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Page 1: Lord Rayleigh - The Theory of Sound Vol 2

The theory of sound / byJohn William Strutt,baron Rayleigh,...

Source gallica.bnf.fr / Ecole Polytechnique

Page 2: Lord Rayleigh - The Theory of Sound Vol 2

Rayleigh, John William Strutt (1842-1919 ; 3rd baron). The theory of sound / by John William Strutt, baron Rayleigh,.... 1877-1878.

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

Page 3: Lord Rayleigh - The Theory of Sound Vol 2

c 3

THE

THEOU Y OF SOUND.

Page 4: Lord Rayleigh - The Theory of Sound Vol 2

i

Page 5: Lord Rayleigh - The Theory of Sound Vol 2

THEORY OF SOUND.

JOHNWILLIAMST.RUTT,BARONRAYLEIGII,M.A.,F.R.S.FOttMRKLYFULLOW0FTKIXITVCOLLEG):,CAMr!):inGE.

MACMILLAN AND CO.

1878

TRK.1..1..1 .ln J

VOLUME M.

bonbon;

[~.Z~/t~MMn'cf~ ]

DY

Page 6: Lord Rayleigh - The Theory of Sound Vol 2

<E'n)nbWt«!t:

i'~tffT)!t)nY(').(').AY.t!A A~T')'UKttf[YH)tS)TV)'ttKSS.

Page 7: Lord Rayleigh - The Theory of Sound Vol 2

CONTENTS.

CHAPTER XI.t'AOU

§§23G–254.1

Aerial vibrations. E.pmlity uf prcswuroin n)l directions. Equations of

motion. Equation of contimiity. Spécial form for incompressible ihnd.

Motion in two dimouBioua. Stroam function. Symmotry abont nu axis.

Volocity-potontial. Lagrango's thcorom. Stokes' proof. l'itysioU in-

tcrprotntiou. ThonMOti's iuYCfiti~tion.Circuhttinn. Equfttion of cnn.

tinuity iu torma of vcL.city-poifmti~. Expressioniu po)nr co-ordiu~teH.

Motion of meomprossibio Unid in fiimplyeonncctedHpMMis dotcnninod

hy bouudn.)? conditionn. Exteusiou touinttiply connectcd spMcs. Sphcro

of u-rotittionally moviug iluid suddeuly soliditiod wouid havo no rotation.

In-otn.tiotml motiou bas tho Jeast possible euergy. AtmioHy with thoorioff

of heat nud ctuctricity. EquatioMof pressure. Gênerai oqu~iou for

sonorous inotiou. Motion m ouo dimension. I'ositi\-u auduc~tiYc

pro-

grcssivo wt~vea. Rolatiou bobveou yclocity aud coudeiMatiou. Har-

monie type. Euorgy propat~tod. Haïf tho ouo~y is potoutia), aud

htdf Junotio. Nowtou'H ealcutatiou of yclucity of Hound. Laplaec~ cor-

roction. Expression of volucity iu tutius of rutio of spécifie hcats.

Experimout of CIcmout and Dosormcs. liaukine's ualeulation trou)

Joute')) equivalout. Possible cilcct of radiation. Stokos' invusti~tio)).

Rapid HtifHng of tho sound. It appearsthat communiHation of Itcat ha.s

no Beusibk oSect iu practico. Voloeity dopondunt upon toupcraturo.

Variation of pitchoforgan-pipos. VclueityofMuudiuwatb)-.Exact

diCcreutial cquatiou for pluuo WttVM. Aiq)iicatioiito wayes of tlicory

of Htcady motion. Ou]y 01 ouo suppositionns to tho law conucctuiH

presHuro aud donsity Otn n. wavo maintain ils fo)-!u without tho assi.st-

ttncsof nu improsscd force. ExpiauatiottofchanHooftype. Puisson's

cquation. Relation botwoou Yotoeity and condensation in a pro~'ssivo

wavo of tinito ampUtndo. DifKcu]ty of ultiniatu disccntinuit.y. Harn.

shaw's intL~rats. Rx!Uiann'Niuvestit;atiun.Limitud initial dist.urbanc<j.

Expcrimontal deturmixationH of thu -v'dooity of sound.

Page 8: Lord Rayleigh - The Theory of Sound Vol 2

Y) CONTENTS.

CIIAPTER XII.

r.\<in

§§255–2Rf)-H.

f.

Vibratiousintnbcf). Goocrnifonn for Hinipbharmo~io type. Nodoannd

loops. Condition for (in opcn ond. lu Ktfttionnry vibrations tboro nn)st

ho nodûH nt intct')t)s of Rottcction of putsCH nt doned ;md opou cu()s.

rrob]f!)ui]icnmpouudvibMttons. VibratiouiufttuboduotucxtoronI

sourcfs. H(~hc))dsf)poi). ]'rf'K''ûB3i\'o~vod))ûtodiHt))rh(H)ccntn)'t!)t

end. ~fotio)iorif.;i"~ti))Hinthotnboit.sLdf. yot'codTibrfttiuno! pistou.

Kuudt.'ticxpfriment.). Stxnn~u'ycij'fHuits. Vibrations ofthoeoJumu

of air in an o]'~n)]-pi))o. Ru)atiou of lon~tb of \vn.o to lougth of pipo.

Ovcrtoncs. Froquoncy of an organ.pipo d(.'pci)dH upon t)to ~s. Com-

parifiou of YcloeitiûH of Houud iu Vfn'ioun RttsoH. Exnminntton of

vi)))'nti))R oolumn of nir by motubrnno and mutd. Dy Kiini~s i))nucfi.

Cm'Ttid pipes. Ih'nnehud pipes. CouditiottH to ho Kntisnod at titt)

junctioi].ofco!)noct<id]iipc!H.Yftn~bIotiGct.ic'n. Approxinmtoc~cuta-

tiottofpitch for pipes ofvfn'inMoMctio)]. I])))nc)tooofYaritttio]Jtof

soctiou on prof.ft'e.ssivo wavoH. Varitttiou of denffity.

CHAPTER XIII.

~2<ji7–65

Acrin) vibration.') in nrectans" chmubo'. Cuhicmbox. Pr.sf~tnnt'oof

rooma. liectan~ufar t))bo. CouipoHttion of two e<pt)d trainH of wnvoM.

I!(;f!octio)i)'y)i)'i~;idp!ftnûWft)I. Ct'con'HinvestigatiotiofrcOccUonfmdMfrnction of pfnno W)t\'M nt n [duno surface. Law <~f !'inM. Casu uf ttir

nnd watt;)-. Hot]t ixcdin ~scuus. J''rL'fi))<!l's cxprcsHion. J!('f!cuti(~n at

KU)'fneoofnir)mdhydrof{t'n. Honeftion fj'omwn.nunir. Ty))d)L)I't)

oxpct'itHcnts. Total t'(!i)uctiou. RnUcctmn from a pln.to of ~luito

ihiHkucss.

U1IAPTER XtV.

§§273–SD.~85

Aj'Liti'dry initial disturbnneo in an uulimited .at.)uobpLL')'o. J'oiHKon's aolu-

tion. Verification. Limitod initiul disturbanco. Cnf<o of two dimeu.

aiotts. Doductiou of Bnhttion for n. disturbauco contiiiiially ronûwcd.

Sources of Hound. Hnrtttojuo type. Vorifiottinj~ of soJutiou. Sources

diiitriLutcdovcrn.Rurfaeo. hifi!)itûp)n)icw)(l). Shoot ofdottbio

Hourcef). Wn.vcs in threo dimnjtax~t' symjuttrieni about n point. Har-

monie ty]ic. A coudfnsf'd or mre'ffcd wnvo ctuntot exist niono. Cot)-

tiuuitytbro))(;hpo)o. Inititdcirc~jnstn))cc.s. ydoeity-potentinlofa. a

givou h'ourco. CtHeubttion of eno~y Mnittcd. SpcaMnt? trompet.

Theory of conicnl tubes. Position of nodcH. Coinpositiou of vibrations

fronitwosiuJpiofiûurc'csofliiiopitL'b. Interférence ofsoundufrom

cjcetriodiy jmtiutftinc'd tunin~ fot'hH. l'oints of (iikueo. Existoicu

ottu!) to be infcrrcd from eoutiidcrutious of hy)!)mctry. CuBe of bd).

Page 9: Lord Rayleigh - The Theory of Sound Vol 2

CONTENTS. vu

Expérimental motliods. Mayor'a oxporimont. Sound shadowa. Aperturo

rAa~

:np!)-Gcn. Huy~htinri'~o~rs.Ccner.T.:f-.).ud~h~(dow!

Obliqua Rcroon. Conditions ofapproximatoly compictorofloction.

DivorginR Wavos. 'Variation ofintcnsity. Foei. RoOoction from

cnrvod snrfacoH. Elliptical nnd parabolic reflectors. Pormat's prin-cipto. Whisporing e~ories. Obsorvationa inStranl'aenthodraLProbaMo oxplnnntion. Rosonaneo in buildings, Atmosphorio réfrac-tion of H0)md. Convoctivo tiquitibrium of tcmporn.turo. Diilorectialoqnntion to path of ray. Rofractiou of sound by wind. Stokos'

cxpiMmtion. Law of rcfra.ctio)]. Total reflection from wind oïcrhenc!.In tho case of rofraotion by wind tito eonrHo of n. Sound ray ia uotrovorsiblo. Observations by RoyjMids. Ty~Il's observations on fogfiRnaIs. Law of diverROt~o of Mund. SpoakinR tmmpot. Diffrnotionof sonnd through a small nporturo in nn inCuito screea. Extension ofGrcon'a thoorom to

vcIocity-potoutiKis. Hoimboitz'a thcorom of reci.

procity. AppHcntion to double Bonrcoo. VnnatioB of total onorgywititiu ft doscd spaco.

CHAPTER XV.

§§~C-302 .13,

Sceondury wnvos duc to a VM-intion in tho modium. Botativo importanco of

Mcondury wnvoa dcpondH npnn tho w~n-Iength. A région of a]terod

comprosaibUity acts lilo [t simple sourco, 0. rogion of n.Jtorod donsity likofutoubio sourco. Law of inverse fourth powers infcrred by method ofdimonsions. Exp])umtion of harmonie cchos. Altération of cimmotorof

compound sound.Scoondfu-y sourcos duo to excessivo nmpHtudo.

Alteration of pitch hy ro]ntivo motion of source M)d récipient. Expori.mental IHnstrationH of Dopp)er'H principto. Motioti of a simple source.Vibrations in a

rectanguiar chambor dnû to intcmal sources. Simpiosource situatud in au unHmitcd iubo. Enorgy ornitted. Comparisonwith conicnl tubo. Further discussion of tho motion. Calcuia.tion oftho réaction of tho air on a vibrating cireular plate, whoBo piano is corn.ploted by n. fixed Bango. Equation of motion for tho plato. Caso ofcoincideuco of uatural and foi-eod periods.

CHAPTER XVI.

§§303–322..1UU

Thcory of rosouatora. RoMnatorcomposod of n piston and air rGscrvoir

rotontml onergy of compression. Poriodio timo. In a largo dMB of airrosonatorH thé compression is

sensiblyuniform thronghout tho réservoir,nnd tho kmetio onergy if.

senaibty eonfined to tho noiHhbourhood of thoair pasMRca. Expression of kinetic one~y of motion through pasmgosin tcrrns of cloctrieal

eonductivity. C~)cu)ation of nnturnl pitch. Caseof sovoral eliannols. Snperior and inferior Jimits to

conductivity ofchannols. Simple aperturos. Eltiptic apertnro. Comp.LriHon witli cir.cular aportnro of cqual nroa. In

many caseB a catculatiou bascd on arca

Page 10: Lord Rayleigh - The Theory of Sound Vol 2

1CONTENTS.

PAOEoniyi.ss.tû.ciuut.

S.tponorandmfonor!im:t.stothuœudnctivityof

PAULP

noehs. Correction to )un~t)t of passage ou acconnt. of opon end. Con-

~<'t'y~~M~nu..tcdi)yhL.ar:ycyiiih!rIcaItim-faoosofr(..Yo!ution.Co.npar.son of <-a)cn)atcd and observer pitch. M..)tiplo resonaneo.Oatc~at.on of periods foi- doubto rcsonator. Communication of eno~yto cxtornai atmospi.crc. Hato of dissipation. N.uucrical exam~u.Lorced vibrat.ons duo tu an cxtornal source. Hulmitoitz's th~ory of

opcn pipes. Con-ectioutoJcngth. Hateof dissipation. Inf)uoucoof

HanHO. Experitnontal mott.od.s ofdctcrmininR tlio pitch of resoiintors.

DMcuMton of motionoriHiuntioH within au op<u pipo. Motion duo to

oxtcrual Boureos. Effoct of cn]a~omcut at a closod end. Absorption ofSound byresonators. Qnmcho'. tnbM.

Opomtionofaro.souatoreto.soto a sourco of sound. Rcitiforeomeut of sound ),y re.4oi-intors. Idea)resonator. Oporatiou of a rosonator eJoao to a double source. Savart'Hoxporimcnt. Two or more rcsona.toM. Qu~tiou of formation of jetsuunug souorona motion.

UHAPTER XVU.

§§323-~5

Ai)plieatious of Lapiacu'H functiuns to ncuusticat prohiems. Cunc.ral .sohition

mYotv.ug tho terni of tho order. Expre.s.-jioa for mdLd vclocity. Di.

vergent wavcs, Ori~iu at aspJ.L.rieal Hurfaco. TLû formation of HonorontiwaYCH rcquu-cs in ~nM-al a eortaiu arca of movinH Hurfaco; othcrwiso thomcchauictd couditions uro HatiKti<d ly a )ucat tmnsfercucu of air withont

appMcinh)ocondc))H,ttiou or raréfaction. StokcH'discussion ofthoeffectof JatoruI motion.

Lo.siio'saxpc.rimf.nt. C.den)ntiou of numcrica] rosult~..Lho tc.rm of zcro ord~.r is i.sua[Iy dutieicut w~n i].a sound ûi-iHinatoa intho vibration of a 8olid )jody. licaction of thc

snrroundinH air on a

Dg.dv.brnt.nHSphoro. Incroascofotïcctivoiuurtia.W).cnth~p)tcru

'asma)Imcomparisou~iththowaYo-Ienf;th.t))croiabutlittlL.connuu.nicntion of euorHy. Vibration of an eHi]Moid. MuMipIo Hourc~. Incases of symmotry Laplaco's f.mctions reduec to L~endrG's functionsCaleulat.ou of tho encr~ M.uttcd from a vibmtinn sphcrical surfaceCaso whou tho disturbaneo ia limited to a f,)naH part of tho 8phcric.UBurf~o. Numorical rcsu)tH. Effcct of a sma)) f,p]~.ro Mtuatcd doso to aBom'co of sound. Auatyiical tranMformati. Caso of coutinuitythrongit polo. Aualyiieal MpressiouH for tl.o -~loeity.potcntial Ex-pression in torms of Bes~I'H fnnctious of fractionai ordM-. Particntarcasos. Vibrations of f~s confinf.d within a rin:d tipherical envciopoI:ad)a)vi),rnL<i(.s. Diauh:tral vibrations. Vibrations

(..xpresscdby aLapiaeo's fonction of tho Hccond ordM'. Ta))]o of wavc.i~~th.s. li~ativopitch of varions toiles. Général motion oxpressiblu by Himpio vibrationsCase ofuniform initia) ve)ocity. Vibrations ofKasiNchtdodbft.wocu

eoncottricsphurica) surfaces. Spitcricatsitcctof~as. Investigation oftho dtsturbauce prodncod whe)i ].)anc wnvcs of sound inipinge npon a

spiiorical oi)stactc. Expansion of tho vetocity-potentiat of p)ano waves.

Splicro tixcd aud ri{;id. Intensity of seeoudary wavcs. rriuuu-y wavcs

originatinn in a sourco at a nnito distance. Symmetriod oxprossioufor

socondtu'y wavcs. Case of a f~seons obstacle. E<iual conipreosi.bitities.

Page 11: Lord Rayleigh - The Theory of Sound Vol 2

CONTENTS. Ix

C'HAPT.-t XVin.?'~m

§§33G–343.?~3

rroUcmofn.Rpbt'rieatiaycroftdt'. l', Expansion cfYtdoeity-potcntifd in

Fouricr'Hsorif.'s. Din'orcntm.t équation Bfttisficdbyc.ichtorm. Ex-

prcHscdintormsof~n.ndof~. Solution fur thoc~stiof symmetry.

Conditious to ho Rn.tisdcd whon tho pôles arc uot sources. Réduction

tot'egcndro'Hfnnetions. Conjugntopropcrty. Transition fromnphn-

ricnl to p)n.nn hyor. Densel'H functionofzoroordcr. Sphorioal

tayor boundcd by pnrallola of Itttitndo. Solution for sphorictd layor

bonuded by smnH circlo. l'nrticular enscs HoluhJo Ly Lcgondro'H fune-

tiouH. Conomi prohicm for unsynnDctricat motion. Transition to

two dimcnfiiona. Comptoto sointinu for ontirc Hphoro in tcrma of

Lftpiftfo'sfunctiins. Expansion of fin nrhitrfn'y function. Fonnuin.

of dérivation. Corrospott<lin(; formula in Dcssol'a fnnotions for two

dimensions. Indupcndcnt invcstit;n.tiou of pifmo proLJom. Tranavorsc

vibrations in a cylindricat ûnvûbpo. Cn.8û of uniform initia] volocity.

Soctor hounded by mdini w~ts. Application to watc-r w~vcn. Vibnt-

tionn,not ncpcss)u'i)ytraMs\'t~'Hc',witi)in n. circnl~u' eylindur witi) piMio0

end~. Cn)np)oto Hointion of diff'it'cntial ('(jn~ti~u without restriction

n.Hton.bsoncnnf potarRourco. l''uriunIn.KfdcriYn.tio)). Expression of

vulooity-potentift) by df'sconding ffcnn-convorHcnt xericH. Cnso of pure').

divorcent wf~vo. Stokcs' n.pp)ication to \i))rntinR Htrinsa. Importn.ueoof Boundin~-bon.rds. l'rovcntion of latéral motion, Volocit-y-potcntitti

of n lincn.r Houreo. Siguificanef! of l'etardfttiou of rrobloui of

)))M)û wavoa impinf.;i))f; upon n. cy)indrica.I obstacle. Fixcd. nnd ri(;id

nylindor. I\rftthcnmtic[~!y annicH" probicm rolnting to tho trftnavûMc

vibrations of nn dMtic soHJ. Application to thoory of light. Tyndfttl's

oxperimonta shewin~ tho sm~IIncHs of tho ohstmction to pound nitorcd

t'y hbricH. whnso porps f~rr' 0))cn.

CIIAPTER XIX.

§§344–3~8 280

Fluid Friction. Kfituru of viscocity. Cocn'tcient of viscocity. ludcpendont

of tho density of tho gff. ~raxwell'a oxpcrimonts. Cotnpn.rifion of

équations of vincous motion with thoso fipplicnbto to tin cin.stio fiolid.

Assnmption thut ft motiun of ~niform dihtatiou or contrfteHon ia not

opposed hy viacons force. hitoTtcs' expression for dissipation fonction.

Appheation to theory of ]~utû wnYos. Craduni dceay of harmoniewaves maintainctl fit thn ori~i)). To n. first approximation thovolocity

of propagation is nnn.u'c'cted t'y viscosity. Kumcricn.~ cnleuhttion of

cocûtciont of decity. Tbo ciTeot of viscosity nt rttmosphcric prcHsuro ia

scusitilo for vory hiRh notes only. A hiss boooinoa inaudibto nt n, mode-

rato distunec fron) ils h'curcc. lu rnroncd n.ir tho hû'ect of viscosity ia

muo!) ineroftscd. Transvorso 'vibrations duo to 'viscosity. Application

to caictunto cffcctH of viscosity on Yibmtious iu nfurow tubea. Holm-

holtii's nnd Kirehhon''s rcaults. Obson'ntions of Schuoebeli nnd Sccbcck.

Principio of dynamien.1 simiifn'ity. Thcory of shipa fmd rnodets. Ap-

plication of prineipio of Eimi]Mity to dnstic plates.

Page 12: Lord Rayleigh - The Theory of Sound Vol 2

CONTENTS.

Correction to Opcn End 9~

Noteto§ 273

Note on ProgrcssiYc Wavcs ~'7

APPENDIX A.

l'AOP,

Page 13: Lord Rayleigh - The Theory of Sound Vol 2

J:.Jf.J

CHARTER XI.

Ai~lUAL VIBRATIONS.

23(!. StNCH t))catrnosp))erc is thc abnost miivc'rsa] vehicic of

Sound, tbci)t\'cstig'atio)toft.bc vibrations of a

gascm).s mcdium

bas alwuys boc'n con.sidcrcd tbc pcculifu- problon of Physic:~

A<'o))st,ics; Lut m ni), (.'xcL'pt :). fcw.sp~-i:dly simple qucstiott.s,

cit)('f!y n'Lt.ing tu Lhcpt-(~):)g:Ltin)t of sumul iu ouc dilnbnstO)), t!t0

]!t:Lt.hc)n:tt,tc.d dinicuitie.s ;u'u such that. pro~rc's.s bas bL'cn vcrystow..f~vu!) when a Utcm'utical rc.sult is oLtuinc~, iL ofto)

])appcns that

]t cannut bcsubmit.(('d(.t)t)tctcsbofuxpcrimc))t,indcf:Utkcf

;u;cun).~ nic't.)hj<).sofmL';tMuri)~ thcintcnsityof vibrations. Iti

.tncj~rts oi'thcMubjuctnHUt!~ woctm dois tu suive thuso

~'robh'm.s \vh~Me mathon~ticu) conditions :n'L!.suf)iui(.-)iLty sitn)))L- to

:'d))utof solution, :).))([ to trust t" thona.n(.lt()~cn).r:t.tp)'i)tei))t<js

""t to fcavc u.s (juitci in titc d:n'k wit)t respect to ot)tur (~ucsiiousm~Licit wu

n~y bu ititurcstcd.

Ja thc prcs~nt c!)aptcr wc shiU! rc'g-:u-d f)ui(!.s nspùrfL'ct, H)at is

tosay, wc sh~![ aHsntnc th:tt t!)C mutu:U action bct\VL'cti any two

port)un.ss(jpamtu()by auu)<).)surfaœ!s);o~i«~o~Hi'ce.

itcrc:Lftcr wc Hha)t say souicthing' about Unidfrietio)i; but, in

~ncra], acoustica) pl)cn(nn<-))a :u'c not mat<t-ia)]y di~turbct! hy.('h (]uviation from pcripct ftoidity as cxists ))i tbc' case of air

a)xtot!)crg:tscs.

Thccqu;)j:ty of prcsfini'L: in a!) diructtons about a givcn point

is !), ])'jccMsary cuns(.'<ptcncc ofpermet Ouidity, ~hutbcr thcrc bc

rcstor]Moho)t,a.sisprovc<}byconside)'in~t)ic<(pu)ibi-iumcfasmal) tctrahcdron uudct- tbc opf.'ration uf tbc fbm) pressures, t])c

Page 14: Lord Rayleigh - The Theory of Sound Vol 2

0

EQUATJOX.S 0F FLUJD ~o-f-fux.t~,L-

~r~ 'r 'r-1"

P~un. "y O. <)

thcir w I 'r ~au.

~o~)Lht'f<L.

Lcdc~.U~,

'h.r.int. ..inOc dUIIOLud

by l,,

~<lil/'('{'s

~nc.nt,~r, ,). ,~j,,J¡riUIll is

~=~(.Y< r~+~)

~t'i' "<c.clmtya cl, cly, cl.= iu tlte co-ordill:tlL's uf tlm ]lUill!, ut whielt l:ltc

¡;idl'l'illg 1)1{'cl/ililil¡ri¡llll of a small cylill(1c-1' ",illt i~;tt L'llds, tllU

(ltmsu vf l'U-Ol'¡Jilla(l:S ¡lI'c~1'1'l'il'celi\'ldycl,c, clJ, cl.r. Tu uhlaill tlm l'(I"atillll~ u1' IlIO/iulI \e ¡IaVU, ilt :,U:<:l Il"dall eu

~'t). D'AJu.nLerL-.sPriucipf., ,~r,)y r.pi~.c Ac.

by Y-

) -L~Ct~'crc

~.d~th.~c.-atiuuscni~~idc .f~uideun-

~rcd. Thus

~r'-p~u.,un, ))u).f ~,t. ,,j t(..nn.s

of~. f.

})fll'ticll', ",ltidl(,l~1' it muy 1..Jl" tli:vt aLtlie t.illlu t is fouud a tlmpuiut rc·, ,1/, 1l.f~ur a HIJlflll illtcl'd uf

t.

~s,.f.bvc)~ity~.U~<.n, (i,,t

,j,~thu otlicr hand

eXIJI'usscs thechauge in Il tlw

\'I,luei ty of tin; umjimcl

EHL~u jM&p~M, buti.iutcs w.t).t).<;))tn,j. 'Jutf.is

which is uut 1ixeù itlslrtcc, Lat IIluVOS wit.h tll() ilnicl. Tu tjtis

..ot..U.~,dt.i.

;hcd.a,

Page 15: Lord Rayleigh - The Theory of Sound Vol 2

237.] EQUATION 0F CONTINUITY. 3

position in spacc (dutû!'min(jd by thc vn-incs of .T, y, z) is rctn-incd

1.] ]]).

l f 1. 1 1 1inv.u'i:).biL!,w)t!i<ji)i

itIs:tCL')'t:dnpnrtic!eofthci)ui(lo)iw))Ich1

intention i.sfixcd. T)njru)!t.t,io)itbotwucntlK't\okIndsoi'dn't'ui'-

uttLiatiun wi()t :'(-'s{)cct tf titnu is uxpru.s.su~) by

;H)())nu.stbt~')e:u'!yc")iCL'i\'cd,t.)")nn']ti~!).I:).)'~cctas.s<)fi)t)porta))t

pru))!c)))swith \\)ti<j)t wcs1tanhuucc'upic(tint.huM(.)tn.'I,t!K!()i.

tiucUun pt'acticaDy di.-iappL':Lt'.s.~Yhcnu\'cr thû tmjtiott. i.s vcry

sm:).]!, tlic tut'ni.s «,- ~c.d)'Ish in !'L']..ttiv<.i importance, n.nd(f.~

7)

utUmatcly~=~.

2~S. Wc havo i'urt,))cr to expruss tt)C condition that therc is

n()Ct'catiunnt-:umihi):).t.i))n<jf]n:)tturittthcintL!riu!'<)ft'!tu<!uI(L

H'K, /3,'y bcthuud~~ (tt'a sni:dl r~ct:Ui~'))!:u' para!)u!cpipc;d

{'.u'idiL']. (.<)th('axt'S(~'c<)-urdinat.(;.s,t)tu<(U:m<.it.yùf]n!ttturw))ic)t

p!is.s(.s (~)t uf t,))u Inc)udud sp:L<t.' inthne (~ iii exce.s.s of ttmt whidi

f'))t~r'<i'<

thu sn-c:)Hc<) uqu~tion of continuity. WLcn /3is constiint (wit))

n's])C(;L tu butti timu and .sp~'t'),t!)< ('((H:)t.io)i ansumus titc simple

forui

lu prohicms conncctcd with suund, thé vclocitics and the \'fu'ia-

tionuf(1cMsityarcusua]!ytr(.LtL!d:)LHsm:t.![qu:mtitic.s. Putting

p =~ (1 +A'), whurc cailcd ttto co~M~'o~, i: small, and ncg~ct-

iug 1 1 zi ~L .Cc.,we 1

i!igt)ieproduct!M-y-,Ac.,wefind

Page 16: Lord Rayleigh - The Theory of Sound Vol 2

.ST!!t.:A~[-r)rx<'Trnv r~nn c>~t~u,

~ci.) ~s<).

fil

~Ln ..sc.t.iy

p~anututhcphmGuf~

~1'°'"Lid,f.

fll'lJitl'rtl'Y. TIlt! 1'nuctiinl is call1,t! the.strcrmi-l'mnctiml, HiIJl'<: tlll!

~i:l.r~ cnrves't = ('UlIstnIJt, ~1'Imn tlm mnticul is

Htl'ady, tlmt is, Hlways t,lo;

"~<h~HII1'-ti) £11'

:llnl5~tic~ully, llre Huh,tit.lltioll of ouc f'm~c.tir~n t

Amutllur ('rUi!! of111111()l'ttlll(' iv \ll'lI tlmr~: i,~

Hj'llIlnctl')' J'olilldtllat nf ,(',

li:l'I'ythjll, is tlH~n('xllJ'l)ssiJ,!o il

tlm 1»I)tioIl ln kt.'sp!al'Ûil pl:IIIl'S pnsHillg tliruylr tlm uxis

of'SYIIlIIIl'tI'Y. II' llm VI.h:iti"<" m\

)'!));t.i)d;t))().)(.rp(;)), r)~rf,,f)n.v;. <'Sj'IlIIIIC'tI',)' 110

~rcun.'i.ity i,–y

In ?)h])n.sf,a]) (it~c. ,<) i- we s]¡:dl ll:we fu

rr

in \'iltlll":

'f'))U))mh..n.rml,r-1-mlr~_l_.i~,rh

III' al. I¡IIO I¡W/III"¡/'u pel'fect ditr(~l'eIJtiaIclc~, it, will rurrt:lin so tur ull

Sld¡sl.'III/I'Il/'

alld Le tlmn

Page 17: Lord Rayleigh - The Theory of Sound Vol 2

:23f).] LAfiHANCE'~)s THEORE~r. 5

Hf't in motionLy

cnn.scrvntive furcc.s axdpressures

transnuLt.cd

trL))ntj]n.:c.\t.c't'i<'r,t)K!~)t:Lnt.ittL's

(whiuh Ave s)):t]L dénote, h y ~) c:u) MLiVO' (bpiu-t i'ron zero.

Wt.;ass)nu<jth:Lt.pi.s!).iunctio!t()rF~,andwcH))!)))writc fur

hrcvitv y

~.)'h('t.'<[Uatit))tS(')'in"tit'n<)ht:uuedf)'uni(I),(2),§~37,tn'e

\nt.h t\vo ot.hcr.sof tlie f~rxt r~f.t.tmg to y:md .?. Hy

ttyputhc.si.s,

</Y~r

~y"

s~ th:tt ))y(!i(1'c)'(!)ni~tm~'thf <u's),uft!)C:~)0\'c cqu~tit~ts wit.h

respect tu 7/:UHtL))L'St'r(m() wit,L respect U').nd .subtt'acting,

i.ûc)i)nn):)t(! cj !)))'! t!ic hoprcssc'd i')i'ccs,(jb).:Li)UtJ~u'~t:(.tiuus

\\)Hc)tju:).y).)CputI))t.uth('r<))'in

1 two ullici;s 1 s,wu; furm ~iviy 1~ J\Ytt)i t\vo oLhcrs oi' thc s:uiiu furni givin~-L/C JL/C

Jnthuca~ of.'u) incotnp)'cs.sib!cfh)id,wcmaysuL.st.itut(jfnr

</« f/o < i,

+ ](..spqmv!).iL't)t,!m(tthn.sùt)t:)n)

«~' <<

which ?u'c thc cquatiott.susft! hyH~hnhoItxtLstitcfunndation

uN)i.stt)C<)rL'msn'sj)t;t't.m~'v'))'tiC(.s.

