lord rayleigh - the theory of sound vol 1
DESCRIPTION
includes chapters on the vibrationsof systems in general, followed by a more detailed consideration of special systems,such as stretched strings, bars, membranes, and plates.TRANSCRIPT
The theory of sound / byJohn William Strutt,baron Rayleigh,...
Source gallica.bnf.fr / Ecole Polytechnique
Rayleigh, John William Strutt (1842-1919 ; 3rd baron). The theory of sound / by John William Strutt, baron Rayleigh,.... 1877-1878.
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THE
THEORY OF SOUND.
MARS~a~
THEORY OF SOUND.
JOHN WILLIAM STRUTT, BARON RAYLEIOH, M.A..F.R.8.
FORMEKÏ.Y FELMW OF TRINITV COLLEGE, CAMH)UnOE.
MACMILLAN AND 00.
1877
T H F
&onlton: +
[~~ ~t~~ r~trt'ed.] J
HY
VOLUME 1.
OEambtitgc:
r)t)M):Y'J'
ATT)tKUS)VRM)TV'')t'<H~.
PREFACE.
IN thé work, of winch théprésent volume is an instal-
lent, my endeavour bas been to lay before the readera connected
exposition of thetheory of sound, which
should include ~he moreimportant of thé adv~nces made
in modem times ùy Matliematicians andPhysicists.
Theimportance of the object winch 1 have had in view
will not, I tinnk, bedisputed Ly those competent to
judge. At thé présent timemany of thé most valuable
contributions to science are to be found onjy in scattered
penodicala and traiMactions of socletles, pubMied invanous parts of thé worid and in several
languies, andarc often
practically inaccessible to those vvho do not
happen to hvc in théneighbourhood of
large publichbraues. In sucli a state of
things the mechanical
impedimonts tostudy entail an amount of unremunera-
tive labour andconséquent hindrance to thé advanco-
ment of science which it would be dimcult to over-estimate.
Since the wcH-known Article on Sound in the .E~-c'~)~« ~)~M~(~ by Sir John Herschel
(1845),no
complète work lias beenpublislied in wilich tlio
suhject is trcatcdjnatl-cmatically. Ly thé promature
death of Prof. Donkin tlie scientific worid was deprivedof onc w!)oso mathcmatical attainments in combmation
with a{n-ctic.-d kr~Judge of mua:c
quahned hifn in a
VI PREFACE.
special manner to write on Sound. The first part of hisAcoustics
(1870), though little jnm~ tbsm a fragment. issumcient to shew that
my labours would l~ave been un-
necessary had Prof Donkin lived to complète his work.In tlie choice of
topics to be dealt with in a. workon Sound, 1 have for thé most part foUoAved thé
exempleof
my predecessors. To a great extent thé theory of
Sound, ascommonly understood, covers thé same ground
as thétheory of Vibrations in général but, unless some
limitation were admitted, thé consideration of such sub-
jects as the Tides, not to speak of Optics, would have tobe included. As a général ruie we shall confine ourselvesto those classes of vibrations for which our ears afford a
ready made andwonderfully sensitive instrument of in-
vestigation. Without ears we should hardiy care muchmore about vibrations than without eyes we should careabout light.
The present volume includes chapters on thé vibra-tions of systems in
gênerai, in which, 1 hope, will be
recognised somenovelty of treatment and results, fol.
lowed by a more detailed consideration of special systems,such as stretclied
strings, bars, membranes, and plates.The second volume, of which a considérable portion is
already written, will commence with acrial vibrations.
My best thanks are due to Mr H. M.Taylor of
Trinity Collège, Cambridge, who bas been good enoughto read thé proofs. By his kind assistance several errorsand obscurities have been eliminated, and thé volume
generally has been rendered lessimperfect than it would
otherwise have been.
Any corrections, orsuggestions for improvements, witli
whichmy rcaders
may faveur me will be highly apprc-ciated.
TEttUNO PLACE, WmiAi.,
.~n7, 1877.
CONTENTS.
C'HAPTER I.
MOE
§§1-27 i
Sound duo to Vibrations. Finite volocity of Propagatiou. Yelooity indo-
pondent of Pitcb. Depmult'a oxporimonts. Sound propagated in water.
'Witoatatono'a experiment. Enfoeblomont of Sound by distance. Notes
and Noisos. Musical motea duo to poriodio vibrations. Siren of Cagniarddo la Tour. Pitch dopendont upon Poriod. Eelationahip between
musical notes. Tho samo ratio of perioda corresponds to the samo
intorval in all parts of tho scale. Harmonie sca-tes. Diatonio soaloa.
Absolute Pitoh. Neoossity of Temperament. Equal Tomperament.Table of FroqnonoioB. Analyais of Notes. Notes and Tones. Quality
depandont upon harmonie overtonos. Resolution of Notes by efu; un.
certain. Simple tcnoa correspond to aimpla pondidona vibra.tiona.
CHAPTER II.
§§28-4.2 18
Composition of harmonio motions of like period. Harmonie Curvo. Com.
position of two vibrations of nearly equal period. Béats. Fourier'a
Theorem. Vibrations in porpendionlar directions. Lissajous' Cylindor.
Lissajous' Figures, Bin.ckburu's poudulum. Kaieidophone. Opticalmethods of composition and analysis. Thé vibration microscope. In.
termittont Illuminntion.
viil CONTENTS.
§§~-c.s
Systumswithon~d~rouoffrt~dom. Indopcndottcu of ampiitudu~nd
pfriod. Frictiomdfurccpruportimudtovotouit.y. l''))t'cnd\i)))'ati<n)s.
)'riuci)do~fSnpurpositiun. J;L'at.S(]uoto H)tpLTp()sitionoff.,rcud~nd
fn~vibrutions.Yan<'UHd(~n.c.sofdmnpin(;Stri))Kwit)tL<Hn). M~
thodofDimcnsionH. Id<~dTtminn-furk. I''('r)iSKivon(.-ar)yp)tr(iio])()M..f)!-kn<~ standards ofpiich. Scbcibicr'snjutb~dsoftuni; Suh~ib-
Itir'sTonomutM's. Cofnpnnndi'cndtdmn. l''()rhsdnvuubycluctru.
t)mK"ft"u. !or]tIntt!rn)ptM-. Itcsonfmuo. Guuurftimtlutionforohc
deHt~ooffrueJott). TM-mHofthotiMondur~orj.;ivcrisetctterivod
toucu.
§§~ (,7
(icnorniixedco-ortiitmtcs. Expression for jMjLontudcncrKy. Stftticaithco-
rMua. Itiitml motions.Hxprcsnionforkinet!<:(!)~ur~y. Hcciprocat
thoornm. ThM))Ho))'sthcorc:r)i. L~'n))HO't!('qutttionn. ThodiH.si~ftti~nfunetion. Couxi.stt'ncoofHnMtUinotioxH.
l''t'cc\-i).mtion.swt.])out.fri<
tion. Nnrmnico-nrdinntos. Thch'eoponodHfuXtt~Htttt.ioj~u'y condi-
tion.AnncccHsiottofinertijtmcrottsc.sthefrucpoiods. ArctttKfttif'~
('fHprmf;i))()rL'!Lsc~ti~fMû)"i[)dH. Tim(;<tt''stfroopcrindi.s n)tttbtiotntu maximum. Ifypcthut.n'fd types nfvntmtinn. Hx)unj))ofr<))untrinn. Approximntoty simpto Hy~touM. StrioK of vn!-itth)o dcnsity.Normal fnnctions. Conju~to propnrty. ]')ctcrnn)Mtion of cf.nstnotM tu
nuit M'hitrary initiât conditions. Stotios' thcurom.
§§~17 <~
CascH in wluch tho thrce funetiotM Y', r aroHunu[t!U)oous)y rcjucibio tu
nmus of fiquarLi.s. Ut;noM)i.tion of Y'uun~'H thcorcut ou tho undft)
pointu of Htrin~H. H(iui)ibriu)u titeoi-y. SyatomH tit.n'tcd fruin rest as
denectcd by n força appiied at ono point. 8yntbtUH Ht~rtud froui tho
equinbrium confjgumtiun by an impniHu applicd nt ono point. Syatcmastartod from rcst na dcftocted by n. force
uuifonuiy distributed. JuUu-oico of sw< frictiona) forcoN on titû vibr~tious of ft «yntom. Solution oft)m ancrai oqn~tioDH for freo vibMtiotM. Impres.~d Forces, rrinciptc"f tho porsititonco of poriods. Inoxontbto motions. MceiprocttI TJ)uo.
ru)!]. Applicution to freo vibmtiouB. Stutemcnt of Mciprooa) t)tcnrcmfor itttnnonic forcos. AppHcationH. Extension to cases in wbich thuconstitution ofthu systcm isn.functionuftLûpuriud. Equations for
two dcgrcus of fruedom. HoutH of dctornnmmttti cquittiun. lutct-niit-
tcntvibmtiomj. Marchofporiods ttHim.rtin.is~mdu~Hy incro~od.
Heaction of n dcpunduut f.vHtun).
<HAPTER IH.
TA<!t:t.
C'frAPTER IV.
CHAPTER V.
CONTENTS. Ix
§§118–14S. 127
Lnw of extension of n, titritig. Transvurso vihrntiona. Solution of tho pro.
bJcm for n string whoso masH is concontrutcd in cquidistant points. Dé-
rivation of no)ution for continuouo string. Pfu'tittI diffcrentitti équation.
Hxpt't.'snious for nnd y. Hoat gonorat form of simpio harmonie mo-
tion. StritiRS with ËxoJ extronitios. GcnoDt) motion of n stnng pcri-
odio. Mcrsonuc'H L)tws. Sonomotur. Kurtnn.1 ntotlua of vibration.
Dctunninat.ion of eonattuits to suit nrbitt'M'y initiât ciroumfitttncus. Ca.Mo
of pluoknd HtrinR. Expression.-) for ï' and y in torms of noriun] co-ordi-
n<tte!i. Nonntd cqutttionH of motion. Strin(; oxcitod by plucMn~.
YoHug'a thoorou. Htiing cxcitud l)y tm impulso. rroblem of pifuio-
forto strinH. Friction j~'oport.ionni tu vulooity. CoBtptn'ison with oqui-
Iibritttn tLeory. l'oriodic force uppliud nt onu point. Modificationa duu
to yiuldiu~ of tho cxtremit.iea. Proof uf Fouriur't) thcorcm. EGuets
of a nnitu loud. Correction for rigidity. ProUûm of violin strin~.
Striuss stretchod on curved (iurfaeca. Solution for tho caso of tho
iiphoro. Correction fur irrcguJaritioa of donsity. TheoMinH of Sturm
and LiouYiDo for n strin(~ of vtu'inbio donsity. rropng~ioa of Wt).vcs
tdonj~ an uniimitod Htring. Positivo fmd nc~~tivo wavos. Stn.tiona.ry
Vibrations. Hcnootion at )L uxod point. Déduction of solution for
tinito strit)R. Grn.phiod mothod. Progressivowawwitli friction.
§§ 149–1~ 188
Ciftsxincttt.ion of tho vibrations of Bars. DiuM'cntin! équation for longitu.
dimd vibt'jttiottH. Numorieal valuos of tho constunta for steûl. Solu-
tion for a Lar frco at both uodH. Déduction of tiûlution fur a Lar with
ona end fret), nnd onc lixod. l!oth ends iixcd. InUuoncu of Mniull Inad.
Solution of problom for Ltu- with ItH'gu Joad n.ttMued. Corrcctiou for
In.tcïft.I motion. SavM't'H "Hou rauqno." Differentini équation fortor-
Hioniti vibrations. Comp~risou uf vclocities of longitudinal a.nd tor-
tiiomd wa.'vos.
§§IGO–1U2 201
PotcutiftI energy of bcndhtg. Expression for kinctio energy. Dérivation
of diSercutitd equ)ttion. Termimtl conditions. G encrât solution for
tt hnrmouic vibrittiou. Conjuguto property of tho normal functiona.
YatucB ufintrKratcd sq~n'CH. ExprcsHi).]) of r in tcrms of not'nml cu-
CHAPTER Vni.
CHAPTER VI.
pAnx
CHAPTEK Vif.
CONTENTS.
ordinatos. Normal equatious of motion. Détermination of constant))
l'AUI~
to suit initial conditions. Caso of rod etartod bv a b)ow. I~od start~from rest as dofloctod by ~t~I ~~c. lu CL.~m ~os tho BorioH ofnormal funoticus coanos tu coj.vcrëo. Form of H~ norma! hmettons
li(:u-fiue bur. Lfnv9 of Jopotj.Icuoo of frofjuoucy ou )ongt)i und tLiok-noM. Caso whou both oudH ~rc clampod. Normal fuuctions for ,)clampod.Ireo bar. Caleulaticu oï puriud.s. CompnrinonH of pitch. Dis-oussiou of tho gravost modo of vibration of a freo-freo bar. Threunodos. Four ~oJoa. Gravost mojo for clampod-froo bar. Position utnodos. Supportcd bar. Calculation of poriod for clamped-froo bar fromLypothetleal typo. Solution of problom for n bar with a loaded ond.Euuct. uf adtUtious to a bar. lufluonco of irrogularitios of donsity.CorreottOM for
rotatory iuortia. L:oots of functioua dorived iinoarly fromnormaJ fuuetioM. Formation of ~uatiou of motiou ~heu thoro is por.Mauout tousiou. Spoeial tûrmiual couditioua. Itosultaut of two trainsof wavcs of iic-iirly cqua) poriod. Fourior'H solution of problom for iniï.nito har.
CHAPTER IX.
§§1~–213.
Tension of a motubrano. Equation of motion. Fixod reotangular lonud-ary. Expression for ~aud iu tenus of normal co-ordiuatos. Normalouations of vibratiou. Examplos of improssed forces. Frequoncy foran olongatod rectangle dépends maitiJy ou tho shortor sido. Casoo iuwhioh difTerout modes of vibration havo tho samo poriod. Dorivodmodes thence arising. Effeet of 6li(;ht irrcgulanties. An
irregu!aritymay rontovo nidoturmiuatonosa of normal modoa. Solutions applicableto a triaugle. Espro~ion of tho Honorai diilorontml eqnation by polarco-ordm~tes. Of tho two functions, w),idt oceur in tho solution, ono iacxcluded by tho condition at tlio polo. Expressions for Bossel'a func-tions. Formutm
rdating theroto. Tublo of tho first two functionsFixod eiroilar boundary. Conjugato proporty of tho normal functionswithout restriction of
boandary. Values of integrntod squares. Ex-proHMoun for T nnd F in tcnua of normal functions. Normal oqua-tions of vibration for ciroular mombraue. Spoci.d easo of froo vibra-tions. Yibratioua duo to a harmonie force
uniformty distributedUtohos of tho varions shnpto tonoH. Tabio of tlio roots of Bosscl'o func-tions. Nodal Fiëur~. Circular mombrano with ono radius fixed.Bessel's B onctions of frnctional ordcr. Ejloct of sma'I lond. Vibrationsof a mombrano whoso boundary is
approximatoiy ciroutar. In manycasos thé pitch of a mombrano mny bu calculated from tlio aroa alonoOf aU .nombraues of equal aroa t)Mt of oireular form l.M tlio gravostpitch. l'itch of a mcmbrano whoso boundary ia au eDipso of smaltceeHntricity. Motliod of obtai)iii)g limits in casos that oumot bo dealtwitli
rigorouf3ly. Comp~rison of fruqueucioa iu varions ca.sc.s of mcm-braues of eqna) arc.a. Histury of tho probion.. Bourh'ot'8 oxperi.ïaonta)
invostigfttiouB.
XICONTEN-TS.
§§214-235
Vibrations of PIatos. Potontial Enorgy of Bending. Transformation of 5~.
Superuoial diiïorontial equation. Dou.ndary conditions. Conjugato
proporty of normal functions. Transformation to polar co-ordinates.Form of gonorni solution continuons through polo, Eqnations doter-
mining tho poriods for a froo ciroular p!nto. EirohhoC'a catouhtions.
Comparison with observation, mdii of nodal cirolos. Irreguln.ntiesKivo riso to boats. Gonoralizution of solution. Cnso of cJampod, or
Hupportod, cdgo. Disturbn.uce of Chiadni's figures. Hifitory of proUom.Mn.tl.iou's critieiamo. DoetfmguiM phtto with aupportoJ edgo. Itoct-
nnguhn- plato with freo edgo. Boundary couditionH. Ono Hpocial cnso
(~ = 0) iH funonablu to mfttttomaticfd tro~tmont. Investigation of codaifigures. WItcntatoue'H application of tho mothod of HnporpositionCompariMU of Whoat~tono'f. liguros wit]. thoso reaUy n.pp)io~!o to n
pMo in tho cnso = 0. Gravost modo of a squnro plate. Caiouhttionof poriod on hypothotica! type. Nodal ~igurcH inferrod from considor.atlona of symmetry. Hoxngon. Comparison hotweon circle nnd squnre.Lnw connooting pitch and thicknoas. In tho cnso of a elfunpod odgonny contraction of tho boundary raisos tLo pitch. No gravest form fora free plato of givon aron. In similar plates tho poriod is as tho linoardimension. Whoat.stono'a expérimenta on wooden plates. Kœnig'aoxperimontN. Vibrations of cylindor, or ring. Motion tangentinl aswoll as normal. Bolation betwoon tangoitial and normal motiona. Ex-prossinna for Mnetio and potontial énergies. Estions of vibration.L'requoncios of tonos. Comparison with Chiadni. Tangential frictionexcites tanguntiat motion. Expérimental vérification. Béats duo to
irregularities.
CHAPTER X.RAOP
CIIAPTER I.
INTRODUCTION.
1. Tim sensation of sound is a thing s:M ~e~eW~, not com-
parab]e with any of our other sensations. No one can express
thé relation between a sound and a colour or a smcil. Directly
or indirectiy, ail questions connected with this subject must
comc for decision to thc car, as t!tG organ of hcaring; and
from it thct'c can be no appea!. But wc are not thcrefore to
infci' that ail acoustical investigations arc conducted with thc
unassistcd car. Whcn once wc have discovercd thc physical
phenomena which constitute thé foundation of sound, our ex-
plorations arc in great mcasurc transferred to another nc!d lying
within thc dominion of thé pi-mciples of Mcchanics. ImportMit
laws arc in this way ai'rivcd at, to which the sensations of thé car
canuot but conform.
2. Very cursory obscrvatioo. often succès to shew that
sounding bodics arc in a statc of vibration, and tha.t thc p)ic-
nomena of sound and vibration are closcly connected. WIicn a,
vibrating bell or string is touched by the finger, thé sound cea~cs
at thé same moment tha.t thc vibration is damped. But, in order
to affect thé sensé of hearing, it is not enough to have a vibrating
instrument t!icre must also be an uninterrupted communication
between thc instrument and thc car. A bcll rung in ~ac!<o, with
proper précautions to prevent thé communication of motion,
rcmains inaudible. In thé air of thé atmosphère, howevcr;
sounds have a univcrsal vehicic, capable of conveying thcin
without break from thé most var)ous~y constituted sources to
thé rccesses of the ear.
3. Thc passage of sound is net instantancous. Whcn a g)in
is jn'cd at a distance, a very perceptible interval séparâtes thé
y 1
2INTRODUCTION.
[3.
report from the flash. This rcpresents the time occupied bysouud in traveUIng from thé gun to thé observer, the rotardatinnof the nash duo to thé finite velocity of light bcing altogethernegligible. Thé first accuratc experiments wero mado by somo
members of the French Academy, in 1738. Cannons were nrc-d,and thé rctardationof thé reports at different distances ohscrvcd.Thé principal précaution -necessary is to revo-se alternatcdy tliedirection along which the sound travels, in order to cllminatc thoinfluence of tlie motion of thé air in mass. Down t!ic wind, for
instance, sound travelsreJativeJy to thé carth faster than its
proper rate, for the velocity of thc wind is added to that properto the propagation of sound in still air. For still dry air n.t a
température of 0"0., thc French observera found a velocity of 337metres per second. Observations of tho samo character wercmade by Arago and others in 1822 by thé Dutch physicists Moll,van Beek and Kuytcnbrouwer at Amsterdam by Bravais andMartins between thc top of the Faulhorn and a station bclowand by others. Thé gênerai result bas been to give a somcwhatlower value for tbc velocity of sound-about 332 mètres persecond. Thc effect of altération of température and pressure on the
propagation of sound will be best considered in connectiou withthé mechanical theory.
4. It is a direct consequence of observation, that within wide
limits, thé velocity of sound is independent, or at least very ncarlyindependent, of its intensity, and also of its pi tel). Wcre this
otherwiso, a quick piece of music would be hcard at a littledistance hopelessly confused and discordant. But when the dis-turbances are vcry violent and abrupt, so that thé altérations of
density concerned arc comparable with thé whole density of the
air, the simplicity of this law may be departed from.
5. An claborate séries of experiments on tlic propagation ofsound in long tubes
(watcr-pipes) has been madc by Rcgnault\He adopted an automatic arrangement similar in principle to thatused for me~suring thé speed of projectiles. At thc moment whena ptstol is fired at one end of tlie tube a wire conveying an electriccurrent is ruptnrcd by thc sliock. Tins causes thé withdrawai of a
tracing point which wasprevionsly marking a line on a revolving
drum. At tho furthcr end of thc pipe is a stretched membrano so
arranged that whcn on thé arrivai of the sound it yields to thé
~MofrM <?<:rjca(~;);«. ~e ~-«/tc< t. xxxvn.
35.]
VELOCITY OF SOUND.
impulse, the circuit, which was ruptured during tho passage of thé
soun< i3 rccumpietfd. At thc sa.mc moment tho tracing point
faits back on tlic drum. Tho blank space loft uumarked corre-
¡ sponds to thc thuc occupied by thé Sound in t~aking the joumcy,
and, wltcn thé motion of thé drum is known, givcs the means of
dctcrmining it. Tho length ofthe journoy hctwccn thé first wiro
and the membrane is fouud by direct mcasurcmcnt. In thcsa
cxperimcnts the velocity of sound appcarcd to hc not quitc indc-
pendent of thé dl~meter of the pipe, whieh vn.)'Icd from 0'108
to 1'100. Tho diso'cpancy is perhftps duo to friction, whose
j innucuco would hc greater in smaller pipes.
G. AIthough, in practice, air is usually the vehicio of sound,
otiicr gases, liquids and solids are equally capable of conveying
it. In most cases, I)owever, thé means of making a direct mcasure-
ment of the velocity of sound are wanting, and wo M'e not yet in
a position to consider tlie indirect methods. But in thc caso of
water tho same diniculty does not occur. In thé year 182G,
S Colladon ami Sturm investigated thé propagation of sound in thc
Lake of Geneva. Tlie striking of a bell at one station was
simultaneous with a nash of gunpowder. The observer at a.
t, second station mcasured the interval between tho flash and the
arriva! of thé sound, applying Itis car to a tube carried beneath
thé surface. At a température of 8°C., thé velocity of sound In
water was thus found to bo 14-35 metres per second.
7. Thc conveyancc of sound by solids may bc IHnstrated by a
pretty experiment due to Wheatstone. One end of a metallic wiro
is connectecl with tho sound-board of a pianoforte, and thé other
taken through thé partitions or floors into anothcr part of thé
building, where naturally nothing would be audible. If a reso-
nancc-board (such as a violin) bc now placcd in contact with the
wire, a tune p]ayed on thé piano is easily heard, and thé sound
seems to cmanatc from thé resonance-board.
8. In an open space thc intensity of sound falls off with grcat
rapidity as tho distance from thé source increases. Thé saine
amuunt of motion bas to do duty over surfaces ever Increa~ing as the
squares of the distance. Anything that confines the sound will
tend to dimini.sh ttte falling off of intensity. Thus over thé flat
surface of still watcr, a sound can'Ies furthcr than over broken
ground thc corner between a smooth pavement and a vertical wall
is still botter; but the most crtcctive ofaU is a tubc-likc enclosure,
1–S
[8.INTRODUCTION.
which prevents spreading altogether, Tlie use of speaking tubesto faciMtate communication between thc dirent parts of abuHdir)<ris wcll known. If It were not for certain crfects (fnctionat and
.other) due to thé sides of thé tube, sound might be thus conveycdwith little loss to vcry great distances.
CD
9. Bcforeprocecding furUicr wc must consider a distinction,
w!uc!t is of grcat unportance, though not frce from dimculty.Sounds
maybc ciassed as musicn.! a)jd
unmusica] thc former for
convcaicnco may bc caHed notes and titc lattur noises. Tho(,,
extreme cases will raiso uo dispute; every one rccngniscs thcdifférence betwecn thé note of a pianoforte and t)ic ereaidng of nshoo. But It is not so casy to draw t]ic line of séparation. Li thefirst place few notes arc frcc from a!i unmus:c:d accompanimcnt.Wit)i organ pipes especially, thc hissing of thé wind as it escapesat thc mouth may bc Iteard beside the proper note of tlie pipe.And, second]y, many noises so far partage of a musical character asto hâve a definite pitcb. T!tls is more easily recognised in a
sequence, giving, forexampJe, tite common chord, than by continuedattention to an individual instance. Thé experiment may Le made
by drawing corks from bottles, previously tuned by pouring waterinto them, or by throwing do\vn on a table sticks of wood of suitabledimensions. But, although noises are somctimes not entirelyunmusical, and notes arc usually not quite free from noise, thcre Isno diniculty in recognising which of thé two is thé simpler pheno-mcnon. Titerc is a certain smoothness and continuity about thomusical note. Moreover
bysounding together a variety of notes-for example, by striking simultaneousiy a number of consécutive
keys on a pianoforte-we obtain an approximation to a noise;while no combination of noises could evcr bicnd into a musical note.
10. We arc thus led to give our attention, in ttic first instance
mainly to musical sounds. Thèse an'angc themselves naturallin a certain order according to pitch-a quality which ail can
appreciate to some extent. Tralned ears can recognise an enormons
.numher of gradations–more than a thousand, probably, within
the compass of the human voice. Thèse gradations of pitch arenot, like the degrees of a thermometric scale, without specialmutual relations. Taking any given note as a starting point,musicians can
single out certain others, which bear a definiterelation to thc first, and are known as its octave, fifth, &c. The
corresponding di~i-ences of pitch arc cal!ed intervals, and arc
10.] piTcn. 5
spokcn of as always thé samc for thé same relationship. Thus,
"horov.'r th~yMn.ynccm' lli thé Hcale, a. note '~d ita. octave arc
sep~u'ated by </tc ~~o'uf~ of ~te oc~~e. It will be our object later
to cxplain, so far as it can be donc, tho origin and nature of the
consonant intervals, but we must now turn to consider thé physical
aspect of tlie question.
Since sounds are produced by vibrations, it is naturel to supposethat tho simpler sounds, viz. musical notes, correspond to ~e/~o~'c
vibrations, that is to sa.y, vibrations which after a certain interval
of timc, called thé per~~ repcat themselves with perfect regularity.And this, with a limita-tioM prcseutly to bo notioed, is true.
11. Many contrivances may bo proposed to illustrate tlic
gencratlûn of a musical note. One of thé simplest is a revolvingw)icol whoso milled cdge is presscd against a card. Each
projection as it strikes the card gives a slight tap, whose regniarrécurrence, as the whee! turns, produces a note of definite pitch,
7't'A-t'yt~ the scale, fM velocity of p't~b?!. MM?-casea. But thé most
uppropriatc instrument for the fundamcntal experiments on notes
is undouhtediy tlie Siren, inventcd by Cagniard de la Tour. It
cousists essentially of a stiff dise, capable of' revolving about its
centre, and pierced with one or more sots of holes, arranged at
cqual intcrvals round thé circumfcrcnce of circles conccntric with
thé dise. A windpipe in conncction with bellows is presented
perpendicularly to thé dise, its open end bcing opposite to one of
thé circles, which contains a set of holes. When thé bellows are
worked, the strcam of air escapes frcely, if a hole is opposite to tlieend of tlie pipe but othenvise it is obstructed. As thé dise turns,a. succession of puffs of air escape throngh it, until, when the
vclocity is sufncicnt, they btond into a note, whoso pitch rises
continually with the rapidity of thé pun's. \Vc shall have occasion
later to describe more claborate forms of thé Siren, but for our
immédiate purpose thé présent simple arrangement will sunice.
12. One of thé most important facts in thé whole science is
cxemplincd by tlie Siren–namciy, that thé pitch of a note dépends
upon thé pcriod of its vibration. Tho size and shape of thé holes,
the force of tlie wind, and other. éléments of tlie problem may be
varicd but if thé number of puffs in a given time, such as one
second, romains unchanged, so also does the pitch. We may even
dispense with wind altogethcr, and produce a note by allowing thé
corner of a card to t~p against the cdges of the holes, as they
6 INTRODUCTION,[12.
revolvc tho pitch will still be thé same. Observation of othcr
sources of sound, such as vibrating solids, leads to the samo con-
clusions, though thé difficulties arc often such as to render
necessary rather rcnned expérimental mothods.
But in saying that pitch depends upon. period, there
lurks an ambiguity, which dcscrves attentive consideration,
as it will lead us to a point of gréât importance. If a
variable quantity is periodic in any time -r, it is also periodic
in the timos 27-, 3ï, &c. Conversely, a recurrence within a given
period r, docs not exclude a moro rapid reourrence within
periods which are tho aliquot parts of r. It would appear
according!y that a vibration really recurring in thé time ~r (for
example) may be regarded as having the pcriod -r, and therefore by
tlie lawjust laid down as produciog a note of the pitch defined by
T. Thc force of this consideration cannot be entircty evaded by
defining as tho pcriod thé least time rcquired to bring about a
répétition. In tlie first place, thé necessity of such a restriction is
in itsc!f almost sufHcient to shcw that we have not got to thé root
of the matter fur although a right to thé period r may be dcuicd
to a vibration rcpeating itself rigorousiy within a time ~T, yet it
must bc auowcd to a vibration that may differ indefinitely little
thcrcfrom. In thc Siren cxperimcnt, suppose that in one of thc
ch'cles of holes containing an cvcn number, every alternate hole is
disp]accd along thé arc of the circle by the same amount. The
déplacement may bo made so small that no change can be detected
in tlie resulting note but the periodic time on whieh thé pitch
dépends lias bccn doubled. And secoudly it is évident from thé
nature of pûl'iodicity, tliat thé superposition on a vibration of period
T, ofothurs having pcriods ~T, ~T.&c., docs not disturb the period r,
while yet it caniiot be supposed that thé addition of thé new clé-
ments bas left thcqualityofthe sound unchangcd. Moreover.sinco
thc pitch is not affectcd hy their présence, how do we kuow that
clcmcnts of the sliorter periodswere not tbercfromt)ie beginnin"'?
13. Thèse considérations lead us to expectrcmarkable rcJations
between thé notes whose periods are as thé reciprocals of thé
natural numbers. Nothing can bc easicr than to invcstigate thé
<tucstion by meaus of tlie Sirot). Imagine two circles of holes, the
inner containing any convcnicnt number, and thé outer twice as
many. TIien at. wfiatcvcr specd thé dise may turn, thé period of
the vibration engendcred by blowing the first set will necessarily
13.] MUSICAL INTERVALS. 7
be thé double of that belonging to thé second. On making the
experiment the two notes are found to stand to cach other in
thé relation of octaves; and we conclude that in passing from any
?M<e its octave, the ~'c~c~/ of vibration is doubled. A similar
method of experimenting shews, that to thé ratio of periods 3 1
corresponds the interval known to musicians asthe<we~ made
up of an octave and a fifth to thé ratio of 4 1, thé double
octave; and to thé ratio 5 1, thé interval mado up of two octaves
and a major </Mr~. In order to obtain tho intervals of the fifth
and third thcmselves, the ratios must be made 3 2 and 5 4
respectively.
14. From those experiments it appears that if two notes
stand to one another in a fixed relation, then, no matter at what
part of thé scale they may bo situated, their periods are in a
certain constant ratio characteristic of thé relation. The same
may be said of thcir /?'e~Me?tc~ or tho number of vibrations
winch they exécute in a given time. Thé ratio 2 1 is thus
characteristic of tho octave intcrval. If wo wish to combine
two Intcrvals,–for instance, starting from a given note, to take
a step of an octave and then another of a fifth in thé same
direction, the corrcspondine ratios must be compounded
Tlie twelfth part of an octave is represented by the ratio !V2': 1,
for tins is thé stcp which repeated twelve times leads to an
octave abovo the starting point. If we wish to have a measure
of intervals in thé proper sense, we must take not the character-
istic ratio itself, but thé logarithm of tliat ratio. Then, and then
only, will the mcMuro of a compound intcrval bc the SM~ of thé
ïucasurcs of thé compouonts.
15. From the intervals of thé octave, fifth, and third con-
sidered above, othcrs known to musicians may be derived. Thé
difference of an octave and a fifth is called a fourth, and ha~ the
3 ératio
2–~=~.This process of subtracting an interval from
thé octave is called ~uer~M:~ it. By inverting the major third
Asingle word to donoto tho numbor of vibrations oxccuted in tho unit of timois indi~ensabio: I know no butter than froquoncy,' which was nsod in this sonso
by Young. Tho sMto word is omployod by Prof. Everott in bis excellent oditionof Doscbanol'a ~atw<t! P/(t'!osop/t~
INTRODUCTION. [15.8
we obtain thé minor sixth. Again, by subtraction of a major
third from a fifth we obtain thé minor third; aud from this by
inversion tho major sixth. The following table exhibits side by
side thé names of the intervals and the corrcsponding ratios of
frcqucncies
Octave 2
Fifth. 3
Fourth. 4
M~jorThird. 5
MiuorSixth. 8
Miner Third. G
M~jorSixth. 5
Thcfjo are ail thc consonant intervals comprised witttin thc
limits of thé octave. It will be remarked tliat tite correspondingratios are ail expressed hy means of ~M~t~ whole numhers, and
t!tat tliis is moreparticularly thé case for thé moro consonant
intervals.
The notes whosc frequencics arc multiples of that of a given
une, are called its AM~M~M, and the whole scries constitutes
a /«M'7/io?!c scctle. As is well known to violinists, they may ail
bo obtaiued from the samc string by touching it lightiy with the
imgcr at certain points, whilo thé bow is drawn.
Tlie establishment of thé conncction between musical intervals
and défunte ratios of frequcncy–a fuudamcutal point in Acoustics
-is duo to Mersennc (J63C). It was indeed known to thé
Grceks iu what ratios tlie Iougtlis ofstrings must bc chaagcd
in ordcr to obtain tlie octave and rifth; but Mcrsenne duntou-
strated tlie Jaw connecting thc length of a string with the ponodof its vibration, and madc thc first détermination of the actual
rate of vibration of a known musical note.
16. On any note takcn as a kcy-notc, or <o?n'c, a d!'M<omtc
scale may bc foundcd, whoso dérivation wc now proceed to ex-
plain. If thé key-note, whatevûr may bc its absolute pitch, be
called Do, thc fifth above or dominant is Sol, and thé fifth helow
orsuhdominantisFa. TIie common cliord on any note is pro-duced hy combining it with its major third, and fifth, giving thé
5ratios of frequency
1or 4 5 6. Now if wo take thé
common chord on titc tonic, on thc dominant, and on the sub-
dominant, and transpose thcm whcn neccssary into the octave
16.]NOTATION. 9
lying immediately above thé tonic, wo obtain notes whose fre-
quencies arranged in order of magnitude are
Do Re Mi Fa Sol La. SI Do
1,9 5 4 3 5 la
9 2.1,8' 4' 3' 2' 3' 8'
2.
Hcro the common cbord on Do is Do-Mi-Sol, with thc
5 3ratios 1 thé chord on Sol is Sol–Si–Re, with thé ratios
T~
~2x~=l:and thc chord on Fa is Fa-La–Do,0 0 T X
tlie c 101' on Fa 18 i a- a- 0,
still with tlie samc ratios. Thc scale is completed by rcpeating
thcsc notes above and bebw at intervals of octaves.
If we take as our Do, or key-note, the lower c of a tcnor
voice, thé diatonic scale will be
c d e f g a h c'.
Usage diffcrs slight~y p.s to thé mode of distinguishing the
different octaves; iu wllat follows I adopt thé notation of Helm-
hoitz. TIic octave below thé one just referred to is written with
capital letters-C, D, <&c.; thé next below tliat with a sufHx–
C,, D,, &c.; and thé onc beyond that with a double su~x–C, &c.
On thé other side acceuts dénote élévation by an octave–c', c",
&.c. The notes of thc four strings of a violin are written in this
notation, g–d~–a'–e'\ The iniddie c of thé pianoforte is c'.
17. With respect to an absoluto standard of pitch therc bas
bcen no uniform practice. At thé Stuttgard conférence in 183-1',
c' = 2G4 complète vibrations per second was recommended. Tilis
corresponds to a.' = 440. Tlie French pitch makes a' = 4-35. In
Handc!l's time the pitch was inuch lower. If e' were taken at 256
or 2", ail thé c's would have frequencies represented by powers
of 2. This pitch is usually adopted by physicists and acoustical
instrument makers, and t)as thé advantagc of simplicity.
Thc détermination ft!) tMt~o of the frequency of a given note is
an opération requiring somo care. The simplest method in prin-
ciple is by means of thé Siren, which is driven at such a rate as to
givo a note In nnison 'with thé given onc. Thé number of turns
cncctcd hythe dise in one second is given by a counting apparatus,
which can be thrown la and out of gear at thé bcginning and end
of a mcasured interval of time. This multiplied by thé number of
cn'ective holes gives thé required frotuency. Thé consideration of
othcr methods admitting ofgreater accuracy must be deferred.
10INTRODUCTION,
f~g.
18. So long as we keep to thé diatonic scale of c, thé notes abovewritten are ail that are required in a musical composition. But itis
frequentiy desired to change thé key-note. Under thèse circum-stances a singer with a good natural car, accustomed to performwitliout
accompanimcnt, takes an entirely fresh departure, con-structing a new diatonic scale on thé new key-note. In tbis wayafter a few changes of key, tho original scale will be quite departedfrom.and an immense
varicty of notes he used. On an instrumentwith fixed notes like tho piano and organ such a
multiplication isimpracticahle, and some
compromise is necessary in order to allowthé same note to perform different functions. This is not théplace to discuss the question at any length, wc will thcrefore takeas an illustration thé simplest, as wcn as thé commonest case-modulation into thé key of thé dominant.
By donation, thé diatonic scale of c consists of thé commonchords foundcd on c, g and f. Jn like manner thé scale of g con-sists of tlie chords founded on g, d and c. Thé chords of c and garc then commôn' to thé two sca!cs; but thé third and fifth of dintrodnce new notes. Thé thu-d of d written
f#has a
frcquency3 5 4a t J
8 4 32 removed from any note in thé scale of c.
But thé fifth of d, with a
frequc.cydiffers but
little from a, whosefrcqucncy
isIn
ordinary keyed instruments
thé interval betwecn the two, represented by and called a
c~ is ncglectcd and thé two note. by a suitable compromiseor ~?~-Hwc~ M-e identined.
19. Various systems of tomperament have been used thésimplest and tliat now most
generally used, or at least aimed at, isthé equal tempérament. On
referring to the table of frequencies fortlie diatonic sealc, it will be secn that the intervals from Do to Refrom Re to Mi, from Fa to Sol, from Sol to La, and from La to Si,are nearly thé same, being rcpresented
byor while tjj
intervals from Mi to Fa and from Si to Do, represented by arela
~1~' ?~equal ~mperament treats
~cs~'ap.proximate relations as exact, dividing the octave into twelve eqnal
19.] EQUAL TEMPERAMENT. 11
parts called mean semitones. From thèse twelvo notes thé diatonic
scalc belonging to any key may be selected according to tho fol-
lowing rule. Taking the key-note as the first, fill up the series
with thé third, fifth, sixth, eighth, tcnth, twelfth and thirteenth
.notes, counting upwards. In this way ail dKScultIes of modulation
arc avoided, as thé twolve notes serve as weU for one key as for
anothcr. But this advantagc is obtained at a sacrifice of true in-
tonation. Thé equal tempérament third, being thé third part of
an octave, is rcprescnted by thé ratio ~2 :1, or approximately
].'2a99, wliile thé true third is 1-25. The tempercd third is thus
higher than thé truc by thé interval 126 125. The ratio of thé
tempered fifth may be obtained from thé consideration that seven
ficmitoncs makc a fifth, wliile twelve go to an octave. Thé ratio is
thcrëforc 2 1, which = 1-4.983. The tempered fifth is thus too
]ow in thé ratio 1'4!)83 1-5, or approximately 881 883. This
cn'or is msignificaut; and even thé error of thé third is not of
much conse<~uencein quick music on instruments like the piano-
forte. But whcn thé notes arc /teM, as in thé harmonium and
organ, thé consonance of chords is materially impaired.
20. The foltowlng Table, giving the twelve notes of the chro-
matic scale according to thé system of equal tempérament, will be
convenient for reference'. Thé standard employedis a' = 440 in
order to adapt thé Table to any other absolute pitch, it is only
necessary to multiply throughout by thé proper constant.
C, 0, C c c" c~ c""
0 10-35 32-70 C5-41 l30'8 261-7 5233 104G-6 2093-2
C~ 17-32 34-G5 C9-30 138'6 277'2 544'4 1108-8 2217-7
D 18-35 3G-71 73-43 14G-8 293-7 587"i 1174-8 2349-G'
D~ 19-41 38-89 77-79 155-6 311-2 G23'3 1244-G 2480-3
E 20-GO 41-20 82-41 1G4-8 329-7 G59'3 1318-G 2G37'3
F 21-82 43-G5 87-31 174-G 349-2 C98'5 1397-0 2794-0
F~ 23-12 L) 4G-25 92-50 185'û 370-0 740'0 1480-0 29GO-1
G 24-50 49-00 98-00 19G-0 392-0 784-0 15G8'0 313G-0
0~25-95 51-91 103-8 207'G 415-3 830-C 1GG1-2 3322-5
A 27-50 55-00 110-0 220-0 440-0 880-0 17CO-0 3520-0
A~ 29-13 58-27 11G-5 333-1 4CG-2 933'3 1864'G 3729-2
B 30-86 61-73 123-5 346-9 493-9 9877 1975-5 3951-0
Zammiuor, Die J/tMf'~ «tx! <!<c MtMtA'<t<t<cyfc?t DutnottCHte. Giessen, 18CS.
INTRODUCTION.[20.
Thé ratios of tho intcrvals of the equal teinpûra.ment scale are
gtvcn bclow (Zaunuluer)
21. Rcturning now for a moment to thc pbysical aspect of t!ie
question, we will assume, what wc shall af'terwards prove to bctruc within wide lim its,–that, whcu two or more sources of sound
agitate thé airsunultaneousiy, thé
resulting disturbance at anypoint
ni the external air, or In thécar-passage, is thé
simple sum
(ni the extendeJ gcomotncal scuse) of what would be caused bycach source acd~g- separately. Lot us consider the disturbancoduc to a simultancous sounding of a note and any or ail of its]iarmouiës. By durmition, thé eompiex wholo forms a note havingt)ic same pcriod (and thcrefore pitch) as its gravcst element. Wc0Iiavo at present no criterion hy which thé two can bc distmguishcdor thc présence of thé highcr harmonies
recognised. And'yet–inthé case, at any rate, where thé componcnt sounds have ail inde-
pendent origin-it is usually not difncult to detect them hy thécar, so as to cnect an analysis of the mixture. This is as much asto say tliat a strictly periodic vibration may give risc to a sensa-tion which is not
simple, but susceptible offurthcranalysis Inpoint of fact, it Ims lon~ been hnowu to musicians that undercertain circumstancus the harmonies cf a note may Le heard alongw.t!t it, uven w!~n thc note is due to a single source, such as avibrato strier, but tl.e sig.lincancc of thé fact was not undcr-stood. Since attention ]~as bccn <1rawn to the subject, it bas becnproved (.nainly by thé labours of Ohm and
Hchnho~) that almosta)t musical notes are
higtdy compound, consisting in fact of thcnotes of a harmonie scale, from which in particular cases onc ormore members may be
missing. Thé rcason of theunccrtaintyand
di~culty of thé aualysis will bc touchod upon prcsontiy
Note. Froquoucy.
c =1-00000
c# 2~'=I-0594G
d 2 ~=1-122-1 G
d# 3'~ 1-18921
o 2~=1-25992
f 2~=1-3348.1
Noto. Froqnonoy.Il
f~ 2~'=1-41421 1
7
g 2'~ =1-49831Ii
ë#2'~= 1-58740
Il
2~=1-68179
100
2~~=1-781801 1
L 2~~=1-88775
c' = 2-000
22.]NOTES AND TONES. 13
22. That kind of note which thé car cannot furthcr resolve is
c:i))ed hv Hehnhoitz in Ccrmn.n a ')!o?t.' Tyndall and other recent
writcrs on Acoustics have adoptcd 'tone' as an Enghsh équivalent,
–a practice which will bc followed in thé présent work. Thc
thing is so important, that a. convenient word is almost a matter
of nccessity. ~<~ thcn are in général made up of tones, thé
pitch of the note being that of thé graves! tone which it contains.
23. lu strictness thé quality of pitch must bc attachecl m the
ih'st instance to simple toncs only; otherwise thé difîlcult.y of dis-
continuity before referred to presents itself. Tlie slightcst change
in thé nature of a note may lower its pitch by a wholo octave, as
was oxcmplined in the case of thé Sircn. We should now rathcr
say that thé effect of thé slight displacement of thu alternate
hules in that experiment was to Introduce a, ncw fceble tone an
octave Jowcr than any previousiy present. This is surHcIent to
altcr tho pcriod of thé wholej but thé great mass of tlic souud
remains vcry nearly as before.
In most musical notes, howcvcr, thc fundamental or gravent
tone is présent in sunicient intensity to impress its cliamctcr on
thé whole. Tho eÛect of thé harmonie overtones is then to
modify thc ~ua~~ or c/t(M'ac<er 1 of thé note, iudcpendently of piteli.
Tliat such a distinction exists is wcll known. Thé notes of a violin,
tuning fork, or of thé hufnan voice with its dincrent vowel sounds,
&c., may aU hâve thé sanie pitch and yet differ indepcndent~y of
ioudness; and though a part of this ditl'erellce is due to accompany-
ing noises, which are cxtraneous to thcir nature as notes, still there
is a part winch is not thus to be accounted fur. Musical notes may
thus be classified as variable in threc ways First, ~t'<c/t. This we
have already sumcicutly considered. Secondly, c/tHrf(c<e)', depend-
ing on the proportions in which the harmonie ovcrtones are com-
bined with the fundamcntal: and thirdly,~oMc~eM. Tins lias to bc
taken last, because thé car is not capable of comparing ('with any
precision)tlie loudness of two notes which differ much in pitch or
character. We shall indeed in a future chapter give a mechanical
measure of thé intensity of sound, including in onc system ail
gradations of pitch; but tins is nothingto thé point. We are hère
concerned witli thé intensity of. thé sensation of sound, not with a
mcasure of its physical cause. Thé dinerence of loudness is,
howcvcr, at once recognised as one of more or less so that wc
Gcrnmn, 'Klaugfarbo' –Frcnch, 'timbre.' Tho word 'chfu-Mter' iHnscd iH t!)is
Moso by Evcrett.
14 INTRODUCTION,f'23.
have hardly any choice but to regard it as dépendent ccc~?'~
~a.rt'&M~ on the magnitude of thé vibrations concerned.
24;. Wu Luve seoi that a musical note, as such, is due to a
vibration which is necessarily pcriodic but thc converse, it is
evident, cannot be truc without limitation. A periodic repetitioMof a noise at intervals of a second–for instance, tlie ticking of fi.
clock-would not result in a musical note, be thé repetition ever
so perfect. In such a case we may say tliat thé fundamentai tone
lies outside the.limits of hcaring, and although some of thé
harmonie overtoues would fall within them, thèse would not ~iveriso to a musical note or ovcn to a chord, but to a noisy mass of
sound likc that produced by striking simultaneousiy tbe twelve
notes of thc chromatic scale. The experiment may be jnadc witit
thé Siren by distributing tho holes quite irregularly round the
circumferenco of a circle, and turning tho dise with a moJcrato
velocity. By tho construction of tho instrument, everything re-
curs after each complote revolution,
25. The principal remaining dimculty in tlie theory of notes
and tones, is to explain why notes are sometimes analysed by thc
ear into toncs, and sometimes not. If a note is reallv comulcx
why is not the fact immediately and certainly perccived, and t)te
compononts disentang!ed ? The feebleness of thé harmonie over-
tones is not thé reason, for, as ~ve shall sec at a later staf-c of our
inquiry, titcy are often of surprisiug loudness, an(.1 play a promiucnt t
part in music. On thé other hand, if a note is sometimes perccivedas a wholo, why does not this happen always ? Thèse questionshâve been carefully considered by Hcimboitz', with a
tolcrabiy
satisfactory result. The difHculty, such as it is, is not peculiar to
Acoustics, but may be paralleled in tlie cognate science of Pitysio-logical Optics.
Thé knowledgo of external things which wo derivo from théindications of our sensos, is for thé most part thc result of inference.When an object is beforc us, certain nerves in our rctinœ arc
excited, and certain sensations arc produced, which wo areaccustomcd to associate with thé objcct, and we forthwith infer its
presence. In the case of an unknown object thé process is muchthe samc. We interpret thé sensations to which we arc subjcct soas to form a pretty good idea of their exciting cause. From thé
sliglitly dincrcnt perspective views reccived by titc two cycs we
infer, oftcn by a liglily claboratc process, thé actual relief and
~<'m~;t(!)ty)yctf, 3rj oditioH, p. 98.
25.] ANALYSIS 0F NOTES. 15
distance of thé object, to which we might otherwise have had no
~np. Thcse inferences are madc with extrême rapidity a.~d quite
UitCunsciousiy. Tbu 'it~l& life ui' bacii ono of us is a continued
lusson in intcrpreting tho signa presented to us, and in drawingconclusions as to the actualitics outside. Ouly so far as we succeed
in doing tins, arc our sensations of any use to us in thé ordinaryaffairs of hfe. TI)is being so, it is no wonder that the study of our
sensations themselves falls into thé background,andthat subjective
phenomena, as they are called, becomc exceedingly difficult of
observation. As an instance of this, it is suNdeiil to mention the.
'blifid spot' on thc retina, which might a ~'K))-~ have been
expectcd to manifest itself as a conspicuous phenomenon, thoughas a fact prohahly-not one person in a hundred million would nnd
it out for themselvcs. Thé application of these i-emar'ks to thc
question in hand is tolerably obvious. In tho daily use of our ears
our object is to disentangle from the whole mass of sound that
may rfach us, thc parts c&mlng from sources which may interest
us at thé moment. 'Whcn we listen to thé conversation of a friend,wc fix our attention on thé sound procecding from him and
cndcavour to grasp that as a whole, while wc ignore, as far as
possible, any other sounds, regarding them as an interruption.Therc arc usually sufilcient indications to assist us in making this
partial analysis. Whcn a man spcaks, thé whoJe sound of his
voice rises and falls together, and wc have no dirnculty in recog-
nlsiug its uoity. It would bc noavantage, but on thc eontrary
a grcat source of confusion, if we werc to carry the analysis furthcr,and résolve thc whole mass of sound présent into its componenttones. A] though, as regards sensation, a resolution into toncs
might be expectcd, tho necessities of our position and thé practicoof our lives lead us to stop tho analysis at thc point, beyond
which it would ccase to bc of service in deciphering our sensa-
tions, considcrcd as sigus of extcrnal objccts\But it may sometimes liappcn. that however much wc may
wish to form ajudgment, thé materials for doing so arc absolutely
wanting. When a note and its octave are sounding close together
and with perfect uniformity, there is nothing in our sensations to
cnahic us.to distinguish, whctiicr thé notes have a double or a
single origin. In thc mixture stop of tlie organ, the pressing down
of each keyadmits thé wind to a group of pipes, giving a note and
Méat prohubty tho powor of nttonding to tho inipût-tant nnd ignoring tho
nnimportant part of our seusationa is to ft great oxtont iuhcntod–tQ how great air
oxtont wo shftt) perha.pa Bovcr know.
1C INTRODUCTION.[25.
its first three or four harmonies. Tho pipes of each group aiwayssound together, and thé result is usually pei-ceived as a singlenotfj n!though .h~'H <ujt
pt'oecuj fron acingle sourco.
26. Thé resolution of n. note into its componcnt toncs is n.matter of very din'crent dimculty with différent individuals. Aconsidérable effort of attention is
rcquired,particu!a~yt).t first;and, until a h~bit bas been formcd, somc cxtcrn:d aid in the slia.pcof a. suggestion of what is to bc Jistoned for, is very désirable.
Thé difliculty is altogethcr vcry similar to that ofIcarning to
draw. From tlicmacitinery of vision it might have hcen expectcd
that nothing would bc easicr than to make, ou a plane surface, a
représentation ofsurrounding solid objccts; but expérience shows
that much practicc is gencrally requircd.We s!ia)I rcturn to the question of tlie analysis of notes at a
later stage, after we hâve treated of thé vibrations ofstrings, with
thé aid of which it is bcst elucidated but a very instructive
expcnment, ducoriginaHy to Ohm and improved by Helmholtx,
may bc givcn hère. Helmitohz' toolc two bottles of thé sliapcreprcsented in the figure, onc about twice as )argc as thc other.ihcsewcrc blownby strcams ofair dirccted acro.ss
thé moutti an<tissuing from
gutta-pcrd)a. tubes,whosc ends had been softcnud and prcsscd flat,so as to rcducc thc bore to the form of a narrow
slit, thé tubes bchig in conncction with thé samc
bellows.By pouring
in wn-ter when thé note is too
low and by pa.rtin.Hy obstructmgtlie mouth whcn
thc note is too high, thé bottJcs may bo made to
give notes with thc exact interval of an octave,such as b and b'. fhe larger bottic, blown a!onc, gives a somcwhatmunlod sound similar in character to tlie vowcl U; but, when bot]ibottles are blown, thé character of thc resulting sound is sharpcr,rcsemb)ing rathci- thé vowel 0. For a short time after thé noteshad bcen heM-d separately Hchnhoitz was able to distinguish themin thc mixture; but as the
mcmoryof thcir scparatc impressions
ff)dcd,<thc Itighcr note scemod by degrecs to amaJgamatc withthc lowcr, which at thé same time bccamo budcr and acquireda sharper charactcr. This bicnding of the two notes may takc
place cvcn whcn thé t)igh note is thé louder.
27. SeGing now that notes are usuaDy contpound, and that
or~y a particular sort caUcd toncs arc nicapabic of further analysis,
7'r'?~M)~/?))~t)yf);, p, tf);).
27.] PENDULOUS VIBRATIONS.
I
17
we are led to inquirc what is thc physical characteristic of tones,to winch they owe their pecuHarity ? What sort of periodic vibra-
tton it, whiciiprod~ces a.
simple tone ? According to wha.t
matl)cmatical function of t)ic time does tlie pressure vary in
thépassage of thc car ? No cluestion in Acoustics can be more
important.
The simpicst periodic functions with which mathcmaticians arc
acquainted are the circular functions, expressed by a sine or
cosine; indecd t!)cre are no otJiers at aU approaclung them ia
.simphcity. TIiey may bc of any penod, aud a<tmitt!ng of no
other variation (except magnitude), secm well adaptcd to producc
simple toncs. Morcovcr it lias been proved by Fouricr, tha.t tho
most gênerai singic-vit.hicd pcnodic function can bo rcsolvcd into
a sories of circular functions, Laving periods winch arc submu!tipies')f that of tho givcn function. Again, it is a conséquence of thc
guttural thcory of vibration that the particular type, now suggcstcdas corrcsponding to a simple tone, is t!te omy one capabjc of
pt-cscrving its intcgrity among thé vicissitudes which it mayItave to undcrgo. Any othcr kind is iiabic to a sort of physieat
analysis, ono part being di~crontly an'ected from anothcr. If thé
analysis within the car procceded on a dinercnt principle from that
cnucted according to thc laws of dead mattor outside the car,tho consequence would Le that a sound originally simple mi~htbecomo compound on its way to thé observer. Thcrc is no i-caMnto suppose that anything of this sort actually happons. When it
is added thataccording to ail thé ideas we can form on the subject,t)tc analysis within t!tc car must takc place by means of a physical
machinery, subject to tlie same laws as prcvail outside, it will boscen tliat a strong case has Lccn madc out for rega.rding tones asduc to vibrations exprcsscd by circular functions. We arc notttowevcr left eutirely to thc guidance of gênera! considérations like
thèse. lu tho chapter on thé vibration of strings, we shall sec
that in many cases theory informs us beforehand of the nature ofthe vibration executcd by a string, and in particular wliether any
specined simple vibration is a. component or not. Hère we havea décisive test. It is found hy experiment that, whcncvcr accord Ingto thcory any simple vibration is présent, thé
corresponding tonecan bc hcard, but, whcnever tho simple vibration is absent, thcnthe tonc cannot be heard. \Ve arc thercforc justined in asscrtinn-that simple toncs and vibrations of a circular type are indissoluh)yconncctcd. This law was discovcrcd by Ohm.
n.
CHAPTER II.
IIARMONIC MOTIONS.
28. TllE vibrations expressed by a circular function of the
time and variously designated as simple, ~w~t~M~OM~ or /mr?)M)n'c,
are so important in Acoustics thatwc cannot do botter thaii (levote
a cha.pter tu thcir consideration, Lefore cntcring on tlic dynamical
part of our subject. Thc quantity, whose variation constitutcs
thé 'vibration,' ma-ybc tlie displacement of a particle mcasured
in a given direction, thé pressure at a fixed point in a iluid
médium, and Bu on. In any case denoting it by M, wo have
in which a dénotes tho ûMHp~<(i~, or extreme value of u; r is
the periodic <M~e, or jperto~, after thé lapso of which thé values
of u recur; and e détermines thé phase of thc vibration at thé
moment from which t is measured.
Any number of harmonie vibrations of ~e same ~j<M~ affect-
ing a variable quantity, compound into anothcr of thé same type,
wliose clements arc dctcrmined as follows
=rcos(~-A.(2),1
if?'=(($acose)'+(SHsin6)~(3),
i).ud tau = 2 (t siu e–~M ces e.(4).
38. j COMPOSITION. 19
so tliat if K'=~, ~f vanishcs. In tliis case thé vibrations arc often
s:ud to t'~cr/b-e, but theexpression is rather
misleading. Two
soundsmay vcry propeny bc said to interfère, when thcytogethcr
cause silence; but thé mere superposition of two vibrations
(whcthcr rest is the consequence, or not) cannot properly bc so
called. At Icast if tbis bc Iiitei-furence, it is difficult to say what
non-intcrforenco can bc. It will appcar in thé course of this
work that whcn vibrations exccetl a, certain intensity tucy no
longer compound by more addition; <AM mutual action mightmore properly bc called Interférence, but it is a pbenomcnonof a totally diiTorent nature from that with which we are now
dcaling.
Again, if tho phases dîner by a quartor or by tbree-quarters of
a pcriod, cos (e e') = 0, and
~=~"+~.
Harmonie vibrations of given pcriod may be reprosented
by linos drawn from a pole, tlie lengths of tlio lincs being pro-
portional to tho amplitudes, and tlie inclinations to tlie phasesoi' thé vibrations. Tbc résultant of any number of harmonie
vibrations is then represented by the geomutrlcn.1 résultant of
thécorresponding Unes. For cxample, if they arc disposcd
synuuctricaHy round thc polo, tlie résultant of the Unes, or
vibrations, is zéro.
2!). If we mcasure off along an axis of x distances pro-
portional to tlie timc, and takc u for an ordinale, we obtain tlic
Iiarmonic curve, or curvc of sincs~
2-2
20 HARMONIC MOTIONS [29.
whcre called the wavc-!cngt]i, is written in place of r, both
quantities dcnoting tho range of tlic indcpendcnt varia.bic corre-
sponding to a complète récurrence of thc fonction. The harmonie
curvc is tlius thc locus of a, point subject at once to a uni-
form motion, and to a ha-rmonic vibration in a perpcndicuta.r
direction. In thé next chapter we shall sec tha.t the vibration
ofn. tuning fork is simple harmonie; so that if an excited tuning
fork is movcd with uniform velocity parallcl to thé lino of its
handio, fL tracing point attached to thé end of onc of its prongs
dcscribes a harmonie curve, which ma.ybc obtained in a permanent
fonn by allowing the tracing point to bcar gently on a piece of
smokcd paper. In Fig. 2 the continuons linos arc two harmonie
curves of thc same wavc-lcngth a,nd amplitude, but of diSercnt
phases thé dotted curve represents haïf thcir rcsu~tant, bcing
<he locus of points midway bctween those in which tlie two
curves are met by any ordinate.
30. If two harmonie vibrations of différent periods cocxist,
Thé résultant cannot here be reprosented as a simple harmonie
motion with oti~cr cléments. If r and r' bc inccmmcnRurabIc, tho
value of ?t never recurs but, if r and T be in thé ratio of two
who!c numbers, M recurs after the lapse of a. time equa.1 to tbo
least common multiple of T and r'; but tbe vibration is not
simph harmonie. For exampic, whcn a note and its fifth are
sounding together, tho vibration recurs after a time eqnat to
twicc the period of tho graver.
30. JOF NEARLY EQUAL PERIOD. 21 1.
One case of the composition of harmonie vibrations of dinereut
periods is worth special discussion, na.me!y, when the dinerenco
ci' the periods is small. Ii' we nx our attention on the course
of thiugs during an interval of time including mcrcly a fcw
poriods, wc sec that the two vibrations are nearly t!ie same as
if their periods were absolutely equa!, in whic]t case they would,as wc know, bc cquiva!cnt to another simple harmonie vibration
01 tho samc poriod. For a fcw periods thcu tho résultant
mution is approximatcly simple harmonie, but tho samc har-
monie will not continue to rcprescnt it for long. Thé vibration
having thé stiorter period continuaDy gains on its icilow
thm'cby altering thé dittcrcncc of phase on which thé éléments
of thé résultant dépend. For simplicity of statement let us
suppose that tho two components Iiave oqual amplitudes, fre-
quencies rcpresentcd by ??~ and ?!, wlicre ??t–?!. is small, and
that when first obsorvod their pitases agrée. At this moment
thuir cn'ccts conspire, and thé résultant ha.s an amplitude double
of that of the components. But after a time 1–2 (M–~) thc
vibration ?~ will hâve gaincd ha)f a period rclatively to thé
othcr; and thc two, boing now in comptete disagreemcnt, ncu-
trahze cach other. After a furtiicr intcrval of time equal to
that abuve named, Mt will hâve gained altogether a who!e vibra-
tion, and complète aceordancc is once more rc-establishod. T!)e
résultant motion is thcrcfore approximately simple harmonie,
wiLh an amplitude not constant, but varying from zero to twicc
that of thc componcuts, thc frcqnency of thèse altérations beingM-M. If two tuniug f<;rks with frequcnelcs 500 amI 501 bc
cqu~ty excited, tho'e is every second a risc and faU of sound
corrcspnnding tu t)m coincidence oropposition of their vibrations.
Tinsphcnontenon is ca))ed béats. We dn not hbre f~dty discuss
thé question how t)tc ear behaves in thé présence of vibrations
having )icar]y etjual fre'~K-ncie.s, butit is obvions tiiat If thc motion
ni thé nelg)ibonr!)ood of thé car almost ccascs for a considérablefractiu)! of a second, thc sound must appcar to fall. For rcasons
that will afterwards appear, béats are best hcard wl)en thé in-
tcrfcring sounds are simple toncs. Consécutive notes of thé
stoppcd diapason of thé organ shc\v thé phcnomcnon very
wcii, at least in thé lower parts of thé scale. A permanent Inter-
férence of two notes may be obtained by mountingtwo stopped
crgari pipes of similar construction and identical pitch sitic
Ly sido on thc same wiud clicat. Thé vibrations uf thé two
2222 HARMONIC MOTIONS. [30.
pipes adjust thcmselvcs to complete opposition, so tliat at a
little distance nothing can be heard, except thé hissing of thc
wind. If by a rigid w:dt bctwecn thé two pip~s one souud
could bc eut off, thé othcr would bc Instautly restored. Or tbo
balance, on which silence dépends, may bc upscb by connecting
thé car with a tube, whose other end lies close to tlie mouth of
eue of the pipes.
By meaus of béats two notes may be tuned to unison with
gréât cxactncss. Tlie object is to make thé béats as slow as
possible, siuce thé numbor of be~ts in a second is oqual to thé
diScrcnce of t]te frcqucnei.os of thc notes. Under favourable
circumstanccs béats so slow a.3 onc in 30 seconds mn,y be re-
cognised, and would indica.te th~t thé highcr note gains only
two vibrations a ?~M:M<0 on thé lower. Or it mighb Le dcsited
mercly to ascertain thé diiTcl'ence of thc froqucncios of two notes
nearly in unison, in which case nothing more is necessary than
to count the number of bca,ts. It wili be rcmcmLcred that t)iG
Jifïcrcuco of frcqncncics docs uot determine tite tM~erua~ bctwccn
tlie two notes; tliat df'pcnds on thé ?'(t<M of frequoncics. T!tU3
thé rapidity of thé bca,ts given by two notes ncariy in unison·
is doubicd, when both arc takcn an exact octave highcr.
AnalyticaUy
M= a cos (27r~< e) + a' cos (2?! e'),
wlicre Mt is small.
Now cos (27r?~ e') may bc writtcn
aud wc hâve
cos 2?~ 27r ()? ~) t e },
M=r cos(2-7rw<– 0) .(1),
whcre = + a." + 2aat' cos [Spr (~ ?~) t + e e] (2),
ft sin e + a' sin {Spr ('~ M) + e'{ ,n.tan c = .(3).tan e
a cos € + (t COS{27T (~ ~) t + € )1
Thc résultant vibration may tLua bc considcred as harmonie
with clements r and which arc not constant but slowly varying
functions of the time, having thé frequency w –M. Thé amplitude
r is at its maximum when
cos {2-7r (?n. ?~ t + €' e} = + 1,
and at its minimum whcn
cos {2-n- (w n) e' e} == 1,
thc corrosponding values beiDg a + a' and a <t' respectively.
31.]FOURIER'S THEOREM. 23
31. Anothcr case of gréât importance is the composition of
vibrations corresponding to a tone and its harmonies. It is known
that thc most gericml single-valued nuito periodic function can
bc expressed by a séries of simple harmonics-
a theorem usually quotcd as Fourier's. Analytical proofs will be
fouud in Todhuuter's J~~e~ra~ Calculus aud Thomsou and Tait's
~~M)Y~ r/~7oso~/ty and a line of argument almost if not quite
amounting to a démonstration will bo given later in this work.
A fcw remarks arc ail tliat will bo required bore.
Fourier's thoorem is not obvious. A vague notion is not un-
common that tlie innnitudc of arbitrary constants in tho séries
of necessity endows it witli the capacity of ropresenting an arbi-
trary pcriodic function. Tha,t tbis is an error will be apparent,
wlicn it is observed tliat the samo argument would apply equally,
if one term of tbe series were omitted in which case thé expan-
sion would not in general be possible.
Another point worth notice is that simple harmonies are not
thc orily functions, in a series of which it is possible to expand
one arbitrarily given. Instead of the simple elementary tcrm
formed by adding a similar one in thé samc phase of half the
amplitude and period. It is évident that thèse terms would
serve as wcU as tlie others for
–a~t?!
so that eacli term in Fourier's sories, and thereforc the sum of
tho séries, can be expressed by means of the double elementary
24HARMONIC MOTIONS ~31.
'¡;¡,t.t:r.v.
terms now auggcstcd. This is mentioncd hero, becausc students,
not, b~in? nc't"aintcdwit~' ~thf) expansions, m~y imagine that
simpic h~rmonio functions arc 1by nature tiio only oncs (tu:;Jitied
to bo thc clements in t!ic dcvclopmont of a periodic function.
Thc rcason of thé prccmincnt iinport.a.nceof youncr's scries in
Acoustics is thc mccha.uic:U onc rcfcrrcd to in thc proceding
ch~pter, and to bc cxp~incd more fuHy ))cre:U'tcr, namciy, th:).t,
in guncrfd, simple harmonie vibrations are thé oniy kind titat arc
propagatcd through a vibrating systcm without sun'ering decom-
position.
32. As in other cases of a similar character, c.g. Tay~or's
thcorcm, if thé possibility of thc expansion be known, thé co-
cfncicnts may bc determined by a. comparativcty simpio process.
\Vc may writc (1) of § 31
Multip)ying by cesor sin and Intcgratmg over
;). complète period from <=C to t = T, wc find
indicating thn.t ~to is thé wcaM value of 1t throughout the period.
Thc degrec of convcrgency in tho expansion of u dépends in
~cnerfd on thc continuity of thé function a.nd its derivatives.
Thc scries formcd hy successive diiïercutiations of (1) converge
k'ss and loss ra.pidty, but still remniM couvergcut, and arithnietical
représentatives of the diH'erential coefficients of it, so long as
thèse lutter arccvcrywhcrc
finite. Thus (T)iomsonand Tait,
§ 77), if aM thc dcrivativcs up to thé M'" inclusive arc frue
from innnitc values, tlic sories for u is more convergent than
onc with
] &c~< ();))'
nm)ni )'
for coc(ncic)tts.
33.]IN PERPENDICULAR DIRECTIONS.
25
33. Another ~MS of compotin(ledvibrations, intcresting
from
tlic facilitywith which they
Icnd themsel-~cs to opticalobserva-
bion, ~cur wt~i harino~c vib~doi~ U."
r~-
ticlc arc exccutcd ~e;~e;~tCM~r~rcc~ons, more cspecialty
whcn thé pcriodsare not oniy
commensur~bic, but in the ratio
oi' two SM~tM whoïc uumbcrs. Thé motiorL is thcn compléter
pcriudic,with pcnod
not manytimcs grcatcr
th~u tliosc cf thé
co.nponents,and thc curve dcscribcd is re-cutrant. If M and v
ho thc co-oi-dmatcs, wc maytakc
reprcscnting in général an ellipse, whose position and dimensions
dépend upon tlie amplitudes of thé original vibrations and upon
tlie dincrcncc of thcir ph~es. If thé phases ~er by a quarter
poriod, co3€=0, and thé équation becomes, ·
In this CMC the axes of thé ellipse coinci'dc with't~osc of
co-ordinatcs. If furthcr thé two cfjmpMieuts ha.vc ecju~! ampli-
tudes, thé locus (JcgenGmtcs into thc cirete..
which is described with uniform velocity. This shows how a
uniform circuiM' motion may bo analyscd into two rcctilmca.r
hn-monicmotions, whosc directions arc pori~enJicula.r.
If thc phasesof thc components agrée, E=0, and the cl!Ipsc
dc"'cncrates iuto the coiticident stmight liucs "r"j
When the unison of the two vibrations is exact, thc cUiptic
path remains pcrfeetly stcady, but in practiccit will ahnnst
:dways happcn th:i.t there is n sli~it ditTo-cnec bctwocu thc
periods. TI~o consequeucc ie timt though a f~xcJ eHipse rcprcscnts
IIARMONIC MOTIONS.26[33.
thc curve described with sufHcient accuracy for a fcw perlods,tho ellipse! itsc)f iyradually cttangos in -jon'cspondence with t)io
'cr«,noB m t,hujua~uiLudo of e. It becomcs thcroiorc a matter
of interest to cojisider thc system of ellipses rcprescntcd by (2),
supposing a and b constants, but j variable.
SInec tho extreme values of u and are i a, t b respcctivcly,thc cHipse is iti all cases insct-ibcd in thc rectangle whose sidcs
arc 2(ï, 26. Sterling with tlio pitascs in agrcemcnt, or 6=0, wc
havo tlic cHipsc coïncident with tliedia"'ona.l
= 0 As°
emcrcascs from C to ~-n-, thc ellipse opcns out until its equationLeçon) es
From tins point it closes up ngam, ultimutely comciding with thc
otherdiagonal +
=0, eon-csponding to thc incrcMc ofe from ~Tr
to 7r. Aftcr t!iis, as e mngcs from vr to 2~ thé dHpsc retraces
Its course untU it again coincidcs with t!ie first diagonal. TIio
sequoice of changes is exhihitcd in Fig. 3.
Thc ellipse, having a,lrca.dy four given tangents, is compictclydctcrmiucd by its point of contact P (Fig. 4) with thc linc ~=&.
33.] ]LISSAJOTJS' CYLINDER. 27
In order to connect this with e, it is Rufîlcient to observe tha.t
when ~=6. cos27r?:<==l; and thercfore !t=acos€. Now if thé
GJIiptic paLhs bc tl~ ~tit- uf thé BUpcrposHion of tvvo hn.rmc:nc
vibrations of ncarly coincidont pitch, e va.ries uniformiy with the
timc, so that 7~ itself cxccutcs a. l)n.rmo)uc vibration a.lorg ~J.'
witit' a fi-cqucney equal to thé differenco uf thc twu givcn frc-
qucncics.
34. Lissn.]ous'bas shown that this systcm of ellipses may be
rcr'-arded as thc différent aspects of onc and tlic Sfimc enipso
d~cnbcd ou thc surface of a. transparent cylinder. In Fig. 5
~Z~T! represents thc cylinder, of which ~1J3' is a plane section.
