The Theory Of Sound 1 1h586i

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PREFACE. IN thé work, of winch thé présent volume is an installent, my endeavour bas been to lay before the reader a connected exposition of the theory of sound, which should include ~he more important of thé adv~nces made in modem times ùy Matliematicians and Physicists. The importance of the object winch 1 have had in view will not, I tinnk, be disputed Ly those competent to At thé présent time judge. many of thé most valuable contributions to science are to be found onjy in scattered penodicala and traiMactions of socletles, pubMied in vanous parts of thé worid and in several languies, and arc often practically inaccessible to those vvho do not happen to hvc in thé neighbourhood of large public hbraues. In sucli a state of things the mechanical impedimonts to study entail an amount of unremunerative labour and conséquent hindrance to thé advancoment of science which it would be dimcult to overestimate. Since the wcH-known Article on Sound in the .E~c'~)~« ~)~M~(~ by Sir John Herschel (1845), no complète work lias been publislied in wilich tlio suhject is trcatcd jnatl-cmatically. Ly thé promature death of Prof. Donkin tlie scientific worid was deprived of onc w!)oso mathcmatical attainments in combmation with a {n-ctic.-d of mua:c quahned hifn in a kr~Judge

VI

PREFACE.

special manner to write on Sound. The first part of his Acoustics (1870), though little jnm~ tbsm a fragment. is sumcient to shew that my labours would l~ave been unnecessary had Prof Donkin lived to complète his work. In tlie choice of topics to be dealt with in a. work on Sound, 1 have for thé most part foUoAvedthé exemple of my predecessors. To a great extent thé theory of Sound, as commonly understood, covers thé same ground as thé theory of Vibrations in général but, unless some limitation were itted, thé consideration of such subjects as the Tides, not to speak of Optics, would have to be included. As a général ruie we shall confine ourselves to those classes of vibrations for which our ears afford a ready made and wonderfully sensitive instrument of inWithout ears we should vestigation. hardiy care much more about vibrations than without eyes we should care about light. The present volume includes chapters on thé vibrations of systems in gênerai, in which, 1 hope, will be recognised some novelty 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 enough to read thé proofs. By his kind assistance several errors and obscurities have been eliminated, and thé volume generally has been rendered less imperfect than it would otherwise have been. Any corrections, or suggestions for improvements, witli which my rcaders may faveur me will be highly apprcciated. TEttUNO

PLACE, .~n7,

1877.

WmiAi.,

CONTENTS.

C'HAPTER I. MOE i

§§1-27 Sound duo to Vibrations. Finite volocity of Propagatiou. Yelooity indopondent of Pitcb. Depmult'a oxporimonts. Sound propagated in water. 'Witoatatono'aexperiment. Enfoeblomontof Sound by distance. Notes and Noisos. Musicalmoteaduo to poriodiovibrations. Siren of Cagniard do 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 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. Optical methods of composition and analysis. Thé vibration microscope. In. termittont Illuminntion.

18

viil

CONTENTS.

C'frAPTER IV. §§~

(,7

(icnorniixedco-ortiitmtcs. Expression for jMjLontudcncrKy. StftticaithcorMua. Itiitml motions. Hxprcsnionforkinet!<:(!)~ur~y. Hcciprocat thoornm. ThM))Ho))'sthcorc:r)i. L~'n))HO't!('qutttionn. ThodiH.si~ftti~n funetion. Couxi.stt'ncoofHnMtUinotioxH. l''t'cc\-i).mtion.swt.])out.fri< tion. Nnrmnico-nrdinntos. condiThch'eoponodHfuXtt~Htttt.ioj~u'y tion. AnncccHsiottofinertijtmcrottsc.sthefruoiods. ArctttKfttif'~ ('fHprmf;i))()rL'!Lsc~ti~fMû)"i[)dH. n)t Tim(;
CHAPTER V. §§~17 CascH in wluch tho thrce funetiotM Y', r aro Hunu[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. Syatcma startod from rcst na dcftocted by n. force JuUuuuifonuiy distributed. oico of sw< frictiona) forcoNon titû vibr~tious of ft «yntom. Solution of t)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)tcnrcm for itttnnonic forcos. AppHcationH. Extension to cases in wbich thu constitution ofthu systcm isn.functionuftLûpuriud. Equations for two dcgrcus of fruedom. HoutH of dctornnmmttti cquittiun. lutct-niittcntvibmtiomj. Marchofporiods incro~od. ttHim.rtin.is~mdu~Hy Heaction of n dunduut f.vHtun).

<~

CONTENTS.

Ix

CHAPTER VI. pAnx 127

§§118–14S. Solution of tho pro. Lnw of extension of n, titritig. Transvurso vihrntiona. bJcm for n string whoso masH is concontrutcd in cquidistant points. Dérivation of no)ution for continuouo string. Pfu'tittI diffcrentitti équation. Hxpt't.'snious for nnd y. Hoat gonorat form of simpio harmonie moGcnoDt) motion of n stnng pcrition. StritiRS with ËxoJ extronitios. Kurtnn.1 ntotlua of vibration. odio. Mcrsonuc'H L)tws. Sonomotur. 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-ordin
CHAPTEK Vif. 188

§§ 149–1~ Ciftsxincttt.ion of tho vibrations of Bars. DiuM'cntin! équation for longitu. dimd vibt'jttiottH. Numorieal valuos of tho constunta for steûl. Solution 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 Differentini équation fortorSavM't'H "Hou rauqno." In.tcïft.I motion. Hioniti vibrations. Comp~risou uf vclocities of longitudinal a.nd tortiiomd wa.'vos.

CHAPTER Vni. §§ IGO–1U2 PotcutiftI energy of bcndhtg. Expression for kinctio energy. Dérivation Termimtl conditions. G encrât solution for of diSercutitd equ)ttion. tt hnrmouic vibrittiou. Conjuguto property of tho normal functiona. YatucB ufintrKratcd sq~n'CH. ExprcsHi).]) of r in tcrms of not'nml cu-

201

CONTENTS. ordinatos. Normal equatious of motion. Détermination of constant)) to suit initial conditions. Caso of rod etartod bv a b)ow. I~od start~ from rest as dofloctodby ~t~I ~~c. lu CL.~m ~os tho BorioHof normal funoticus coanostu coj.vcrëo. Form of H~ norma! hmettons li(:u-fiue bur. Lfnv9of Jopotj.Icuooof frofjuoucyou )ongt)i und tLioknoM. Caso whou both oudH ~rc clampod. Normal fuuctions for ,) clampod.Ireo bar. Caleulaticu oï puriud.s. CompnrinonHof pitch. Disoussiou of tho gravost modo of vibration of a freo-freo bar. Threu nodos. Four ~oJoa. Gravost mojo for clampod-froobar. Position ut nodos. cd bar. Calculation of poriod for clamped-froobar from Lypothetleal typo. Solution of problom for n bar with a loaded ond. Euuct. uf adtUtious to a bar. lufluonco of irrogularitios of donsity. CorreottOMfor rotatory iuortia. L:ootsof functioua dorived from normaJ fuuetioM. Formation of ~uatiou of motiou ~heuiinoarly thoro is por. Mauout tousiou. Spoeial tûrmiual couditioua. Itosultaut of two trains of wavcs of iic-iirlycqua) poriod. Fourior'Hsolution of problom for iniï. nito har.

CHAPTER

IX.

§§1~–213. Tension of a motubrano. Equation of motion. Fixod reotangular lonudary. Expression for ~aud iu tenus of normal co-ordiuatos. Normal ouations of vibratiou. Examplos of improssed forces. Frequoncy for an olongatod rectangle dépends maitiJy ou tho shortor sido. Casoo iu whiohdifTerout modes of vibration havo tho samo poriod. Dorivod modes thence arising. Effeet of 6li(;ht irrcgulanties. An irregu!arity rontovo may nidoturmiuatonosa of normal modoa. Solutions applicable to a triaugle. Espro~ion of tho Honoraidiilorontml eqnation by polar co-ordm~tes. Of tho two functions, w),idt oceur in tho solution, ono ia cxcluded by tho condition at tlio polo. Expressions for Bossel'a functions. Formutm rdating theroto. Tublo of tho first two functions Fixod eiroilar boundary. Conjugato proporty of tho normal functions without restriction of boandary. Values of integrntod squares. ExproHMounfor T nnd F in tcnua of normal functions. Normal oquations of vibration for ciroular mombraue. easo of froo vibraSpoci.d tions. Yibratioua duo to a harmonie force Utohos of tho varions shnpto tonoH. Tabio of tliouniformty distributed rootsof Bosscl'o functions. Nodal Fiëur~. Circular mombrano with ono radius fixed. Bessel's Bonctionsof frnctional ordcr. Ejloct of sma'I lond. Vibrations of a mombrano whoso boundary is approximatoiy ciroutar. In many casos thé pitch of a mombrano mny bu calculated from tlio aroa alono Of aU .nombraues of equal aroa t)Mt of oireular form l.M tlio gravost l'itch of pitch. a mcmbrano whoso boundary ia au eDipso of smalt ceeHntricity. Motliodof obtai)iii)g limits in casos that oumot bo dealt witli rigorouf3ly. Comp~rison of fruqueucioa iu varions ca.sc.sof mcmbraues of eqna) arc.a. Histury of tho probion.. Bourh'ot'8 oxperi. ïaonta) invostigfttiouB.

l'AUI~

CONTEN-TS.

XI

CHAPTER X. RAOP §§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 dotertho mining poriods for a froo ciroular p!nto. EirohhoC'a catouhtions. Comparison with observation, mdii of nodal cirolos. Irreguln.nties Kivo riso to boats. Gonoralizution of solution. Cnso of cJampod, or Hupportod,cdgo. Disturbn.uceof Chiadni's figures. Hifitory of proUom. Mn.tl.iou's critieiamo. DoetfmguiM phtto with aupportoJ edgo. Itoctwith freo edgo. Boundary couditionH. Ono Hpocialcnso nnguhn- plato (~ = 0) iHfunonabluto mfttttomaticfdtro~tmont. Investigation of codai figures. WItcntatoue'H application of tho mothod of Hnporposition CompariMU of Whoat~tono'f. liguros wit]. thoso reaUy n.pp)io~!o to n pMo in tho cnso = 0. Gravost modo of a squnroplate. Caiouhttion of poriod on hypothotica! type. Nodal ~igurcHinferrod from considor. atlona of symmetry. Hoxngon. Comparison hotweoncircle nnd squnre. Lnw connooting pitch and thicknoas. In tho cnso of a elfunpod odgo nny contraction of tho boundary raisos tLo pitch. No gravest form for a free plato of givon aron. In similar plates tho poriod is as tho linoar dimension. Whoat.stono'a expérimenta on wooden plates. Kœnig'a oxperimontN. Vibrations of cylindor, or ring. Motion tangentinl as woll as normal. Bolation betwoon tangoitial and normal motiona. Exfor Mnetio prossinna and potontial énergies. Estions of vibration. of L'requoncios tonos. Comparison with Chiadni. Tangential friction excites tanguntiat motion. Expérimental vérification. Béats duo to irregularities.

CIIAPTER I. INTRODUCTION. 1. Tim sensation of sound is a thing s:M ~e~eW~,not comparab]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 explorations 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)icnomena 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!
2

INTRODUCTION.

[3.

report from the flash. This rresents the time occupied by souud in traveUIng from thé gun to thé observer, the rotardatinn of the nash duo to thé finite velocity of light bcing altogether negligible. Thé first accuratc experiments wero mado by somo 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 tlie direction along which the sound travels, in order to cllminatc tho influence of tlie motion of thé air in mass. Down t!ic wind, for instance, sound travels reJativeJy to thé carth faster than its proper rate, for the velocity of thc wind is added to that proper to the propagation of sound in still air. For still dry air n.t a température of 0"0., thc French observerafound a velocity of 337 metres per second. Observations of tho samo character werc made by Arago and others in 1822 by thé Dutch physicists Moll, van Beek and Kuytcnbrouwer at Amsterdam by Bravais and Martins between thc top of the Faulhorn and a station bclow and by others. Thé gênerai result bas been to give a somcwhat lower value for tbc velocity of sound-about 332 mètres per second. Thc effect of altération of température and pressure on the propagation of sound will be best consideredin connectiou with thé mechanical theory. 4. It is a direct consequence of observation, that within wide limits, thé velocity of sound is independent, or at least very ncarly of its independent, intensity, and also of its pitel). Wcre this otherwiso, a quick piece of music would be hcard at a little distance hopelessly confused and discordant. But when the disturbances 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 of sound in long tubes (watcr-pipes) has been madc by Rcgnault\ He adopted an automatic arrangement similar in principle to that used for me~suring thé speed of projectiles. At thc moment when a ptstol is fired at one end of tlie tube a wire conveying an electric current is ruptnrcd by thc sliock. Tins causes thé withdrawai of a tracing point which was previonsly 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
5.]

¡

j

S

t,

VELOCITY OF SOUND.

3

impulse, the circuit, which was ruptured during tho age of thé soun< i3 rccumpietfd. At thc sa.mcmoment tho tracing point faits back on tlic drum. Tho blank space loft uumarked corresponds 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 indendent of thé dl~meter of the pipe, whieh vn.)'Icdfrom 0'108 to 1'100. Tho diso'ancy is perhftps duo to friction, whose 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 mcasurement of the velocityof sound are wanting, and woM'enot 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, 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. 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 14-35metres per second. 7. Thc conveyanccof sound by solids may bc IHnstrated by a pretty experiment due to Wheatstone. One end of a metallic wiro is connecteclwith 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 resonancc-board (such as a violin) bc now placcd in 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 oversurfaces ever Increa~ingas 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 smoothpavement and a vertical wall is still botter; but the most crtcctive ofaU is a tubc-likc enclosure, 1–S

INTRODUCTION.

[8.

which prevents spreading altogether, Tlie use of speaking tubes to faciMtatecommunication between thc dirent parts of abuHdir)
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'atedby
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 conclusions, 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 reating 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 sosmall 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 cléments 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 considérationslead 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
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 ing fromany ?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 made corresponds the interval known to musicians asthe<we~ 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
Tlie twelfth part of an octave is represented by the ratio !V2':1, for tins is thé st 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 characteristic 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 considered above, othcrs known to musicians may be derived. Thé difference of an octave and a fifth is called a fourth, and ha~the 3 é ratio This process of subtracting an interval from 2–~=~. thé octave is called ~uer~M:~it. By inverting the major third Asinglewordto donotothonumborofvibrations oxccuted inthounitoftimo is indi~ensabio:I knownobutterthan froquoncy,' whichwasnsodin thissonso byYoung.ThosMtowordis omployod byProf.Everottin bisexcellentodition ofDoscbanol'a ~atw
8

INTRODUCTION.

[15.

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 4 Fourth. 5 M~jorThird. MiuorSixth. 8 Miner Third. G 5 M~jorSixth. Thcfjo are ail thc consonant intervals comprised witttin thc limits of thé octave. It willbe remarked tliat tite corresponding ratios are ail expressed hy means of ~M~t~whole numhers, and t!tat tliis is more particularly 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?!cscctle. 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, whilothé 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 of strings must bc chaagcd in ordcr to obtain tlie octave and rifth; but Mcrsenne duntoustrated tlie Jaw connecting thc length of a string with the ponod of 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
16.]

NOTATION.

9

lying immediately above thé tonic, wo obtain notes whose frequenciesarranged in order of magnitude are Do Re Mi Fa Sol La. SI Do 5 5 9 4 3 la 2. 1, 8' 4' 3' 2' 3' 8' 9 with thc Hcro the common cbord on Do is Do-Mi-Sol, 5 3 thé chord on Sol is Sol–Si–Re, with thé ratios ratios 1 T~ thc cchord tlie 101' on Fa 18 is iFa-La–Do, a- a- 0, ~2x~=l:and T X 0 0 still with tlie samc ratios. Thc scale is completed by reating 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 a h c'. g Usage diffcrs slight~y p.s to thé mode of distinguishing the different octaves; iu wllat followsI adopt thé notation of Helmhoitz. 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 The iniddie c of thé pianoforte is c'. notation, g–d~–a'–e'\ 17. With respect to an absoluto standard of pitch therc bas bcen no uniform practice. At thé Stuttgard conférence in 183-1', c' =2G4complète vibrations per second was recommended. Tilis correspondsto a.' =440. Tlie French pitch makes a' =4-35. In Handc!l'stime 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~oof the frequency of a given note is an opération requiring somo care. The simplest method in principle 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'ectiveholes gives thé required frotuency. Thé consideration of othcr methods itting ofgreater accuracy must be deferred.

10

INTRODUCTION,

f~g.

18. So long as we keep to thé diatonic scale ofc, thé notes above written are ail that are required in a musical composition. But it is frequentiy desired to change thé key-note. Under thèse circumstances a singer with a good natural car, accustomed to perform witliout accompanimcnt, takes an entirely fresh departure, constructing a new diatonic scale on thé new key-note. In tbis way after a few changes of key, tho original scale will be quite departed from.and an immense varicty of notes he used. On an instrument with fixed notes like tho piano and organ such a multiplication is impracticahle, and some compromiseis necessary in order to allow thé same note to perform different functions. This is not thé to place discuss the question at any length, wc will thcrefore take as an illustration thé simplest, as wcn as thé commonest casemodulation into thé key of thé dominant. By donation, thé diatonic scale of c consists of thé common chords foundcd on c, g and f. Jn like manner thé scale of g consists of tlie chords founded on d and c. Thé chords of c and g g, arc then commôn'to thé two sca!cs; but thé third and fifth of d introdnce new notes. Thé thu-d of d written has a frcquency t f# J 3 5 4a removed from any note in thé scale of c. 8 4 32 But thé fifth of d, with a frequc.cy little from a, whose frcqucncy

is

differs but

In ordinary keyed instruments

thé interval betwecn the two, represented by and called a is ncglectcd and thé two note. c~ by a suitable compromise or ~?~-Hwc~ M-eidentined. 19. Various systems of tomperament have been used thé simplest and tliat now most generally used, or at least aimed at, is thé equal tempérament. On referring to the table of frequencies for tlie diatonic sealc, it will be secn that the intervals from Do to Re from Re to Mi, from Fa to Sol, from Sol to La, and from La to Si, are nearly thé same, being rresented or while tjj by intervals from Mi to Fa and from Si to Do, represented by are la ~1~' ?~ equal ~mperament treats ~cs~'ap. relations proximate as exact, dividing the octave into twelve eqnal

11

EQUAL TEMPERAMENT.

19.]

parts called mean semitones. From thèse twelvonotes thé diatonic scalc belonging to any key may be selected according to tho following 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 wayail dKScultIesof modulation arc avoided, as thé twolve notes serve as weUfor one key as for anothcr. But this advantagc is obtained at a sacrifice of true intonation. Thé equal tempérament third, being thé third part of an octave, is rrescnted 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 ficmitoncsmakc 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 pianoforte. But whcn thé notes arc /teM, as in thé harmonium and organ, thé consonanceof chordsis materially impaired. 20. The foltowlng Table, giving the twelve notes of the chromatic scale according to thé system of equal tempérament, will be convenient for reference'. Thé standard employed is 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~ D D~ E F F~ G 0~ A A~ B

0,

10-35 32-70 17-32 34-G5 18-35 3G-71 19-41 38-89 20-GO 41-20 21-82 43-G5 23-12L) 4G-25 24-50 49-00 25-95 51-91 27-50 55-00 29-13 58-27 30-86 61-73

C C5-41 C9-30 73-43 77-79 82-41 87-31 92-50 98-00 103-8 110-0 11G-5 123-5

c"

c l30'8 138'6 14G-8 155-6 1G4-8 174-G 185'û 19G-0 207'G 220-0 333-1 346-9

261-7 277'2 293-7 311-2 329-7 349-2 370-0 392-0 415-3 440-0 4CG-2 493-9

5233 544'4 587"i G23'3 G59'3 C98'5 740'0 784-0 830-C 880-0 933'3 9877

c~

c""

104G-6 2093-2 1108-8 2217-7 1174-8 2349-G' 1244-G 2480-3 1318-G 2G37'3 1397-0 2794-0 1480-0 29GO-1 15G8'0 313G-0 1GG1-2 3322-5 17CO-0 3520-0 1864'G 3729-2 1975-5 3951-0

DutnottCHte. Giessen, 18CS. Zammiuor, Die J/tMf'~«tx!
INTRODUCTION.

[20.

Thé ratios of tho intcrvals of the equal teinpûra.ment scale are gtvcn bclow (Zaunuluer) Note. Froquoucy. c =1-00000

Noto. f~

7

c# 2~'=I-0594G d

g 2'~=1-49831 Ii

2 ~=1-122-1G

d# 3'~

ë# 2'~= 1-58740 Il 2~=1-68179 100 2~~=1-78180 11 L 2~~=1-88775

1-18921

o

2~=1-25992

f

2~=1-3348.1

Froqnonoy. Il 2~'=1-41421 1

c' = 2-000 21. Rcturning now for a moment to thc pbysical aspect of t!ie question, we will assume, what wc shall af'terwards prove to bc truc within wide lim its,–that, whcu two or more sources of sound agitate thé air sunultaneousiy, thé resulting disturbance at any point ni the external air, or In thé car-age, is thé simple sum the extendeJ gcomotncal scuse) of what would be caused (ni by cach source acd~g- separately. Lot us consider the disturbanco duc to a simultancous sounding of a note and any or ail of its ]iarmouiës. By durmition, thé eompiex wholo forms a note having t)ic same pcriod (and thcrefore pitch) as its gravcst element. Wc0 Iiavo at present no criterion hy which thé two can bc distmguishcd or thc présence of thé highcr harmonies recognised. And'yet–in thé case, at any rate, where thé componcnt sounds have ail independent origin-it is usually not difncult to detect them hy thé so as to cnect an analysis of the mixture. This is as much as car, to say tliat a strictly periodic vibration may give risc to a sensation which is not simple, but susceptible offurthcranalysis In of it Ims point fact, lon~ been hnowu to musicians that under certain circumstancus the harmonies cf a note may Le heard along w.t!t it, uven w!~n thc note is due to a single source, such as a vibrato strier, but tl.e sig.lincancc of thé fact was not undcrstood. Since attention ]~asbccn <1rawnto the subject, it bas becn proved (.nainly by thé labours of Ohm and Hchnho~) that almost a)t musical notes are higtdy compound, in fact of thc notes of a harmonie scale, from which inconsisting particular cases onc or more may be missing. Thé rcason of the unccrtainty and di~culty of thé aualysis will bc touchod upon prcsontiy

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.yof discontinuity 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 surHcIentto 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 1 of thé note, iudendently of piteli. modifythc ~ua~~ or c/t(M'ac<er Tliat such a distinction exists is wcll known. Thé notesof 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'erellceis due to accompanying noises, which are cxtraneous to thcir nature as notes, still there is a part winch is not thus to be ed fur. Musical notes may thus be classifiedas variable in threc ways First, ~t'
14

INTRODUCTION,

f'23.

have hardly any choice but to regard it as dépendent ccc~?'~ on the magnitude of thé vibrations concerned. ~a.rt'&M~ 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 repetitioM of 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 ~ive riso 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 recurs 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 componontsdisentang!ed? The feebleness of thé harmonie overtones is not thé reason, for, as ~veshall sec at a later staf-c of our inquiry, titcy are often of surprisiug loudness,an(.1 play a promiucntt part in music. On thé other hand, if a note is sometimes perccived as a wholo, why does not this happen always? Thèse questions hâve been carefully considered by Hcimboitz', with a tolcrabiy result. The satisfactory difHculty,such as it is, is not peculiar to Acoustics, but may be paralleled in tlie cognate science of Pitysiological Optics. Thé knowledgo of external things which wo derivo from thé indications 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 are accustomcd to associate with thé objcct, and we forthwith infer its presence. In the case of an unknown object thé process is much the samc. We interpret thé sensations to which we arc subjcct so as 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 3rj oditioH, ~<'m~;t(!)ty)yctf, 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.~dquite UitCunsciousiy. Tbu 'it~l& life ui' bacii ono of us is a continued lusson in intcrpreting tho signa presented to us, and in drawing conclusionsas to the actualitics outside. Oulyso far as we succeed in doing tins, arc our sensations of any use to us in thé ordinary affairsof hfe. TI)is being so, it is no wonderthat the study of our sensationsthemselvesfalls 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, though as a fact prohahly-not one person in a hundred million would nnd it out for themselvcs. Thé application of these i-emar'ksto 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&mlngfrom sourceswhich may interest us at thé moment. 'Whcn welisten to thé conversationof 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 recognlsiug its uoity. It would bc no avantage, 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 component tones. A] though, as regards sensation, a resolution into toncs might be expectcd, tho necessities of our position and thé practico of our lives lead us to stop tho analysis at thc point, beyond which it would ccase to bc of service in deciphering our sensations, 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 keyits thé wind to a group of pipes, giving a note and Méatprohubtytho poworof nttondingto tho inipût-tant nnd ignoringtho is to ft greatoxtontiuhcntod–tQhowgreatair nnimportant partofourseusationa oxtontwoshftt)perha.pa Bovcrknow.

1C

INTRODUCTION.

[25.

its first three or four harmonies. Tho pipes of each group aiways sound together, and thé result is usually pei-ceived as a single notfj n!though .h~'H
I 27.]

PENDULOUS

VIBRATIONS.

17

we are led to inquirc what is thc physical characteristic of tones, to winch they owetheir pecuHarity? What sort of periodic vibratton it, whicii prod~ces a. simple tone ? According to wha.t matl)cmatical function of t)ic time does tlie pressure vary in thé age of thc car ? No cluestion in Acoustics can be more important. The simpicst periodicfunctions 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
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, Leforecntcring 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 Buon. 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~ affecting a variable quantity, compound into anothcr of thé same type, wliose clements arc dctcrmined as follows

=rcos(~-A.(2), if i).ud

?'=(($acose)'+(SHsin6)~(3), tau = 2 (t siu e–~Mcese.(4).

1

38. j

COMPOSITION.

19

so tliat if K'=~, ~fvanishcs. In tliis case thé vibrations arc often s:ud to t'~cr/b-e, but the expression is rather misleading. Two sounds may 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;
20

HARMONICMOTIONS

[29.

whcre called the wavc-!cngt]i, is written in place of r, both quantities dcnoting tho range of tlic indendcnt varia.biccorresponding to a complète récurrence of thc fonction. The harmonie curvc is tlius thc locus of a, point subject at once to a uniform 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, fLtracing point attached to thé end of onc of its prongs dcscribesa 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
Thé résultant cannot here be reprosented as a simple harmonie motion with oti~crcléments. If r and r' bc inccmmcnRurabIc,tho value of ?t never recurs but, if r and T be in thé ratio of two who!cnumbers, M recurs after the lapse of a. time equa.1to 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. J

OF 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 harmonie will not continue to rrescnt 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, frequencies rresentcd 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, ncutrahze cach other. After a furtiicr intcrval of time equal to that abuve named, Mtwill hâve gained altogether a who!e vibration, and complète aceordancc is once more rc-establishod. T!)e résultant motion is thcrcfore approximately simple harmonie, wiLhan amplitude not constant, but varying from zero to twicc that of thc componcuts, thc frcqnency of thèse altérations being M-M. If two tuniug f<;rkswith frequcnelcs 500 amI 501 bc cqu~ty excited, tho'e is every second a risc and faU of sound corrcspnnding tu t)m coincidenceor opposition of their vibrations. Tins phcnontenon is ca))ed béats. We dn not hbre f~dtydiscuss thé question how t)tc ear behaves in thé présence of vibrations butit is obvions tiiat If thc motion having )icar]yetjual fre'~K-ncie.s, ni thé nelg)ibonr!)oodof thé car almost ccascs for a considérable fractiu)! of a second,thc sound must appcar to fall. For rcasons that will afterwards appear, béats are best hcard wl)en thé intcrfcring 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 Interférence of two notes may be obtained by mounting two stopped crgari pipes of similar construction and identical pitch sitic Ly sido on thc same wiud clicat. Thé vibrations uf thé two

22

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 recognised, 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'enceof 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<Mof 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), = + a." + 2aat'cos [Spr(~ ?~)t + e e] whcre (2), ft sin e + a' sin {Spr('~ M) + e'{ ,n. tan ec = .(3). 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 corrospondingvalues beiDg a + a' and a

31.]

FOURIER'STHEOREM.

23

31. Anothcr case of gréât importance is the composition of vibrations correspondingto 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 uncommon that tlie innnitudc of arbitrary constants in tho séries of necessity endows it witli the capacity of ropresenting an arbitrary 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 expansion 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 would serve as wcUas 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

'¡;¡,t.t:r.v.

24

HARMONIC

MOTIONS

~31.

now auggcstcd. This is mentioncd hero, becausc students, not, b~in? nc't"aintcd wit~' ~thf) expansions, m~y imagine that nature tiio only oncs (tu:;Jitied simpic h~rmoniofunctions arc 1by 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:Uonc 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 without sun'ering decoma vibrating systcm through propagatcd 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 cocfncicnts may bc determined by a. comparativcty simpio process. \Vc may writc (1) of § 31

and Intcgratmg over or sin Multip)ying by ces ;).complète period from <=C to t = T, wc find

indicating thn.t ~to is thé wcaMvalue of 1tthroughout 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 lossra.pidty,but still remniMcouvergcut, and arithnietical représentatives of the diH'erential coefficients of it, so long as thèse lutter arc cvcrywhcrc finite. Thus (T)iomson and 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

]~< ();))' for coc(ncic)tts.

nm)

ni )'

&c

IN PERPENDICULAR

33.]

DIRECTIONS.

25

from intcresting ~MS of compotin(led vibrations, observatlic facility with which they Icnd themsel-~cs to optical U." harino~c vib~doi~ bion, ~cur wt~i r~more cspecialty ~rcc~ons, ticlc arc exccutcd ~e;~e;~tCM~r but in the ratio whcn thé pcriods are not oniy commensur~bic, Thé motiorL is thcn compléter oi' two SM~tM whoïc uumbcrs. timcs grcatcr th~u tliosc cf thé not with many pcnod pcriudic, If M and v curve dcscribcd is re-cutrant. thc and co.nponents, ho thc co-oi-dmatcs, wc may takc 33.

Another

and dimensions whose an position in ellipse, général reprcscnting vibrations and upon thé of tlie original amplitudes upon dépend tlie dincrcncc of thcir ph~es. If thé phases ~er by a quarter · poriod, co3€=0, and thé équation becomes,

In this CMCthe axes of thé ellipse coinci'dc with't~osc of co-ordinatcs. If furthcr thé two cfjmpMieuts ha.vc ecju~!amplitudes, thé locus (JcgenGmtcsinto thc cirete.. which is described with uniform velocity. This shows how a uniform circuiM' motion may bo analyscd into two rcctilmca.r whosc directions arc pori~enJicula.r. hn-monic motions, If thc phases of thc components agrée, E=0, and the cl!Ipsc "r"j dc"'cncrates iuto the coiticident stmight liucs

When the unison of the two vibrations is exact, thc cUiptic in practicc it will ahnnst but remains stcady, pcrfeetly path bctwocu thc ditTo-cnec n is th:i.t there sli~it :dways happcn periods. TI~oconsequeucc ie timt though a f~xcJ eHipserrcscnts

26

IIARMONIC MOTIONS.

[33.

thc curve described with sufHcient accuracy for a fcw perlods, tho ellipse! itsc)f iyradually cttangos in -jon'cspondencewith t)io 'cr«,noB m t,hujua~uiLudoof e. It becomcs thcroiorc a matter of interest to cojisider thc system of ellipses rrescntcd 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 tlie = 0 As ° dia"'ona.l emcrcascs from C to ~-n-,thc ellipse opcns out until its equation Leçon)es

From tins point it closesup ngam, ultimutely

comcidingwith thc

other

= 0, eon-cspondingto thc incrcMcofefrom ~Tr + diagonal 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.dyfour given tangents, is compictcly dctcrmiucd by its point of P (Fig. 4) with thc linc ~=&.

33.] ]

LISSAJOTJS'

CYLINDER.

27

In order to connect this with e, it is Rufîlcient to observe tha.t Now if thé when ~=6. cos27r?:<==l;and thercfore !t=acos€. of tvvohn.rmc:nc uf thé ~tittl~ bc BUpcrposHion GJIiptiaLhs vibrations of ncarly coincidont pitch, e va.ries uniformiy with the vibration a.lorg ~J.' timc, so that 7~itself cxccutcs a. l)n.rmo)uc witit' a fi-cqucneyequal to thé differenco uf thc twu givcn frcqucncics. 34. Lissn.]ous'bas shown that this systcm of ellipses may be of onc and tlic Sfimc enipso rcr'-arded as thc différent aspects 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 is into a tangent at J. to tlie plane sections, tlie cylinder projcctcd now that thc into its thc Suppose and diagonal. ellipse rectangle, thc plane section with it. cylindcr turns upon its axis, cai-rying Its own projection romains a constant rectangle in which thé pro-

6 represents the posiis inscribcd. of thc Fig. ellipse jcction tion of tlic cylindcr after a rotation through a right angle. It tlio cylinder round we obtain in uppcars thereforc that by turning succession ail thé ellipses corrcsponding to thc pa-thsdescribed by iixcd a point subjcct to two harmonie vibrations of equal pcriod and be turned continuously amplitudes. Moreovcr if tho cylinder 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 rrcsuutation 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 composition of 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'dexpcrimeutn.))y. A Ii(j:Lvypondulum-bob, hung from a iixud point by a long wirc or string, descrihes cliipscHundcr t))c action of gravity, which may in particular cases, according to thé circumstunc'e.sof projection, 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 circumstances 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

1 Dy a vibration will oyclo of aiwaya ho mcaut iu this work a comj)~<
3G.]

CONSONANT INTERVALS.

29

the rectwhich for ~1 values of e représenta a curvc inscribed in angle 2ct, 2&. If e = 0, or 7r, wo ha.vc

shcws thc various curves for U~c FIg. 7 représenta p~bolas. iutcrvals of tLc octave, twuifth, aud ûfth.

To aU thèse systems Lissajous' method of represontation by relative thé transparent cylinder is applicable, and whcn the thé différent circ~mstanccs of from whctber is altcrcd, phase or continuously owing to a sbght déin diiferent cases, projection will viatior. from exMtness in tho ratio of tbe poriods, thé cylinder of thé digèrent to the aspects as to so eye to turn, présent app~r line traced on its surface. sa.DiO so 37. There is no dinicutty in arranging a vibrating system vibrations that thé motion of a point shall consist of two harmonie in any assigued ratio. in perpendicular planes, with their periods A wire The simplest is that known as Blackhurn's pendnlum. two nxcd points at tbe samc Icvel. ~t C-Bis fastcncd at ~1 and . Tbe bob P is attached to its middle point by another wirc For vibrations in thé plane ofthe diagram, thc point of suspension iH practically C, provided that thé wires are sunIcicQtIystretched

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 and DP. Thus if ~C=36'~ the bob describc.s thé figures of thc octave. To obtain tito séquence of curvc.s correspondin~ to ~pproxnnatc unison, Y~ must bc so ncarly tight, tiiat is rdativeJy small. 3S. Another contriv~ncocalled thc kalcidophonc was originally invented by Whcatstoiie. Astraight tllin bar of steel carry'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 circulethé poriod of vibration is indepeudcnt of thc plane in which it is pcrformcd. But let us suppose that the section is a rectale with unequal sidcs. Tlie stress of tl.c bar-tho force with which it rcsists Lcndin~-is thcu grcater in t!te plane of mc.ater Huc~nc.ss,aud tlie vibrations in this phuie have thé shortcr pcriod By a suitable adjustmcnt of tho thickncsses, the two poriods of vibration may bc brought into any required ratio, aud thé eorresponding curve cx]iibitûd. Thc defeet in this arrangement is that thc samc bar will r.Ivc only one set of figures. In ordcr to ovurcome tins objection thé fullowlng modification lias bccn deviscd. A slip of steci is takcn whosc rectangular section is very ciongated, so tliat as regards bcnding in onc plane the stiHhcssis 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 thc gi'L'utthujkMt~ ûi' tho ùti~i-. W: oi' ojf:, ~<j.,tt'.inK LitickucfiM tlie compound rod is clamped in a vice at a point bolow the junction, thé period of thé vibration in one direction, dépending 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/eis thrown into apparent motion by means of successive reflection from two vibrating mirrors. A smaUhole 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.ctas 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~ecyc so that, if it be required to know tlie character of thé vibration, somo indirect mcthod must be adopted. Thé simplest, thcoretically, is to compound thé vibration undcr examination with a uniform motion of translation in a perpcndicuhu'direction, as when a tuning fork dra-wsa harmonie curve on smoked paper. Instead of moving tlio vibrating body itself, we may make use of a revolving mirror, w!iich provides us with an M~K~ein motion. In tins way we obtain a. représentation of tlic function charactcristic of tiLe vibration, with thc abscissa proportional to timc.

33

UARMONICMOTIONS,

[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'

a.~tDu~ui~

vibnt.t'ofi i

'fbU~i'bL'

))nri.).1

in

i.h<'

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 parallcl to itsedgcsdraw thé curve rrescnting 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 running 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. Conversely thcrofore, in order to obtain tho représentative curve of tho vertical vibrations, the cylinder containing t]ic apparent path must 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 adjustment of tho periods is not quite exact, for thon tLe cylinder appears to turn, and the contrary motions serve to distinguisb those 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 a vibration-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 ed. 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 rigour and universality of the law connecting pitch and ~ep't'o~. TIms it will bc found that any point of a vibrating body -\v)uc!)gives a 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 dépendsupon 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 afforcledby 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'aIIclto thc prongs of thé fork, and so placed as to aAbrd a fj'cc agethroug)i thé plates whcn thé fork is at rcst, or ing through thé middte point of its vibrations. On thé opening so formed,a beam ofsunHght is concentrated by means of a burningglass, 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 exct when the fork is ing through Us position of equilibrium, or nearly so. The nashcs of light obtained by this method arc not so instn.nta.nouusas 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 hodythrough a séries of holes arranged in a circlc on a-revolving (tisc. Several séries of holes ma.y be providcd on the same
34

HARMONIC

MOTIONS.