1)-' thc motinn bc continuons, thc cocfMcicntsof ~,?;, ~in

t))C !~)()vnc())t!'Ltio)).s!u'(' !)]!)iuitc. LctZ'1u!tut(;thuir~)'c:LtL'sL

]nt)nG)'ic:dY!t)uL',f).))ttntLL'HU)uof thé nuiUL'ric:d vaincs (~

By i~yputhc.sis,Hisiniti:diyzcro; tItC<)Uc.sLi(;a is w!)L'thcL' in

Page 18: Lord Rayleigh - The Theory of Sound Vol 2

LACRAXOE'S THEOREM.r~Sf).

<~ c. ,,f time it can b,Tho, c.~~y~

'r'is

"'° ~i~'TT its

..c~ ~-°~" "°~tlie sulution of the c(I1l:1tion

~l ,~irrlimv in t1m :u't.n;tl casc~, n CHIlllot dep;ut fl'ùlllZeru,

;t!e'U!nse.j.si.j.),t h.t.),.)

fnrrosal'!illg mn e;mlt

lr.lrtinle hro-}lurtioll:d tu ilsveloeit)', ns Illn.)' IH, sel'II

}I)' sltbstitlltillg' .l-rcn1'-K i'~ K ac~, f"r .1~, .).; ir1 (~>j'. '1' it i;J 1ilie,

J IC10, ur oh III:').

'lit II. iii it

i. H!rWISU \VIt 1cxist in

IJllids, mul vrc clc-llcndellt o0 the oclolinc \Iocities uf their

parts.

~H~

.<)<c.tdw.d.~v<sdMt.,S..'y ~u,c).y;[,ut th,tt~cw ,r, is

~f.t~mc,t .y tosllcw that il-, aud"en t., .E, {'v,u, t),t;,r ,)i~rcntMt

is vpoint tlmt is

.n

.v~w, lcticm

~.r~r'

Y.

tll;tt

'?

itto P.~s ,r; Yalljslll.'s ia tllc limit, nnt

to tlo: first orcler, Lut

~r~'<)on<t).cc,

~t

~"<' '~n ~) t),c ~m"°nbody.

~t'.ocf,tio,,Iml 1>ccn 8 CF..s, aH the ditli:1'vlltial cocflicimts of s witlll'o¡.;pectw'illt if' (li(l HO, and tllcn il,

migl¡t hc in-fClTcd1\'gitilllateJy that s cOldeI HOn'1'

var)' t'1'01l1 zc:ro.

By H tlll!orel!l rIl/(' to8tll/('S, tllc II1f~111C11t5 of 1iIOIIIcntuIll :t)mat

.~F: jllfillitesill1aJsj:,lmric:tl hurtiutlnf' Ilui~i

0(til~ti tui;, ~7, r, lI1uJti¡dil'd hy tllc II1t)J}W/lt of

"L~='s.r:

=~

nf'gl"C:1ing tlie torms 11t'pl'IJ']"llt nn iuertin, \0olitill'il 14j11:Iti~llH111~1~IiC7ll~IU to tIlL. 11)(-tioli of L'JL'ell'il'ity tlwuuglr 111Jiflll'Ill¡;olldlldol'R.

-(~?;7/7'] un. p.(,7.~.A.I~r,nir~mie.)7.

Page 19: Lord Rayleigh - The Theory of Sound Vol 2

230.] -1HOTATOtTY VELOCITIES. 7

ns HK' c~mpc'ncut rotatory vclocit.tcs of thf: HuIJ n.t thé point to

wijichtheyrufcr.

If vanish thrnu~hont, a. spn.cc occupicd ~y moving

<)ui() :n~y.s)n:dlsphcrica) portion ofthcttuid if suddcn!ys<jitdiflc(l

wonM rct:i.in on)y motion of trimstation. A prdûf of this

pn'p~.sitiuuin r).)K'r:L)iscd

iona wi)l he givcna, littic iatcr.

i,:)~r:u)"c;'stt)Cormnthus <'onsi.st.sin thé assertion t)):ttp:),rti.c!cs

ut' tinid ut :my ti'nc dt'.stit.utcuf rotation CiUi ncvur ac~uirc it.

2K). A st)nn;]):tt()it't'(.-rcttt )not)ciof'in\'c;st.ig'!t(ion]):tshuen

:nL.ptcd hy T!)o)nsu)t, whic)i at'tut'ds :i hi~-)t)y uistructivo vicw

t~'t)nj\v)H)t('s))'!)jt'ct'.

J!yt)tui\un]:m)Lcut:de'p)atiuusW T 'I1

IntoH'n'tIn~t,his c'juidiu'i ~!ong !ny fmitc !U'c

7\ moving'

wiLh thé Unit), wch.tvc

in which sufïixcs (~nntc ti'o vahœ.s of thc bmckctcd function

:)); t1.<; puiats ~nd r~spucti\'c)y.If thé arc bu a. complète

· ch'cuit,

Page 20: Lord Rayleigh - The Theory of Sound Vol 2

CIHC'ULATfO~.r.QL-

L")))WO)'(!s,

~7<f/~)~/t;r,/

,') C~r~~y;tpt! ;Ll(Ia;ci~ 7'UrlIrtG

nernuins con,tunttlrrurr~hout nlltinte.

isnppmpl'iatt'Ij' ca.l/l'i! t,llc cincu-

lcction, :ual theprllposil ¡l'" IlIn,)' )lu, st.utucl

-morrinr~ vcritlr thc,llrticl ~'e-'1/1((ÍII8 cumv'tmot.

L.

as~.f~.t).

'.y~s tr.iU~ Lu").

~hy c.v.<,n.j.j, ail)'.s~ .c. .),~

''caco)np)..<G(hn'<rc)!<i;,).'y-t-

~td.nin..j~i.)

c. c.u~c.r..u~i.n,

u-.)~iU..h~~t, ..si.

~o~a ,,f,i,

I.h. cn.sc a)) U.~can h. is

tuI.nf. vt.uut

p..ss.<.uf.st~.r.o.c.pi..) Ly In.o~tionaHv

<i:î

nn~utu.,fy.cii~

;lru s;liclto hc

r~i~jc

..I..n L,f.n<h..t

~=;tiunally 1I1ovillg' flnicl.

'\Vit).in.nova)sp.cc..s,.d.s<hat

i~)~h.)byanc.))msoi<) ~)c.rc.n.s

,-cc.n<.it.)c.i U.f. if .f ul' j~

~vc.n..t.uiu,.diy.U.crec. bc

,drc..).tir.Lic](I~(~II C\\l'I'U (1rawlI wiillill it. HlIC:1tspa,('cs ~re C:LJIed

simply-

~r~j~r~surliwu d' :1I1 ;tnclmr rils" a clmmi <!1Jl'ugoillg rmmci tltc ring is

reclucille tu a point, and tIlcl'cfol'o t}I()!'üma)' lie

t'n-cn]:)t).)nn!ot)< t cvoi

.i)t~

.tth~vh.!cv.),c .dn~ jBut Un- c.ircH).n

~oeveryc~) c.rve

~).p~ roundthcrin. an.

'c.~nccun.s~t~iuu~a])<)~<.).at..Io.°'

24L When

~+~s~c~ct.h-n<~ (,

~cL.~y

ina.y .h..c.ctl.n is

.pr~d hy théc..n.)i

cha.f~ .)~h i.sc.)M~.vd.ci y-p~f, L<)

°

Page 21: Lord Rayleigh - The Theory of Sound Vol 2

~]YHLOCrrY-rOTEXTIAL.

9

It\S''k')t<)tcn.nyc]<.sc(Ls)))'fnc~,t)œ)-at.co('Howf'utw~.sMros.stlie

ute.nont <? Is expresse by f~S', whcrc is thc r~ of va.

ti<)n()f~inr'c'it)~<'ntwn.]-(1s:).]'gtLun")-n):Ll.~nti~ocu.st-ut'

constant d~)).sity,t))ut.jt:d!nssut'Omdiutttuc~i.sthos

th<! intcgmtion ran~ing'ovcr thu w)to)c surr;)cc of 6. If thn spf~œ

<S' hcfn~ buth :ttthcb'~i"nu'g:L~'c cudDi'thctitnc~,

t)K')~ss]nustv:u)ish;!m(mmH

-whc'nitiH'l~irud towurk \tthp"U- co-ordmatcfi, thc trans-

funnL-.lc.))~ti..))is ~~crcadi)yf.Lt~)t.~<nrucUyby:q~yh~(i)

tu t~! C(n-rc.s))un<U)~uk')n<;nt. < \'ohm" titan hy tnmsfurtnm~ (2)

in a~'onhu~-c wib)) t.)'c :u):ttyLie:U rntcsi~)' uUccting ch~u~'os

m thu

in(.k'})(;ndL'ntv:u'):th)L's.

Thus, if wu ttdœ poiarct)-"n1inat.s ia thc pt:u)C A'y, so t)~t

Page 22: Lord Rayleigh - The Theory of Sound Vol 2

r~OPERTy opIKp.

r :,2.1 L

~1.

~L~.r

~!nl tlir, m,~tln!1 ~Ilnll file

"<y'.sby(~'c~t(,u..uf~

'1~ cunvcniuntfur t!I(~IJ1'llld('11I il! Ilancl.

-y~p~ flllid witllinallY ,~iIlJpl.r-c()lIlll'dl'd ('Iusl'd1'11'(' 8 i,

i~wnlilc~ful,S~ vlutmwnimnll,y

r: ~y~n'j.,)),j,

~cri,d"j.~ ~.s.s,f..

,y.yt.~.i.i,n.)h. in

')

'<~JUH. .)'i.J.L~ ,.tl't'st, it (';IIJ

al'llllil'e III) IIIO/I'cillal' l'otatioll 1111l1(,)' tlH: ulu,rutiun uf)~

.f

thespaeu

;u~~ value

'b~~thcc.

.-P–in 7'Imnu~oynul

.S ~<~<~M~y

.L'~J:f.?,

~t..h.r.1'1'u1.JJVIJl i, jlossil¡Je.

~<Mtf)('<jrcni.)rr7~-n

tlie iut(-~t;t,~ j;iL.r(-h.j.~

J'1l.illg OV('I' IIi(}\'111111110,'t.

<-"c. s~~ry;

JiU tll'Ol'lIlIelillns,

'~J~<,juat. am!

'L~r. Ju,

L. S

1

Page 23: Lord Rayleigh - The Theory of Sound Vol 2

242.] 1 MULTIPLY-CONNECTED SPACES. 11

over thc s'i)-f:).cc of .S'. Undcr thoso ch-cumst:inccs t)~ douDe

.n~r.dih ~) .~t.ItC.s,:md~uiii'r:):.n. :)t")y p~ntof~'

</A~ </A~ In ~hcr wonl.s A~;<; <

<~Zmast ho u(lnal j,o zc:l'l). ln oth cr won Is Â~

tnust hc cunstnnt, and thc twom')tio)'s i<k-nticn.L As npnr-

t.icu):n- <<sc, thd-cc:u)bc no motion oft~c Ir)-ut:).tiun:d Mnd

withnt D'u votunu! <S', indupc))')unt)yofamotion of thc surface.

T)tc rcstrictiuu t~ shn]))y con))ect.c(! Hp~CL's is r~-ndc-t-ud ncccss:uy

bythci'.u)))~ofUn~ns t))corcn),w)ucl), !t.swas fit-stpuintud

uttt hy I!c])uh<dtx, is othot-wisc possi)))u.

V~~u t)K: s[):x'cJ.s nu))t))')y-<n)f'ct<(),

thc nïotn.tiona.t

ntoticit is still (h'tummKttc, if h~ido.s thc nunmd vuhteity n.t.

cvcry p<jintuf 'S thcrc bc ~'ivun titu v:du('s uf thc'. constanL cir-

odidinn.s h) id) thc p~.s.sihh' iiTC'co)it'i):ddc cit'outs. Fur :).

comptctu discussion '.fthis ()m.'sti('))tW(3))U)st,t\'rtt)Thomsun'ss

ori'dx.'d ~u'u~'i)', :ux) coxtrttt, ourscivcs ht'rc whh thf c:L.sc of

adu))h)y-conttL'(-t(.'ds})a('c,w))i''h\i)l.suthc(j!furI]lustr~tiu)t.

Lct J/)'C'D I"! :)n(!t)d)cs.s tuho within which i)ui<tmovM

irrot.LtiutKdty. Fur this m'~nj~ t)~i-c nmsL cxLst a Ycitjcity

Fin.

put.uht.t:d,w!n)sc<!ift'un'))t[:tl cocrHcic)tts,cxpt-(;ss)ng-, :Lsthcyd'-),

t)n! cump~m'nt vdocitK'.s, :u-c )K!(;c.s-i:u'i)y sin~lc-value'),but

wL~h t)C('-]nf)t itsc)f]K! siu~ic-Y!~))C().Thc si)np)cst wayof

:tac).U).n' t))C (linicu)typr(-suntc(thytht' mnt-igulLyof~,is to

coh~ivn :L ]):u-ncr~i~ta)<rn:K'r<LSs))mn))~,son.-<toc)nscths

p:~s:).n'c.T)~- sp~c ~)/~7~~tA' is t)h-)). sin)p)y cnnti)mo"s,

an()<!n'cn'.st))(~)!-c)n:)ppii('sto1t.witI)~utjn("1ific:~tion,if!L))ow-

;m~!bu tn:uh; f.))-:t])~s.siL)cfn)it'(tit'turt'ncoin tho value cf~

~nthctwonidcsofU)~ t'arricr. Ttti.s~in't.'t-cnc~if iLcxIst, is

Page 24: Lord Rayleigh - The Theory of Sound Vol 2

<0

~Y-Cr,xx,,(.T, ,p,

=~all<l il tilt'

h,Jl'o-dYllaIllÍ(';¡/ :1111di(,iltillll ('XPI'('S,s(:s lllu circolulic,;l l'olilld i11; l'jll~''ri''j"i'cc.;t,;ui.,u

<h.()“)““),

0\1' illn i11'o 1;11'1' cd'

;i:r: J\IIIVsincc

clc~Jms tlm ¡-lIlIu \'allll' uu the t\o sid"

~o.~sU.).

thct~.sid.,s.

th.sL.u.L .iiO-.n.~of T). if rc

\'<I/lisll,Lc.u.cuh~i.n ,)

-J

~Lc~cl~e

~T:

be giV011, 1!~ur,ifep amlcj~-1-V~, IJU twa 1'mnclimns

snlisfJ'illgLnplaee's l'Cjuation

~i=~

alld the snrnc l1uI'lIlal

.t~.LaplaCl!'S l'(IIIIL-

t iUIl :lIId ilm cOllditio)1 tIlaL tl)(,l'u s]¡;¡J/ I.JU III.itl,CI'('il'cula/ioll~s~cr .S' p \1:

~<cu)..<i..n

~-d.~n~

~J.

J~~sin.p. (at;.nvp,v!

~s n,,rhy

't,. ~rf~,

circ.).t.i., -ya.

'y-<~cd n.s °" I~c.. f.,IlItlll il'ly-cIIIJIJ('dl.d as \('/1 asl;illlp/('IIIII}('ct(~d

1;1':1('(' if vsd in .ionhy.

tllc wJJilh! 11I:ISScumes to l'cst so 80UII vs tltc IJJt¡tioll ut' thu LUIIJJ-'y<.ua.s~. ~L.fthL.Lu~-

H'Lnm'd'~vI~~ith~Lf.r.).ti~ .H,,

olltsidurcei~ut t.,bc-!iko surjhœof,nr~out.si.~

~s)y su~, t).bec. i.t,

~iD.i. tJ.c.tubccn.nc..s~ <witlliu t1IU taLe comes in l't'st. '1.'llis IlIech:lIlil'alilltc'j>ctati01I,L.

tùlIuLlcl'sLand ) Ilu!'c

Page 25: Lord Rayleigh - The Theory of Sound Vol 2

~)~.] J.\XAL()(:Y W)'rif H~AT AX)) EH':(")'):r('!TV. 13

(.'))':n']\'w)'!tt ismoanthy a(tuit) havit'gnocircuhttion, axdit

h~~s <t)t <'xh.;))M)')norSt.uhcs''h<t~)~ witht'rspt'< tnxx'if-

cuhtr n'tation. i'or, H ait. thc Hmd ())iovh)j.; suhjt.'ct tu n,

v~h"it.y-p~<)'ti:'t) cnLsidu n.spho'io:)) f':(\'ity(~f :t))y r:n)ms bc-

c<n))L' M)nh!untyso)itt,t]tL! fhud in.sitk' Un.' c:Lvity cnur<jt:nnno

motion. 0' aswu m:ty :so.st:)hj it,a!)y sp)K'rica[ portion oF

:t)) i)T')t:)tionn)ty n~vrn~ Ouid ht'cumin~' suti(!t'!)ty .sotid wou)d

))()SSt.'s.so)ttyatUt)ti())i(jft)':m.i1:~it)n,<'i'<ru<<~<'o~

A Mi)t)i!!0'proposition w)))app)yt')!L(.'ircul!))'t1iHC,f))'cy1in(1~r

~it)ti)at.<'utls,i!).t.ho(.)..s(jofih)i<t ]uo\'iu~it'rot.:ttio))!(Hyintwo

(iitm'n.sit'jisonty.

Thu )))')(!')!) ofn.n i)tComp)'('i.si))tL'fhnd\)tic)t)n)s~)CL'nf)UCC!

atrL'.stj):))'tak('.sot'<.lK'TL')n:n'kaL)cp)'()p(')'ty(§7'))~')""n<)utot)):t.t

ot':))t Hysk')))s'))ich:n'o set n) motion \it))j))'('st'ri))c<tvcL)('iti('s,

nauu'ty, that thé uncr~'y is <))u icast, possib)c. R'!)ny uthcr

motion 1)L' jToposml H:(tistyhi~' thc ('qu:t.ti"nofc(mti;)uity nn'l

thchotnx.hu'y cumtitioo.s, itst.'Ot'r~y isncccssat'ity~rc'ittcr t)):)n

thatofth'jmotio))'\vhichwoutdb~.L;'<)'ci';Ltcdn'o)nrust.

2t- Du; f:).ct that thc irrot:ttiu));~ motion of mcomprt-ssihh'!

jt!)ud (tcpcnds upon V(.-]o'-it.y-potcntir).] Sittisfyi))~ Lap)nc(;'s

L'(~n:t.t.io)t, isthc foum):<.tion of:). F!U'-i'u:L(.)i))~)))!))o~y butwccn

t)~! motion of'.such~ thtid.and U~tof uluctricityorituatni

:).))t))tot')n cum)u('tor,v))ic)[ iti.soft.t'not'~rc.'ttso'viceto hc:t)'

in mind. L Th~ s:un(i )n:).y he s:ud <jt' thc conncf'ticn bc'twccn

att. thc b[':UiL'))C.s ofDiysics wttich dc'pcH(tmat))(j)n!).tical)yo)i

potuntia), i'"r ItoftL')). happuns tl~t thu :i.n:tto~ous thuorcms

:).ru f:n' from cqnidiy uhviou~. For (.'xampic, thc ~na)ytie:dt)tcui'cni that, if \7~ = 0,

nvcr a do.SL'd surface. is inost r(.u]i]y sn~'cstcd hy t!)C Ouid

intc]']')'ct!).ti"!), Lut oneû ohtaitK'd nniy bu intc')'pret(.'d i'ur cIc'cLric

or)n:).nctic

iorcos.

Ag'am. in thc thui'ry of t)ic con(h)etion of hcat or cicctricit.y,

it is obvious th:it tlK'rc can hc uost.L'ady motion in tho IntL'rIor

()('fS',withoub tnu)H!n)H~i<'u across sontLi pin't of thc Luundin~Hur-

f~cc, but Uns, whuu iuturp)'L-t.(jd for incompres.'jiD.u itnids, ~ivcs :ni

I)npot'tanta)id raLhci'rcctjttditc t:L\v.

T))~)mon en r~e .Vot;), !or. ctY.

Page 26: Lord Rayleigh - The Theory of Sound Vol 2

EQUATION 0F 1-KESSUHH.f~L-

2.). t. W)u..nv..).)city-p.(.nti..d c.xisLs, Die cquation t., .X.tcr-

!n')t0t)i(.prus.sm-ct,):!y

!<ut:)'i)L!L.i `.},jj.M

~§~iJ,'J~-t'\JJ

r.

Th.samcc.n.iu.s.un~ybc .-u.n..d~by~ din.ctappH~tinnuf.nL.d.mca)

prn«.p).s Lu the circu.n.s~ucc.s ofinipul.iv~uotlun.Jf~= c({u:Ltton (~) fakcH thc' f(.i-)n

If thc n~t.on hc suc), tt.at thucn.nponcnt v~)oci<.i.s

arc ~)ways ti.u.s.un.L ~s. p.int ..f

spac., iL i.s c.))<.d.) ~eo,~

"n.L.nt.f d~tun,,Tj.uc.p..tiuu ufprc.s.s.uci.st).cn

uc.~pncahon, of

(2). thc vctoeities and eondcnsa-t.o.

~d) t.

siit~c

furif

~j;part

Page 27: Lord Rayleigh - The Theory of Sound Vol 2

rf.AKE~VA.VEM. 152.N.)

~)'). T)K!simph;ntkin() uf'\vfi\'c'-)nn<iut)isth!).tinwhK'ht)!C

(.'xcur.sit)n.s()i'u\'uryp!H't)ck':n'c])ar:dtott~:t))X(.)i)m'n't:)rc())c

s:u)tui)t:t!)])):uu'.spt'rpuHt)icu!;n'tu),)t:)t. tinc. Lut tis t)!L-)'L'f~rc!

(i)s.s))n)i))L;'t)mt7)'=0) sup])usf t)):tt~i.s:(.funcLiuiiut'~(:md<)

ot)]y. Out'L'<~uatiun(!))§:i'l'))L'co));.c'H

)'t'pn:'sunting thc pn)p~).ti")) oi' mdcpcutk'nL W!(.ca in t,])u positive

:)ndnug:tti\'L;(Ih'L'ut.iu)tSwit.)t<.))L:cu!)Utnjnvci()cityn.

\Vit-hmsu(')))unitsns:dtt)Wt.ht;~p~tic:Lti<)nf)f'th(!n}'p)'uxi)i)a<c

(p<))):)Ltinn (!),~K;\r)ucity()f.sound i.St'))tiru)yim1(.'}K'ndc)itot'thc

i'urttt uf thc w:n'c, bcm~, i'orc-x:unp)t. ttiu .samc fui' snnptu WtLves

vlic~tcver tl.ie vwve-lcn~tl> >na.y b~ 't'ln; comlil,i~m s~,tisfic~l l~y tlu,whatcvcr tbc w~vc-k'ngt!) mny bu. Tix; condition Hati.sficd by thc

pusiLivc Wi~ve, :)n<[ LhL'rutbrc hy thé ixitiai disturbimcc ii' n. posi-

tive Wt).vc ulunu bc gctict'aLud, is

Page 28: Lord Rayleigh - The Theory of Sound Vol 2

l'L.\XH I~HOaRKS.SSJV~ WA\ )-~t5.

Whatcvf.r tlieinit.iat.t)Ht.urb.'U)cuniayhc: (axd Mand.sanibuth

arhjt.rary), it c-anniway.s bc <))\'i(!,jd into two

parts, .s~LiHfyi)~

n:-s).ucUvdy(:{) :u.(-t.), whicharcj.mpagat.udundistm-hc(t. In

')'(-)t);)..nuntw:tvcthc(H )~ct.i.)n«i'pr«{);~a<i~nisd)C~)))C:t.s

thatuftjju!n(~)<.nu)'t)tL-c~/i«'p:u'Lsui't)~f)ui().

TiK'r;L<(;atw!tic))L-n('r~yi.str:).ns))HLt.(.acru.s.s()niLof:u'c?),()f!t)'):nt(-p:t)-ati(.t t,)~),c

f't-unt(~).prt~)-CH.sivcwavG))~yhurc-~u'dud as

t))u)ncdtani(;;dut~surcuft)ici))L(.-n.sit.y(,ft))C radia. t.iun.

.)))t]itjc;t.sooi'a.si]i)j))c\u'c',i(jr~))ich

Jf t).c int~ratiun wit].rcHpcct to thnc cxtcnd ov<

nny Mumhcr of

comjttutu j.criu(]s, or}.r:Lctica))y ~hcncvcr its r:in~c i.s .softicicuUy

tung, thupunudic L'rm.s may hu (ntiitted, and \ve mny tnhc

Page 29: Lord Rayleigh - The Theory of Sound Vol 2

~5.]E~EUGY 0F FLANE WAVES. 17

or by (H), ifj9 dénote thc maximum value of~

Thus the work consumed in gcnerating wavcs of harmonie type

is t)'f! sa.tnc f~i would bcrcquil'cd

to givc the maximum vclocity ~3

to t))c whole mass of air through which thé wavcs extend 1,

In tenns of the maximum excursion 0: hy (7) and (D)

whcro T-(=\–<ï) is thc pcriodic timc. In a ~)e)~ ~e~tM?~ thc

H)t'<i:uncal mcasm'e of tlie intensity is pi\)port'iona.l to tho squareoi'tho amplitude direetly, and to the square of the periodic time

Invcrsc)y. Tiie rcadur, howevGr, must be on Lis guard against

supposing that thc mechanical measure of Intensity ofundulations

of din'ut-cnt wave lengtfts is a propcr measure of the Joudness of

the con'cspouding souuds, aa pcrceivcd by thé ear.

In any p!:uic progressive wave, wLetIier the type he hn-rmonic

or not, tho whole cncrgy is cqually divided between the potentialand Rinctic fom)s. Purhaps the sunpk-st roa.d to this rcsult is

to consider the formation of positive and negative waves from an

initia! disturbancc, whose energy is wholly potcntial~. Thé total

énergies of thé two derived progressive waves are evidently equal,and nmke up together the energy of the original disturbancc.

Moreftvcr, In cach progressive wavc the condensation (or rare-

i'actiojt) i.s one-))alf of tliat which existed at the corrcsponding

point InitiaUy, so that the po<e~(~ energy of cach pro"Tessivewuve is o!!C-~<M~er of that of tlie original disturbance. Since, as

we !u).ve just seen, thé whole energy is o?te-/<a~' of thé same

quantity, it follows that in a progressive wave of any type ono-

haïf of thc energy is potential and one-haïf is kinetic.

Thé same coTidusion may aise be drawn from the general

expressions for thc potential and kinetic énergies and thé relations

betwoen velocity and condensation expressed in (3) and (4).

Thé potential energy of the clément of volume c~Fis the work

Tho endic.-it statoment of tho principio ûmbo<Ued in cqnniion (10) tirnt 1 htivomot with is in )i pftper by Sir W. Thotuson, "On tho possiblo deusity of tho

ImuhnforoHs mctUnm, and on tho mcchMuctil value of a oubic mile of suu-Hght."7~t'V<tf;. ix. p. 3f!. 18;

Uoxan<[)tut. ~/tt<. ~/<t~. xLv. p. 17! 1873.

/«/. ~fy. (;) p. 2CO. 187C.

R. ir.

Page 30: Lord Rayleigh - The Theory of Sound Vol 2

18 NEWTON'S INVESTIGATION.[245.

that would bc gained during the expansion of thc con'cspondin~

qua-ndtyofga;: from ita .tctualto its normal volume, thc expansion

bcing opposcd t])ronguout by tlic uonnalpt'cfisurc At any

stage of thc expansion, whou the condcnsntit)!i is s- the cn'ccdvc

pressure ~) is by § 2.J4 f~~s', wl)icl) pressure bas to bc muItipHcd

by thc corresponding incrément of voh)mc fn'c/s'. Thc wholo

work ga.Incd dunng thc expansion from ~~toJr(l+s) is

thcrcforc ~p,~F. or ~F. s' Thc gcncra! expressions1 n

for thopotential and Mnctic énergies arc accord i;)f!y

If the p!ano progressive wavoa be of Itarmonic type, and sat any moment of time are eircuiar functions of ono of the spaceco-ordinates (a.-), and titorcfore thc mefui value of tlieir squaresis ono-half of thc maximum value. Hcncc thc total cuc!y oftho waves is equal to thé kinetic energy of the whole mass ofair concerned, moving with titc maximum vclocity to bc fouud inthé waves, or to tho potcntial cncrgy of tlie same masa of air

wl)cn conclensed to the maximum dcnsity of thc wavcs.

246. Tiie first theoretica~ investigation of thc vclocity of

sound was made by Newton, who assumed that tho j'dution bc-

tweon pressure and deusity was that fornudatcd in Boyle's law. Ifwe assume p==A- wc .sec that the vulocity of sound is oxpressed

by V/f, or in which t!ic dimensions of p (= force-area)arc [37] [Z]-' [2']" and thoscof (= mass voJume) arc [.Vj [Z]-Newton Gxprcssed t))c rcsult in terms of thc 'e~ of ~e/iOM:o-

~eHeo:~ n/M~.f~Aerc,' dcnncd hy the équation

wherc~) and p rcfcr tu thc pt-cssun; a)i(t t)m dc-nslty {tt thc carth's

surface. Thc velocity of sound is thus or thcvclocity which

would be :tC()nirud )~y body i:d[ing frudy under thu action of

gravlty tin'ougt) ludfthc h~ht uft)t(.' homogcncous atmosphore.

Page 31: Lord Rayleigh - The Theory of Sound Vol 2

2-] LAPLACE'S CORRECTION.

Toobtmnanumurical )-csu)t "/c r~'uiretcju~w ~pnir<fsimuftancous v.uc5

of f~d It i.s foundby cxpcru.tent ti)at

fit (~ Cent. undor a pressure of 1033 grammes per squ~-c ccnti-

tnctrc, thcdcn.sity of

dry air i.s '001293 grammes pci- cubic œnti-

metr~. If wc takc thoccntimctrc, grannnc, f~d second as tho

fmidamcuta! unit.s t)tc (c.O.s. System), tl~sc dat:imvc

.sothat tho vclocityof sound at()"v'o)dd bc27.')'f)5 mètres pt.rscœn.), faHing short of t)ic rc.su!t of direct nbsur~tinn Ly abonL

:).hixt)jpart.

Ncwton'sInvc.stig~tit.n c.stab])Mhc.) tl.at tho vetocit-y of .suund-should be indcpcndcnt of thé amplitudu of tho vibration, and aisoof the pitch, but thc

discrc~ney b~wccn InH ca)cu]atcd v~m.