Seen from n.u infiultc distance in thé direction of tlie common
tangent at J. to tlie plane sections, tlie cylinder is projcctcd into a
rectangle, and thc ellipse into its diagonal. Suppose now that thc
cylindcr turns upon its axis, cai-rying thc plane section with it.
Its own projectionromains a constant rectangle in which thé pro-
jcction of thc ellipse is inscribcd. Fig. 6 represents the posi-
tion of tlic cylindcr after a rotation through a right angle. It
uppcars thereforc that by turning tlio cylinder round we obtain in
succession ail thé ellipses corrcsponding to thc pa-ths described by
a point subjcct to two harmonie vibrations of equal pcriod and iixcd
amplitudes. Moreovcr if tho cylinder be turned continuously
1 ~tHM~s de CAtM~ (3) LI, 147.
28 HARMONIC MOTIONS. [34.
with uniform velocity, which insurcs a harmonie motion for .P,
wc obtain a complète rcprcsuutation of thc varying orbit
dcscribcd by thc point wh~n Lhc periods uf thc two compunents
differ slightiy, eacli complète revolution answoring to a gain or
loss of a single vibration'. Thé révolutions of thé cyliuder arc
thus synchrouous vlt)i thé béats which woutd rcsult f)'om thc
compositionof thc two vibru-tious, if they wcrc to act in thc s.uuc
direction.
35. Vibrations of thc Mnd hère considercd arc very easily
rcn.Hxt'd expcrimeutn.))y. A Ii(j:Lvy pondulum-bob, hung from a
iixud point by a long wirc or string, descrihes cliipscH undcr t))c
action ofgravity,
which may in particular cases, according to thé
circumstunc'e.s of projection, pass into straight lincs or circles.
But in order to sec thé orhits to thc best advantagc, it is necessary
that thcy sliould be described so quic)dy tl~at thé 'Itnprcssio!i
ou thé retina madc by thé moving point at any part of its course
bas not time tofade materially, heforc tl)e point cornes round again
to its action. This condition is fulfilled by thé vibration
of a silvered bead (giving by reflection a luminous point), winch is
att~ched to a straight mctaUic wire (such as a knitting-necdie),
firmiy clamped in a vice at the lower end. When tiie system is set
into vibration, the luminous point dcseribcs ellipses, which appear
as fine lines of light. Thèse ellipses would gradually contract in
dimensions under thé influence of friction until t!iey subsidcd
into a stationary bright point, without undergoing any othcr
change, wcre it not that in ail probability, owing to somc want
of symmetry, the wire lias s)ightly ditiering puriods according to
thc plane in which thé vibration is cxceutcd. Undcr thèse cir-
cumstances thé orbit is sceu to undcrgo t!io cycle of changes
already cxplaincd.
3G. So far we Itavc supposcd tho periods of thé component
vibrations to be equal, or nearly cqual; thc next case in ordcr of
sitnpiicity is when one is the double of tho othcr. Wc have
M=acos(4~7!-<–e), ~=Z'cos2?!7~.
Tlie locus resulting from thc élimination of t may bc written
1Dy a vibration will aiwaya ho mcaut iu this work a comj)~<<! oyclo of
chfUtgOB.
3G.]CONSONANT INTERVALS.
29
which for ~1 values of e représentaa curvc inscribed in the rect-
angle 2ct, 2&. If e = 0, or 7r, wo ha.vc
représenta p~bolas. FIg. 7shcws thc various curves for U~c
iutcrvals of tLc octave, twuifth, aud ûfth.
To aU thèse systems Lissajous' method of represontation by
thé transparent cylinder is applicable,and whcn the relative
phaseis altcrcd, whctber from thé différent circ~mstanccs of
projectionin diiferent cases, or continuously owing to a sbght dé-
viatior. from exMtness in tho ratio of tbe poriods, thé cylinderwill
app~r to turn, so as to présentto the eye digèrent aspects of thé
sa.DiO line traced on its surface.
37. There is no dinicutty in arranging a vibrating system so
that thé motion of a point shall consist of two harmonie vibrations
in perpendicular planes, with their periods in any assigued ratio.
The simplest is that known as Blackhurn's pendnlum.A wire
~t C-B is fastcncd at ~1 and two nxcd points at tbe samc Icvel.
Tbe bob P is attached to its middle point by another wirc CP.
For vibrations in thé plane ofthe diagram, thc point of suspension
iH practically C, provided that thé wires are sunIcicQtIy stretched
30 IIARMONIC MOTIONS.J37.
but for a motion porpendiculin- to this plane, thé bob turns about
D, can-ying tho wire ~O'j9 witli it. TIic pc.ri~s of vibration in
thc principal planes arc in the ratio of t!.c square roots of CPandDP. Thus if ~C=36'~ the bob describc.s thé figures of thcoctave. To obtain tito séquence of curvc.s
correspondin~ to~pproxnnatc unison, Y~ must bc so ncarly tight, tiiat is
rdativeJy small.
3S. Another contriv~nco called thc kalcidophonc wasorigin-
ally invented by Whcatstoiie. A straight tllin bar of steelcarry'i~a bcad at its uppcr crid is fastcncd in vice, as cxpMncd in a
previous p~ragraph. If the section of thé bar is square, or circule-thé poriod of vibration is indepeudcnt of thc plane in which it ispcrformcd. But let us suppose that the section is a rectalewith unequal sidcs. Tlie stress of tl.c bar-tho force withwhich it rcsists
Lcndin~-is thcu grcater in t!te plane of mc.aterHuc~nc.ss, aud tlie vibrations in this phuie have thé shortcr pcriodBy a suitable adjustmcnt of tho thickncsses, the two poriods ofvibration may bc brought into any required ratio, aud thé eor-
responding curve cx]iibitûd.
Thc defeet in this arrangement is that thc samc bar will r.Ivconly one set of figures. In ordcr to ovurcome tins objectionthé
fullowlng modification lias bccn deviscd. A slip of steci istakcn whosc
rectangular section is very ciongated, so tliat asregards bcnding in onc plane the stiHhcss is so gr~t as to amount
practically to rigidity. Thc bar is divided into two parts, and the
38.]OPTIOAL METIIODS. 31
broken ends reunited, the two pièces bcing turned oa one another
throush a rigtit angle, so that tho plane, which contains thé small
LitickucfiM oi' ojf:, ~<j.,tt'.inK thc gi'L'utthujkMt~ ûi' tho ùti~i-. W:
tlie compoundrod is clamped in a vice at a point bolow the junc-
tion, thé period of thé vibration in one direction, dépend ing alinost
cntircly on thé Icngth of tho uppcr pièce, is nearly constant; but
that in t)]C second direction may be controlled by varying thé
point at which thé lowcr pièce is clamped.
39. In this arrangement thé luminous point itself exécutes
thc vibrations which are to bc obscrvcd but in Lissajous' form of
the experimont, the point of light remains rcaiïy fixed, while its
M~Mf/e is thrown into apparent motion by means of successive
reflection from two vibrating mirrors. A smaU hole in an opaque
scrcen placed close to the iiame of a lamp giycs a point of light,
which is observed after reneetion in thé mirrors by means of a
small télescope. The mirrors, usually of polished steel, arc attMhcd
to thé prongs of stout tuning forks, and thé whole is so disposed
that wlieu thé forks are thrown into vibration thé luminous point
appears to describe harmonie motions in pcrpendicuhn' directions,
owing to tho angular motions of the renccting surfaces. Thé
amplitudes and periods of these harmonie motions dépend upon
thoso of tho corrcspnnding forks, and may bo made sucli as to give
witli cnhanced brill.ianey any of thé figures possible witli tlic
kalcidophonc. By a similar arrangement it is possible to project
tho ri~ures on a scrcen. In cither case they gradually contra.ct as
the vibrations of the forks die away.
40. Thé principles of this cliapter Itavc reccived an important
application in the investigation of rectilinear periodic motions.
Whcn a point, fur instance a particio of a sounding string, is
vibratiug with such a period as to give a note within thc limits of
hearing, its motion is much too rapid to be followed by tl~e cyc
so that, if it be required to know tlie character of thé vibration,
somo indirect mcthod must be adopted. Thé simplest, thco-
retically, is to compound thé vibration undcr examination with a
uniform motion of translation in a perpcndicuhu' direction, as when
a tuning fork dra-ws a harmonie curve on smoked paper. Instead
of moving tlio vibrating body itself, we may make use of a revol-
ving mirror, w!iich provides us with an M~K~e in motion. In tins
way we obtain a. représentation of tlic function charactcristic of
tiLe vibration, with thc abscissa proportional to timc.
33 UARMONIC MOTIONS, [40.
But it often happons that the application of this mcthod would
ho dimcult or inconvénient. Jn such cases we may substituts for
thc uniform n'<u a.~tDu~ui~ vibnt.t'ofi 'fbU~i'bL' ))nri.).1 in i.h<'i
same direction. To fix our ideas, let us suppose that thé point,whose motion we wish to invcstigatc, vibratos vertically with a
period T, and let us examine thé result of combining witli ttus a
horizontal harmonie motion, whose period is somc mu]tip)o of 7-,
say, M/r. Take a rectangutar pièce of paper, and with axes parallclto itsedgcsdraw thé curve rcprescnting thé vertical motion (hy
sctting off abscissa3 proportional to thé timc) on such a scale that
tLc papcr jnst contains ?~ repctitions or waves, and then bend tlic
paper round so as to form a cylinder, with a re-entrant curve run-
ning round it. A point dcscribing this curve in sucli a manno'
that it revolves uniformly about thé axis of thé cylinder will
appear from a distance to combine thé given vertical motion of
punod T, with a horizontal harmonie motion of pcriod ~T. Con-
versely thcrofore, in order to obtain tho représentative curve of
tho vertical vibrations, the cylinder containing t]ic apparent pathmust bc imagincd to he dividcd along a gencrating Une, and
developcd into a piano. Thcre is less difnculty iu couceiviug thc
cylmdcr and thé situation uf thc curve upon it, \vitcn thc adjust-ment of tho periods is not quite exact, for thon tLe cylinder
appears to turn, and the contrary motions serve to distinguisbthose parts of thé curve which lie on its nearer aud further face.
41. Thé auxiliary harmonie motion is generally obtained
optically, by means of an instrument called avibration-microsc-opc
invented by LIssajoua. One prong of a large tuning fork carries
a lens, whose axis is perpendicular to thé direction of vibration
and which may be used cithcr by itself, or aa t!tc object-glass of
a compound microscope formed by tho addition of an eye-pieco
independently supported. In either case a stationnry point is
thrown into apparent harmonie motion along a lino parallcl to
that of tho fork's vibration.
The vibration-microscope may be appHcd to test thé rigourand universality of the law connecting pitch and ~ep't'o~. TIms
it will bc found that any point of a vibrating body -\v)uc!) givesa pure musical note will appear to describe a rc-entrant curve,
when examincd witb a vibration-microscope \\hosc note is in
strict unison with its own. By thé same means thc ratios of
frequeucies characteristic of the consonant intervals may be
41.]INTERMITTENT ILLUMINATION. 33
verified; though for this latter purpose a more thoroughly
acoustical méthode to be described in a future chapter, may be
prcfcncd.
42. Another method of examining thc motion of a vibrating
body dépends upon thc use of intermittent illumination. Suppose,
for exampic, that by mcans of suitable apparatus a series of
cleetric sparks are obtained at regnfar intcrvals T. A vibrating
body, whose period is also T, cxamined by thc light of thc sparks
must appear at l'est, because it can be sccn only in one position.
If, Itowcvcr, thé period of thé vibration differ from T cvcr so
little, the iHuminatcd position varies, and the body will appear
to vibrato slowly ~ith a frequcncy which is thc diffcrcncc of that
of the spark and tliat of the body. Thé type of vibration can
thon be observed with facility.
The séries of sparks can bc obtained from an Induction-coih
whose primnry circuit is periodicauy broken by a vibrating fork,
or by somc othcr intcrruptcr of snrRcient regularity. But a bette)'
rcsult is afforcled by sunlight rendered intermittent with tlie aid of
a fork, whosc prongs carry two small plates of meta], parallel to
the plane of vibration and close togethcr. In each plate is a slit
pM'aIIcl to thc prongs of thé fork, and so placed as to aAbrd a
fj'cc passage throug)i thé plates whcn thé fork is at rcst, or passing
through thé middte point of its vibrations. On thé opening so
formed, a beam ofsunHght is concentrated by means of a burning-
glass, and thc object undcr examination is placed in thé cône of
rays diverging on thc furthcr sidc'. When tlic fork is made to
vibrato by an cicetro-magnetic arrangement, thc illumination is eut
off exccpt when the fork ispassing through Us position of equi-
librium, or nearly so. The nashcs of light obtained by this method
arc not so instn.nta.nouus as clectric sparks (especially when a
jar is connected with thc sccondary wire of thé coil), but in my
expérience thé rcguhu'ity is more perfect. Carc shoultl bc takcn
to eut on' extrancous ]ight as far as possible, and thu cnect is thon
very striking.
A similar result may bc arrived at by looking at thé vibrating
hody through a séries of holes arranged in a circlc ona-revolving
(tisc. Several séries of holes ma.y be providcd on the same
<tisc, but thé observation is not satisfactory without some pro-vision for sceuring uniform rotation.
Ti~ier, 2'/ti/Vn~. Jtm. 1807.
H. 3
HARMONIC MOTIONS.[43.
34
Except with respect to the sharpness of definition, the result is
the samf when the pcriod of thé light is any multiple of tt~t of.
thé vibmtin~ ~c'y. Tiiis pouit. ~HHt bu att,ciided tu ~i)eu thé
revolving wheel is used to determine an unknown frequency..
When the frequency of intermittence is an exact multiple of
that of thé vibration, t!te object is seen without apparent motion,
but generally in more than one position. Titis condition of thingsis sometimes advautageous.
Similar effects arisc when thé frcquencies of thé vibrations
and of thé flashes are in thé ratio of two smaU whole numbers. If,
for example, thé number of vibrations in a given time be half
as gréât again as the number of flashes, thé body will appear
stationary, and in general double.
CHAPTER Iir.
SYSTEMS IIAVING ONE DEGREE 0F FREEDOM.
43. THE matcrial systems, with whosc vibrations Acoustics is
concerned, are usually of considérable complication, and are sus-
ccptible of very varions modes of vibration, any or a!l of which
may cocxist at any particular moment. Indeed in some of thé
most important musical instruments, aa strings and organ-pipes,
thé number of independent modes is theoretically infinite, and
the consideration of several of tliem is essential to the most prac-
tical questions relating to the nature of tho consonant chords.
Cases, however, often present thcmselvcs, in which one mode is
of paramount importance and cvcn if this were not so, it would
still be proper to commence thé consideration of thc general pro-
blem with thé simplest case-that of one degrce of frcedom. It
need not be supposed that thé mode treated of is thé only one
possible, because so long as vibrations of other modes do not occur
their possibility under other circumstances is of no moment.
44, TIte condition of a system possessing one degree of frec-
dom is denncd by thé value of a single co-ordinate M, whose origin
may be taken to correspond to thc position of cquilibrium. TIie
Mnetic and potential énergies ofthc system for any given position
arc proportional respectively to and
r=~~ F=~(i),
whcre w and are in general functions of M. But if we Hmit our-
selves to tlie consideration of positions M!. the ?'y~:e~'<~e ?:eK/A-
&~u)7iOOfZo/</Mt< con'M~on~t')~ e~x~t~, u is a small quantity,
and m and are sensibly constant. On this understanding wo
3-2
ONE DEGREEOF FREEDOM. [4~.36
now proceed. If there he no forces, cither rcaulting from internai
friction or viscosity, or imprcss'~d on the systcm from without, the
\vhole energy remains constant. Thus
y+ 1~= constant.
Substituting for T and V their values, and differentiating with
respect to tho time, wc obtain tlie e~ua-tion of motion
~m + /tW= 0 (2)
of which thé complète integral is
~=(tcos(?)< a) (3),
whcrc ?~=/7):, rcprcscnti))~ a ~Muc vibration. It will bo
sccû that thc pcriod alone is detemuned by thé nature of the
system itself; the amplitude and phnse dépend on cothttcral cir-
cumstances. If tlie difrercutial equation wcrc exact, that is to
say, if T werc strictly proportional to and F to thon, without
any restriction, thé vibrations of thé system ahont itsconDguration
of equilibrium would bc accuratc)y harmonie. But in thé majority
of cases tlic propoi'tionaHty is only approximate, dcpending on an
assumption that tlie displacemeut ?< is always small–how small
depends on thé nature of the particular system and tlie degree of
approximation required and thon of course we must be careful
not to push thé application of thé intégral beyond its proper
limits.
But, although not to be stated without a limitation, the prin-
eipic that thé vibrations of a system about a configuration of
cquilibrium have a period dcpending on thé structure of thé
system and not on the particular circumstances of tlie vibration,
is of suprême importance, whcthcr regarded from thé theoretical
or thé practical sidc. If thc pitch and thé loudness of thé note
givcn by a musical Instrument wcre not within wide limits in-
depcndcnt, thc art of thé pcrformer on many instruments, such
as thé violin a.nd pianofortc, would bc revolutionized.
Thé periodic time
so that an increase in w, or a decrease in /t, protracts thc Juration
of a vibration. By a generalization of the kuguage employed in
thé case of a matcrial particle urged towards a position of eqnHi-
brium by a spring, ?~ may be called thé inertia of thé system, and
44.]DISSIPATIVE FORCES. 37
u. thé force of thé équivalent spring. Thus an augmentation of
mass, or a rc!f).xation of spring, incrcas<?s thé perK'dic t.imc. By
means of this principlc wc may somctimes obtain limits for
the value of a, period, which cannot, or cannot easily, he calculated
cxact)y.
415. Thé absence of atl forces of a frictioual character is an
idéal case, never reahzcd but only approximatcd to in practice.
Tho original cnergy of a vibration is aiways dissipated sooner or
latcr by conversion into leat. But there is another source of loss,
whichthough not, properly speaking, dissipative, yet produces
results of much thc same nature. Consider the case of a tuning-
fork vibrating in ~fMMO. TIic internai friction will in time stop
thé motion, and thé original energy will bc transformed into
heat. But now suppose that thé fork is transferred to au open
space. In strietness tlie fork and the air surrounding it consti-
tute a single system, whose parts cannot be ti'catcd separately.
In attempting, Ilowcver, tlie exact solution of so complicated a
prohicm, wc sliould gencrally bc stopped by mathematical dini-
cultics, and in any case an approximate solution would be de-
sirable. Thc crfect of thc air during a few periods is quite insig-
nincant, and hecomes important only by accumulation. We are
tbus led to considcr its effect as a ~s~<r~?:ce of the motion which
would take place t'~ ~acKO. Ttie disturbing force is periodic (to
thé same approximation that thé vibrations are so), and may he
dividcd into two parts, one proportional to tite accélération, and
the other to the velocity. Thé former produces thé same offect as
an altcration in thé mass of thé fork, and we have nothing more
to do with it at present. Thé latter is a force arithinetica.Hy pro-
portional to thc velocity, and aiways acts in opposition to the
motion, and thcrefore produccs enccts of thc same character as
those duc to friction. In many similar cases thé loss of motion by
communication may bc trcatcd undcr thé same head as that duo
to dissipation proper, and is reprosentecl in thé diScrential équa-
tion with a degrce of approximation sumcicnt for acoustical pur-
poses by a tenn proportional to thé velocity. Thus
M-T XM+ H"M ==-0. (1)
is tlie équation of vibration for a system with one dcgreo of
frcedom subject to frictional forces. The solution is
M=~e'~ cos (~i~. <(}.(2).
38 ONE DEGREE 0F FREEDOM.[45.
If thc friction be so gréât that > thc solutionchanges its
fonn, and no lorger f'orrRsp.nds to nn os<-Hiatnry motion; but In.di acousticai applications A: is a small qu~ntit'y. 'Under Dicsocircumstances (2) mny bc r~u'ded as cxprcssing a harmonie vibra-
tton, whoscMnpiitudc is not constant, but dimiuishcs m
gco-mett-ical
progrc.SHio]), wlicn consi~o-cd aft-crcqu~l iutcrv~Is of
time. Thc difïercncc of thé logarithms of successive cxtronu
excursions Isnc:u-)y consent, :md is ca)!ed t]tc Logar:t!imlc Ducre-
mfut. It is cxpresscd by ~r, if T bc thu puriodie timc.Titc
frcquotcy.dcpcnding on ?~- ~~Invo!vG.s on]y tite sccotd
powcr of A:; so that to thc rir.st order of approximation ~e/c~'o~/t(M ?!0 e~ec~ o~ ~c y)en'o~a principe ofvo-y gênera! appiicatiun.
Tho vibra~on iicrc consideœd is ca!]ed thc/y-ce vibration. Itis tbat cxccutcd hy thc System, when disturbcd from cquiHbrium,and tbcn to itself.
4G. Wc must now turn oui- attoition to anothcr problem, notJcss
Important,–thé bchaviour ofthc systcm, whan subjuctud to aibrcc
varying as a harmonie funetion of thc timc. In ordcr tu savc
rcpctition, wc may takc at once the more gcncral caseijicludinn-
friction. If tho-c be no friction, wc bave on)y tu put in oui- rcsults/< = 0. Thé dincrential équation is
['
This is caDed a. /M-c<~ vibration; it is thc responsc of thc Systemto a force Imposbd upon it from wititout, and in mainta.iued by tho
coutinued opcratioa of that force. T]ic amplitude is proportional
3946.] FORCED VIBRATIONS.
to ~–thc magnitude of tlie force, and the period is the same
as that ofthe force.
Let us now supp<jHu gi~uu, ahd trace tLe effuut on a given
system of a variation in tlie period of thé force. The effects
produced in dinfcrent cases are not strictly similar; hecause tlie
frequency of thé vibrations produced is always the samo as that of
t)ie force, and thcrefore variable in thé comparison which we are
about to institute. Wc n~ay, however, compare thé cncrgy of the
system in different cases at thé moment of passing through the
position of equilibrium. It is necessary thus to specify thé moment
at which the energy is to be computcd in each case, because the
total energy is not invariablo througitout tlie vibration. During
one part of the period tho systcm reçoives energy from the
impressed force, and during thé remainder of thé period yields it
back again.
From (4), if u = 0,
cncrgy ce ce shi~c,
and is thcrefore a maximum, when suie==l, or, from (5), p=n. If
thé maximum kinetic energy bc denoted by wc bave
T=~sm~(6).
The kinctic encrgy of the motion is therefore the grcatest possible,
when the period of the force is tliat in which thé system would
vibrato fruciy undcr the influence of its own ela-sticity (or othcr
internai forces), ~t0i<t ~h'c~'o?! Thé vibration is then by (4)
and (5),
and, if bo small, its amplitude is very grcat. Its phase is &
quarter of a period bohind that of tlie force.
Thc case, where = ?!, may also be treated IndGpendentIy.
Since tho period of tlio actual vibration is the same as that
natural to thc system,
ONE DEGREE 0F FREEDOM.40
[46.If p bc Jcss tha.n ?;, the rctardation of phase relatively to tho
force lies betwech xeru and a qu:u-te!- pcriod, audwhcn is ~reater
tit:m}.[.,butwcchi(.~U!u'~t'~(.'i:m!n.i,.bntfuut~d.In t!)c cusc of a systcln devoid of i'riction, tlie solution is
When is amaller ttian ?~ thc pl.ase uf tiiu vibration agrées withtliat of thc force, but
whcn~ Is thé grever, the sign of thé vibra-tion is clianged. Thé change of phase from complète agreementto complote disagrcemeut, which is graduai wlien friction acts,hère take~ place abruptty as pa.sses through t!ic value 7t. At thésamc tune thc expression for thé amplitude bccomes inanité. Ofcourse this oniy means that, iu thc case of cqual periods, friction7~<~ he taken into account, Ijowever smali it may be, aud liowevcr
insigniricaht its rcsultwben and ?t are not
approximatc!y cqua).Thc limitation as to thé magnitude of thc vibration, to which weare all along subject, must a)so bc borne in mind.
That thé excursion shouid bc at its maximum in one directionwhi!e thé generating force is at its maximum in tho oppositedu-eetion, as happons, for
cxampic.in the canal theory oft!ic tiftc.s,is somcti.ncs considcred a paradox. Any dimculty that may befc)t will bu ronovcd by considering the extrême case, in which thé
".spring vanishes, so t!.at thc natural period is Innnitety lono-. Infact we nced oïdy consider the force acting on the bob of a'com-mon pendutum swinging frecly. in which case t]ic excursion on onesicle is
greatest w)tcn the action of gravity is at its maximumm thc opposite direction. When on thc other hand the inertia ofthé system is very sma)I, we hâve the otticr extrême case in whichthé so-c.Ued equiHbrium theory bccomes applicable, tlie force andexcursjou being in tlie samc phase.
Wi~en t]te pcrioJ of thc force is longer than the nature period,thc cncet of an increasing friction is to introduee a retardationin thé ph:Lsc oft)tc
dispiacementvaryingfrom zero up to nquarterpenod. If, ),owever, the period of thé natural vibration bc tho
longer, thé original retardation of haïf a period is diminished bysomethmg short ofa quarter period; or thé cn'eet of friction is toMc~e tlie phase of thc disphccment cstimatcd from that eon-c-
spond.ng to thc absence of friction. In cither case thé influenceof fr.ct~on i to cause an
approximation to thc state of things thatwou!d prcva)I tffrictioTi wcre paramount.
46.] PRINCIPLE 0F SUPERPOSITION. 41
If a force of nearly equal period with thé free vibrations
vnry s1o\v1y to a maximum and then slowly décrète, thc dis-
pjacement docs not rcach its maximum untd aftcr thé force lias
bcgun to diminish. Under thc opération of the force at its
maximum, thc vibration continues to increaso until a certain limit
is approachcd,and this incrcase continues for a time cven att))ouglt
tlie force, having passed its maximum, begin.s to diminis)). Ou
ttds principic tlie t'utardation of spring tidcs bchiud tlie da.ys of
ucw and full luoou lias bccn cxp]ained'. 1.
47. From tlie linearity of the cquations it follows that the
motion rcsulting from thc simuItanGOus action of any numbcr of
forces is thc simple sum of tlie motions duc to the forces ta~en
scparate!y. Each force c:uises tlie vibration proper to itself,
wthout regard to tlie presoicc or absence of any othos. Thc
peculia-rities of a force arc thus in a manner transmitted into thé
motion of tho system. For example, if thc force be periodic in
timc T, so will be thé resulting vibraLion. Each ])armonic ele-
ment of tlie force will call forth a corresponding harmonie vibration
in tlœ system. But since tlie rctardation of phase e, and the ratio
of amplitudes M is not the samc for thé different components,
the resulting vibration, though periodic in thé same time, is dif-
férent in c/t<t7'KC<c?'from the force. It may happcn, for instance,
that one of thc components is isocbronons, or ncurly so, wit)i thé
frce vibration, in whicli case it will mauifcst itself in thc motion
out of al] proportion to its original importance. As another
example we may consider the case of a System actcd on by two
forces of nearly cqual period. Thé resulting vibration, bcing com-
pounded of two ncarly in unison, is intermittent, accordiug to the
pt'inciples cxphuned in thc last chapter.
To the motions, which arc tlie Immédiate effects of t])c im-
pressed forces, must always be added thc tcrm expressing frec
vibrations, if it be desired to obtain the most gencral solution.
Thus in thc case of one impressed force,
48. Thc distinction betweenybrce~and~'ee 'vlbra.tioQS is very
~Airy'B2'(~t'<n))~n'at'f~Art.328.
ONE DEGREE OF FREHDOM.42 [48.
important, and must bu olearly understood. Thc pcrioJ of t))c
former is detenniucd solcly by thc force whicli is supposed to act
.u Umh~ lic-ni ~rdiQut:, ~hHu Lii:t ut' thc htttcr
dépends un)yon the constitution ofthe system itself. Anothcr point of din'er-
cnce is that so long as the extcrnal influence continues to opcratc,a forced vibration is permanent, being rcpresentcd strictly by a
harmûnic function; buta frec vibration graduallydies away, be-
coming ncghgibic aftcr a timo. Suppose, fur cxample, that the
systcni is :),t rcst when thc force7~ cos ~j{ bcgins to operate. Su.ch
rinitc vaincs must bc givcn to thé constants jd and a iti (1) of § 47,that buth and ii arc initiatty zéro. At first tllen tiiere is a f
frec vibration not less important than its rival, but after a time
friction rednces it to insignificanee, and the forced vibration is left
ill complète possession of the nc!d. Tins condition of things will
continue so long as the force opérâtes. Wlien thc force is removed,thcrc is, of course, no
discontimuty in the valucs of M or !<, but
tho forced vibration is at once convcrtcd into a frce vibrationand the poriod of thc force is cxchangcd for that natural to the
system.
Dm'ing thc coexistence of the two vibrations lu thc earlier partof thc motion, tho curious phc'nomcnon of beats may occu)', in
case the two periods diiicr but siight)y. For, ?! and being nearly
equa), and smali, tlie initial conditions arc approximately satis-
fied by
!<= a cos (~< e) e' cos ~1- ej.
Thcrc is thus a risc and faU in the motion, so long as e' remains
sensible. TI)is intermittence Is vcry conspicuous in the earlier
stages of thc motion of forks driven by cicctro-magnetism (§ G3).
49. Vibrating Systems of one degree of freedom may vary intwo ways according to t)tc values of the constants M and K. Thédistinction ofpitch is
sumcIe)tt!yIntG!!igibic; but it is worth w]nleto examine more closcly the con'sefjncnccs of a grcatcr or less
dcgree of damping. Titc most obvions is the more or lessrapid
extinction of a fi-ce vibration. The enbct in this direction may be
mcasurcd by the numbcr of vibrations wliich must e)apsc bcforethe amp)Itudc is reduced in a given ratio. Initiât )y tho amplitude
may be takcn as unity; after a time <, lot it be 0. Then 6 = c'
40.]VARIOUS DEGREES OF DAMPING. 43
2If = ~T, wc have a; =
log la a, system subject to on)y axT
nnjucru-Ludu~reu
ut'dampmg,
\vctua.y tn.kc
upprox.UTmtcly,
Thi.s gives thc number of vibrations which arc performed, heforc
thea.)nplituduf!iih)to0.Thc inuucncc of damping is aiso powcrfu~y Mt in a, forccd
'Ibra.tion, wlicu thcre is a. :uear approach to isochronism. In the
case ci' an exact equality betwcen a.nd ?~ it is thc damping alone
witich prcvcnts thc motionbecommg m~nite. We might casily
auticipate thatwheu tUc damping is small, a. compara.tively slight
dcvia-tion from perfect isoein'onism wûuld cause a large fa.Hmg off
in thc Magnitude of thc vibration, but that with a larger damping,
thc s:uuc precision of adjustmcut would not bc rcquired. From
tlie enuatious
so that if bc sm~l!, must bc very nearly equa.1 to 7)j lu. ordcr to
producc a, motion ]iot grea.tly Icss than thé maximum.Thé two principal eScets of damping may be compared by
climijiating betwecu (1) and (2). Thé result is
where thé sign of tlie square root must be so cliosen as to make
the right-hand sidc négative.
If, when a system vibrâtes frcely, tlie ampUtude be reduced in
the ratio after x vibrations then, wben it is acted on by a force
(p), thc energy of the resulting motion will bc less than in thé
case of perfect isochronism in the ratio T T~. It is a mattcr of
mdiifcreucc whcthcr thé forced or tlie free vibration bc thé higher;
all dépends on the M/erua~.
In most cases of interest thc intcrval is small; and then, putting
p= ~+8~ tlie formula may be written,
44 ONE DEGREE 0F FREEDOM.['49.
The following table c~culatcd from thcso formulte haa been
givcn by Hcimlioltx'
Ijttcrvfd con-cspon.Ung to a réductionv'Lmtif.nM nfter whiuh tho
of Uto ruM.ttauco to ouc-touth.i'itcnHity of a frco vibrntiou is ro-
y y ~Qducudtoono.tunth.
~=A.
tonc.~.oo'~
19'00
9-50
? G-33
Whuif! tonc. 4-75
tuno. 3.~0
y tono= minor thu-d.
g. jy
7 toile.2-71
Twu whuit'tonea~ major third. ~'37
Formula(4) shcws that, w!ien i.s small, it varies c~~M
'1. asMtt)~~ asa:
50. From observations of forced vibrations due to known
forces, tlie natural period and dampiug of a system may Le deter-tniûGd. TIio formuhu are
~ifH~/ntJf~fyc~ p. 221.
If tlie equilibrium thcory Le known, tlie comparison of ampli-
tudes tells us tlie value of sav
4550.] ] STRING WITH LOAD.
and e is also kuown, whence
51. As bas been ah-cady stated, the distinction of forccd and
frcc vibra.tions is important but it may be remarked that most of
thé forced vibrations which we shall Lave to consider as affecting
a system, take tbcir origin ultimately in the motion of a second
system, which influences thc first, and is innuenccd by it. A
vibration may thus have to be reekoned as forced in its relation
to a system whose limits are fixed arbitrarity, cvcn when that
pystem lias a share in dctcrmining thc period of the force which
acts uponit. On a wider view of thc matter embracing both thé
Systems,thc vibration in.
questionwill be recognizcd
as free. An
example ma.y ma~c tliis clca.rer. A tuning-fork vibrating in air
is part of a compound system including thé air and itself, and
in respect of this compound system the vibration is free. But
although thc fork is influenccd by thc réaction of thé air, yet thc
amount of such innuence is smaU. For practical purposes it is
convenieat to eonsidcr the motion of the fork as givcn, and Lhat of
thé air as forced. No crror will be committed if thé f<c<:ta~ motion
of the fork (as innucnccd by its sun'oundings) be takcn as tbe
basis of eaicutation. But thé peculiar adva.ntagc of tlils mode of
conception is manifcstcd in thc case of an approximatc solution
bcing rcquired. It may then sumce to suhstitute for thé actual
motion, what would bc tbc motion of thé fork in the absence of
air, and afterwards introduce a correction, if uecessary.
52. Illustrations of the principlesof this chapter may bc
drawn from ait parts of Acoustics. Wo will give bc're a few
applications which deserve an early place on accouut of their
simplicity or importance.
A string or wire J.CJ3 is stretched bctwccn two nxed points
~1 and and at its centre carries a mass J~ which is supposed to
bc so considérable as to rcndcr thé mass of the string itself ncgli-
gibic. WIten is pulled asidc from its position of equilibrium,
and thcn Ict go, it exécutes along thé lino C~ vibrations, whicb
are the subject of inquiry. C= 6'~ = M. C'.V= x. Thé tension
of thc string in the position of equIHhrium dépends on the amount
of the stretchiug to which it has been subjected. In any othcr
4G ONE DEGREE 0F FREEDOM.[52.
position the tension is ~reatcr but we limit ourscivcs to the case
of vibrations so small that tt~e additiona! strotching is a ncgJigibJefraction of the who)c. On th~ (~ncHii~n thc ~<i)i rn:~ bc
treated as con&tant. We dénote it by y
Thus, Idnetic cncrgy=
~ï;
Thé amplitude and phase dépend of course on thé initial cir-
cumstances, being arbitrary so far as thé dinforcntial équation is
conccrned.
Equation (2) expresses thé ïnanner in which 7- varies with eachofthe Independent quantities V.~a: resultswhich may all bcoutained by considération of the (~MCHs~~ (in the tcchnica! sensc)of the quanti tics involved. T!~G argument from dimensions is sooften of importance hi Acoustics tliat it may bc wcll to considerthis first instance at Icngtit.
In the first place wc must assure ourselvcs tliat of all thé
quandties on which T may dépend, thé only oues involving a
53.JMETHOD 0F DIMENSIONS. 47
référence to thc three fundamental units–of length, time, and
mnss–ure a, and T. Let thé solution of the problem bo
wnLLeu-
This equation must rcta.in its form unchanged, whatever may
l)e the fundamcntal units by means of which thé four quantities
arc nnmerically expressed, as is évident, when. it is considered
that in deriving it no assumptions would be made as to thé mag-
nitudes of those units. New of all tlie quantities on which f
dépends, T is the oniy onc involving time and since its dimen-
sions arc(Mass) (Length) (Ti.me)'
it follows tl~at whe!i ? and ~f
arc constant, ïoc.T' otherwise a change in thé unit of time
would necessarily disturb the equation (3). Tins being admittcd,
it is ca~y to see that in order that (3) may be independent of the
unit of Icngth, we must Imve r ce T"~ n~, when Is constant and
finally, in order to secure indcpcndence of the unit of mass,
Therc must be no mistake as to what this argument does and
docs not prove. Wc Iiave nMKMte~ that thcrc is a deHnitc
periodic time dcpeuding on no other quantities, having climen-
sions in spacc, time, and mass, t!ia.n thosc aLove mcntioncd. For
example, we hâve not proved that r is indépendant of thc ampli-tude of vibration. That, so far as it is truc at ail, is a consé-
quence cf thc linearity ofthe approximate dinercntial équation.From the neccssity of a complète cnumeration of all the
quantifies on which thé required rcsult may dépend, thc method
of dimensions is somewhat dangerous but when used with properCt~re it is
unqucstionably of great power and value.
ONE L.MCREE 0F FREHDOM.F~.
.'3 1 lie solution of thé présent problem might bo made thcfoun~tion of a ,nethcd for Lhe absolute n~asurerncnt of pitch.
pnncip~J impedunc-nt toaccuracy would prubabjy.be t!to
difBculty oiu~ku~ suf!ictcutfy i~ iu relation to thé m~ of
tlie ~u.c, without at tlie samc timclo~crin~ thé note too much in
thé musical scalc.
T)ic wirc may bo strctc)ied by wcight ~t~chcd te itsfur hcr en<) beyo~i bndgo or pulfey at Thé pcnodic timewouidbcc:dcu);),tcdfrom
T).c ratio of = ~i, t).e balance. If r. be ,no..suredin fect, aud~= ~.2. tl~c pcriodic timc is exprcs.sed in seconds.
~n~ anmusical the .vcight, Instead of beingconcentratcdin thé centre, is
uniformtydistnbuted over its !cn~
~evertiK.Icsst)ic
présent problem gives some ide~ of thé n~tu.'c oi-the gr~vest vibration of snch string. Let t..
compare thé two
c~cs
morec osoJy, supposingthc amplitudes of vibration t)te same
at thé jmddie point.
-When thé uniformstring is .straight, thé moment of passin~tLro~h thé position of
cquilibruua, its dirent parts are a~
54.] COMPARISON WITH UNIFORM STRING. 49
mated with a. variable velocity, increasing from either end towards
thc centre. If we attribute to thé whole mass tho vclocity of tho
centre, it la évident that thé kinetic cnergy will bGcousidcrab!y
ovcr-estimated. Again, at the moment ci' maximum excursion,
thé uniformstring i.s more stretched than its suhstitutc, winch
foltows thc straight courses ~1~ and accordingly the poteu-tial cncrgy is dumnished ùy t)to substitution. TIiG concentration
"i thé mass at the middie point at once increascs tho kinctic
cncrgy whcn a;= 0, and decreascs ttte potential energy when ~-= 0,and thercforc, according to the principle explained m 44, prolon'~the pcriodic timo. For a string thon the period is less than that
catcuiatud from the formula of the last section, on the suppositionthat ~1/ dénotes thé mass of the string. It will afterwards appeart)jat in order to obtain a correct result we should !)avc to takc in-
4 4stcadof.Von!y-~V. Of thefactor-~ hy far thc more import-TT TT
ant part, viz. is duc tu (,he difTcrcnce of tlie kinetic énergies.
55. As another example of a System possessing practicu.Hy but
one dcgree of freeclom, let us considcr tlie vibration of a spring, one
end of which is clamped in a vice or otherwise held fast, wliile thc
otiter carries a heavy mass.
In strictncss, this System !iko tho last lias
an innnite numbcr of Indcpcndent modes of vi-
bration but, whcn thc mass of t!tc spi'mg is
)-e!ativc!y sn-i:d), ttiat vibration which is ncarly
indcpcndont of its inci'tla. buconics so much thé
mostimportant t!)at tho othcrs may bo ignored.
Pusinng this idca, to it.s limit, we may regard the
spring merety as tite origin of a force urging thé
attaehed mass towards thé position of equilibrium,
and, if a certain point be not excecded, in simple
proportion to thc disp!acement. Thc result is a
harmonie vibration, with a period dépendent on
thé stinhess of tho spring and the mass of the
toad.
56. In conséquence of tho oscillation oi' the centre of inertia,H~ci-e is a, constant tendency towards the communication of motion
to tlie supports, to resist which a.dequate!y thé latter must be
very ni'm and massive. In ordcr to obviate this inconvenience,
R. 4
~0 ONE DEGREE 0F FREEDOM.[5G.
two prccisely similn-r springs and lo~ds m~y bc mountcd outlie same frame-work in a symmctrical manncr.
If thc two loadspcrform vibrations of cqual amp)i-
tude in such a, manner that the motions arc a.Iwn.ys
opposite, or, {m it may otherwise bc e.xprcsscd, with
a phasc-tiiHcrcucc of !m]f a period, thc centre of
inertia of thc whole system rcmains at rcst, and
thcro is no tendency to set thc fra.mc-work into
vibra.tion. We shaU sec in a future chapter that
this peculiar relation of phases will quiddy esta-
b)ish itself, wt~tever may be tho original disturb-
a.nce. In fact, any part of tho motion winch does
not conform to the condition of Icaving thc centre
of inertia unmoved is soon extinguished by damp-
ing, unless indccd thc supports of tbe system arc
more than usually nrm.
57. As in our first exemple wc found a rough illustration oftho fundamental vibration of a musical string, so hère with tlie
spring and attachcd load wc may compare a uniform slip, or bar,of elastic material, one end of which is securejy fastencd, such forinstance as the ~:<e of a )~e~ instrument. It is truc of coursethat tlie mass is not coucentmtcd at onc end, but distnbutcdover thé whole Icngth; yet on account of tlie smallness ofthc motion ncar the point of support, thé inertia of that partofthe bar is of but little account. Â~e infer that thc fundamentalvibration of a uniform rod cannot be very dincrcnt in cbaractcrfrom that which we ])ave bcen considering. Of course for pur-poses rcquiring précise calculation, the two Systems arc sufncientlydistinct but where t!ie object is to form clear idcas, precision mayoften be
advantagcously cxchanged for simplicity.In the same spirit we may regard tlie combination of two
springs and loads shcwn in Fig. 13 as a représentation of a
tuning fork. This instrument, which bas been much improvcdof late years, is indispensable to the acoustical investigator. Ona large scale and for rough purposcs it may bc made by wcidinga cross piece on the middle of a bar of steel, so as to form a T, and
then bending the bar into t!io shape of a horse shoe. On thé
handle a acrew should be eut. But for thé botter class of tunmgforks it is préférable to slape thé whole out of one piece of stecL
A division running from one end down the middic of a bar is first
S~J TUNJXGFORKS. 5j I
madc, thctwo parts opcned ont to form the prongs of the fork,and thé whole workcd by tho iiammer and n!u into thc rcquircdshape. T)ic two prongs must bc cxactiy symmctricat with respectto a
plane passing through the axis of thc liandie, in ordcr that
during t!ie vibration thé centre of incrtia may remainunmoved
–unmoved, tiiat is, in thc direction in which thc prongsvibrato.
Thc tuning is cnected t)ms. To make thé note higher, thé
équivalent incrtia of thcSystem must bc rcduccd. This is donc
hy nling away t)ie ends of thc prongs, cithordiminishing their
thickncss, or actuaiïy sliortening thcm. On the other hand, toJowcr the pitch, tlic substance of the prongs ncar thc bcnd maybe rcduced, the effect of which is to diminish thé force of the
.spring, Icaving t)te inertia pmctically unchangcd or the inertia
may be increased (a mcthod which would be préférable for tcm-
porary pm-poscs) by loading thc ends of thc prongs with wax, or
othcrmaterial. Large forks arc somctimus provided with movc-
able weights, which slide along thc prongs, and can be nxcd in
any position by screws. As thèse approach thc ends (whcro thc
vetoeity isgreatcst) the équivalent incrtia of thc
System incrcascs.
Inthis way a considérable range of pitch may bo obtained from
one fork. TJ)c number of vibrations per second for any position
ofthe weights may be markcd on thé prongs.Tite relation bctwcen the pitch and thc''size of tnnin~ forks is
rcmarkablysimple. In a future chapter it will be provcd that
provided the material remains thc samc and tho shape constant'tt.c period of vibration varies, dircctty as t)te linear dimensionTIrns, if t!ic linear dimensions of a tuning fork be doubicd, itsnote falls an octave.
58. Thc note of a tuning fork is a ncarly pure tone. Imme-
diateJy after a fork is struck, high tones may indccd be hcard,
con-espondingto modes of vibration, whosc nature will bc subse-
qucnHy considered; but thèse rapidiy die away, and cven whilc
s they exist, they do not b!cnd with thé propcr tone of the forkpart~y on account of thcir very high pitch, and partly bccause
ihey do not bchng to its harmonie scale. In the forks examincd
~.byHelmhoitz the first of thèse overtones had a frequcncy from 5-8
to n-G timcs titat of the proper tone.
Tunmg forks are now generaUy supplied with résonance cases,whosc effect is
greatly tu augment the volume and pnrity of the
4–2
53 ONE DECREE OF FREEDOM. [58.
sound, according to principles to be hcreaftcr dcve!opcd. In
oiJer to excite thon, a -viotin or ccHo bow, wcll supp)icd with
')~ :~dr~t.t .Cr~~ rh< prongs'u~'u dit'<<t)uuof\'b''ai.u)r'.
Thc souud so prothccdwIU last n minute or more.
R~. As standards of pitch tuningforksarcinvaluabic.T)~
pitclt of organ-pipcsvaries with tlic température and with thé
pressure of t!ic wind; th~t of strings with thé tension, wltio]) cnn
nuvcr be rctaincd constant for long; but n. tuning fork kcpt ctc.m
and not subjccted to violent changes of températureor magnct-
ixation, prcscrvcs its pitch with gréât fideUty.
By means of bcats a. standard tuning forl. may bc copicd with
very gréât précision.Thé nnmbcr of béats !)card iu a second is
t))u dinurencc cf thé frc()uencicsof thc twu tcncs which produce
thcm; so that if thc bcats can 1)0 madc so s)ow as to occupy hah'
a minute cach, ti)C frequcncicsdiH'cr hy on)y l-3()th of a vibra-
tion. Still grcatcr precision might be obtaincd by Lissajous
(~ptic:tl incthod.
Very sh)\v bcats bcing dimcult of observation, In consc<)ucncû
uf tho unccrtainLy whcthcr a faHing ofi in thu sonnd is duc to
interférence or tu thc graduât dying away of tho vibrations,
Schcib)cr adoptcd a sonicwbat modihcd plan. Ho took a fork
~ightiy différent in pitchfrom tho standard–wbcther highci- or
lo~cr is not materia!, but wc will say, tower,–and countcd tbc
'Tmmber of bcats, when they were soundcd togctbcr. About fuur
béats a second is thé most suitab)c, and thèsemay
be countcd for
perbaps a minute. Thé fork to bc adjustcd is then made sligbt]y
higbcr than the auxiuary fork, and tuncd to givc wit)t it prccisdy
tlie samc numbcr of beats, as did thé standard. lu tins way ft
copy asexact a~ possible
is secured. To facilitate Ute counting
of thc béats Scbcibk'r cmployed pendulums, whose periods of
vibration could bc adjusted.
60. T)ie mcthcd of bcats was aiso employed by Scheibler to
détermine tbe al)so]ute pitch of lus standards. Two forks were
tuned 'to an octave, and a number of others prcparcd to bridge
ovc-r thc lotcrval by stcpsso smaU tliat cacii fork gave with its
immédiate ncighbourshi t!œ séries a numbcr ofbcats that could
be casily couutcd. T!tC din'urencc of frcqucncy con'csponding to
each stcp was observcd with aU possible accuracy. Thuir sum,
being tlie din'crencc of fi'cquencies for the intcrval of thé octave,
was 'quai to thc frcqnency ofthat fork which formcd thé starting
(;0.'jSCHEIBLER'S TONOMETER. 53
point at thé bottom of tho séries. Thé pitch of thé other forks
couldbc dcduccd.
If consécutive forks givc four béats per second, C.5 in a.ll will
bc rnquirod to bridge over thc intcrval frora c' (2.')G) to c' (5L2).
Un thisaceountthc mctitod is laborious; but it is probably thé
most accm-atc for tl)C original dctcrmina.tion of pitch, as it Is liabtc
to no ct-rors but such as care and repetitioti will clhninatc. It
mn.y bc obscrvcd tliat thc cssctYtiat thingis tho mcasurcmcnt of
t)tc ~er~ce of frcqucncics for two notes, whosc ~o of frcqucn-
cics is UKlcpcndcnt]y known. If wo could be sure of its accm-a-cy,
thc Intci-v:d of thé nfth, fourt)i, or cvcn. major third, might bc suh-
stitutcd for thé octave, with thé advantagc of rcducmgttie number
of thé ncccssary interpolations.It is proba.b!c tttat with thc aid
of optic!d mcthods t))i.s course might bc succcs.s(ut!y adoptcd, as
thc con'csponding Lissajous' ngurcs a.rc casily rccognised, and
thcit- stcadinoss is a vcry sovcrc test of t!ie accm'acy with whicb
tt'e ratio isattainud.
Thc frcqnency of large tuning forks may bc detcrmincd by
aHowiug them to trace a harmonie curve on smokud papcr, which
tnny couvcnicnHy bc mountcd on thc circumicrenco of a rcvo)ving
drmn. Thu muubci' of wavcs cxccutcd in a second of thnu givcs
thcfrcqucncy.
In many cases tbc nsc of Ittterniittcnt Hturnination duscribcd
in § 4-2 givcs a convcniunt )net))odof dctcrmining an nnknown
frcqucncy.
(il. A scrics of forks ranging at snndi int.crv:us over an octave
is vcry uscf\d for thé dcturtnination of thc frcqucncy of any
)nusic:d note, and is caUcd Schuibtcr's Tonomctur. It may a~o
bc nscd for tnuing a note tu any desirct). pitch. In cilber case
thu f')-u(Utcncy of thé note is dctermincd hy tl)e nuinher of beats
\vhic)i it givcs with thc i'orks, which lie aearest to it (on cach
sidu) in pitch.
For tuning pianofortcsor organs, a. set of twelvc forks may be
uscd giving thc notes of thc cbromatic sealc 0)1 tho equal tempé-
rament, or any dcsircd system.Tbc corrcsponding notes are
adjusted to unison, and t])C otbcrs tuned hy octaves. It is betto-,
I~owevcr, to prépare thé forks so as to givc four vibrations per
second k-ss than is above proposed. Eacli note is thcn tuncd
little higher than tlie corresponding fork, until thcy givc when
sounded togcthor cxactiy four béats in thc second. It will be
54 C~K DEGREE 0F FREEDOM.[61.
ubservcd that tho addition (or subtraction) of a constant number
to thc frcfptencicsis not the samc thing as a more displaccnicut
ofthescatcinabsolutcpitch.
In thé ordinary practicc of tuners a' is takcn from a fork, and
tlie other notes dctermiued by cstinm.tion of (Iftt)s. It will bc
rcmcmbcrcd that twefve truc ~fths arc slightiy in excess of seven
uct.ivcs, so thitt on tho equal tcmpcranicut System cn.ch f)ft)~ is :).little fiât. Thc tuner procccds upw:irds from Ly succci-isivu
fifths, coining down au octave aftcr about every altet'ttate stop, m
ordor tu reimutt in nearly the same part of the scfdo. Twcivc
Hfths should britig ])itn back to «. If this Le not thc case, the
wurk must bc ruadjustud,unt,iJ all the twe)vc ftfths arc too fhtt by,
as nearly as can bcjndgcd, thu samo sma!) amount. Thu incvita-
biu o-ror is thcnhnpartiaUy di.stributed, an<t rotdcrcd as little
sensible as possible. Tt)c octaves, of course, arc all taned truc.
Thé fo!Iowii]g numbers indicatc thc order in whic)t the ilutes maybc takcn:
c'c'~
e'y' a'
M~b' c" c~ c"
JJ
13 51G 81911 314 6 17 9 1 12 415 7 18 10 3
In practicc thc cqual tempérament is only approximatcly at-
taincd but this is pcrhaps not of muc)t conséquence, cousidering
titat the systcm ainied at is itself by no mcatis pcri'uction.
Violins and other Instruments of that class arc tuncd by truc
nfthsfrom«'.
G2. In illustration of/o;'C6(Z vibration let us consider the case
of n. pendutum whosc point ci
xouta! harmotuc jnution. is
thcboba.ttachedbya.fincwu'c
to a movcn.btc point 7~. 07'*=
7'() = and .r is thé honxol-
tal co-ordiniitc of (). SInco tlie
vibrations arc supposed sina.)!,
thc vertical motion !n:).y Le
~cgiccted, and tho tension of
thc wlrc Cfjuatcd to thc wcight
of (,). Hunce for t))c Itorizonta!
support is subjoct to n small huri-
motion;e+~+.(.t;J=0.
C2."]COMPOUND PENDTJLUM. 55
New oe cos~<; so that p)itting~=)~, our équation takes
tlie form ah'cady trea-tcd of, viz.
.v + A:~ + )~ = cos~
If~) l)e equa.1 to ')!, thc motiou is limited o!i1y by the friction.
Thc a~sumed horizontal harmonie motion forP maybe rcajized by
mc:ms of a, second pcndulum of massive construction, which can'Ies
.P witli it in its motion. An cfncicntarrangement is shewa in
t)tc ngnrc. /t, .H arc iron rings scrowcd into a beam, or other nrm
support; C', D similar rings attachcd to a stout bar, which carrics
cqua! hcavy weights A', :tttac!K!d ncar its ends, and is supportcd
in a hurizo)it,al position at riglit angles to thé beiuri by a wirc
passing through thc fuur rittg.s in thc tnanner shcwn. Whcn tlie
pcndulutii i.s )nndc to vibratc, n. point m thc rudmidway
bctwcot
C' and D exécutes a hiu'mouic motion ni a direction paridtcl to
6'D, and jnn.y bo nmdo thé poitit of a.tta.chmcnt of auother pcn-
dutunt -Z~. If ttte wcights A~ and be vcry grcat in relation
to Q, t)jc uppur penduhun swings vury ncn.r)y in ils 0~1 propur
poriod, and induccs in () a. furccd vibr:<.tion of titc s!t.nic period.
\Vhcn thc ]c!)gth ~Q is so adjusted that thc nattu'id pc!'i<j(!s oftite
two pcnduimns arc nearly t)ic s;unc, Q will bu tLrown into viuk'ttb
motion, evun t!)&u~h thc vibration ot' j! bc of but niconsidura.bln
ampHtudc. ln this case the diHcrencc of phase is about n (~)artcr
5G ONH DEGREE OF FRKEDOM.[62.
of a. pcriod, by winch amount thé uppcr pcndulum is in a.dvancc.If the two pcriods hf vnry dHTo'cnt, thc vibrations ~either a.g)'ec
or arc compictcly opposed in p!)asc, accordin~ to équations (4)
and (5) of § 4C.
63. A vo'y good cxa)np)c of {t furccd vihruttcn i~i aÛbrdcd by
n. ibrk under thc iunucnec of tui intermittent ctcctric cui'rcnt,
~hoso period is ncarly cqual to its own. ~).CZ? is the fork; 7?a
sma!) c)ectro-magnct,formed by winding insula.tcd wire on nn iron
corc of tho shape shcwu ni E (simila.r to titat known as 'Sioncn.s'
armature'), ~nd supportcd betwccu tho prongs ofthc fork. Whcu
an intermittent current i.s sent through thé wire, a periodic force
acts uponthe fork. This force is not cxprcssibic by a simpic cir-
cular fonction; but mn-y bc cxpandcd by Fouricr'.s theorcm lu a
scrics of sucli functions, ha.vlng poriods T, T, T, &c. Ifnny of
thcsc, of not too small amplitude, bc ncarly isochronous with the
furk, thc latter will be canscd to vibrato othcrwisc t]tc effect is
insigninca.nt. In wbat follows wc will suppose that it is the com-
plete pcriocl T whicb ncarly ngrcc.s witlt tliat of the furk, and cou-
scqucntly rega.rd thc séries expressing thé pcriodic force as reduccd
to its first term.
lu order to obtain t))C maxitnum vibration, thc fork must be
cai'cfuHy tuncd hy a small siiding pièce orby w:LX', uutit its j~turat
pcriod (without friction) is cfpud to that ot' thé force. Dus is bcst
cloue by actual trial. Witen tho desired c~uidity is approacticd,
and thé fork is a)!owcd to start from rc'st, thc ibrccd and com-
ptctncntary frce -vibration arc of nearly cqual amplitudes a.nd
frequencics, and therefore (§ 4-8) in thc bcginning of thc motion
produce ~ef< whose stowncss is a measuro of the accm'acy of
y"r Uu~ j'urposc \\nx mny <'onvp))ifnt]y Lo fioftcncd Ly )nc'IUnK it wiU) a )itt)<i
txrjK'ntino.
G3.]RELATION 0F AMPLITUDE AND PHASE. 57
tho adjustmcnt. It is not until a-fter tl)c froc vibration lias bad
time to subside, that thé motion assumes its peru.anent ch~'acter.
T))C vibrations ofa tuning fork properly constructed and inounted
arc subject to very little damping; consequcntlya vcry slight
déviation from perfect isoclironism occasions a markcd falittig off
in thé intcnsity of the résonance.
The nmpHtudo of thc forccd vibration can bc obsci-ved with
sufïicicnt accuracy by thc car or cyc but thé expérimenta! verifi-
cation of thc relations pointed out by thcory bctwccn its phase
and that of thé force which ca.uscs it, re<~ures a modined at'rangc-
mcnt.
Two similar cicctro-magncts acting on similar forka, and in-
cluded in thc samo circuit, arc excitcd by the same Intermittent
current. U])dcr thcsc circumstances It is ctear tha.t thé Systems
will bo thrown intn sunDar vibrations, becausc thcy arc actcd on
by cqual forces. Tliis similarity of vibrations rcfcrs both to phase
at)d amplitude, Lot us suppose now that the vibrations arc
effected iu pcrpendicula.r directions, and by mcans of one of
Lissnjous'mcthods arc opticallycomponndcd.The resulting ngure
is ncccssaritya. straight
lino. Starting from tho case in which thc
o.mpHtudes are a maximum, viz. whoi tbo natural pcriods of both
forks arc tbc same as that of thc force, lot onc of them bc put a
little out of tnnc. It must bc rcmonbercd that whatevcr their
natural periods may be, the two forks vibrato in perfect unison
with thc force, and thcrcfore with onc another. Tho principa.1
Ciffcct of thc dift'urcnce of tbc natural periods is to destroy the
synchroïlismof phase.
Thc straight hue, which prcviousiy rcprc-
scnted the compound vibration, bccomcs an ellipse, and this
i-cmaius perfccHy steady, so long as thé forks arc not tonchcd.
Originally thc forks arc botb a quartcr period behind thc force.
~Vhcn thc pitch of one is slightiy ]owcred, it falls still more bchind
the force, and at thc samc timc itsamplitude
diminishcs. Let titc
diifcrcncc of phase betwccn thc two forks bc e', and tlie ratio of
amplitudes of vibration (t: (t.. Thcu by (H) of § 4C
M = Mycose'.
ONEDEGREE0F FREEDOM.58[C3.
It appears tbat a. considérable altération of phase ni either
direction may be obtaincd without very materialty reducin"' thé
amplitude. Whcn one furk is vibrating at its maximum, thc
othcr may be made to dinfcr from it on either sido by as muc)t as
CO" in phase, without lo.sing moro than t)alf its amplitudu, or by as
much a.s -I<5",without losing more tha)i Iiaïf its e?M)'~y. By aHow-
ing one fork to vibratc 45" in advance, and tbc othcr 45" in arrcarof t)te phase corresponding to t]ic c:~c of maximum
résonance, wo
obtain a phase diScrcncc of 90" in conjonction with an cquality of
amplitudes. Lissajous' ngurc then bccomes a cir~e.
G4. Tbc intermittent current is best obtaincd Ly a fork-
interrupter invented by Hchnbottz. T)tia may consist of a fork
and cicctro-magnet mountcd as before. TIie wires of thc ma~nctarc connected, ono witb ono po!c ofthcbattcry, and thé othcr with
a mcrcury cup. Thc ot]ier pôle of tbc battcry is connectod witha second mcrcury cup. A U-shapcd rider of insulatcd wirc is
carried by t!)c lower prong just over thc cups, at sucb a Iieigbttha.t during the vibration thé circuit is
altcrnatejy made and
brcken by titcpassage of one end into and out of thc mercury.
T)ie other end may bc kept pcrmancntiy immcrscd. By mcans
of t!tc pcriodic force t)tus obtaincd, thc cnuct of friction is com-
pensatcd, and thc vibrations of thé forkpcrmancnciy maintamed.
In order to set anotbcr furk into forced vibration, its associatcd
ctcctro-magnct maybc includcd, either in tbe sanicdrivix'Y-circuit
?'<)«'))t;~?)t<~o~t't), p. li)0.
The following table shows thé simu1ta,neous values of a c<a.nde'.
e 0
e
1-0 0
-!) 25°50'1
-8 3C° 52'
.7 4.T' 3-t'
'C 53°7'
-5 GO"
'4. 66"25'
'3 72° 32'
-2 78° 27'
-1 84.° 15"
G4.]FODK INTEBRUPTER. 59
or m a, second, whose periodic interruption is effected by another
rider dipping luto mci'cury cups'.
Tho ??~(~<& ~«/ of tm.s kind ui seti'-act-ing instrument is
often imperfcctiy apprehcudcd. If the force acting on thé fork
dependcd only on its position–on whetlier tlic circuit were open
or eloscd–tbû work donc in pressing ttirough any position wouid
bc undono on tlie return, so that aftcr a, complète period therc
would be nothing outstanding by wliieh ttie effect of thc frictional
forces could bc compcnsa.tcd. Any explanatiol whic!i docs not
take accouut of' thc rctardation of thc currcnt is wholly bcside the
mark. Thc causes of retfM'datiou arc two irregular contact, and
scJf-mduction. Wltcn the point of thé rider nrst touches thé mer-
cnry, thc cicctnc contact is imperfcet, probahly on account of
adhcring air. On thc other ha.ud, in leaving tlie mcrcury tho
contact is prolonged by the adhésion of tlie hquid in the cup to
thu amaigama.tcd wire. On botli accounts thé currcnt is retarded
behiud wliat would correspond to thc mcrc position of the furk.
But, evcn if the resistance of the circuit dcpended only on the
position of thé fork, thc current would still be rctarded by its self-
hiduction. However perfect thé contact may be, a finite current
efumot bo gencrated until aftcr the lapse of a finite time, any
more ttian in ordinary mechanics a finite vclocity eau be suddenly
impressed on an tuert body. From whatcvcr causes arising", the
effect of thé rctardation is that more work is ga.iued by thc fork
during the retreat of tlie rider from tlie mcrcury, tlian is lost
durin<T its entrancc, and thus a, balance remaitis to be set off
against friction.
If t!)C magneticforce depcuded onlyon t]tc position of the fork,
thé phase of' its first harmonie component nught bc considcred to
be ISO" in advance of that of tlie fork's own vibration. Thc re-
1 1 Lnvo arr<mgc<l aoveral iutcrruptora on tho nbovo pJfH),un t)io componont
n)trtn being of homo manufacture. Tho forks woro mado by tho vilittgo blucksmith.
Tho eupn conxiat.od of iron thimbloa, (ioldored on ono omi uf copier slips, tho
further entl being ticrowod down ou tho bo.so board of tho instrument. Scmo
tuoms of adjuating tho IcYcl of tho morciu-y surfaco ia necosMry. lu Hcimholtx'
intcrruptor a horso-.shoo cloetro-magnot embraemg tho fork in adoptcd, but I nul
inctmod to profur tho prosent arranHcmcnt, nt auy rate if tho pitch bo low. In
somo cases a greater motive powor iHobtuinod by n horffo-fihoo magnot acting on n.
Kuft iron Mmftturo carried horizontally by tho uppor prong aud porpoudicuhtr to it.
1 h<woususUy found a singlo Smco cull suûicieut buttery puwor.
Any desired rctardtt.tion might bo obttdued, in dcffmH of ûthor mcans, by
attnching tlio rider, not to tho prong itscJf, but to tho fnrthor oud of n liglit
Hirnight spriug cnrrieù by tho prong and Bet iuto forccd vibration by tho motion of
its point of nttttclnuent.
60 ONE DEGREE OF FREEDO~f.[G4.
taxation apoken ofrcdnccs this advance. If thé phasc-diu'crcncebe rcdueed to 90", thé force acts in thé most favourable manner,
~tU.'t.x~t p'-K-~bh.~Yibmuon.i.sptudtiL-cfJ.
It is important to notice t))at (cxccpt in thc case just, rcfcrrcd
to) the actual pitch of ttie mterruptcr dKFcrs to some cxtent from
tbat natur.'d to thc fork according to thé hnv cxprcsscd in (5) of
§ 4G, e being in thc présent case a. prescribcd pbase-difïcrenco
depcnding on t!)c na.turo of thc contacts :ind <Lo jnagnitudc of thc
selt'-uiducti.on. If thc Intermittent currcnt hc empioycd to drive
a, second ibr]<, thc maximum vibmtion i.sobiained, wlien thc fre-
'jucncy of thc fork coincides, not with thc natural, but with tbc
modHic-d frcqn<jncy of t)te inten'ttptcr.
Thc déviation of a. tunmg-fork intcrrupter from its natur:d
pitch is practica.Hy very smitt); but thé fact that such a déviation
is possible, is a.t nrst sight rather surprising. Tho explanation (Inthé case of a. sma,H rctarda.tion of current) is, that during t)u),t, iia-If
of thé motion in whieb thé pt'ongs tu-c thû most scparatcd, thé
eicctro-magnet acts in aid of thc proper recovering powcr duc to
rigidity, and so natnrally mises Hie pitc)). Wha.tc'vcr tlie relation
of phases may be, Hic force of thu magnct n):Ly be dividett into
t\vo parts rc.spectivc)y proportional to tho vclucity and (tisn)ace-mcnt (or acculcration). To ti)c nrst exclusi-vety is dnc t]ie sostain-
ing powcr of thé force, and to thc second the atteratioti ofpitch.
G5. TI)e gênerai pbcnomenon of résonance, thnugh it cannot
bc exhaustively considcrcd undcr tbc hcad of onc dcgrco of free-
don), is in thé main referab!e to the same goncral prineipic.s.AVhen a forced vibration is cxcitcd in onc part of a. system, all
the other parts are aiso Innucnccd, a vibration of thc same pcriod
bcing cxcitcd, whose amplitudo dépends on thc constitution ofthe
systum eonsidercd as a whote. But it notunfrcquently happons
tliat intcrcst centres ou thé vibration of an outiying part whose
conncctio)i with thc rest of théSystem
is but Joosc. In such a case
the part in question, provided a certain limit of amplitude bc
not exccedcd, is very inuch in thc position of a. systcm possessinfonc
dcgrceof frccdoni and acted on by a force, \vhich may bo
regarded as ~e~, indepcndcntty of thc natural pcriod. T)ic
vibration is accordingly governed by thé ]a\vs we bave ah'cady
investigated. In thé case of approximatc cfpudiry of pcriods to
which t)ie name of résonance is gencra))y restnctcd, thé ampli-tude may be very considcrahic, cvcn titough In other cases it
might bp so sma]! as to lie of !itt)c account; and thc précision
C5.]RESONANCE. 611
required in thé adjustment of thé pcriods in order to bring out
thé effect. dépends un tlic degrcc of damping to winch thé systcm
Is subjcctcd.
Among bodics winch resound without an extrême précision of
tuning, may be mentioned strctched membranes, and strings asso-
ciated withsounding-boards,
as in tho pianoforte and thc violin.
\Vhcn thé propcr note Is sounded in thcir neighbourhood, thcsc
bodies arc caused to vibrato in a very perceptible nianner. Thc
cxperimcnt may bo made by singing into a pianoforte tho note
giveu by any of ils ttrings, Iiaving nrst raised tlie con'csponding
dampcr. Or if onu of tbo Mtrings beionging to any note bc plu.ckcd
()ikc a Itarp string) with tlie nnger, its feHows will be set Into
vibration, as may immediatcly bc proved by stopping thc nrst.
T)tC piienotncnon of résonance is, howover, mo.st striking in
cases ~'hero n. vo'y accm'atoc([uality
ofpcriods
isnccessary
in
order to cHeit t))c full cfrcct. Of tins class tuning forks, 'muuntcdon résonance boxes, are a. conspicucus example. Witen thc UMison
is perfect thc vibration of ono fork wIH be taJ~cn up by anothcr
across thc width of a room, but thc slightcst déviation of pitch
is sumcicnt to l'cnder thc phenomcnon almost insensible. Forks
of 25C vibrations pcr second arc commonly used for thc purposc,
and it is found that a déviation from unison giving oniy one bcat
in a. second makcs ail thc dincrencc. Whcn thc forks arc '\vcU
tuncd and ciose togcthcr, thé vibration may be transferred back-
wards and furwards bctwcen thcm scvcral times, by damping thcm
a!ternatc!y, with a toucit of thé nngcr.
IMustrutions of tho powerfui c~ccts of isochronism must bc
vitinn t))e expérience of every onc. Tticy are often of importance
in very dinerent neld.s from any with which acoustics isconccrned.
For cxample, few things are more dangerous to a ship than to lie
in thé trough of thé sea. undcr thc innucncc ofwavcs whose pcriod
is ncarly that of its own natural ro)Hng.
(iG. Thé solution of thc équation for frcc vibration, viz.
M+ /C!t + )ï'M = 0 (1)
may be put into another form by cxprcssing tlie arbitrary con-
stants of intégration J- and a in ternis of tlie initial values of !<
and M, which we may dénote by and Wc obtain at once
ONE DEOREE OF FREEDOM.[f!G.
Thé cfTectof ~7i.s to gcncr~to in time f~' a vdocit.y r< w))o.scrc8u!t a.L tune < will thcrcforc be
U bcing the force at time <
The lowcr limit of tlie intégrais is so far arbitr.-uy, but it will
gencrally bc eonvunicnt to make it zero.
On this supposition u aud as givcn by (G) vanish, wlient = 0, and the complete solution is
Whcn t Issumdently grcat, thé
complementary tcrms tcnf] <oY~Ish on account oftl)c factor e-~ and mny ti~en hc omittc<1.
C7.]TERMS 0F THE SECOND ORDER. G3
G7. For most acoust.ical purposcs it is sufRcicut to consider
tho vibrations uf t)i0 systons, with which wc may ha.vc to deal,
:m m~nituly sm!).H, or i-iLthcr as simil.n' to Infiuitcly srn:Jl vibra-
ttons. This rust.nctiu]i is thc i'omuhttion of thc important lim's
oi' isoclu-ontsm fur t'r<jG vibrations, and (jfpc-rsistcucc
of pcriod
fur forecd vibrations. Thcru arc, nowevcr, ph<jnunicn:t,of a sub-
ordina.tc but not insigniricant charactcr, winch d<jpc!td csscutiidiy
on the s<trc and highcr ptjwc'rs of thc motion. Wc will thcrcforc
dcvutc t]ic rcmaindur of this chuptcr to thu discussion of thc
motion of asyton
of onc dcgrcc of freuttom, thc motion not bemg
so smaU that thé souarcsatid Ilighor powcrs
can bea~togcthcr
ueglefted.
The approximate expressionsfor the potcntlal and kinctic
énergies wIM be of thc form
G4 ONE DEGREE OF FREEDOM.[G7.
shewing tha.t thc propcr tone (?;.) of thc system is accotnpauicd
by its octave (2y:), whosc ?'e~e importance lucrotses with tho
!'fti~Htud(.' uf Yibr:d.io!t. A t'tihc(! (.~r c~.ii guncr.diy pcrccuc thô
octave 111 tho suu))() of a. tunitig fork causcd to vibrn.te strongty by)H(;:t.ns of a bow, imd wit)i thc :ud cf i).pp!i:).uccs, to bo cxpl!uuc(t
):(,tu]', t))c cxist,unco of the octave may bc niadc manifcst to anyonc. By foUowhtg thc same inethod the appmxnnatioa ca)t
hu ca)-)-iud furthcr but wc pa,.ss on now to the case of :). syston)n which thc recovo-mg power is synmiotncal
with respoct to
)hu position ofcqmtibrmni. T])c équation of motiou is t!tcn
app)'oxi)))n.tu!y
winch may be uuderstood to i-ufc-r to Mio vibrations uf a hcavy
p'n(h)!un~ or oi'a )u:).<1c:u-n<jd at tho end ofa, sti'iu~ht spring.If wu t:tkc an a jh'st npj~'oxitnatmn M=-~ cos?~, cotïespondi)]g
to /9 = 0, a.ud substitatc in tttc tcnn muhiplicd hy /3, we get
Corrcsponding to the lasttcrmofthiscqnation, wc shonid
obtain ni tho soh~tion a tcrm oftiie form <sin~, becominggrcatcr without Jnnit with t. Tt.is, as in a paraUd case in t]ic
Lun:n- Thcury, indicatcs that our assumcd iirstapproximation
is not rc!).]!y an approximation at a]), or at Icast docs not coH~eto bc such. If, Ilowcvcr, wo bikc as our starting point u =~4 cosM~,~ith a, suitaUc vaillo for M?, wc sitaïï find that titc solution
tnaybe cotnplutc() with thc aid of perio(]ic tcnns on!y. lu fact it is
evident buforchand that all wc are entiticd to assume is that thc
motion isapproxinuttely simple harmonie, with a pcriod ah-
~M'o.-n~n<< the sanic, as if /3=0. A very slight cxaminationis sn~cicnt to s)tcw that the terni varying as M", not
on!y may,
but ~~M< afîcct tho period. At tlie saine time it is évident
tlmt a solution, in which thc pcriod is assumed wrongly, no
n)!)ttcr by tiow little, must at Icngth ccasc to rcprcsent thc motionwith any approach to accuracy.
