v/n) (3),
in any arbitrary position depending upon the point of application
of the bow. A similar fixity of the nodal system occurs, in spite
of the variable €, when a is so chosen that cos na » or sin na =s 0.
But in general there is no fixed nodal system. When e is a
multiple of 2v, that is when the two vibrations are restored to
the same phase, there is a nodal system represented by (3). And
when € is an odd multiple of v, so that the two vibrations are in
opposite phases, we have in place of (2)
8inn(tf +a)cosp^ (4),
with a nodal system
^ = -.a + m('7r/n) (5).
In these cases there is a nodal system, and in a sense the system
may be said to oscillate between the positions given by (3) and (5) ;
but it must not be overlooked that at intermediate times there is
no true nodal system at all Thus, when e = Jtt, (1) becomes
cos na sin nO co& pt + sin na cos nO sin pt
The squared amplitude of this motion is
cos' na sin' n0 + sin* na cos* nO,
a quantity which does not vanish for any value of 0. In general
the squared amplitude is
cos* na sin* nO -f sin* na cos* wtf - 2 cos na sin na cos n0 sin n0 cos 6,
or, as it may also be written,
^ — ^cos2na cos'2n0 — ^sin2na sin 2n0 cose (6).
This quantity is a maximum or a minimum when
tan 2n0 = cos€ tan2na (7).
The minimum of motion thus oscillates backwards and forwards
between tf = + a and 5 « — a ; but as we have seen, it is only in
these extreme positions that the minimum is zero.
A like phenomenon occurs during the fi^ee vibrations of a
circular membrane, or in fact of any system of revolution such
that the position of nodal lines is arbitrary so long as the
symmetry is complete.]
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366 VIBRATIONS OF PLATES. [221.
The two other cases of a circular plate in which the edge
is either clamped or supported would be easier than the preceding
in their theoretical treatment, but are of less practical interest on
account of the difficulty of experimentally realising the conditions
assumed The general result that the nodal system is composed
of concentric circles, and diameters symmetrically distributed, is
applicable to all the three cases.
221a. The use in the telephone of a thin circular plate
clamped at the edge lends a certain interest to the calculation of
the periods and modes of vibration W such a plate. It will suffice
to consider the symmetrical modes.
By (5) § 218 we may take as representing the motion in
this case
w = Jo(*r) + \Jo(i*r) = Jp(tr) + \7o(A?r)^ (1),
from which
^^Jo'{kr) + %\J,'{ikr)^^J,(kT) + \I,(kr) (2>,
where we write
/oW = /oW = l+| + 2rV» + W.
z ^ ^
Ii{z) =iJo'(tz) = 2 ■*" 2«T4 ■*■ 2«T4V:5 "*" ^^^
Since the plate is clamped at r^a, both w and dw/dr must
there vanish. Hence, writing ka — z, we get as the frequency
equation
^i^U[^^,^0 (5).
In (5) /i and Iq are both positive, so that the signs of Ji and J^
must be opposite. Hence by Table B § 206 the first root most
lie between 2*4 and 3*8, the second between 5*5 and 7*0, and
so on. The values of the earlier roots might be obtained without
much difficulty from the series for /© and /i by using the table
§ 200 for Jo and Ji ; but it will be convenient for the present and
further purposes to give a short table ^ of the functions /« and /i
themselves. For large values of the argument descending series,
analogous to (10) § 200, may be employed.
1 Calculated by A. Lodge, Brit. Ass. Rep., 18S9.
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221a.]
••J
CLAMPED
BOUN
DABY.
AW
X
JoW
AW '
z
AW
OO
i-oooo
0-0000 1
3-0
4-8808
3-9534
•2
1-0100
-1005 ,
3-2
5-7472
4-7343
•4
1-0404
-2040
3-4
6-7848
5-6701
•6
l-0»20
•3137
3-6
8-0277
6-7927
•8
11665
•4329 ;
3-8
9-5169
8-1404
1-0
1-2661
•5662 '
4-0
11-3019
9-7595
1-2
1-3937
-7147
4-2
13-4425
11-7056
1-4
1-6534
-8861
4-4
16-0104
14-0462
1-6
1-7500
1-0848 '
4-6
19-0936
16-8626
1-8
1-9896
1-3172 ,
4-8
22-7937
20-2528
2-0
2-2796
1-5906 '
5-0
27*2399
24-3356
2-2
2-6291
1-9141
5-2
32-5836
29*2543
2-4
3-0493
2-2981
5-4
39-0088
35-1821
2-6
3-5533
2-7554 ,
5-6
46-7376
42-3283
2-8
41573
3-3011
5-8
56-0381
50-9462
i
6-0
67-2344
61-3419
367
The first root of (o) is z = 3-20. This then is the value of ka
for the gravest symmetrical vibration. The next value of 2^ is
about 6*3. Since the fi-equency varies as A;* (§ 217), the interval
between the tones is nearly two octaves.
Returning to the first root, we have for the frequency (n)
§217,
^ p ^ (3-2)«c' ^ (3-2) V g. A
27r 27ra* 2iraW3p (1 - /a«) ^^^'
This is the general formula. For rough calculations fi* in the
denominator may be omitted. If for the case of iron we take
p = 7-7, gr = 20xlOi«,
2-4xlO».2A _
— r 0)>
we find
n =
2A and a being expressed in centimetres.
A telephone plate measured by the author gave
a = 2-2, 2A = -020.
According to these values
n = 991 vibrations per second
222. We have seen that in general Chladni's figures as traced
by sand agree very closely with the circles and diameters of
theory; but in certain cases deviations occur, which are usually
attributed to irregularities in the plate. It must however be re-
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368 VIBRATIONS OF PLAT£S. [222.
membered that the vibrations excited by a bow are not Btrictif
speaking free, and that their periods are therefore liable to a
certain modification. It may be that under the action of the bow
two or more normal component vibrations coexist. The whole
motion may be simple harmonic in virtue of the external force,
although the natural periods would be a little different. Such an
explanation is suggested by the regular character of the figures
obtained in certain cases.
Another cause of deviation may perhaps be found in the
manner in which the plates are supported. The requirementB of
theory are often difficult to meet in actual experiment When
this is so, we may have to be content with an imperfect compari-
son ; but we must remember that a discrepancy may be the fiuilt
of the experiment as well as of the theory.
[In the ordinary use of sand to investigate the vibratioDS of
flat plates and membranes the movement to the nodes is irregular
in its character. If a grain be situated elsewhere than at a node,
it is made to jump by a sufficiently vigorous transverse vibration.
The result may be a movement either towards or from a node;
but after a succession of such jumps the grain ultimately finds its
way to a node as the only place where it can remain undisturbed.
Grains which have already arrived at a node remain there, while
others are constantly shifting their position.
It was found by Savart that very fine powder, such as lyoo-
podium, behaves differently from sand. Instead of collecting at
the nodes, it heaps itself up at the places of greatest motion.
This effect was traced by Faraday^ to the influence of currents of
air, themselves the result of the vibration. In a vacuum all
powders move to the nodes.
In some cases the movement of sand to the nodes, or to some
of them, takes place in a more direct manner as the result of
friction. Thus, in his investigation of the longitudinal vibrations
of thin narrow strips of glass, held horizontally, Savart' observed
the delineation of nodes apparently dependent upon an aooom-
paniment of vibrations of a transverse character. The special
peculiarity of this phenomenon was the non-correspondence of the
lines traced by sand upon the two faces^ of the glass when tested
^ On a Peoaliar CIebb of Aconstioal Figures, Phil. Trcau., ISSl, p. 299.
s Ann. d. Chim,, vol. 14, p. 118, 1820.
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I
1
222.] SAV art's obsebvations. 369
in succession, a fistct sufficient to shew that the transverse motion
was connected with a failure of uniformity. In consequence of
this there are developed transverse vibrations of the same (high)
pitch as that of the principal longitudinal motion, and therefore
attended with many nodes. These nodes are of course the same
whichever &ce of the glass is uppermost, and it might be supposed
that they would all be indicated by the sand, as would happen if
the transverse vibrations existed alone. But the combination of
the two kinds of motion causes a creeping of the sand towards the
alternate nodes, the movements of the sand at corresponding
points on the two sides of the plate being always in opposite
directions. On the one side an inwards longitudinal motion (for
example) is attended by an upwards transverse motion, but when
the plate is reversed the same inwards longitudinal motion is
associated with a transverse motion directed downwards. If there
were no transverse motion, the longitudinal force upon any
particle resulting from friction would vanish in the long run, but
in consequence of the transverse motion this balance is upset, and
in a manner different upon the two sides of the plate. The above
considerations appear to afford sufficient ground for an explanation
of the remarkable phenomenon observed by Savart, but an attempt
to follow the matter further into detail would lead us too
fer\]
223. The first attempt to solve the problem with which we
have just been occupied is due to Sophie Germain, who succeeded
in obtaining the correct differential equation, but was led to
erroneous boundary conditions. For a free plate the latter part of
the problem is indeed of considerable difficulty. In Poisson's
memoir 'Sur T^uilibre et le mouvement des corps ^lastiquesV
that eminent mathematician gave three equations as necessary to be
satisfied at all points of a free edge, but Eirchhoff has proved that
in general it would be impossible to satisfy them alL It happens,
however, that an exception occurs in the case of the symmetrical
vibrations of a circular plate, when one of the equations is true
identically. Owing to this peculiarity, Poisson's theory of the
symmetrical vibrations is correct, notwithstanding the error in his
view as to the boundary conditions. In 1850 the subject was
^ See Terqnem, (7. R., xlyi., p. 776, 185S.
' MSm. de VAcad. d. Sc. ft Par. 1829.
B. 24
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870
VIBRATIONS OF PLATES.
[25
resumed by Kirchhoff ^ who first gave the ttvo equations approprii
to a free edge, and completed the theory of the vibrations ol
circular disc.
224. The correctness of Elirchhoff's boundary equations 1
been disputed by Mathieu*, who, without explaining where
considers KirchhoflTs error to lie, has substituted a different set
equations. He proves that if u and u be two normal functions,
that w = u cos pt, w = u'co9p't are possible vibrations, then
(p^ — p'*) 1 1 uu'dxdy
, dV^u -, du* ^^ ,du ^ dV»w'
an dn an a<
rl-
.(1).
This follows, if it be admitted that u, u ssatisfy respectiv<
the equations
Since the left-hand member is zero, the same must be true
the right-hand member; and this, according to Mathieu, cani
be the case, unless at all points of the boundai^ both u and
satisfy one of the four following pairs of equations :
u «0
du
dn
=
V»tt =0
dn
=
u =0
P ^0
dn
d^^u_
dn "
The second pair would seem the most likely for a free edge, b
it is found to lead to an impossibility. Since the first and thi
pairs are obviously inadmissible, Mathieu concludes that the foui
pair of equations must be those which really express the conditi
of a free edge. In his belief in this result he is not shaken by t
fact that the corresponding conditions for the free end of a I
would be duldx = 0, d^u/da:^ = Oj the first of which is contradict
by the roughest observation of the vibration of a large tunin
fork.
^ Crelle, t. xl. p. 51. Ueber das Gleichgewicbt und di« Bewegang eiuer eli
tiscben Scheibe.
' LiouviUe, t. xiv. 1S69.
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224-] HISTORY OF PROBLEM. 371
The &^t is that although any of the four pairs of equations
would secure the evanescence of the boundary integral in (1), it
does not follow conversely that the integral can be made to vanish
in no other way; and such a conclusion is negatived by KirchhofiTs
investigation. There are besides innumerable other cases in
which the integral in question would vanish, all that is really
necessary being that the boundary appliances should be either at
rest, or devoid of inertia.
226. The vibrations of a rectangular plate, whose edge is
supported^ iMiy be easily investigated theoretically, the normal
functions being identical with those applicable to a membrane of
the same shape, whose boundary is fixed. If we assume
w = sm- sm -r^ (io^pt (1),
we see that at all points of the boundary,
w = 0, d^wlda^ « 0, d^wfdf = 0,
which secure the fulfilment of the necessary conditions (§ 215).
The value of p, found by substitution in c^V^w^^p^w^
- ^-«^nf + (2).
shewing that the analogy to the membrane does not extend to the
sequence of tones.
It is not necessary to repeat here the discussion of the primary
and derived nodal systems given in Chapter ix. It is enough to
observe that if two of the fundamental modes (1) have the same
period in the case of the membrane, they must also have the same
period in the case of the plate. The derived nodal systems are
accordingly identical in the two cases.
The generality of the value of w obtained by compounding
with arbitrary amplitudes and phases all possible particular solu-
tions of the form (1) requires no fresh discussion.
Unless the contrary assertion had been made, it would have
seemed unnecessary to say that the nodes of a supported plate
have nothing to do with the ordinary Chladni's figures, which
belong to a plate whose edges are free.
24—2
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372
VIBBATIOJrS OF PLATES.
[2:
The realization of the conditions for a supported edge
scarcely attainable in practice. Appliances are required capa
of holding the boundary of the plate at rest, and of such a nat
that they give rise to no couples about tangential axes. We u
conceive the plate to be held in its place by friction against
walls of a cylinder circumscribed closely round it.
The problem of a rectangular plate, whose edges
free, is one of great difficulty, and has for the moiSt part resis
attacks If we suppose that the displacement w is indepeud
of y, the general diflferential equation becomes identical with tl
with which we were concerned in Chapter vrii. If we take
solution corresponding to the case of a bar whose ends are ii
and therefore satisfying dhu/da^^O, d*w/dx^ = 0, when ^ = *
when a? = a, we obtain a value of w which satisfies the gene
di£ferential equation, as well as the pair of boundary equatious
A.
dx
d^w .« .d^w] ^\
dhv
d^
d^w ^ \
.(1)
which are applicable to the edges parallel to y ; but the sect
boundary condition for the other pair of edges, namely
d^w d*w ^
da^'
A^]
will be violated, unless /a==0. This shews that, except in
case reserved, it is not possible for a free rectangular plate
vibrate after the manner of a bar; unless indeed as an approxii
tion, when the length parallel to one pair of edges is so great t
the conditions to be satisfied at the second pair of edges may
left out of account.
Although the constant fi (which expresses the ratio of lat<
contraction to longitudinal extension when a bar is drawn c
is positive for every known substance, in the case of a few s
stances — cork, for example — it is comparatively verj* smalL Th
is, so far as we know, nothing absurd in the idea of a subsU
* [The oase where two opposite edges are firee while the other two edges
supported, has been diaoussed by Voigt {Gdttingen Nachncht^n, 1893).]
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226.] RECTANGULAR PLATE. 373
for which /a vanishes. The investigation of the problem under
this condition is therefore not devoid of interest, though the results
will not be strictly applicable to ordinary glass or metal plates,
for which the value of /i is about ^}
If 111, t£„ &c. denote the normal functions for a free bar inves-
tigated in Chapter viiL, corresponding to 2, 3, nodes, the
vibrations of a rectangular plate will be expressed by
w^Ui {xja), w = U2 (x/a), &c.,
or w = Ml (y/6), w = ii, (y/6), &c.
In each of these primitive modes the nodal system is composed
of straight lines parallel to one or other of the edges of the
rectangle. When b^a, the rectangle becomes a square, and the
vibrations
w = Un (x/a), w^Un (jz/a),
having necessarily the same period, may be combined in any pro-
portion, while the whole motion still remains simple harmonic.
Whatever the proportion may be, the resulting nodal curve will of
necessity pass through the points determined by
Un {xja) = 0, On (y/a) = 0.
Now let us consider more particularly the case of n = 1. The
nodal system of the primitive mode, w — u^ (xja), consists of a
pair of straight lines parallel to y, whose distance from the nearest
edge is '2242 a. The points in which these lines are met by the
corresponding pair for w = tij (y/a), are those through which the
nodal curve of the compound vibration must in all cases pass. It
is evident that they are symmetrically disposed on the diagonals
of the square. If the two primitive vibrations be taken equal,
but in opposite phases (or, algebraically, with equal and opposite
amplitudes), we have
w= t*i (x/a) - Ui (y/a) (3),
^ In order to make a plate of material, for which ft is not zero, vibrate in the
manner of a bar, it would be necessary to apply constraining couples to the edges
parallel to the plane of bending to prevent the assumption of a contrary curvature.
The effect of these couples would be to raise the pitch, and therefore the calcu-
lation founded on the type proper to /tsQ would give a result somewhat higher in
pitch than the truth.
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374
VIBRATIONS OF PLATES.
[22
from which it is evident that w vanishes when a? = y, that is alo
the diagonal which passes through the origin. f%, n.
That w will also vanish along the other diagonal
follows from the symmetry of the functions, and
we conclude that the nodal system of (3) comprises
both the diagonals (Pig. 41). This is a well-known
mode of vibration of a square plate.
A second notable case is when the amplitudes are equal ai
their phases the same, so that
w = Ui{x/a)'^tti(y/a) ,„(4).
The most convenient method of constructing graphical
the curves, for which t(; = const., is that employed by Maxwi
in similar cases. The two systems of curves (in this install
straight lines) represented by t^ (x/a) = const., t(i (yfa) = canst,, a
first laid down, the values of the constants forming an arit
metical progression with the same common diflFerence in the u
cases. In this way a network is obtained which the requir
curves cross diagonally. The execution of the proposed pb
requires an inversion of the table given in Chapter VUL, § 17
expressing the march of the function Ui, of which the lesult is
follows : —
M|
X : a
M,
X : a
+ 1O0
•5000
- -25
•1871
•75
•3680
•50
•1518
•50
•3106
•75
•1179
•25
•2647
l^OO
•0846
•00
•2242
125
•0517
-1-50
•0190
The system of lines represented by the above values of jr (C4)i]
pleted symmetrically on the ftirther side of the central line) an
the corresponding system for y are laid down in Fig. 42, Froi
these the curves of equal displacement are deduced; At tl
centre of the square we have w a maximum and equal to 2 on tl
scale adopted. The first curve proceeding outwards is the locits <
points at which w = 1. The next is the nodal line, separating th
regions of opposite displacement. The remaining curves taken i
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226.]
CASE OF A SQUARE PLATE.
375
order give the displacements — 1, — 2, — 3. The numerically great-
est negative displacement occurs at the comers of the square,
where it amounts to 2 x 1-645 = 3-290.*
The nodal curve thus constructed agrees pretty closely with the
observations of Strehlke '. His results, which refer to three care-
fully worked plates of glass, are embodied in the following polar
equations :
-40143 01711 00127^
r = -40143 + -01721 cos 4t + '00127 > cos 8i,
-4019 -Oiesj '0013 J
nr~^
Fig
.42.
r""5r
- -■ J2 _,Z
_s
^ !i- '/\
- A^ -,Z
N. W
.-Zt JZ.
^c-^/"^.--
V- ^ ^
/^
V,
\/ \/
T ^'^~ 7^
S. Z^^ 7^
5/ S/ -
7 2\~"
^--'■^
_7S^ /S^.
7 J- ^-
/
N
Z H \
f 1 7
\
/
\ /\ c
I '
\
\
/
^ Y
1
/
}\ A
s___S_ S
\
Z \Z..v
.S± ^ 1
\
/^
\. Z\ Ti
- S S^
^
\^ ^^ -
- s^^zS^
-S2.\ J2S .
V 5^ ^
Z :^7 "S,i
-S zs
X
y
yv y^
- ^2 S^
^**N,^
JZ. SZ .
.J2.\^ S_
-^Z ^1\
^L^^=^
:i^
^
^-l^lz-Ai
the centre of the square being pole. From these we obtain for
the radius vector parallel to the sides of the square (t = 0) -41980,
1 On the nodal lines of a sqoare plate. Phil. Mag. Angast, 1878.
« Pogg. Ann, Vol. cxlvi. p. 819, 1872.
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376 TIBRATIONS OF PLATBSi [:
'419S1, '4200, while the calculated result b '4154. The rs
vector measured along a diagonal is '3856, ^S855, '3864, an*
calculation 3900*
By crossing the network in the other direction we ohtaiE
locus of points for which w^ (^r/a) — lii (y/a) is constant, which
the curves of constant displacement for that mode in which
diagonals are nodal. The pitch of the vibration is (accordit
theory) the aame in both cases.
Fig* 43.
V
^
^
/
/
\
\
N,
\
'■■•.
\
\
/
V
/
/
\
The primitive modes represented by w — w^ (jaja) or w = «,
may be combined in like manner. Fig 43 shews the nodal c
for the vibration
m - n^ixfa) ± u^ (y/a) .,•*,.,*....., ,.(
The form of the curve is the same relatively to the other dia^
if the sign of the ambiguity be altered.
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227.] wheatstone's figures. 377
227. The method of superposition does not depend for its
application on any particular form of normal function. Whatever
the form may be, the mode of vibration, which when /a =
passes into that just discussed, must have the same period,
whether the approximately straight nodal lines are parallel to
a? or to y. If the two synchronous vibrations be superposed,
the resultant has still the same period, and the general course
of its nodal system may be traced by means of the considera-
tion that no point of the plate can be nodal at which the
primitive vibrations have the same sign. To determine exactly
the line of compensation, a complete knowledge of the primitive
normal functions, and not merely of the points at which they
vanish, would in general be necessary. Doctor Young and the
brothers Weber appear to have had the idea of superposition as
capable of giving rise to new varieties of vibration, but it is to Sir
Charles Wheatstone ^ that we owe the first systematic application
of it to the explanation of Chladni's figures. The results actually
obtained by Wheatstone are however only very roughly applicable
to a plate, in consequence of the form of normal function implicitly
assumed. In place of Fig. 42 (itself, be it remembered, only an
approximation) Wheatstone finds for the node of the compound
vibration the inscribed square shewn in Fig. 44.
This form is really applicable, not to a plate vi-
brating in virtue of rigidity, but to a stretched
membrane, so supported that every point of the
circumference is firee to move along lines perpen-
dicular to the plane of the membrane. The
boundary condition applicable under these circumstances is
, «s ; and it is easy to shew that the normal functions which
involve only one co-ordinate are
w = cos (mTTir/a), or w^ cos (wiry /a),
the origin being at a comer of the square. Thus the vibration
2'n'X 2'7rt/ ,,.
WssCOS +COS — ~ (1)
has its nodes determined by
7r(a?H-y) 'rr{x — y) ^
cos ^ cos ^- = 0,
a a
1 PUL Trans. 1838.
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378
VIBRATIONS OF PLATES,
[22^
whence X'\-y = ^a or |a, or x — y=^±^a, equations whi(
represent the inscribed square.
If
2irx 2iry
= ooa — nna — ^
t(; = cos
COS
(2X
the nodal system is composed of the two diagouals. This resu
which depends only on the symmetry of the normal functions,
strictly applicable to a square plate.
When m = 3,
w =5 cos h cos — ^
a a
and the equations of the nodal lines are
.,43),
^ + y=g, a, g,
^-y = ±g,
Fig. 4^,
shewn in Fig. 45. If the other sign be taken, we
obtain a similar figure with reference to the other
diagonal.
When m = 4,
4BTrx 4nry
w = cos + cos — ~ .
a a
.(4)^
giving the nodal lines
_a 3a ba 7a
"^^^""4' T' T' T'
a.-y = ±i, ±x(Fig^*C).
With the other sign
we obtain
w =s cos cos — -
a a
a;+y = |. a.^, ar-y = 0, ± | (Fig. 47)
representing a system composed of the diagonals,
together with the inscribed square.
These forms, which are strictly applicable to the membran
resemble the figures obtained by means of sand on a square pkl
more closely than might have been expected. The sequence (
tones is however quite different. From § 176 we see that, if fi wei
zero, the interval between the form (43) derived from thre
primitive nodes, and (41) or (42) derived from two, would b
^
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227.] GRAVEST MODE OF SQUARE PLATE. 379
1"4629 octaves; and the interval between (41) or (42) and (46) or
(47) would be 2*4358 octaves. Whatever may be the value of jj, the
forms (41) and (42) should have exactly the same pitch, and the
same should be true of (46) and (47). With respect to the first-
mentioned pair this result is not in agreement with Chladni's
observations, who found a difference of more than a whole tone,
(42) giving the higher pitch. If however (42) be leffc out of
account, the comparison is more satisfactory. According to theory
(m = 0), if (41) gave d, (43) should give ^r'-, and (46), (47)
should give /' + . Chladni found for (43) g't + , and for (46),
(47) g'% and ^'t-^ respectively.
2128. The gravest mode of a square plate has yet to be consi-
dered The nodes in this case are the two lines drawn through the
middle points of opposite sides. That there must be such a mode
will be shewn presently from considerations of symmetry, but
neither the form of the normal function, nor the pitch, has yet
been determined, even for the particular case of /Li => 0. A rough
calculation however may be -founded on an assumed tjrpe of
vibration.
If we take the nodal lines for axes, the form w^xy satisfies
V^w = 0, as well as the boundary conditions proper for a free edge
at all points of the perimeter except the actual comers. This is
in fact the form which the plate would assume if held at rest by
four forces numerically equal, acting at the comers perpendicu-
larly to the plane of the plate, those at the ends of one diagonal
being in one direction, and those at the ends of the other diagonal
in the opposite direction. From this it follows that w^xycospt
would be a possible mode of vibration, if the mass of the plate
were concentrated equally in the four comers. By (3) § 214, we
see that
^-30T^)'''P' ^^>'
inasmuch as
d^w/dx^ = cPw/dy^ = 0, d^wjdxdy = cos^^^
For the kinetic energy, if p be the volume density, and M the
additional mass at each comer,
T=^f sin*^ U^ ""T " iphahj^dxdy + Jifa*l
■.W^.^pt[^\ + %M] (2).