(pubhshcd in 1G87) and tlie expérimenta! v;due w~ not cxplaineduntil replace pointcd out tbat tbo use of Boyic's law involvedthc

assumption tliat in tho cotidensatinns ~ud rarofactions ac-

companying sound thotempérature renmins constant, in contra-

diction to thé known fact that, wltcn air is suddcnly comprefiscdits tc.npcmturc ri.ses. Thc ]aws of Boyic and Charles supply only«ne relation hctwccn the tliree

quantities, prc.ssurc, volume, andtempérature, ofn.ga.s, viz.

wnerc thé température Is tnca.su.-ed from the ~ero of U.c. ~sthermotnutcr, and thcrcforc wit!iout some auxiliary assumptio~itis nnpussiHe to specify tlie conuecHon

bctwccn and v (or p)Lapiacc con.sidcrcd that thc condensations and raréfactions cou-cerncd in thé propagation of sound take place with such mpiditythat thé I~cat and cold produced hâve not time to pass away, andtf.at thcrofore tho relation betwccn volume and pressure is sensibiythé sa.nc as if thc air were confiucd in an absoh.tcty non-con-

duct.ng vossch Undcr thèse cireumstanecs thccl.angc of

pressure

currespondmg to a given condensation or raréfaction is greaterthan on thé hypothesis of constant tenipcraturo, and thé vcloci.yoi sound is

aceordingty iucrcased.

In équation (2) !ct dénote the volume and the pressure ofthé unit uf mass, and Ict be expressed in centigrade dcgrees

n_ n

Page 32: Lord Rayleigh - The Theory of Sound Vol 2

20 LAPLACE'S CORRECTION, [346.

rec~oncd from thé a.bsohttc xcro'. Tho conittion of thc gn.s (if

uniform) is de~ned by any two of :tt0 thrcc quantitiGs~), a.nd

i.h(t Uth'd t~tt.y bc -t~)Ld n. td'ins ûi tiicn!. Thc rc~tiou

between thé simuku-Dcous varin-tions of thc tfjrce quantitics is

In ordcr to cffect tLo change spccificd by f~) and dv, it is

in general nccess:n'y to communic!ttc hcat to tlie gn.s. C~IHn"'

thé nccessM'y qua.utity of hcat ~Q, wc may writc

Page 33: Lord Rayleigh - The Theory of Sound Vol 2

246.]EXPERIMENT 0F CLEMENT AKD DESORMES. 21

if, as usua!, tlie ratio of thc spécifie Iicats be denoted by 'y.

Lapl~cc's vaiuc of titc velocity of Sound is thcrcfore grcater t)ta.n

Newton's in thc ratio ùf~y 1.

By Intégration of (8), we obtain for thc relation bctwccn

M aud p, on thc supposition of uo communication of hcat,

whcrc )),, o,; arc two smuuta.neous values. Unuer the same

circumstaaccs tlic rci~tion 'betwecu pressure and température is

Ly(3)

Thé magnitude of 'y cannot bc dctermined with accuracy by direct

cxpcrhncnt, but an approximate value may be obtained by a

iiietliod of which tlie following is tlie principle. Air is compressed

into a resci'voir capable of being put into communication with

thc external atmosphere by opening a wide valve. At first thé

tcinperatui'o of tlie compressed air is raised, but after a time

tite superiluous hcat passes away and the whole mass assumes

tlie température of thc atmosphère 0. Lct thc pressure (measured

by a manometer) be p. Thc valve is now opened for as short

a time as is sufHcieut to permit thc equilibrium of pressure to

be compictcly estabhsbed, that is, until thc internai pressure

bas become cqual to that of thc atmosphère P. If thé experiment

bc properly arrangcd, this opération is so quick tliat tlie air in

thc vcssel lias not sufncient time to reçoive heat from tlie sides,

and thercfore cxpands ucarly aceording to the law expressed in

(9). Its température at tlie moment thé operation is complete

is thcrefore detcrmincd byV

Thé cnclosed air is uext. a.!lowcd to absorb heat until it bM re-

g:Lincd thc a.tmosphcnc température 0, and its pressure (jp') is

thcii obscrved. During tlie last c!~nge thc volume is constant,

:uid thereforo tlie relation bctwecu pressure and température

c~vos

It Is horo nssnmod that is const.nut. This équation appears to havo beou

eh'on iirat by roisson.

Page 34: Lord Rayleigh - The Theory of Sound Vol 2

~2 RATIO 0F Sl'ECIFIC IIEAT.S. [2-iG.

s(jU)at,hydhnu)nticnof~:("),

By cxpurnnontsuf thisnat.)))~ (.')c)))c))t. and Dc'sonn~s dc-

~))tinL:d'y=l~t!bt)t,).JtcïnuLhudiso))\'iouHtynots))sc(-ptih)cof

:my grcat iK-cm-i~y. Thc v~hnj uf 'y j'C((uir<jd tu ~ccmcittjt))L: catcn!atud:n)doL.S(j)-rdvc)uc~iL!.s«t'.som)d isl--K)S,oft)tu

su[).st,antt!t.!currcuLuc.ssufw)ticht.)tcruc:m))<jiitt.teduuht.

\Vc!L)-cnot,))owc\-t;'r,(tcpc)i(L'ntoi)th<Jt)hun<)niun:iofi-nndfut- unr

knowlud~c of thc!nagnit))()c of 'y. Thu ViLtuc (.F /<

thc spécifie liuat at constant prcs.sorc–hn.s Lccti dt.;tu)inincd

L'\}n'nnie)tt:d)y by Rc~n:udL; :md :dt))ough 0)1 account uf in.

lurent <)ifiicu)tic.s tho c'xpcrimcntid mu(.]njd tnay i:Lil tu yic)d;t Matisfac~ory rosult iur /< tho infurtnatiutt sought fur may bc

uhtaincd indircetly by niGana of n. rehdioti bctwcoi tl)< two .spc-

oitc ])cats, Lruu~))t tu Ji~ht Ly t]iu mudum science of Thcrnio-

dynamics.

Iffroiat.hccquatiun.s

Lût as suppose th:Lt (~ = Q, or that thcre in no communication

of))C;it. Itisknowu that Hic )te:t.td'j\-c)')p(!d(ht)-)ngt)]c com-

pression ofa.napproximatc!y pcrfcct~s,.su<j)iaHair,isatinost

ux~cDy thc thcDnul c'~tivatent of tlie work donc lu comprcs.sin~It. This nnportant principtc w.s assuined by Mayer iu his

ccicbt-atud liiemoir oa H)L' dynfLnuca~ thcory of hea.t, titough(')t g'ru)))xts wtuch ean hai'ttiy be cousidured mtcqua.tc. Howcvcr

th:Lt nmy bc, t)te priucipic itscif is vcry nûariy truc, as bas since

been proved by thc cxpennieuts ufJmdu aud TItonison.

If wo mcasm-ù Le:i.t in dynamical nnits, Maycr'H principic

may bc cxpresscd ~=~~u o~ tlic understaudiug that there

Page 35: Lord Rayleigh - The Theory of Sound Vol 2

2.1 G.]RANKIN-E'S C~LCUJjATION. 23

is no communication of hcat. Coniparing this with (15), wc see

tha.t t

By Rcgnault's cxperniicnts thc spécifie hcn-t of air is -2379

of that of watcr; and in ordor to raise a gramme of watcr onc

<)~rcc Cent., 423.')0gratnnie-ccntimeti'ca of work must be donc

on i t. Iluncc with thc same units as for ZP,

= -2379 x 42350.

Culculating from thosc d~tn, we find ~y= 1'410, ~grecing almost

cxacUy with thé value dcduccd from thû vclocity of sound. This

investigation is duc to Ra.nkluo, who cmploycd it in 1850 to

c:dcula.to tlie spécifie hoat of air, t:i.M;ig Joule's équivalent and

the obscrvcd volocity of sound as data. In this way he antici-

patcd tlie result of Rcgna.ult's cxpcriments, which were not

publislicd uutil 1853.

247. Laplacc's thcory bas oftca bccn thé subjcct of mis-

apprchcnsion among studcnts, aud astumblingb~ock to those

rcmn.rkabte persoi~, caDcd by De Morga,n, pM'ildoxcrs.' But therc

Ciui be 110 i'L':i.suna.b).c donbt t)ia,t, antccedcntly to ail calculation~

t!)c Ilypothesis of no communication of hca.t is greatly to bc

prct'urrcd to t)ie cquaHy spécial hypothcsis of constant temporature.

TitCii'c wotdd bc a. reaL di.HlcnIty if tlie velocity of sound were

not dccidcd)y in excess of Nuwton's 'vainc, a.nd t!ie wondcr is

l'~thcr that tho cause of thc cxccs.s rcmained so long undiscovcred.

Tlie on)y question which can possibly Le consiclered open,is wliother n small part of the Iicat and cold dcveLopcd ma,y not

escape by conduction or radiation bcfoi'e producing its full effect,

Kvcrything must dépend on thc rapidity of thc altern~tions.

Hc)ow a certain, limit of slowness, thc hcat in exccss, or dcfeet,

would have time to adjust itself, aud tho tonpcrature woutd

remainscusiUy constant. In ttlis case thc relation betwcen

Page 36: Lord Rayleigh - The Theory of Sound Vol 2

24 STOKES' INVESTIOATJON[247.

pressée and density wouid bo th:tt w)uc]t I~~d.s to Ncwton's value

of titc vulocity ofsound. On 'hc othcr !~m], Mhnvc a. cci-t:un Hmit

~iqmckm.s, Lhc!,a.-i Y.'uufd .~Jt.vc~.jifc~nhcdm.~no)i-L:~n-

dxcttng vc.ssci, a.s suppo.scd in L~pt~œs Uicory. Nuw fdthoughtito circumstiu~cc.s of thc autoa-t pi-obtf!n anj hctt(.;r rcpi-c.scatcd

by titc lattct' t)~n Ly t))c formut' supposition, theru ]n:Ly still

(it m:)y bu sn-id) bo n. sensible d(;vi:tt.io)i fnjni t)ic law of pressureand dcnsity invulved in Laplace'.s theory, ent:u)i))g so)newli:).t

slower velocity of propag:tt.iun of sound. Thi.s <juustiun h:i.s bccn

carefully discusscd by Stokes In a p~pcr publishcd in 1851',of wbich titû fullowin~ is :ui outlinc.

Thé meclianical cqua.tions for tho SH;a~ motion ofuir arc

Thc tcmpct~turc is supposcd to bc u)ufor)n cxccpt il, so f:).ras It is disturbud by tho vibrations thcmselvc.s, so that if dcuotet!)e e~ceM of tcmpcratnrc,

ihe cncct of a smati surdon cundcnsfition s Is to producc an

elovation of tcmpcmturc, which m~y be dcnutcd by /3~. Let

f~be tlie

quantitéof heat cntcnng ttic dément of volume in

time dt, mcasuredby thc risc of tcmporit.turc thfLt it wou!d

produce, if thcre were no condensation. Thon (t.hc distinction

betwcenand Leing ncglectcd)

bcing a function of and its di6crcnttal cocûicœuta with

respect to 6pacc, dépendent on the spécial character of thc

dissipation. Two extrcnie cn.sc.s may bc mcntioncd, thc Urst

whcn tho -tcndency to equaHsation of température is due to

conduction, thc second whun tho opei~ting causo is rudiatio])and the

tra.nsparcncy of the nicduun suc.h that radiant licat is

jP/t<y.Af< (i) i. 805.

Page 37: Lord Rayleigh - The Theory of Sound Vol 2

247.']0F EFFECT 0F RADIATION. 25

not scnsibtyabsorbcd within distruice of sévère wavc-lcngths.

Li thc l'unnur c~c ~~x~~ ;n.d in t.hc lutter, winch :s that11 t l\ ùl'lnur Cli:'JC

<tCv-a, ;LIll JJI 1.1~IJ \lUCI', W IIC 1 1:'>t lat

sctcct.ed by Stokes for :u)a)ytienl investigation,(- Ncwton's

h~v uf radiation bcing ftssumcd a.s a, sufïicicnt approximationto

Ihc trutli. We i~ve thoi

lu thc c~sc of pl~no wn.vcs, to which wc shdl cunHnu our

:~tu!ttion, u :Uid Vfuush, wlule «, ~), & arc functions of (:uid <)

unty. Eiuuiuatiug~ and M bctwœu (1), (2) a,ud (3), wcHM~

if y be written (in the sa-mc sense as beforc) for 1 + a~.

If the vibrations be IiM-nionic, we may suppose that & varics

as e" aud thc cqutttioii bceomes

but (~ being positive, fmd Icss tbMi ~7r) if wc wish for thc

expressionof t))û wave travelling iu tlie positive direction, wc

must take tits lowcr sign. Discarding thc ima.giuary pM't, we

fiud as tlie appropriatc solution

Page 38: Lord Rayleigh - The Theory of Sound Vol 2

TII~ AMPL~'UDE 1S ~fOUE r~.i~.

Thc ~rstt))in~ tu bc noticed is thf).t t)io Honm! ca.nnot 'bu

i't'upa~tttjd tu dis(.;)ncL' untc.snMm Le insensible.

Thuvcioc'iLy afpro~ag.Ltion (F) in

i\ow irom (O)w(j suc t)<at cannot bcinsensible, un!css

is cithcr vcry grcat, ur vcry smaiï. On tiic first suppositionfrum (11), ut- dircctiy from (7), wc hâve

apj)roximatcly, ~=~

(Newton), and on t]tc second, F=~, (Laphice), ns oughtc-vjdunt)y tu bc t)i(i case, w])0i titc

meaningof y in (.-)) is con-

sxicrud. W)):~ we now Icaru is t!):it, if and M wcrccompamb)e,

thu c<!c:ct. wou)(] be not mcrdy a dcviaiion of from eititer oi'tho

limiting values, but a rapi.! stiiling of tbu sound, w!iich wcJ<nu\v docs uot takc place in nature.

Of tins theorctical rcsult wc may convince ourscives, asStuhcs

cxptiuns, witl.ont tbo use of analysis. Imagmc a m~s

of air to be conimcd witbin {t dosed cylinder, in which pistonis workcd wit)i a rc-ciprocating motiou. If thc period of tiicmotion be vcry long, tbe température of the air rcinains ncar]yconstant, tlic hcat dcvclopcd by comprcssicn n~ving time to

cscape by conduction or radiation. Uudcr tliesc circumstauccst))u prcssm-u is a fnnctiou of vohtnic, and whatcver work basto Lu cxpcnded in

producing a. givcn compression is rcfundedwhcn ihc j)i.ston passes Dn-ough tite samc position in thc reverse

du'L'cttun; nowurk is constuned in the Lng run. Next supposeO'at thé motion is so rapid that Hicrc is no time for thc hcatand cu!d duvchjped hy t!iu condensations and raréfactions to

c-seapc. 'i~cpressure is stiU a function of volume, and no work

'sdissipated. Tiic ou)y din'ercnce Is t)iaL nuw thc variations

uf pressure arc niorc considcrabfc t!ian hcforc I)i comparison~itht))c variations of volume. ~Vesechowitisthathotho)!Newton s and on Laptac~s hypoU.L'sis, tJK! wavcs travcl without

dissipation, t))ongh with difïcrcnt vc'Iocitics.

But in inturmediatccas~s, whcn thc motion of thc piston

Js ncithcr so s!ow that ti.ctempérature romains constant nor

.so quick titat t])c ])cat lias no time to adjust itsu)f, thé rcsuJtis diHurent. Thé work cxpundcd in

produciug a, sm~I! condcnsa-

Page 39: Lord Rayleigh - The Theory of Sound Vol 2

2t7.~ JINrLUEXCRD TIFAX TIÏH VELOCITY. 27

tion is no longer con)p)ct(dy rcfnodod during thucorrcsponding

rart.'t'action on accunnt of thu ditninishcd tutnporaturo, part of

thc I~'at dcvulopud by t))c compression havin~ in Utc tnciUttime

csc~pL'd. ln f:Lcb t))u pa.s.su~'c of ]x.tt by conduction or nuti~tion

front n- wiu'nn't' to a nnitcty co)dur Lody :d\V!tyH invoh'cs dissipa-

tion, a p)'I)K'ip)'j whiutL occupius a futuhunoit.al positi<jnin thc

SL'iunce! of Tt)ct'!)io<)yn:unics. In order thci'(.;for<j to m:Li)~tain thc

tnot.iunot' t))o piston, cncr~'y nm.st bc supptiL'd front -\vithont,

and if t.))(.'ro bc on)y a liniitL'd store to Le <]r:).wn fron, titu motion

inust. ultitti<<.tofy subside.

Atlothcr point to hc noticcd is that, if f/ and worc coin-

p;u':d.)lc, Kwon)d depund upon x, vix. ou thc pik-h of t!i<j soum~

a st:).to of thin~'swhidi frotn

cxporirnej)twc h.'t.vc no ]'(,'ason to

suspect. On thc contrary thu cvidcncc of observation gocs to

provo that thurc is no such eonncction.

From (10) wc sec that thc faUlng off in thc intcnsity, cstl-

tnatcd pcr wavc-lcn~tl), is a maximum \it)i tan~, or and.

by (!)) is a m~xininm, whcn=

~Y. In this case

Calcula.tmg from titesc Jut~ wc fhtd th~t for cach wavc-

lun~h of :).dvnncc, thc innptitudc of the vibration would Le

dn)T.i)iishudinthcrntIu'C172.

To tnkc a. iiumerical cxampic, Ict

la 20 yards thc intcnslty would bc dhnmi.sitcd in thé ratio

ofitbout 7 ~'Hions tu (me.

Cun'cspondi.ngtotins,

If* the vahic of y were ~ctnaUy that just written, smtnds of

thcpttc)tiu<~n-:stiouw(ju!dbcY(jryr!Lpid)ystifiud. Wtjthcre-

iurcmfL'i'th:tt(/isitiii).ctcithurrnucI)gr<~t(;ror(;Lscmueh!L'ss.

But cvcM so large a Y.duc as 2000 is utt(-!r)y inudmissib)' as

we m~y couvince our~ctvcs by cousidcrin~ thé si~ntficancc of

équation (5).

Page 40: Lord Rayleigh - The Theory of Sound Vol 2

38 EFFECT 0F CONDUCTION.[247'.

Suppose th~t by a rigid envdopc t.i-anspa.t-cnt to raJiiUit huât,thu vohnuc of small ])):),.ss of ga.s wcru n):u))t:unc<t con.st!mt,L!u

t.hL-c~);).}-)'~) to ..L-ni~.iu k..s tht.ri~:u ~idiiiun nt!j

t..neis

-whcre ~1 dénotes thc Init,i:U cxcc.ss of toupuratnru, provittg th:tt

tLt'ter a time t)tc cxcus.s of tcmpcr~urc wouht fait to les~ tLun

t~tfits 0)-ig:u:),l value. To.suppose thatt]ii.scou)d happeningtwo thousan<!th of second of time would he In contradicLion totho most supcrficial observation.

We arc thcreforc justihcd in assuming th:tt <7 is voy smallin cotnparison witli and our

cquntions then bucoine ap-

])rcxiniateiy

Thc effects of a srnall radiation of Iicat arc to Le sougbtfor rati.cr ni a damph.g of the vibration than in au altcred

vclocity of propagation.

Stukes calculatcs that if Y =1-414, F=ll()0, H~c ratio(A 1) ni which tlie intcnsity is diminis])c<t in

pa.ssiu~ over a

<)I.stancc.-r, is given by Io~=-0001156~ in foot.scco'ndmca-surc. Altho~h we are not able to

makc prccisc measuremûMtsof the iutensity of sound, yet the tact t)mt audible vibrationseau bc propagatcd fur many miles cxcludcs any suc!i value of

q as couldapprcciabiy affect t!tc velocity of tmusmissiou.

Ncitbcr is it possible to attributc to thc air sucl a conduetin<Tpawcr as couid

niatcriaUy disturb thc application of LapJacc'~theory. In order to trace tho en-cct.s of

conduction, wc haveouly

to rcp!ace iu (5) by Assuming as a particu]ar solution

Page 41: Lord Rayleigh - The Theory of Sound Vol 2

247.]VELOCITY DEPENDENT UPON TEMPERATURE. 29

teaving thc velocity of propagation to tbis order uf approximation

still equal to ~/x'y.

From (18) it appears tliat thc nrst cffect of conduction, as

of radiation, is on tlie amplitude ratbcr than on tlle velocity of

propagation.In truth the conductmg powcr of g~ses is so

fec~c, Md in the case of audible sounds at any rate thc time

durin"' which conduction can take place is so short, that dis-

turbance from this cause is not to bc looked for.

In thc prcccding discussions the waves arcsupposed

to bc

propagatcd in an opcn spacc. Whcn thc air is confined wlthin

a tube, whosc diamctcr is small in comparison with thc wavc-

Icngth, thc conditions of thé problem are aitcrcd, at least in

thé case of conduction. Wliat we have to say on this hea.d

will, however, comc more conveniently in n-noUier place.

24-8. From the expression \/(~y) ~p, wc sec that in thé

same gn.s thc velocity of sound is independent of tlie denslty,

becausc if the température be constant, varies as p (~ =~p0).

On thé other hand thé vclocity of sound is proportional to the

square root of the n-bsolute température, so that if f~ be ils

value at 0" Cent.

wherc thé température is mcasurcd in thc ordinary manncr from

the freezing point of watcr.

Thé most conspicuous effect of thé depondence of thc velocity

of sound on temperature is thc variability of tlie pitch of orga.u

pipes. We shn-ll sec in thc following chapters that thé period

of thc note of a flue organ-pipo is thé time occupicd by a. pulse

in runnin~ over a distance which is a dennitc multiple of thé

Icngth of thé pipe, and therefore varies inversely as the velocity

of propagation. Thc inconvenience arising from this altération

Page 42: Lord Rayleigh - The Theory of Sound Vol 2

30 YELOCITY 0F SOUXI) IN W.ATKH. [248.

of pitch is nggravatctt ))y thc tact ti)at t))c rccd pipc.s arc not

snodarly af'fuctud; so that a change of ttjmpcraturc puis fu)

o)'g!)))0))tcftu))H~'ithit.sc!f.

l'rof. Alaycr' ttaspropo.scd tomakuthcconncction bctwcc'n

tutnperattn-c and wa~c-l~ngt,)~ thé fonn<)ation ci' a pyromctric

mctitod.bnti amuotawarc \v))ct,ho-t))c o.xpcrunott h~sc~r

Lccn cm'ricd out.

Tlie con'cetncss of (1) as rcgiu'ds nir nt tlie tcmpcndurcH ofO"

and 1()U" i~s Ltjcnvurificd L~pcnmcutaXy by Kundt. Sec § 2GO.

In difï'crcnt gascs at givun tc-mporaturu ant] prc.ssm-G rt is

inversû)y proportiona! to thc square rùnts nf t)ic dchsitics, a),

IcaHt if Y bc cn)tstant=. 2, For thcnon-cptKhjusahtc gases ry doc.s

not scnsibiy vary froin ils vatn< for air.

Thc velocity of sound is not elltirely indcpcndcnt of thc

do~-ec cf drytic.ss of thc air, sinec at n- givcn prossurc moist air

is somcwhat Ji~))tct- than dry air. It is caicidatcd t]jat at 50° F.,air satnrated wlt!t moisturc wou!d

propagate sound bctwGun

2 and :{ fcct pcr second fastcr t)iau if it \verc po'fucDy dry.

T))c fnnnu)a~=~

may bc applied to c:dcu)atc thc vciocity

of tionnd in nquids, or, if thut bo I{nown, to infcr convcrsc]ytho coufHcIcnt of comprcseibiHty. In. the case of watcr It is

found by expcri)ncnt, titat thecompression per atmosphère is

-0000457. Thus, if (//j = 103:; x US], in absoiutc f.f:.s. units,

IIeucc~=-0000457, siucc p = 1.

~= 1489 nictrcs pcrsccou(),

which docs notdifïur much from thc observcd va)uc (143~).

~4:). In thé preceding sections t))C thcory of plane wavcs

bas bccn dcrivcd from t))C gênera! c'() nations of motion. Wc

nuw proccc'd to an indepoident investigation in which the motion

iscxprcsscd in tenus of the actual position of thc layers of air

instcad of by mcan.s of thc vulocity potentia!, whose aid is no

fungur ncecssmy inasniuc)i as iu oac dimension thcrc eau b~!

no question of moiccuhu' rotation.

'OnntiAcousttcryronctcT. 7'/<xLV. )).])-<. 1873.

Accnrdin~ to thc tiinctic theory of ~sca, t1o voiof.ity of onun~ is detorminol

M)!t').y).y,ttndis))roportin)mlt.i,t)R~unmv(.)ocityafthn)))n]<.fu]('f). )'rc!-t<),

.)/(5)tn.;)..tU. ]~77.

Page 43: Lord Rayleigh - The Theory of Sound Vol 2

249.] EXACT DIFFERENTIAL EQUATION. 31

If V)2/+-

(Ic-nnc thc a.ctua!pnsitions n.t timc < of

nt'!Q'bot!n!g i~ym'~ of :!)r ~'huso u'jUiiibt'iuttt positions :u'c (tt~no)

by .f and a;+~, thc dcnsity of t!)C Ittchn'cd s)ico is givc'n hy

thé expansions a.nd condcnsntinns bci))~ sxpposcd to tn.kc nhicc'

;iccurdit)g to tlie adiabatic titw. 'l'lic tn:s of uniL cf :u-~ of

J 1.. l 1 1 1" r ~l!' 1tlie slicc i8 and thc corruspondin~ ]no\'ing force IsfZj;

~iving for tlie cqnation of motion

Equation (~) is a.n e.wtc~ equ!Ltnj)i dctmin~ thc actual absci.ss:i

intci-tns of thc

cquitibrium abscis.sn. ruid thc timu. If t))e

motion bo assumcd to bc smal], ~vcmny rc))!acu

f~)wltic))

oceurs as tho coc~icicnt of thé sni:dt qnn.ntlty by its np-

proxunatc va.!uc unity; and (-t) thcn bccomcs

thc orclinary approximatc equation.

If the expansion hc isot))cr)na!, n.s in Newton's titcory, thn

équations con-csponding to (~) !tn<1 (5) ;n'c cbt:uu(.'(! hy tnurcty

putting 'y = 1.

Whn.tcvcr may bc the relation betwoon n.nd(k'pcndi))~ o~

thé coustitutioh of thu médium, t!tc cquatiou(~f motion is

Ly

(1) :m(l(3)

Page 44: Lord Rayleigh - The Theory of Sound Vol 2

32 \VAVE8 0F PERMANENT TYTH. [249.

from which p, occurringin is to be eHminatcd by mcans nf

tllu rc~muiut: 1)c";vw;n

.f/ uit~ ~1j.therci:(.:ionbc:)VC(;!)~anL),cxpreHS(.(tirt(l~.cl:c

2.'i0. lu thc prcccding investigations of acria! wf~es wc

h:ivc HU}'pf)sc<lthat thc air is at t'est cxcfpt in s') f:n' as it i.s

distnrbcd by thc vibrations of sound, but we arc of course at

Itbcrty to attribntc to thé who)e mass of fur conccmc'd any

comnion inotiou. If wo suppose that tbc air i.s moving in thc

direction contrary to that of thé wavcs and with the s:unc actuat

velocity, thc wavc form, if permanent, is stationary in spacc,

and thé motion is .~eftf~ In thé présent section we will con-

sidcr thé prol)lcm under this aspect, as it is important to ohtain

a)! possible dcarness in our vicws on t!)e mechanics of wave pro-

pagation.

If p~ dénote respect! vcly the velocity, pressure, and

density of thc nuid in its nndisturbed state, and if p bc

thé currcsponding q~antitics at a. point in the wave, wc !iavc

fur thé equation of continuity

dctcrmining thc law of pressure ~nder whi~i n.]onc it is possible

for a stutionn.ry wavc to ma.inta.in itscif in Huid moving with

vc)ocit.y Frum (3)

Smcc thc relation between. thé pressure Mi'I thc <1enslty of

fictuat ga.scs is not t)tat cxpresscd in (5). wc cnnehidc that f). sdf-

!n:nnt:uning stn.Liunnry ao'ial wnvc is un iinpu.s.sihitity, wha.tcver

Page 45: Lord Rayleigh - The Theory of Sound Vol 2

250.] WAVE 0F PERMANENT TYPE. 33

may bcthcvcJocity?~ ofthc gênera) currcnt, or in othcr wor~s tha.t

a w:),vc Cfmnot hc jn'n])!~at(;d ru)ativc')y to thé undi.sturbcd parts

of thcg~s withriut uttdurguitt~ !in :)ltc)-ation of ty[)0. Ncvcrtixjtc.s,

wh~n Lho ch:U)~us 01 (jcnsity concenn-tt are sma!], (;')) niny ))G

satisfied:).ppruxi)natc)y; arxt wc sec fron (-t) th:tt. tho vubcity

oistruatu neeussiuy to kou? tho w~vc stit.tiun~t'y is ~cn Ly

which is thc same as thu velocity uf titc v'uvc cst.iinatcd 1'da.tivdy

tut)ic(ini(l.

Tins ))i('t))()d of rc~.i,)'<tin~ tho sn1)jfct sliews, pcrhaps more

clL':).)-)y(.hana.)]y<)L))('r, thc !)i)turuoft!)< rotation hc<v(!cnve)oci<y

andcoudcnHation § 2-t.') (~), (')<). In !L st:U,i<jt):u'y w!).vo-f"r)n :), )(jsa

of vducity iiceompauic.s !m au~tnc'ntt't) dcnsity accor<]in~ to t))c

prmcipic ofcnm'gy, and tticrc-f'orc thc fLnd cotopo.sing the con-

dcnscd parts ()fa '\va.vc movcs f<u'\va,)'() more slow]y than tho

)in<]istu)'bc(t purt.iuti.s. RL'Iativciy to t)tû Huid thcrcfurc tho

niotiun of t)m c(j)t(h;t)Hû<! parts is in tlic sa.mc dh'uctiuil as that

iu winci) thc wavcs arc prcpagatcd.

Whcn thc reh~tion bct,wccn pressure a.)id density is otho' than

thatcxprcs.scd in

(~),a

stationtu'ywavc catt bo m~mtan.icd

oniy

by thu aid of an hupru.ssct! force. By (1) and (2) § 2S7 wc hâve,

ou thé supposition that thc moti'.m is stcady,

shcwing that an hnprcsscd force is uccc.ssary at every place whcrc

M is variabtc and uncqual to M.

2.')1. Thé reason of thc chimie of type v'hich cnsucs when a,

waA'e is Ict't, to it.suJF is nnt <)i~unft to midcr.stimd. ;From t)te

ordinary thcory wc know that an infitutciy Stnali (!ist)t)')):mcc is

propagattid with a. certain vdocity \))ic)t vdoL-lty i.s !-<-)at:o

to tlie parts of thé mcdmni urxti.sturhud hy thc wavt;. ]~t us

considor now tito case of a. wavc so long ti~at titc variations of

R. II.

Page 46: Lord Rayleigh - The Theory of Sound Vol 2

SUPERPOSITION 0F PARTICLE VELOCITY. [251.34

V(')oc!tyand dunsity are insensible fur a. considérante distante

a)o))~it,:tnd:tt.:),p)acc w])('rcthcvc!oeity(«)isnrntf'Jct )).s

i' ts!)).!t!s~co')'r.t\~ t'h'~)r. TL'v 1

with w)tich thcsccondary wavc is

pn.pnniUcd <hn)))~h t.he

tnediuolis~, !))[). on :'c-('HUt)t<.t't))(;h~:dt)t(,tit)nf,f'th(-]n<j<!iu)n

its<')t't))Cwht')t)V(']ucityot'!it]v:).n'-t' i.s~+M,an(I<)u))('))ds upmi

t!!C p!U'tf)t'ti)e]t.H' W!L\(! !tt, w)))(j)t tftCS))):))) ~:)V(.-isp]:K;).