Wc takc thun for the approximate cqnation
ofwilichthc solution wilibe
67.]TERMS 0F THE SECOND ORDER. 655
Thé tcrm in /3 thus produces two cS'ccts. It altcrs thc pitch
of thé fundamcntal vibration, a,ud it introduecs thé <MeM!~ as
a uccessary accomp~nimcnt. Thc altération of pitch is in most
ca~cs excccdiugly small–dcpcuding on thé square of the amplitude,
but it is uot altogether insensible. Tuning forks gencrally risc
a little, though very little, in pitch as thé vibration dics away.
It may be remarkcd that thc samo slight dcpendence of pitchon amplitude occurs wlien tlie force of restitution is of thc
form M'M+mM°, as may be seen by continuing the approximation
to thé solution of (1) onc step furthcr than (3). Thc result in tbat
case is
Thc difference w" is of the same order in J. in bot)i casesbut in one respect there is a distinction worth noting, namely,that in (8) m" is always greatcr than while in (7) it dcpcudaon the sign of /3 whethcr its effect is to raiso or lowcr the pitch.
However, In most cases of the unsymmctricat class the changeof pitch would depend partly on a term of tho form «M' and
partly on another of the form /3 and thcn
C8. We now pass to the considération of the vibrations
forced on an unsymmetrical system by two harmonie forces
Thc cq~a.tion of motion is
CG ONE DEGREE 0F FREEDOM.[68.
Substit.uting this in tLc termmuJtiplicd by wc~ct
Thc addition~ tcrms rcprcscnt vibrationshaving frcqucncic~
which arc scvcndty thc d.u.bius ~.<I tl.c sum nud di~n-nec oft)iose of thc prin~ncs. Of thé two latter tlie iUT.phtudcs ~e
proportion~ to thc product of the origine ~nplitudcs, s)icwingth~t t!iu derivcd toncs incrcasc ni relative
impcrt:tuco withtho
intensity of theirp:irunt toucs.
lu a future chitptL-r wc shidt have to consider thc importantcousequeuecs which Hulmlioltz lias dcduced from this thcory.
CMAPTER IV.
V1HRA.T1NU SYSTEMS IN Gi;NEl{.AL.
G!). WH ha.ve now cxamitied m some dctfn! the osciH:t,tions
of f),systcm posscssed
of oncdcgrce
of frccdom, nnd thc i'esu)ts,
at whicit wu have an'ivud, hâve a vcry widc apphca.tion. But
m:Ltt.;ri:dSystems cnjoy iu
guticndmore than ouc dL'grcc of
frcudoui. In o!'(!cr to (tufinc thcir cou(1gur:ttio)i at. any moment
scvcnd uxhipcmtott vin'mbic qn~tttidcs must bc spccificd, whici),
by :t ~(.'))ut':dix:t,tif)t) of ):u)gU!~c ori~hin.Hy cm]))oyc<l for a ponit,
arc caUutt thu co-or~t'~f~es et' thc systcm, thc uumbcr of indcpcu-
dcnt co-ordin:tt(js bumg tho MK~ q/rce<?o?~. Strictiy spc:dting,thc disphtccmuuts possibtc to {t n:).tm'a,l systcm arc infmitcly
Viu'ious, and caunot hu l'cp~'cs~ltc(~ as m:)dû up of a finitc numbcr
of déplacements of sp<jcifiu(1. type. To thc cicmcntary pru-ts of
a. so)Id body !uiy nrhiti~ry dispt~ccmcnts may bc givcn, subjcctto coti()Itioi)s of cotitituuty. It is oïdy by a pt'ocL'ss of idjstraction
of t]tu kind so constiUttty pr.LctIsctt in N~tuml ThMosophy, th:it
so!i<)s aru trc:t.tcd as )'i~'i.d, fluids n~ incompressible, n.nd othcr snn.
phdctitions mtroduecd so tli:).t thé position of a, System cornes to
depoid on :), finite numbur of co-ordin:).teH. It is not, however,our intentiou to cxcludc thc considération of Systems possessin<'f
infirntely various freedom oti thc contnu'y, somc of thc most
mtcresting appHcatiûus of t!ic results of this eh:mtc]' will lie in
that direction. But such Systems arc most conveuicutiy conccivcd
as limits of othcrs, wl)osc frcednm is of a, more rcstncted Mnd.
Wc sh:).ll accordi))g)y commence with systcm, wtiose position
is spccincd by a, finite uumber of independunt co-oi'diitatcs -~r,
t~ &c.
70. Thc ma,In prohicm of Acoustics consists itt t!io investi-
gation of thé vibrations of a, system about a position of stable
cquihbrium, but it will bc eonvenient to commence with the
st.itica.1 part of thc subjcct. By thc Frinciple of Virtual Vc-
K_o
C8 VIBRATING SYSTEMS IN GENERAL.[70.
locities, if we rcckoïl thé eo-oi'dmatcs &c. from tho
configuration of equilibrium, tlie potentiel energy of any othcr
cuufigumLtuu will bu :(.h~~m'nc-ous qud.cr~i.ic function of t))C
co-ordiuatcs, provided t!~t the displacemcnt be sufHcicnDy smdt.
Tins quantity is ciUlcd and reprcseuts thc work thf~t may bc
gfdncd in passing from the actuel to tlie equilibrium configuration.We mny write
Since by supposition thc equilibrium is thoruughiy stable, tho
quantitics c,c~, c, &c. must bc such that V is positive for
all real values of thé eo-ordiaa.tes.
71. If tlie system bc Jisplaccd from tho zero configuration
by thc action of given forces, thc new configuration may ho
found from thc Prineipic of Virtual Velocities. If thé work done
hy thc given forces on thé hypothetical dispkcement 8~, S~,&c. be
this expression must bc cqu:vn,!cnt to 8F, so thfttsmcc 8~, 8~,&c. nro ludcpcudcnt, the new position of cquilibrium is doter-
mincd by
-where there is no distinction in value bctwecn c,, and c,From thèse équations the co-ordinatcs may bc dctermmcd in
terms of the forces. If ~7 bc thc dctûrmIuMt
71.] RECIPROCAL RELATION. G 9
Thcsc équations détermine ~r,, &c. uniquely, slaco doca
ilot ] ~u 'iLb~til t.Mût Y:in)Hh, .'i~ t).ppcm. ~ont t! coï.~idi.'t'Htiuit ~h' Lh' 'f~f;{'.L'uh
===0. &c. could othcrwiso be s~tisficd by fmitc values of tha
co-ordina.t.cs,provided oniy ttmt tho ?'(t~'os wcre suitable, winch ia
contnu'y to thé hypothcsis timt t!ie systcni is tboroughiy stable
iu t)ie xct'o conHgura.tioa.
If thc forces ~F, a.nd be of the same Mnd, we may suppose
them equal, aud wu then recoguiso that a force of any type acting
alone produces :idisplacetneut
of f), second type cqual t.o tlie
displiicement of tlie first type duc to thc action of au cqnid force
of thc second type. For example, if and R 'be two points
of n, rod snpported horlzont:dty in ~ny maunc! the vertical de-
ricction at jl, whcn a wcight }F is ~ttachcd at is tl)u s:t.me as
the détection at 7?, wlien ~F is appiied at ~t\
73. Since F is a homogeueous qua.dra.tic function of thc co-
ordinates,
If + ~~). ~+ ~~) ~c. rcprcscnt auniticr (Iisp!act'mcnt for
wbich thc neeessaryforcus n.ro ~+/ ~+A~,&c.,thecor-
Ou thiB eubjoct., sco 7~tt~. J/< Deo., 1874, nud MMeh, 1875.
70 VIBRATING SYSTEMS IN GENERAL.f73.
ruspouding potentud cncrgy is givcu by
whL'rcA'FistiKi'n~n.'ncuojf'thupntcntI.dcnL'rgicsinttxjtwo
ciLScs, :unt wu must p:u'<.Icnl:n')y uoticu tIi:Lt by tlie i-eeinroc:~
rui~iuu, § 72 (I),
From (~) and (~) wu may deduectwonnport:u)t Dicorc'tns
rclating to t!ic vainc of fur a systeni subjeetcLt to <Ivcn dis-
pitt.cement.s, and tu given forces respect! ycty.
7~. Thé first thcorem is to t!tc eNfuct t!):tt, if given ()isn]acc-
mcnts (llot su~iclunt hy ttionscivos to dûtcrmhio thc C())tti~u)':)tK)ii)b(.' produœd in a.
systcm by f'urccs uf con'c'spundixg typ~s, t)tc rc-
Hulting vaJuc of ~for thc .sy.stcmso
displaccd, :uid m u~uHi))rium,
is ns sin:dt as it can bc u))(icr thé givcn di.spinccmoit couditiun.s'and that the vainc of fur :Uty othcr couhgurattou excuc-ds tins
by thc potcntia! uncrgy of thu cunHguratioR wincli is thé (tiSurcnce
of t)m two. Thc on)y diHioLdty In thu abovc statcmcnt consists
in undurshuidit~g what is ntcant by 'forées of coi'r<spo]]di!)"' types.'
Suppose, for cxampic, that thc systum is a. strutchcd stri))" of
which agivcn point jf-* is to bu subjcct to an cbligatory dispJacc-
!n(U)t; thu force of corrc.sponding type is Itère a. force applicdut thc ])oint .P itself. And gun(.'r:dty, thc forces, by which thé
proposcd displacumt.-nt is to bc tunde, must bc such as woul(i do
no work on Hic systum, proyidud on!y tiuLt thut disptuccmcutwurctio~made.
By a suitabic choicc of co-ordinatcs, ttic givcn displaccmcnt
cotditicustnaybe cxpt-L-ssud by ascribmggiven vaincs to thc first
?' co-ordinatcs nud thu conditions fm to thc forces
wdl thcn bc rcpr<j.s(jntcd by inaking thc foroja of thc rcmaini))~
typL's &c. vanish. ïf -+A-~ rcfur to any ot)~cr con-
hgnratiou of thc systum, and ~+A~ bc thocorrcsponding forces,
we are to suppose that A- A~, ~c-. as f:n' as A~ aH vanisli.
TIiusfor tite first r suifixes vauishcs.aud fur thé remaimD~0
74.] STATICAL TIIEOREMS. 71
sufHxcs ~Fvanishcs. AccordinglyST.A- is zero, n.nd therefore
.P'r i.~ aise xcro. Hon~c
2A~=~A~.A~(1),
which provcs that if thc givcn déplacements bo niadc in any
othur than tinj prc'scribcd way, thu potcntial cncrgy la incrcased
by t)ic encrgy of thc différence of the configurations.
By means of t!i!s t))corcm we may trace thc cH'cct on T'of any
l'cl~xation m t!)e sttH'ucss ofa. System, suhjcctto
given displacemcnt
conditions. For, ifaftcr tlic altération m stitTness thc original cqui-
librium connguration Le considut'cd, thcvidnc of Vco)'ruspon()ing
t))crcto is by supposition Icss t)i:m bcforc; :md,as wc h~vc justseoi, therc will be n. still furthcr dinunution in tbe value of F'
whctt tlio Hystcm passesto cqnilibrimu undtjr the niterud con-
ditions. Henco wc condudc titat a. diminution I)i as a functiou
of thc co-ordin:t.tcs cntails also n diminution in the actual vatuo
of F' whcn asystcnt is subjcct
togiven disp!:).cemcnts.
It will
bo undur.stood tluit in pa.rticuhu.' cases thc dinunution spokcn of
may vanish*. l.
Forcxample,
if a point J' of a bar dampcd at both ends be
disph~cud latcndiy to a given small antountby
a force tbm'c ap-
piicd, thc potentiel cnurgy of thc dcfui'mation will be diminished
by :).ny relaxation (however loc:d) in tite stiiïhess ofthe bar.
75. Tlic second theorem relates to f), system displaccd ~tM~
forces, and asscrts that in this case tho value of V in eqnilibriuni
is gi'cater than it would be in any other conngurationin \vhich
thu syst~'m coutd bc maintained at rest undur t))c givcn furecs, bythe opération of mcre constraints. We will shew that tho )'c?/MM~
ofconstt'aints increascs t!)c vainc of
TIio co-ordinatcs may bc so choscn that thé conditions of con-
straint arc cxprcs.scd by
~=0, ~=0,=0.(1).
Wc hâve thon to provc that whcn ~P~, ~P~ <c. arc givcn, tho
va)nc of V is Ica.st whun t!tc conditions (1) !~))d. Thu second
configuration bcing dcnutud as bufot'u by + A~, &:c., wc seo
that fur snrRxcs up tu ')' inchtsive vanishcs, and fur higinjr
sunixcs A~F vanislics. Hunce
S~A~=SA~P=0,
Soo n. imper on Goncrfit Thcorcma rchting to Eqnilibrium aud luititd nt)d
Stuady Motiouij. 2'/«7. Af~ Mure! 1H7S.
~2 VIBRATING SYSTEMS IN GENERAL.[75.
and therefore
shewing that thc incrcase in F duc to thc rcmoval of the con-stramts is cqual to tlic potcntial encrgy of tlie din~rcnce ofihc two
configurations.
7G. We now pass to the luvesti~tion of thc initial motion ofa systcm which starts from rcst undor thc operation of givcnimpulses. The motion thus ~equired is Indepcndcitt of anypotuutm encrgy .vhicl~ the system n~y possess .vhcu actu~y
disptaccd, siueo by tho nature cf impulses we h.wc to do onlywith thé mitml configuration itself Thc initial motion Is also
mdependcnt of any forces of Huitc kind, whethci- imprcsscd ontlie system from without, or of the nature
of viscosity.If Q, 7i' bc the component impulses, parallel to thc axes, on
~partie e ~vhosorcct.nguhr co-ordinates
are h.vc byDAlGmbei't'sPj-iucip!o
whcrc dénote thé vclocities aequircd by the particle in virtucof the impulses, aud
correspond to auy arbitrary dis-
placcmcnt of thc system which docs not violate thc councction of itsparts. It is required to transform (1) iuto an
cquatiou cxprcsscdby thc independerit gcncralizcd co-ordinn.tcs.
For thé first side,
whcrc the kinetic cnergy of the system, is supposcd to be ex-presscd as a function of &c.
76.] IMPULSES. 73
On tlie second side,
whcrc 8~, S~, &c. arc now eompletely iudcpcudcut. Hcuco te
détermine tlie motion wc ha.vo
whcrc &c. may bc constJcrcd as tlie gGneraItzcd componcntsS
ofi;i)putsc.
'77. Since y is a homogcncous quadratic fuuction of the gene-ralized co-ordiuatcs, we may takc
whcrc there is no distinction in value between 0,, and
Again, by the nature of T,
The theory of initial motion is c!osc]y analogous to that of thé
displaccmcut of a. system froni a configuration of stable cquitibrinm
by steadily a,pp!Icd forées. lu thé présent tucm-y thé initial kinetic
encrgy T bears to the vclocities aud impulses tho same relations
as in thc former F' bcars to thé displacements and forces respcct-
74 VIBRATING SYSTEMS IN GENERAL. [77.
ively. In one respect the thcory of initi:d inot!ons is the more
complet~ in!).s)nuc)i. as is cxactty, w)ti!c L in gt'ncmt oniy
approximatcty, a,itomogcuc'jus (~uadrntic fuuction of thé variables.
If' dunotc nnc set of vclocitics and impulses
for n. systuin st:u'tcd f'run) rcst, :utd a. second
sct~ wu iua,y pt'uvc, as in § 72, thé fuDuwing recipt'uca.l t'L'ta.tiun
This thcdn'm ndniits cfitttcresting' :)pplie:ttl<)!i to f)~)i<~ motion.
It is kuu\v)), :utd will bc provcd I~t'jr in thu coin'sf.! of tins work,
th:)t thu tnoti~n ui' !), inctiun!uss ineotnprcs.sibtu liquit), which
starts i'rota rc'.st, i.s cf such :t. kind t))!~t its cumpom'nt vutoeitics
~t nny point aru thc con'L'spondiï)~ dit'fu)'c'nLi:d cocHicicnts uf n,
('(.')'t,:nu fnncti"n, c~Hm] thc vctocity-potcntiid. Let t))c fh)i() bc
sut In )n(jt[on by :t prcso'ibu)! tn'bitt'in'y ()L'fo)m:).tiu)i of th(j surface
/S' of :t c)')SL'() spucc describud within it. Tiiu rcsniti))~ mution is
(h.'tL'nnincd hy thc normid vctocitics of thé cloucnts of winch,
bL-in~ s)t:n'L'd by t!ie Hnid in contact witti thcm, m'c duuotcd by
if M be tho vc'ioeity-potcuti.t.], \vltich luto'prctcd phy.sica!)y dé-
notes tnc ijnputstvc pressure. Hunce by thc t]iC(H'cm, If bc t]io
VL'Iucity-potuntlid uf u, secoud motio;), corruspuuding to unother
set uf arbitrary suifacc vclocitics
–an équation immcdiatoly foiJowing from Grccn'a thcorcm, if
bcsi()<s~'thurc be ou)ytixcd soli<[s inunur.scd in tho ftuid. Thé
prusunt ]m;t)n)d unabius us tu attributc to it a much Itighur gcnu-
ruHty. yur (.x:unp)u, t)ic untm'rscd soHd.s, mstuad of buing Hxc<
m:i.y t)c irc-L', :dtogct))cr or ni part., to takc tho motion iinposcd
upo)j tl)u)n by thL! Huid prcssm'
78. A paTtk'ular cnsc nf t)ic gcticra! thcorem is wortl)y of
spécial notice. In thu nrst motion Jet
78J THOMSON'S TIIEOREM. 75
In words, if, by mcans of a suitabic impulse of the correspond-
ing type, n givcn arbitrary vclocity of onc co-ordinatc bc imprcsscd
on a system, the imputse corresponding to M second co-ordinatc
nccessaryin ordcr to
prcventit from
changing,is t)ie samc M
would bc rc'ptircd for the first co-ordiuatc, If titc given velocity
\vo'chnprMS.sud
on ttic second.
As :t simple uxampic, tedœ the eMC of two sphères and J~
nmncrHcd in aliqnid, wliusc ccntrca arc f)'L'c tu !)iovL; along ccrtiun
lines. Jf ~t bc sut in motion with givcu vulf~'ity, -B will
natnndiy bL'gin to movc also, Thc thcorcm :LS.surts th:Lt the
i]))pt))su rctmin.'d to prcvent thc motion uf if) thc s:mic as if
thc functions of yt !md 7? wo'c cxchimgcd :uul this cvcn thuug)i
thcrc Le ot])cr rigid bodius, C', D, &c., in the ituid, citl~cr fixcd, or
frcc ill whulc or i)t part.
Thc case of cicctric cnrrcnts mutually i)iflnencing cach othcr by
induction is prccisciy simihu-. Lct thcru bc two circuits and
m titc ncig!ibour)tood of which thcrc may be a.ny numbcr of othcr
wirc circuits or sohd condnctors. If a unit cnrrent bc snddcidy
duvulopecl in thc circuit J, tho clectromotive Impulseinduced ill
is the slulc as there would have bccii iu ~1, hn.d tlic currcnt been
furcibly dcvclopcd in
79. Thc motion of a system, on which given a.rbitrary vclocitios
are nnprcs.scd by mcans of thcncecssfu'y Itnp)dscs
of t)ic corrc-
sponding types, posscsscs a rcmarkabtc prnpcrty discovcrcd byTiiom.son. Thc conditions arc that arc givcn,
vanish. Lct &c. currcspoiid to
thc actu.d motion; and
~+A~, ~t.A~ ~+A~, ~+A~
to anothcr motionsatistying thc saine velocity conditions. For
cach snmx cithcr AT~- u)' vailislics. New for t)iu kiuctic cnergyof thc supposcd motion,
2(~+Ay)=~+A~)(~+A~)+.
=2~'+~A~+~+.
+ A~ + A~. +. + A~A~+ A~A~+.
But by thé rcciprocal rctatioa (4) of§'77
~A~.+. =A~+.
of \vbich tlic former by ItypoUtcsis is zéro; so that
2A2'=A~A~+A~A~,+. (1), J
VIBRATFNGSYSTEMSIN GENERAL.7G
[79.shewing that the encrgy of the snpposcd motion excceds that offthé actual motion by thé energy of that motion winch would hâvec bc
c.nrlod
..ith },t. rcp,.d.~ ihc fbnner. Thé
motionactualtymduced in thc
System bas tf~.s Jcssoucrgytlm,i
~y others~. yn.g tho same velocity conditions. In a
snbs~.cntch. ptcr we shall make use of this
propcrty to find a supenor Jinutto the cncrgy of a system set in motion with prcscribcd vc-Iocitics~ny dnnmutiou be made in thé inertie of
any of t)je parts ofa system, t)ic motion
corresponding to prescribcd velocity conditionswu iu genem undorgo a
change. Thc value of will nece.ss.riiybe less than before for t)~ere wouM be a decrease cven if tliemotion rc.nained
unchangc<I, and tl.crcforc /b7~ w]~en théniot.on ~s such as to make 7' an absoJute mim.num.
Converselvany incre~c m tlie inertia increascs thc initia! value of T.
lu. thcorcm Isanalogous to that of § 74. Thé
analogue forinitial mot.ons oi thé thcorem of § 75, relating to t].c potential~~gy of
a.system d~.ced by given forces, is that of Bertrandand may be thus stated -If ,y, start from rest under théopera.on of givcn nnpu!scs, the kinetic encrgy of tl.e actual motionLxcceds that of any otlier motion which thé system might I~.vebeen gu.)ed to takc with the a.ssistance ofmere
constrain~ by thekinetic encrgy of the din-crence of t)to motions' 1.
80.WcwiIluotdwcUatanygreaterIengthonthemcd.anicsof a system subjcct to
impulses, but pass on toinvestie
Langesequations for continuous nation. Wc .shalt supposethat the connections
bniding togcthcr thé parts ofU.c..svstc.nare not o.plicit functions of t). tune; sucli ca~sof H
motion as we shall have te consider will buspeciaily .shcwn toue wiHun thé scope of the
investigation.
YdoÏt~combination with that of Virtual
Vc10ci tics,
(~~ + y8~ + ~~) = S (.Y~ + F~ +~)
~herc 8~dénote a
d:sp!acemont ofthe system of thé most
~r r'~t~ -nection. of":f
parts. Sn.cc théd,sp)acemcnt.s of thé individu.-d partides oft system arc
~nutuaHy relatcd, are not indcpen~t. T)ohjec .ow is to transfonn tu other variahJc.s whichs!tatl bc indcpcndent. We hâve
ThomBou auj 'fuit. § ~il. ~,7..V.y. Mareh, 1875. J
7780.] LAGRANGE'S EQUATIONS.
so tha,t
if T ho cxprcsscd as n. quadmtic function of whose
cocfHcicntsorc ni gcuerfil functions of AIso
Since ~F8~ denotes Hie work donc on tho system during a
disp~cemcnt ma.y bo recoarded as thc gcncralized com-
ponent of force.
In thé case of a, conscrvativc system it is convûnient to
separate from thosc parts which dépend only on thc connTurfi-
tion of tho system. Thus, if V dénote thc potential encrgy, wc
may write
whcre ~P is now limited to tlie forces acting on thc system which
~F'are not aIrGady taken account of in thc tcrm
a~
78 VIBRATING SYSTEMS IN GENERAL. [81.
81. Tilcrc is also another group of forces whoso existence
it is ofton a~/fmt~gcous to rccngnizc spccifdty, namoty thosc
?;r..s)r~ 'n.i fn.-n~tyip.Hy. if y~
.pp~o L),)H c~h
piu'ticlc of thc syston is rctiu'dcd by forces proportion~ to its
eomponbnt velocitics, t)ie cH'ect will bc sitown in thc fu)jd:uncnt:U
équation (1) § 80 by t!tc addition to tl)c Jcft-Jtiuu! mcnibur of
thc terms
whcrc A-y, nrc cocfHcicnts indûpOKicnt of t))o ve!(jcit.Ics,but
pos.stbiy dcpcudcitt on t)tc configuration of thc syston. T))c
tr:msibr)n:itn)u to thc indc'pcndunt co-ordinutus &c. is
cHucted iu a. sirnihu' manner to tJKtt of
7~ it will bu obscn'ed, is hl.c a honngcncons quadraticfucetiou of t!tc vuioctLies, po.siUvu for :dl rL-:d v.ducs of thov:u-)ab)cH. It
!~pruscnt.s hait thu r~tu ~t whidt cncrgy i.s (hs.sij~~cd.Thc abovc itivcsti~tiua i-ufcrti to t~tarding iornus propordonat
to thc absolute vclucitics but it is equaUy important tu cuusidursucb as dupend ou tho p-e~~c vulocitics of thc parts ci' titc
system, and furtuuately tins eau bc done witiiout auy incrcaso01
complication. For cxampic, if a furcc aet ou the partielc xiproportiona! to thc-rc will bo at thc samo momont an
cqu~I and opposite force acting ou thé partide a- T!ie additioualterins in the faudamental eqoatioti wi)l hc of the furm
and so on for any numbcr of pairs of mutually ]nfhtctic;ngpfirticics. TIic only effect is thé addition of ncw tcrms to 7~whicli still appears iu the form (2)'. We silall sec- prcscntly t)iat
Tho difforecoes rûferred to iu tho toxt may of course pass iuto djilcrcntia!eoefUcients in tho case of a body oontiuuouBly deformed.
81.] THE DISSIPATION FUNCTION. 7!)
thc existence of tho fonction 7~ which may bc cailed thé Dis-
sipation Funetion, implics certiLin rctations among thc coenicicnts
ut' tho gcncralizcd cqn:t.tio!]s of vibration, which ctu'ry with Utem
Iniportaut couscqucnecs'. l,
Butalthougli
In animportant c]~ss of c~ses thc cffccts of
viscosity arc l'ept'cscuted by thc function théquestion romains
opcn whctitcr snch a method of rcprcsGntation is apptic:).b)c in aU
cases. 1 think it pTobable th~t it is so; but it is cvidcnt that wc
cannot cxpect to provc any gûncmt propùrtyof viscous forces
Î!) t'hc absence of n strict (L'nnition \v!ncb will cnable us to duter-
minc wit)). certainty wha.t forcus are viscous !H)d what n.rc not. In
sono CMCS cons!dc;['!Ltio)is ofsymmetry arc sun~'Icnt to shcw
tbat thé retardmg forces ma.y bu rcprcsoitcd as dunvcd from a
disHipatioti fnnction. At any rate whuucvcr tbc rctarding forces
arcproportional to thc absolute or relative vcloeittcs of thc
parts uf tlic systuia, wc slutti liavc équations of motiun of tlic form
82. Wc mny now mtroduce tho condition that t))0 motion
takcs place iu tho nn)nc()i:).tc neigh'b(n)i'hoo(L of a. conH~u'tt.tIonof t)iorou~I)ly stable cquHibnum 7' and F' arc then homogcncous
qmuh'atic functions of ti~c vclocitics witli coufHciunts winch aro
to bc tœatcd as constant, !ui(l i.s a snnUar fuucttou of thé
co-ordina.tcs tticnisdves, provided that (as we suppose to bo
t!io case) the origin of CMh co-ordmatc is taken to con'esponjwith the couhgura.t.ion of
cquilibrium. Moreovcr all threo
~Vfuuctious arc ossentiaUy positive. Since ternis of tho form
f/:n-c ofthc second ordcr ofstnMil quantities, the equations of motion
heconic h)iear, assumiug the form
whcrc under ~P arc to bc mciudcd ail forcesn.cting
on thcSystem
notalœady provided for
by tlie diffcrcutial coefficients of Faud
Tho Dissipation Funetion ttppoMs for tho Rrat timo, so far as 1 am nwnrc, iu
a pnpor on Gonoral TItooroma retatmg to Vibratious, publishod m tlio 2~ocfe~HM<o/' the ~~tCMNttca! Soete~ for Juno, 1873.
80 VIBRATING SYSTEMS IN GENERAL.[82.
Thc threo quadra.tic functions will be expressed as foUows
whcrc thc cocfHeicnts c are constants.
1
From équation (1) wc may of course fait back on prcviousresults by supposing ~and F; or .Fand T, to vanisii.
A thin! set of thcorcms of intcrcst in tlie appHcation to E)~-
tnc.tymayboobtaiucd byomittlng~and F; wliile ~isrctaincd,but it is
uuneccssai-y to pursue the subject hcrc.
If we substitute thc values of T, F and F; and write D for
we obtain a system of equations which may bc put into tlie forni
dt1
83. Beforeproccoding further, we may draw an important
inference from thé of our equations. If correspondincrespectivoly to tho two sots of forces
\Pi l't~tt"' H\ 0motions dcnoted by be possible, thon mustalso be possible thc motion
~,+~ ~+~ in conjunet:onwith hc forces~+~ ~+~ Or, a p.rticuL case,when there arc no impressed forces, thé
superposition of any twonatural vibrations constitutes also a natural vibration This is thcccJcbrated principle of thc Coexistence of SmaU Motions, firstclcar)y cnunciatcd by Daniel Bernoutli. It will be uuderstoo.!that its truth dépends in gênerai on tlie justice of thé
a.s.sumptionthat the motion is so small that its square may be neglectcd
84.] COEXISTENCE 0F 8MALL MOTIONS. 81
84-. To invcstig~tc thé free vibrations, wo must put
l~qu:¡JtoI' lui'~1~ .n:d we 4Yjjl P-01H11IPl!eÜ ~jt.l! Il f.:y¡¡tt~.m On which n0
equ;)J t.~ forces -n!~ v-'c wiH cotmttt'ttce~'itLany'ttf'mouwhic]). arefrictioual forces ~ct, for which therefore thc coefRcienta &c. are
M)~ functioDS of thé symbol We havo
From those équations, of which thcrc arc as many (??t) as thé
system possesses degt'ces of liberty, lot all but onc of thc variaMes
bc climin~tcd. Thc result, wliieh is of the samc form whichcvcr bc
the co-ordinate ret<uucd, may bc writton
~=0.(2),
where \7 denotca thé determinant
and is (if there bc uo friction) an even function of D of degrec 2M.
Let i\ ±\ ±\t roots of V=0 coueidered as au
équation in D. Then by the theory of dliferential equations thé
most genera.1 va,!uc of is
whcrc the 2w quantities ~4, J/, J?, J~, &c. are a.rbitrn.ry constants.
This fonn hoids good for eMh of the co-ordinatcs, but tlie consta-nts
in the différent expressions arc not indcpendcnt. In fMt if a
particular solution bo
~=~ ~=~' &c.,
the ?'a~M ~t~ -~a. M'c complete]y determined by thé
équations
where in each of the coefficients such as is substituted for D.
Equations (5) arc necessarily cc~upa,tible, by the condition that
is a. root of \7=0. Thé ratios ~1/ =-~3' correspouding to
thé root arc tho samc as the ratios ~1~ ~1, but for
thé othcr pairs of roots X~, &c. titcrc are distinct Systems of
ratios.
R. G
82 VIBRATING SYSTEMS IN GENERAL.fgg.
85. Tho nature of thé system with which wo arc doalinrr
imposes an importât restriction on the possible values of Ifwcro .6a}, elthci-
or woutd bc re~I and positive, and wosho-Jd obtain a particular solution for which tho co-ordinatos, audwith them thé kinetic energy denoted bv
incrc~c without limit. Such a. motion is obviousiy Impossible fdra conscrvative system, wbose whoJc energy can uever di~cr fromthe sum of tho poteutial and kinotic energies with which it wasMimatcd at starting. This conclusion is not cv~cd by takmgnégative, beeausc we arc as much at liberty to trace thé motioubMkwards as forwards. It is as certain that t!te motion ncvcr ~sinfinite, as tliat it nover will The same argument excludcs t).c
possibility of a complex value ofX.
Wc infer that aU ttie vaincs of are purely imacrniary cor-
rcspondmg to~a~e values of
Ana)yt:caHy, t)ie tact thatthc roots of = 0, considered as an équation iu are at! real and
negative, must bc aconséquence of thc relations subsisting bctwecn
thé coefficientsvirtuo of fact for
all real values of the variables 2' and F arc positive. Thc ca~e oftwo degrees of
liberty will be afterwards worked out in full.
86. Tho form of tlic solution may now be~IvMta~cousIy
changcd by wnting for &c. (wherc .=~1), ~d ~dngnewarbitrary constants. TIius
where &c. are thé roots of thé equation ofdecrec111n' found by writing -M" for in = 0. For each value of
thé ratios~1, ~1, are dctcrminatc and real.
This is thc complète solution of the problem of tho frce vibra-tions of a conscrvative system. We sec that thé whole motionmay be resolved mto normal harmonie vibrations of (in général)difforent période each of which is entirely indepeDdcnt of tbcothers. If tbe motion, depending on thc original disturbance, bcsuch as to reducu itsdfto onc of thèse ~.), wc hâve
] NORMAL COORDINATES. 83
t'
where thé ratios AI dépend on the constitution of thé
system, and only thc absoluto amplitude and phase arc arbitrary.Thé several co-ordinatcs arc always in similar (or opposite) phasesof vibration, aud the whole system is to be found m the configura-tion of equilibrium at thé same moment.
We perçoive hère the mechanica.1 foundation of tlie suprcmacypf harmonie vibrations. If the motion be sufHcientIy small, tho
diffcrential équations becomc Iluear with constant coefficients~hi]e circular (and exponentia)) functions arc thé ouly oncs which
reta-in their type on diffcrentiation.
87. Thé 7~ pcriods of vibration, determined by t!ic équation
= 0, are quantities Intriusic to thé system, and must corne outt.he same whatever co-ordinatcs may be choscn to define the con-
n~uratton. But there is one system of co-ordinatcs, which is
especially suitable, thatnamely in which the normal types of
vibration arc defiued by thé vanisbing of aU tlie co-ordinates butonc. In the first type the original co-ordinatcs &c. Iiave
given ratios let the quantity nxing thc absolute values be < sothat in tliis type each co-ordinate is a known multiple of < Soin thc second type each co-ordinate may be regarded as a known
multiple of a second quantity and so on. By a suitable deter-mination of thé quantities &c.. ~y configaration of tite
system may bu rcpresentcd as compoundcd ofthc ~t configurationsof these types, and thus tlie
quantifies <~ thcmselvcs may b'c Jookcd
upon as co-ordinates denning tite configuration of thé system.Titcy are called tlie ttor~a~ co-ordinatcs.
When expressed in terms of thc normal co-ordinates, ?' and Varc reduced to sums of squares; for it is easily sccn that if the
products also appcarcd, the resulting équations of vibration wouldnot be satisned by putting any ~-1 of the co-ordiuates cqual to
zero, whilc thcrcmaining one was finite.
We might hâve commenced with this transformation, assumin~ZD1 0
from AJgebra that any twohomogcncous quadratic functions can
bo reduced by linear transformations to sums of squares. Ttms
whcrc thc cocnicicuts (in which thé double sufHxe.s arc no tono-crrequired) are ncccssarily positive,
G–2
84 VIBRATING SYSTEMS IN GENERAL.[87.
88. The interprétation of thc équations of motion leads to atlleorem of considcrabio importance, which may bc thus statcd'. t.Thé period of cousorvittivu
system vibrating i)i a,const)-!nned typeabout a position of st<).h)c
cquiHbrium isstationary in v:Uuc when
thé type is norm: We might provc this from the ori~inid cqua.-tions of vibmtion, but it will bc more convcnicnt to
cmploy thenormal co-ordinatcs. Thc constnunt, w]nc)i may bo snpposcd tobc of such a cha.racter as to ic:ws only onc dcgrc'e of fj-cedom, is
represcuted by taking théquantittes in
givcn rutios.
If wc put
This gives thc period of thé vibration of tlie constrained typeand it is évident tliat thc period is stationary, when a!l but one ofthé cocfncients ~l,, ~1, vanish, that is to say, -when thé typecoincides with one of those natural to the system, and no constraintis necdcd.
By means of this tlicorem wc may provc that an iucrease inthe mass of nny part of a
vibrating system is attendcd by a pro-longation of all tho natural periods, or at auy rate that no pcriodcan be diminished. Suppose tlie incrernent of mass to bc infi-nitesimal. Aftcr thé altération, the types of free vibration will in
général be changed; but, by a suitable constraint, thé system may
r~~c~t')).?)! of ~;f;~~<fma()ra! ,9of«'~)/, Juno JH73.
8588.]
PERIODS OF FREE VIBRATIONS.
bo made to rctain any one of tlio fonner types. If this be donc,
it is certain thnt any vibration which involves a motion of thé part
whosu inass lias been increased will I)ave its period prolonged.
Only as a particula.1' case (as, for exampic, whcn a load is placed at
the nodc oi' a vibrating string) eau thé period romain unchangod.
Tlic tlicorem now allows us to assert that tho removal of tlie con-
straint, and tlie conséquent change of type, can only aScet thé
period by a quantity of thc second order; and that therefore in thé
limit the free period cannot bc Icss than before the change. By
intégration wc infcr that a imite incrcasc ofinass must proloiig the
period of' every vibration which Involvcs a motion of thé part
aliected, and that in no case can tlie period bc diminishcd but in
order to sec the corrcspondcnce of thé two sets of periods, it may
be necessary to suppose the altcrations madu by stcps.
Couvcrsely, thé efïect of a rcmoval of part of thc mass of a
vibrating system must bo to shorten the pcriods of all thé froc
vibrations.
In iike manner we may prove that if the system undergo sucli
a change that the potential energy of a given configuration is
diminislied, while thé kinctic energy of a given motion is unaltered,
the periods of thé free vibrations arc aU increased, and convcrscly.
This proposition may sometimes be used for tracing the effects 6f
a constraint for if we suppose that thé potential energy of
any configuration violating the condition of constraint gradually
incrcases, we shall approach a state of things in which tl]e
condition is observed with any desirod degree of completeness.
During each stop of thé process every free vibration becomes
(in général) more rapid, and a number of thé free pcriods (equal
to thc degrees of liberty lost) become infinitely small. Thé
same practical result may be rcached without altcring thé po-
tential energy by supposing the kinetic energy of any woftOM
violating the condition to incrca~e without limit. In this case
one or more periods become infinitely large, but thé finite
periods are ultimatcly thé same as those arrivcd at whcn tlie
potential energy is increased, although in one case the pcriods
have been throughout increasing, and iu tlie other diminishing.
This example shews the nocessity of making thé altérations by
steps; otherwise wc sliould not understand tl)C eorrespondcnce
of tlie two sets of pcriods. Furtlier illustrations will bc given
under thé head of two degrees of frecdom.
86 VIBRATING SYSTEMS IN GENERAL.[88.
By me~s of théprincipe that tho value of tho frec
periodsstationary wc
n..y ea.i!y calculato eorrcctions duo toany
<!cv~n
~h.
J~a
hypothct~ type of yibr.tion thatprope~ to thé
sl.nplusystc~ thé
punod so found wiH di~r from the truthby quan-
tit.esdcpcndmg ou tho
~uares of ti.eu.rcguh.-Itic.s. Scvcral
exaiaplcs cf suci~ c..dcu!atiuns will Legtvcn in thé course of
tins work.
80. Anothcr po.nt ofunpor~ncc reJ.ting to thc period of
y.steiu
vihr~ng
,n .n~rbitrary type rcn~ins io be noticedt .ppcars from (2 § 88 that thc p.riod of fhe vibration o
~c.sp n. u~to ~ny hypothcti~I type is inciu.Icd bctwocn thc
~.tcst
and Ic~t of thosc n.tur.I tu t!~ system. In thc c~
o c ntuu~.sdeior.n.t~n, thcrc is no I~t uatu~
pericd;
h "Y~i any hy-puthet c.d type c.uinoL cxcccd that
bclo~l,~ to thé Gr~esttyp. Whe. tLer.f..c ti.
cLject°i.J~I~
~cdr'of calculons
resultwill como out t tao small,
usc~h~~1 type jadgn~nt must bc
uscd t)~ ohjcct ben,g to approach thé truth asnearly as can
he donc w~thout toogrc.t sacrinco of
.hnpHcity.ypcor ~g hc.vily ~i,ht.d ~htLe tdœu froin thc extrême case of an innnite Joad ~hen thotwo
p~
of thést.~ .ould Le
str~ht. AsJe.~pl~cale..tion of tins Jun~ of which the rcsult is known, wo
will t~Tj~~~h:dw~th tcusion 7 anj mquirc what the period would be oncertam
supposions as to thé type of vibration.
Taking the origin of .r at t)io ~idd!o of tho string, lot thecurvc of vibration on thc positive sidc bo
~ul on thc ncg~vc side the Im~c of tins in the axis of ybc~g not !c.ss than .nity. This form satires thé condid~
ihat y vanishes whcn ~.=1 Wc h~vo now to form the ex-prcs~.s for 2' aud a.d it will Le su~icicut te c~t~
89.]PERIODS 0F FBEE VIBRATIONS. 87
positive lialf of tho string only. Thus, p being thc longitudinal
(tcuHtty,
and
Hcucc
If M==l, thc string vibratos as if tho mass were concentratcd
in its middie point, and
TT-TTho truc value of p" for the gravest type is
–,r,so that
plutho assumption of a para-boUc form gives a pcriod which is too
small in thc ratio 7r ~/10 or '993G 1. Tlie minimum of p",
VG +1as givcn by (2), occurs when
~=–=l'72-t74,and gives
It will he seen that there is considérable latitude in thé
choicc of a type, even tho violent supposition that thé string
vibratos as two straight pièces giving a period less than ton
pcr cent. in error. And whatever type wc choose to take, tlie
period calculated from it cannot be greater than the truth.
90. The rigorous determination of thc periods and types of
vibration of a given system is usually a matter of gréât diË&culty,
arising from thé fact that thé functions necessary to express tho
modes of vibration of most continuons bodies are not as yet rccog-
nised in analysis. It is therefore often ucccssa.ry to fait back on
methods of approximation, referring t!io proposed system to somo
VIBRATING SYSTEMS IN GENERAL.88
[90.
other of a character more amende to analysis, andcalculatiugcorrections
depending on the supposition that the differerce be-tween the tv.'o sy.ste~.s .aU. Th. r~ ~proxi.c!vsimple systems is thus one of great importance, more
especiallyas it is impossible in practice actually to realise tlio simple forn,sabout winch wc eau most casily reason.
Let ussuppose then that tho vibrations of a simple System arc
thoroughly known, and that it is required toinvestigate tho.sc
of a systcm derived from it by introducing small variations inthc mechanical factions. If &c. bc the normal co-ordi-nates of tho original system,
and for thé varicd system, rcferrcd to the sameco-ordinatcswhich arc how only approximtttciy normal,
in which&,
small
quan ~cs.
In eert.m cases new co-ordinates may appe~ but
if
so t!.cir coe~cnts must bc small. From (1) ~c obtam for theijagrangian equa,tious of motion,
.In theoriginal systcm the fondamental types of vibrationare thosc .h.ch
corrc.spondto the variation of buta single co-erd~na e .1 a timc. Let us fix our attention on one of
them,involving say variation of while a!I thcremnining co-ordinates vanish. Thc change in tlie system ,vi!l in ~1cntail an altcration in tlie iund~c.tatcr normal types; butunder tlie cu.cumstanccs
contemplatcd tlie alteratio~ small.ne normal type is e.pre~cd by the synchronous variationof h' other in to but ratio of anysmall.known, ~e normalmode of the aftered systcm will be dc-tei-mincd.
1 90.]APPROXIMATELY SIMPLE SYSTEMS. 89
Since thé wl)olc motion is simple harmonie, we may suppose
tha) cn.ch o~-ordinato va-ucs a~ cos~, a"<! f.~Lft.it.utu thc
diff'erential équations for D' In thé a"' équation occurs
with tiio Snitc coefEcient
Thé otlier tcrms a.rc to be neglected in a first approximation,
sincc both the co-ordma.te (rcla.tivcty to ~) and its coefficient arc
small quantities. Hcnce
Now
andthus
tlie required result.
If thé kinetic energy alone undergo va.ria,tion,
The correctcd value of the period is determined by tlie ?'t!)
equation of (2), not hitberto used. We may write it,
Thé first term gives tlie value of p/ calculated without allow-
ance for thé change of type, and is sufficient, as wc have aiready
proved, wheu thc square of thc altération in the system may
he neglectcd. The terms included under thc symbol S, in
which the summation extends to ail values of s other than r,
give thc correction due to thé change of type and are of the
second order. Since ?, and a,, are positive, thé sign of any term
depends upon that of –p~. 2* If > p~ that is, if the mode
s be more acute than the mode r, the correction is négative,
and makes tlie calculated note graver than beforc; but if the
mode s be thc graver, thé correction ra-ises the note. If t' refcr
90 VIBRATING SYSTEMS IN GENERAL.[90.
to the gravest mode of thé system, tho whole correction is
negative; and if r refer to thé acutest mode, tho whole coiïectionis positive, as we have aJrea.dy seen by another method. g
91. As an example of the use of these formulae, we maytake thé case of a stretched string, wliose longitudinal density pis not
quite constant. If x ho measurcd from oue end, andhc tho transversc displaccmcnt, t!ie configuration at any time twill he exprèssed by
being the longth of thc string. arc tlie normalco-ordiuatcs for p== constant, and t)iough hcre p is not strictlyconstant, tlie configuration of tbc systcni may still bo expressedby means of the same quantités. Since the potential cnergyof any configuration is tlie samc aa if/)= constant, 8~=0. For itlie kinetic cncrgy we liave
If p wero constant, thc products of tho velocities would dis-
appear, since &c. arc, on that supposition, the normalco-ordm~tcs. As it is, tlie mtcgml cocaicicnts, thoug!i uot actuallyevancscont, arc small quantities, Lot p=p.+~; thcn in our
previous notation
Thus thc type of vibration M expressed by
or, since
t 01.]EXAMPLES. 91
Let us apply this result to calculato tho disp~cemcnt of thé
nodu.1 point of the second mode (?'=2), which would bc iu the
iniddte, if tlie string woro uniform. In the neighbourhood of
this point, if x == + &c, tho approximate value ofy is
Hcncc when~=0,
approximately, where
To show the n.ppUca.tioji of these formula, wc may suppose
the Irrcgularlty to consist in a. small load of mMS p~ situatcd
at x =though thc result might bc obtained much more easUy
JIrectIy. We have
from which the value of Sa; may bc calculated by approximation.
'l'lie rcal value of 8x is, however, very simple. Thc series within
bmckcts may bc written
The value of thc definitc intégral is
and thus
Todliunter'a f)t(. C'tt~c. 255.
92 VIBRATING SYSTEMS IN GENERAL.[91.
as may also be rcadily proved by equ:ttin~ thé periods of vibra-tion of the two parts of thé
string, tha.t of the loadcd part buingca.Icutatcd
:),pp!-oxim:ttc]y on the assumption of' unchanged type.As ~u cx:unp)G of tlie formu!:), ((!) § 90 fur thé pcriod, wo
may tn.ko tho case of a. striug c:u-rymg a, smaH lo~d at its
middie point.. Wc havo
and t)ms, if P, bc thé value corrcsponding to = 0, wc g'ct whcu
?' is evcu, = 7~ and wheu r is odd,
whcrc thc summation is to bo extendecl to all t!)e odd vducs
ofNot,herthan?'. If?'=],
g!vlng t~o pitch of the gravcst tone accuratcly as far as thc
square of thé ratio À.
In the gencml case the value of p, correct as fur as thc
rtrstorJcriu~p.wiIIbc
02. Thc thcory of vibrations throws grcat Hght on expansionsof arbitrary functions in séries of other <\mctlo)is o(' spccif]cd
types. Thé best known cxamptc of such cxpansioDs is th~t
gencrally callod after Fourier, in which an arbitrary periodic
92.]NORMAL FUNCTIONS. 93
function is rcsolved into a. séries of harmonies, whose periods
arc submultiples of that of the given function. It is well known
that thé diniculty of thc question is confined to thc proof of tho
~OMtM~y of the expansion if this be assumcd, thé détermination
of thc cooHjcieuts is casy cnough. Wlia.t 1 wtsh now to draw
f~ttentio)). to is, that in this, aud au immense varicty of similar
cases, thé possibility of the cxptuisioli may bc infcrred from
physica.1 considerations.
To fix our ideas, let us consider the small vibrations of a
tmif'u)')astring strutc)~ed bctwceu rixc<t points. We know from
the gcncnd thcorythat thé wludc motion, wha-tever it may
hc, c:tn bc aua.)ysG(t iuto a. scries of componcnt motions, each
rcpresuntcd by a, harmonie function of tho time, and capable
ofcxisting by itscif. If we can discover thcsc normal types,
wc sh:dl bc in a position to rcprcscnt thc most général vibration
possible by combinmg thcm, assiguing to cach an arbitrary
amplitude and phase.
Aasuming that a motion is Iiarmonic with respect to time,
wo gct to détermine tlie type an equation of thé form
We infer that tlie most gencral position which tho string can
assume is capable of rcprcseuta.tion by a scrics of tlie form
which is a particular case of Fourier's theorem. There would
bc Jio dirHculty in proving thé tlicorem in its most general form.
So far the string has bcen supposed uniform. But we ha.ve
only to mtrojucc a variable density, or cven a single load at
any point of thé string, in ordcr to altcr compictely the ex-
pansion wliose possibility may be inferred from thé dy~amical
tlicory. It is unnecessary to dwc)l hère on this subject, as
wc stmil liave furtlier examples in thé chaptcrs on the vibrations
of pa.rticular Systems, such as bars, membranes, and connned
masses of air.
94 VIBBATING SYSTEMS IN GENERAL.f93.
93. The détermination of the cncnicicnts to suitarbitra
initial conditions may always hc rcadily enfected bv the funda-mental
property ofthc normal functions, and Itmay be convenicntto sketch the process Iicre for systems like strings, bars, mem-branes, plates, &c. in which thcre is only one dépendent variable~tobe considcred. If
~be tlie normal functions, and
~t, ~j, thécorrespondtng co-ordinatcs,
and thc problem is to dctcnninc so as to
correspond with arbitrary values of and
If p dx bc tite mass of the eicmcnt dx, wc have from (1)
But the expression for T in tcrms of~, &c. cannot containtiie products of tlie normal gcnGraI.zed velocities, and therefore
cvery iutcgra.1 of tlie form
Hcnce to determine 7?, wc have only tomultiply thc first
of équations (4) by pu, and intcgratc over tlie system. Wo thusobtain
Similarly,
93.] CONJUGATE PROPERTY. 95
The process is just the saine whether tho élément dx be a line,
area, or volume.
The conjugate property, expressed by (5), depends upon the
fact that the functions are normal. As soon as this is known
by the solution of a diSercntia.1 équation or othcrwise, we may
infer the conjugate property without further proof, but thé pro-
perty itself is most intimntely connected with thé fundamental
variational equation of motion § 04'.
94. If be the potential cnergy of déformation, thé
displacement, and p thc density of the (line, area, or volume)
clement dx, thé equation of virtual velocitics gives immediatety
lu this équation ~F is a symmctnca.1 function of and 8~,
as may bc rca.dily provcd from the expression for V in terms
of gencralizud eo-or<U)ia.tcs. In fa.ct if
Suppose now that refera to tho motion corresponding to
n. normal function so tha.t ~+?:~=0, whilc 8~'is idontinod
with another normal function M, then
Agtuu, if wc suppose, as we arc cqudiy c~tit!cd to do, that
varies as M, fu)d 8~ as K~, we gct for thé same quantlty ~V,
from which thé conjugate property folln-ws, if thé motions rc-
presentcd rcspectively by a.ùd M, have différent pcriods.
A good example of tlie connection of the two methods of
treatment will be found in the chapter on the transverse vibrations
of bars.
9G VIBRATINO SYSTEMS IN GENERAL.[95.
95. Professor Stokes' lias drawn attention to a vcry général
law connecting thon. p:~t~ &f t!ie f.ue mut.iun which dépend
on the initial cKsp~ce~M?t<s of a system not subject to fnction~l
forces, with titosc which depend on tlie initial velocities. If
a velocity of any type bo communicated to a system at rest,
and then after a small intcrvnl of time thé opposite velocity
ho communicated, tlie effoct in t)ie limit will be to start thé
system without velocity, but with a displacement of thé corre-
sponding type. We may rcadily prove from this that in order
to dcduce thé motion depending on initial displacements from
tbat depending on tlie initial vclocities, it is only necessary to
diSerentiate with respect to thé time, and to replace thé arbitrary
constants (or functions) which express thé initial velocitics by
thosc which express thé corrcspouding initial displacements.
Thus, if ~) bc any normfti co-ordinatc satisfying the equation
of which thc first term may bc obtaincd from.tlic second by
Stokes' rule,
Dynamical y/t<'or;/ of Dt~'racft'on, Can~rtf~e rraM. Vol. IX.
CHAPTER V.
VIBRA.TING SYSTEMS IN GENERAL
CONTINUED.
<)C. WlfEN dissipative forces act upon a system, the charactcr
of the motion is iu général more complicated. If two only of thé
functions 7', and be finite, we may by a suitable lincar trans-
formation rid our.setvcs of the products of thé co-ordinatcs, and
obtain t)te n<jrm:d types of motion. In the preceding chapter we
h:).vc conHidcrcd thé ca.so of ~= 0. Tho same theory with obvious
modifications will apply whcn 7'=0, or F=0, but these ca.ses
thougb of impurtance in othcr parts of Physics, such as Heat and
Electricity, scarcoly belong to our présent subject.
Thc'prcscjice uf friction will not interfuEC with the réduction of
T and to sums of squares'; but thé transformation proper for
them will not in general suit also the requirements of The
général équation can thcn only he rcduccd to thé form
~+~~+~+- +~=~. &c. (1),
and not to t!te simpler form applicable to a system of ono dcgrce
of frecdom, viz.
~+~+cA=~i. uc. (2).
Wc may, howcver, choosc whieli pair of functions we shall
rcduce, though in Acousties tlie choicc would almost always fall on
l' and Y.
97. There is, however, a not unimportant class of cases I)i
which the réduction of ait thrce functions may be effccted and
tlie theory then assumes an exceptiona.1 simplicity. Under this bead
U~e most important are probably those when j~is of thé same form
as T or V. The first case occurs frequently, in books at any rate,
when thc motion of cach part of thé system is rcsistcd by a re-
tarding force, proportional both to the mass and velocity of thé
R. 7
D8 VIURA.TINC 6YSTHMS IN GENERAL.[97.
part. Thc same cxccptioual réduction is possible whcn J~ is a
Iinear fun.) of T'Mf'! K cr wbon 7' is itself oft~.r' ~mp form
K J.n any of tliese cases, t)io équations of motion are of thc samc
form as for a. system of onc degrcc of frccdfnn, and tlie theory
possGsscs certiua pccuHarities which m{tke it wortLy of scparatocousidGra.tiou.
Thc équations of motion aro obta-incd at once froin F
~nd
in which thc co'ordinatcs arc scpa.rated.
For the froc vibrations we Itavc oniy to put <= 0, &c., and
tlie solution is of the form
and and are thc initia! values of<~ n.nd <
The whoïc motion may thcreforc bo analysod into component
motions, each of wltich corresponds to thc variation of but one
normal co-ordinate at a tinis. And tlie vibration in eacb of thèse
modes is altogcther similar to that of a systcm with only one
dcgt'cc of libcrty. After a certain thnc, grcatcr or less a.ceording
to the nmount of dissipation, tbc free vibrations become insignifi-
cant, and tlie system returns sensibly to rest.
Simuttn.ncous1y with thc frce vibrations, but in pcrfcct indc-
pen<)encc of thon, thcre may exist forccd vibrations dcpending on
tho quantitics tl\ Precisuly as lu. tlie case of ouc dc-groc of frec-
dom, thc solution of
To obtain thé cor)p!cte expression for (~ wc must n.Jd to thé
right-hn.ud member of (4), which makes the initud values of
and (~ vanish, thé terms given in (2) which rcprcscnt thé rcsidue
97.] GENERALIZATION 0F YOUNG'S THEOREM. 99
at time t of tho initia,! values and If there be no friction,
th<Y-<!(.i(.'of'~i.u(~)rcducc;,I.j
98. The complète indepcmlence of thé normal co-ordinates
leads to au interusting theorcin concerning the relation of tho
subsequent motion to thé initi.d disturhancc. For if tlie forces
whicii act upon thé system bc of such a clmracter ttiat' thcy do no
work ou thc Jispiaeemunt indieatcd hy tlicn = 0. No such
forces, huwe~er long continucd, eau produce any cn'uct on tho
motion If it cxist, thcy cannot destroy it; if' it do not cxist,
they cannot gcncratc it. TI)c most important application of thé
theorcm is wt~cn tlie forces apphcd to t!)G system act at a nodo of
tlie uormi],! component tliat is, at a point which thc componcntvibration in question does not tend to set in motion. Two extrême
cases uf such forces may bc specially noted, (1) whcn tho force is
an impulse, starting tlie system i'rom rest, (2) \vhen it lias acted so
long that the systum is agai)i at rest under its influence in a dis-
turbed position. So soon as tho force ceascs, natural vibrations
set in, and in tlie absence of friction would continue for an in-
dennite time. We infer that whatevcr in other respects their
charactcr may be, thcy contain no component of thc type Tliis
conclusion is limited to cases w!tcre T, F, F'admit of simultaneous
réduction, ineludmgof course tlie case of no friction.
99. The formutni quoted in § 97 are applicable to any Mnd of
force, but it will oftcu Itappen that wo have to deal only witli the
cnccts of impressed forces of tlie harmonie type, aud we may then
advantageous]yemp)oythe more spécial formu)u3 applicable to such
forces. In using normal co-ordinates, we iiave first to calculate tlieforces cl\, (1~, &c. corrcsponding to eacli period, aud thence deducethc values of the co-orclinates titcmselves. If among tl)e natural
periods (calculated without allowance for friction) there be anynearly agreeing in
magnitude with the pcriod of animprcsscd
force, tliecorresponding componcnt vibrations will be abnormaHy
large, ultless indecd tlie force itself bo grcat)y attenuatcd ni tlie
preliminary résolution. Suppose, for example, that a transverse
force of harmonie type and given pcriod aets at asingle point of
a stretched string. Ail the normal modes of vibration will, in
gênerai, be excited, not however in their own propcr periods, but
7-2
~0 VIBRATIN~ SYSTEMS IN (-.ENEKAI..['~9.
in thc period of tbe Imprcssed force but any normal component,
v.hich b:t~ u. nojj aL thep!j:nt o!'
n,pp!iu.tuu~wit! not bt; cxcited.
Thc magnitude of cach componcnt t)ms dépends on two tbings:
(1) on thé situation of its notics with respect tn the point at which
thc force is appHed, and (2) on thé denrée of agrccmcnt betwccn
its own proper period and that of thé force. It is import.fuit to
remembor that in respousc to a simp]u h:u')nomc force, thc syst.on
will vibra-tc in gcnera.) in «~ its modes, :dthong)i in pfn-tK'uhu'
cases it ma.y somctimcs be snOicicnt to nttc-nd to only onc of thcm
as bcirig of paramount importance.
100. When tho pcriods of tho forces oporating a.rc vo'y long
rc)~tivc!y to thé free pcriods of thc systcm, :n] cqui!ibriumthcoryis sometimes ad~uate, but in such n. ca.sc tlie solution could
gcnc!Lt!y Le fuund more casily without thc use of thé nonnn)
co-ordina.tcs. BcrnoulH'.s Dicory of thc Tides is of this class, :Lnd
proceeds on thcassumption that thc frcc pcriods of' thc masses of
watcr found on tbe globe are s!n:d) rdativdy <.o thc pcriods of thc
operative forces, in whicli case thc incrtia, of thc water might bc
Icftoutofaccount. As a matter of fact this supuosition is on]y
vcry rougidy and pa.rtialty applicable, and we arc conseqnc'ntiy
still in tbe dark on many important points relating to thc tides.
Thc principal forces have a scmi-diurnal pcriod, whicb is not sufn-
ciently long in relation to tbe natural pcriods concerned, to a)!o\v
of thé Incrtia of Ibc water buing ncgiccted. But if thé rotation of
the cartb bad bccn much slower, tbecquilibrium theory of the
tides migbt !)ave bccn adc(ptatc.
A con'cctcdcquDibrium t!)cory is sometimes uscfuL w])en thc
pcriod of tbe imprcsscd force is sumcicntiy long in compar!son
witb most of the natund poriods of aSystem,
but not so in thc
case of onc or two of thom. It will bc sufDcient to ta){c thc case
\vherc tucre is no friction. In thc équation
f?~ + c~)= <ï~, or + ?t~ =
suppose tbat t)ic imprcssed force varies as cos Theti
100.] EQU1LIHHIUM THHORV. 101
S)!ppo'.<)o\v <'))f~thi~t'u'c'sj~stif!;(,bk\f'xcfpti)(:p''ct
uf thu sin~te normal co-ontinatc ~),. Wc )):LVC tho) only to :uid
to thé rcsult uf thc cquitibrium thcary, the diircrcncc betwcou
the truc and thc tliere ;),snut)ied v~luc uf (& viz,
Thc other extrême case ought aiso to bo noticcd. If thc
ftu'ccf! vibrations bc cxtrcmciy rn.pid, they may becoino ne:u')y
iodupendunt of thé potential enei-gy of the system. Instout
of ne~cctin~ in comparison with wc ]i:wc thcu to ncg!cet?; iu comparisofi with wlucti ~ivcs
If tbere Le onu or two co-ordinatcs to w)iic)i this trcatnicnt
is not i~pplic~bic, wc may suppicniott thc result, calcuintcd on
Lhc hypothu.sis th:).t is !t)t.ogct!tcr nc~tigibic, with con'cet.ious
fur thèse particular co-ordinates.
101. Beforepassing on to t))c ~encml theory of thc vibrations
of .Systems snbjcct to dissipation, it may bc well to point out
Home pcculiaritics uf thc free vibrations of onntinuons Systems,
startcd bya force applicJ at a single point. On thc suppositions
aud notations of § ~8, tbe con6guration at any time is detcr-
jnincd bv
Suppose now that tho System is held n.t rest by a, force applied
at thc poijtt (?. T))C value of is detjcrmincd by thé considem-
tion tha,t <I\8< reprcsGnbs thc work donc upon thé System hy tlie
itnprcssed forces d'n'ing a hypothetical disptaecmcnt S~=S6
that is
102 VIBRATINQ SYSTEMS IN GENERAL.[101.
If thé system hc let go from this Cuiif!?ura.tion at<=0,vc
twe a.t any sub.uu't t:nie <
and a.t the point P
neither converges, nor diverges, with r. Thé series for ~thercfore
converges wltb t)~
Again, suppose that thc system is started by an impulse
from thé configuration of equilibrium.In this case initially
Dus gives
shcwing that in Uns case the series converges with n, that
is more slowly tha.u in thc prcvious case.
101.]SPECIAL INITIAL CONDITIONS. 103
In both M3. it mny br* nbsnrvcd thah tho value of is
symmctrical with respect to 2-' aud proving tliat tho disptacc-
ïncnt n.t time t for thé point 7-* when the force or impulse is ap-
pitpd at < is the Sinne as it would bo at () if tlie force or impulse
h:td bueu {),pplicd at -P. This is an example of a vcry general
reciprocal theorcm, which we shall consicler at !eugt)i pt'csc!itly.
As a thit'jd case wc may supposa thé body to start from rcstas dcfur)ned by a force M)!bn~y f~M~M~c~, over its lcn~t.1),
arca, or vuluinc. \Ve rcadily Hud
The series for will hc more convergent than whcn thc force
is conccntrated i)i a siugtc point.
In exactly tlie sa.mc w~y wc may trcat thé case of a con-
tinuous body whonc motion is Eubjcct to dissipn.tion, pruvidod
tliat thé tlirce futictions 2~ J~ bc simulta.ncousiy reducible,
but it is not necessary to write dowu tlie formuJœ.
102. If thé three mccha.nica.I functions T, -F' and V of any
system be not simultancousiy reducibic, tlie natural vibrations
(as has aiready bcen observed) arc moru complica.tcd in tlicir
charactcr. Whcn, lowever, thé dissipa.tion is small, the mctttod
of réduction is still usofnl; and this class of casusbcsidcs being
of sonc importancu in Itscif will form a good introduction to
tlie more gcncrat theory. We suppose thcu. that 2' and V arcc
cxprcsscd as sums of squares
Thc équations of motion a.re accord!ng]y
in which the coefficients & &c. arc to be trcatcd as small.
]f tlicrc were no friction, ttic abovc systcmof c(}uations wuuld
JU4 VIBRATING SYSTEMS IN GENERAL. F 10 3.
~e!u:uimg' part ofthc tenus mdudcdundcrS bcingrcit),thc con-uctiun lias no uifuct un Litc ruai p.n-t uf oa w}dchtitc r:T.te uf dceay (lej)cnds.
bc satisfied by supposing onu co-ordinate to vf~' suitahly,while t,!ic other co-ordinatc-s vanish. In the actual case thero
will be a corrusponding sohttion in which the value of any ot)icr
co-ordi!mtc will bu small rclativcty to 6,
Hence, if wc omit tc'rm.s of the second ordc-r, thc ?' equation
bceomes,
from which wc infcr tliat variesn-pproxim~tuly :LS if tliere
were no c!)angc duc to friction in thc type of vibration. If (A
v:u'yase'wcubt:utitodct(;rmiuc~
Thc roots of this équation arc comptux, but tho real partis small in eomparison with tlie imaginary part.
From thc équation, if wc introduce tlie supposition that
a-M tho co-ordinates vary as e" we gut
This cquatinn dctcrminesapproxiniatdy thc altcrcd type
of vibration. Sincc thc chief part of laima~hary, wo sco
tliat thc co-ordinn.tcs arca.pproxi!natc!y in the sa.me phMc,
~<~ </tC6<~j~Me f~y'e~ (î ~MM?'/er per~o~ /?'o!~ </<e ~aMq/' Hcnec wttcn thc function F docs not rcduee to a sumof squares, thc chamctcr of thc
c]cmentary modes of vibrationis ic.ss simp)u th~n othei-wisc, aud thc Y~rious parts of tlie Systemarc no
longer simuttanconsly in thu samc phase.
We provcd abovu that, w)tC!i titc friction is small, the value
of y?, may bc calodatcdapproximatuty without aUowancc for
thc change uf tyj)e but hy means of (6) we may obtain a stillclosur
approximation, in winch thusquares of thé small quaritities
are i-ctahied. Thu ?- équation (3) givea
103.]SMALL DISSIPA TIVE FORCES. 105
103. Wc now returu to ihb cunsidùratioti uf thc g.'n<t..J
cquations of § 84.
If &c. bc thé co-ordinates an<I &c. tlie forces,
wc !ia.vc
For thc free vibt'atiuus ~F,, &c. va.msh. If \7 bc thc dc-
tcnninant t
thc result of elimiuating from (1) aU t!tc co-ordinates but onc, is
V~=0. (4).
S~ncc \7 nnw co))taius odd powcrs of' D, t!)G 2?~ roots of tho
cquahon = 0 no jouter uccur in equal positive :uht ncgativc
piurs, Lut cot)ti).in !). ruai as wu!! as an imagmary p~rt. TIte
compJutu intégra! may ]n)w<;vcr stiïl bc writtun
= ~c~ + J'g~~ + Z?e~ + 7/e' +. (5),
where thc pairs uf cunjug-u.tc roots are uc. Corru-
Mpoiding to cach roût, thcro is a. particular solution such as
~=~~ ~=~ ~,=~ &c.,
in which thc ?Yt~'os j'l, arc determined by thc equa-tions of motion, and oniy thc absohjtc value ronains arbitrary.In t!te présent case ]iowcvur (wlicre contains odd powers of Z))thèse ratios aru not in gcncral i'c:d, and therefore thé variations
oithcco-oi'din:Ltes'&c.:u'c not
synchronous in phase. If
we put /~=a,+t/3,, ~=a,-t/3~, &c., wc sec tha.t none of thé
quantifies a can bc positive, since in that case thc energy of
thc motion would Incrcase with the time, as we know it cannot
do.
Enoug)i bas now beeti said on thc snbjcct of the froc vibra-
tions of aSystem in general. Any further illustration that it
may rcqnirc will bc anorded hy t!)c discussion of the case of two
dugrees of frccdom,§ 112, and by the vibrations of strings and uthur
spécial bodics with whicli \vc shaU soon beoccnpicd. We résume
ti)e équations (1) with thc view ofinvcstigating
further tbc
nature of forced ~m~to;
VIBRATING SYSTEMS IN GENERAL.[104.
10G
104. In or~r to cnmu.atc from thc crjnnH~ns ~!L t'~ .)-
ot-dinatcs but onc (~), oper~tc ou tilcm in succession with the
minor dctenuinimts
and a.dd the results togcthcr; and in IHœ manncr for thc othcr
co-ordinates. We ttius obtain as the cquivalcut of thc urigina.!system of équations
in which the dincrentiations of ~7 are to be made without re-
cognition of the cquaHty subsistmg botwecn e, and e
Thc forces &c. arc any whatcver, subject, of course,to tlie condition of not producing so grcat a displacement or
motion that tlie squa.res of thé small quantities become sensible.
If, as is ofteu t!ie case, the forces opcrating he !nade up of two
parts, one constant with respect to timc, and tlie other periodic,it is convenicnt to separatc in hn~ginn.tion tlic two classes of
cncets produced. T!ie effect duc to tlie constant forces is exactlythe same as if they acted alonc, and is found by thé solution
of a statical problem. It will therefore gcneraHy bc sufficicnt
to suppose thé forces pcriodic, tlie effects of any constant forces,such as gravity, being mcrcly to altcr t!tG configuration about
which tlie vibrations proper arc exccutcd. Wo may thus without
any rcat loss of gcnera]ity confine ourscives to perlodic, and
therefore by Fourlor's thcorcm to harmonie forces.
Wc might thereforc assume as expressions for ~P,, &c. circular
functions of thé tune but, as we sliidi have fréquent occasion
to recognise in thé course of this work, it is usualty more con-
venicnt to employ an imaginary exponential function, such as
~'c' where~Is a constant which may bc complex. When thé
corrcsponding symbolical solution is obtained, its real and
Imaginary parts may be separated, and belong respectlvc!y to
tiie real andImaginary parts of thc data. In thia way tlie
104.] FORCED VIBRATIONS. 107
~n!)Jy-t['! gam.s consider~biyiu
brevity, Inn.smuch ~s ditfcrpntmtton!!
anfl altérations of phnsc a.rc expresscd by mcrcly modifying
thé compicx coûfHcicnt without chang-ing thc form of thé functiou.
We therefore write
Thé minor dctcrmmanta of the type arc rational intégra.!re
fonctions of the Bymbo! 7), and operatc on &c. according to
thelaw
where &c. are certain complex constants. Aud thc sym-
bolical solutious arc
wherc (~) dénotes the rcsult of substituting for D in
Considcr f!t'st the case of a System exempt from friction. ~7and its (liHereutial coefHcieuts arc titen c~M functions of D,
so that ~7 (~) is rca!. Tbrowiug a.way thé imaginary part of
thc solution, writing ~°' for ~t~ &c. wc hâve
If we suppose tliat the forces &c. (in thc case of more
than one goiera.Uzcd component) liave ail thc same pliasc, they
may be cxpressod by
and then, as is casily sccn, thé co-ordiuatcs themsolvcs agrcc
in phase with tlic forces
Thé amplitudes of tlie vibrations dépend among othcr tlungs
on the magnitude of \7(?'~). Now, if thc period of thé forces
bc the same as one of those bctonging to tlie frce vibrations,
(ip)= 0, a.ud tlie amplitude becomcs iniiiiite. This is, of
VIHRATINC: SYSTEMS IN GENERAL.[104.
J08
course, just thc casu iu which it js essential to introduce the
Ct'nHtdct'f~tiun of friction, from wltich no natural system is ruaHy
exempt.
If thcrc bc friction, ~7 (ip) is eompiex but it may bc dividcd
into two pa.rts–onc rca.1 a)id thc otho' puruly Itn~ginary, ofw!iic)i
tlie latter dépends entirely on the friction. Thus, if wc put
V (~)=~ (~) +~ V.(~). (7), 1
\7.~ are eveu functions of and therefore real. If as bcforc
J,=Vt\e~ our solution takes thc form
Wc ha.ve said that ~(~) dépends entircly on thc friction but
it is not truc, on the othur hand, th:).t 7,(~) is cxa.ctiy thc s~me,
as ift)tcre had been no friction. Howcver, this is approximatciythe ca~e, if the friction ho sma!]; bccausc any pn-rt cf ~(~), which
dépends on thc first power of thé coe~cients of friction, is noces-
sari)y imaginary. W!)cncvcr there is n, coincidence between tho
period of thc force and tliat of onc of thé frcc vibratious, \7~;))
va.Dishcs, f~nd we ha.ve tan y = ce :).u<~thcrcforc
indicating a vibration of large amplitude, only Hmitcd by the
friction.