[43.

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 things is 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. IIAVINGONEDEGREE0F FREEDOM. SYSTEMS 43. THE matcrial systems, with whosc vibrations Acoustics is concerned, are usually of considérable complication, and are susctible 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 practical 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 problem 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 frecdom 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 ourselves to tlie consideration of positions M!.the ?'y~:e~'<~e?:eK/Au is a small quantity, &~u)7iOOfZ o/< con'M~on~t')~ e~x~t~, and m and are sensibly constant. On this understanding wo 3-2

36

ONE DEGREEOF FREEDOM.

[4~.

now proceed. If there he no forces,cither rcaulting frominternai friction or viscosity,or imprcss'~don 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-tionof motion ~m + /tW= 0 (2) of which thé complète integral is ~=(tcos(?)<

a)

(3),

vibration. It will bo whcrc ?~=/7):, rrcscnti))~ a ~Muc sccû that thc pcriod alone is detemuned by thé nature of the system itself; the amplitude and phnse dépend on cothttcral circumstances. If tlie difrercutial equation wcrc exact, that is to and F to thon, without say, if T werc strictly proportional to any restriction, thé vibrations of thé system ahont its conDguration of equilibrium would bc accuratc)y harmonie. But in thé majority of cases tlic propoi'tionaHtyis only approximate, dending on an assumption that tlie displacemeut ?
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 eqnHibrium by a spring, ?~ may be called thé inertia of thé system, and

44.]

DISSIPATIVEFORCES.

37

of u. thé force of thé équivalent spring. Thus an augmentation mass, or a rc!f).xationof spring, incrcas
<(}.(2).

38

ONE DEGREE0F FREEDOM.

[45.

If thc friction be so gréât that rel="nofollow"> thc solution changes its fonn, and no lorger f'orrRsp.nds to nn os<-Hiatnrymotion; but In .di acousticai applications A:is a small qu~ntit'y. 'Under Dicso circumstances (2) mny bc r~u'ded as cxprcssinga harmonie vibratton, whosc Mnpiitudc is not constant, but dimiuishcs m gcomett-ical progrc.SHio]), wlicn consi~o-cd aft-cr cqu~l iutcrv~Is of time. Thc difïercncc of thé logarithms of successive cxtronu excursions Is nc:u-)yconsent, :md is ca)!edt]tc Logar:t!imlc Ducremfut. It is cxpresscd by ~r, if T bc thu puriodie timc. Titc frcquotcy.dcnding on ?~- ~~Invo!vG.s on]y tite sccotd powcr of A:;so that to thc rir.storder of approximation ~e/c~'o~ /t(M?!0e~ec~o~ ~c y)en'o~a principe ofvo-y gênera! appiicatiun. Tho vibra~on iicrc consideœd is ca!]ed thc/y-ce vibration. It is tbat cxccutcd hy thc System,when disturbcd from cquiHbrium, and tbcn to itself. 4G. Wc must now turn oui-attoition to anothcr problem, not Jcss Important,–thé bchaviour ofthc systcm, whan subjuctud to a ibrcc varying as a harmonie funetion of thc timc. In ordcr tu savc rctition, wc may takc at once the more gcncral case ijicludinnfriction. If tho-c be no friction, wc bave on)y tu put in oui- rcsults /<= 0. Thé dincrential équation is

[' This is caDeda./M-c<~vibration; it is thc responsc of thc System to a force Imposbdupon it from wititout, and in mainta.iued tho by coutinued opcratioa of that force. T]ic amplitude is proportional

46.]

FORCED VIBRATIONS.

39

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 effuuton 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 samoas 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 ing through the position of equilibrium. It is necessary thus to specifythé 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 ceshi~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
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,

40

ONE DEGREE

0F FREEDOM.

[46.

If p bc Jcss tha.n ?;,the rctardation of phase relatively to tho force lies betwech xeru and a qu:u-te!pcriod, aud whcn is ~reater tit:m}.[.,butwcchi(.~U!u'~t'~(.'i:m!n.i,.bntfuut~d. In t!)c cuscof a systcln devoid of i'riction, tlie solution is

When is amaller ttian ?~thc pl.ase uf tiiu vibration agrées with tliat of thc force, but whcn~ Is thé grever, the sign of thé vibration is clianged. Thé change of phase from complète agreement to complote disagrcemeut, which is graduai wlien friction acts, hère take~ place abruptty as pa.sses through t!ic value 7t. At thé samc tune thc expression for thé amplitude bccomes inanité. Of course this oniy means that, iu thc case of cqual periods, friction 7~<~he taken into , Ijoweversmali it may be, aud liowevcr insigniricaht its rcsult wben and ?t are not approximatc!y cqua). Thc limitation as to thé magnitude of thc vibration, to which we are all along subject, must a)so bc borne in mind. That thé excursion shouid bc at its maximum in one direction whi!e thé generating force is at its maximum in tho opposite as for du-eetion, happons, cxampic.in the canal theory oft!ic tiftc.s, is somcti.ncs considcred a paradox. Any dimculty that may be fc)t will bu ronovcd by considering the extrême case, in which thé ".spring vanishes,so t!.at thc natural period is Innnitety lono-. In fact we nced oïdy consider the force acting on the bob of a'common pendutum swinging frecly. in which case t]ic excursion on one sicle is greatest w)tcn the action of gravity is at its maximum m thc opposite direction. When on thc other hand the inertia of thé system is very sma)I,we hâve the otticr extrême case in which thé so-c.Ued equiHbrium theory bccomes applicable, tlie force and excursjou 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 retardation in thé ph:Lscoft)tc dispiacementvaryingfrom zero up to nquarter penod. If, ),owever,the period of thé natural vibration bc tho longer, thé original retardation of haïf a period is diminished by short ofa somethmg quarter period; or thé cn'eet of friction is to Mc~e tlie phase of thc disphccment cstimatcd from that eon-cspond.ng to thc absence of friction. In cither case thé influence of fr.ct~oni to cause an approximation to thc state of things that wou!d prcva)I tffrictioTi wcre paramount.

46.]

PRINCIPLE 0F SUPERPOSITION.

41

If a force of nearly equal period with thé free vibrations vnry s1o\v1yto a maximum and then slowly décrète, thc dispjacement 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 increasountil a certain limit is approachcd,and this incrcase continues for a time cven att))ouglt tlie force, having ed its maximum, begin.s to diminis)). Ou ttds principic tlie t'utardation of spring tidcs bchiud tlie da.ysof ucw and full luooulias bccn cxp]ained'.1. 47. From tlie linearity of the cquations it follows that the motion rcsulting from thc simuItanGOusaction of any numbcr of forces is thc simple sum of tlie motions duc to the forcesta~en sarate!y. Each force c:uises tlie vibration proper to itself, wthout regard to tlie presoicc or absence of any othos. Thc peculia-ritiesof 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 element of tlie force will call forth a corresponding harmonie vibration in tlœ system. But since tlie rctardation of phase e, and the ratio is not the samc for thé different components, of amplitudes M the resulting vibration, though periodic in thé same time, is diffrom the force. It may happcn, for instance, férent in c/t
48.

Thc distinction betweenybrce~and~'ee 'vlbra.tioQSis very ~Airy'B2'(~t'

42

ONEDEGREEOF FREHDOM.

[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)y on the constitution ofthe system itself. Anothcr point of din'ercnce is that so long as the extcrnal influence continues to opcratc, a forced vibration is permanent, being rresentcd strictly by a harmûnic function; buta frec vibration graduallydies away, becoming ncghgibic aftcr a timo. Suppose, fur cxample, that the systcniis :),trcst when thc force 7~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 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 possessionof 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 Mor !<,but tho forced vibration is at once convcrtcd into a frce vibration and the poriod of thc force is cxchangcd for that natural to the system. Dm'ing thc coexistence of the two vibrations lu thc earlier part of 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 satisfied 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 in two ways according to t)tc values of the constants Mand K. Thé distinction ofpitch is sumcIe)tt!yIntG!!igibic; but it is worth w]nle to examine more closcly the con'sefjncnccs of a grcatcr or less dcgree of damping. Titc most obvions is the more or less rapid extinction of a fi-cevibration. The enbct in this direction may be mcasurcd by the numbcr of vibrations wliich must e)apsc bcfore the 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'

f

VARIOUS

40.]

If = ~T, wc have a;= nnjucru-Lu

du~reu

DEGREES OF DAMPING. 2

xT log

ut' dampmg,

\vc

43

la a, system subject to on)y a tua.y

tn.kc

upprox.UTmtcly,

Thi.sgives 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 motion becommg m~nite. We might casily auticipate thatwheu tUc damping is small, a. compara.tivelyslight dcvia-tion from perfect isoein'onismwûuld cause a large fa.Hmgoff 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.1to 7)jlu. ordcr to producc a, motion ]iot grea.tly Icssthan 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 cliosenas 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 casesof 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 réduction of Uto ruM.ttauco to ouc-touth. y y ~Q

v'Lmtif.nM nfter whiuh tho i'itcnHity of a frco vibrntiou is roducudtoono.tunth. ~=A.

tonc.

~.oo'~ 19'00 9-50

? Whuif! tonc.

G-33 4-75

tuno.

3.~0

y tono = minor thu-d. 7 toile. Twu whuit'

tonea~

g. jy 2-71

major third.

Formula (4) shcws that, w!ien

~'37

i.s small, it varies c~~M

as '1. Mtt)~~ a: 50. From observations of forced vibrations due to known forces, tlie natural period and dampiug of a system may Le detertniûGd. TIio formuhuare

If tlie equilibrium thcory Le known, tlie comparison of amplitudes tells us tlie value of sav

~ifH~/ntJf~fyc~

p. 221.

50.]]

STRINGWITH LOAD.

45

and e is also kuown, whence

51. As bas been ah-cady stated, the distinction of forccd and frcc vibra.tionsis 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 upon it. On a wider view of thc matter embracing both thé as An Systems,thc vibration in. question will be recognizcd free. 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 motion thé air as forced. No crror will be committed if thé f <:ta~ of the fork (as innucnccd by its sun'oundings) be takcn as tbe basis of eaicutation. But thé peculiar adva.ntagcof 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 principles of 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 ncgliis pulled asidc from its position of equilibrium, gibic. WIten and thcn Ict go, it exécutes along thé lino C~ vibrations, whicb C= 6'~ = M. C'.V=x. Thé tension are the subject of inquiry. of thc string in the position of equIHhriumdépends on the amount of the stretchiug to which it has been subjected. In any othcr

4G


[52.

position the tension is ~reatcr but we limit ourscivcs to the case of vibrations so small that tt~e additiona! strotching is a ncgJigibJe fraction of the who)c. On th~ (~ncHii~n thc ~
Thus, Idnetic cncrgy = ~ï;

Thé amplitude and phase dépend of course on thé initial circumstances, being arbitrary so far as thé dinforcntial équation is conccrned. Equation (2) expresses thé ïnanner in which 7-varies with each ofthe Independent quantities V.~a: resultswhich mayall bc outained by considération of the (~MCHs~~ the tcchnica! (in sensc) of the quantitics involved. T!~G argument from dimensions is so often of importance hi Acoustics tliat it may bc wcll to consider this 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.J

METHOD 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 wnLLeuThis equation must rcta.in its form unchanged, whatever may l)ethe 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 magnitudes of those units. New of all tlie quantities on which f dépends, Tis the oniy onc involving time and since its dimensions 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 ittcd, it is ca~y to see that in order that (3) may be independent of the unit of Icngth, we must Imver ce T"~ n~,when Is constant and finally, in order to secure indcndence 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 deuding on no other quantities, having climensions in spacc, time, and mass, t!ia.nthosc aLove mcntioncd. For example, we hâve not proved that r is indépendant of thc amplitude of vibration. That, so far as it is truc at ail, is a conséquence 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 proper Ct~reit is unqucstionably of great power and value.

ONE L.MCREE 0F

FREHDOM.

F~.

.'3 1 lie solution of thé présent problem might bo made thc foun~tion of a ,nethcd for Lhe absolute n~asurerncnt of pitch. pnncip~J impedunc-nt to accuracy would prubabjy.be t!to oi difBculty u~ku~ iu relation to thé m~ of suf!ictcutfy tlie ~u.c, without at tlie samc timc i~ thé note too much in lo~crin~ thé musical scalc.

T)ic wirc may bo strctc)ied by wcight fur hcr en<) beyo~i bndgo or pulfey at wouidbcc:dcu);),tcdfrom

~t~chcd te its Thé pcnodic time

T).c ratio of = ~i, t).e balance. If r. be ,no..sured in fect, aud~= ~.2. tl~c pcriodic timc is exprcs.sedin seconds. musical the .vcight, Instead of ~n~ an being concentratcdin thé centre, is uniformtydistnbuted over its !cn~ t)ic présent problem gives some ide~ of thé n~tu.'coi~evertiK.Icss the gr~vest vibration of snch string. Let t.. compare thé two more c osoJy, supposingthc amplitudes of vibration t)te same c~cs at thé jmddie point.

-When thé uniform string is .straight, thé moment of in~ 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 bG cousidcrab!y ovcr-estimated. Again, at the moment ci' maximum excursion, thé uniform string i.s more stretched than its suhstitutc, winch foltows thc straight courses ~1~ and accordingly the poteutial 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 supposition that ~1/ dénotes thé mass of the string. It will afterwards appear t)jat in order to obtain a correct result we should !)avc to takc in4 4 Of stcadof.Von!y-~V. thefactor-~TT hy far thc more importTT ant part, viz.

is duc tu (,hedifTcrcnceof tlie kinetic énergies.

55. As another example of a Systempossessing practicu.Hybut 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 Indcndent modes of vibration but, whcn thc mass of t!tc spi'mg is )-e!ativc!y sn-i:d),ttiat vibration which is ncarly indcndont of its inci'tla. buconics so much thé most important 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-eis a, constant tendency towards the communication of motion to tlie s, to resist which a.dequate!y thé latter must be very ni'm and massive. In ordcr to obviate this inconvenience, R. 4

~0


[5G.

two prccisely similn-r springs and lo~ds m~y bc mountcd ou tlie same frame-work in a symmctrical manncr. If thc two loads pcrform vibrations of cqual amp)itude in such a, manner that the motions arc a.Iwn.ys {m opposite, or, 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-workinto vibra.tion. We shaU sec in a future chapter that this peculiar relation of phases will quiddy estab)ish itself, wt~tever may be tho original disturba.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 damping, unless indccd thc s of tbe system arc more than usually nrm. 57. As in our first exemple wc found a rough illustration of tho 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 for instance as the ~:<e of a )~e~ instrument. It is truc of course that tlie mass is not coucentmtcd at onc end, but distnbutcd over thé whole Icngth; yet on of tlie smallness of thc motion ncar the point of , thé inertia of that part ofthe bar is of but little . Â~einfer that thc fundamental vibration of a uniform rod cannot be very dincrcnt in cbaractcr from that which we ])ave bcen considering. Of course for purposes rcquiring précise calculation, the two Systemsarc sufnciently distinct but where t!ie object is to form clear idcas, precision may often 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 improvcd of late years, is indispensable to the acoustical investigator. On a large scale and for rough purposcs it may bc made by wciding a cross piece on the middle of a bar of steel, so as to form a and T, 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 tunmg forks 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.

5jI

madc, thctwo parts opcned ont to form the prongs of the fork, and thé whole workcd by tho iiammer and n!u into thc rcquircd T)ic two shape. prongs must bc cxactiy symmctricat with respect to a plane ing through the axis of thc liandie, in ordcr that during t!ie vibration thé centre of incrtia may remain unmoved –unmoved, tiiat is, in thc direction in which thc prongs vibrato. Thc tuning is cnected t)ms. To make thé note higher, thé équivalent incrtia of thc System must bc rcduccd. This is donc hy nling away t)ie ends of thc prongs, cithor diminishing their thickncss, or actuaiïy sliortening thcm. On the other hand, to Jowcrthe pitch, tlic substance of the prongs ncar thc bcnd may be 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 tcmporary pm-poscs) by loading thc ends of thc prongs with wax, or material. Large forks arc somctimus othcr provided with movcable weights, which slide along thc prongs, and can be nxcd in any position by screws. As thèse approach thc ends (whcro thc vetoeity is greatcst) the équivalent incrtia of thc System incrcascs. In this way a considérable range of pitch may bo obtained from one fork. TJ)c number of vibrations per second for any position the weights may be markcd on thé prongs. of Tite relation bctwcen the pitch and thc''size of tnnin~ forks is In a future chapter it will be rcmarkably simple. provcd that the material remains thc samc and tho provided shape constant' tt.c period of vibration varies, dircctty as t)te linear dimension TIrns, if t!ic linear dimensions of a tuning fork be doubicd, its note falls an octave. 58. Thc note of a tuning fork is a ncarly pure tone. ImmediateJy after a fork is struck, high tones may indccd be hcard, to modes of vibration, whosc nature will bc subsecon-esponding qucnHy considered; but thèse rapidiy die away, and cven whilc s they exist, they do not b!cnd with thé propcr tone of the fork part~y on 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 overtoneshad a frequcncy from 5-8 to n-G timcs titat of the proper tone. Tunmg forks are now generaUysupplied with résonance cases, whosc effect is greatly tu augment the volume and pnrity of the 4–2

53

ONE DECREEOF FREEDOM.

[58.

In sound, according to principles to be hcreaftcr dcve!opcd. oiJer to excite thon, a -viotin or ccHo bow, wcll supp)icd with :~dr~t.t .Cr~~ rh< prongs'u ~'u dit'<
(;0.'j

SCHEIBLER'STONOMETER.

53

Thé pitch of thé other forks point at thé bottom of tho séries. couldbc dcduccd. If consécutive forks givc four béats per second, C.5in 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-atcfor tl)C original dctcrmina.tion of pitch, as it Is liabtc to no ct-rors but such as care and repetitioti will clhninatc. It is tho mcasurcmcnt of mn.ybc obscrvcd tliat thc cssctYtiatthing t)tc ~er~ce of frcqucncics for two notes, whosc ~o of frcqucncics is UKlcndcnt]yknown. If wo could be sure of its accm-a-cy, thc Intci-v:dof thé nfth, fourt)i, or cvcn.major third, might bc suhstitutcd 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!yadoptcd, 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 which aHowiugthem to trace a harmonie curve on smokud papcr, of a rcvo)ving tnny couvcnicnHybc mountcd on thc circumicrenco drmn. Thu muubci' of wavcs cxccutcd in a second of thnu givcs thcfrcqucncy. In many cases tbc nsc of Ittterniittcnt Hturnination duscribcd an nnknown in § 4-2givcs a convcniunt )net))odof dctcrmining frcqucncy. (il. A scrics of forks ranging at snndi int.crv:usover 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 of thé note is dctermincd hy tl)e nuinher of beats thu f')-u(Utcncy \vhic)i it givcs with thc i'orks, which lie aearest to it (on cach sidu) in pitch. For tuning pianofortcs or organs, a. set of twelvc forks may be uscd giving thc notes of thc cbromatic sealc 0)1tho equal tempérament, or any dcsircd system. Tbc corrcsponding notes are is adjusted to unison, and t])C otbcrs tuned hy octaves. It 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 frcfptencics is not the samc thing as a more displaccnicut ofthescatcinabsolutitch. 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.chf)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 incvitabiu o-ror is thcn hnpartiaUy di.stributed, an
;e+~+.(.t;J=0.

C2."]

COMPOUND PENDTJLUM.

oe cos~<; so that p)itting~=)~, New tlie form ah'cady trea-tcd of, viz. .v+ A:~+ )~ = cos~

55

our équation takes

If~) l)e equa.1to ')!,thc motiou is limited o!i1yby the friction. Thc a~sumed horizontalharmonie motion forP maybe rcajized by mc:ms of a,second pcndulum of massive construction,which can'Ies .P witli it in its motion. An cfncicnt arrangement is shewa in t)tc ngnrc. /t, .Harc iron rings scrowcd into a beam, or other nrm

; C',D similar rings attachcd to a stout bar, which carrics cqua! hcavy weights A', :tttac!K!dncar its ends, and is cd in a hurizo)it,al position at riglit angles to thé beiuri by a wirc ing through thc fuur rittg.s in thc tnanner shcwn. Whcn tlie pcndulutii i.s)nndc to vibratc, n.point m thc rud midway bctwcot C' and D exécutes a hiu'mouic motion ni a direction paridtcl to 6'D, and jnn.ybo nmdo thé poitit of a.tta.chmcntof auother pcndutunt -Z~. If ttte wcights A~and be vcry grcat in relation to Q, t)jc uppur penduhun swings vury ncn.r)yin ils 0~1 propur poriod, and induccs in () a. furccd vibr:<.tionof titc s!t.nieriod. \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~hthc vibration ot' j! bc of but niconsidura.bln ampHtudc. ln this case the diHcrenccof 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~iaÛbrdcd 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.tcdwire on nn iron corc of tho shape shcwu ni E (simila.r to titat known as 'Sioncn.s' armature'), ~nd cd betwccu tho prongs ofthc fork. Whcu an intermittent current i.ssent through thé wire, a periodic force acts upon the fork. This force is not cxprcssibic by a simpic circular fonction; but mn-ybc cxpandcd by Fouricr'.s theorcm lu a scrics of sucli functions, ha.vlngporiods T, T, T, &c. If nny 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 inca.nt. In wbat follows wc will suppose that it is the complete pcrioclT whicb ncarly ngrcc.switlt tliat of the furk, and couscqucntly rega.rdthc séries expressing thé pcriodic forceas reduccd to its first term. lu order to obtain t))C maxitnum vibration, thc fork must be cai'cfuHytuncdhy 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 comptctncntary 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\\nxmny<'onvp))ifnt]y it wiU)a )itt)
G3.]

RELATION

0F AMPLITUDE

AND PHASE.

57

tho adjustmcnt. It is not until a-ftertl)c froc vibration lias bad time to subside, that thé motion assumes its peru.anent ch~'acter. T))Cvibrations ofa tuning fork properly constructed and inounted arc subject to very little damping; consequcntly a vcry slight déviation from perfect isoclironism occasionsa markcd falittig off in thé intcnsity of the résonance. The nmpHtudo of thc forccd vibration can bc obsci-vedwith sufïicicnt accuracy by thc car or cyc but thé expérimenta! verification of thc relations pointed out by thcory bctwccn its phase and that of thé force which ca.uscs it, re<~uresa modined at'rangcmcnt. Two similar cicctro-magncts acting on similar forka, and included in thc samo circuit, arc excitcd by the same Intermittent current. U])dcrthcsc circumstances It is ctear tha.t thé Systems will bo thrown intn sunDar vibrations, becausc thcy arc actcd on both to phase by cqual forces. Tliis similarity of vibrations rcfcrs 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 ncccssarity a. straight lino. Starting from tho case in which thc of both o.mpHtudesare a maximum, viz. whoi tbo natural pcriods 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 Ciffcctof thc dift'urcnce of tbc natural periods is to destroy the synchroïlism of phase. Thc straight hue, which prcviousiy rrcscnted 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. ~Vhcnthc pitch of one is slightiy ]owcred,it falls still more bchind the force, and at thc samc timc its amplitude 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'.

58

ONE DEGREE 0F FREEDOM.

The following e table shows thé simu1ta,neousvalues of a a.nde'.

[C3. c< 0

e 1-0 -!) -8 .7 'C -5 '4. '3 -2 -1

0 25°50'1 3C°52' 4.T'3-t' 53°7' GO" 66"25' 72° 32' 78° 27' 84.°15"

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 dinfcrfrom it on either sido by as muc)t as CO"in phase, without lo.singmoro than t)alf its amplitudu, or by as much a.s-I<5", without losing more tha)i Iiaïf its e?M)'~y. By aHowing one fork to vibratc 45"in advance, and tbc othcr 45"in arrcar of 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 forkinterrupter invented by Hchnbottz. T)tia may consist of a fork and cicctro-magnet mountcd as before. TIie wires of thc ma~nct arc connected, ono witb ono po!c ofthcbattcry, and thé othcr with a mcrcury cup. Thc ot]ier pôle of tbc battcry is connectod with a second mcrcury cup. A U-shapcd rider of insulatcd wirc is carried by t!)c lower prong just over thc cups, at sucb a Iieigbt tha.t during the vibration thé circuit is altcrnatejy made and brcken by titc age 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 compensatcd, and thc vibrations of thé fork pcrmancnciy maintamed. In order to set anotbcr furk into forced vibration, its associatcd ctcctro-magnct maybc includcd, either in tbe sanic drivix'Y-circuit ?'<)«'))t;~?)t<~o~t't), p. li)0.

G4.]

FODK

INTEBRUPTER.

59

or m a, second, whose periodic interruption is effected by another rider dipping luto mci'cury cups'. of tm.s kind ui seti'-act-inginstrument is Tho ??~(~<& ~«/ 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 wouldbe 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'datiouarc two irregular , and scJf-mduction. Wltcn the point of thé rider nrst touches thé mercnry, thc cicctnc is imperfcet, probahly on of adhcring air. On thc other ha.ud, in leaving tlie mcrcury tho is prolonged by the adhésion of tlie hquid in the cup to thu amaigama.tcdwire. On botli s thé currcnt is retarded behiud wliat would correspond to thc mcrc position of the furk. But, evcn if the resistance of the circuit dended only on the position of thé fork, thc current would still be rctarded by its selfhiduction. However perfect thé 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<Tits entrancc, and thus a, balance remaitis to be set off against friction. If t!)Cmagnetic force 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 reun t)iocomponont 1 1 Lnvoarr<mgc
60

ONE DEGREE OF FREEDO~f.

[G4.

taxation apoken ofrcdnccs this advance. If thé phasc-diu'crcnce be 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 (cxct in thc case just, rcfcrrcd to) the actual pitch of ttie mterruptcr dKFcrsto 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 s :ind
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 associated with sounding-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 Mtringsbeionging to any note bc plu.ckcd ()ikc a Itarp string) with tlie nnger, its feHowswill 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'ato c([uality of pcriods is nccessary in order to cHeit t))c full cfrcct. Of tins class tuning forks, 'muuntcd on 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 backwards 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.sfrom 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 constants of intégration J- and a in ternis of tlie initial values of !< Wc obtain at once and M,which we may dénote by and

ONE DEOREEOF FREEDOM.

Thé cfTectof ~7i.sto gcncr~to in time f~' a vdocit.y r< rc8u!t a.Ltune <will thcrcforc be

[f!G.

w))o.sc

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, wlien = t 0, and the complete solution is

Whcn t Is sumdently grcat, thé complementary tcrms tcnf]
C7.]

0F THE SECONDORDER.

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-iLthcras simil.n' to Infiuitcly srn:Jl vibrattons. This rust.nctiu]i is thc i'omuhttion of thc important lim's oi' isoclu-ontsmfur t'r<jGvibrations, and (jf pc-rsistcucc of pcriod fur forecd vibrations. Thcru arc, nowevcr,ph<jnunicn:t,of a subordina.tcbut not insigniricant charactcr, winch d<jpc!td csscutiidiy on the s
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(! (.~rc~.iiguncr.diy pcrccuc thô octave111tho suu))()of a. tunitig fork causcd to vibrn.testrongty 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,uncoof the octave may bc niadc manifcst to any onc. By foUowhtg thc same inethod the appmxnnatioa ca)t hu ca)-)-iudfurthcr but wc pa,.sson now to the case of :). syston )n which thc recovo-mg power is synmiotncal with respoct to )hu position of cqmtibrmni. T])c équation of motiou is t!tcn app)'oxi)))n.tu!y winch may be uuderstood to i-ufc-rto Miovibrations uf a hcavy p'n(h)!un~ or oi'a )u:).<1 c:u-n<jdat tho end ofa, sti'iu~ht spring. If wu t:tkc an a jh'st npj~'oxitnatmnM=-~ cos?~, cotïespondi)]g to /9 = 0, a.ud substitatc in tttc tcnn muhiplicd hy /3, we get

wc shonid Corrcsponding to the last tcrmofthiscqnation, obtain ni tho soh~tion a tcrm oftiie form <sin~, becoming without Jnnit with t. Tt.is, as in a paraUd case in t]ic grcatcr Lun:n- Thcury, indicatcs that our assumcd iirst approximation is not rc!).]!yan approximation at a]), or at Icast docs not coH~e to bc such. If, Ilowcvcr, wobikc as our starting point u=~4 cosM~, ~ith a, suitaUc vaillo for M?,wc sitaïï find that titc solution tnay be 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 is approxinuttely simple harmonie, with a pcriod ah~M'o.-n~n<
ofwilichthc solution wilibe

67.]

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~csexcccdiuglysmall–dcuding 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 dendence of pitch on 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 cases but in one respect there is a distinction worth noting, namely, that in (8) m" is always greatcr than while in (7) it dcuda on the sign of /3 whethcr its effect is to raiso or lowcr the pitch. However, In most cases of the unsymmctricat class the change of pitch would depend partly on a term of tho form «M' and partly on another of the form /3 and thcn

C8. We now to the considération of the vibrations forced on an unsymmetrical system by two harmonie forces Thc cq~a.tionof motion is

CG

ONE DEGREE0F FREEDOM. Substit.uting this in tLc term muJtiplicd by

[68.

wc~ct

Thc addition~ tcrms rrcscnt vibrations frcqucncic~ which arc scvcndty thc d.u.bius ~.
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 onc dcgrce 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 guticnd more than ouc dL'grcc of frcudoui. In o!'(!cr to (tufinc thcir cou(1gur:ttio)iat. any moment scvcnd uxhipcmtott vin'mbic qn~tttidcs must bc spccificd, whici), of ):u)gU!~c ori~hin.Hycm]))oyc
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 sufHcicnDysmdt. Tins quantity is ciUlcd and reprcseuts thc work thf~t may bc gfdncd in ing 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,!cntto 8F, so thfttsmcc 8~, 8~, &c. nro ludcudcnt, the new position of cquilibrium is dotermincd by

-wherethere is no distinction in value bctwecn and c, c,, From thèse équations the co-ordinatcs may bc dctermmcd in of the forces. If ~7bc thc dctûrmIuMt

71.]

RECIPROCALRELATION.

G9

Thcsc équations détermine ~r,, &c. uniquely, slaco doca ] .'i~ ~u t).ppcm. ~ont t. ~h' Lh' 'f~f;{'.L'uh ilot Y:in)Hh, Mût 'iLb~til t! coï.~idi.'t'Htiuit === 0. &c. could othcrwiso be s~tisficd by fmitc values of tha co-ordina.t.cs, provided oniy ttmt tho ?'(t~'oswcre 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 :i displacetneut of f),second type cqual t.o tlie displiicement of tlie first type duc to thc action of au cqnid force and R 'be two points of thc second type. For example, if of n, rod snpported horlzont:dty in ~ny maunc! the vertical dericction at jl, whcn a wcight }F is ~ttachcd at is tl)u s:t.meas the détection at 7?,wlien ~F is appiied at ~t\ 73. Since F is a homogeueous qua.dra.ticfunction of thc coordinates,

If + ~~). ~+ ~~) ~c. rrcscnt auniticr (Iisp!act'mcntfor wbich thc neeessary forcus n.ro ~+/ ~+A~,&c.,thecorOuthiBeubjoct., sco7~tt~.J/< Deo.,1874,nudMMeh, 1875.

70

VIBRATING

SYSTEMS IN GENERAL.

f73.

ruspoudingpotentud cncrgy is givcu by

whL'rcA'FistiKi'n~n.'ncuojf'thupntcntI.dcnL'rgicsinttxjtwo ciLScs,:unt wu must p:u'<.Icnl:n')yuoticu tIi:Lt by tlie i-eeinroc:~ rui~iuu, § 72 (I), From (~) and (~) wu may deduectwo nnport:u)t Dicorc'tns rclating to t!ic vainc of fur a systeni subjeetcLt to
STATICALTIIEOREMS.

74.]

sufHxcs ~Fvanishcs. AccordinglyST.A.P'r i.~aise xcro. Hon~c

71

is zero, n.nd therefore

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'ccton T'of any l'cl~xationm t!)esttH'ucssofa. System,suhjcct to given displacemcnt conditions. For, ifaftcr tlic altération m stitTnessthc original cquilibrium connguration Le considut'cd,thcvidnc of Vco)'ruspon()ing t))crcto is by supposition Icss t)i:m bcforc; :md,as wc h~vc just seoi, therc will be n. still furthcr dinunution in tbe value of F' whctt tlio Hystcmes to cqnilibrimu undtjr the niterud conditions. Henco wc condudc titat a. diminution I)i as a functiou of thc co-ordin:t.tcscntails also n diminution in the actual vatuo of F' whcn a systcnt is subjcct to given disp!:).cemcnts. It will bo undur.stoodtluit in pa.rticuhu.'cases thc dinunution spokcn of l. may vanish*. For cxample, if a point J' of a bar dampcd at both ends be disph~cudlatcndiy to a given small antount by a force tbm'c appiicd, thc potentiel cnurgy of thc dcfui'mation will be diminished by :).nyrelaxation (howeverloc:d)in tite stiiïhess ofthe bar. 75. Tlic secondtheorem 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 connguration in \vhich thu syst~'mcoutd bc maintained at rest undur t))c givcn furecs, by the 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 constraint arc cxprcs.scd by ~=0,

~=0,=0.(1). Wc hâve thon to provc that whcn ~P~, ~P~
~2

VIBRATING

SYSTEMS IN GENERAL.

[75.

and therefore shewing that thc incrcase in F duc to thc rcmoval of the constramts is cqual to tlic potcntial encrgyof tlie din~rcnce ofihc two configurations. 7G. We now to the luvesti~tion of thc initial motion of a systcm which starts from rcst undor thc operation of givcn impulses. The motion thus ~equired is Indepcndcitt of any potuutm encrgy .vhicl~ the system n~y possess .vhcu actu~y disptaccd, siueo by tho nature cf impulses we h.wc to do only with thé mitml configuration itself Thc initial motion Is also mdependcnt of any forces of Huitc kind, whethcion imprcsscd tlie system from without, or of the nature of viscosity. If Q, 7i'bc the component impulses, parallel to thc axes, on ~partie e ~vhoso rcct.nguhr co-ordinates h.vc by are DAlGmbei't'sPj-iucip!o

dénote thé vclocities aequircd the whcrc by particle in virtuc of the impulses, aud correspond to auy arbitrary displaccmcnt of thc system which docs not violate thc councction of its parts. It is required to transform (1) iuto an cquatiou cxprcsscd by thc independerit gcncralizcd co-ordinn.tcs. For thé first side,

the kinetic cnergy of the whcrc system,is supposcd to be expresscd as a function of &c.

IMPULSES.

76.]

73

On tlie second side,

whcrc 8~, S~, &c. arc now eompletely iudcudcut. Hcuco te détermine tlie motion wc ha.vo

whcrc S

&c. may bc constJcrcd as tlie gGneraItzcdcomponcnts

ofi;i)putsc. '77. Since y is a homogcncousquadratic fuuction of the generalized 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]yanalogous to that of thé displaccmcut of a. system froni a configurationof stable cquitibrinm by steadily a,pp!Icdforées. lu thé présent tucm-ythé 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

VIBRATINGSYSTEMSIN GENERAL.

[77.

ively. In one respect the thcory of initi:d inot!ons is the more L in gt'ncmt oniy complet~ in!).s)nuc)i.as is cxactty, w)ti!c approximatcty, a, itomogcuc'jus(~uadrnticfuuction of thé variables. dunotc nnc set of vclociticsand impulses If' for n. systuin st:u'tcd f'run)rcst, :utd a. second sct~wu iua,ypt'uvc,as in § 72, thé fuDuwingrecipt'uca.l t'L'ta.tiun This thcdn'm ndniits cfitttcresting' :)pplie:ttl<)!ito 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.sibtuliquit), which starts i'rota rc'.st,i.s cf such :t.kind t))!~tits cumpom'nt vutoeitics ~t nny point aru thc con'L'spondiï)~dit'fu)'c'nLi:d cocHicicntsuf n, ('(.')'t,:nufnncti"n, c~Hm]thc vctocity-potcntiid. Let t))c fh)i() bc sut In )n(jt[onby :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'nnincdhy thc normid vctocitics of thé cloucnts of winch, bL-in~s)t:n'L'dby t!ie Hnid in witti thcm, m'c duuotcd by if Mbe tho vc'ioeity-potcuti.t.],\vltich luto'prctcd phy.sica!)ydénotes tnc ijnputstvc pressure. Hunce by thc t]iC(H'cm,If bc t]io VL'Iucity-potuntliduf 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<[sinunur.scd in tho ftuid. Thé prusunt ]m;t)n)dunabius us tu attributc to it a much Itighur gcnuruHty. yur (.x:unp)u,t)ic untm'rscd soHd.s,mstuad of buing Hxc< m:i.yt)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

THOMSON'STIIEOREM.