'V)):).t,)~asbc;t'n~)(](~T,.st'c<))))]!))yw;LV~n))))i)('s:)).-i()tnt))Op!)]'ts

"t't))c)L'))~Yt.' itst.)f')))d<.))usw('H(-c t))!)t,!tf'(.c)-a.t,imc itD~n

pt!K.hcr(':t.(-t')'t:un \)oci<ytfist<'b(. fomnt.isin :u)\)ncc"f

i<so)'i~in:J p~.s)<icj)])y;t<)i.st:)nœc')n!(),~()<,t()<l)ut.t<)(~-+~)<;

")')s W(']<t;!y(~)))'c.ssi),); i.spn)[)~)tf(twit!t:t.vc)()(;ityn.+?/.

!').sym))n)i(-:dnt)t:)tin))"=-=.(.+((~-)-)~,wJH.ru/iMa.n!H'bitr:ny

functiun, anCt~))ati~))(i)'.st<~tai))L'()by Poisson'.

Froni ihc!L)~)U))<'))tj))st(.')~))]oy(-di<.n)i~)(. f)ppc:u-!)tf))'st

si~)ttt));tt:).))<)-:)ti!~))()t'<y))(.-w;).s!t)K'L't-.ss:))'yit)('i()(;))tint))L'])rog)'~sH

"t'!).w:)\-c,inth~)('Ht)(.'Ht)yofanyp:u-ticn)iu'supposition as totho

)'c);t<iottL('hvt-cH pressure :)))()dotsity.nndyct, it \).spro\-('(I m

§~)() t)):)t))tt))'(::)St'()i'on(Jp;uLit.'u);n' !:tw

ut'pressure tht'rc

wo))H he no !ttter!)t)o)t ut' type. AVe ),a\'e, ito~'ever, tacitty:).ssH)))L'd intitu présent.se('io)) (hâtais <-o))s):)))t, w))ie)~is t:)nta-

momt.toa restrictiuttto ]3oy)e'.s !a\v. Un~'rnuyc'titerifLWcf

pressure~)

isa iuncLiun of/3,a))d theref'ore.~s wu shaH soc

presen<)y,of/f. In thé ca.scct't.hchuvexpn.ssp<1 in (;'))§ 2.'i(),t)iC'

rotation Letw-'L-u Ma.n'tpfora, progressive \vaveis sxch that

A/(; j+~

is existant, asnu~h :ul\'anee heing !o.stby s)o\vcr

propa~atit)))()nct<)nugniente(t()cn.si<yas is~'ainc'ttby superposi-tion

ot'thevetucity~.

So far as t))c constitution ofth<' médium Itscif'isconccrncd

thc;)'t'isnot)nn~topr(;vent,oHras('ribi))~arbit.rnryva))K'stt)bf)th

?/arn)/), hut in a progressive wave a rotation ~et.\yeenH)csc<<)

<jUa))tit.ics]nust)'c'sa.tis)itj(). ~Vckuo\v:u)-ea(]y~2-)-))tha.t<his

is thc case whon thc 'Ust.urbancc i~ smaO.and tl)cfoUowing

arj.;)nn('ntw')H))oton]y:,1)cw<))at,snch:).r(;)atiu)iistobccxp('ctt'd

in cases whurct))cs(piarcoft))c motion inust.bcrcta.incd,but

will cvun derinc thu forn) of't))C rutn.tiun.

~rrn)('i)CM)u-)aT)h"'f.]i('~))Sn)). Jo~))~ ~N~t. vn.

p.)U. ]sos.

Page 47: Lord Rayleigh - The Theory of Sound Vol 2

251.] RELATION BETWEEN VELOCITY AND PEXSITY. 35

W)):)t(;vcr may bc t!)c Luvofpre.ssm-e, tlie

vaiccity ofp]-cpa"!i-

'n.H.!is~uL. !.s.'jy~ ..<

1

tn.=.

I¡"II n 1IIl,:¡ j Il,'itlt!.J;III(' 1.bJ", :1.)

('I¡'WI.

'V~,'P¡II n

j~.s.(.vf'p)-~r.ssn-uwLvuthcrc)atinnbctwcc..vc.)oeityan.!cnn-

d<)!s;Ltioni.s

T<],s n-I.-d.if.nbc vi.tcd.t:)y pnm~it w:vcwi!) ..n~r~,

'r.-u')!.)~int),c. n~~i\-c <h'r~-tim..L~usno~pk-hn-cto.

~<c~cof~)~vcpm~n~~vcw~~in~~n~,Uœch~Hat

YL.!uc.t.y~n<h)..nsit,y :m.v..ry ~nu)u..t) h,,tbc.-on~.i.nj.ntL~

nc<unn,!atjm,))..t,i).smq~n-w).Ltc.dit,i.,u.smu.stbc.s~!sfi..di.L

"r<t.rt.o],rcvc..t),h<jf.,r,u:,t,n<,f..L,)~t.ivuw~vu. [ti.sc)(.)-0.atth<.nn.s~rt..

<c<)U~tiu.h.-t).r,not,:Ln..g..ttiv~wavc~-i)tbc

~-n~.<L.<hL(,anyp.,intwi[),)..j~n.tupu.iM<c.st.~c of't,!m)~i.,t).c

i~m~].nLci~)~uur)n,ud thor't,a.hh.tt,p,~ t.hc.stator

thu~.s.-Lt~<)..sL-L.K.cfr.nni(,n.twm t! bud~cnmn.j<tby

thucn).L.n~nnpp)ic.)~. <.us,na))di.st.rb~~s. ln

ap].)yi,rt).i.s

c~nuuwc~rcto~.m~h-r D.c

v..)uciti..saHdeund~t,im.s, nut

~)ut~y,l,trd~.ivdy h,<).pmvai[ms h,

thcnci~.bnuri,~f.rt..s oi ti.c m..hui~ .s.j tJuLt tliu furni uf (1) jn.upL- fur t.).c présentpm'po.sujtj

wh!d. )sthc dation lx.t.woc.1 «and~ ncec~aryfura p~.sitivoim~rcs.,ve ~vc. E~u~iou (~ .ya.s obtai,,L.d

a..aiytic.d)y hyi'.in'n.sfiaw'.

1.~ the case of Bcy!c.s h~v,y~

isconstant, and t).c re!

lion bchv~.uvciucity ~nd

den.sity~ given fir.st, 1 LeHcve, byHuhnholt/iij

n bc tiic()c)ts)t.y co)Tcspon(]ing to = 0.

In this c~c Poissfjn'.sintègre aHow.s n.s to ~.rm .-L<).fmitc I.)c.-t

et U.cchang-c oi-

type ~ccompanying tlie c.-u-Iierst~cs uf t!io

7'/N7. ~'nf)).<. l.s~:), p. Hf;.=

J~r~r)~. (~ y' ,y. p. ioc. 18~2.

3–2

Page 48: Lord Rayleigh - The Theory of Sound Vol 2

ULTÏMATE DISCONTINUITY.[25L

3G

progressofthcwavc,n.nd it finaUyIcads us to n. diMcultywhich

has not a~i yct bccn surmonntcd'. 1. If wc dra.w a curvc to rcprescnt

thé distribution of vclocity, taking .'<; for aLscissa a.nd ?' for

ordinatt. wc inay find thc con'cspondin~ curyc after H)C )ap.sc of

timc by the fotiuwh)~ cunstruction. Thron~h nny point on t)'c

Ot-igitia) curvc draw a st.raight ]me in thé positive direction para1)''t

to a;, and of k'n~th cqmd to (f7.+ ?~) or, as wc n.)').; co))cc)')u''d with

thc shapc of thc on'vc on!y, cqual to Il <.t. Thu ]ocns of tbc ends ut'

thèse linc's is thé vulocity on'vc aftur a, time <,

But thi.s Ia.w ofdcrivation caxnot ))o)d gond iudcnoitciy. Thc

crcsts of thc velocity cnrvc g'ain conth)U:d)y on thu t)'(m~'hs and

must nt last nvcrtakc! thcm. Aftcr this t))c cm'vc woutd imticatt!

two vaincs of ?/ fcf onc Y.dno of cca.siog to rcprcst.'nt anythi))~

tha.t eould actuaHy t:)~~ p]acc. Ju fact wc an' not at lihcrty to

push the application of thc intégra! htyorni thc point at which thf

velocity becomc's discontinuons, or thé Ych'city cnrvc bas n, vertical

tangent. In ordcr to nnd wlicn this happons !ct us ta~c two

ncighbonring points on any p:u't of thc Ctu'vf which sh)pcs down-

wards in thc positive direction, andinouirc

attc-r what time this

part of thc curvc beconcs vcrtica]. ]f thc différence of ahscissa!

bc ~.r, thc hindcr point will ovcrtake thc forward pointin thc

timc ~(–f/~). Thus thc tnotion,as dctcrmincd by Poisson's

e<ptation,bccomcs discoitinuons aftcr n, titnc C(p)at

to thé ruci-

procahtakcn positivciy, of tl~c grca.tc.st négative value of

ff~

For cxa-mple, lot u.ssuppose tha.t

whcrc !7is thc grcatcst Initia) vdocity. Whcn < = 0, thé grcatest

négative-value of Is so th~t discoutinuity will com-

mence nt thc time < = X 2??' !7.

Wi)cn ~iscnntinnit.y sots in, stritc of things cxists to which

thcus)tn.l<!iH'crunti:d uqu:).t.i~))Sn.)'c inappHcabtc'; fLUt) thé suhsc-

qucnt prognjHS of thc tnotion ~s nf't hccn duterniincd. It is

probable, as suggcstcd Ly Stores, that s(j)nc soi'tofrpOcctioawouid

ensue. In regard to this ma.ttcr wc must be carctul to kcep

StokM, Ou a dimeulty m thé T)icory of Soumi." .P/< ~ny. Nov. 1818.

Page 49: Lord Rayleigh - The Theory of Sound Vol 2

251.]] EA.RNSIlAW'S INVESTIGATION. 37

purcly mathcnmtica!questions distinct from physical oncs. In

practtuc wu havu to do with sphcricat waves, witusc divcr~encyinay ofitsdt'bc sufMcicut to ho)d la chuck tl~c tcndcncy tu d~con-

tmnity. In actuel ga.sc.s tuo it is curkun t!t:).t: bufut-c discontinuitycould enter, thé hnv of

pru.ssurc won!dbc~'in to

citanteils funn,

~ndDtcutOnenccof'vi.sco.sityconid nu lunger ije ncg)cctcd. Hnt

H'csc eon.sidur~tions hâve nothing to <)o wit]~ tho mathumatlc:il

pt'obton of(tutunninin~ wh:).t wuuld ))appcn tu wa.vcs of nnitc

a)np]ttu<)u in :t, nicdiun), frce froni viscosity, wlioso pressure isundur~ cireumstauccs cxactty proportionn! to its duusity; audtlus probJunt hus not bucu sulvud.

It is worti~y of rcinark tiiat, althou~'h wc may ofcout-se conçoivea w.Lvoof Unitc disturhance to cxist at aoy inomunt, thcrc is ai'tmt tu tho dun).ti~n ut' its prcvious indcpcndcnt cxistc])co. By

dmwing Unes in t!)u ncgativu instcad of in thc pusiLivc dit'ectioa~'c may trace t))u Jtisturyof ttie vuiocitycurvc; and wc .sce thatas wc pus)i our lutjulry furth~t- an(t f))rt)nir int.) past timu t))c fur-

ward ~lupus bucunic oasiur a)td t)ic back\v:u-d slopcs stccpci-. At a

timc.cquai tu thc grcatcst positive valueof~,

antceedunt to that

at which thc curve is nrst coiitcmplatcd, thé vclocity would bc

diseuutinuous.

2.~2. Thé cotnpicto Intégration ofthc exact cquations (4) aud

(Ci) § 2-t.!) lu t))u ca.su uf a progrc.ssivc wavo was iirst utfcctcd byEarnshaw'. Findiug ruason fur tifinking that iu a sound wavu t])o

équation

ea)i by jncfuis of tliearbitrary function bc nnutc tu coiucide

witli a!)y dynanuc:d équation in wliicli t])C ratio of aiidt~

i.scxpresscd

in tcrm.s of Tlie f'orm of thc fnnct.ion F Lc-in~M~C °

7'ruct'~t~ u/' </<e~«! ,S'~t\ Jau. C, 1850. /<t' 2')~. 18CO, p. 1~;(.

Page 50: Lord Rayleigh - The Theory of Sound Vol 2

~8HARNSHAW'S INVESTICATICN.

[252.

thus dctormin~ t))e8o)ut,io))maybccumpicicd bytLc usuat

prucc.ss upplicablu Lo sncb cases'.

Writing for Lrevity et in pince of- ~'e h~.vco Jcl.c

as might aiso ha.vc bccn info'rcd from (.).) § 2.')1. Thc cûnstant (7

vanisi)CH,if~(o!),viz.<,V!U)i.s)tw!)(j)iet=I,û)-jp=~; othcnvisc

itrc{-)renent.iavu!uuityot'thc)n<;(ih[)u:Lsa\Y)tut(j,!)avin~])()L!)in"'tu do wit.h ti)c w.t.vo as such. Fur a~oA~~e pro~russivu \vayc t)~

)<vcr si~tts in ttie :uubig(ntics :u-c to Le uscd. Titus in ptaœ <jf

(!{),wc;ha\'e

fi-uni which by (8) wc .sec ti.aty-(a+ «) t is an arbitra~ function

1Loulu'h' Z~t')i<t<;< A'</t<ff<<(~f.<, Ch. xt\

Page 51: Lord Rayleigh - The Theory of Sound Vol 2

253.] MEMANN'S EtJUATtONS. 39

of &, or of ll. Conver.s.-Iy thcreforc M is fu.fu-bitrary funetiuu ut'

,?/–(a+;f)<,aiidwc]n!).ywri).u

E<;))ation(9) is Poi.s.son'.sInt~m), con.si.tcrud ht thc

pr~cc()i))g.section, whurc t)<c sytnbu! x htLs thu sa.)nc

mu.tni))~ :Ls hure ~t~ch~

tt)~.

2.'i3.T)'c]))-uh]mn<)ftj!:).))cwtt\'(;.s<)ft)))it(!!un)))it.()(!(.).tt]-!)ctc<!

a)so t))c !~tcnt~)t uf Rictn:mn, wttu.se niutnou- wns connnutucat~ttu t)ic Ruyid Sociuty(jt'UuLt.i))~~n on t)te 2Kt.i)

ut'Novuntber, LS.T)'.

Riu)n:mn'sinvnstig:)tio)t i.siomxk~ontttu~.ncrathydrodytuuniciLt

e.t.Ltion.s Invc.st.i.n.-d.cd in ~:{7, 2:~8, :Lnd is )mt. rcst.ict.c.t tu :my

p:n'tic)ti:n-):).wot']~.ssurc.Inon)(;)-,))uwuvur,]H)tn)t(h)iytuHx-

to~t thé di.scu.ssio)~ of thi.s }):n-t. uf onr.subj('(.-t,, a)t-c:KJy po-h:U).s

trc:U.cd n.tgr~Ltcr Icn~t)) t)):).n itsphysicid imp..rt:t.nc(i wonM

w;u-t':mt, wn si)a)l hu)'u confnKi uur.sutvu.s tu thc e~su ofBoytu'~ hw

of pressure.

Appfying c~u:Uions (i), (2) of § 2.~7 and (1) uf § 238 to t]iccu'cumstimccs ut'thu presunt pruLk-m, wc "'ut

'Uub~dicrurtpfJnnxunH ebencrL"fbv(4)ûnvnne))d]i(..hcrSrLwin~u,)~-

witc. (iiittixn~n, /)~/«nt~/)n<t,t.vm. 1~0. S<:e)d.un.ucxc(;Hûnt nb.itt'tMt.

i~the/«.<r/~t'<<r)'<XY.p.l~

Page 52: Lord Rayleigh - The Theory of Sound Vol 2

40 IJMITED INITIAL DISTURBANCE.[253.

Thèse équations arc more général than Poisson's and Earnshaw'3

ill t)tattheyfu-c not!imitudtoi]te ciuseofa single positive, or

négative, progressive wavc.. From (5) wc Jearn tfiat \v})atevci-

may bc Lhe v:dae of7~corrc.sponding to thé punit a; and t]ic timc

t, thé s:une vahtc of l'corresponds tu thé point .c+(?;.+a) at

thé time <+~; am) in thcsiunc w~y from (<i) wc sec that Q re-

m:Li))su])cI):U)~() \Y)K-n.a.~d ac<)uirc! t)~ n)C[-(j)no)t.s (~-a)f~n-nd </< rcspuctivL-ty. If 7' and Q Le given at a. ccrtiun instant of

ti)no ns fonctions ci'.r, aud thc i-cpt-esott.fttivc! curves bc drawn, we

may duducu Lhccot'i'c.spondii)~ vatue < ;< hy (-t), and thus, as in

§~51,cunstructthecu[-vcsrup)t.i~(j)tti))~t])evatucsufj!and (~aftc-rthc sma)! intcrvaloftunu~, ft-un)w))ie)t thcnuw values

of <t au<] p in their turu bcemac knowu, a)id thc pruccss can bc

l'cpuatcd.

T))C ctcmcnt of the fluill, to wLich the v:ducs of T~and (? at

any moment Lctong, is itHc]f nioviug tvith tite vcfucity ;<, so that

thé velocitie-s of~and Q rch~tively to tho dûment arcmunuricaHy

thc satnc, and (.-quai tu a, th~t of 2' huiog in tlio positive direction

aud titat of in thcncgativc direction.

Wo aro now in a position to trace theconséquences of an

initial disturhancc which is confincd to a finite portion of tho

jncdimn, e. bctwcon .x= c( jmd s-=~, ont.side wldch t)ic médium

is at rcst and at its norma)(icnsity, so that thc values ofjP and

arc ci ]o~ ]!ch valuc uf P propagatus itsuif in turn to thc dc-

mcntsof nui.) whicitliu in front «fit, and cad)Yah)ûof~tot]foscthat lit: Lcitindit. Thuimniorlimitof the région In-\vhi<jhPIa

variabfc, viz. thu p)acc witcro first attains t)tc constant valueft

log~, cotnes into contact rirst \Yit)i thc variahfc va)ucs ofQ, and

movcsaccurdingiywith a Yariah!c'velocity. At a ddinito timu,

rc()uiri))~ fur ils dL-tL-rmination :t Sf~utioa ofthu din'urcutiatéqua-

tions, t))c )tin()cr(L'ft ham!) iinut of thc rL-gion throug)) whicii

varies, tuccts thu ttindcr(rig)it iiand) iinut of thé région throuTh

wtti(di~ varies, afterwhic)) thé t\vorégions scparatet!)em.se!vcs

and indudu hchYCL'n titem a portion uf ihtid in its etputihrium

condttion, as appears from thé tact t])at thévatucs of~and are

hot]j (tlugp.. in ttie positive ~avo () lias thé constant value

M log~, so that !~= et log~ as ill(é) § 251 in the

négative waveC)r~

At t)na pnint no f'rrf))' sccmsto haYO cr~pt into Diem~uu' work, winch iB cor-rcetLd iu tlie t~tract uf the 7'u/<t/t«<' ~< ~t'A-.

Page 53: Lord Rayleigh - The Theory of Sound Vol 2

253.] POISSON'S INTEGRAL. 41

P bas tlie samc constant value, giving as tlic relation botwcen u

andp,

M = log Since in cach progressive wavc, wben iso-Po

latcd, a law prcvaits connccting thequantitics M and p, we sec

that in the positive wavc f~t vanishos with and in tlie négativewavc ~<t vanishcs with < Tims from (5) wc k-.arn that in a

positive, progressive wave vanishcs, if thc incroncnts of a? and

t bu sucii as to satisty thu équation <~ ()t + f~= 0, froni winch

Poissun's intégrât Immodiatuly foltows.

It wou)d Icad us too far to foUow eut titc ana)ytical dcvclop-mcnt of Kiemaun's method, for which the readcr must Le refet-rcd

to tho original mcmoir; but it would bc ijnpropcr to pass over iu

sUcucG an en-or on the suhject of discontinuous motion into which

Rionann and otiier writcrs iiave faUcn. It bas bcen bc!d that a

statc of motion ispossible in wbich thc nuid is dividcd into two

parts bya surface of discontinuity propagating itscif with constant

vclucity, ail t)~ nuid on onc sidc of the surface ofdiscontinuity

hcmg in onc unifonn condition as to density and vclocity, and on

the othcr sidu in a second unifonn condition in thé sanm respects.

Now, if this motion werepossible, a motion of thc same kind

in which thc surface of discontinuity is at l'est would a)so ho

possible, as we may sec by supposing a vclocity cqua! and

opuosite to that with wl)i(.-h the surface of diseontinuity at nrst

mo\-c.s, to bc impres.sed upun tite whoie mass ofnuid. In ordcr to

nnd thc relations that must subsist betwcen thé velocity and

density on t]ic onc sidc (~, ~,) and t))e velocity and density on the

othe)- sidc (M,, p.J, wc notice in thc nrst place that by tho principtoof conservation of matter

p,=p,?< Again, if wc considcr tho

momentum ofa siicc bounded by par:d!el pianos and including the

surface of (U.scontinuity, we sce t))at the momcntum Jeaving thé

stice in thu unit of time is for cach unit of area(/J.=ptM,)~,

wbiie thc momentum cnto-ing it is p,!< T)ie dincrence of'm~

moitum mnst t)C ba)anced by thepressures acting at thc boumhu-ies

of thc slice, so titat

Thé motion thus dctct'uuueJ is, howevcr, not possible; it satisfics

Page 54: Lord Rayleigh - The Theory of Sound Vol 2

EXPERIMENTAL DETERMINATIONS [253.42

tlie con()it,io))s <~ n~ss :md motnottum, but it viutatus thccututitiou of cnc~y ~) cxpru.sHud by tiic c~xatioa

i'I.fs fu-~mcnt ha.s b~-caah-~dygivo.i In anothcr for.n in 2.~

whicit wuu)datone jus) ify us in

rL.JMt.n~ t!.ca.ss..mod rnoti.-n.sincoit a]))ic:u'.s (.[)~t no

stcady motion ispo.ssibfccxcoptundcr theiawof

dcn.sity D.crc dctcrxnnc.t. Fmm~~u.-Ltiun (.S) of t).t .s~ctiun wu

can fuxt wliat unp)-('ss~) furcc.s woui.t benccc.ssary (.) nia:).t:))i t)jo

jnuLio..(tcHned ).y (7). It f.p].c:)r.s that th.; force .Y, tho.~h c~n-

~))0() to tllc pkcc ut'~cuuti.mify, i.s ))):u)u up of two ~rL.s of

app.).sitc si~ns, sincu Ly (7) !tp!t.s.s. t.),u y:,),)u ~'h..

whoïcmoving fu)~,vi~p~v;u)i.s].c-.s, !u.d tiji.s cxphun.s i.ow

it is t))at tho con<)itio))n..)~ti))~ to muj.K'ntum i.s satis~!

by (7),thoug]) thc iu)-co A' bu i~nurL'd :dt.o~ht.-t-.

25~. Tho cx~tcxp~ritncnta] .~crnnxation nf tlte

VL-toci~yof sound i.s a m:~u.-

of~ro:Ltcr di<)~u!t.y ti.:u.nn~ht i.:Lvc hcL.n

cxpectcd. OL.sc.-v;K.iun.s in thc u]K-.i .nr arc ti~Hc to crror.s f.outhc ctt'uct.s of wind, am] frum

nnccrtfunty wi~irc.spcct to thé

exact eon<titio;t of thcntmo.sphcrcasto tonpcnt.tureand drync.s.s.

On thc uthci- Jtand wht~i sunnd isp)-pa~!)tcd throug]i air cc.~

tained in pipes, disturhance ~risos fron fricLimi :md fr.nn tt-iu).sf.;rof Iicat; and, aithou~h in) ~t'cat o-rot-s frmu tht-sû .sources arctu bo fearcd i)i thu case of Luhu.s of con.sidcrabic

di:u))ut(.'r f,m;h:ts Hutnc of thosu

ctnptoycd Ly R~xatdt, it is dimcuit to f(;ejsure that tt~c iduat ptanc wavcs of

tt~-ory aruiicarty cno~h

rcalixcd.

T)~c foDuwin, Table' nontain.s a list of theprincipe cxpcri-

meutai dutermiim.tiona \iuc)t )mvo bcen madc hithcrto.

Nnmos of Observera.v. of

<J"Cuut.i)iMutruH.

AcadomcdesScicnccs (1738).

Lcnxcubcrg(lMlJ)

(33~3f~33'7

(~l()ing-ham(182l) s~i~ur~u des Longitudes (1822) 3~o.('MuHandv~nBbdk

3~')")

'Lusanrjnet,U~.Apri), 1877.

Page 55: Lord Rayleigh - The Theory of Sound Vol 2

254.j0);' TilK VELOCITY 0F SOUND. 4:3

KaincsufObsorvM~. VctocityofSnuudu.tU"C(jut.u)Mct.run.

St.'U))j'J'c)'andA!yt'b;K'k.

;)h-!LV:Lisan<[AhtrL)ns(lS4-).). M2~

W<jrt.hcnn H~l'C

~tunu (IM71) :4

Lu i~ux. :j~()'7

R~-)t:mJtt 3!:3()'7

In Stonc's cxpo'nnGnts' thc course ovcr ~']nch thc Sound

wa.stitnud connnunccdatib distance of (!-t()fcetfr<)nth<so))rc(-

sot)tat :niyetTorsarisIt~ fi'oa uxccssi.ve disLurbancu werc to

a ~rcat uxtunt avuidcd.

A mcthod hasbcoi proposcd by BusHc)):rf"t- d(jteniu!)in~

thc vctocity ut'sound wit)K~tt t!<c nsuoi'gruat distances. It

dépends ~pu))thc!prccisiotiwiU)w)nc]tt)iu car is!).b)c to décide

~')~(jt.))ernh()rttiekHar('sinndt:U)eouH,ornot. InKonig''s"f()rni<jf

thc cx])cnincnt, two sn)!LH c)ectt'o-]nagnettc countcrs arc controllcd

Ly :). foj-k-intCD-uptcr (§ C~), who.sc pcriod is ono-toith of n. scconf~

and givc syncftnmous t.i('.k.s of thc ~atnc pcrtod. \V!)en thc

conntcrs :u'u c)o.sc to~ctitur thu mtdibtc tid(s eoincido, but as otic

c'onxtur is gmdmdiy rcittoved from t])(i on.)', thé two scrics of tic)<s

f;d!. !Lsuud<r. AVficn t))u dii!'cruncc of distances is ~bout 3-)< jnctres 1coincidc-ncc

a~aintakcs

place, proviog Diat ~-t mctt'es is about

thc distance travcrscd by sound iu a tcntli part of~, second.

'J"/tt;«M.187~p.l. ~7'<w/)));.xcfj. 1~.1851.

~tM~.CXVIU.UlU.lHC~.

Page 56: Lord Rayleigh - The Theory of Sound Vol 2

CHAPTER XII.

VIBRATIONS IN' TUi~ES.

255. Wn hâve an-e~dy (§ 245) considcrcd tne solution of our

fundamcntal c'pm.t.ion, whcn tiie yciocity-potuntia. in au utilimitcd

Huid, is a. fuuction of onc spaec co-urdmate on]y. Ja thu absunco

uf n'ictioti noctm.ugc

woutd be ciUt.scdby tlie introduction of a.ny

munber of iixed cylituh'icit.I surfaces, w)ioso geucrating lincs a.re

parallul to thc eo-ordumt.c lu questiou for evcn whmi t)ie Hurfa.cca

arc absent thu ihud ba-s no toidcncy to move across tbeni. Ifons

of t)ic cyHtKh'ica.1 surfu.CLj.s bc ctosed (in rc'.spcct to its transverse

section), wc ftavc tbu impot-t:uit prubicm of tlic axitd motion of air

wltinn :). cylindricat pipe, whici), wncn once tho tnechanical condi-

tions at thc cnd.s arc givcn, is iudcpcudcnt of anything that may

happcn outsidc thu pipe.

Considerin~ a simple harmonie vibration, wc kuow (§ 2-t5)

that, if <~ varies as (~

of winch hnniïy on;y ti)C rcn.1 parts will bc rct.:uncd. Tho first

f~rm will bc must (.-uuvcuiclit whcu t!ie vibration is ëtationm'y, or

Page 57: Lord Rayleigh - The Theory of Sound Vol 2

255.] HARMONIC WAVES IN ONE DIMENSION. 45

nc:u-!y so, and the second whcu thn motion rcduces itself to

pn.stLive, or ncg.~ivc, progressive undulation. Thé const:).nts

and m tlic syinLùiiciL! sotution may bc co]np!px, a.nd thus the<in:d

expression in ternis of rcfd qufmtit.ics wi)Iinvoive/arbl-

trary cousta.nts. If wo wish to use re.d ()H!UttHics throughout, wemust takc

but thc fmatytic~] work wou]d gt:.])er!)Hy be longer. W])cn no

:unhi~uity e:).n anse, we sl)a)) sotnetin~s for thc s~hc of brevityd)-up,urr<st<n'c, tbo

t~torinvolvmg t.)ie tuncwithout oxpt-c.ssmention. E(~u:ition.s such as (ï) are of course

cqu:t))y truc whethcrthc f~ctor be undcrstood or uot.

Taking thc Urst form in (3), wc h~vo

If therc bc any point at which citLcr <,&or is pcrmancnUyzcro,

thc mtio 7~ must bc rc~, ~nd thcn t)ic vibration is ~i'OMury,t))at i.s, thc samc in p])n.sc~t fdt points sinmttiLncousfy.

Let us suppose t)tat H)crG is a nodc at thé origin. Thcn whenf/d)

~cs, thc condition of which is 23 = 0. Titus

From thcsc équations we tcarn that vanis!)Gs wh crever(!.CG

sn] ~ï:=0; th~t is, th~t. hésites thf: ori~in t.hcrc are nodes at tho

points .B=~?~X., M bcm~any positive or ucgit.tivc: intcgcr. At anyof thcsc piaccs inHnite)y thiu rigid ptanc b:u'ricrs normid to a;

mt~ht be stretched across the tube without in any way alter-

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4G NODES AND LOOPS.[255.

Ing thc motion. Midw~y hct.wccn p:~h pnir of conRccutive nfx~'s

thm'u is :). /w/),o)' p)!t.e(.' of no prcssm'c v:)ri;)tion, sincc ~) =–~

(())§:2't"t-. ALïmy ofthc'sc h~ps~ c'tuttnihiciLtion '\Y)L)) tho

cxtcrna) ntmuspiicrc tnight, ht.'npencd, wittmut c:m.sin~ :).))y ()ist:)))'h-

fmcecftLc'tnotKjnJ'romit.it'passing'inoront. 'J~)u)nopsa)'cthc

ptitccsof maximum V(.')"cit,y,n))(l t)tc nodestitthscof maximum

pressure v:u'i:).tiun. Àt iutcrv:ds of uvL'ryUtit)~ is cx:)ct)y rc-

pcaLud.