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380
VIBRATIONS OF PLATES.
[221
Hence
l_p(l +M)a «/, ^qM\
(3).
where J/' denotes the mass of the plate without the loads. Th
result tends to become accurate when M is relatively great ; othe
wise by § 89 it is sensibly less than the truth. But even whi
M — Q, the error is probably not very great. In this case s
should have
,^ 96qh^
p(l+M)a* "
P"-
■(4).
giving a pitch which is somewhat too high. The gravest mo
next after this is when the diagonals are nodes, of which the pitx
if ^ = 0, would be given by
,,_qh^ (4-7300y
|)- =
(see §174).
pa*
■(5),
We may conclude that if the material of the plate were su
that ft=0, the interval between the two gravest tones woi
be somewhat greater than that expressed by the ratio VZ
Chladni makes the interval a fifth.
u
Fig. 48.
G
229. That there must exist modes of vibration in whi
the two shortest diameters are nodes may be
inferred from such considerations as the following.
In Fig. (48) suppose that GH is a plate of which
the edges HO, GO are supported, and the edges
QC, GH free. This plate, since it tends to a
definite position of equilibrium, must be capable
of vibrating in certain fundamental modes. Fixing
our attention on one of these, let us conceive a
distribution of w over the three remaining quadrants, such that
any two that adjoin, the values of w are equal and opposite
points which are the images of each other in the line of separati
If the whole plate vibrate according to the law thus deterrain
no constraint will be required in order to keep the lines GE, i
fixed, and therefore the whole plate may be regarded 2^ free. T
same argument may be used to prove that modes exist in whi
the diagonals are nodes, or in which both the diagonals and t
diameters just considered are together nodal.
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229.]
PRINCIPLE OF SYMMETRY.
381
The principle of symmetry may also be applied to other forms
of plate. We might thus infer the possibility of nodal diameters
in a circle, or of nodal principal axes in an ellipse. When the
Fig. 49. Fig. 50. Fig. 51.
boundary is a regular hexagon, it is easy to see that Figs. (49),
(50), (51) represent possible forms.
It is interesting to trace the continuity of Chladni's figures, as
the form of the plate is gradually altered. In the circle, for
example, when there are two perpendicular nodal diameters, it is a
matter of indifference as respects the pitch and the type of vibra-
tion, in what position they be taken. As the circle develops into
a square by throwing out comers, the position of these diameters
becomes definite. In the two alternatives the pitch of the vibra-
tion is different, for the projecting comers have not the same effi-
dencj in the two cases. The vibration of a square plate shewn in
Fig. (42) corresponds to that of a circle when there is one circular
node. The correspondence of the graver modes of a hexagon or
an ellipse with those of a circle may be* traced in like manner.
230. For plates of uniform material and thickness and of
invariable shape, the period of the vibration in any fundamental
mode varies as the square of the linear dimension, provided of
course that the boundary conditions are the same in all the cases
compared. When the edges are clamped, we may go further
and assert that the removal of any external portion is attended
by a rise of pitch, whether the material and the thickness be
uniform, or not.
Let AB be a part of a clamped edge (it is of no consequence
whether the remainder of the boundary be clamped, or not), and
Fig. 52.
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382 VIBRATIONS OF PLATES. [230.
let the piece ACBD be removed, the new edge ADB being also
clamped. The pitch of any fundamental vibration is sharper
than before the change. This is evident, since the altered
vibrations might be obtained from the original system by the
introduction of a constraint clamping the edge ADB. The effect
of the constraint is tq raise the pitch of every component, and
the portion ACBD being plane and at rest throughout the motion,
may be removed. In order to follow the sequence of changes
with greater security from error, it is best to suppose the line
of clamping to advance by stages between the two positions
ACB, ADB. For example, the pitch of a uniform clamped plate
in the form of a regular hexagon is lower than for the inscribed
circle and higher than for the circumscribed circle.
When a plate is free, it is not true that an addition to
the edge always increases the period. In proof of this it may be
sufficient to notice a particular case.
AB IB B, narrow thin plate, itself without inertia but carrying
loads B,t A, By C. It is clear that the addition to the breadth
Fig. 58.
•A •€ B*
indicated by the dotted line would augment the stiffness of the
bar, and therefore lessen the period of vibration. The same
consideration shews that for a uniform free plate of given area
there is no lower limit of pitch ; for by a sufficient elongation
the period of the gravest component may be made to exceed
any assignable quantity. When the edges are clamped, the
form of gravest pitch is doubtless the circle.
If all the dimensions of a plate, including the thickness, be
altered in the same proportion, the period is proportional to the
linear dimension, as in every case of a solid body vibrating in
virtue of its own elasticity.
The period also varies inversely as the square root of Young s
modulus, if /^ be constant, and directly as the square root of the
mass of unit of volume of the substance.
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231.] CYLINDER OF RING. 383
231. Experimenting with square plates of thin wood whose
gfain ran parallel to one pair of sides, Wheatstone^ found that
the pitch of the vibrations was different according as the ap-
proximately straight nodal lines were parallel or perpendicular
to the fibre of the wood. This effect depends on a variation
in the flexural rigidity in the two directiona The two sets of
vibrations having different periods cannot be combined in the
usual manner, and consequently it is not possible to make such
a plate of wood vibrate with nodal diagonals. The inequality
of periods may however be obviated by altering the ratio of the
sides, and then the ordinary mode of superposition giving nodal
diagonals is again possible. This was verified by Wheatstone.
A further application of the principle of superposition is due
to Eonigl In order that two modes of vibration may combine,
it is only necessary that the periods agree. Now it is evident
that the sides of a rectangular plate may be taken in such a
ratio, that (for instance) the vibration with two nodes parallel
to one pair of sides may agree in pitch with the vibration having
three nodes parallel to the other pair of sides. In such a case
new nodal figures arise by composition of the two primary modes
of vibration.
232. When the plate whose vibrations are to be considered
is naturally curved, the difficulties of the question are generally
much increased. But there is one case in which the complication
due to curvature is more than compensated by the absence of a
free edge ; and this case happens to be of considerable interest, as
being the best representative of a bell which admits of simple
analjrtical treatment.
A long cylindrical shell of circular section and uniform thick-
ness is evidently capable of vibrations of a fiexural character
in which the axis remains at rest and the surface cylindrical,
while the motion of every part is perpendicular to the generating
lines. The problem may thus be treated as one of two dimensions
only, and depends upon the consideration of the potential and
kinetic energies of the various deformations of which the section
is capable. The same analysis also applies to the corresponding
vibrations of a ring, formed by the revolution of a small closed
area about an external axis (§ 192 a).
1 Phil Trans. 1833.
> Pogg. Ann. 1884, cxzn. p. 238.
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384 VIBRATIONS OF PLATES. [2{
The cylinder, or ring, is susceptible of two classes of vibrati<
depending respectively on extensibility and flexural rigidity, b
analogous to the longitudinal and lateral vibrations of strai^
bars. When, however, the cylinder is thin, the forces resisti
bending become small in comparison with those by which <
tension is opposed; and, as in the case of straight bars, 1
vibrations depending on bending are graver and more imports
than those which have their origin in longitudinal rigidi
In the limiting case of an infinitely thin shell (or ring), i
flexural vibrations become independent of any extension of 1
circumference as a whole, and may be calculated on the s\
pasition that each part of the circumference retains its nain
length throughout the motion.
But although the vibrations about to be considered
analogous to the transverse vibrations of sti-aight bars in resp
of depending on the resistance to flexure, we must not fall i
the common mistake of supposing that they ai-e exclu&iv
normal. It is indeed easy to see that a motion of a cylinder
ring in which each particle is displaced in the direction of
radius would be incompatible with the condition of no extensi
In order to satisfy this condition it is necessary to ascribe
each part of the circumference a tangential as well as a non
motion, whose relative magnitudes must satisfy a certain diri
ential equation. Our first step will be the investigation of i
equation.
233. The original radius of the circle being a, let the eq
librium position of any element of the circumference be defii
by the vectorial angle 0. During the motion let the polar
oi-dinates of the element become
If da represent the arc of the deformed curve corresponding to a
we have
(ds)» = (ad0y = (dSry + r« (dtf + dhOy ;
^vhence we find, by neglecting the squares of the small quanti
Si , B0,
Sr dh0 ^
7 + W=^- (^
as the required relation.
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233.] POTENTIAL AND KINETIC ENERGIES 385
In whatever manner the original circle may be deformed at
time t, Sr may be expanded by Fourier's theorem in the series
-^ A,coa 80 '\- B,8m 80 + ...} (2),
and the corresponding tangential displacement required by the
condition of no extension will be
A B
B0 ==-Aiam0 + B,co80 +...--- sin 80 -¥ — coa 80 - (3),
O
the constant that might be added to S0 being omitted.
If (rad0 denote the mass of the element ad0, the kinetic
energy T of the whole motion will be
+ (l + i)(i/ + 5/) + ...| (4).
the products of the co-ordinates A,, B, disappearing in the
integration.
We have now to calculate the form of the potential energy V.
Let p be the radius of curvature of any element ds ; then for the
corresponding element of V we may take i^Bds {S (1/p)}*, where
£ is a constant depending on the material and on the thickness.
Thus
Now
and
r=^Bar(B-yd0 (5).
u =s - =3 - {1 — ilj cos ^ — 5i sin ^ — ... },
for in the small terms the distinction between ^ and may be
neglected.
Hence
8 - = - 2 {(«» - 1) {A, cos 8^ + B, sin a<^)},
r
B. 25
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r-
386 VIBRATIONS OP PLATES, [2
■m
and
F= ^ p {2 («» - 1) {A, COS80 + B, sin s$}Y d6
= ^^2(«'-l)'(^' + 5.') (6
in which the summation extends to all positive integral val
of 8.
The term for which « = 1 contributes nothing to the poten
energy, as it corresponds to a displacement of the cLrcIe as a wh
without deformation.
We see that when the configuration of the system is defined
above by the co-ordinates -4i, -Bi, &c., the expressions for Taiic
involve only squares; in other words, these are the n&ritwl
ordinates, whose independent harmonic variation expresses
vibration of the system.
If we consider only the terms involving cos s8, sin s8, we hi
by taking the origin of 6 suitably,
A
8r = ail. cos «tf, Stf = -misd (7)
s
while the equation defining the dependence of A^ upon
time is
<ra'(l+^)i>^(«'-l)'A = (8J
from which we conclude that, if A, varies as cos (pi — c)»
P^ = ^-«'T1- <^'
This result was given by Hoppe for a ring in a memoir p
lished in Crelle, Bd. 63, 1871. His method, though more conipl
than the preceding, is less simple, in consequence of his not
cognising explicitly that the motion contemplated corresponds
complete inextensibility of the circumference.
[In the application of (9) to a ring we have, § 192 a,
^-p.... ...(10)
where q is Young's modulus, p the volume density, and c i
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233.] EQUATION FOR THE FREQUENCY. 387
radius of the circular section. For the cylindrical shell, (18)
§ 235 g,
a '*3(m + n)p ^ ^'
2h denoting the thickness, and m, n the elastic constants in
Thomson and Tait's notation.]
According to Chladni the firequencies of the tones of a ring
are as
3» : 5» : 7« : 9«
If we refer each tone to the gravest of the series, we find for
the ratios characteristic of the intervals
2-778, 5-446, 9, 13-44, &c.
The corresponding numbers obtained from the above theoretical
formula (9), by making 8 successively equal to 2, 3, 4, &c., are
2-828, 5-423, 8-771, 1287, &c.,
agreeing pretty nearly with those found experimentally.
[Observations upon the tones of thin metallic cylinders, open
at one end, have been made by Fenkner \ Since the pitch proved
to be very nearly independent of the height of the cylinders, the
vibrations may be regarded as approximately two-dimensional.
In accordance with (9), (11), Fenkner found the frequency propor-
tional to the thickness directly, and to the square of the radius
inversely. As regards the sequence of tones from a given
cylinder *, the numbers, referred to the gravest (« = 2) as unity,
were 2*67, 5-00, 800, 1200, &c. The agreement with (9) would
be improved if these numbers were raised by about -j^ part,
equivalent to an alteration in the pitch of the gravest tone.
The influence of rotation of the shell about its axis has been
examined by Bryan*. It appears that the nodes are carried
round, but with an angular velocity less than that of the rotation.
If the latter be denoted by o), the nodal angular velocity is
^-1 T
1 Wied, Arm. vol. S, p. 186, 1879.
s Melde, Akustih, Leipzig, 1883, p. 228.
8 Proc. Camb. Phil. Soc. vol. vn. p. 101, 1890.
26—2
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"1
388 VIBRATIONS OF PLATES. [234
234. When s^l, the frequency is zero, as might have been
anticipated. The principal mode of vibration corresponds to ^^2,
and has four nodes, distant from each other by 90^. These so-
called nodes are not, howeyer, places of absolute rest, for the
tangential motion is there a maximum. In fact the tangential
vibration at these points is half the maximum normal motion.
In general for the ^ term the maximum tangential motion is
(l/s) of the maximum normal motion, and occurs at the nodes of
the latter.
When a bell-shaped body is sounded by a blow, the point of
application of the blow is a place of maximum normal motion
of the resulting vibrations, and the same is true when the
vibrations are excited by a violin-bow, as generally in lecture-
room experiments. Bells of glass, such as finger-glasses, are
however more easily thrown into regular vibration by friction with
the wetted finger carried round the circumference. The pitch of
the resulting sound is the same as of that elicited by a tap with
the soft pait of the finger; but inasmuch as the tangential motion
of a vibrating bell has been very generally ignored, the production
of sound in this manner has been felt as a difficulty. It is now
scarcely necessary to point out that the effect of the friction is in
the first instance to excite tangential motion, and that the point
of application of the friction is the place where the tangential
motion is greatest, and therefore where the normal motion
vanishes.
236. The existence of tangential vibration in the case of a bell
was verified in the following manner. A so-called air-pump re-
ceiver was securely fastened to a table, open end uppermost, and set
into vibration with the moistened finger. A small chip in the rim,
reflecting the light of a candle, gave a bright spot whose motion
could be observed with a Coddington lens suitably fixed. As the
finger was carried round, the line of vibration was seen to in-
volve with an angular velocity double that of the finger; and
the amount of excursion (indicated by the length of the line of
light), though ^'ariable, was finite in every position. There was,
however, some difficulty in observing the correspondence between
the momentary direction of vibration and the situation of the point
of excitement. To effect this satisfactorily it was found necessair
to apply the friction in the neighbourhood of one point It then
became evident that the spot moved tangentially when the bell was
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235.] TANGENTIAL MOTION. 389
excited at poiDts distant therefrom 0, 90, 180, or 270 degrees ; and
normally when the friction was applied at the intermediate points
corresponding to 45, 135, 225 and 315 degrees. Care is sometimes
required in order to make the bell vibrate in its gravest mode
without sensible admixture of overtones.
If there be a small load at any point of the circumference,
a slight augmentation of period ensues, which is different accord-
ing as the loaded point coincides with a node of the normal or
of the tangential motion, being greater in the latter case than
in the former. The sound produced depends therefore on the
place of excitation; in general both tones are heard, and by
interference give rise to beats, whose frequency is equal to the
difference between the frequencies of the two tones. This phe-
nomenon may often be observed in the case of large bells.
236 a. In determining the number of nodal meridians (2^)
corresponding to any particular tone of a bell, advantage may be
taken of beats, whether due to accidental irregularities or intro-
duced for the purpose by special loading (compare §§ 208, 209). By
tapping cautiously round a circle of kUitude the places may be in-
vestigated where the beats disappear, owing to the absence of one
or other of the component tones. But here a decision must not
be made too hastily. The inaudibility of the beats may be &voured
by an unsuitable position of the ear or of the mouth of the re-
sonator used in connection with the ear. By travelling round,
a situation is soon found where the observation can be made to
the best advantage. In the neighbourhood of the place where the
blow is being tried there is a loop of the vibration which is most
excited and a (coincident) node of the vibration which is least
excited. When the ear is opposite to a node of the first vibration,
and therefore to a loop of the second, the original inequality is
redressed, and distinct beats may be heard even though the
deviation of the blow from a nodal point may be very small. The
accurate determination in this way of two consecutive places where
no beats are generated is all that is absolutely necessary for the
purpose in view. The ratio of the entire circumference of the
circle of latitude to the arc between the points in question is in
fact 48. Thus, if the arc between consecutive points proved to
be 45'', we bhould infer that we were dealing with the case of « = 2,
in which the deformation is elliptical. As a greater security
against error, it is advisable in practice to determine a larger
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390 VIBRATIONS OF PLATES. [235 a.
number of points where no beats occur. Unless the deviation
from symmetry be considerable, these points should be uniformly
distributed along the circle of latitude \
In the above process for determining nodes we are supposed to
hear distinctly the tone corresponding to the vibration under
investigation. For this pui-pose the beats are of assistance in
directing the attention; but in dealing with the more difficult
subjects, such as church bells, it is advisable to have recourse to
resonators. A set of v. Helmholtz s pattern, as manufactured bj
K5nig, are very convenient. The one next higher in pitch to
the tone under examination is chosen and tuned by advancing the
finger acrdss the aperture. Without the security afforded by
resonators, the determination of the octave is very uncertain.
The only class of bells, for which an approximate theory can
be given, are those with thin walls, §§ 233, 235 c. Of such the
following glass bells may be regarded as examples : —
I. c', g"b, c'"«.
II. a, c'% V\
III. f% V\
The value of s for the gravest tone was 2, for the second 3,
and for the third tone 4.
Similar observations have been made upon a so-called hemi-
spherical bell, of nearly uniform thickness, and weighing about 3
cwt. Four tones could be plainly heard,
the pitch being taken from a harmonium. The gravest tone has a
long duration. When the bell is struck by a hard body, the
higher tones are at first predominant, but after a time they die
away, and leave eb in possession of the field. If the striking body
be soft, the original preponderance of the higher elements is less
marked.
By the method described there was no difficulty in shewing
that the four tones correspond respectively to s = 2, 3, 4, 5. Thus
for the gravest tone the vibration is elliptical with 4 nodal meri-
dians, for the next tone there are 6 nodal meridians, and so on.
^ The beUs, or gongs, as they are sometimes caUed, of striking clocks often giro
disagreeable beats. A remedy may be found in a suitable rotation of the beU roond
its axis.
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235 <X.] BELLS. 391
Tapping along a meridian shewed that the sounds became less
clear as the edge was departed from, and this in a continuous
manner with no suggestion of a nodal circle of latitude. A question
to which we shall recur in connection with church bells here
suggests itself. Which of the various coexisting tones characterizes
the pitch of the bell as a whole ? It would appear to be the third
in order, for the founders gave the pitch as E natural.
In church bells there is great concentration of metal at the
" sound-bow " where the clapper strikes, indeed to such an extent
that we can hardly expect much correspondence with what occurs
in the case of thin uniform bells. But the method already
described suffices to determine the number of nodal meridians for
all the more important tones. From a bell of 6 cwt. by Mears
and Stainbank 6 tones could be obtained, viz. :
e', c", r+, h"\^, d!'\ r\
(4) (4) (6) (6) (8)
The pitch of this bell as given by the makers is d!\ so that it
is the fifth in the above series of tones which characterizes the
bell. The number of nodal meridians in the various components
is indicated within the parentheses. Thus in the case of the tone
e' there are 4 nodal meridians. A similar method of examination
along a meridian shewed that there was no nodal circle of latitude.
At the same time differences of intensity were observed. This
tone is most fully developed when the blow is delivered about
midway between the crown and the rim of the bell.
The next tone is c". Observation shewed that for this vibra-
tion also there are four, and but four, nodal meridians. But now
there is a well-defined nodal circle of latitude, situated about a
quarter of the way up from the rim towards the crown. As heard
with a resonator, this tone disappears when the blow is accurately
delivered at some point of this circle, but revives with a very small
displacement on either side. The nodal circle and the four meri-
dians divide the surface into segments, over each of which the
normal motion is of one sign.
To the tone /" correspond 6 nodal meridiana There is no
well-defined nodal circle. The sound is indeed very faint when
the tap is much displaced from the sound-bow; it was thought
to fall to a minimum when a position about half-way up was
reached.
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392 VIBRATIONS OF PLATES, [235 a.
The three graver tones are heard loudly from the sound-bow.
But the next in order, ft'^l?, is there scarcely audible, unless the
blow is delivered to the rim itself in a tangential direction. The
maximum effect occurs about half-way up. Tapping round the
circle revealed 6 nodal meridians
The fifth tone, d'", is heard loudly from the sound-bow, but
soon falls off when the locality of the blow is varied, and in the
upper three-fourths of the bell it is very faint. No distinct circular
node could be detected. Tapping round the circumference shewed
that there were 8 nodal meridians.
The highest tone recorded, /'", was not easy of observation,
and the mode of vibration could not be fixed satisfisustonly.
Similar results have been obtained from a bell of 4 cwt, cast
by Taylor of Loughborough for Ampton church. The nominal
pitch (without regard to octave) was d, and the following were the
tones observed : —
el? -2, d"-6, /"-h4, 6'1?— 6", d'\ f,
(4) (4) (6) (6) (8)
In the specification of pitch the numerals following the note
indicate by how much the frequency for the bell differed fix)m
that of the harmonium employed as a standard. Thus the gravest
tone e^ gave 2 beats per second, and was flat. When the number
exceeds 3, it is the result of somewhat rough estimation, and
cannot be trusted to be quite accurate. Moreover, as has been
explained, there are in strictness two firequencies under each
head, and these often differ sensibly. In the case of the 4th tone,
6"b — 6" means that, as nearly as could be judged, the pitch of the
bell was midway between the two specified notes of the
harmonium.
Observations in the laboratory upon the above-mentioned bells
having settled the modes of vibration corresponding to the five
gravest tones, other bells of the church pattern could be sufficiently
investigated by simple determinations of pitch. The results are
collected in the following tabled and include, besides those already
given, observations upon a Belgian bell, the property of Mr
Haweis, and upon the five bells of the Terling peal. As regank
1 On Bells, PWZ. Uag.y vol. 29, p. 1, 1890.
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1
CHURCH BELLS.
393
235 cl]
the nominal jHtch of the latter bells, several observers concurred
in fixing the Dotes of the peal as
/#» 5*' «ft ^ ^'
no attention being paid to the question of the octave.
Mean,
1888.
Ampton,
Belgian
BelL
Terling (5),|Terlmg (4), Terling (8), Terling (2), Teriing (1),
Osborn, Hears, Graye, Gardner, Warner,
1788. ' 1810. 1628. 1728. 1868.
Actual Pitch by Harmonium.
«'
<?'l?-2
<f-4
g^Z
a+a
atf+3
(f-6
ef+2
e'
ef'-6
<^'«-ef'
^^4
/«-4
a'+6
a'lt-5
6'+2
r+
/'+4
r+i
o'+e
6'+6
c"«+4
(f'+8
fl"
V'\>
6"t?-ft"
a"-6 1 ef'-3
^'«-«"
d"+6
i7"«+ao)
/'«+4
a"
<f"
/'»-2
<7"«-e
«"«
6"+2
^"«+3
r
r
Pitch referred to fifth tone as c.
d
cJ-2
clt-3
c«+3
(j+3
et^-6
c«+2
h\>
c-6
cJt-4
c-4
61^+6
6-6
6t^+2
*l? +
et?+4
el^+6
e|?+6
cb + 4
6t^+8
erl^
ab
at? -a
ab-3
9-9%
/«+6
a+8
5^+4
c
c
c-2
c-6
c
c+2
cH-3
Examination of the table reveals the remarkable fact that
in every case of the English bells it is the 5th tone in order
which agrees with the nominal pitch, and that, with the exception
of Terling (4), no other tone shews such agreements Moreover,
as appeared most clearly in the case of the bell cast by Mears and
Stainbank, the nominal pitch, as given by the makers, is an octave
below the only corresponding tone.
The highly composite, and often discordant, character of the
sounds of bells tends to explain the discrepancies sometimes
manifested in estimations of pitch. Mr Simpson, who has devoted
much attention to the subject, has put forward strong arguments
for the opinion that the Belgian makers determine the pitch of
their bells by the tone 2nd in order in the above series, so that
for instance the pitch of Terling (3) would be a and not at. In
subordination to this tone they pay attention also to the next
(the 3rd in order), classifying their bells according to the character
^ In this comparison the gravest tone is disregarded.
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394
VIBRATIONS OF PLATES.
[235 a.
of the third, whether major or minor, so compounded. Thus
in Terling (3) the interval, a' to c" , is a major third. The com-
parative neglect with which the Belgians treat the 5th tone,
regarded almost exclusively by English makers, may perhaps be
explained by a less prominent development of this tone in Belgian
bells, and by a difference in treatment. When a bell is sounded
alone, or with other bells in a comparatively slow succession,
attention is likely to concentrate itself upon the graver aud more
persistent elements of the souud rather than upon the acnter
and. more evanescent elements, while the contrary may be
expected to occur when bells follow one another rapidly in a peaL
In any case the false octaves with which the Table abounds
are simple facts of observation, and we may well believe that their
correction would improve the genei*al effect. Especially should
the octave between the 2nd tone and the 5th tone be made true.
Probably the lower octave of the gravest, or hum-note, as it is
called by English founders, is of less importance. The same may
be said of the fifth, given by the 4th tone of the series, which
is much less prominent. The variations recorded in the Table
would seem to shew that no insuperable obstacle stands in the
way of obtaining accurate harmonic relations among the various
tones.