On thc hypothesis ofsmaiï friction, is in general smaU, and
so also is T, except in case of approximatc (.-qnality of pcriods.
With certain exceptions, thereforc, the motion bas nearly thé
samo (or opposite) phase with tlie force that excites it.
Wlicn a. force expressed by a harmonie term acts on a system,
the resulting motion is everywttcrc harmonie, and rcta.ins tlie
original period, providcd always that thé squares of the displace-
ments and velocitics may bc neg]cctc<1. This important principle
wa~ cnuuciatcd by Laphicc and a.pplicd by him to the theory of
104.]J INEXORABLEMOTIONR. lOf)
the tidus. Its.grcftt gcncratlty was atso rccogt)!scd hy Sir John
Hcrsche!, to witom wo owc a formai domonstr~tion of its truth*.
If thé force bc not a. harmonie function of the time, thc types
of vibration 'ni dtfferent parts of the system are in. général différent
from each other and from that of thé force. Thc harmonie
fonctions are thus thû on!y oncs winch préserve their type nn-
changed, wldcii, as was rcmn.rked in thc Introduction, is a strong
rcason for anticipating that thcy correspond to simple toncs.
105. We now tnrn to a. somewhat différent Idnd of forccd
vibration, where, instcad ofgiven forces as hitherto, given inexora-
ble wo~t'o~s are prescribed.
If we suppose t)ta,t the co-ordinates arc givcn
ftinctions of the thno, while thé forces of thc rcmaiuing types
vanish, thc équations of motion divide thein-
selvcs Into two groups,viz.
In cach of the ~–?' équations of thc latter group, thc first r
tcrms are known cxphcit fnnctions of thc time, and hâve thc sa.tne
cH'ect as know)i furccs a.cti)ig on the system. Thc Ct~uations of
this gronp are thcrcfore su~icicnt to (tt.'t.crminc thc uuknowu
quautities; after whic)i, If rct~uircd, thc forces ucccss:u'y to maLin-
ta,m tlie pi'cscribcd motion may bc Jetermiucd from thc rirst
group. It is obvions tit~t thcrc is no esscutial différence betwciCti
the two classes of prohtcms of forecd vibnitions.
10G. The motion of a systcm dcvold of friction and cxccuting
slinpicharmonie vibrations in conséquence of prescribed variations
of sorne of thc coordinat.es, posscsscs a pcculiarity paraUelto thosc
considcred in §§ 74, 7~. Let
= J~cos =
-jeos
~<,&c.
in whicit thc quantitics ~l,.arc regarded as givcn, whi)c thc
7~)cye. ~<~ro~. art. 823. AJHO 0)<f/t'))f.< < ~"<rf'))fMy, § fino.
110VIDRATING SYSTEMS 1~ GENERAL.
[lOG.
~~§T
from tlie expressions forl' and V, § 82,
2(y+ =~(~+~,)~/+.+(~+~)j~+.,
+~(~u)~+.+(~J~~+.jcos2~
from whic)i we see that tlie équations of motion express tlie con-dition th.t A', tlie variable part of y+ r, ~i,ich is proportional to
Hc,J~+.+(~)~~+,~
shall bcstation~-y in v~uc, for variations of tlie claautitics
~r., ~I, Lcb bc ttie value of~ n~turat to thc System wlieuvjbratiug under tlie restraitit dcHned Ly tlic ratios
Krom this we sec that if beecrtainfy less than tbat is,if the prc.scr.bed pcriod be grcatcr than any of tllose natural to
the system uuder the partial constraint rcprescuted by
~t.J,
then is necessarlly positive, aud tl~e.tationary val~e-t!~re canbc but ouc-~ an absolute minin~un. For a similar rea.so~ if the
près nbed ponod be less tiuL.i any of tliose natural te thcpa,-tia)iyconstraincd System is an absolute ~xhu~
a~braica!Iy, butarit)nuctieat!y an absoluto rnini.num. But whcu lies witbin thcrange of possible vaines of~, n.ay bc positive or négative, andthé actual value is not thé greatest or least possible, Wi~enever anatura! vibration is cor~sistent with the hnposed conditions thatwill ue thc vibration assumed. Tj.e
y.,l.Uc part of ?'+zero.
For convenience of treatment .ve hâve considered apart t),etwo gréât cl~es <.f forced vibrations and f~vibrations; but hc c
is, of course, noth.ng to prcvcnt their coexistence. After tl.e lapse oof a .sumc~nt interval of time, the frce vibrationsahvays
appcar, howcver small thc friction .nay be. The case of abso-lutely uo fnchon is purc!y idca!.
Ti.crc is onc caution, lowever, ~Lich may not bc supGrrIuou.in respect to thc case whcrc givon ~Jare forcj~
106.] RECIPROCALTHEOREM. 111
System. Suppose, as before, that the co-ordinates arc
givou. Thf'u ~t); .frc't) \'ibt'it.{.iui')~ wh' '~L-n<x' or .non-cxistcnco
is a matter of indiffurenco so fur as thé forced motion is concerned,mnst Le understood to be such as thé system is capable of, when
the co-ordinates are not aKo~ <o MtryJ~'o~?~ zero. In
order to preveut their varymg, forces of thc corresponding typesmust bc iutroduced; so that from one point of view thé motion in
question may be regarded as forced. But tlie applied forces are
mercly of the natu.rc of a constraint; and their ctï'ect is the same
as a limitation on the frecdom of thc motion.
107. Very rcmurkable reciprocal relations exist between tlio
forces aud motions of different types, which may be regarded as
extensions of thé cerrespondi.ng theorems for systems in winch
only For T bas to be considered (§ 72 and §§ 77, 78). If we sup-
pose that ail thc component forces, except two–F and ~F –arc
zero, we obtain from § 104,
We now considcr two cases of motion for the same system first
whcn~ vanishes, Mtd secondiy (with da.shcd Ictters) whcnva.nisitc.'i. Ii"~K=0,
In thèse équations ~7 and its dtiïcrentia! coefficients arc rational
Intégral functions of tlie symbol D; and sincc m cvury case
~r.=
~.r' V is a. symmetrical detcrminaut, and thercfurc
Hcucc wo sec that if a. force act on the system, tlie co-
ordinatc is rclatcd to it in the same way as the co-ordinate
is retatcd to tlic force whcn this latter foi-ce is supposed to act
a.Lone.
lu addition to thc motion hère contcmplated, thcre may be
frcc vibrations dépendent on a. disturbance ah-eady cxistin~ at thé
112 VIBRATINO SYSTEMS IN GENERAL.[107.
moment subséquent to which ai! new sources of disturbance are
included in ~F; but thèse vibrations are thcmsc)ves tho e~-ct ofiorc~s \t'hicii acLed previousiy. However sinail thé dissipationmay be, tliere must be an interval of time after which free vibra-tions die out, and beyond winch it is
unnecessary to go in takingaccount of the forces wbich hâve acted on a system. If thereforewe include undcr forces of sumcicnt reinotcness, there are no
independcnt vibrations to be considered, and in this way tlietheorem may be cxtendcd to cases which would not at first sigbtappeM- to corne within its scope. Suppose, for example, that the
systcm is at rcst in its position of equilibrium, and then begins tobc acted on by a force of the first type, graduatty Increasing in
magnitude from zero to a finite value at which point it ceasesto incrca.sc. If now at a given epoc)i of timo the force be sud-
dcn!y dcstroyed and reinain xcro c-vur afto-wards, frcû vibrations oftI)G systcm wIU set in, and continue until destroyed by friction.At any tirne t
snbsc()ucnt to thé given cpoch, tlie co-ordinatehas a vatnc dépendent upun t proportional to T))c tiicoremallows us to assert that this value bears the same rcJation to
as~outd at t))c same mnmcnt hâve borne to~ if thc originalcause of the vibrations bad been a force of thé second type in-
creasing g.-aduai)y from xcro to and thcn suddenly vanishingat thé given cpoch of timc. Wc ))avc ah-cady had an example oftins in § 101, and a like result obtains wiien thé cause of thé
origin:).! disturbance is nn Impulse, or, as in the problem of the
pianofortc-string, a variable force of finite though short duration.ln tljesc app)ications of our theorem we obtain results rclating toh'ee vibrations, considercd as t]jc residual effect of forces whoscactual opération may ]tavc becn long bcforc.
~08. In an important e)ass of cases thé forces and are
harmonie, a)td of thé samcpcriod. We
may rcprnscnt themby
J,e~ wherc J, and J~ may be assumcd to bc ?- If tbcforces be in thc same t-hase at the moments compared. Titoresults may then be written
108.]RECIPROCAL THEOREM-. 11~
SIncc tlie ratio Y~ is by hypothesis real, thé same is
true of thé ratio which signifies tha.t the motions
represcnted by those symbols are iu tlie same phase. Passing
to rcal quantifies wu tU~y state the thcorem thus
If a force=
A~ cos pt, c(c<t'H~ o~ ~e s~/s<e~ ~n'6 rise to
<<? ~to~t'o~ = 0A, cos (pt e) ~cn wt'~ (t force=
A~ cos pt
~ro~t;cc </tC ?~o<K)?t '= ~A~' cos (pt c).
If thcre Le no friction, e will Le zero.
If J,= thcn '=~. But it must be remembcrcd that
thc forces ~F, and are not nccessarily comparable, any more
than thc co-ordinates of corresponding types, one of wluch for
example may represent a linea-r and another an angula.r dis-
p!accmcnt.
Thc reciprocal theorem may bc statcd in sever~l ways, but
before proceeding to thèse we will give another investigation,
not requiring a knowlcdgc of déterminants.
If and be two sets
of forces and corrcspuuding disptacctnents, the équations of
motion, § 103, give
New, if aiï the forces vary as e' the cfFect of a symboUc
opcrator such an e~ ouany of thc quantttics is mcrcly to
multiply that quanti ty Ly thc constant found Ly substituting
?'/) for j9 iu Supposing this substitution juade, and havicg
regard to tlie rclationae~
=e~, we may write
wltich is tlie expression oi'tlie reciprocal ruiatiuii.
~9. lu thc applications that we arc abuut to makc it
will be snpposcd throughout that the forces of aiï types In:t
two (whicli wc mn.y as well take as the first and secoud) are
zero. Thus
VIBRATING SYSTEMS IN GENERAL. [109.114
Thé con'icquences of this equation may be cxhibited in thrce
dinerent ways. In tlie first we suppose that
~=0, ~/=0,
whcnco
~=~' ~(2),
shewing, as before, that the relation of to in thé first
case when ~==0 is tlie same as thé relation of to in
tlie second case, when = 0, tlie identity ofrelationship' ex-
tending to phase as well as amplitude.
A fe\v examples may promote the comprehension of a law,whose extrême
generality is not unilkely to cuuvey an impressionof vagnencss.
If .P and Q bc two points of a horizontal bar supported in
any manner (c.g. with one end clamped and tlie other frce), a
givcn harmonie transverse force applied at P will give at anymoment the same vertical dencction at Q as would have been
found at had the force acted at (~.
If we take angular instcn.d of lincar displacements, the
theorcm will l'un :-A given harmonie couple at P will give the
same ?'o~~o~ at as the couple at would givc at P.
Or if one dispJacemcnt bc Jincar and the ot))er angular, thércsult may be stated thu.s –Suppose for thc first case that a
harmonie couple acts at .P, and for thé second that a verticalforce of the same pcriod and phase acts at Q, thon thé linear
displacement at Q in t!tc first case bas at cvcry moment tho
same phase as the rotatory displacement at in thé second,and tbe amplitudes of thé two déplacements are so related that
thé maximum couple at P would do the same work in actingover thé maximum rotation at P due to thé force at Q, as thé
maximum force at <~ would do in acting through tlie maximum
displacemcnt at Q due to the couple at P. In this case thé
statement is more compHcatcd, as the forces, being of different
kinds, cannot be taken equa!.
If we suppose thc period of thé forces to be excessivoly long,tbe momentary position of the system tends to coïncide withthat in which it would be mamtained at rest by thé then acting
forces, and tlie equilibrium theory becomes applicable. Our
theorem thcn reduces to thé statical one proved in § 72.
As a secondexample, suppose that in a space occupied by
air, and either whol]y, or partly, connncd by sotid boundaries,
109.J APPLICATIONS. 115
thcre arc two sp)tercs and whosc centres have one denrée
of freeclom. Thoi a. periodic force acting on ~4 will producotho same motion in j?, as if the parts werc
intcrdtangcd and
thi.s, wliatevcr mcmbra.ncs, strings, forks on résonance cases, or
other bodie.s capable of bcing set into vibration, may be present in
their neighbourhood.
Or, if A and dénote two points of a solid etastic bodyof any shape, a force paraUcl to acting at A, will produccthe same jnotion of the point parallcl to Oras an cquaL force
pn.ra!Iel to Oy acting at would producc in the point ~1,
pM'a!!cl to ~J~.
Or aga,in, lot A a)nl Le two points of a space occupied byair, between which arc situatcd obstacles of any kind. Thcu asound originating- at A is perccived at B with the samc
intensity
as that witit which an cqual soundoriginating at jS would be per-
ccived at ~i/ Thc obstacle, for instance, might consist of a rigidwall picrcecl witli one or' more holcs. This example correspondsto the optical law that if by any combination of renectin~ or re.
fracting surfaces one point can be seen from a second, the secondcan also bc seen from thc first. Fn Acoustics the sound shadows
arc usually only partial inconséquence of the not insignificant
value of thé wave-Iength in comparison with thc dimensions of
ordinary obstacles and tlie rcciprocal relation is of considerable
interest.
A further example may be taken from electricity. Lct therebe two circuits of insulated wire /1 and B, and in their neigh-bourifood any combination of wu'c-eircuits or solid conductorsin communication with condensers. A periodie electro-motiveforce in thé circuit A will give rise to thé same currcnt inas would be excited in il if the cicctro-motivc force opcratet)inR
Our last example will bc takcn from thé theory of conductionand radiation of heat, Ncwtou's la,w of cooling being assuvnedas a basis. Thé température at any point ~t of a conducting and
radiating system due to a steady (or harmonie) source of hcatat is thé same as thé température a.t due to an equal sourceat Moreover, if at any time tlie source at B be removed théwhole subsequent course of température at A will be the sameas it would be at B if thé parts of.D and A were interchanged.
1Helmhoitz, C'r<~< Bd. Lvn. Tho Bonnes must be Rnch as iu the absence of
obstacles would diffuse thernselyos eq~Iy in ~11directions.
8–2
VIBRATINH SYSTEMS IN CHNKRAL. [110.n~
HO. The second way of stating the reciproc~ theorcm is
fu'rived at by t~king in (1) of § 1()9,
shewing that thc relation of to ni thé nrst case, wlien = 0,
is the samc as the relation of to in tlic second case,
wheit ~=0.
Tlius in tite cxampio of the rod, if thc point P be held at
rcst wliilc agivcu
vibration isimposed upon (by a force thcrc
applicd), thé reaction at jP is thc same hotli in amplitude and
phase as it would bc at if that point were beld at rest and
thc givcn vibration were imposed upon 7~.
So if J- a.nd bc two electric circuits in thé ncighbourhood
of any uumber of othcrs, C, D, whether closed or terminating
in condensers, and a givcn periodic enrrcnt bo cxcitod in ~1 by
thé necessary cicctro-motive force, thé Induccd cicetro-motive
force ii) is thc saine as it wou!d be In ~t, if thc parts of ~1
and wcre Intcrchangcd.
TItC tinrd form of statemcnt is obtaincd by puttingin (1)
of 5 109.
proving that thé ratio of to in tho first case, \vhpn nets
abne, is tlie négative of tlie ratio of to in tlie second
case, when tho forces arc so rclatcd as to kecp cqual to zero.
Thus if thé point P of the rod be held at rest while a
periodic force acts at Q, tho rcaction at P bears tho samc numeri-
cal ratio to the force at Q as thc disptaccment at Q would bcar
to thé displa.ccmcut at P, if thé rod wcre causcd to vibrate by
a force applied at .P.
111. Thc reciprocal theorem bas been proved for ait Systems
in which the frictional forces can be represented by tlie function F,
but it is susceptible of a further and an important generaHzation.
We have indeed proved thé existence of the function F for
a large class of cases whcrc thé motion is resisted by forcf's
proportional to thc absotut.u or relative velocities, but theru arc
11L] TWO DECREESOF FRËEDOM. 117
oth~r sources ofdissipation not to be brought under this hcad,
whose effects it is eqnally important to include for exemple, thé
dissipation due to the con(htction or radiation of hcat. Now
f).tt))ough it bc truc that the forces in thèse cases arc not for ~M
~)~6'~e ~ùhuns in a constant ratio to the velocitics or displace-
ments, yct in any actual case of pcriodic motion (T) tliey arc
ncccssarity periodic, and tttcrcforc, wliatevur tlicir phase, ex-
pressible by a sum of two tonns, one proportional to thé dis-
placement (absolute or relative) and tho other proportional to the
vulocity of thé part of the system aneetcd. If thé coemcicnts
bo thc same, notncccs.sarity for ail motions whatever, &br a~
motions u/e~ T, the fmiction ~exists in thc only sonse
requirud for our présent purposc. In fact since it is exclusivelywith motions of pcriod T titat t)ie Dtcorcm is concerncd, it is
p):un!y a matter of indiiTcruncc whct)icr tlie fonctions Y; F
are dépendent upon T or not. Thus cxtendcd, tho theoi-em is
pct-haps sufliciently gênera) to covo.- tho wtiole ricld of dissipativeforces.
It is important to remember Hiat the Prnicipio of Reciprocityis ilmited to systems which vibratc about a configuration of e~t-~M~)t, and is therefore not to bo apptied wititont reservation tosuch a problem as tliat presented by thc transmission of sonornus
wavcs through tbe atmosphère wi)C)i clisturbed by wind. Thcvibi-iLtions must also bc of such a charaeter that tlie square of themotion can bo ncglectcd througitout; ot))crwise our démonstra-tion wou!d not hotd good. Other apparent exceptions dépend ona
misunderstanding of thc principle itsclf, Carc mnst be takcnto observe a propcr corrcspoudcnce between the forces and dis-
placements, the ruie being that thé action of thé force over tho
disphccmcnt is to represent wo~ ~ne. T)ms co!~)~ correspondto )'oMw:s, ~re~M?'M to inercmcnts of ~'o~trne, aud so on.
112. In Chapter III. we considered thé vibrations of a
system with onc degrec of frccdom. TIie remainder of the pré-sent Chapter will he devoted to sonic detaits of the case whcre the
degi'ccs of freedom arc two.
If and y dénote the two co-ordinatcs, t)ic expressions for 2'and F are of the form
118 VIBRATING SYSTEMS IN GENERAL.[112.
so that, m thc absence of friction, the cquations of motion arc
Thc constants L, J)f, -~V; ~1, 7~, C, arc not entu'ely arbitr~u'y.
Since l' and F arc essuutially positive, thc foHowing mequa)itics
juust be satis~icd
Zy> ~1C>~ .((!).Moreovcr, L, N, ~t, C must thcmsdvcs be positive.
We procecd tu examine thé ciTuct of tliese restrictions on the
roots of (5).
Iti thé first place t!)e tin'ec coefficients in the equation are
positive. For the first and third, this is obvions from (G). Tlie
cocMcicnt of
m winch, as is sccn from (6), ~/ZV~6~ is ncccssarily grcatcr than
J/7- Wu concludc thttt tlie vulucs of if rca!, arc both négative.
It rem:t.Ins to provc that tlie roots are m ihct rcal. Dm eoi-
(.Uti<j)t to bc satisHcd is that thc i'olluwing quantity be not néga-tive
whieh shows that thc condition is satisncd, since ~A~lC-Jt/~is positive. This is titc au~ytica! proof th:tt thc vaincs uf are
hoth rca! and négative a fact tlu~t might I~vc bccn anticipatud-iUtout :uiy an:dysis from titc pltysica.1 constitution of thc systum,whosc vibrations
thoyserve tu
express.
113.]ROOTS 0F DETERMINANTAL EQUATION. 119
Thc two values of are different, uniess &o~
Thé common spherical pcndulum is an example of this case.
By mcans of a suitable force F the co-ordinatc may be pre-
veuted from varying. T]to systcm thon loses one dugrcc of frcc-
<!o:n, and thc purnjd eon'cspouding to thc rcmaining onc is i)i
general diUbrent from cither of thosc possible beforc thu introduc-
tion. of 3~. Suppose tliat tlie types of the motions obttUtied by
titus preventing iu tuni the variation of and x are rcspectively
e~ Tlien are thc roots of the équation
(L~ + A) (~V~ -t. C) = 0,
bcuig that obtained from (4') by supprcssing and R Hclice
(4) may itscif bc put into the form
Zy(~) (~ -~)= (~+ B)' (8),
wltich shews at once tliat ncitlicr of tlie roots of X" can be inter-
modiatc in value betwoeu a.ud A little fui'thcr examina-
tion will provc that onc of the routs is grciLtcr than hoth the quan.-
tities and the othcr loss tha!i both. For if wc put
/(~) = L~(\' /) (~ ~) (.V~ + ~)=,
wc sec tbttt whcn is vcry smati, f is positive (J~–J~); when
deercasca (id~cbr&icaDy)to f dingos sign and bccomcs
ncgativc. Bctwcoi 0 a.nd there is thcreforc a root; imd :t.)so
by sintii:).)' rcasoning bctwccn aud ce. Wc conchtdu tha.t thc
tones obt:t.iued by subjecting t)ic systcm to tbe two kinds of con-
stnuut in qnustio)i arû bot)) intunncdia.te In pitc)i bctwecn tbc
tonos giveu by t)~G nntuml vibrations of tlic system. lu p:ii'ticu!a.r
cases /t may bc cqua.1, and then
This proposition mn.y bc gcnerulixcd. ~h?y ~~nd of constr~mt t
wLidi Icuvcs thc System still in posscssiolof onc (tcgrcc of frce-
dom may be rcganicd ~s thc impositiou of n, fot'ccd relation
bctwccu t)ic co-ordiuates, such as
120 VIBRATING SYSTEMS IN GENERAL. [112.
Now if Cta? + a.nd any othcr homogeneous Hncar func-
tiult r.f .'s rmd .y,~o tfd!cn ~s n~w vnrinUcp~ t.ijo eftinc argument
provcs tha.t tlic single pcriod possible to thc systc'm after t)io
introduction of tho constmint, ja intermc(U:Ltc in va.!nc bctweoithosc two in which tlic natural vibrations wcrc prcviousty pcr-
ibrmcd. Convcrs~Iy, t!)C two periods which bccomc possibJc
whcn constraint is rcmoved, lie ouc on cn.c!t si()c of tho original
period.
If tlie values of À." be cquat, winch can on)y iMLppen when
Z ~=.1
thc introduction uf :L constmint h~s no crrcet on t)~c pcriod fur
instanco, thc !imit~ti(jn of a sphcrical pundutum to one vertical
phme.
113. As a. simple cxampte of a sysLem wittt two (tegrecs of
H'CL'doni) wc may take a. strctehcd string of ]en~t)i itsdf with-
out inortin, but cnrryiug two uqua.) nasses /?t nt distfuiccs a a.nd
6 froin onc end (Fig. 17). Tuusion =
rig. 17.
Sincc T and F are not of thc satne form, it fullows t)tat thc
two periods of vibration aru in cvcry case nncqua!.
If tl)e loftds be symmct.ricn.Hy a.ttactio(~ thc cLa.mctcr of thc
two componcnt vibrations is évident. In the first, which wil! Itave
t)Le longer period, titc two weights move togcthcr, se that a' and yrcma.in equ:d throughout the vibration. In tho second x n.nd arc
nmncriea!Iy cqua!, but opposcd in sign. Thé middie point of the
string thon rcmains at rest, and tlie two masses arc aiways tobc found on a straight Une passing through it. In the first case
= 0, and in thc second x + = 0 so tliat x and + y
arc thc ncw vanahies winch must he assmncd in ûrdcr to rcducc
the functions T and Fsimultancousty to a sum of squares.
113.']INTERMITTENT VIBRATIONS. 121
For example, if thé masses bc so attaclied as to divide thé
string into three equal parts,
fro.n which we obtain a.s thé complete solution,
where, as usual, thc constants a, 7?, j8 arc to be dctcrmincd by
the initial circumstances.
114. Whoi thc two Jiatural periods of a systcm are nearly
equal, the phe~omcnon of intermittent vibration sometimes prc-
scnts itself in a very curions manuer. In order to ittustratc tins,
wo ma.y recur to ttic string loadcd, we will now suppose, with two
equal masses at distances from its ends cqun.1 to one-fourth of thé
length. If thc middte point of tlie string were absolutcly iixcd,
thé two sinuhu' aystonson eitlicr side of it would hc compictcly
independent, or, if thc whole be considered as one system, the two
periods of vibration would bc cqnal. Wc now suppose that
Instead of bcing absolutely nxed, tlie mid(Uc point Is a.ttachcd to
sprints, or other machincry, dcstitute of mcrtia, so that it is
capable of yichling s~/t~y. The reservation us to incrtia is to
avoid the introduction of a third dcgrce offrocdom.
From thé symmctry it is évident that thc fundamcntal vibra-
tions of tlic system arc thosc rcprcscnted by a;+y and a?-y.
Thcir periods arc shghUy différent, bccause, on account of the
yieldin~ of thc centre, thc potential energy of a déplacement
\vhcn œ and v are equal, is less than t)iat of a disp!acemcnt
whcn x and y are opposite; whcrcas ttie kinctic énergies arc
the samc for the two kinds of vibration. I)i thé solution
wc arc theroforc to regard M'! as near)y, but not f~ntc, cqu:d.
Now let us suppose that initially a? aud n: vanisi). Thc condi-
tions are
122 VIBRATING SYSTEMS IN GENERAL. [114.wiuch give approximatcly
Thus
Thc va)uo of thc co-ordinatc .T? is bercapproxitoatL'iy cx-
prcsscd by I~u-rnouic tcrm, whosc an.phtudc, being proportiona!
tosin-
t, is aslowly varying harmonie function of t!ic thne.
TIic vibrations of thc co-ordinates are tbcrcforc Intermittent andso adjnstcd t]~t each iuuplitudu v~)ns)ics at tbe moment tl~at theothcr is at its !uaximum.
T)us phenomenon may bc pret.Uiy shcwn by a tunin~ fork of
vcry low pitcb, hcavUy wui~htud )Lt tbc ends, i~d fh-m!y'hdd byscrewing t!ic staik iuto a massive support. W!tcn tLc fork vibratoin thc normal ïnanncr, thé rigidity, or want 01 rigidity, ot' théstalk dœs not comc into p!ay; but if' tbc di.spj~ccmcnts of'the twoprongs Le m t)ic samc direction, t~c .s!ig)it yidding of Hie sta!kcntails a small
change of pcriod. If t)ic furk be excitcd by strikh~oneprong,t)ic vibrattons are
intermittent, and appcar to transfeî-tl~emscives back~-ard.s and forward.s bctwecu thé
prongs. U.dc.sa,
howcvcr, t)iesuj~port bu vcry firm, Utc abnorm:d vibration, which
involvcs a .notion of thé centre of Inertia, is soon dissipated andthon, of
course, tbe vibration appcars to bccomcstcady If thc
iork be mcrc)y hctd in thc hand, t!ic p!ienomeuon of' mtennitteneccannot bc obtaiucd at a!
115. TItc strctclicd string with two attaclt(jd nasses May bcuscd to ~lustrale somc gênerai principies. For example, ths periodof t!tc vibmt.ou which remalus
pos~ibte wbcu onc mass i.s !te)dat rcst, is Intermedi~te between thé two frec pericds. Any in-crcase ni eithcr Joad depresses t!ic pitcb of both thé naturalvibrations, and
co~vcrsciy. If t)ie new load be situated at a pointci théstring not,
cuinciding witb tlic places whci-c t)ie other !oa()sare attiiched, nor with tho uodc of one of' thc two prcviousiypossible frcc vibrations (thc othcr lias no nodc), thé efî'cct is stillto
prolong both thc periods alrcady prc.scnt. With regard to thethird nnite period, w]iicli becomes possible for thc first time afterthe addition of the new load, it must be rcgardcd as denvcd from
115.] IMPRESSEDFORCES. 123
one of infinitely smalt magnitude, of which an iudennitc number
may be 8t)pposcd to form part of the system. It is instructive
to trace tlie enect of the introduction of a new load and its graduai
increase from zero to infinity, but for tins purpose it will be
simpler to take thc case where there is but one other. At the
connueiicejncnt thcre is one finite pcriod T~and another of in-
nnitcsimal tuagnitudo T~.As t!)e load increascs T~ bccomcs finite,
and both T. and T.. continually increase. Let us now considur
wliat happciis when thé load becomes vcry grcat.Onc of thc
puriods is nccessarity largo and capable of growing bcyond ail
limit. The otiicr must approach a fixcd iinite Innit. T!ie first
bcloags to a motion in which thc largûi- mass vibratos nearly as
if tlie other were absent thé second is tlie period of thé vibration
of tlie smiUler mass, taking place mucb as if the larber werc fixed.
Now sincc ï~ and T~ can nover bc equaL must be aiways thé
gruatcr a~d we infer, that as tlie load becomes cot~tinually larger,
it is ï~ tliat met-cases iudennitcly, and T~ that approachcs a iiuite
limit.
Wc uov pass to tlie consideratio!i of forccd vibrations.
116. Thé général équations for a system of two degrecs of
frecdom includingfriction arc
If thc conncction between x and bc of a loose character, thé
constants Jt~, ~3, are small, so that tlie tcrm (J9 ~W +1'/3~)"
in thé denominator may in général bc ncglected. 'When this
is pcrmissiblc, thc co-oi-diiiate y is thé same as if x had been pre-
vented from varying, and a force V had bcen introduced whose
tna~itude is independent of N, y, and C. But if, in conséquence
of an approximate isoclironism between the force and onc of the
motions which beeome possible whcn x or is constraincd to bc
zero, eitlier ~+~~orC'+~be smaU, then tlie
term in the dcnominator coutaining tlie coefficients of mutual
innuencc must be retained, bcing no longer ?'e~~ue~ unimportant;
and thc solution is aecordingly of a more complicatcd charactcr.
1~ VIBUATI~O SYSTEMS IV GENERAL,Hl~
~'mmetry~hcw.s that If we had ~sumed A~=0 y=~" w.
~o.Id have fou..d the .ne va).c fur ..s no~vobtah.fur. This
R~.p, a
uc lurcncd to as .u exiunpjo.°
th.tTi~ -suppose
v- 1 r <
~<~==~) ~t is p.-c.scnbcd,
vhi~1=0, and for g~ter .ImpHcity we .shalt coufinc ourscIvcJto t!.c case ~hcrc /3 = U. TI~ vaiuc of~ is
.n. ''?'~"7 P~ of t)'6 co.mcl.nt of Le rc
spcetivejy ~j~~y
~e ic-
and
Itappears tlmt tlle effect of tho reaction
of (over and above~hat:
ffc'ct of(–– -1
into .4
be causcd
ci'= is ~J~~scnted by chang.-ng
~~1"the
tl. ~1to
~n intho coefficients ofspring and friction, 'l'liese
a1tcmtiolls, howcver,=~ of tlce peniod of tlee ~~aotion cou-tenrylccted, whose cllaracter ive now pl'occed to cousicler.
Por~ the valuecorrcsponding to ~e n.tnnd frictiun!ess
of (., be~g n~u~aincd at zéro); so that ~V=o''l'jJ(3ll
J J1 most cases with IVhich we arelrvctically C0J1ccrno(1 'Y is
?:?=~
of y nut l11ucI. ditJer..Wo sh.JI
acconhng!y Jeavc out of account ihc
117.] REACTION0F A DEPENDENTSYSTEM. 125
variations of thc positive factor (ZF- J! and in thc small tcrm
'y~, substitutc for~) its approximatc vainc ?!. Witcn p uot
nearly equal to M, the tcrm lu question is of no importance.
As might be anticipatctt from thé gênerai pnncipic of work,
&' is aiways positive. Its maximum. value occurs wlicu p =
ncarly, and is thcn proportiollal to which varies ~e~e~/ with
y»'y. Tins might not hâve bcen cxpected on a
supernclalview of the
mattcr, for it sccms rather a paradox that, thé grcatcr thé friction,
thé !c.ss Hho)t)(! hc its resn!t. But it must bc remonhci'cd tha.t 'y
is on]y tiie co~'c~e/!< of friction, and that whcn y is small t)io
maximum motion is so much incrcascd thf~t thc whoïc work spent
against friction is gi'catcr tilan if'y were more considurahle.
But thc point of most Interest is the dcncndeiicc of ~1' on
If ~) bc less than x, ~1' is négative.As p passes through thé va.Iuc
?:, ~1' vanisitos, am~ changes sign. WI)on J.' is négative,thé in-
Hncncc ofy is to diminish thé rccovcring powcr of tbc vibration a?,
aud wc sec that this happons whcn thc furccd vibration is slowcr
tliau t)iat natural to Thé tenduncy of thé vibration y Is thus
to retard thé vibration x, if tho latter be ah'cady thc slower, but to
accelcratc it, If it bc ah'cady thc more rapid, ou!y vanistting in tbc
critical case ofpcrfect isochronism. TI~c attempt to makc .B
vibrate at thu rate detcrmincd by n is beset with a peeuHar
difnculty, anaiogous to that met with in balancing a hcavy
body with thé centre of gravity above thé support. Ou whicb-
cvcr sido a shgtit departure from précision of adjustmcnt may
occur thé innucncc of thé dépendent vibration is aJways to incrcasc
thc error. Hxatnph's of thc Instabihty of piteh accompanying a
strong résonance will comc across ns hercafter; but undoubtcdly
thc most intcrcsting application of thc results of this section is to
thé explanationof the anomahius réfraction, by substances posscss-
ing a, very markcd sclectivo absorption, of thé two kinds of light
situated (in a normal spcctrum) Immetnatuty on citbcr sidc of tbc
absorption band*. It was obsc~'vc(~ by Christianscn and Kundt,
thc discovcrcrs of this rcmarkalde phenomenon,that média of the
kind in question (for example,/MC/~MC in a!coho)ic solution) rcfract
thé ray immcdiatcly ~~o~ thé absorption-band abnorma.Uy tM
e.ïCMs, and that above it in <e/ec<- If we suppose, as on othcr
grounds it would be natural to do, that thé intense absorption is
J'/u't..1~ M~y, 1872. A)so SoUm~inr, r~y. /i)t)). t. cxliii. p. 272.
<
126 VIBH.ATING SYSTEMS IN GENERAL.[LIT'.
the rcsult of an agréeront bctwccn. the vibrations of thc kiml of
light affecte d, and somc vibration proper to tbc mo]eeu)es of thc
absorbing age~t, oui- theory would in(!ica.tc tb~t for light of some-
w]t~t gi-(i:ttcr poriod t!ie cH'cct inust bc thc saine as a relaxation of
tho natural clasticity of the cthur, rnanifustuig itscif by a slowcr
propagation aud incrùasud réfraction. Oit t))c otitor sidc of tbc
absorptioM-band its rnHucucc must bc iu thc opposite direction.
lu ordcr to trace tlie law of conncction hctwecn ~1' and takc,for brevity, 'y~ = f/, jV~ /r)
=x, so t)t:Lt
Whcn. the sign of .<-is chan~'d, /t' is rcverscd with it, but pré-serves its muncricai value. Whun a;=0, or ±M, ~1' vanislies.
Hcnce thc origin is on thc représentative curvc (Fig. 18), and thé
axis of x is an asymptote. Thé maximum n.nd minimum vaincs of
~t' occur wtien x is respectively eclual to + ce, or –a a.n(t thcn
Hencc, the smallcr thé value of or 'y, thé grcatcr will bc thc
maximum alteration of tuni tliocorrcspohding vainc of will
approach uearcr a.nd nearer to n. It may be well to repeat, that intlie optical application a (liminishcd is attend cd by an ~crpf<M~maximum absorption. When the adjustment of periods is such asto faveur ~t' as much as possible, thc
corrcspondijig value of a' is
one hn.lf of its maximum.
CHAPTER VI.
TRANSVERSE VIBRATIONS 0F STRINOS.
118. ÂMONG vibrating bodies tliere are none tliat occupy a
more promineut position than Stretclied Strings. From tlic
earliest times thcy have bcen employed for musical purposes,
and in thé présent day thcy still form thé essentielparts
of such
important instruments as tlie pianoforte and the vioHn. To tho
mathematician they must always possess a peculia.r interest as tho
battle-neld on which wcre fouglit out tlie controversics of D'A)cm-
bert Euler, Bcruoulli and Lagrange, relating to the nature of tho
solutions of partial difTerential équations. To tlie studcnt of
Acoustics thcy arc doubly important. In conséquence of thé com-
parative simplicity of their theory, they are the ground on which
difncult or doubtful questions, such as those rclating to the nature
of simple toncs, can bc most advantageousiy faccd while in t!]o
form of a Mouochord or Sonomcter, thcy afford tlie most gcnc-
ratty available means for thc comparison. of piteli.
Thc'string'
of Acoustics is a perfectly uniform and floxible
clament of solid matter stretched between two fixcd points–in
fact Ml ideal body, never actually realizcd in practico, though
closely approxima.ted to by most of thé strings emptoyeJ in music.
We shaU afterwards sec how to takc account of any small devia-
tions from complete ncxibility and uniformity.
Thé vibrations of a string may be dividcd into two distinct
classes, which are practically independcnt of one another, if the
amplitudes do not exceed certain limits. In thé first class t!tc
displacements and motions of the particles are ~o~tf~ so
that thé string always ret~ins its straightness. The potential
energy of a déplacement depends, not on the whole tension, but
on tho changes of tension which occur in thé various parts of tho
string, due to thc increased or diminished extension. In order to
TRANSVERSE VIBRATIONS 0F STRINGS. [118.128
calculate it we must know the relation between the extcn.slon ofa
strmg and tlic stretching force. Thé iipproxim~tc hw (giveti byRccku) may be exprcsscd by s~'ing tliat thé extension variesas thc tension, so that if aud dénote tlie uatural and t!icstretched Jengths of a string, and 7'tlie tension,
whcre is aconstant, dépendit on thc m~tcn:d and thé action,
~ncti m~y bc intcrpreted to meaa tl.e tension th.tt would bc
necess~y to strctcii t].estnng to twice its natuml !cngth, if t).c
law apphed to so grcat cxteu.sions, whicl., in gcnem!, it is farirom
douig.
119. Thé vibrations of U.e second kind arc ~YtH~~e; that isto say, the particles of thé .string movo sensibly in planes perpen-diclllar to the Ime of t), c
string. In tliis case t)~e potential ener.-yof a déplacement depends upon the genend tension, and thé
~aUvariations of tcnsion
accompanying t!.e additionalstretcl.iur.duc to the dLsp]accmcnt .nay bu Icft out of account. It is he~
as.suincJ ti.at the.s~ching duc to ~c inotioa rnay 1~ nc~c.cted
in co.npar~on with tl.at to ~)uch thostring is
aircady subject il)its
position ofe<tuilibrium. Once assured of thé futnimcntof t).is
condition, wc donot, iu thé
investigation of tmnsverse vibrationsrcqnu-e to know
anyt)ung further of the huv of extensiou.
The most gênera! vibration of thé transver.se, or latéral, kind.y bc resolvcd, a~ve shal!
presently prove, into two sets of com-P .ent nor~
v~rat.ons, executcd in perpcndicu)ar pL~s.b.nc. it is only ill tho initial circumstances that there can be anyd.st.n.tion, pèsent to the question, bctw~ ono plane ande~c ? sufHc~nt for
.nostpurposes to regard the motion ascntndy couhned to a single plane passing tbrough thé line of theMrin~
Intreating of tlle
theory of strings it is usual to commencewith two particular solutions ofthe partial di~rential équationrepresenting the transmission of waves in the positive and ne~tive directions, and to combine thc.se in such a manner as to suit
theeaseofannitestring, w).ose ire maintained at rest;ne~ther of the solutions taken by itself' boing consistent with theexistence of or places of permanent rest. This aspect of thoT'cst.on .svery emportant, and we shaU fully consider it; but it
119.] TRANSVERSEVIBRATIONS0F STRINOS. 139
aecms scarcely désirable to found thc solution in tlie first instance
on a property so pecu)iar to a MMt/b?'7H string as the undisturb~d
transmission of waves. Wc will procced by thc more gencral
mcthod of assuming (in conformity with what was provcd in thc
last chapter) that the motion ma.y bc resolvcd into normal com-
poncnts of thc harmonie type, and dutorminingthcir pcriods and
chajactcr by the special conditions of thé system.
Towards carrying out tliis design thé nt'st stop would naturally
bo tlie investigation of thc partial din'ercntial equation, to which
thc motion of a continuons string is subjcct. But in order to
throw liglit on a point, which it is most important to understand
cicarly,–tho connection bctwccn finite and Innnite freedom, and
the passage corrcsponding thereto between arbitrary constants
and arbitrary functions, we will commence by following a some-
what different course.
120. lu Chapter in. it was poiatcd out th~t thc fundamental
vibration of a string would not be entircty altered in charactcr,
if tho mass wcro concentratcd at thé middic point. Followin~
out this idea, wu sec tbat if ttte whole string werc divided into a
uumbsi' of small parts and tho mass of cach concentrated at its
centre, we might by sufficicntly mulbip~yiu~ tttc numbcr of parts
arrive'at a system, stiiï of finite frecdom, but capable ofreprcsent-
ing the continuous string with any dcsired accuracy, so far at
lc:~t as tlie lower component vibrations arc conccrncd. If thé
analytical solution for any numbcr of divisions can bc obtained,
its limit will givc thc result correspoudiug to a uniform string.
This is thc mcthod'followcd by Lagrange.
Lot be the Icugtb, pl tho whole mass of the string, so that
p dénotes the mass per unit Icngth, T, thc tension.
Fig. M.
Thc Icngth of tlie string is divulcd into w+1 equal parts (<t),
so that
R.
130 TRANSVERSE VIBRATIONS 0F STBINGS. fiSO.
At thé ?? points of division equal in~sc.s arc supposed con-centratcd, which arc tl.e représentatives of' thé mass of thé por-tions (~ of thc string, .vlucii tl~cj ,.vur.y LLsect. TI~e mass ofcach
term~Iportiou of' lengLh is
suppose.! to be concoutratcdat thé flua.1 pomts. On tlus
understand.in~ we hâve
Wc procecd toinvcstigatc tho vibrions of a
strin~, itself
dcv~d
o
mcr~,
but Icadcd at e.ch of points ,ant(a) from thcmsolvc. aud from thc euds, witli a mass
If <~notc thc ~tcral displaccmcnts of thcloadcd pon~t.s. mciud.ug tlic initial aud ~nal poin~ wo h.vc thcfuUowmg expressions fur F mid F
with the conditions tliat ~nd y~ish. These givc byLugranges Mcthod the équations of motion
whcrc
Supposing now t),at the vibrat:ou under consideration is onoci normal type, wc assume that &e. arc atlpropor io a~
cos~-e).where .c.uain.s to bc dctcnniu.d.tlien bc rcgardcd constants, with a .suL.titution of -7~ for
If for thc Rakc ofbrevity wc put
tLc ..iucs of ..uu~cstlie form
120.]MASS CONCENTRATED IN POINTS. 131
From this équation tho values of the roots might bo found.
It may bc provcd th~t, if C= 2 cos t!ic déterminant is equivalcnt
to sin (?~ + 1) sin but \ve shall attain our o1)]Gct with grea.tcr
GfMCdircctly from (5) by acting on a hint dcrivcd from the known
results rclating to a continuons string, and assuming for trial a
particular type of vibration. Titus lut a solution be
'where s is an intoger. Substitutingtho assumed values of ~r in the
uquations (5), we find that thcy arc satisfied, provided tliat
A normal vibration is thus roprcsented by
whcre
and P,, 6, dcnote arhitrary constants indcpcndcnt of the genernl
constitution of tlie systcm. Thc w a.dmissibte values of ?! arc
found from (14) by n-scribiug to N in succession thc vatucs 1, 2,
3.W, and arc all diHcrent. If wc tnlce .s'=Mt+l, ~vttnishcs,
so that this ()oes not correspond to n, possible vibration. Grcatut'
values of s give only tbc same periods over a.ga.in. If ni + 1 bc
evcn~ one of thc values of M–that~ uame)y, con'cspondiug to
9–3
132 TRANSVERSE vmRATJONS 0F STRT~GS. [1~0.
a=~ (~ + l),–is thc same as woutd bo found in thc case of un)y
n. single load (~== 1). Thc interprétation is obvions. Jn tho kmd
of vibration considcred every n-lterruite partictc rona.ins nt rcst, so
that the intermediate oncs rca.Hy movo as titough thcy wo'u
a.tta.ched to tlie centres of struigs of Icngth 2c< fMtcncd at
the ends.
Thé most general solution is funnd hy putting togcthor a,Ii tlie
possible particular solutions of norinul type
and, by ascribing suitabic values to tbc ar1.)Itrary cn~~iants, can bn
identificd with thc vibration resulting from arbitrary Initiât cir-
cumstanccs.
Let a; dénote tbc distance of the partic~o f from thc cn(~ of the
striug, so that ()'–l)ct=x;; then hy substituting fur~. unda
from (1) and (2), our solution may he written,
In order to p:ms to the case of n. continuous strittg, we hâve
oniyt') put ?~ induite. Thé fn'st Qqnn.t.iou rctains its form,)') 1
RpcciHes thé disptacumcitt at any point a*. ThL: tiMiiting furm oi
ttie second is simply
The periods of tlie compnncnt toncs arc thus alicpiot parts uf
that of tiie gravcst of the series, fuund by puttin~ N=1. Thc
whole motion is in a.U cases periodic; and thé pcriod is 2~t/
This statement, however, must not bc undcrsiuod as cxcludi!)~
a shorter pcriod for in particular cases any uumber of tl~
Jower compoueuts may bc n.bsciit. Ail that is asscrtcd is that, ti)u
120.J MASS CONUENTRATED IN POINTS. 13~
above-mentioncd interval of timc is 6'M~tCte~ tobring aboutacont-
p!(.'tcr(:cu!<)~?'\ W';t!<f")'.))'thé prusentanyfurthpr discussion
oftttc import:t.~t formuJn, (1!)), but it is ititerusting to observe the
approach to a limit iti (17), as ?~ is madc Hucccssively grca.ter and
~rcuttjr. For tins pm'poso it will bu suHiciunt to takc thc gr~vest
tuuc for w))ich s=l, f).nd according)y to trace the variation of
2(w+1) ?!-
–n/' ,–1~'TT 2(M;+1)
Thc fuilowing arc a séries of simnitancoua values of tho func-
tion aud variable
?~) I 2 3 4 9 19 39
~)sni– .9003 .9549 .9745 -983C
.995U .9990
-9997-T ~(~t.+l)
It will bc sccu that for very inodcrate values of m thé limit is
closely approachcd. Sinco ?~ is tlie rnnuber of (!novca.ble) loads,
the case ?;= 1 corresponds to thc probbiti uive.sti~ated in Chap-
ter 111., but in comparing thé results wu must rememher thn.t we
tliere supposed the w/~e m~ss of tlie string to bc concentrated at thc
centre. In thé prcscnttCascthc h):).d n.t thé cuntre is oniy haïf as
grca.t; thc reina.indcr bcing supposed couecutrated at tlie ends,
wbere it is witliout cf~ct.
Froni thé fn.ct that our solution is general, it follows that any
initial form of the string c:UL bu niprcscntcd hy
And,su)co auy furm pos.-ilbio fur thc stringataU mn,ybc
rc-g~rdml as initud, we infur thut any iini.t.e singlu valued functioti
uf te, wltich v:).)M.s)ies at ~=0 ~nd a;=~, c:ni be exp~nded withiu
those Ihnits in n. scries of sincs of a.nd its mu)tiplcs,–whtc)i
is a. Citsc of Fouriur's t)icorum. ÂVc sliall prcscutly sitcw how the
more gcncml furm cnn bc duducud.
121. We might now détermine the constants for a. continuous
string by Intcgrntio!i a~ lu § !)3, but it is instructive to solve the
probicm first in tho gcnct':d c:tso (~ finitc), aud afterwards to
procecd to the limit. TIic iuitial conditions are
TRANSVERSE VIBRATIONS 0F STRINGS. [121.134
where, for Lrevity, ~~=~cosf,, and ~-(r~), '(2a) i~(mM)
arc tlic iuitial displaccments of thé w p:u'tictcs.
To dotcrnano a.ny constant muttiply t!tc first équation by
sins~,
thc second by sin 2s &c., aud <t.dd tlie results. Thcn,
byTrigojiomctry,thc coefficients of a!l the constants, cxcept J,,
vanisli, wliile tliat of = (~~ + 1) Henco
'Wc ncc'd not stay !)orc to write down the values of 7?, (cqu~l
to jf~,sin e,) ibs deponding on the initial vcincitics. W!tcn becomes
I)i~nite]y smaU, )'~ under tho sign of sutumation ranges by in<i-
nitcsinial stcps from zero to At tlie same time = a i'??t + i t
so tliat writing ?'M= x, fï = (1,,v,we Iiavc u!tima.tc!y
cxpressing d, lu tcrms of thc iultial displaccmcnts.
122. Wc wi)t now invcstigatc indcpcndently the partial difFercn-
ti:i] équation govcrnin~thctt'ansvo'.sumotiottofa.po'fcctiyHcxiLfc
strin~, on thé suppositions (t) thatthe jnagnitudc ofthe tension
mny bo cunsiucrcd constant, (2) t!)at thc square of H)c inclination
of any part of thc string to Its itntial diruction may bc ticgicctcd.
As befure, dcnotus thc lincar dcnsity at any point, and y'~ is the
constant tension. Let rcctat~nlarco-ordinatcs bc takcn pandie!,
and pcrpcm~cuhtr to thu stril~, su t))at x: givcs tite cquilibnum
and .c, y, z thc disptacud p<jsiti<'n uf any partictu at tinic t. Thc
forces acting on thu clément (/.c :u-o thé tensions at its two cuds,
und any impresscd forces .)~ ~p< ByD'AIcmbcrt's Pnn-
ToJhuutor'H J)t<. C«;c., p. 267.
122.] DIFFEBENTIAL EQUATIONS. 135
cipin thèse form an equilibrating system with thé réactions
against accélération, p p At tlie point x thu com-CLu MC
ponents of tension arc
If thc squares of Le ncgiccted so that ttic forces acting(a; c~x
on thc clément arising out of tlie tension arc
IIonco for tho equations of motion,
from winch it appo~rs that thé dépendent variables y and z arc
attogethcr indupundont of onc another.
Tho student should compare tlicso équations with the corrc-
spoudmg cquations ofHuitc diM'crcncGs in § 120. Thc latter maybe written
which nmy :).!su bc provcd dircctiy.
13G TRANSVERSE VIBRATIONS CF STRINGS. [123.
Thé nrst is obvicuii from thc deHnition of 2~ To prove the
second, it is sufHcipnt ~o notice tha.t thc potcnti~ cncrgy in a.!iy
configuration is the work requiro~ to producc tlic nceessary
stretching against thé tension T,. Ruckoning from tlie conHgura.-
tion of equihbrium, wc ha.ve
and, so far tta tlic third power of 1
123. In most of thc applications that we sha.H have to mako,
tho dctiatty p Is constant, there arc no imprcssed forces, and the
motion may bu supposed to take pheu in onc plane. We may
thon convenicBtIy write
n.nd tlie difrerentia.! cqua.tion is expressed by
If we now assume t]iat y varies as cos ?)M~ our equation
bccomcs
of which the most gcnera.1 solution is
This, howcvcr, is uot thc most goieral ha-rmonic motion of
thc period in question. lu ordcr to obta,in the lattcr, ws must
assume
\vhcro ;?/ M'c fuuctiotis of a*, not ucccssarUy thc samc. On
substitution in (2) it appca.rs thn.t y~ a.ud arc subjuct to cqua.-
tions uf thé fut'm (3), so tlia.t Hnally
:ut expression conta-himg four a.rbitra.ry constants. For any con-
tiuuous tcu~tL of string sa.tisfyiug without iutcrruptiou the differ-
123.]PIXED EXTR.EMITIES. 137
ential cquaticn, this is tlie most gcnera.1 solutioR possible, under
thé condition, th~t thé motion at every point shaH be simple har-
monie. But whenever thc string forms part of a. system vibrating
frccty n.nd withoub dissipation, wo know from former chaptcrs
t,)):it :).I1 parts aru simuit~neousty in thc same phase, which
t'cfjuircs that r\
12-t. Thc most simple as wcH as t)ic most nnpoï-tant problem
connected with our présent subjccb is the investigation of thé free
vH))-~tions of a fnntc sti-iug of Icngth held fast at both its ends.
If we takc thc origiti uf a.- at ono und, tlie tcrmirnd conditions a,rc
that when a:=0, Mid wheM a!=~ vanishes for ~11 values of t.
T)ie nrst i-cquit-es tha.t in (G) of § 123
and thé second that
or that ~==.S7r, wlicre s I.s ~n intcger. We IcM-n that thc only
hM-mouic vibrations possibleare such as mnkc
iUtdthuu
and then thc most goncra.1 vibration of simple harmonie type is
Now wc know M./)Worz th:tt whn.tcvcr thé motion may bc, it
CMt be rcprusoitcdas a suni of simi'ic htu-mouie vibrations, a.nd
wc thurcfurc coucludc th:tt thé luost gcncra.1 solution for a string,
TRANSVERSE VIBRATIONS 0F STRINGS. [124.138
so that, as has bccn aiready statcd, the whoïc motion is under ail
circumstanccs pcriodic in the t:mc r~. Thé sound cmitted con-
stitutes in gênerai a musical 7:0~, acconUng to our dennition of
that term, whose pitch is nxed by the period of its gravest
component. It may happen, however, in special cases that the
gravest vibration is absent, and yct that the whoïc motion is not
periodic in any shorter tune. This condition of things occurs, if
~/+~/ vatjish, while, for examp)c, ~l./+7?./ and ~t~+~ are
finite. lu such en.ses the sound could hrn'dty be called a note;
but it usuiJIy h~ppcns in practicu that, w]tcn tho gravcst tone is
absent, .some othcr takcs its p)acc in the cbaractcr of fundamcntal,
and the sound still constitutes a note in the ordiltary sensc,
though, of course, of c!cvaLcd pitch. Asimple
case is wheu ail
the odd compollcnts beginning with thc first are missing. Tho
whote motion is thcn periodic in the tit-nc ~Tp and if the second
component bc présent, thé sound présents nothing nnusual.
T]~c pitch of thc note yicidcd by a string (C), aud thc character
of the fundainenta! vibration, werc first invcstigatcd on meclianical
principics by Brook Taylor in 171-5 but it is to Daniel Bernouni
(175.')) tbat wc owe the générât solution containcd in (5). He
obtained it, as wc bave donc, by the syutbusis of particnlar solu-
tions, pcrnussibic in accordancc with his Principtc of the Co-
existence of Sniat! Motions. In bis time tbe gcncrality of the
result so arrived at was opcn to question; in tact, it was tlie
opinion of Eu!er, and aiso, strangdy cnough, ofL:t,grange',that
thé scrics of sincs in (;")) was notcapabte
of rcprescnting an
arbitrary function; and Bcrnouln's on the other side,
drawn from the iunnitc nuinber of thc disposabic constants,
was certaiu!y inadéquate~
Most of the ]aws embodicd in Taylor's formula (C) had been
discovcred experinientaHy longbefore (1G3L!) by Mersennc. Thcy
may bc stated tbus
SoQRiGmfU)D'ajP«r<<f~<; D~/c'rctXtn! O/t'tc/tftN~c~, § 78.
Dr YounK, iti Lia momou' of 1800, HC-ûiHH to liave understood this matter quito
<orrcct)y. Ho s~'H, "At tlio samo timo, ns M. DernfXtUi tma ]HHtIy obsorvod, Rinoo
nvory ligure may bo iu~uitoty approxinxited, by cûtt.sidunnt; its ortiinutofj as
<'u)nposoJ of tho ot'dinfttos of au iniinitc mnuber of tmcix'id.s of (liFfcrcnt tun~ni-
tUticH, it may bo demonstrntod thttt aU tbcsû cunstitnont ou'ves woulJ revert to
tLicir initia) Htato, in tho samo timo tbat a Rimiln.r choni bcnt into a trochoida!
curvc wouhi purforn) a sinn)o 'vibration aud this is in ttoinc retipecta a couvomoat
oud eumyoudious mothod of consideriug tho problom."
124.] MERSENNE'SLAWS. 139
(1) For a, givcn string and a givcn tension, the time varies as
the length.
This is the fundamcntal principle of thé monocbord, and ap-
pears to hâve bccn understood by thé anciects*.
(2) Whcn tho length of the string is given, the time varies
inverseiy as thc square rout et' tho tension.
(3) Strings of thé same length and tension vibrato in timcs,
w~~ich arc proportiona) to t)tc Stmare roots of thc lincar dcnsity.
Thcseimportant
rcsultsmay
aH bc obtained by the mcthod of
dimensions, if it be assumud tha.t T dépends on]y on p, and 2'
Fur, if thc units of length, time and mass be denoted rc-
spectivcly by [Z], [2'J, [~j, thé dimensions of thèse symbols are
givcn hy
~=M, p=[~Z-'], ~=[~L~],
and thus (see § 52) the onty combination of thcm capable of re-
prcscnting a time is T, Thé oniy thing left uudetermined
is Uic numeriea.1 factor.
125. Merscnnc's laws are cxcmphfied in a!l stringed instru-
ments. In playin~ thé violiu din'ercnt notes are ubtaincd from
thc same string hy shortening its cnicient Icngth. la tuning tho
vioun or the pifmuforte, an adjustment uf pitch is cûectcd witli
a constant !engt.h by varylng t!ic tension but it must ho re-
mcmbercd tliat /) Is not quite invariable.
To secure a prescrihcd pitch with a string' ofgivcn materiaL it is
rcquisitc that onc rctation only bc satisficd bctwccn the Icngth, tiie
thickness, and thé tension; but in practice thcrc is usuaUyno grcat
latitude. Thé length is often limited by consi<turations of con-
Vùnicncc, and its curtaiimcut cannot idways be compensatcd by
an incrcase of thickness, bccausc, if thc tension he not increascd
proportionaDy to thc section, thcro is a loss of HcxihiHty,
whUcif'thc tension bc so incrcascd, nothing is cH'cctcd towards
lowering the pitch. T!ic dirricuity is avoidcd in t!tc )owcr strings
ofUic pianofortc and violin by thc addition of a coil of fine wirc,
whose cU'ect is to Impart Inc'rtia' wiLhout too much impairing
ncxibility.
Aristono "hncw t.tmt a pipo or (t ohnrd of dnohiû Jen~th pt'oduco'l )t ftonud of
which tbovibmt.iousoccupitid a JuuHû timo; [md timt tho propcrtics of coiteords
JopeudoJ on tho pmport.muH of tho thnes occnpiod by tl)0 vibrations of tho
soparftto sounds.Youuë's Lcetu~M o)t Ntt<xnft~/tt<uM~y, Vol. i. p. ~01.
TRANSVERSE VJBRATJONS 0F STRINGS.[125.
140
For quantitative investigations into the laws ofstrings, the
aonoïncter is emphjycd. Hy mcans of a, weight lianging over a
puUey, a catgut, or a mctaHic wire, is stretcijed across two bridf-cstnounted on a résonance case. A moveable bridge, whose positionis cstimated by a sca!c
running parahel to thc \vire, i-ivos thc
means ofshortcning
tite cfHcicnt portiott of tlie wire to anydcsit'cd extunt. TIie vibrations may bc cxcitcd by p!uckin" as
in thc harp, or witli a. bow (well suppiicd with rosin), as in titû
violiu.
If the moveable bridge be placed ha!f-waybGtwecn the Dxcd
nnes, thc note is raiscd an octave; whcn thc string is reduced to
one-third, thé note obtained is tt)C twclfth.
By means of the law of lengths, Mcrscnnc determined for thc
nrst time thc frequencics of knowu nmsicul notes. He adjusted the
Icngtil of a string until its note was one of assurcd positiuu in thé
musical scale, and then prolonged it under t!)e same tension until
thû vibrations were slow enough to bc couuted.
For expérimental purposes it is convenient to hâve two, or
more, strings mounted side hy sidc, and to vary in turn theh-
Jcngt!i3, their masses, and tlie tensions to winch they aresubjucted.
Thus In order that two strings of equa! length may yle!d t))c in-
tcrva! of t)te octave, their tensinns mnst be In thc ratio of 1 4.if thé masses be tlie samc; or, if thc tensions be the same, thé
masses must bc in thc reciprocal ratio.
Thc sonomctcr is very uscfut for thc nmnerleal détermination
ofpitch. By varyiug the tension, tlie string is tuned to unison
with a fork, or other standard of known frcf~ucncy, and thcu by
adjustment of thé moveable bridge, thu Icngttt of thestrin"' is
determined, whieh vibrâtes in unison with any note proposed for
mcasuremcnt. Tito ]aw of Icngths tticn givcs thé mcans of
cn'eeting t]ic dL-siredcomparison of frequcncies.
Anotiicr application by Scheib)er to tho détermination of
absente pitch is Important. Thé priucipiu is tlie samc as that
cxptainud in CIiapter ni., and thc mcthod dépends ondeduchifr
tiiu absulute pitcii of two notes from a knowlcdge of both t)ie
?'a~o and thé (/~(;7-e7ice of their frequencies. Thé Icngths of t)ie
souometerstrmgwhen in unison with afot'k.andwhengivin~with
it four béats p'u- sucond, are caœfuUy mcasured. Thé ratio of thé
lungths is thé iuversc ratio of thé frcqueucies, aud thc difierence
125.]NORMAL MODES. 141
ofthe frcquoncies isfour. From tbcsc data thc absolute pitclt of
thé fork can bc ea.leula.ted.
Thc pit.ch of a string may be calcnlatcd a!so by Taylor's for-
mu]a from tlie mcchamcal eicmcuts of tlie system, but grcat pré-
cautions are necessary to secure a.ccuracy. Thc tonsio)) is producc<)
by a.welght,whosemass (cxprcsscdwith tl)o samc unitasp) m:t.y bo
called P; so that y,= whGi-e
= 32'2, if thé units oficngth a-nd
timc bc the foot aud thé second. In order to securc that tho who)e
tension acts on t!tc vibrating segment, no bridge must bc intf.;]--
poscd, a condition only to bc Siltisfied by suspending tlie string
vcrtiea.lly. After thc weight is !Lttachcd, a portion of thé string
is isola.ted by dumping it nrmiy at two poitits, and tlic length is
mea~urcd. The mass of the unit of longth refers to tbc strctchcd
statc of the string, and may bo found in<Urcct!y by obscrviug thc
elongation due to a tension of the same order of magnitude as
and calculating what -\vou!d be produced by T, a-ccording to
Hooke's law, and byweighing a. known length of thé string in its
normal stato. After the clamps hâve bccn sccurcd grca.t carc
is rcqnircd to avoid fluctuations of' tonpcra.turc, which wotdd
scriousty inftucnce thé tension. In tliis way Sccbcck obtaiued very
~cenratc resttits.
126. Whcn a string vibrâtes in its gravcst normal mode, tlie
'7T.7;excursion 13 at any moment proporttouf).i to
stnincrc~smg
nurnerically from eithor end towards the centre; no intcrmcdiato
point ofthu string rcina.tns pcrmaucntly nt rest. But it is othcr-
wiso in tlie case of thc ingher normn.l componcnts. Thus, if the
vibration bc of thc mode cxprossed by
t] S7rX1.'
l lle excursion is proportional tosin.
whichvanishesat~–1 1
points, dividing thc string into s cquat parts. Thèse points of no
motion arc caticd nodes, and rna-y cvidcntiy Le touci~cd or Iield
fast without in any way disturbinp; the vibration. T)tC produc-
tion of harmonies' by iightty toucliing thc string at thc points of
aliquot division is a well-known rGsource of thc violinist. Ail
component modes are excludcd which hâve not a node at thc
point touched; so that, as regards pitch, tlic cuuct is the same as
if tho string werc securely fastened thcrc.
TRANSVERSE VIBRATIONS 0F STRINGS. [127.142
127. Tho constants, which occnr h) the gênerai value ofv, § 124,
dépend on thc epcci.i! cii'cn)nnt:mces of t~ \)i.t.t!ot), ~))~ :~t;y b'
exprcsscd in tcrms ofthc initial valuus of~ a.ud
Putting t = 0, we fmd
Multiplying hysin
and intcgrating from 0 to wo obtain
Thcsc rcsults cx(jmr!i(y Stokes' ifiw, § 95, for tha.t part of~,which
dépends on thc mitia-! vctocities, is
and from tl)is thc part dcpcnding on initial displaccmcnts may bc
infcrref!, by diH'ci-eutiating wit!i respect to the tirne, and sub-
stitutmg~for~.
Witcn thc conditiou of the string at some one moment la
thoronghiy known, thcsc formu!:L! allow us to c{dcula.te the
inotiou ibr ait subséquent timc. For exemple, ]ct tho strLng bo
initiany at rest, aud so displaced that it forms two sidcs of a
triangle. Then= 0, and
onintégration.
Wc sec that vanishes, if'sin~ =0,
that if thcre be a
nodc of thc componcnt iu question situatcd at j~. A more com-
prefensive view of tlie subjcct will bo aitbrdcd by another modeof solution to bc given prcsently.
128.] POTENTIAL AND KINETIC ENERGY. 143
128. In tlie expression fer thc coemdcnts of sin arc
tiio normal co-orfimatcs of Chn-piers iv. M]d v. We wi)I de-note thcm thcrcfofG by so thfit the conjuration an<] motionof tho System at any instant arc dcfined by the values of d~ tmd
according to the équations
Wc procced to form thc expressions for aud 1~ aud Lhcneoto dcducc thu no)'ma.I eqnatious of vibration.
For thc kinctic ciiergy,
thé product of cvory pair of tcrmsvanishing by the gcnend
proporty of normal co-ordiua.tcs. Hence
0 b
Thcse expressions do notpresuppose any paj-ticular jnotion, either
natural, or othcrwisc but wemay apply thcm to calculate tho
wi)o!e energy of string vibr~ting nattirally, as follows :–If j)~'bc tlie whoïc mass of tlic string (pl), and its cquiv~cnt (n~) busubstituted for we find for the smu of thé cuergies,
144 TRANSVERSH VIURATIONS 0F STUINCS. [128.
If the motion bc not connncd to the p):u)c of wc havn
nK'rc'Iy to add thé cno'gy of thc vibrations in tbc pcrpcndicuiai'
plane.
Lagra,nge's metbod givos immcdia.tcly thc équation of motion
which hn~i hecn ah'cn.dy considcrctl in § GC. If <~) a.ud bc tho
initial values of 6 and tlie guncral solution is
By dc~nitio!i <I~ is such that <I~ 5~ ]'cprcsc!)ts titû work do])c
by thc imprcssud forces on thc dispI.Lcemcnt 8~. Hcncc, if thu
fut'cc acting at tirnc ou an cioneut of tlie strmg p bc p 1~
In theso équations is a. tincar qnaatity, as \ve scefrom (1); and
<I~ is thct'cfore a force of thé ordinary kind.
129. In tlie a.pplica.tious that wc Ii~vc to make, the only
unprcsscd force will be supposcd to act in the immediate neigh-
hourilood of one point .K=6, and may usually be rcckoned as
a whoïc, so that
If thé point of application of thc force eoincidc witli a. node of
tbc mode (~), <I~,=<), and wc Icarn that the force is aRogether
without influence on tho componcnt. in question. Tins principle
is of grcat importance it shcws, for exarupic, that if a string bc
at l'est in its position of cquilibrium, no force applied at its centre,
whether in thé form of plucking, striking', or bowitig, can generate
auy of thé even normal componcnts'. 1. If aftcr tlie opération of
the force, its point of application be datuped, as by touehiug it
1 The obaBrvation that a. harmonie !s uot gencratcd, whûti ono of its uodnl
poluta ia plucked, iti duo to Youug.
129.] YOUNG'S TIIEOREM. 145
with thé finger, aH motion must forthwith cease for those com-
ponents which have not a. node at t! point, ht q~stion a.re
stopped Lyttie dumping, and tl~oso wbich hâve, are absent from
thcbcginumg'. More gencraHy, by damping any point of a
sounding string, wc stop :dl the composent vibrations which have
not, aud Jeave cntirely unaifueted those which ha.ve a nodu at tlie
point touched.
The case of a string puticd aside at one point and afterwards
let go from rest may Le regard cd as includcd in thé preceding
statements. Thé complete solution may be obtained thus. Let
the motion commence at thé time <=0; from which moment
= 0. Thé value of at time t is
where (<~), (~)~ dénote tlie iuitial values of the qua-ntitiesaffected with thé suiBx N. Now in tlie problem in ha.nd (~ = 0~
and (~). is determined by
if y dénote thé force with which the string is he!d aside at the
point b. Hclice at time t
..(5),
where = s~ra
Thesymmetry of the expression (5) in x and b is an examplo
of thé principle of § 107.
The problem of determining the subsequent motion of a stringset into vibration by an impulse acting at thc point b, may be
treated in a similar manner. Integrathig (6) of § 128 over tlie
duration of thc impulse, we find ultimately, with thé same nota-
tion as bcforc,
A liko rosutt ensuos whon thé point which ia dampod i.-iat tho samo distanceïrom ono eud of tho string as the poiut of excitation ia from tho othor on!.
R. 10
146 TRANSVERSE VIBRATIONS OP STRINGS. [139.
if~y~Le denoted
by 3~. At the samc time (~).= 0, so that hy
(2)atHme< t
The séries ofcomponcnt vibrations is less convergent for a, struckthan for a plucked string, M the prcceding expressions shcw.
The reason is that in thc lattcr case t!ic initial value of y is
continuous, and only cliscontinuous, wlu!e in tlie former it isIV
y itself that makes a sudden spring. Sce §§ 32, 101.
The problem of~string set in motion hy an impulse may also
be solved by tho gênera! formuJœ (7) and (8) of § 128. Tlie force
<mds tLc string at rest at < = 0, and acts for an infinitely short
time from ~=0 to ~=T. Thus (~.). and (~). va~ulsh,and (7)
of § 128 reduces to
Hithcrto we hâve supposcd tho disturbing force to be con-
centrated a.t a. single poi)it. If it be distributcd over a distanceon citlier side of we l)avc only to iutcgratc thé expressions (C)aud (~) with respect to substituting, for cxample, in (7) in
r tT-place of .1, sin
-y–,
Tho principal effect of thé distribution of t])C force is to render
tbe series for y more convergent.
130.] PIANOFORTE STRING. 147
130. Thé problem which will next engage our attention is
that of th~ p!ancfnrtc wit'û. Thc causu of t!ic vibration la hcrothc blow of a hammcr, wiiich is projeeted against tlie string, and
after thé impact rehounds. But we should not bc justified in
assuming, as in thé iast section, that the mutual actionoccupies
so short a time that its duration may be ncg)ccted. Mea.surcd bytlie standards of ordinary life tlie dnration ofthe contact is Indecd
very small, but hère thc propcr comparison is with tlie natural
periods of tlie string. Now tlie hammcrs used to strike thé wires
of a pianofoi-to arc covcrcd with sûvcral layers of eloth for tho
express purposcof making them more yielding, with the effect of
prolonging the contact. The rigorous treatment of thé problemwould bc difficult, and thé 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 thé actual
yielding of the string must bc slight in comparison with that of
the covering of tlie hammer, tlie law of tlie force called into play
during the contact must be ncarly thc samc as if thé string wero
absolutely nxcd, in which case thé force would vary very noarly as
a circulai' function. We sliall tlicrcforc suppose that at the time
t = 0, whcn there are neither velocities nor displacements, a force
.Fsin~ betiins to act on thé string at a:=~ and continues throughhalf a period of the circuiar function, tliat is, nntil <="7r-jp, after
which thé string is once more frce. Thé magnitude of ~) will
dépend on thc mass and clasticity of the hammcr, but not to any
grcat extcnt on thé vulocity with which it strikes tlie string.
T)io i-cquired solution is at once obtamcd by substituting for
in thc gênerai formula (7) of§ 128 its value given by
148 TRANSVERSE VIBRATIONS 0F STRINGS.[130.
and thé final sohttion for becomes, if we snbstitute for M and ptheir Ya.}nus,
We sec tha.t, a)!componcnts
vanish w]uch ha.vc' n. nodc at the
point of excitctnent, but this cnnctusion does not dépend on a.ny
particu): !:(.w of force. Thé Intcrest uf thé présent solution lies
in thc infortnation H)!).t may be ctieitcd frnm it a.s to the depcnd-
enco of thé rcsulting vibrations un thc duration of contact. If
wo dénote the nitio of tliis q~antity to tlie fundamcnta! period of
tlie stnng by so tha.t = Tra 2~ thé expression for thé ampli-tude of the cumponcnt s is
Whcn in nnitc, those components disa-ppcar, wbosc perlons
§' ?' t.~c duratinn of contact; and wltbn .s is vcry
grcat, thc séries coivo-ges wit)i N' Some tUbwancc rnust at.so
bc ))i!idG for tho (hnte breadt)~ of thc btunnicr, thc cHect of whichwill a!so bc to faveur thé convergence of thc séries.