78J

75

In words,if, by mcans of a suitabic impulse of the corresponding type, n givcn arbitrary vclocity of onc co-ordinatc bc imprcsscd on a system, the imputse corresponding to Msecond co-ordinatc nccessary in ordcr to prcvent it from changing, is t)ie samc M would bc rc'ptircd for the first co-ordiuatc, If titc given velocity \vo'c hnprMS.sud on ttic second. As :t simple uxampic, tedœ the eMC of two sphères and J~ nmncrHcdin a liqnid, wliuscccntrca arc f)'L'ctu !)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.surtsth:Lt the if) thc s:mic as if i]))pt))su rctmin.'d to prcvent thc motion uf thc functions of yt !md7? 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)toodof which thcrc maybe a.ny numbcr of othcr wirc circuits or sohd condnctors. If a unit cnrrent bc snddcidy duvulopeclin thc circuit J, tho clectromotive Impulse induced ill is the slulc as there wouldhave bccii iu ~1,hn.dtlic currcnt been furcibly dcvclopcd in 79. Thc motion of a system, on which given a.rbitraryvclocitios are nnprcs.scdby mcans of thc ncecssfu'y Itnp)dscs of t)ic corrcsponding types, posscsscs a rcmarkabtc prnpcrty discovcrcd by Tiiom.son. Thc conditions arc that arc givcn, vanish. Lct &c. currcspoiid to thc actu.d motion; and ~+A~, ~t.A~ ~+A~, ~+A~ to anothcr motion satistying thc saine velocity conditions. For cach snmx cithcr AT~u)' vailislics. New for t)iu kiuctic cnergy of 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 ItypoUtcsisis zéro; so that 2A2'=A~A~+A~A~,+.

(1),J

7G

VIBRATFNG SYSTEMS INGENERAL.[79.

shewing that the encrgy of the snpposcd motion excceds that off thé actual motion by thé energy of that motion winch would hâve

c

bc

..ith

},t.

rc

p,.d.~

ihc

fbnner. Thé motionc.nrlod actualtymduced in thc System bas tf~.s Jcss ~y other s~. yn.g tho same velocity conditions. In aoucrgytlm,i snbs~.cnt ch. ptcr we shall make use of this propcrty to find a supenor Jinut to 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 of a system,t)ic motion correspondingto prescribcd velocity conditions wu iu genem undorgo a change. Thc value of will nece.ss.riiy be less than before for t)~ere wouM be a decrease cven if tlie motion rc.nained unchangc
ThomBou auj 'fuit. § ~il.

~,7..V.y.

Mareh, 1875.

J

80.]

LAGRANGE'S

EQUATIONS.

77

so tha,t

if T ho cxprcsscd as n. qutic function of cocfHcicntsorc ni gcuerfil functions of

whose AIso

Since ~F8~ denotes Hie work donc on tho system during a disp~cemcnt ma.y bo recoarded as thc gcncralized component of force. In thé case of a, conscrvativc system it is convûnient to thosc parts which dépend only on thc connTurfiseparate from 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 aIrGadytaken of in thc tcrm a~

78


[81.

81. Tilcrc is also another group of forces whoso existence it is ofton a~/fmt~gcous to rccngnizc spccifdty, namoty thosc if y~ .pp~o ?;r..s)r~ 'n.i fn.-n~t yip.Hy. L),)H c~h piu'ticlc of thc syston is rctiu'dcd by forces proportion~ to its eomponbnt velocitics, t)ie cH'ectwill bc sitown in thc fu)jd:uncnt:U équation (1) § 80 by t!tc addition to tl)c Jcft-Jtiuu! mcnibur of thc whcrc A-y, nrc cocfHcicnts indûpOKicnt of t))o ve!(jcit.Ics, but pos.stbiydcudcitt 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 tJKttof

7~ it will bu obscn'ed, is hl.c a honngcncons quadratic fucetiou of t!tc vuioctLies, po.siUvu for :dl rL-:d v.ducs of tho v:u-)ab)cH.It !~pruscnt.shait thu r~tu ~t whidt cncrgy i.s(hs.sij~~cd. Thc abovc itivcsti~tiua i-ufcrtito t~tarding iornus propordonat to thc absolute vclucitics but it is equaUy important tu cuusidur sucb as dupend ou tho p-e~~c vulocitics of thc parts ci' titc system, and furtuuately tins eau bc done witiiout auy incrcaso 01 complication. For cxampic, if a furcc aet ou the partielc xi thc-rc will bo at thc samo momont an proportiona! to cqu~I and opposite force acting ou thé partide a- T!ie additioual terins in the faudamental eqoatioti wi)l hc of the furm

and so on for any numbcr of pairs of mutually ]nfhtctic;ng pfirticics. TIic only effect is thé addition of ncw tcrms to 7~ whicli still appears iu the form (2)'. We silall sercscntly t)iat Thodifforecoes rûferredto iu tho toxt mayof courseiutodjilcrcntia! eoefUcients in thocaseof a bodyoontiuuouBly deformed.

81.]

THE DISSIPATION

FUNCTION.

7!)

thc existence of tho fonction 7~ which may bc cailed thé Dissipation Funetion, implics certiLin rctations among thc coenicicnts ut' tho gcncralizcd cqn:t.tio!]sof vibration, which ctu'rywith Utem l, Iniportaut couscqucnecs'. But althougli In an important c]~ss of c~ses thc cffccts of thé question romains viscosityarc l'ept'cscutedby thc function opcn whctitcr snch a method of rrcsGntation is apptic:).b)cin aU cases. 1 think it pTobableth~t it is so; but it is cvidcnt that wc cannot cxpect to provc any gûncmt propùrty of viscous forces Î!) t'hc absence of n strict (L'nnition \v!ncbwill cnable us to duterminc wit)).certainty wha.tforcus are viscous !H)dwhat n.rcnot. In sono CMCScons!dc;['!Ltio)is of symmetry arc sun~'Icnt to shcw tbat thé retardmg forces ma.y bu rrcsoitcd as dunvcd from a disHipatiotifnnction. At any rate whuucvcr tbc rctarding forces arc proportional 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:).tcneigh'b(n)i'hoo(Lof a. conH~u'tt.tIon of t)iorou~I)lystable cquHibnum 7' and F' arc then homogcncous qmuh'atic functions of ti~c vclocitics witli coufHciuntswinch 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'esponj with the couhgura.t.ion of cquilibrium. Moreovcr all threo ~V fuuctious arc ossentiaUy positive. Since ternis of tho form f/ :n-cofthc second ordcr ofstnMilquantities, the equations of motion heconic h)iear, assumiug the form

whcrc under ~P arc to bc mciudcd ail forces n.cting on thc System not alœady 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 GonoralTItooroma retatmg to Vibratious, publishod m tlio 2~ocfe~HM< the ~~tCMNttca! for o/' Soete~ Juno, 1873.

80

VIBRATING SYSTEMS IN GENERAL.

[82.

Thc threo quadra.ticfunctions will be expressed as foUows

1 whcrc thc cocfHeicnts c are constants. From équation (1) wc may of course fait back on prcvious results 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-yto pursue the subject hcrc. If we substitute thc values of T, F and F; and write D for 1 dt we obtain a system of equations which may bc put into tlie forni

83. Before proccoding further, we may draw an important inference from thé of our equations. If correspondinc to tho two sots of forces respectivoly H\0 i l't~tt"' \P motions dcnoted by be possible, thon must also be possible thc motion in conjunet:on ~,+~ ~+~ with hc forces ~+~ a p.rticuL case, ~+~ Or, when there arc no impressed forces, thé superposition of any two natural vibrations constitutes also a natural vibration This is thc ccJcbrated principle of thc Coexistence of SmaU Motions, first clcar)y cnunciatcd by Daniel Bernoutli. It will be uuderstoo.! that its truth dépends in gênerai on tlie justice of thé a.s.sumption that 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:¡J

toI' lui'~1~

.n:d

we

4Yjjl P-01H11IPl!eÜ

~jt.l!

Il f.:y¡¡tt~.m On which

n0

are for which therefore thc coefRcienta &c.are frictioual forces-n!~ equ;)Jt.~ ~ct, v-'c wiHcotmttt'ttce~'itLany'ttf'mouwhic]). We havo M)~ functioDSof thé symbol

From those équations, of which thcrc arc as many (??t)as thé system possessesdegt'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-ordinateret
and is (if there bc uo friction) an even function of D of degrec 2M. roots of V=0 coueidered as au ±\t ±\ Let i\ équation in D. Then by the theory of dliferential equations thé most genera.1va,!ucof is whcrc the 2w quantities ~4,J/, J?,J~, &c. are a.rbitrn.ryconstants. This fonn hoids good for eMh of the co-ordinatcs,but tlie consta-nts in the différent expressions arc not indendcnt. In fMt if a particular solution bo &c., ~=~ ~=~' the ?'a~M ~t~ -~a. M'c complete]y determined by thé équations

is substituted for D. where in each of the coefficientssuch as Equations (5) arc necessarilycc~upa,tible,by the condition that is a. root of \7=0. Thé ratios ~1/ =-~3' correspouding to but for thé root arc tho samc as the ratios ~1~ ~1, thé othcr pairs of roots X~, &c. titcrc are distinct Systems of ratios. G R.

82


fgg.

85. Tho nature of thé system with which wo arc doalinrr imposes an importât restriction on the possible values of If wcro .6a}, elthciwoutd bc re~I and positive, and wo or sho-Jd obtain a particular solution for which tho co-ordinatos, aud with them thé kinetic energy denoted bv incrc~c without limit. Such a. motion is obviousiy Impossible fdr a conscrvative system, wbose whoJc energy can uever di~cr from the sum of tho poteutial and kinotic energies with which it was Mimatcd at starting. This conclusion is not cv~cd by takmg négative, beeausc we arc as much at liberty to trace thé motiou bMkwards as forwards. It is as certain that t!te motion ncvcr ~s as tliat infinite, 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 corto rcspondmg values of ~a~e Ana)yt:caHy,t)ie tact that thc roots of = 0, considered as an are at! real and équation iu negative, must bc a conséquenceof thc relations subsisting bctwecn thé coefficients virtuo of fact for all real values of the variables 2' and F arc positive. Thc ca~eof two 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 ~dng new arbitrary constants. TIius

where &c. are thé roots of thé equation of decrec 111n' found by for writing -M" 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 vibrations of a conscrvative system. We sec that thé whole motion may be resolved mto normal harmonie vibrations of (in général) difforent période each of which is entirely indepeDdcnt of tbc others. If tbe motion, depending on thc original disturbance, bc such 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) phases of vibration, aud the whole system is to be found m the configuration of equilibrium at thé same moment. We perçoive hère the mechanica.1foundation of tlie suprcmacy harmonie vibrations. If the motion be sufHcientIysmall, tho pf diffcrential équations becomc Iluear with constant coefficients ~hi]e circular (and exponentia)) functions arc thé ouly oncs which reta-intheir type on diffcrentiation. 87. Thé 7~ pcriods of vibration, determined by t!ic équation = 0, are quantities Intriusic to thé system, and must corne out t.he same whatever co-ordinatcs may be choscn to define the conn~uratton. But there is one system of co-ordinatcs, which is especially suitable, that namely in which the normal types of vibration arc defiued by thé vanisbing of aU tlie co-ordinates but onc. In the first type the original co-ordinatcs &c. Iiave given ratios let the quantity nxing thc absolute values be < so that in tliis type each co-ordinate is a known So multiple of < in thc second type each co-ordinate may be regarded as a known and so on. By a suitable determultiple of a second quantity mination of thé quantities &c.. ~y configaration of tite system may bu rresentcd as compoundcd ofthc ~t configurations of 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 of thc normal co-ordinates, ?' and V arc reduced to sums of squares; for it is easily sccn that if the products also appcarcd, the resulting équations of vibration would not be satisned by putting any ~-1 of the co-ordiuates cqual to zero, whilc thc rcmaining one was finite. We might ZD hâve commenced with this transformation, assumin~ 1 0 from AJgebra that any two homogcncous quadratic functions can bo reduced by linear transformations to sums of squares. Ttms

whcrc thc cocnicicuts (in which thé double sufHxe.sarc no tono-cr are required) ncccssarilypositive, G–2

84

VIBRATING

SYSTEMS IN GENERAL.

[87.

88. The interprétation of thc équations of motion leads to a tlleorem of considcrabio importance, which may bc thus statcd'.t. Thé period of cousorvittivusystem vibrating i)i a,const)-!nned type about a position of st<).h)ccquiHbrium is stationary 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 the normal co-ordinatcs. Thc constnunt, w]nc)i may bo snpposcd to bc of such a cha.racter as to ic:ws only onc dcgrc'e of fj-cedom,is represcuted by taking thé quantittes in givcn rutios. If

wc

put

This gives thc period of thé vibration of tlie constrained type and it is évident tliat thc period is stationary, when a!l but one of thé cocfncients ~l,, ~1, vanish, that is to say, -when thé type coincideswith one of those natural to the system, and no constraint is necdcd. By means of this tlicorem wc may provc that an iucrease in the mass of nny part of a vibrating system is attendcd by a prolongation of all tho natural periods, or at auy rate that no pcriod can be diminished. Suppose tlie incrernent of mass to bc infinitesimal. 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;~~
88.]

PERIODS

OF FREE VIBRATIONS.

85

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 constraint, 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 sts. 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, whilethé kinctic energy of a given motion is unaltered, the periods of thé free vibrations arc aU increased, and convcrscly. This propositionmay 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 potential 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 By me~s

SYSTEMS IN GENERAL.

of thé principe wc n..y ea.i!y ~h.

[88.

that

tho value of tho frec periods stationary calculato eorrcctions duo to any J~ a
80. Anothcr po.nt of unpor~ncc reJ.ting to thc period of ,n .n ~rbitrary y.steiu type rcn~ins io be noticed vihr~ng t .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 and Ic~t of thosc n.tur.I tu t!~ system. In thc c~ ~.tcst o c ntuu~.s deior.n.t~n, thcrc is no I~t uatu~ pericd; ~i any hyh c.d"Ytype c.uinoL cxcccd that puthet bclo~l,~ to thé Gr~est typ. Whe. tLer.f..c ti. cLject°i.J~I~ of calculons ~cdr' result will como outt tao small, ~1 type jadgn~nt must bc usc~h~ t)~ uscd ohjcct ben,g to approach thé truth as nearly as can he donc w~thout too grc.t sacrinco of .hnpHcity. ypcor ~g hc.vily ~i,ht.d ~ht Le tdœu froin thc extrême case of an innnite Joad ~hen tho two of thé st.~ .ould Le As Je.~pl~ str~ht. p~ cale..tion of tins Jun~ of which the rcsult is known, wo ~~h:d will t~Tj~ w~th tcusion 7 anj mquirc what the period would be on certam supposions as to thé type of vibration. Taking the origin of .r at t)io ~idd!o of tho string, lot the curvc of vibration on thc positive sidc bo

~ul on thc ncg~vc side the Im~c of tins in the axis of y bc~g not !c.ss than .nity. This form satires thé condid~ ihat y vanishes whcn ~.=1 Wc h~vo now to form the exprcs~.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-T Tho truc value of p" for the gravest type is so that –,r, plu tho assumption of a para-boUcform gives a pcriod which is too small in thc ratio 7r ~/10 or '993G 1. Tlie minimum of p", VG+1 as givcn by (2), occurs when and gives ~=–=l'72-t74,

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 modesof vibration of most continuons bodies are not as yet rccognised in analysis. It is therefore often ucccssa.ryto fait back on methods of approximation, referring t!io proposed system to somo

88


[90.

other of a character more amende to analysis, and calculatiug corrections depending on the supposition that the differerce between the tv.'o sy.ste~.s .aU. Th. r~ ~proxi.c!v simple systems is thus one of great importance, more especially as it is impossible in practice actually to realise tlio simple forn,s about winch wc eau most casily reason. Let us suppose then that tho vibrations of a simple System arc thoroughly known, and that it is required to investigate tho.sc of a systcm derived from it by introducing small variations in thc mechanical factions. If &c. bc the normal co-ordinates of tho original system,

and for thé varicd system, rcferrcd to the same co-ordinatcs whicharc how only approximtttciy normal,

in which

&, small In eert.m cases new co-ordinates may appe~ but quan ~cs. coe~cnts must bc so t!.cir small. From (1) ~c obtam for the if ijagrangian equa,tiousof motion,

.In the original systcm the fondamental types of vibration are thosc .h.ch corrc.spondto the variation of but a single coerd~na e.1 a timc. Let us fix our attention on one of them, involving say variation of while a!I thc remnining coordinates vanish. Thc change in tlie system ,vi!l in ~1 cntail an altcration in tlie iund~c.tatcr normal types; but under tlie cu.cumstanccs contemplatcd tlie alteratio~ small. ne normal type is e.pre~cd by the synchronous variation of h' other in to but ratio of any small. known, ~e normal mode 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"
Thé otlier tcrms a.rc to be neglected in a first approximation, sincc both the co-ordma.te(rcla.tivcty to ~) and its coefficientarc 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 maywrite it,

Thé first term gives tlie value of p/ calculated without allowance 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 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 rel="nofollow"> p~ that is, if the mode depends upon that of –p~.2* If s be more acute than the moder, the correction is négative, and makes tlie calculated note graver than beforc; but if the mode s be thc graver, thé correction ra-isesthe note. If t' refcr

90


[90.

to the gravest mode of thé system, tho whole correction is negative; and if r refer to thé acutest mode, tho whole coiïection is positive, as we have aJrea.dyseen by another method.

g

91. As an example of the use of these formulae, we may take thé case of a stretched string, wliose longitudinal is not quite constant. If x ho measurcd from oue density p end, and hc tho transversc displaccmcnt, t!ie configuration at any time t will he exprèssed by

being the longth of thc string. arc tlie normal co-ordiuatcs for p==constant, and t)iough hcre p is not strictly constant, tlie configuration of tbc systcni may still bo expressed by means of the same quantités. Since the potential cnergy of any configuration is tlie samc aa if/)= constant, 8~=0. For tlie kinetic cncrgy we liave

If p wero constant, thc products of tho velocities would dis&c. arc, on that supposition, the normal appear, since co-ordm~tcs. As it is, tlie mtcgml cocaicicnts, thoug!i uot actually evancscont, arc small quantities, Lot p=p.+~; thcn in our previous notation

Thus thc type of vibration M expressed by

or, since

i

t

01.]

EXAMPLES.

91

Let us apply this result to calculato tho disp~cemcnt of thé nodu.1point 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.tiojiof 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


[91.

as may also be rcadily proved by equ:ttin~ thé periods of vibration of the two parts of thé string, tha.t of the loadcd part buing ca.Icutatcd:),pp!-oxim:ttc]y on the assumption of' unchanged type. As ~u cx:unp)Gof 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 ?' is evcu, =7~ and wheu r is odd,

= 0, wc g'ct whcu

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 expansions of 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~yof the expansion if this be assumcd, thé détermination of thc cooHjcieutsis 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.1considerations. To fix our ideas, let us consider the small vibrations of a tmif'u)')astring strutc)~ed bctwceu rixc<tpoints. We know from the gcncnd thcory that thé wludc motion, wha-tever it may hc, c:tn bc aua.)ysG(tiuto a. scries of componcnt motions, each rresuntcd by a, harmonie function of tho time, and capable of cxisting by itscif. If we can discover thcsc normal types, wc sh:dl bc in a position to rrcscnt 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 rrcseuta.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 expansion 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 suit arbitra initial conditions may always hc rcadily enfected bv the fundamental property ofthc normal functions, and Itmay be convenicnt to sketch the process Iicre for systems like strings, bars, membranes, plates, &c. in which thcre is only one dépendent variable ~tobe considcred. If tlie normal functions, and ~be ~t, ~j, thé correspondtng 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 &c.cannot contain of~, tiie products of tlie normal gcnGraI.zed velocities, and therefore cvery iutcgra.1of tlie form

Hcnce to determine 7?, wc have only to multiply thc first of équations (4) by pu, and intcgratc over tlie system. Wo thus obtain

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 property 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 velociticsgives 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 of gencralizud eo-or
refera to tho motion corresponding to Suppose now that 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 rresentcd 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.iunwhich 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 corresponding 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;/ ofDt~'racft'on, Can~rtf~erraM.Vol.IX.

CHAPTER

V.

VIBRA.TINGSYSTEMSIN GENERAL CONTINUED. <)C. WlfENdissipative forcesact 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 transformation rid our.setvcsof the products of thé co-ordinatcs, and obtain t)te n<jrm:dtypes of motion. In the preceding chapter we h:).vcconHidcrcdthé ca.soof ~= 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, scarcolybelong to our présent subject. Thc'prcscjice uf friction will not interfuECwith the réduction of T and to sums of squares'; but thé transformation proper for The them will not in general suit also the requirements of 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. uc. (2). ~+~+cA=~i. 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.1simplicity. Under this bead U~emost 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 retarding force, proportional both to the mass and velocity of thé 7 R.

D8

VIURA.TINC6YSTHMSIN GENERAL.

[97.

part. Thc same cxctioual 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 possGsscscertiua pccuHarities which m{tke it wortLy of sarato cousidGra.tiou. Thc équations of motion aro obta-incd at once froin F ~nd

in which thc co'ordinatcs arc sa.rated. For the froc vibrations we Itavc oniy to put < = 0, &c., and tlie solution is of the form

and

are thc initia! values of<~n.nd< and 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 insignificant, and tlie system returns sensibly to rest. Simuttn.ncous1ywith thc frce vibrations, but in pcrfcct inden<)enccof thon, thcre may exist forccdvibrations dending on tho quantitics tl\ Precisuly as lu. tlie case of ouc dc-groc of frecdom, 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 given in (2) which rrcscnt thé rcsidue

97.]

GENERALIZATION 0F YOUNG'STHEOREM.

at time t of tho initia,! values and th
99

If there be no friction,

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~cntlie forces apphcd to t!)G system act at a nodo of tlie uormi],!component tliat is, at a point which thc componcnt vibration 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 disturbed position. So soon as tho force ceascs, natural vibrations set in, and in tlie absence of friction would continue for an indennite time. We infer that whatevcr in other respects their charactcr may be, thcy contain no component of thc Tliis type conclusionis limited to cases w!tcre T, F, F'it of simultaneous réduction, ineludmg of course tlie case of no friction. 99. The formutniquoted 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)u3applicable to such forces. In using normal co-ordinates, we iiave first to calculate tlie forces cl\, (1~,&c. corrcspondingto eacli period, aud thence deduce thc values of the co-orclinates titcmselves. If among tl)e natural periods (calculated without allowance for friction) there be any nearly agreeing in magnitude with the pcriod of an imprcsscd force, tlie corresponding 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 a single 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 Imprcssedforce but any normal component, v.hich b:t~u.nojj aL the p!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 forceis 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')nomcforce, thc syst.on will vibra-tc in gcnera.) in «~ its modes, :dthong)i in pfn-tK'uhu' cases it ma.ysomctimcs 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!yto thé free pcriods of thc systcm, :n] cqui!ibriumthcory is 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'.sDicory of thc Tides is of this class, :Lnd proceeds on thc assumption that thc frcc pcriods of' thc masses of watcr found on tbe globe are s!n:d) rdativdy <.othc pcriods of thc operative forces, in whicli case thc incrtia, of thc water might bc As a matter of fact this supuosition is on]y Icftoutof. 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 sufnciently 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, tbe cquilibrium theory of the tides migbt !)ave bccn adc(ptatc. A con'cctcd cquDibrium t!)cory is sometimes uscfuL w])enthc pcriod of tbe imprcsscd force is sumcicntiy long in compar!son witb most of the natund poriods of a System, but not so in thc case of onc or two of thom. It will bc sufDcient to ta){cthc 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.]

EQU1LIHHIUMTHHORV.

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)iedv~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 comparisofiwith 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.sisth:).t is !t)t.ogct!tcrnc~tigibic, with con'cet.ious fur thèse particular co-ordinates. 101. Before ing on to t))c ~encml theory of thc vibrations of .Systemssnbjcct to dissipation, it may bc well to point out Home pcculiaritics uf thc free vibrations of onntinuons Systems, startcd by a force applicJ at a single point. On thc suppositions aud notations of § ~8, tbe con6guration at any time is detcrjnincd bv

Suppose now that tho System is held n.trest by a, force applied at thc poijtt (?. T))Cvalue of is detjcrmincd by thé considemtion tha,t < reprcsGnbsthc work donc upon thé System hy tlie itnprcssed forces d'n'ing a hypothetical disptaecmcnt S~=S6 that is

102

VIBRATINQSYSTEMSIN GENERAL.

[101.

If thé system hc let go from this Cuiif!?ura.tionat<=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, is more slowly tha.u in thc prcvious case.

that

101.]

SPECIAL

INITIAL

CONDITIONS.

103

In both M3. it mnybr* nbsnrvcd thah tho value of is symmctrical with respect to 2-'aud proving tliat tho disptaccïncnt n.ttime t for thé point 7-*when the force or impulse is appitpd at < is the Sinneas it would bo at () if tlie forceor impulse h:td bueu {),pplicdat -P. This is an example of a vcry general reciprocal theorcm, which we shall consiclerat !eugt)i pt'csc!itly. As a thit'jd case wc may supposa thé body to start from rcst as 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 conccntratedi)i a siugtc point. In exactly tlie sa.mc w~y wc may trcat thé case of a continuous 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.Ifunctions 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

&c. arc to be trcatcd as small. in which the coefficients & ]f tlicrc were no friction, ttic abovc systcm of c(}uationswuuld

JU4


F 103.

bc satisfied by supposing onu co-ordinate to vf~' suitahly, while t,!ic other co-ordinatc-svanish. In the actual case thero will be a corrusponding sohttion in which the value of any ot)icr co-ordi!mtc will busmall rclativcty to 6, Hence, if wc omit tc'rm.sof the second ordc-r,thc ?' equation bceomes, from which wc infcr tliat varies n-pproxim~tuly :LSif 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 part is small in eomparison with tlie imaginary part. From thc équation, if wc introduce tlie supposition that a-Mtho co-ordinates vary as e" we gut

This cquatinn dctcrmines approxiniatdy thc altcrcd type of vibration. Sincc thc chief part la ima~hary, wo sco of tliat thc co-ordinn.tcs arc a.pproxi!natc!y in the sa.me phMc, ~<~ < ~j~Me f~y'e~ (î ~MM?'/er per~o~ /?'o!~
~e!u:uimg' part ofthc tenus mdudcdundcrS thc con-uctiun lias no uifuct un Litc ruai p.n-t uf titc r:T.teuf dceay (lej)cnds.

bcingrcit), oa w}dch

103.]

SMALL DISSIPA TIVE FORCES.

105

103.

Wc now returu to ihb cunsidùratioti uf thc g.'n
For thc free vibt'atiuus ~F,, &c. va.msh. If \7 bc thc dctcnninantt

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!)G2?~ roots of tho = 0 no jouter uccur in equal positive :uht cquahon ncgativc piurs, Lut cot)ti).in!). ruai as wu!! as an imagmary p~rt. TIte compJutu intégra! may ]n)w<;vcrstiïl bc writtun = ~c~ + J'g~~ + Z?e~+ 7/e' +. (5), where thc pairs uf cunjug-u.tcroots are uc. CorruMpoiding to cach roût, thcro is a. particular solution such as ~=~~ ~=~ &c., ~,=~ in which thc ?Yt~'osj'l, arc determined by thc equations 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 oi thcco-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 vibrations of a System 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 be occnpicd. We résume ti)e équations (1) with thc view of invcstigating further tbc nature of forced ~m~to;

10G

VIBRATING

SYSTEMS

IN GENERAL.

[104.

104. In or~r to cnmu.atc from thc crjnnH~ns ~!Lt'~ .)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 recognition 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 exactly the 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 convenicnt 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 and Imaginary parts of thc data. In thia way tlie

104.]

FORCED VIBRATIONS.

107

gam.sconsider~biy iu brevity, Inn.smuch~s ditfcrpntmtton!! ~n!)Jy-t['! anfl altérations of phnsc a.rc expresscd by mcrcly modifying thé compicx coûfHcicntwithout chang-ingthc 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 thelaw

&c. according to

&c.are certain complex constants. Aud thc symwhere bolical solutious arc

wherc (~) dénotes the rcsult of substituting for D in Considcr f!t'st the case of a System exempt from friction. ~7 and 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

&c. (in thc case of more If we suppose tliat the forces 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

J08

VIHRATINC:

SYSTEMS IN GENERAL.

[104.

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.1a)id thc otho' puruly Itn~ginary,ofw!iic)i tlie latter dépends entirely on the friction. Thus, if wc put V (~)=~ (~) +~ V.(~). (7),1 and therefore real. If as bcforc \7.~are eveu functions of J,=Vt\e~ our solution takes thc form

Wc ha.vesaid that ~(~) dépends entircly on thc friction but it is not truc, on the othur hand, th:).t 7,(~) is cxa.ctiythc s~me, as ift)tcre had been no friction. Howcver, this is approximatciy the ca~e,if the friction ho sma!]; bccausc any pn-rtcf ~(~), which dépends on thc first power of thé coe~cients of friction, is nocessari)y imaginary. W!)cncvcr there is n, coincidence between tho period of thc force and tliat of onc of thé frcc vibratious, \7~;)) thcrcforc va.Dishcs,f~ndwe ha.ve tan y = ce :).u<~

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 displacements and velocitics may bc neg]cctc<1. This important principle wa~ cnuuciatcd by Laphicc and a.pplicd by him to the theory of

INEXORABLE MOTIONR.

104.] J

lOf)

the tidus. Its.grcftt gcncratlty was atso rccogt)!scdhy Sir John Hcrsche!, to witomwo owc a formai domonstr~tionof 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 nnchanged, 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 inexorable wo~t'o~sare prescribed. arc givcn If we suppose t)ta,t the co-ordinates ftinctions of the thno, while thé forces of thc rcmaiuing types vanish, thc équations of motion divide theinselvcs 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'ectas know)i furccs a.cti)ig on the system. Thc Ct~uationsof this gronp are thcrcfore su~icicnt to (tt.'t.crmincthc uuknowu quautities; after whic)i, If rct~uircd, thc forces ucccss:u'yto maLinta,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 slinpic harmonie vibrations in conséquence of prescribed variations of sorneof thc coordinat.es,posscsscs a pcculiarity paraUel to thosc considcred in §§ 74, 7~. Let =

= J~ cos in

whicit

thc 7~)cye.

quantitics ~<~ro~.

~l,. art. 823.

arc

-j

eos ~<,

regarded

&c. as

givcn,

AJHO 0)
whi)c

§ fino.

thc

110

VIDRATING

SYSTEMS

1~ GENERAL.

from tlie expressions for

l' and V, § 82, ~~§T 2(y+

[lOG.

=~(~+~,)~/+.+(~+~)j~+., +~(~u)~+.+(~J~~+.jcos2~

from whic)i we see that tlie équations of motion express tlie condition th.t A', tlie variable part of y+ r, ~i,ich is proportional to Hc,J~+. +(~)~~+,~ shall bc station~-y in v~uc, for variations of tlie claautitics ~r., ~I, Lcb bc ttie value of~ n~turat to thc System wlieu under tlie restraitit vjbratiug dcHned Ly tlic ratios

Krom this we sec that if be ecrtainfy less than tbat is, if the prc.scr.bed pcriod be grcatcr than any of tllose natural to the system uuder the partial constraint rrescuted by ~t.J, then is necessarlly positive, aud tl~e .tationary val~e-t!~re can bc but ouc-~ an absolute minin~un. For a similar rea.so~ if the près nbed ponod be less tiuL.iany of tliose natural te thc pa,-tia)iy constraincd System is an absolute ~xhu~ but a~braica!Iy, an arit)nuctieat!y absoluto rnini.num. But whcu lies witbin thc range of possible vaines of~, n.ay bc positive or négative, and thé actual value is not thé greatest or least natura! vibration is cor~sistentwith the possible, Wi~enevera hnposed conditions that will ue thc vibration assumed. Tj.e y.,l.Uc part of ?'+ zero. For convenience of treatment .ve hâve considered apart t),e two gréât cl~es <.fforced vibrations and f~ vibrations; but hc c of is, course, noth.ng to prcvcnt their coexistence. After tl.e lapseo of a .sumc~nt interval of time, the frce vibrations ahvays appcar, howcver small thc friction be. The case of abso.nay 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.]

RECIPROCAL THEOREM.

111

System. Suppose, as before, that the co-ordinates arc wh' '~L-n<x' or .non-cxistcnco givou. Thf'u ~t); .frc't)\'ibt'it.{.iui')~ is a matter of indiffurencoso 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~
We nowconsidcr two casesof motion for the same system first whcn~ vanishes, Mtd secondiy (with da.shcd Ictters) whcn va.nisitc.'i. Ii"~K=0,

In thèse équations ~7and its dtiïcrentia! coefficientsarc 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 coordinatc is rclatcd to it in the same way as the co-ordinate is retatcd to tlic force whcn this latter foi-ceis supposed to act a.Lone. lu addition to thc motion hère contcmplated, thcre may be frcc vibrations dépendent on a. disturbance ah-eadycxistin~ 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 of iorc~s \t'hicii acLed previousiy. However sinail thé dissipation tliere must be an interval of time after which free vibramay be, tions die out, and beyond winch it is unnecessary to go in taking of the forces wbich hâve acted on a system. If therefore we include undcr forces of sumcicnt reinotcness, there are no independcnt vibrations to be considered, and in this way tlie theorem may be cxtendcd to cases which wouldnot at first sigbt appeM- to corne within its scope. Suppose, for example, that the systcm is at rcst in its position of equilibrium, and then begins to bc acted on by a force of the first type, graduatty Increasing in magnitude from zero to a finite value at which point it ceases to incrca.sc. If now at a given epoc)i of timo the force be suddcn!y dcstroyed and reinain xcroc-vurafto-wards, frcû vibrations of tI)G systcm wIUset in, and continue until destroyed by friction. At any tirne t snbsc()ucnt to thé given och, tlie co-ordinate has a vatnc dépendent upun t proportional to T))c tiicorem allows us to assert that this value bears the same rcJation to at t))c same mnmcnt hâve borne as~outd to~ if thc original cause of the vibrations bad been a force of thé second type increasing g.-aduai)yfrom xcro to and thcn suddenly vanishing at thé given och of timc. Wc ))avc ah-cady had an example of tins 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)icationsof our theorem we obtain results rclating to h'ee vibrations, considercd as t]jc residual effect of forces whosc actual opération may ]tavc becn long bcforc. ~08. In an important e)ass of cases thé forces and are harmonie, a)td of thé samc pcriod. We may rrnscnt them by and J,e~ wherc J, J~ may be assumcd to bc ?- If tbc forces be in thc same t-hase at the moments compared. Tito results may then be written

108.]

RECIPROCALTHEOREM-.

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. ing to rcal quantifies wu tU~ystate the thcorem thus = A~cos pt, c(c
New, opcrator multiply ?'/)for j9 regard to

if aiï the forces vary as e' the cfFect of a symboUc such an e~ ou any of thc quantttics is mcrcly to that quanti ty Ly thc constant found Ly substituting iu Supposing this substitution juade, and havicg tlie rclationa e~ = 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

114

VIBRATING

SYSTEMS IN GENERAL.

[109.

Thé con'icquences of this equation may be cxhibited in thrce dinerent ways. In tlie first we suppose that whcnco

~=0,

~/=0,

~=~' ~(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 of relationship' extending 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 impression of vagnencss. If .P and Q bc two points of a horizontal bar ed in any manner (c.g. with one end clamped and tlie other frce), a givcn harmonie transverse force applied at P will give at any moment 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, thé rcsult may be stated thu.s –Suppose for thc first case that a harmonie couple acts at .P, and for thé second that a vertical force 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 acting over 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 with that 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 second example, 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 produco tho same motion in j?, as if the parts werc intcrdtangcd and thi.s, wliatevcr mcmbra.ncs, strings, forks on résonance cases, or other bodie.scapable of bcing set into vibration, may be present in their neighbourhood. dénote two points of a solid etastic Or, if A and body of any shape, a force paraUcl to acting at A, will producc the same jnotion of the point parallcl to Oras an cquaL force would producc in the point ~1, pn.ra!Iel to Oy acting at pM'a!!clto ~J~. Or aga,in, lot A a)nl Le two points of a space occupied by air, between which arc situatcd obstacles of any kind. Thcu a sound originating- at Ais perccived at B with the samc intensity as that witit which an cqual sound originating at jS would be perccived at ~i/ Thc obstacle, for instance, might consist of a rigid wall picrcecl witli one or' more holcs. This example corresponds to the optical law that if by any combination of renectin~ or re. fracting surfaces one point can be seen from a second, the second can also bc seen from thc first. Fn Acoustics the sound shadows arc usually only partial in conséquence 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 there be two circuits of insulated wire /1 and B, and in their neighbourifood any combination of wu'c-eircuits or solid conductors in communication with condensers. A periodie electro-motive force in thé circuit A will give rise to thé same currcnt in as would be excited in il if the cicctro-motivc force opcratet) inR Our last example will bc takcn from thé theory of conduction and radiation of heat, Ncwtou's la,w of cooling being assuvned as a basis. Thé température at any point ~t of a conducting and radiating system due to a steady (or harmonie) source of hcat at is thé same as thé température a.t due to an equal source at Moreover, if at any time tlie source at B be removed thé whole subsequent course of température at A will be the same as it would be at B if thé parts of.D and A were interchanged. 1 Helmhoitz, Bd.Lvn. ThoBonnesmustbe Rnchas iu the absenceof C'r<~< obstacles woulddiffusethernselyos directions. eq~Iy in ~11 8–2

n~

VIBRATINH

SYSTEMS IN CHNKRAL.

[110.