If thcrc Le :t. undc n.t.T=~ ns wH as a.tthc nrigin, si))K~=(),

nr À.=~ wht.')'c )~ isit, posit.ivc intL'~r. Thc ~')';L\'ust tono

~'))ichc!H]bcs<'un(]('d~)y:),ircunt:LinL'di)t!Ld')n))tyc)nm;d pipe

of [cngLh i.s th<)'~f'o)'(; thut winc)) h:ts :L 't.Vt'u))g't.)t op):~ tu

TJii.sstitt.umott, itwiilhu observa), )t(')').s~'()J\]):~<vc'rbctJ)c

g:).swlthwhic))t)tcpipnisfi))cd; h))t<~cin~)))<tcy,o)'th(; place

«)' thL! tune' i)t t)tu nm.sio:).! Hc:()c, dcpL'thts n.).so on thc nature; uf

t))u p:),rti<jutar g:t.s. Th<! pernjdic tiuK! is givun by

Thcothci'ton's pns.sib)c foradoubiyclonu()pi])c i~vcporiods

whic)t :u'< sub)))))tti))iL'sof t]):Lt of thu gr:LVcst t'~tc, :Utd thc whotc

sy.st,('nifo]'msah:n'tnu)ticsc:t)c.

LutttsnnwHUpjto.sc, witLout, htnppm~forthcnn~ncnt. toin-

f[nn't.;howsn(;)t:t. condition ()fthin~sc':U)~t''sc'f'u)'(.t)):)tt))C)'ciH

:).It)opinstt.K)ot':).])<)()c!tttnej"~mt~=/. 1. )'u:t<i<n)((i)~ives

cn.s /< = 0, whc'nco = -)-~ (2/~ + J), whf'rc ?)! is zero <')' :), pusit~iv'

iutc'gcr..)n t~his c:).se thc ~'t'nvc.st to))(; hit.s w:n'c-tu))~L)t oq'):)l

lof<'n]'ti)ncst))c]t')in't)tr)fthcpipci'u('k')))''(It't'<))<) thu nodctu

t))o Joup, in]ttthcf)t))c['t~nc.sf())'tnwit.hit,:L!):n'n)0)tic.sc:L)(.ft'om

w))ic)), huwuvcr, :d) thc nicnibcr.s ufevoi untcr :u'e ini.s.si))~

25f!. By incans of a ri~ift han'icr titcre is no(tifUculty in

scou'i))~ a]t<"1'tt:uiy<k~i!'u<)po!nt()f:),tub< but. thc condition

for a h'op, i.c. t!)!tt mxtfr uo circumstaucus sha)) tl)c pt'c.ssurc va.ry,

ci\n on!y bG rcit.lixct) n.pproxi)natu)y. In inost cases tbc vnriatiou

oi' press)u'e at n.uy point of n. pipe mny Le nuide stnal)by aHowinn'

a h'ce coinntuuicution with t))G externiLl ah'. T)ms Eu)cr und

L:).grangc nssumodconstn.ncy of pressure n,s t!)C condition to be

satisficd.rLtthccndofanopcnpipc. WeshilUaftcrwn.rdsrcturn

to the pro1))cnt of thc open pipe, aad investi~a.tc hy n. rigorous

Page 59: Lord Rayleigh - The Theory of Sound Vol 2

25G.] 1 CONDITION FOR AN OFE~ END. 47

proccss thc conditions to bc satisficd a.t thc en(). For our im-

tnctiiato purpnse it will 1w su~cicut to kuow, w)):)t is imIcL-d

t<)!c)-;i.b)yobvi.jus,t))!).t tlle ope;) cndofft.pijtCjn~y i~ L'uat.ùd~

:).f""p,if thc diiunutcruf thcpipe bc n<~)<ctcd iKcotnpitrison

W)th<.))c~tvc-ju~t,]t,p)'uvi(tedthucxtct')]:Ltp)-c.sm'ûi~t.)t(.!]tc!r'-h-

Lu).)t'))U(t(!<jft))copeitC))<)henotitsu]i'v:n'i:djJ(jf)'<))ns())nc cause

J!'t)<-p(;)i<)(.-ntofth'))i[)ti~n\vitltinthc! pipe. Wi~nthcruisfm

m<))t.'n()c)ttH<)U)'cc(~'sunnd,t))c pressurent the c))duf'(.])c pine!).stho~n)0f).sitwou)d

)'t.'])tt))(;Ha.n)cp)acc',it't))Gp)pewcre

!t\ny. Thé in)pC()i)))C))tto.secun))~t))nfnHn))t(;!)tot').))(_! conditio)ii"r

a )~<)))~t:),ny(]('sircd point fies in théinertie ofth(jn):K;])iuc!y

)'r(pti)'n()t<).su.stai))t.h('p)-(.!H.su)'(i.FurthcorL'tif.-idptn'jto.st.'i-iwetnay

('vc-r]<)t~t])isdif!i('.u!ty,!UKU))t.!)~inua. 'nasstc.s.spintonh~kc'dhya conp~Hscd sn)-it)~a!so withuuttnaM.s. T!)C

a.s.sumption ofa,

foopatanopcncndofapipc ji.stant.a)nonntto)ic")cc[.in"'t)tc

mc'rtiaofthcoutHidcnir.

Wcha\-csc'('nt))a<ifa))odccxi.st:)tnnypninLoF apinc

tL(.-runn)stheasurk.s,r:ni~date(jU!di))t(.'rv!L)s~t.ttniid\vay

L<t~.ju)c:u.:)tpairoff-o))H~c))t.i\'tj;)~)dc.sth('rc)i)t).st])(;ah)op, a)t(t

t)tatthcwho!cvibratu)a))msth(.sta),i(')ia)-y. l''i)C.samoc(i)u])).siu;t

f~')].)ws ifthercbcatanypuiut a )<)op; hutit

)nay])L-rf'L'('tiywc]l

h:t,pcn thattho-o arc n~ititcr Utilesnorioop.s, as

jt)rt..x:mii))c inthL! case whcutitu motion i'(j()))~s to a positive or

~cg-ativu pro-gressive wavc.

In.st.atiu)):u-yvi))rati(jnthcrci.s)]ntra)).sfc)-(;nccof

('nL!r~y atun~ t])c tube in ei[,)tcr direction, forenurgy canjtot pass

anodt.'ora.loop.

2;'i7. Thc !'r-]a.ti(~)S bctwfCt) thc ]c)i~'ths nf an nnen or d~cd

pii)G attd t))c Avavc-icn~ths of (,])c itictudctt (-()]u~]i of au-rnny ~so

l'cinvostigato] hy it.Jtowi])~ t])c inotioa nf n, ~?/7. by ~1,ie]) is

umkTstuod a wavc c<~)(i)tc<! within n;UTow )nnits andcompo.scd

ofuniformiy cun<I<;))su<) nr r!U'cfi<j<! finie). Jn

]<x)hiu~ at thé ])):),t,t~ri'rom this point of vi<j\v it i.s

nccc.ssfu-yto takc intoncœunt e:u'c-

i'nily t]io cil-curn.sta.xccs nndcr Avhic]) tftc v:u'iuus t'cftuct.ious takc

])):tcc. Lc't us Ht-st sttppo.se tiiat cundutt.sct! pu]sc tr.ivcis in tho

positive dirccbion to~u-ds a b;(.n-icr fixct! acro.ss thc tuLc. Sinccthc

CDO-gy couta.mct) in thé \va.vc ca)ni"tcscapc h'om t!tc t)tbc

thcrc must bc n. rcficctcd wave, ami thiT.t this rcftcctcd wave isa).so n. w:i.vc of condcnsntion nppcar.s from t)tc fact th:).t tl)crc is no

!oss uf ttuid. Thc same conclusion may bo arnvcd at in another

way. 'J'hc cfîuet of thc hu'ricr may bc inutatcd by thc introduc-

Page 60: Lord Rayleigh - The Theory of Sound Vol 2

48 REELECTION AT AN OPEN END. [257.

tton of a. sinn)ar and cquidistant wavc of condcnsat!on moving in

thc négative direction. ~ncû thé two wavos are bot)) condenscd

ai'td J.C):Op;!g~i.~(.i Ut<<{t'!i.y <!i!'<'C!.io~ !<<'(.'h~'it.i~sut tho

ibtid composing tlicm arc C(jna) and opposite, and iherutbrc neu-

tralise one another wltoi tlic wavcs arc snpcrposcd.

If thc progrcss of t])C négative rcf)cctcd wavc bc intcn'nptcd

t)y a second barricr, a. shnilar rcitectiou t:).kcs p)acc, and thé ~'avu,

still ronainin~ (;ondui)S(;<1, regains i(.s positive chiu'af'tur. Whcn a

(tistancc bas bncu tt'avc))('d C()ual to twic<j t)ic ]L'))gtb f'f tbc pipe,

tbo ori~)n:d statc of thii~s i.s cotnp)ctt;)y rcstorcd, ?u)(! thc snmo

cycle ofcvcutsrf'pcats itsc)fin(h'finitdy. Wc Ica)'~ t]n!r('<'f)'c that

thé pcriud within a <)')<th)y doscd pi])<j is t))c tinic! cccxpiud by a

pu]sc in travdHng twiec t)tc Icngtl) ofthc pipe.

Thc case of an opcn end is Homc'what différent,. Thc suppic-

Jncntary Jtf'~ativci wav(; nuccssary to imitatc thc cHuct of thc <~)Cti

end umst cvid(;nt]y hc a wavu of rarcfaction c:tpahtc uf ncntral~in~

thé positive pressure of tbc eondcnscd primary wavc, and t)n)s in

the act (jf rcf)(jction a wavc chao~c.s it.s c!):trac).('r fro)n eonthjuscd

to rarc'ticd, or fnou rarcfx.'d to c<n)d(.')).scd. Anotbur way of con-

8id(.-i-ingthc)n:ttt.('r is to observe that in a positive condeuscd

pu)nc thc momcntum ofthu motion is forw:u'<)s, am) in thc

absence of thé neR~ssary forces cannot bc changL'd by thé rcrtcc-

tion. But forward motion in ti)c rc-uccl~t négative wavc is

indis.so!ub)y connected with ttic rarcficd couditioti.

Whcn both ends of a tube arc open, a puise tr.T.vc'Ding bar'k-

wards and forwards within it is compiL't'dy rcstor<d to its original

sta-tc aftcr travo-sing twicc tbc ]cngt)i of tbc tube, .sufft'rixg in thé

proecss two rcucctions, and t))us tbc rctation bct\vcen Icngth and

period is thc sarnc as in thé case of a tubu, whosc bnds aro both

closed but whcn onc end of a tube is opt'n and thc othcr cloncd,

a. double passage is not Huniciunt to c)osc thu cyctc of chaxgc's.

Thé origif~a) con<)e))S('d or rarencd ch!tract<;r eatmot bc rccovcrcd

until aftor two )'(.'H(j(;tionH from thc opcn cm!, and aoco)'')i))g)y in

thc case contomp)at(jd t))c p~'riod is t)n; tinic n'fptircd by thu puise

to travcl overyu; titnc.s thc k'ngth oft)'c pipe.

258. Afiur t1)c fid) discussion of thc rorrcsponding prnb]cms

In thc chaptcr on Strings, it wiii not ht) m'ccssary to say mucit on

thc cornpound vibrations ofco]umnsof air. Asasimp)ccxat)tp)o

we may take tl)c case of a pipe opeu at ono end aud ctosed at thc

Page 61: Lord Rayleigh - The Theory of Sound Vol 2

258.] PRODLEM. 49

othe)-,whichisM)(Mcn]yhrougLttorcstn,tthctimc =0,-<('t(jr

Lcing for sonc tune I]) motion vi).h a uniformvelocity pand)~! te

its ieu~t.)). Thé Initia! statc of t)te cont:uncd air is tiK-ti onc 0!'unif'orni vducity p:u-nHu] to .-K,{md offrocdom from

compression:ui(! r:u'<jfactic)i. If wc

suppose t)):Lt thc ori~iu i.s at tl~e clu.sett

un<), Du! gf;ne]-it.I sofutiuu is by (7) § 25;'j,

con.st:).nt,.s.

S!ncc~!st<)LcKcroirnbi;).])yfor:tnv;L)u(-Sfjf~ thccocfn-cio.t.s nnt.st vanish; thccocfHcient.s arc to Le (IcLcTinl)icd

t'y thc condition that for~H vidu~ of.~ 'bctwccn 0 :m<]

~'ho-c Lhc.sum)n:tLioucxtcndstc~fintc~-al vaincs of?' frotu

to œ. Thc ~cte)'mi))!itinn of thc cocfficiotts jd fron (2) is(-fîbctc<) int).c

usu:dway. Muitipfying hy sin~d inté-

gra t)ng fron 0 tn we gct

Jn U)ccascofatubestnp])(.<]att.)tcorigi)tan(]opcnat

.7; = l, lut ~= cos ?i< Lh tlie value of t),c potcnt.ia) :tt thc open en.) 1(h~ to !Ut cxt.<nfL) .scmrœof so.nxL

Dcf.onnit.i.~ r and 6' ili

<)!ahon(7)§25~, \vff!)t.)

Itappcar.sthatt)~ vibration within thé tube is aminimum

whencos~=~l,thati.swhe.~i.samu]tip]oof.inwhk.hca.sc

Ho4

Page 62: Lord Rayleigh - The Theory of Sound Vol 2

50 FORCED VIBRATION.[259.

thcrc Is a nodc at a:= Whcn Is an odt} multiph of ~X, cos /<~vanishes, and thcn aceording to (1) the motion wcuid bccomo

n.f. lu f.)ns (w;)' the sujtpùstt,icn thut Lhc pressure at thc

openend is in(tcpendcntofw!)!).t h.q)pcuswithin thé tube bruaks

down; and we c:m oniy mf'cr that the vibration is vo-y large;, in

eouscqueucu of thc isoc!n-o))i.sm. Sincc thcre is a. ï]ode at .r=0,thurc must be a loop w!icn is an odd !nultip]c of and wo

concludc that in thc ca.sc of isochrouism t)ic variaLioti ofpressure

at the opcn end of thc tube duc to the externe! cause is cxact)yncutr~Ised by thc variation of pressure du<; to thé motion withinthe tube itsdf. Jf thcrc were rca))y itt thc opon end a variationof pressure on thc whoïc, thé motion Must increasc without limit

tn thc absence of dissipativc forces.

If wo .suppose that the ori~in Is a loop instead of a node, H)c

solution is

whc)'c<~)=eos~ is thc givcn vainc of at tlie open end .~=~.Iti this case thc expression bccomes ItiHt-nte, \v)tc]i /c~=~t7r,or

= ?~X..

Wo will ncxt considcr tho case of a tube, whoso ends arc both

opcn and cxposcd to (!i.stut-))ancesoft.hc saxicpenod,n)nkin"-6

cqual to 7/c'"<, 7~ rc.speetivdy. Un)css -t.hc dist.urbanccs at t))C

ends arc in thc s:une phase, one at Icast of thc coc-mcients 7f, 7~

must becomplex.

Taking the nrst form in (3) § 255, wc Iiave as thé gcnera!

expression for

If wc take thc o'Igin in thc midd]e ofthc 'tube, and assume that,

t.hc vatucs /e" 7~c'"< con-ospond rcspcci.ivcly to ~=~, a;=- l,wc ~nt to détermine ~1 a.nd 7~,

Page 63: Lord Rayleigh - The Theory of Sound Vol 2

DOTH ENDS OPEN.259.]51

This resutt n..ght .Iso bo dcduccd from (2), if we con.sider t!h.tthc rc.tun.Gd motion ariscs fro,nt).o superposition of thc motionwinch 13 <h,c to t)~ disturbance /ca!cul.tcd on thé hypnthosist'~Lt tl.co~or end ~=-~ is a !oop, on thc motion, which isduc to A c'"< on ttic Ilypothesis that thé end .e= ~is ]oop.

Thé vibration cxprc.s.scd by (.t) c~nnot bc~y, un!c.s.s thc

raho 7. be r~t, th.~t is unicss tho di.stu.-b~necs at t)~ ends bcin s.nnhu, or ni opposite, pba-scs. Hcncc, cxccpt in t).e casesrcscrved, therc is no

loop anywftcrc, ~nd thcrcforc no pf.~c nt

~h.cha bmnch t.ibc ca.i bc cunncctcd along which sou.Kl will not

L'opropi~ated'. 1.

At thé Tniddic ofthc tube, for which ?= 0,

~inr,that

thé variation of pressure (proportion~ to vanishcs

ifyy +A =(), tj~t j~ ;(- di.stm.b:tncM n,t thc ohi.s bc cqu~l un.!

"i

~o.s~cphases. U.i)css this condition bc .s~Is~d, t)~ expres-sion bccomcs infini (,o, wiicu 2/ = (2~ + 1)

At a point disant from tlie middte of tho tube t!)oexpress) on for is

v~nisïnng wh,n 7/=7f, that is, wlicn thé di~turbancea at the end.scq.ud ,,nd ni thc ~7~ phase. In

gcncnj Lccomcs infiuitc,w'icn

s.n~=(), oi-2/=~.

If ~t onc end of an un]i,uitcd tube thcre Le a variation ofP~arc duc to an cxternal

source a train of progrc.s.sivc wavu.sw.H ~e

pror.~atc<) inward.s from that end. Thns, if thé Jcn~tht)ic tube Mcasnrcd froiu t!,e opun end bo t).e velocity-

Potential i.sexpre~d Ly

~=co.s(~).c

corrcsponding tu

~"J' of

(Phil.I.itcbE~ co.np.nn~ the ~cn.itic. of .ourcc.s of .onnd of the ,~nu

hitcli. of tho iHthé .~u,.ce. to bc

v thenntil -'1" "ibr.t.0..

't~ "?' ~° '° "tric c.su)ca .c 'of~y'"R point of J""cti.n tho dist.n.h.this al, °

tho test it nppcars tL.t""Mnsbmnptton is 110t

thooretic)t])y cornet.

4–3

Page 64: Lord Rayleigh - The Theory of Sound Vol 2

52 l-'ORCHD VIBRATION 0F PISTON.[25~.

~)=cos7~at~=0;sut)):)t,ii'thcc!tusRoftLL'di.stm1j:))]cc\YitJtiH

thc tubu bc t))c p:ts.s:(~(j of :t. train of' progressive WiLVus aen'.ss (.hu

opcitcnd, thci)i.tt'nsity\YiUnnt]iL'tu1juwinb(;thcs:uneii.s]nt))~

space outsidc. It nmst nut bc for~'Lt.cu thfLt. thé di~tneto' of thu

tnL(iIs.s))ppos(;(t tobùI)tfihitc)ysm~niucu)ttu:t.riHunwit))t)tL:

k'))gt))of:Lwavc.

Lct us ncxt suppose thnt thc sonrec of t))C motion is within thc

tnl)uitsc)f,'h)u ~(.-xinnpteto thc inexorable tnotiunof;). piston

~t the origin'. Tho constants in (~) § ~5.') :).)'c to hc detenninctl

by thc couditions t!ia.tw!)uu ,t;=0, ~'=cos?)< (s~y), and th:Lt,1.)y tlie COI1<ttIons t lat W len ~,c= 0,il:lJ.IJ

cos ?lt .111(l th:lt,

-hcn ~=~, ~=0. T)ms ~=-t:m~, ~7~=1, :nid thc ex-

t))'<rjt)f<)t'f~i';

T))c motion i.s a.)nini]num,whcncosA~=i!,t1)at is, whcnttie

!t.))<'ft))CtnbHis.'(.!nu)tipIcuf~

Wi'cn is ah oddmu)tij))c uf t]ic place ocenpicd hy thc

pi.st()nw()n!()buano<.k',ift)Ki<)pcncu(twcrci-ea))yaIo()p,hnti)t

ttti.sc~sctiics~ntiun i'ai)s. T)tc cscapenfoncr~y iront t))c tuhc

p)-C!VO)ts t]iccncr~y frumaccumulatmg beyond a ccrt~m puint-;

but no account [-:ui Le tiL~oi of <)ns so !on~ as UtC opoi end is

trca.tudri~orousiyasfLioop. Wcshidt rc'sutne t))u question n)'

résonance at'Lct- wc hâve conHido-cd in i~)\d<;r (tt.'tail thu thoury ot'

t))(- opcn end, whcn wc shatt bu ab)~ U~ duid with it )nor(j satis-

factoj-ijy.

j!i )i)<(- mnnnc')- if thé point ~'== bc a nodc, instcad of :), h.-op.

thc cxprcHsion for is

~t)"'st.hc)))ut)<)nI.s:t.mi)ii)nu)n\v!tt;n~isnn(K]d)n))!t.i[))c;of~,

inwhichca.set.iicori~in i.siUoop. Whcun.snn cvcmnuttipicut'.t))(;o)-i~i)i.st)C)u)dbc:). notk', wh~h isrurbid.tcubyUtucotif))-

tiotisot't.hc'.jucstiun. Int)tis e:Lsc~ecor()it)gto(~)thc motion

Luc()H)C.sin()nit.c,w!ncIt)t)c:)n.stt)atinthc!d)S(;'ucoofdis.sipativcfurcus thu vibmtiuti wou)d Incrc:~o witituut limit.

'TitMcpruUc)tLsarucotMiJct'cdbyl'ui.ssu)),.VJw.<)).<t.n.t'.M)j.

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2GO.J KUNDT'S EXPERIMENTS. 5:3r

s

2(!0. Tbc cxpcrintcnta.! Investigation of acria) wavcs within

ptpus ))as bucn e~'cctot with œnsi(iorab!c .success by M. Kundt'.

To gcncratc \av(;s iso:L.sy cnougb; but it is not so

ca.sy to iuveut a

tnctbud by wbich thc.y can bucf~ctnatiy examina. J\[. Knndt

(H.scttv~rcd t));Lt t)tc uu(!u.sof.st:Lti(juary \v:L\'c's e:ui b<j tnadc cvidcnt

by ttx.st. A iitt!u iinc s:).nj «riycoptxtimn .sL'(j(t, .shakot over t)jc

ititcriur oi' ;). ~ss tube cotitaiumg- av!b)-iLt.i)t~ cobnnn of :).ir,

(H.s~u.su.s itscti' in t-ccun'ij)~- p;Ltt.urns, by means fi' wbich it is e~syto dut~nninu t)i(; po.sit.i<)t).s of tbe Dottcs amt to measm'c thé

it)t(;)-L)s ))ctw<x-n t))0)). h) Kun(]t'.scxp~'itncnts t)te ori'nn of

(bu sumid w!~ iu thu iut~ItmUmtt vibnLtion of~giass tttbc~dbjd

tbusounding-tubt. :u)d tbu

(tust-fi~u)-cs were furfncd i]La.scco!idand far~r tube, c.d)cd t)tc w~vc-tubu, t))u fattL-r bctng provutcd\(!) :t !U()VC:).b)u .stoppur fur tbn

purposc ufadju.sting Itsicnc.'t)).

Tbu otbo- end of tLu wave-tubc was fittcd witit a cork t])ruu")t

wbicb t)tu Muun()i))~-tabc pa~s~d h:df w.y. By snitab!o fncti~it)~

.soondm~-tubcwa.s can.scd to vibratc in its

~ra.vc.st mode, so

that tbo œntral jn.'ittt \vas nod:J, and it.s ititeriurextremity (doscd

Avith n. cork) cxcitud act'h'd vibratiffn.s in titc Avave-tubc. Bymcansof tbe stojtpfjr Un- k'n~tb of tbc coftunn ofair c-oubi bu adjustcd so

as to ui~ku t)t(i \'i))r)Ltio))s as vi~jroos as po.s.sib)c-, whicitbappcns

-~bcn tijc intorvat bctwcan tbcstopper and t)te end of tbc

sounding-tubc is a. iuu]tiptc of hati' t)icwavc-!c)]gt!i of thé

.Sound.

With t)tis ~ppamtus Knndt wa.s abio to compare thc w~ve-

Jungtb.s uf t))c same sound in various gasc.s, ft-om wbicb t!)e rc-h).-tivu vcbcitius cf propagation are at once dcducibic, but tbo rcstdts

wct-o not cntirc-Iy satisfactory. It was found that tbe Intcrvfdsof i-ccurrencc of tlie dnst-patteDi; were !iot

strictly cqua], and,what wa,s worse, that titu pitch of t.be sound Dot constantft'om onc

cxpL'rinieut to another. Thc.se dcfeets wcrc traced to a

communication of motion to thé waye-tubetbrougb the cork, by

w))ich thcdn.st-f]gures were di.sturbcd, and t!]C pitch madû

in-c~uur)n conséquence uf unavoidab!c variations in tbc mountinn- of tbc

apparatu.s. To cbviatc tllem, Kundt replaccd tbc cork, wlnc)ifunned too stiff~ conncction betwecn the tubes, by layers ofsbcet

iodiarubber tied runnf) with sitk, obtai)~ in this way a ncxibleand

pcrfectfy air-tight joint; and in ordcr to avoid any risk of tbc

coniparison of wave-Jengtbs bcing vitiatcd by an altcrationofpitcb,

7'f~. ~otf. t. cxx. p. 3:)7. 1H(!8.

Page 66: Lord Rayleigh - The Theory of Sound Vol 2

54 KUNDT'S EXPERLMENTS. FSGO.

tho a.pp!U-atus was modined so as to makc it possible to excite

thn t.w.' sysh~ns ofdus'-f~tros :i::u)'n!u;!y :t)! u)

r~p~~c'. tu

thc .jiu'tc souod. A coitatcraJ adviuttage of thé ncw metl~od Cûn-

sisted ui thc élimination of temporature-corrections.

lu thc improvcd "DoubleApp{t)-iit~s" thé

sounding-tube was

cau.scd tu vibratc its seco?;(/ ~;o~ by friction appticd nGar

thé )nid()ie; n-nd thu.s tl~ nodcs wun- fonnc<t at thc points di.st:uitfrom thé cn(is by ottc-fom-t.h of thé Jength of thc tube. At cac)tof thcsc

points conncetion wns madc wit]t an imh'pcndcnt wa.vc-

lubc, providcd wit.h iin ndj)tstab)c stoppa', and witi) brancb tubesaïK! stop-cocks snit~bic for admittin~ t)tc varions gasc.s to bc

cxpLTimuntcd upo)). Jt is (-vident )hat dust-tigures fonnct! in thctwo tubus

corru.spond ri~oron.s)y to t)te salnc pitd), mid th~t t)~.rc-forc a. compansou (~' thu intorvais of rccurrcncu iund.s to a. correct

dutcrtninatiou of thé veiocitius of propi~-atiol, undur t])(! circmn-Ht:Luccs of tbc uxpurintcut, fur thc two ~ascs -\vit)) ~-)nch t!)G tubesarc tiHed.

'J'bc rcsnits at which K~ndt arrivcd wcrc as fo))o\ys:_

(«) Tho vdocity of .sound m a tubu din)i!))Hhcs with tbedi:nnct<jr. Abovc a eùrtain ditunutcr, Ituwuvur, thu citangu is uot

pL:recptibIc.

(/') Thc dntlinution of vclocity iacrcascs wit!i thé \va.vc-

I(j])g-t.h ofthc totic cmp!oycd.

(c) Powdur, sciittcrcd in a. tube, duninisbcs thevelocity of

sound in narrow tubes, but in widc oncs is without cH'cct.

(~ lu n:u-row tubes thé effuct of powder iticrcases, whenit is very rniety dividud, aud is strong)y agitated in

conscfjuoice.

(c) Rou~bc-ni)~ tho intct-Ior of a uarrow tube, orinerGasiu~

its surface, di)uini;jhcs t!)u velucity.

°

(/) Iti widc tuhcs thèse changes ofvclocity arc of no im-

portance, so that thc mcthod may be uscd in spite of thcni furuxact dctenninations.

(~) Tiie inHuoicc of tho intcnsity of .sound on thévelocity

eannot bo prov'ud.

(/<) With thc exception of thc îh-.st, thcwavodcngths of a

to!)c as shcwn by dust arc not an'uctcd hy thc mode of excitation.

(i) Jn wido tubes thc vdudty is indt'pcndott ofpressure

Lut ia sn)a)t tubc-s thc vcluclty iucrca.sus with tlie pressure.

Page 67: Lord Rayleigh - The Theory of Sound Vol 2

2GOJ KUNDT'S EXPERIMENTS. 55

(D AU the obscrved eimngcs hi t.hc veloc~ty worc due to

fncUon, .uK' cttpecj.Jfy to exchangc of béat betwceu thé air andthc sidcsofUto tube.

(/.) Thé velocity of sound at 100' agrées cxactiy witli that

given by thcory'.

We stiaU rcturn to thcquestion ofthc propagation ofsound in

narrow tubes a.s aiï'cctcdby

thc causes mcntioncd aboyé (~'), ands))a!) ti)cu

investigate tlic formula givcn by Hchnholtx and

Kirchlioir.

2(il. In thé Gxpcrimcnts dcscnbed in thc prcceding section theact'ial vibrations arc j~ee(/, t))e pitch bcing dctcrmiucd by tbccxt~rnal source, an() not (in any appréciable dcgree) by tbc Icngtbof tbe column of air. Indcud, strictly spcaking, aU .snstain'cdvibrattons are forccd, as it is nut in tl~c puwer of frue vibrationsto inaintain

thum.selves, cxcept in t])e idcal case whcn tbcrc is

absutntL-]y no fi-ictiun. Ncvcrthc)css tborc is an important pmc-t)cal distinction bctween thé vibrations of a colunui of air asexcitcd by a

lougitndiniLHy vibrating rod or by a tuuing-fork, andsucit vibrations as thosc of thé

organ-pipc or cttcn)Ical))annonicon.

I)i thu latter cases t)to pitch of' tho sound dépends prineipa)!y onthc Icngttt of thé acriat cotumn, thc function of tbc wind or of tbcHanio' buing mcrc)y to rcstore t)~ cncrgy lost by friction, and uycummunication to thu cxtcrnal air. Tbc air in an orgau-pipu is tobc considcrcd as a column

swinging almost frecly, thc Jowcr end,ncroHM wbie]i t!.c wind swccps, bclug trcatcd roughiy as opcn, a)i(tthc upper end as closcd, or open, as thé case may hc. Tbus the

v-avc-Iengtb of tl.e principal tonc of a stoppcd pipe is four timest))c tungth of thc pipe, and, cxcept at thé cxtrcmities, tbcrc isncjthcr nodc nor Juop. Thé ovcrtoncs of thé pipe are thc ofM

bai-tnontcs, twctftb, bigl.er tbird, &c., corresponding to t)ic varions ssubdivisions of thé column of air. lu thc case of tbe twcifth, for

exan)plc, tbere is a node at tbc point of trisection Hearcst to thc

Fron. Mn~o expressions in tho momoir aiready eited, from wLich t].o uotiec

.nthotoxtis,-ri..cip,d]ydcriYc.d, M.Juu.dtappears to L.Yo cuntcu.phttod con-

.uuat.ou of Lisiu~ti~ious, butlamu~bio to ~ind M.yl~cr publication ou

thosuLj(jet.