Thc laws of tl)c vibration of strings Tnay be veriHcd, at least
in their main featm'cs, by opticiil mcthods of observation–cither
with thcvibration-tnicroscope, or by n. trn.cing point rccoi-ding tlie
characteroftttc vibration on a revolvmg drum. This character
dépends on twotbings,–thc mode of cxcitement, a,nd the point
whose motion is se)eetcd for observation. Titosc components do
not appear winch bti.'ve nodes either at the point of cxcit.cmcnt, or
at tbc point of observation. Thé former are not gcno-atcd, and
t!)C latter do ])ot mfmifust. thcmselvcs. Thus t!ic himpicst motion
is obtaincd by ptucking thc string at the centre, andobscrving
une of tbc points of trisection, or vice w?'M. In this case t!te
first harmonie wbich contaminâtes thé purity of thc principalvibration is thc nf'Lh cornponcnt, wbose intcnsity is usuaUy in-
sunictcnt t.o prudnco nmch disturhancc. In a future chaptcr wc
shall compare t)ic results of tiic dynamica.! tlicory with aurai
130. jFRICTION PROPORTIONAL TO VELOCITY. 14S
observation, but rathcr with tlie view of disc~vcring and tcstingthu [aws of iiûmm~, LiuUi uf confirming' Lhu theury Itseli'.
131. Thé case of a. penodic force is Included in t!)û generalsolution of § 12!S, but we prêter to foUow a somcwha-t dirEcrent
jnethod, lu ordcr to m:Lkc fui cxtcusion in anc'thcr dircetion. We
have hithcrto takoi no account ofdissipativc forces, but wc will
now suppose that thé motion ofca.ch élément of thé string is resistcd
by a force proportional to its velocity. TIte partial dinercntia!
équation becomes
by means of whieh the suhjcct may bc trea.tcd. But it is still
simptor to avail oursetves of thé rcsults uf thé last chiipter, re-
ma.rkmg that in tho présent case the fnctiun-function is of
the s:unc form as T. In fact
wherc < < are thc normal co-ordinates, by means of which
y fmd are reduccd to sums of sq)t:u'o.'j. Tho equntiot)s of
motion are thei'cfore simpfy
~+~.+~.=~(3),
of thc samc form as obta.ins for Systems with but one dcgrcc of
frcc<)o)n. It is only ncccss:u'yto add to what was said Iti Citap-
ter ni., that sincu K is indupcndunt of thc jmtural vibnt.tionssubside in suc!) a manuer that t!tc
amplit.udcs manitidu thcir rcla-
tive values.
If a periodic force .Fcos~ act at a single point, wc liave
If among thc natural vibrations thcrc bc any one ncarlyisochronous with cos~< tLcn a large vibmtio)i of th:ht ty})o will
bc forcer, unless Indecd thc point of Gxcitcment s)iould happcn to
150 TRANSVERSE VIBRATIONS 0F STRINGS.[131.
fall nea.r a node. In the case of exact coincidence, thc componcntv:brn'-) ~n
.i'j<<)~ VMmshc.s; n< foœapp]iûd !tt a. nu'-iu ca.u
gcneratc it, under thc présent law of friction, whieh howcvcr, it
may be rcmarkc(), is very special in character. If tliere be no
friction, /<:= 0, and
132. The preceding solution is an example of the use of
normal co-ordinatcs in a probicm of forced vibrations. It is ofcourse to free vibra.tions that titcy are more cspcciaHy applicable,and they may gcncrally bo uscd witli advantagc throughout,whcncvcr the system after thé operation of various forces is
ultimately left to itself. Of this application we have already had
examples.
In tlie case of vibrations due to periodic forces, one advantagcof the use of normal co-ordinates is the facility of comparison with
thé efir:<(?~ ~<?o?-~ which it will Le remcmbercd is the theoryof thé motion on the supposition that thé inertia of the system
may bc left out of account. If the value of thc normal co-or-
dinate on thc cquilibrium theory bc A, cos~, then thc actualvalue-wiH bc given by the équation
so that, whcn thc result of thé equilibrium theory is known andcan rcfidiiy bc cxprcssed in terms of thc normal co-ordinatcs, thetrue solution with thc effects of inertia included cn.u ~t once bcwritten down.
In the présent instance, if a force .Fcos~ of vcry long periodact at tlie point b of thc string, tho result of the equilibrium
theory, in aecordaûcc with whieh the string would a.t any momentconsist of two straight portions, will bo
132.] COMPARISON WITH EQUILIBRIUM THEORY. 151
from which the actual result for all values of p i~ derivcd bysimply
writing in place of
Thc value of in tins and similar cases ma.y Itowcver be
cxprcsscd in finito tcrms, and thé difHculty of 0'btaltnng tlie
funte expression is usua.IIy no greater than that of findin"- thé
forni of the normal functions wlien tlie systein is frec. Thus in
tlic équation of motion
and a subsequent détermination of ?~ to suit thc boundary con-
ditions. In thc probicm of forced vibrations ??t is given, and we
havo only to supplemont any particular solution of (3) with' thé
compicmentary function co~taining two arbitrary constants. This
function, apart from tlie value of and thé ratio of tho constantsis of the same form as thc normal functions; and a.11 that remains to
be enected is the détermination of the two constants in accordanco
with thé prcscribcd bounda-ry conditions whicli tlie completesolution must satisfy. Similar considérations 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 considcred; the first, whcn at thé
point œ= &, a given periodic force acts; tlie second, when It is thé
actual motion of tho point that is obligatory. But it will bc
convenient to treat theni together.
Thc usual differentia.1 equation
is satisfied over both thc parts into which thc string is (UvIJcJ at
b, but is viola.tcd in crossing from one to thé othcr.
TRANSVERSE VfDBATIONS 0F STRINGS. [133.152
In order to allow for a change in thc arbitrary constants, wo
must thcrofore assume distinct expressions for and a,ftcrwa.rd8
introduce tlie two conditions whidi must bc satisfied at thc point
of junction. Thèse arc
(1) Tha.t there is no discontinuons change in thc value of
(2) That thc résultant of the tensions acting at b balances the
imprcsscd force.
Thus, IfFcos~ bo tho force, thé second condition gives
where A(")
dénotes the altération in the value of Incurrcd\.a~/ f~'
in crossing the point x = in thé positive direction.
We sha! however, Hnd it advantagcous to replace cos?~ bythe complex exponential e" a.nd 6tia!Iy disc~rd tho imagiuary
part, when t!iesymhoHcal solution is
completed. On the assump-tion timt~ varies as e" thc differential équation becomes
The most genera.1 solution of (3) consists of two tcrms, pro-
portionn.irespcctively to 8ui\a;, and cosÀa;; Lut thc comlition to
be sa,tishcd a.t ~= 0, shcws tliat thc second ducs not occur here.
Hence if y e' be tlic value of at x = b,
is the solution n.pplying to the first part of tlie string from a;=0
to x;= In likc manner it is évident that for ttte second part wc
sttaJtlia.vo
If y bc given, thèse équations constitute the symbolica.1 solution
of thc problem, but if it be thc force that bc given, we requirefurther to kuow thc rcla.tion betwecn it and
133.J PERIODIC FORCE AT ONE POINT. 153
D~erentlution of (5) aud (G) and substitution in thc cquation
analogous to (2) givcs
Thus
Thcso équations excmplify thé général law of reciprocity
proved in the last chaptor; for it appcars that tlie motion at x
duc to thé force at & is thé same as would have been found at
had thc force acted at x.
In discussing thé sohition we will take first the case in which
there is no friction. Tfjc coenieicnt is then zero while is
rca. aud equal to p a. Thc rca.1 part of thé solution, correspond-
inb to thé force .Fcos~, is found by simply putting cos~)< for
in (8), but it sccms scarcely nccessary to write thé équations againfor the salœ of so small a change. Thé same rcmark applies to
the forced motion given in terms of y.
It appears that thc motion beco'mcs infinite in case the force
is isochronous with one of thé natural vibrations of the entire
string, unicss thé point of application be a node; but in practice
it is not easy to arrange that a string shall be subjcct to a force
of given magnitude. Perhaps thé best method would be to attach
a. s)nall mass of iron, attractcd perIodicaUy by an elcctro-magnet,
whose coils are travcrscd by an intermittent currcnt. But unless
some means of compensation wcre deviscd, the mass would have to
bc vcry small in order to avoid its Iiiertia Introducing & new com-
phc:).tion.
A better approximation may he obtained to the imposition of
an obligatory motion. A massive fork of low pitch, cxcited by
a bow or sustained in permanent operation by electro-magnetism,
exccutcs its vibrations in approximate independcnce of the re-
actions of any light bodies which may be connecte(l with it. In
order tbei-cforc to subjcct any point of a string to an obligatory
Donlnu'3 ~co)M<<M,p. 121.
TRANSVERSE VIBRATIONS OF STRINGS.154
[133.
traverse motion. it is only necessary to attach it to thé extremityof one prong of such a fork, whose plane of vibration is perpendicularto the length of théstring. This method of
cxhihiting thé forcedvibrations of a string appels to hâve beou first used by Meldc.
Another arrangement, hetter adapted for aurai observation,bas been employcd by Helmholtz. Tj~ end of thé stalk of apowcrfui tuning-fork, set into vibration with a bow, or othenviseis pressed against thé
string. It is advisable to ~e the surface,which cornes into contact ~ith t),e
string, into a suitable (.saddie-shaped) form, tho botter to prevcut slipping and jarring.
Referring to (5) we sec that, if sin X& vanished, thé motion(according to this équation) would hecome Infinité, which may betaken to prove that in thc case
eontempiated, the motion wouldreal!y become great-so grcat tl.at corrections, previousiy insi~u-ficant, rise into importance. Now sin vanishes, when the forceis isochronous with one of thc natural vibrations of thé first partof tho string, supposed to be )tdd nxed at 0 and b.
When a fork is placed on théstring of a ~onochord, or other
instrumentproperly providcd with a
sound-board, it is casy tofind by tnal thé places of maximum résonance. A very slightdisplacement on eitlier side entails a considerable falling o~In~evolume of tlie sound. Thé points thus determined~i~ thestring into a of
parts, of length that thenatural note of any one of them ~hen nxed at both ends) istlie same as thé note of thé fcrk, as may readily be verified, Theimportant applications of resonance .vhieh Helmholtz lias made topurify a simple tone from extraneous
accompaniment willoccupyour attention later,
134.Returning now to the case
complex,o have to extract thé real parts from (5), (R), (~ of§ 133. For~f~~T~ occur as
reduced tothe form Beie. Thus let
134.] FRICTION PROPORTIONAL TO VELOCITY. 155
corresponding to thc obligatory motion =~y cos~ at
By a similar process from (8) § 133, if
correspondmg to tho impresscd forco .Fcos~ at b. It remains to
obtaiu tlie forms of ex, &c.
The values of a and /3 are dotorminecl by
while
This completes thé solution.
If thc friction be very small, the expressions may be simpli-
Hcd. For instance, in this case, to a sufEcicut approximation,
TRANSVERSE VIBRATIONS 0F STRINGS.[L34.
15G
so tha.tcorrcsp~nding to tlie obligatory motion at & ?/ =-yeosp!; tLc
nmphiudt'' of :uu!j~ bt'i~c~n 0 ;ut.t is, )tpp~Xit)jft.tu!y
-which bccumcs grcat, but not inanité, wheu sin = 0, or thcM
point of application is a node.
If thc hnposed force, or motion, bo ])ot exprcsscd hy a single
harmonie term, it must first bc rcsolvcd into such. Thc precedingsolution may then be applicd to each componcnt separately, and
thc resuits addcd togcther. T!ic extension to thé case of more than
one point of application of thc imprcssed forces is atso obvions.
To obtain tho most gcnera) solutions~tisf'ying the
conditions thc
expression for the i)fitur:d vibrations must also Le addcd bnt
thèse become reducecl to Insignincancc after tnc motion lias been
in progress for a sufïiciunt timc.
Thé !n.w of friction !msumcd in the prcccding investigation is
thé only one whoso resuits can bc ca.si)y fuiloweddoductivety, and
it is sunicient to givc a gênerai idca. of t))C effects of dissipativcforces on tlic motion of a string. But in other respects thé con-
clusions drawn from Itpossoss a nctitioua
simplicity, dcpcndinr'- on
the fact that 7'tl)e frictinn function–is similar in form to 7'which makcsthe normal co-ordinatcsindepcndent of cach other.In ahnost any other case (for oxample, when but a sit)g)c point of
the string is rctardcd by friction) tttcrcarc no nonnfd co-ordinates
propcriy so called. Tho-c exist itutocd ctcmcntary types of vibra-tion into which the motion may bc rosolved, a)id which arc
perfectly indcpendcnt, but thèse are essentially different in cha-
racter from thosc with which wc have hecn conccrncd hithcrto forthe varions parts of the system (as affected by onc
dcnicntary
vibration) arc notsimu!t!).neous!y in thc samc
phase. Spécial cases
cxcepted, no lincar transformation of thé eo-ordinatcs (with real
coefficients) can rcduce T, and F togcther to a sun) of squares.If wc suppose that tho striug lias no itx.'rtia, so that ~==()
-~and F may tbcn be reduced to sums of squares. This probfemis of no acoustical importance, but it is
Intercsting as bcingmathcmaticaMy analogous to that of thc conduction :utd radiationof lipat in a bar whuse ends arc maintaiucd at a cojtstaut tem-
pérature.
135.] EXTREMITIES SUBJECT TO YIELDING. 157
135. Thus far wc have supposcd that at two fixcd points,
Œ = 0 and .c = tttû string is hcld at rest. Since absolute Hxtty
c:),nnot bc fLttu-uibd m prit-cticc, it is ~ot without iuterest to inquire
in whut ma.nncr t))Ci vibrations nf a string are liable tu bo modiried
hy a yichHr)~ cf t.bc points of attactuncnt; and tlie prob!cm
wiU fm'tush occasion for onc or two remarks of importance.
For t))C sahc of simplicity wc shaU suppose that thc System is
synunctrical witti rcfurcncc to tho centre of thc string, a.rid that
cactt cxtrundty is a-ttachcd to a mass (trcatcd as uncxtendcd in
spacc), and is urgcd by a spring (~t) towards thc position of cqni-
iibrium. tf uo frictionat forces act, thé motion is nccessa.rity
rcsolvabic iuto normal vibrations. Assume
~= (~ sin )Ha;+~3 cos MM'}cos (wa~ e).(l).
Tho conditions at thé ends arc that
whiehgivc
two cquaitons, sufîicicnt to détermine and tlic ratio of~S to a.
E)inun:tt.i)tg titc lattur ru.Lio, we Hud
Equation (3) has an infinite number of roots, which may ho
fnund by writing tan for so tliat tan ?/~ = tan 2~, and the result
of adding togcthcr thé corrcspouding particular sohttions, each
with its two arbitrary constants et and c, is necGssfu'ity thé most
guncralsoiution of winch thé prublem is capable, and is thercforc
adéquate to rcprcsunt thé motion duc to an arbitrary initial dis-
tribution of dispiacemeut and velocity. Wc infcr tbn.t any function
of x may bc cxpanded bctwucn x = and a;=~ in a-scrics of terms
~,(~,sin~)- cos?)!) + ~~(~s!nm.c+cos~) + (5),
?~ H~, &c. bclug thc roots of (~) and &c.thc coi'rcspouding
TRANSVERSE VIBRATIONS OF STRINGS.158
values ~p~" arc co-ordinatesof' t] i syst.em,
From thcsymmctry of thc system it follows that in each
"y thc s.,nc at pointsdistant––P~. thé
twoends,wherc~=0~d. l. Hcncc ~sm.t-co~ =+1 1,as may bc proved also from (4-).
8 4
T!.e Mncdcenergy y of thc ~I.ole nation Is made
up of thcMergy of thc
string, and that of the masses T!u~r;
y=~p){S
SUl + cos M!a-)~ f~
+~+~+.r+~~(~sin~+cos~~+.Buthy theenaracteristic
property of normal co-ordinates, terms
~I~~tr'~
cannot~lypr~eut in t.. c.prc.s-sion for l' 80 that
pf"
1
(vr .9ili 9?bX+,coq ili,x)p~(~ sin + cos ?M~) (~ sin M~.t-+ ces M,?;)
+ + (~ sin m~ + cos 7~) (~ sin M/ + ces M/) = 0. (G),if 7' and g Le differcjit.
This
t~orems~gcsts how to détermine thé
arbitrary con-stMts sothatthe senc-s (5) mayrcprcscnt au
arbitraryfunctiony. Takc thé expression
p~y(.sin ~~+cos~)~.+~+ sin cos
~(7)
.~d
~titutc
in it thé scrics (.) exprc.ssi.,g Thé rc.nit is ascries of tcrms of thc type
p~~(~ sin + cos
~) (~ sin ?x,~ cos ~~)
+ + (., sin 7~? + cos ,) (~ s; + eosail of
whi~
vanish hy (6), cxccpt thc onc for whieh 7. = Henccjs equal to thé expression (7), dividcd by
p~.sin~+cos~+~+~
and thus thé
coc~icntscftho series arc detcrnnnod. If ~=0even
althc.g~
bo rinitc, thc p,but thc unrcstnctcd prob~ is Instn,ctive. So nu,ch strc.~
135.] FOURIER'S THEOREM. 159
often laid on special proofs of Founer's and Lapiacc's séries, tliat
tho st~doit M apt, <,o ~cquirc' tor* contra<;tcd a vic~v of thc na-torc
ofthosc important rcsults ofanaly~Is.
We shall now shew bow Fouricr's thcorom in its ~encrai form
can bc deducpd from our présent investigation. Let ~=0; thcn
if /t= -X), the ends of thé string arc fast, and thééquation
de-
tcrniining ?~ becomcs tan M~= 0, or m~ =~Tr, as we k~ow It must
bu. lu this case t!)o séries for y becomes
which must bc gênerai cnongh to reprcscntn.ny arbitra-ryfunctions
of.K, vanisitingat 0 an'l ?, betwccn thosc Innits. But now suppose
th~t~ is zéro, ~/8ti!l v!t,nishmg. Thc ends of thc string may be
supposed capable of slidiug on two smaotit m:ls perpendicuiM- to
its length, Mid the tcrmina.l condition is the vanishing of
Thc cquf~tion in is thé same OM~e/~rc; and wc Ic~rn. that any
fnnction y' whose rates of variation vauish a.t a? =0 and a?= can
be expanded In a scriua
Tbis series remains unan'cctcd when thé sign of a; is changcd,
and thé first series mcrcly changes sign without altcring its
numcnc:il magnitude. If tlierofore y' ho an even function of x,
(10) represciits it n'om to + And in the samc wa.y, if y bc
au odd funetion of x, (9) roprescuts it betwecn thc samc limits.
Now, whatcvcr funetion of a; ~) (:r) may bo, it can bc divided
into two parts, one of w!iich is even, and the other odd, thus
so tha.t, if (x) be such tha,t (- = (+ Z) and < (- ?)=
~)' (+ ~),
it eau bc rcpresetitcd betwccn thé limits ± by the inixed séries
This series is penodic, with tlie pcnod 2?. If thcrcforc (x)
possess thc samo property, no matter what in othcr respects its
160 TRANSVERSE VIBRATIONS OF STRINGS. [135.
charactcr may be, the series is its complete équivalent. This is
Founcr's thcorcm*.
Wc now procccd to cxa.minc titc c~ccts of a slight yielding of
thc supports, in tlic cfu-:c (.)f :). striog whosc onds are approximat~'ty~xcd. Titc quantity tnay Lu g)-(.),t, cit!)c;)- through or throngj)
Wc shn)l conHne oursulvos to thu two principal cases, (1)
-whcn is ~rea.L aud vanishes, (2) wlicn vauishes and is
gréât.
and the équation in isapproxhnatdy
and
To this ordur of approximation thc tones do not cease to forma harmonie scalc, lmt tlie pitch of tlie whule is slightiy loweredTho effect
oftftcyiciding is in fact thc same as that of an increase
in tlie length of thé string in the ratio 1 1+
as might
have beenanticipatcd.
Thé rcsult is otherwisc if /t vanish, while Al is great. Hcre
and
Hcnce
Thé cfTcct is thus cquivfileut to a dccrcMc in l in tlie ratio
Tho ~t System' forprovi~g Fonrier's t]iMrem from dynamic~I considéra.
tio~ is au cmUess chain ~tchod round s,u.~th cyiiudcr (S M!)), or thiDro-outraut culumn of uir eucluecd iu a
nitH.sIfnpcd tube
135.] ] FINITELOAD. l(j~1
andeonscqucntiy thcrc 1. a nso in pitch, t]ie rise bcing thé
grcatcr thc lowcr tho couiponcut tone. Tt nngl.b bc thoughttttfit any kind
ofyiL-!di!)g wou)(t deprcss thc ptieti of thc string,but thc preceding uivestigation sficws that tins is not tiie case.
Whet))cr thc pitch will be raiscd or lowcrcd, dcponds on thc
sign of and this agam dcpcnds on whcthcr tlic nn-tura.! note ofthc mas-s urgcd by t!)c spriug I.s lowcr or !iig]tcr th:ui th~t ofthc component vihra.tion In
(question.
136. Thc proDcni of an ot))crwise unifonn string c~n'yinga nuite load ~at .ï;= ciui ho .sutvcd by thc formutœ InvcstigiLtett!n § 13:}. Fur, if thc force 7''cus~< be duc to the réaction againstaccuIcraHon of thc mass
which comhined with équation (7) of§ 13~ gives, to determine thc
possible vaincs of (or p r/),
Thc vfihtc of y for any normal vibrationcorrcsponding to is
whcre P :uid e arc fu-bitrfu'y constants.
It docs not rcfjuh-c anaiy.si.s tn provc thn-t any normal cojn-
ponents which have a, no.h at t!)C pnint of attachment are mi-.ectcd hy tlie présence of tlie load. For Instance, if a stnng be
wei~hted at the centre, its componcnt vibrations of evcn ordersrcin~in unchanged, \vhi)c a)i thc odd components arc dcpresscd in
pitch. Advantagc m!Ly somctitnc-s hc takcn of t)tis effect of a.load, wi)cu it is desircd for anypurp~cto distnrb the harmonierelation of thc comptjncnt tones.
If bc voy g-reat, titG gravest component is wideiy sepa-ratcd in pitc)i from !i]I t).u others. We will take thc case whent)te !oad is at thc cfnt rc, so t.hat = b = U.l. Thc équation in
t])cn hceomos
where 3/, dcnnting thé ratio of t)te masses nf the stritig andth.' )oad, is a sma])
quantity which may bc caltud Th<~ <ir~
K.1
162 TRANSVERSE VIBRATIONS 0F STRINGS.~:3G.
root corresponding to tlie tonc of lowest pitch occurs v'hcn ~~is
sma,!l,andsucht)i~t
Thc second term constitutes a correction to tho rcngh vn.luc
obLn.incd in a previous chn.ptcr (§ 52), by ncglecting th(; Ino-tin. of
thc st.ring :dtogcthcr. Thit.t it won)d bc i~htitivc might. I)n.vc
hecn cxpcctm), at)d indecd thé formuia. a.s it standsjua.y bc ob-
ta.incd frorn thc considcra.tion that in thc actutd vibration tlie two
~:u'ts of tlie string in-c ncn.riy str&ight, and may bc n.ssmncd to hc
<'xnct,Iy so inconipnting
titc kitK'tic n.nd potcntifd énergies, -\v!th-
<'ut C!)t)ufi!)g fmy apprcci:d)Ic crror in t)tc cn.lcoh~tcd pcriod. On
<J)i.s s~ppu.sition thc rctcntion of t))e incrti:t of thc string incrca.scs
thc kinctic cnf'rgy corresponding to {t givcn vclocity of t))C Jond in
thc mtio cf ~)7'+ whic)) icads to thc nhovc rcs)dt. This
mothod t):is indced thca.dvantagc
In oncrcspuct,
aa it )ni'd)t bc
npplicd whcn is not nnifortn, or ncarly uniform. ~)] th:tt is
ncccss.iry is t.ha). ),hc )oad -/)/ shoufd he su(H(.-icnt)y prcdouioant.
13C.] CORRECTION FOR RIGIDITY. 1G3
Thcrc is no othcr root of ('t), until sin~X~=0, which gives
thc second component of thc .string,–a, vibration indcpcndent of
the load. T!ie roots aftcr thc first occur ni closely contiguous
pairs; for oue set is givcn hy ~X~==S7r, ~nd tho other approxi-
mn.tc!y by ~=N7r+- in which tho second tcrm is sma.&'7T./)/
Thc two types of vibration for N= 1 are shcwn in thé Hgurc.
The goncralformula (2) may a)so be applicd to find the cifcct
of a small load on thc pitch of the various components.
137. Actua.1 strings and wh'es arc not perfectly flexible.
Thcy oppose a. certain résistance to bcnding, which may bc divided
into two pa.rts,p)'oducing two distinct enccts. The first is called
viscosity, and shcws itself hy df~nping thé vibrations. This part
produces no sensible efTcct on thé poriods. The second is con-
servative Iti its chtu'a.ctcr, an<t contributes to the potcntia.1 cnorgy
of thc system, with thc effect of shortening thc pcriods. A eom-
phjte investigation cannot convcnioltiy bc givcn hcro, but thc
case 'which is most intcrcsti))g in its application to musical instm-
mcnts, adinits of a sufficicntly simple treatmeut.
Whcn rigifhty is takcn intn account, somctiling more must ho
specined with respect to thc terminal conditions tha.u that y
vanisties. Two cases may hc particularly noted
(1) Mrlicii tii(, eiids arc so tli~it q = 0 tt tll(" C.,n(ls.(1) Whcn t)ic ends are clamped, so that= 0
at thc ends.
(2) Whcn thc termina) dircftions are pcrfcctiy free, in which
case
= 0.
f/.C'
Itis thé laLLct'whichwc propose nowtocnnsxtcr.
Jf tho'c wo'c no ngi'tity, t)tc t.ypc of vibration wouht hc
,c~r,r,f '1 1 1,
yx si~L–
p satt.sfymgt!ic second cnn'Ution.
Thc ciïcctofthc ri~iditymight bc slighUyto distnrh the type;
hnt whethcr such a rcsult occur or not, thc pûriod calcntatud
from thc potcntiiU a!)d kinctic énergies on thc supposition that
the type rc)n:uns mudtcrcd Is nccc.ssarilycorr'-ct as f:n' ~s thc first
ot'dcr of.stna)) qu:)nLit.ic.s (§ US).
N')\v Dit' potc))ti:d pocrgy duc to thc stiiïncs.s isexpresse'! by
~64TRANSVERSE VIBRATIONS OF STRINCS.
[137.
where /e is aquantity dcpcnding on the nature of thc mntcrial
n.nd on t.he form ofthn ~cticu in a pir.infr thn.t. nrc. nnt. ncw
prepai .;(i K, u.nh.j.Ti.e/u. urë~' is évident, bcc-msc ttic f~i-cc
required to bcnd any clément Is proportion. to and to thcamount of bcuding a]rcady c-iTcctcd, t!)at is to Thc whoicwork w))icli mu.st bo donc to producc a curvaturc 1 p in dsis thcrcforc proportional to ~-p'; whl)c to thc
app)-o.ximatnm to
1 1which we work = and =
p M.<,
if T. dénote what tl)c pcriod wouM bccomc if t])c st.nng wcrccndowcd with pc.rfect fi~ibiHty. It nppears that t).c cffcct of thc.st~ncss n.crc.ascs
rapidiy wilh t].o ordcr of thc couinent vibra-ttons, which cea.sc to
bc)o.~ to !L I~nnnnic scalc. Ho\vcvu! i.t t!.c
Htrn~s cmpjuyc..) in niu.sic, t).c ton.si.m i.susn~Dy sufîicicnt to
rcducc thu inHucnceofrigi()it,y
toinsignifiance.
T))G )neH)od offhis section cannot hc~ppjicd without inodin-
c.~ion to t).c ot).e.- case of t~nina) comiition, n:unc)y, whcu t).ccnd.s arc c-hunpcd. In thcir immudiatc nci~hbuurhood t)ie type ofvibration must dm' from that a.ssuincd by a po~cHy Hcxible
stnng byaquantity, w),.d. is no !o)~c.r s,n:dt, and w).osc squarethcrcforc cannot be nog]~t~d. Wc sha)) rcturn to this suhject,wttcn
ti-cattog of thc transvcrsc vibrations of rods.
J38. TLct-G i.s oncp.-obicm rdating <o t).u vihratiut.s of.strin.ïswhtdi wc I.ave not yut considcrcd, but which 1~ of .s.~nc practi~i
intcrcst, na.ndy, thc cluu-acter of thé nation of a vioiin (or ccHo)stnng undor thc action of thc bow. In this prob]e.n thé ~o~s~W!~ oft!)c bow is not
.sufHcicntJy u).()c.stood to aUow us tofoHow
cxeh.sivdy thc M ;)~~ mcthod thc indications of thoorymo.st bc .supp)emuntL~ ).y spccia) observation.
By a dextc-rou.s
combitiationof cvidcncedrawn frorn both Kourcc.s !Id)nho!tz)tas-snccccdcd m
d.r.nining thé principe tcaturus of thc. cas~ butsomc of thc détails arc .stii) obscure.
138.]VIOLIN STRING. 165
Since thu note of a. ~ood Instrumr'.nt, well ha.ndlcd, is musicn],
wc infcr thaL Lh~ yibraLiuurt a.i'c stricHy pcriodiu, or at least that
strict periodieity is thc td'td. Morcover–and this is very import-
ant–t!ic note clicitcd by the bow lias nctu'Iy, or qnitc, tite sn.me
pitch a.s tho n:itu)':d note of thé string. TIic vibra.ttons, although
ibrccd, arc thus iti sutuc sensé frcb. Thcy are whony dépendent
for tbcn' mn.intcnn.ncc on thé energy drf),wn from thé bow, and yet
tho how doos not dctcrniine, or cvcn sensibjy mod)fy,their pcriods.We arc rcmindcd of thc scif-aeting clectricaL intcrrnptcr, whosc
motion is Indûcd furccd in thc tochnica.! sense, but haa t!ia.t kind
offrcedom which consi.sts indcturjnitnng (who))y,or in part) undci'
what influences it sha,ll coinc.
But it docs not at once fullow from thé fuct thn-t tho string
vibrâtes witti its na.tura.1pcriods, that it confortns to its naturnl
types. If thc coefHcients of tlie Fourier expansion
be takcn as tlie independcnt co-ordhiatcs by wlticb thc conngura.-tion oftiie system is at any moment de~ned, we kuow that whcn
tliere is no friction, or friction such tliat oc titc na.tur:U vibra.-
tiu)is arc cxpresscd by ma.king cach co-ordin:tte n. s~e harmonie
(or quasi-harmonie) Hmction of thé timc; while, for a.l). that h:m
hitticrto appeitred to t))e contrary, eacii co-ordin.~to in the présent
c:mc nii~ht bc M?t~/function of tim time periodic in time'T. But a
Httle examiua.tion will show that tlic vibrations must hc sci)sib)y
natural in their typos as wcti as in thcir periods.
Tho force excrciscd by the bow at its point of application may
bc exprcsscd by
so tha.t tlie equation of motion for tlie co-ordin~tc is
& being thc pouit of appHcn.tion. Each of the componcnt parts of
will give a corrcspondin~ tcrm of its own pci-tod in thé solu-
tion, but tbe ono whosc period is thé same as tho natuml po-Iod
of~ will risc cnoi'tnousiyin relative importance. Pra.ctienUy then,
if tlic damping bo suialt, wc uccd only rctain tha,t p:~rt of
TRANSVERSE VIBRATIONS 0F STRINGS.[138.
166
whicL d' pends on J, c.~i" e~
thut I., tu ~y, w~ may rogm-dW 11(:" ( 1.
j"!en! S 011 ij. c,¡
¡¡~) tua!' ¡ci tu s.ty, Wu ~ün.y regM'
the vibrations as natural in their types.
Anuther matent fact, supported by cvidûncc drawn both from
theory and aurai observation, is tins. AH component vibrations
are absent which have a node at the point of excitation. In
ordcr, however, to extingui.sh thèse tones, it is neccssfa-y t!)at the
coincidence of the point of application of the bow with tho nndc
shonid bo vcry c.mc<. A very small déviation rcproduces t)tc
jnis.si)ig tones with considorabJc strcnf-tb'
Tlic rcm~inder of tho évidence on w)nch He!mboltx' theoryrcsts, vas dcrived from direct observation with thc vibration-
microscope. As explained in Chapter n., t)tis instrament affurds
aview oftbe curvercprcscnting thé motion of thc point undcr
observation, a~ it would ùe seen traced on tlic surface of a trans-
parent cylinder. In ordcr to Jcduec t!te représentative curve in
its ordinary form, the imaginary cylinder must be conceived tu
be uni-otiud, or developed, into a. p)anc.
The simp]cst results arc obtaincd whcn the bow is a.ppl!cd at a
nodc of one of tho higher componcnts, and thé point obscrved is
onc of the otlier nodes of the samc system. If thé bow works
fairly so as to draw out tho fundamcntal tono cicarly and strongly,thc représentative curve is tha.t she\vn in figure 22; where tho
abscisse correspond to tho tirne (~173 hcing a conpicte period),and t]io ordinates reprcsent the displaccment. The rcmarkabio
fact is discloscd t))at. thc whotc po-iod T ma.y bc divulcd inin two
parts ï~ nnd r-T., during c~ch of whieh thcvdocityof thet)b-
sct-vcd point is consent,; but thé vulucitics to and fro ;).re in
goict'a! uncqua].
Wc Iiavc now to rcprc.scnt this curvc by n so-ies ofiiarmnnic
terms. If t)tc ori~in of timccorrospond to t))c point J, and
Donkin'<~cf)~.<f)~, p. 13).
138.] VIOLIN STRING. 1GT
J 7' T'Y?== Fourier's t)teorcm E;ivea
With respect to thc value of T., wc know that ail those com-
pouents of~ must vauish for whichsm-°=0
(xa being thc
point of observation), 'bccn.usc under thé circumstances of the case
the bow cannot gcnGrato them. Tliere is thcreforc reason to
suppose thn.t T. T= a', l; and in fact observation proves tbat
J.C' C~ (iti tho figure) is cqual to the ratio of the two parts iuto
which tlicstring
is divided by the point of observation.
Now thc i'rec vibrations of thc string are rcprcsentcd in
geucral by
=sia cos
+ sin
and Uns at thc point a; = must agrec with (1). For convenience
of euu)parison, we inay write
2S7T< 2S7T< 2S7T/. T~A
287rt+ B~ Sin ~~7rt =
C~ Co. 2s7rt 19
~î, cos–– + R sin =(7, ces < ~)
T T T 2/
~(<),D
Ir ( 1'0)
and it thon a.ppears that C,= 0.
We find a.Iso to détermine D,
whcucc
In thc ca.se reserved, thecomparison
!c:wcs DH undetcrmincd,
but wc know ou otlier groundH tliat DH then vanistics. Howcver,
for the sakc of simplicity, we sh:dl suppose for tlie pt'cscnt that.
D~ isahvitys givcn by (2). If the point of application of tlie bow
do not coïncide with a nodc of any of the lowcr componcuts, thc
error comtnittcd will bc of nu grcat cousof~uencû.
On tliis undGrsta.uding tlie complote solution ofthc problem is
168 TRANS VERSE VIBRATIONS OF STRINGS.['138.
The a.mpl:tudea of thé componcnts fu-e t.tjercfore proportions) to
int~u!p!!t~,)dMt!-l; ~)U;tdfu!fi'JC<;)t'C!t)~)"'y A 0
futictious'sin'S~
L, 1 l is l, 'J If J stringi'uuctiou
s-"sln'whic)i is sonicwhat sinnJfn-. Iftijc string
ho ptucked at thé mnhilc, thé cvcn components v~nish, but thcOth! oncs foHow thc same )!tw as obta.hjs fur a vioiin
strincr. T))c
c()ua,tiûn (3) ulcHcatcs t]):tt thcstrmg is
:Jways m t))e fun~oftwo
.st.mi~ht Unes mcetittg a.t an angtc. In order inore convettiontivto shew this, !ct us
change thc origin of tftc tune, a.ud t)ic constat
mu)t,ip!ier, so that
will hc thc équation cxprcssing thé form oft))c string :tt any titnc.
Now wo know (§ 127) that thcC(tU!ttio.i of thé p.ur of lines
proceeding from thc fixod em]s of thc .string, and mcet.nrr at a
puint wliosc co-ordinates arc or, /9, is
Thèse équations indicatc that thc projection on thé axis of:Bcf thc point of intersection moves
uniforndy backwards andforwanis bctwecn .~=0 an.! ~=Z, an.) t.)iat t).c point of inter-.suction itscif i.s situatc.d on onc or ot).< uf t~-o p;u-abo]ic arcs,'~t' whieli thc equilibrhun positon of thé .string is a connnonchofti.
Since the motion ofthc string as th~ d~nu<i hy tl.at of thc
point of intersoction of its two straight parts, bas no cspccia!rctatioti to (Lhc point of observation), it. fo)h~-s that, accordin.to t])c.se
ouations, titc sa.ne J<ind of motion m.gbt Le obscrved a't
any otho- point. And t)iis isapproximatciy trnc. But tbc thco-
rctica) rL-.suk, it wil! bo romonbcrud, was o)i)y obtaincd by as-
H)))ni))g tbc présence incertain proportions ofcomponent vibrations
ha\'in~ nodc.s atthongh in tact thuir abscucc is )-<j(p)ircd by
"chanica) !aws. Thc présence or absence of thèse components is
138.] STRINGS STRETCHED ON CURVED SURFACES. 109
a mattcr ofindifïercncG when a, node is thc point of observation,
but not in nny otho' c-f. Wh~n thc nid.' i.-i doparted from, thc
vibration curvc shews a séries of ripples, duc to thc absence of
thc conponcntsIn question.
Somc furt!icr dctails will be fuund
in Hcitnhoitz and Donhin.
Thc sustaining powcrof thc bow dcpeuds upon the fact that
so)id frictiuu is Ic.ss at modéra te t))an at smalL velocitics, so thf).t
w)tût] t)tc part of thé stri!~ actudnpou
is movingwitil thé bow
(nut imprububly at thc s:~mc vulocity), thc mutual action is greater
titmi wi'eu thc string is moving in tho opposite direction with
:L greatcr relative vulucity. Thé ~ccctcrating eH'cct in tl~c first
part of thc motion is thus not cntirdy ncutratiscd by thé sub-
séquent rctfu-da.tion, and an outstanding accctcra.tion rcmains
cap:(.b)cof ]n:untaining
thé vibration in spite of other losscs of
~ncr"-y. A cm-ious cncct ofthc samc peculiarity of solid friction
bas becn obscrved by Mr Froudc, who found that tl)0 vibrations
uf a, pcndulnm swinging from a sbaft mightbc maintained or
cvcn Incrcascd by causingtt~c shaft to rotate.
139. A strin"' stretched on a. stnooth cui'vcd surface will in
cquilibrium lie along a gcodcsic Une, and, subject to certain con-
ditions of stubitity, will vibrato about tilis eonnguratiun,if dis-
ptaced.TI)C simplest case that call bc proposcd is when. tlic
surface is a cyHnder of any form, and thé cquitibrium position
of tlie string isperpcndicular
to titc gûnerating hncs. Thé studcnt
will casUy provc that tbc n~otion is indcpcndcnt of thc curvature
of tlie cylinder, and that thc vibrations arc in :dl essential respects
thc samc as if thé surface wcrc developcd into a plane. Thé case
of an endiess string, funning a nccidace round thé eylindcr, is
v/orthy of notice.
In oi'tter to Ulustratc tlic charactcristic features of this class of
problenis, we will tako thc conparatively simple cxample of a
stringstretched on thé surface of a smouth sphère, and lying,
v/hcn incquilibrium, :dong a grcat circle. Tite co-ordinatea to
which it will be most convcnicnt to refer thc system are thé
Jatitudc mcasurcd front thc grcat circle as equator, and thé
~n~itudcmeasured alung it. If thc radius of thc sphère be
wb bave
170 TRANSVERSE VIBRATIONS 0F STRINGS. fl39.
Thc extension of thé string is denoted by
J(~i)~.
Now
so tliat~=(~f~+(ocos~
sothat
f~f/ 1/=
{(~~ +
=2 (~~ 2 ~PP'tc]y.
Thus
a.nd~(y-(~'
and liCltp
-8-dtp.(2);1
~)~.
ô V= aTl.i-~ -10Q ose ~(to
+ e Jcp.
If thc ends Le fixed,
~=0'L~J. 0
and thc equation of virtual velocities is
8~+ = o,
0 0
0 se dtpo 0 (10
+ 8 dcfJ= 0,
whence, since S~ is a-rbitrary,
"(~)–
This is thc cquatiou of motion.
If wc assume oc cos~<, wc get
_rl'B 0 cc'p 22
(4),~,+~0.
cf \vl)ieh the solution, subject to t)ic condition that vanisheswith is
~=~sinj~~+l~.cos~ .(5).
Thorcmaining condition to bc satisfied is that vanislics whcn
«~ = or <j& = et, if a = <! K.
Tiiis givcs
I\
~h' -~=p'(~1)
a p a -1 p ( ¿:I- cG~ J .G
~herc ?~ is an iutcger.
CambrMHOMathcmaticftt TritMB Exnmination, 187G.
139.] VARIABLE DENSITY. 171
Tho normal functions arc thus of ~'c samc form a,a for a.
stnufht strmf. viz.
but thc series of periods is digèrent. Thc effect of thé curvature
is to makc cach tone graver thao. the corresponding tonc of a
straigbt string. If a> 7r, 0110 at least of tho values of p2 is néga-
tive, mdica.ting tha.t the corrcspouding modes are unstable. If
a =='7r, is zéro, tlie string bcing of tlie same length iu tlie dis-
placed position, as whe!i = 0.
A similar method might be applied to catculatc the motion of
a striug strctched round tlie equator of any surface of révolu-
tion.
140. The approximate solution of the problem for a vibrating
string of ncarly but not quitc uniform longitudinal dcnsity bas been
fully considcred in Chapter IV. § 01, as a convenierit cxampic of
thc general thcory of approximately simple systems. It will bc
sufficient hère to repeat thc result. If tlie density bc ~+ thc
pcriod ï, of thc ?' component vibration is given by
Thèse values of r" arc correct as far as thc first power of thc
small quantifies 8p and ?~, and give the incans of calcul~ting a. cor-
rection for such slight dcpartures from uniformity as must always
occur iu practice.
As might be expecte(l, thé effect of a small load vanishes at
nodes, and rises to a maximum at tlie points midway bctw<;cu
consécutive nodes. WIien it is dcsircd mcrcly to make a rough
Gstimato of thc effective dcnsity of a ncarly uniform string, thc
formula indicatcs tliat attention is to Le given to the neighhour-
hood of loops rather than to that of nodcs.
1-tl. The dinerential équation determining thé motion of a
string, whose longitudinal dcnsity p is variable, is
TRANSVERSE VIBRATIONS 0F STRINGS.F 141.
172
from which, if wcassume oc cos wc obttuu to dctcrmu~ th
ï.'<))';n:u iuucti~
~yhcre ,swntt.nfor~-?', This~uationis of thé second
o'randhncar, b,.thasnothit)icrtobccn.so!v.d:n huitotcrms.
Cun.s.dcrc.tdcH.ling thé curve ~su.ucd by tho .st.rin.r In thc
uonna! mode uih!cr considération, it dutcnnincs thc c~rc atany pou.t, nnd
accordin~jy cinbodic~ a ru)c I.y whidi thé c-n-vocan bc construct.cd
~phic.Hy. Thns in thunpptic~iun to
string nxcd both cnd.s, if st.t from c.ithcr end nn ~rbitraryinclination, and wit). zéro
curvatm-c, ~-0 are ahvay.s directcd by tbe
équation w.Lh what eurvaturc to p.-uceed, and in tins way wcjaay trace out thé cutirc curvc.
If thc assumcd value of be rigbt, the curvc will crossth. axis of at thc rc.uircd distance, aud thc ]aw of vibrationwill hc
con~tctdy dctcrndncd. If Le nul known, ditterentvaincs may bc tri.d untii thc curvc ends rightiy; a sufncientapproximation to tho value
cf~m.~u.su~iyho ~-nvcdathy~c~cul~tion founded on an as.smncd type (§§ 88, 90).
Whcthcr t!.clongitudinal density be uniform or net thé
pcncdic timo of any simple vibration varies c~~ as thcs<(u~e root cf thc den.sity aud
Invcr.s.ly us thé .square root of thetension undur w)nch t]io motion takcs piace.
Thc eonvcrsc prob)cm ofdct~mining thc <icnsity, w!,c.u thé
pcnod and H,c type of vibration arc gi vcn, is always sutuhic Fortins purposc is oïdy necessary <o substitutc thc givcn vah.c of vand of its second di~brontial cocmcient in équation (2). Unksstbedcns.tybo innuitc, thé extrunutics of a string arc points ofzero curvature.
W!tcn agivcn string is s)iortencd, every componcnt tono is
ra..scd ,n p.tc)L For tho new stato ofthings may bc rcgarded as
dcnvcd from thc old by intradnction, at t!ic proposed point ofhxturo, of a spring (without inc.rtia), ~vhose stifFncss is
gradua]!yincr~scd without limit. At cac)..stc.p of thc proccss tho potcntia!cncrgy ofa givcn déformation is angmentcd, and t).c-rcforû (§ 88)thé intch of every tone is raiscd. In likc manner an addition tothc length ofa
str.ng dcpresscs thc pitc! cven though thc addedpart bc dcstitutc ofiucrtia.
142.]VARIABLE DENSITY. 173
14-2. Atthongh a gênerai Intégrationof équation (2) of§141
~c' <<pr,ui'
v' m.T\p~yt'~ t!n)-~k'n .u- c'ft.h'Sh'
1.
many intcresti! propcrties of thé solution of thé hnuar équation
of'thc second order.which liavc hcen detnonstrated LyMM. Stnrnt
and LiouviHu'. It Isimpossible
in tins work to give anythiug
hkc n. compictu ~ccomit of titeir invc.st.i~Lions; bot :), sketch, in
which tho te:tdi))~ fca.tm-cs n.ru inctudcd, m~y be found intcrcst-
incr, and will tin'uw li~ht on sone points comicctud with thé
gcncnd thcory of the vibrntions of continuons bodics. 1 hâve not
thought it ticccs.s:u'yto adhère vcry c~oscly to t)'c mcthods adoptcd
io thc origina.)tncmon's.
A.t no point of t!t0 curvc satisfying thc cquation
rl'r/ 2) (1)
~+~~n.(D,
can both y a.nd'(
vanish togctiicr. By sucœsstvc diHcrcntin.tio~s
of (1) it is c~sy to prove that, If n,nd vanish simnitancousiy,
aU thc highcr dnïcrcntial coemcicnts &c. musta!soCl'`
tC~:Cs
vanish at thé samc point, and tilo'cfore by T~ytor's theorcm tho
curve must eoincidc with thé axis of a;.
Whatevcr \duc be ascnbcd to thc cn)-vc satisfying (1) is
suu~ beingconc:wc tbt'oughout
to\vard.s tlio axis of a-, sinec
p is cverywllere Ilositive. If at l 1 ana~Lxis cvoywhcrc positive. If at thé origin y vanish, and
Lu positive, thc ordinatc will rcnmin positive for aU vaincs of a;
bclow a curtain limit dei'cndcnt on thc vainc ascribed to
If bc vu'y smaH, thc cm-vaturc is slight,artd thc curve will
remain on t]tc positive sidc of thu axis for a gi'cat distance.
Wc hâve now to provo that as incrcascs, aU tho vahtcs of a;
which satisfy thc cotation = 0 gradua)Iy diminish in magnitude.
Lct Le thé oi-dinatc of a second curve sati.sfyingthé équa-
tion
~+,=0.(2),
cl;c'+ Il p?J
as weil as thc condition that vanishes at thc origin, and lut us
suppose t)iat is somcwhat grcatcr than Multiplying (2) hy y,
T)to )i)c;]))"i)'n rnferrc~ to ui tho tcxt nrt: euittttmcd iu tito first volume of
LiouyiUu's .yuto'/t'~ ()'S!it!j.
174 TRANS VERSE VIBRATIONS 0F STRINGS.[143.
n.nd (1) by subtmcting, andintegn).th]g with respect, to x
bctwecu ttic limita 0 and x, wc obtn't), sincti and ~ot!) vnmsh
wii,L.'t!,
.1
If wc furthersuppose thftt .c
corrcspomiH to n. point :).t winch
y vcmi.sfK's, ;u]d th~t tlie di~rcucc betwccn and is vcry small,we gct ultimately
TIic right-hand mcmber of (4) bcing csscntially positive,
wo Jcaru that y and arc of thc samcsign, and thcrcforc t!)at,
w])ethor be positive 01- négative, is ah-eady of thc samc sign
as that to which y is changing, or in o~K-r words, thé value of a;
for which vanishcs is less t)ian that for winch vanishcs.
If wo Hx our attention on thc portion of the curvc !yin~betwccn ;K=0 and .r= thc ordinatc contitmcs positive t)n'ou"'h-ont as thé value of
incrcases, until a curtain v:duc is attaincd,
which wo will call Titc function is now idcntical in form
with thc first normal functionM,
of a string of dcnsity /) (ixod
at 0 and a~d lias no root cxccpt at thosc points. As a'~ain
iacreases, thc first root movc.s inwards from a;=~ unti), when a
second special value is attaincd, thé curve again crosses thé
axis at thé point a'=~, and thcn rcprcscnts t]tc sccon(t norma!
functiou M,. This function bas thus onc internat root, and onc
ot~y. In likc manucr corrcsponding t.o a hi~hcr value wc
ohtain titc third nonna! functiot ?~ with two interna! roots, andso on. Thc ?"' functiou M,, bas thus cxactiy 1 intcrnid roots, and
sinco its ih'st dinurcntial cocfticient ucvcr vanis))cs sinndtancoustywith thc function, it
changes sign cach titnc a root is passcd.
Frn)n équation (3) it app~u-s that if nnd hc tw.) di~'rcnt
normal functions,
A bciUttifu! thcorum bas bccndi.scuvc-rcd by Shmn rc)!~in<r
to the mnnhur uf Uic routs ci' funcLio;) (k'rivcd by addition
from Hnitc tiumbur of nurnud fuuctious. If bu thc eompoucnt
143.] STURM'S THËOREM. 175
of lowest order, and M~thc component of highost order, tho functtou
whcrc~), <
&c. arc arbitrary coefHcicnts, has a< ~ecM< m–1 1
internai roots, and ~os~ M–l intcrna.1 roots. The cxtrenutics
f~t ~=0 aud at .~=~ con'cspon.d of course to roots in a.U cases.
The following démonstration bca.rs somc rcsonbluncc to that givcn
hy Liouville, but is considcrn.bly simpicr, aud, 1 bellcvc~ not less
rigorous.
If 'wc suppose that /(.E) ~as cxact]y Internai roots (any
number ofwincli may bo cq)ia.l), tho derived functionj~) cannot
Iiavo less tl)an + 1 internai roots, sincc therc mnst bc at ]cast
onc root of/'(.'c) bctwccn cach pair <~fconsccntivc roots ofy(a;), and
t.Itu whoïc numbcr of roots of~(.~) eoncurnud is ~.+2. 1); liko
manncr, wc sec that thcrc must bo at Icast roots ofy'(a:),
bcsides tlie cxtrenutics, which thcmselvcs necessarlly correspond
to roots; so that in passmg from _/(~) to y"(~') it is impossible
that any roots can bc lost. Now
bas at luast /t interna.! roots; and thé proccss tnay bo continnc'd
tu fmy uxtcttt. la this w~y wc obtai)i a scrics of' ftmctions, :t.])
with intct'n:d roots at !en,st, whieit dUrur from the origina!
imtCtioM/(:)') by tho continua]]y menjasin~ relative !))iport!incc of
the componL'uts of thc hi~Lcr oniurs. Wi~cn t!i(i procL'ss I):~s bcot
ciu'i'iudsufficicnt.Jy fur, we sh~H :),n'ivc !tt a function, whosu iorm
ditturs as )itt)c :)H we p!t.'asc ft-om that of t))c normal fonction uf
hi~))cst ordcr, viz. M, and w)iic)i ])as thcnjforc )t– 1 intcrn:d roots.IL funows thi~t, sincc no roots can 1je lest in passit)g down thc
so'ics uf fu))ctio!)s, thc m))ub(.'r uf Int(.'t'))a) ruuis ufy(;<') c;U)n<jt
UXC')) ;) t.
TRANSVERSË VIBRATIONS 0F STUINOS.17G
[142.
Thc other ]~!f of tho thcorcm is proved in n..simDar mannor
hycc!tt:t~:u~th~8';)-Ic:~if'un.cti.< h.kw.'n~fl'u~i' h.
Utiswaywcobt:).i)i
:;i ll
arriving- t~t last at a functiunsc.n.sib]y coiuc.d~nt in form with thc
normal fu])ctionof ordcr, v~ ~ndhavin~ Lhei-cforo
M-lnitcrnali-oots.Sinccnoroutsc.-mbctastinp~sin~upthc
senesfrom <.his functinn to/(..), it full~y.st].at/(.r) ~nr~ot Lave
fewertntcrn~ roots tl.nn ~-1; but it must bc und~-stood thatauy number oft)~ w 1 roots
mny be cqu.d.
Wc wi!! now prove tl.at,/(.) cannot beide..t:ca!!y zcro un)c~
a!I tliccocfHcicnts va.n.sh. Supposa t).at is not 7ûro
Muhiply (G) hy p and intégrée wit), respect to~ betwccn t!tcJhmts0aud/.
Thc.tby(5)
~nnc
from wl.ich, since t)ic intègre on thc rigbt-hand sidc i.s ~nitc wesec t)iat/(.r) cannot vanish fui- aU vah.c.s of Incli.dud withuAhc
t'fmgcofintogratio)).
LIouviiIe ].M inadc u.sc of Stur.n'.s thco-ent to sliew i)ow ascriGS of normal f.u.ctions n~y be eo)np<,u)h)c<I su as to have an
at-bitrary sign atatt puint.s iymg bL.twccn ~=0 and a;=~. Hismethodi.ssunK.'wItatasfoDuws.
Thc va]u~ of~ fur windt thé n.nc~iun is tochangr. sign bfi.)<ï
&,c, (.juantitic.sw).)) wiDxmt lossofgcnera.Hty~G m~
suppose tu bc aU (tiOcrent, fut u.scon.sidc.r tl~suries of détermi-nants,
T.c~.snsa!.nc..u-funct,onof.,(.)~.) ~andbyStunn'st!K..o,-cm h.~th~forc onci.iU.rna! n,o< atn.ost,whidi roulis
cvidc.ht)y ]\rcover t).o dL.tcrunnant is notidcntica))y zcrosn.ce thé cu~ciont of «,(..), viz, ~~), .)“ ~.t ~)~ ,,]~tevcr
bcth.v~ho ut' -\Vc hâve thus oht.in.d a function, ~h:ch
chan~s~natauarLiLrarypuiut.r/ui.!thcreon)yiuL.rn:t))y
143.]EXPANSION IN SERIES 0F NORMAL FUNCTtONS. 1~7
The second déterminant vanishes when ~;=~, and witen ~=&,
und, ~incû it canuoL hnvc tnure tb~n tv.u iatcmai ~oms, it chu-u~ca
sign, wheu x passer through t)ieso values, and tl)ere on!y. Thc
coefHcient of ~(a;) is tlie value nssumod by the fii'st dcterminn.ut
whcn x = &, and is thcrefoi'e finite. Hcnco thc secoud dcterminaut
is not identically zéro.
Simila.rly thc third dctermma.nt in thé series vanishes and
changes sign whcn x = a, when a; = and wilcn = c, and a.t those
internai points only. Thc coefficient of ~(;E) is funtc,bei!)gthc
value of the second déterminant wheu .E= c.
It is evident that by continu Ingthis process we can form
functions compounded of thé normal functions, whieh s!)all vanish
aud change sign for any arbitrary values of a*, and not eisewhere
internally; or, in other words, we can form a function whose sign
is arbitrary over thé wliote range froin..B= 0 to x =
On this theorem Liouville founds his demonstration of the
possibility of representing an arbitrary function between x = 0 and
.c =by a series of normal functions. If we assume the possibility
of the expansion and take
/(.c) = 2j'~) f p i').)~ j p !t;(~) r~-}.(11).(. ~o -'a U
If tlie séries on thc right bc dcuotud by ~'(.~), it ron.Lms to
cst.abtisli thc idcutHy of/(..t') and ~(x:).
Iftiie right-hmjd mcmbhr of (11) he tunttiplicd by pi<~(~) and
Intcgra.tud with respect to from a:= 0 to x = wc sec that
or, as we nmy :tlso writc it,
t!)c necessary values of < < &c. arc determiued by (9), a.nd wc
fiad1 J f8l
where M~(-c) ts ct~y nonna.) function. Frutu (12) it follows that
w~crc the coefHcicnts &c. are !n'bitmry.
it. ~),
TRANSVERSE VIBRATIONS 0F STRINGS. [143.178
Now if F(~)-/(.c) bo not Idcntic~Hy xcro, it will bc pnssiMoso chonsc t.hc
c<.tis~)t.s .1. J,. t~ <M -t .< .r~tuLs
throughout thc sa).tcsign ~'Y.'(.<:) -), in' \v])ic). CMc'cvc.-y
eloncnt ofthe jutcgmt woul(! bc positivu, :m() cqu~~ion (13) cuu)dnot bu tn.o. It fuUuw.s Lh.Lt F(.) -/(.<.) cannot <)i~r imn) zéro,or th:),t t])e s(;i-ics of sonnât fnnciions fortning tho right-hatxtnimber of (11) is idcntie:d
wit)i/(.r) for ~11 vaJue.s of .t- from tc= 0to =
Thc arguments and rc.suits of Uns snctinn arc of course ~~p-ptic-aDo tu thc ]):u-ticu).-u- case of a unifonn stmig fur wi~ch titenormal functious arc circuhu'.
14.3. Whcn Lhc vibmtions of a string arc not con~icd to cno
pJanc, it is usua)]y ninst e.mvenient to rcsnivo thon into two setscxGcutc-d in
perpendicuL-u- ~imc.s, which m.~y be trcatcd indc-
pcndonUy. Thcro is, howcvcr, onc case of t!.i.s description worth
p~sing notice in which thu niotion i.s most casity cuuccivcd audtt'catcd witliout résolution.
Suppose tha.t
Thcn
aud
-shuwin~ <),<. thc~-hojc-stri.~I.sata~'tnonu.ntin one j.Jano
wh.cii ~volves uni(ur.n)y, :uni tliat cachpa.-Ucic dc..scrib~ circi~
with radius siu~ Intact, t!.c wh.dc ~tcni turns witi~ut t
rdativc <]isp)ace.ncnt. ahuut. its position <c.tui)ib.-iuni, c.mi{~tin<r
cachrevuludunin H.ctimuT- Th~n~nic.s uf-t.hiscn.sci.s
quitc assise aswh~t).cnmt.H)u!sc<~finu.)
t.,o.K.phuK.,thcrésultant uf)L].c tensions
arti.~ at i!H.L.xt.-cn.iti~<,f any s'n~t)
p..rtiuu oi' thcstriu~ iu~Ht bL-ing Latancud by tlic
cuntrifu.~]furce. °
144.] UNLIMITED STRING. 179
144. Thcgênerai cHScrendal
equation for a uniformstril)"-
'ix.
cxprcssingfho rciation bctween a;andy, reprcscntsthe form of thc
string. Achange
In tho value of t is mercly cquivaleut to an
attcra.tion in thu origm of x;, so t!)at (4) indicates that a certain
/or?~ is propn.gatcd aiong thc string witli uniform velocity ft in tho
positive direction. WImtcvcr thc vainc of maybcat thc pointa; and at thc tuue t, thé samc value of y will obt:uu at thc pointa: + a A< at ti~e time + A<.
Ttio form thus perpctua.tcd may ho any \vlu).tcvcr, so long as it
docs not viotatc thu rustrietioas on whici~ (1) dcpcnds.
Whcn titc motion consists of thc propagation of a wave in thc
positive direction, a certain relation subsists betwccn thc iuchna-
tion and thc velocity at any point. Difï'ercntiatinn' (4) wc find
Initia [y, und ïn:).y buth bc gLvcn arbttrariiy, but if tho
a.bovc relation Le not sati.sfied, t)ic motion cannot bc rcprc.scntcd
by(4).
Inasmultu'nmnucrthcuquatiun
y=~+~).(G),dénotes the propagation of :), wave in tho ?<e~(t<tM direction, and
t!)C relation butween :Lnd corresponding to (5) is
]2_2
180 TRANSVERSE VIBRATIONS 0F STRINGS. []4.t.
lu tho gênerai case tlie motion consists of thé simuttaneous
propagation of two waves with vciocity ?, thc one in the positive,anu iL~ cthcr in die
uugative du-cc~L'u; aud tiiesu wa,ves arc
entirely indepcndcnt of one a.nother. lu the first~= ft~
and<c'
m thé second=
T)ie initial values of and must hom t e secon
Mt,=
n~-10 Illltw, va \lCS an must ue
conecived to be divided ioto two parts, which satisfy rcspcctivctythe relations (5) and (7). The nrst ccn.stitutcs the wavc whichwill adva.nce in thc positive direction without change ofform the
second, the negative wave. Thus, Initia)]y,
whence
If the disturbance be origina)!y confined to a rmite portion ofthe string, the positive and ncgative wavcs sep:L:-atc after t))0interval cf time required for each to traverse bulf the disturbod
portion.
Suppose, for example, that is thé part initialiy disturber).
A point P on the positive side remains at rest nntil thé positivewave has travelled ft-om A to P, is disturbed during thé passngoof the wave, and ever after remains at rost. Thé negative wave
never affects P at ail. Similar statements apply, ?!H~M ~M~iA',
to a point <3 on thé negative si de of~4Z?. If thé character of thé
original disturbance he such tha.t vanishesinItiaUy. tho-~f<.c o a<
144.] POSITIVEANDNEGATIVEWAVES. 18L
is no positive wfi.ve, and thé point P is never disturbed at all;
and if + vanisti initially, there Is no négative wave. IfCf~ (tM<t
C)K<
vanish iiiitially, the positive and the négative waves are similar
and e(~un.l, and tlien ucither can vanish. In cases whei'e eittier
wavc vanishes, its cvanesccnce may be considered to be due to the
mutual destruction of two componeiit waves, one depending ou
thé Initi:d di.sptaccments, and tlie other on the initial velocities.
On thé one side thèse two wavcs conspire, and on the other
thcy destroy one anotlicr. Ttiis explains thé apparent paradox,
that P can fail to bo affectcd soonct' or later aftcr -~jB Las been
disturbcd.
Thé subséquent motion of a string that is initially displaced
without vutocity, mayhe readHy traced
by graphical mcthod.s.
Sinco tllC positive aud thenégative
wavcs are equaL it is on)y
ncccssaryto dividc t)iu original disturbance into two equal parts,
to ()i.spi:(cc thèse, onc to tho right, and thé ot)ier to the left,
through a spacc equal to at, and then to recompuund them. We
shall present)y apply this method to tho case of a plucked string
of nnite tongth.
]-t5. Vibrations are called N~o?M)' when thé motion of each
partidc of thé system is proportional. to some functiou of thé time,
tlie same for a!l thé particles. If we endeavour to satisfy
Ly iissuming y=~Y', Avhore J~ dénotes a. function of a? on!y, and
a. function of t ou [y, wc H)id
1 ~T Id~Y
Y'=A'=~~constant),
sothat
proving that thé vib:).tio)is must bosimple harmonie, though of
arbitrary pcriod. Thc value ofy mny be written
y= cos (~t~ e) cos (/a; a)
= PCOS (~(~ + M.T; e Ct)+ ~7'' cos (Mf;< ?~.K e -t- a).(3),
shcwing that thé most gcner:).l kind of stationary vibration may
be regarded as due to the superposition ci' cqual progressive vibra-
~82 TRANSVERSE VIBRATIONS 0F STRINGS.[145.
tions, whoso directions ofpropagation arc opposcd. Converscly,
two stationary vibrations may eotnbinc into a progressive one.
Thé solution~=/(~)+~~+~ in tlic in-
stance to an infinite string, but may bc interprcted so as to
give thc solution of thc prol)le.n fur a nnitc string in certainca~cs. Let us
.suppose, forcxamp!c, tl~t tlœ string terminâtes
at ~=0, a)id is he!d fast thcrc, whilc it cxtcmLs to inHnity inthé positive direction on)y. Nuw so long as thc point .-c= 0
Mtua]!y rcmains at rcst, it is a mattcr of indinfcrcnco whctherthe string Le
prulongcd on t!to ncg~Ive hidc or not. Wcarc thus Icd to regard thé
gh'cn string as ibrnung part of one
doubly inrin.tc, aud to scck ~iicthcr and how thc initial disp]acG-mcnts a.nd vetocities on tlie ucgativc side can hc taken, so that onthc wltolo t!)crc shidi ho no dispfaccmcnt ~=U throughout t!ic
subséquent motion. Titc initial values ofy and y on thc positivesnic détermine thc
corrc.sponding parts of t!.c positive and négativewa.vc.s, into which wc kuow that thc whulc mution can bc resolvcd.Thc former bas no influence at thc point .7-= 0. On thé négativeS)de thc positive and thé négative waves are hntiaHy at our disposa!,but with thc latto- we arc not concerned. TI.c problem is todétermine thé
positive wave ou thé négative .side, so that in
conjunct.ion with tliegivcn négative wave on t).c positive side
of tlic origin, it sh:dt Icavu that point undisturbed.
LctM~
bc thc line (of any form) i-cprcscnting théwavc m wluch advanees in ttie négative diruetiou. It is
evident that thé reqmrcmcnts of thc case arc met by taidng onthc uthur side cf 0 what may be caUcd t!)c cû?!<a?-~ wave, so that
is tlie gcumctncid centre, biscctmg every chord (such as TV)which ])a.s.s~ tijrough it.
Au:dytlc:d)y, If =/(.c) is thc équationof O~ =/(-) is thu equatiou of O~'Q'7);
145.]REELECTION AT A FIXED POINT. 183
Whon after a, timc t the curves M'e shifted to the loft !md to
thé right rcspectivcly throttgh a, distance at, the co-ordinatca
cut'rcspojiding to ? = 0 arc necessa-nty cqual and opposite, and
tlicreforc when conipoutidcd give zero rcsultant displacomont.
Thc efïcct of the coustrahit at 0 may tttcrcforc bo reprcsented.
by supposingt!):T,t thé négative
wavc tnoves through undisturbed,
but that apositivo
wnvc n.t tho s~mc timcémerges
from (9. This
l'cfL'ctcd w~vu may a,t auy timc be fouud from its pa.rcut by tbe
iulluwing ru le
Lcit ~7- bo tho position of the pM-cnt wave. Thon the
rcflectcd wavc is ttic position which this would assume, if it werc
turncd thro-ugh two nght angles, fn'st about OJC as an axis of
rotation, and then thrungh thu samc angle about OY. In other
words, t!)G rutuni \vavc is thc itnagc of ~P()~~ formcd by
successive optical rciteetion iu O~Y aud OY, regarded as piano
mirrors.
Thé same rcsult may aiso bc obtamcd by a more analytical
process. lu the guneral solution
y=/(a;)+F(~+ft<),
tito functious /'(~), J~(s) arc dutcrmincd by the initial cn'cumstances
fur al! positive values ot' z. Thé condition at œ = 0 requircs that
/(-~)+(F(~)=0fur al!
positive values of or
/(-~)=-F(.)
fur positive values of z. Thc functions and .F are thus dc-
tc'nnincd ibr aH positive values of and
Thcrc is now no difnculty in tracixg thc course ofcvcnts wbcn
~o points of thc strmg -/i and J? are hctd fast. Thc initial dis-
turbance in ~17~ dividcs itself Into positi.vc and négative wavcs,
which are l'cnuctcd backwards and furwards bctwucii tlic nxed
184 TKANS VERSE VIBRATIONS 0F STUINGS.['145.
points, ch~nging their character from positive to negative, andvice M)' at each renection. Aftcr an even numhf~ pf )~tions in each case tite original ibrm and motion is
compictelyrecovcrcd. The proccss is most casiiy followej in imaginationwhen thc iniLiaI Ji.sturb.mcc is connncd to a .sma!l part of the
iitring, more particularly when its charactcr is suc!t as to give riseto a wave propagatcd in ouc direction on!y. The ~«~ travels withuniform velocity (f() to and fro along thé Icngth of the .string, andafter it has rcturned ? ~eco?~ time to its starting point' the
original condition of things is exa.et]y rcstorcd. The period ofthé motion is thus tha time requircd for the pulse to traversethe length of the striug twicc, or
Thc s~unc iaw cvidcntty ho]ds good wlmtcver may be the characterof the original disturbnncc, only in tlie gcnera! case it may
happen that thé s/io?'~ period of récurrence is some aliquot partof T.
14G. TItc metliod of the !a.5t fcw sections may boadvantage-
ons!y applied to thc case of a plucked string. Since the initial
velocity vanishes, haïf of tho displacemcnt belongs to the positiveand haïf to thé negative wavo. The ma.nner in which thé wavemust be complotcd so as to produce the same effect as tlie con-
straint, is shewn in thé figure, wliere thé uppcr curve rcpresents
tho positive, and thc lower the negative wave in their initial
positions. In order to find the connguration of the string at any
185146.] GRAPHICAL «L%IETI-IOD. 185
fotm'f time, the two curves :nust be superposed, after the upper
has been sluftcd to thé right and tbe lower to the left through a.
space e(~ual to at.
TI)G resulta.nt curve, like its components, is made up of stra.ight
pièces. A succession of six at intervals of a, twcifth of thé period,
shewiug tho course of thé vibration, is given in the figure (FIg. 27),
taken from Helmholtz. From the string goes back aga.In to il
throughthc same stages'.
It will be observed that thé inclination of thé string at tho
points ofsupport alternates bctween two constant values.
147. If a small disturbance be madc at thé time t at the
point x of an infinite stretched string, tlic effect will not be fcit at
0 until aftcr thc lapso of the timc a, and will be in ail
respects the same as if a like disturbance had bccn made at
the point a; + Ax at time t- A.c-r a. Suppose tliat similardisturb-
an ces are communicated to thc string at intervals of time r at
pointswhosc distances frorn 0 incrcase each time by ctSï~ then
1 This mothod. of troittUf; tho vibration of a plackod string is duo to Yonng.
J~tt!. 2'Mf~ 1800. Tho studcnt is Tecommonded to mnkc Idmself fftmi!iar with it
by actuaHy constructiug thé forms of Fig. 27.
18 G TRANSVERSE VIBRATIONS 0F STRINGS.('147.
it is évident that thc result at 0 will be the sarne as if tlic dis-
~nc~w<~anniauuatth.!san)upoint,j)rovtdu(t~iatti!Ctii:i~
intorvais bc incrca.sed fru.n T to T + 8r. This rcmark contaiu.s tlic
t!)tjoiy of t)to altoi-atiun of pitcit duc to motion of tixj s(n)i-ce of
disturbance; a subjcct w)uc)t will corne uuder our notice a~aiuin connecti'Jti with acrial vibrations.
148. AViten onc point cf an innnitcstring i.s
subjcct to a forccd
vibration, trains of wavcs procccd frorn it ill bot!)'directions ac-
oord.n~ to hws, wf.ic)iarc roadity invost.i~ated. Wc shall snjipo.sc
tb~ thc or~in is thc point of excitation, théstring bcing thcro
subject to t!~ forccd motion y=~ and it will bu suniciunt tocon.sidcr tl.c positive sido. If tbc motion of cach dc.ncnt boresistcd by t)ic frictional force tlie dinTercutial équation is
148.] DAMPIN&0F PROGRESSIVEWAVES. 187
If wc suppose that /<: is small,
:uid
This snhttion sttcws thft-t thcrc ia propa~tcd a!oi)g the string
a wavc, '\v))ost; funptitmtu stowty duninishus oti nccunut of ttie
cxponcntiaHactur. If <=(), t)iis factor disappca.rs, iuid wc hâve
simpty
This rcsult stands in contradiction to t)ie général !aw H)a.t,
whcn thcre is no friction, thc forccd vibrations uf a. System (due
to a sing)e snnphharjnonic force) must bc
synchrouousin ptiase
tbrougbont. According to (9), on tho contrM'y,t)iu phase varies
cuntiuuuusty in passin~ ft'om une point to a.uother along tiie string.
Thu fuct is, tl)a.t wc M'o not a.t liburty to suppose /e==0 in (8),
Inasmucti as timt cqmt.tion was obtalucd on tuu assumption that
thc rca,l part of X in (3) is positive, aud not zéro. Howcvcr long
a nulte stnng may 'bc, t!m coenicicut oi' friction may Le ta~en so
stnaM that the vibrations are not dampcd bufurc readiing t!t0
Hirthcr end. Ai'Ltjr tliis poitit of smaUness~ reficctcd waves bcgin.
to compHca.tc thé result, and when thc friction is dinilnistied
indonuitely, an iunnite séries of such inust be takeu iuto accoun.t,
and wcuid give a. résultant motion of thc samc phase throughout.