HO. The second way of stating the reciproc~ theorcm is fu'rivedat by t~king in (1) of § 1()9,

to ni thé nrst case, wlien = 0, shewing that thc relation of in tlic second case, to is the samc as the relation of wheit ~=0. Tlius in tite cxampio of the rod, if thc point P be held at rcst wliilc a givcu vibration is imposed upon (by a force thcrc applicd), thé reaction at jP is thc same hotli in amplitude and if that point were beld at rest and phase as it would bc at thc givcn vibration were imposed upon 7~. So if J- a.nd bc two electric circuits in thé ncighbourhood whether closed or terminating of any uumber of othcrs, C, D, 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 putting in (1) of 5 109.

nets to in tho first case, \vhpn proving that thé ratio of in tlie second to abne, is tlie négative of tlie ratio of cqual to zero. case, whentho forces arc so rclatcd as to ke 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 numerical 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 of dissipation not to be brought under this hcad, whoseeffects it is eqnally important to include for exemple, thé dissipation due to the con(htction or radiation of hcat. Now f).tt))oughit bc truc that the forces in thèse cases arc not for ~M ~)~6'~e ~ùhuns in a constant ratio to the velocitics or displacements, yct in any actual case of pcriodic motion (T) tliey arc ncccssarity periodic, and tttcrcforc, wliatevur tlicir phase, expressible by a sum of two tonns, one proportional to thé displacement (absolute or relative) and tho other proportional to the vulocity of thé part of the system aneetcd. If thé coemcicnts bo thc same, not ncccs.sarity for ail motions whatever, a~ &br motions u/e~ T, the fmiction ~exists in thc only sonse requirud for our présent purposc. In fact since it is exclusively with 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-hapssufliciently gênera) to covo.-tho wtiole ricld of dissipative

forces.

It is important to Hiat the Prnicipio of Reciprocity is ilmited to systems which vibratc about a configuration of e~t~M~)t, and is therefore not to bo apptied wititont reservation to such a problem as tliat presented by thc transmission of sonornus wavcs through tbe atmosphère wi)C)i clisturbed by wind. Thc vibi-iLtionsmust also bc of such a charaeter that tlie square of the motion can bo ncglectcd througitout; ot))crwise our démonstration wou!d not hotd good. Other apparent exceptions dépend on a misunderstanding of thc principle itsclf, Carc mnst be takcn to observe a propcr corrcspoudcnce between the forces and displacements, the ruie being that thé action of thé force over tho disphccmcnt is to represent wo~ ~ne. T)ms co!~)~ correspond to )'oMw:s,~re~M?'Mto 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 présent Chapter will he devoted to sonic detaits of the case whcre the degi'ccsof 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 essuutiallypositive, thc foHowing mequa)itics juust be satis~icd

Zy>

~1C>~ .((!).

Moreovcr, L, N, ~t, C must thcmsdvcs be positive. We procecd tu examine thé ciTuctof tliese restrictions onthe 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.Insto provc that tlie roots are m ihct rcal. Dm eoito bc satisHcd is that thc i'olluwing quantity be not néga(.Uti<j)t 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.1constitution of thc systum, whosc vibrations thoy serve tu express.

113.]

ROOTS 0F

Thc two values of

DETERMINANTAL

EQUATION.

119

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 preveuted from varying. T]to systcm thon loses one dugrcc of frcc
This propositionmn.ybc gcnerulixcd. ~h?y ~~ndof constr~mtt wLidi Icuvcs thc System still in posscssiol of onc (tcgrcc of frcedom may be rcganicd ~s thc impositiou of n, fot'ccd relation bctwccu t)ic co-ordiuates,such as

120


[112.

a.nd any othcr homogeneous Hncar funcNow if Cta?+ tiult r.f .'s rmd .y,~o tfd!cn ~s n~w vnrinU~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:Ltcin va.!nc bctweoi thosc two in which tlic natural vibrations wcrc prcviousty pcribrmcd. Convcrs~Iy, t!)C two periods which bccomc possibJc whcn constraint is rcmoved, lie ouc on cn.c!tsi()cof tho original period. If tlie values of À."be cquat, winch can on)y iMLppen when Z

~=.1

thc introduction uf :Lconstmint 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 wc may take a. strctehcd string of ]en~t)i itsdf withH'CL'doni) 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.Hya.ttactio(~ thc cLa.mctcr of thc two componcnt vibrations is évident. In the first, which wil! Itave t)Lelonger period, titc two weights move togcthcr, se that a' and y rcma.inequ: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 to bc found on a straight Une ing 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.sthé complete solution,

where, as usual, thc constants the initial circumstances.

a, 7?,j8 arc to be dctcrmincd by

114. Whoi thc two Jiatural periods of a systcm are nearly sometimes prcequal, the phe~omcnon of intermittent vibration scnts itself in a very curions manuer. In order to ittustratc tins, wo ma.yrecur to ttic string loadcd, we will now suppose, with two of thé equal masses at distances from its ends cqun.1to one-fourth length. If thc middte point of tlie string were absolutcly iixcd, thé two sinuhu' aystons on eitlicr side of it would hc compictcly two independent, or, if thc whole be considered as one system, the periods of vibration would bc cqnal. Wc now suppose that Instead of bcing absolutely nxed, tlie mid(Uc point Is a.ttachcd to it is sprints, or other machincry, dcstitute of mcrtia, so that to incrtia is to capable of yichling s~/t~y. The reservation us avoid the introduction of a third dcgrce offrocdom. From thé symmctry it is évident that thc fundamcntal vibrations of tlic system arc thosc rrcscnted by a;+y and a?-y. Thcir periods arc shghUy différent, bccause, on 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 conditions are

122

VIBRATING

SYSTEMS IN GENERAL.

[114.

wiuch give approximatcly Thus

Thc va)uo of thc co-ordinatc .T?is berc approxitoatL'iy cxprcsscd by I~u-rnouictcrm, whosc an.phtudc, being proportiona! to t, is a slowly varying harmonie function of t!ic thne. sinTIic vibrations of thc co-ordinates are tbcrcforc and so adjnstcd t]~t each iuuplitudu v~)ns)icsat tbe Intermittent moment tl~at the othcr is at its !uaximum. T)us phenomenon may bc pret.Uiyshcwn by a tunin~ fork of vcry low pitcb, hcavUy wui~htud )Lttbc ends, i~d fh-m!y'hdd by t!ic staik iuto a screwing massive . W!tcn tLc fork vibrato in thc normal ïnanncr, thé rigidity, or want 01 rigidity, ot' thé stalk dœs not comc into p!ay; but if' tbc di.spj~ccmcntsof'the two Le m t)ic prongs samc direction, t~c .s!ig)it yidding of Hie sta!k cntails a small change of pcriod. If t)ic furk be excitcd by strikh~ oneprong,t)ic vibrattons are intermittent, and appcar to transfeîtl~emscivesback~-ard.sand forward.sbctwecu thé prongs. U.dc.sa, howcvcr, t)ie suj~port bu vcry firm, Utc abnorm:d vibration, which involvcs a .notion of thé centre of Inertia, is soon dissipated and of thon, course, tbe vibration appcars to bccomc stcady If thc iork be mcrc)y hctd in thc hand, t!ic p!ienomeuon of' mtennittenec cannot bc obtaiucd at a! 115. TItc strctclicd string with two attaclt(jd nasses May bc uscd to ~lustrale somc gênerai principies. For example, ths period of t!tc vibmt.ou which remalus pos~ibte wbcu onc mass i.s !te)d at rcst, is Intermedi~te between thé two frec pericds. Any incrcase ni eithcr Joad depresses t!ic pitcb of both thé natural vibrations, and co~vcrsciy. If t)ie new load be situated at a point ci thé string not, cuinciding witb tlic places whci-c t)ie other !oa()s are attiiched, nor with tho uodc of one of' thc two prcviousiy possible frcc vibrations (thc othcr lias no nodc), thé efî'cct is still to prolong both thc periods alrcady prc.scnt. With regard to the third nnite period, w]iicli becomes possible for thc first time after the addition of the new load, it must be rcgardcd as denvcd from

115.]

IMPRESSED FORCES.

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 innnitcsimal 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 smiUlermass, taking place mucb as if the larber wercfixed. must be aiways thé Now sincc ï~ and T~can nover bc equaL gruatcr a~d we infer, that as tlie load becomes cot~tinuallylarger, it is ï~ tliat met-casesiudennitcly, and T~ that approachcs a iiuite limit. Wc uov to tlie consideratio!i of forccd vibrations. 116. Thé général équations for a system of two degrecs of frecdom including friction 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-diiiatey is thé same as if x had been prevented 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 be smaU, then tlie zero, eitlier ~+~~orC'+~ 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 complicatcdcharactcr.

1~

VIBUATI~O

SYSTEMS IV GENERAL,

Hl~

~hcw.sthat If we had ~sumed ~'mmetry A~=0 y=~" w. ~o.Id have fou..d the .ne va).c fur ..s no~v Th obtah.fur. is a R~.p, uc lurcncd to as .u ° exiunpjo. -suppose th.tTi~ ~< ~==~) ~t is p.-c.scnbcd, v- 1 r 1=0, < and for g~ter we .shalt coufinc ourscIvcJ .ImpHcity vhi~ to t!.c case ~hcrc /3 = U. TI~ vaiuc of~ is

.n. ''?'~ spcetivejy ~j

"7

P~ ~~y

of t)'6 co.mcl.nt of

Le ~e icrc

and

It appears tlmt tlle effect ffc'ctof of tho reaction and above of (–– (over -1 becauscd ~hat: = is ~J~~scnted by into .4 chang.-ng ci' the ~~1" to ~n in tho tl. coefficients ~1 of spring and friction, 'l'liese a1tcmtiolls, howcver, =~ of tlce peniod of tlee ~~aotioncoutenrylccted,whose cllaracter ive now pl'occed to cousicler. Por~ the value corrcspondingto ~e n.tnnd frictiun!ess of (., be~g n~u~aincd at zéro); so that ~V=o' 'l'jJ(3ll

JJ1 most cases with IVhich we are lrvctically C0J1ccrno(1 'Y is of y nut l11ucI.ditJer.. Wo sh.JI ?:?=~ acconhng!y Jeavc out of ihc

0FADEPENDENT SYSTEM.125 117.] REACTION and in thc small tcrm variations of thc positive factor (ZF- J! Witcn p uot 'y~, substitutc for~) its approximatc vainc ?!. 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

y»

which varies ~e~e~/ with

'y. Tins might not hâve bcen cxpected on a supernclal view of the mattcr, for it sccms rather a paradox that, thé grcatcr thé friction, hc its resn!t. But it must bc remonhci'cd tha.t 'y thé !c.ssHho)t)(! is on]y tiie co~'c~e/!
< 126

VIBH.ATING

SYSTEMS

IN GENERAL.

[LIT'.

the rcsult of an agréeront bctwccn. the vibrations of thc kiml of light affected, and somc vibration proper to tbc mo]eeu)es of thc absorbing age~t,oui- theory would in(!ica.tctb~t for light of somew]t~t gi-(i:ttcrporiodt!ie cH'cctinust 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-bandits 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 préserves 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.ndminimum vaincs of ~t' occur wtien x is respectively eclual to + ce, or –a a.n(tthcn

Hencc, the smallcr thé value of or 'y, thé grcatcr will bc thc maximum alteration of tuni tlio corrcspohding vainc of will approach uearcr a.nd nearer to n. It may be well to repeat, that in tlie optical application a (liminishcd is attend cd by an ~crpf<M~ maximum absorption. When the adjustment of periods is such as to faveur ~t' as much as possible, thc corrcspondijig value of a' is one hn.lfof its maximum.

CHAPTER

TRANSVERSE

VIBRATIONS

VI.

0F

STRINOS.

118. ÂMONGvibrating 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 essentiel parts 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)cmbert 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 comparative 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 gcncratty 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 closelyapproxima.ted to by most of thé strings emptoyeJ in music. We shaUafterwards sec how to takc of any small deviations 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 so displacements and motions of the particles are ~o~tf~ 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

128

TRANSVERSE

VIBRATIONS

0F

STRINGS.

[118.

calculate it we must know the relation between the extcn.slon of a strmg and tlic stretching force. Thé iipproxim~tc hw (giveti by Rccku) may be exprcsscd by s~'ing tliat thé extension varies as thc tension, so that if aud dénote tlie uatural and t!ic stretched Jengths of a string, and 7'tlie tension,

whcre is a constant, 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].e stnng to twice its natuml !cngth, if t).c law apphed to so grcat cxteu.sions, whicl., in gcnem!, it is far irom douig. 119. Thé vibrations of U.e second kind arc ~YtH~~e; that is to say, the particles of thé .string movo sensibly in planes perpendiclllar to the Ime of t), c string. In tliis case t)~e potential ener.-y of a déplacement depends upon the genend tension, and thé variations of tcnsion accompanying t!.e additional ~aU stretcl.iur. duc to the dLsp]accmcnt bu Icft out of . It is he~ .nay 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 tho string is aircady subject il) its position of e
VIBRATIONS 0FSTRINOS.139 119.] TRANSVERSE 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.ybc resolvcd into normal componcnts of thc harmonie type, and dutorminingthcir pcriods and chajactcr by the special conditionsof 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 age corrcsponding thereto between arbitrary constants and arbitrary functions, we will commence by following a somewhat 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 ofreprcsentat ing the continuous string with any dcsired accuracy, so far 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 (
130

TRANSVERSEVIBRATIONS0F STBINGS.

fiSO.

At thé ?? points of division equal in~sc.s arc supposed concentratcd, which arc tl.e représentatives of' thé mass of thé portions (~ of thc string, .vlucii tl~cj ,.vur.y LLsect. TI~e mass of cach portiou of' lengLh is suppose.! to be concoutratcd term~I at thé flua.1pomts. On tlus understand.in~ we hâve

Wc procecd to invcstigatc tho vibrions of a strin~, itself o but Icadcd at e.ch of points ,ant dcv~d mcr~, aud from thc (a) from thcmsolvc. euds, witli a mass If <~notc thc ~tcral displaccmcnts of thc loadcd pon~t.s.mciud.ug tlic initial aud ~nal poin~ wo h.vc thc fuUowmgexpressions fur F mid F

with the conditions tliat ~nd y~ish. Lugranges Mcthod the équations of motion

These givc by

whcrc Supposing now t),at the vibrat:ou under consideration is ono ci normal type, wc assume that &e. arc atl propor io a~ .c.uain.s to bc dctcnniu.d. cos~-e).where tlien bc rcgardcd constants, with a .suL.titution of -7~ for If for thc Rakc of brevity wc put

tlie form

tLc ..iucs of

..uu~cs

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)]Gctwith grea.tcr GfMC dircctly 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

'wheres is an intoger. Substituting tho 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 indcndcnt of the genernl constitution of tlie systcm. Thc w a.dmissibtevalues of ?! arc found from (14) by n-scribiugto N in successionthc vatucs 1, 2, 3.W, and arc all diHcrent. If wc tnlce .s'=Mt+l, ~vttnishcs, so that this ()oesnot 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-lterruitepartictc 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,Iitlie possible particular solutions of norinul type

and, by ascribing suitabic values to tbc ar1.)Itrarycn~~iants, can bn identificd with thc vibration resulting from arbitrary Initiât circumstanccs. Let a; dénote tbc distance of the partic~o f from thc cn(~of the then hy substituting fur~. unda striug, so that ()'–l)ct=x;; from (1) and (2), oursolution 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.iourctains its form,)') 1 RpcciHesthé disptacumcitt at any point a*. ThL: tiMiitingfurm oi ttie second is simply

The periods of tlie compnncnt toncs arc thus aliiot parts uf that of tiie gravcst of the series, fuund by puttin~ N=1. Thc whole motion is in a.Ucases 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 compoueutsmay bc n.bsciit. Ail that is asscrtcd is that, ti)u

MASSCONUENTRATED IN POINTS.

120.J

13~

above-mentioncd interval of timc is 6'M~tCte~ tobring aboutacontp!(.'tcr(:cu!<)~?'\ W';t!
)

I

2

3

4

9

19

39

sni– .9003 .9549 .9745 -983C -9997 ~(~t.+l) .995U .9990

It will bc sccu that for very inodcrate values of m thé limit is closelyapproachcd. Sinco ?~ is tlie rnnuber of (!novca.ble)loads, the case ?;= 1 corresponds to thc probbiti uive.sti~ated in Chapter 111.,but in comparing thé results wu must rememher thn.t we tliere supposedthe w/~e m~ssof tlie string to bc concentrated at thc centre. In thé prcscnttCasc thc h):).dn.t thécuntre 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:ULbu niprcscntcd hy

And,su)co auy furm pos.-ilbiofur thc stringataU mn,ybc rc-g~rdmlas initud, we infur thut any iini.t.esinglu valued functioti at ~=0 ~nd a;=~, c:ni be exp~nded withiu uf te, wltich v:).)M.s)ies those Ihnits in n. scries of sincs of

a.nd its mu)tiplcs,–whtc)i

is a.Citscof Fouriur's t)icorum. ÂVcsliall prcscutly sitcw how the more gcncml furm cnn bc duducud. 121. We might now détermine the constants fora. 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

134

TRANSVERSE

VIBRATIONS

0F STRINGS.

[121.

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 sin

thc second by sin 2s

&c., aud
s~, 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,)ibsdeponding on the initial vcincitics. W!tcn becomes I)i~nite]y smaU, )'~ under tho sign of sutumation ranges by in
cxpressing d, lu tcrms of thc iultial displaccmcnts. 122. Wc wi)tnow invcstigatc indcndently the partial difFercnti:i] équation govcrnin~thctt'ansvo'.sumotiottofa.po'fcctiy HcxiLfc strin~, on thé suppositions (t) thatthe jnagnitudc ofthe tension mny bo cunsiucrcd constant, (2) t!)at thc square of H)cinclination of any part of thc string to Its itntial diruction may bc ticgicctcd. As befure, dcnotusthc lincar dcnsity at any point, and y'~is the constant tension. Let rcctat~nlar co-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<'nuf any partictu at tinic t. Thc forces acting on thu clément (/.c :u-othé tensions at its two cuds, und any impresscd forces .)~ ~p< ByD'AIcmbcrt's PnnToJhuutor'H J)t<.C«;c.,p. 267.

122.]

DIFFEBENTIAL

EQUATIONS.

135

cipin thèse form an equilibrating system with thé réactions against accélération,

p CLu p MC At tlie point x thu com-

ponents of tension arc

If thc squares of Le so that ttic forces acting (a; c~x ncgiccted 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 corrcspoudmg cquations ofHuitc diM'crcncGs in § 120. Thc latter may be written

which nmy :).!subc provcddircctiy.

13G

TRANSVERSE

VIBRATIONS

CF STRINGS.

[123.

Thé nrst is obvicuiifrom 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.Hhave 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 convenicBtIywrite

n.nd tlie difrerentia.!cqua.tionis expressed by

If we now assume t]iat y varies as cos ?)M~ our equation bccomcs

of which the most gcnera.1solution 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 M'c fuuctiotis of a*, not ucccssarUy thc samc. On \vhcro ;?/ substitution in (2) it appca.rsthn.t y~ a.ud arc subjuct to cqua.tions uf thé fut'm (3), so tlia.t Hnally

:ut expression conta-himgfour a.rbitra.ry constants. For any contiuuous 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 harmonie. But whenever thc string forms part of a. system vibrating frccty n.nd withoub dissipation, wo know from former chaptcrs t,)):it :).I1parts aru simuit~neousty in thc same phase, which t'cfjuircs that r\ and then thc most goncra.1vibration of simple harmonie type is

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))-~tionsof a fnntc sti-iug of Icngth held fast at both its ends. If we takc thc origiti ufa.- at ono und, tlie tcrmirnd conditions a,rc vanishes for ~11values of t. that when a:=0, Mid wheM a!=~ T)ie nrst i-cquit-estha.t in (G)of § 123 and thé second that or that ~==.S7r, wlicre s I.s ~n intcger. We IcM-nthat thc only hM-mouicvibrations possible are such as mnkc

iUtdthuu

Now wc know M./)Worzth:tt whn.tcvcr thé motion may bc, it CMtbe rrusoitcd as a suni of simi'ic htu-mouievibrations, a.nd wc thurcfurc coucludc th:tt thé luost gcncra.1 solution for a string,

138

TRANSVERSEVIBRATIONS0F STRINGS.

[124.

so that, as has bccn aiready statcd, the whoïc motion is under ail circumstanccs pcriodic in the t:mc r~. Thé sound cmitted constitutes 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.sesthe sound could hrn'dty be called a note; but it usuiJIy h~ppcns in practicu that, w]tcn tho gravcst tone is absent, .someothcr 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. A simple 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]~itch 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 solutions, pcrnussibic in accordancc with his Principtc of the Coexistence 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 not capabte of rrescnting an the other side, arbitrary function; and Bcrnouln's on 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< <;D~/c'rctXtn! O/t'tc/tftN~c~, § 78. DrYounK, iti Lia momou' of 1800, HC-ûiHH to liave understood this matter quito
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 appears 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~~icharc proportiona) to t)tc Stmare roots of thc lincar dcnsity. Thcse important rcsults may 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 rcspectivcly 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 reThé oniy thing left uudetermined prcscnting a time is T, is Uic numeriea.1factor. 125. Merscnnc's laws are cxcmphfied in a!l stringed instruments. 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 remcmbercd 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
140

TRANSVERSEVJBRATJONS0F STRINGS.

[125.

For quantitative investigations into the laws of strings, the aonoïncter is emphjycd. Hy mcans of a, weight lianging over a puUey, a catgut, or a mctaHic wire, is stretcijed across two bridf-cs tnounted on a résonance case. A moveable bridge, whose position is cstimated by a sca!c running parahel to thc \vire, i-ivos thc means of shortcning tite cfHcicnt portiott of tlie wire to any dcsit'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 thehJcngt!i3,their masses,and tlie tensions to winch they are subjucted. Thus In order that two strings of equa! length may yle!d t))c intcrva! 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 of thé moveable bridge, thu Icngttt of the strin"' is adjustment 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 on deduchifr 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 a fot'k.andwhengivin~with it four béats p'u- sucond,are caœfuUymcasured. 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 formu]a from tlie mcchamcal eicmcuts of tlie system, but grcat précautions are necessary to secure a.ccuracy. Thc tonsio))is producc<) by a.welght,whosemass (cxprcsscdwith tl)o samc unitasp) m:t.ybo called P; so that y, = whGi-e = 32'2,if thé units oficngth a-nd timc bc the foot aud thé second. In orderto 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.tedby 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
1.' l l 1 whichvanishesat~–1 S7rX to excursion is proportional sin. points, dividing thc string into s cquat parts. Thèse points of no motion arc caticd nodes, and rna-ycvidcntiy Le touci~cd or Iield fast without in any way disturbinp; the vibration. T)tC production of harmonies' by iightty toucliing thc string at thc points of Ail aliquot division is a well-known rGsourceof thc violinist. at thc component modes are excludcd which hâve not a node point touched; so that, as regards pitch, tlic cuuct is the same as if tho string werc securely fastened thcrc. t] le

142

TRANSVERSE

VIBRATIONS

0F

STRINGS.

[127.

127. Tho constants, which occnr h) the gênerai value ofv, § 124, dépend on thc epcci.i!cii'cn)nnt:mcesof t~ \)i.t.t!ot), ~))~:~t;y b' exprcsscd in tcrms ofthc initial valuus of~ a.ud

Putting t = 0, we fmd

Multiplying hy sin

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 dcnding on initial displaccmcnts may bc infcrref!, by diH'ci-eutiatingwit!i respect to the tirne, and substitutmg~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

on intégration. Wc sec that

vanishes, if' sin~

if thcre be a =0, that nodc of thc componcnt iu question situatcd at j~. A more comprefensive view of tlie subjcct will bo aitbrdcd by another mode of 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 denote thcm thcrcfofGby so thfit the conjuration an<] motion of tho Systemat any instant arc dcfined by the values of d~ tmd according to the équations

Wc procced to form thc expressions for to dcducc thu no)'ma.Ieqnatious of vibration.

aud 1~aud Lhcneo

For thc kinctic ciiergy,

thé product of cvory pair of tcrms vanishing 0 by the bgcnend proporty of normal co-ordiua.tcs. Hence

Thcse expressions do not presuppose any paj-ticular jnotion, either or natural, othcrwisc but we may 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~) bu substituted for we find for the smu of thé cuergies,

144

TRANSVERSHVIURATIONS0F STUINCS.

[128.

If the motion bc not connncd to the p):u)c of wc havn nK'rc'Iyto add thé cno'gy of thc vibrations in tbc pcrpcndicuiai' plane. Lagra,nge's metbod givos immcdia.tcly thc équation of motion

which hn~ihecn ah'cn.dyconsidcrctl in § GC. If <~)a.ud initial values of 6 and tlie guncral solution is

bc tho

By dc~nitio!i
In theso équations is a. tincar qnaatity, as \ve scefrom (1); and
If thé point of application of thc force eoincidc witli a. node of tbc mode (~), <),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,whûtionoof its uodnl itiduoto Youug. polutaia plucked,

129.]

YOUNG'S

TIIEOREM.

145

with thé finger, aH motion must forthwith cease for those components which have not a. node at t! point, ht q~stion a.re stopped Lyttie dumping, and tl~oso wbich hâve, are absent from thc bcginumg'. 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 regardcd 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-ntities affected 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

The symmetry 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 string set 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 notation as bcforc,

Alikorosuttensuoswhonthépointwhichia dampodi.-iat thosamodistance ïromonoeudofthostringas the poiutof excitation iafromthoothoron!. R. 10

146

TRANSVERSEVIBRATIONSOP STRINGS.

[139.

Le denoted by 3~. At the samc time (~). = 0, so that hy if~y~ (2)atHme< t

The séries ofcomponcnt vibrations is less convergent for a, struck than 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 IV cliscontinuous,wlu!e in tlie former it is 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 <mdstLc 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 concentrated a.t a. single poi)it. If it be distributcd over a distance on citlier side of we l)avc only to iutcgratc thé expressions (C) aud (~) with respect to substituting, for cxample, in (7) in r tT-sin of place .1, -y–,

Tho principal effect of thé distribution of t])Cforce is to render tbe series for y more convergent.

130.]

PIANOFORTESTRING.

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 hcro thc 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 action occupies so short a time that its duration may be ncg)ccted. Mea.surcd by tlie standards of ordinary life tlie dnration ofthe 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-toarc covcrcd with sûvcral layers of eloth for tho express purposc of making them more yielding,with the effect of prolonging the . The rigorous treatment of thé problem would bc difficult, and thé solution, when obtained, probably too complicated to be of use; but by introducing a certain simplification 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 coveringof tlie hammer, tlie law of tlie force called into play during the 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, whcnthere are neither velocities nor displacements,a force .Fsin~ betiins to act on thé string at a:=~ and continues through half 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-cquiredsolution is at once obtamcd by substituting for in thc gênerai formula (7) of§ 128 its value given by

148

TRANSVERSE

and thé final sohttion for their Ya.}nus,

VIBRATIONS

0F

STRINGS.

[130.

becomes,if we snbstitute for Mand p

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): !:(.wof force. Thé Intcrest uf thé présent solution lies in thc infortnationH)!).tmay be ctieitcd frnm it a.sto the depcndenco of thé rcsulting vibrations un thc duration of . 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 amplitude of the cumponcnt s is

Whcn in nnitc, those components disa-ppcar, wbosc perlons t.~c duratinn of ; and wltbn .s is vcry §' ?' grcat, thc séries coivo-ges wit)i N' Some tUbwancc rnust at.so bc ))i!idGfor tho (hnte breadt)~of thc btunnicr, thc cHect of which will a!so bc to faveur thé convergenceof thc séries. Thc laws of tl)c vibration of strings Tnaybe veriHcd, at least in their main featm'cs, by opticiil mcthods of observation–cither with thc vibration-tnicroscope,or by n.trn.cingpoint rccoi-ding tlie characteroftttc vibration on a revolvmg drum. This character dépends on two tbings,–thc mode of cxcitement, a,nd the point whose motion is se)eetcd for observation. Titosc components do not appear winch bti.'venodes either at the point of cxcit.cmcnt, or at tbc point of observation. Thé former are not gcno-atcd, and t!)Clatter do ])ot mfmifust.thcmselvcs. Thus t!ic himpicst motion is obtaincd by ptucking thc string at the centre, and obscrving une of tbc points of trisection, or vice w?'M. In this case t!te first harmonie wbich contaminâtes thé purity of thc principal vibration is thc nf'Lh cornponcnt, wbose intcnsity is usuaUy insunictcnt t.o prudnco nmch disturhancc. In a future chaptcr wc shall compare t)ic results of tiic dynamica.! tlicory with aurai

130. j

FRICTION

PROPORTIONAL

TO VELOCITY.

14S

observation, but rathcr with tlie view of disc~vcring and tcsting thu [awsof iiûmm~, LiuUiuf confirming' Lhutheury Itseli'. 131. Thé case of a. penodic force is Included in t!)û general solution of § 12!S,but we prêter to foUowa somcwha-tdirEcrent jnethod, lu ordcr to m:Lkcfui cxtcusion in anc'thcr dircetion. We have hithcrto takoi no ofdissipativc forces, but wc will now suppose that thé motionofca.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, rema.rkmg that in tho présent case the fnctiun-function is of the s:unc formas 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 Citapter ni., that sincu K is indupcndunt of thc jmtural vibnt.tions subside in suc!) a manuer that t!tc amplit.udcs manitidu thcir rclative values. If a periodic force .Fcos~ act at a single point, wc liave

If among thc natural vibrations thcrc bc any one ncarly isochronous 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 componcnt v:brn'-) ~n .i'j<<)~ VMmshc.s; n< foœ app]iûd !tt a.nu'-iuca.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 of course to free vibra.tions that titcy are more cspcciaHyapplicable, 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 advantagc of the use of normal co-ordinates is the facility of comparisonwith thé efir:<(?~ ~
so that, whcn thc result of thé equilibrium theory is known and can rcfidiiy bc cxprcssed in of thc normal co-ordinatcs, the true solution with thc effects of inertia included cn.u ~t once bc written down. In the présent instance, if a force .Fcos~ of vcry long period act at tlie point b of thc string, tho result of the equilibrium in aecordaûcc with theory, whieh the string would a.t any moment consist 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 in place of writing 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 conditions. In thc probicm of forced vibrations ??tis 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 constants is of the same form as thc normal functions; and a.11that remains to be enected is the détermination of the two constants in accordanco with thé prcscribcd bounda-ry conditions whicli tlie complete solution must satisfy. Similar considérations apply in the case of any continuous system. 133. If a periodic forcebe 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.1equation

is satisfied over both thc parts into which thc string is (UvIJcJat b, but is viola.tcdin crossingfrom one to thé othcr.

152

TRANSVERSE VfDBATIONS 0F STRINGS.

[133.

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?~ by the complex exponential e" a.nd 6tia!Iy disc~rd tho imagiuary part, when t!ie symhoHcalsolution is completed. On the assumption timt~ varies as e" thc differential équation becomes

The most genera.1solution of (3) consists of two tcrms, proportionn.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 ye' 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.1solution of thc problem, but if it be thc force that bc given, we require further to kuow thc rcla.tionbetwecn 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.1part of thé solution, correspondinb to thé force .Fcos~, is found by simply putting cos~)
154

TRANSVERSE VIBRATIONS OF STRINGS.

[133.

traverse motion. it is only necessary to attach it to thé extremity of one prong of sucha fork, whose plane of vibration is perpendicular to the length of thé string. This method of cxhihiting thé forced vibrations 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 a powcrfui tuning-fork, set into vibration with a bow, or othenvise is pressed against thé string. It is advisable to ~e the surface, which cornes into ~ith t),e string, into a suitable (.saddieshaped) 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 be taken to prove that in thc case eontempiated, the motion would real!y become great-so grcat tl.at corrections, insi~upreviousiy rise into ficant, importance. Now sin vanishes, when the force is isochronouswith one of thc natural vibrations of thé first part of tho string, supposed to be )tdd nxed at 0 and b. When a fork is placed on thé string of a ~onochord, or other instrument properly providcd with a sound-board, it is casy to find by tnal thé places of maximum résonance. A very slight displacement on eitlier side entails a considerable falling o~In~e volume of tlie sound. Thé points thus determined~i~ the string into a of parts, of length that the natural note of any one of them ~hen nxed at both ends) is tlie same as thé note of thé fcrk, as may readily be verified, The important applications of resonance .vhieh Helmholtz lias made to a tone purify simple from extraneous accompaniment will occupy our attention later, 134. Returning nowto the case complex, o have to extract thé real from parts (5), (R), (~ of§ 133. For ~f~~T~ occur as reduced to the form Beie. Thus let

134.]

FRICTION PROPORTIONAL TO VELOCITY.

155

corresponding to thc obligatory motion =~ycos~ 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 simpliHcd. For instance, in this case, to a sufEcicut approximation,

15G

TRANSVERSE VIBRATIONS 0F STRINGS.

[L34.

so tha.t corrcsp~nding to tlie obligatory motion at & ?/ =-yeosp!; tLc :uu!j~ bt'i~c~n 0 ;ut.t nmphiudt'' of is, )tpp~Xit)jft.tu!y

-which bccumcs grcat, but not inanité, wheu sin = or thc M 0, 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 preceding solution 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) solution s~tisf'ying the conditions thc expression for the i)fitur:d vibrations must also Le addcd bnt thèse become reducecl to Incancc after tnc motion lias been in progress for a sufïiciunt timc. Thé !n.wof friction !msumcd in the prcccding investigation is thé only one whoso resuits can bc ca.si)yfuilowed doductivety, and it is sunicient to givc a gênerai idca. of t))C effects of dissipativc forces on tlic motion of a string. But in other respects thé conclusions drawn from It possossa nctitioua simplicity, dcndinr'-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 vibration into which the motion may bc rosolved, a)id which arc perfectly indendcnt, but thèse are essentially different in character from thosc with which wc have hecn conccrncd hithcrto for the varions parts of the system (as affected by onc dcnicntary arc not in thc samc phase. Spécial cases vibration) simu!t!).neous!y 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 probfem is of no acoustical importance, but it is Intercsting as bcing mathcmaticaMy analogous to that of thc conduction :utd radiation of lipat in a bar whuse ends arc maintaiucd at a cojtstaut température.

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:),nnotbc fLttu-uibdm prit-cticc,it is ~ot without iuterest to inquire in whut ma.nncr t))Civibrations 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))Csahc 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 cqniiibrium. tf uo frictionat forces act, thé motion is nccessa.rity rcsolvabiciuto normal vibrations. Assume cos (wa~ e).(l). ~= (~ sin )Ha;+~3cosMM'} Tho conditions at thé ends arc that

whiehgivc

two cquaitons, sufîicicnt to détermine E)inun:tt.i)tgtitc lattur ru.Lio,we Hud

and tlic ratio of~S to a.

Equation (3) has an infinite number of roots, which may ho fnund by writing tan for so tliat tan ?/~= tan 2~, and the result thé corrcspoudingparticular sohttions, each of adding togcthcr with its two arbitrary constants et and c, is necGssfu'itythé most guncralsoiution of winch thé prublem is capable, and is thercforc adéquate to rrcsunt thé motion duc to an arbitrary initial distribution of dispiacemeut and velocity. Wc infcr tbn.t any function of x may bc cxpanded bctwucn x = and a;=~ in a-scrics of ~,(~,sin~)cos?)!) + ~~(~s!nm.c+cos~) + (5), &c.thc coi'rcspouding ?~ H~,&c. bclug thc roots of (~) and

158

TRANSVERSE VIBRATIONS OF STRINGS.

values ~p~" of' t] i syst.em,

arc

co-ordinates

From thc symmctry of thc system it follows that in each thc s.,nc at points "y distant thé ––P~. twoends,wherc~=0~d. l. Hcncc ~sm.t-co~ =+1 1, 8 4 as may bc proved also from (4-). T!.e Mncdc energy y of thc ~I.ole nation Is made up of thc Mergy of thc string, and that of the masses T!u~ r; SUl + cos y=~p) {S f~ M!a-)~ +~+~+.r+~~(~sin~+cos~~+. Buthy theenaracteristic property of normal co-ordinates, cannot ~lypr~eut in t.. c.prc.ssion for ~I~~tr'~ l' 80 that 1 .9ili9?bX sin cos?M~) (~ (vr +,coq + ili,x) (~ sin M~.t+ ces M,?;) pf" p~ + + (~ sin m~ + cos7~) (~ sin M/ + ces M/) =0. if 7' andg Le differcjit.

(G),

This s~gcsts how to détermine thé t~orem senc-s arbitrary constMts sothatthe (5) mayrrcscnt au arbitraryfunction y. Takc thé expression y(.sin ~~+cos~)~.+~+ sin cos p~ ~(7) .~d in it thé scrics (.) Thé rc.nit is a exprc.ssi.,g scries~titutc of tcrms of thc type (~ sin

p~~ + +

+ cos~)

(~ sin ?x,~ cos ~~)

(., sin 7~?+ cos ,) (~ s; + eos ail of vanish hy (6), cxct thc onc for whieh 7.= whi~ to js equal thé expression (7), dividcd by

Hencc

p~.sin~+cos~+~+~ and thus thé cftho series arc detcrnnnod. coc~icnts If ~=0 even bo rinitc, thc p, althc.g~ 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 Mapt, <,o~cquirc' tor*contra<;tcda vic~vof thc na-torc ofthosc important rcsults ofanaly~Is. We shall now shew bow Fouricr's thcorom in its ~encrai form can bc dedud from our présent investigation. Let ~=0; thcn if /t= -X),the ends of thé string arc fast, and thé équation detcrniining ?~ 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,vanisiting at 0 an'l ?, betwccnthosc Innits. But now suppose th~t~ is zéro, ~/8ti!l v!t,nishmg. Thc ends of thc string may be supposedcapable of slidiug on two smaotit m:ls perpendicuiM-to its length, Mid the tcrmina.l condition is the vanishing of Thc cquf~tionin is thé same OM~e/~rc; and wc Ic~rn.that any fnnctiony' 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, to + And in the samc wa.y,if y bc (10) represciits it n'om 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!iichis even, and the other odd, thus

so tha.t, if (x) be such tha,t (- = (+ Z)and < (- ?) = ~)' (+~), it eau bc rresetitcd betwccn thé limits ± by the inixed séries

This series is penodic, with tlie pcnod 2?. If thcrcforc (x) possessthc 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 s, in tlic cfu-:c (.)f:).striog whosconds 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.Laud is vanishes, (2) wlicn vauishes and gréât.

and the équation in

is approxhnatdy

and To this ordur of approximation thc tones do not cease to form a harmonie scalc, lmt tlie pitch of tlie whule is slightiy lowered Tho effect oftftcyiciding is in fact thc same as that of an increase in tlie length of thé string in the ratio 1 1 + have been anticipatcd. Thé rcsult is otherwisc if /t vanish, while Al is great.

as might Hcre

and Hcnce

Thé cfTcctis thus cquivfileut to a dccrcMc in l in tlie ratio

Tho ~t System' for provi~g Fonrier's t]iMrem from dynamic~I considéra. tio~ is au cmUess chain ~tchod round s,u.~th cyiiudcr (S M!)), or thiD ro-outraut culumn of uir eucluecdiu a nitH.sIfnpcd tube

135.]]