No adequate explanation has been given of the form adopted
for church bells. It appears both from experiment and from the
theory of thin shells that this form is especially stiff, as regards the
principal mode of deformation (s = 2), to forces applied normally
and near the rim. Possibly the advantage of this form lies in its
rendering less prominent the gravest component of the sound,
or the hum-note.
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CHAPTER Xa.
CURVED PLATES OB SHELLS.
236 6. In the last chapter (§§ 232, 233) we have considered
the comparatively simple problem of the vibration in two dimen-
sions of a cylindrical shell, so far at least as relates to vibrations
of a flexural character. The shell is supposed to be thin, to be
composed of isotropic material, and to be bounded by infinite
coaxal cylindrical surfaces. It is proposed in the present chapter
to treat the problem of the cylindrical shell more generally, and
further to give the theory of the flexural vibrations of spherical
shells.
In considering the deformation of a thin shell the most
important question which presents itself is whether the middle
surface, viz. the surface which lies midway between the boundaries,
does, or does not, undergo extension. In the former case the
deformation may be called extensionaX, and its potential energy is
proportional to the thickness of the shell, which will be denoted
by 2A. Since the inertia of the shell, and therefore the kinetic
energy of a given motion, is also proportional to A, the frequencies
of vibration are in this case independent of A, § 44. On the
other hand, when no line traced upon the middle sur&ce under-
goes extension, the potential energy of a deformation is of a
higher order in the small quantity h. If the shell be conceived
to be divided into laminse, the extension in any lamina is pro-
portional to its distance from the middle surface, and the con-
tribution to the potential energy is proportional to the square
of that distance. When the integration over the thickness
is carried out, the whole potential energy is found to be propor-
tional to h\ Vibrations of this kind may be called inextensional,
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396
CURVED PLATES OR SHELLS.
[235 6.
or flexural, and (§ 44) their frequencies are proportional to A, so
that the sounds become graver without limit as the thickness is
reduced.
Vibrations of the one class may thus be considered to depend
upon the term of order A, and vibrations of the other class upon
the term of order A', in the expression for the potential energy.
In general both terms occur ; and it is only in the limit that the
separation into two classes becomes absolute. This is a question
which has sometimes presented difficulty. That in the case of
extensional vibrations the term in A' should be negligible in
comparison with the term in h seems reasonable enough. But
is it permissible in dealing with the other class of vibrations to
omit the term in h while retaining the term in A* ?
The question may be illustrated by considei-ation of a statical
problem. It is a general mechanical principle (§ 74) that, if given
displacements (not sufficient by themselves to determine the
configuration) be produced in a system originally in equilibrium
by forces of corresponding tjrpes, the resulting deformation \&
determined by the condition that the potential energy shall be
as small as possible. Apply this principle to the case of an elastic
shell, the given displacements being such as not of themselves to
involve a stretching of the middle surface. The resulting defor-
mation will, in general, include both stretching and bending, and
any expression for the energy will be of the form
Ah (extension)* + Bh^ (bending)' (1).
This energy is to be as small as possible. Hence, wlien the
thickness is diminished without limit, the actual displacement
will be one of pure bending, if such there be, consistent with
the given conditions.
At first sight it may well appear strange that of the two terms
the one proportional to the cube of the thickness is to be retained,
while that proportional to the first power may be neglected. The
fact, however, is that the large potential energy that would
accompany any stretching of the middle surface is the very reason
why such stretching does not occur. The comparative laigeness
of the coefficient (proportional to h) is more than neutralized by
the smallness of the stretching itself, to the square of which the
energy is proportional.
\
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235 6.] FLEXURAL VIBRATIONa 397
An example may be taken from the case of a rod, clamped at
one end A, and deflected by a lateral force ; it is required to trace
the effect of constantly increasing stiffness of the part included
between A and a neighbouring point B, In the limit we may
regard the rod as clamped at By and neglect the energy of the
part AB, in spite of, or rather in consequence of, its infinite
stiffiiess.
It would thus be a mistake to regard the omission of the term
in A as especially mysterious. In any case of a constraint which
is supposed to be gradually introduced (§ 92 a), the vibrations
tend to arrange themselves into two classes, in one of which the
constraint is observed, while in the other, in which the constraint
is violated, the frequencies increase without limit. The analogy
with the shell of gradually diminishing thickness is complete if
we suppose that at the same time the elastic constants are in-
creased in such a manner that the resistance to bending remains
unchanged. The resistance to extension then becomes infinite,
and in the limit one class of vibrations is purely inextensional, or
flexural.
In the investigation which we are about to give of the
vibrations of a cylindrical shell, the extensional and the in-
extensional classes will be considered separately. It would
apparently be more direct to establish in the first instance a
general expression for the potential energy complete as far as
the term in A*, from which the whole theory might be deduced.
Such an expression would involve the extensions and the curva-
tures of the middle surface. It appears, however, that this method
is difficult of application, inasmuch as the potential energy (correct
to A') does not depend only upon the above-mentioned quantities,
but also upon the manner of application of the normal forces,
which are in general implied in the existence of middle surface
extensions\
236 c. The first question to be considered is the expression of
the conditions that the middle surface remain unextended, or if
these conditions be violated, to find the values of the extensions in
terms of the displacements of the various points of the smface.
^ On the Uniform Deformation in Two Dimensions of a Cylindrical Shell, with
Application to the General Theory of Deformation of Thin Shells. Proe. Math,
Soe., vol. XX. p. 872, 18S9.
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398 CURVED PLATES OR SHELLS. [235 C.
We will suppose in the first instance merely that the surface is of
revolution, and that a point is determined by cylindrical co-ordi-
nates z, r, <l>. After deformation the co-ordinates of the above
point become Z'\-Sz, r'\-Sr, ^ + S^ respectively. If ds denote
an element of arc traced upon the surface,
(ds + dB&y = (dz + dSzy + (r + Sr)« (d4> + dS<^)« + (dr + dSry,
so that
dadBd^dzdSz + r'd^dSit) + rSr {d4>y + dr dSr (1).
In this we regard z and (f) as independent variables, so that, for
example,
while ^^^(F^^dS ^^'
in which by hypothesis dr/d<l> = 0. Accordingly
dB8_(dzy(dSz drdSrj {d^^f ( dSif> I
ds "{dayidz ^dz dz)^{dsy X d<t> ^ ^)
dzd<f> ( dSz dS^ dr dSr ) .a\
"^ (ds)> \d<l>^ dz'^dzd<f>) ^ ^'
in which dSs/ds represents the extension of the element ds. If
there be no extension of any ai'c traced upon the surface, (2) must
vanish independently of any relations between dz and d^. Hence
diz dr d£r _ .^.
dz dz dz
r^ + Sr = (4),
dSz d5^ ^^^ — Ci)
d<f> dz dz d<f>
From these, by elimination of Sr,
dSz^drd^r dB(f> \ ^
dz dzdz\ d<f> J '
dBz dS<l> ^drd^B<t>^Q^
d<f> dz dz d<f>^ '
and again, by elimination of iz^
£(r^m_r^d'^^0 (6).
\
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235 c]
CONDITIONS OP INEXTENSION.
399
If the distribution of thickness and the form of the boundary
or boundaries be symmetrical with respect to the axis, the normal
functions of the system are to be found by assuming S<f> to be
proportional to cos 9^, or sin 8(f>. The equation for B<l> may then
be put into the form
ef)-
«.r^ga0 = O
(7).
It will be seen that the conditions of inextension go a long way
towards determining the form of the normal fonctions.
The simplest application is to the case of a cylinder for which
r is constant, equal say to o. Thus (3), (4), (5), (7) become simply
_=0. Sr + a^ = 0. 5^ + a'-^ = (8).
^-0
(9).
By (9), if S^ oc cos 8<l>, we may take
aB<l>=:{Ata + Btz) cos 8<f> (10),
and then, by (8), Sr = 8 (Aga + Bgz) sin 8<l> (11),
Sz = - 8"^ B^a sin 8<f> (12).
Corresponding terms, with fresh arbitrary constants, obtained by
writing 8(t> + Jtt for 8<f>, may of course be added. If 5, = 0, the
displacement is in two dimensions only (§ 233).
If an inextensible disc be attached to the cylinder at j? =r 0, so
as to form a kind of cup, the displacements Sr and Btj) must vanish
for that value of z, exception being made of the case « = 1. Hence
Ag = 0, and
aS<l> = BtZ cos 8<f>, Sr = 8 BfZ sin 8<l>, Sz = — «~*-B,a sin s^... (13).
Again, in the case of a cone, for which r = tan 7 . z, the equa-
tions (3), (4), (5), (7) become
dBz ^ ^ dBr ^ , dB<l> ^ ^ ^
d^+^^'^d^^^' ;.tan7^+Sr=0
dBz
d<l>
dB^
dz
+ z^ tan*7 ^^ + tan 7 ^ =
dBr
I4
.(14),
U-^)'" a^'-
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400 CURVED PLATES OR SHELLS. [235 C
If we take, as usual, 3^ oc cos 8^, we get as the solution of (15)
S<^ = (^ + 5,-er-i)cos«<^ (16),
and corresponding thereto
Br = 8 tan y(AtZ'\-Bg)am8<f> (17),
Sz = tan« 7 [s'^ 3,-8 {A^z + S,)] sin «^ (18).
If the cone be complete up to the vertex at jgr = 0, -B, = 0, so that
S^ = -4,cos«^ (19),
Sr = tfil, rsin«^ (20),
8^ = — «-4,tan7rsin«^ (21).
For the cone and the cylinder, the second term in the general
equation (7) vanishes. We shall obtain a more extensive class of
soluble cases by supposing that the surface is such that
r*-T-5 = constant (22),
an equation which is satisfied by sur&ces of the second degree in
general. If
z^ r^
^ + 6i = l (23).
we shall find 7^-=-^ = — r (24);
dz^ a» "^ ^'
and thus (7) takes the form
d^-^«* = (25).
if i<f> X cos 8^t Aiid a is defined by
a^jr^dz (26),
or in the present case
' = W>«^-z <27).
The solution of (25) is
^♦-Ksr-K:-^r]-*- (»>
The corresponding values of Br and S^ are to be obtained from (4)
and (5).
1
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235 c] NORMAL MODES. 401
If the surface be complete through the vertex z^a, the term
multiplied by B must disappear. Thus, omitting the constant
multiplier, we may take
^ = (Sr^'* <29);
whence, by (4), (5),
*"-a(^T^""'* <3^>'
*"=(**+"> SStS^'"^'* •^^'>-
If we measure / from the vertex, z ^a — z^ and we may write
S* = .(^)'co8*^ (32),
Sr=«r(^ysin«<^ (33),
Sz = -g/=:^I(jj + l)a-«/U^ysin«<^ (34).
For the parabola, a and 6 are infinite, while b^/a^2a\ and
r*=^4ia'z\ Thus we may take*
S^ = r»cos«^, Sr = ^+i8in«<^, S-2' = -2(« + l)aV8in^...(35).
We will now take into consideration the important case of the
sphere, for which in (23) b = a. Denoting by the angle between
the radius vector and the axis, we have z^^a cos 0,r^a sin 0, and
thus from (29), (30), (31)
S<^ = co8«<^tan'i^ (36),
Sr/a = « sin «<^ sin ^tan'i^ (37),
Sz/a = {1'\- 8 cos 0) sin 84i ton' i0 (38).
The other terms of the complete solution, corresponding to
(28), are to be obtained by changing the sign of «.
In the above equations the displacements are resolved parallel
and perpendicular to the axis 0-0, It would usually be more
convenient to resolve along the normal and the meridian. If the
components in these directions be denoted by w and aS0, we have
w==Brsm0'\-Sz cos 0, aS0 ^ Br cos — Sz sin 0;
1 On the Infinitesimal Bending of Sorfaces of Bevolution. Proc, Math. Soc.,
vol. xui. p. 4, 1S81.
R, 26
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402 CURVED PLATES OR SHELLS. [235 C.
SO that altogether
S<^ = co8d<^[^,tan'i^ + 5,cot«itf] (39),
S0 = - sin 8(f> sin [A, tB.n' ^0 -- B,cof ^0] ..(40),
w/a = sin 8<f) [Ag {s + cos 0) tan* ^0 + 5, (« - cos 0) cot' ^0] . . .(41).
To the above may be added terms derived by writing «^ + fir
for 8<f>, and changing the arbitrary constants.
235 d. We now proceed to apply the equations of § 235 c to
the principal extensions of a cylindrical surface, with a view to the
formation of the expression for the potential energy. The axial
and circumferential extensions will be denoted respectively by €i,
€2, and the shear by w. The first of these is given by (2) § 235 c,
if we suppose that d<f> = 0, dz/ds = 1. Since in the case of a
cylinder dr/dz = 0, we find
dSz .
'^-Tz <^^
In like manner
'^-H^'dif, ^'^
The value of the shear may be arrived at by considering the
diflference of extensions for the two diagonals of an infinitesimal
square whose sides are dz and ad<f>. It is
1 d8z , dS<^ ,^.
'' = ad<l>+''-di ^^^
The next part of the problem, viz. the expression of the potential
energy by means of €1, Cj, bt, appertains to the general theory of
elasticity, and can only be treated here in a cursory manner. But
it may be convenient to give the leading steps of the investigation,
referring for further explanations to the treatises of Thomson and
Tait and of Love. In thfe notation of the former {Natural
Philosophy, § 694) the general equations in three dimensions are
na=^S, nb = T, nc=U (4),
Jlfe=P-c^(Q + i^)^
Mf^ Q^a(R + P)[ (5),
Mg = R-a{P + Q))
where <^ — -a — (6)*-
2m '
1 M is Young's modulus, 0- is Poisson's ratio, n is the constant of rigidity, and
(m - ^n) that of compressibility.
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235 d] CYLINDRICAL SHELL. 403
The energy lu, corresponding to unit of volume, is given by
+ 2(m-n)(/5r + 5'«+e/) + n(a^ + 6» + c») (7).
In the application to a lamina, supposed parallel to the plane
ary, we are to take -R = 0, iS = 0, T = 0, so that
g <^f^> « = 0, 6 = (8).
Thus in terms of the extensions e, f, parallel to x, y, and of the
shear c, we get
«, = „|e. + /. + ^--^(e+/)» + ic'| (9).
This is the energy reckoned per unit of volume. In oixier to
adapt the expression to our purposes, we must multiply it by the
thickness {2h), Hence as the energy per unit area of a shell
of Ihickness 2A, we may take in the notation adopted at the com-
mencement of this section,
2nA|e,» + e,» + iw> + ^^(6, + 6,)j (10).
This expression may be applied to curved as well as to plane
plates, for any modification due to curvature must involve higher
powers of h. The same is true of the energy of bending.
235 e. We are now prepared for the investigation of the
extensional vibrations of an infinite cylindrical shell, assumed to
be periodic with respect both to z and to ^. It will be convenient
to denote by single letters the displacements parallel to z, if>,r\
we take
Sz^u, aB<l> = v, hr^w (1).
These functions are to be assumed proportional to the sines or
cosines o( jz/a and «^. Various combinations may be made, of
which an example^ is
u=U cos 8<l> cos jz/Uj t; = F sin 8<l> sin jz/a,
w = TT cos s4> sin jz/a (2) ;
so that (1), (2), (3), § 235 d
a. €i^— jU cos 8<t) sin jz/a (3),
a,€i^(W + sV)cos8<l>sinjz/a (4),
a.w = (— sU -\- jV) sin 8<l> cos jz/a (5).
^ Additions of Jt to $</>, or io jz/a, or to both, may of coarse be made at pleasure.
26—2
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404 CURVED PLATES OR SHELLS. [235 e.
The potential energy per unit area is thus (10) § 235 d
2n/ia-»reos«5^sin«j^/a|j»Cr>+(Tr + «F)« + ^^(Tr + «F-jir^
+ ^Qm^8<l> coa^jz/ai-sU + jVy] (6).
Again, if p be the volume density, the kinetic energy per unit
of area is
ph\ (-it) cos!^ s<I> cos* jz/a-^(-jT-] sin^ 86 sin* jz/ a
8<f> sin*jz/a ,
.(7).
In the integration of (6), (7) with respect to z and 0, ^ is the
mean value of the square of each sine or cosine.* We may then
apply Lagrange's method, regarding [7, F, TT as independent
generalized co-ordinates. If the type of vibration he cospt,
and p*p/n = A;^, the resulting equations may be written
{2(iV + l)j« + 5»-Ar^a«}Cr-(2iV + l)>F-2i\rjTr = 0...(8),
-2i\rjcr+2(i^+i)5F+{2(iv+i)-iM>*^=o...(io),
where N ^"^ (11).
The frequency equation is that expressing the evanescence of
the determinant of this triad of equations. On reduction it may
be written
[ifc^a* - j» - «*] {^^a« [^^a« - 2 (iV^ + 1) (j» + fi« + 1)]
+ 4(2i\r + l)j^} + 4 (2iVr+ i)jV = (12).*
These equations include of course the theory of the extensional
vibrations of a plane plate, for which a = oo . In this application
it is convenient to write o^ = y, s/a = /9, jja = 7. The displace-
ments are then
w= UcosPyc6syz, v = Vsmfiysinyz, w — W cos fiysinyz
...(13).
^ In the physical problem of a simple cylinder the range of integration for ^ is
from to 2t ; but mathematicaUy we are not confined to one reTolation. We may
conceiye the sheU to consist of several superposed conYolations, and then s is not
limited to be a whole number.
' Note on the Free Vibrations of an infinitely long Cylindrical Shell. Proc.
Ray, Soc„ vol. 46, p. 446, 1889.
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235 e. J PLANE PLATE, 405
When a is made infinite while i8, 7 remain constant, the
equations (10), (8), (9) ultimately assume the form TT «= 0, and
{2(2«r+i)y + ^-.A;»}[r-(2iV^4.1)^y9F=0...(14),
-(2iV^ + l)7/9l7 + {7« + 2(iV^+l)/3»-A»)F-0...(15);
and the determinantal equation (12) becomes
^[*'-y-/9*][**-2(iV^H-l)(7» + )S«)]«0 (16).
In (16), as was to be expected, 1^ appears as a function of
(i8* + 7^). The first root A:» = relates to flexural vibrations,
not here regarded. The second root is
A'^/S' + T^ (17),
or l>^ = ^08«+y) (18).
At the same time (14) gives
7l7-/8F=rO (19).
These vibrations involve only a shearing of the plate in its own
plane. For example, if 7 = 0, the vibration may be repre-
sented by
tt = cos)9y cosp^, t;=«0, t£; = (20).
The third root of (16)
A:» = 2(i\r+l)(;8« + y) = ^(/8» + y) (21)
gives p> = J^?!^^±0^ (22).
The corresponding relation between U and V is
^t7 + 7F=0 (23).
A simple example of this case is given by supposing in (IS),
(23), )8 = 0. We may take
14 = 00872: coQptf v = 0, w = (24),
the motion being in one dimension.
Reverting to the cylinder we will consider in detail a few
particular cases of importance. The first arises when j = 0, that is,
when the vibrations are independent of z. The three equations
(8), (9), (10) then reduce to
(«*-A:»a«)Cr=0 (26),
{2(iV + l)««-A;«a»}F+2(iV+l)«TF = (26),
2(JV'+l)«F+12(i\r + l)-A«a«}Tr = (27);
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406 CURVED PLATES OR SHELLS. [235 e.
and they may be satisfied in two ways. First let F= Tr = ; then
U may be finite, provided
fi»-jfc»a» = (28).
The corresponding type for u is
tt =cos«0 co^pt (29),
where p' = — ^ (30).
In this motion the material is sheared without dilatation of area
or volume, every generating line of the cylinder moving along
its own length. The frequency depends upon the circumferential
wave-length, and not upon the curvature of the cylinder.
The second kind of vibrations are those for which {7=0, so
that the motion is strictly in two dimensions. The elimination of
the ratio VjW from (26), (27) gives
A»a«{Jfc»a»-2(iV^ + l)(l+«»)}=0 (31).
as the frequency equation. The first root is A:* = 0, indicating
infinitely slow motion. The modes in question are flexural* for
which, according to our present reckoning, the potential eneigy
is evanescent. The corresponding relation between V and W is
by (26)
«F+ Tr = (32).
giving in (3), (4), (5),
€i = 0, €, = 0, «r = 0.
The other root of (31) is
A;«a» = 2(iV + l)(l+5») (33),
or ;)» = — ; (34) ;
while the relation between V and W is
F-«TF«0 (36).
The type of the motion may be taken to be
u ss 0, v « « sin tf cos pty w = cos 8<fi cos pt (36).
It will be observed that when 8 is very large, the flexural
vibrations (32) tend to become exclusively radial, and the exten-
sional vibrations (35) tend to become exclusively tangential.
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235 e.] EXTENSION AL VIBRATIONS. 407
Another important class of vibrations are those which are
characterized by symmetry round the axis, for which accordingly
« = 0. The general frequency equation (12) reduces in this case to
{ifc«a«-j«}{ifc»a«[ifc«a«-2(iV^+l)(i«+l)] + 4(2iV+l)j'}=0
...(37).
Corresponding to the first root we have 17=0, TT^O, as is
readily proved on reference to the original equations (8), (9), (10)
with « = 0. The vibrations are the purely torsional ones repre-
sented by
w = 0, v = sin {jzja) cos p^, w; = (38),
where 1>" = ^! (39).
The frequency depends upon the wave-length parallel to the
axis, and not upon the radius of the cylinder.
The remaining roots of (37) correspond to motions for which
F=:0, or which take place in planes passing through the axis.
The general character of these vibrations may be illustrated by
the case where j is small, so that the wave-length is a large
multiple of the radius of the cylinder. We find approximately
from the quadratic which gives the remaining roots
3m-' + (ir-+l> <«)•
>^a>-^-^i^- ,«>
The vibrations of (40) are almost purely radial. If we suppose
that j actually vanishes, we fall back upon
A«o' = 2(iyr+l) (42),
and p. = i-^«-l_ (43)S
obtabable from (33), (34) on introduction of the condition «=s0.
The type of vibration is now
wrsO, t; = 0, w^cospt (44).
1 This eqaation was first given by Love in a memoir *'0n the small Free
Yibrations and Deformation of a thin Elastic Shell," Phil. Trans., vol. 179 (18S8),
p. 528.
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408 CURVED PLATES OR SHELLS. [235 e.
On the other hand, the vibrations of (41) are ultimately purely
axial. As the type we may take
u = 008 jz I a . cos pty t; = 0, w =* -= — j sin jz/a .cospt... (45),
where »^ = —^ (461
Now, if q denote Young's modulus, we have, § 214,
q = n (3m — n)/m,
80 that ;>»=^ (47).
Thus u satisfies the equation
d*u _ q d^u
dt*'"p d?'
which is the usual formula (§ 150) for the longitudinal vibrations
of a rod, the fact that the section is here a thin annulus not
influencing the result to this order of approximation.
Another particular case worthy of notice arises when « = 1, so
that (12) assumes the form
Jk^a«(ifc»a«-j«-l)[ifc»a«-2(iV'+l)(j« + 2)]
+ 4j»(ifc»a»-jO(2iV+l) = 0...(48).
As we have already seen, if j be zero, one of the values of jfc*
vanishes. If j be small, the corresponding value of A:' is of the
order J*. Equation (48) gives in this case
^""'"W+i^* <*^^'
or in terms of p and g,
^' = 2$ (^«)-
The type of vibration is
M = ^
t; = 8in^8inJ2:/a.co8jt>t \ (51)^
w = — cos <l> sin jz/a . cos pt j
and corresponds to the flexural vibrations of a rod (§ 163). In
(51) t; satisfies the equation
dt''^~2^d?'''
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235 e.] EXTENSIONAL VIBRATIONS. 409
in which J a* represents the square of the radius of gyration of the
section of the cylindrical shell about a diameter.
This discussion of particular cases may sufSce. It is scarcely
necessary to add, in conclusion, that the most general deformation
of the middle surface can be expressed by means of a series of such
as are periodic with respect to z and ^, so that the problem con-
sidered is really the most general small motion of an infinite
cylindrical shell
The extensional vibrations of a cylinder of finite length have
been considered by Love in his Theory of Elasticity^ (1893), where
will also be found a full investigation of the general equations of
extensional deformation.
235/ When a shell is deformed in such a manner that no
line traced upon the middle surface changes in length, the term of
order h disappears from the expression for the potential energy,
and unless we are content to regard this function as zero, a
further approximation is necessary. In proceeding to this the
first remark that occurs is that the quality of inextension attaches
only to the central lamina. Consider, for example, a portion of a
cylindrical shell, which is bent so that the original curvature is
increased. It is evident that while the middle lamina remains
nnextended, those laminae which lie externally must be stretched,
and those that lie internally must be contracted. The amount of
these stretchings and contractions is proportional in the first place
to the distance from the middle surface, and in the second place to
the change of curvature which the middle surface undergoes. The
potential energy of bending is thus a question of the curooJtures of
. the middle surface. Displacements of translation or rotation, such
as a rigid body is capable of, may be disregarded.
In order to take the question in its simplest form, let us refer
the original surface to the normal and principal tangents at any
point P as axes of co-ordinates, and let us suppose that after
deformation the lines in the sheet originally coincident with the
principal tangents are brought back (if necessary) so as to occupy
the same positions as at first. The possibility of this will be
apparent when it is remembered that, in virtue of the inexten-
sion of the sheet, the angles of intersections of all lines traced
1 Also ¥m. Trans, vol. 179 a, 188S.
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410
CURVED PLATES OR SHELLS.
