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.

1/ Les contenus accessibles sur le site Gallica sont pour la plupart des reproductions numériques d'oeuvres tombées dans le domaine public provenant des collections de laBnF.Leur réutilisation s'inscrit dans le cadre de la loi n°78-753 du 17 juillet 1978 :  *La réutilisation non commerciale de ces contenus est libre et gratuite dans le respect de la législation en vigueur et notamment du maintien de la mention de source.  *La réutilisation commerciale de ces contenus est payante et fait l'objet d'une licence. Est entendue par réutilisation commerciale la revente de contenus sous forme de produitsélaborés ou de fourniture de service. Cliquer ici pour accéder aux tarifs et à la licence 2/ Les contenus de Gallica sont la propriété de la BnF au sens de l'article L.2112-1 du code général de la propriété des personnes publiques. 3/ Quelques contenus sont soumis à un régime de réutilisation particulier. Il s'agit :  *des reproductions de documents protégés par un droit d'auteur appartenant à un tiers. Ces documents ne peuvent être réutilisés, sauf dans le cadre de la copie privée, sansl'autorisation préalable du titulaire des droits.  *des reproductions de documents conservés dans les bibliothèques ou autres institutions partenaires. Ceux-ci sont signalés par la mention Source gallica.BnF.fr / Bibliothèquemunicipale de ... (ou autre partenaire). L'utilisateur est invité à s'informer auprès de ces bibliothèques de leurs conditions de réutilisation. 4/ Gallica constitue une base de données, dont la BnF est le producteur, protégée au sens des articles L341-1 et suivants du code de la propriété intellectuelle. 5/ Les présentes conditions d'utilisation des contenus de Gallica sont régies par la loi française. En cas de réutilisation prévue dans un autre pays, il appartient à chaque utilisateurde vérifier la conformité de son projet avec le droit de ce pays. 6/ L'utilisateur s'engage à respecter les présentes conditions d'utilisation ainsi que la législation en vigueur, notamment en matière de propriété intellectuelle. En cas de nonrespect de ces dispositions, il est notamment passible d'une amende prévue par la loi du 17 juillet 1978. 7/ Pour obtenir un document de Gallica en haute définition, contacter reutilisation@bnf.fr.

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