FINITELOAD.

l(j~1

and eonscqucntiy thcrc 1. a nso in pitch, t]ie rise bcing thé grcatcr thc lowcr tho couiponcut tone. Tt nngl.b bc thought tttfit 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, donds on thc sign of and this agam dcnds on whcthcr tlic nn-tura.!note of thc mas-s urgcd by t!)c spriug I.slowcr or !iig]tcrth:ui th~t of thc component vihra.tion In (question. 136. Thc proDcni of an ot))crwise unifonn string c~n'ying a nuite load ~at .ï;= ciui ho .sutvcd thc formutœ by InvcstigiLtett !n § 13:}. Fur, if thc force 7''cus~< be duc to the réaction against accuIcraHonof 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 vibration corrcsponding to

is

whcre P :uid e arc fu-bitrfu'yconstants. It docs not rcfjuh-c anaiy.si.s tn provc thn-t any normal cojnponents 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 orders rcin~in unchanged, \vhi)c a)i thc odd components arc dresscd in pitch. Advantagc m!Lysomctitnc-s hc takcn of t)tis effect of a. load, wi)cu it is desircd for anypurp~cto distnrb the harmonie relation of thc comptjncnt tones. If bc voy g-reat, titG gravest component is wideiy separatcd in pitc)i from !i]I t).u others. We will take thc case when t)te !oad is at thc cfntrc, so t.hat = b = U.l. Thc équation in t])cn hceomos

where 3/, dcnnting thé ratio of t)te masses nf the stritig and th.' )oad, is a sma]) quantity which Th<~
162


~: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.incdin 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 stands jua.y bc obta.incdfrorn thc considcra.tionthat in thc actutd vibration tlie two ~:u'ts of tlie string in-c ncn.riystr&ight,and may bc n.ssmncd to hc <'xnct,Iyso in conipnting titc kitK'tic n.nd potcntifd énergies, -\v!th<'ut C!)t)ufi!)gfmy apprcci:d)Ic crror in t)tc cn.lcoh~tcdpcriod. On <J)i.ss~ppu.sitionthc rctcntion of t))e incrti:t of thc string incrca.scs thc kinctic cnf'rgy corresponding to {tgivcn vclocity of t))CJond in thc mtio cf whic)) icads to thc nhovc rcs)dt. This ~)7'+ mothod t):is indced thc a.dvantagc In onc rcspuct, aa it )ni'd)t bc npplicd whcn is not nnifortn, or ncarly uniform. ~)] th:tt is ncccss.iryis t.ha).),hc )oad -/)/shoufd he su(H(.-icnt)y prcdouioant.

CORRECTION

13C.]

FOR RIGIDITY.

1G3

Thcrc is no othcr root of ('t), until sin~X~=0, which gives thc second component of thc .string,–a, vibration indcndent 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 approxiin which tho second tcrm is sma. mn.tc!yby ~=N7r+- &'7T./)/ Thc two types of vibration for N= 1 are shcwn in thé Hgurc. The goncral formula (2) may a)so be applicd to find the cifcct of a small loadon 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. Thefirst is called viscosity, and shcws itself hy df~nping thé vibrations. This part produces no sensible efTcct on thé poriods. The second is conservative Iti its chtu'a.ctcr,an
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, y

p x

si~L–

f '1 satt.sfymg

t!ic

second

1, 1 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.sis expresse'! by

~64

TRANSVERSE

VIBRATIONS

OF STRINCS.

[137.

where /e is a quantity dcnding 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-mscttic f~i-cc required to bcnd any clément Is proportion. to and to thc amount of bcuding a]rcady c-iTcctcd,t!)at is to Thc whoic work w))icli mu.st bo donc to producc a curvaturc 1 p in ds is thcrcforc proportional to ~-p'; whl)c to thc app)-o.ximatnmto 1 which we work = and = 1 p M.<,

if T. dénote what tl)c pcriod wouM bccomc if t])c st.nng wcrc cndowcd with pc.rfect fi~ibiHty. It nppears that t).c cffcct of thc .st~ncss n.crc.ascsrapidiy wilh t].o ordcr of thc couinent vibrattons, which cea.scto bc)o.~ to !LI~nnnnic scalc. Ho\vcvu! i.t t!.c Htrn~s cmpjuyc..) in niu.sic, t).c ton.si.m i.s usn~Dy sufîicicnt to rcducc thu inHucnce ofrigi()it,y to insignifiance. T))G)neH)odoffhis section cannot hc ~ppjicd without inodinc.~ion to t).c ot).e.- case of t~nina) comiition, n:unc)y,whcu t).c cnd.s arc c-hunpcd. In thcir immudiatc nci~hbuurhood t)ie type of vibration 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 thcrcforc cannot be nog]~t~d. Wc sha)) rcturn to this square suhject, wttcn ti-cattog of thc transvcrsc vibrations of rods. J38. TLct-Gi.sonc p.-obicm rdating
138.]

VIOLIN STRING.

165

Since thu note of a. ~ood Instrumr'.nt, well ha.ndlcd,is musicn], wc infcr thaL Lh~yibraLiuurta.i'c stricHy pcriodiu,or at least that strict periodieity is thc td'td. Morcover–and this is very important–t!ic note clicitcd by the bow lias nctu'Iy,or qnitc, tite sn.me pitch a.stho 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.nccon thé energy drf),wnfrom 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.stsin dcturjnitnng (who))y,orin part) undci' what influences it sha,llcoinc. 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 coefHcientsof tlie Fourier expansion

be takcn as tlie independcnt co-ordhiatcs by wlticbthc 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:Uvibra.tiu)is arc cxpresscd by ma.king cach co-ordin:tten.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.~toin the présent c:mcnii~ht bc M?t~/ function of tim time periodic in time'T. But a Httle examiua.tionwill 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-todin thé solution, but tbe ono whosc period is thé same as tho natuml po-Iod of~ will risc cnoi'tnousiyin relative importance. Pra.ctienUythen, if tlic damping bo suialt, wc uccd only rctain tha,t p:~rt of

166

whicL W 11(:"

TRANSVERSE

( 1. d' j"!en! pendsS 011 on J, ij.

VIBRATIONS

c,¡ c.~ i"

e~ ¡¡~

0F

thut ) tua!'

STRINGS.

I., tu ~y, ¡ci s.ty,

[138.

w~ ~ün.y Wu may regM' rogm-d

the vibrations as natural in their types. Anuther matent fact, ed by cvidûnccdrawn both from theory and aurai observation, is tins. AHcomponent vibrations are absent which have a node at the In point of excitation. ordcr, however, to extingui.sh thèse tones, it is neccssfa-yt!)at the coincidenceof the point of application of the bow with tho nndc shonid bo vcry c.mc<. A very small déviation rroduces t)tc jnis.si)igtones with considorabJc strcnf-tb' Tlic rcm~inder of tho évidence on w)nch He!mboltx' theory rcsts, vas dcrived from direct observation with thc vibrationmicroscope. As explained in Chapter n., t)tis instrament affurds aview oftbe curve rrcscnting thé motion of thc point undcr observation, a~ it would ùe seen traced on tlic surface of a transparent cylinder. In ordcr to Jcduec t!te représentative curve in its ordinary form, the imaginary cylinder must be conceivedtu be uni-otiud,or developed, into a. p)anc. The simp]cst results arc obtaincd whcn the bow is a.ppl!cdat 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 (~173hcing a conpicte period), and t]io ordinates reprcsent the displaccment. The rcmarkabio

fact is discloscd t))at. thc whotc po-iod T ma.ybc divulcd inin two parts ï~ nnd r-T., during c~ch of whieh thcvdocityof thet)bsct-vcd point is consent,; but thé vulucitics to and fro ;).re in goict'a! uncqua]. Wc Iiavc now to rrc.scnt this curvc by n so-ies ofiiarmnnic . If t)tc ori~in of timc corrospond to t))c point J, and Donkin'<~cf)~.
138.] J 7'

VIOLIN

1GT

STRING.

T'Y? == Fourier's t)teorcm E;ivea

With respect to thc value of T., wc know that ail those com(xa being thc pouents of~ must vauish for which sm-°=0 point of observation),'bccn.uscunder 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 tlic string is divided by the point of observation. Now thc i'rec vibrations of thc string are rrcsentcd in geucral by = + sin sia cos and Uns at thc point a;= must agrec with (1). For convenience of euu)parison,we inay write 2S7T/. T~ 2S7T< 287rt 2S7T< = A Co.2s7r < ~) ces Sin~~7rt (7, C~ ~î, cos––T + R B~sin T T t 2/ 19 D ~(<),

Ir (

1'0)

and it thon a.ppears that C, = 0. We find a.Isoto détermine D,

whcucc

In thc ca.se reserved, the comparison !c:wcs DHundetcrmincd, but wc know ou otlier groundH tliat DHthen vanistics. Howcver, for the sakc of simplicity, we sh:dl suppose for tlie pt'cscnt that. D~is ahvitysgivcn 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.udingtlie complote solution ofthc problem is

168

TRANS VERSE VIBRATIONS

OF STRINGS.

['138.

The a.mpl:tudea of thé componcnts fu-et.tjercfore proportions) to int~u!p!!t~,)dMt!-l; A ~)U;tdfu!fi'JC<;)t'C!t)~)"' 0 y futictiou i'uuctiou 1 l is sonicwhat whic)i l, 'JsinnJfn-. If Iftijc J string s'sin'S~ L, s-"sln' ho ptucked at thé mnhilc, thé cvcn components v~nish, but thc Oth!oncs foHowthc same )!twas obta.hjs fur a vioiin strincr. T))c c()ua,tiûn (3) ulcHcatcst]):tt thc strmg is :Jways m t))e fun~oftwo .st.mi~ht Unes mcetittg a.t an angtc. In order inore convettiontiv to shew this, !ct us change thc origin of tftc tune, a.ud t)ic constat mu)t,ip!ier, so that

willhc thc équation cxprcssing thé form oft))c string :tt any titnc. Now wo know (§ 127) that thc C(tU!ttio.iof 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:B cf thc point of intersection moves uniforndy backwards and forwanis 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 connnon chofti. Since the motion ofthc string as th~ d~nu
138.]

STRINGSSTRETCHEDON CURVEDSURFACES. 109

a mattcr ofindifïercncG when a, node is thc point of observation, but not in nny otho' c-f. Wh~n thc nid.' i.-idoparted from, thc vibration curvc shews a séries of ripples, duc to thc absence of thc conponcnts In question. Somc furt!icr dctails will be fuund in Hcitnhoitz and Donhin. Thc sustaining powcr of thc bow deuds upon the fact that so)id frictiuu is Ic.ssat modérate t))an at smalL velocitics, so thf).t w)tût]t)tc part of thé stri!~ actud npou is moving witil thé bow action is greater (nut imprububly at thc s:~mcvulocity), thc mutual titmi wi'eu thc string is moving in tho opposite direction with :L greatcr relative vulucity. Thé ~ccctcrating eH'cct in tl~c first not cntirdy ncutratiscd by thé subthus is motion of thc part accctcra.tion rcmains séquent rctfu-da.tion, and an outstanding vibration in spite of other losscs of cap:(.b)cof ]n:untaining thé of solid friction ~ncr"-y. A cm-iouscncct ofthc samc peculiarity bas becn obscrved by Mr Froudc, who found that tl)0 vibrations uf a, pcndulnm swinging from a sbaft might bc maintained or cvcn Incrcascd by causing tt~cshaft to rotate. A strin"' stretched on a. stnooth cui'vcd surface will in to certain concquilibrium lie along a gcodcsic Une, and, subject ditions of stubitity, will vibrato about tilis eonnguratiun, if disthat call bc proposcd is when. tlic ptaced. TI)C simplest case surface is a cyHnder of any form, and thé cquitibrium position of tlie string is perpcndicular to titc gûnerating hncs. Thé studcnt will casUyprovc that tbc n~otion is indcndcnt of thc curvature of tlie cylinder, and that thc vibrations arc in :dlessential 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'tterto Ulustratc tlic charactcristic features of this class of of a problenis, we will tako thc conparatively simple cxample and lying, string stretched on thé surface of a smouth sphère, v/hcn in cquilibrium, :dong a grcat circle. Tite co-ordinatea to which it will be most convcnicnt to refer thc system are thé mcasurcd front thc grcat circle as equator, and thé Jatitudc of thc sphère be ~n~itudc measured alung it. If thc radius wb bave 139.

170


fl39.

Thc extension of thé string is denoted by

Now sothat so tliat f~

J(~i)~. ~=(~f~+(ocos~ = f/ {(~~ +

= 1/ 2 (~~

2 ~PP'tc]y.

Thus ~(y-(~'

andli a.nd

Cltp

ô V= aTl. Q i-~ -10 ~)~. If thc ends Le fixed,

-8-

dtp.(2);1

ose ~(to

+ e J.

~=0' L~J. 0 and thc equation of virtual velocitiesis 0 se dtp 0 0 8~ whence,since S~ is a-rbitrary,

o0

+ 8 dcfJ= = o, 0, (10

"(~)– This is thc cquatiou of motion. If wc assume occos~<,wc get _rl'B 0 cc'p 22 ~,+~0. cf \vl)ieh the solution, subject to t)ic condition that with is

(4), vanishes

.cos~ .(5). ~=~sinj~~+l~ Tho rcmaining condition to bc satisfied is that vanislics whcn «~= or <j&= et, if a =
~h' -~=p'(~ 1)

a p a -1 p J ~herc ?~ is an iutcger. CambrMHO Mathcmaticftt TritMB 187G. Exnmination,

.G

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 is straigbt string. If a> 7r,0110at least of tho values of p2 négative, mdica.ting tha.t the corrcspouding modes are unstable. If a =='7r, is zéro, tlie string bcing of tlie same length iu tlie displaced 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 révolution. 140. The approximate solution of the problem for a vibrating bas been string of ncarly but not quitc uniform longitudinal dcnsity of fully considcred in Chapter IV.§ 01, as a convenierit cxampic thc general thcory of approximately simple systems. It will bc thc sufficienthère to repeat thc result. If tlie density bc ~+ 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.correction for such slight dartures 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 neighhourhood 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

172

TRANSVERSE

from which, if wc assume ï.'<))';n:u iuucti~

VIBRATIONS

occos

0F STRINGS.

F 141.

wc obttuu to dctcrmu~ th

~yhcre

,swntt.n for~-?', of thé second This~uationis hncar, b,.thasnothit)icrtobccn.so!v.d:n o'rand huitotcrms. Cun.s.dcrc.t dcH.ling thé curve ~su.ucd by tho .st.rin.r In thc uonna! mode uih!cr considération, it dutcnnincs thc c~rc at nnd any pou.t, accordin~jy cinbodic~ a ru)c I.y whidi thé c-n-vo can bc construct.cd Thns in thu npptic~iun to ~phic.Hy. string nxcd both cnd.s,if st.t from c.ithcr end nn ~rbitrary inclination, and wit). zéro curvatm-c,~-0are ahvay.sdirectcd by tbe équation w.Lh what eurvaturc to p.-uceed, and in tins wc way trace out thé cutirc curvc. jaay If thc assumcd value of be rigbt, the curvc will cross th. axis of at thc rc.uircd distance, aud thc ]aw of vibration will hc con~tctdy dctcrndncd. If Le nul known, ditterent vaincs may bc tri.d untii thc curvc ends rightiy; a sufncient approximation to tho value cf~m.~u.su~iyho ~-nvcdathy~ c~cul~tion founded on an as.smncd type (§§88, 90). Whcthcr t!.c longitudinal density be uniform or net thé pcncdic timo of any simple vibration varies c~~ as thc root cf thc s<(u~e den.sity aud Invcr.s.ly us thé .square root of the tension undur w)nch t]io motion takcs piace. Thc eonvcrsc prob)cm of dct~mining thc
173

VARIABLEDENSITY.

142.] 14-2.

a gênerai

Atthongh <<

Sh' ~c'

v'

pr,ui'

of

Intégration t'~

m.T\p~y

(2)

équation

.u-

t!n)-~k'n 1.

of§141 c'ft.h'

many intcresti! propcrties of thé solution of thé hnuar équation of'thc second order.which liavc hcen detnonstrated LyMM. Stnrnt and LiouviHu'. It Is impossible 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-csn.ruinctudcd, m~y be found intcrcstthé incr, and will tin'uw li~ht on sone points comicctud with not gcncnd thcory of the vibrntions of continuons bodics. 1 hâve thought it ticccs.s:u'yto adhère vcry c~osclyto t)'c mcthods adoptcd io thc origina.)tncmon's. A.t no point of t!t0 curvc satisfying thc cquation rl'r/ 2 ~+~~n.(D,

)

(1)

can both y a.nd '( vanish togctiicr. By sucœsstvcdiHcrcntin.tio~s vanish simnitancousiy,

n,nd

of (1) it is c~sy to prove that, If

aU thc highcr dnïcrcntial coemcicnts

t

Cl'`

&c. musta!so C~:Cs

vanish at thé samc point, and tilo'cfore by T~ytor's theorcm tho curve must eoincidcwith thé axis of a;. Whatevcr suu~

being

\duc

be

conc:wc

ascnbcd

to

tbt'oughout

thc

cn)-vc

to\vard.s

tlio

satisfying axis

of

(1)

is

a-, sinec

ana l 1 and origin y vanish, Ilositive. If at thé cvoywhcrc positive. p isis cverywllere ~Lx Lu positive, thc ordinatc will rcnmin positive for aU vaincs of a; bclow a curtain limit dei'cndcnt on thc vainc ascribed to bc vu'y smaH,thc cm-vaturc is slight, artd thc curve will If remain on t]tc positive sidc of thu axis for a gi'cat distance. incrcascs, aU tho vahtcs of a; Wc hâve now to provo that as which satisfy thc cotation = 0 gradua)Iy diminish in magnitude. Lct Le thé oi-dinatc of a second curve sati.sfyingthé équation + Il p?J

~+,=0 cl;c'

.(2),

vanishes at thc origin, and lut us as weil as thc condition that Multiplying (2) hy y, suppose t)iat is somcwhat grcatcr than iu titofirstvolumeof T)to)i)c;]))"i)'n rnferrc~to ui tho tcxt nrt: euittttmcd .yuto'/t'~ LiouyiUu's ()'S!it!j.

174

TRANS VERSE VIBRATIONS

0F

STRINGS.

[143.

n.nd (1) by subtmcting, and integn).th]g with respect, to x bctwecu ttic limita 0 and x, wc obtn't), sincti and .1 ~ot!) vnmsh wii,L.'t!,

If wc further suppose thftt .c corrcspomiHto n.point :).t winch ;u]d th~t tlie di~rcucc betwccn and is y vcmi.sfK's, vcry small, we gct ultimately

TIic right-hand mcmber of (4) bcing csscntially positive, wo Jcaru that y and arc of thc samc sign, 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-rwords, 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"'hont 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 function M, of a string of dcnsity (ixod /) at 0 and a~d lias no root cxct at thosc points. As a'~ain iacreases, thc first root movc.sinwards from a;=~ unti), when a second special value is attaincd, thé curve again crosses thé axis at thé point a'=~, and thcn rrcscnts 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, and so on. Thc ?"' functiou M,,bas thus cxactiy 1 intcrnid roots, and sinco its ih'st dinurcntial cocfticientucvcr vanis))cs sinndtancousty with thc function, it changes sign cach titnc a root is cd. Frn)n équation (3) it app~u-s that if nnd hc tw.) di~'rcnt normal functions,

A bciUttifu! thcorum bas bccn di.scuvc-rcdby Shmn rc)!~in
143.]

STURM'S THËOREM.

175

of lowest order, and M~thc component of highost order, tho functtou &c. arc arbitrary coefHcicnts,has a< ~ecM< m–11 whcrc ~), < 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.Ucases. The followingdémonstration bca.rssomc rcsonbluncc to that givcn hy Liouville, but is considcrn.blysimpicr, aud, 1 bellcvc~not less rigorous. If 'wc suppose that /(.E) ~as cxact]y Internai roots (any number ofwincli may bo cq)ia.l), thoderived functionj~) cannot Iiavo less tl)an + 1 internai roots, sincc therc mnst bc at ]cast onc root of/'(.'c) bctwccn cach pair <~fconsccntivcroots ofy(a;), and t.Itu whoïc numbcr of roots of~(.~) eoncurnud is ~.+2. 1); liko roots ofy'(a:), manncr, wc sec that thcrc must bo at Icast bcsides tlie cxtrenutics, which thcmselvcs necessarlly correspond to roots; so that in mg 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~ywc obtai)i a scrics of' ftmctions, :t.]) intct'n:d roots at !en,st, whieit dUrur from the origina! with imtCtioM/(:)')by tho continua]]y menjasin~ relative !))iport!inccof the componL'utsof thc hi~Lcr oniurs. Wi~cnt!i(i procL'ssI):~sbcot ciu'i'iud sufficicnt.Jyfur, we sh~H :),n'ivc!tt a function, whosu iorm ditturs as )itt)c :)Hwe p!t.'ascft-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 it)g down thc so'ics uf fu))ctio!)s, thc m))ub(.'ruf Int(.'t'))a)ruuis ufy(;<') c;U)n<jt UXC'));) t.

17G

TRANSVERSË

VIBRATIONS

0F

STUINOS.

[142.

Thc other ]~!f of tho thcorcm is proved in n..simDarmannor hycc!tt:t~:u~th~8';)-Ic:~if'un.cti.< h.kw.'n~ :;i fl'u~i' h.ll Utiswaywcobt:).i)i

arriving- t~t last at a functiun sc.n.sib]ycoiuc.d~nt in form with thc normal fu])ctionof ordcr, v~ ~nd havin~ Lhei-cforo M-lnitcrnali-oots. Sinccnoroutsc.-mbctastinp~sin~upthc senesfrom <.hisfunctinn to/(..), it full~y.s t].at/(.r) ~nr~ot Lave fewertntcrn~ roots tl.nn ~-1; but it must bc und~-stood that auy number oft)~ w 1 roots mny be cqu.d. Wc wi!!now prove tl.at,/(.) cannot be ide..t:ca!!y zcro un)c~ a!I tlic cocfHcicnts va.n.sh. is not 7ûro Supposa t).at and intégrée wit), Muhiply (G)hy p respect to~ betwccn ~nnc t!tc Jhmts0aud/. Thc.tby(5)

from wl.ich, since t)ic intègre on thc rigbt-hand sidc i.s ~nitc we sec 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 a scriGSof normal f.u.ctions n~y be eo)np<,u)h)c
~andbyStunn's T.c~.snsa!.nc..u-funct,onof.,(.)~.) t!K..o,-cm h.~th~forc onci.iU.rna! n,o< atn.ost,whidi roulis ]\rcover t).o dL.tcrunnant is not cvidc.ht)y idcntica))y zcro sn.ce thé cu~ciont of «,(..), viz, ~~), .)“ ~.t ~)~ ,,]~tevcr bcth.v~ho ut' -\Vchâve 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 er through t)ieso values, and tl)ere on!y. Thc coefHcientof ~(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 continuIng this process we can form functions compounded of thé normal functions, whieh s!)allvanish 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 t!)c necessary values of < < &c.arc determiued by (9), a.nd wc fiad 1 J f8l /(.c) = 2 j'~) (.

f~o

p i').)~

j-'a Up !t;(~) r~-}.(11).

If tlie séries on thc right bc dcuotud by ~'(.~), it ron.Lms to cst.abtislithc 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,

where M~(-c) ts ct~y nonna.)function. Frutu (12) it followsthat

w~crc the coefHcicnts it.

&c. are !n'bitmry. ~),

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VIBRATIONS

0F

STRINGS.

[143.

Now if F(~)-/(.c) bo not Idcntic~Hyxcro, it will bc pnssiMo so chonsct.hc t~ <M c<.tis~)t.s.1. J,. -t .<.r~ tuLsthroughout thc sa).tc in' \v])ic). CMc'cvc.-y sign ~'Y.'(.<:) -), eloncnt ofthe jutcgmt woul(!bc positivu, :m() cqu~~ion(13) cuu)d not bu tn.o. It fuUuw.sLh.Lt F(.) -/(.<.) cannot <)i~r imn) zéro, or th:),t t])e s(;i-ics of sonnât fnnciions fortning tho right-hatxt nimber of (11) is idcntie:d wit)i/(.r) for ~11vaJue.sof .t-from tc= 0 to = Thc arguments and rc.suits of Uns snctinn arc of course ~~pptic-aDo tu thc ]):u-ticu).-ucase of a unifonn stmig fur wi~ch tite normal functious arc circuhu'. 14.3. Whcn Lhcvibmtions of a string arc not con~icd to cno pJanc, it is usua)]y ninst e.mvenient to rcsnivo thon into two sets cxGcutc-din perpendicuL-u-~imc.s, which m.~y be trcatcd indcndonUy. Thcro is, howcvcr, onc case of t!.i.sdescription worth p~sing notice in which thu niotion i.smost casity cuuccivcdaud tt'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 cach pa.-Ucic dc..scrib~ circi~ with radius siu~ t!.c wh.dc ~tcni Intact, turns witi~ut t rdativc <]isp)ace.ncnt. ahuut. its position < c.tui)ib.-iuni, c.mi{~tin
144.] 144.

UNLIMITED Thc

gênerai

cHScrendal

STRING. equation

179 for a uniform

stril)"-

'ix.

cxprcssing fho rciation bctween a;andy, reprcscntsthe form of thc string. A change 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.gatcdaiong thc string witli uniform velocity ft in tho positive direction. WImtcvcr thc vainc of maybcat thc point a; and at thc tuue t, thé samc value of y will obt:uu at thc point a:+ a A
Initia [y,

und

ïn:).ybuth bc gLvcn arbttrariiy, but if tho

a.bovcrelation Le not sati.sfied,t)ic motion cannot bc rrc.scntcd by(4). Inasmultu'nmnucrthcuquatiun

y=~+~).(G), dénotes the propagation of :),wave in tho ?<e~(t
]2_2

180

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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

m tthée secon second

= T)ie 10Illltw, initial va values \lCSof an and must ho ue = Mt, n~conecived to be divided ioto two parts, which satisfy rcspcctivcty the relations (5) and (7). The nrst ccn.stitutcs the wavc which will adva.ncein 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 of the string, the positive and ncgative wavcs sep:L:-atcafter t))0 interval cf time required for each to traverse bulf the disturbod portion.

is thé part initialiy disturber). Suppose, for example, that A point P on the positive side remains at rest nntil thé positive wave has travelled ft-omA to P, is disturbed during thé ngo of 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 <3on thé negative si deof~4Z?. If thé character of thé original disturbance he such tha.t f<.c o a< vanishesinItiaUy. tho-~

144.]

AND NEGATIVE WAVES. 18L POSITIVE

is no positive wfi.ve,and thé point P is never disturbed at all; vanisti initially, there Is no négative and if + wave. If C) t Cf~ (tM< 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 cvanesccncemay 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, may he readHy traced by graphical mcthod.s. Sinco tllC positive aud the négative wavcs are equaL it is on)y ncccssary to dividc t)iu original disturbance into two equal parts, to ()i.spi:(ccthèse, 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 weendeavour to satisfy

Ly iissuming y=~Y', AvhoreJ~ dénotes a. function of a? on!y,and a. function of t ou[y, wc H)id

sothat

1 ~T Y'=A'=~

Id~Y ~constant),

proving that thé vib:).tio)is must bo simple 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:).lkind of stationary vibration may be regarded as due to the superposition ci' cqual progressive vibra-

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VIBRATIONS

0F STRINGS.

[145.

tions, whoso directions of propagation arc opposcd. Converscly, two stationary vibrations may eotnbinc into a progressive one. Thé solution in tlic ~=/(~)+~~+~ instance to an infinite string, but may bc interprcted so as to give thc solution of thc prol)le.n fur a nnitc string in certain ca~cs. Let us .suppose, for cxamp!c, tl~t tlœ string terminâtes at ~=0, a)id is he!d fast thcrc, whilc it cxtcmLsto inHnity in thé positive direction on)y. Nuw so long as thc point .-c= 0 Mtua]!y rcmains at rcst, it is a mattcr of indinfcrcncowhcther the string Le prulongcd on t!to ncg~Ive hidc or not. Wc arc 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]acGmcnts a.nd vetocities on tlie ucgativc side can hc taken, so that on thc wltolo t!)crc shidi ho no dispfaccmcnt ~=U throughout t!ic subséquent motion. Titc initial values ofy and y on thc positive snic détermine thc corrc.spondingparts of t!.c positive and négative wa.vc.s,into which wc kuowthat thc whulc mution can bc resolvcd. Thc former bas no influence at thc point .7-=0. On thé négative S)de thc positive and thé négative waves are hntiaHy at our disposa!, but with thc latto- we arc not concerned. TI.c problem is to détermine thé positive wave ou thé négative .side, so that in conjunct.ion with tlie givcn négative wave on t).c positive side of tlic origin, it sh:dt Icavu that point undisturbed. Lct bc thc line (of any form) i-rcscnting thé wavc m M~ wluch advanees in ttie négative diruetiou. It is

evident that thé reqmrcmcnts of thc case arc met by taidng on thc uthur side cf 0 what may be caUcd t!)c cû?!
REELECTION

145.]

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-ntycqual and opposite, and tlicreforc when conipoutidcd give zero rcsultant displacomont. Thc efïcct of the coustrahit at 0 may tttcrcforc bo reprcsented. by supposing t!):T,tthé négative wavc tnoves through undisturbed, but that a positivo wnvc n.t tho s~mc timc émerges from (9. This l'cfL'ctcd w~vu may a,t auy timc be fouud from its pa.rcut by tbe iulluwing rule Lcit ~7bo 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~Yaud 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(~)=0 fur

al!

positive

values

of

or

/(-~)=-F(.) fur positive values of z. Thc functions tc'nnincd ibr aH positive values of and

and .F are thus dc-

Thcrc is now no difnculty in tracixg thc course ofcvcnts wbcn ~o points of thc strmg -/i and J? are hctd fast. Thc initial disturbance in ~17~dividcs itself Into positi.vc and négative wavcs, which are l'cnuctcd backwards and furwards bctwucii tlic nxed

184

TKANSVERSEVIBRATIONS0F STUINGS.

['145.

points, ch~nging their character from positive to negative, and vice M)' at each renection. Aftcr an even numhf~ pf )~ tions in each case tite original ibrm and motion is recovcrcd. The proccss is most casiiy followej in compictely when thc iniLiaI Ji.sturb.mcc is connncd to a .sma!l imagination part of the iitring, more particularly when its charactcr is suc!t as to give rise to a wave propagatcd in ouc direction on!y. The ~«~ travels with uniform velocity (f()to and fro along thé Icngth of the .string, and after it has rcturned ? ~eco?~ time to its starting point' the original condition of things is exa.et]y rcstorcd. The period of thé motion is thus tha time requircd for the pulse to traverse the length of the striug twicc, or

Thc s~unciaw cvidcntty ho]ds good wlmtcver may be the character of the original disturbnncc, only in tlie gcnera! case it may that thé s/io?'~ period of récurrence is some aliquot part happen of T. 14G. TItc metliod of the !a.5tfcw sections may bo advantageons!y applied to thc case of a plucked string. Since the initial velocity vanishes, haïf of tho displacemcnt belongs to the positive and haïf to thé negative wavo. The ma.nner in which thé wave must be complotcd so as to produce the same effect as tlie constraint, is shewn in thé figure, wliere thé uppcr curve rresents

tho positive, and thc lower the negative wave in their initial positions. In order to find the connguration of the string at any

146.]

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(~ualto at. TI)Gresulta.nt curve, like its components, is made up of stra.ight pièces. A successionof six at intervals of a,twcifth of thé period,

shewiug tho courseof thé vibration, is given in the figure (FIg. 27), taken from Helmholtz. From the string goes back aga.Into il through thc same stages'. It will be observed that thé inclination of thé string at tho points of 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 similardisturban ces are communicated to thc string at intervals of time r at points whosc distances frorn 0 incrcase each time by ctSï~ then 1 Thismothod. of troittUf;thovibrationof a plackodstringis duoto Yonng. fftmi!iar withit to mnkcIdmself J~tt!.2'Mf~ 1800. Thostudcntis Tecommonded théformsofFig.27. byactuaHyconstructiug

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.sedfru.n T to T + 8r. This rcmarkcontaiu.s tlic of t!)tjoiy t)to altoi-atiun of pitcit duc to motion of tixj s(n)i-ceof disturbance; a subjcct w)uc)t will corne uuder our notice a~aiu in connecti'Jtiwith acrial vibrations. 148. AVitenonc point cf an innnitc string i.ssubjcct to a forccd vibration, trains of wavcs procccd frorn it ill bot!)'directions acoord.n~to hws, wf.ic)i arc roadity invost.i~ated. Wc shall snjipo.sc tb~ thc or~in is thc point of excitation, thé string bcing thcro subject to t!~ forccd motion y=~ and it will bu suniciunt to con.sidcrtl.c positive sido. If tbc motion of cach dc.ncnt bo resistcd by t)ic frictional force tlie dinTercutialéquation is

WAVES. 187 0FPROGRESSIVE 148.] DAMPIN& If wc supposethat /<:is small,

:uid This snhttion sttcws thft-tthcrc 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 snnph harjnonic force) must bc synchrouous in ptiase tbrougbont. According to (9), on tho contrM'y,t)iu phase varies cuntiuuuusty in in~ ft'omune 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 Xin (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.tcthé 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 problemmay be soivedfor a. string whosemass is supposed to be concuntratcd at C(p)idistantpoints, 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 maya.11 be satisned by taking whcrc is a comptex constant dctemuucd by a quadratic cquatn)!i. Thc result for a. cuntmuous string )nn-ybc afterwards deduecd.

CIIAPTER VII. LONGITUDINAL ANDTORSIONAL VIBRATIONS 0F BARS. 1-tJ. Tin, next sysL.m to thé string in order of simnjicity is tlie bar, by winch terni is usuaUy undcrstood in Acoustics a of natter of uniform substance and mass orm. At t!ic c..ds thu cylinder is eut oH' c)ongatud cytindrical to tlie gc~-atu~ lincs. Tho centres of by p]anes pcrpcndicuJar 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 throe kind.s-iongitndina! the ~-rorJ: but ~T~'at t.c same time the most diHicu)t in tt.cory. Tbcy are considered by thc.n.sctvcsm thc next chapter, and will on]y bc rcferrcd to hcrcsofarasis ncce.ssaryfor and contrast W)th thc othcr two ktnd.s of vibrations. co~parison Long.tndu.at votions arc those in which thc axis romains nnmoved. whde t)~ transversc sections vibrato to and fro in the direction pcrpendieuL-u.to their planes. Thc moving powcr is tho r~stancc o~red by thc rod to extension or compression. OuccuH~ityofthIs class of vibrations I.s at once évident Since the force neccssary to produce a given extension in a bar is proportional to tho area of the section. ~hHe thé ,na.ss to be moved a!so in the same proportion, it fo)!ows t!mt for a bar of given length and inatcrial ti.epcriodic tunes and the modes of vibration arc ~dépendent of thé area and of tlie for.n of thé traverse sect.on. A.sinufar law obtain.s, as we shaU prcsentty in tite case ot torsionat -sce, vibrations.