[235/
upon it remain unaltered. The equation of the original surface in
the neighbourhood of the point being
\pi pJ
\PI P2
that of the deformed surface may be written
.(1).
^"■*i>i + Sp/
•2Ticy|
.(2).
Pi + Bp^
In strictness (pi + Spi)""S (p^ + Bp-)"^ are the curvatures of the
sections made by the planes a:, y ; but since the principal curvatures
are a maximum and a minimum, they represent in general with
sufficient accuracy the new principal curvatures, although these
are to be found in slightly different planes. The condition of
inextension shews that points which have the same x, y in (1)
and (2) are corresponding points ; and by Gauss's theorem it is
further necessary that
^^i + ^* = (3).
Pi P2
It thus appears that the energy of bending will depend in
general upon two quantities, one giving the alterations of principal
curvature, and the other r depending upon the shift (in the
material) of the principal planes.
The case of a spherical surface is in some respects exceptional
Previously to the bending there are no planes marked out as
principal planes, and thus the position of these planes after
bending is of no consequence. The energy depends only upon
the alterations of principal curvature, and these by Gauss's theorem
are equal and opposite, so that, if a denote the radius of the
sphere, the new principal radii are a + Sp, a — Bp. If the equation
of the deformed surface be
Zz^Ax' + iBxy-k^Cy- (4),
(a + Sp)-i + (a-Sp)-» = il + C,
(a + Sp)-^(a-Sp)-* = il(7-5*;
{s^y^H^-cy+B^ (5).
we have
so that
We have now to express the elongations of the various laminff
of a shell when bent, and we will begin with the case where t = 0,
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235/] ENERGY OF BENDING. 411
that is, when the prmcipal planes of curvature remain unchanged.
It is evident that in this case the shear c vanishes, and we have to
deal only with the elongations e and /parallel to the axes, § 235 d.
In the section by the plane of zx, let s, s' denote corresponding
infinitely small arcs of the middle surface and of a lamina distant
h from it. If -^ be the angle between the terminal normals,
s = pi^^, s' = (/?! + A) -^j s' — s = A-^. In the bending, which leaves
8 unchanged,
Ss'«ASn^ = AsS(l/pi).
Hence e = Ss'/s' = AS(l/pi),
and in like manner /= AS(l/pj). Thus for the energy U per unit
area we have
[\ pJ \ pJ m + 71 \ pi p,/ )
and on integration over the whole thickness of the shell (2A)
This conclusion may be applied at once, so as to give the result
applicable to a spherical shell; for, since the original principal
planes are arbitrary, they can be taken so as to coincide with the
principal planes after bending. Thus t = 0; and by Gauss's
theorem
S(l/pOi-S(l/p,) = 0,
so that u^^^B-J (7),
where S(l/p) denotes the change of principal curvature. Since
c = — / 5r = 0, the various laminae are simply sheared, and that in
proportion to their distance from the middle surface. The energy
is thus a function of the constant of rigidity only.
The result (6) is applicable directly to the plane plate; but
this case is peculiar in that, on account of the infinitude o( pi, p^
(3) is satisfied without any relation between Bpi and Sp,. Thus for
a plane plate
'^-r\^--h^i^.{y^'} ""■
where l/pi, l/ps, are the two independent principal curvatures after
bending^
1 This wOl be found to agree with the value (2) § 214, expressed in a different
notation.
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'1
412 CURVED PLATES OB SHELLS. [235/
We have thus far considered r to vanish; and it remains to
investigate the effect of the deformations expressed by
S^ = T^y = iT(p-i7') (9),
where f, 17 relate to new axes inclined at 45*^ to those of x, y. The
curvatures defined by (9) are in the planes of f, rj, and are equal
in numerical value and opposite in sign. The elongations in these
directions for any lamina within the thickness of the shell are At,
- At, and the corresponding energy (as in the case of the sphere
just considered) takes the form
U'=^*^ (10).
This energy is to be added ^ to that already found in (6) ; and
we get finally
as the complete expression of the energy, when the deformation is
such that the middle surface is unextended. We may interpret r
by means of the angle x* through which the principal planes are
shifted; thus
-^<-p-.) • *"'■
235 g. We will now proceed with the calculation of the
potential energy involved in the bending of a cylindrical shell
The problem before us is the expression of the changes of prin-
cipal curvature and the shifts of the principal planes at any point
P (z, <l>) of the cylinder in terms of the displacements u, v,w. As in
§ 235 /, take as fixed co-ordinate axes the principal tangents and
normal to the undisturbed cylinder at the point P, the axis of s
being parallel to that of the cylinder, that of y tangential to the
circular section, and that of ^ normal, measured inwards. If, as it
will be convenient to do, we measure z and <f} fix)m the point P, we
may express the undisturbed co-ordinates of a material point Q in
the neighbourhood of P, by
x^z, y=^a<l>, ?=ia0' (1).
I There are clearly no terms involving the products of r with the cfaanges of
principal curvature d {pi~^)t S {p2'^^) ; for a change in the sign of r can have so
influence upon the energy of the deformation defined by (2).
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2355^.] APPLICATION TO CYLINDER, 413
During the displacement the co-ordinates of Q will receive the
increments
u, to8m<f>-\-vcos<f}, — i(;cos0 + t;sin^;.
so that after displacement
or, if u,v,whe expanded in powers of the small quantities z, <f>,
du du , ,^.
.=. + «.+ ^^. + ^^+ (2).
dv dv , ,^.
y = a<f> + w,<l> + v, + ^^z + ^^^<f>+ (3),
. , ., 1 dhu , dhu . . dhv .,
dv . dv ., ...
+d./*+d^/ W'
i£o, Vo, . . . being the values of u, v at the point P.
These equations give the co-ordinates of the various points of
the deformed feheet. We have now to suppose the sheet moved as
a rigid body so as to restore the position (as far as the first power
of small quantities is concerned) of points infinitely near P. A
purely translatory motion by which the displaced P is brought
back to its original position will be expressed by the simple
omission in (2), (3), (4) of the terms Uq, v©, w^ respectively, which
are independent of z, <f>. The effect of an arbitrary rotation is
represented by the additions to x, y, f respectively of yoa^ — (Icdj,
(Joh — ^o>8» xto^ — ytoi ; where for the present purpose Wi, Wj, ©3 are
small quantities of the order of the deformation, the square of
which is to be neglected throughout. If we make these additions
to (2), &c., substituting for x, y, f in the terms containing d their
approximate values, we find so far as the first powers of z, ^
du du
. , , dv , dv .
^ dw dw . , . ^ ,
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414
CURVED PLATES OR SHELLS.
[235 ()f.
Now, since the sheet is assumed to be unextended, it must be
possible so to determine cdi, Wji ©s that to this order a? = z, y = a^,
f=0. Hence
du ^
dv ^ dv
«9»
— Vo + axoi = 0.
The conditions of inextension are thus (if we drop the suflSces
iij^ BO longer requii-ed)
du
dz
^ , dv ^ du ^ dv ^
.(5).
which agree with (8) § 235 c.
Returning to (2), &c., as modified by the addition of the trans-
latory and rotatory terms, we get
x^z + terms of 2nd order in z, ^,
dv
, d^w ,^dv . dv ..
or yince by (5) d^w/dz^ = 0, and dv/d<l> = — «;,
The equation of the deformed surface after transference is thus
y- (1 dp 1 (^ ) f 1 1 _l_dHo)
Comparing with (2) § 235/ we see that
fin s 1 1 / j_ <^'«'^ 1 /d» d»w \ ,_.
^^ = ^' ^^=-^r + df'J' "^^afc-did^j-^^)'
so that by (11) §235/
'^ ~ 3a» ti^^Mi a? V ■•" d<^V "'' \dz dzd<f,) J ^^
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235 gr.] APPLICATION TO CYLINDER. 415
This is the potential energy of bending reckoned per unit of
area. It can, if desired, be expressed by (5) entirely in terms of v\
We will now apply (8) to calculate the whole potential energy
of a complete cylinder, bounded by plane edges z^±l, and of
thickness which, if variable at all, is a function of z only. Since
u, V, w are periodic when 4> increases by 2'7r, their most general
expression in accordance with (5) is [compare (10), &c., § 235 c]
t; = 2 [(il,a + B,z) cos 8^ - {A,' a + B^z) sin «^] (9),
w = 2 [« {A^a + Bgz) sin «^ + « {A^a + B^z) cos 8^] .... (10),
tf — 2[— «~^5#asin«<^ — «~^5/acos«<^] (11),
in which the summation extends to all integral values of 8 from
to ». But the displacements corresponding to « = 0, 5=1 are
such as a rigid body might undergo, and involve no absorption of
energy. When the values of u, v, w are substituted in (8) all the
terms containing products of sines or cosines with different values
of 8 vanish in the integration with respect to <^, as do also those
which contain cos 8if> sin 8<fi. Accordingly
Jo - 3a [_m + na* ^
{{A,a + B,zy -f (A/ a + B.^zf] + 2 («^ - 1)» (5,« + 5/»)1 . . .(12).
Thus far we might consider A to be a function of z ; but we will
now treat it as a constant. In the integration with respect to z
the odd powers of z will disappear, and we get as the energy of the
whole cylinder of radius a, length 21, and thickness 2h,
+ ^,(5.»+5;«)|+£.^+£;»] (13),
in which «=2, 3, 4,....
The expression (13) for the potential energy suflBces for the
solution of statical problems. As an example we will suppose
that the cylinder is compressed along a diameter by equal forces
F, applied at the points z^Zi, ^ = 0, <^ = 7r, although it is true
that so highly localised a force hardly comes within the scope of
1 From the general equations of Mr Love's memoir already cited a concordant
result may be obtained on introduction of the appropriate conditions.
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416
CURVED PLATES OR SHELLS.
[235 g.
the investigation, in consequence of the stretchings of the middle
surface, which will occur in the immediate neighbourhood of the
points of application ^
The work done upon the cylinder by the forces F during the
hjrpothetical displacement indicated by SAg, &c., will be by (10)
- Fls (a£A/ + zMs) (1 + cos sir),
so that the equations of equilibrium are
-rj-i = -(l + cos 8ir) saF,
dAg
dB. ^'
dv
dBs
V = — (1 4- COS 87r) 8ZiF.
Thus for all values of s,
and for odd values of 5,
But when s is even,
m + n
^' = -
Ssa^F
SsaZiF
..(14),
.(15):
\m + n 3a« ■** j ' SirnhH («^ - 1)*
and the displacement w at any point (z, <f>) is given by
w^2(A^'a-\- B^z) cos 2^ + 4 {A^a + B^z) cos 4<^ + . . .(16),
where A^, JS/, -4/,... are determined by (14), (15).
A further discussion of this solution will be found in the
memoir' from which the preceding results have been taken.
We will now proceed with the calculation for the frequencies
of vibration of the complete cylindrical shell of length 21, If the
volume-density' be p, we have as the expression of the kinetic
energy by means of (9), (10), (11),
T=i.2V.[j(i' + t)» + w»)arf^d-2
= 2TrpUa 2 {a» (1 + s") {A, + i/')
+ [i^ni +«') + «-*«'] W + ^/')} (17).
^ Whatever the corvature of the surface, an area upon it may be taken so small
as to behave like a plane, and therefore bend, in violation of Gauss's conditioD,
when subjected to a force which is so nearly discontinuous that it varies sensibly
within the area.
9 Proc. Roy, Soc, vol 45, p. 105, 1888.
' This can scarcely be confused with the notation for the ourvatore in the
preceding parts of the investigation.
I
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235 gr.] FREQUENCY EQUATION. 417
From the expressions for V and T in (13), (17) the types and
frequencies of vibration can be at once deduced. The fact that
the squares, and not the products, of A,, B,, are involved, shews
that these quantities are really the normal co-ordinates of the
vibrating system. If -4„ or ^/, vary as cosp^, we have
P''^^m-f~n'^^ ^TT ^^^>-
This is the equation for the frequencies of vibration in two
dimensions, § 233. For a given material, the frequency is pro-
portional directly to the thickness and inversely to the square
on the diameter of the cylinder*.
In like manner if Bg, or B,, vary as co&pt% we find
8a' m -f n
^' ^m+npa* «» + l , 3a' ^^^^-
(«* + ««)?
If the cylinder be at all long in proportion to its diameter, the
difference between p/ and p, becomes very small. Approximately
in this case
,, - 3a' /m + n 1 \ ,^^^
P./i>.=i+2^(---^ (20);
or, if we take m — 2n, « = 2,
236 A. We now pass on to the consideration of spherical
shells. The general theory of the extensional vibrations of a
complete shell has been given by Lamb', but as the subject is
of small importance from an acoustical point of view, we shall
limit our investigation to the veiy simple case of Sjrmmetrical
radial vibrations.
If w be the normal displacement, the lengths of all lines upon
the middle surface are altered in the ratio (a + w): a. In calcu-
lating the potential energy we may take in (10) § 235 d
6i = €a = w/a, w = ;
1 There is nothing in these laws special to the cylinder. In the case of similar
shells of any form, vibrating by pore bending, the frequency will be as the thick-
nesses and inversely as corresponding areas. If the similarity extend also to the
thickness, then the f^qnency is inversely as the linear dimension, in accordance
with the general law of Cauchy.
* Proe, Land. Math, 8oe. xiv. p. 50, 1882.
R. 27
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^
418 CURVED PLATES OR SHELLS. [235 A.
80 that the energy per unit area is
. , 3m — n w*
4fnh ; 1,
or for the whole sphere
F=47ro».4nA?''^^^ (1).
ni + n a'
Also for the kinetic energy, if p denote the volume density,
T^^.*7ra'.2h.p.w' (2).
Accordingly if w = TT cos pt, we have
4n3m-n
^ a^p m + n ^ ^'
as the equation for the frequency (p/^ir).
As regards the general theory Prof. Lamb thus summarizes his
results. "The fundamental modes of vibration fall into two
classes. In the modes of the First Class, the motion at every
point of the shell is wholly tangential. In the nth species of
this class, the lines of motion are the contour lines of a surface
harmonic Sn (Ch. xvii.), and the amplitude of vibration at any
point is proportional to the value of dSn/de, where de is the angle
subtended at the centre by a linear element drawn on the surface
of the shell at right angles to the contour line passing through the
point. The frequency {p/^ir) is determined by the equation
k»a» = (w-l)(rH-2) (i),
where a is the radius of the shell, and k"=p*/o/n, if p denote the
density, and n the rigidity, of the substance."
" In the vibrations of the Second Class, the motion is partly
radial and partly tangential. In the nth species of this class the
amplitude of the radial component is proportional to 5^, a surface
harmonic of order n. The tangential component is everywhere at
right angles to the contour lines of the harmonic Sn on the surface
of the shell, and its amplitude is proportional to AdSn/de, where
A is a certain constant, and de has the same meaning as before."
Prof. Lamb finds
. k^a»-47 ....
^ = -2ri(n + l)7 ^""^^
where k retains its former meaning, and 7 = (1 + <r)/(l — <r), a
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235 A.]
COMPLETE SPHERE.
419
denoting Poisson's ratio. " Corresponding to each value of n there
are two values of k*a*, given by the equation
k* a*- k»a*{(n« + nH-4)7H-n«H-n- 2} +4 (n» + n- 2)7 = 0.. .(iii).
Of the two roots of this equation, one is < and the other > 47. It
appears, then, firom (ii) that the corresponding fundamental modes
are of quite diflferent characters. The mode corresponding to the
lower root is always the more important."
" When n =s 1, the values of k»a* are and 67. The zero root
corresponds to a motion of translation of the shell as a whole
parallel to the axis of the harmonic Si. In the other mode the
radial motion is proportional to cos d, where is the co-latitude
measured from the pole of Si ; the tangential motion is along the
meridian, and its amplitude (measured in the direction of in-
creasing) is proportional to i sin 0."
*' When n = 2, the values of ka corresponding to various values
of a are given by the following table : —
<r =
<r = J
<^ = A
«r = ^
«r = i
1-120
3-570
1-176
4-391
1-185
4-601
1190
4-752
1-215
5-703
The most interesting variety is that of the zonal harmonic.
Making 5=^(3 cos' 6^—1), we see that the polar diameter of
the shell alternately elongates and contracts, whilst the equator
simultaneously contracts and expands respectively. In the mode
corresponding to the lower root, the tangential motion is towards
the poles when the polar diameter is lengthening, and vice versd.
The reverse is the case in the other mode. We can hence under-
stand the great diflFerence in frequency."
Prof. Lamb calculates that a thin glass globe 20 cm. in
diameter should, in its gravest mode, make about 5350 vibrations
per second.
As re^rds inextensional modes, their form has already been
determined, (39) &c. § 235 c, at least for the case where the
bounding curve and the thickness are sjrmmetrical with respect
to an axis, and it will further appear in the course of the present
investigation. What remains to be effected is the calculation of
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420 CURVED PLATES OR SHELLS. [235 h.
the potential energy of bending corresponding thereto, as depend-
ent upon the alterations of curvature of the middle surfSaoe. The
process is similar to that followed in § 235^ for the case of the
cylinder, and consists in finding the equation of the deformed
surface when referred to rectangular axes in and perpendicular
to the original surface.
The two systems of co-ordinates to be connected are the usual
polar co-ordinates r, 0, ^, and rectangular co-ordinates a?, y, j;
measured from the point P, or (a, 0o, ^o)i on the undisturbed
sphere. Of these x is measured along the tangent to the
meridian, y along the tangent to the circle of latitude, and ^
along the noimal inwards.
Since the origin of ^ is arbitrary, we may take it so that
^0 — 0. The relation between the two systems is then
a? = r {— sin (d - ^o) + sin cos 0o (1 — cos ^)} (4).
ysrsindsin^ (5),
?= — r {cos (0 — ^o) — sin dosin d (1 — cos ^)} + a ....(6).
If we suppose r = a, these equations give the rectangular
co-oitlinates of the point (a, 0, <f>) on the undisturbed sphere.
We have next to imagine this point displaced so that its polar
co-ordinates become a + Sr, + h0, 4> + ^> ^^^ to substitute these
values in (4), (5), (6), retaining only the first power of Sr, S0, S^
Thus
a?=:(a + Sr) {— mi (0 — 0o) + sin cos 0q{1 — cos^)}
H- aB0 {- co8{0 - 0o) + coa0 cos 0o (I - cos <^)}
+ aS^ sin 5 cos do sin ^ (7),
y = (a H- Sr) sin d sin ^
-{■ aB0 COB sin <f> + aB^ sin 0co8(f> (8),
^^a-ia + Sr) [qos(0 - do) -sin 0o sin d (1 - cos <^)}
+ aB0 {sin (0 - 0^) + sin do cos d (1 - cos if>)}
+ aS^sindosindsin^ (9X
These equations give the co-ordinates of any point Q of the sphere
after displacement ; but we shall only need to apply them in the
case where Q is in the neighbourhood of P, or (a, do, 0), and then
the higher powers of (d — d©) and <f> may be neglected.
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235 A.] INBXTENSIONAL MODES. 421
In pursuance of our plan we have now to imagine the displaced
and deformed sphere to be brought back as a rigid body so that
the parts about P occupy as nearly as possible their former
positions. We are thus in the first place to omit from (7), (8),
(9) the terms (involving S) which are independent of {6 — d©), ^•
Further we must add to each equation respectively the terms
which represent an arbitrary rotation, viz.
determining to^, a>„ co, in such a manner that, so far as the first
powers of {0 — do), ^, there shall be coincidence between the original
and displaced positions of the point Q,
If we omit all terms of the second order m {0 — 0q) and ^, we
get ft^m (7) &c.
a; = - a (d - do) - ^o(0- Oo)
-a|[S«o] + ^(d-do)-h^<^| + aS^8indoC08do.*... (10),
y = a sin do- ^ + 8n sin 0o.<f> + aB0o cos 0o . ^
+ a sin d<
:{m*'^/e.e.y^Z*\
+ aB<l>, COB 0,(0-0,) (11),
f-[-«'.]-g(»-*)-t*
+ a80o{0-0o) + aBif>oBm*0o.if> (12),
Sto &C. representing the values appropriate to P, where (0 — 0o)
and <f> vanisL The translation of the deformed surface necessary
to bring P back to its original position is represented by the
omission of the terms included in square brackets. The arbitrary
rotation is represented by the additions respectively of
a sin do . ^ . 0),, a (d — d©) co,, — a (d — do) a)j — a sin do . ^ . c»i ;
and thus the destruction of the terms of the first order requires
that
Sr/a + dSd/dd = (13),
-dSd/d<^ + 8indcosdS^+sindo), = (14);
sin d dS<^/dd + COB dS<^ + 0)5 = (15),
(Sr/a) sin d + Sd cos d + sin dd8^/d^«0 (16);
-d8(r/a)/dd + Sd-a), = (17),
- dS (r/a)/d<^ H-8in«d8<^- sin d 0)1=0 (18);
the suffixes being omitted.
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422 CURVED PLATES OR SHELLS. [235 A.
These six equations determine c^i, 6),, co,, giving as the three
conditions of inextension
Br/a + dS0/d0^O (19),
dB0/d<f> + sin^edB<l>/de^O (20),
Br/a + cote 80 + ^ld(l>=^0 (21).
From (19), (20). (21), by elimination of Br,
»(^.)--*w-° <^>.
^«*--'a(^)'« <^>^
or, since sin 6 djdd = djd log tan i^,
d^Un^y^dlogtanid'"" ^^*^'
d<f> dlogtani^Und/ ^ ^
From (24), (25) we see that both B^ and B6/sin satisfy an
equation of the second order of the same form, viz.
d^U . *^ _ A /«fiX
diiogtan^ey'^ d4>^^ ^ ^•
If the material system be symmetrical about the axis, ti is a
periodic function of ^, and can be expanded by Fourier's theorem
in a series of sines and cosines of ^ and its multiples. Moreover
each term of the series must satisfy the equations independentlj.
Thus, if t* varies as cos«0, (26) becomes
diio^lw -'''''' ^''^''
whence u = A'taa*^e + B'cof^ (28).
where A' and R are independent of 0. If we take
S^ = co8«<^[il,tan'id + 5,cot»i^] (29),
we get for the corresponding value of Sd from (24)
S^/sin ^ = - sin «0 [A, tan* ^6 - B, cot* ^ef\ (30) ;
and thence from (21)
h-ja = sin «0 [A, (« + cos d) tan' ^d + B,{8-ws 0) cot* i^. . .(31),
as in (39), (40), (41) § 235 c.
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235 A.] INEXTEXSIONAL M0DE8. 423
The second solution (in Bg) may be derived from the first (in Ag)
in two ways which are both worthy of notice. The manner of deri-
vation from (27) shews that it is sufficient to alter the sign of s,
tan* ^6 becoming cot*^^, sin*^ becoming — sin*^, while cos«^
remains unchanged. The other method depends upon the con-
sideration that the general solution must be similarly related to
the two poles. It is thus legitimate to alter the first solution by
writing throughout (tt — 0) in place of 0, changing at the same
time the sign of hd.
If we suppose « = 1, we get
sin 0hit> = cos ^ [ill + 5i - (^1 - A) cos 0\
Sd = - sin <^ [ill - jBj - (ill + 50 cos ^],
Sr/a = sin <^ [(^i + B^) sin 0],
The displacement proportional to (ilj — B^ is a rotation of the
whole surface as a rigid body round the axis — ^, ^ = ; and
that proportional to {Ai + Bi) represents a translation parallel to
the axis — \ir, ^ = ^7r. The complementary translation and
rotation with respect to these axes is obtained by substituting
^ + |7r for ^.
The two other motions possible without bending correspond to
a zero value of 8, and are readily obtained from the original
equations (19), (20), (21). They are a rotation round the axis
^ = 0, represented by
Sd = 0, S^ = const., Sr = 0,
and a displacement parallel to the same axis represented by
or S^ = 0, S^ssysind, Sr = — yacos^.
If the sphere be complete, the displacements just considered,
and corresponding to « = 0, 1, are the only ones possible. For
higher values of 8 we see frt>m (31) that Sr is infinite at one or
other pole, unless Ag and Bg both vanish. In accordance with
Jellet's theorem^ the complete sphere is incapable of bending.
If neither pole be included in the actual surface, which for
example we may suppose bounded by circles of latitude, finite
1 «<0n the Properties of Ineztensible Surfaces/* Irith Acad, Tram.^ vol. 22,
p. 179, 1855.
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424 CURVED PLATES OR SHELLS. [235 A.
values of both A and B are admissible, and therefore necessary for
a complete solution of the problem. But if, as would more often
happen, one of the poles, say ^ = 0, is included, the constants B
must be considered to vanisL Under these circumstances the
solution is
S^=sil«tan'^^cos«^ \
Sd = -il,sindtan'idsin5^ I (32),
Sr = ^,a (« H- cos ^) tan* ^^ sin «^ J
to which is to be added that obtained by writing %^ + \ir for «^,
and changing the arbitrary constant
From (32) we see that, along those meridians for which
sin 8^ = 0, the displacement is tangential and in longitude only,
while along the intermediate meridians for which cos«^ = 0, there
is no displacement in longitude, but one in latitude, and one
normal to the surface of the sphere.
Along the equator » ^,
S^ = Ag cos 8<^, BO — '-Aa sin 8(f>, &r/a = AgS sin 8^,
so that the maximum displacements in latitude and longitude are
equal.