-'Tho subjcct of ~nsitivc fiâmes witb aud without pipos i.s tre~ted in con-

..d..i~lc

dut.,1 by Pr.,f. Tyn.hdt in )~ ~-r-rk ouSuund; but tho nu.c).ani<f

t as ch~ of ph.,no.non,~ JH still vcry impcrfL.cUy uudc.r.stoud. Wo .)u~ retnm tuttmnsubseqtK'ntchai'ter.

Page 68: Lord Rayleigh - The Theory of Sound Vol 2

5 GEXPERIMENTS 0F SAVART AND KUNm.

[2G1.

"pcn end. and a,]o.,p at the othcr point of trisection

midwvy

'tw'thcnrst.a)'dth~s(o'~rd(')-d~fth~p:p.

ru thc case of thé <.pcn or~an-pipo both end.s are)nops, and

thL-rc .nnst bc at luast cnc interne n~.dc. Tj.cwave-fcr~~H. of tho

r'nc,pattonc~twice thé icngthof thepipe, ~uch is dividcd

into two sniularparts by a nodu I)i thu iniddic. Fnj!n tt.i.s wc sec

<~ foundatiun ut' thu ordi.~uy r.ttc t)uit Lhu pitch of an open pipeis

'csatnc~th~ufa.pp~[pip.d')Laifit.s]o,~th. For rasonsto bc

~norc f.,))y e.xpi.i.d in a sub.s.[Uunt ch~pt. co.n.ectc.dwith onr

prc.sL.,it i.npc.rfL.cL freinent uf t),c opcn cm), thc ru)c is

.y appruxinmtdy eon-.ct. Thé opc.n pipe, ditrurin~ in th:.s ru-~t irmn U,c

stuppud pipe, i.scnp.)c of

soundinq thu whu]c .sc.ricsut

ton~h.nnin~ thc )~r.jiunicsc~fuuudedupun its

~~c. In t)~ case of t)~ octave t)K.rc is :t )nop at t!.c ccnLro of U.u

t'.pc :u)d nodc.s at t).c pun.ts tnidway b~twcun thc cuntrc and the(.trc]mt)cs.

Since t).ufrequcncy of thc vibration in a

pipe Is proportionrdto tlte

vdoc.ty ofpropagation of .sound in thc ga.s with which tl.c

f")"~ is r.iicd, thccompari.son of thc pitd.cs of thc nutes ohtaincd

'nthosarno

pipe indinbruutgasc.s i.s

auobviou.sn.ethodof

dGtenn.nn., thuvclucity r~f

pn.pa~ati.n, in c~scs whcrc th.impos-

~L'lLy of ..hta.uiDg a.snHicicntiy long co)nrnn of thc ~as prccludcs

tho

~.sc<,i thc dn-cct moD.od. In this

appiicntion C'fdadnl ~it], his"sna

.s~city i~H.o way. T).c

suhjcct ~.s rc.snnK.d at iatcrd.~c by D~on~~ and by Werthein~, w)~ obtainud

fair)y satisfac-toryrcsnit.s.

2<'2. Thc condition of tlle air in the inicrior of anor~n-pinc

-t~~,,Sa..rt~ho I~dIntJ

1"~

~i .strctd.cd m.n.bmncon ~Uch a htHo sand .vas~ttcrud. In ~o

nc-ighbou.i.uud u)- anode thesandrc.maincd

~~biyund.sturb.d, but, a.s

a fo.p wa.sapproac!.cd, It danccd .vi~

.c and muruv,go..r. But by far th. n..st

striki,~ funn of tin.

~r~.n.nt

:.s thatinvcntcd hy K.ini~. In Dus nK-th~ thc v:bra~

"c.tc.d),ya.sn.dt ,a.s fianK.fud t).r.gh a tnbc .hich

n.cu,nnn.,ucati.n ~It). a

cavity caHcd ~nanon~triccapsule.

i..L~c~ ~C7~1. xr,c. p, 11 a.

'-f'~<.<~(~tw.ii.'l.]~,t.xx)l.p..t;(f. t,'O.f/fC/ftM., t. xxtY. ),); ~j

Page 69: Lord Rayleigh - The Theory of Sound Vol 2

2G2.] CURVED riPE. 57

This cavity is boundcd on onc sidc by n. mouhra.ne on which

t)~ih"I:)g:nr :)oN.A:ifhc~i~nbr;inc~ih)'itt.i-c:idcri))gthc i C

capacity of thccapsule Y!u-iah]L!, thc suppiy of gas bccon~'s un-

sit-iKiyand t))c Hamcir~o'jnit.tcnt. Thc pcriodisof course too

~)na!t<or<!)cint<n))ith.-)tœtotun))i~stit.sdfn.ssuchwhcn tho

i)!unc'ish)('l-:t.'(1nt.st(;a<H!y. Bys)t!t~in~'t)m])t':u),()r~-itttt)m!ii(!

c)t':L]tin\'caL)ctnir)'(n',ti)crusu[ut[on int.onturcoricssd~t.achct~

ini:tg'('.sn)aybccf'i'L'cte;d; but<jvc!twithoutrcs()h)tiunt!iecdtcred

<h:D-:)ctcr of' thc f):unc i.s cvi()L')tt irom its gcncml ftppc~r~ncc. In

thu app)ic!Ltiou toorgiUl-pipes, one or niorc

cap.s<))cs arc tnounted

on a pipe in su<Ii !i. umuncr that U)e tuonhrancs nrn in contact

\it)t t))e vil)rati!)g culunui ufair; nud' thc diffurcnRc in thé Hamc

is vury markcd, iLCcordmg- as the associated ca,psu)c is sttua.tcd at

n,)iodeorat:).!o')p.

2G3. H!t.h(;t.() weItn.vc .snpposcd thc p!pc to hc stmight, but

itwi)) rcaditybc anticipât~) t].t, whctt thoernssHecLion i.s.smaU

n))d does xot vin-y in a.rc:L, stnughtnc.ss is tiot n nmttcr of impor-tanœ. C'ouccivc a curvc([ axis of running n.)"))g thc jniddfe of

<)K- pipe, and ici tlle constant scctjOtt pc']-pct)dicuiar to this axis

bc ~S'. Wtien t])c grc~tc.st diatn~tcr of <S'is voy smaU incomparison

with thé wnvc-iungth of tho sonnd, thuYclucLty-pctcnti!),!

huconn's nea)-!y inviu'iahJc ovcr t))u section; applying' Grccn's

thcorL'n) to thc sp~cc houndcd hy thé iutcnor of thu pipe andhy

two crosssections, we gct

shcwing that dépends upon .r in tlie samc wnyas if t))0 pipewo'c strai~ht.. By means of uquation (1) t.))e vibrations of itir hi

Page 70: Lord Rayleigh - The Theory of Sound Vol 2

~S8 DRANCIIED PIPES.f3G3.

curved pipes ofun.form sectionjnaybc c~I)yinvc!,t!~tcd,a.nd the

rc.s.dt.s !U-et).c ri~orous consc-quences ofuurftnKL-uncnt~) oquations

~Y!t-L{,m,<j,ji~ .C'~pi.sjj-th~nitcfy smaH. In t!K' case of t!nn tubes such as wuu!d ~0

use.! in e.xpcnmcnt, t).cy s~iœ at any ratc to givu !L vury goodreprcsentft.tiun of \viiat actu:J)y Iu)ppuns.

26- Wc now p~ss on to tlie c.jnsidcmtmn of ccriinn cases ofconnc.otcd tu!s. In t].e

~cco.npanyi.~ f~uru J~ rpp.o.t.s at)'t p)pu, wh:di divi.]c.s !(.t J9 iuto two b~nchc.s Z)/~ 7~ At 7~t-hcbranc!s rcunitcnnd fonn a.

si.~)u t.ubu7~ Thc sciionsthc .si.~)c tubes and uf thu L.-aucfiu.s ~rc .Lssumcd tu bu unifurm

as ~'c)t as vury stn.d).

In thc first instance ]ct nssuppose that a positive wavc o<

arhitnn-y typuis

advancing in ~1. On its arriva) at thc fork D, it

witt~ivuriscto positive wavcs in .Z~md C, and, unjcss a ccrtam

condittun hc .satisfic.), tu a n~ativu rcf!uctc(] wavc lu Lut tlie

putcntlal ofthc positive wavc.s hc denotcdby~ /bcing in

cach case afunctiou uf ~<; i and let tlic !~HcctL-d wavc'bo

~'(~+<-<<). Thcn thé conditions to bc satisfiud at D aru nr.st thatthc pressures .shaH be t! same for thc threc pipes, and second)ythat tlie wlh.ic vclocity of thc Huid In s)udl bc equal to thé sumof t))e who!c vdocitics of t)ic Huid in 7~ and C'. Thus, usinrr-J, Z?, C' to dcnotc thé arcas of t]ie sections, wc hâve, § 2 i4,

whcnco

~of'n'h.\nsa).pn~tn~.tonninot).o refluer nn~~frncteJwaYcsthc junc~-n of ~o tKL.s uf MC'Juns 7. + r, and rc.spc.ctivcfy, arc b'iv~ Ly

Page 71: Lord Rayleigh - The Theory of Sound Vol 2

2G4.] DRANCHEDriPES. 5!)

It appcars tha.t/, ~nl/ are always ttie Siunc. Thcre is no réduc-

tion. if

iho wavc thcn auvanccs in 7) and C cxact!y as it wouh! ])ave

donc iu ~1, ha([ titere bccn no break. If thu lengtb.s of tho

branches bc'twecn and be cquai, and thé section of bc uqualto t)tat of/), thé waves on arrivai at JE' combine into a wavc

pro-

p:Ltu(t :L)u))~ !Uida~:un thurc is no ruftccdou. Thu division

uf L))c tulx.: bas thus Lcun ab.sohttcly witbout ufïect; ~nd smcc t!ic

s:nno wontd bo truc fur a négative w:t.vc p:sing froin J~to ~1,

wc nt!t.y condudc gettcndty ti~at a tube may be (tividcd into two,or more, branches, :dl of thc M:unc hngth, without in auy w~y

infiuoicin~ tl)u law of acri:d ~'ibnUio)), providcd ~I)~t thé whole

socHoti rc!ii:u)i c(jnst:).nt. If thé Icn~.hs of titc bra-nchcs from 2)

to 7~' bc unuquid, t!iu rcsult is difTercut. Bcsn]us thé positive wavo

in titcrc will bu in gênerai négative rcHectud waves in 7j' and C.

Ti)e tno.st intcrcstittg casu is whun tho wave is of harmonie typea.nd onc of t)iu br:u)chcs is longer thftn thc other by :). multiple of

If t)tc diHcrcncc bc an CMH. multiple of thc rcsnit will bc

t]ic samo as if t!tc branches wci-c of cqual Jcngth, and no rcncction

wIH cnsue. But suppose that, wltile and (7 arc cqual in section,onc of them is longer than tbe othcr by an o~ multiple ofSincc t)to waves arrive at J~ Ln

opposite phases, it foHows from

synunctry that tlie positive wavc In ~must vanish, and that thé

pressure at whieh isneccssa.rily tbe same for a!l thc tubes,

must bo constant. Thé waves In 7~ aud 6~ are thus rencctcd as

frum au open end. T))at thc conditions of thé question arc tbus

satisned may aiso be sccn hy supposing a barriur takcn across tlic

tube 7~ in thé neighhour)iood uf jE' in suc!i a way that tbe tubes

J~and C'communicato~'ithout a change of section. Thc wavc in

cach tube will thcn pass on into thé otbci- without Interruption,and thc prossurc-viu-iation at j~ heiug thé résultant of equal and

opposite componeuts, will vanish. Tins bcing so, t!ie barrier maybc rcmoved without altcring tt~e conditions, and no wave will be

propagatcd along F, wbatcvci- its section may be. Thé arrangc-

J'nisso)], Jt/eM. r~)f!ft'<H~,t. i!. p. 305. Thé rcrutor will not forgot that bothdittmetcra must be smnU in comptu'isou witli tho Wt).vo-lci)~th.

Page 72: Lord Rayleigh - The Theory of Sound Vol 2

~0 BRA~CHED PIPES.r-?(;L

~ntnuwundcr considérationv.asinvcutcdhy Rc~dK~andh~s

~'uuncmp)..y(.d Ly<~)in(-kcaud otLur.sforcxpt.ritm.nta) purposcs–

:tj,p)!~<.tiu'tt!,i..)~s)):)rh.r~;).(),i. i-, .].

H)ep~o.on)ono)Y.tsc]t .s<,ft.r~.n~d'.oa.s:u)..x;).np)c(,ri,,tcr-

~ru.c.t..)~)t.d.t)HT..c-hufH)uLj..(.tiu.I,u).<).(..san)cc.n,(,t!'<- .snit) w))(.). O.~ru~Io-is

]~)tu.su).pu. t).;)t.L)n-}H,sit.ivcwavt..sn~utmii.sccachoth~-iti

7'nh!t).:).tL)n.n.t!,m);)tt..rc.nds Jt,inust

.K~.Tbcf.~<~hjn))..tdhTuisnoh,ss<,fc.n~inint.~fcruncu

b))t.)iya.)itr(.rc))<.)i.-<tnbu<iun; ~i..)K.r.;yi.s diverti fro.,)'

""cp!itn.)pj[,car.si)tanut)KT.I"t!h.})n,st.t(.asuthcp.).sit!vu

~vcin~c.m~ys(..H~ywithi<. n't.).)-is nu

wavea)u..rr~'O~rc.

~hvo],s.s.),)u alternative.-it!r.ya(.cnnm~t.s

'n Débranches, or dscit passes La<ka)<t in't.),c f-onant-a a

n~at.vewav. In.~urtos~ whatr<.uiy]~pj~.s, lut u.s trace

t))epro~ress of thu wavt'.s rcfkctûtt back at 7'

Thèse wav(.s arc-~qua] !na~tittxic a)u) .start fn.tn 7;:n

oppnsitc p).a.s~;i.)< pa.s.sa~ihjin7;tu~<)~~ ~rav..]

!atLT .)i.sta..œti.anthuoth~-Ly au (,<)

,uu)tip)uof' andthc.rdurc ou arriva! at t).y~ij) bo

iuc.anp!u(u ac.-ur.i'ancp

Ut.(h-rth(.sc<.irc)..n.stanc~t)~.ycun.b.ncInt~.si.)L;Icw:u'(. whirh

tr.~(.su~ati~.)y a]o~.t,an.itl.c.rc i.snn rcf)ccti.,n. W).~t),cne~atn-c ~avc ruac)K..s tr,c c.nd of thc tu),G J, or i.s .,t.hurw..sc <!is-t.trbc.d ni its euur.se, t).c ~hoie or a part may !~c rdJcctud, and thentl.c procès .s rupoatcd. But h.,wcvcr <t.c~ this

~ay haj<pcn thcrcwill bc i)o wavu

a)u,~ F, un!c..s.sl,y ac.uunda.tam in

oon.sc.qucnce of.'L

c.an~)cnœur pcri.~s, thc. vibration ia thc hranehc.s bccon.G so

g-rL-t that a .sma)I fraction ofit eau notonner bc

u~k-ctcd.

Fig.CG.

Or wc n.un U.ns. Suppose t).c tube.FeuLuH'Lya a

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2G4.J BRANCHED PIPES. GlL

bu.n'icr as bcfurc. Thc motion in thé ring bcing duc to furcc.s

actixgat D is nL-ccss:t.niy Mytnmot.ricfd wii)t t'unjK'ct to 7~, and

thu point \Yh:ch(!ivid'jt!7)/~<)i)tt.oc~))!t.l parts. J~ncc'j~'Ls:).

n<'(t(.audt)ic vibration i.s.st~ti(jna.)y. Tinsbein~, t))ocasc,ata,

])<)i))tA'tnh!t:utt.fro))iJ'/<))tclL)K.')-si(]c,th(jrunm.stbc!T.]o(m;

an~tt'thcb;u-r)ui-Lorctn()Y(-!dih(.'rcwi)Isti![bcnotcmiL'ncyto

produccvibration in 7'

H'tttepf.'rimct~roft~cring-bca.inultiplu

ot' thcrc may bc vibration wit))i)~ it oi' tho ncriod iaquestion,

i)u)cpu)n)<j))tiyuf:t.ny)at(;rat(ip(.'ttings.

Anyc'nnijination nfcfjnncct~d tuhc-s maybo trcit-tcd in .T,si)ni!:u-

~t:mm.'r. Tho gênera), principfc is t)tat at ai)y jmictiun a spauecan bL; takoi lar~u enougtt tu inctude n)I t)ic

région t)u-ou<')i wiucit

Fij;. 57.

thc \vant of uniformity afTccts thé huvof thé wa-vcs, and yct so smaU

that itsiongcst dimension

ma.y bc ncg~ceted incomparison with

Undt-r t.hcsc circtunstancc's t)te nuid within thc space in (tucstion

jnny be trcatcd ns if t)tc: wiLYu-Icn~th wcrc infinité, or thc nuid

itsc-if incutnprGs.sibJc, in wiuch case its velocity-potcntial wou)()

satisfy ~<~ = 0~ fu!!owing t,hc 8:une I~vs as ctectricity.

265. Whcn thc scetiou of a pipe is variabtc, thc probtcni of thc

vibrations of air within it canuot gcncraHy be solved. T)ic case

of conic~ pipes will be tre~tcd on a, future page. At présent wc

will invosti~te an approximate expression for thc pitch ofancarty

cylindrical pipe, takin~- first thé case whci'e both ends are closcd.

Thc metiiod tha,t will be cmp!oycd is sixiilar to that nsed fur a stringwhose

dcnsityis not

(nutc constant, §~ !)I, J-tn, depending on thc

pnnciple that thc pcriod of a. iroc vibration fu)n!s thé stationary

condition, and may tLerefore be calcuiated froni thé potcutial and

kinetic cncrgics of any hypothctica) motion notdcparting far from

tbe actua! type. In accordance wit]i this p]:(.n we shall assunte that

tbe velocity no)-)nal to any section ~S' is constant over thc scctioti,as most be vcry ncarly t)tc case wlien tho variation of is slow.

L<'t ~V reprcsent thc tota) tmnsfcr of ftnid at tune across the

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VARIABLE SECTION.f2G5.

Rr~ntir\ti nt ~?. <~section at x, rcckoned f.-om thcG()ui)ibnum condition thcn

reprosents the total vclocity of tlle currcnt, .nd .Y- rcprc.scntsactu~}.ty .f. ~i,~t!~ thu kiucuc

enorgy of t)ic motion Avithin thc tube iscxprcs.sc-d hv

fhc rcsult mayhc cxprossc<! convenicntly in tcrms of A~ thc cor-rcct.on t .at rnust bc ,n.dc. to in ordcr ti~.t U.e pitch ,nay Le-a!cu)ated

~.n

thco..hnary fur.uul~ as if ~e,.c con.st..at. y.r

)."c value of A~ wc I)avc

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2G5.] VARIABLE SECTION. 63

Tho cn'cct of a \'ai-iation of section is greatest near a nodc or neara loop. An

cnhirgRmcnt of sucMon in t)~ Ursb case lowers tt~c

p!tc)t, fu)d in thc second OLsc mises iL At thé points midwaybctwecn thé nodc.s and loops fi .slight variation of section is withoutcf~ct. Thc pitch is thus dccido(Hy:dtcrcd by an enlargement 01-contr~tion uc~ t]tc middtc of tl.e tube, but tho iaftucnce of a

.s!i~))t cunic:dity woufd bc inuch less.

Thc expression for A~ in (8) is applicable as it stands to the

gmvcst tono on)y, but wc n)ay apply it to thc tone of thc har-

monie scalc, if wc modify it by thé substitution of cos

2~for cos

6

In ttte case of a tuhc o~eH at both ends (~) is rcplaccd by

instead of (8). T),c piteh of thc sound is now raiscd by an

cn)argcnic.nt at thc end.s, or hy a contraction at thé middie, of the

tubu aud, as bcfure, it is unaifected by a slight gcnct-al couicality

2GG. Thc case of processive w~vcs movh~ in a tube of vari-ab)c scctio)i i.s a!.so

intcru.stmg. 1~ its ancrât form thé probicmwontd bc onc of grcat dimc.dty; but wherc tho change of sectionis vo-y graduai, so that no considérable altération occm-s wit!un adistance of a grcat niany wavc-Icngths, t)ic princip)c of cnergywill guide us tu an

approxin~tc .solutioi). It is not difncult to seethat in thc case supposed t!tcrc will bc no sensib)e rcncction ofthowavc at any part ofits course, and that thcrcforc t)ic cnergy of tliemotion must romain

uncha.ngcd'. 1. Now wc know, § 24;-), that for

a givpn area of wavc-front, tiiccnergy of a train of simple wavcs

is as the square of thc amphtudc, from which it foltows tliat asthc wavcs advancû tlic amptitudeuf vibration varies Invcrscly asthc square root of t!)c sectiun of thc tube. In nJt othcr respects t!)c

typc of vibration remains fdjsdutcdy undtangcd. From thèse re-suiLs wc mayget a général idca of tlic action of an ear-trumpct.

P/ ~«y. (5) i. p. 2('.i.

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G-~ VARIABLE DENSITY.F 2 G G.

It appears that according to théordinary approximate cquatious,

t!)erc is uo limit to t)tc concentration of sound produeibie in a

t!'b<o!n!ua[!yt!r.uiiHh)]tg.jf:t].~j.

TJic saioe mctimd i.s app)ic;ib)c, w~cn titc dcnsity of the)ncdiuni varies

.siowiyfroni point to point. Fur cx:u)))))< thc

amplitude ofa sound-wuve movins- upwu-dniM thcatmo.sphcœ]nay bo detcnniucd hy t)tc condition that titu cn(;)~y ronaina

unc)i:u)ged. From § 2.t.-i it ~ppcars th~t thû a)np)i~de is in-

vct-i-iuly as t)ic square root of thc Jcnsity'.

A de]i(.nto qu~ti~n nrisos ns to tho u]ti)!))itc hto of fio)]nrot)Hwavcs ))rnpaRat(.dupwardH. It sL.mId bu ro.tarhcd thut tu rnrc fur thc (teadoui~ infiucneo of viscomtyiHtimehmcrcnseJ.

Page 77: Lord Rayleigh - The Theory of Sound Vol 2

CHAPTER XIII.

SPECIAL PROBLEMS. REELECTION AND REFRACTION 0F

PLANE WAVES.

2G7. BEFdRH undcrtaking tho discussion of the gcncra.1 équa-

tions for acria) vibrations wcmay eunvcnicntty turn our attention.

to a fcw spuci:d problumS) rc):Lting princip~Uy to motion in two

<)in)L')).sions, which arc susccpt,ib!u of rigorous fu~d yct cotnpan).-

tivuiy Hi)np!c solution. In tins way ttic l'eadcr, tu whotn thc

suLject is ncw, will aequirc soae famiU.u'ity wit)i thu I()cas aud

tnuthuds cniptoycd bcfui'c attacking more fot'miJtt,ble difïicultics.

In thc prcvions cha.ptcr (§ 255) wc hivestigatcd thc vibra.ttons iu

one dimension, whichmay tnkc p!:tce piu-nnul to thé axis ofa. tube,

of \vbi(.;h both ends arc c)oscd. \Ve wiH now i)j<[nirc wbat vibrations

!t)-L' possible wiL!)in :t closedrect:Lng)dnr box, dispcnsing witb thé

restriction tb:).t thé motion is to he in one dimension on]y. For

ctu'hsimple vibt':Ltion, ut' whicit t!)<j systcm is cap:).b)u, ~) varies as

a circufat' i'unction of thu time, .say cos/<:«~ whcrc /c is somo

coostant huncc = aud therci'ore by tbu gcncral difrurea-

tia) équation (9) § 2-M

Equation (1) must ho Sfitisficd throughuut t!hj w))u!e of thc

"'cindud Yûtunm. Thc surface coudit.iun to bc KLtisfiud ovur thc

si\!sldc.s<jfthuboxissnupty

~'huret'bprcsott.s au cjoncnt. of thé normn.! to t,!ic sut'fn.cc. It

uuly fur spécial v~ducs of A: t.)iat. it is possible to s;.).tisiy (1) :).nd

(~)si)nu]t:LtK!un.s)y.

ï!.U. 5

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AEMAL VIBRATIONSf2G7.L-

Taking three edgcs which meet as axes ofrcctungular co-ordi-

~ates, .~dsupposing th~b tho lengths of the edges are

respeetivelva, p, 7, wo kuow (§ 255) t)):~

~hcre~,y, r are Intcgers, arc particul~r solutions of thc prob!cm

~y any of thc.se forms équation (2) is satisned, and providcd'

that be equal to or r thc case may be, (1) is also

satisf~d.It is cquaHy évident that thé

boundary cquatiou (2) Issatjsnud over aH tho surface Ly tlie furm

whcre a..s bcforc 7' are intcgcrs.

T!~~ncml soh.tinn, cl.tained by compounding ail particule-

solutions incinded undcr (~), ia°

i~ch.land~arc arbitrary constants, and tlie summation is

L-xtcnded to ail mtcgrai values uf~, ?..

This solution issufncientiy générât to covcr the case of anynut.al stato of thln.s ,vit)un thé box, not

involving rn.]ccutarrotation, ihc initia! distribution of vdocitics dépends upon théinitial value

of or/+~~+~~), and by Founei-s

theorem can bc ropresented by (5), .suitab!e vahics bcii~ ascrih.dto thé eoc-mciunts ~1. In likc nianncr an arbitrary initiât distribn-tion of

~nduisation(ur raréfaction), <)c.pending on the initial

~e~nt r"1" ~scribing suitabic vahies to thé

coefflcicnts 13.

Théinvestigation might be prcscnted somewhat

dirfurent]yby connnencii.g wit!i

a.ss~ing in accordance witb Fo~-icr'.

Page 79: Lord Rayleigh - The Theory of Sound Vol 2

IN A RECTANGULARCIIAMBER.267.] 67tleorem that tlie gênera! value of~at time t c.-ui bc cxprcssed mtitofurm

0 bc il,

.a wh~ch the cocfïic.cnts C n..y dépend upon <, but not upony. i).c c.prc.s.sion.s iur y and woaM thcu bc funned an<t-shown to .nvuive onJy t).c .squ~-es of t!~ coemcicut.s 6', ~,d f.-<ntLc.se expressions ,vou!d foHow thc nrn.na)

c.qu..Hons of n~iou

counectmg each uormd co-urdin:~c (7 wit)t t])c tune.

Thé gravcstn~deof vibration is th~t i.~ ~.hieh tl.o entircn.ot.on ..s

P~))dtothc)~c.st<Ii,ne,,sinnofthuLnx .nd thcre..s no ,nt.crnal nodo Thu.s, if !,c thc g.-catest of tlic ti.rec .sidcsa, 'y, WC :U-C tu t:).kc

= 1, = (~ = ()

In thc e~c of a cubie~bo.x~=~=~, and t!.cn i..stea<) of

(t) WeL~VC

As in thc case of thc membrane(§ H)7), w).cn two or more

pmn.t.vc modus hâve t)~ .s~mc pcriod of vi),n..ti~, otiior n.odcsof like pcnod mn.y bc derivcd by composition.

ThctrcDy incite .Grics of po.ssihfe .si,np]e componont vibra-

tions isnotnc.ccssartiycomp)ete]y rcprcscntcd in pa.-ticularc~es of

compnund v.b~dons. if, forc~.npju, wc

suppose thu contents ofthe box in its .niti..d condition to bc nc.it!.cr condcnscd nor ra.-cficd

~yp~-t~nd to hâve unifor.nvelocity, whose

componcntsj)araf)ci to t!)c axes of co-ordinates fu-e

rcspcctivcfyno.s.mptc vibrations arc gcncrated for whicii more than' o~

t'K. thrccuu.nhcrs r is finite. In tact each

component initiavdocny rnay hc con.sidcrud

sopamtdy, a.nd thc pr.~Jo.n is sin.if.u-to t])at

solvoi in § 258.

5-3

Page 80: Lord Rayleigh - The Theory of Sound Vol 2

NOTES OF NARROW PASSAGES.f2G7.[~f.

In future chaptcrs we shaH meet wlt)i other examples of tticvibri)tions of tir within cotnpIuLdy c!oscd vusscis.

Some of thc natuml no~-s oi- thc air ccntiiincd witbin a rootn

may gencra))y Le detected onsinging thé sc~Ic.

Prob~y it issomewiiat in this way that blind pcople are able to estimée thosize of rooms'. 1,

In long and narrow p~sa.gcs thc vibrations pamUcI to thc

)cngt)i are too slow to affect tho car, but notes duc to transveraevibrations m:Ly often bc hcard. Ti~ relative proportious of thcvarions overtones dépend upon the place at winch tlie disturbaiiceis crcatcd".

Insome ca.scs of this kind thé pitch of the vibrations, whosc<hrectio!i is principally transvur.su, is infiuuiicod by thc occurrence

iongitudinal motion. Suppose, forcxamp!c, in (3) and (4), that

<y= 1, )- = U, and thM a is much groater than /3. For theprincipal

transversevibration p

= 0, and = 7r Bat besidos this thcroarc other modes of vibration in which thc motion i.s

principaHytrausvcrsc, obtainûd hyascribing to small Intégral vaines. Thuswhcn))=l.

shewing tha.t thé pitcb is ncariy the samc as bc'forc'.

2fi8. If wc suppose ry to bucomeinHnitcJy grc: t)ic box of

the p~ccding suctiun is tr.-uisformcd into an Infiintcrc.ctan.~fa.-

H.bo, wbosc sides ~-0 a and /3. Wfh-itevcr nmy be thé motion uft)~ an- w.tbm tbis tube, its

vclocity-potcntial may be cxprcs~dby Fuunur's t!tcorem in the scrics

A n-markaUo iustnuco is qnotod in Young'n A'~Nr~J'/n'~o~y<y. u. p 272,

rom Darwiu-.~M.< d87. Tho !atc bHnd Jn.tico Fiold, walhed for t]~

f.rst tnnu into .ny r.oM, whcn ho on.u vi.itcd me, an<l aftor spc.a)<inK a few won)..<n. This r..<.a is about M leet long, 1~ ~ijc, 12 high ail which lie gucs.c.c''y tho efu- wiHt ~rent nccurncy.

Oppe), J~ /<~M~ ~<ya,-«~~e )r)7u)~ <.rr~<e)t J~-

./f<<f))t~ut; 7''<))'~r/;r/«< ~.r /~)/.<;A', xx. p. l;i0.

"H' t~ss, in n,y hœ.M in whieh it is p~iMo. hy~nf; t).. r~ht note, to c..cite fr.o vihr.tiun.s of ,n.uy ,<cun~' Jurati~ ~,d it

t. L~ h.pj.~H that (Lu n~n.nt uuto in aficctcd ~it). di.tinct bcnts. TI~. bn.~Hhuf thu j~s~~r is ,t)joHt f~t, and tho ),<.i[;ht nt.uut fc.ct.

Page 81: Lord Rayleigh - The Theory of Sound Vol 2

G9268.] RECTA.NGULA.R TUBE.

whcre thé coenkicnts arc independent of .r and y. By tho usef.f this fonn wc sccui'e the fu)f1))ncnt of th~ h.')unf!)!.ry cn~fHt~jt

t)~t LhcrG is to he no velocity across tho sidcs of the tube; the

niitm-c of as a, fnnction of 2! and < dépends upon tlic other

cunditions oftho problem.