This problem may be soived for a. string whose mass is supposed
to be concuntratcd at C(p)idistant points, by thé mcthod of § 120.
Thc co-oi'din:Ltc may be supposc<) to hc givcn (= ~le""), and
it will bc found that thc systcm of équations (5) of § 120 may a.11
be satisned by taking
whcrc is a comptex constant dctemuucd by a quadratic cqua-
tn)!i. Thc result for a. cuntmuous string )nn-y bc afterwards de-
duecd.
CIIAPTER VII.
LONGITUDINAL AND TORSIONAL VIBRATIONS 0F BARS.
1-tJ. Tin, next sysL.m to thé string in order ofsimnjicityis tlie bar, by winch terni is
usuaUy undcrstood in Acoustics a
mass
of natter of uniform substance and c)ongatud cytindricalorm. At t!ic c..ds thu cylinder is eut oH' by p]anes pcrpcndicuJarto tliegc~-atu~ lincs. Tho centres of u.c.-tia of t).e tmnsvcrse
sections lie ou a stra.ght ]inc whic!. is calk-d t!.e
Thc vibrations <-t- a bar arc of throekind.s-iongitndina!
~T~'
the~-rorJ:but at t.c same time the most diHicu)t in
tt.cory. Tbcy areconsidered by thc.n.sctvcs m thc next chapter, and will on]y bcrcferrcd to hcrcsofarasis
ncce.ssaryfor co~parison and contrastW)th thc othcr two ktnd.s of vibrations.
Long.tndu.at votions arc those in which thc axis romainsnnmoved. whde t)~ transversc sections vibrato to and fro in thedirection pcrpendieuL-u. to their planes. Thc moving powcr istho r~stancc o~red by thc rod to extension or compression.
OucpccuH~ityofthIs class of vibrations I.s at once évidentSince the force
neccssary to produce agiven extension in a bar
is proportional to tho area of the section. ~hHe thé ,na.ss to bemoved a!so in the same proportion, it fo)!ows t!mt for a bar ofgiven length and
inatcrial ti.epcriodic tunes and the modes ofvibration arc ~dépendent of thé area and of tlie for.n of thétraverse
sect.on. A .sinufar law obtain.s, as we shaUprcsentty-sce, in tite case ot torsionat vibrations.
Itisothcrwiscwhen the vibrations arc latéral. Thc pcriodictunes are mdecd i.~ependent of t!.e thickness of tbc bar in thédirection perpendicular to ~o plane ofuexurc. but the motive power
14!).] CLASSIFICATION OF VIBRATIONS. 189
in this cttse, viz. tlie résistance to bcnding, incrcases more rapidiy
th~i the thickness in that plane, and therefore an incr~lae in
tinckuess ]s accompa.uicd by n risc of pitch.
In thc case of Iongi.tudiun.1 and latéral vibradons, ttic mcchan-
ical consta.uts coticcrncd a.rc thc dcnsit.y of thc m~terud nad tho
v:due ofYoung's tnodulus. For sm:d) extensions (or compressions)
Hookc's Ia.w, according to w!dch thc tension v:n'ics a.s thù extension,
Tf ,i aetnid Icngth nntural ]ongthhoids good. If tLe extension, viz. -n ",i–°
nittundtengtn 1
bc callecl c, we liave y=~, whet'o isYoung's nioduius, and T
is thé tension per unit m'en,ncccssary
to producc thc extension e.
Young's moJnhis maythercforc be dcancj as tlie force whieh would
ha-ve to bc appHcd to a bar of unit section, in oi'dcr to doub]c its
length, if Hooke's law contiuncd to hold good for so grea.t exten-
sions; its dimensions are a.ccol'd.ing~y those of a force divided by an
area.
The torsional vibrations depend aiso on a second clastic con-
stant IL, whose interprétation will be considered in the proper
place.
Although in tlleory the threc classes of vibrations, depcnding
respectively on résistance to extension, to torsion, and to ncxurc
are quitc distinct, and independent of one another so long as thé
squares of the strains may be neglectcd, yct in actual expérimenta
with bars which are ncititer uuiform in matcria). nor accuratcly
cyliudrical in figure it is often found Impossible to excite longi-
tudinal or torsional vibrations withont tlie accompaniment of some
measure of latéral motion. In bars of ordinnry dimensions tbû
gravest lateral motion is far graver than tbo gravest. longitudinal
or torsional motion, and consequently it will generally happcn that
thc principal tonc of either of thé latter kinds agrées more or less
perfectly in pitch witli some overtone of thé former kind. Under
such circumstances thc rcgidar modes of vibrations becomc
uustabic, and a small irregularity may prcduce a great effect, Thc
dimculty of exciting purely longitudinal vibrations in a bar is
similar to that of getting a string to vibrato in one plane.
With this explanation we may proceed to consider tbe threc
classes of vibrations independently, cominencingwith longitudinal
vibrations, which will in fact raise no mathematical questions
beyond those aiready clisposecl of in thé previous chapters.
190 LONGITUDINAL VIBRATIONS 0F BARS.D.50.
150. Whcn a rod is stretchcd by a force parallel to its tcngth,the stretching is in général accompanied by latéral contraction insu~-h a manner thaï. thé ~<) of v<Juiuo I~ss than ifthé déplacement of cvcry particlc wcrc paraHc! to thc axis. In thecase of a short rod andof a partiel situated ncar tlic cyliudrical
bonndary, this L~tcral motion would bucomp~ble iu
jn~nitudcwith thu longitudinat motion, and eou]d not bc ovorlookcd withoutrisk of considérable crror. But where a rod, whosc Io)gth is grc~tin proportion to tho lincM- dimensions of its.section, is subjcct toa
strctching of onc sigu tliroughont, t)ic longitudinal motion accu-mulâtes, and thus In thé caso of ordinary rods
vibratmg lon.d-
tu()in:d!y in thc graver modes, thc inertia of thc hter~motionmay bc negicctcd. Morcover wo shall sco lator how a correction
may bc introducud, ifnecessa.ry.
Lct bc thé distance of théhyer of
particles composino- anysection from thc cqnHibrium position of onc end, whc.i thc~-od isunstrctchcd, cithcr
by pcrtnancnt tension or as thc rusuit ofvibrations, an<I !ct bc t]ie <Hsp]accment, so that thé actuat
position is givcn by + T)~eqnihbrium and actuai
position
of a, nuighbounng laycr bcing a;+~~-+~+~+~~
rc-f~
spoctivcly, tl.c e~~ is und thus, if T be thc tension per
unit arca, acting across thc section,
Considcr now the forcesacting on thc s)ice boundcd by a:
and + 8~. If tl.o arca of tho suetion bu the tension at .c is
by (1)y~ actiug In thc négative direction, and at a:+~
tho tension is
~~+~ J
acting in thc positive direction; and thus thé force on the sHccdue to thé action of thé adjoining parts is on Die whutc
Tho mass ofthc clément is If p bc tl.e original densityand thcreforc if ~be titc acce!erating force acting on it.thc equa-~
150.] GENERALDIFFERENTIALEQUATION. 191
tion of oquilibrium is
In what foHows wc shaH not rcquirc to cojtsidcr thc opération
of:ui itnprcsscdforce. To find thé équation of motion wc hâve
ouly to replace by thé réaction ~gaiust accélération aud
thus if p =a~, wo hâve
Tins équation is of thé same form as tbat applicable to tho
transverse displacuments of a, strctched string, an<) itidicatcs thc
undistnrbud propagation of waves ofany type
in ttie positive and
négativedirections. Ttie velocity <t is rotative to thc UH~'e<c/<er/
condition of thc bar; thc apparent vuloeity witb which a disturb-
n.uce is propugatud lu spacc wilt bc gruatcr in thé ratio of thé
strctched aud uustrctched Icngths of any portion of thc bar. Tho
distinction is matcrial oniy in t!ic case ofpermanent
tension.
151. For tho actual magnitude of thc vclocity of propagation,
wc liavc
f~ = </ p = ~M ~)M,
which is the ratio of the wholo tension necessary (according to
Hoo~c's law) to double thc length of thé bar and t)t0 longitudinal
density. If tho samc bar wcrc strctchfd wit)t total tension T,
and wcrG ncxibic, thé velocity of propagation of wavcs alung it
would hc ~/( 2' /3M). In order titcn that thc vclocity inight bo
thc 8:nnu in titc two cases, Tmust Le ~M, or, in othcr words, tlic
tension would hâve to bc t)iat thcorcticaUy nccessary in ordcr to
double tho Icngth. Thc toncs of longitudinaUy vibrating rods
arc thus very high in comparison with tilose obtainable froin
strings ofcompiu'abtc Icngttt.
In titc case of stcel thc viduc of q is about 22 x 10" grammes
weight pcr s<p)arecentimètre. To express this In absotnte utnt.s
of force on thc c. f!. S.' systmn, wc ninst mnttipty by 9SU. In
thc same syston thc dcnsit.y of stcct (Identical witb its spécifie
gravity rcferrcd to water) is 7'8. J~-nec fur steel
1Centimètre, Gramme, Second. Tliis System is recommeuded by n. Coinnutteo
of tboDritiBliAasociatiûu. Brit. Ase. Report, 1873.
102LONGITUDINAL VIBRATIONS OJ BARS.
fiS].
velocity of -~el is
~ut. 0,000cent.n.etre.sper second, or about 1G ti~es grc.ter
L "TLhc samc as in .stec].
It ought to bc nicntioncd th<it in strictnoss t.hc valuenf dctcr-
minedby.statua! expc.ri.ncnta is not that wiuch o~ht tu be ~scdhère As in thé
c~c
ofga.cs, .vbi.~ will bc. trcntod lu a
.subscq~ntchaptcr, thc mp,d altcration.s of state co.~crnc-d iu t)~
pr.p~-tion ofsounJ arc étende.! witli ther,~ e~cts, onc rusul~of
~nch
to n~e thc active va!ne cf bcyond tl.at obtaluod
from cbscrvat.~s
on c.xtcn~. co.luct.cd at a constant tc,np~turc But tho d.ta arc notprécise enoug]~
to m..d<c this con-cctionci any consGqucncc ni tlie c~sc of solids.
v~ '<~tud;na]vibrations ofau uniimited bar, n:unc]y
~=7(~-a<)+~(~+~),
bcing the same as t~t appHc~Ie to a string, need not be furti~reonsidered hcre.
Whcn both ends of a bar are fre~ titere is of course no pcrmi-nent tcns.on, and at the ends the.n.sdvc.s titerc is notcn~rvtension.
Thoconditiouforaf.-cec.ndisthcrefore
~=0.le doter~nc t)~ nor.ual n,odc.s of vibration, wo must assume
th~t~vanes as a harmonie function of tho timc-cos7i~ TI~n
as a function of .r, ,nustsatisfy
Nowsinco~vanishcs al~ys ~hcn ~=0, we get j3=0; an<!
again smce
gvanishcs ~~cn ~=/-thc u.turat iungth of tlic
bar, sin ~~=0, wbich sl~cws tih.Lt is oftiie form
t'bcingiutc'graj.
152.] BOTH EXTREMITIES FREE. 193
Accordingty, the normal modes arc given by équations of thc
form
in which of course auurbitrury constant may bc nddcd to < if
<!esh'cd.
Thc complete solution for Il bar with both ends frce is thcro-forc cxprcssed by
whcrc and arc arbitrary constants, wliiehmay bc detcrmincd
in the usu:d mauncr, whun thc Iuiti:d values of aud arc
givcu.
A zcro vainc of i is admissible it gives a termrcprcsentmg a.
dispIn.ecmGnt constant with respect both to spn.cc and tuno,aud amounting in fact only to an altération of the origin.
TIic period of the gravcst component in (6) corresponding to
t=I, is 2~ which is thctinlc occuhied by a. disturhanee in
travelling twice the Icngth of t)io rod. The other toncs fonnd
Ly ascribing integral values to i form a complète harmonie scale iso that according to tliis theory tl)c note givcn hy a rod in
longitudinal vibration would bc in aU ca~cs muslca.1.
In thc gravest mode thc centre of the rod, whcro /c= is a
place of no motion, or nodc; but thc periodic elon~ation or com-
f~pression is thcrc a maximum.
153. The case ofa bar with onc end frec and the other fixed
may be deduecd from thé gcncral solution for a bar with both
ends froc, and of twice the Icngth. For whatever !rmy be ttie
initial statc of thc bar froc ut .B=0 n,nd ftxcd at x = l, sucii dis-
placements a.ud velocitius ma.y a.!ways bo ascribed to the sections
ofabarextending from 0 to 2~ and frce a.t both ends as shaH
make thc motions of thé parts from 0 to Identical in thé two
cases. It is only ncccssary to suppose that from to 2~ the dis-
placements and vclocitics arc initially cqual and opposite to thosc
found in thc portion from 0 to at an cqnal distance from thc
ccutre x = Uiidcr thcsc circumstanccs tho centre must bytl)c
symmr-tryrcm!).in at rest throughout t)ie motion, and thcn thc
R. 13
194LONGITUDINAL VIBRATIONS 0F BARS.
H 5 3.
portion from 0 to s~tisnes all tl.e required conditions. We con-c udc that the vibrations of a bar frce at one end and fixed at thoothcr arc ~dcnfj.d ~L .h. of bar uf twice thelongth of which both ends arc free, thc latter
vibrating only nunevcn modes, obtained by ~king in succession a))~ i~~T!~ tones of tlie bar stillbelong te a hannomc ~.h, b~'cvcn toncs (octave, &c. of thé
fund.menta!) are.vauting.
Thé period of the gravest tone is thé timeoccupicd by a pulsein travelling/b~ timcs the length of thé bar.
154. ~)cn both ends of a bar are fixed, the con(litions tobe satisfied that the value~t~At K-0, we may suppose that ~=0 At
& is a small 1constant .h is zero if j no permanent tension. "'1:"dep.n.).nt)y of' th. vibrations w.),av.
~idcnt)y f~T<we should obtain our result mostsimply bt~M' this tenuat once. But it
maymethod.
Assumingthat as a fonction ofthetimc varies as
~COS7!f~ + 7?sinK~,
we sec that as a function of x it mustsatisfy
of which the général solution is
JBut
since vanishcs with x for ~11 values of t, r n ~)we may write<. C'=0, nnd thus
154.] BOTH EXTREMITIES FIXED. 1!)5
The series of tones form a, compote harmonie scalc(ft-on
which ])owcvcr any of the mcmbo-smay bo
mi.ssiugni
any
actua! case of vibration), and tho period of the gravest com-
poncnt is the tinic takcn by a pulse to travc! twice tho Icnn-t.bof thé rod, thc sa)nc thcroforo as if both ends wcre frec. Itnnist be observed thnt we hâve bore to do with thc MH~r~c~~
length of thé rod, and that thé period for a givcn natural lengthis ludependent of the permanent tension.
The solution of the problcm of the doubly fixed bar in the
case of nopermanent tension
might aiso bc derived from t)iat
of a doubly free bar by mcrc ditfcrfmtiation with respect to .c.
For in thc latter problem satisfics thenecessary diËfereutial
équation, viz.
(I' d
= a2(le~.E~ <
masmuch ns satis~cs
and at both ends vanishcs.According!y iu this problem(lx dx
satisfies ail tl~e conditions prcscribed for in the caso whenboth ends arc ~xcd. The two séries of toncs are thus identicul.
155. Thc effect of a small ioad ~f attac])cd to any point ofthe rod is rcadi]y ca!cu)ated
approxnnatc!y, as it is sufncientto assume thc type of vibration to bc uuaitcrcd (§ 88). \Vcwill takc the case of a rod nxed at .~=0, and free at .t'= The
kinetic cncrgy is proportinnal to
or to
~G LONGITUDINAL VIBRATIONS OFDARS. [155.
Since the potentia.1 cncrgy is uudtcred, we sec by t!ic prm-ciples of Chapter iv., th~t tho cfrcct of ti~) sma!) !o~ at a'h~ucc &: u'u:ti Lhe iixcd cud is to inci-cMC the period of' tho
compouent toucs in thc mtiu
Tho snrnHquantity p~ is thc ratio of tlic !o:td to t!ic
wholo mass of thé rod.
Iftheload bcatt~chcd at thc frec end,sm'~=l,
and tlic
effect is to dcprcss the piteli of cvery tone by t!ic s~mc smallmtcn'd. It will bc rcmembei-ud t!)at i is hei-c an MMC~t mtcgcr.
If the point of .~chmcut of J!f bo nodc of any componcnt,tlie pitch of that eojupouont ronams uualtcrcd by thé addition.
150. Another problem worth notice occurs whcn t]to load ~tthc frcc end is grc~t iu
compiu-isoi with thc masa of thc rod.In th)3 CMC wo
inay assume as thc type of vibration, a. conditionof Utntbrm extension along tlic Icngth of the rod.
If bc tttc displaccmcnt of thc load 3/, tho kinetic cncrgy is
The correction duc to the incrtia of the rod is thus cquivalentto t)ic édition to ~ofone-third of thc mass of the rod.
1.~7. Our mathem~tic~ discussion ofJongitudinfLl vibrations
nmy close with an estimatc of thc cn-or invo!vcd in ncgieetinrrtttc latc.ral n.ut.ou of thc parts of thé rod net situatcd on tlic
157.] CORRECTION FOR LATERAL MOTION. 197
~xis. If the ratio of latéral contraction to longitudinni cxtoisinndenutcd bj. i,inj huerai disp~acoment of a. particie distant
?- from thc axis will bc ~re, in the case of cquilibnum, where e isthe cxtcMsion. Altiiougli in strictiless this relation will bc modi-iicd by tho Incrtia of thc httcl-al motion, yct for thc présent pur-pose it may bc supposed to hold good.
TIic constant /t is uumericalquantity, lying between 0 and
If/~worc ucgativc, a longitudinal tension would produce a latéral
swelting, and if were greater tbau tlie lateral contractionwould bc grcat oicugh to ovcrbalance thé elongatiou, and causea diminution of volume on the whole. Thé latter statc of thin~would be mconsistclit witli stability, aud tlie former can scarccîybe possible in ordinary solids. At one time it was supposedthat was nccessarily equal to so tliat thcro was only one
independoiit clastic constant, but oxperimcuts have since shcwu.
that is variable. For glass and brasa Wcrthchn found expcri-nicutally /t =
If dénote tlie lateral displacement of thc particlc distant rfrom the axis, and if thc section bo circular, thé kinetic encrgyduc to t]ie lateral motion is
Thc effect of tlie incrtia of thé latcra! motion is thct'cfoi-c <oInercasc the poriod m thc ratio
This correction will bc nearly insensible for tlie graver modes ofbars of
oi-dinary proportions of length to thickness.
198 LONGITUDINAL VIBRATIONS 0F BARS.[158.
158. Expérimenta on longitudinal vibrations may be made
v.'itii icds of dc~l ûr ot' g!a.~s. Tho vibmtbns arc cxcitcdby
friction, with a wet doth in the case of glass; but for métal or
wooden rods it is neccssa.ry to use Icather charged with powdered
rosin. "T!tc longitudinal vibrations of a pianofortu string may bc
cxcited by gcnt)y rubbing it longitudinaUy wIHi a piece of india
rubber, and those of a violin string by p!a.cing the bow obliqucJyacross the string, and moving it aiong thc string loogitndina.Hy,
kecping tire same point of thc bow unon thé strirtg. Thc note is
unpjcasn.ntiy sin'ill in bot!i ca.sus."
"If t]te peg of thc vioini bc turncd so as to attc't' thc pitcb of
thé lateral vibrations vcry considcrabty, it will be found tba.t t))u
pitch uf' thé ]ongitudina.i vibrations )ias :dt(ired vcry shghtty. Tim
rca~on uf this is tha.t in thc case of t)tc lateral vibrations thc
ehtuigc of vclocity of wavc-transmission dépends cbicny on t)io0
change of tension, which is considérable. But in thc case of thc
longitudinal vibrations, thc change of vclocity of wavc-transniis-
sion dépends upon thé change of extension, which is comparativcfy
sligttt'
In Savart's expci'Imcnts on longitudinal vibrations, a, peculia.r
sound, calted hyhim a "son rauque," was occasionaMyobservcd,whosc pitcii was an octave below tl)at of tbc longitudinal vibra-
tion. According to Terquem" thc cause of this sound is a trans-
verse vibration, whuse appcarance is due to an approximatc
agrecmcntbetwee)i Itsown pcriod and that of the sub-octave of thc
longitudinal vibration. If this view be correct, the phenomenonwuld be one of thé second order, prubabiy referable to the fact
that longitudinal compression of a bar tends to produce curvature.
15!). Thc second class of vibrations, ca)ted torsional, whic!i
dépend on t!te résistance opposed to twisting, is of very small
importance. A solid or hoi)ow eylindricat rod of circular section
may be twistcd by suitable forces, applied at the cuds, in suctt a
nianuer that cach transverse section remains in its own plane.But if thc section be not circular, thé cneet of a twist is of a
]norc compticated cliaractcr, the twist being necessarUy attendcd
by a warping of thé layers of matter originally composing tho
nornud sections. Altijough tho enccts of thé warping might pro-
Doukin'H ~c')t«~t'M, p. ~i.
~'«f. C'Anott-, Lvn. 12U–1!)U.
159.] TORSIONAL VIBRATIONS. 199
bably be dctermiucd in any particular case if it wero worth
'.vhi!c, v~jh:I c~nnti. our~ivc~ iicrc t,o dm c~e ut' c~-cutai
section, wheu there is uo motion pa.r:~tel to the axis ofthe rod.
Thé force with which twisting is resisted depends upon anclitstic constant different from q, ca.Hed thé rigidity. If we de-note it by n, tlie relation between q, m, a.nd may be written
shewing that n lies betwcen and In the case of ~=~M=~.
Lot us now suppose that we hâve to do with a. rod in the formof a thin tube of ra.dius r a.ud thickness ~r, and Ict dénote t]io
angular displacement of any section, distant a: from the origin.
Thc rate of twist at a: is reprcsentcd by and thé shear of the
materialcomposing the pipe by
r~.The opposing force per
umtof areais~
and since thc area is 27n-~ the moment
round the axis is
Since this is independent of r, the same equation appUca to a
cyliader of fmitc thickness or to one solld throughout.
The velocity of wa,ve propagiition la
A/and the wholo thoory
is prccisely similar to that of longitudmal vibrations, the condition
TIiornson aod Tait. § 683. This, it ahould bo remarkcJ. applies to inotropicmntcria! on]y.
LONGITUDINAL VIBRATIONS 0F BARS. [159.200
7/]for a free end bcing = 0, and for a fixed end =
0, or, if a'C
permanent twist be coutempla.tcd, = constant.
The vclocity of longitudinal vibrations is to tliat of torsional
vibrations in tlie ratio or ~/(3 + 2~) I. Thé samc ratio
applics to the frcqucncics of vibration for bars of cqna! Icngthvibra.ting in
corresponding modes undercorrcsponding terminât
conditions. If == the ratio of frequencies would bc
:=~/8 :3=1'G3,
correspond ing to an interval ratitcr grcatcr than a nftb.
In any case tbc ratio of frcqucncics must lie between
V2 1 = 1-414, aud ~/3 1 = 1-732.
Longitudinal and torsional vibrations were nrstinvcsti"-atcd bvCbladni.
CHAPTER VIII.
LATERAL VIBRATIONS 0F BARS.
160. IN tho present chapter wc sliall consider the lateral
vibrations of thin ctastic rods, which in thcir natural condition arc
straight. Next to those ofstrings,
this class of vibrations is per-
haps tlie most amenable to thcoretical and expérimental treatment.
Thcre is dimculty sufncieut to bring into prommenco somc im-
portant points connected with thé gênerai theory, which thé fami-
liarity of thé reader with circular functions may lead him to pass
over too Hghtiy in thé application to strings; while at the same
time the difficulties ofanalysis arc not such as to engross attention
which should be devoted to general matliematical and physical
principles.
Daniel Bernoulli' scems to have been tlio first who attae~ed
thé problem. Euler, Riccati, Poisson, Cauchy, a.nd more reccntly
Strehiko", Lissajous", a.nd A. Scebeck~ arc foremost among thoso
who have advanced our knowledge of it.
161. Thc problem divides itsolf into two parts, according to
the presence, or absence, of a permanent longitudinal tension.
Thc considération of permanent tension entails additional compli-
cation, and is of interest only in its application to stretchcd
strings, whose stiffiiess, though small, cannot bc neglecteù al-
together. Our attention will therefore bc given principally to the
two extrême cases, (1) whcn there is no permanent tension,
(2) when the tension is thc chief agent in the vibration.
C'oHUMn<./<M< J'<'<r~). t. xnt. rogg. ~;t)!. Bd. xxvu.
~;)~. f!. Chimie (H), xxx. !}85.
~h/«!))~~f~<'M d. ~/<!< J'/ty~. Classe fL /C..S'<M'/«. CMC~M/t~/t d. !rf«j!C;t-
sc/t<fc)t. Leipzig, 1852.
LATERAL VIBRATIONS 0F BARS.[161.
202
WIth respect to thc section of thc rod, wc sha)! suppose thatone principal axis lies m thc phuic of vibration, so that t!ic bendino-at cvcry part takcs p!acc iu a direction of maximum or miuimm~
(01 st,atiûnary) rtcxunU ri~idity. For uxample, thc surface of thcrod may bc onc of révolution, cach section bdug circular, thoun-hnot ncccssarily of constant radius. Under t!icsc circumstances thc
potentiaJ cncrgy of thé bending for each clément of lungth is pro-portional to the square of thc curvaturc multiplied by a qnantitydcpcnding on thc matcriid of t))e rod, and on thc moment ofinertia of thc transvcrsc section about an axis
t))rough its centre ofinertia pcrpeudicuhu- to tlie plane of bending. Jf be thc areaofthe section, its tnomcut of
inertia,~ Young's moduius,~ théclonent of icugth, and ~F' t)ic
corrcspouding poteutial energy fora curvature 1 of tlie axis of the rod,
This resuit is readily obtained by coDsidermg the extension ofthé varions filaments of whicli the bar may Le supposed to bomade up. Lot be tlie distance from the axis of thc projectionon thé piano of bending of a nl&ment of section ~M. TIien thc
length of the niament is altered by the bending in thé ratio
-K being thé radius of curvature. Thus on thé side of thc axis forwhidi~ is positive, viz. on thé o~c~ side, a filament is extended,while on thc other side of thé axis there is compression. Tho
force necessary to produce thé extensionis (~ by the deûiii-
tion of Young's modulus; and thus thé whole couple by which thé
bending is resisted amounts to
if &) bc thé area of thé section and < its radius of gyration abouta Imc through tlie axis, and perpendicular to the plane of bending.The angle of bcuding corresponding to a length of axis ds isaud thus the work rcquired to bend o~ to curvature 1 Ti! !qc
~t
siucc thé Mea?; is hdfthc~~ value of tlie couple.
161.] POTENTIAL ENERGY OF BENDINQ. 203
For a circular section ? is onc-ha.If t!te radius.
Th~t thé potential Ct'icrgyof thcbcndingwoutd bc proportionn.1,
cc~e?'tN ~ft~tfs, to the squ:).ro of thc cut-Yidure, is évident bcfore-
hand. If wc en!! tho couificiotit J9, wo may tako
in which y is tlie lateral dispiffcmcnt of tliat point on thc axis of
thc rod w!iosc abseissa, mc'asurcd paraltel to thé undisturbed posi-
tion, is x. In thé case of a rod whose sections arc similar and
siniiladysituated 7~ is a constaiit, and may bc removed from under
the intégral sign.
Tho kinetic cncrgy of thc moving rod is derived partly from
tlie motion of translation, parallel to of thé éléments composing
it, and partiy from tlie rotation of thé same elements about axes
through thcir centres of inertia perpendicular to thé plane of vibra-
tion. Thé former part is expressed by
if p dénote the volume-Jensity. To express the latter part, we hâve
only to observe that thé angula.]' displacement of thc élément dx is
",a.ndtherefore its angu!a.r vclocity Thé square of this
(~uantity must bc multiplied by haïf t!ie moment of inertia. of tho
clement, tliat is, by ~m < We thus obtain
1G2. In ordcr to form thc equation of motion we may avail
ourselvcs of tlie principle of virtual velocities. If for simplicity we
confine ourscivcs to the case of uniform section, we have
LATERAL VIBRATIONS 0F BARS. [162.204
where thé terms free from the Intégrât sign are to be t~cn hetwœuthé I.nuts. This expression inctu.ics ouiy t])e internat forces dueto tlie ben<hng. In what futlowa ~û sh.U! .s..pposc ti.at there areno forces Mting from wltlinut, or ra-thcr none that <)o work upont))c System. A force of
con.stramt, suc)i as th!itncccssary to ]iotd
any pu.nt of the hn.r at rc.st, need not bc rcgn.rded, it do~ nowork and therctore cannot appcar in t!~c équation of virtual veio-ettics.
Thc virtual moment of tlie accélérations is
Thus tlie variational équation of motion is
in which thé tcrms free from the mte~-al sign arc to h.. takcnbetween thé limits. From this we Jer-
~edatallpo~ofthel~t~f~~
..J~.Jongi tud inl11 lVilVl'S.
1G2.] ] TERMINAL CONDITIONS. 205
Thc condition (5) to be satisfied at thc ends assumes different
forma according to thc circumstances of tlic case. It is possible to
conceive a constraint of such a nature that thc ratio 8 ( ") 8v has
a prescribed finitc value. Thé second boundary condition is then
obtained from (5) by introduction of this ratio. But in aH the
cases that we shaH hâve to consider, there is either no constraint
or thé constraint is such that eithcr 8[-")
or Sy vanishes, and
thon thé boundary conditions take the form
We must now distinguish the special cases that may arise. If
an end be frcc, 8y and S( ~)
are both arbitrary, and the eonditiona
becomc
the first of which may bc regarded as expressing that no couple
acts at thc frec end, and tlie second that no force acts.
If thé direction at thé end be frec, but the end itself he con-
strained to romain at rcst by the action of an applied force of the
necessary magnitude, in which case for want of a botter word the
rod is said to be supported, thé conditions are
by which (5) is satisfied.
A third case crises w!)cu an cxtrcmlty is constramcJ to main-
tttin its direction by a.n applied couple of the necessary magnitude,
but is free to take any position. We ha-vc thcn
Fourth)y, thc extrcnuty may bc constrained both as to
position and direction, in which case thc rod is said to be c~n~ec~.
Thc conditions arc plainly
LATERAL VIBRATIONS OF DARS. [162.206
Of these four cases thé first and last are the more important
tlie third we shall omit to consider, as there are no expérimentât
means by which the contomplated constraint could bc rcaHzed.
Even with tins simplification a considérable varicty of problems
romain for discussion, as cither end of thc bar may bc frco,
clamped or supportcd, but the complication thencc arising is not
so groat as might have hccn expected. We shaH find that
difforent cases may be treated togethcr, and that thc solution
for onc case may sometimes bc dcrivcd immediately from that of
another.
In cxperimcnting on thc vibrations of bars, thc condition
for a clamped end may bu rcahzcd with thc aid of a vice of
massive construction. In thc case of a frec end thero is of course
no difilculty so far as thc end itself is concerned but, whcn both
ends are free, a question arises as to how thé weight of the bar
is to be supportcd. In order to Interfère with the vibration
as little as possible, thé supports must be connned to thé ncigh-bom'hood of thé nodal points. It is sometimcs surHcicnt mcrelyto !ay thé 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 wou!d perliaps bc prcferabJc to carry thé weight of
thé bar on a pin travcrsing a holc driHed through thé middie of
thé thickness in thc plane of vibration.
Whcn an end is to ba 'supported,' it may be pressed into
contact with a fixed plate whoso plane is perpendicular to the
longth of the bar.
1G3. Before procccding fnrthcr we shall introducc a sup-
position, which will greatly simplify thc analysis, without set-iolisly
intcrfcring with thé value of tlie solution. We sliall assume that
thé terms depending on théanguhu' motion of the sections of
thé bar may be neglected, which amounts to supposing the
tHer~ of' each section conccntratcd nt its centre. We shall
afterwards (§ 180) investigate a correction for thé rotatory in-
ertia, and shall provo that under ordinary circumstances it is
émail. Tho équation of motion now becomcs
')M.
163.] HARMONIC VIBRATIONS. 207
Thé next step in conformity with thé gênera,! plan will be
thc assumption of the harmonie form ofy. We may convenicntly
take
where l is the Icngth of thé ba.r, and w is a.n abstract number,
whose value ha.s to be deternimed. Substituting ni (1), wo
obtain
p"*If M == e be a, solution, we see that p Is one of tlie fourth
roots of unity, viz. +1, –1, +t, –t; so that the complète
solution is
containing four arbitrary constants.
Wc have still to satisfy thc four boundary conditions,-two
for each end. These detcrmino thé ratios A (7 -D, and
furnish besides an equation whieh '?~ must satisfy. Thus a series
of particular values of w a.rc alone admissible, a.nd for cach ?~
thé coiTcsponding ic is determincd in everything except a constant
multiplier. Wc shall distinguish the different functions u be-
longiug to the sa.mcsystcm by suffixes.
Thc value of y at any time may bc cxpanded in a series of
the functions (§§ 92, 03). If < &c. be tho normal co-
ordiDates, we have
and
We arc fully justified in asserting n.t this stage that each
intcgrated product of the functions vanishes, and tl]crcforc thé
process of thé followiiic, section need not bc regarded as more
than a Mr(/?ce[<t'o?t. It is however rcquircd in order to determine
thé value ofthc intcgra.ted squares.
LATERAL VIBRATIONS 0F BARS. [164.308
IC't. Lot M, <)cnotc two of thc nonna.1 functious cor-
respOQding respectivciy to ?~ and ?/ Thcn
If wc snbtmct cqn:ttiot)S (2) aftcr multiplyiog tl)cm hy M~
rcspcctivcly, and thcn ixtugmtc over thc lungHi uf thc biu',
we!ia.vc
the Intcgratod tcrms bcing takcn bctwecn tlie limits.
Now whcthcr thc end in question bo cla.mpcd, supportctt, or
free', eacli terni vanishcs on account of one or other of its
factors. We may therefore couchtdc that, if M~, ?< rcfcr to two
modes of vibration (corrcsponding of course to thé same terminal
conditions) of -winch a rod is capable, then
providcd ?)t and Mt' bc (Useront.
The attentive rcader will perçoive that in theproccss mst
foiïowcd, we ha.vc in fact rctraecd thcstops by w)nch t))c fnnda-
mcntd diiïcrcntialéquation
was itsctfprovcd iu
§ 1G2. It is the
Tho ronder ahonitl obscrvn t)nit tho eftscs hcro Rpoeificd MG pa.rticuJn.r andVthnt tho right-hand monbt'r of (;!) Vtuushcs, provided t.)t)tt
<~
~j ~m-<t:Llll
<~ <<j: ~.E f/
Thoso conditions incindo, for ittstfmco, tho Ctlao of n rod whofiO end is urpodtowm'Js its position of L'quili)')-iu)u Ly n terce pr~portional to thc dispUtcc'ment, as
by n spring witimut inertia.
164.] CONJUGATE PROPERTY. 209
originat MM'M!<MK<ïi!cquatio!) that ha.s the most nnmcdin.te con-
ncctiun with tho conjug~tc propcrty. If we dcnotc by M aud Sy
by~.
and this proof is cvidentiy as direct and général as cou)d bc du-
sired.
TIie reader may investigate the formula corresponding to (6),
whcn thé term l'cprescnting tlie rotatory inertie is retnined.
By !ne!ms of (G) we m!iy verify thn.t tlie admissible varies of n2
aro rca.1. For if 7~ were complex, and 1t = a + !3 were a normal
function, thcn a i,8, thé conjugatc of u, would bc a normal
fonction also, corresponding to tlie conjugate of ?~, and thon tlie
product of the two functions, being a. sum of squares, would not
vanish, whcn Ititcgratcd
If in (3) w. and ?~' hc the samc, thc équation hecotncs Idcn-
tica.lly truc, and wc cannot at once Infcr the value ofn~
This mcthofl is, I bdicvo, t!no to roisson.
R. 14
LATERALVIBRATIONS0F BARS. [164.210
We must take ?~' equal to M + 8w, and trace thé limiting form of
thé equation as 87~ tends to vanish. In this way we find
betwecn the limits,
Now whether an end be clamped, supported, or free,
M~"=0, ~V=0,
and thus, if we take the origin ofa; at one end of the rod,
==~(~-2~V+~),(8).
Thé form of our integral is independent of thé terminal condi-
tion at x =0. If thé end œ= b& free, M" and u"' vanish, and ac-
cordingly
that is to say, for a rod with one end free the me~n value of u' is
one-fourth of tbe terminal value, and that whether the other end
l)e clamped, supported, or free.
1G4.] VALUES 0F INTEGRA/FED SQUARES. 211
Ag-a.in, if wc suppose that thé rod ia c){nnpcd ~t == n.nd M
vanisL, a)id(8) givc's
Since tins must huld good whatevcr be the termina.l condition n.t
the other end, wc sce tliat for roJ, one end of which is fixed and
theot.itcrfree,
shewing thnt in t)iis case M' at the frec end is the samc as M"' a.t
thc c!ampe() end.
TIiea.!i!)cxed
table gives t)ic vahies of four times thé mea.n of M*
in thc différent cases.
c!tunped,frpf.M°(ft'ccend),ot'M"'(cIu.mpedend)
free,ft'eo M'(ft-Rccnd)
clf~mped, c]!i)npcd M' (clampcd end)
supportcd, supported 2~ (supportcd end) = 2~"
supported, ft-eo M" (freo end), or -2M'M'" (supported end)
snpported, chmpL-d M"' (damped end), or 2M'M'" (supported end)
By thé introduction of these values thé expression for T
assumes a. simpler form. In thé case, for example, of a clamped-free or a frec-frec rod,
where the end <c=~is supposed to befrce.
165. A similar method may be applied to investigate thé
values ofjM"~c,
and In the derivation of equation (7) of the
preceding sectionnothing WM assumed beyond thé truth of thé
equation M""=M, and since this équation is equally true of anyof thé derived functions, we are at liberty to replace M by M' or u".
Thus
14–2
LATERAL VIBRATIONS OF BARS. [1G5.212
ta.Iœn between tlie limits, since tlie tcrm M' vanishcs in ail threo
c~es.
For Il frec-frec rod
for, as wc shall sec, thc values of ~M' must bc cqn~I and opposite
at thc two ends. WhcUtcr u bc positive or négative at a' =~, ')(t/
is positive.
For a rod which is clamped at a: = 0 n.nd free at = l
We ])a.vc ah'cadysccn that ~"=~ a.nd it. will appca.r (§ 173) t)):).t
M"'=–u/, so that
Il rcsult thf),t wc sliall have occusion to use latcr.
By n.pplying thc same équation to tlie cva.inn.tion of ~M' wc
find
sinco M'u" a.cd Mu'" vanish.
Comparing tins with (8) § 1G4-, wc sec that
whatcvcr tlie termina! conditions may hc.
Tho samc result maybc arnvud at more dirccMyby intcgmting
by pa.rts thé equation
1CG.] NORMAL EQUATIONS. 213
16G. We may now form thé expression for V in tcrms of tlie
normal co-ordina.tcs.
If thc fonctions :t bc thosc propcrto :). rod frce nt ~= t)us expres-
sion reducesto
In any case thé équations of motion arc of tlie form
and, since ~~t is by définition the work done by the Imprcsscd
furcus during tlie dispin.ccmcnt 8~
if YpM~<; Le thc lateral force actmg on thceictncnt of mass pax~c.If thcre be no impresscd forces, the cquatiou reduces to
~+
n.s wc know it ought to do.
1C7. Thé signidc~ucc of the réduction of the Intégrais
~(~te dcpoidcnco on thc terminal values of thc function aud
its dcnvativcs may be p!a.ccd in n clearcr light by thc foHowing
!me of fu'gmncnt. To fix tlie ideas, considur tho case of a
rod chunpcd at x=(), and free at A-=~ vibrating in the normal
mode cxpressed by u. If a sm:di addition A~ bu madc to the
rod at ttic frec end, thé form uf K (cons~ered a~ a function of
~) is ehanged, but, l!i accordaucc with thc gmicral principlG
CHtabii~hcd iti Chaptcr iV. (§ SH). wc ma.y calcntatc tlic period
LATERAL VIBRATIONS 0F BARS. [167.214
under thé altcred circumstances wititout aUowance for the change
of type, if we are content to ncglect tbe square of thcchange.
In conséquence of thé strajghtness of thc rod :).t thc place where
the addition is made, therc is no altération in thé potentiel
energy, and therefore the altoration uf period dépends eiitirelyon thé variation of ?'. This quantity ia incrcased in the ratio
which is also tlie ratio in wliicli tlie square of thc period is
augmcutcd. Now, as wc sliall suc preseutly, tlie actuat pcr!ot!varies as f, and thet-cforc the change in the square of tlie periodis in the ratio
A con)p:u'isou uf thé two ratius shcws tlint
M;'ft(W~
= -i.
TIie above rcasoning is not inalsted upon as a démonstration,but it serves at least to exptain tlie réduction of which ttie in-
tégral is susceptible. Other cases in winch sucli Intégra]~ occur
tnay be treated in a sunitar manncr, but it would often requirecare to predict with certainty what atnonnt of discontnunty in the
varicd type might be admitted without passiug out of the rangeuf the principle on which the argument dépends. The reader
may, if lie p)cases, examine tlie case of a string Iti the jniddic
of whieh a small piece is Ititerpolated.
168. In treating problems relating to vibrations t!)o usnal
course bas bceu to détermine in thé first place the forms of the
nornial functions, viz. the functions represeuting thé normal
types, and afterwards to investigate the intégral formuim by<neans of which thé particular solutions may be conibined to
huit arbitrary initial circumstances. 1 have prefen'ed to follow
a dinercnt ordcr, t)ie bcttcr to bring out the generality of thé
jnethud, w/~cA (/oes not depend M~o~ (t knowledge of the 7:o?'?~a~
yM/tc~'c~s. In pursuance of thé same plan, 1 shali now investigate
168.] INITIAL CONDITIONS. 315
the conncction of thé arbitrary constants with thé initial circum-
stiUlces, and solvc oue or two problems analogous to those treated
uiidcr thé head of Strings.
Thé gcnend value of~ ma.y be written
formuloe which détermine the arbitrary constants B,.
It must be observed that we do not need to prove analytically
thc possibility of thé expansion expressed by (1). If a~ the
particular solutions arc iucludcd, (1) necessarily represents thé
most general vibration possible, and may therefore be adapted
to represent any admissible initial state.
Let us now suppose that thé rod is originally at rest, in its
position of cquilibrium, and is set in motion by a blow which
imparts velocity to a small portion of it. lilitially, that is, at
thc moment whcn tlie rod becomes free, = 0, and differs from
zero only in thé ncighbourhood of one point (x =c).
From (4) it appeurs that the coeiHcients vanish, and from
(5) that
216 LATERAL VIBRATIONS 0F BARS. [1G8.
CiUling~~<u~,
thc w!)ule momcntum of tbc blow, Y, wc
ha.vc
If thc blow bc app)ied at a no(te of onc of thc normal com-
poncnts, tha.t conponcnt ismissing in thé rcsutting motion. Tlie
prcsunt ca.)cu!atiun is but a. pai'Licular c~c of thuinvestigation
uf§101.
ICf). ~a another examplG we may take the case of a bar,which is initially at rcst but dcHected from its natural positionhy a latéral force acting at .'c=c. Undor thèse circumstances
the coefficients B vanish, and tlie others arc given by (4), § 1G8.
Now
!)i which titû tenus frce from thc iutcgml sigii arc to 'bc takcn
bctwccu tlie inuits; by t))e nature of thé c:).sc satisdes tlie
169.] SPECIALCASES. 217
same tcrminal conditions <m docs and thus a.ll thèse tcrms
vanisli a.t both limits. If tlie external force initially applied
to thc .cicmeut bc yi~c, thé cq~a.tion of equilibrium ci' tlie
har tri vos
If wc nowsuppose
thn.t thc initia dispiaccmcntis duc to
a. force applicd in thc immédiate mjighbuurhuod ut' t))c punit
a; = c. wc tiave
a.nd for tlie complète value of y at time t,
In Jenving the above expression we have not hitherto made
any special assumptiunsas to the couttitions at thé ends, but
if we now confine curscivus to ttte case of fi ba.t' which is c!cnnpcd
at a; = 0 aud irec at x = l, ve may replace
Ifwc supposefurthcr that the force to whk-h thc Initial dcHcetio))
is duc acts at thé end, so that c= wc get
Whcn t=0, this cquadonnuist represent t)'c initiid dispjacc:-
tncnt. Iti cases of this Idnd di~culty tnay pi-cscnb itMu[f as
to !)0\v it is possible for the series, cvery terni of which satisfics
thé condition y"'= 0, to rcprcs~nt
au initial displacement Iti
which tins condition is violated. Thé iact is, that after triple
diH'crentiation wiHt respect to tlic series no longer converges
for a~, and accurdingly the value of y" is not to be ~rrived
at hy making the diHerentia.tioas first and summing the terms
LATERAL VIBRATIONS 0F BARS. [1G9.218
aftcrwards. Thé truth of tins Rta.tement will be a.ppa,rent if
<ve cousiJer a point distant dl from the end, and replace
For thé solution of tlic présent probtcm by normal co-ordiiiatcs
the reader is referrecl to § 101.
170. Thc forms of tlie normal functions ni the varions p!u'-ticutar cases arc to bc obtained by deterimuing thu ratios uf thc
four constants iu thc générât solution of
If for thc sako of brcvity be written for thé solution may
heputintothofonn
cosh x and sinh x arc tlie hypcrbollc cosine and sine of x, defined bythé équations
1 hâve foltowed thc usual notation, though thc introduction ofa special symbol might vcry weH be dispcnsed with, since
cosha'==cos/.c, sinha;=-.t'siu.(3)
w]~t-e t== y- 1, and then tI)G conncctioti between tlie formuh~ ofcircular and hypcrbo)ic tngttnoniet.ry wou)d Le moi-c apparent. Théruics for diiTurcntiation arc cxprcs.sed in ttic cquatiuus
In diiïcrentlating (1) any number oftimes, the same four com-
pound funetions as thcrc occur are contmuaUy reproduced. Thé
on!y one of them which does not vanish with is cos a;' + cosh ic,wbose value is thon 2.
170.1 ]NORMAL FUNCTIONS FOR FREE-FREE BAR. 219
Let us take ~h'st the case in which both cnJs are free. Sincc
This is the équation whose roots are tttc admissible values of ?/t.
If (7) be satisfied, tlie two ratios of C given in (5) ~rc e(tual,
:m(l either of titcm ma.ybe substituted ia (4-). The constaut multi-
plier bciug omittud, wc have for thc !iorm:d function
171. The frcqucncy of thé vibration is~M~,
in whiuh & is
a velocity dcpcnding only on the material of which thé bar is
formcd, a-nd Ht. is an abstract number. Hence for a given material
and mode of vibration tlie frcqucncy varies clirectly as K–thc
radius of gyration of tlie section about an axis perpendicular to thé
~0 LATERAL VIBRATIONS 0F DARS. fl~I.
plane ofbending–and iin'erseiy as thc square of the Jcngtit. Thcsc
rusults might hâve bccu anticip~tcd by thc argument f'rum dimcn-
Stons, tf if worc considérât that t))C frcqnency is Mcccssari)ydctci-tninc-d Ly tl.e v.d)!e of togeUtcr widt th~t of ~–thouuly qu.mtity (]cpendh)g on sj~cc, timc and mass, which occurs int)to diH'crcntiid cqn~ion. If cvcrytiting eonccnnng a bar be given,cxcept its absolutc
m:~It.udc, tiicfrettuency vanes invcrscly as
thc iincar dimension.
Thcsc I~ws find anImportant application In thc case of tuning
furks, w))oscprongs vibratc fm rods, Hxcd at t))c ends wherc thcy
juin the stal!~ and frce at tiie othcr cuds. Thusthc pcriod uf vibra-tion of furks of t.hc samo tnatcrinl and shapc vancs as thu lincar
dimension. Thé period will Le approximatdy indcpcndent of thé
thickncss po-pG!)dicn)ar to thé plane of bending, but will vary in-
vcr.scly with thc thickness in thcplane of
bcnding. WIien thctliiekncss is givcn, tlic penod is as thc square! of t]ie length.
In ordcr to ]owcr thé pitch of a fork wejnay, for tonporary
purposcs, load thc cnd.s of thé prongs witli soft wax, or file awaythc métal near thu base, thcrcby wcakcnitig thc sprinn'. To raisuthc pitch, thc cuds of titû prongs, which act by inertie may bcfilod.
Thc value of b attains its maximum in tho case of steel, forwhich it amouuts to about 5237 mutt-cs per second. For b'msst)ic vcloclty would Le less in about thé ratio 1'5 1, so that a
tunni~ fork n~tc of bt-~s woutd bc a.bout a. Hfth lower iu pitchth~ti ii'thc miLtcrial were stcd.
172. TIie solution for the ense w!)cn buth ends arc dampcdmay
bcun!ncdi)ttu]y dcrived from thc prcccding by a double dif-
~rcutiation. Since y satisnc~ at both ends tiic terminal con-ditions
~'hich arc thc conditions for :Lchunpcd end. I~torcovcr t))C gêneraidtRerctit.iid eqtmtiot) is a]so s~tisticd hy y". Titu.'j wo
may talœ,
oniitting cunst:u]t multiplia-, as bcfur~,
173.] NORMAL FUNCTIONS FOR CLAMPED-FREE BAR. 221
while 7K is given by thc same équation as hcforc, namely,
cos H: cosh??t=l.(2).
We conctudc that the component tones hâvethc samc pitch in tho
two cases.
In each case therc arc four systoms of points determincd bythe evanesccncu of amt its dcrivativcs. WIien vanislies, thore
is a nodc ~hcrc ~anisites, a. loop, or place of maximumdisplacc-
mcnt wlicro y" vnnis!)RH, n. point of inftection and whcru
Vimishes,:). placeuf maximum curvaturc. Whcre thercaru in thé ni-st
case (frec-frœ) points of iuncction and of maximum culture, there
arc in thc second (chunpcd-chunpcd) nodcs and loops rcspcctivcly;and vice ~er~, points of inftcctiou and of maximum curvature for
a douhly-clampud rod correspond to nodcs aud loops ofa rod whoso
ends arc free.
173. We will now consider thé vibrations of a rod clamped at
a;=(), and frec at a;=~. Rcvcrting to the guncral intégral (1)
§ 170 we sec that ~1 and C' vanish in virtuo of the conditions at
a;=0, so that
=~ (cos;r' cosh~) + D(sin .r –sinh ~') .(1).
The remaining conditions at x =givc
.D ( cos ?)!. + cosh ?M) + D (sin ni + sinh w) = 0
}J?sin ?~ + sinh ?M) + D (cos ?K+ eosli ~t)
= 0 )
whence,omitting the constant multiplier,
) \f i ")= (sm M+sinh?~)
~cos–coshu (sin ne + si n
( t t J
( i "~1(cos?H+
cosh?~wsm–srnii (2),
or
i ~~)M = (cos Mt+ cosh
-;cos ( t cosh t J
( Ma: ??!.v]+(smm- smhw~sm –sinh .(3),+ smm-Sllll1n
6 J
wherc M: must bc a root of lf
cos M cos!) 'w -)-1= 0 (4).
Thc pcriods of the componcnt toncs in thé présent proLIcm arc
thus dincrent from, though, as wc shall see presently, nearly rc-
httcd to, thosu of a rod both wtiose ends are clamped, or frcc.
LATERAL VIBRATIONS OF BARS.[173.
222
If thé value of !( in (2) or (3) be diftcrentiatcd twice, the rc-
sult (!) satisfies uf course t!ic fundamcntal diffcrential cqua.tion.
At .u=0, ,t", )i"ani.sh, but nt.<;=~ M"a.nd -r- va.nish.< ~.c" ~.r
The function ?<" is therefore applicable to a. rod clumpcd ft.t and
free at 0, proving that thé points of inncction and of maximum
curv~ture in thé origina.1 curve :u'c at thé samc distances from the
clampcd end, as thc nodes and loops respcctiydy arc from tiic free
end.
174'. In dcfault of tn.htcs of tho hyperLoHc cosine nr its !oga-
rithin, thé admissible vaincs of Mmay
bc ca)cu!atcd as follows.
Ta.lun~ nrst ttic cquation
we see tha.t ~t, when Jfu'gc, must a-pproximate in value to
~(2t +1) Tr, i being auintcgcr. If we assume
;8 will bo positive and comp~ra.tivcly small in magnitude.
Substituting in (1), wc find
eot~=~=~ y
an équation which may bc solved by successive approximation aftcr
cxpamHng tan~ and e in ascending powers of thé small
<;uantity /3. The result is
which is sufficiently accur~te, cven whcn t= I.
By cn.lcula.tion.
/3, = -OI79CGG -0003228 + 0000082 -0000002 = -017C518.
~t. ~g, arc found still more easily. Aftcr thé first term of
ttie séries gives ~3 correctly as far as six significant ~g~ires. Thé
Thia prncoRs ia aomewhat Himi!ar to that adopted by Strehikp.
174.] CALCULATION OF PERIODS. 223
table contains the value of ~3, théangle whoso circu]n.r measurc is
/3, and tlie value of siu ~/3, which will be required fm'ther on.
-F'ee-Free j~~)'.
1 10'' x -17G518 1" 0' 40"-9.t 10-' x -88258
2 10-777010 2'40"-2G99 10-'x-38850
3 10-" x-335505 C"-92029 10-1G775
4 10-'x-144989 -2MOG2 10-" x-72494
5 10-'x-C2C55G -0129237 10"'x-31328
Thé values of whichsatisfy (1) are
where however a' = e~
From this it appea-rs that thé series of values of a is the samo
as that of /3, though the corresponding sufHxes are not the same.
In fact
so that we ha-ve nothing furthcr to calculate than c(~ for which
however the series (4) is not sufHcientIy convergent. Thé value
Thia connexion between a and ~3does not a.ppeM to have been hitheïto
noticed.
.q oxproHsed in dcgroea, ~3minutcH.tmdtiCcondt). ~2'
M,= 4-7123890 + /3, = 4-7300408
M,= 7-8539816 /3, = 7-8532046
= 10-9055743 += 10-995C07S
= 14-137~1669 = 14-1371655
= 17-2787596 + /3, = 17'2787596
e"=cot~=~e~ 1
~=~, a,=/3,cf~=/3.
LATERAL VIBRATIONS 0F BARS.[174.224
of a, may bu obtaincd by trial and crror from the équation
log~cot a, 'C821S82 --t342:)-<-48 a,= 0,
a.nd will bc found tu bc
a.=-304.3077.
Another method by wbieh /)!, may bc obtamcd (Hrcct)y will bc
givcn prescntty.
Thc vaincs of ?~ wtiich satisfy (5), arc
M,= 1-57079M + a,
= 1-S7.~04
w, = 4'7123890= 4-C94737
M, = 7-85M.S1C + aa= 7'85-)<758
=10'!)!)55743
= ]0'9!).554.1
7)!, = 14'137tCC9 + a6 = 14'137JU8
7M, = 17-278759G = 17'278759, 1
aftcr which 7H =~(2t l)'7r Rensibly. Thc frcqncncics are propor-
tional to ?M", and ~re thcrtifurc for the highcr tones ncariy in thc
ra.tio of thc squares ofthe odd nmnbers. Howcvcr, in thc ca~c of
ovcrtcnes of vcry high order, thc pitch may bo stightiydisturbed
by thc rotatory inertia, whosc effect is Itcre ncglectcd.
175. Since thé componcnt vibrations of a system, Dot subject
to dissipation, arc nccessa.rily of tlic harmonie type, n!! the values
of Mt", which satisfy
cosM cosh m = ± l.(l),
must be reah We sce furthor that, if M bc a root, so arc also
–w, w 1, –??i. 1. Hence, taking nrst thé lower sigt], wo
hâve
If we takc thc logarithms of both sidus, cxpand, and cquate co-
cfHcicuts, we gct
This is for a clatnpcd-frcc rnd.
175.] COMPARISO~ 0F l'J/J'CH. 225
F''om thc known value of S?)~, thé value of )?~may Le dcrivbd
witli tlie aid ofapproximate values of?y~, w, Wc fmd
2~ =-0065t7C2J,
~ndM~
= -OOOOO.i-237
~=-000000069
~=-000000005.
whencc w,='OOG5i.33)0
j~iving = '187510;'), n.s befo)-(\
In likc manner, if but.)) ends of t))u Lar bc el.nnpcd or free,
~W. n1
9It4
+
l 7)r
I~n
.Cc. (4),'-Ï~+-=~)('
(4),
whcnce S =&c, whct'e nf crmrsc t))p sumDin.tion Is exclu-wliciice
~~a' :2,tJ ¡)J%vliet-e of entil-se tilt, ,;Ill)lnl~itioli is
sive ofthc zero value of??{.
17C. The frcq~cncics of thc sorics of toncs are proportional to
w". Thé interval between any tonc and the gra.vcst of thé séries
may con'vcnientiy bc expressed h) octaves and fractions of an
octave. Tins is effected by dividing the diffurencc <if thc logarithms
of w' hy thé logarithm of 2. Thc rcsults are as fotlows
r4G2<) 2'C4788
2't3.')8 4-1:~2
~'1590 .IO!}G6
3-7382, ~c. ~-8288, &e.
wliere the first column relates to the toncs of a rod hoth whose
ends are clamped, or free; and the second column to the case of a
rod clampcd at ottc end but free at thé other. Thus from the
second column we find that tlie first overtone is 2'()-t78 octaves
higher than thé gravest tone. The fi-actioiial part may be rcduced
to mean semitones by multiplication by 12. The interval i.s then
two octaves + 7'7736 mean scmitonçs. It will be seen that thé
rise of pitch is inuch more rapid than iu thé case of strings.
If a rod be clamped at one end and free at thé other, thé pitchof the gravest tone is 2 (log 4'7300 log 1'87.51) log 2 or 2-G698
octaves lower than if both ends were clamped, or both free.
R. 15
22G LATERAL VJBRATIONS OF BARS. [177.
177. In ordcr to examine more closely the curve in which thc
rod vibmtes, wc will transfbrm the expression for M into form
more convcnicnt for nutncrieal odeuladou, takin~ fu'st thc case
when both ends arc free. Sinco w=~(2t+l)7r–(–l)'/3,
cosM=sin/3, siu?~=cos'7rxeos~3; and thoreforc, bcing n,
root of cos M ccsh ?/=!, ccsh Mt = coscc /?.
AlijO
Hinh" w = coHii" ~t 1 = tau2 ut = cot' ~3,
or, smcc cot/3 ispositive,
sit)h~ =cot/3.
Thus
slM?):–8[nh)~l–cos!'7rsin/3
cosMt–cosh?~ eus/3
(cos ~/3 cos 't'Tr sin ~/3)°
(COS COStTT Hit) ~/3) (COS ~/3 + COS ZTr Si)! ~)
ces~/3
cos tTr sin A/3
ces cos ï'Tr + sin
Wc may thcrcfore take, omittiog the constant nudtiplio',
SiIl(~~ 7T i~f/3)
=~cos<7r.s.n~ -~+(-])'~
Mj t~
+SHl~e'-COSZ7TCOS~C'' .(~.
If wc furthcr throw out the factor ~/2, an(tput~=l,wc
may ta.ko
M =~+~+7-whcrc
= cos ï'-n-sin {;); ~7r + ~( –1//3J
}
!og7~= ~cloge+Iogsin~-logys '(2),
!og ± ~,=Mi~
log e + log coslog ~/3
from which 7~may be ealculated for dUTercut values of i an'!
177.] GHAVEST MODE FOR. PREE-PREE BAR. 227
At thc ecutro of thc bar, = and are numcrically
cqual i)i ~irtue of e'" = cot ~3. Whcn i is ~fc~ thcso tcrms ea.ncel.
For.F~weha.vc ~=(-l)'siu~7r, which is cqn:~ to xcrowhcn
i is evcn, tuni to i 1 whoi i is odd. WItcn is even, t)io'cfoi'c,
<hc! sumof thc threctcrms'v:mishcs, and thorc is accordin~y n,
nodc in tlic mi<!d[c.
Whcn = 0, M reduccs tn 2 (- l)'sin (.~ 7r (- 1)'/3}, winch
(since Isa.Iw!).ys smal!) shows that for no vfttuc of i is t)icre a.
nndo at thc end. If a long ]):u' of steel (hcl(), fur exempte, ut, thé
centre) bc gcnt)y t~pped with n, ita.mtncr whilo vtn'ying points of
its length !u-c damped wit)i thé nngcr.s, n.n unu.su:d dcaducss in
thc souud will bc uoticcd, as the end is cluscly approacttcd.
178. Wc will now t:).kc somc p:u'ticn!ar cuses.
F~n~'o): w~/t. <wo HOf~M. i = 1.
If -t'= 1, thé vibration is thé ~ravcst of which the rod is capa-btc.Our fonnuhe bueotnc
=sin (270° + 1" C' 40" '94) -M" 30' 2()"'4.7}}
h'g 7~ = 2 054231 a; + 3-7!)52301
log= 2-054231 a; + 1-8494G81,
from whicit is calculatcd titc fuDowlng table, giving thc values of
M for a; equal to 'OU, '05, '10, &c.
Thc values of M :M('~) for thc intcrmcdiatc values ofa; (in tlic
last column) werc found by iutcrpoht.tion formulK. If o, ~,?', N, t
be six consccntivc terms, that intcrmcdiatc between aud r is
228 LATERAL VIBRATIONS 0F BABS.[i78.
7~ ?;. M:~(-5)v
I~ I n I~a at. ~c 7c(')
-000 +-7133200 +-OOG3408 +-7070793 !+l.42GC401'+l-G45219
'025 1-45417G
-050 -5292548 -0079059 -5581572 1.0953179 I-2G3134
'075 r0721<!3
'075. 'O1001o3 '~140G005 I '7GGJ401 1:0721G2-100 -3157243 -0100153 -440GOC5 -7GG3401 -8837528
'125 -G9<!9004
'150 +-084GIGC -012G874 -3478031 -4451071 -5133028
'17~ -3341(!25
'200 --1512020 -01G072C -2745503 + -1394209 + -1G07819
'225 -0054711
'250 -3786027 -0203G09 -2IG7256 -14151C2 -1G31982
'275 -3109982
'300 -5849255 -0257934 -1710798 -3880523 -44750GG
'325 -5714137
'350 -7586838 -0326753 -1350477 7 -5909G08 -GR15Q32
'375 .7766G2!)
'400 -8902038 -0413934 -10GG045 -7422059 -8559210
'425 -9184491
-450 -9721G35 -0524376 G -0841519 -8355740 -9G35940
'-175 -9908730
'500
-1-000000
+-OGG4285 -OGG4282 -8G7I433 -1-0000000
Since thé vibration curvc is symmctnca,! with respect to t)ie
middie of thé rod, it is unneccssary to continue the table bcyond
~='5. Thc curve itself is shewa in Hg. 28.
To Hnd thé position of tlie node, we bave by interpolation
~1 G(i2530
178.] FREE-FREE BAR WITH THREE NODES. 229
which is thé fraction of tlie whole Icngth by which tlie uodc is
distant from thé Huarer end.
Vt&)Yt<MM ?~~A </<reë nodes. i = 2.
FI =s:n ( (450° 2'40"-27) .B-4.5" +1' 20"-135) }
log~= 3'410604.c+4'438881G
log (- F,)= 3-410G04 +1-8494850.
iC ~M(0) XM:-ït(0)
-000 -l'OOOO -2500 +-5847
-025 -8040 -275 -6374=
-050 -G079 -3ÛO -6620
-075 -41477 -325 -6569
-100 -2274 -350 -6245
-125 -0~87 -375 -5653
-150 + -1175 -j00 -4830
-175 -2G72 -4255 -3805
-200 -3973 -450 -26277
-225 -5037 -475 -1340
-500 -0000
In this table, as in thé prcecding, thé values of !( were calcu-
]:t.tcd directiy for x = -000, '050, '100 &e., and intcrpotated for thc
ititcrmediate values. For thé position of thc nodc tlie table gives
by ordinary ititerpolatioM a; ='132. C:T.lculatiug from thé above
formulœ, wc fiud
~(-1321) =--000076,
M(-1322)=+-OOU88Î,
\vhen.ce x = '132108, agreeiug with the result obta-ined by Strehike.
The place of maximum excursion may be found from the derived
function. We get
('3083) == + -00~6077, (.~081)= -0002227,
whence u' (-308373)= 0.
Hcnce is a maximum, when a; = -308373 it then attains
the value -6636, which, it should be observed, is mnch less than thé
excursion at the end.
230 LATERAL VIBRATIONS 0F BAHS.[178.
Thc curvc is s!)ûwn in fib. 2~).
Fig.Si).
r~t'tt~M M~/t ybtO' )!O~CN. i = 3.
7'~= sm [ (G30" + G")2) 45" 3"-4G],
if~- = 4-33~ .e + 5-0741.~7,
Io~ 7~= 4-77.'i!~2 + I-S-~4850.
From t))is ~(())=1'4M24., M (.~)= 1-00570. T!icpositions
of
thé uodcs are ruadi)y foumt by trial and crror. TIms
u (-3558) = -C()()037 M (-3-')59)= + -001047,
whcncc M (-35.~S03) = 0. Thc \t)ut; of :r ibr thc nodc nca.r the (nid
is -09~ (Scebeck).
Thc position of the loop i.s he.st fuund from t!te dc'rived
function. It ~ppc~rs thut ~'=0, w)ju!i a;=':22UO, ard thc!i
M =–34-9. Tibère is a)so a tuop at thu centre, whcre I)o\vûVt.;r
t!tc excursion is not so grc:t.t as at thu two uUters.
Wu sa.w t))!tt at thc centre of thc bar 7'~ :H)d ~u'c n)t)]icric:d]y
eqn:U. lu thc nci~hbom-hood of t])c )))i<!([)c-, 7'~ is L'vidcntly vury
sni:)]!, if bcmodurn.tciy ~rc~t, fmd thus t))C c~uation fur tbu ïnjdcs
rcducus approximateiy to
?!. bcing f),u int~cr. If wu tr~tisf~nn thc ot'igiu to thc centre of
thc ru(.),~nd rcptacc 7?tby its approximate vainc K2~+]) Tr wc
Hnd
178.]GRAVEST MODE FOR A CLAMPED-FREE DAR. 231
shcwing tha.t ncM' thc middic of thc bar thc nodcs are uniformiy
spac~d, thc intm'vit.t bctwecn consceutivo nodes bcing 2~– (2t+ 1).
Tins t))corct,icn.t rusult lias bccu verifiud by titû mca.surcmcuts of
Strchtkc and Lissajous.
F(.'t' mutliods ofn.pproxiin:(.tio)i npp)ic:)bic:
to thc nudcs nc~r
thc cud.s, whcn i is gre~tur th:n) 3, thc l'cadcr is rcf'~rœd to t)tc
mcinnir by Scubcck :di'eu.dy tnoutioued § 160, :uid to ]3on~in'!t~tcu:cs (p. 194').
179. Thc ca.lculn.tions n.rc vcry simitar for tho case of a. bar
clamped at onu end aud frcc n.t thc uthcr. If ïto: a.u.d
~'=~+7~+7~ wc hâve in gcncrui
Ift= ], we obtain for thc culculation of tlie gravest vibratiou-
cm'vu
Thèse givcou ealcut.i.ti'ti
( 0) = -OOUOOO,
~(-2) =-10297-
(.-t)= -370G25,
frutn which fig. 31 was cot~tructcd.
~( -G)= -7-t3~2,
~( -8)=riCUO:32,
F(l-())=l-G1222-t,
LA.TEHAL VtBRATIONS 0F J3ARS. [170.232
Thé distances of tlie nodus i'rom tlie free cud in tlie case of a
rod clamped at the other eud are given by Secbcck aud by Donkin.
2"tonc -22G1.
~'i.(Htu -132<, -4ij!')i).
4.to!tu -0!)-t-4., -3-').')8, -04.3!).
~hm~ ~3 4~-7~)754~
–t,
"Thé last row in this table must be understood as meanincr4/-3
°
that may bc takcn as the distM)cc of thej)' uodc from thé
froc (.'nd, cxecpt for t!tu first tin-ce aud thc last two nodes."
Wlmn buth ends are ft-ce, tlie distances of the uodes from the
ncarcr end are
1" tone '2242.
2'tonc-1321 -a.
3"' tone -0!)44 '3.')58.
~tone- ~i~ -3i"''t-<+2 4t+2 2 ~t'+2 4t'+2'
Thé points of inUcction for a h-ec-frcc rod (corresponding to
thc nodcs of a chunpcd-dampcd rod) arc atso givcn by SccLeck–
1~point. 2"~
point. «t''point.
l~tf)no No inaecdonpoint,
2"tone. -f)f)OÛ
3'tone -03
~"tone..S.9!)!)3 4.+1 1
i tltone
4t+2
i~
-in-2 LI ~+~
1
Exccpt in thé case of thé extrême nodes (\vh!ch have uo cor-
responding infieettou-point), thé nodes :mdInHection-poiuts alw~ys
uceur m closeproximity.
180. Ttiu case whcre onc eud of:). rod is ft-ce and the other s~-~o~eJ dous u~t ubcd an indcpcndent investigation, as it may be
180.] POSITION 0F NODES. 233
rufcrrcd to that of a rod with both ends free M'M~ in an e~?t wof~,
that is, with anode in themiddie. For attitc central node
y aud v" vanish, winch are precisely thc conditions for a supportcd
end. In hkc nianner the vibrations of a clamped-supportcd rod
are the saine as tliose of one-haïf uf a rod both wliosc ends are
c)amped, vibrating with a central nodc.
181. The last of tlic six combinations oi' tenninal conditions
occtu's whcn both ends arc supported. Refcrring to (1) §170, we
sec that tlie conditions at x = 0, give ~1 = 0, -D = 0 so that
=(<7 + D) sin .e' + (C D) sinh
Since M and M" vanish when a:' = C' D = 0, and sin Ht = 0.
Hencc the solution is
'J'TT.'r ~TT~X~
y=sin -cos–~< (1),
wltcrc i is an intcgcr. Anarbitrary
constant inultiphcr may of
course be prcnxcd, and a constant may be addcd to t.
It appcars that tlie normal curves arc tlie sanie as in thc case
of a string stretchcd bctwuen two fixed points, but the scqncncc of
toncs is altogcther dirt'crcnt, tlie frcqucncy varying as tlie square
cf i. Thé uodes and InnccLton-points coïncide, and thu loops
(which arc also the points of maximum curvature) biscct thc dis-
tances between thc uodes.
182. Thé theory of a vibrating rod mn.y be appHcd to illtistrate
tlie gcnera.1 principle that thé natura] periodsof a
systemfulfil the
maximuni-ininilnum condition, and that the greatest of thé natural
periods exceeds any that can be obtained by a variation of
type. Suppose tliat thé vibration curve of a clamped-free rod is
that in whieh thc rod would dispose itself if dcnected by a force
appHcd at its free extrcrnity. The équation of thé curve may be
taken to bc
y=-3~+~,
which satisfics = 0 throughout, and makes y and y vanish at<<
b J J
0, and at Ttius, if thc configuration of thé rod at time t be
~= (-3~+~) cos~ (1),
thc potcntial cncrgy is by (1) §].61, C~cos~X, while thé
LATERAL VIBRATIONS OFDARS.[182.
234
J"33
l7n 2
1]2] 40 7
9kinetic cncrgy I.s
~sin'and thas
~=~
~ow (U)c truc v~tuc of ;) fur tlic gravest tonc) is cqual to
~(~~J.
suthat
shewing that thc i-cal pitch of tho gravest tonc is rather (but
coniparativcfy IitUc)!owerthan t)iatca!culated from the I)ypotheti-cal type. Jt is to bc observed tbat thc hypothctic:d type in
question violâtes thc terminal condition y" = 0. Thiscircumstancc,
however, (tocs not intcrfcro with (hu application of' ti)e pnncipi~for the assumed typu niny bu :my whicii wouid bu admissibie as anunti:d
couf~m-atiou but it tends to provcnt a very dose ngrcc-Jnent of pcriods.
Wc )nny cxpcct a bottur approxitnatiot), ifwc found our calcu-
I~tioa on thc cnrvc in whici) thu rod wou)d bc d~flectcd by a force
actiug at somo litttc (ti.stancu frutn thu frcc c-nd, butwcen whicti
and the point of action of the force (.c= c) thc rod would bo
strai~ht, and tbcrcforc witiiout putential cncr~y. Thns
potential eno-gy = (J y~M~ cos'
Ti)C kinetie cno-~y can bc rcadify found by intégration from
t))c ~'atuu ofy.