Itisothcrwiscwhen the vibrations arc latéral. Thc pcriodic tunes are mdecd i.~ependent of t!.e thickness of tbc bar in thé direction 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.uicdby n risc of pitch. In thc case of Iongi.tudiun.1 and latéral vibradons, ttic mcchanical consta.uts coticcrncda.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,accordingto w!dch thc tension v:n'icsa.sthù extension, aetnid Icngth nntural ]ongth hoids °good. Tf If ,i tLe extension, viz. -n ",i– nittundtengtn1 bc callecl c, we liave y=~, whet'o is Young'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 forcewhiehwould ha-veto bc appHcd to a bar of unit section, in oi'dcrto doub]c its length, if Hooke's law contiuncd to hold good for so grea.t extenthose of a forcedivided by an sions; its dimensionsare a.ccol'd.ing~y area. The torsional vibrations depend aiso on a second clastic constant 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 longitudinal or torsional vibrations withont tlie accompanimentof 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 classesof vibrations independently, cominencingwith longitudinal vibrations, which will in fact raise no mathematical questions beyond those aiready clisposeclof 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 in su~-ha manner thaï. thé ~<) of v<Juiuo I~ss than if thé déplacement of cvcry particlc wcrc paraHc!to thc axis. In the case of a short rod andof a partiel situated ncar tlic cyliudrical this L~tcral motion would bu comp~ble iu bonndary, jn~nitudc with thu longitudinat motion, and eou]d not bc ovorlookcd without risk of considérable crror. But where a rod, whosc Io)gth is grc~t in proportion to tho lincM-dimensions of its.section, is subjcct to a strctching of onc sigu tliroughont, t)ic longitudinal motion accumulâtes, and thus In thé caso of ordinary rods vibratmg lon.din thc tu()in:d!y graver modes, thc inertia of thc hter~motion may bc negicctcd. Morcover wo shall sco lator how a correction may bc introducud, if necessa.ry. Lct bc thé distance of thé hyer of particles composino-any section from thc cqnHibrium position of onc end, whc.i thc~-od is unstrctchcd, cithcr by pcrtnancnt tension or as thc rusuit of bc t]ie
Considcr now the forces acting on thc s)ice boundcd a: by 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 sHcc due to thé action of thé ading parts is on Die whutc

Tho mass ofthc clément is If p bc tl.e original density and thcreforc if ~be titc acce!erating force acting on it.thc equa-~

DIFFERENTIAL EQUATION.191 150.] GENERAL tion of oquilibrium is

In what foHowswc shaHnot rcquirc to cojtsidcr thc opération of:ui itnprcsscd force. To find thé équation of motion wc hâve aud ouly to replace by thé réaction ~gaiust accélération 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 of any type in ttie positive and négative directions. Ttie velocity
1 Centimètre, Gramme, Second. Tliis Systemis recommeuded by n. Coinnutteo of tboDritiBliAasociatiûu. Brit. Ase. Report, 1873.

102

LONGITUDINAL

VIBRATIONS

OJ BARS.

fiS].

velocity of -~el is cent.n.etre.sper second, or about 1G ti~es grc.ter ~ut. 0,000 Lhc L samc as "T in .stec]. It ought to bc nicntioncd th
'<~tud;na]

~=7(~-a<)+~(~+~), bcing the same as t~t appHc~Ie to a string, need not be furti~r eonsidered hcre. Whcn both ends of a bar are fre~ titere is of course no pcrminent tcns.on, and at the ends the.n.sdvc.s titerc is no tcn~rv tension. Thoconditiouforaf.-cec.ndisthcrefore

~=0. le doter~nc t)~ nor.ual n,odc.sof vibration, wo must assume vanes as a harmonie function of tho timc-cos7i~ th~t~ TI~n as a function of .r, ,nust satisfy

Now

sinco~vanishcs

again smce

al~ys ~hcn ~=0, we get j3=0;

an
u.turat iungth of tlic g vanishcs ~~cn ~=/-thc bar, sin ~~=0, wbich sl~cws tih.Lt is oftiie form

t'bcingiutc'graj.

152.]

BOTHEXTREMITIES FREE.

193

Accordingty, the normal modes arc given by équations of thc form

in which of course au urbitrury constant may bc nddcd to if <
whcrc and arc arbitrary constants, wliieh may bc detcrmincd in the usu:d mauncr, whun thc Iuiti:d values arc of aud givcu. A zcro vainc of i is issible it gives a term rrcsentmg 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 which is thc tinlc occuhiedby a. disturhanee in t=I, is 2~ travelling twice the Icngth of t)io rod. The other toncs fonnd Ly ascribing integral values to i form a complète harmonie scale i so 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 comf~ is thcrc a maximum.

pression

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 displacements a.ud velocitius ma.ya.!waysbo 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 displacements 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 by tl)c symmr-try rcm!).inat rest throughout t)ie motion, and thcn thc R. 13

194

LONGITUDINAL VIBRATIONS 0F BARS.

H 5 3.

portion from 0 to s~tisnes all tl.e required conditions. We conc udc that the vibrations of a bar frce at one end and fixed at tho othcr arc ~dcnfj.d ~L .h. of bar uf twice the longth of whichboth ends arc free, thc latter vibrating only n unevcn modes, obtained by in successiona))~ ~king i~~ T!~ tones of tlie bar still te a hannomc ~.h, b~' belong cvcn toncs (octave, &c.of thé fund.menta!) are .vauting. Thé period of the gravest tone is thé time occupicd by a pulse in travelling/b~ timcs the length of thé bar. 154. ~)cn both ends of a bar are fixed, the con(litions to be satisfied that the value ~t~ At K-0, we may that suppose ~=0 At & is a small1 constant .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 most simply bt~M' this tenu at once. But it may method. that as a fonction ofthe Assuming timc ~COS7!f~+ 7?sinK~,

varies as

we sec that as a function of x it must satisfy

of whichthe général solution is J

But since vanishcs with x for ~11 values of t, <. rC'=0, n nnd ~) thus we may write

BOTH EXTREMITIES

154.]

FIXED.

1!)5

The series of tones form a, compote harmonie scalc (ft-on which ])owcvcr any of the mcmbo-s may bo mi.ssiug ni any actua! case of vibration), and tho period of the gravest componcnt is the tinic takcn by a pulse to travc! twice tho Icnn-t.b of thé rod, thc sa)nc thcroforo as if both ends wcre frec. It nnist 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 length is ludependent of the permanent tension. The solution of the problcm of the doubly fixed bar in the case of no permanent 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 the necessary diËfereutial

équation, viz.

masmuch ns

(I' ~.E~ d satis~cs

= a2

(le <

and at both ends

vanishcs. According!y iu this problem (lx dx satisfies ail tl~e conditions prcscribed for in the caso when both ends arc ~xcd. The two séries of toncs are thus identicul. 155. Thc effect of a small ioad ~f attac])cd to any point of the rod is rcadi]y ca!cu)ated approxnnatc!y, as it is sufncient to assume thc type of vibration to bc uuaitcrcd (§ 88). \Vc will takc the case of a rod nxed at .~=0, and free at .t'= The kinetic cncrgy is proportinnal to

or to

~G

LONGITUDINALVIBRATIONSOFDARS.

[155.

Since the potentia.1 cncrgy is uudtcred, we sec by t!ic prmciples of Chapter iv., th~t tho cfrcct of ti~) sma!) !o~ at a 'h~ucc &: u'u:ti Lhe iixcd cud is to inci-cMCthe period of' tho compouent toucs in thc mtiu

Tho snrnH quantity wholo mass of thé rod.

p~ is thc ratio of tlic !o:tdto t!ic

Iftheload

bcatt~chcd at thc frec end, and tlic sm'~=l, effect is to drcss the piteli of cvery tone by t!ic s~mc small mtcn'd. It will bc rcmembei-udt!)at i is hei-can MMC~t mtcgcr. If the point of .~chmcut of J!f bo nodc of any componcnt, tlie pitch of that eojupouontronams uualtcrcd by thé addition. 150. Another problem worth notice occurs whcn t]to load ~t thc frcc end is grc~t iu compiu-isoi with thc masa of thc rod. In th)3 CMCwo inay assume as thc type of vibration, a. condition of Utntbrmextension 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 cquivalent to t)ic édition to ~ofone-third of thc mass of the rod. 1.~7. Our mathem~tic~ discussion of JongitudinfLlvibrations close with an estimatc of thc cn-or invo!vcd in nmy ncgieetinrr tttc latc.ral n.ut.ou of thc of thé rod net situatcd on tlic parts

157.]

CORRECTION

FOR LATERAL

MOTION.

197

~xis. If the ratio of latéral contraction to longitudinni cxtoisinn denutcd bj. i,injhuerai disp~acoment of a. particie distant ?-from thc axis will bc ~re, in the case of cquilibnum, where e is the cxtcMsion. Altiiougli in strictiless this relation will bc modiiicd by tho Incrtia of thc httcl-al motion, yct for thc présent purpose it may bc supposed to hold good. TIic constant /t is uumerical quantity, lying between 0 and If/~worc ucgativc, a longitudinal tension would produce a latéral were greater tbau swelting, and if tlie lateral contraction would bc grcat oicugh to ovcrbalance thé elongatiou, and cause a diminution of volume on the whole. Thé latter statc of thin~ would be mconsistclit witli stability, aud tlie former can scarccîy be possible in ordinary solids. At one time it was supposed was nccessarily equal to that 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 r from the axis, and if thc section bo circular, thé kinetic encrgy duc to t]ie lateral motion is

Thc effect of tlie incrtia of thé latcra! motion is thct'cfoi-c
This correction will bc nearly insensible for tlie graver modes of bars of oi-dinaryproportions 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 cxcitcd by friction, with a wet doth in the case of glass; but for métal or wooden rods it is neccssa.ryto 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 obliqucJy across the string, and moving it aiong thc string loogitndina.Hy, keing tire same point of thc bow unon thé strirtg. Thc note is unpjcasn.ntiysin'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.ivibrations )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-transniission 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 vibration. According to Terquem" thc cause of this sound is a transverse 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 phenomenon wuld 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 proDoukin'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:~telto the axis ofthe rod. Thé force with which twisting is resisted depends upon an clitstic constant different from q, ca.Hed thé rigidity. If we denote it by n, tlie relation betweenq, m,a.nd may be written

shewing that n lies betwcen M=~.

and

In the case of ~=~

Lot us now suppose that we hâve to do with a.rod in the form of 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 material composing the pipe by r~. umtof area

is~ round the axis is

and thé shear of the The opposing force per

and since thc area is 27n-~ the moment

Since this is independent of r, the same equation appUcato a cyliader of fmitc thickness or to one solld throughout. The velocity of wa,vepropagiitionla

and the wholo thoory

A/ is prccisely similar to that of longitudmal vibrations, the condition

TIiornson aodTait.§683. This,it ahouldbo remarkcJ.appliesto inotropic mntcria!on]y.

200

LONGITUDINAL

7/]

VIBRATIONS

0F

BARS.

for a free end bcing and for a fixed end 'C = 0, permanent twist be coutempla.tcd, = constant.

[159.

= 0, or, if a

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! Icngth vibra.ting in corresponding modes under corrcsponding terminât conditions. If == the ratio of frequencies would bc :=~/8 :3=1'G3, corresponding to an interval ratitcr grcatcr than a nftb. In any case tbc ratio of frcqucncicsmust lie between V2 1 = 1-414, aud~/3 1 = 1-732. Longitudinal and torsional vibrations were nrstinvcsti"-atcd bv Cbladni.

CHAPTER

VIII.

LATERALVIBRATIONS0F 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 of strings, this class of vibrations is perhaps tlie most amenable to thcoretical and expérimental treatment. Thcre is dimculty sufncieut to bring into prommenco somc important points connected with thé gênerai theory, which thé familiarity of thé reader with circular functions may lead him to over too Hghtiy in thé application to strings; while at the same time the difficultiesofanalysis 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 complication, and is of interest only in its application to stretchcd strings, whose stiffiiess, though small, cannot bc neglecteù altogether. 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. t. xnt. /<M<J'<'
202

LATERAL VIBRATIONS 0F BARS.

[161.

WIth respect to thc section of thc rod, wc sha)! suppose that one principal axis lies m thc phuic of vibration, so that t!ic at cvcry part takcs p!acc iu a direction of maximum or bendinomiuimm~ (01 st,atiûnary) rtcxunUri~idity. For uxample, thc surface of thc rod may bc onc of révolution, cach section bdug circular, thoun-h not ncccssarilyof constant radius. Under t!icsc circumstances thc potentiaJ cncrgy of thé bending for each clément of lungth is proportional to the square of thc curvaturc multiplied by a qnantity dcnding on thc matcriid of t))e rod, and on thc moment of inertia of thc transvcrsc section about an axis t))rough its centre of inertia pcrpeudicuhu-to tlie plane of bending. Jf be thc area ofthe section, its tnomcut of inertia,~ Young's moduius,~ thé clonent of icugth, and ~F' t)ic corrcspouding poteutial energy for a curvature 1 of tlie axis of the rod,

This resuit is readily obtained by coDsidermgthe extension of thé varions filaments of whicli the bar may Le supposed to bo made up. Lot be tlie distance from the axis of thc on thé piano of bending of a nl&ment of section ~M. projection TIien thc length of the niament is altered by the bending in thé ratio

-Kbeing thé radius of curvature. Thus on thé side of thc axis for whidi~ 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 necessaryto produce thé extension by the deûiiiis (~ tion of Young'smodulus; and thus thé whole couple by which thé is resisted amounts to bending

if &)bc thé area of thé section and < its radius of gyration about a Imc through tlie axis, and perpendicular to the plane of bending. The angle of bcuding correspondingto a length of axis ds is aud thus the work rcquired to bend o~ to curvature 1 Ti!!q c ~t siucc thé Mea?;is hdfthc~~

value of tlie couple.

161.]

POTENTIAL ENERGY OF BENDINQ.

203

For a circular section ? is onc-ha.Ift!te radius. Th~t thé potential Ct'icrgyofthcbcndingwoutd bc proportionn.1, cc~e?'tN ~ft~tfs, to the squ:).roof thc cut-Yidure,is évident bcforehand. If wc en!!tho couificiotitJ9, 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 position, 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 vibration. 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 Thé square of this therefore its angu!a.r vclocity ",a.nd (~uantitymust bc multiplied by haïf t!ie moment of inertia. of tho We thus obtain clement, tliat is, by ~m <

1G2. In ordcr to form thc equation of motion we may avail ourselvcs of tlie principle of virtual velocities. If for simplicity we confineourscivcs to the case of uniform section, we have

204

LATERAL

VIBRATIONS

0F

BARS.

[162.

where thé free from the Intégrât sign are to be t~cn hetwœu thé I.nuts. This expression inctu.ics ouiy t])e internat forces due to tlie ben
Thus tlie variational équation of motion is

in which thé tcrms free from the between thé limits. From this we mte~-al sign arc to h.. takcn Jer~edatallpo~ofthel~t~f~~

..J~. tud

Jongi

inl11

lVilVl'S.

1G2.]]

TERMINALCONDITIONS.

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 ( ")

8vhas

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 constrained 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 ed, thé conditionsare

by which (5) is satisfied. A third case crises w!)cu an cxtrcmlty is constramcJ to maintttin its direction by a.n applied couple of the necessary magnitude, but is free to take any position. We ha-vcthcn

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

206

LATERAL VIBRATIONS OF DARS.

[162.

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 cd, 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 cd. In order to Interfère with the vibration as little as possible,thé s must be connned to thé ncighbom'hood of thé nodal points. It is sometimcs surHcicntmcrely to !ay thé bar on bridges, or to 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 'ed,' it may be pressed into with a fixed plate whoso plane is perpendicular to the longth of the bar. 1G3. Before procccding fnrthcr we shall introducc a supposition, which will greatly simplify thc analysis, without set-iolisly intcrfcring with thé value of tlie solution. We sliall assume that thé depending on thé anguhu' 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 inertia, 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 issible, a.nd for cach ?~ thé coiTcspondingicis determincd in everything except a constant multiplier. Wc shall distinguish the different functions u belongiug 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 coordiDates,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 It is however rcquircd in order to determine than a Mr(/?ce[
308

LATERAL VIBRATIONS 0F BARS.

[164.

M, <)cnotctwo of thc nonna.1functious correspOQdingrespectivciy to ?~ and ?/ Thcn IC't. Lot

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,ctt, or free', eacli terni vanishcs on of one or other of its factors. We may therefore couchtdc that, if M~,?< rcfcr to two modes of vibration (corrcspondingof course to thé same terminal conditions) of -wincha rod is capable, then

providcd

?)t and Mt' bc (ont.

The

attentive rcader will perçoive that in the proccss mst foiïowcd, we ha.vc in fact rctraecd thc stops by w)nch t))c fnndamcntd diiïcrcntial équation was itsctf provcd iu § 1G2. It is the Tho ronder ahonitl obscrvn t)nit tho eftscs hcro RpoeificdMGpa.rticuJn.r andV thnt tho right-hand monbt'r of (;!)Vtuushcs,provided t.)t)tt <~ ~j t:Llll

<~

~m<<j:

~.E

< f/

Thoso conditions incindo, for ittstfmco,tho Ctlaoof n rod whofiOend is urpod towm'Jsits position of L'quili)')-iu)uLyn terce pr~portional to thc dispUtcc'ment,as by n spring witimut inertia.

209

CONJUGATE PROPERTY.

164.]

originat MM'M!<MK<ïi! cquatio!) that ha.s the most nnmcdin.te conncctiun with tho conjug~tc propcrty. If we dcnotc by Maud Sy by~.

and this proof is cvidentiy as direct and général as cou)d bc dusired. TIie reader may investigate the formula corresponding to (6), whcn thé term l'rescnting tlie rotatory inertie is retnined. By !ne!msof (G)we m!iy thn.t tlie issible 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 Idcntica.lly truc, and wc cannot at once Infcr the value of n~ Thismcthofl is,I bdicvo,t!notoroisson. R.

14

210

LATERAL VIBRATIONS 0FBARS. [164.

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, ed, 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 condition at x =0. If thé end œ= b& free, M"and u"' vanish, and accordingly

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, ed, or free.

1G4.]

VALUES0F INTEGRA/FED SQUARES.

Ag-a.in,if wc suppose that thé rod ia c){nnpcd ~t vanisL, a)id (8) givc's

==

211 n.ndM

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. TIie a.!i!)cxedtable gives t)ic vahiesof four times thé mea.nof M* in thc différent cases. M°(ft'ccend),ot'M"'(cIu.mpedend) c!tunped,frpf. M'(ft-Rccnd) free,ft'eo M' (clampcd end) clf~mped,c]!i)npcd 2~ cd, ed (cd end) = 2~" M"(freo end), or -2M'M'" (ed end) ed, ft-eo snpported, chmpL-d M"' (damped end), or 2M'M'"(ed end)

By thé introduction of these values thé expression for T assumes a. simpler form. In thé case, for example, of a clampedfree or a frec-frec rod,

where the end
A similar method may be applied to investigate thé

values of

and In the derivation of equation (7) of the jM"~c, preceding section nothing WM assumed beyond thé truth of thé equation M""=M, and since this équation is equally true of any of thé derived functions, we are at liberty to replace Mby M' or u". Thus

14–2

212

LATERAL

VIBRATIONS

ta.Iœnbetween tlie limits, since tlie tcrm c~es.

OF BARS.

[1G5.

M' vanishcs in ail threo

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.ndfree at

We ])a.vcah'cadysccn that ~"=~ M"'=–u/, so that

=l

a.ndit. will appca.r (§ 173) t)):).t

Il rcsult thf),twc sliall have occusionto use latcr. By n.pplyingthc same équation to tlie cva.inn.tionof ~M' find

wc

sinco M'u"a.cd Mu'" vanish. Comparing tins with (8) § 1G4-,wc secthat

whatcvcr tlie termina! conditions may hc. Tho samc result maybc arnvud at more dirccMyby intcgmting by pa.rtsthé equation

213

NORMAL EQUATIONS.

1CG.]

16G. We may now form thé expression for V in tcrms of tlie normal co-ordina.tcs.

If sion

thc

fonctions

:t

bc

thosc

propcr

to

:). rod frce

nt ~=

t)us

expres-

reducesto

In any case thé équations of motionarc of tlie form

and, since ~~t is by définition the work done by the Imprcsscd furcusduring tlie dispin.ccmcnt8~

if YpM~<;Le thc lateral force actmg on thceictncnt of mass pax~c. If thcre be no impresscd forces,the cquatiou reduces to ~+

n.swc know it ought to do. 1C7. Thé signidc~ucc of the réduction of the Intégrais te doidcnco 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~hcditi Chaptcr iV. (§ SH). wc ma.y calcntatc tlic period

214

LATERAL

VIBRATIONS

0F BARS.

[167.

under thé altcred circumstances wititout aUowance for the change of type, if we are content to ncglect tbe square of thc change. In conséquence of thé strajghtness of thc rod :).tthc place where the addition is made, therc is no altération in thé potentiel energy, and therefore the altoration uf period dépends eiitirely on 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 period is in the ratio

A con)p:u'isouuf 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 intégral is susceptible. Other cases in winch sucli Intégra]~ occur tnay be treated in a sunitar manncr, but it would often require care to predict with certainty what atnonnt of discontnunty in the varicd type might be itted without iug out of the range uf 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
168.]

INITIAL

CONDITIONS.

315

the conncction of thé arbitrary constants with thé initial circumstiUlces,and solvc oue or two problems analogous to those treated uiidcr thé head of Strings. Thé gcnend value of~ ma.y be written

formuloewhich 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 issible 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

CiUling~~
VIBRATIONS

0F BARS.

[1G8.

thc w!)ule momcntum of tbc blow, Y, wc

If thc blow bc app)ied at a no(te of onc of thc normal componcnts, tha.t conponcnt is missing in thé rcsutting motion. Tlie prcsunt ca.)cu!atiun is but a. pai'Licular c~c of thu investigation uf§101. ICf). ~a another examplG we may take the case of a bar, which is initially at rcst but dcHected from its natural position hy a latéral force acting at .'c=c. Undor thèse circumstances the coefficientsB 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.]

SPECIAL CASES.

217

same tcrminal conditions <mdocs and thus a.ll thèse tcrms vanisli a.t both limits. If tlie external force initially applied bc yi~c, thé cq~a.tion of equilibrium ci' tlie to thc .cicmeut har tri vos

If wc now suppose thn.t thc initia dispiaccmcnt is 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 thé ends, but any special assumptiuns as to the couttitions at if we nowconfine curscivus to ttte case of fiba.t'which is c!cnnpcd at a;= 0 aud irec at x = l, ve may replace

Ifwc suppose furthcr that the force to whk-h thc Initial dcHcetio)) is duc acts at thé end, so that c= wc get

Whcn t=0, this cquadon nuist represent t)'c initiid dispjacc:tncnt. Iti cases of this Idnd di~culty tnay pi-cscnb itMu[fas to !)0\v it is possible for the series, cvery terni of which satisfics thé condition y"' = 0, to rrcs~nt au initial displacement Iti which tins condition is violated. Thé iact is, that after triple tlic series no longer converges diH'crentiation wiHt respect to for a~, and accurdingly the value of y" is not to be ~rrived at hy making the diHerentia.tioas first and summing the

218

LATERAL

VIBRATIONS

0F

BARS.

[1G9.

aftcrwards. Thé truth of tins Rta.tement will be a.ppa,rent if
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

coshx and sinh x arc tlie hypcrbollccosineand sine of x, defined by thé équations 1 hâve foltowed thc usual notation, though thc introduction of a special symbol might vcry weHbe dispcnsed with, since cosha'==cos/.c,

sinha;=-.t'siu

.(3) w]~t-et== y- 1, and then tI)Gconncctioti between tlie formuh~ of circular and hypcrbo)ic tngttnoniet.rywou)dLe moi-c apparent. Thé ruics for diiTurcntiationarc cxprcs.sedin ttic cquatiuus

In diiïcrentlating (1) any number oftimes, the same four compound funetions as thcrc occur are contmuaUy reproduced. Thé on!y one of them which does not vanish with is cos a;' + cosh ic, wbosevalue 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 issiblevalues of ?/t. C given in (5) ~rc e(tual, If (7) be satisfied,tlie two ratios of :m(leither of titcm ma.ybe substituted ia (4-). The constaut multifunction plier bciug omittud, wc have for thc !iorm:d

171. The frcqucncy of thé vibration is ~M~, in whiuh &is a velocity dcnding only on the material of which thé bar is material formcd, a-ndHt.is an abstract number. Hence for a given 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 VIBRATIONS0F DARS.

fl~I.

plane of bending–and iin'erseiy as thc square of the Jcngtit. Thcsc rusults might hâve bccu anticip~tcd by thc argument f'rumdimcnStons, tf if worc considérât that t))C frcqnency is Mcccssari)y dctci-tninc-dLy tl.e v.d)!e of togeUtcr widt th~t of ~–tho uuly qu.mtity (]endh)g on sj~cc, timc and mass, which occurs in t)to diH'crcntiidcqn~ion. If cvcrytitingeonccnnng a bar be given, cxcept its absolutc m:~It.udc, tiic frettuency vanes as invcrscly thc iincar dimension. Thcsc I~wsfind an Important application In thc case of tuning furks, w))osrongs vibratc fm rods, Hxcd at t))c ends wherc thcy juin the stal!~ and frce at tiie othcr cuds. Thusthc pcriod uf vibration of furks of t.hc samo tnatcrinl and shapc vancs as thu lincar dimension. Thé period will Le approximatdy indcndent of thé thickncss po-pG!)dicn)arto thé plane of bending, but will vary invcr.scly with thc thickness in thc plane of bcnding. WIien thc tliiekncss is givcn, tlic penod is as thc square!of t]ie length. In ordcr to ]owcr thé pitch of a fork we jnay, for tonporary purposcs, load thc cnd.s of thé prongs witli soft wax, or file away thc métal near thu base, thcrcby wcakcnitig thc sprinn'. To raisu thc pitch, thc cuds of titû prongs, which act by inertie may bc filod. Thc value of b attains its maximum in tho case of steel, for which it amouuts to about 5237 mutt-cs per second. For b'mss t)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 pitch th~ti ii'thc miLtcrialwere stcd. 172. TIie solution for the ense w!)cn buth ends arc dampcd bc may un!ncdi)ttu]ydcrived from thc prcccding by a double dif~rcutiation. Since y satisnc~ at both ends tiic terminal conditions

~'hich arc thc conditions for :Lchunpcd end. I~torcovcrt))C gênerai dtRerctit.iid eqtmtiot) is a]so s~tisticd hy y". Titu.'jwo may talœ, oniitting cunst:u]t multiplia-, as bcfur~,

173.]

NORMALFUNCTIONSFOR CLAMPED-FREEBAR. 221

while 7Kis 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 by the evanesccncu of amt its dcrivativcs. WIien vanislies, thore is a nodc ~hcrc ~anisites,a. loop, or place of maximum displaccmcnt wlicro y" vnnis!)RH,n. point of inftection and whcru Vimishes,:). place uf 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 correspondto nodcs aud loops ofa rod whoso ends arc free. 173. We will nowconsider 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 =~ The

remaining

(cos;r'

+ D

cosh~) at

conditions

(sin

.r

–sinh

~')

.(1).

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 ") sin u (sin ne + –cosh = (smsinh?~) M+ t (~cost J (cos?H+ or

( cosh?~wsm

i "~1 –srnii

(2),

i ~~) -;cos( t cosh tJ Ma: ( smm-Sllll1n + –sinh ??!.v].(3), +(smm- smhw~sm 6 J wherc M:must bc a root oflf M= (cos Mt+ cosh

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 rchttcd to, thosu of a rod both wtiose ends are clamped, or frcc.

222

LATERAL

VIBRATIONS

OF BARS.

[173.

If thé value of !( in (2) or (3) be diftcrentiatcd twice, the rcsult (!) satisfies uf course t!ic fundamcntal diffcrential cqua.tion. At .u=0, ,t", but nt.<;=~ M"a.nd-rva.nish. < ~.c")i"ani.sh, ~.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.1curve :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 hyperLoHccosine nr its !ogarithin, thé issible vaincs of M may 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 au intcgcr. If we assume

;8 will bo positive and comp~ra.tivclysmall in magnitude. Substituting in (1), wc find y

eot~=~=~

an équation which may bc solvedby successiveapproximation aftcr in ascending powers of thé small cxpamHng tan~ and e <;uantity/3. The result is

which is sufficiently accur~te, cven whcn t= I. By cn.lcula.tion. /3, = -OI79CGG -0003228+ 0000082

-0000002 = -017C518.

arc found still more easily. Aftcr ~t. ~g, 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.

CALCULATION OF PERIODS.

174.]

223

table contains the value of ~3,thé angle whosocircu]n.rmeasurc is /3, and tlie value of siu ~/3, whichwillbe required fm'ther on. -F'ee-Free j~~)'. in dcgroea, oxproHsed minutcH.tmdtiCcondt).

.q 1 2 3 4 5

10'' x -17G518 10-777010 10-" x-335505 10-'x-144989 10-'x-C2C55G

Thé values of

1" 0' 40"-9.t 2'40"-2G99 C"-92029 -2MOG2 -0129237

~2'

~3

10-' x -88258 10-'x-38850 10-1G775 10-" x-72494 10"'x-31328

which satisfy (1) are

M, = 4-7123890 + /3, = M, = 7-8539816 /3, = = 10-9055743+ = = 14-137~1669 =

4-7300408 7-8532046 10-995C07S 14-1371655

= 17-2787596 +/3, = 17'2787596

e"=cot~=~e~ where however

1

a' = e~

From this it appea-rsthat thé series of values of a is the samo as that of /3, though the corresponding sufHxesare not the same. In fact ~=~, a,=/3,cf~=/3. so that we ha-ve nothing furthcr to calculate than c(~for which however the series (4) is not sufHcientIyconvergent. Thé value Thiaconnexionbetweena and ~3doesnot a.ppeMto havebeenhitheïto noticed.

224

LATERAL

VIBRATIONS

0F

BARS.

[174.

of a, may bu obtaincd by trial andcrror 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)ywill bc givcn prescntty. Thc vaincs of ?~ wtiich satisfy (5), arc M, = 1-57079M+ a, = 1-S7.~04 = 4-C94737 w, = 4'7123890 M, = 7-85M.S1C+ aa = 7'85-)<758 = 10'!)!)55743 = ]0'9!).554.1 7)!, = 14'137tCC9+ a6= 14'137JU8 = 17'278759,1 7M,= 17-278759G aftcr which 7H= ~(2t l)'7r Rensibly. Thc frcqncncics are proportional 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.rilyof 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 cocfHcicuts,we gct

This is for a clatnpcd-frcc rnd.

COMPARISO~

175.]

0F l'J/J'CH.

225

F''om thc known value of S?)~, thé value of )?~mayLe dcrivbd witli tlie aid ofapproximate values of?y~,w, Wc fmd 2~ ~nd

M~

=-0065t7C2J, =

-OOOOO.i-237

~=-000000069 ~=-000000005. whencc

w,='OOG5i.33)0 = '187510;'), n.sbefo)-(\

j~iving

In likc manner, if but.))ends of t))u Lar bc el.nnpcd or free, 9It 1

4

~W.

+

'-Ï~+-=~)('

l

7)r

n

I

.Cc. ~n

(4), (4),

= crmrsct))p ,;Ill)lnl~itioli whct'e of %vliet-e nf entil-setilt, sumDin.tionis Is exclu~~a' :2,tJ¡) J &c, sive ofthc zero value of??{. wliciice whcnce S

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
LATERAL

22G

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 cosM ccsh ?/=!, ccshMt= coscc/?. AlijO Hinh"w = coHii"~t or,

smcc

1=tau2ut = cot' ~3,

is positive,

cot/3

= cot/3.

sit)h~

Thus slM?):–8[nh)~

l–cos!'7rsin/3

cosMt–cosh?~

eus/3 (cos

(COS ces

COStTT Hit)

~/3

ces

cos

~/3

cos

tTr

sin

cos

ï'Tr

+ sin

't'Tr sin

~/3)°

~/3) (COS ~/3 + COS ZTr Si)!

~)

A/3

Wc may thcrcfore take, omittiog the constant nudtiplio',

(~~ 7T i~f/3) SiIl -~+(-])'~ =~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=~+~+7whcrc = cos ï'-n-sin {;); !og7~= !og

± ~,=

~7r+ ~( –1//3J '(2),

~cloge+Iogsin~-logys Mi~ log

e + log cos

log

~/3

}

from which 7~may be ealculated for dUTercutvalues of i an'!

GHAVESTMODEFOR. PREE-PREE BAR.

177.]

227

At thc ecutro of thc bar, = are numcrically and cqual i)i ~irtueof e'" =cot ~3. Whcn i is ~fc~ thcsotcrmsea.ncel. which is cqn:~ to xcrowhcn For.F~weha.vc ~=(-l)'siu~7r, i is evcn, tuni to i 1 whoi i is odd. WItcn is even, t)io'cfoi'c,
sin

(270°+ 1"C'40" '94)

-M" 30' 2()"'4.7} }

h'g 7~ = 2 054231a; + 3-7!)52301 log = 2-054231a; + 1-8494G81, from whicit is calculatcd titc fuDowlng table, giving thc values of Mfor 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.tionformulK. If o, ~,?',N,t be six consccntivc , that intcrmcdiatc between aud r is

228

LATERAL

v

I~

In

VIBRATIONS

7~ I~a

0F

BABS.

?;. at.

[i78.

~c M:~(-5) 7c(')

-000 +-7133200 +-OOG3408+-7070793 !+l.42GC401'+l-G45219 1-45417G '025 -050 -5292548 -0079059 -5581572 1.0953179 I-2G3134 '075 r0721
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 ~1G(i2530

FREE-FREE BAR WITH THREE NODES.

178.]

229

which is thé fraction of tlie whole Icngth by which tlie uodc is distant from thé Huarerend. ?~~A
iC -000 -025 -050 -075 -100 -125 -150 -175 -200 -225

~M(0) -l'OOOO -8040 -G079 -4147 7 -2274 -0~87 + -1175 -2G72 -3973 -5037

X -250 0 -275 -3ÛO -325 -350 -375 -j00 -425 5 -450 -475 -500

M:-ït(0) +-5847 -6374= -6620 -6569 -6245 -5653 -4830 -3805 -2627 7 -1340 -0000

In this table, as in thé prcecding, thé values of !( were calcu]:t.tcddirectiy 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.lculatiugfrom thé above formulœ, wc fiud ~(-1321) =--000076, M(-1322)=+-OOU88Î, \vhen.cex = '132108, agreeiug with the result obta-inedby Strehike. The place of maximum excursion may be found from the derived function. We get ('3083)==+ -00~6077, (.~081)= -0002227, u' (-308373)= 0. whence 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!)ûwnin fib. 2~). Fig.Si).

r~t'tt~M M~/tybtO' )!O~CN.i = 3. 7'~= sm [ (G30"+ G")2)

45" 3"-4G],

= 4-33~ .e + 5-0741.~7, Io~7~ = 4-77.'i!~2 + I-S-~4850. if~-

From t))is ~(())=1'4M24., M(.~)= 1-00570. T!ic positions 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.rthe (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.tas at thu two uUters.

Wu sa.wt))!ttat thc centre of thc bar 7'~:H)d~u'c n)t)]icric:d]y eqn:U. lu thc nci~hbom-hoodof t])c )))i
?!. bcing f),u int~cr. If wu tr~tisf~nn thc ot'igiu to thc centre of thc ru(.),~nd rtacc 7?t by 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.tbctwecn consceutivonodes bcing 2~– (2t+ 1). Tins t))corct,icn.trusult lias bccu verifiud by titû mca.surcmcutsof Strchtkc and Lissajous. F(.'t' mutliods of n.pproxiin:(.tio)inpp)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.dytnoutioued § 160, :uid to ]3on~in'!t ~tcu:cs (p. 194'). 179. Thc ca.lculn.tionsn.rc vcry simitar for tho case of a. bar a.u.d clamped at onu end aud frcc n.t thc uthcr. If ïto: ~'=~+7~+7~ wc hâve in gcncrui

Ift= ], we obtain for thc culculation of tlie gravest vibratioucm'vu

Thèse

givc

ou

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,

232

LA.TEHALVtBRATIONS0F J3ARS.

[170.

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!). ~3

~hm~ 4~

4~-7~)75 –t,

"Thé last row in this table must be understood as meanincr ° 4/-3 that may bc takcn as the distM)ccof thej)' uodc from thé froc (.'nd,cxet 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.

i"' ~tone't-<+2

~i~ ~t'+2

4t+2 2

-3 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.

l~tf)no

No inaecdon

2"tone.

-f)f)OÛ

3'tone

-03

tl tone i~"tone..

2"~ point.

i~ 4t+2

S.9!)!)3 -in-2LI

«t''point.

point,

4.+1 1 ~+~

Exct in thé case of thé extrême nodes (\vh!chhave uo corresponding infieettou-point),thé nodes :md InHection-poiutsalw~ys uceur m close proximity. 180. Ttiu casewhcreonc eud of:). rod is ft-ceand the other s~dous u~t ubcd an ~o~eJ indcndent 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~, For attitc central node that is, with anode in themiddie. y aud v" vanish, winch are precisely thc conditions for a cd end. In hkc nianner the vibrations of a clamped-cd 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 ed. 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 Mand 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. An arbitrary 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 distances between thc uodes. 182. Thé theory of a vibrating rod mn.ybe appHcdto illtistrate tlie gcnera.1principle that thé natura] periods of a system fulfil 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 of curve may be appHcd at its free extrcrnity. The équation thé taken to bc y=-3~+~, and makes Jy and Jy vanish at which satisfics << = 0 throughout, b Ttius, if thc configuration of thé rod at time t be 0, and at ~= (-3~+~)

cos~

thc potcntial cncrgy is by (1) §].61, C~cos~X,

(1), while thé

234

LATERAL

J" kinetic

~ow

VIBRATIONS

OFDARS.

[182. 9

33

n 40 7 2 I.s l7 cncrgy and 1] thas 2] ~sin' ~=~ (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)ypothetical type. Jt is to bc observed tbat thc hypothctic:d type in question violâtes thc terminal condition y" = 0. This circumstancc, however,(tocs not intcrfcro with (hu application of' ti)e pnncipi~ for the assumed typu niny bu :my whicii wouid bu issibie as an unti:d couf~m-atiou but it tends to provcnt a very dose ngrccJnent of pcriods. Wc )nny cxpcct a bottur approxitnatiot), ifwc found our calcuI~tioa on thc cnrvc in whici)thu rod wou)d bc d~flectcd a force by actiug at somo litttc (ti.stancufrutn 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 = (Jy~M~ cos' Ti)C kinetie cno-~y can bc rcadify found by intégration from t))c ~'atuuofy. From

0 to c

y =

:}~

+

i

amt from c to L

y = (c 3.<'), as may bc sccn frutn the considération that yand y' nn)St not sudd<jn)ychange at :c= c. Thércmt)t. is kinctic cnc.rgy=

sin'

+

(~- r) (.' + 3f-)1

\yhcncc "12 70'3.1Ga

`~

(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)c second nornud compuncnt vibration. If' wc takc c =~, ~vo obtaini a result wllich is tw fngh o in the m~hicat scatc by thc intun-a)

182.]