Let us coisidcr thc case in which the motion nt every point is

!t:u')nunic, and due to a normal motion nnposed npon a ba.n-ier

ntrct(;)ting across t]tc tube at = 0.Assumin~ to be proportiun:U

to e"~ :Lt ali pohtts, wc hâve thc usn:d dltrcruuti.d équation

winch by tlie conjugatc pt-opcrty of tho functions must be s.itis~d

scparatcly by cach term of (1). TIms to dotermiuo M aiuncttu)i of z, \vc get

Thc solution of this équation diffcrs in form according to thé si~nof thc coefHeicnt of J~. Whcn and y arc both zero, the eocfH-eicnt is ncccssariiy positive, but as and q incrcasc thé coefficient

c!)angcs sign. If thé coefficient bG positive and bc calledtlie gênerai value of may be written

whcre, M thc fa.ctor e"~ is expressed, are ~solutc

constants. However, the first terni in (4) expresses a motion

prop:~g:tted in the ncgn.tive direction, which is excludcd by the

comHtions of tho probicm, and thus we are to take simply M tlieterm

corresponding to y,

In this expression C~ may bc complex p~ssing to rca.1 qua.jititicsand t~king two ncw rca.1 arbitrary constants, we obta.Ia

Wc hâve now to considcr thé form of thé solution in cascawhcre the cocfHcicnt

of~ in (3) is négative. If wc caH ittlic solution

con'esponding to (')-) is

Page 82: Lord Rayleigh - The Theory of Sound Vol 2

70MMCTANUL'LAR TUBE. FSCS.)-~U.

of witich thé first term is to be rqjceted asbceoming In~initc with z.

We tbua obtuincun-uspoidn)~ to f5)

Thc solution obtaincd by combining ~} the part:cu!ar sohitions

givcn by (5) and (7) is the gênera! solution of t!.e prob~m, ~d

atlows of value of over thé section ~=0, arbitrary at every

point lu both amplitude and piiaso.

At a gréât distance frmn thc source the tcrms given in (7)become insensibje, and thc motion is repre.scnted by thc tcrms of

(.~ done. Thc cnect of thu tcrmsiuvolviog high values of~ am) y

is t).us connuud to thc neighbourhood of thc source, and atn.odcratc (ti.staucc.s any suddcn van~tiuns or discontmuitics in tilcn.ot.ou at ~=0

areg.-adua))y cased oif a..d oblitcrated.

Jf wc nx our attention on any particular simple mode of vibra-tion (for which and do not bot,], vanis).), and conçoive tho

iruqucncy of vibration to increasc from ~i-oupwards, we see that

t)te eHeet, at first connned tu t!.o ncighbourhood of thc source,graduaiiy cxtcnds furthcr and furthL.r, nnd after a certain valueis passc-d, propngatcs it.seif to an inimité distance, thu criticat

h-c.p.cncy bcing that of tho two di.nensional free vibrations ofthe

corrcsponding modo. Below thé critical point no work is requircdto 7~«.~ thé .notion abovc it as much work must be doue at

= 0 as ..s carncd otf toinnnity in tiie samc time

2of). We will now examine the rcsalt of thé composition oftwo tra.n.s of

p)anc wavc.s ofi~rmonic typc, wiioscampUtudcs ~nd

wave-lun~ths are equa!, but whosc directions of proj~gation ~ciuch)~) to onc anothcr at an ang]e 2~. T)ie probtcm is one oftwo dnncnsK.ns o.dy, In~much as

everything is thé same inptanes pcrpcndicular to thc H.tcs of i.iturscetiou uf tlie two sets of\avu-ft'onts.

At any moment of time the positions of ttto p]anes of maximumcondensation for cac), train of wavcs rn.~y bo rcprc.scntcd by pa-r.dte) hncs drawn at equat intervais on thc plane of tlie papcr,and t)~so fines inust Le suppose J to move wit), a vdocity hi a<hrcL-ti..n

pc~L.ndic.,)ar toD.cir length. If- buth sets of lines Le

drawn, t)ic p.~pcr will be divi~cd into a System of equa! parailuio-

Page 83: Lord Rayleigh - The Theory of Sound Vol 2

2 G 9.]TWO EQUAL TRAINS OT WAVES, Tl

grams, which advance in tlie direction of onc set of diagonals. At

cacli corner of a paraUeIogra,m thé eonduns~tion is doubled by the

superposition of tlie two trains of waves, and in thé centre of each

paraHuIogrnm thc rarefaction is a, maximum for tlie same rcason.

On cach diagonal there is therefore a series of maxima. and minima

condensations, a.dvaneing without change of relative pusitioti and

with vclucity ft cos a. Bctweeu eacli adjacent pair of lines of

tnaxima and minima thcre is a parallel Iine of zero condensa.tion,

ou which thc two trains of waves neutralize one another. It is

uspceia.Hy remarkable titat, if tlie wave-pattern were visible (like

tfie corrcsponding water wave-pattern to which thc whole of tlie

prcceding argument Is- appticabhj), it would appear to move for-

w:u'ds without citange of type in a direction dincrent from that of

cithci- component train, aud with a velocity ditîcreut from that

with wliieh bûth coniponent trains move.

In ordcr to express the result analyticfdty, let us suppose that

thc two directions of propagation are C(p)ally incHncd at an anf)Gc<

to tlie axis uf x. Tlie condensations themsetvcs may be dcnoted by

It appears from (1) that thc distribution of on the plane a~advauces pa.ml)el to thé axis of unchanged in type, and with a

uniform vclocity a– cos a. Considered dcpcnding on is a

maximum, wbcn sin a is equal to 0, 2~ 3\ &c., while for tlie

iutermediatc values, viz. A., &c., s vamshcs.

If a = 7r, so that thc two trains of waves meet one another

dircctiy, tlie velocity of propagation paraUel to x becomes iuHuite,aud (1) assumes thé form

Page 84: Lord Rayleigh - The Theory of Sound Vol 2

72 REELECTION FROM FIXED WALL. [2G9.

Thc problem that wc hâve just hccn considcring Is in rcality

thc same as titat of thé runcction of a train of plane wavcs hy an

infinitc p):mc walh Sincc thé expression on thc right-hâjt.! sidc

of équation (1) is fin evcn fnnction of y, s I:i symmetrical '\vit)i

respect to t)te axis of ?, n.nd consequcntty there is no motion

a.cruss t)tn,b nxis. Undef thèse ch'cttmst:u)ccs it is évident th:tt thc

motion cou)d in no way bc aKcrcd by thc intr(j(]uct,ion ~lon~ thé

~xis of a; of a.n. absotutdy immov~b!c w:U). If a bc thé angtc

betwcûn tho Murfttcc und thc dircetioti of propng'n.tion of thc Inci-

dent wttvcs, tbc vciocity with winc)~ thc pièces of !n~xinn)!'i con-

densation (con'CHpondingto the g)'c;itc.st dévotion of w:T.tcr-W!L\'cs)

movc a.)ong thc w~H is ft– cos a. It may ho noticcd t!):t.t thc n.cn:d

prcs.sm'cs])a.ve no tcttdcncy to move thc wa.)t ns n. who)c, cxccpt in

titC case of nbsuhttely pcrpcndiculur incidence, since thcy m'o at

any moment us mnch ncg:).tivc as positive.

270. So ion~ as thé médium which is t!ie vchiclc of soun(t con-

tinues of unbrn~cn unifonnity, phmc wavcs !na.y bcpropagatcd in

any directiou with constant vclocity and with type unchangud Lut

a disturhancc! cnsucs wlten tho wa.vcs ruach any part whcrc thc

tucchanical prcpcrtics of thc médiumundergo

a.change.

Thû

gcttcral proDeni of thc vibrations of a vai'iabic modimn is probabty

quite bcyond thc grasp of our présent mathcmatics, but n~a~~y of

thé points of physical intcrcst arc misod in thc case of phmc

wavGS. Let us suppose that tttc rncdium is uniform abovo and

bctow a certain innnitc plane (~=0), but that in crossing that

p!ano t!)erc is an abrupt variation in thé mccbanical propcrtics on

which tho propagation of sounddépends–nam~y

thc cf~t~'eN.s't-

!)t7~y nnd ttte ~C! On thc nppcr sidc of thc plane (which for

distinctness of conception we may suppose horizontal) a train of

plane wavcs advanccs so as to meet it more or less cbUquety thc

nrobtnni is to détermine tho (rcfractcd) wave which is propagatcd

onwards within thé second médium, and aiso that tlu'owu back

into tbe nrst nicdium, or reftected. Wu bave In thé nrst p]ace

to form t!ie cquatioua of motion aad to express tite boundary

conditions.

In t)ic uppcr médium, if p bc thc natural Jensity and s thc

condensation~

density=

p (1 + A'),

and pressure= J* (1 + ~1&'),

Page 85: Lord Rayleigh - The Theory of Sound Vol 2

270.]] REFRACTFO~ OP PLANE WAVES. 7:3

whcre ./) )H cncfificicnt(Icpcnding 01 thc compres3:M)ity, f~n(~ P

is L).c uadistm-bud pressure. 1~ ii)~ !n:u)ncr ni tlic Jowc)- jnudium

thcnndisturhcd pressure b~ng thé s~mc on both si(]cf! of ~=0.

T;~i)~thc:~is ut'~p:u-!Ut(jlt.)t))o]incof intm-sueti..n.,t't,hc

p):t))<t ~f U.c Av:wcs witli t!.c surfhcc uf .scparatiuu.x=0, wo hâve

i'<.))'Lhcuppcrnicdiu)u(§~44),

Thcscchâtions m~st bc s~tisHcd a!l points of thc fini.). FnrUicr

thcb<)un~!HycondiLion.s rc~)u-e(~) th~t aH points oi'tho

Murf~cc of HCp:u'atio)i thu vulociticspet-pcndicu~r to tlic suiTacc

must bc t)ie s~mu for thc two ituuls, or

lu onicr to rcprcsent a. tmm of waves of harmonie type, wc

m:t.y nsHumo and <~ to bc proportional to e'<~+~+') whcrc

+= cmt.st. ~:vcs thc direction of t)ic ptane of thc wavcs. If

wc :iti.su)nc for t))u incident wavc,

Page 86: Lord Rayleigh - The Theory of Sound Vol 2

GREEN'SINVESTIGATION

Fs~Q.1 '1

V.

tlie rcficcted and refractcd waves may be repr~nted respect:vc!yby

Thécoemc.cnt of < isncccs.sariïy tlic same in ait thrcc waves

on ~ccount of tlieperiodicity, and ti~e coemcicut of y nu.st be ti.c

samc, .~ncct).c tracus of a)l thé waves on thc p!.nc ,f sectionmust n.ovc togctl~ With regard to ti.c coefHcicnt of if ap-pc~ by substitution in thc diHcrentud

équations that It.ssi~n

~h~ged

inp~Ing i-ro~ thc lucidcnt to thé rcncctcd wavc'. In

fact

Now&- V(..+ ~) ,s thé sine of tlie angle Included between theaxis of x and tlie norn~! tu thc plane of t]~ w~vcs-in optic.1an~u, t)~ sine of the a~]c of incidence, ~d

& ~(. "+ is inT7~ of'

1~ anglesbe c.I!cd (~ asserts th.t sin~: sin~ is cqual to the con-stant rat.o

= ~-the.cU-J.ncwn law of sincs. TI.c )~ of re-

f.act~n .nd réaction ~!cw simply from tlie fact that the vc]o-city of propag~n normal to tlie wave-fronts is constant in caclin~dunn that to

say, indcpcndcnt of the ~c~ of thc wave-front, t.ken in conucetiou with thc equ.! velocities of tlie traces ofaH thc waves on the phtnc of séparation ( sin = F sin )It renoms to satisfy thé

boundary conditions (7) and (8).'

Thèse mvo

This ccmp!ctes the syn,bo)Ica! solution.If (auj bc rca!, wc

sue that. if the incident wave be'v~

Page 87: Lord Rayleigh - The Theory of Sound Vol 2

270. jJOF BEFLECTION AND REFRACTION. 75

is ho-c obtaincd on thc supposition t!i~t t)ic w~ves arcof harmonie

t.ypu; but sincc itdocs not. involve and t))<j)-u iH no change of

phase, it may bc cxtenjcd by Fuuncr's thcorcm to waves oFauy

typu whatuver.

It' thL-rc bu no rcficctcd w~vc, cot cot = from whichaud (1 + cot' ~) (1 + eot' ~) = wc dcduce

which shcws that.providcd thc refractivc index F Fbc inter-)UL~)iatc in value bctwccu unity aud p tiicrc is aiways au

:u)~c of incidoicc at which t)ic wavc is cun~jlutcly intrujuittedbut otho'wiso titcrc i.s no such tUT'c.

Smco (18) is not altercd (cxccpt as to sign) by an i))tcrch{).ngc

of<9, &c., wc infer that a wavo incident m tlie secondjncdimn at :).n angle is refiectcd in t)io same prupurtioa as awave incident in thc first médium at an a.n'du

As a numerical cxampic Jet us suppose that thcuppcr medimu

is air at atmospho-ic pressure, and thc luwcii- médium watcr.

Substitnti)]g fbr eut its value in tcnns of and thc rufractivo

Index, wc gct

Page 88: Lord Rayleigh - The Theory of Sound Vol 2

FRESNEL'S EXPRESSIONS. [270.

whieh shows that thé ratio ofcotangcnts dhnini.shcs to xcro, M

h.crc~os from xcro to ~out 13", afLM-whic!. it htconc.sim:~i'),:u.y,

m<)ie:).t,ij)g tutal rcf!cct:on, tm wo shafi sec prcscntty. It n~st buronc.tnbcrcf) thut in

~Jying optiez! tcrms tu acoustics, it is thuw~er t!jftt nutst Le concuivcd to be thé 'rare' medimn. Thc ratiooi'duusitic.s is abuut 77U 1; so that

Evcn at pcrpcndicular Incidence thé réfection is scnsihiy pcrfcct.

If both mc-di~ bcgascott.s,

= If thétempérature Le c.m-

fitant; fu)d evcu if thcdcve]op)ncut of ]tc.tt Ly compression be

takca intoaccount, thcrc will bc no sensible difîcrcttcu bctwucn

and in t])o case of thé si)np]c g~sus. Now, if =~p, /) =siu~ siu' and thc fonnult~ fui- thc

intcnsity of tliorenceted wa.vc bcconics

comcidmg with that givcn by Fre.snc! for light polarized pcrpcn-'hcufarly to t].e plane of incidence. In nccordimco witli Brcw.stcr'shuv tlie rdicetion vanishes at tlie angle of incidcuce, wl~osc

tangent is F'–

But, if on thc othc!- hand'/),=p, tho cause of disturbanco

buing thc change ofcomprcssihiJity, we I)~ve

agrccing~h Frcsncl's fonnu)a for !ig).t pohmzcd in t).c p~ncof !nc.(h;ncc. In tUs c~sc t!to rL.ficctcd wave docs not vauisfi atany angle of incideucc.

In geuGi~J, wlicn = 0,

Page 89: Lord Rayleigh - The Theory of Sound Vol 2

370.]REFLECTION DUE TO TEMPERATURE AND MOISTURE. 77

so that thcrc is no rcfMon, if~=

~ascs F' =p~ p, and theu

Suppose, forcxamp]c, that aftor p(.rppndicu)ar incidence rc-

~'cdott takcs place at a surface scp~-atin~ air au() ]iydro~ou. Wuiiavu

Thc mtio of intcnsitics, which Is as the squft,-G of tho ~p]itudcs,is ~-t-02 1, so tliat about onu-t)[ird part is ruficctcd.

If thc di<ÏL.rcncc betwccn the two mcd~ bc very sni~)), and wowritc ~=F+~, (24.) bccumcs

If the f~rst médium bc air nt 0" Cent., and thc second jnedium boair ~t C'eut., r+ ~F=

r~H--003CC<; so th~t

Tho ratio of thé intcn.sitics of thé reHccted and incident sounds ist))~rcforu-83x]0~x~:l.

As annU~r examptc of tlie sfunc ]dnd wc may ta~c Mie case mwhich t))c <h-.st mudium is dry air and t))c second is air of thcs.unu

tonpcmture satm-atcd wit)~ moisture. At ]U° Coït. :ur

.s:tt)tmtcd with moi.stui-0 i.s li~htcr t)~n dry air Ly abuutono p!trbv

iu 2~0, so t])nt~~=~~ J)c:u-)y. Huncc wc conclude from (25)

tlmt th(! ruHcctcd sound is on!y about onc 77~,000"' part of thuincidunt suund.

Frotn thèse calculations wc sec that rcf!cetiuns froni warm ormoist ail' jnust

gcncrany be very smalt, tinjugh of course thu entbct

jnay accomulatc by répétition. It mn.st aiso Le rumonbcrcd thatill practicu t!)L: transMiun frum one state uf thin~s to the ot)tcrwould bo gradl1:tI, anll tint abrupt, as thc prcHunt thcnry supposes.Jt' t)K-

sjtaœ occxpicd byt))Ctr!U)sitiun amfmntto a considérable

Page 90: Lord Rayleigh - The Theory of Sound Vol 2

TYNDALL'S EXPERIMENTS.r'270.

fraction of' thc wavc-Iength, t))c réaction v-ouh) bematpria))y

iesscued. On tins account wc rnight expect grave sound.s to travelthrough a

heterogeneous médium lessfn-<y t)t;,n ..)ctjt.c ya~n.h,.

Thc rcnection of sound from sur~ccsscpamti,~ portions of

~s ot d~crcrt dcns.tics hn.s cngaged the attentionof Prof Tynd-d!who h~ dcvi.cd suvcm) striking c.pcri.nonts in iHu.stmtion of thc

.sul.jcct Bor cxa.np)., sound fro.n ahigh-pitchcd rccd was con-

ducted t)mn,gh a tin tube tcwant.s a sensitive f!,une, w).ic]~ servitns an n.dicator. By thc

intuition of a c.al-ga.s ~a.no issui,~from au ord.nary bat's-~iny humer Lutwecn thc tuhc and th~~.ns~ve f!a.nc, t)~ grcatcr part of th~ cHect couJd bc eut oifNot œdy so, but by holdillg thc uamc at a suitaUo an.de thcsound cou!d he rui!cetcd thmugh anuther tube in .sumdcnt

nua'ntityto excite a second sensitive ~une, which but fur theinterpositionof the

rcnoctuig Hamo wuuh! havu rcinainsd undi.sturhcd.

Thé prGccdiug expressions (JG), (17), (18) hold good in cverycase of rc.ffcct.ou from a 'dc~er'medinm; but if thé

vclocity ofsound bc grcator in the lo~vc-r médium, and ti.c angle of Incidencecxcccd the critical

ang)c, becornesimaginary, and t).e formuh,.

require modification. In thc latter ca.~ it isi.npossib)c that a

rcfracted~avc should exist, sincc, cvcn if th. aug)e of réfraction~-c UO its trace on th. p~o ofscparation rnu.st

neccssardyoutrun the trace of the incident wavc.

If bc written in p)aœ of thé symboilcal équations are

T~e~e?~ t~ite

from whieh by discarr!ing théin~ginary parts, wc ohtain

'yo~/)(f,3rd édition,p. 282.

Page 91: Lord Rayleigh - The Theory of Sound Vol 2

270.] TOTAL REELECTION. 79

~ese formu~ indicée total reficction. Thc disturbance in t).o

sec~d médiumis uot a ~vc at a)t in t).o o.-dina.y scn.sc, ~n<[ at

a .sl.ort d~anco from thc surface of sopu-aticn (.. ncgative) be-comes msc~bJe.

C~IcuI~ting from (12) andcxprc.sing it iu

tcrms of and wc Aud

shcwing t!mt ti~c distm-bance does not penetrate into the secondmcdtum more thfui a. few

wa.ve-Ienn'ths.

Thc difFercncc of phase bctweeli the rcHc-ctcd and thé inddcntwaves is 2e, wlicrc

Since thcro is no loss of energy in reflection and réfraction, théwork transirutted in any time across any aroa of the front of thcincident wave must be cqmd to thé work transmitted in the sametn.ie across

co.-rcsponding areas of the rcrieetcd and refractcdwaves. TItesc

con-csponding areas areplainly in tlie ratio

Page 92: Lord Rayleigh - The Theory of Sound Vol 2

L~W 0F ENERGY VERIFIED. f~O.

-~n u.e.n.rgy cun~~ou, ~d agrées with the rcsult of nu.iti-

pfy.ng togethcr tbu two bonudary équations (13),WJien tho

vcloeity of propagation is grever ia thé lowor t!~uthé uppcr médium, aud the angle of incidence excecd. thc

critical ~g!c noenorgy i.s tr~mitteJ into the second inediu.n.othor words thc reficctiou is total.

Tlie method of tho present invcsti~tiou issubstantia))y ~c

a.ne asth.t~pjoyed by Grccn :n p.per on the ReHectJand

Icract.onof Sound T). caseofpcrpcndicu]. incidence

~.tu.vc.st.g.ted byPoi.s.~ who cbtained

fonnui~corrc.sp.n,);n.(3) and (2.t). ~).eh I~d i.wcvcr bc.n

airc.dy givcu tj Y.utho rcf)cct.cn of Ligi.t. lu a sub~~cnt ~oi/Poi~

c.n.s.dcrcd t!.e gênera! c.scofobtiquc incident H.nitinghimscif

.owcvc, tog.cous n..di. for ~ich Boyie-s law

hoids~od, d.y a

.cryccmpi.c.tc. ana]ysis an-ived at a rcsult cqui~icnt to-'). Hc a!so vor.hod th~t t),c énergies of the rcftected aud ro-~-acted wavcs make up that of Htc hicidunt wavo.

271. If'to]y cxtcnded do~i-

v~d. w,thcomp],.to .nlfon.ity in its ~ch. parties do

tr.ns.n, ted wave isprop.g~d onw.rd.

eonti,I)y. 'B~jf at

c'gc i~ thcco.sihi)ity, densityo both p.r of thé w.vo wHI bc throwll back, .nd ou .riv~ttlie b~

~.=0; will hc divid.) iuto t.o parts, o e11~ ~t u..di..n, .ud eue r.ficct.d b~/to b. ag.~d.k.d at ..= ,d .su .n. Hy f.JIo~i~. thc pr~re.s.s of thcse

c ,.t.u .f the pr.hi.. may be.bt.i.L;t,

~ctodand traus.uttcd ~.s bcing c.,npoundcd of an incite

~"th~ /r''i" 'c.is tlie .~hodu.s..diy ad.pt.d In ()p,ics for thc

c~poud'i~ ;?

-s .s~huu~tiy c.p)anK.d but it ducs not

appc.r to hâve any ad~nt..m.a ~cr.straightf~rd auaJy.si.s. r. t.f.c

f.Uo~. ri

~~n

.Ld cu~nc ou~dvc. te thu ça.hcre th J."cdu.,n is .u.njar ill its

p, thu

~< 7'fu<.«~))).~ ]~jg=

~rM;.~7, t. Jt. p. iJOg IQjf)

<~ ~r'~i"rlr

l'In,tilrrt, 1. X'l', ;i17, 1,-j;Jl,

Page 93: Lord Rayleigh - The Theory of Sound Vol 2

271.] PLATE OF FINITË TIIICKNESS. g).

Jn or(]cr to pass to rea! quantities, t)tcse expressions must ))c

put into t).c form TPe" 7/'c~ &e )-e< we fmd corresponding tuthu incident \va.vc

R. If.(;

Page 94: Lord Rayleigh - The Theory of Sound Vol 2

S2REELECTION FROM A PLATE r-27l

s).cwingt)~L cxccptf.thc .hcr~i.n r,fp].e, tho ,vho)c of t).c!ncd)U)n m~ht as wut! ijavc Leen unifonn.

If bc small, wc h~-capprcxHnatcfy for L),c rcf)ccto.] wavc

a~-mu]aapp)yu.g~hcnthop!at.cist)uuin

eom~n.son ~ith

tliew~c-!cngt).

Sinco =~cos~. it appears t)~t for a givc.

ang!c uf incidence thé a)np!itud~ variesim-c.-scly as or as

Jn any case t!.c rcHection vanishe.s, ifcot~</ = t)~t is, if

~bc.n~n intc.g.r. Thewnvei.sthcnwhu))yt,-au.smittc.).

At i.crpe..<)icL,lar incidcnœ, thcI~cusity or t].c rcftc.etion is

express) by

Let us nc.vsuppôt t!.it thé .ecuud médium i.s

ii.eu~prcssibic, .so

Page 95: Lord Rayleigh - The Theory of Sound Vol 2

371.]OF FIXITE TlifCKNESS. 83

t}):Lt =x oui' cxprcssKtM bceomc.s

shcwing ))ow thc amonnt of roHcctiou dupcuds upon thé )'uhttiv.

xut.s.sc.sot'HucIt t~)!U)tities of Ute média as itavc vu)utncs m thc r~tiu

of It is obvinus tt):).t tho {icœud médium huhave.s fiku n.

y rigid body !md act.s on]y in virtuc of its ino'tiu.. If thi.s bc suf-

ticient, thc ruf)uetio)i inay bcMmc! scn.sibty tot~).

Wc hi~vo )it)W to cousidcr thc ca.su iu whicti(~

is i)nagin:uy.I)i thc symbo)ic:d expressions (5) aud (U) co.sft~ amt t si~n~ :u'c

rca], wltitu a, a+-, a–- arc pure hnagiuanus. Thus, if we sup-

pose that r/=~ 0=~ :u)(t introducc tho not~tt'm of thc hypcr-buhc sine and cosine (~ 170), w~ gut

Page 96: Lord Rayleigh - The Theory of Sound Vol 2

0 j81~0 LOSS 0F ENEUGY.

f~

–H'S.X encrgies oftransl11ittcd

(Lecotilit flll" tllc wholoeller~y of tlle incident

=~F"

cil' waÏiL:-front aroc(lual fur ail threc it is

ully )lOCrssnry to uclcl tlic

~h" cquati0118 (7), ol7 in e(ll1a-tiulls(12), (B),

272.'l']¡esc calcnlalions of rcilcction fln(1 refr<~etion umler

~L- hecarricr1 fllrtlJer, Lut tllcir intcrcstlic rat/1er oytical tll:llI licotisti(-,11. It is important to beal'iiiii](1 tkü 110

l'tlLlyy 1~ destroyed I)y ail)' IlIlJnber of i,oflectioll,4

S~tiullulwa~~s l'l.:al'l)(!al'ing iu HllOthcr,

011 aCC01lllt of tlm ;~rc.,tt dit}'ercl1ccuf' (1C11SlLICS l'efluction is

liclnitl l11attl!r, ~oululs 1)1'o(IticL-(l iu ;lir arc )lotcvsily coml11l1lli-

I:WUI)(ls,IYIIUSI: U1'1bj11 il> 1Iudel' water,wit.c.c<. );

(liflictilt,y iu air.~L~"iofwooL),ora. mctaJiic

distances with very littlo loss.

Page 97: Lord Rayleigh - The Theory of Sound Vol 2

g

CHAPTER XIV.

GENERAL EQUATIONS.

273. 1~ conncetion with 1)~0générât probicm of aurial

vibrations in thrcc dImcusiujiH one of thc first questions, whic]~

natu!U)y of~rs itsdf, is titû dctern)i;iaLio)i of thc motion in anuniimitud

atmosphère conséquent upon arbitt-ary initia.1 dis-turbances. It will be assumcd t))at thc disturbancc is small, sothat thc ordinat-y ~pproxiniatu équations arc applicable, aud furt'hcrthat the initial vu!ocltics are snch as cnn bc dcrived from a vclocity-])otcntial, or (§ 240) that tllere is no CM-c«/ If thé Jatter con-dition bc violntell, the

probtcMi is onc ofvortexmotion, on whieh

wc do not enter. \Ve s]iaH idsosuppose in the m-st

place that no

cxtcrna! f..rccs act upon the uuid, so t)iat tlie motion to bcinvcstigatcd i.s duc soldy to a disturbanco actuaDy cxistin~ ata titnc (<=0), prcviou.s to which wc do jiot push our inrp~-iusThc mct.hod that wc s!.a!l

c.npjoy is not very dinTo-ent from thatof Poisson hy whom thc proHon was first

succcssfn!!y attackcd.

If M., bc tlie initial velocities at the point a-, z, and 80thc initial

condensation, wc hayc (§ 2-),

°

by which t!.G ituLi~l values of thcvolocity-potcntial and of its

ditrereutlal coefficient with respect to tunG arc Jetermined.

1 Sur l'int~~tin.. quc.]qncs <juation~ lindairos aux di~rcnccs pM-ti~Met p,u'hcul~remcnt do l'équation ~n.rato du luouvcmeut de. fluides

6iaBtiq~~~i. <!t; <Y)~ft<x~ t. m. p. 121. 1820.

Page 98: Lord Rayleigh - The Theory of Sound Vol 2

8G ARBITRARY INITfAL DISTURBANCE.f273.

.jiujmjM~tjjt,.)Z/,),

TLc pmUc,n L~furc us is todctcrminc at tm.e <!from thc ab.~

rnt.iat v.du.j.s, and tho ~ncnd c.juaLiott ~pp)ica).!<! .t !) t!m~ .-m<!pinces.

Whcn is ]<nown, it.s dcrivativcs~ivc t)tcccrnponcnt; vetocitics at.

:nyponit.

ZD

Th~ symhuHca) sohnion of (:3) ,,my hc ~-ritten

~hcrc~and ~fu-c twofu-hit.nu-y fun<;t.ions of.~ ?/ nn<1 ~=/-T)'

Tuc.,nnc.c~an.Lh the init;)vah~of~u.t~~hi~

.shaJi < c.tc ~d 7''rc.spuetive)y, it isoniyncec~rytuub.scrvc.'

th.~twhen<=0,(-l.)~i\'cs

in

which

equ~on

thc question of theInto~rctation of od.) po~.s

Hlly"syllibulic

\lol]y evcl.

In the

c.~hc.-cwas a faction of .r c~y, .vo s.w (§ 245)t!.t its ~uc fur

.nyp.int..t tin.c~.pe,.dcd on thc nitvalllcs .f and at thcp.i.ts ..h~ cc-n~ we~ at-I. +~, and .s w.Iiy i. ~J

a)I uthcr points, In thc pru.sent ca.s. Lho.si.np)~ suppositionis

point 0 clepencls ont..nacs of nu.t at points .situatcd on f)~ ,f,~ of t~

spf.crc .-I.o.sc c.nLrc i.s Oand radius~ ,n< as tLcrc eau Le no!)~

r'ccovc.ra~her,wc.rc's Jcd to~nvcst. t),,

cxpr..ssioa for ti.c ~can vainc cf.c.onovcr a sphuri.a) surface In tcr~ of ~csuccc-.iv r.tud coc~cicnts of the funetion at thc ccutrc.

By the syn.boJieat f~ of Mac!ri.'s tj.ccrcm the value of.t~~ ~:7 point P on tlic l'lay be writtell

Page 99: Lord Rayleigh - The Theory of Sound Vol 2

273.'J1AR13ITRAP.Y INITIAL DISTURBANCE. 87

tho centre of thc .spftcrc 0 bcing thc ori~ui of co-ordmatcs. In

tlie mt.c~t'ft.tion over tlic sm'fuce of tlie sphèrelm~o

I)chavn as c<)t)nt;tnts; wc may dénote them tcmpora.rity by 7~, ?~

so t]):it ~=~+~+?r.