From 0 to cy = :}~ + i
amt from c to L y = (c 3.<'),
asmay bc sccn frutn the
considération that yand y' nn)St not
sudd<jn)y change at :c= c. Thé rcmt)t. is
kinctic cnc.rgy = sin' + (~- r) (.' + 3f-)1
\yhcncc
~=~[~+~]-"12 70'3.1 Ga `~
(c2 e3Gt)
The jnaxinmm vainc hf 1-~wiM occnr wttcn t)tc point of
application of thc force is ill thu ueighbour)tood of the nodc of tl)csecond nornud compuncnt vibration. If' wc takc c =~, ~vo obtain i
a result wllich is tw fngh in the m~hicat scatc by thc intun-a)o
182.]LOADEDE~D. 235
cxprcsscd by thé ra.tio 1 '9977, a.nd is ~ccordingly cxtremdy nea.r
t,he trut)h This cxampio may givc un idca. how uciu'ty thc pci'iod
of a. vibr~tin~ systort may bu catcntittcd by simplerncans without
thé solution uf diHurcMti:d or tra-uscuudenttd cqu:),tious.
Thc type of vibration just cousidered wout<t be tliat actua.ily
~ssumcd by a. bar whicii is itscif dcvoid of inertie but can'ics :t
lu.td J/n.t its frec end, providcd that tbe rotatury inurtin ofJ/could
bu ]mg!ccted. Wc sliould h:n'c, in i'act,
F= Cf/~N~ eus' 7' = 2~ si)i'
:<sothat
V~.(.<).
Evcn if thc i))Grti!i of tbe bar bc not attogcthcr n~gligibic in
eomp:u'isoM with jV,v'c may still tid~ titc saniu typu as tticbasi.sut'
:m appruxijaatuc'idcutation
th:Lt is, J/ is to bc incrcascd by n.hont onc quartcr of tLc mass of
thu )'od. Mincu titis rcsuk is accm'!).k! whcn is mfimt.e, atld dous
hot ([m'ur nmch (ruiu t!ic trnt)), cvun whcu~V=0, Itrn:).ybu rc-
~n.rdudas gutiuraHy a.pplic:).b!u !).s a.u
a.pj'roxhn~tiun.Thé cn'or
will ahv~ys Le on tlie sidu of cstimatm~ ttiu pitch tuo liigL.
183. But thcncglect
of thé rotatory Inortia of ~f could not
bcjustiiicd midd' thc ordi)i:u'y couditi~us of cxpenmoYt. It is as
unsyto Im:)ginu, thou~h ~ot to construct,a.c:).sc m whie!) tlic inertia
of translatioji s))un)(i bL;ncgligIDc
incomparisou
with thc iucrtia of
rotation, as t)~ opposite uxtrutne wtuch bas just bccn considcrcd.
If both kinds of incrtia. in thu !na.ss ~f bp iuctudcd, cven thougli
that of t)ic l):ti' bc nc~jectcd ft!to~ctiicr, thé systum possesscs two
distinct aud indupendoit po'iods of vibration.
Lct z and dénote thc vaincs of and ut .B= Then tLe
cquatiou of thc cm'vc of thu b:).r is
~+
S3G LATERAL VIBRATIONS 0F UARS. [l83.
and
whileforthûkIneUcctiu'gy
~=~+ L~ .(2), J
If~ be the nutius of gyration ofJLTabout an axis pcrpendicular to
t!)ep)an(;!ui'vibnLtioti.
Tbc equa.tions of motiou are theruforc
whciicc, if z and vary as cos j;)<, wc find
con'cspo)idingto tlic two penods, which arc aiways difïci'cnt.
If wc negluct thc rotatory iucrtia by putting /e'=0, we fall
back on our prcvious rcsuit
3f7~"MoT
~f
Tite ot!)cr value of~ is thon infinite.
If ?' bc merc!y sma. so tliat its Iti~Itcr powcrs may be ueg-
lectcd,
If on the other hand A:" be vcry gréât, so tha.t rotation is pre-
vcntud,vented,
12<7A'~ <~M
~=-77r-or
thé lattcr of which is vcry sma]]. It appc~rs thn.t when rotation
is prcvc!'t.c<),tlie pitch is an octave iu~Itci' than if therc were no
rotatory inertia at a!). T))cse cundusions might also be derivcd
183.] EFFECT0F ADDITIONS. 237
dit'cctty from tlie diH'crentiat équations; for if/c'=~, 0=0,a.nc
tlieii
butif/<=0, ~=~ by thc second of équations (3), and in
thatcase
184. If any addition to a bar bc made at thc end, thé periocl
of vibration is prohjnged. If tlie encl in question bc frce, supposenrst that thc pièce addcd is wit)iout inertia. Since thcrc would bo
]t0 altération in eithcr tho potcntia! or kinetic énergies, thé pitch
would be nncliangcd but in proportion as the a.dditiona.t part a.c-
quires inertia, the pitcli Mis (§ 8S).
In the sa.mo way :), smiUL conthiun.tiun of a. har bcyond a
clumpcd end wonid hc wiLhout nu'ect, ns it wou)d ac()ui)'c no
motion. No change will cusue if tlie ncw end bc a.tso c):).mpcd
but as thc first chunping is rc!a.xcd, thc pitch faits, In conséquence
of thé diminution in thc potential cucrgy of a givcn dutormation.
The case of a supportai oïd is not quitc so simptc. Lct tlie
original tjn(L of thé rod bc and let tlie added piccu whieh is at
nrstsupposed to hâve no incrtia., bc ~t/?. InitiaNy thc end ~1 is
fixed, or held, if we )ikc so to l'cgiu'd it, by a spring of inrinitc stin'-
ncss. Suppose tbat this spring, which )ias no ino'tia,, is graduaHy
rclaxod. During this proccss thc motion of thc ncw end
diminishcs, and at a certain point of relaxation, -D cornes to rcst.
During this proccss tlie pitth falls. 7~, being now at rest, may bc
snpposed to become nxcd, and the abolition of thé spring at ~1
cntails anothcr f:d! of pitcli, to Le further increased as ~J3 acqnircs
inertia.
18.5. Thc case of a rocl whieh is not quitc utufonnmay
bc
treated by the gencr:d method of§ 90, We ))ave in thc notation
thcre adoptcd
238 LATERAL VIBRATIONS OF J!AHS. [185.
whcucc, P,. bc'iog t))C uncu)-)-cct,cd value oi'
For examplu, if the rod bc e!:unpu(! at 0 and frœ :tt
Thc samc fonnu!~ appiics to a doubly frcu bar.
T)ie e~uct ofa. smaH lo~d (~V is thus givctt Ly
who-c dénotes tl.c mass of thc whoïc har. If thu load bc at
t!iocn<],it,s cficctJst])esa)nGnsa.iL-]igth('))i))g'ofthcb!U-mt~c
ratio ~+~J/: (Compte §1U7.)
~8G. T!)c samc prineipte jnay hc appticd to estimatc t!)C
corroctio)! duc to ti)c rotatory inertin of n. ~ttifoi-in rod. Wc havo
on!yto <md what additton to m;)kc tothGkineticcncrg'y, sup])osing
tha.t tho bur vibrâtes accordin~ t.o thc samu !:tw as wou]d oblai)~were Uierc no rotatory ioc'rtia.
Lctu.s take, far cxmnpic, thu case uf a L:u- c!a)npc() at Oaudfrcc at a.nd assume tftat thé vibration is of thé type,
.V= !<cus~
whcre M is one of thc func-tionsinvosti~atud in § 170. Thu ].i))(-tic
f'no-gy cftttc rotation is
18C.] CORRECTION FOR KOTATORY INERTIA.
Tothismustbca.ddt.id
23U
ti0 that tlie lunctic encrgy Is mcrcascd in thc ratio
Thc atto'ed frcqnency Lcnrs to thi~t calcnh~tcd without allow-
anee for rotatury inertie n mlio '\v!uch is thc square root of thé
rcctprociti ofthe! prcce~ing. Thus
?~/c' ,?</ M~\7'= ~=
1-(~,+~(1).
By use of thc retat.ions cosh ?~ == suc M, sitilt ?~==cun<'7r.t:).)t7~,
wu m~y cxprcusK' « A\'L<j~ .-<;= ill thc furjn
sin M eus a
;t. eus <7r + ces ?~ 1 ces /7r siu a'1
if wc substitntc fut' ?~ front
~=~(2t-l)7r-(-l)'a.
In thc c~su of thé ~ra.YCSt tune, ot='3()43, or, in dL'grccs and
nunutcs, K==17°2C', wlicncu
Thus
which ~ivcs tlic corrcctitn) fur rotatory incrtin. in tlic case of thc
gt'avcst tonc.
WtK'u thc ordcr of thé tone is modoratu, a is vcry small,
andtheti
'u=l sc'nsibly,
n /w\?)~atld r=l-fl +
~-)(3),
shcwing tl~at thc correction incrcases in importance witi) thc
order of tlic component.
In a.ll ordina.ry bars K Is verysma.!), and thc tcnn dcpcnding
on its square ma.y be ncgluctcd wit))out sensible error.
LATERAL VIBRATIONS 0F BARS.[187.
240
187. Wben thc rigidity and dunsity of a bar are variaDc
from point to point aJong it, t)ie nonna.1 functions cunn~t in
gênera.! be expresse) tumiyticaUy, but tticir nature tnay be invcsti-
gated by tlie method.s oi'St.ur<n and LiouviHe oxpituned in § 14.2.
If, as in § 1G2, 7~ <tcnotc thc vanahie flexunU rigidity at anypoint of the bar, and pM~ the mass of the clément, whosc Ien"-this wc nnd as tlie gcncral dUTo'entiiJ équation
tho effets f'frotat'ry ixo-tiubcmg onuttcd. If wcnssumc- t]):)t
c< cos! wo ot)tain as the équation to (]ctL')-i))C thc i'orm of' thu
nonnat fonctions
in whieh is IImited by thé termina! conditions to bc one of an
induite series ofdcnnitc quantitles )~,
Let ussuppose,
for cxamp!c, that thc har is cliunpc'd at both
ends, so tliat thé termina! values of and v:mi.st). TI)c first~.<;
normal function, for which Las its lowest vatnc Jms no
inturnal root, so that t)tc vibration-curvc lies cntirdy on nnc sidc
of the eqnilibrinm-po.sition. T)nj .second nonnid function bas onc
intcrna.1 root, thu t))ird function has two interna) roots, an'),
gcncra!)y, t!ie )' function bas t- 1 internat roots.
Any two dincrent nor)n:d fnncti~)).s arc conju~a.tc, tliat is to
say, their product wiH vanish when mu)tip)icd by ~Mt7~, and
Intc'grated over tho Icngt,h of thc bar.
Let us uxn.)nine thé nurnber of rf'ts uf a funetion /'(.) «f
thé fonu
/M =~M + M-t- +~ (.).(3),
compoundcd of a Hnite number of normal functions, of \\hich tho
function of lowest ordor is ':<(.) and that of highest ordct- is
(-<"). If tl'c numbcr of internai roots of/(~) be so that thcro
arc ~+4 roots in all, thc dcrived functiou (.?') cannot hâve )css
than + 1 Internai roots besides two roots at thu extremitles, and
thc second derived fonction c-annot hâve Icssthan~+2 rûots
187.] ROOTS OF COMPOUND FUNCTIONS. 24L
No roots can bc lost whcn the latter function is multiplicd by 7~,and another double din'ercntiation with respect to x will ]cave at
least internal roots. Hcncc by (2) and (3) wc conclude that
M +M, M +
+ < M M. (4<)
bas at least as many roots as /(.). Since (4) is a function of the
same form as/(.), thc same argument in~ybe ropcated, a.nd{), a
series of functions obtaincd, every mcmber of whic)) lias at least
aa many roots as/(~) lias. When the operation by wliieh (~) was
derivcd ft-otn (3) bas hccn rcpcatcd su<Rcient)y ofteo, a function is
arrived at whose fo'm differs as !itt!c as wc picasc from that of thu
component normal function of highest order ?<“(?'); and we con-
cludothat/(;c) cannot have more than~-l Intcrual roots. In
likc manncr wc may provo t);at/(.r) cannot hâve less than w-1 1
internf).! roots.
The application of this thcorcm to deMonstratû thé possibility
ofexpa.nfiinga.narbitraryfanctioninan infinité series of normal
functious would procecd cxact)y as in § 14'2.
188. When. thé bar, whose latéral vibrations are to bc considered,
is subject to longitudinal tension, thc potential energy of any con-
figuration is composed of two parts, the nrst (ieputtding on the
stinness by which thé bonding is directly opposcd, and the second
on thé reaction against thé extension, which is a neeessary accom-
paniment of tho bending, w])en thé ends arc nodcs. Thé second
partissimUa-r to the potential energy of a dcnectcd string; the
first is of thc same nature as that with wtuch \vc have becn
occupied hithcrto in this Chapter, thongh it is not entirely inde-
pendent of thé permanent tension.
Consider thé extension of a filament of thé bar of section f~u,
whose distance from the axis projected on thé plane of vibration
is Since thé sections, which were normal to the axis originally,remain normal during thé bending, thé length of thé niament
bears to thé corresponding élément of thé axis the ratio Tt* + JT,
7~ being tbe radius of curvature. Now thé axis itself is ex tend cd
in thé ratio q :y-~y, reckoning from thc unstretchcd state, if
7'ùj dénote the whole tension to which the bar is subjected.
Hence the actua! tension on thc filament is-~+~(7'+~)~M.
R. J (j
from whieh we find for thc moment of t))c couple acting across tho
section
and for thc who)e potcntud cno'gy due to stithtcss
an expression din'cring from that previousiy uscd (§ 1C2) t'y tho
substitution ofy+7'fory.
Sincc is ttic tension ruquircd to strc-tch a har of unit arca to
twicc its natnmt loi~'th, it is cvidunt th~t m most pra.cticat cases
Y'would bc nc~tigibic iticomparison
with
T!te expression (1) dénotes thc work th~t \vou!J ne ~iocd
dnring thc strai~htcning'oftLe bar, if thé luugth of c:).ch dément
ofthc axis W(.'['o prc'scrvcd constant dut'ing t])(; proccss. But
whcn a. strctchcd L~r or strin~ is attowcd to p:LSS frotn iL disp!a.ccd
to thé nutura! position, thc to)~t]) uf t)tc axis is dccrcascd. Tho
illllolll'It of tho clocrcascis :~f G?fY
-cl.r, and the corrospol.lding gainamount of thé dco'casc Is ( .") << and thu corrcspouding gainj\~t<
cfworki.s
~(~.:¡ T(ù
cl,cd.c.
Thus
r=< (~.(~)'
&+ï-~(~,)'(a).
T)ie Yariation of the first part duc to n. hypothc'ticfd dispiacc-
ment is givon in § 1G2. For thc second part, wc hâve
icf/7 f~Sy (~y~l f~-Vc 7 /o\
~8~c= =
~y~r.. (3).
J V~ j ( J f<~
In aH <hc cases that wc hâve to consider, ~y vnnishcs at thc
limits. Thcgcncra) diircrentia) équation iHa.cconiingty
or, if ~Yc' put -t- T = ~'= «~),
~+~p.
rlx cl.~c 1. r!t .t. cl.c clt ..l..t.~0. (4).<:t~.v «'fc/</ cAc.U fM
Fora. more dctailcd investigation of this equation tlie readcris
rcfcrrcd to thé writings ofCtebsch' n.nd Don~i)i.
~«'()r<t'~t'r7'<(ts<)'ct<«</M<fr7~[ir/)<'r. Leipxig, 16G2.
189.] PERMANENTTENSION. 24:3
18'). If thc ends of thc rod, or wire, bc chnnpcd, = 0, !ind
tite tc)-)i)!)i:L) conditions fu-c saLisfied. ]f t])c nature of Lhe supportbe such that, wlutc thé cxtrutnity is coii.stnuuud to he a, node, tiio'o
is no conp!c itctmg on thc b:u-, must vanish, thilt i.s to say, tho
on! nmst Lestrfught. T)tis
suppo.sition i.su.sua.Hy t~(cn to
rcprc-
scnt thc c:i.su ofa.string strctchcd ovcr
hritt~cs, a.s i)imanytxu.sic:).!
i)Lst)'nmcnts; but it is cvidoit that titc pfu't beyotid thc bridgemust pa.rt~kc ofthc vibration, !ui(l that thcrcforc its lo~tit cannotLe altogcther n ]n:).ttcr of Ijiditfcruncc.
n'in thc ancrai dif1b)'c:nti;t.l cqu.Ltiou wctit-ke~pi-oportional
to cos wcgct
whi<hiscvident)ysatiH(n'dhy
if bc suit:d)ty dcto'mif~d. T)tc sanic solution a)sn makcs
yat)dy" vimisha.tthccxtt-etnitic.s. By substitution wcnbtflin
for??,
n ~+~7!
"'=~' -~+/W (3),
which détermines thcfrcquc~oy.
If wesoppose t))C Aviru innnitt'Iy thin, ?r=~7r~ thc same
as wn.s i'ound in OtaptcrVt., by startin~ from thé supposition of
perfcct m-xibUity. ]f wc t)-e:tt ns a vcry sma])qnantity, thc
approximatc vah)c of?; is
,rr<'7r</f -;7T~rc )
"= 1'+'
2~ (rr-~}-
For a. \vit'(.; of circulât' scetion of radius r, ~= and if wû
rcpht.cc a)i([ f< hy thcir va)ups in tc'rtns of y, 7', an()
whic!) gtvcs t!)c corruction fur ri~idity'. 1. Since t))û expression\vithin brackets invoivcs ?', It appc'ars that t])C harmonie rclatinti
of thé componcnt tones is (Ust~rbed by thc stiitncss.
'Dont{in's.-frn)f.f'f~,Art.im.
]f!–S
LATERAL VIBRATIONS OF BARS.[190.244
190. The investigation ofthe correction for sti~ncss when the
ends ofthewirearc ctanipcd is not so simple, In conséquence of
thc change of type which occnrs ncn.r tlie ends. In on)< to puss
from the cftsc of thé preceding section to that now undcr con-
sIdGration au ~hiitional consti-:unt must be introduced, with the
eHcct of attti fm-ther raising the pitch. Die fu!tow!ng is, in the
ma.in, thc investigation of Scobcck and Donkin.
If the rotatory incrtia be neglected, thé differential équation
becomes
where a and /3 are fonctions of ?t determmed by (2).
Thé solution must now be nm<!e to sattsfy tho four boundary
conditions, which, as therc are only three clisposable ratios, tca.d
to an equation connecting a, ~3, This may be put into thé form
190.j PERMANENT TENSION. 245
Thus far our equations are rigorous, or ruther as rigorous as
the dincrential equation on which thcy are founded; but we sha.11
now ititroduce the supposition that tlie vibration considered is but
slightty aSected by tlie existence of rigidity. Tttis being thé case,
t!te approximate expression for y is
uearty.
Thé introduction of thèse values into thé second of equations
6~(G) proves that H'
<ur
.jis a stna]] quftntlty under thé cir-
cumstn-nces contempiatud, a.nd thei'cforc tli:tt a'~ is a l:u'gc (~tfnitity.
Siucc cosha~, sinha~ are both I~i'gc, ('(~uation (5) rcduccs to
According to this équation thc component tones are ail raised in
pitch by tlie same smaU interval, and thcrcforo the harmonie rela-
tion is not disturbed by thé rigidity. It would probably be other-
wise if terms involving f were reta.incd it does not therefore
follow that thc harmonie relation is botter preserved in spite nf
rigidity when the ends are ctamped than when they are frec, but
only that tbcre is no additional disturbance in thé former case,
though tlie absolute altération of pitch is much greater. It should
be remarked that b (t or ~/(<y + l') \/7', is a large quantity,and that, if our rcsuit is to be correct, A: rnust. be small enoughto bear multiplication by b a and yct romain small.
346 LATERAL VIBRATIONS 0F BARS. [190.
Thé theoretica-1 rcsult cmbodicd in (8) ha.s been eompared with
experimoit by Seebeck, who found a. s:d.infactory agruemcnt. Thu
constant of stirfness \vas dmtuccd frum observations of tite rapidity
oi't)io \'ibr~tions uf n smaU piuco of thu \vii'C) wttcn one end was
(.'tutiipud lit tt. Vice.
191. It lias hcon shewn ni t)ns c])apter tliat thc theory of bars,
cvcn whcnsh)]pUHud to thc utrnost by t)tc omission of
uniniportant
quantifies, is (tceu~dty morecumpticated
t!t:ut t.hat of po'ftjct!yfiuxibtc
stnugs. The ruasun of thu extrême snnpiicit.y of thc
vibrations of .strings is to Le fcund in thc flet titat \),v(.'s of tho
luu-monic type arc propagated with a velecity. indcpL'nd(;)it of tho
wave Iun~t](, so tti~t an a.rbitnu'y wa.ve is aHowcd to travut Avithout
décomposition. But whcu wc pass from string's to b;u's, t!ic con-
stmt iu tlle (litlèrential C( lliLtloll ~'1Z.`l
-I- `l = U is nostant in thc din'crGntifd équation, vix. ~t-/<=(), is no
longer cxprus.sihiu as a veh'city, and thcrufm-c t])(j V(.d"city of
transmission of a train of harmonie wavc.s c:mnotdépend on thé
dif)'cr(;utial ontution ,'dom', but must vary with tiiu wa-vc lungth.
Indccd, if it bt.' admittud t])at t))e train uf harmonie wavcs can
bc propagatcd a.t a)), titis considuration is sufHck'nt by itscif to
provc that thu velucity must vary inycrscly a.s thc wavc tcngt)).Thu samo titing may bc scen front thu soJution npj)]ieab!c to
C)wavcs propagatcd in onc direction, vlx.=cos"- (H–~),
À.
which satisfies thc diH'urcuti.d C(p(ation if
Let u.s suppose that titcrc tu'c two tminn of wavcs of equa.1
amp)itudL's, but. uf diftbruut w~vc ]c))gt)).s, trnv'L'HI))~ m t.hc samc
directujn. Tiius
If T r~ bc .smn.1), we ha.ve a train ofwavc-s, Avitose
nmpti-
tu()c s!(jw)y vancs from ouc: point to anothur IjctwGOl thc vatucs
0 amt 2, ft)!')ning- a so'ic.s of group.s S(-)):).r:).ted from onc aufjther by
]'egiot]s cojnparativ-cly frcu ironi distm'baucc. In t)tc case of u.
stringor of a co]um!i ofair, v:n-ics as T.and t!)cn thc gt'oups move
~91.] RESULTANT0F TWOTRAINS0F WAVES. 347
forw:u'd with titc same velocity n.s thc compone~t trains, :t.nd t!ierc
is no change of type. It is ot.ttcrwise whcn, as in tiic case ot' a bar
vibrating t.ransvcrse!y, thé vdoctty of' propagation is a fmictton
ot'thc wave Icogth. Titc position at ti)nu t of thc middia of t)t0
grnup which was initiatty at thu origiiiis
givoi. hy
In thc j'rcscnt cn.SL-)!.== 1, an<t accordingly t)in vc!ocity oftiio
gt'oup.s is <t'ce that ot' <,)nj compoount w.~ves*.l,
H)2. On account of tho (tt.'ppn<)cncc' of thc! vclocity of propaga.-
tion on tho wave Icngth, tin.! cutHution of :ui infhnto bar at :u)y
time subsc'j'tcnt to an initm.t (tistnrbancc f'ontuu'tt tu a, lunitcd
purtiott, will h:tve n<jnc of t.hc simplicity wttich chanicteri.sGS tho
cort'cs~ondmg pt'obtt.'m ior a .sLriug'; bt'.t ncvL'rLhutcHS Fouricr's
i)ivcstig:).tion ofthis qncstton umy property <)ttd :t.p):).cu hci'c.
It I.s rcquircd to dutcrmmc a. function of :nid t, so us to
sfitisfv
and !U:~kc initiaity = (.), ~=' (~').
A solution of (1)is
~/=cos~ cos~(.<x).
whcrc and a arc constants, irom '\vluch we conclude t))at
In tho c<in'csponJh)f} pr')1))om fur wfivcs 0)i thû surfaco of Jcfp water, tho
\'dot;it.y of prf)p!ts'~t't"~ Yarit~ dh'(;Kt)y as tho square root nf tho W!t.vo Icut;) su
that M=A. Tho vetocity û( tt group of such Wftvcfi is tLcrefuro f~<; of thttt of
tbe component trains.
LATERAL VIBRATIONS 0F BARS. [193.248
is n.)so a sohttion, where j~('x) is an arbitrary function of a. If
!iowweput<=(),
which shcws that ~(a) must be takcn tobe (a), for then by27r
Founer's double intègre thcorcm ~j,=~(A'). Murccvcr, y=0;
lience
By Stokcs' t!morc!n (§ f)5), or iudependently, we mny now
suppty t])ei-(;m:umng p:u-t of thc sulution, which I)ns to Ha.tisfy tlie
(liU'ercutml équation whilc Ib makcs initi:d)y =0, = (.); it is
Thc Hnal result is obtained by a.dding thc right-I~ud members
of (3) aud (4.).
Jn (3) thc intégration with respect to q may bc c~ected bymctuis of thc formula.
which may Le proved as follows. Ji' in thc wcM-known intégralformula
Now suppose that~=<'=< whcre !:=VI-l, and rctain
only thé ren.1 p:).rt of tho equation. TI)us
193.] FOURIER'S SOLUTION. 249
whencc
from -which (5) foUows by a. simple ct~nge of va.nab!c.
ecma.tiou (3) may bc wi'tttc]i
Thus
CHAPTMR IX.
ViïiHATtONS 0F MHMtiItANES.
U)3. Tm-: tlicorctica! monbranc is a pcrfcct]y f)cxib!c and in-
nnitctythin Jnmina ofsotid )nattcr,of nnifoDn materiat and thick-
ncss, whicb is strctcbcd in :dt directions by a tension so grcat as torem~in scusibly unidtcre.t during thé vibrations at)d di.spfaccmonts
eontcmpjated. If fuiimagioary Une bc drawti across t!ie mem-
brane inany direction, t))omut)):d action betwccn thc two
portions
separatcd byan eiemunt uf Uœ !inc is proportionn! to thc len'rth ofthc dûment and pcrpcndicutar to its direction. 1. If t)ic for~c in
gestion bti l' ~.9, 7', mnybo caifcd tbc <o<uM of ~e??te~6~He-
it i.s a quantity ofont: ditncn.siun in tnas.s and–2 in time.
Ti)c principat probfon in conncction with tinssnbjoct is tlie
investigation uf tbc trau.svo-sc vibrations of mcmbratic.s of dirïbrcnt
shapc.s, whosc boumhu-iu.s arc nxcd. Otbcr questions ind~cd rnaybc proposcd, but thcy arc of
compat-ativuty Htt!(j intcrcst; and,niorcovur, t))e tuutttod.s prop~'r for
sulvin~ thcm wi)[ be 'suff~
cicnHy iitustratûd in otticr parts of this work. Wc may titcroforû
procuud at unce to the con.sidcration ofa membrane strctchcd ovcrthc arca inc!)tdcd witbin a nxcd, closed, ptanc bound:u-y.
10~. Taking t)tc phinc of tbcbonndary as t)iat of a'y, let M
dénote thé smail disp!aco;ncnt tbcret'roni of any point 7~ of thomonbranc. Round takc fi sma)t an~ amt consido- thc forces
acting upon it parattcl to z. T)~roso)ved part of the tension is
cxprcsscdbym f~~'j~
wltero (~ dcnotc.s an ch-mont of tbcbound~y of and r/~ nnt/ ) "m~ tf/t tm
cfomjnt ot thc normal to thc cnrvc drawn out\ar<).s. This isbalanccd by the reaction against accctcration mcasnred by ~v
194.] EQUATION OF MOTION. 251
p buin"' a symbot uf onc dimension in mass a.nd 2 hi length
dcnotmgtlie supui'Hci:d density. Nuw by Grcen's theorem, if
;S' ukimatufy,
f).ud thus thé cf~un.tiott of motion is
~) J.d).
Thc condition te bu s:).tisficd at tlio bomidm'y is of course w= 0.
Thc diU'orential équation ma.y a!so bc invcsti~atcdfrom thc
expression for thc putcntin.! cncrgy, winch is fouud by muttiplyingthc totisiou Ly thé supediciid strctclimg. T)ic :dtcred a.rca. is
from which 8~ is casily ibund Ly au intL'gratIou by parts.
If wc writc ?~ /]=c' thun c is of t)ic nature of tt.vc!ocity,aud
tlie diH'cruutial con~tion is
!!)!'). We sha!l now suppose that tho boundary of thé mem-
brane is thu rcchuig!~ formcd by Llic cnordinatc axus and thc linc.s
te = n, y= for ovcry point withhi tlic arc:). (:}) § 104 is satisiicd,
fmd fur cvory point ou tnc boundary 'w=().
A particuttu' It~tegral is cvidcntiy
,7)~ ?~\l 11 C l'C ?l- C-7T'
2 /))Lz
)l21CI»)where
+~(~'
and M~and Harc intcgcrs n.nd from this thc gênerai solution ina.y bc
dcrivcd. Thus
w=~ )<=~o ??;7TT' );'77'w=S
M-i
S
M=t siu sln-(~~cos~<+7?~sin~}.{:Imn COSI)~ -j- En," Sll1
2)t} (3).
252 VIBRATIONS 0F MEMBRANES.F 19 5.
That this result is really gênerai may be proved a posteriori,by shcwu~ tliat it n~y be ad~p~.d to express arbitrary initialcircumstanccs.
WI~tevcr fiiiietioii of thc co-ordinatos may hc, it can bs ex-
presscd for all vaines of bctwccn thc limits 0 and by thé séries
where thé coemdents Y., &c. are Indcpeudont of Againwhatever function of~nnyoac ofthe coc~cicnts YmayLe, it canbe expanded betwecu 0 and & iu t)ie series
where C, &c. arc constats. From this we conclude that anyfunction of x and y can bc expressed within thé limits of the rect-
angle by thé double series
and thcrcforc that tbe expression forain (3) eaubeadapted to
arbitrary initial values of w and In fact
.(4.).
Thc dmmctcr of tlie normal functions of a given rcctang!c,
as depending on and is easily undcrstood. If and n be bothun.ty, w retains thc same sign over thé whole of thé rectanclcvamshing at thé edge only but in any other case there arenodal lines running parallel to the axM of coordinates. Thénumberofthc nodal lines paraHetto is n -1, their equationsbeing
195.]RECTANGULAR BOUNDARY. 253
In thé sa-me waythé équations
of thé noda.1 lme3 pMa.Hcl to
n)'f.
being w 1 in number. The nodal system divides thé rectangle
into ??~ equal parts, in ea.ch of which thé numcnca.1 value of w is
repeated.
106. Thé expression for w in terms of thé normal functions
1s~q
whcrc 6,&c. are the normal coordinatcs. We proceed
to fonu
the expressionfor Fin terms of We hâve
v V v
In integrating these expressions over thé area of tlie rectangle
the productsof thé normal coordinates disappear, and we find
the summation being extcnded to <d! intègre vahtes of w and ?!.
The expression for thé kinetic cncrgy is proved in thé sMne
wr,v to be
if Zf~cf~ dénote the tmnsvcrse force acting on the element ~.t.-<
254 VIBRATION 0F MEMBRANES.[1()Q.
Let ns suppose that, tl.e i.iitial condition is one of resb undcrt)tc opération uf a consent, force .s.ie], as nmy Le
supposed toansu from gascons pressure. At thc tin.c <=0, tlic i.nj~dforce is rc.novcd, ~nd thc mo.nbmuc Jeft to itsc)f.
IniMativ thccquation ofcquiiibrium is
i"C('njunctinnwit,h(~.
In on]cr to cxpn.s.sv.-Lh.c ,n (..), or ,n t.).is case
si,nr]y to ren.uvc f,.o,n undcr thcJ'iLpgr;d.s)gn. Thus
Thi.si.s an cxamph.of (.S), {;)()!
If tlie ,nen~r.n.. p,i.s)y at. n.sL in its po.s.Uo. ofcn.i),-settlie
solution is
4 W7TX M~S/Y
=~ .sin ~< .(~).
197.] CASES OF EQUAL rERIODS. 255
197. The frpqucncy of thc na-tural vibra-tions is fouud by
ascribing diiTerent intégral values to M and in tho expression
For n, givcn. mode of vibration thc pitch faUs whcn cither
Hido of ttie rectangle is incrca-scd. In thé case of thc gravest
mode, when w=], ~=~ additions to thc shorter Hidc iirc titc
more effective; n.nd whcn thé iurni is very clo!)~ttcd, additions
tu thé longer sidc {u'c a-tmost wlthout c~uct.
WitCH a~d are inconnnensur:ddc, uo two pairs ci values
of w and ); can gi\'c t))C sa-mc frcqncncy, and cach fuodamcntal
]t)cdc of vihratiuu bas ils own ch!Li'actc)'istic pcriod. Uni whctt
ft" a)Kl arc coonnmisurabtC) two or more fut)d:uncnt:d modes
may hâve t)tu samc pcriodic ti)nc, and may tlien cocxist in any
proportions, w)fi)e thc motion sti)) rctains ilssimpte
harmonie
charaetcr. In suc)) casus thc sp~'incxtion of thc pc'riod docs
not co)np)ctc)y detenninc Lhc type. Thc fuM cnnsidcration of
thc prohion now prcscntin~ it~c]f n'nnircs thc aid of thc thcory
of numhbr.s; Lut it will bc sufUcicnt for thc purposes of this
work to considcr a few of thc sunpk'r cases, which arisc whcn
thc membrane is square. Thé rcadcr will find fnHci' information
in Ricmann's lectures on partial diUbrential équations.
If f; =
Thé lowest tone ]S foun<t by putting n~ and )z cqual to unity,
which givcs only onc funda)ncn<d )nodc
Next suppose that one of the numbo's ?)!, ?; is cqu:~ to 2, and
the other to unit.y. In this way two (tistinct types of vibra-tion
are obtfuncd, '\v!fosc po'iod.s arc thé s:unc. If Utc twr) vibrations
be synctironous in phase, thc wt)ole motion is exprc'sscd by
so that, although every part vibrâtes synchronousiy with a
liarmomc motion, thc type of vibration is to somc cxtcnt arbitrary.
35G VIBRATIONS 0F MEMBRANES. f'197.
Four particular cases may be especially noted. First, if 2? =0,
which mdicn.tcs a vibration with one node along thé line a?==~.
Sinailarly if C'=0, we have a node parallel to thé other pair of
edges. Ncxt, howevcr, suppose that C' fuid D are ûnitc and
equal. Thon w is proportional to
which mn,y Le put into thé furm
This expression vauishcs, whcu
or ngfun, \vhnn
The first two équations give the edges, which wcre originaHy
assumed to be nodal while the third gives ~+a*= a, representing
one diagona.1 uf thé square.
In thé forn'th case, when C= D, we obtain for thé nodal
lines, thé cdgca of ttte square together with the diagonal ~=.r.
The figures represent t!]e four cases.
c+~=o.
For other relative values of 6' and 7) thc interior nodal Iine
is corvcd, but is always a.nalytica)]y expressed by
and may he casily constructed with thé help of~ table oflogfu'ith-
mic cosines.
197.J CASES OF EQUAL PERIODS. 257
Thc next case in ordcr of pitch occurs w!icn = 2, = 2.
Tiie values of M~ and n being equal, no altération is caused bytheir mtcrchangc, -\v]nlc no ottter pair of values givcs the samc
ft-equcncy of vibration. Thé oniy type to bo considered is
accordIn~Iv
whose nodcs, Jetct'mincd by thé equation
arc (in addition to the cdgcs) thé straight lines
T~)c next case winch we shaH consider is obtained by ascribm"-
ta w, n thé values 3, 1, and 1, 3 successively. Wc have
f~. 37ra; Try Tra; 3~~M) = Usin sin + D sin sin cos M<.
( a a o o J
The nodes arc given by
or, if we reject the first two ,{a.ctoys, which con'cspond to thé cdges,
which represent thé two diagonals.
R. 17
258 VIBRATIONS 0F MEMBRANES. [107.
Last)y, if C'= tlie équationof thc nodc is
Jn ca~c (4-) wlicn a: = a, y = ft, or and similarty whcn
y = a, ? = «, or TL'us oue ha!f of Ctich of tlic lines julning
tlie xuddie points of opposite cd~cs is intcrccpted by thé curve.
It should bc noticcd th~t in wha-tever !'n,tio to one another
Mid D may Le t~kcn, thé nodfd eurvc always passes through
thc funr points of mtcrscctio!i of thc nod~I lines of tlie Urst two
cases, C'=0, D=0. If the vibrations of thèse cases 'bc com-
pounded with correspon~ing phfmcs, it is évident tha.t in thc
shaded compnrtmcuts of Fig. (3.')) tlio directions of disph~cment
n.rc thc s~nc, und that thcrcfore no pM-t of the nocM curvc
ia to bc found thcrc; whn.tevcr thc ratio of amplitudes, thc
curvc Jnust bc drawn tlu-ough thc utish~dcd portions. When
on the othcr hand the phases ~]-G opposcd, tlie nodal curvc will
p:uis Gxelusivcly through thc shadcd portions.
When w =3, ?t=3, tlie nodcs M-e thc straight lines par:illct
tu thé ed~cs shown in Fig. (3G).
197.] EFFHC'T0F SLIGIITIRREGDLARITIES. 259
Thc iMt ca~c which we shd! consider is obtaincd by putthi~
or, if thé factorscon'esponding to thé edgcs be rejected,
c(4co~l)cos~+Deos~~c~l)=0.(0)
o 4cof:l(4
-1cos-.+Deos-- 4C08 ce
-1 -0.(0).M o <x\ a
If C or D vania!), wc feU! back on tl)e nodai Systems of thé
eomponent vibrations, consisting of straight lincs paraUel to titc
edgcs. If (7=~, our équation may bc written
of whieh the nrst factor rcpresputs the diagonal ~-)-~=~ a,nd
the second a hyperboHc curve.
If (7=-7), wc obtain the same figure re]ati.vc!y to t!)e othcr
diagonal'.
~98. Titc pitch of tlie natural modes of a. sqaa.re membrane,
which is nearly, but not quite aniform, may be nn'estigatcd byt he geucra] method of § 90.
We will suppose in. thc first place tha.t w a.nd ? M'c equal.In. this case, when thc pitch of a umform membrane is givcn,the mode of its vibration is comp!etc!y determiued. If we now
conecive a variation of dcnsity to eusue, the natural type of
vibration is in gênera! modincd, but thc period may be calcutated
approximatcly without aHowanco for thé change of type,
Wc have
of which thé second terni ifl the increment of T due to 8p. Hence
ifwoecos~ n.nd P dénote thé v~lue cf~ previousiy to variation,
we have
9. T) s i 4ff"8p ~H7T.T .B~~V, 7p~ 1 ,,en=1-
4
rnÕp m~r_r. ~in2 ~~z~r,y
~,r,rly. (1)~=~ n ~s' ~a ~y.(l).(1 o. PO (1 a
'It()n)~,J')).<<tfrt'c~.<<'<<{?,p.l29.
17--2
2GO VIBRATIONS 0F MEMBRANES. [198.
For exemple, if thcrc bc a small load Jtf attached to thc middie of
the square, 1r
in which sin~ ~~Tr vanishes, if be cvcn, and is cqual to unity, if
bc odd. la thé former case thc centre is on thé nodal line of
thé unloaded membrane, and thus thé addition of thé load pro-
duces no result.
When, however, M and n arc uncqnal, the problem, though re-
maining subject to the same gencral principles, presents a pccn-
liarity different from anything we have hithcrto met with. Ttie
raturai type for thé unloaded membrane corresponding to a speci-
fied period is now to some extent arbitrary; bnt the introduction
ofthe load will in général removc the indeterminate élément. In
attempting to calculate thé period on tlie assumption of thé undis-
turbed type, the question will arise how the selection of tho undis-
turbed type is to be made, secing that there are an indefinite
number, which in thé uniform condition of thé membrane give
identical periods. Thé answer is that those types must be chosen
which differ Innnitely little from thé actual types assumed under
thé operation of thé load, and such a type will bo known by thé
criterion of its making thé period calculated from it a maximum
or minimum.
As a simple example, let us suppose that a small load Jt~ is
attached to thé membrane at a. point lying on the line x = and
that we wish to know what periods are to be substituted for t!ic
two equal periods of tlie unloaded membrane, found by making
= 1, M== 2, or ?~ = 2, M= 1.
It is clear that the normal types to be chosen, arc those whose
nodes are represented in thé first two ca~es of Fig. (32). In tlie
first case thé incroase in thé period due to tbe load is zero, which
is the least that it can bc; and in thé second case the increase
is the gréâtes possible. If /3 be thé ordinate of Jf, the kinetic
energy is altered in the ratio
198.] SOLUTIONS APPLICABLE TO A TRIANGLE. 261
whilo e ?)'=P'-2
Tito ratio eLaractcristic of thc interv~l betweoi t!)c two uaturnl
toucs of thé loadcd membrane is thus approxnna.tcty
If = ~<ï, ncither pcriod is aHected by the load.
As another example, thc case, where thé values of w and
are 3 and 1, considered in § 197, may Le referred to. With a. !oad
in the middie, ttie two normal types to bc seleetcd are those
corresponding to thc last two cases of FIg. (3't), in thé former
of winch the load has no efTect on tlie period.
The probleiii of determhung the vibration of a square mem-
brane winch carries a relativcly heavy load is more dIiHcuIt, and
we shall not attempt its solution. But it may be worth while tu
rccali to metuory thc fact that the actual period is greater than
auy ttiat can hc calculatcd from a hypotlictica.1 type, winch dinars
froui tlie actual one.
199. The preceding tlicory of square membranes ine!udcs :).
good dcal more than was at first iutcudcd. Wheuevcr in a vibrat-
ing systom certain parts remam at rest, t!iey may be supposcd to
be absohitelynxed, and \ve thus obtain solutions ofothcr questions
than t!)osc origmaUy proposed. For example, in thé present case,
'whGrcvcr a diagonal of thé sqnaro is nodal, we obtain a sohttioti
apphcabte to a membrane whoso fixed boundary is an isoscelcs
right-angled triangle. Morcovcr, any mode of vibration possible to
tho triaugle corresponds to sotno natnnd mode of tlie square, as
may ho scen by supposing two triangles put togcther, tlie vibra-
tions being equal and opposite at points which are thé images of
each other lu thc common hypothcnu.se. Undor thcsc circum-
stances it is evident that thé bypothenuse wou!d remain at rest
witttont constraint, aud tl~crcfbrc tlie vibration in question is iu-
cludcd among those of wttich a complète square is capable.
Thc frequency of thc gravcst tone of tlie triangle 1s found by
puttiug ?~ == I, n= 2 in the formula
r/~and is thercforc coud to'1
2ft
2G2 VIBRATIONS 0F MEMBRANES. [199.
Thc next tone occurs, whcn M =3, ?: = 1. lu this case
as might also bc seen by uoticing that thé triangle dividcs itself
into two, FIg. (37), whose sidca arc Icss than those of thé whoïc
triangle in the ratio \/2 1.
For tho tlicory of thc vibrations of a membrane whose bound-
ary is in thc form of an cquilatend triangle, thé reader is refcrrcd
to Lamd's 'Levonssur l'élasticité.' It is provcd that thé frcquency
of thc gravest tone is c /t, wlicrc A is tlie hcight of thé trianghi,
which is thc same as thc frequeucy of tlie gravest tone of a square
whosc diagonal is A.
200. Whcn thc fixcJ boundary of thc membrane is circular,
thé first step towards a solution of the probicm is thc expression
of thc général diHcrcntiaI cquationin polar co-ordinates. This
may be effected analytically but it is simpler to form the polar
cquation de novo by considering thc forces whicli act on thé potar
etemcnt of arca ?' dO t~ As in § 194- the force of restitution acLing
on a small arca of tho membrane is
and thus, if TI p = c" as before, tlie equa-tiou of motion is
The subsidiary condition to bo satisncd at the bouadary is that
w=0,whcn?'=f/.
In order to invcstigatc thé normal component vibrations we
ha.vc nn\v to assume that is n harmonie fonction of thc time.
't'hus, if ~cc cos(~<–e), and for thc sakc of brcvity we writu
/) c = /< the rhfï'crcntia! cquation appcars in the form
2G3200.]
l'OLAR CO-ORDINATES.
In which is thc ruciproca.1 of a liucar quantity.
Now whatevcr ma.y bc tlie nature of as & functiou of ?' and
it eau be cxpMiJed lu Fonrier's series
M =w. + cos (~ + al) + M~ cos 2 (~ + a.~) +.(3),
in which &c. arc fuuctions of but not of The result
uf snbstitutiug froni (:;) In (2) may be written
thc summation cxtcnding to all mtcgra.1 values of ?:. If wc
multiply this équation by ces M(~+ aj, and integrate witli respect
to betwuen thé limits 0 and 27r, wc sce thttt each term must
vanish separately, and we thus obtain to dotermmG as a
function of r
in which it is a mattcr of indirfcrcnce whcther the factor
cos n (~ + a,,) bc supposcd to be includcd in or not.
Thé solution of (4) involvca two distinct functions of r,
cach multiplied by an arbitrary constant. But one of thèse
functions becomes Infi nite when )' vanishes, and the corresponding
particular solution must be cxctuded as not satisfying the prc-
scnbed conditions at thé origin of co-ordinates. This point may
bc illustratcd by a roforeiice to the simpicr equation derived from
(4) by making K and ?!. vanish, when the solution in question
ruduccs to to=Iog?', which, however, does not at tlie origin
satisty \7~ = 0, as may bc scen from the value of inte-
grated round a small circle with the origin for centre. In like
tna.uner the comptctc Intégral of (4) is too gencral for our
présent purpose, since it covers thé case in which thé centre of
tlie membrane is subjected to an exteriial force.
Thé othcr function of )', which satisfies (4), is the Bessel's
function ofthc border, dcnoted by (~?-), and may bc cxpressed
i)i several ways. Thé asccnding sories (obtained nnmcdiately
from thc difrerential équation) is
264 VIBRATIONS OF MEMBRANES.[200.
which is Pessel's ongiua! form. From this expression it is évident
t!)at J,, and its differchtia! coe~eicnts with respect to z are aiwaya
less than umty.
Ttie aseending séries (.5), though InHnitc, ia convergent for all
values of~ aud z; but, -\vhen is grca,t, the couvergcncc does not
Lcgin forn. long time, and then thé séries bccomes useless as a basis
for nuincrical calculation. In such cases anot)ter series procecding
l)y desconding powcrs ofmay
Le suLstituted with ttdvantagc.
This séries is
it terminates, if 2~ bc cqual to an odd Intcger, but otherwise, It
runs on to innnity, and becomes ultimately divergent. Neverthelcaa
wlten z is grent, thé convergent part may be employed in ca~cula-
tion for it can be proved that thé smn of auy nuinber of term~
differs from the true value of thc function by less than thé last
tûnn inctuded. Wc sba,U ha.ve occasion later, in connection with
anothcr problem, to consider thé dérivation ofthis descending series.
As Besscl'sfunctiohs are of considérable importance in thcoreti-
cal acoustics, I have thougbt it advisahie to give a table for thc
functions J,, and extracted from LommcI's' work, and due
Lommd, $<;<(~'< «&cr clic /?M~'t-c';<) FtOtc~fn. Leipzig;, 1868.
200;]DESSEL'S FUNCTIO~S.
2G5
~) ~(~=
~(=) ~.(~')
~(~_
0.0 1.0000 0.0000 4-5 -3205 .2311 9-0 -0903 -2453
0-1 -9975 -0499 4.G -2i)Gl 1 6 f; 9-1 -1142 ~) -2324
0-2 .9900 -0095 4-7 9 3 -2791 1 9-2 -1367 -2174
0-3 -977C -14834-8 -2404 4 -2985 9-3 -1577 -2004
0-4 -9604 -I960 4-9 -20!)7 -3147 9-4 -17G8 -I81G
0.5 .93~5 -2423 5-0 .1776 -3276 (-) 9-5 -1939 -1G13
0-6 -9120 -28G7 5-11 -1443 -3371 9-G -2000 -1395
0-7 -88)2 -3290 5-2 -1103 -3432 "~) 9-7 -2218 -116G
U-8 -84C3 -3(!88 5-3 -0758 -34GO 9-8 -2323 -0928 8
0-9 -8075 r) -4000 5-4 -0412 ~? -3153 9-9 -2403 -0684
1-0 -7~2 2 -4401 1 5-5 --OOG8 -3414 10-0 -2459 -0435
1-1 -71!)C -4700 .6 6 +.0270 0 -3343 10-1 -2490 +-0184
1.3 -C7)l 1 -4983 5-7 -0599 -3241 10-2 -24% --OOGG
1.3 -6~1 -5220 C-8 -0!)17 -3110 10-3 -2477 7 -0313
1.4 -MG9 9 -541U 5.9 .1220 -2951 10-4 -2434 4 -0555
1.5 -~118 -5579 G.O -150G -27G7 7 10-5 -23GG6 -0789
1.6 -4554 -5C99 G-l -1773 -2559 10-6 -2276 6 -1013
1-7 -980 -5778 G.2 -2017 i -2329 10-7 -2164 -1224
1.8 -MOO -5SI5 6-3 -2238 8 -2081 10-8 -2032 '1422
1.9 -~818 -5812 6.4 -2433 3 -1816 10-9 -1881 -1604
2-0 -2239' -57C7 6-5 -2601 -1538 11-0 -1712 -17~8
2-1 -!66C -5C83 6.6 -2740 -1250 11-1 -1528 -1913
2-2 -1104 -5560 6.7 -2851 -0953 11-3 -1330 -2039
2-3 -0555 -5399 6-8 -2931 -0052 11-3 -1121 -3143
2-4 +-002;') -5202 6.9 .2981 -0349 11-4 -0003 -3225
2-5 --0484 -4971 7-0 -3001 --0047 11-5 -OG77 -2284
2-6 -09G8 -4708 7-1 -2991 1 +-0252 11-C -044G -2320
2.7 -1424 -4416 7-2~), -2951 -0543 11-7 --0213 -2333
2-8 -1850 -4097 7-3 -288~ -0826 11-8 +.0020 -2333
2-9 -2~43 -3754 7-4 -278<! -1096 11-9 -0250 0 -2290
3-0 -2601 -3391 7-5 .2663 -1352 12-0 -0477 -2234
3-1 -2921 -3009 7-6 -251G -1592 12-1 -OC97 -3157 7
3-2 -3202 -2613 7-7 -2346 -1813 12-2 -0908 -3060
3-3 -3443 -3207 7 7-8 -2154 -2014 12-3 -1108 -1943
3-4 -M4:) -1792 7-9 -1944 -2192 12-4 -129G 6 -1807
3-5 -3801 -1374 8.0 -1717 -2346 12-5 -146U -1655
3-6 -3918 -0955 8-1 -1475 -3476 6 12-6 -1626 -1487
3-7 -3902 -0538 8-2 -1222 2 -3.580 12-7 -1766 -1307
3-8 -402G +.0128 8-3 -0960 -3M7 7 12-8 -1887 -1114
3.9 -4018 -.0272 8-4 -0692 -2708 8 12-9 -1988 -0913
4-0 -3973 '~) -0660 8-5 -0419 -2731 1 13-0 -2069 -0703
4-1 -3887 -1033 8-6 +-014G -2728 13-1 -3129 -0489
4.2 -37GG -1386 8.7 --0135 -2G97 13-3 -3167 7 -0271
4-3 -3610 -1719 8-8 -0392 ~), -2G41 13-3 -2183 --00.~2
4-4 '3423 -2028 8-9 -0653 -2559 13-4 -3177 7 +-01G6
originatly to Hansen. Thc functions J, and J, arc conncctcd by
thc rela-ticii = J~.
266 VIBRATIONS 0F MEMBRANES.[201.
201. lu accorda.nce with thc notation for Bcsscl's functions
the expression for a nurin:),! component vibration may thercfure bo
written
?(/=P~(/t)') cos~~+cf) cos(~+e).(1),
a.]ul tlie boundary condition requircs that
~(~)=0.(2),an équation -whose roots givu the admissible values of /c, am
tli ère fore of~).
The complete expression for w is obta.ined by combitung th(
particular solutions embudicd in (1) wit)i all admissible values u:
und M, and is ncccsstn'Hy general enough to cove).' any initif),
circumstanccs thatmay
beimagiucd. We conclude tliat an~
i'Huction of r and 0 may be exp:mdcd within tlic limita of thc
circle ?' = a in the series
~=S~~ (/er) (~ cos 7~+-~sm~).(3).
For overy intégral -aluc of ? thcrc are a series of values of
given by (2) and for cach of these tlic constants <~ and arc
arbitrary.
Thc détermination of the constants is effected in thc usual
way. SInce tlœ energy of the motion is cqua.1 to
and whcn expressed by mca.ns ofthe normal co-ordinatcs can onlyinvolve their squares, it fullows tliat thc product of any two of tlie
terms in (3) vnnisbes, when integra.tcd over tlie area of tho circle.
Tims, if wc multiply (3) by ~(~')cos~ and integratc-, wc
find
-7r~[,7:.(~)r~(5),
by w)iich is dctermincd. Thc corrcsponding formula, for -~r is
obtaiMcd hy writing sin~ for cos?: A mctiiod of cvaluatingthc lutcgral on the right will be givoi prcscntly. SiucG and
cacii contam two terms, one varying as eos~~ and thc other a~
sui~ it is now évident how t)ic solution may be ad~ptcd so as to
:'grec with arbitrary initial values of w and w.
202.] ORCULARBOUNDARY. 267
202. Let us now examine more pa.rticukriy tlie character of
thé fondamental vibrations. If ?t=0, iv is a function ofrouly,
that is to say,the motion is synnnetrical with respect to thé centre
of thé membrane. Thc nodus, If any, are thé concentric circlus,
wliose équation is
~(~-)=0.G).
Whcn has an integral value dinurcut froin zéro, w is a func-
tion of 0 as well as of 7', and thé équation of tlie nodal system
takes thé form
J,(~?-) cos n (~-c<)=0.(2).
Thé nodal system is thus divisible into two parts, thé iirst eon-
sisting of tlie concentric circles represciited by
J,(~.)=0.(3),
and thé second of tlie diameters
wherc is an integer. Thèse diametcrs arc ?!. in number, and
are ra.ngcd uniformly round tlie centre in other respects thcir
position is arbitrary. The mdn of tlie circular nodes will bc in-
vustiga.ted further on.
203. Thc important interal formula
wliere /t and ?' arc different roots of
~(~)=0.(2),
may be verified analytically by mcaus of thé differential equations
s:Ltisned by <7,.(~), J,.(K')-); but it is both simpler and more
instructi-ve to begin with thc more gênerai probicm, whcre the
boundary of thé membrane is not restricted to be circular.
Thc variational equation of motion is
where
2G8 VIBRATIONS 0F MEMBRANES.[203.
and thcrcforc
In thèse équations w refera to thé actual motion, and to a hypo-tbctical dispiacetnent consistent with thc conditions to which tlie
system is subjcct.cd. Let us now suppose that tlie system is exe-
cuting one ofits uormal component vibrations, so that w = M and
while 8w is proportional to anottier normal function v.
Siucc =p c, we get from (3)
whcrc /< bears the sa.mc relation to that /< Lcars to u.
Accordingly, if thc normal vibmtions rcprescnted by M and
hâve diQcrcnt penods,
In obtaining this rcsult, we hâve madc no assumption as to the
boundary conditions bcyoad w!~t is impjicd in thé absence of ré-
actions uga-inst a-ccelcration, which, if they existcd, would appearin thé fundamental équation (3).
If in (8) we suppose /c' =A-, thé equation is satisfiedidenticallv,
and we cannot iufer thc value of~i~cfZy.
In ordcr to evaluatc
this intégral wc must follow a ratlier différent course.
If u and v be functions sa-tisfyiug within a certain contour the
equations \7"M + = 0, + A: = 0, wc have
203.] VALUES0FYNTEGRATEDSQUARES.2G9
byGreen'athcorcm.Letusnowsupposethat'Uisderivedfromubyslightiyvarying/C,sotliat
v=it ~tc.8K,=K bic;~=~+-,0~~=~+0~;a/<substitutingm(10),wcHnd
or,if u vanish on the boundn.ry,
For tlie application to a circular arca of radius r, we have
and thus from (10) on substitution of polar co-ordinatca and integra-
tion with respect to 6,
Accordingly, if
and /e and ?' be different,
an equation first proved by Fourier for the case when
Again from (12)
dashes denoting differentiation with respect to Kr. Now
97f)VIBRATIONS 0F
MEMBRANES.[20:}.
and thus
0
s'hl~
as tlie towitli fixed
bouu~laries,
tosi 1111)li fy tlic
cxpl'ossions for 'L'fLIlcl Inrom
~=~(~-)cos~+~7,
(~-)sm~j. n)wcfind
Md a suuihu- equation for The vn.).w. nf
~=-
the work ~lone by t]mimpressec] forc:es cluring a ]iypothetic.vl clisplacement 8~«~ so t],at tif Z be thcnnpresscd for~ reckoucd por unit of area,
't
lvhe" 0 aua a.re amalgamatod. We 0 thcn have
~.rs~ tliat the initial velocities are ,<,r.that assllmed
influence 1constant pressure Z; thusn 1
20 4. jJSPECIAL PROBLEMS. 2711
No\v by thc difforential équation,
andthus
thc sommation extchding to all the admissible values of/c,
As an example of/b?'ce~ vibrations, wc may suppnsc tnat sti)t
constant with respect to spacc, variGS a~ a harmonie function of the
timc. Titis tnay bc takcn to reprosent roughiy thé circumsta.nœs
of a small mumbrane set in vibration by a train of aerial wavcs.
If Z= cos wc nnJ, ncarly as bef'ore,
Thé forced vibration is of course independcnt of It will bc scen
that, while none of thésynunetrical
normal componcntsare
missiug,
thcir relative importance mny vary grca.tly, especially if there be :).
ncar a.pproachin value bctwccn y a.nd onc of thé séries of quauti.
tics If thc approach be vcry close, tlie cScct of dissipativo
forces must be included.
205. Thé pitches of the various simple tones Mtd thù radii or
thé nodal circles depend on the roots of tlie équation
(~)=
J,. (.)= 0.
If thèse (exclusive of zero) ta~eu in order of magnitude be
calledd z(~)
2; then thé admissible values of~ca e z", z" w. 1011 10 a n118SI e ues 0 p
272 VIBRATIONS 0F MEMBRANES. [205.
are to bc found by multiplying thc quantitios by c a. Th.c
particular solution may thcn be writteu
=~)
cos + sin ?~) cosf~ 6,'4 (1).
Thé lowest tone of the group 7t con'csponds to 2! a.nd since in
this case J,,(~)
does not vanish for any value of r less than a,ca
there is no interior nodal circle. If we put s = 2, <7,, will vanish,
when
<s) ")
ar
2
that is, when r = a
~n
which is the radius of thé one interior !iodal circle. Simi!arJy
if we take tho root wc obtain a vibration witli 1 nodal
circles (exclusive of the boundary) whosc radii are
AU the roots of tho equation J,, (~a)= 0 arc re< For, if
possibtc, let Ka = X + bc a root then ?'0 == t~ is also a root,
and thus by (14) § 203,
Now (~r), J~ (~) arc conjugate complex quantities, whose
product is necessarily positive so that theaboveequa-tion requircs
tha~t either X or /t vanish. That X cannot vanish appears from
the considération that if rca were a pure ima.gmary, cach term of
thé ascending series for .7,, would bc positive, and thcrefore t!~o
sum of thé series incapable of vanishing. We conclude that
/n=0, or that /tis real'. Thé same result might be arrived at
from thé considération that only circular functions of tho time
ca.n enter into the analytical expression for a normal component
vibration.
The equation J" (z)= 0 bas no equal roots (exccpt zero). From
equations (7) and (8) § 200 we get
205.] ROOTS OF BESSEL'S FUNCTIONS. 273
whcnco we see tha.t if <7,/ vanished for thc same -va.hie of.s:,
would a.)sovanish for U~t ~atuc. Butinv)i'tueof(8) §200
this wou)d rcquu'e tti;Lt H~ ttiu functions vtmi.s)i fm' ttic va.lnc
uf iu (question'.
20G. Thé actu:)! Yalucs of .3~ m~y bc found by into'pola.tion
fron Hansen's t:).hl<j.s su f:t.r a.s thèse uxtou) ) or furmuRc ma.y be
catcutatcd froni t!tu duscunding séries by t)tc niettiod of suceuH.stve
a.pproximatiu[), cxprcHsiu~ thé routs dirccLiy. For t!)c it~portant
case of thc sytiunctricat vibrations (~=
0), t)[C values of J~ay bc
found frutn thé fuliuwin~, ~iveti by Stokcn'~
Thé lutter séries is convergent enough, even for tho first root,
con'csponding tos== 1. 'J'!tC series(1)
will sunice for values of
grcate;' t.h:munity; but thc firsb root must ho cn.Icutn.tcd
indepcndcntly.Thc
:K'co]np~)yingt!tb)e
(A)is t~cn from
Stokcs' pa.pcr, with a sti~ht dittLit-ence of notation.
It wHI be secn cither frum tho fo)'mul:H, or t)ic t:iUe, that tho
tlifTcreocG of successive mots of)ngh erder i.s
n.pproxinmtcly 7r.
Tiu.s is truc fur all vaincs of ?~ as is évident from t))e dcscending
series(10) § 200.
M. Bourgct hn.s gtven in his tncmoir vcry claborate tab)c3 of
the frcqucncies of thé diiTcrent sirnptc toncs and of tho rn.dii. ofthe nodal circles. Table J3 i)t.cludes tlie values ofz, whicii SH.tis(y
J,.(.!),for~=0,l,5,s=].,2, 9.
BourRet, "M~mnircsurIotnnnvGmcntYibrntoiro des mf'mbrMtfs eircu!Mrea,"
~inn. de !fo~ onrwft~, t. tu., 1H(!(!. In ono j~nssnRo DonrHet implifs t)int ho1)M provod thnt nn two Hessefti functtons of intf~r'~ order cnn havo thf.' HMnn root,
buticannotfmtithat La hns donc ho. Tho thf'oron), howovcr, is pr<)t))t)))y truc;
in thti cnso of functioxn, wlioso ordurs JiOur t'y 1 or 2, it mny bo easity provud frotn
t)tofnrtnn])nf'f§2()f).
C'~Mt~. ~t<7. 7'M); Vt)I. tx. On thé num'jno~ C)t!cu]&ti):t of ct~ss of dof!-
nitc intégrais and infinitc Rerics."
n. 18
274 VIBRATIONS 0F MEMBRANES. [206.
ÏABLR B.
s )t.=0 M==l ~~2 ~=3 3
M=4
M=5 r
1 2'40t :~832 f''I3.') G'370 7~SG C) 8'780
2 n'O 7'0!G fi 8'4!7 î 9'7M H-r'G.t I2'{M
3 8-G54-1 10-173 3 11-G20 13-017 î l-t-373 3 1;~70()
4 11-792 13-323 3 H-7UG 1G-2244 17-G1G 18-983
5 14~:il 1 lG-t70 17-OGO H)'.tl0 20-827 22-220 0
C 18-071 1 19-G1C 21-117 22-f!83 21-018 2~-431 1
7 21-212 22-7f:0 2t-270 25-74!) 27-200 28-~28
8 24-353 25-903 27-421 28-909 30-371 1 31-813
9 27-4U44 29-047 3U-5711 32-050 33-512 34-983
Wth'n is consido'ahic thc calculatinn of tlie carlicr mots
becomes troubicsono. Forvcry hig!) v:du(;s of approxi-
nia.tcs to ratio ofo<~)f~it.y,
as xmy bc scL'n frmn t])u consittumtion
thut thc pitch of thc gm.vcst tonc ofn. very acutc sector tnust tend
to comcidu wit)i th:).t nf a. tong pamiiul strip, whosc width ici c~ua.1
to tttC grcatcst 'idt)i of thc scctor.
TAI!LE A.
a ~fur.)=0. 0 DUF. ~fM-(:)-0. Diû'.7T )df TTOI'.Ti(z) O. Difl:
1 '7'S .n. 1'2~7
2r7~7L ~0 i;~3
~1:
C)
·1
3'75;H'J!)~1
:R:3l'()(lii:3
.) 4 '¡ fi~7'!J!)!¡33
4'2111 1H)()~~ î
~1~
7l
7
(j'7fd!)~!I!1!lï
6 .139l'(I OU3
~Sî
S
-7516'!J!J~¡8
8'2.I,j.1l'll(I():ï
S~,E
10 J'î;rl3'!ln!i9
1(1~`?.IG33lUU.1
t~ irt'i~JJ''J iin~~ Iww~
~~9 1.000312 n'7~1 12-2.1G!)
20G.~ NODAL FIGURES. 275
Thc ~gnrcs rcprcscnt thc more Important normal modus of
'vibnLtIon, ami the uumbci's afHxcd givc thu ft'cquenRy r(jfut')'cd to
18–2
2 7 G VIBRATIONS OF MEMBRANES. [206.
the gravest as unity, to~ether with thc radii of thc circu~ar nodes
cxpresscd as fractions of thc radius of thc mcmhra.nc. Iti the case
cf six noda.1 diiuncters t!)C frcqucncy statcd is the rcsult of a. rough
calculation by myscif.
Thé tones eoi'rc.spoïldingto tlio varions fund:unental modes of
thé circular monbr.mc (!o not bclong to a, htu'momc scale, but
therc are one or two n.pproxima.iety Imrmouic relations winch may
bc worth notice. Thus
x l-5!)t = 2-125 = 2-136 ne:u')y,
x 1-59.1. = 2-G57 = 2-65:; nearly,
2x 1-59~=3-188=3-156 nearly;
Ho that thc four gra.vest modes with nodal diamctcrs oniy would
give a consonant chord.
Thé arca. of tho membrane is (lividcd into serments by the
nodalsystcm m snch a manncr that thé siga of thé vibration
changes whencver a. Ticdciscrosscd. In those modes of vibration
which hâve nndal diameters thcre is Gvidcntlyno displaceme])t of
the centre ofinertia of thé memitrane. In thé case of symmfttri-
cal vibrations t))c disp]aceinent of tbe centre of inertia is propor-
tiona.! to
an expression which does not vanish for any of the admissible
values of /c, sincc (.?) ami '~(~) can)!ot vanish simuitancousiy.
In all thcaymmct.)'ic:).l
modes thcrc is thcreforc a. dispia.cctncu't of
thé centre of incrtia of thc membra.nc.
207. Hithcrto wc ha.vc supposcd thc ctrcu!ar a.rca of thé
mcmbr.'me to hc complutc, and thé circumfcrcncc on!y to he
nxcd but it is évident that our thcory virtually includcs thé
solution of ûther prohicms, fur exampic–some cases of a mem-
brane boundcd by two conccntric circles. Thc cow/~e theory
for a. membrane in thé form of a ring requircs tbc second Besscl's
fnnction.
Thé probtem of thc membrane in the fonn of a scmi-circle
inay ht.' re~ardfd as ah'uady so]ved, since any mode of vibration
uf whkh thc soni-circlc is capable !nn.st be app!ieab]c to thé
207.]] FIXED RADIUS. 377
complète circle a!so. In order to sce this, it ia on]y necessary
to attribute to any point in ttje conpicmenta.ry semi-circle ttic
opposite inotiou to th~t whic)i o!)tn.ins at its opti.ca.l image in
thebounding diameter. Dus line will ttien requu'e no constraint
to kccp it no(h).l. Simila.)' cotisidcrations apply to auy sector
whoso angle is an atiquot pru't of two right angles.
Whe]i thé opening of thé sector is arbitrary, the prohlem
may be soh'ed in terms of Bess~l's fonctions of fractional order.
If the fixed radii are 0=0, = /3, thé particular solution is
who'c is an intogcr. Wc Pcc th:).t if /9 bc an a.liqllût part of 7r,
f7r /3 is integr:). :un.t thé suJutiou is inctuded amoug those a.lready
used for tlie complète cirelc.
Au Intcre.stmg ca.su is when /3=27r, which corresponds to thé
pl'oblum of cotti[)!(jtc circle, uf whici) thé radius ~=0 is cou-
str:uned tu bo nodal.
Wc Lave
w =Pt7)(. (/f)') sui eoa
(~ e).
When v la even, this gives, as might be expected, modes of
vibration possib]e without the coustraitit; but, -\vhen v is odd,
new modes make their appearance. lu fact, in tlie latter case
thé dcscc~~di~)g séries for V terminâtes, so tliat tl)e solution is
cxprcssibic in nni.tc ternis. Thus, whcn ~=1,
The values of /< are given by
siu /<<t =0, or /M =7/t-n-.
VIBRATIONS OF MEMBRANES. [207.278
Thusthc circula)- nf)th;sdiv!(tc thc ~xcd radins into equat
parts, tunithe MCt'ic'sot'tuntj'~ ~nn hfn'mcnic scittu. Int~e
e~suut'tho~r:LVust,tm)dc,thL!wt)()luoi'thûn)u)nbrau' is~ta~y
)nu)ncntdcHcctcdo)t thus.uuc sideof its c(~!i)ibriuta positon.
Ibis runuu'kubtcLh~t, t,)tc:Lpp[i(.Lt,i"n ut' L)m cuntit.t'aiuLtuthc
radins ~=0 innkcs Lhuprobtetn casier t)t!Ui buturu.
If wc t~kc t~= 3, Hic solution is
In this case thc nodal l'asti arc
2-n-
"='3' '=~T'
and tlie possible toncs arc givcn by thé cqun.ti.on
ta.n/ca=Kf!(4).
To caleulatc thc roots of tan = x wc may assume
a;=(m-)-~)7r-y=Jr-~
whcrc y is a positive quuntity, winch is smaU \v!ien is large.
Substitutiug this, we find cot = JV M,
wLcuce
1/1 V° Sy" I7'/~x(~~+r~)-'3--ii-J ~L x x 2 -'3-15-31¡-
This cquatioti is to bc solved by successive approximation.
It will rcadity bu found tha.t
2/=Y-'+~~+~~+~~Y-J = ,rl +3
Al]ij
X-510~
i-~ +.
207.] EFFECT0F SMALLLOAD. 279
so that thc roots of tMi <c= x a.re given by
whcro J\"==(M+~)77-.
In thc Hrst quildrn-nt tho-c is no root aftcr xcro since tana; > a',
n.nd in thc sccottd <p)adnn)t thcrc is noue ))cc:m.sc thé signs of
a;:uitl L:ui.Ba)-eopp"si~. Thé fii'stroot.iit'tc)-zéro is thus in
thut)nrdqu:u1)-:L))t, c(.rrcspondin~to~=l.Evcn m tins case
thu sft-ic.s convoies suHicicntty to ~ivc thc v:t)'tc of' Die root
wiMi œnsitierahic :t.ccur:Lcy, whitu fur I)i~hcr va.lucs of' ?~ it is
a)i t)~ con)d )'e (h;sh-c'L Thé ~-tn:~ vahtcs of~ 7r m'o l--t303,
2'-i.~0, 3'-t.70U, 't'-i74.7, 5'4Mi~ C-4H-H-, &c.
208. Thc cH'cct on thc pcriods of n. sU~))t incqnd~y in thc
dunsity of tlic circular tTtC-nthnmu ))t~y hu invc.stig~tctt hy thé
gcnut'a.) xictinxt § !)0, <'f ~hich scvcrid ux!LU)ptt;s h:w(j :Urca<)y
Lcun '(.'n. IL wtH hu snH'iciunt heru to considct- tlie case of a
s)n:dl io:Ld ~:Ltt:).c)tcd to thc monhnuie at a, point wim.se radius
vector i.s )'
We wi)l t:dœ first the symmctncal types (-M=0), which n~y
sti)t hc supposai to :i.pp~y notwit.hs~ndiogtlic prcseuec of Thc
knictic cnergy2'is (C) § 20-t :dtcrcd from
p7T~ J;' (~) t0 ~TT~ J~ (~) + (~).
whcrc P, dénotes thc value of~ whcn thcre is no !oad.
Thé unsymmctt-ic:U nnrt~a.1 types are not ful)y dctcrmhmtc for
the unioadcd moubrane Lut foi- thc présent purpd.se tttcy must
bo tfdœn so as to nmko thc i-c.sultin~ pcnodsn. ma-xinunn or
minimum, tliat is to s:).y, so th.Lt thc cH'cct of the load is thé
greatest ~td lc:tst possible. Now, since a. !oad can ncvcr r~isc
thc pitch, it is c)ca.r th:Lt thé inthmnce of tho !oad is tlie lu~st
possible, viz. xcro, whcii tlic type is such that a uod:d diamctcr (it
is mdiHcrcnt winch) passes t!n-ough the point nt which t!)0 lo~d is
ahtachcd. Thc untoadcd mcmbmnc must bc supposud to h:ivu two
couLCidoit pcriods, of which o;~ is untdtcrud by tho addition of thc
280 VIBRATIONS 0F MEMBRANES. [208.
load. The other type is to be chosen, so that the fdtcration of
pcriud is os ~rca.t as possible, which will ovidoiHy be t))c case
whun t)ic r.t.dms vecto!- )-' bisucts thc futg!c bctwcon two <uij:LCcntnud~l diatnutcrij. T)tus, if correspond to = 0, wc are tu take
'~=~~ (~)cosH~; i
so tha.t (2) § 204.
Of course, if r' bc such t!mL titc lu~d tics on one of tbc tiodal
circles, ncithcr pcrio<t is af!'uct.cd.