Reverting now to the expressions for x, y, f in (7), (8), (9),
with the addition of the translatory and rotatory terms by which
the deformed sphere is brought back as nearly as possible to its
original position, we know that so fiu* as the terms of the first
order in (d — 0o) and <f> are concerned, they are reduced to
a? = -a(d-^o), y = asindo.^, ?=0 (33).
These approximations will suffice for the values of x and y ; but
in the case of (f we require the expression complete so as to
include all terms of the second order. The calculation is straight-
forward For any displacement such as Br in (9) we write
The additional rotatory terms are by (17), (18)
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235 A.] POTENTIAL ENERGY OF BENDING. 425
In these we are to retain only those terms in x, y, which are of the
second order and independent of B, so that we may write
X = ia^» sin do cos ^o, y = a(0- 0^) if> cos 0^.
In the complete expression for f as a quadratic function of
{0 — 0^) and ^ thus obtained, we substitute x and y from (33).
The final equation to the deformed surfiwe, after simplification by
the aid of (19), (20), (21), may be written
r= — II- — -1*^4-— ^J-1 ^^ cotddSr|
^ 2a( a ad^J asindj ad0d4 » #J
y^L_Sr_cotddSr_ I cP8r ]
'^2a( a a" dd asin»d d<^»J ^"^ ^'
the suffixes being now unnecessary.
Taking the value of ^/a from (32) we get
"a~ad^~=^"8i5^^'*^^*^«"^^* (3o),
1 (?8r , cosd dSr «•-« . , .,^
^/i T/ijvH- . i/i-Ti =— .-^r7i-^,tan'+dcos«<p ....(36),
Sr cotddSr 1 d^hr «»-* . ^ .,^. ^ .^^^
- -J/, 7-T-^ ,—- = -.— ^ il, tan* \0 sm «6. ..(37).
a a d0 a sm* d d^» sm* ' ^ i' \ /
To obtain the more complete solution corresponding to (31), we
have only to add new terms, multiplied by B„ and derived from
the above by changing the sign of 8. As was to be expected, the
values in (35) and (37) are equal and opposite.
Introducing the values now found into (5) § 235/, we obtain
as the square of the change of principal curvature at any point
' ^^f ^ [Af tan" ^0 + 5,» cot" \0 - iA,B, cos 28if>] . . .(38).
It should be remarked that, if either A^ or B, vanish, (38) is
independent of ^, so that the change of principal curvature is the
same for all points on a circle of latitude, and that in any case
(38) becomes independent of the product AtB^ after integration
round the circumference. The change of curvature vanishes if
« = 0, or 8 » 1, the displacement being that of which a rigid body
is capable.
Equations (35) &c shew that along the meridians where S^
vanishes (cos^^^sO) the principal planes of curvature are the
cj)-
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426 CURVED PLATES OR SHELLS. [235 A.
meridian and its perpendicular, while along the meridians where
Sr vanishes, the principal planes are inclined to the meridian at
angles of 45^
The value of the square of the change of curvature obtained in
(38) corresponds to that assumed for the displacements in (29) &c.,
and for some purposes needs to be generalised We may add
terms with coefficients Ai and Bi corresponding to a change
of 8^ to (^^ + ^), and there is further to be considered the
summation with respect to s. Putting for brevity i in place of
tan^d, we may take as the complete expression for [£(1/^)]*,
[2^*^ {{A,i? + 5.rO sin s^ + {Aii? + 5;r») sin («^ + }9r)} j"
+ [2 ~^-Q K^.** - 5«^~0 cos 8^ + {Ai^ - £/r*) COS («<^ + i^)l I* .
When this is integrated with respect to ^ round the entire
circumference, all products of the generalised co-ordinates A^^Bt^
Ai, Bi disappear, so that (7) § 235/ we have as the expression for
the potential energy of the surface included between two parallels
of latitude
F = 27r 2 («» - bJ (h sin-» {(A,^ + A/^) t«
+ {B.' + B.'*)tr^}d0 (39),
where H^^nh^ (40).
In the following applications to spherical surfaces where the
pole d = is included, we may omit the terms in B; and, if
the thickness be constant, H may be removed from under the
integral sign. We have
dO — ^ — -, sm^ = ;
so that
]]siD-'d<»d^=i|(i+«.)»<»-«d«»=i(-*;;;i+?^+^j...(4i>.
In the case of the hemisphere ^=1, and (41) assumes the value
2«» — 1
4^ <*2).
Hence for a hemisphere of uniform thickness
F=fn-fr2(«'-«)(2««-l)(^« + ^'») (43).
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235 A.] STATICAL PROBLEMS. 427
If the extreme value of d be 60°, instead of 90°, we get in
place of (42)
S^ + ^8-3 ....
4.3'+n^-«)
and V = iyrHl. 3-<'+») («» - «) (8«« + 4« - 3) (^l,* + ^;«). . .(45).
These expressions for F, in conjunction with (32), are suflScient
for the solution of statical problems, relative to the deformation of
infinitely thin spherical shells under the action of given impressed
forces. Suppose, for example, that a string of tension F connects
the opposite points on the edge of a hemisphere, represented by
d = ^, ^ = i^ or fw, arid that it is required to find the deforma-
tion. It is evident from (32) that all the quantities A/ vanish,
and that the work done by the impressed forces, corresponding to
the deformation BAg, is
— SAgOS {sin ^STT + sin ^sir} F.
If 8 be odd this vanishes, and if 8 be even it is equal to
— 2SAga8 sin ^87r.F.
Hence if « be odd Ag vanishes ; and by (43), if 8 be even,
dVldAg'^irH(8'-8){28'-l)Ag==:-2a8Qm^S7r.F;
whence ^, = _ _^___^__^ (46).
By (46) and (32) the deformation is completely determined.
If, to take a case in which the force is tangential, we suppose
that the hemisphere rests upon its pole with its edge horizontal,
and that a rod of weight W is laid symmetrically along the
diameter 0'^^v, we find in like manner
A _ aW sin ^87r ..^
^•"7rff(^-5)(2^-l) ^*'^
for all even values of 8, and Ag^O for all odd values of 8.
We now proceed to evaluate the kinetic energy as defined by
the formula
in which <r denotes the surface density, supposed to be uniform.
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(fy^(-
428 CURVED PLATES OR SHELLS. [235 A.
If we take the complete value of S<f> from (29), as supplemented by
the terms in A/, £/, we have
^^^ = 2 [cos 8if> (A,if + Bstr*) + cos (««^ + ^w) (A/f + £/(-•)].
When this expression is squared and integrated with respect
to <f> round the entire circumference, all products of letters with a
different sufiSx, and all products of dashed and undashed letters
even with the same suflSx, will disappear. Hence replacing cofi**^
&c. by the mean value' i, we may take
sin»d(^^*y = isin»dS(i,» + i/0«»'
+ isin»d2W + B;»)r« + 8in»d2(i,5. + i;5,').
The mean value (30) of {dZOjdty is the same as that just
written with the substitution throughout of — B for B, so that we
may take
^?^|^y = sin«^2(4.»-h^'»)^«
H-sin»d2(4' + Mt-^ (*9),
as the mean available for our present purpose. In (49) the
products of the symbols have disappeared, and if the expression
for the kinetic energy were as yet fully formed, the co-ordinates
would be shewn to be nomud. But we have still to include that
part of the kinetic energy dependent upon d8r/dt As the mean
value, applicable for our purpose, we have from (31)
+ J 2 (4' + 5.'') (» - COS d)* r"
+ 2 (A.B, + A,'B,') («» - cos' e) (50).
The expressions (49) and (oO) have now to be added. If we set
for brevity
jtaa''^0{(8 + co6ey + 2Bia'0] sin edd=/(8) (51),
or putting x — l + cos 0,
f{8)^j*(^)'{(8-iy + 2x(s + l)-a*}dx (52),
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235 A.] CALCULATION OF KINETIC ENERGY. 429
we get
T = i^cra* {2/(«) (i.« + i/«) + 2/(- 8) (4' + 5;«)
+ 22f(«»-co8'd)8m^dd(i,iB. + i;j^/)} (53).
It will be seen that, while V in (39) is expressible by the
squares only of the co-ordinates, a like assertion cannot in general
be made of T. Hence A^y B, &c. are Tiot in general the normal
co-ordinates. Nor could this have been expected. If, for example,
we take the case where the spherical surface is bounded by two
circles of latitude equidistant from the equator, symmetry shews
that the normal co-ordinates are, not A and £, but {A -h B) and
{A - B). In this case /(- s) =f(s).
A verification of (53) may readily be obtained in the particular
case of « = 1, the surface under, consideration being the entire
sphere. Dropping the dashed letters, we get
T = iTTcra^ {^ (i,» -h A') + f ii A}
= i7rcra*{ 4(i, -h A)* + f (ii - A)"} (54).
In this case the displacements are of the purely translatory and
rotatory tjrpes already discussed, and the coiTCCtness of (54) may
be confirmed.
Whatever may be the position of the circles of latitude by
which the surfisuje is bounded, the true types and periods of
vibration are determined by the application of Lagrange's method
to (39), (53).
When one pole, e.g. d = 0, is included within the surface, the
co-ordinates B vanish, and il,, A/ become the normal co-ordinates.
If we omit the dashed letters, the expression for T becomes
simply
r=iircra*2/(«)i,» (55).
From (43), (55) the frequencies of free vibrations for a hemi-
sphere are immediately obtainable. The equation for A, ia
<ra'f{8)As'{-H(8'-8)i2^^1)A,^0 (56);
so that, if Ag vary as coapat,
^,_fl^(^-^)(25»-l) 2nh' (^■^^)(2^-l) , ..
p' — ^v(.) ^s^^' m ^ ^'
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430 CUBVBD PLATES OR SHELLS. [235 A.
if we introduce the value of H from (40), and express a by means
of the volume density p.
In like manner for the saucer of 120°, from (44),
^,_ g(^-^)(8^ + 4g-3)
P' craV(.).3*+^ ^^^^-
The values of /(«) can be calculated without difficulty in the
various cases. Thus, for the hemisphere,
/(2)= I ar-«(4 - ^x •\' a?){\ +Qx -- a?)dx
= 20 log 2 - 12^ = 1-52961,
/(3) = 57^ - 80 log 2 = 1-88156,
/(4) = 200 log 2 - 136i = 2-29609, &c. ;
so that
i>,=-^-^x 5-2400, ^,= ^x 14-726, p^ = -xT" x 28462.
In experiment, it is the intervals between the various tones
with which we are most concerned. We find
jt)8/pa = 2-8102, jt)4/p2 = 5-4316 (59).
In the case of glass bells, such as are used with air-pumps,
the interval between the two gravest tones is usually somewhat
smaller ; the representative fraction being nearer to 2-5 than 2*8.
For the saucer of 120°, the lower limit of the integral in (52)
is |, and we get on calculation
/(2) = -12864, /(3) = -054884,
S^^^^S P^-St^^^''''^^ ^' = 0^:^^20-911,
;)8:jt),= 2-6157.
The pitch of the two gravest tones is thus decidedly higher than
for the hemisphere, and the interval between them is less.
With reference to the theory of tuning bells, it may be worth
while to consider the effect of a small change in the angle, for the
case of a nearly hemispherical bell. In general
^Hif-syi sin-»dtan*'iddd
i>.^ n -.(60).
aV tan" \e [{a + cos 0^ + 2 sin* 0] sin 0d0
Jo
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Thus
235 A.] FREQUENCY EQUATION. 431
If 0^^ + B0, and P, denote the value of p^ for the exact hemi-
sphere, we get from previous results
K=P.' [i + s^ Iff -r^el] = ^«* (^ -2««^)'
shewing that an increase in the angle depresses the pitch. As to
the interval between the two gravest tones, we get
shewing that it increases with 0. This agrees with the results
given above for = 60°.
The fact that the form of the normal functions is independent
of the distribution of density and thickness, provided that they
vary only with latitude, allows us to calculate a great variety of
cases, the difficulties being merely those of simple integration. If
we suppose that only a narrow belt in co-latitude has sufficient
thickness to contribute sensibly to the potential and kinetic
energies, we have simply, instead of (60),
4g(^-.)'Bin-^g
P' a^a{(8 + coa0y + 28m^0] ^^^^'
whence ^' = 4 /r 6 + 4cosg-cos«g )
Whence p."" V tll + 6cosd«cos«d{ ^^^^•
The ratio varies very slowly from 3, when = 0, to 2*954, when
^ = j7r.
If 2h denote the thickness at any co-latitude 0, Hcch\ aoc h.
I have calculated the ratio of frequencies of the two gravest tones
of a hemisphere on the suppositions (1) that hoc cob 0, and (2) that
A oc (1 + cos 0). The formula used is that marked (60) with H and a-
under the integral signs. In the first case, p^ipt^ 17942, differing
greatly from the value for a uniform thickness. On the second
more moderate supposition as to the law of thickness,
Pz'.pt^ 2-4591, P4:pi = 4-4837.
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432 CURVED PLATES OR SHELLS. [235 A.
It would appear that the smallness of the interval between the
gravest tones of common glass bells is due in great measure to the
thickness diminishing with increasing 0,
It is worthy of notice that the curvature of deformation S(p~0»
which by (38) varies as sin"* d tan* Jd, vanishes at the pole for
« s 3 and higher values, but is finite for 8 — 2,
The present chapter has been derived very largely from
various published memoirs by the author^ The methods have
not escaped criticism, some of which, however, is obviated by
the remark that the theory does not profess to be strictly
applicable to shells of finite thickness, but only to the limiting
case when the thickness is infinitely small. When the thickness
increases, it may become necessary to take into account certain
" local perturbations " which occur in the immediate neighbourhood
of a boundary, and which are of. such a nature as to involve
extensions of the middle surface. The reader who wishes to
pursue this rather difficult question may refer to memoirs by
Love", Lamb', and Basset*. From the point of view of the present
chapter the matter is perhaps not of great importance. For it
seems clear that any extension that may occur must be limited to
a region of infinitely small area, and affects neither the types nor
the frequencies of vibration. The question of what preciselj
happens close to a free edge may require further elucidation, but
this can hardly be expected fix>m a theory of thin shells. At
points whose distance from the edge is of the same order as the
thickness, the characteristic properties of thin shells are likely to
disappear.
1 Proc, Land. Math. Soc, xiii. p. 4, 1881 ; xx. p. 372, 1889 ; Proe. Ray. Soe., toL
45, p. 106, 1888; Tol. 45, p. 448, 1888.
» Phil Trans., 179(a), p. 491, 1888; Proc. Ray, Soc., vol. 49, p. 100, 1891;
Theory of Elasticity, ch. xxi.
> Proe. Land. Math. Soc, voL xxi. p. 119, 1890.
« Phil. Trans. 181 (a), p. 488, 1890 ; Am. Math. Jaum., vol. xvi. p. 254, 1894.
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CHAPTER Xb.
ELECTRICAL VIBRATIONS.
235 i. The introduction of the telephone into practical use,
and the numerous applications to scientific experiment of which
it admits, bring the subject of alternating electric currents
within the scope of Acoustics, and impose upon us the obligation
of shewing how the general principles expounded in this work may
best be brought to bear upon the problems presenting themselves.
Indeed Electricity affords such excellent illustrations that the
temptation to use some of them has ah^ady (§§ 78, 92 a, 111 6)
proved irresistible. It will be necessary, however, to take for
granted a knowledge of elementary electrical theory, and to abstain
for the most part from pursuing the subject in its application to
vibrations of enormously high frequency, such as have in recent
years acquired so much importance from the researches initiated
by Lodge and by Hertz. In the writings of those physicists and in
the works of Prof. J. J. Thomson^ and of Mr O. Heaviside* the
reader will find the necessary information on that branch of the
subject.
The general idea of including electrical phenomena under those
of ordinary mechanics is exemplified in the early writings of Lord
Kelvin ; and in his " Dynamical Theory of the Electro-magnetic
Field' " Maxwell gave a systematic exposition of the subject from
this point of view.
^ Recent Researches in Electricity and Magnetism^ 1898.
s Electrical Papers, 1892.
' Phil, Trans. Yol. 155, p. 459, 1865 ; CoUeeted Works, vol. 1, p. 526.
R, ' 28
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434 ELECTRICAL VIBRATIONS. [235;.
236/ We commence with the consideration of a simple
electrical circuit, consisting of an electro-magnet whose terminab
are connected with the poles of a condenser, or Uyden}, of capacity
(7. The electro*magnet may be a simple coil of insulated wire, of
resistance i2, and of self-induction or inductance L. If there be an
iron core, it is necessary to suppose that the metal is divided so as
to avoid the interference of internal induced currents, and further
that the whole change of magnetism is small*. Otherwise the
behaviour of the iron is complicated with hysteresis, and its effect
cannot be represented as a simple augmentation of L. Also for
the present we will ignore the h3rstere8is exhibited by many kinds
of leydens.
If X denote the charge of the leyden at time t, a: is the
current, and if i^icosj^^ be the imposed electro-motive force, the
equation is
Lx'hRx'\-x/C^EiCOBpt (1).
The solution of (1) gives the theory ot forced electrical vibrations;
but we may commence with the consideration of the free vibra-
tions corresponding to Ei^O. This problem has already been
treated in § 45, from which it appears that the currents are
oscillatory, if
R<2^(L/C) (2).
The fact that the discharges of leydens are often oscillatory was
suspected by Henry and by v. Helmholtz, but the mathematical
theory is due to Kelvin*.
When R is much smaller than the critical value in (2). a lai^
number of vibrations occur without much loss of amplitude, and
the period r is given by
T = 27rV((7Z) (3).
In (2), (3) the data may be supposed to be expressed in CG.s.
electro-magnetic measure. If we introduce practical units, so
that L\ R\ C represent the inductance, resistance and capacity
reckoned respectively in earth-quadrants or henrys, ohms, and
microfarads*, we have in place of (2)
iJ'<2000V(Z7C") (2-),
1 This term has been approved by Lord Kelvin (" On a New Form of Aii Lejdefl
<fec.*' Proc. Roy, Soe., vol. 62, p. 6, 1892).
« Phil Mag., vol. 23, p. 226, 1887.
s ** On Transient Electric Currents," Phil. Mag., Jane, 185S.
< Ohm=:10», henry =10*, microfarad =10"".
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235 y.] CALCULATION OF PERIOD. 435
and in place of (3)
T = 27r.lO-V(C7'2;0 (30.
With ordinary appliances the value of t is very small ; but by
including a considerable coil of insulated wire in the discharging
circuit of a leyden composed of numerous glass plates Lodge ^ has
succeeded in exhibiting oscillatory sparks of periods as long as
^second.
If the leyden be of infinite capacity or, what comes to the
same thing, if it be short-circuited, the equation of free motion
reduces to
Z« + jRi«0 (4);
whence x^di^er<^l^^* (5)»,
io representing the value of x when ^ = 0. The quantity LjR is
sometimes called the time-constant of the circuit, being the time
during which free currents fall off in the ratio of 6 : 1.
Returning to equation (1), we see that the problem falls under
the general head of vibrations of one degree of freedom, discussed
in § 46. In the notation there adopted, w' = (CZ)-^ k^R/L,
EssEJL; and the solution is expressed by equations (4) and (5).
It is unnecessary to repeat at length the discussion already given,
but it may be well to call attention to the case of resonance,
where the natural pitch of the electrical vibrator coincides with
that of the imposed force (p^LC^l). The first and third terms
then (§ 46) compensate one another, and the equation reduces to
Rx^E^coapt (6).
In general, if the leyden be short-circuited ((7= oo ),
E
^ "" Dj^Xb^ ^^ cos j>^ -I- J)/; sin pt] ..(7);
so that, if p much exceed JR/X, the current is greatly reduced by
self-induction. In such a case the introduction of a leyden of
suitable capacity, by which the self-induction is compensated,
results in a large augmentation of current*. The imposed electro-
motive force may be obtained from a coil forming part of the
circuit and revolving in a magnetic field.
1 Ptoc. Roy. ImU, March, 1SS9.
' Helinholtz, Pogg. Arm., lzxzzii., p. 505, 1851.
» Maxwell, «* Experiment in Magneto-Electric Induction," Phil. Mag., May,
1868.
28—2
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436 ELECTRICAL VIBRATIONS. [235/.
In any circuit, where vibrations, whether forced or free, pro-
portional to cos pt are in progress, we have x s —p^x, and thus the
terms due to self-induction and to the leyden enter into the
equation in the same manner. The law is more readily expressed
if we use the stifftiess /a, equal to 1/(7, rather than the capacity
itself. We may say that a stiffness fi compensates an inductance
£, if fi^p^L, and that an additional indtlctance AX is compensated
by an additional stiffness A/t, provided the above proportionality
hold good This remark allows us to simplify our equations by
omitting in the first instance the stiffness of leydens. When the
solution has been obtained, we may at any time generalise it
by the introduction, in place of Z, of L — /fP"^, or Z — (p^CyK In
following this course we must be prepared to admit negative
values of L,
235 k. We will next suppose that there are two independent
circuits with coefficients of self-induction L, N, and of mutual
induction M, and examine what will be the effect in the second
circuit of the instantaneous establishment and subsequent main-
tenance of a current x in the first circuit. At the first moment
the question is one of the function T only, where
T^^Ldi' + Mxy + ^N^ (1);
and by Kelvin's rule (§ 79) the solution is to be obtained by
making (1) a minimum under the condition that x has the given
value. Thus initially
y.--f* (2);
and accordingly (§ 235 j) after time t
y = -^ie-(«W (3),
if S be the resistance of the circuit. The whole induced current,
as measured by a ballistic galvanometer, is given by
'.,. Mx ...
y* = --^- (4),
in which N does not appear. The current in the secondary circuit
due to the cessation of a previously established steady current x in
the primary circuit is the opposite of the above.
A curious property of the initial induced current is at once
evident from Kelvin's theorem, or from equation (2). It appears
/;
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235/;.] SECONDARY CIRCUIT. 437
that, if if be given, the initial current is greatest when N is least.
Further, if the secondary circuit consist mainly of a coil of n turns,
the initial current increases with diminishing n. For, although
ifxw, Nocn*; and thus yoal/n. In fact the small current
flowing through the more numerous convolutions has the same
effect as regards the energy of the field as the larger current in the
fewer convolutions. This peculiar dependence upon n cannot be
investigated by the galvanometer, at least without commutators
capable of separating one part of the induced current from the
rest ; for, as we see from (4), the galvanometer reading is affected
in the reverse direction. It ia possible however to render evident
the increased initial current due to a diminished n by observing
the magnetizing effect upon steel needles. The magnetization
depends mainly upon the initial maximum value of the current,
and in a less degree, or scarcely at all, upon its subsequent
duration. ^
The general equations for two detached circuits, influencing
one another only by induction, may be obtained in the usual
manner from (1) and
F^^Rx'^^Sy' (5).
Thus Lx + My-^Rx^X) .
ifir + i\ry + Sy=F| ^^^•
These equations, in a more general form, are considered in
§ 116. If a harmonic force X ^e^^* act in the first circuit, and
the second circuit be free from imposed force (F=0), we have on
elimination of y
shewing that the reaction of the secondary circuit upon the first is
to reduce the inductance by
P* ^*^ (8)«
P p'N^ + S' ^°''
and to increase the resistance by
^ p^N'^8- ^^^•
1 PhU, Mag,, vol. 8S, p. 1, 1869 ; vol. 89, p. 428, 1870.
< MazweU, Phil, Tram,, vol 155, p. 459, 1865, where, however, Af* is mis-
printed M.
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438 ELECTRICAL VIBRATIONS. [235 i.
The formula (8) and (9) may be applied to deal with a more
general problem of considerable interest, which arises when (as in
some of Henry's experiments) the secondary circait acts upon a
third, this upon a fourth, and so on, the only condition being that
there must be no mutual induction except between immediate
neighbours in the series. For the sake of distinctness we will
limit ourselves to four circuits.
In the fourth circuit the current is due ex hypothetn only to
induction from the third. Its reaction upon the third, for the rate
of vibration under contemplation, is given at once by (8) and (9) ;
and if we use the complete values applicable to the third circuit
under tliese conditions, we may thenceforth ignore the fourth
circuit In like manner we can now deduce the reaction upon
the secondary, giving the effective resistance and inductance of
that circuit under the influence xrf the third and fourth circuits ;
and then, by another step of the same kind, we may arrive at the
values applicable to the primary circuit, under the influence of Wl
the others. The process is evidently general; and we know by
the theorem of § 111 6 that, however extended the train of circuits,
the influence of the others upon the first must be to increase its
effective resistance and diminish its effective inertia, in greater
and greater degree as the frequency of vibration increasea
In the limit, when the frequency increases indefinitely, the
distribution of currents is determined by the induction-coefficients,
irrespective of resistance, and, as we shall see presently, it is of
such a character that the currents are alternately opposite in sign
as we pass along the series.
236 1. Whatever may be the number of independent currents,
or degrees of freedom, the general equations are always of the
kind already discussed g 82, 103, 104, viz.
ddTdFdV^y .
didx'^di^d^^^ ^ ^'
where T,F, Fare (§ 82) homogeneous quadratic functions. In(l)
the co-ordinates a^, ajj, ... denote the whole quantity of electricity
which has passed at time t, the currents being £i, ^, &c. When
F«0, it is simpler to express the phenomena by means of the
currents. Thus, in the problem of steady electric flow where all
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235 Z.] INITIAL CURRENTS. 439
the quantities X, representing electro-motive forces, are constant,
the currents are determined directly by the linear equations
dFjdik^Xu dF/dx^^X^,Sic (2).