235

LOADEDE~D.

cxprcsscdby thé ra.tio 1 '9977, a.nd is ~ccordinglycxtremdy nea.r t,hetrut)h This cxampio may givc un idca. how uciu'ty thc pci'iod of a. vibr~tin~ systort may bu catcntittcd by simple rncans without thé solution uf diHurcMti:dor tra-uscuudenttdcqu:),tious. Thc type of vibration just cousidered wout
:< V~

.(.<).

Evcn if thc i))Grti!iof tbe bar bc not attogcthcr n~gligibic in with jV,v'c may still tid~ titc saniu typu as tticbasi.sut' eomp:u'isoM :m appruxijaatu c'idcutation

th:Ltis, J/ is to bc incrcascd by n.hont onc quartcr of tLc mass of thu )'od. Mincutitis 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 gutiuraHya.pplic:).b!u!).sa.u a.pj'roxhn~tiun. Thé cn'or will ahv~ysLe on tlie sidu of cstimatm~ ttiu pitch tuo liigL. 183. But thc ncglect of thé rotatory Inortia of ~f could not bcjustiiicd midd' thc ordi)i:u'ycouditi~us of cxpenmoYt. It is as unsyto Im:)ginu,thou~h ~ot to construct,a.c:).scm whie!)tlic inertia of translatioji s))un)(ibL;ncgligIDc in comparisou 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

cquatiou of thc cm'vc of thu b:).ris

~+

and

ut .B=

Then tLe

S3G

LATERALVIBRATIONS0F 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.tionsof motiou are theruforc

whciicc, if z and

vary as cosj;)<,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 oT 3f7~"M ~f Tite ot!)cr value of~ is thon infinite. If ?' lectcd,

bc merc!y sma. so tliat its Iti~Itcrpowcrs may be ueg-

If on the other hand A:"be vcry gréât, so tha.t rotation is prevented, vcntud, 12<7A'~ or <~M ~=-77rthé lattcr of which is vcry sma]]. It appc~rs thn.twhen 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

EFFECT 0FADDITIONS.

183.]

dit'cctty from tlie diH'crentiat équations; for if/c'=~, tlieii

butif/<=0,

~=~

237

0=0,a.nc

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, suppose nrst 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.tpart a.cquires inertia, the pitcli Mis (§ 8S). In the sa.mo way :),smiULconthiun.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 chunpingis rc!a.xcd,thc pitch faits, In conséquence of thé diminution in thc potential cucrgy of a givcn dutormation. The case of a ai oïd is not quitc so simptc. Lct tlie and let tlie added piccu whieh is at original tjn(Lof thé rod bc nrstsupposed to hâve no incrtia., bc ~t/?. InitiaNy thc end ~1 is fixed, or held, if we )ikc so to l'cgiu'dit, 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, -Dcornesto 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 Lefurther increased as ~J3 acqnircs inertia. 18.5. Thc case of a rocl whieh is not quitc utufonn may 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))Cuncu)-)-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. smaHlo~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 cficct Jst])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-inrod. Wc havo on!yto <md what additton to m;)kc to thGkineticcncrg'y, sup])osing tha.t tho bur vibrâtes accordin~ t.o thc samu !:twas wou]d oblai)~ were Uierc no rotatory ioc'rtia. Lctu.s take, far cxmnpic, thu case uf a L:uc!a)npc() at Oaud frcc at a.nd assume tftat thé vibration is of thé type, .V= !<cus~ whcre Mis one of thc func-tions invosti~atud in § 170. Thu ].i))(-tic f'no-gy cftttc rotation is

18C.]

FOR KOTATORY CORRECTION INERTIA.

23U

Tothismustbca.ddt.id

ti0that tlie lunctic encrgy Is mcrcascd in thc ratio

Thc atto'ed frcqnency Lcnrs to thi~t calcnh~tcdwithout allowanee for rotatury inertie n mlio '\v!uch is thc square root of thé rcctprociti ofthe! prcce~ing. Thus ?~/c' ,?
sin M ;t.

eus <7r + ces

eus ?~

1

ces

a /7r siu a'

1

if wc substitntc fut' ?~ front ~=~(2t-l)7r-(-l)'a. In thc c~suof thé ~ra.YCSttune, ot='3()43, or, in dL'grccsand 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, atld

n r=l-fl

/w\?)~ + ~-)

(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 dcnding on its square ma.y be ncgluctcd wit))out sensible error.

240

LATERAL

VIBRATIONS

0F

BARS.

[187.

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 invcstigated by tlie method.soi'St.ur
tho effets f'f rotat'ry ixo-tiubcmg onuttcd. If wcnssumc- t]):)t c< cos! woot)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 us suppose, 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 .secondnonnid 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~)).sarc 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.)ninethé nurnber of rf'ts uf a funetion /'(.) thé fonu /M

=~M

+

M-t-

+~

«f

(.).(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 rcatcd su
J (j

from whieh we find for thc moment of t))c couple acting acrosstho 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.cticatcases Y'would bc nc~tigibic iti comparisonwith T!te expression (1) dénotes thc work th~t \vou!J ne ~iocd dnring thc strai~htcning'oftLe bar, if thé luugth of c:).chdément ofthc axis W(.'['oprc'scrvcd constant dut'ing t])(; proccss. But whcn a. strctchcd L~r or strin~ is attowcd to p:LSSfrotn iLdisp!a.ccd to thé nutura! position, thc to)~t]) uf t)tc axis is dccrcascd. Tho and the thu corrospol.lding illllolll'Itof amount of thé tho dco'casc clocrcasc Is -cl.r, << and corrcspoudinggain gain ( .") is :~f j\~t< G?fY cfworki.s :¡ T(ù ~(~.

cl,c

d.c.

Thus r=< (~

&+ ï-~(~,)'(a). .(~)' T)ie Yariationof the first part duc to n. hypothc'ticfd dispiaccment is givon in § 1G2. For thc second part, wc hâve icf/7 ~c= f~Sy ~8 J V~ j

(~y~l f~-Vc 7 /o\ r.. (3). f<~ J ( ~y~ In aH
or, if ~Yc'put

-t-T =

=

~'= «~),

rlx<:t~ .vcl.~c 1.~+~ r!t .t. «'fc/
p. 0. .UcltfM..l..t.~ cl.ccAc

(4).

Fora. more dctailcd investigation of this equationtlie readcris rcfcrrcd to thé writings ofCtebsch' n.nd Don~i)i. ~«'()r <(ts<)'ct<«
Leipxig, 16G2.

189.]

PERMANENT TENSION.

18'). If thc ends of thc rod, or wire, bc chnnpcd,

24:3 = 0, !ind

tite tc)-)i)!)i:L) conditionsfu-csaLisfied. ]f t])c nature of Lhe be such that, wlutc thé cxtrutnity is coii.stnuuudto he a, node, tiio'o is no conp!c itctmg on thc b:u-,

must vanish, thilt i.sto say, tho

on! nmst Le strfught. T)tis suppo.sition i.su.sua.Hyt~(cn to rrcscnt thc c:i.suofa. string strctchcd ovcr hritt~cs,a.s i)i manytxu.sic:).! but it is cvidoit that titc pfu't beyotid thc bridge i)Lst)'nmcnts; must pa.rt~kc ofthc vibration, !ui(l that thcrcforc its lo~tit cannot Le altogcther n ]n:).ttcrof Ijiditfcruncc. n'in thc ancrai dif1b)'c:nti;t.l cqu.Ltiou wc tit-ke~pi-oportional to cos wc gct

whi

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! (3), "'=~' -~+/W which détermines thcfrcquc~oy. If we soppose t))CAviruinnnitt'Iy thin, ?r=~7r~ thc same as wn.si'ound in OtaptcrVt., by startin~ from thé supposition of ns a vcry sma]) qnantity, thc perfcct m-xibUity. ]f wc t)-e:tt approximatc vah)c of?; is <'7r
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

244

LATERAL

VIBRATIONS

OF BARS.

[190.

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 consIdGrationau ~hiitional consti-:unt must be introduced, with the eHcctof 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
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~ is a stna]] quftntlty under thé cir(G)proves that H' < ur .j cumstn-ncescontempiatud, a.ndthei'cforctli: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 relation is not disturbed by thé rigidity. It would probably be otherwise if 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 ~/(
346

LATERALVIBRATIONS0F BARS.

[190.

Thé theoretica-1rcsult cmbodicd in (8) ha.sbeen eompared with experimoit by Seebeck, who found a. s:d.infactoryagruemcnt. Thu constant of stirfness \vasdmtuccd frum observations of tite rapidity oi't)io \'ibr~tions uf n smaU piuco of thu \vii'C)wttcn one end was (.'tutiipudlit tt.Vice. 191. It lias hcon shewnni t)ns c])apter tliat thc theory of bars, cvcn whcn sh)]pUHudto thc utrnost by t)tc omission of uniniportant quantifies, is (tceu~dty more cumpticated t!t:ut t.hat of po'ftjct!y fiuxibtc stnugs. The ruasun of thu extrême snnpiicit.y of thc vibrations of .stringsis to Le fcund in thc flet titat \),v(.'s oftho luu-monictype arc propagated with a velecity.indL'nd(;)itof tho wave Iun~t](,so tti~t an a.rbitnu'y wa.veis aHowcdto travut Avithout décomposition. But whcu wc from string's to b;u's,t!ic con`l `l = Uisis no stmt in stant iu thc tlle din'crGntifd (litlèrential C(lliLtloll ~'1Z. -Iéquation, vix. ~t-/<=(), longer cxprus.sihiu as a veh'city, and thcrufm-c t])(j V(.d"cityof transmissionof a train of harmonie wavc.sc:mnot dépend on thé dif)'cr(;utialontution ,'dom',but must vary with tiiu wa-vclungth. Indccd, if it bt.' ittud t])at t))e train uf harmonie wavcs can bc propagatcda.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!cto C) wavcs propagatcd in onc direction, vlx.=cos"À. (H–~), which satisfiesthc diH'urcuti.dC(p(ationif

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.vea train of wavc-s,Avitosenmptitu()c s!(jw)yvancs from ouc:point to anothur IjctwGOlthc vatucs 0 amt 2, ft)!')ning-a so'ic.sof group.sS(-)):).r:).ted from onc aufjther by ]'egiot]scojnparativ-clyfrcu ironi distm'baucc. In t)tc case of u. stringor of a co]um!iofair, v:n-icsas T.and t!)cn thc gt'oups move

0FTWO TRAINS 0FWAVES.347 ~91.] RESULTANT forw:u'dwith titc same velocity n.sthc compone~t trains, :t.ndt!ierc is no change of type. It is ot.ttcrwisewhcn, 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 origiii is givoi. hy

In thc j'rcscnt cn.SL)!.== 1, an
and !U:~kc initiaity A solution

=

of (1)

(~').

is

~/=cos~

whcrc

(.), ~='

cos~(.<x).

and a arc constants, irom '\vluch we conclude t))at

In tho c
248

LATERAL VIBRATIONS0F BARS.

[193.

is n.)so a sohttion, where j~('x) is an arbitrary function of a. If !iowweput<=(),

which shcws that ~(a) must be takcn to be 27r (a), for then by 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])e i-(;m:umngp:u-tof thc sulution, which I)nsto Ha.tisfytlie (liU'ercutmléquation whilc Ib makcs initi:d)y = 0, = (.); it is

Thc Hnal result is obtained by a.dding thc right-I~ud of (3) aud (4.). Jn (3) thc intégration with respect to q may bc c~ected by mctuis of thc formula.

which may Le proved as follows. formula

Ji' in thc wcM-knownintégral

Now suppose that~=<'=< whcre !:=VI-l, only thé ren.1p:).rt of tho equation. TI)us

and rctain

193.]

FOURIER'S

SOLUTION.

249

whencc

from -which (5) foUows by a. simple ct~nge of va.nab!c. Thus ecma.tiou(3) may bc wi'tttc]i

CHAPTMR IX. ViïiHATtONS

0F MHMtiItANES.

U)3. Tm-: tlicorctica! monbranc is a pcrfcct]y f)cxib!c and innnitctythin Jnmina ofsotid )nattcr,of nnifoDn materiat and thickncss, whicb is strctcbcd in :dt directions by a tension so grcat as to rem~in scusibly unidtcre.t during thé vibrations at)d di.spfaccmonts If fui eontcmpjated. imagioary Une bc drawti across t!ie membrane in any direction, t))omut)):d action betwccn thc two portions separatcd byan eiemunt uf Uœ !inc is proportionn! to thc len'rth of thc dûment and pcrpcndicutar to its direction.1. If t)ic for~c in gestion bti l' ~.9,7', mnybo caifcd tbc
m f~

~'j~ wltero (~ dcnotc.s an ch-mont of tbc and tf/t r/~ tm bound~yt/ of ) "m~ nn cfomjnt ot thc normal to thc cnrvc drawn out\ar<).s. This is balanccd by the reaction against accctcration mcasnred by ~v

EQUATION

194.]

251

OF MOTION.

p buin"' a symbot uf onc dimension in mass a.nd 2 hi length dcnotmg tlie supui'Hci:ddensity. Nuw by Grcen's theorem, if

;S' ukimatufy, f).udthus thé cf~un.tiottof motion is ~) J

.d).

Thc condition te bu s:).tisficdat tlio bomidm'yis of coursew= 0. Thc diU'orential équation ma.y a!so bc invcsti~atcd from thc expression for thc putcntin.!cncrgy, winch is fouud by muttiplying thc totisiou Ly thé supediciid strctclimg. T)ic :dtcred a.rca.is

from which 8~ is casily ibund Ly au intL'gratIouby 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 membrane is thu rcchuig!~ formcd by Lliccnordinatc axus and thc linc.s for ovcry point withhi tlic arc:).(:}) § 104 is satisiicd, te = n, y = fmd fur cvorypoint ou tnc boundary 'w=(). A particuttu' It~tegralis cvidcntiy

l 11C l'C where

,7)~

?l-

2 /))Lz C-7T'

?~\ +~(~' )l21

CI»)

and M~and Harc intcgcrs n.nd fromthis thc gênerai solutionina.ybc dcrivcd. Thus w=S

w=~ )<=~o S M=t siu M-i

??;7TT'

);'77' {:Imn COSI)~ sln-(~~cos~<+7?~sin~}.

-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 initial circumstanccs. WI~tevcr fiiiietioii of thc co-ordinatos may hc, it can bs expresscd for all vaines of bctwccn thc limits 0 and by thé séries

where thé coemdents Y., &c. are Indeudont of Again whatever function of~nnyoac ofthe coc~cicnts YmayLe, it can be expanded betwecu 0 and &iu t)ie series

where C, &c. arc constats. From this we conclude that any function of x and y can bc expressed within thé limits of the rectangle by thé double series

and thcrcforc that tbe expression forain (3) eau beadapted to arbitrary initial values of wand 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 both un.ty, w retains thc same sign over thé whole of thé rectanclc vamshing at thé edge only but in any other case there are nodal lines running parallel to the axM of coordinates. Thé numberofthc nodal lines paraHetto is n -1, their equations being

RECTANGULAR

195.]

In thé sa-me way thé équations

BOUNDARY.

253

of thé noda.1 lme3 pMa.Hcl to

n)'f.

divides thé rectangle being w 1 in number. The nodal system is into ??~ equal parts, in ea.chof which thé numcnca.1value of w repeated. functions 106. Thé expression for w in of thé normal 1s ~q

to fonu whcrc 6, &c.are the normal coordinatcs. We proceed We hâve the expression for Fin of

v

V

v

In integrating these expressions over thé area of tlie rectangle we find the products of thé normal coordinates disappear, and

of wand ?!. the summation being extcnded to
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 undcr t)tc opération uf a consent, force .s.ie], as Le supposed to ansu from gascons pressure. At thc tin.c nmy <=0, tlic i.nj~d force is rc.novcd, ~nd thc mo.nbmuc Jeft to itsc)f. IniMativ thc cquation ofcquiiibrium is

i"C('njunctinnwit,h(~. In on]cr to cxpn.s.s v.-Lh.c,n (..), or ,n t.).is case si,nr]y to ren.uvc J'iLpgr;d.s)gn. Thus

f,.o,n undcr thc

Thi.si.s an cxamph.of (.S),{;)()! If tlie ,nen~r.n.. p,i.s)y set solution is 4 =~

at. n.sL in its po.s.Uo. ofcn.i),tlie

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 Hidciirc titc more effective; n.nd whcn thé iurni is very clo!)~ttcd, additions tu thé longer sidc {u'ca-tmostwlthout c~uct. WitCH a~d are inconnnensur:ddc, uo two pairs ci values of w and ); can gi\'c t))Csa-mc frcqncncy, and cach fuodamcntal ]t)cdc of vihratiuu bas ils own ch!Li'actc)'isticriod. 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 ils simpte 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
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

VIBRATIONS0F 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,yLe 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.1uf 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 nodalIine is corvcd, but is always a.nalytica)]yexpressed by

and may he casily constructed with thé help of~ table oflogfu'ithmic 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 by their mtcrchangc, -\v]nlcno 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~)cnext 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~~ Usin sin D sin + sin cosM<. 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 Last)y, if C'=

[107.

tlie équation of thc nodc is

and similarty whcn Jn ca~c (4-)wlicn a: = a, y = ft, or TL'usoue ha!f ofCtichof tlic lines julning y = a, ? = «, or tlie xuddie points of opposite cd~cs is intcrcted by thé curve. It should bc noticcd th~t in wha-tever !'n,tioto one another Mid D may Le t~kcn, thé nodfd eurvc always es through thc funr points of mtcrscctio!i of thc nod~I lines of tlie Ursttwo cases, C'=0, D=0. If the vibrations of thèse cases 'bc comtha.t in thc pounded with correspon~ing phfmcs, it is évident 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 ~]-Gopposcd, tlie nodal curvc will p:uis Gxelusivclythrough 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).

0FSLIGIIT 197.] EFFHC'T IRREGDLARITIES. 259 Thc iMt ca~c which we shd! consider is obtaincd by putthi~

or, if thé factors con'esponding to thé edgcs be rejected, o 4cof:l -1 -1 -0.(0). (4 M o cos-.+Deos-ce a c(4co~l)cos~+Deos~~c~l)=0.(0) <x\4C08 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 rresputs the diagonal ~-)-~=~ a,nd the seconda hyperboHc curve. If (7=-7), diagonal'.

wc obtain the same figure re]ati.vc!yto t!)e othcr

~98. Titc pitch of tlie natural modes of a. sqaa.remembrane, which is nearly, but not quite aniform, may be nn'estigatcd by t hegeucra] 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!ydetermiued. 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 aHowancofor thé change of type, Wc have

of which thé second terni iflthe increment of Tdue to 8p. Hence ifwoecos~ n.ndP dénote thé v~lue cf~ previousiy to variation, we have 9. 1T) s,,en=i1- 44ff"8p Õp ~H7T.T .B~~V, ~~z~r,y 7 m~r_r.~in2 p~ ~a (1 nPO =~ (1 o.rn ~y.(l). a ~,r,rly. (1)~ ~s' 'It()n)~,J')).< <<'<<{?,p.l29.

17--2

2GO


[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 produces no result. When, however, M and n arc uncqnal, the problem, though remaining subject to the same gencral principles, presents a pccnliarity different from anything we have hithcrto met with. Ttie raturai type for thé unloaded membrane corresponding to a specified 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 undisturbed type, the question will arise how the selection of tho undisturbed 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 and attached to thé membrane at a. point lying on the line x = 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.] whilo e

SOLUTIONSAPPLICABLETO A TRIANGLE. ?)'=P'-

261

2

Tito ratio eLaractcristic of thc interv~l betweoi t!)c two uaturnl toucs of thé loadcd membrane is thus approxnna.tcty

= ~<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 efTecton tlie period. The probleiii of determhung the vibration of a square membrane 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.1type, winch dinars froui tlie actual one. If

199. The preceding tlicory of square membranes ine!udcs :). good dcal more than was at first iutcudcd. Wheuevcr in a vibrating systom certain parts remam at rest, t!iey may be supposcd to be absohitelynxed, and \ve thus obtain solutions ofothcr questions than t!)osc origmaUyproposed. For example, in thé present case, 'whGrcvcra 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 vibrations being equal and opposite at points which are thé images of each other lu thc common hypothcnu.se. Undor thcsc circumstances it is evident that thé bypothenuse wou!dremain at rest witttont constraint, aud tl~crcfbrctlie vibration in question is iucludcd 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 '1 to 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 boundary is in thc form of an cquilatend triangle, thé reader is refcrrcd to Lamd's 'Levons sur l'élasticité.' It is provcd that thé frcquency of thc gravest tone is c /t, wlicrc Ais 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 cquation in 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 ?' dOt~ 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-tiouof 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.vcnn\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

l'OLAR

200.]

In which

2G3

CO-ORDINATES.

is thc ruciproca.1of a liucar quantity.

Now whatevcr ma.ybc tlie nature of it eau be cxpMiJed lu Fonrier's series M = w. +

as &functiou of ?' and

cos (~ + al) + M~cos 2 (~ + a.~)+.(3),

in which &c.arc fuuctions of but not of uf snbstitutiug froni (:;) In (2) may be written

The result

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 as a vanish separately, and we thus obtain to dotermmG function of r

in which it is a mattcr of indirfcrcnce whcther the factor or not. cosn (~ + a,,) bc supposcd to be includcd in Thé solution of (4) involvca two distinct functions of r, cach multiplied by an arbitrary constant. But one of thèse functions becomes Infi nitewhen )' vanishes, and the corresponding particular solution must be cxctuded as not satisfying the prcscnbed conditions at thé origin of co-ordinates. This point may bc illustratcd by a roforeiiceto 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 intesatisty \7~ = 0, as may bc scen from the value of 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 bccomesuseless as a basis for nuincrical calculation. In such cases anot)terseries procecding l)y desconding powcrs of may 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 becomesultimately divergent. Neverthelcaa wlten z is grent, thé convergent part may be employed in ca~culation for it can be proved that thé smn of auy nuinber of term~ differsfrom the true value of thc function by less than thé last tûnn inctuded. Wc sba,Uha.veoccasion later, in connection with anothcr problem, to consider thé dérivation ofthis descending series. As Besscl'sfunctiohs are of considérable importance in thcoretical acoustics, I have thougbt it advisahie to give a table for thc functions J,, and extracted from LommcI's' work, and due «&crclic/?M~'t-c';<)FtOtc~fn.Leipzig;, Lommd,$<;<(~'< 1868.

2G5

DESSEL'S FUNCTIO~S.

200;]

and J, arc conncctcd by Thc Hansen. to functions J, originatly thc rela-ticii = J~. ~)

~(~

0.0000 0.0 1.0000 -0499 -9975 0-1 -0095 0-2 .9900 0-3 -977C -1483 -I960 -9604 0-4 -2423 0.5 .93~5 -28G7 -9120 0-6 -88)2 -3290 0-7 -3(!88 U-8 -84C3 r) -4000 -8075 0-9 -44011 1-0 -7~22 -4700 1-1 -71!)C -C7)l1 -4983 1.3 -5220 -6~1 1.3 -541U -MG9 9 1.4 -5579 -~118 1.5 -5C99 -4554 1.6 -980 -5778 1-7 -MOO -5SI5 1.8 -5812 1.9 -~818 -57C7 -2239' 2-0 -5C83 -!66C 2-1 -5560 -1104 2-2 -0555 -5399 2-3 -5202 2-4 +-002;') -4971 2-5 --0484 -09G8 -4708 2-6 -4416 -1424 2.7 -4097 -1850 2-8 -3754 -2~43 2-9 -3391 -2601 3-0 -3009 -2921 3-1 -2613 -3202 3-2 -32077 -3443 3-3 -M4:) -1792 3-4 -3801 -1374 3-5 -0955 -3918 3-6 -0538 -3902 3-7 -402G +.0128 3-8 -4018 -.0272 3.9 -3973'~) -0660 4-0 -1033 -3887 4-1 -1386 -37GG 4.2 -1719 -3610 4-3 -2028 '3423 4-4

=

~(=)

~.(~

') ~(~_

-2453 -0903 9-0 .2311 4-5 -3205 -2324 -1142~) f; 9-1 6 -2i)Gl1 4.G -2174 9-2 -1367 -27911 9 3 4-7 -2004 9-3 -1577 -2985 4-8 -24044 -I81G 9-4 -17G8 -3147 -20!)7 4-9 -1G13 9-5 -1939 -3276(-) 5-0 .1776 -1395 9-G -2000 -3371 -1443 5-11 -116G -2218 -3432"~) 9-7 5-2 -1103 -09288 -2323 9-8 -34GO 5-3 -0758 -0684 -2403 9-9 -0412~? -3153 5-4 -2459 -0435 10-0 -3414 5-5 --OOG8 -2490 +-0184 10-1 -3343 .6 6 +.0270 0 -24% --OOGG 10-2 -3241 -0599 5-7 -0313 10-3 -24777 -3110 -0!)17 C-8 -0555 10-4 -24344 -2951 .1220 5.9 -23GG 6 -0789 10-5 -27G7 7 -150G G.O -1013 10-6 -22766 -2559 -1773 G-l -1224 -2164 10-7 -2329 -2017i G.2 '1422 -2032 10-8 -2081 6-3 -22388 -1604 10-9 -1881 -1816 6.4 -24333 -17~8 11-0 -1712 -1538 -2601 6-5 -1528 -1913 11-1 -1250 -2740 6.6 -2039 -1330 11-3 -0953 -2851 6.7 -3143 -1121 11-3 -0052 -2931 6-8 -3225 -0003 11-4 -0349 .2981 6.9 -2284 -OG77 11-5 -3001 --0047 7-0 -044G -2320 11-C -29911 +-0252 7-1 -2333 11-7 --0213 -0543 ~), -2951 7-2 -2333 11-8 +.0020 -0826 -288~ 7-3 -02500 -2290 11-9 -1096 -278
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.]ultlie boundary condition requircs that

~(~)=0.(2), an équation -whose roots givu the issible values of /c, am tli èreforeof~). The complete expression for w is obta.ined by combitung th( particular solutions embudicd in (1) wit)i all issible values u: und M, and is ncccsstn'Hygeneral enough to cove).'any initif), circumstanccs that may be imagiucd. We conclude tliat an~ i'Huction of r and 0 may be exp:mdcd within tlic limita of thc circle ?'= a in the series (/er) (~ cos7~+-~sm~).(3). ~=S~~ For overy intégral -aluc of ? thcrc are a series of values of arc given by (2) and for cach of these tlic constants <~ and arbitrary. Thc détermination of the constants is effected in thc usual way. SInce tlœ energy of the motion is cqua.1to

and whcn expressed by mca.nsofthe normal co-ordinatcs can only involve their squares, it fullowstliat thc product of any two of tlie 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 -~ris obtaiMcd hy writing sin~ for cos?: A mctiiod of cvaluating thc lutcgral on the right will be givoi prcscntly. SiucG and cacii contam two , 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.]

BOUNDARY. ORCULAR

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 function of 0 as well as of 7', and thé équation of tlie nodal system takes thé form J,(~?-) cosn (~-c<)=0.(2). Thé nodal system is thus divisible into two parts, thé iirst eonsisting 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 nodes will bc inposition is arbitrary. The mdn of tlie circular 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


[203.

and thcrcforc

In thèse équations w refera to thé actual motion,and to a hypotbctical dispiacetnent consistent with thc conditionsto which tlie system is subjcct.cd. Let us now suppose that tlie system is executing one ofits uormal componentvibrations, so that w = Mand while 8wis proportional to anottier normal function v. Siucc

=p c, we get from (3)

whcrc /< bears the sa.mcrelation to

that /< Lcarsto u.

Accordingly, if thc normal vibmtions rrescnted by Mand 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 réactions uga-inst a-ccelcration,which, if they existcd, would appear in thé fundamental équation (3). If in (8) we suppose /c' =A-,thé equation is satisfied identicallv, and we cannot iufer thc value of In ordcr to evaluatc ~i~cfZy. this intégral wc must follow a ratlier différent course. If u and v be functions sa-tisfyiugwithin a certain contour the + A: = 0, wc have = 0, equations \7"M+

2G9 YNTEGRATED 0F VALUES SQUARES. 203.] from is d erived 'U that Let us n ow thcorcm. G reen'a suppose by sotliat /C, ubyslightiy varying = =it ~tc. v~=~+-,0~ ~=~+0~; 8K, a/< K bic; 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 integration with respect to 6,

Accordingly,if

and /eand ?' be different,

an equation first proved by Fourier for the case when Again from (12)

dashes denoting differentiation with respect to Kr. Now

97f) and

VIBRATIONS

0F

MEMBRANES.

[20:}.

thus

0 witli fixed s'hl~

as

to

bouu~laries,

cxpl'ossions for 'L'fLIlcl ~=~

tlie

to si111 1)lifytlic

Inrom

(~-)cos~+~7,

(~-)sm~j.

n)

wcfind

Md a suuihuequation for

The vn.).w.nf the work ~loneby t]m impressec]forc:escluring a ]iypothetic.vl clisplacement8~«~ so t],at if 't t ~=-Z be thc nnpresscd for~ reckoucd por unit of area,

lvhe" 0 aua

a.re amalgamatod. We 0thcnhave

~.rs~ constant pressure Z; thus

tliat the initial velocities are ,<,r. that assllmed n influence 11

204. Jj

SPECIAL PROBLEMS.

2711

No\v by thc difforential équation,

andthus

thc sommation extchding to all the issible values of/c, sti)t As an example of/b?'ce~vibrations, wc maysuppnsc tnat constant with respect to spacc, variGSa~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,

It will bc scen Thé forced vibration is of course independcnt of that, whilenone of thé synunetrical normal componcntsare missiug, thcir relative importance mny vary grca.tly,especially if there be :). ncar a.pproach in value bctwccn y a.nd onc of thé séries of quauti. If thc approach be vcry close, tlie cScct of dissipativo tics forcesmust 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 ues 0of~ n118SIe values then 1011thé 10 aissible ed zz", z"(~) w. p called ca 2;


272

are to bc found by multiplying thc quantitios particular solution may thcn be writteu =

~)

cos

sin ?~) cos f~

+

Thé lowest tone of the group 7t con'csponds to 2!

[205.

by c a.

6,'4

Th.c

(1).

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) ")r a

that is, when

r=a

2

~n which is the radius of thé one interior !iodal circle. Simi!arJy wc obtain a vibration witli 1 nodal if we take tho root 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 rcawere 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 (exct 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.hieof.s:, would a.)sovanish for U~t ~atuc. Butin v)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.ssu 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

con'csponding t.h:m grcate;' indepcndcntly. Stokcs' pa.pcr,

séries tos==

is convergent 1. 'J'!tC series but

unity; Thc with

a

thc

enough, (1) will firsb

root

even sunice must

for tho first for ho

t!tb)e is :K'co]np~)ying (A) dittLit-ence of notation. sti~ht

values

root, of

cn.Icutn.tcd t~cn

from

It wHI be secn cither frum tho fo)'mul:H,or t)ic t:iUe, that tho tlifTcreocGof successive mots of )ngh erder i.s n.pproxinmtcly7r. Tiu.sis 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.of the nodal circles. Table J3 i)t.cludestlie values ofz, whicii SH.tis(y 9. J,.(.!),for~=0,l,5,s=].,2, des mf'mbrMtfs eircu!Mrea," BourRet, "M~mnircsurIotnnnvGmcntYibrntoiro ~inn. de !fo~ onrwft~, t. tu., 1H(!(!. In ono j~nssnRo DonrHet implifs t)int ho 1)M provod thnt nn two Hessefti functtons of intf~r'~ order cnn havo thf.' HMnnroot, buticannotfmtithat La hns donc ho. Tho thf'oron), howovcr, is pr<)t))t)))ytruc; 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

VIBRATIONS 0F MEMBRANES.

274

[206.

TAI!LE A.

a

0 ~fur.)=0. 7T

1

'7'S

2

r7~7L C) ~1: 3'75;H 4 '¡ fi~7 7 ~1~ (j'7fd!)

·1 .) 7

-7516 ~S S 10 t~ 12

J'î;rl3 S~ irt'i~ n'7~1

)df ~fM-(:)-0. Difl: (z) O.Diû'. DUF. TTOI'.Ti 1'2~7

.n.

~0 'J!)~1 '!J!)!¡3 3

:R:3 4'21111 6

~!I!1!l ï î '!J!J~¡8 '!ln!i9 JJ''J ~~9

.139

8'2.I,j.1 3 1(1~`?.IG3 iin~~ 12-2.1G!)

i;~3 l'()(lii:3 H)()~~ î l l'(I OU3 l'll(I():ï lUU.1 ,E Iww~ 1.0003

is consido'ahic thc calculatinn of tlie carlicr mots Wth'n becomes troubicsono. Forvcry hig!) v:du(;s of approxias xmy bc scL'nfrmn t])uconsittumtion nia.tcsto ratio of o<~)f~it.y, thut thc pitch of thc gm.vcst tonc ofn. very acutc sector tnust tend to comcidu wit)i th:).tnf a. tong pamiiul strip, whoscwidth icic~ua.1 to tttC grcatcst 'idt)i of thc scctor. ÏABLR B. s

)t.=0

M==l

~~2

M=5r

~=3 3 M=4

1 2 3 4 5 C 7 8 9

2'40t n'O -1 8-G54 11-792 14~:il1 18-071 1 21-212 24-353 4 27-4U4

:~832 7'0!Gfi 3 10-173 3 13-323 lG-t70 19-G1C 22-7f:0 25-903 29-047

f''I3.') 8'4!7î 11-G20 H-7UG 17-OGO 21-117 2t-270 27-421 1 3U-571

G'370 9'7M 13-017 î 4 1G-224 H)'.tl0 22-f!83 25-74!) 28-909 32-050

C) 7~SG H-r'G.t l-t-373 3 17-G1G 20-827 21-018 27-200 1 30-371 33-512

8'780 I2'{M 1;~70() 18-983 22-220 0 2~-431 1 28-~28 31-813 34-983

20G.~

NODAL FIGURES.

275

Thc ~gnrcs rrcscnt thc more Important normal modus of 'vibnLtIon,ami the uumbci'safHxcd givc thu ft'cquenRyr(jfut')'cdto 18–2

27G

VIBRATIONS

OF MEMBRANES.

[206.

the gravest as unity, to~ether with thc radii of thc circu~arnodes cxpresscd as fractions of thc radius of thc mcmhra.nc. Iti the case cf six noda.1diiuncters t!)Cfrcqucncy 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.ietyImrmouic 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;

Hothat 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 nodal systcm 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 symmfttrical vibrations t))c disp]aceinent of tbe centre of inertia is proportiona.!to

an expression which does not vanish for any of the issible values of /c,sincc (.?) ami '~(~) can)!ot vanish simuitancousiy. In all thc aymmct.)'ic:).l modes thcrc is thcreforc a. dispia.cctncu'tof 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 membrane 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

377

RADIUS.

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.limage in the bounding diameter. Dus line will ttien requu'e no constraint to kc 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 of Bess~l's fonctions of fractional order. = /3, thé particular solution is If the fixed radii are 0=0,

who'c is an intogcr. Wc Pcc th:).t if /9 bc an a.liqllût part of 7r, f7r /3 is integr:). :un.tthé suJutiouis inctuded amoug those a.lready used for tlie complète cirelc. Au Intcre.stmg ca.suis when /3=27r, which corresponds to thé pl'oblum of cotti[)!(jtccircle, uf whici) thé radius ~=0 is coustr: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 valuesof /<are given by siu

/<
or

/M =7/t-n-.

278

VIBRATIONS OFMEMBRANES.

[207.

Thusthc circula)- nf)th;sdiv!(tc thc ~xcd radins into equat hfn'mcnic scittu. Int~e parts, tunithe MCt'ic'sot'tuntj'~ ~nn 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 Lhu probtetn 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 V° Sy" I7'/ 1/1 J~x(~~+r~)-'3--ii~L x x 2 -'3-15-31¡This cquatioti is to bc solved by successive approximation. It will rcadity bu found tha.t = ,rl + Al X-5 J2/=Y-'+~~+~~+~~Y3 ]ij 10~

i-~ +.

LOAD. 0FSMALL EFFECT

207.]