For cxampic, lut ~V Le at thn contre of t)tc membrane. J,. (0)
vanishcs, cxœpt whcn )t=0; ~ud (<))=!. It is on)y thé
symmctricai vibrations whosc pitc)) isin!)uc[]ccd by a central load,aud furthclu by (1)
~(~)~fil ni
0 ( ~no) P
By(G)§2()0 ~(.)=-~(.),
so that the application of t)tc funrmJarc()uircs oniy a ktlowlo~c of
thé va)ucs of' (2). whun (.2) viuushes, § 200. For thu gravcstmndc thc value of
J/(A-~) is -5190:}'. Whcn ~.0 is cousidor-
abic,
~~o~)=2-7r~
approximatoly so that for thc tngacr components thc influence of
thc !oad inaltcring t!tc pitcb incrua.scs.
Tbc it]f)uence of a smaUirreguiarity in disturbing the nodal
systcni may be ca!cutatcd froïn thé formula of § 90. Tbe mostobviuus cn'ect is thc brcakin~ up of nod:).! diamutcrs into curvesof hypcrbolic form duc to tbe introduction of suhsidiary sym-mctrical vibrations. In many ca.scs thc disturbance is favoured
by close agreerucut betweeu some ofthc natural puriods.
20!). We will next investi~ato how t])c natnral vibrations ofa unifonu metubrane are auected by a s)ight departurc from thé
exact circular form.
ThoBUfoeedingTa!nosnrc~proximnte]y -341, -271, -23~, '20(!, -187, A'p.
209.] NEARLYCI&CULAR DOUNDARY. 281
Wttatcver ma.y Le thé nn.ture of thé bounda.ry, t~ sa.tisfies the
cnuntion
whcre /c is a, constant to be dcto'mmcd. By Fouricr's thcorcm M
inny bc cxpundcd in tlie series
whcrc ~,w~ &e. arc functions of r only. Substituting i)i (1), wc
sec that must sa.tisfy
ofwhich thc solution is
M.~ ~.(~');
for, as in § 200, the otlier function ofr cannot appear.
Tbe général expression for M may thus bc writton
~==J.J.(/<-r)+~(/<:r)(~,cos0+7?,sin0)
+ + J.. (/~) (.1, cos H0 + 7?, sin )~) +. (2).
For all points on the boundary M is to vauisb.
In thc case of a nearly circulai' mombranc thé radius vector is
nearly constant. Wc may ta~e r=ft+8?', ~)' bcing a small
function of Hence thc boundary condition is
0=~[.7.,(~)+~(~)]+.
+ [' (~t) + t/ (/<:ft)] [J~ cos + sin tt0]
+. (3),
which is to hold good for aH values of
Let us considcr first those modes of vibration wMcb are nearly
symmutrical, for which therefore approximately
~=~.J.(~-).
A)) tbc rcmaining coefficients arc small relativcly to j~, since
thé type of vibration can only differ a little from w!iat it would
VIBRATIONS0F MEMBRANES. [209.282
bn, wcrc the boundnry an exact circic. Hcncc if thé squares of
th!'s)n:)H(jU:u)titn'"b'~)mittr.L,~)!jf'co!ri<
o L~. (~~) + '~o' (~)] + 'A (~~) [~i cos <9+ .B, sin <9]
+.)- (~) [. co.s ?t<9+ 7~. si n
/~j +.=(). (.t).
If wcintt'gratc
thi.scquattun widi
respect to butwcc!) titc
limits 0 aud ~7r, we ubtf).i)i
or
which shcws that thc piidt of tlie vibration is n.pp)-nx!)natu]y thc
s:unc as if thé radius v~ctot- ha<!uui~rtnty its ~e~/t ~<e.
This t-csnit idiuws us to f'm-m :i rou~h csLimatc of t)tc pitch of
any mcnil.triutc whuscboundary iij nuL cxtmva~mUy cjon'ratud.
]f o- dénote thu fu'c:t, su t)t:).t po- is t)tc )naHH of Ute whutc muni-
Lraue, tho frcqucjtcy of t)ie gt~vost {.une is approximatdy
2~
2-40.i.x~(6). ~P
In ot~cr to invcst.tc thealtcred type of vibration, wc m~y
mnltilly (-t) by eus y~, or sm and thoi int~-m-atc as beforuThu.s
Witen thc vibration is notnpproxim~cly symmctriciil, thc
question bocomes tuorccoinp)ic:).tcd. Tlie nor))i;d tn0()cs io;- t).
truly circuler mcmbrMG are to somcextcut Indetcnniuatc, but tin'
209.] NEARLYCIRCULAR.BOUNDARY. 283
irrc~Iarity in thé bonndru'y will, in gênerai, rcmovc tlie indcter-
minatcn<'SH. T!tC position ufthc uudal difuncters nmst ))utukcn,
no that Ute resulLin~ pct'io~s !n!Ly h:LV<! maxinium or minimum
values. Lot us, howuvcr, snpposL- t!u).t thc approxiinato type is
w=~t~<7,, (/fr) cos ~(9),
a.nd aftcrw~rds invcsti~atc how thé initia.! linc must bc ta~en in
ordcr that this form may )io)<! good.
A!l thcrcmaming coe~cients bo!ng ti'cated as small in compa-
nson with Jt., wc gct froin (4)
winch shcws tha.t tlic effective ra.dius of the mcmbra.ne is
or
Thc rn-tios of ~t,, a.ud 7?,, to A,. mn.y bo found as before by in-
tcgraiin~ équation (10) a.ftarinulLiptic~tion by cos sin )!0.
But the point of~rcatcst intercst is thé pitch. Tt~c initial line
is to bc so t:).kun as to )ti:~œ thcexpression (11)
a maximum or
minimum. If we refer ta a, lino fixed in spacc byputtin~a
instc:).d of we liave to consider t)~c dcpeudencc on a of the
quanti ty
r~eos~(~-a)~,
J0 8rcos' v (B u) clfl,
J o
which may aiso bc writtcn
VIBRATIONS0F MEMBRANES. [209.284
andisof thé form
J. cos~ fx + 2Z?cos t/x sin va + (7sin'x,
2?, (7 buing indepundcnt of a. Thure are according]y two
admissible positions for tho nodal diamctci's, one ofwhich makes
thc period a. maximum, and tl)e othci' a minitnum. T!)c- dianietcrs
ofonû set bisuct the angles bctween tlie diameters of thc other
set.
Thcrû are, howcvcr, cases whcrc thé nortna! modes remfun InJc-
tC!'minatc;,witich happcnswhcn thc expression (12) is mdcpendent
ût'a. This is t!ie case whcn S/' is constant, or whoi is pronor-
tional to cos For exa.mpic', if wcrc proportional to cos 2~,
or in ot))erwor<]s thc hnund:u'y wcro s]ight)ye!liptica),thc uodal
system corrcspunding to )t=2 (that consistingof a pair of pcr-
pondicular dinmctcrs) would ho arhitrary in position, at Icast to
this onict' ufapproxunation. But thé single diamctcr, con'cspond-
ing to !t=l, must coincide witit one of thé principal axes of
thc ellipse, and tlic pcriod.s will be diircrcnt for thc two a.xes.
210. Wc hâve SGGn that tho gravcst tone of a membrane,
whose houndary isappruxhnately circular, is ncarly the samc as
that ofa mcchanicaHy simil.'t.r membrane in the form of a. circle of
tlic samc mcan radius or area. Ii' thc arca of a membrane hc
givcn, thcre must evidenHy bu some furm of boundary for wltich
thé pitch (of thc principal tonc) is thc gravest possible, and this
form can he no othcr than the circle. Ju thé case of approximate
circuhu'ity an analytical demolistration may Le givcn,ofwhich thc
foUowlng is an outhnc.
The gênerai value of~ being
~~=~1,<(/<) +. +J,. (xr) (~cos~+J9sin~) + (1),
in which for the présent purpose tliecoenicients~ 7? arcsmaM
rc]ativcty to J, we nud from thc condition that M vanishes
wltcn ?' = ft + 8r,
J. (~) + J.' (~a) + ~J, (~). (~.)' +
+S [(J,(~)+ (~) 8?' + .l~eos ?~ + J3, sin ~)]= 0. (2).
Hence, if
~'= ~cos~+/3~in ~+ + ~cos/+ /3~sin/i~+ (3).
210.]FORM OF MAXIMUM PERIOD. 285
from wLieh we soc, as hcforc, that if tlic squares of tt)C small
(}uantitk's bc nc~1cct(~, <(/ca)=0, 01- that to tLis ordcr ofa.p-
proxima-tioti thé )nc:i.n radius is :).]so thc L'Huctivu radius. In
ontur to obtailt :L ctoscr n,pp)'oxi)n;ttiun \ve <h'stdutcnniuc ~1~
and ~o l'y multiplyin~ (2) by cus~ sin?;~ aud thoi in-
tcgr~ting butween <hc limits 0 aud 277-. Thus
286 VIBRATIONS 0F MEMBRANES. [210.
T))C <-)))cst.icnis i~ow us to thc sign
of thu i-hL-h:UKl muinbur.
If M= 1, :L'~t -ï Le wnttcu fur A:<(,
sothat
vani.shcs approximatdy by (7), .since in gcncml ,~=-and
In the p~sc.nt case ,(~)= 0 n~u-)y. Tt'ns f~<' = 0, as shou)d
cvidunUy b~ Lhc case, .sitjcc thc term iufjuu.stion rcprcMcnts mcrcly
disphtucmcntof thé cirdc wifLont an i~turatiun in thé f~nu uf
t)~ buuudary. Whcu = 2, (M) § ~UO,
is pnsitn'c for mtc-gra.l values of M grever than 2, whoi .!= 2-401.
For this purpnsuwn n):)y nvail ourseh'cs of a. thojron givcn in
Kiuma.nti's jf'«r<~e D~rc;c/N~<~e/i, to thc cft'cct th~t
ubithur "or< has a r<)"t (t)<hcr ttt:ni xo'(t) l(.'ss than ?t. Thc
di~reuti~l c~u~Lujn for may bc put lato thc furm
whilu luitiaUy J, und J,'(~s
wcll as.)
~~c positive. Accord-
in'~v- "-Lc~insLv increa.sing a.nd docs not cca.se to do so
°-log~°
before .:=?:, from whieh it is cicar tliat within tlie range= 0 to
210.] ELLIPTICAL DOUNDARY. 287
.3= M, ncither ,7~ nor can vanish. And siuco t/~and J' are
hothp'it~u:]tii~~M.itJ'('vsth~t,ii5anittt~gc'rgr('atcr
tha.n 2'4U~, f~t is positive. \Ve ccmdude tha.t, nn!cns c' /9~
0, all va.ni.s)), f~ is gycn-tcr than winch shews tlmt in thé
c)t.sc of nny mumbranc of appmxim:)tc)y ci)'cu)ar outHnc, tho circle
ofc~na.tat'cacxccedsthocu'cIcoi'ctjualpiLtdt.
Wc ttave seen that a good cstimatc of thé pitch of an npproxi-
matùly circulai' monbrancmay
bc oLtaincd frutn its arca a!onc,
but by tucims of c~nat.iott (~) a stil) ctoscr approximation ntay Le
cn't-'etL'd. Wc will apply tilis method tu thu case uf an ellipse,
~hosc sciai-axismaj~r
is Tt! anfl ucccntricity e.
'J'!mp(;)ar équation ofthubonndaryis
In which the term coutaining e* shouM bc correct.
Thu result may also bc expressed m tei'ms of c and the arca o-
Wc have
and thua
from wLidi we sec how smal! is the influence of a moderato ecccn-
tricity, whcn the arca, is givcn.
288 VIBRATIONS 0F MEMBRANES. [211.
211. Whca thc nxed boundary of a. membrane is ncitbcr straight
j)or<'h\!dar, tbcpr~b~'i.'t ofdck't'i-'i'jh.~ir.\iL)'.i<i~ prc.-cnts
difficultics witich iM gênera! could not bc cvc'reotnc wltliont thé
intro()uction of functions not hit!)crto discusscd or tabniatcd. A
partiat exceptionmnst bc ma()c in faveur of an uttiptic bonndnry
but for thc purpoHcs of t))i.s trc'ittisG thc i)npm't)).ncu of t-bu probton
is scarc<)y sufHciunt to warrattt thc i))tr(j(hK;t.ioti of compHcatcd
an;dy.sis. 'J'h(jr(':K)uri8thurnf"ru!'cfurrc<ltot)K'()ri~iu:d invusd-
gatx.n ufM. ~!athi(;u'. l, It will bH su~Hc'icnt to n)C))t,U))i l'f't'o that
t))C txjdtd systetu is composcd of tl)0 confocal cUipscs a.nd hypcr-
ho]as.
Solubtc cases m~y bc invonted by meaus of thc gcncnd
solution
!o=~l.J.(A:r)+.t- (.l..cos~+7?,.sin~).7,.(~-) +.
For exa.mp!c we might take
?~ = (~r)X.
J, (/<-r)cos
and attacinng dif1!rcnt v:duc;s to X, trace thc vfn-ious forms of
bonnd:u'y to which thc solution will thcu app~y.
U.scfui infortnation ~aysonictinics bc obtaincd from thé
theoron of§ 88, whicb aDows us to provc that anycontractioa of
thc iixcd bouudary ofa vibrating niembnmc tnu.st cause an éléva-
tion ofpitc!), because tbe ncw state of thin~s may bu conccivcd to
diffcr from the o)d mcrc)y by tbc introduction of an additional
constraint. Springs, wlthout incrtia, arc s~ppnscd to urgc thé
linc of thé prnposcd boundary towards ilscquitibrimn position,
and graduaHy to bicorne stin'cr. At c:).cb stcp thc vibrations
becomc more rapid,until tbcy approach
a linut, corrcHpondingto
infinitc stiH'nuss "f thc nprings and abso1ut,c (ixity of thc-ir points
of application. Itisnotn(.'c~ssarythat t)io p:n'tcntoffshou!d
hâve thé H:unc dcnsity as thc l'est, or cvcn at~y dcnsityat a.)L
For instance, the pitch of a reg~dar polygon is intcrmcdiate
bctwecn tboseofthe inscribcd and circmnscribcd circles. doser
Umits won!d bowcvcr bc obtained by substituting for the circuni-
scribed circle that ofequa~ arca according to ttic rcsult of § 210.
In thé case of thc hcxagon, thé ratio of tbc radius of the circle of
crp)al arca to that oft!te circle Inscribcd Is l'OaO, so that the tnean
'I.intni)].]M8.
2UJ MEMBRANES OFEQUAL AREA. 289
of the two limita cannot differ from thé truth by so much as 2~ percent. Li t.he~iic w:ty we migh~conc)'dc(.h.').tthesect.<)rcfacircle of G0° is n graver form than tlic equilateral triangle obtained
by substituting thé chord for thé arc of thé circle.
Tho following table giving the relative frequency in certain
calculable cases for thé gravest tone of membranes under similar
mecbanical coadi fions and of equal cn-e~ (o-), shews tho effect of a
greater or less departure from the circular form.
CIrcIc. 2-404.=4-261.
Square. ~2.-n-=4'443.
Q_1
f. 1 5'135./ 45w1Quadrant of~circle. f~?.~=4.~i~~s
Sector of a circle 60°.6-379 A/~=4'616.
/13Rectangle 3x2. A/7r=4'G24.
Equilatcral triangle. 27r. ~/tan 30" = 4~74.
Semicircic.3832A/~=4'803.
Rectangle 2x1.1 /5
R~ctangle 2x,l. ,} 'T'~2= 4'067.
Right-angled isosceles tna-ngle.J ~y~=~'967.
Rectangle 3x1. 7!-A/~= 5-736.3 3 1~r1/
-.) G u,
For instance, if a square a.nd a. ch'c]c have thc same area, thc
former is thé more acutc in thé ratio 4-443 4'2C1.
For thé circle thé absolute frequency is
In thé case of similar forms thé frequency is inverscly as tho
linear dimension.
212. The thcory of thé frce vibrations of a membrane was
first succcssfut)y considered by Poisson'. l, His thcory in thé
case of thé rectangle left little to be desired, but his treatmeut
1 Af~m. (le r~e(!(MMt'< t. vm. 1829.
R. 19
~0 VIBRATIONS OF MEMBRANES. [~312.
of the circular membrane vas restrictcd to thé Rymmetncfd
vibrations. Kirci-.h"~ ~~di.mortbc~'m~r,-.ut -h i'c
dimcult, problem ofthc circular plate w~ publisbcdin 1H;)0 aud
Ctebsch'a y/~op-y q/Y~ (18G2) givcs thc gênerai thcory of tho
circular memt)ranc induding thc efFects of stiil-ness and oi rotatory
incrtia. It will thercfore be sccn that tticro was not much left
to Le donc m 186G; ~evertheless tlie mcmon- of Bourget aircady
refon-ed to contains uscfui discus.sbn of thc problemaccom-
paHicd by very complète Dumcricid results, thc whoïc of which
howcver wcrc not nc\v.
213. In his cxpcnntcnta! mvcsti~tions M. Bourget made uso
of various m~terials, of winch papcr provedto hu as
goodas any.
Tl)c papct- is immerscd in wato-, and aftcrTonova! ofti'c snperHuons
~noisturc by blotting papcr is piac~d upon a framc of woud wbose
edges havo bcGn prcviau.sty coatcd with gtuc. Thc contraction of thé
papcr in drying produccs thc ncœssfu-y tension, but manyfaihu'cs
mny be met wUlt bufurc a satisfactory rcsult is cbtaincd. Evcn
a wcll strctchcd mcmbmne refiuircs cottsidci-abic pt-ecautionsui
use, bcing Uabic to gréât variations in pitch in consc.tuencc of thc
varying niuisturc of tho atmosphère,'i~hc vibrations are cxeltcd
hy organ-pipcs, of which it is necessary to tiavc a scrics procecdiug
by sma!! intcrvals of pitch, :uid they arc mado évident to thc cyc
by means of a littic sand scattcrcd on H'c mombranc. If tho
vibration be sufHcicntty vigorous, thc s!uut accumulâtes on thé
nodal lincs, whosc fortn is thus dcHneJwit)~ more or less prcciston.
Any Ine'jUidity in thc tension shcws itsclf by thé cire-les beeoning
elhptic.
Thé principal results of experimentarc the foUowing
A circulât- membrane cannot vibratu in unison with cvcrysonnd.
It can ouly place itself in unison with sounds more acute than
tliat Iicard whcn thé membrane Is gcnt)y tapped.
As theory Indicates, thèse possible sounds are separated by Icss
aud Icss intervais, tho highcr thcybceomc.
Thé nodal lines are oniy formed distinctiy in rcsponseto
certain deunite souuds. A littie above or Mow confusion cnsues,
and when d~e piteli ofthe pipe is decidcdly altcred, thc membrane
re.nains un.aoved. Thero is not, as Savart supposai,a continuons
transition from one System of nodal Uncst') auother.
213.] OBSERVATIONS 0F M. BOURGET. 291
Tho nodal Unes arc circlos or diamcters or combinations of
cu'c~os an ~~Tn.tors, an ~ipory rcrju~-f~, ITo~'cvcr, tvh'ju thc
number of diamcters excecd.s two, thc s~nd tends to hea.? itself
eonfuscdly toward.s t!ic iniddie of the membrane, and the nodos
are not well dcfincd.
The sa.me gcncra.1 laws wcrc vcriHcd Ly MM. Bernard and
Bourgct in thc case ofsquare membra.ncs'; a.nd these authors con-
sidcr that the rcsn)ts of theory arc Elecisively established in oppo-
sition to thé vicws of Savart, who hc!d that a membrane was
capable of i'<jspondin~ to any sound, no matter what its pitch
might be. But 1 must tierc remark that the distinction between
forccd and free vibrations docs not secm to have been suniciently
borne in mind. Whcn a membrane is set in motion by aerial
wavcs having tLcir origin in an orgau-pipc, the vibration is
propcriy spcaking /(j;'ce~. Theory asscrts, not that thé membrane
is only capable of vibrating with certain denned frcqueneieH, but
that it is on!y capable of so vibrating j~'e~y. When however thé
period of thé force is not approximately equal to one of thé
natural periods, the rcsulting vibration mny be insensible.
In Savart's cxpcnmcnts the sound of thé pipe was two or three
octaves higber than t)~e gravest tone of thé membrane, and was
aceordin~y ncvcr fnr from unison with eue of thé séries of over
tones. MM. Bourget and Bernard made thé experiment under
more favourable conditions. Whcn they sounded a. pipe somew!~a,t
lower in pitch than thé gr~vest tone of thé membrane, tlie sand
rema.ined nt rest, but was thrown into véhément vibration as unison
was approached. So soon as the pipe was decidedly higher than thé
membrane, titc ~and returncd again to rest. A modification of the
cxperimcntwa.s madc by first tuning a pipe about a tliird higher
than thé membrane whon in its natural condition. Thé membrane
was then heatcd until its tension had increased sumciently to
bring tbc pitch above that of tlie pipe. During the process of
cooling thé pitch gradually fe! and the point of coincidence
manifcstcd itself by thé violent motion of thé sand, which at the
bcghmiug and end of thé experiment was scnsihiy at rest.
M. Bourget found a good agreement between thcory and obscr-
v:).tion with rcspuct to t)]C radii of thc circuler nodcs, though the
test wns not very prccisG, in conséquence of tlie scusibic width of
~n;. C~tt'w. M. 449–47f, 1860.
~:)–3
VIBRATIONS OF MEMBRANES. [213.S93
the bands of sand; but thc relative pitch of thc various simple
tones deviated considerahly from thé theoretica.1 estimâtes. Thé
committee of tlie Frcnch Acadcmy appointed to report on
M. Bourgct's memoir suggcst as thé explanation thé want of
perfect fixity of thé boundfu'y. It should also be remcmbered t))at
the thcory procccds on thé suppositionof perfect HcxibiHty–a
condition of tbings not at ail closely approached by an ordinary
membrane sti-etchcd with a comparatively small force. But
perlaps thé most important disturbing cause is tlie resistance of
thc air, which aets with much grcater force on a membi-a.ne than
on a string or bar in conséquence of thé large surface cxposcd.
The gravest mode of vibration, during which tlie dtsplacement is
at ail points in thc same direction, might bc affccted very
differcntiy from tlie highci- modes, which would not roquire so
grca.t a transference of air from one side to tlie other.
CHAPTER X.
VIBRATIONS 0F PLATES.
214. IN order to form according to Green's method thé équa-
tions of eduilibrium and motion for a thin solid plate of uniform
isotropic material aud constant thickness, we require thé expressionfor thé potential encr~y of bending. It is easy to sec that for each
unit of area the potential cjiergy is a positive homogeneous
symmetrical quadmtic function of thc two principal curvatures.
Thus, if p~, bc tlie principal radii of curvaturc, the expressionfor V will be
where A and arc constants, of which J. must be positive, and
/n inust be numerically less than unity. Moreover if thc matcrial
be of such a character tha.t it undergoes no lateral contraction
when a bar is pulled out, the constant must vanish. This
amount of information is almost ail that is recaured for our
purpose, aud wc may thcrcfut'c content ourselves with a mere
statcnicut of tlie relations of thé constants in (1) with those by
mcans of ~hich t)io elastic properties of bodies are usually de-
nncd.
From Thomson and Tait's -Mra~ Philosoplty, §§ G30, 642,
720, it appears that, if b be tlie thickness, y Young's modnius,
and thc ratio of latcral eoutraction to longitudinal elongation
when a bar is puited out, thé expression for V is
294 VIBRATIONS0FPLATES. F314.
If Mbo the small dispiMcmcnt pcrpendicular to thc plane
of tho plate at tlie point wliusc rectangular coordinates in tho
plane of tlie pta,tc arc ?, y,
and thus for a unit of area, wc have
which quantity bm) to bc integrated ovcr the surface (~9) of thé
plate.
215. We procccd to find the variation of F, but it should bc
prcviously noticed that tlie second tcrm in V,uame!yj< P,P~
représenta thé <o~ cMruct~tp'e of the p]:ttc, and is thereforc de-
pendent only on thé state of thinga at thc edgc.
so that ve have to consider tlie two variations
1 Tho following comparison of tho notations uscd by tho principal writers may
iinvo trouble to thoso who wish to conault the oriH'H'H mouuira.
'hx
Youae's moda!uB=F (Clcbseh)=~ (Thon)aon)=:(TIiouison)
~K+~t
~"(~) (Tbomso!J)=? (~rckLoS tmdDoDkiu)=2~(Hirchhofï).
Ratio of latcral contraction to longitudinal elongation
=~ (Clobseh aud Douldu)=<r (Thomson)="~ (Thomson)=~(Eircidtoff).
Poiasou MiituaeJ this ratio to Lu and Werthuim
215.]l'OTENTIAL ENERGY 0F BENDING. 295
Now by Grecn's theorem
in which f~s donotes un clément of thé boundary, and dénotes
diH'crcntIation with respect to thc normal of tbe boundary drawn
outwards.
Thc transformation of the second part is more difficult. Wo0
have
Tho quantity under the sign of integration mn.y be put into
tlie furm
'wherc is tho angle bctween aud tlie normal drawn outwards,
and tlie intégration on the right-ha.ud side extends round the
boundary. Using thèse, wc (tnd
If ve ~8w ~8M; tZ~If wc substituto ior thcir values in termsf/;<; <<y M~
from tlie équations (sce FI~. 40)
VIBRATIONS OF PLATES.29G [215.
wc obtain
Collecting and rearra.ugmg our results, we ~Ind
r~ f~w.- .cfw <~M\
-s-~+(1-~)
jcos~sm~(~ "'(7~\ cos8smO \,t/y''TZ)\
+(eos~-sui~) 1(cos' 8
't~/y/J y
( ~~M .f~w+ f~- ~'C7"M + (1 ~) cos' -i-sin"J~/i. (' d~' a~
Tliere will now bo no difficulty in forming the equa-ticns of
motion. If p bc the volume deusity, aud Z~ the transverse
force acting on thc c!cment c?6',
215.]CONDITIONS FOR A FREE EDf.E. 397
8F-ff~8w~+f~wSM~M=0.(7)'
1
is thc gcno-al v:u'iation!U équation, which must bc true wha.tevur
fmiction (consistent with tlie constitution of ttic system) mn.y
bosupposcd to be. Hcocu by tlie principles
of thc Cidculus ot'
Variations
at evcry point of thc ph(.tc.
If thc cdgcs of thc pMe bo froc, therc is no restriction on thé
hypothctic&l bound~ry va)ucs of 8w and a'id thei-cfore thé
cne~cicnt.softhcsoquautities in thé expressionfor SFmustvanisIt.
Thé conditions tu Le: s~isncd at a, frcu cdgc arc tlms
If thé whole circumfercucc of the pla,te be clampcd, 5w = 0, = 0,cln
and tlie satisfactiou of thc boundary conditions is already sccured.
If thé cdge bc 'supportcd", ~=0,hut~ia
tn-bttrary. Thé0 cl~a
sccoud of thé cqua.tious (9)must m tins case bc s~tisfied by w.
216. The bound:n-y équations may be simplified by getting
rid of thé extrinsic élément involved in tho use of Cartesian co-
ordinates. 'l'aking the axis of a: pM~Ielto thé normal of tlie
buuuding curve, wc sec that we may writo
Aiso
Tho rotatory inortia ia l'cre uc~locted. CoinpMe § 1G2.
VIBRATIONS 0F PLATES. [216.298
whcreo-is a,fixcdaxiscoincidiugwith thc tangent at t!t0 point
d l '1 C12w.1'œ (I"wT bunderconsideratiou.Ingenci'a.l-diH'ei'=ih'om Toobtain
M~* <M'thé relationbctweenthem,we may proceedthus. Expn.ndw byMa.cta-unn'sthcorcniin ascoudingpowcrsof thc smaUqua.ntitiesn and o',and substitutoforMand o-thcn.'valueslu terms ofa, théarc ofthc curve.
Thusin gênera).
fF~ ~"w ~<;w= + + o- +Aj–j ?~+ ?:o-+ -r 0-'+
(/)!~ ~0-~ ~o-ft
s'"whUcon thé curvc o- = s + cubes, == + whcrc/? Is thé
radiusofcurvature. Accordinglyforpointson thé curve,
and thcrcforc
whencefrom(l)2 ~"W. 1~!0 0~~
~"tc=-+-+. (3).v Iop (~'
We concludcthn.tthé secondbouoda.ryconditionin (9) § 215
mn.ybe put iuto tlic form
In the sa.mo way by putting == 0, we sec th!).t
is équivalent to wherc it is to be undcrstood that the axescht cl~
of M ûnd cr a.rc Rxcd. Thé (h'st boundary conditiou now becomes
If wc apply thèse Ct~uations to thé rectangle whose sides arc
200216.1 CONJUGATE PKOPERTY. 299
pa.mllol to the coordinato axes, wc obtahi as thé conditions to bo
sutisncd :).long tho cdges pa.ra).iel to
In this ca.se the distinction betwecn o- and s disa.ppcars, and p, thc
radius of curvaturc, is inlinitely gt'ea.t. Thé conditions for tlie
other pair of edges are found by mterchanging x aud y. Thèse
rosults may be obtained cquaUy well from (0) § 215 directly, with-
out thé prelilninary transformation.
Auy two values of w, K and con'csponding to thc same
boumi:ny conditions, arc co~x~e, that is to say
provided tha.t tlie periods bo différent. In order to prove this
from thc oi-Jiuat-y diU'erentia!. équation (3), we should ha.vc to
retrace thc stops by which (3) was obtaincd. Tins ia the method
!~dopted by Kirclihoff for thé ch-cular dise, but it is much aimpicr
Mid more direct to use thé va.rin.tiond équation
in whick w refurs to the actual motion, and 8~ to %ti arbitrary
displacemcnt consistent with thé nature of tlie system.SF'Isa-
symmctricalfunetiou of w and ~M, as may be seen from § 215, or
from thc general character of V (§ 04'.)
300 VIBRATIONS 0F PLATES. [217'.
If we now suppose in tlie first place th~t w = :t, 8~ = wc
hn.vc
~~=~~f:tu~;
and i)i Hkc nia.tuier if we put w = v, 8~ = u, which wc are equally
oitittcdtodu,
gr=~f~~s',
'\vbencc
Tins démonstration is valid wl~tcvcr may be thc form of thé
boundary, and whcthcr thé cdge be cla.mped, supportcd, or frec, in
'\vltolcori!ipa.rt.
As for thc case of mcmbmnes in the la-st Chapter, equation
(7) may bu onpiuycd to prove that thc admissible v:duus of arc
ruai; but tins is évident from physieal cousidcrations.
218. For thc application to a circular dise, it is necessary to
express the équations by means of polar coordinates. Taking
titc ccnti-c uf tlie dise as polo, wc hâve for the gcncral uquatiun to
bc satisnud at ail points ofthe arca
To cxprcsa the boundary condition (§ 21G) for a frcc ~d~o
()-=(t),we!m.vo
p = radius of curvatnre =M; and thus
AfLcr tl)C diiTcrcut~tions are pc-rfurmed, r is to Le made cqu~I
to«.
218.] POLAR CC-ORDINATES. 301
If w bc cxpa.nde<l in Fourier's series
w =?t~ + + + +.
each term sepM'atcly must satisfy (2), and thns, since
!~<x cos(M0–a),
/r~ l~A ,/2-~r~ 3-~ \Q'
~~p+r~'r~d1'
(l'zo"+
1(_luu"(¿2
nt
USIl
(3).
~=0
0
The superficial difïerentia.1 équation may bc written
(V'+~)(~)~=0,
which becomes for the general tei-m of thé Fourier expansion
.1 M'~_z' .1~ n_' ,Y,
f –+--T-i+~('n+''7 -a
K M
=
~f~ )'f~' ?" ?'~)' )-'
shewing that tlie complete value of will bc obtained by ndding
togcther, with arbitrary constants preHxcd, thé genera.1 solutions of
The equation with the npper sign is the samc as that which
obtains in thé case of thé vibrations of circular membranes, and
as in the last Chapter wc conclude that thc solution applicable
to thc problem in hand is ce J.. (/o-), the second function of r
bcing hère inadmissible.
In tlie same way the solution of tl)c équation with the lower
niguis
Wnx
,7,, (~r),whcre t == s/ 1 as usual.
The simple vibration is thus
M), =cos ?t0
{o( J,. (~-) + /3~, (~?-)}+ sin {'yJ, (~-) + SJ,. (t/<?-)}.
Thé two boundary équationswill détermine tl)e admissible
values of and the values which must bo given to the ratios
a ~3 and y 8. From the form of thcsc équations it is evident
that we must have a /3 = 'y 8,
and thus «'“ may bc expressed in the form
t., = P cos (~ a) (J, (~-) + (~)) cos (~ e).(5).
VIBRATIONS 0F PLATES. [218.302
As ni thé case of a, membrane the nodal system is composed of
tlie diametcrs symmetric~Iy (UstribuLed round tlic centre, but
othenviso arbitrary, dcnoted by
cos(~-a)=0 .(~,
togethcr with tlie conccutric circles, -\vhose ctiu~tion is
J~-)+XJ,(~-)=0.(7).
219. In order to dctcrmiMC ?L a.nd we must ultrcduce thé
bouudary conditions. Whcn tue edge is free, we obtain from
(3) § 218.
in which use has been nmde of tl<e diU'crcntiid équations satisncd
by (/c-), J,.(~)'). In e~ch of thé fractions on thé right Hie dct)o-
minator ma,y be dcrh'cd from tLc numcrator by writing in place
of BycHnunatioaofXthc cquntiou is obtuiucd wliose roots givc
tlic admissible vidues of /c.
Whon = 0, tlie rcsult assumes aj simple form, viz.
Jn('A:a) ~(~'t) rt /'9')
2(l-~)+~~+~y~=0.(2).
This, of course, could ha.ve beGH moro easily obta.incd by neglecting
M from tticbcgiumng.
The calculation of the lowest root for each value of is trouble-
some, and in the absence of uppropri~c tables must 'be cÛceted
by menns of thc asccnding séries fur thefunctions ~(~'), .y,.(!).
lu the case of tho higher roots recunrsc ~n~y Le h~l to thé semi-
convergent descend ing séries fur thc s~ne functions. Kirc))hoff
finds
~L-+8~(8~)" (8m)"
tan (~Tr) = –T–
~~8~~(8~t)~
whcrc
~=~=(1-~)-
~=ry(l-4~) -8,
C = 'V (1 4~') (9 4~) + 4.8 (1 + 4~),
7) = ((1 4~) (f) 4~) (13 4n')] + 8 (9 + 136~ + 30~).
219.] KIRCMHOFF'S THEORY. 303
where~isanintcger.
It appears by a numcrical comparison that /t Is idcntical with
the uuinbcr of circulai' nodcs, n.nd (4.) cxprc.ssGHa, law discovei-cd
by CIdadni, that tlie ft-e<tucncics eorrespouding to figurcs with a
given number of nodal diameters arc, with the exception of the
lowest, approximately proportionalto tlie squares of consécutive
cven or uncvcn uumbers, accordingas thc number of the diamGtcrs
is itself cvcn or odd. 'Within thé limits of application of (4), we
sec also that thé pitch is approximately unaltercd, when any
number is subtracted froni A, provided twice that number be
addcd to ?:. This law, of which traces appear ill the following table,
may be expressed by saying that towards raising thé pitch nodal
circles have twice t!ic eScct of nodal diameters. It is probable,
however, that, strictly spealtiug, no two normal components have
exactiy thé same pitch.
/t ?=0 ~=-1 1
Ct~ P. W. Cn. P. W.
1 Gis HiH+ A-t- b h- c-
2 g:s'+ h'- b'-(- o"+ f"+ fts"+
/ti
?t=3 ~=3 3
'cnr*'p.~ w. Cil. P. w.
0 C C C d (tis- d)s-
1 g' gis'+a/- d".dm" di8"+ c"-
Thé table, extracted from Kirchhoff's mcmoir, gives thé pitch
ofthe more important overtones of a free circular plate, thé gravest
being assumed to bc C' The three columns under tlie heads
Ch, refer respectivclyto tlic rcsults as observed by Odadui
and as calculated frou theory with Poissou's and Werthenu's
values of A signdeMotes that tlie actual pitch is a little
higher, aud a 7)~)t!<s signthat it is a little lowcr, than that written.
1 Gis corresponds to (~ "t the EnK)ish notation, and ~t to b natural.
VIBRATIONS 0F PLATES. [219.304
Thé disercpancics between theory and observation are considérable,
l)ut perhaps not greater than mny bc attributcd to jrrcgularity in
thé plate.
220. Titc radil of the noda! cit'dus in tlie symmctric:d case
(t;=
0) were calcuiated by Poisson, and comparcd by him with
results obtait~ed expci'ime!)tf)J]y by Savart. The following numbers
arc taken from a papcr by Strehikc', who made somû careful mea-
suremcnts. The radius oftho dise is taken as unity.
Obsorvnt.ion. CnIeuJation.
One circle 0-67815 0-68062.
TwofO-39133 0-39151.
Iwo
cu-cles. ~.g~~
fO-25631 0'25679.
Thrcc circles 0-50107 0-59147.
~0-893GO 0-89381.
Thc ca.!culated rcsuttsappcn.t-
to refur to Poisson's value of but
would vary very little if Wert.Itcim's v~luc were substituted.
The foUowing titb~givcs
a,comparisou of Kircilhoffs theory
()! not zéro) with measuremcnts by Strebikc m~dc on less accurate
dises.
7?~~M q/' ~'CM~ft)' 2Vo<~<M.
Obser~tbu.Œ]cu!a<on.
~==nP.). ~=~(W.).
?t=l, A=l 1 0-781 0'783 0-781 0-783 0-7M136 0-78088
~=2, /t=l 0-70 0-81 0-S3 0-82194 0-82274
~=3, ~=1 1 0-838 0-843 0-8.1523 0-84G8I0-488 0-493 0-40774 4 0-49715
M-i, /t~Q.g~ Q.g~ 0-87057 0-87015
221. WLcn thu plate is truly symtnctrictd, whctherunifonn
or not, theory indicntes, and exporiment veri~GS, th:).t tlie position
of the nod:U diameters is fu'bitnu'y, or ra.ther dcpcndcnt only on
thc manner in which tlie pl~tc is supportcd. By varying thc
place of support, any dcsircd (liamctcr mny be made nodal. It is
goncraUy othcrwise wlien t!)crc is a.ny sensible dcpartui'c from
exact symmctry. Ttic two modes of vibration, whicli originany,
1Pc~e- /iH".xcv. p. 577. 185S.
221.] BEATS DUE TO IRREGULARITES. 305
in consequence of the equa.lity of pcriods could be combined in
any proportion without ceasing to be simple harmonie, are now
separated and anected with different periods. At the same time
tlie position of the nodal diameters becomes determinate, or rainer
limited to two alternatives. The one set is derived from the other
by rotation tlirough haïf the angle included between two adjacent
diameters of thé s:nnc set. This supposes that thé deviation from
uniformity is small otlierwise tlie nodal system will no longer be
composed of approximate circles and diameters at al!. Thc cause
of the deviation may be an irrcgularity either in thc material or in
the thickness or In the form of thc boundary. Thé effect of a small
load at any point may be investigated as in the parallel problem
of thé membrane § 208. If thc place at which thé load is attached
does not lie on a nodal circle, tho normal types are made deter-
minate. Thé diamétral system corresponding to one of the types
passes through the place in question, and for this type the period
is unaltered. Thé period of thé other type is Iiiereased.
The most gênerai motion of thé uniform circular plate is
expressed by thé superposition, with arbitrary amplitudes and
phases, of the normal components already investigated. Thé
détermination of the amplitude and phase to correspond to
arbitrary initial displacements and velocities is effected precisely
as in the corresponding problem for thé membrane by thé aid of
the characteristic property of thé normal functions proved in § 217.
Thé two other cases of a circular plate in which the edge
is eit,her clamped or ~)o?'~ would be easier than thé preceding
in their theoretical treatment, but are of less practical interest on
account of thé difficulty of expcrimcntally realising the conditions
assumed. The général resuit that thé nodal system is composed
of concentric circles, and diamctcrs symmetrically distributed, is
applica.ble to all thc tin'ee cases.
222. Wc have seen that in general Chiadni's ligures as traced
by sand agrée very closcly with thé circles and diameters of
theory but in certain cases déviations occur, which are usually
attributed to irregularities in tlie plate. It must however be re-
membered that the vibrations excited by a bow are not strictly
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 mnre normal component vibrations coexist. The whole
J!. 20
306 VIBRATIONS 0F PLATES. [223.
motion may be simple harmonie in virtue of tho external force,
althougli the natural periods would be a little différent. Such an
explanation is suggcsted by thé rogular charactcr of thé figures
obtained in certain cases.
Another cause of deviation may perhaps Le found in thé
manner in which tho plates are supported. Tho rcquirementsof
theory are often difficult to meet in actual cxperimcnt. WheM
this is so, we may have to be content with an imperfect compari-
son but we must remember that a discrepancy may bc thc f:~u!t
of the experiment as well as of thé theory.
223. The first attempt to solve thé problem with which we
have just been occupied is duc to Sophie Germain, who succccded
in obtainiag tho correct differential equation, but was led to
erroneous boundary conditions. For a frec plate tlie latter part of
thc problem is indeed of considérable dimculty. In Poisson's
mcmoir Sur l'équilibre et le mouvement des corps diastiques'
that eminent mathematician gave ~7'ee equations as necessary to be
satisfied at aU points of a free edge, but Kirchbon* bas proved tht~t
in général it would be impossible to satisfy thcm aU. It happons,
however, that an exception occurs in the case of tlie symmctrical
vibrations of a circular plate, whcn one of tlie équations is true
identically. Owing to this pcculiarity, Poissou's theory of tho
symmetrical vibrations is correct, notwithstanding the error in his
view as to the boundary conditions. In 1850 thé subjcct was
resumed by Kirelihon' who first gave thc two équations appropriate
to a free edge, and completed the theory of thc vibrations of a cir-
cular dise.
22~. The correctness of Kirchho6''s boundary équations bas
bcen disputed hy Mathieu", who, without explaining whero lie
considers Kirchhoff's error to lie, bas substituted a dinEcrent set ui
équations. He provcs that if M and u' be two normal functions, so
that w=~cos~, w=«'eos~'< arc possible vibrations, thcn
m~. de !4Md. d, Se. <t Par. 1829.
Crelle, t. XL. p. 51, Ucber dus CIcitihgowicht und die Bcwcgung cincr c]~-
tichenScttcibc.
~Z,~)f)-~t'.t.xtY.J8G9.
224.]HISTORY 0F PROBLEM. 307
This follows, if it bc admitted that u, satisfy respectively
the e<~uations
c* ~7~ =~, c" ~7~<t'==
~/V.
Since thé left-hand member is zero, the same must be true of
the right-hand member; and this, according to Mathieu, cannot
bc thc case, uuless at ail points of thé boundary Luth u a.nd u'
satisfy onc of t!ic four following pairs of equa,tions
Thc second pair would seem the most likely for a free edge, but
it is found to lead to an impossibility. Since thé first and third
pairs arc obviouely inadmissible, Mathieu coneludes that the fourth
pair of equations must be those which really express thé condition
of a frec edge. In his belief in this result hc is not shaken by the
fact that thé corresponding conditions for thé free end'of a bar
would be
the first of which is contradicted by thé roughest observation of
tlie vibration of a. large tuning fork.
The fact is that although any of the four pairs of équations
would secure thé evancscence of the boundary integral in (1), it
does not follow conversely that the integral eau be made to vanish
in no other way; and such a conclusion is negatived by KirchhofPs
investigation. There are besides innumera/bla other cases in
which thc integral in question would vanish, a.11 that is really
necessary being that the bounda.ry appliaBCCs sbould be either at
l'est, or devoid of inertia.
225. Thc vibrations of a rectangular plate, whose edge is
.suMWteJ, mny bc casily investigated theoretically, the normal
functions being identical with those applicable to a membrane of
tbf same shapc, whose boundary is fixed. If we assume
2D–2
VIBRATIONS OF PLATES. [225.308
we sec tbat at ail points of thc boundary,
~-n~=0,
~=0, ~-0,
whicit secure thc fu~hnent of tlie. ncccssa-ry conditions (§ 215)
The value of p, found by substitution in c*o=~'M,
sbewing that the anatogy to thé membrane docs not cxtcnd to thé
séquence of toncs.
It is not necessary to repcat bero the discussion of the prnnary
and derivcd nodal systems given in Cbaptcr IX. It is enough to
observe that if two of tlie fondamental modes (1) hâve thé same
period in tlie case of thé membrane, thcy must ah-so hâve thc same
period in tlie case of tlie plate. The dcrived nodal systemsare
accordingly idontica! i)i tlie two cases.
The freucratity of tlie value of w obtaincd by compounding
with arbitrary amplitudesand phases
ailpossible particular
solu-
tions of tlie form (1) i-cquircs no fres!t discussion.
Unless thé contrary assertion I~ad bcen madc, it would bave
seemed unnccessary to say that the nodes of a ~M~w?' plate
bave nothing to do with the ordinary Cbladni's ng'n-es, which
belong to a plate whose cdges arc frec.
The realization of the conditions fur a snpportcd edge is
scarcely attainabic in practice. Appliances are required capable
of holding t!)e boundary of tlie plate at l'est, and of sucb a nature
that they give rise to no couplesabout tangential axes. Wc may
conceive the plate to be hc!d in its place by friction against thc
watts of a cylinder circumscribed closcly round it.
226. The problemof a rectangutar plate, whosc cdges are
frec, is onc ofgréât dimeulty, and bas for tbe most part
rcsisted
attack. If we suppose that tlie displaccmentis independent
of?/, thc général differcntial équation bocomes identieal with that
with \vbich we werc concerned in Chapter Vin. If we take t)te
solution corresponding to the case of a bar whose ends are frec.
and tbci'cfore satisfying
0 o< <
22G.] UECTANGULAR PLATE. 30!)
when .c=U and when a;=~, we obtain a value of?o which sa~tisfies
t!fe getierai (liiïerenti~l cqua.tion, M well as thé pair of boundat'y
cqua.tiou.s
which :u'c a.pplica,b}c to tho cdges parallel to y; but tlie secotu)
boundary condition for thc ottior p:ur of edges, namely
~M f~t?
(?)~+~~=0.(2),CI;C
will be violated, uniess ~.=0. This shews that, exccpt in the
case reserved, it is not possible for a frce rectangular plate to
vibrato after tlie manner of a bar; uuless indeed as a.n approxima-
tion, when the Jcngth paraHcl to one pair of edges is so grcat
that thc conditions to bo satisfied n-t thc second pair of edges
may be left out of account.
Although the cottstaut fk (which expresses thc ratio of lateral
contraction to longitudinal extension wbeti a, bar is drf).\va out)
is positive for every known substance, in tlie case of a. few sub-
stances–cork, for cxampic–it is comparatively very smaIL Therc
is, so far as we know, nothing absnrd in the Iden of a substance
for which vanishcs. Thc investigation of the probtem undcr
this condition is tilercforc not devoid of interest, though the results
will not be strictly applicable to ordinary glass or meta.1 ptatcs,
for which tiie value of is about 1
If &c. dénote the normal functions for a frce bar invcs-
tigatcd in Chapter vin., corresponding to 2, 3, tiodcs, thé
vibrations of a rcctangular plate will be expressed by
1 In M'dor to rnuko n. pinte of mfttorial, for wluch is not xero, vibrato m tho
mnuner of a bar, it would bo noecfiHfu'y to apply conHtnutling couples to tLe edgea
pnraUû! to the p)anp of bondinn to provent tlio aasumption of a contmry earvfttuTe.
Tho oficct of thcsocouples wouH bo to rnise tho pitch, und thorofora tho calon-
intion founded on thé type propnr to ~=0 would give )t rosutt fiomowhat higbcr in
pitch tlxm the truth.
310 VIBRATIONS OF PLATES.[236.
In each of these primitive modes thé nodal system is composer
of straight lines parallel to one or other of tho cdges of thé
recta.ng]c. Whc)i b = o~ thé rectangle becomes a squa-rc; aud the
vibrations
/a;\
~=~u' "=~a1 Il a
having nocessarily tlie same period, may be combined in any pro-
portion,while thé whole motion still remains simple harmonie.
Whatever thé proportion may be, the rcsulting nodal curve will of
necessity pass through thé points detcrmined by
Now Ict us consider more particularly tlie case of = 1.
The nodal system of thé primitive mode, w =M,
[ ),consists
a
of a pair of straight lines parallel to y, whose distance from the
nearest cdge is '2242 a. Thé points in which thèse lines arc met
by the corresponding pair for w=u1
('),
a,rc thosc through wlticli
thé nodal curve of thé compound vibration must iu a.ll cases pass.
It is évident that they are symmetrically disposed on thé diagonals
of tho square. If tlie two primitive vibrations bc taken equal,
but in opposite phases (or, algebraically, with equal and opposite
amplitudes), we have
from which it is evident that w vanishes whcn a:==~, tha.t is along
thé diagonal which passes through tlic origin. That w will also
vanish along thé other diagonal follows from thc symmetry of
thé functions, and we conclude that tho nodal system of (3) com-
FiR. 41.
prises both the diagonals (Fig, 41). This is a well-knowu mode of
vibration of a square plate.
326.] CASE 0F SQUARE PLATE. 311
A scccnd notable cage is when the amplitudes arc cqual and
their phases tlie sa.rne, so that
Tho most convenient method of constructing graphically
thc curves, for which M=const., is that employed by Maxwell
in similar cases. Tho two systems of eurves (in this instance
straight Unes) represented by ~j= const.,
)~J
= const., a.rc
first laid down, thé values of thé constants forming an arith-
metical progression with thé sa.me common différence in the two
cases, In this way a network is obtained which thé required
eurves cross diagonaUy. The execution of tlie proposed plan
re<[uires an inversion of thc table given in Chapter yllL, § 178,
expressing thé march of t!ie function M~ of which thé result is as
follows
Thé system of lines representcd by the above values of x (com-
pleted symmetrically on thé further side of thé central line) and
tlie corresponding system for y arc laid down in the figure (42).
From titcse thb curves of cqual displaeement are deduced. At the
centre of tlie square we h:Lve w a maximum and equal to 2 on thé
séide a-dopted. The first curve proceeding outwards is thé locus of
points at which w= 1. Thé ncxt is tlie nodal line, scparating thé
regions of opposite disphcement. Thé remaining curves taken in
order give thc displacements 1, 2, 3. The numerically great-
est négative dispt~cement occurs at tlie corners of the square,
where it amounts to 2 x l'G-to = 3'290.'
na a
M~ a:: M M, ~:0
+1-00 '5000 '25 -1871
-75 -3680 -50 -1518
-50 -3106 -75 -1179
-25 '2647 1-00 -0846
-00 -2342 1-25 -0517
-1-50 -0190
On tbo nodat linos of squnro plate, Phil. Angust, 1873.
312 VIBRATIONS OF PLATES.[226.
The nodal curve thus conatructed agrees pretty closely with t!]c
observations of Strehike Hia results, winch refcr to three care-
fully worked plates of glass, are embodied in tlie following polar
équations:
-40143 -017H-00127)
r= -40143 + '0172 cos4< + -00127~ cos 8~,
-4019 -OJC8J '0013 1
the centre of the square being pole. From these we obtain for tire
radius vector parallel to thc sidcs of the square (<=0) '4-1980,
-41981, -4.200, whilo the c:deulatcd rcsult is -4154!. Thc radius
vector mefmm'cd along a diagona.1 is '3S;')C, -3855, -38C4, and bye~culation -3900.
rf'g~. -hfM. Yol. CXLVt.p. :<lf.
226.] NODAL FIGURES. 313
By crossing thé network in thé other direction wc obtain the
locus of points for which
is constant, winch are tlie curves of constant displaccmeut for that
ttmdc in wti!c)i the (Uagonals M-e nodal. Thc ~<c/t of thc vibratiou
is (accordi ng to theory) thc samc lu both cases.
/.K\
The primitive modes represented by w =ï~
~tor :t) =
M,)~)
may be combined in likc manner. FIg. 43 shews the noda.1 curve
for thc vibration
.(~(~<.).
Thé form of the curve is thé same rciativcly to tlie othcr diagODa),
ii' thé sign of the amhtgmty bc altcrcd.
VIBRATIONS OF PLATES. [227.314
227. Thc method of superposition docs not dépend for its
application ou any particular for)n of norma.1 function. Whatever r
t!ie form may bc, thé modo of vibra.t.Ion, winch wben = 0
passes into thatjnst discussed, must have the same period,
whethcr thc approximately straight nodal lines arc par~nd to
x or to If the two synchrouous vibrations bc superposed,
thc rcsultaut lias still Hjc sa.iuc ])criod, and the gênerai course
of its nodalsystem may
bu tra.ced by mcans of thc considéra-
tion tlmt no point of thé plate ca.)i bc nodal at w)nch tho
primitive vibrations hâve the sa.mc sign. To dotcrmiuc exact)y
thc line of compensation, a complete knowledge of thé primitive
normal functions, and not mercly of thé points at whicti thcy
vanish, would in gencra.1 be necessary. ])octor Young and thé
brotllers Weber appear to have had thc idea of superposition as
capable of giving risc toncwvarleties of vibration, but it is to Sir
Charles Wheatstone' that we owe tlie first systematic application, of
it to thc cxplanation of Chiadni'a figures. Thc results actually ob-
taincd by Wheatstonc arc however only very roughiy applicable to
a plate, in conséquence of thc form of normal function implicitly
assumed. In place of Fig. 42 (itself, bc it remcmbcrcd, only an
approximation) WIieatstone nnds for the node of thé compound
vibration thé inscribcd square shcwn in Fig. 44.
Fig. 44.
This form is rcally apptica.b!c, not to a, plate vibrating~ In. virtue
of rigidity, but to a. sti'ctched mctnbranc, so supported th~t cvery
pomt of thé ch'cumfcrcncc is free to move n.lon~ li.ncs perpendi-
cular to thc p!:uie of thé membrane, Thé boundary condition
ahplicable 1 1 circumstances is ~h° 0 0 and it is easyapplica-bic undci' thcsc circumstfLnccs Is~ln
= 0, and ib is ca.sy
to shew tbat tbo normal functious whieb involveoniy one co-
ordinale a.rc == cos
( 7M
or w = ces
??t), thc orjgui beingorùmate arc 10 =
ttor W = m
(xt lU ongm omg
a.t a corner of thé square. Thus thc vibration
31522~.]
WHE, ATSTONE )S FIGURES.
thc noda.1 systom is composed of the two diagonals. This rcsu~t,
which dépends ouly on thc symmetry of the normal fonctions, is
strictly applicable to a square plate.
shcwn in Fig. 45. If tlie other sign bc takcn, wc obtain a similar
figure with rei'ercnec to tlie other diagonal.
VIBRATIONS 0F PLATES. 227.316
Withthcothcrsign
wcobttun
ruprcscntinga. systum eouiposcd of thé diagonale tcgcthcr witb thc
inscribcd square.
Thcsc foDns, which aro strictly appHcab)c to t)tc membrane,
rescinble thé ngurcs obtained by mcans of sand 0)1 a, square p1atu
more closu)ythan might hâve bccn expcctcd. Thé séquence of
toncs is howcvcr quite durèrent. Frum § 176 wc sce that, if /t were
zo-o, thc interval bctwec!). thc furm (4.3) dcrivcd from thrcc
primitive nodcs, and (41) or (42) durived i'rom two, woutd bc
l-4-(i29 octave and thû interva.1 between (41) or (42) a.ud (4M) or (47)
wou!d be 2-43.')8 octa-vcs. Wbn.tcvcr may bc thé value of tbc
furms (4!) !U)d (42) shouki have exacte tlie same pitcb, and tbn
samc sbould be true of (4(i) a.nd (47). Witb respect to tbeHrst-
moitionod pair this resuit is not in a.grecmetit witb CbLidrit's
observa.tionH, wbo found a dirt'crencc of more than a whoïc tone,
(42) giving thc higbcr pitd). If bowcvcr (42) bc Icft: out of
account, thc cumparisonIs more satisfactory. Aecording to thuory
(~=0), if (41) gave (43) should givc fmd (4(i), (47)
sbould give~"+. Cbhubu tuund for (43) ~)-, andfor(4G),
(47)and + respectively.
228. Thc gravest mode of a. square plate bas yet to bc consi-
dered. Tbc nodus in tbis case arc tbc two Hues dra.wn througb tbc
middio points of opposite sides. That thcre must Le sueh a mode
will 1)G shewn prcscntly from considerations of symmctry, but
neither tbc fonn of Hic normal function, nor tbo pitch, bas yct
beeu dctcrnnucd, cveM for tlie particidar case of= 0. A rongh
calcnlatioli howcvcr mny bc founded on an as.sumed type of
vibration.
228.]GRAVEST MODE OF SQUARE PLATE. 317
If wc take tlie nodal lines for axes, thc form !o = a; satisfies
\7*M = 0, as wcll as the boundary couditions propor for a free edge
at ail points of the porimeter cxcept thé actual corners. This is
in fact tlie foi'tn which thc plate wuuid assume if hold at l'est by
four forces uumericidiy equa!, acting at thc corners pcrpendicu-
larly to tlie plane of thc plate, thosc at tlie ends of eue diagonal
beh)"' in one direction, and those at the ends ofthe other diagonal
in the opposite direction. From tins it follows that w=~cos~~
would bc a possiblemode of -vibration, if thc mass of the plate
werc concentratcd equally lu tlie four corners. By (3) § 214, we
sec that
For thé kinetic enorgy, if p be thé volume density, and ~)/ thc
:ulditionn.I m:tss at eacli corner,
whcrc dénotes tlie mass of tlie plate without the loa.ds. This
result tends to become accurate whcn~jf is re~tLvuiy grcut; other-
wi.sc by § 8f) it is scusibly less than tlie truth. But even when
jtf=0, thé error is probably not very gréât. In this c~e we
should have
2-~2 4 q b~
~=p(l~
giving a. p'Lc!) which is somewha.t too high. Thé gra.vest mode
next a.fter this is whcn tlie diagona.~ a.re nodes, of which the pitch,
if= 0, would Le given by
(sec §174).
VIBRATIONS OF PLATES. [228.318
Wo may conclude that if thc m~tcri~ of thc plate wcrc sud.
,-(/themterv~bctwecnthc two gravest
tones would
be somewlmt greaterthan that express où by the ratio l'SU;.
Chludni makestho intcrval t~ Afth.
-P––d by thc ratio 1.1..
2~9 That therc must cxi.t modes of vibratiou in which
thc two shortcsb .~cters ~-o ncdcs maybc infcrrcd from
su~
~Lldcrationsas thc following. lu Fig. (~) suppose that
Pig. 18.
is a plateof which the cdgcs Jf~, CO arc ~or~, and thé
i s 6'CT~~ P'
~t;onof cqu~brium,
must be capable of vibmting in certam
uuLhuncnt.1 modes. Fixing our attention on one of thèse, let us
conceive a distribution of over the th~c rcmammg quadrants
such that in any two that adjoin,tl~c values of .<; .u-c cqual aud
oppositeat points
which arc the inmges of each other in thé line
of séparationIf thé whulc pl~tc vibrato accordmg to thc law
thus detcnnincd, no eoustmiat ~ill be rcquircdin ordcr to kccp
the lines C~, ~cd, and thereforc thé ~hoh plate may be
rc~rdcdas free. Thc samc argument may be uscd to prove that
modes exi.t in ~hich thc diagonals arc nodcs, or in ~!uch botb the
Ji~onals and thc diamctcrs just considered are togethcrnod~.
Thé principleof symmetry may aiso be applicd to other forms
of plateWo might thus iufcr thc possibility of nodal diameters
in a eirele or of nodal principalaxes in an ellipse. Whcn tlie
boan~ry is a rcgular hexagon, it i.s e~ytn sec that Fi~. (4f)),
(~0), (;) rcprc'scnt pnssib!ofonns.
229.] PRINCIPLE0F SYMMETRY. 319
It i.s intcrcsting to trace thc continuity of Chiadni's figures, as
tho form of tlie plate is graduaDy altered. In the circ1e, for
cxa)nplc, whcn thcre are two perpendicular nodal diameters, it is a.
mattcr of indiifo'cncc as respects the pitch and thé type of vibra-
tion, in what position thcy bo tnken. As the circlc develops into
a. square by throwing out corners, tlie position of thèse diamctcm
becomes clefiuite. In the two alternatives tho pitch of tlie vibm-
tinn is dinercnt, for the projccting corners have not t))C Sfunc cfïi-
cicncy i)i the two cases. TIis vibration of a square plate shcwn in
Fig. (42) corresponds to that of a circlc whcn thcrc is ouc circular
nodc. rite con'cspondcncc of tho graver modes of a hexagon or
an cHipsc witli tliose of a cirele may bc traced in likc manner.
230. For plates of uniform material and thiclcness and of
invariable shapc, thc period of the vibration in any fondamental
mode varies as tlie squareof the linear dimension, providcd of
course ttiat tlie boundary conditions are thc same in aU tl~e cases
comparcd. Whcn thé edges fn'e clamped, wc may go further
and assert that the removal of n~y external portion is attcnded
hy a risc of pitch, whethcr tho inatcrial and the thickncss bc uni-
form, or not.
Let ~4~ bo a part of a clamped edgc (it is of no consequence
whethcr the rcununder of thc boundary be clamped, or not), ami
let thc pièce ~4C'J3D be remoYed, the ncw edgc ~173B being also
cla.mpcd. TIie pitch of any fuad{nuenta.l vibration is sbarpcr
than beforc tlie change. This is evident, since thé altered
vibra.tions might be obtained from the original system by thc
introduction of a constra.mt clamping thc edge ~4-DR The effect
of thc constt'Mut Is to raise ttio pitch of evcry componcnt, and
thc portion ~IC~Z) being plane and at rest throughout thé motion,
may bc rcmovcd. In order to follow thc séquence of changes
with greater security from error, it is best to suppose thé Une
of clamping to advanee by stages betwcen the two positions
jr' ~1/)/ For pxampic, the pitch of a ~niform chmpcd ptuto
VIBRATIONS 0F PLATES. [230.320
in thé form of a régulai' hexagon is lower than for thc inscribed
circle and higher tlian for tlie circurnscribed circle.
WIien a plate is free, it is not true that an addition to
tlie cdgc always incrcases the period. In proof of this it may
be sufncicnt to notice a particular case.
~7~ is a na.n-owthin plate, itscifwithoutinertin. but cn-rrylng
Ion.ds at A, C. It is clear that thc addition to the hrcadth
indicated by thc dotted line would augment the stifrncsa of thé
bur, and tlierefore ~Ot thc period of vibration. Thc same
consideration shews that fur a uniform free plate of givcn area
therc is no lowcr limit of pitch for by a sufficicnt elongation
tho period of thé gravest component may be made to exœcd
any astiignabic quanti ty. W!ten thc cdges are clamped, thé
form ofgrn.vest pitch is doubtless the cirele.
If an tlie dimensions of a plate, including the thickness, be
altered in the same proportion, t!tc period is proportional to thé
linear dimension, as in cvery case of a solid body vibrating in
virtue of its own elasticity.
The period also varies inversely as thé square root of Young's
modulus, if be constant, and directiy as the square root of tlie
mass of unit of volume of thé substance.
231. Experimenting with square plates of thin wood whose
grain ran parallcl to onc pair of sidcs, W heatstone found thut
thc pitch of thé vibrations was difforent according as the ap-
proximatcly straight nodal Unes were paraUel or pcrpendicular
to thé fibre of thé wood. This effect dopends on a variation
in thé flexural rigidity in the two directions. Thc two sets of
vibrations having djfferent periods cannot hc combincd in tlie
usual manner, and conscquently it is not possible to mal~e such
a plate of wood vibrato with nodal diagonals, The inequality
of periods may however bc obviatcd hy altcring thé ratio of the
sides, and tlien thé ordinary mode of superposition giving nodal
diagnnals is again possible. This was verified by Wheatstonc.
'J~.T'r~j'.lHM.
231.] CYLINDER OR RING. 321
A furthcr application of the principle of superposition is duc
to Konig 1, In order that two modes of vibration may combine,
it is only neccssary that thé periods agrée. Now it is evident
that thé 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 sidcs may agrcc in pitch with thé vibration having
thrce nodes paralhl to t!)e other pair of sides. In such a caso
new nodal figures arise by composition of thé two primary modes
of vibration.
232. W!icu the plate whose vibrations are to be considered
is naturaUy curvcd, thc difficulties of tbe question are gcnerally
nmch incrcascd. But thcre is one case ia which thc complication
due to curvature is more than compcnsated by tho absence of
a free edge; aud this case happens to be of considérable interest,
as being thé best représentative of a bell which at présent admits
of analytical treatmcnt.
A long cylindrical sitell of circular section and uniform tluck-
nesa is evidently capable of vibrations of a flexural character
in winch thé axis remains at rest and the surface cylindrical,
'while thé motion of every part is perpendictilar to the generating
lines. The problem may thus be treated as one of two dimensions
only, and dépends upon the consideration of thé potential and
kinetic energies of thc various deformations of which tho section
is capable. Tlie same analysis also applies to thé corresponding
vibrations of a ring, formed by thé revolution of a small closed
area about an external axis.
Thc cylindcr, or ring, is susceptible of two classes of vibrations
depcnding rcspectively on extensibility and flexural rigidity, and
analogous to thé longitudinal and lateral vibrations of straight
bars. When, however, the cylinder is thin, the forces resisting
bcnding become small in comparison with those by which ex-
tension is opposed; and, as in the case of straight bars, thé
vibrations depcnding on bcnding are graver and more important
than those which have their origin in longltudina.1 rigidity,
In thé limiting case of an ilifiiiitely thin shell (or ring), thc
flexural vibrations become independent of any extension of tho
circumfcrencc as a whole, and may be calculated on thé sup-
position that each part of the cii'cumfcrence retains its natural
length throughout tho motion.
rnRt!)));). 186i, cxxti. p. 238.
R. 21
VIBRATIONS OF PLATES. ~333.322
But although tho vibrations about to be considercd are
analogous to thé transverse vibrations of straight bars in respect
of depending on tlie résistance to nexure, we must not fall into
thé common mistake of supposing t)~at t)iey are exctusively
normal. It is indeed casy to sec that a motion of a. cylinder or
ring in which each pf).rtictc is displaœd in the direction of the
radius wou!d be incompatible with the condition of no extension.
In order to satisfy this condition it is neccssary to aseribe to
cach pa.rt of the circnmfcrence a ta.ngentia.1 as wcU !ts a. normal
motion, whose relative inagnitudes must satisfy a certain di~'er-
cntial équation. Onr nrst stop wi)l be the investigation of this
équation.
233. The original radius of tlie circlc 'being a, let thé equi-
)Ibrium position of any clément of the cireumfcrcncc be dcnncd
by the vcctorlal angle During the motion let the polar co-ordi-
natcs of the cl émeut beeomc
?'=ft+8r, ~=6+M.
If ds rcprcscnt the arc of the deformed curve corresponding to
we have
(~)" = (af~)' = (~8r)'+ (~ + f~)'
\vl)cncG wc nnd, by negiccting the squares of the small qnantitles
~?-,
.(y,.(1),
as the required relation.
In whatcver manner the original circle may be deformed at
time t, 8r may be cxpandcd by Fourier's theorem in the séries
8r = ft {~1. cos + J9, su) <?+. cos 2~ + 7?, sin 2~ +
+j4~cosM~+~sin~+.}.(2),
and the corresponding tangcntia! disptaccmcnt required by the
condition nfno extension will be
~=-~l,s:n~+73,cos~+.smM0+-"eos?t0- .(3),?t M.
tho constant that might be added to 86 being omittcd.
233.] POTENTIAL AND EINETIC ENERGIES. 323
If o-< denote the mus of thé clement the kinetic
energy T of the whole motion will be
thé products of the co-ordinatcs disappea.nng ia tho
intégration.
We havo now to cn.kula.tc tho form of tho potent!al energy K
Lct be thé ra.dms of curvaturc of any eletncnt f~, thcn for tho
1\"
coi-responding clément of F~wc may take~f~(8-j,
whero ~Is a
constant dcpcuding on tho materia,! and on the thickncss. Thus
Now
and
for in the small terms tl)c distinction bctwcen and <? may bc
neglected.
Hencc
aod
in \vhich thc summation extcnJs to ail positive intégral values
of~.
324 VIBRATIONS OF PLATES.[333.
The tenn for which n = 1 contributes nothing to thc potential
energy, as it corresponds to a. displaccmcnt of the circle as a whoïc,
without déformation.
Wc sec that when thé configuration of tlie system is defined as
above by thé eo-ordinates J,, ~t, &c., t]ic expressions for f7'a.nd V
involve only squares in otlier words, tlicso are tlie ))or)~ co-
ordinates, whose Independent I)armo)ilc 'variation expresses thc
vibration of tlie system.
Ifwcconsidcr only thc terms invûlving cos?:~ sin~ wc have
by taking the origiu of suitably,
8r = a.A cos 110, îO 'n r, siii ?td (7),~=~~cos?~, 8~=--n"sin~(7),
This resuit was given by Huppe for ring in a mcmoir pub-
Hshcd in CrcIIc, Bd. 03,1871. His mcthod, though more comptctc
than thé preceding, is less simpJe, in consequence of his not rc-
eognising cxplicitiy that the motion contempla.tcd corresponds to
complete inextensibility of thc circumfcrence.
According to Chiadni the frc(~icnclca of the toncs of a ring
arc as
3' 7' 0'
If we rcfcr cach touc to thu gravcst of thc series, wc Dnd for
the ratios chara.ctcristic of the iuturvaJs
2'778, 5-44.5, 9, 13-4.4, &c.
Thc corrcHponding numhcrscbt~iticd from thca.hove thcorctic:Ll
formula?, by making7t
succcssivcly cqual to 2, 3, 4, are
2-828, 5-423, 8-771, 12-87, A'c.,
agrccing prctty nc:u'!y ~'it.h titosc- fonnd cxpcrimcutaDy.
234.;] POSITION 0F NODES. 325
234'. When = 1, the frequency is zéro, a.s might have been
anticipated. TIie principal mode of vibration corresponds to ?! = 2,
and Ims four nodcs, disant from each other by UO". Thèse so-
called nodes arc not, however, places of absolute rest, for the
tangentiat motion is ttiere a maximum. In fact tlie taugentia).
vibration at thèse points is hali' thc maximum normal motion.
In gênerai for t)ic ?t"' turm the maximum tangcntial motion is
of t!ic maximum normal motion, and occurs at the nodes ofM
thc lattcr.
Whcn a bu!I-s)tapcd body is sounded by a blow, thé point of
application of thu blow is a place of maximum normal motion
of thc resutting vibrations, and tlie same is truc when thc
vibrations are excitcd by a violin-bow, as gcneraHy in Iccturc-
room cxperimcnts. Bu!!s of glass, such as nnger-glasscs, arc
howcvcr more casily thrown iuto j'cgular vibration by friction with
thc -wctted migci' carried round the circumfcrcncc. Ttic pitch of
the rcsulting sound is the same aa of that chcitcd by a tap with
tlie soft part of tho finger; but inasmuch as the tangential motion
of a vibrating beU bas been very gonerally ignorcd, thé production
of sound in tliis manner bas been fc!t as a difficulty, It is now
scarccly necessary to point out that the cffect of the friction is in
the first instance to excite tangential motion, and that the point
of application of thé friction is the place wherc thc tangential
motion is grcatest, and therefore where the normal motion
vanishes.
235. The existence of tangential vibration in tlic case of a bell
was verified. in thc following manner. A so-called air-pump rc-
cciver was sccureiy fastened to a table, opcn end uppermost, and set
into vibration with thé molstencd nnger. A small chip in tlie rim,
reflecting the light of a. candie, gave a bright spot whose motion
could be observed with a Coddingtou lens suitably nxcd. As the
nngcr was cai'ricd round, the hne of vibration was scen to rc-
volvc with an angu!:u' veloeity double that of the nnger; and
the amount of excursion (indicatcd by the length of thé line of
light), though variable, was rinitc in cvery position. There was,
however, somc difficulty in observing thé correspondence bctwccn
thc momcntary direction of vibration and thc situation of the point
of cxcitoncnt. To crfeet thissatisfactoriiy
it was found nocessary
to apply thé friction in the ncighbourhood of one point. It thc'n
326 VIBRATIONS OP PLATES. [235.
C.\MDKUXm: rtUNTKD !jY C. J, CLAY, M.A., AT TOE U!f!Vii:!(H!'[Y rttHSS.
bccamc évident that the spot moved tangentially whon thc boll was
excited at points distant thcrefrom 0, 90,180, or 270 degrees and
norma.Iiy when tho friction was a-pplied ai the intormediate points
corresponding to 45, 135, 225 and 315 dcgrecs. Carc is somctimes
required iu order to ma.ke the bell vibrato in its gravest mode
without sensible a.dmixture of overtoncs.
If tliere be a smn.U load at any point of tho c!rcumferencc,
a slight a.ugmcnta.tion of pcriod cns~cs, which is different accord-
ing as the Ioa.ded point coincides with a node of the normal or
of the tangcntiai motion, being greater in thc latter ca.so than
in the former. Thé sound produccd dépends therofore on the
p!a.ce of excitation in gcncral both tones arc hcard, and by
interférence give rise to beats, whose frequency is equal to the
diffurence between tlie frc(~)encies of the two toncH. This phc-
uomeuon may often bc obscrvcd in thé case of largo hells.
END OF VOL. I.
1