On the other hand when the question under consideration is
one of initial impulsive effects, or of forced vibrations of ex-
ceedingly high frequency, everything depends upon T, and the
equations reduce to
d dT ^ d dT «. p .^v
didAT^'' didi,'^*'^ ^^>-
As an example we may consider the problem, touched upon at
the close of § 235 &, of a train of circuits where the mutual induc-
tion is confined to immediate neighbours, so that
+aiaa?iJJi+aa,ajja:a-|-a,4^sa?4-|- (4)^
coefficients such as au, 014, o^ not appearing. If ^ be given,
either as a current suddenly developed and afterwards maintained
constant, or as a harmonic time function of high frequency, while
no external forces operate in the other circuits, the problem
is to determine x^, x^, Sic so as to make T as small as possible,
§ 79. The equations are easily written down, but the conclusion
aimed at is perhaps arrived at more instructively by consideration
of the function T itself. For, T being homogeneous in Xi^x^, &c.,
we have identically
''■-'•'£*'•'£,* <»)•
And, since when T is a minimum, dT/dx^, dT/dx^, &c., all vanish,
dT
But if a?2, a:,, &c., had all been zero, 2T would have been equal to
OriiXiK It is clear therefore that chiXiX^ is negative ; or, as a^ is
taken positive, the sign of x^ is the opposite to that of a^i.
Again supposing Xi, x^ both given, we must, when T is a
minimum, have dT/dx^, dT/dx^y &c., equal to zero, and thus
^T^iTL = Oiia?!* + iOiiX^x^ + aj,a?8« -h 2a„ir,a?,.
As before, 27 might have been
' The dots sure omitted as unneoeBBaiy.
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440 ELECTRICAL VIBRATIONS. [235 /.
simply. The minimum value is necessarily less than this, and
accordingly the signs of x^ and x^ are opposite. This argfument
may be continued, and it shews that, however long the series may
be, the induced currents are alternately opposite in sign^ a result
in harmony with the magnetizations observed by Henry.
In certain cases the minimum value of T may be very nearly
zero. This happens when the coils which exercise a mutual
inductive influence are so close throughout their entire lengths
that they can produce approximately opposite magnetic forces at
all points of space. Suppose, for example, that there are two
similar coils A and B, each wound with a double wire (ili, -4,),
{Bu -Ba), and combined so that the primary circuit consists of -4,,
the secondary of A^ and Bi joined by inductionless leads, and the
tertiary of B^ simply closed upon itself. It is evident that T is
made approximately zero by taking x^^-^x^ and 5?, = — a:, = a-,.
The argument may be extended to a train of such coils, however
long, and also to cases where the number of convolutions in
mutually reacting coils is not the same.
In a large class of problems, where leyden effects do not occur
sensibly, the course of events is determined by T and F simply.
These functions may then be reduced to sums of squares ; and the
typical equation takes the form
ax + hx=^X (6).
If X = 0, that is if there be no imposed electro-motive forces, the
solution is
x^d^e-^f^ (7).
Thus any system of initial currents flowing whether in detached
or connected linear conductors, or in ^olid conducting masses, may
be resolved into " normal " components, each of which dies down
exponentially at its own proper rate.
A general property of the "persistences," equal to a/6, is
proved in § 92 a. For example, any increase in permeability, due
to the introduction of iron (regarded as non-conducting), or any
diminution of resistance, however local, will in general bring about
a rise in the values of all the persistences*.
In view of the discussions of Chapter v. it is not necessary to
dwell upon the solution of equations (1) when X is retained. The
* PMl Mag., vol. 38, p. 18, 1869.
' Bnt. Assoc. Report, 1886, p. 911.
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235 /.] TWO CONDUCTORS IN PARALLEL. 441
reciprocal theorem of § 109 has many interesting electrical appli-
cations ; but» after what has there been said, their deduction will
present no difficulty.
235 m. In § 111 6 one application of the general formulae to
an electrical system has already been given. As another example,
also relating to the case of two degrees of freedom, we may take
the problem of two conductors in parallel It is not necessary to
include the influence of the leads outside the points of bifurcation;
for provided that there be no mutual induction between these parts
and the remainder, their inductance and resistance enter into the
result by simple addition.
Under the sole operation of resistance, the total current a^
would divide itself between the two conductors (of resistances jR
and S)in the parts
8_ . R
R + S""' ^"""^ RVS""''*
and we may conveniently so choose the second co-ordinate that
the currents in the two conductors are in general
70:1 + a^a and t>T"c»^"'^2'
Xi still representing the total current in the leads. The dissi-
pation-function, found by multiplying the squares of the above
currents by ^R, ^8 respectively, is
^ = i^a'i' + K-B + 'S)^,' ^^)-
Also, L, M, N being the induction coefficients of the two
branches,
■'"* (R+sy '^^
Thus, in the notation of § 111 b,
_ LS*+2MR8 + NR* (L-M)8 + (M-Jfr)R
">'" {R + Sy ' "^'^ R + S
ar,=^L-2M + N;
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442 ELECTRICAL VIBRATIONS. [235 ffl.
Accordingly by (5), (8) § 111 6,
^,_L S' + 2MRS+NR^ {(£^itf)g + (if-iy^ii}«
{{L^M)8-^(M-N)RY ...
These are respectively the efiective resistance and the efifaetive
inductance of the combination ^ It is to be remarked that
(L — 2M + N) is necessarily positive, representing twice the kinetic
energy of the system when the currents in the two conductors
are + 1 and — 1.
The expressions for R and L' may be put into a form' which
for many purposes is more convenient, by combining the com-
ponent fractional terms. Thus
J., ^ RS(R + S)-^p^{R(M^Ny + S{L^My} . ,.
(R'\'Sy+p^(L^2M^Ny ^ ^'
in which {LN-^M^) is positive by virtue of the nature of T.
As p increases from zero, we know by the general theorem
§ 111 6, or from the particular expressions (3), (4), that B! con-
tinually increases and that L' continually decreases.
When p is very small,
^, R8 J, ZS« + 2Jlfi2iSf + -yiJ« ,.,
^^R^^ ^ (RVsy ^^^-
In this case the distribution of the main current between the
conductors is determined by the resistances, and (§1116) the values
of R: and L coincide respectively with 2F\x^, 2Tlx^\ The resist-
ance is manifestly the same as if the currents were steady.
On the other hand, when p is very great,
p, R{M^Ny^^8{L-My LN^M-
^ {L^2M-^Ny ' ^'L^2M+N-^''^'
In this case the distribution of currents is independent of the
resistances, being determined in accordance with Kelvin's theorem
1 PhiL Mag,, vol. 21, p. 877, 18S6.
3 J. J. Thomson, loc, eit. § 421.
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.235 m.] CONTIGUOUS wires. 443
in such a manner that the ratio of the currents in the two con-
ductors is (N—M) : (L — M). As when p is small, the values in
(6) coincide with 2F/xi\ 2,Tlx^.
When the two wires composing the conductors in parallel are
wound closely together, the energy of the field under high fre-
quency may be very small. There is an interesting distinction to
be noted here dependent upon the manner in which the con-
nections are made. Consider, for example, the case of a bundle
of five contiguous wires wound into a coil, of which three wires,
connected in series so as to have maximum inductance, constitute
one of the branches in parallel, and the other two, connected
similarly in series, constitute the other branch. There is still an
alternative as to the manner of connection of the two branches.
If steady currents would circulate opposite ways {M negative), the
total current is divided into two parts in the ratio 3 : 2, in such a
manner that the more powerful current in the double wire nearly
neutralises at external points the magnetic effects of the less
poweifiil current in the triple wire, and the total energy of the
system is very small. But now suppose that the connections are
such that steady currents would circulate the same way in both
branches {M positive). It is evident that the condition of mini-
mum energy cannot be satisfied when the currents are in the same
direction, but requires that the smaller current in the triple wire
should be in the opposite direction to that of the larger current in
the double wire. In feet the currents must be as 3 to — 2 ; so
that (since on the same scale the total current is unity) the
component currents in the branches are both numerically greater
than the total current which is algebraically divided between
them. And this peculiar feature becomes more and more strongly
marked the nearer L and N approach to equality^
The unusual development of currents in the branches is, of
course, attended by an augmented effective resistance. In the
limiting case when the m convolutions of one branch are supposed
to coincide geometrically with one another and with the n convo-
lutions of the second branch, we have
L\M\ N^m^ : mn : n\
andfrom(6) ^''"S^- 01
1 Phil. Mag,, vol. 21, p. S76, 18S6.
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444 ELECTRICAL VIBRATIONS. [235 m.
an expression which increases without limit, as m and n approach
to equality.
The fact that under certain conditions the currents in both
branches of a divided circuit may exceed the current in the mains
has beeu verified by direct experiments Each of the three
currents to be compared traversed short lengths of similar German-
silver wire, and the test consisted in finding what lengths of this
wire it was necessary in the various cases to include between the
terminals of a high resistance telephone in order to obtain sounds
of equal intensity. The variable currents were derived from a
battery and scraping contact apparatus (§ 235 r), directly included
in the main circuit.
The general formulae (3'), (4') undergo simplification when the
conductoi*s in parallel exercise no mutual induction. Thus, when
Jlf = 0,
_ RS( R + S) +i^ (RN^ + SD)
(8),
(9).
If further K^O, (8) and (9) reduce to
8[R(R + S)-^1^ D} LS^
^~ (R-^-Sy-^p^L' ' ^^{R + Sy + p^L^'-'^^^^-
The peculiar features of the combination are brought out most
strongly when 8, the resistance of the inductionless component, is
great in comparison with 12. In that case if the current be steady
or slowly vibrating, it flows mainly through fi, while the resistance
and inductance of the combination approximate to 12 andZ respec-
tively ; but if on the other hand the current be a rapidly vibrating
one, it flows mainly through 8, so that the resistance of the combi-
nation approximates to 8, and the inductance to zero. These
conclusions are in agreement with (10).
If the branches in parallel be simple electro-magnets, L and N
are necessarily positive, and the numerator in (9) is incapable of
vanishing. But, as we have seen, when leydens are admitted, this
restriction may be removed. An interesting case arises when the
second branch is inductionless, and is interrupted by a leyden of
1 PhU. Mag,, vol. 22, p. 495, 1SS6.
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235 771.] COITDUCTORS IN PARALLEL. 445
capacity (7, so that i\r= — ((7p')"*^ while at the same time -B = fif.
The latter condition reduces the numerator in (9) to
Thus L vanishes, (i) when LCp^ = 1, and (ii) when CB^ = L. The
first alternative is the condition that the loop circuit, considered
by itself, should be isochronous with the imposed vibrations.
The second expresses the equality of the time-constants of the two
branches. If they be equal, the combination behaves like a simple
resistance, whatever be the character of the imposed electro-
motive forced
235 n. When there are more than two conductors in parallel,
the general expressions for the equivalent resistance and induc-
tance of the combination would be very complicated ; but a few
particular cases are worthy of notice.
The first of these occurs when there is no mutual induction
between the members. If the quantities relating to the various
branches be distinguished by the suffixes 1, 2, 3, ..., and it Ehe
the difference of potentials at the common terminals, we have
E^{ipL, + It,)xj^{ipL^ + IL)x^^ .'...(1);
by which R and L' are determined. Thus, if we write
2-
we have from (2)
^F+7F~^' ^W+p^L^'^ ^^^'
^'A'+fB" ^ ~ A*+p*JEP ^*^'
Equations (3) and (4) contain the solution of the problem'.
When p = 0,
""~2(ij-')' {2(ie-)i ^''''■
When on the other hand p is very great,
^""{2(1-0}"' ~2(X-^) ^^^'
* Chxystal, "On the Differential Telephone," Edin. Trans., vol. 29, p. 615,
1880.
« Phil. Mag., vol 21, p. 879, 1886.
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446 ELECTRICAL VIBRATIONS. [235 n.
Even when the mutual induction between varioos membeis
cannot be neglected, tolerably simple expressions can be found for
the equivalent resistance and inductance in the extreme cases otp
infiuitely small or infinitely large. As has already been proved,
(§111 b), the above-mentioned quantities then coincide in value
with 2i7(a?, + ic, + ...)', and 2r/(iCi-|-a^ + ...)", and the calculation
of these values is easy, inasmuch as the distribution of currents
among the branches is determined in the first case entirely by F
and in the second case entirely by T. Thus, when p is infinitely
small, i^ is a minimum, and the currents are in proportion to the
conductances of the several branches. Accordingly, if the induction
coefficients of the branches be denoted, as in § 111 6, by Ou, o^, ...
Oia, ttui •••> and the resistances by IZx, iZ,, &c., we have
^■~'(i7ij,+i/ie,+..:)» ~i/ij,+i/ij,+ ... ^'^'
^ " (l/iJ, + 1/B, 4- l/fi, + ...)» •••^^^•
A similar method applies when p — oo, but the result is leas
simple on account of the complication in the ratios of currents due
to mutual induction \
235 0. The induction-balance, originally contrived by Dove
for use with the galvanometer, has in recent years been adapted
to the telephone by Hughes^ who has described experiments
illustrating the marvellous sensibility of the arrangement. The
essential features are a primary, or battery, circuit, in which
circulates a current rendered intermittent by a make and break
intennipter, or by a simple scraping contact, and a secondary
circuit containing a telephone. By suitable adjustments the two
circuits are rendered conjugate, that is to say the coefficient of
mutual induction is caused to vanish, so as to reduce the telephone
to silence. The introduction into the neighbourhood of a third
circuit, whether composed of a coil of wire, or of a simple con-
ducting mass, such as a coin, will then in general cause a revival
of sound.
The destruction of the mutual induction in the case of two flat
coils can be aiTived at by placing them at a short distance apart,
^ J. J. Thomson, loc. ciU § 422.
« PhxL Mag, vol. vra., p. 60, 1879.
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1
235 0.] hughes' induction balance, 447
in parallel planes, and with accurately adjusted overlapping. But
in Hughes' apparatus the balance is obtained more symmetrically
by the method of duplication. Four similar coils are employed.
Of these two Ai, A^, mounted at some distance apart with their
planes horizontal, and connected in series, constitute the primary
induction coiL The secondary induction coil consists in like
manner of JBi, -Bj, placed symmetrically at short distances from
^1, il„ and also connected in series, but in such a manner that
the induction between Ai and Bi tends to balance the induction
between A^ and B^. If the four coils were perfectly similar,
balance would be obtained when the distances were equal. This
of course is not to be depended upon, but by a screw motion the
distance between one pair, e.g. Ai and Bi, is rendered adjustable]
and in this way a balance between the two inductions is obtained.
Wooden cups, fitting into the coils, are provided in such situations
that a coin resting in one of them is situated symmetrically
between the corresponding primary and secondar}' coils. The
balance, previously adjusted, is of course upset by the introduction
of a coin upon one side, but if a perfectly similar coin be intro-
duced upon the other side also, balance may be restored. Hughes
found that very minute differences between coins could be ren-
dered evident by outstanding sound in the telephone.
The theory of this apparatus, when the primary currents are
harmonic, is simple ^ especially if we suppose that the primary
current a?i is given. If a?i, a?,,... be the currents; 61,62,... the
resistances; an, (hi, Oij, ... the inductances, the equations for
the case of three circuits are
We now assume that cci, a?s, &c. are proportional to 6*^*, where
p/2'7r is the frequency of vibration. Thus,
ip (OajflJa + as,a7,) -h 6ja?a = - iporu^i*
ip ((hsCCi + a^a?,) + 6,a?, = — ipa^ooi ;
whence by elimination of ic,
a:Aipa^ + b. + J^\^^ipa..x,-P>^^ (2).
1 Brit, Assoc. Rep. 1880, p. 472.
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448 ELECTRICAL VIBRATIONS. [235 0.
From this it appears that a want of balance depending on a^
cannot compensate for the action of the third circuit, so as to
produce silence in the secondary circuit, unless bt be negligible
in comparison with pa», that is unless the time-constant of the
third circuit be very great in comparison with the period of the
vibration. Otherwise the effects are in different phases, and
therefore incapable of balancing.
We will now introduce a fourth circuit, and suppose that the
primary and secondary circuits are accurately conjugate, so that
ais ~ 0, and also that the mutual induction o^ between the third
and fourth circuits may be neglected. Then
ip (a42^s + ^^44^4) + &4^4 = — tpcti4^ 5
whence
= — 0"^i i- ,- + ^ -xf W
^ bpd^ + o^ tpa44 + 04)
It appears that two conditions must be satisfied in order to
secure a balance, since both the phases and the intensities of the
separate effects must be the same. The first condition requires
that the time-constants of the third and fourth circuits be equal,
unless indeed both be very great, or both be very small, in com-
parison with the period. If this condition be satisfied, balance
ensues when
?^ + 2^^ = (4):
and it is especially to be noted that the adjustment is independent
of pitch, so that (by Fourier's theorem) it sufiices whatever be the
nature of the variable currents operative in the primary.
As regards the position of the third and fourth circuits, usually
represented by coins in illustrative experiments, it will be seen
from the symmetry of the right-hand member of (3) that the
middle position between the primary and secondary coils is suit-
able, inasmuch as the product OuOn is stationary in value when
the coin is moved slightly so as to be nearer say to the primaiy
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235 O.] GENERALIZED RESISTANCE. 449
and further from the secondary ^ Approximate independence of
other displacements is secured by the geometrical symmetry of the
coils round the axis.
236 p. For the accurate comparison of electrical quantities
the "bridge" arrangement of Wheatstone is usually the most
convenient, and is equally available with the galvanometer in the
case of steady or transitory currents, or with the telephone in the
case of periodic currents. Similar effects may be obtained in most
cases without a bridge by the employment of the differential
galvanometer or the differential telephone*.
In the ordinary use of the bridge the four members a, b, c, d
combined in a quadrilateral Fig. (53 a) are
simple resistances. The battery branch/ ^\^ ^'
joins one pair of opposite comers, and the
indicating instrument is in the "bridge"
e joining the other pair. "Balance" is
obtained, when ad — be. But for our
purpose we have to suppose that any
member, e.g. a, is not merely a resistance,
or even a combination of resistances. It may include an electro-
magnet, and it may be interrupted by a leyden. But in any case,
so long as the current x is strictly harmonic, proportional. to e^^S
the general relation between it and the difference of potentials V
at the extremities is given by
F=(ai + iaa)a; (1),
where Oi and io, are the real and imaginary parts of a complex
coefficient a, and are functions of the frequency p/2ir. In the
particular case of a simple conductor, endowed with inductance L,
ai represents the resistance, and a, is equal to pL. In general, Oi
is positive; but a^ may be either positive, as in the above ex-
ample, or negative. The latter case arises when a resistance R is
interrupted by a leyden of capacity (7. Here ai = ii, a, = — l/pG,
If there be also inductance i,
a, = R, at^pL-l/pC (2).
As we have already seen, § 235 j, Os may vanish for a particular
frequency, and the combination is then equivalent to a simple
> See Lodge, Phil. Mag., rol. 9, p. 123, 1S80.
' Chrystal, Edin. Trans., loc. eit.
R. 29
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450 ELECTRICAL VIBRATIONS. [2S5 p.
resistance. But a variation of frequency gives rise to a positive
or negative a,.
In all electrical problems, where there is no mutual induction,
the generalize quantities, a, 6, &c., combine, just as they do when
they represent simple resistances^ Thus, if a, a be two complex
quantities representing two conductors in series, the corresponding
quantity for the combination is (a + a). Again, if a, a represent
two conductors in parallel, the reciprocal of the resultant i« given
by addition of the reciprocals of a, a. For, if the currents be x, x ,
corresponding to a difference of potentials V at the common
terminals,
so that a: + a?' = F(l/a + 1/a').
In the application to Wheatstone*s combination of the general
theory of forced vibrations, we will limit the impressed forces to
the battery and the telephone branches. If x, y be the currents
in these branches, X, Y the corresponding electro-motive forces,
we have, § 107, linear relations between x, y, and X, F, which may
be written
X^Ax-^By
7^Bx + Cy
} (3),
the coeflBcient of y in the first equation being identical with that
of a? in the second equation, by the reciprocal property. The three
constants A, J5, C are in general complex quantities, functions o(p.
The reciprocal relation may be interpreted as follows. If
F = 0, 5a:+(7y = 0, and
y^B^^Ac <^>
In like manner, if we had supposed X « 0, we should have
found
BY ,.,
^'^n^ic ^""^^
shewing that the ratio of the current in one member to the electro-
motive force operative in the other is independent of the way in
which the parts are assigned to the two members.
^ For a more complete disonBsion of this rabject see Heaviside '* On Besistanee
and CJonductance Operators,*' PHt Mag., vol. 24, p. 479, 1887; EUctrietd Papm,
Tol. n., p. 856.
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235 j9.] wheatstone's bridge. 451
We have now to determine the constants A, B^ C in terms of
the electrical properties of the system. If y be maintained zero
by a suitable force T, the relation between x and X is X^Ax.
A therefore denotes the (generalized) resistance to any electro-
motive force in the battery member, when the telephone member is
open. This resistance is made up of /, the resistance in the
battery member, and of that of the conductors a'\-c, b + d,
combined in parallel. Thus
^./+(?_+^klh-4) (6).
In like manner
^^ (a + m£±d)
a + 6 + c + a ^ ^
To determine B let us consider the force F which must act
in e in order that the current through it may be zero, in spite
of the operation of X. We have T=Bx. The total current x
flows partly along the branch (a + c), and partly along (6-l-d).
The current through (a-^-c) is
^Ka-^c) ^ (b-^djx
l/(a + c) + l/(6 + d) a-^b + cVd ^ ^'
and that through (b + d) is
(a + c) a?
(9).
a-l-6-i-c + d '
The difference of potentials at the terminals of e, supposed to
be interrupted, is thus
c (6 + d) X — d (a + c) J? ^
oTft+c+d '
and accordingly 5=._J|r^ (10).
By (6), (7), (10) the relationship of Z, Yto x,y is completely
determined.
The problem of the bridge requires the determination of the
current y as proportional to X, when F»0, that is when no
electro-motive force acts in the bridge itself; and the solution is
given at once by the introduction into (4) of the values of A, JB, C
from (6), (7), (10).
If there be an approximate " balance," the expression simplifies.
For (be — ad) is then small, and JS" may be neglected relatively to
29—2
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452 ELECTRICAL VIBRATIONS. [235 p.
AC in the denomiuator of (4). Thus, as a sufficient approximation
in this case, we may write
y,^^A (L^) (11)
X AC (6)x(7) ^ ^'
The following interpretation of the process leads very simply
to the approximate form (11), and is independent of the general
theory. Let us first inquire what electro-motive force is necessary
in the telephone member to stop any current through it. If such
a force act, the conditions are, externally, the same as if the
member were open ; and the current x in the battery member due
to a force equal to X in that member is X/A, where A is written
for brevity as representing the right-hand member of (6). The
difference of potentials at the terminsds of e, still supposed to be
open, is found at once when ic is known. It is given by
cx(8)-dx(9) = fiar,
where B is defined by (10), In terms of X the difference of
potentials is thus BXjA. If e be now closed, the same fraction
expresses the force necessary in « in order to prevent the genera-
tion of a current in that member.
The case with which we have to deal is when X acts in /and
there ia no force in e. We are at liberty, however, to suppose that
two opposite forces, each of magnitude BX/A, act in e. One of
these, as we have seen, acting in conjunction with X in/, gives no
current in c ; so that, since electro-motive forces act independently
of one another, the actual current in 6, closed without internal
electro-motive force, is simply that due to the other component.
The question is thus reduced to the determination of the current
in e due to a given force in that member.
So far the argument is rigorous ; but we will now suppose that
we have to deal with an approximate balance. In this case a force
in e gives rise to very little current in / and in calculating the
current in e, we may suppose /to be broken. The total resistance
to the force in e is then given simply by C of equation (7), and the
approximate value for y is derived by dividing — BX/A by C, as
we found in (11).
A continued application of the foregoing process gives y/X in
the form of an infinite geometric series : —
., . 5» , B' ) _ B
[ "^ AC^ A'C''^ '"I " B'-AC'
X Acy
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235 J9.] APPROXIMATE BALANCE. 453
This is the rigorous solution already found in (4) ; but the first
term of the series suffices for practical purposes.
The form of (11) enables us at once to compare the effects of
increments of resistance and of inductance in disturbing a balance.
For let ad = be, and then change d to d + d', where d' = d/ + idj'.
The value of y/X is proportional to d\ and the amplitude of the
vibratory current in the bridge is proportional to mod. d\ that is,
bo V(di'* + dj'*). Thus di', d,' are equally efficacious when nu-
merically equaP. In most cases where a telephone is employed,
the balance is more sensitive to changes of inductance than to
changes of resistance.
In the use of the Wheatstone balance for purposes of measure-
ment, it is best to make a equal to c. The equality of b and d can
then be tested by interchange of a and c, independently of the
exactitude of the equality of these quantities. Another advantage
lies in the fact that balance is independent of mutual induction
between a and c or between b and d.
236 q. In the formulae of § 235 p it has been assumed that
there is no mutual induction between the various members of the
combination. The more general theory has been considered very
fully by Heaviside*, but to enter upon it would lead us too far.
It may be well, however, to sketch the theory of the arrangement
adopted by Hughes, which possesses certain advantages in dealing
with the electrical properties of wires in short lengths'.