279

so that thc roots of tMi
whcro

J\"==(M+~)77-. In thc Hrst quildrn-nttho-c is no root aftcr xcrosince tana; rel="nofollow"> a', n.nd in thc sccottd

whcrc P,

t0 ~TT~J~ (~)

dénotes thc value of~

+

(~).

whcn thcre is no !oad.

dctcrmhmtc for Thé unsymmctt-ic:U nnrt~a.1 types are not ful)y the unioadcd moubrane Lut foi-thc présent purpd.se tttcy must bo tfdœn so as to nmko thc i-c.sultin~pcnods n. ma-xinunn or load is thé minimum, tliat is to s:).y,so th.Lt thc cH'cct of the !oad can ncvcr r~isc greatest ~td lc:tst possible. Now, since a. thc pitch, it is c)ca.r th:Lt thé inthmnce of tho !oad is tlie lu~st that a uod:d diamctcr (it possible, viz. xcro,whcii tlic type is such is mdiHcrcnt winch)es t!n-oughthe point nt which t!)0 lo~d is ahtachcd. Thc untoadcd mcmbmnc must bc supposud to h:ivu two couLCidoitpcriods, 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.dmsvecto!- )-' bisucts thc futg!c bctwcon two
(~)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
0 ( ~no) P

By(G)§2()0 ~(.)=-~(.), so that the application of t)tc funrmJa rc()uircsoniy a ktlowlo~c of thé va)ucs of' (2). whun (.2) viuushes, § 200. For thu gravcst mndc thc value of J/(A-~) is -5190:}'. Whcn ~.0 is cousidorabic, ~~o~)=2-7r~

approximatoly so that for thc tngacr components thc influence of thc !oad in altcring t!tc pitcb incrua.scs. Tbc it]f)uenceof a smaU irreguiarity in disturbing the nodal systcni may be ca!cutatcd froïn thé formula of § 90. Tbe most obviuus cn'ect is thc brcakin~ up of nod:).!diamutcrs into curves of hypcrbolic form duc to tbe introduction of suhsidiary symmctrical 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 of a 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.]

281

DOUNDARY.

NEARLYCI&CULAR

Wttatcver ma.yLe thé nn.ture of thé bounda.ry, t~ sa.tisfiesthe cnuntion

By Fouricr's thcorcm M

whcre /cis a, constant to be dcto'mmcd. 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 expressionfor 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 wMcbare 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

282

VIBRATIONS 0F MEMBRANES. [209.

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'(~)] + 'A (~~) [~i cos <9+ .B,sin <9] +.)(~) [. co.s?t<9 + 7~.si n/~j +.=().

oL~.(~~) +

If wc intt'gratc thi.s cquattun widi respect to limits 0 aud ~7r,weubtf).i)i

(.t).

butwcc!) titc

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
Witen thc vibration is not npproxim~cly symmctriciil, thc question bocomes tuorc coinp)ic:).tcd. Tlie nor))i;d tn0()cs io;- t). truly circuler mcmbrMGare to somc extcut Indetcnniuatc, but tin'

209.]

NEARLY BOUNDARY.283 CIRCULAR.

irrc~Iarity in thé bonndru'y will, in gênerai, rcmovc tlie indcterminatcn<'SH. T!tC position ufthc uudal difuncters nmst ))utukcn, no that Ute resulLin~pct'io~s !n!Lyh:LV
or

winch shcws tha.t tlic effective ra.dius of the mcmbra.neis

Thc rn-tiosof ~t,, a.ud 7?,,to A,. mn.ybo found as before by insin )!0. tcgraiin~ équation (10) a.ftar inulLiptic~tionby cos But the point of~rcatcst intercst is thé pitch. Tt~cinitial line is to bc so t:).kun as to )ti:~œ thc expression(11) a maximum or minimum. If we refer ta a, lino fixed in spacc byputtin~a instc:).dof we liave to consider t)~c deudencc on a of the quanti ty r~eos~(~-a)~, cos' v (B u) clfl, J o0 8r which may aiso bc writtcn

284

VIBRATIONS 0FMEMBRANES. [209.

andisof thé form J. cos~fx + 2Z?cost/xsin va + (7sin'x, 2?, (7 buing indepundcnt of a. Thure are according]y two issible 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 remfunInJctC!'minatc;,witichhappcnswhcn thc expression (12) is mdendent ût'a. This is t!ie case whcn S/' is constant, or whoi is pronortional to cos For exa.mpic',if wcrc proportional to cos 2~, or in ot))erwor<]sthc hnund:u'y wcro s]ight)ye!liptica),thc uodal system corrcspunding to )t=2 (that consisting of a pair of pcrpondicular dinmctcrs) would ho arhitrary in position, at Icast to this onict' ufapproxunation. But thé single diamctcr, con'csponding to !t=l, must coincide witit one of thé principal axes of thc ellipse, and tlic pcriod.swill be diircrcnt for thc two a.xes. 210. Wc hâve SGGnthat tho gravcst tone of a membrane, whose houndary is appruxhnately circular, is ncarly the samc as that ofa mcchanicaHy simil.'t.rmembrane 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 foUowlngis 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. (~) + +S [(J,(~)+

J.' (~a)

+

(~) 8?'+ .l~eos

~J,

(~). (~.)' +

?~ + 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.pproxima-tioti thé )nc:i.n radius is :).]so thc L'Huctivuradius. In ontur to obtailt :Lctoscr n,pp)'oxi)n;ttiun\ve
286

VIBRATIONS

0F

MEMBRANES.

[210.

sothat

muinbur. T))C <-)))cst.icnis i~ow us to thc sign of thu i-hL-h:UKl If M= 1, :L'~t-ï Le wnttcu fur A:<(,

and vani.shcs approximatdy by (7), .since in gcncml ,~== 0, as shou)d In the p~sc.nt case ,(~) = 0 n~u-)y. Tt'ns f~<' cvidunUy b~ Lhc case,.sitjcc thc term iu fjuu.stionrrcMcntsmcrcly in thé f~nu uf disphtucmcnt of thé cirdc wifLont an i~turatiun t)~ buuudary. Whcu = 2, (M)§ ~UO,

is pnsitn'c for mtc-gra.lvalues of Mgrever than 2, whoi .!= 2-401. For this purpnsu wn n):)y nvail ourseh'cs of a. thojron givcn in to thc cft'cct th~t Kiuma.nti's jf'«r<~e D~rc;c/N~<~e/i, ubithur "or< has a r<)"t (t)
wcll as whilu luitiaUy J, und J,' .) (~s

~~ositive. Accord-

Lv increa.sing a.nd docs not cca.seto do so in'~v- "-Lc~ins° °-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~ all va.ni.s)),f~ is gycn-tcrthan winch shews tlmt in thé 0, c)t.scof 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 npproximatùly circulai' monbranc may 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-axis maj~r is Tt!anfl ucccntricity e. 'J'!mp(;)ar équation ofthubonndaryis

In which the term coutaining e*shouMbc correct. Thuresult may also bc expressed m tei'ms of c and the arca oWc have

and thua

from wLidi we sec how smal! is the influence of a moderato ecccntricity, whcn the arca,is givcn.

VIBRATIONS 0F MEMBRANES.

288

[211.

211. Whca thc nxed boundary of a.membrane is ncitbcr straight \iL)'.i
Solubtc cases m~y bc invonted 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 ~ay sonictinics bc obtaincd from thé theoron of§ 88, whicb aDows us to provc that any contractioa of thc iixcd bouudary ofa vibrating niembnmc tnu.st cause an élévation 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 ils cquitibrimn position, and graduaHy to bicorne stin'cr. At c:).cb st thc vibrations becomc more rapid, until tbcy approach a linut, corrcHponding to infinitc stiH'nuss "f thc nprings and abso1ut,c (ixity of thc-ir points t)io p:n'tcntoffshou!d of application. Itisnotn(.'c~ssarythat hâve thé H:unc dcnsity as thc l'est, or cvcn at~y dcnsity at a.)L For instance, the pitch of a reg~dar polygon is intcrmcdiate inscribcd and circmnscribcd circles. doser bctwecn tboseofthe Umits won!d bowcvcr bc obtained by substituting for the circuniscribed 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.

MEMBRANES

2UJ

AREA.

OFEQUAL

289

of the two limita cannot differ from thé truth by so much as 2~ per cent. Li t.he~iic w:ty we migh~conc)'dc(.h.').tthesect.<)rcfa circle 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 coadifions 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.

_1 f. 1 Quadrant of~circle. Q

5'135./ 45w1 f~?.~=4.~i~ ~s

Sector of a circle 60°.

6-379 A/~=4'616. /13 A/7r=4'G24.

Rectangle 3x2.

27r. ~/tan 30"= 4~74.

Equilatcral triangle. Semicircic.

3832A/~=4'803.

Rectangle 2x1.1 isosceles tna-ngle.J R~ctangle 2x,l. Right-angled

/5 = 4'067. ~y~ =~'967. 'T'~2

,}

3Rectangle 33x1. 1

G u, 7!-A/~= -.) 5-736. ~r1/

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. (ler~e(!(MMt'< t. vm. 1829. R.

19

~0

VIBRATIONS

OF MEMBRANES.

[~312.

of the circular membrane vas restrictcd to thé Rymmetncfd i'c -.ut -h vibrations. Kirci-.h"~ ~~di.mortbc~'m~r, in 1H;)0 aud w~ circular ofthc publisbcd plate dimcult, problem of tho thc thcory gênerai givcs Ctebsch'a y/~op-yq/Y~ (18G2) oi circular memt)ranc induding thc efFectsof stiil-nessand rotatory not much left was tticro that sccn be will thercfore It incrtia. to Le donc m 186G; ~evertheless tlie mcmon-of Bourget aircady uscfui discus.sbn of thc problem accomrefon-ed to contains thc whoïc of which Dumcricid results, complète very by paHicd howcver wcrc not nc\v. uso 213. In his cxpcnntcnta! mvcsti~tions M. Bourget made of various m~terials, of winch papcr proved to hu as good as any. Tl)c papct-is immerscd in wato-, and aftcrTonova! ofti'c snperHuons wbose ~noisturc by blotting papcr is piac~d upon a framc of woud Thc contraction of thé with coatcd havo bcGn gtuc. prcviau.sty edges but many faihu'cs tension, thc in ncœssfu-y papcr drying produccs rcsult is cbtaincd. Evcn wUlt bufurc a met be satisfactory mny ui a wcll strctchcd mcmbmne refiuircs cottsidci-abit-ecautions in consc.tuencc of thc in variations Uabic to pitch use, bcing gréât are cxeltcd vibrations 'i~hc of tho niuisturc atmosphère, varying tiavc a scrics procecdiug to of it is which necessary hy organ-pipcs, évident to thc cyc mado arc :uid of intcrvals sma!! they pitch, by H'c mombranc. If tho by means of a littic sand scattcrcd on vibration be sufHcicntty vigorous, thc s!uut accumulâtes on thé less prcciston. nodallincs, whosc fortn is thus dcHneJ wit)~more or thé cire-lesbeeoning Any Ine'jUidity in thc tension shcws itsclf by elhptic. Thé principal results of experiment arc the foUowing A circulât-membrane cannot vibratu in unison with cvcry sonnd. than It can ouly place itself in unison with sounds more acute 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. to Thé nodal lines are oniy formed distinctiy in rcsponse certain deunite souuds. A littie above or Mow confusioncnsues, and when d~e piteli ofthe pipe is decidcdly altcred, thc membrane continuons re.nains un.aoved. Thero is not, as Savart supposai, a transition from one Systemof nodal Uncst') auother.

213.]

OBSERVATIONS

0F M. BOURGET.

291

Tho nodal Unes arc circlos or diamcters or combinations of cu'c~osan ~~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 of square membra.ncs'; a.ndthese authors considcr that the rcsn)ts of theory arc Elecisivelyestablished in opposition 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 forccdand 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. Whcnthey sounded a. pipe somew!~a,t lower in pitch than thé gr~vest tone of thé membrane, tlie sand rema.inednt 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 cxperimcnt wa.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 goodagreement between thcory and obscrv:).tionwith rcspuct to t)]Cradii 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

S93

VIBRATIONS

OF MEMBRANES.

[213.

the bands of sand; but thc relative pitch of thc various simple tones deviated considerahly from thé theoretica.1estimâtes. 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 supposition of 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.nethan 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 équations of eduilibrium and motion for a thin solid plate of uniform isotropic material aud constant thickness, we require thé expression for 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 qutic function of thc two principal curvatures. Thus, if p~, bc tlie principal radii of curvaturc, the expression for Vwill 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 denncd. 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 Vis

294

0FPLATES. F314. VIBRATIONS

If M bo 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 tcrm in V, prcviously noticed that tlie second uame!yj< P,P~ of the p]:ttc, and is thereforc dereprésenta thé
so that ve have to consider tlie two variations

1 Tho followingcomparison of tho notations uscd by tho principal writers may iinvotrouble to thoso who wish to conault the oriH'H'Hmouuira. 'hx Youae's moda!uB=F (Clcbseh)=~ (Thon)aon)=:~K+~t (TIiouison) ~"(~)

(Tbomso!J)=? (~rckLoS tmdDoDkiu)=2~

(Hirchhofï).

Ratio of latcral contraction to longitudinal elongation =~ (Clobsehaud Douldu)=
(Thomson)=~ and Werthuim

(Eircidtoff).

215.]

l'OTENTIAL

ENERGY

0F

BENDING.

295

Now by Grecn's theorem

in which f~sdonotes un clément of thé boundary, and

dénotes

diH'crcntIationwith 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.ybe 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

tZ~ If wc ve substituto ior ~8w f/;<; ~8M; thcir values in M~ <
29G

VIBRATIONS

[215.

OF PLATES.

wc obtain

Collecting and rearra.ugmg our results, we ~Ind

r~ -s-~

f~w..cfw +(1-~) cos8smO \,t/y' (~ " (7~\ jcos~sm~

<~M\

'TZ)\ (cos'8 +(eos~-sui~) 't~/y/J ~~M .f~w ( + + -icos' d~' sin" a~ J~/i. f~- ('~'C7"M (1 ~)

y 1

Tliere will now bo no difficulty in forming the equa-ticns of the transverse motion. If p bc the volume deusity, aud Z~ force acting on thc c!cment c?6',

215.]

CONDITIONS

FOR A FREE

EDf.E.

397

1 8F-ff~8w~+f~wSM~M=0.(7)' is thc gcno-al v:u'iation!Uéquation, which must bc true wha.tevur mn.y fmiction (consistent with tlie constitution of ttic system) bo supposcd 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 a'id thei-cfore thé and 8w va)ucs of bound~ry hypothctic&l for SFmustvanisIt. cne~cicnt.softhcsoquautities in thé expression Thé conditions tu Le:s~isncd at a, frcu cdgc arc tlms

If thé whole circumfercucc of the pla,tebe clampcd,5w = 0, cln = 0, and tlie satisfactiou of thc boundary conditions is already sccured. Thé tn-bttrary. If thé cdge bc ~=0, 'cd", 0 hut~ia 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 coordinates. 'l'aking the axis of a: pM~Iel to thé normal of tlie buuuding curve, wc sec that we may writo

Aiso Tho rotatory inortia ial'cre uc~locted.

CoinpMe § 1G2.

298 VIBRATIONS 0F PLATES. [216. axiscoincidiug whcre o-isa,fixcd at t!t0point withthctangent (I"wToobtain .1'œ l '1 Ingenci'a.l-diH'ei'=ih'om underconsideratiou. d C12w M~* <M' b thérelation bctween wemayproceed thus.Expn.nd wby them, inascouding Ma.cta-unn's thcorcni ofthcsmaU powcrs qua.ntities lu nando',andsubstituto forMando-thcn.' values ofa,thé arcofthccurve. ingênera). Thus fF~ ~"w ~<; -r 0-'+ j–j?~+ ~0-~ ?:o-+ ~ow= + + o-+A(/)!~ ft whUc onthécurvco-=s +cubes,== s'" + whcrc /?Isthé radius ofcurvature. forpoints onthécurve, Accordingly andthcrcforc whencefrom(l) 1~!00~~ 2 ~"W. Io v~"tc=-+-+. p (~'

(3).

thésecond Weconcludc thn.t condition in(9)§ 215 bouoda.ry beputiutotlicform mn.y In the sa.moway by putting

==0, we sec th!).t

is équivalent to wherc it is to be undcrstood that the axes chtcl~ of Mûnd cr a.rc Rxcd. Thé (h'st boundary conditiou now becomes

If wc apply thèse Ct~uationsto thé rectangle whose sides arc

216.1

CONJUGATEPKOPERTY.

200 299

to bo pa.mllol to the coordinato axes, wc obtahi as thé conditions sutisncd :).longtho cdges pa.ra).ielto

In this ca.sethe 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, without thé prelilninary transformation.

con'csponding to thc same Auy two values of w, K and boumi:ny conditions, arc co~x~e, that is to say

to prove this provided tha.t tlie periods bo différent. In order 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 it is much aimpicr !~doptedby Kirclihoff for thé ch-cular dise, but Mid more direct to use thé va.rin.tiondéquation

in whick w refurs to the actual motion, and 8~ to %ti arbitrary SF'Isadisplacemcnt consistent with thé nature of tlie system. or symmctrical funetiou of w and ~M,as may be seen from § 215, from thc general character of V (§ 04'.)

VIBRATIONS0F PLATES.

300

If we now suppose in tlie first place th~t w = :t, 8~ = hn.vc

[217'. wc

~~=~~f:tu~; and i)i Hkcnia.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 in boundary, and whcthcr thé cdge be cla.mped,cd, or frec, '\vltolcori!ipa.rt. As for thc case of mcmbmnes in the la-st Chapter, equation (7) may bu onpiuycd to prove that thc issible v:duus of arc ruai; but tins is évident from physieal cousidcrations. For thc application to a circular dise, it is necessary to express the équations by means of polar coordinates. Taking titc ccnti-cuf tlie dise as polo, wc hâve for the gcncral uquatiun to bc satisnud at ail points ofthe arca 218.

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.]

301

POLAR CC-ORDINATES.

If w bc cxpa.nde
+

+.

+

each term sepM'atclymust satisfy (2), and thns, since !~<xcos(M0–a), /r~ l~A d1' ~~p+r~'r~ (l'zo" ~=0

+

3-~ US

,/2-~r~

Il\Q' (3).

1 (_luu" (¿2nt

0

The superficial difïerentia.1équation may bc written (V'+~)(~)~=0, which becomes for the general tei-m of thé Fourier expansion M' .1 ~_z' )'f~' ?" f –+--T-i+~('n+'' ~f~

.1~ 7 ?'~)'

n_' -a )-'

,Y, = KM

will bc obtained by ndding shewing that tlie complete value of of togcther, with arbitrary constants preHxcd, thé genera.1solutions

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 ceJ.. (/o-), the second function of r to thc problem in hand is bcing hère inissible. In tlie same way the solution of tl)c équation with the lower nigu is Wnx ,7,,(~r), whcre t ==s/ 1 as usual. The simple vibration is thus + sin {'yJ, (~-) + SJ,. (t/
302

VIBRATIONS

0F

PLATES.

[218.

As ni thé case of a, membrane the nodal system is composed of diametcrs symmetric~Iy (UstribuLedround tlic centre, but tlie othenviso arbitrary, dcnoted by .(~, cos(~-a)=0 is togethcr with tlie conccutric circles, -\vhosectiu~tion J~-)+XJ,(~-)=0.(7). 219. In order to dctcrmiMC?La.nd we must ultrcduce thé we obtain from bouudary conditions. Whcn tue edge is free, (3) § 218.

in which use has been nmde of tl<e diU'crcntiidéquations satisncd fractions on thé right Hie dct)othé e~chof In (/c-),J,.(~)'). by minator ma,ybe dcrh'cd from tLc numcrator by writing in place of BycHnunatioa ofXthc cquntiou is obtuiucd wlioseroots givc tlic issible vidues of /c. Whon = 0, tlie rcsult assumes aj simple form, viz. ~(~'t) rt /'9') Jn('A:a) 2(l-~)+~~+~y~=0.(2). This, of course, could ha.vebeGHmoro easilyobta.incd by neglecting Mfrom ttic bcgiumng. The calculation of the lowest root foreach value of is troublemust 'be cÛceted some, and in the absence of uppropri~c tables by menns of thc asccnding séries fur thefunctions ~(~'), .y,.(!). lu the case of tho higher roots recunrsc ~n~yLe h~l to thé semifunctions. Kirc))hoff s~ne fur thc séries descending convergent finds ~L-+ 8~(8~)" (8m)" tan (~Tr) = –T– ~~8~~(8~t)~ whcrc ~=~=(1-~)-8, ~=ry(l-4~) C ='V(1 4~') (9 4~) + 4.8(1 + 4~), 8 + 136~ + 30~). 7) = ((1 4~) (f) 4~) (13 4n')] + (9

219.]

KIRCMHOFF'S

303

THEORY.

where~isanintcger. /t Is idcntical with It appears by a numcrical comparison that a, law discovei-cd the uuinbcr of circulai' nodcs, n.nd (4.)cxprc.ssGH to figurcs with a by CIdadni, that tlie ft-e
Ct~ 1 2 /t i

~=-11

?=0

/t

Gis g:s'+

W.

Cn.

P.

W.

HiH+ A-tb'-(h'-

b o"+

hf"+

cfts"+

P.

?t=3

'cnr*'p.~ w. C C 0 C 1 g' gis'+ a/-

~=33

Cil. P. w. d (tis- d)sd".dm"di8"+ c"-

Thé table, extracted from Kirchhoff's mcmoir, gives thé pitch ofthe more important overtones of a free circular plate, thé gravest three columns under tlie heads The C' bc to assumed being refer respectivcly to tlic rcsults as observed by Odadui Ch, and as calculated frou theory with Poissou's and Werthenu's little A values of sign deMotesthat tlie actual pitch is a is a little lowcr, than that written. it that a 7)~)t!<s aud sign higher, notation,and~tto bnatural. 1 Giscorresponds to (~ "t theEnK)ish

304

VIBRATIONS

0F

[219.

PLATES.

Thé diserancics between theory and observationare 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~edexpci'ime!)tf)J]yby Savart. The followingnumbers arc taken from a papcr by Strehikc', whomade somû careful measuremcnts. The radius oftho dise is taken as unity. Onecircle Two Iwo

Obsorvnt.ion.CnIeuJation. 0-67815 0-68062. fO-39133

0-39151.

~.g~~ cu-cles. fO-256310'25679.

Thrcc circles 0-50107

0-59147. ~0-893GO 0-89381.

Thc ca.!culated rcsutts appcn.t-to refur to Poisson'svalue of would vary very little if Wert.Itcim'sv~luc were substituted.

but

The foUowingtitb~ givcs a, comparisou of Kircilhoffs theory ()! not zéro) with measuremcnts by Strebikc m~dc on less accurate dises. 7?~~Mq/' ~'CM~ft)'2Vo<~<M. Obser~tbu.

Œ]cu!a
?t=l, A=l1 ~=2, /t=l ~=3, ~=11 M-i, /t~

0-781 0-70 0-838 0-488 Q.g~

0'783 0-781 0-783 0-81 0-S3 0-843 0-493 Q.g~

0-7M136 0-82194 0-8.1523 0-40774 4 0-87057

~=~(W.). 0-78088 0-82274 0-84G8I 0-49715 0-87015

221. WLcn thu plate is truly symtnctrictd, whctherunifonn or not, theory indicntes, and exporiment veri~GS,th:).ttlie position of the nod:U diameters is fu'bitnu'y, or ra.ther dcndcnt only on thc manner in which tlie pl~tc is cd. By varying thc place of , any dcsircd (liamctcr mny be made nodal. It is goncraUy othcrwise wlien t!)crc is a.ny sensible dartui'c from exact symmctry. Ttic two modes of vibration, whicli originany, 1 Pc~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:nncset. 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 effectof 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 determinate. Thé diamétral system corresponding to one of the types es 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 would be easier than thé preceding is eit,her clamped or ~)o?'~ in their theoretical treatment, but are of less practical interest on 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.bleto 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 ed 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 ed. Tho rcquirements of theory are often difficult to meet in actual cxperimcnt. WheM this is so, we may have to be content with an imperfect comparison but we must 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'eeequations 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 circular 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 dinEcrentset ui équations. He provcs that if Mand u' be two normal functions, so that w=~cos~, w=«'eos~'< arc possible vibrations, thcn

m~. de !4Md. d, Se.
224.]

HISTORY

0F

307

PROBLEM.

This follows, if it bc itted that u, satisfy respectively the e<~uations ==~/V. c" ~7~
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 obviouelyinissible, Mathieu coneludesthat 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 converselythat 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.11that is really necessary being that the bounda.ry appliaBCCssbould be either at l'est, or devoid of inertia. 225. Thc vibrations of a rectangular plate, whose edge is the normal .suMWteJ, mny bc casily investigated theoretically, functions being identical with those applicable to a membrane of tbf same shapc, whoseboundary is fixed. If we assume

2D–2

308

VIBRATIONS

OF PLATES.

[225.

we sec tbat at ail points of thc boundary, ~=0,

~-n ~-0,

~=0,

whicit secure thc fu~hnent of tlie. ncccssa-ry conditions (§ 215) The value of p, found by substitution in c*o=~'M,

to thé sbewing that the anatogy to thé membrane docs not cxtcnd 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 must ah-sohâve thc same period in tlie case of thé membrane, thcy nodal systems are period in tlie case of tlie plate. The dcrived accordingly idontica! i)i tlie two cases. The freucratity of tlie value of w obtaincd by compounding with arbitrary amplitudes and phases ail possible particular solutions 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 are required capable scarcely attainabic in practice. Appliances of holding t!)e boundary of tlie plate at l'est, and of sucba nature that they give rise to no couples about 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 problem of a rectangutar plate, whosc cdges are rcsisted frec, is onc of gréât dimeulty, and bas for tbe most part is independent attack. If we suppose that tlie displaccment with that of?/, thc général differcntial équation bocomes identieal 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}cto 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, exct 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 approximation, 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 . Although the cottstaut fk (which expresses thc ratio of lateral contraction to longitudinal extension wbeti a, bar is drf).\vaout) is positive for every known substance, in tlie case of a. few substances–cork, for cxampic–it is comparativelyvery 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 conditionis tilercforc not devoid of interest, though the results will not be strictly applicable to ordinary glass or meta.1ptatcs, for which tiie value of is about 1 If &c.dénote the normal functions for a frce bar invcstigatcd in Chapter vin., corresponding to 2, 3, tiodcs, thé vibrations of a rcctangular plate will be expressed by

1 In M'dorto rnuko n.pinte of mfttorial, for wluch is not xero, vibrato m tho mnuner of a bar, it would bo noecfiHfu'y to apply conHtnutlingcouples to tLe edgea pnraUû! to the p)anp of bondinn to provent tlio aasumption of a contmry earvfttuTe. Tho oficct of thcso couples wouH bo to rnise tho pitch, und thorofora tho calonintion founded on thé type propnr to ~=0 would give )t rosutt fiomowhathigbcr 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;\ 1 Il a ~=~u'a "=~ having nocessarily tlie same period, may be combined in any proportion, while thé whole motion still remains simple harmonie. Whatever thé proportion may be, the rcsulting nodal curve will of necessity through thé points detcrmined by

Now Ict us consider more particularly tlie case of = 1. consists The nodal system of thé primitive mode, w = M, a[ ), of a pair of straight lines parallel to y, whose distance from the nearest cdge is '2242a. Thé points in which thèse lines arc met a,rcthosc through wlticli by the corresponding pair for w= u1 ('), thé nodal curve of thé compound vibration must iu a.ll cases . 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 es 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) comFiR.41.

prises both the diagonals (Fig, 41). This is a well-knowu mode of vibration of a square plate.

326.]

CASE 0F SQUARE

311

PLATE.

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 = const., a.rc = const., straight n Unes) represented by a )~J a~j first laid down, thé values of thé constants forming an arithmetical 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 M~

a:: M

+1-00 '5000 -75 -3680 -50 -25 -00

-3106 '2647 -2342

M,

~:0

'25 -1871 -50 -1518 -75 1-00 1-25 -1-50

-1179 -0846 -0517 -0190

Thé system of lines representcd by the above values of x (completed 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:Lvew 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, sarating thé regions of opposite disphcement. Thé remaining curves taken in order give thc displacements 1, 2, 3. The numericallygreatest négative dispt~cement occurs at tlie corners of the square, where it amounts to 2 x l'G-to= 3'290.' Ontbonodatlinosof squnro plate, Phil.

Angust,1873.

312

VIBRATIONS

OF PLATES.

[226.

The nodal curve thus conatructed agrees pretty closely with t!]c observationsof Strehike Hia results, winch refcr to three carefully worked plates of glass, are embodied in tlie following polar équations: -40143 -017H -00127) r= -40143 + '0172 cos4<+ -00127~cos 8~, '0013 1 -4019 -OJC8J

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'cdalong a diagona.1is '3S;')C,-3855, -38C4,and by e~culation -3900. Yol.CXLVt. rf'g~.-hfM. p. :

226.]

NODAL FIGURES.

313

obtain the By crossing thé network in thé other direction wc locus of points for which

is constant, winch are tlie curves of constant displaccmeut for that ttmdcin wti!c)ithe (UagonalsM-enodal. Thc ~
/.K\ = w The primitive modes represented by ï~ ~t or :t)= M,)~) the noda.1curve may be combined in likc manner. FIg. 43 shews for thc vibration .(~(~<.). Thé form of the curve is thé same rciativcly to tlie othcr diagODa), ii' thé sign of the amhtgmty bc altcrcd.

314

VIBRATIONS

[227.

OF PLATES.

227. Thc method of superposition docs not dépend for its application ou any particular for)n of norma.1function. Whateverr t!ie form may bc, thé modo of vibra.t.Ion,winch wben = 0 es into that jnst 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 nodal system may bu tra.ced by mcans of thc considération 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 CharlesWheatstone' that weowetlie first systematic application,of it to thc cxplanation of Chiadni'a figures. Thc results actually obtaincd 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 ed th~t cvery pomt of thé ch'cumfcrcncc is free to move n.lon~ li.ncs perpendicular to thc p!:uie of thé membrane, Thé boundary condition ahplicable 1undci' applica-bic to

shew

tbat

tbo

1

circumstances is Is ~h°0 and ibit isis ca.sy = 0,0 and easy

thcsc

circumstfLnccs

~ln

normal

== cos ordinale orùmate a.rc 10= arc

functious = or w or W=

whieb ces

involve

m

(x ), ( a.ta corner of thé square. Thus thc vibration 7Mtt

??t

oniy

one

co-

tthclU ongm orjgui being omg

22~.]

WHE,ATSTONE )S FIGURES.

315

thc noda.1systom 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.

316

VIBRATIONS

0F

PLATES.

227.

Withthcothcrsign

wcobttun

of thé diagonale tcgcthcr witb thc a. eouiposcd systum ruprcscnting inscribcd square. Thcsc foDns, which aro strictly appHcab)c to t)tc membrane, rescinble thé ngurcs obtained by mcans of sand 0)1a, square p1atu more closu)y than might hâve bccn expcctcd. Thé séquence of toncs is howcvcrquite durèrent. Frum § 176 wc sce that, if /t were thrcc zo-o, thc interval bctwec!). thc furm (4.3) dcrivcd from durived i'rom two, woutd bc or and nodcs, (41) (42) primitive l-4-(i29octave and thû interva.1between (41) or (42) a.ud (4M)or (47) wou!d be 2-43.')8octa-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 tbeHrstmoitionod pair this resuit is not in a.grecmetit witb CbLidrit's observa.tionH,wbo found a dirt'crencc of more than a whoïc tone, bc Icft: out of (42) giving thc higbcr pitd). If bowcvcr (42) , thc cumparison Is more satisfactory. Aecording to thuory fmd (4(i), (47) (43) should givc (~=0), if (41) gave Cbhubu tuund for (43) ~)-, andfor(4G), sbould give~"+. and + respectively. (47) 228. Thc gravest mode of a. square plate bas yet to bc considered. Tbc nodus in tbis case arc tbc two Huesdra.wnthrougb tbc middio points of opposite sides. That thcre must Le sueh a mode will 1)Gshewn prcscntly from considerations of symmctry, but neither tbc fonn of Hic normal function, nor tbo pitch, bas yct beeu dctcrnnucd, cveMfor 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'tnwhich thc plate wuuid assume if hold at l'est by four forces uumericidiy equa!, acting at thc corners pcrpendicularly 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 possible mode 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.Im:tss at eacli corner,

dénotes tlie mass of tlie plate without the loa.ds. This whcrc result tends to become accurate whcn~jf is re~tLvuiygrcut; otherwi.scby § 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 24 2-~q b~ ~=p(l~ Thé gra.vest mode giving a. p'Lc!) which is somewha.t too high. next a.fterthis is whcn tlie diagona.~ a.re nodes, of which the pitch, if = 0, would Le given by

(sec§174).

318

VIBRATIONS

OF PLATES.

[228.

m~tcri~ of thc plate wcrc sud. if thc that conclude Wo may thc two gravest tones would ,-(/themterv~bctwecn l'SU;. the ratio 1.1.. thc that by than express -P––d où be somewlmt greater tho intcrval t~Afth. Chludni makes of vibratiou in which modes cxi.t must therc 2~9 That bc infcrrcd from ~-o ncdcs may .~cters thc two shortcsb that su~ lu suppose (~) Fig. thc following. ~Lldcrations as Pig.18.

and thé arc CO ~or~, the Jf~, which cdgcs is a plate of T~~ P' i s 6'C of vibmting in certam be must capable of cqu~brium, ~t;on our attention on one of thèse, let us modes. Fixing uuLhuncnt.1 the th~c rcmammg quadrants conceive a distribution of over values of .<;.u-c cqual aud tl~c that two ad, in any such that each other in thé line of the arc which inmges opposite at points vibrato accordmg to thc law whulc thé If pl~tc of séparation in ordcr to kc be ~ill eoustmiat rcquircd no thus detcnnincd, thé ~hoh plate may be thereforc and ~cd, the lines C~, Thc samc argument may be uscd to prove that free. as rc~rdcd or in ~!uch botb the arc nodcs, thc ~hich in diagonals modes exi.t diamctcrs just considered are togethcr nod~. thc and Ji~onals aiso be applicd to other forms of may symmetry Thé principle of nodal diameters thc iufcr thus possibility Wo might of plate Whcn tlie in an axes ellipse. nodal of or principal in a eirele

tn sec that Fi~. (4f)), it i.s e~y a is hexagon, rcgular boan~ry fonns. (~0), (;) rrc'scntpnssib!o

229.]

PRINCIPLE 0FSYMMETRY.

319

It i.sintcrcsting 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 vibration, in what position thcy bo tnken. As the circlc develops into a. square by throwing out corners, tlie position of thèse diamctcm becomesclefiuite. In the two alternatives tho pitch of tlie vibmtinn is dinercnt, for the projccting corners have not t))CSfunccfïicicncy 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 cHipscwitli 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 square of 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 uniform, 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'J3Dbe remoYed, the ncw edgc ~173Bbeing 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 ~1/)/ For pxampic, the pitch of a ~niform chmpcd ptuto jr'

320

VIBRATIONS0F PLATES.

[230.

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.dsat A, C. It is clear that thc addition to the hrcadth

indicated by thc dotted line would augment the stifrncsaof 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 quantity. 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, Wheatstone found thut thc pitch of thé vibrations was difforent according as the approximatcly 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 its of analytical treatmcnt. A long cylindrical sitell of circular section and uniform tlucknesa 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 extension 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.1rigidity, 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 supposition that each part of the cii'cumfcrence retains its natural length throughout tho motion. rnRt!)));).186i,cxxti.p. 238. R. 21

322

VIBRATIONS

OF PLATES.

~333.

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).rtictcis 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.1as wcU !ts a. normal motion, whose relative inagnitudes must satisfy a certain di~'ercntial 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 During the motion let the polar co-ordiby the vcctorlal angle natcs of the cl émeutbeeomc ?'=ft+8r,

~=6+M.

If ds rrcscnt the arc of the deformed curve corresponding to we have

+ (~ +f~)' (~)"= (af~)'=(~8r)' \vl)cncGwc nnd, by negiccting the squares of the small qnantitles ~?-,

.(1), .(y, 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)
M.

tho constant that might be added to 86 being omittcd.

.(3),

233.]

POTENTIAL AND EINETIC ENERGIES.

If o-< denote the mus of thé clement energy T of the whole motion will be

thé products of the co-ordinatcs intégration.

323

the kinetic

disappea.nng ia tho

We havo now to cn.kula.tcthoform of tho potent!al energy K Lct be thé ra.dms of curvaturc of any eletncnt f~, thcn for tho 1\" clément of take whero ~Is a F~wc coi-responding may ~f~(8-j, constant dcuding on tho materia,!and on the thickncss. Thus

Now

and

for in the small tl)c distinction bctwcen neglected.

and
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 expressionsfor f7'a.nd V involve only squares in otlier words, tlicso are tlie ))or)~ coordinates, whose Independent I)armo)ilc 'variation expresses thc vibration of tlie system. Ifwcconsidcr only thc invûlving cos?:~ sin~ by taking the origiu of suitably, 'n = a.A cos110, îO 8r 8~=-~=~~cos?~,

r,"sin~(7), siii ?td n

wc have

(7),

This resuit was given by Huppe for ring in a mcmoir pubHshcdin CrcIIc, Bd. 03,1871. His mcthod, though more comptctc than thé preceding, is less simpJe, in consequence of his not rceognising 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.ctcristicof the iuturvaJs 2'778, 5-44.5, 9,

13-4.4, &c.

Thc corrcHpondingnumhcrscbt~iticd from thca.hovethcorctic:Ll formula?,by making 7t succcssivclycqual 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 socalled 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 of M thclattcr. Whcn a bu!I-s)tapcdbody 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 Iccturcroom 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 necessaryto 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 rccciver wassccureiyfastened 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 rcvolvc 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 this satisfactoriiy it was found nocessary to apply thé friction in the ncighbourhood of one point. It thc'n

VIBRATIONS OP PLATES.

326

[235.

bccamc évident that the spot moved tangentially whonthc boll was excited at points distant thcrefrom 0, 90,180, or 270 degrees and norma.Iiywhen tho friction was a-ppliedai 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.dmixtureof overtoncs. If tliere be a smn.Uload at any point of tho c!rcumferencc, a slight a.ugmcnta.tionof pcriod cns~cs, which is different according 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 phcuomeuon may often bc obscrvcdin thé case of largo hells.

ENDOFVOL.I. 1

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