The apparatus consists of a Wheatstone's quadrilateral, Fig. 53 b,
with a telephone in the bridge, one of the
sides of the quadrilateral being the wire
or coil under examination (P), and the
other three being the parts into which a
single German-silver wire is divided by
two sliding contacts. If the battery-
branch (B) be closed, and a suitable in-
terrupter be introduced into the telephone-
branch (T), balance may be obtained by
shifting the contacts. Provided that the
interrupter introduces no electro-motive
^ "On the Bridge Method in its Application to Periodic Electric Currents.'*
Proc. Roy, Soc, vol. 49, p. 203, 1891.
2 '^On the Self-induction of Wires," Part VI.; Phil. Mag,, Feb. 1887; Electrical
Papers, 1892, vol. n., p. 281.
» Jotirn, Tel. Eng., vol. xv. (1886) p. 1 ; Proc. Roy. Soc., vol. xl. (1886) p. 451.
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454 ELECTRICAL VIBRATIONS. [235 (/.
force of its own\ the balance indicates the proportionality of
the four resistances. If P be the unknown resistance of the
conductor under test, Q, R the resistances of the adjacent parts of
the divided wire, 8 that of the opposite part (between the sliding
contacts), then, by the ordinary rule, PS^QR; while Q, iZ, S are
subject to the relation
Q + i2 + S=Tr,
W being a constant. If now the interrupter be transferred from
the telephone to the battery-branch, the balance is usually dis-
turbed on account of induction, and cannot be restored by any
mere shifting of the contacts. In order to compensate the
induction, another influence of the same kind must be intro-
duced. It is here that the peculiarity of the apparatus lies. A
coil (not shewn in the figure) is inserted in the batter}^ and another
in the telephone-branch which act inductively ujx)n one another,
and are so mounted that the effect may be readily varied. The
two coils may be concentric and relatively movable about the
common diameter. In this case the action vanishes when the
planes are perpendicular. If one coil be very much smaller than
the other, the coefficient of mutual induction M is proportional to
the cosine of the angle between the planes. By means of the
two adjustments, the sliding of the contacts and the rotation of the
coil, it is usually possible to obtain a fair silence.
Hughes interpreted his observations on the basis of an as-
sumption that the inductance of P was represented by M, irre-
spective of resistance, and that the resistance to variable currents
could (as in the case of steady currents) be equated to QRjS.
But the matter is not quite so simple. The true formulae are,
however, readily obtained for the case where the only sensible
induction among the sides of the quadrilateral is the inductance L
of the conductor P.
Since there is no current through the bridge, there must be
the same current (x) in P and in one of the adjacent sides (say) iZ,
and for a like reason the same current y in Q and 8, The differ-
ence of potentials at time t between the junction of P and R and
the junction of Q and 8 may be expressed by each of the three
following equated quantities : —
' A condition not always satisfied in practice.
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2355.] hughes' arrangement. 455
Introducing the assumption that all the quantities vary har-
monically with frequency j?/27r, and eliminating the ratio y : a?, we
find Bs the conditions required for silence in the telephone
QR^SP^p^ML (1),
3f(P + Q + ii + S) = Si (2).
It will be seen that the ordinary resistance balance (SP = QR)
is departed from. The change here considered is peculiar to the
apparatus and, so far as its influence is concerned, it does not
indicate a real alteration of resistance in the wire. Moreover,
since p is involved, the disturbance depends upon the rapidity of
vibration, so that in the case of ordinary mixed sounds silence can
be attained only approximately. Again, from the second equation
we see that M is not in general a correct measure of the value
ofi'.
If, however, P be known, the application of (2) presents no
diflBculty. In many cases we may be sure beforehand that P,
viz, the effective resistance of the conductor, or combination of
conductors, to the variable currents, is the same as if the currents
were steady, and then P may be regarded as known. But there
are other cases, — some of them will be alluded to below — ^in
which this assumption cannot be made; and it is impossible to
determine the unknown quantities L and P from (2) alone. We
may then fall back upon (1). By means of the two equations
P and L can always be found in terms of the other quantities.
But among these is included the frequency of vibration ; so that
the method is practically applicable only when the interrupter is
such as to give an absolute periodicity. A scraping contact,
otherwise very convenient, is thus excluded; and this is un-
doubtedly an objection to the method.
If the member P be without inductance, but be interrupted by
a leyden of capacity (7, the same formulae may be employed, with
substitution of — 1/p^C for i. Equation (1) then gives a measure
of C which is independent of the frequency.
236 r. The success of experiments with this kind of apparatus
depends very largely upon the action of the interrupter by which
the currents are rendered variable. When periodicity is not
1 " Diflousdion on Prof. Hughes' Address." Joum. Tel. Eng., vol. xv., p. 64,
Feb., lase.
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456 ELECTRICAL VIBRATIONS. [235 r.
necessary, a scraping contact, actuated by a clock or by a small
motor, answers very well; but it is advisable, following Lodge
and Hughes, so to arrange matters that the current is suspended
altogether at short intervals. The faint scraping sound, heard in
the neighbourhood of a balance, is more certainly identified when
thus rendered intermittent.
But for many of the most interesting experiments a scraping
contact is unsuitable. When the inductance and resistance under
observation are rapidly varying functions of the frequency, it is
evident that no sharp results are possible without an interrupter
giving a perfectly regular electrical vibration. With proper appli-
ances an absolute silence, or at least one disturbed only by a slight
sensation of the octave of the principal tone, can be arrived at
under circumstances where a scraping contact would admit of no
approach to a balance at all.
Tuning-forks, driven electromagnetically with liquid or solid
contacts (§ 64), answer well so long as the frequency required
does not exceed (say) 300 per second ; but for experiments with the
telephone we desire frequencies of from 500 to 2000 per second.
Qood results may be obtained with harmonium reed interrupters,
the vibrating tongue making contact once during each period
with a stop, which can be adjusted exactly to the required position
by means of a screw'.
But perhaps the best interrupter for use with the telephone is
obtained by taking advantage of the instability of a jet of fluid.
If the diameter and the speed be chosen suitably, the jet may be
caused to resolve itself into drops under the action of a tuning-
fork in a perfectly regular manner, one drop corresponding to
each complete vibration of the fork. Each drop, as it passes,
may be made to complete an electric circuit by squeezing itself
between the extremities of two fine platinum wires. If the
electro-motive force of the battery be pretty high, and if the
jet be salted to improve its conductivity, sufficient current passes,
especially if the aid of a small step-down transformer be invoked.
Finally the apparatus is made self-acting by bringing the fork
under the influence of an electro-magnet, itself traversed by the
same intermittent current. Such an apparatus may be made to
work with frequencies up to 2000 per second, and it possesses
many advantages, among which may be mentioned almost absolute
i PhiL Mag., vol. 22, p. 472, 18S6.
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235 r.] INTERRUPTEBS. 457
constancy of pitch, and the avoidance of loud aerial disturbance.
The principles upon which the action of this interrupter depends
will be further considered in a subsequent chapter.
2S6 8, Scarcely less important than the interrupter are the
arrangements for measuring induction, whether mutual induc-
tion, as required in § 235 q, or self-induction. Inductometers, as
Heaviside calls them, may be conveniently constructed upon
the pattern of Hughes. A small coil is mounted so that one
diameter coincides with a diameter of a larger coil, and is
movable about that diameter. The mutual induction M between
the two circuits depends upon the position given to the smaller
coil, which is read by a pointer attached to it, and moving over a
graduated circle. If the smaller coil were supposed to be infinitely
small, the value of M, as has already been stated, would be pro-
portional to the sine of the displacement from the zero position
(Jf = 0). But an approximation to this state of things is not
desirable. If the mean radius of the small coil be increased until
it amounts to '55 of that of the larger, not only is the efficiency
much enhanced, but the scale of M is brought to approximate
coincidence, over almost the whole practical range, with the scale
of degrees*. The absolute value of each degree may be arrived at
in various ways, perhaps most simply by adjusting the mutual
induction of the instrument to balance a standard of mutu^^l
induction.
For experiments upon the plan of § 235 q the one coil is
included in the telephone and the other in the battery branch,
but when the object is to secure a variable and meiwurable
inductance, the two coils are connected in series. The inductance
of the combination is then L + iM-^N, of which the first and
third terms are independent of the relative position of the coils.
236 t Good results by the method of § 235 q have been
obtained by Weber", and by the author* using a reed interrupter
of frequency 1050 per second ; but the fact that inductance and
resistance are mixed up in the measurements is a decided draw-
back, if it be only because the readings require for their interpre-
tation calculations not readily made upon the spot.
1 Phil Mag,, vol. 22, p. 498, 1886.
a EUctHeal Review, April 9, July 9, 1886.
' Phil. Mag., loc, cit.
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458 ELECTRICAL VIBRATIONS. [235 L
The more obvious arrangement is one in which both the
induction and the resistance of the branch cont^ning the subject
under examination are in every case brought up to the given
totals necessary for a balance. To carry this out conveniently we
require to be able to add inductance without altering resistance,
and resistance without altering inductance, and both in a measur-
able degree. The first demand is easily met. If we include in
the circuit the two coils of an inductometer, connected in series,
the inductance of the whole can be varied in a known manner by
rotating the smaller coil. On the other hand the introduction, or
removal, of resistance without alteration of inductance cannot well
be carried out with rigour. But in most cases the object can be
sufficiently attained with the aid of a resistance-slide of thin
German-silver wire which may be in the form of a nearly close
loop.
In the Wheatstone's quadrilateral, as arranged for these ex-
periments, the adjacent sides R, 8 may be made of similar wires
of German silver of equal resistance {\ ohm). If doubled they
give rise to little induction, but the accuracy of the method is
independent of this circumstance. The side P includes the
conductor, or combination of conductors, under examination, an
inductometer, and the resistance-slide. The other side, Q, must
possess resistance and inductance greater than any of the con-
ductors to be compared, but need not be susceptible of ready and
measurable variations. In order to avoid mutual induction be-
tween the branches, P and Q should be placed at some distance
away, being connected with the rest of the apparatus by leads of
doubled wire.
It will be evident that when the interrupter acts in the
battery branch, balance can be obtained at the telephone in the
bridge only under the conditions that both the inductance and
the resistance in P are equal in the aggi'egate to the correspond-
ing quantities in Q. Hence when one conductor is substituted for
another in P, the alterations demanded at the inductometer and
in the slide give respectively the changes of inductance and of
resistance. In this arrangement inductance and resistance are
well separated, so that the results can be interpreted without
calculation; but the movable contacts of the slide appear to
introduce uncertainty into the determination of resistaoice.
In order to get rid of the objectionable movable contacts
some sacrifice of theoretical simplicity seems unavoidable. We
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235^.] SYMMETRICAL ARRANGEMENT. 459
can no longer keep the total resistances P and Q constant ; but by
reverting to the arrangement adopted in a well-known form of
Wheatstone's bridge, we cause the resistances taken from P to be
added to Q, and vice versd. The transferable resistance is that of
a straight wire of German-silver, with which one telephone ter-
minal makes contact at a point whose position is read off on a
divided scale. Any uncertainty in the resistance of this contact
does not influence the measurements.
Fig. 53 c.
B
= @ ^ ,.^ :^^=^o>=
The diagram Fig. (53 c) shows the connection of the parts. One
of the telephone terminals T goes to the junction of the (^ ohm)
resistances R and S, the other to a point upon the divided wire.
The branch P includes one inductometer (with coils connected in
series), the subject of examination, and part of the divided wire.
The branch Q includes a second inductometer (replaceable by a
simple coil possessing suitable inductance), a rheostat, or any
resistance roughly adjustable from time to time, and the re-
mainder of the divided wire. The battery branch B, in which may
also be included the interrupter, has its terminals connected, one
to the junction of P and B, the other to the junction of Q and 8.
When it is desired to use steady currents, the telephone can of
course be replaced by a galvanometer.
In this arrangement, as in the other, balance requires that the
branches P and Q be similar in respect both of inductance and of
resistance. The changes in inductance due to a shift in the
movable contact may usually be disregarded, and thus any alte-
ration in the subject (included in P) is measured by the rotation
necessitated at the inductometer. As for the resistance, it is
evident that (R and 8 being equal) the value for any additional
conductor interposed in P is measured by twice the displacement
of the sliding contact necessary to regain the balance.
Experimental details of the application of this method to the
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460 ELECTRICAL VIBRATIONS. [235 <.
measurement of variouB combinations will be found in the paper*
from which the above sketch is derived. Among these may be
mentioned the verification of Maxwell's formulae, (8), (9) § 235 k,
as to the influence of a neighbouring circuit, especially in the
extreme case where the equivalent inductance is almost destroyed,
and of the formula (10) § 235 vi relating to the behaviour of an
electro-magnet shunted by a relatively high simple resistance.
But the most interesting in many respects is the application to
the phenomena presently to be considered, where the conductors
in question are no longer approximately linear but must be
regarded as solid masses in which the currents are distributed in
a manner that needs to be determined by general electrical
theory.
As has already been remarked more than once, a leyden may
always be supposed to be included in the circuit, the stiffness
thereof having the effect of a negative inductance. If there be no
hysteresis in the action of the leyden, the whole effect is thus
represented ; but when the dielectric employed is solid, it appears
that dissipative loss cannot be avoided. The latter effect manifests
itself as an augmentation of apparent resistance, indistinguishable,
unless the frequency be varied, from the ordinary resistance of the
leads. A similar treatment may be applied to an electrolytic cell,
the stiffness and resistance being presumably both functions of the
frequency.
236 u. It was proved by Maxwell* that a perfectly con-
ducting sheet, forming a closed or an infinite surface, acts as a
magnetic screen, no magnetic actions which may take place on
one side of the sheet producing any magnetic effect on the other
side. " In practice we cannot use a sheet of perfect conductivity ;
but the above described state of things may be approximated to
in the case of periodic magnetic changes, if the time-constants of
the sheet circuits be large in comparison with the periods of the
changes."
"The experiment is made by connecting up into a primary
circuit a battery, a microphone-clock, and a coil of insulated wire.
The secondary circuit includes a parallel coil and a telephone.
Under these circumstances the hissing sound is heard almost as
well as if the telephone were inserted in the primary circuit
1 Phil Mag., loc. cit,
^ Electricity and Magneti$m, 1873, § 656.
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235 t/.] ELECTROMAGNETIC SCREEN. 461
itself. But if a large and stout plate of copper be interposed
between the two coils, the sound is greatly enfeebled. By a proper
choice of battery and of the distance between the coils, it is not
difficult so to adjust the strength that the sound is conspicuous in
the one case and inaudible in the other "\
One of the simplest applications of Maxwell's principle is to
the case of a long cylindrical shell placed within a coaxal magnet-
izing helix. The condition of minimum energy requires that such
currents be developed in the shell as shall neutralize at interna)
points the action of the coil. Thus, if the conductivity of the
shell be sufficiently high, the interior space is screened from the
magnetizing force of periodic currents flowing in the outer helix,
and conducting circuits situated within the shell must be devoid
of induced currents. An obvious deduction is that the currents
induced in a solid conducting core will be more and more confined
to the neighbourhood of the surface as the frequency of electrical
vibration is increased.
The point at which the concentration of current towards the
surface becomes important depends upon the relative values of the
imposed vibration-period and the principal time-constant of the
core circuit. If p be the specific resistance of the material, ^i its
magnetic permeability, a the radius of the cylinder, the expression
for the induction (c) parallel to the axis, during the progress of the
subsidence of free currents in a normal mode, is
c^^U.ikr) (1).
where k^^^^^'^J'- (2).
P
and ka is determined by the condition that
/o(Ara) = (3).
The roots of (3) are, § 206,
2-404, 5-520, 8-654, 11-792, &c.,
so that for the principal mode of greatest persistence
c = e^* Jo (2-404 r/a) (4),
, _ (2-404)V .^^
where ^ = - :i \ W-
Aooastical Observations, PhiL Mag., toI. 13, p. 344, 16S2.
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463 ELECTRICAL VIBRATIONS. [235 W.
For copper in c.G.S. measure p = 1642, fi^l, and thus
T = (-X)->.g^nearly\
In the case of iron we may take as approximate values, /i = 100,
p = 10*. Thus for an iron wire of diameter (2a) equal to '33 cnt,
the value of r is about -^^ of a second, and is therefore comparable
with the periods concerned in telephonic experiments.
Regarded from an analytical point of view the theory of forced
vibrations in a conducting core is equally simple, and was worked
out almost simultaneously by Lamb', Oberbeck' and Heaviside^
In this case we are to regard \ as given, equal (say) to ip, where
pjiir is the frequency. If le^^ be the imposed magnetizing force,
the solution is
"■fM)"'-^ »
the value of k being given by (2).
" When the period in the field is long in comparison with the
time of decay of free currents, we have JaQcr)^!, nearly, so that
c is approximately constant and ^ p,I throughout the section of
the cylinder. But, in the opposite extreme, when the oscillations
in the intensity of the field are rapid in comparison with the decay
of free currents, the induced currents extend only to a small depth
beneath the surface of the cylinder, the inner strata (so to speak)
being almost completely sheltered from electromotive force by the
outer ones. Writing A* = (1 - iyq\ where
we have, when qr is large,
Jo (At) = const. X ,
approximately, and thence
This indicates that the electrical disturbance in the cylinder
1 "On the Daration of Free Electric Currents in an Infinite Gondnetifig
Cylinder," BriU Assoc, Report for 1SS2, p. 446.
9 Proc, Math, Soc, vol. xv., p. 189, Jan. 18S4.
3 Wied, Anti.t vol. zxi., p. 672, Ap. 1884.
* Electrician, May, 1884. Electrical Papers, vol. n., p. 853.
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235 U.} CONDUCTING CORE. 463
consists in a series of waves propagated inwards with rapidly
diminishing amplitude\"
For experimental purposes what we most require to know is
the reaction of the core currents upon the helix, in which alone
we can directly measure electrical eflTects. This problem is fully
treated by Heaviside*, but we must confine ourselves here to a
mere statement of results. These are most conveniently expressed
by the changes of effective inductance L and resistance R due to
the core. If m be the number of turns per unit length in the
magnetizing helix, and if SL, BR be the apparent alterations of L
and R due to the introduction of the core, also reckoned per unit
length, we have
Si = 4»i»7r«a«(/iP-l)) ,..
Si2:=47;i»7r2aV.pQ j ^ ^'
where P and Q are defined by
P-iQ^<l>'l<l> (8),
the function <f> being of the form
<^(a:) = J'o(2tV^) = l+a?+-^i-...+ ^^^f^ ^, + (9),
and the argument x being
ipfx.ira^lp (10).
If the material composing the core be non-conducting, a; » 0, and
therefore
P = l, (2 = 0.
Accordingly SZ:=4mVa»(/i- 1), Si2 = (11).
These values apply also, whatever be the conductivity of the
core, if the firequency be suflBciently low.
At the other extreme, when ^ s= oo , we require the ultimate
form of <^7<^. From the value of J© given in (10) § 200, or other-
wise, it may be shewn that in the limit
f/*«^-* (12),
80 that P«Q« ^ (13).
The introduction of these values into (7) shews that in the
limit, when the frequency is exceedingly high^
Si»-4mVa«, hR^O (14),
^ Lamb, loc, ciu, where is also discussed the problem of the currents induced by
the sudden cessation of a previously constant field.
' loc. cit.
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464 ELECTRICAL VIBRATIONS. [235 U.
as might also have been inferred from the consideration that the
induced currents are then confined to the sur£9kce of the core.
An example of the application of these formulae to an Inter-
mediate case and a comparison with experiment will be found in
the paper already referred to^
236 V. The application of Maxwell's principle to the case of
a wire, in which a longitudinal electric current is induced, is less
obvious; and Heaviside' appears to have been the first to state
distinctly that the current is to be regarded as propagated inwards
from the exterior. The relation between the electromotive force
E and the total current C had, however, been given many years
earlier by Maxwell* in the form of a series. His result is equi-
valent to
/c-W^-'-lltS <•>•
in which R denotes the whole resistance of the length I to
steady currents, /i the permeability, and pjiir the frequency. The
function ^ is that defined by (9) § 235 1^, and ^ is a constant
dependent upon the situation of the return current*.
The most convenient form of the results is that which we have
already several times employed. If we writ«
E^RC-^ipLV (2),
in which R' and Z' are real, these quantities will represent the
effective resistance and inductance of the wire. When the argu-
ment in (1) is small, that is when the frequency is relatively low,
we thus obtain
iJ' = ii{l + ^^^'-^£^+...} (3).
i7^=4-.^|i-A^'];?+BM.^v...} (♦)'•
1 Phil Mag., vol. 22, p. 493, 1886.
^ Electrician, Jan., 1885 ; Electrical Papers, vol. i., p. 440.
» Phil, Trant., 1865 ; Electricity and Magnetism, vol. ii., § 690.
* The simplest case arises when the dielectric, which bounds the cylindrical
wire of radius a, is enclosed within a second conducting mass extending outwards
to infinity and bounded internally at a cylindrical surface r=b. We then have
il = 2 log (6/a). See J. J. Thomson, loc. cit,, § 272.
< Phil. Mag., vol. 21, p. 387, 1886. It is singular that Maxwell {loc. eit.) seems
to have regarded his solution as conveying a correction to the self-induction only of
the wire.
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235 v.] LIMITING FORMS. 465
When p is very small, these equations give, as was to be
expected,
R^R^ i' = Z(^+i/i) (5).
If we include the next terms, we recognise that, in accordance
with the general rule, L begins to diminish and -R' to increase.
When p is very great, we have to make use of the limiting
form of ^7^* As in § 235 u,
4./f=(i+i)V(ii)WJJ) (6);
and thus ultimately
R^^dplfiR) (7),
L'll^A^s/{,jLRmi) (8),
the first of which increases without limit with p, while the second
tends to the finite limit A, corresponding to the total exclusion of
current from the interior of the wire.
Experiments^ upon an iron wire about 18 metres long and 3*3
millimetres in diameter led to the conclusion that the resistance
to variable currents of frequency 1050 was such that RjR = 1*9.
A calculation based upon (1) shewed that this result is in harmony
with theory, if ft = 99 '5. Such is about the value indicated by
other telephonic experiments.
2216 w. The theory of electric currents in such wires as are
commonly employed in laboratory experiments is simple, mainly in
consequence of the subordination of electrostatic capacity. When
this element can be neglected, the current is necessarily the same
at all points along the length of the wire, so that whatever enters
a wire at the sending end leaves it unimpaired at the receiving
end. In this case the whole electrical character of the wire can
be expressed by two quantities, its resistance -R and inductance Z,
and these may usually be treated as constants, independent of the
frequency. The relation of the current to the electromotive force
under such circumstances has already been discussed (7) § 235 j.
When we have occasion to consider only the amplitude of the
current, irrespective of phase, we may regard it as determined
by ^/\R'\'P^L% a quantity which is called by Heaviside the
impedance. Thus in circuits devoid of capacity the impedance is
always increased by the existence of L,
1 PhiL Mag,, vol. 22, p. 488, 1886.
R. 30
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466 ELECTMCAL VIBRATIONS. [235 W.
Circuits employed for practical telephony may often be re-
garded as coming under the above description, especially when
the wires are suspended and are of but moderate length. But
there are other cases in which electrostatic capacity is the domi-
nating feature. The theory of electric cables was established
many years ago by Lord Kelvin^ for telegraphic purposes. If S
be the capacity and R the resistance of the cable, reckoned per
unit length, V and C the potential and the current at the point z,
we have
SdV/dt^-dC/dz, RC^-dVjdz (1).
whence RSdCjdt^d^Gldz^ (2),
the well known equation for the conduction of heat discussed by
Fourier. On the assumption that C is proportional to c***, it
reduces to
d»(7/ck«={V(ii>iiS).(l-fi)}»a (3);
so that the solution for waves propagated in the positive direc-
tion is
(7=Ooe-^<*^^J-'cos{pe-V(ii>i2fif).-^} (4).
The distance in traversing which the current is attenuated in the
ratio of to 1 is thus
z = ^(2lpRS) (5).
A very slight consideration of the magnitudes involved is
sufficient to give an idea of the difficulty of telephoning through a
long cable. If, for example, the frequency (p/Zir) be that of a
note rather more than an octave above middle c, and the cable be
such as are used to cross the Atlantic, we have in c.G.S. measure
Vi> = 60, (RSy^ ^2 xlO'\
and .accordingly from (5)
^ = 3 X 10* cm. = 20 miles approximately.
A distance of 20 miles would thus reduce the intensity of
sound, measured by the square of the amplitude, to about a
tenth, an operation which could not be repeated often without
rendering it inaudible. With such a cable the practical limit
would not be likely to exceed fifty miles, more especially as
the easy intelligibility of speech requires the presence of tones
still higher than is supposed in the above numerical example'.
^ Proc, Roy, Soc.y 1855 ; Mathematical and Physical Papers^ vol. n. p. 61.
3 •( On Telephoning through a Cable." Brit, Ass. Report for 1884, p. 682.
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235 X.] heaviside's theory. 467
236 X, In the above theory the insulation is supposed to be
perfect and the inductance to be negligible. It is probable that
these conditions are sufficiently satisfied in the case of a cable,
but in other telephonic lines the inductance is a feature of great
importance. The problem has been treated with full generality
by Heaviside, but a slight sketch of his investigation is all that
our limits permit.
If jB, S, Z, J2* be the resistance, capacity or permittance, in-
ductance, and leakage- conductance respectively per unit of length,
V and C the potential-difference and current at distance z, the
equations, analogous to (I) § 235 w, are
^r..i-=-f, .c.zf.-|E (,,
Thus, if the currents are harmonic, proportional to e^^*,
^^^{R + ipL){K+ipS)C. (2).
with a similar equation for F.
It might perhaps have been expected that a finite leakage K
would always act as a complication; but Heaviside^ has shewn
that it may be so adjusted as to simplify the matter. This case,
which is remarkable in itself and also serves to throw light upon
the general question, arises when RjL = KjS. We will write
XSt;«=l, RIL^KjS^q (2),
where t; is a velocity of the order of the velocity of light. The
equation for V is then by (1)
i;»d«F/d^ = (d/dt-f?yF (3);
or if we take U so that
V^&-^U (4),
v^d^Uldz^^d^UldV' (5),
the well-known equation of undisturbed wave propagation § 144.
"Thus, if the wave be positive, or travel in the direction of
increasing z, we shall have, if /i {z) be the state of F initially,
Fx = e-Vx(^-vO. ^^1= y^lLv (6).
If Fa, Oj be a negative wave, travelling the other way,
y^-^e'^Mz^vt\ C^^VJLv (7).
^ Electrician, June 17, 1887. Electrical Papers, vol ii. pp. 125, 809.
30—2 —
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468 ELECTRICAX VIBRATIONS. [235 X.
Thus, any initial state being the sum of V^ and F, to make F,
and of Ci and (7, to make C, the decomposition of an arbitrarily
given initial state of V and C into the waves is effected by
F, = i(F+t;iO), V,^\{V^vLC) (8).
We have now merely to move V^ bodily to the right at speed
V, and F, bodily to the left at speed v, and attenuate them to the
extent e~^, to obtain the state at time t later, provided no changes
of condition have occurred. The solution is therefore true for all
future time in an infinitely long circuit. But when the end of a
cii*cuit is reached, a reflected wave usually results, which must be
added on to obtain the real result."
As in § 144, the precise character of the reflection depends
upon the terminal conditions. ''One case is uniquely simple.
Let there be a resistance inserted of amount vL, It introduces
the condition F = vLG if at say B, the positive end of the circuit,
and F:=— vZC if at the negative end, or beginning. These are
the characteristics of a positive and of a negative wave respect-
ively ; it follows that any disturbance arriving at the resistance is
at once absorbed. Thus, if the circuit be given in any state
whatever, without impressed force, it is wholly cleared of electrifi-
cation and current in the time Ijv at the most, if I be the length
of the circuit, by the complete absorption of the two waves into
which the initial state may be decomposed."
" But let the resistance be of amount -Bi at say B ; and let Fj
and Fj be corresponding elements in the incident and reflected
waves. Since we have
V.^vLC,, V, — vLC^, F,-f F, = iJ,(C', + C,)...(9),
we have the reflected wave given by
V, R,^vL
V, R,^vL'
.(10).
If Ri be greater than the critical resistance of complete ab-
sorption, the current is negatived by reflection, whilst the electri-
fication does not change sign. If it be less, the electrification is
negatived, whilst the current does not reverse."
"Two cases are specially notable. They are those in which
there is no absorption of energy. If iJi = 0, meaning a short
circuit, the reflected wave of F is a perverted and inverted copy of
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235 a:.] heaviside's theory. 469
the incident. But if ii = oo , representing insulation, it is G that
is inverted and perverted \"
The cases last mentioned are evidently analogous to the reflec-
tion of a sonorous aerial wave travelling in a pipe. If the end of
the pipe be closed, the reflection is of one character, and if it be
open of another character. In both cases the whole energy is
reflected, § 257. The waves reflected at the two ends of an electric
circuit complicate the general solution, especially when the sim-
plifying condition (2) does not hold. But in many cases of
practical interest they may be omitted without much loss of
accuracy. One passage over a long line usually introduces con-
siderable attenuation, and then the effect of the reflected wave,
which must traverse the line three times in all, becomes insigni-
flcant.
In proceeding to the general solution of (2) for a positive
wave, we will introduce, after Heaviside, the following abbrevia-
tions,
»'iS=l, RILp=f, KjSp^g (11).
In terms of these quantities (2) may be written
d^C/dz'^(P + iQyC (12),
where
^ or (? = i {p/vY {(1 +/»)* (1 + 5^0* i (f9 - 1)1 . . . (13).
Thus, if P and Q be taken positively, the solution for a wave
travelling in the positive direction is
(7= Coe'^'cos{pt - Qz) (14),
the current at the origin being Cq cos pt
The cable formula, § 235 w, is the particular case arrived at by
supposing in (13)/= go , ^r = 0, which then reduces to
i>a = (2» = jpiJ£f (15).
Again, the special case of equation (3) is derivable by putting
f=g — qjP' The result is
P = ?/v, Q^pIv (16).
If the insulation be perfect, ^ = 0, and (13) becomes
P« or (?-i(;,/t;)»{(l +/«)»?!} (17).
^ Heaviside, Collected Worki^ vol. n. p. 312.
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470 ELECTRICAL VIBRATIONS. [235 X.
In certain examples of long copper lines of high conductivity,
/ may be regarded as small so far as telephonic frequencies are
concerned. Equation (17) then gives
P=pfl2v^R/2vL, Q=pIv (18).
For a further discussion of the various cases that may arise
the reader must be referred to the writings of Heaviside already
cited. The object is to secure, as far as may be, the propagation
of waves without alteration of type. And here it is desirable to
distinguish between simple attenuation and distortion. If, as in
(16) and (18), P is independent of p, the amplitudes of all com-
ponents are reduced in the same ratio, and thus a complex wave
travels without distortion. The cable formula (15) is an example
of the opposite state of things, where waves of high frequency are
attenuated out of proportion to waves of low frequency. It appears
from Heaviside's calculations that the distortion is lessened by
even a moderate inductance.
The eflFectiveness of the line requires that neither the attenua-
tion nor the distortion exceed certain limits, which however it is
hard to lay down precisely. A considerable amount of distortion
is consistent with the intelligibility of speech, much that is
imperfectly rendered being supplied by the imagination of the
hearer.
236 y. It remains to consider the transmitting and receiving
appliances. In the early days of telephony, as rendered practical
by Graham Bell, similar instruments were employed for both
purposes. Bell's telephone consists of a bar magnet, or battery
of bar magnets, provided at one end with a short pole-piece
which serves as the core of a coil of fine insulated wire. In close
proximity to the outer end of the pole-piece is placed a circular
disc of thin iron, held at the circumference. Under the influence
of the permanent magnet the disc is magnetized radially, the
polarity at the centre being of course opposite to that of the
neighbouring end of the steel magnet.
The operation of the instrument as a transmitter is readily
traced. When sonorous waves impinge upon the disc, it responds
with a symmetrical transverse vibration by which its distance
from the pole-piece is alternately increased and diminished.
When the interval is diminished, more induction passes through
the pole-piece, and a corresponding electro-motive force acts in
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235 yO bell's telephone. 471
the enveloping coil. The periodic movement of the disc thus
gives rise to a periodic current in any circuit connected with the
telephone coil.
The electro-motive force is in the first instance proportional
to the permanent magnetism to which it is due; and this law
would continue to hold, were the behaviour of the pole-piece and
of the disc conformable to that of the " soft iron " of approximate
theory. But as the magnetism rises, and the state of saturation
is more nearly approached, the response to periodic changes of
force becomes feebler, and thus the eflSciency falls below that
indicated by the law of proportionality. If we could imagine the
state of saturation in the pole-piece to be actually attained, the
induction through the coil would become almost incapable of
variation, being reduced to such as might occur were the iron
removed. There is thus a point, dependent upon the properties
of magnetic matter, beyond which it is pernicious to raise the
amount of the permanent magnetism ; and this point marks the
maximum efficiency of the transmitter. It is probable that the
most favourable condition is not fully reached in instruments
provided with steel magnets; but the considerations above
advanced may serve to explain why an electro-magnet is not
substituted.
The action of the receiving instrument may be explained on
the same principles. The periodic current in the coil alternately
opposes and cooperates with the permanent magnet, and thus the
iron disc is subjected to a periodic force acting at its centre.
The vibrations are thence communicated to the air, and so reach
the ear of the observer. As in the case of the transmitter, the
efficiency attains a maximum when the magnetism of the pole-
piece is still far short of saturation.
The explanation of the receiver in terms of magnetic forces
pulling at the disc is sometimes regarded as inadequate or even as
altogether wide of the mark, the sound being attributed to " mole-
cular disturbances " in the pole-piece and disc. There is indeed
every reason to suppose that molecular movements accompany
the change of magnetic state, but the question is how do these
movements influence the ear. It would appear that they can do
so only by causing a transverse motion of the surface of the disc,
a motion from which nodal subdivisions are not excluded.
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472 ELECTRICAL VIBRATIONS. [235 y.
In support of the '' push and pull theory " it may be useful to
cite an experiment tried upon a bipolar telephone. In this
instrument each end of a horse-shoe magnet is provided with a
pole-piece and coil, and the two pole-pieces are brought into
proximity with the disc at places symmetrically situated with
regard to the centre. In the normal use of the instrument the
two coils are permanently connected as in an ordinary horse-shoe
electro-magnet, but for the purposes of the experiment provision
was made whereby one of the coils could be reversed at pleasure
by means of a reversing key. The sensitiveness of the telephone
in the two conditions was tested by including it in the circuit of
a Daniell cell and a scraping contact apparatus, resistance from a
box being added until the sound was but just easily audible.
The resistances employed were such as to dominate the self-
induction of the circuit, and the comparison shewed that the
reversal of the coil from its normal connection lowered the sensi-
tiveness to current in the ratio of 11 : 1. That the reduction was
not still greater is readily explained by outstanding failures of
symmetry; but on the "molecular disturbance" theory it is not
evident why there should be any reduction at all.
Dissatisfaction with the ordinary theory of the action of a
receiving telephone may have arisen from the difficulty of under-
standing how such very minute motions of the plate could be
audible. This is, however, a question of the sensitiveness of the
eai*, which has been proved capable of appreciating an amplitude
of less than 8 x lO^^cra.^ The subject of the audible minimum
will be further considered in the second volume of this work.
The calculation a priori of the minimum current that should
be audible in the telephone is a matter of considerable difficulty ;
and even the determination by direct experiment has led to
widely discrepant numbers. In some recent experiments by the
author a unipolar Bell telephone of 70 ohms resistance was
employed. The circuit included also a resistance box and an
induction coil of known construction, in which acted an electro-
motive force capable of calculation. Up to a frequency of 307
this could be obtained from a revolving magnet of known moment
and situated at a measured distance from the induction coil. For
the higher frequencies magnetized tuning-forks, vibrating with
measured amplitudes, were substituted. In either case the
1 Proc, Roy, Soc. vol. xxvi. p. 248, 1877.
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235 y.]
MINIMUM CUKRENT.
473
resistance of the circuit was increased until the residual sound
was but just easily audible. Care having been taken so to
arrange matters that the self-induction of the circuit was negli-
gible, the current could then be deduced from the resistance and
the calculated electro-motive force operating in the induction
coil. The following are the results, in which it is to be under-
stood that the currents recorded might have been halved without
the sounds being altogether lost :
Pitch
Source
Current in
10-« amperes
128
192
256
307
320
384
512
640
768
Fork
Revolving Magnet
Fork
Revolving Magnet
Fork
2800
250
83
49
32
15
7
4-4
10
The effect of a given current depends, of course, upon the
manner in which the telephone is wound. If the same space be
occupied by the copper in the various cases, the current capable of
producing a particular effect is inversely as the square root of the
resistance.
The numbers in the above table giving the results of the
author's experiments are of the same order of magnitude as
those found by Ferraris^ whose observations, however, related
to sounds that were not pure tones. But much lower estimates
have been put forward. Thus Tait' gives 2 x 10"" amperes,
and Preece a still lower figure, 6 x 10""". These discrepancies,
enormous as they stand, would be still further increased were
the comparison made to refer to the amounts of energy absorbed.
According to the calculations of the author the above tabulated
sensitiveness to a periodic current of frequency 256 is about what
might reasonably be expected on the push and pull theory* At
1 Atti delta Aeead. d. Scu Di Torino, vol. xiii. p. 1024, 1877.
2 Edin, Proc. vol. ix. p. 561, 1878.
' I propose shortly to publish these calculations.
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474 ELECTRICAL VIBRATIONS. [235 y.
this frequency, which is below those proper to the telephone plate
(§ 221 a), the motion of the plate is governed by elasticity rather
than by inertia, and an equilibrium theory (§ 100) is applicable as
a rough approximation. The greater sensitiveness of the telephone
at frequencies in the neighbourhood of 512 would appear to
depend upon resonance (§ 46). It is doubtful whether the much
higher sensitiveness claimed by Tait and Preece could be re-
conciled with theory.
It appears to be established that the iron plate of a telephone
may be replaced by one of copper, or even of non-conducting
material, without absolute loss of sound; but these effects are
probably of a different order of magnitude. In the case of copper
induced currents may confer the necessary magnetic properties.
For a description of the ingenious receiver invented by Ekiison
and for other information upon telephonic appliances the reader
may consult Preece and Stubbs' Manual of Telephony.
In existing practice the transmitting instrument depends
upon a variable contact. The first carbon transmitter was con-
structed by Edison in 1877, but the instruments now in use are
modifications of Hughes' microphone \ A battery current is led
into the line through pieces of metal or of carbon in loose juxta-
position, carbon being almost universally employed in practice.
Under the influence of sonorous vibration the electrical resistance
of the contacts varies, and thus the current in the line is rendered
representative of the sound to be reproduced at the receiving
end.
That the resistance of the contact should vary with the
pressure is not surprising. If two clean convex pieces of metal
ai*e forced together, the conductivity between them is represented
by the diameter of the circle of contact (§306). The* relation
between the circle of contact and the pressure with which the
masses are forced together has been investigated in detail by
Hertz \ His conclusion for the case of two equal spheres is that
the cube of the radius of the circle of contact is proportional to
the pressure and to the radii of the spheres. But it has not yet
been shewn that the action of the microphone can be adequately
explained upon this principle.
^ Proc. Roy, Soc., vol. xxvii. p. 862, 1878.
» Crelle, Jo^im. Math. xcn. p. 166, 1882.
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APPENDIX.
ON PROGRESSIVE WAVES.
From the Proceedings of the London Mathematical Society^
Vol. IX., p. 21, 1877.
It has often been remarked that, when a group of waves advances
into still water, the velocity of the group is less than that of the indi-
vidual waves of which it is composed ; the waves appear to advance
through the group, dying away as they approach its anterior limit.
This phenomenon was, I believe, first explained by Stokes, who re-
garded the group as formed by the superposition of two infinite trains
of waves, of equal amplitudes and of nearly equal wave-lengths, ad-
vancing in the same direction. My attention was called to the subject
about two years since by Mr Froude, and the same explanation then
occurred to me independently*. In my book on the "Theory of
Sound" (§191), I have considered the question more generally, and
have shewn that, if V be the velocity of propagation of any kind of
waves whose wave-length is X, and k = 2ir/X., then U, the velocity of
a group composed of a great number of waves, an3 moving into an un-
disturbed part of the medium, is expressed by
-=T' <».
* Another phenomenon, ftlso mentioned to me by Mr Froude, admits of a similar
explanation. A steam-launch moving quickly through the water is accompanied by
a peculiar system of diverging waves, of which the most striking feature is the
obliquity of the line containing the greatest elevations of successive waves to the
wave-fronts. This wave pattern may be explained by the superposUion of two (or
more) infinite trains of waves, of slightly differing wave-lengths, whose directions
and velocities of propagation are so related in each case that there is no change of
position relatively to the boat. The mode of composition will be best understood by
drawing on paper two sets of parallel and equidistant lines, subject to the above
condition, to represent the crests of the component trains. In the case of two trains
of slightly different wave-lengths, it may be proved that the tangent of the angle
between the line of maxima and the wave-fronts is half the tangent of the angle
between the wave-fronts and the boat's course.
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476 APPENDIX,
or, as we may also write it,
"■■^-'^z^. <^^
Thus, if FocX«, Cr=(l-w) V (3).
In fact, if the two infinite trains be represented by cos k(Vt^x)
and cos k' ( V't — a:), their resultant is represented by
cos A; ( F« - a;) + cos k' ( V't - x\
which is equal to
(k'V'-kV ^ k'--k 1 (k'V'-^kV k' + k \
2cos| ^_^-_-a:| .cosj— 2— «- 2 4-
If k' -ky V - r be small, we have a train of waves whose amplitade
varies slowly from one point to another between the limits and 2,
forming a series of groups separated from one another by regions com-
paratively free from disturbance. The position at time t of the middle
of that group, which was initially at the origin, is given by
{k'V'^kV)t-'{k'-k)x=0,
which shews that the velocity of the group is {k' V - kV) -r {k* — k).
In the limit, when the number of waves in each group is indefinitely
great) this result coincides with (1).
The following particular cases are worth notice, and are here tabu-
lated for convenience of comparison : —
FocX,
u=o,
Reynolds' disconnected pendulums.
r«x*,
u=ir,
Deep-water gravity waves.
r«\»,
u^r,
Aerial waves, Ac.
r«\-»,
u=ir,
Capillary water waves.
V<c\-\
U=2V,
Flexural waves.
The capillary water waves are those whose wave-length is so small
that the force of restitution due to capillarity largely exceeds that due
to gravity. Their theory has been given by Thomson (PhiL Mag.,
Nov. 1871). The flexural waves, for which U=2V, are those cor-
responding to the bending of an elastic rod or plate ("Theory of
Sound," § 191).
In a paper read at the Plymouth meeting of the British Association
(afterwards printed in "Nature," Aug. 23, 1877), Prof. Osborne
Reynolds gave a dynamical explanation of the fact that a group of
deep-water waves advances with only half the rapidity of the indi-
vidual waves. It appears that the energy propagated across any point,
when a train of waves is passing, is only one-half of the energy neoes-
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PROGRESSIVE WAVES. 477
sary to supply the waves which pass in the same time, so that, if the
train of waves be limited, it is impossible that its front can be propa-
gated with the full velocity of the waves, because this would imply the
acquisition of more energy than can in fact be supplied. Prof. Reynolds
did not contemplate the cases where more energy is propagated than
corresponds to the waves passing in the same time ; but his argument,
applied conversely to the results already given, shews that such cases
must exist. The ratio of the energy propagated to that of the passing
waves \a U : V', thus the energy pi*opagated in the unit time \a U : V
of that existing in a length F, or U times that existing in the unit
length. Accordingly
Energy propagated in unit time : Energy contained (on an average)
in unit length =d{kV) : dk^ by (1).
As an example, I will take the case of small ir rotational waves in
water of finite depth ^. If z be measured downwards from the surface,
and the elevation (/*-) of the wave be denoted by
h = H cos {nt — kx) (4),
in which n = kVj the corresponding velocity-potential (^) is
* = - VH'^-^^^Bmint-hx) (5).
This value of <^ satisfies the general differential equation for irrota-
tional motion (v*^ = 0), makes the vertical velocity dif>/dz zero when
« =: Z, and - dh/dt when « = 0. The velocity of propagation is given by
^-i^ («)•
We may now calculate the energy contained in a length x, which is
supposed to include so great a number of waves that fractional parts
may be left out of account.
For the potential energy we have
Vi^gpjj zdzdx=zyp jh^dx^lgpH^x (7).
For the kinetic energy,
■l''/(*ff)„*'-l'"^-' («)■
by (1) and (6). If, in accordance with the argument advanced at the
* Prof. Be^rnolds considers the troohoidal wave of Bankine and Froude, which
inyolves molecular rotation.
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478 APPElffDIX.
«nd of this paper, the equality of Vi and T be assumed, the value of
the velocity of propagation follows from the present expressions. Hie
whole energy in the waves occupying a length x is therefore (for each
unit of breadth) V^ -^ T = \gpH^ , x (9),
H denoting the maximum elevation.
We have next to calculate the energy propagated in time t across a
plane for which x is constant, or, in other words, the work ( W) that
must be done in order to sustain the motion of the plane (considered
as a flexible lamina) in the face of the fluid pressures acting upon the
front of it. The variable part of the pressure (Sp\ at depth Zy is
given by
«p = - p -^ = - nVH — ^ETT^HM — cos (nt - kx),
while for the horizontal velocity
^ = ^^^ ^,g-fa ooB{nt^kx);
so that W=jj^^(hdt = \gpHKVt.h+^i^^'j (10),
on integration. From the value of V in (6) it may be proved that
and it is thus verified that the value of W for a unit time
= — W— X energy m unit length.
As an example of the direct calculation of U, we may take the case
of waves moving under the joint influence of gravity and cohesion.
It is proved by Thomson that
P = f-i-rA; (11),
where T' is the cohesive tension. Hence
"When k is small, the surface tension is negligible, and then U^^^V;
but when, on the contrary, k is large, U=^V, as has already he&a.
stated. When TJ^ — g, U=^V. This corresponds to the miniwyj^m
velocity of propagation investigated by Thomson.
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PROGRESSIVE WAVES. 479
Although the argument from interference groups seems satisfactory,
an independent investigation is desirable of the relation between
energy existing and energy propagated. For some time I was at a loss
for a method applicable to all kinds of waves, not seeing in particular
why the comparison of energies should introduce the consideration of
a variation of wave-length. The following investigation, in which the
increment of wave-length is imaginary^ may perhaps be considered to
meet the want : —
Let us suppose that the motion of every part of the medium is
resisted by a force of very small magnitude proportional to the mass
and to the velocity of the part, the effect of which will be that waves
generated at the origin gradually die away as x increases. The motion,
which in the absence of friction would be represented by cos {nt — kx),
under the influence of friction is represented by e"*** cos {nt — kx),
where ^ is a small positive coefficient. In strictness the value of k is
also altered by the friction; but the alteration is of the second order as
regards the frictional forces, and may be omitted under the circum-
stances here supposed. The energy of the waves per unit length at
any stage of degradation is proportional to the square of the amplitude,
and thus the whole energy on the positive side of the origin is to the
energy of so much of the waves at their greatest value, i.e., at the
origin, as would be contained in the unit of length, as j'^ e-^" dx : 1,
or as (2/a)"^ : 1. The energy transmitted through the origin in the
unit time is the same as the energy dissipated ; and, if the frictional
force acting on the element of mass m be hinVy where v is the velocity
of the element and h is constant, the energy dissipated in unit time is
Ji^mr^ or 2hTy T being the kinetic energy. Thus, on the assumption
that the kinetic energy is half the whole energy, we find that the
energy transmitted in the unit time is to the greatest energy existing
in the unit length as h : 2^. It remains to find the connection be-
tween h and /*.
For this purpose it will be convenient to regard cos {nt - kx) as the
real part of e*"* 6***, and to inquire how k is affected, when n is given,
by the introduction of friction. Now the effect of friction is repre-
sented in the differential equations of motion by the substitution of
(Pjd^ + hdjdt in place of d^/d^, or, since the whole motion is proportional
to e*^, by substituting - w* + ihn for — w*. Hence the introduction of
friction corresponds to an alteration of n from n to n — ^ih (the square
of h being neglected) ; and accordingly k is altered from k to
k-^ihdk/dn. The solution thus becomes e"******/^** e' <"*"**>, or, when
the imaginary part is rejected, e-4*«<»/<*» cos {nt — kx); so that
/A = ^ A dkjdn^ and A : 2/t = dn/dk. The ratio of the energy transmitted
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480 APPENDIX.
in the unit time to the energy existing in the unit length is therefore
expressed by dnjdk or d(kV)jdk, as was to be proved
It has often been noticed, in particular cases of progressive waves,
that the potential and kinetic energies are equal ; but I do not call to
mind any general treatment of the question. The theorem is not
usually true for the individual parts of the medium*, but must be
understood to refer either to an integral nmnber of wave-lengths, or to
a space so considerable that the outstanding fractional parts of waves
may be left out of account. As an example well adapted to give in-
sight into the question, I will take the case of a uniform stretched
circular membrane ("Theory of Sound," § 200) vibrating with a given
number of nodal circles and diameters. The fundamental modes are
not quite determinate in consequence of the symmetry, for any dia-
meter may be made nodal. In order to get rid of this indeterminate-
ness, we may suppose the membrane to carry a small load attached to
it anywhere except on a nodal circle. There are then two definite
fundamental modes, in one of which the load lies on a nodal diameter,
thus producing no effect, and in the other midway between nodal dia-
meters, where it produces a maximum effect ("Theory of Sound,"
§ 208). If vibrations of both modes are going on simultaneously, the
potential and kinetic energies of the whole motion may be calculated
by simple addition of those of the components. Let us now, supposing
the load to diminish without limit, imagine that the vibrations are of
equal amplitude and differ in phase by a quarter of a period. The
result is a progressive wave, whose potential and kinetic energies are
the sums of those of the stationary waves of which it is composed.
For the first component we have Vi = B cos' nt, Ti = E sin* rU ; and
for the second component, V^^ E sin' n<, T^= E cos' rU ; so that
]\ + r, = 7\ + 7*2 = ^, or the potential and kinetic energies of the
progressive wave are equal, being the same as the whole energy of
either of the components. The method of proof here employed appears
to be sufficiently general, though it is rather difficult to express it in
language which is appropriate to all kinds of waves.
* Atrial waves are an important exoeption.
END OF VOL. I.
GAMBBXDOB :^ PRINTBD BT C. J. CLAY, lf.A. ANIi SONS, AT THB UNIVSBSITY PBB88.
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