Thus,)- buhig the mdius of thc sphère, and fui dément of

its surface, sitjcc', by tt)u syntinctry of tlie sphère, wo may repla-ce

/.<;+/)/+/M n.Õ 1 r <-1nny functtonof by tlie s:unc iuncttou of .? wnho~tnllY tlllctlOn

+ +..)y le sa1!lC 11l1CtIOll0 Z WltlOUt

attcring- thc rcsult of tlic intc'gra.tiou,

Thé mea.n value of~ovcr thc surface ofthc sphère of radius 7' is

tttus expt'cssed by tlie l'csutt. uf the opcratioa un 2'' of thc symbol

of,if ~~o-

dctiotc intégration with respect to Mgu!arï\

or, 0

space,

or in words, ~) at any point at timc < is thc mcan of thc initial

vaincs of 6 over tlic surface of' thé spho'c described round t)ic

point niquestion

with mdius r~, thc wlioïc niultiplied by <.

By Stokcs' rdc (§ 95), or by simple Inspection of (5), we sce

thfit t)ic part of dupcmling on t)ic initia values of <~ mn,y bc

(tcrivcd from t))n,t just. writtcn by diffcrcntia,tmg with respect to <

andchanging

thé arhitnu'y fmieti'jn. T)iC comptetc value of at

thuc is thcreforu

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VERIFICATION0F'SOLUTION.

~3

whichi.sPoissoii'sresuIt'.

On aecouutofthe importance of the présent p,.oL!cn. it ,n.v

~r 7?~t pt~t it s.t..sHc.s thé gcncr~ dIHfcrenti~ c~ua.ticn (3). T,~ f~

t~p~ntth~.tt~culy,~d ~.rin, i..iud t IMytnbohc cquatton

o'L[.u

Ncw~~i. t~

~J~satisHcd.

Sincc thé second part of I, cLtaincd from thé Rr.t hy dl~rcnt~un, ,t aiso n~ust

satisfy thc fundamcntaléquation.

Wkl~ respect to thc iuttial ecuditions we sec th.t ~hen is ma.tro~ual to zéro iu (8),

~S-~ in~rchh~mutiacAc

l'h~~ik, 1, 517, 1876.

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273.]LIMITED INITIAL DISTURBANCE. 89

of which the first term boeomcs in the limit 7~(0). Whcn < = 0,

sinec the oppositcly situatcd c!u)nent.s cancct In thé Hmit, \vl)cn

thé radius of thé spherical surface is indefiuitely ditninistted. Thc

expression m (8) thcrefbrc satisfics thé prcscribcd initial con-

ditions as wcl] as thé général din'ercntial équation.

27~. If t))C initial di.sturbancc be couuned to a spacc ?~ thc

Intégrais in (8) § 27~ arc zéro, nnless somc p:t.rt of thc .surface! ot'

tfiu sphère ?'=<~ bc includcd within 7'. Lct ~bc a puint cxtcl'n.'d

to 7', ?'t a.nd ?'~ thé ]':u1ii (~' titc h'n.st and grcn.tcst sphcrcs dcso-ibcd

about C) A\'hic)t eut it. Thoi so lon~ as ft< <)\, rcmn.ins cqual

to xuru. Whcn H< lies butwccn ?\ und )' may bD nnitc, but fur

v.'dues grcatcr tban ~)is

ag'ain zéro. Thc di.sturbance is thus at

aoy monKjnt coinncd to those parts of'.sp~cc for which r(< is lutcr-

ïncdiatc butwccn and ?' T])C Hmit ot'thc wn.vc is thé oivclopcof sphères with radius at, whosc centres arc situatcd on tho surface

of T. Whcn < is smat), ttus System oi' sphcrcs will hâve an

extcrior cnvelopc of two shccts, ti)c outer of thcse shcets bcin'r

exturior, and thc inaer intcrior to tho shcH formcd by thc as-

SL-mh!agcoft])C sphères. Thc outer shect funns thé outer limit

tu t])c portion of thc médium in which thé dUatation is diffurcnt

from zéro. As < Incrcascs, t!ic inncr stjcetcontracts, and at Jast its

opposite sides cross, and it changes its character from bcin"- px-

terior, with référence to the sphères, to interior. It then cxpands,

an<t forms t)ic inner Loundary of thc shull in whieli t!)e wavû of

condensation iscompriscd'

Thé successive positions of the

boundaric's of thé wavc arc thus a scrius ofpiU'aUGi surfaces, and

each boundary is propagatcd normaDy with a vcloeity cqual to

If at tho time < = 0 thorc bc no motion, so that thc initial

disturbancc consistsmerety

in a variation of dcnsity, the subsé-

quent condition ofthings

is expressed l)y thé first terni of (8) § 273.

Lct us suppose that thc original disturbance, still hmited to a

imite région y, consists of condensationonty, without raréfaction.

It might be thought that thc samc pccuharitywould attach to thc

Stokes, "DyHUtUtcul Thcory of Difïraet.iou," C<<w&.ï'y'«f)i:. ix. p. lu.

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CASE 0F PLANE WAVES.f'2~.L-

resulting wavcDn-oughout thc who)c of Its

subs~uent course butas Prof. Stores bas

rcniarked, such a conclusion wouht be erron'eous'

For vah.es of thé tune )ess than r, -a t].c poientia! at is ~ro-it then becoiacs négative (~ being positive), aud continues nc.~a-t.ve unt.I

ttv.uu.shesagain wheu <=?-«, after wl.ie)i itajw~ys

ren~n.s equ.'d to xero.W)u!c is

di.nini.shiug, t!.c médian atis in a .stato of

cmKk.nsatIou, buta~ incroases

a~u.i to.en, th~

statc. of t]ic med.um at is onc of nu-cfactioa. Thc wavcprom

gatcd .-utwants cnnsi.st.s thurcforc of two parts at )cast, of ~-hichthc first is cundc-nscd and thc. !ast nu-efi~. ~~),atcvcr ,nay hc t).ccharactcr of thc. ~mai di.sturhanec. wit)un t),c <i,d y; of6tlt anyextc.ja) poillt (J is t,),c. sa.nu .LS t)h. Initiât Ya]uc, an<[ therc-iurc, sutcu a~=- t)ic n.c.aucon~n.sati.)).

dun)~- thu pas.sa.re ofthc wavc., d~cndiu~ on U.u int~r~ is ,ero. Undcr"'thchcad of sp)~.nca! wavcs wc shali h~vu occasion tu rcturu to this

suhjcet (§ 27!)).

Thc général solution cmbodied in (8) § 273 must of courseembracc the part.cuiar c~su of p)anc ~vc.s, but a few words onthis application may not bc supurfiuous, for it nn~ht appear atfirst si~ht that the cUcct at a ~iveu point ofn di.sturbancc i)iiti:U)yconnnL.d to a sficc of thc tncdiunt oyclosed between two paraHetplanes woutd not pa.s.s oit- in any tinitc titnc, as wc know it ourrj.tto do. Let us

suppose forshnpHcity t!.at is zero throug-In~t

and U.at wit)mi thc. slice in <p~atiou thu initial value -A isconstant. Fron. the

thcory ofp)anu waves we ]n.ow that at

any

~rbitrary point the di.sturb.-incc wi)) fioaDy cca.se aftcr thc iapse ofa time .such that <!< i.s

c<{u:d tu thc distance (~ of thc pointundcr considération froni thé furtller

boundary of theinitially

disturbcd région ~diiie on thc oti.er hand, sincc thc sphcrc ofradius ~< continues to eut the région, it woutd appear D-om tbc

gênera) formuk t))at thc di.sturbancc continues. It is truc indced

that remains tillite, but this i.s nut incon.sistentwith rest. Itwill in fact

appear on cxatnination that t]~c me:m valueof

multipiicd by t)~ mdius of t!.c sphère is thé .samewhatever may

bc t)te position and sixe of tl.e spbcre, provided on]y tjiat iteut con.plcte)y through thé n.gion of original disturbance. Ifa0f/, cp is thus constant v-ith respect both to space and timo,and accordingjy tbe jnedium is at rc.st.

275. J)] two dimc.n.stons, when i~ indepcndent nf~ it mightbc supposcd t)):)t the

corrusponding f..r!nu)a wou)d hc obtained°by

Page 103: Lord Rayleigh - The Theory of Sound Vol 2

275.]TWO DIMENSIONS. 91

smiply substitutingfor thé sp)ici-c of radius thc circlc of cqnd

r:uHus. Tilis, howcvcr, is not ttie C!isc. It mity bo provcd t)iftt

thc mcan value of a function 7~(~ ~) uver thc circu!)ifei')juCtj of a

circlc ofnidius ?' is ~(~7) 7~, wh<jrc z==~T,

difïcnng t'rom what is rcquired to satisfy thc fun(t~mc~t:Uéquation.

Thc correct rcsntt applicn.btn to t\vo dimensionsnifi.y bc obtn.incd

i'rom thc gcncnd fùrtnula. Thc ctoment of sp)tct-ic:U sutf.LCo cM

i ) i ) )'f~'<~ 1/1nmy bc rcpfaced by Ayhc'rc t', 0 arc plane poln.t- co-onh-

COS'1'

na~'s, an~ is thc angle bct\vccn the tangent phmc and that in

which thé utotiut) takcs place. Thus

where thc intégration extcnds ovcr thc arcn, of t!ic circle )'==<

T))e other tcrm might bc obtalued by Stokes' ruie.

This solution is~p))lic:iblc to thc motion of

Jaycrof

gns

bctwccn twopiu'a.lld -phuics, or to th:it of fm unUoutcd stretchc~

mutnbmnc, wtuctt dépends upon thc s:unc fundftmcuta.téquation.

270. From thc sohttion in terms of Initial con<Htions wc May,

as usu:d (§ (!G), t]cdnce t)io eH'nct of' a eont!nn:dly rcnewcd dis-

turhancc. Let us suppose tliat throughout the spa.cc (w)tich

willuttirnatc~y be !i)ade to vanish), a, uniform disturba.ucc

cqua)tu

(~')~, is communicn.tcd at tmic T!)0rcsultiu~vatuc

of 6 at timc t is

whcrc ~S'dcuutcs tlie part of thc surface of the sphcrc )' =«(<-<')

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SOURCES 0F SOUND.f-S/C.j~~Jt'J.

nttc.rccpLcd witinn 7~,a qnantlty which vanishes, unicsa a (t becompresscd bctween tho nan-ow Iimits r, and )~. Ultimatcly

~y Le rcplaccd by 7. and~(<') by

~);and thé rc-

sult of tlie Integ~tion with respect to is found by writinc(tlicvo)umc)fur/«,9< Ifcnce

.shown~

tha thc di.tnrbancco~inating at ~ny point .sprcads itsc!f

.yun..utn~I)y ni ali d.recMui.s witlivotocity and

with an~plitudc

v~y.n~mvcr.sdy thc <)i.s~ncc. Sincc

~ny nutnbei- ofparticuJar

sduL.01~ n~y bu superposcd, tho ancrai .sotutiuu oi-t!ic c.juatiou

?-dcnoting thc distance of thc cicmcnt ~F.situatcd at y~from

0 (at wftich<~ i.s

cstimatcd), and <P1')

thc value of for thc

point at thé ti.nc tCompJc.ncutary terms, satisfying

t].oui,r), ait sj~cu thécqu~tion = u~y of course occur inde-

pcm)c))t)y.

In our prcvious notation (§ 2't-J.)

.Lnd t ,s assumcd t]~atA'~+r~+~i, complote di~rcntiaJ

l'orc~ undcr ~'].o.sc action t).c ~cdiu.u coutd not a.)ju.st itsclf toc~n .bruun, arc c.xcludcd; as for in.stanee, force uniform in

n~rr.ultudc and direction witllin a space Y~ andvani.shing outsidc that

spnce H.c nature of thc disturl.ancc d~K.ted by ispe,.],~sœn by con.sidcring t).c extrcn.c ea.sc ~c.n ,i,

c,through a .s.naH voininc, ~-).ich is

suppo.scd to <!i,nini.sh ~it),out

ht.nt, wh~ thc magnitude of incrca.sc.s in such a. manncr that titewhoïc ctrect retins finite. If thon we integrate équation (~

Page 105: Lord Rayleigh - The Theory of Sound Vol 2

2 7 G.]IIARMONIC TYPE. M

throughfL sma)t space Including thc point at which <P is ulti-

nm.tL'iy concuutrat.cd, -c Und in thé limit

shewing that thc effect of <t* may bc rcprcscntcd by a proportiona)

introduction or abstracti(j)) of nuid at t))C p)ac:u in (~uestio)i. Thc

simptcst source of sound is thus an:i.io~ous to a, fucus lu thé thc'n'y

of conduction of tioat, or to au électrode in thc theoryof cicctricity.

277. Ttiu prcceding oxpressionsn,rc ~cncmt in respect of thc

relatiou to timc of tite functions conccrncd, but in :d)nost idi t))C

applications th.'ttwc s!)a!l Iiave to ma)<c,it will bc convcniL'itt to

analyse thc tnotion by FouricT'.s thcorcm and trc~t scparatciy t))û

sunp)e harmonie tintions of varionspcriods, ai'tcrwards, if

ncccssat'y,

conpounding tii(i rcsult.s. Thc value of < and. if si)np)u Inu'-

!nonic at cvcry ))oint of spacc, may bc cxpresscdin tite form

.7)!coH(?~+e), 7t' and ebcing indcpcudott of timc, Lut variable

from point to point. But as in such cases it of'tun conduccs to

simplieity to add thc term ~sin(M<+€), malung :dtogcthcr

T~e~ or jf~c".e' wc wiM assume si)np)y that aU thc functions

winch hntur into a problon are proportiottat tu c" tite c"ef~i-

cients being in général cotnpiex. After our opérations are coni-

ptcted, titû re:d and imaginary parts of thc expressions can hc

separa.tcd, cither of theni by it.setf constituting a solution of t)te

question.

Since~) is proportiona! to e' ~=–?r< and thcdirfercntia!

cquation becomcs

To adapt (3) of thé prcccdmg section to thc présent case, It is

?*

oniyncccssfn'y torcmark that tlie substitutionof~fbr<Is

1.

(-'iTeetc'd byiutroducing tl)e ftetor c or e' thus

Page 106: Lord Rayleigh - The Theory of Sound Vol 2

VERIFICATION 0F SOLUTION, f'277

and tbc-solution of(I) Is

to winchmay bc addcd

any suiutiun of + ~=0.

Ifthcdi.stu.-bing forces bc ~iin

U.c.samcph.and <),u

r~.o.) thmnghwtnchthcy~buvuty.s)na])i,.co.np!U-i.son with

t!.c~c-]L.n~e-n,aybcro.nov~fn,,n~dcr

ihuint~ds~n, and at n. sufhcicnt. distance wc

may ta~c

°

In ordcr to vcriiy that (3) ~tisfie.s thediHcrcutiat e.n.atioit CI)wc m~y procccd a.s in thc thcory of thé common potcl.ti~I. Con-

Mdcrn~eue cicluent of t).e iut~rat ut a time, wo Ii~c Hrst to

shcw that

-satires~+~=~ ~j,

.s.m~st

course is tocxprc.s.s in

p.]ar co-ordi.i~es referrcd tothc c!cmcnt itsuffas pulc, whcn it

appels tha.t

crt),at (3) .s.tisncs + = (), ,t a!) points for

w!.K.). va.n.shc.s. 1,, t).c case of a puint at w].ich do~ notvan.si. ~e may put out of account thu elc.nen~ situatc.) at a

~H<cd).~ncu(ascont)-ib)[tij)g on)y icnns.satis~ing \7~+~=Q)and io.- t)jc etcmcut at an innuitesin.a!dlst!mccrun)~c c-bv

umty. Thns on thé w)tu)e

cxactfy as in roi.sun'.s t!ieorcni for the cnrnmon potenthd'.

't-!cûTi)o)nso))nnJTmt.'tiA'«<.7'/<f/i;.JtU.

Page 107: Lord Rayleigh - The Theory of Sound Vol 2

278.]SURFACE DISTRIBUTIONS. 95

278. Tho ciï'uct ofa. force <I~ (ti.strihtttcd ovcr a. surface <S'ma.y

})C obtitiaed ~s a. iituitin~ case frotn (!)) §277. (iTis rcpI.LCcd by

~~(~S', (lunoting thc t))icknc.ss of thc l<t.yer; {uid m the Ihuit wù

may writc b = TiLus

Thcvatocûff~ i.sthcsn-nioûu ihctwos[<]cscf~butthcrc la

(ji.scoutimuty I)t its dcriva.Mvcs. If<~ Le dra.wu outwards n'ont ~S'

nortn!tl)y, (4) § 276 ~ivus

Page 108: Lord Rayleigh - The Theory of Sound Vol 2

OC INFINITE PLANE WALL.[278.L-,

Thu siUtiû mcLitmI isapplic:tb)c to thc ~oncra.! case whcn thc

tnotiou is uot rcstrictcd to bu sinipic harmonie. Wc hâve

whcrebyF~j

is dcnoted t!te nornuU velocity at the plane

for thc c)emcnt J~ at the time t (?'- ft), that is to say, at a time

)'–ft antécédent to that atwhidt~iscst-imated.

In orderto complète thé solution of thé probton for t!)e

unntnitedmassoffhndJyingononcsidcofaniniiniteptane,we

hilve toadd thé most gênerai value of<j&, consistent witit F=().

Tins part oftite(juestiun i.s Identical v'it!i tlic ~eucra) probtem ui'

rcileetion from an inHuit.c ri~id plane'.

It is évident that thé cft'cct of tl)c constraint will be reprc.scntcd

by thé ititroduetion on thé other sidc of t))C phuic of fictitious

initial displaccment.s and forces, formin~ in conjunction witit thèse

actually cxl.sting on t!ie nrst side a systcm ped'ccdy symmetrieal

with respect to the plane. Whatcver the initiai values of~ and

<~ may be belon~in~ to any pointon thé first side.thf .same must

bc ascribed to Its ;'M~7~c, and in )ikc mantier whatever function of

thé thne inay be at thc fu'st point, it nmsL be conceived tu t)e thé!

satno function ofthc titneat theother. Underthesu circunistances

it is c!eM' that fur aH future dnic ~) will be synimetrical wiLh

respect to the plane, and thcrefore thé normal vefocity zéro. Su

far then as thc motion on thé hr~t side is concernet), thcre wili he

no ch:)ngc if t))e plane be removed, and t))e Huid continucd

iudcnnit.L'iy in ail directions, provided thc circumstnnces on tht;

second side arc t)ie exact reftection of t))ose on thé first. This

being nnder.st.ood, thc général solution uf thû problem for a

nuid boundcd by an Inrinite p)ane is containod m thc formuiaj

(8) § 273, (3) § 277, and (8) of thé présent section. They give thc

resuit of arbitrary Initial conditions (~ and <~), arbitrary nppticd

forces (<P), and arbitrary motion ufthe plane (J'').

Measurcd bythc resn)ting potential, a source ofgivcn magni-

tude, i.c. a source at which agiven introduction and withdrawai

ofnuid takes place, is thus t\vice as enectivc when close to a rigid

plane, as if itwcrc sitnatcd in thé opcn, and thc t-esult is ulti-

'P('if.n,J~;fnMi'(;)'crn~j)f)~<f'cyut)'~)f<t.vn. 1ROR.

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DOUBLE SHEETS.~7

"E, m. 1 '1..

_u-.

~t~~T source concentrated in a pointcloseh~u a

corresponding norm~ ~n'J.tJ3o~ui.Hccof thé plane itself,

e

Thé operation of the plane is to doubte thé effective pressureswhich oppose thé expansion and contraction at thé .so~.ce'ed~ double totalenergy and since this

energyis diffusedr'~ only space, ~y~

~~(~ ~1~ amplitude,or potential (~ 2.I~),

Wc will now.upposc that instead

of~=0,the prcscribcd

condition at tho infinite pL.ne is th.t ~=0. In this case thefictitious distribution of on the second side ofthe planernust s

If fT"" of on first side, so the sum ofvalues at twocorresponding points is always zero. Tins .ocurcsthat on thé plane ofsymmctry itself shall vanish

throughont.Lot us next suppose th.t there arc two parallel surfaces

~h~r small ~~1 and ~'at théo~n~ second is equal opposite to the valueof on the first. In

crossing thcre is by (2) finite changein the value of to thc amount of but in

crossing théd7& 'esame finite

change occurs in the reverse direction. When is

reduced without Iimit, and replaced by will thédit

same on the two sides of the double shcet, but there will bediseontinulty in thé value of < to thc amount

of At thosame time (1) becomes

po~or~ ""T"' sign ~"rfacc-potentiaLPositive on the one side and native on thé other, due to theR.II.

7

Page 110: Lord Rayleigh - The Theory of Sound Vol 2

98 SPIIERICAL WAVES. [278.

action of the forces at <S. Thé direction of must be under-

stood to bo ~o~e~'f~ the side at which <~ is to be estimated.

279. The probleni of sphefical waves diverging from a p&!nt

has aiready been forced upon us and in some degree considered,

but on account of its importance it demands a more deta.iled

treatment, If tlie centre of symmctry be taken as pôle the velo-

city-potentialis a funetion of )- only, and (§ 241) reduces to

~+~ or to~r.

Thé equation of freo motion (3) §273</7' r a?' ?' ar

As in the case of one dimension, thc first term represents a wavo

advancing in thé direction of r increasing, that is to say, a diver-

gent wave, and tlie second tcrm represents a. wave convcrgiug upon

tlie pole. The latter does not in itself possess much interest. If

we confine our attention to tlie divergent wavc, wc h:t.vc

the same relation as obta.ins in thé case of a plane wave, as might

hâve been expected.

If the type bo harmonie,

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279.] CONTINUITY THROUGH POLE. 99

If a divergent distance bc conflucd to a sphencal sheïïw.t~r.: ~iLhcur. ~L t! udthur condens~ion norveioctt~ tho chamctcr of the w~e M linutcd by remarkable re-lation, first poiuted out hy Stokcs'. l, From équations (4) wc have

shewing th~, thé v.~ueof/(~) I, the s.-unc, viz. zero, both

in.s.de and outside tbc she]] to which t])e w~ve is IImited. Henecby (~ if M and bo radii less aud grcater tliau tlie cxtremoradn ûf tho shel!,

wnich is thc expression of thé relation referred to. As In § 274we sec that a conden.scd or a mrencd wave cannot exist alone.'When thé radius beeomes gréât in comparison with thé thicknesstho variation of m thc intégral may be negleetcd, and (8) thonexpresses tliat thé ~eu~ condensation is zero.

In applying thé general solution (2) to dcduce thé motion

rcsuttmg fromarbitrary initial

circumstances, we must rememberthat in its présent form it is too gênera! for the

purpose, since itcovers the case in which the pôle is itself a source, or place wherefhud is ~ntroduecd or wit)idrawn in violation of thé équation of

contimuty. The total cnrrent across the surface of a sphère ofradius r is 47n- or by (2) and (3)

an équation which must hold good for ail positive values of thé

argument".

By thé known initial ch-cumstances tho values of M and s arcdetermined for thé timo ~=0, and for a!l

(positive) values of r.

J'/t)7. jt/<!f/. xxxiv. p. 52. 1819.

Tho Mtutiou for sphcricat vihrntions mny bo nbt.nucd without tho usa of (1)by superposition of tmins of piano wavGs. reiated similarly to thé polo, Md tra.veUtnK ont.warda iu nJl diroctions

cymmetrioUly.

7–3

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100 I~ITEAL CIRCUMSTANCES.[279.

If thèse initla.1 values be rcprcscntcd by M~, and wo obttun

f~.m(3)M-!d(~)

by which thc function yis dctcrmincd for at! négative arguments,

and thé function for a. positive arguments. T)ic for)Ti of for

positive arguments follows by moins of (0), and then thé whoïc

subséquent motion is dctcrmincd by (2). Thé form of F for

nogative arguments is not rcquired.

Thé initial distnrh~ncc divides itself into two parts, Irn-veHin~

inopposite directions, in cach of which ?'~ is propa.gnted with

co))sta,nt velocity (?, and tlie inwn.rds tr:ivc)!ing wa.vc is cot~tinua.Uy

reflected a.t thc poJc. Since the condition to be thcrc satisfied is

?'<~=0, thc case is somewhat simHfu- to tha,t of a pa.ra.Hel tube

tcrminated by an <T/je~ end, and wo may thus pcrha.ps botter

understand wloy thc condcnsed w;ivc, arising from tlie Hbo'n.tion.

of a mass of eondensed air round the polo, is ibHowed immcdiatcly

by a wave of rarefaction.

280. Returning now to thé case of a train of harmonie waves

travelling outwards continually from tho polo as source, let us

invcstigato thé conncction between the vctocity-potentia! aud thc

quantity of nuid* which rnust bc supposed to be introduced and

withdrawn altcrnately. If thé velocity-potcntial be

wc !)a.vc, as in thc preccding section, for the total cun-cnt crossinga sphere of radius ?',

when ?- is smal) cnough. If thé maximum rate of introduction of

<))!)(! bc dcnot.cd by J, titc corresponding potential is givoi by(l).

It will be obscrvcd that. when tite source, as mca-sured by is

finite, thc potential and thc pressurc-va)-i:tt!on (proportiona! to <&)are inDnttc at thc po]o. But tbis dncs not, as might for a moment

be supposcd, i'npty an inDnitn émission ofcncrgy. Jfthc pressure

Page 113: Lord Rayleigh - The Theory of Sound Vol 2

280 J ENERGY EMITTED PROM GIVEN SOURCE. 101

bc divided into two parts, one of which lias thé same p!tase a.st).c vclocity, aiid tlie other tlie .same phase as thé acceleration, itwill bc found that tl)e former part, ou which thé work dépends,is ~nite. The mRnite part of tlie pressure does no work on thé

who)e, but mcrcly kecps up tlie vibratiuu of tt~e air immediatelyround t!ie source, whose effective inertia is

indenhitely gréât.

We will now investigate thé energy emitted from a simplesource of givea magnitude, supposing for the saké of greatergeneraHty tfiat tlie source is situated at tlie vertex of a rigid coneof aotid angle M. If the rate of introduction of iluid at thé sourcobe A cos we have

Of thc DgLt-hn.nd mcmbcr thc first tcrm is entirely pGriOfJIc, a.nd

ni the sccoud thc mean vatuc of sm" «:(«<–)-) is Thus in ttto

long l'un

It will bu remarked that when the sourco is given, the ampli-tu<Jc varies iuver.sc)y as M, and thei-oforc the intensity Inversetyas M'. Fur an acute cone tite intensity is greater, not on!y onaccouut of the dunimition in tlie soHd angtc through which the

Cmubfid~o MathcmaUcttI Tripus ExMuinatitiU, 187G.

Page 114: Lord Rayleigh - The Theory of Sound Vol 2

~02 SPEAKING TRUMPET.[280.

sound is distributed, but also because the total energy emitted

from thé source is ~t~eJf increaa.d.

When thé source is in tho open, we hâve only to put M = 4.77-,and when it is close to a rigid plane, <u = 27r,

Thc results of this article nnd un iutercsting application ill thé

thcory of thespeaking trumpet, or (by tl)e law of reciprocity

§§ 10}), 294) Itc~nng ti-uinpot. If thé diameter of the large openend be sumil iu comparison with tl)e wa.ve-Iength, thé waves on

arrivai 8uHcr copiuus reflection, and the ultimatu rcsuft, which

must dépend h).rgcly on thc précise relative Jcngths of thé tubeand of the wavc, rcquircs to he determiucd by a diifurent process.But

by suHicicntIy protouging thccunc, this rdtcction inay bo

duninisbct!, and it will tend to ceasc witen tho diameter of thé

open end inchtdcs a large junnbcr of wavc-tongttts. Apart from

friction it would thei-cfore bc possible by diminishing c.) to obtainfrom a given source

any desired amount of energy, and at thosame thue by InHgthening t!ie cône to sccure thé unimpededtransfureucc of this encrgy from thé tube to the surrounding air.

From the thcory of diffraction it appcars that tl)c sound willnot fal! ofï' to any gréât extcnt in a latéral direction, unless t]iediameter at thé largo end exceed hn.If a wave-lcngth. Thé

ordinary explanation of the eftect of a common trumpet, dcpendingon a supposod concentration of

7-~8 in the axial direction, is thus

untenablo.

281. By means of Eulcr's equation,

wc may casily establisli a thcory for conical pipes with open ends,

Midogous to thM of Bernoulli for pam]Iel tubes, subject to the sameInnit~tion as to thé stnaDncss of thé diameter of tlie tubes in com-

p~isoti witli t]teW!ive-Icn~th of the sound. Assuming tliat tho

vibration isstationary, so th~t p-~ is cverywherc proportioual to

cos /M~ wc gct from (1)

Page 115: Lord Rayleigh - The Theory of Sound Vol 2

281.] THEORY Or CONICAL TUBES. 103

Tho condition to be satisfied a.t an open end, viz., that there 13

to be no condensation or ra.reiactiou, gives = 0, so tha.t, if thé

extrême radii of tho tube bo )\ and we have

whencc by elimjnation of ~4 :7?, sin < (~ ?-J= 0, or =

where ??t is an integer. In fact since the form of the generalsolutton (3) and thé condition for an open end are thé same as for

a paraDc) tubo, the rcsult that the length of thé tube is a multipleof thé

halfwave-length is necessarily also the same.

A cone, which is complete as far as tho vertex, may bc treated

a.s if tlie vertex were an open end, sincc, as we sa.w in § 279, the

condition ?'<~= 0 is there satisfied,

The rescmblancc to thé case of parallel tubes does not extend

to the position of thé nodes. In thé case of thé gravest vibration of

a parallel tube open at both ends, thc nodc occupies a central posi-tion, and thé two halves vibrato synchrououslyas tubes open at one

end and stoppod at the other, But if a conical tube were divided

by a partition at its centre, thé two parts would have different

pcrioda, as is ovidcnt, becausc thé one part differs from a paralleltube by being contracted at its open end where the effect of a

contraction is todcpress thé pitch, wItHe the other part is con-

tracted at itsstopped end, \vherc thc effect is to raise thé pitch. In

order that the two periods may be thé sa.me, thé partition must

approach nearer to the narrower end of thc tube. Its actual

position may ue determined analytica!!y from (3) by equating to

zero thé value ofo'

When both ends of a conical pipe are closed, tho correspondin~notes are determined by eHniinating jl between the cquatious,

Page 116: Lord Rayleigh - The Theory of Sound Vol 2

~04 TWO SOURCES 0F LIKE riTCn. [281.

if y, and ?'j, bc very great, tan'' A-?', and tan'' /<r, :t.re botti odd

multiples uf ~7r, so th:t.t ~-r, Is a. niu!tip!c of À., as thc thcory"f par:~)~ tnh~ t.'tp'ires.

282. If there 1)G two distitict sources of sound of the same

p'tctt, situftted at 01 tuid the vclocity-potenti:d at a poiut7'whofie distances f:'oiu arc r, and ?' may

bcexpresscd

whcre A and arc coc~eicnts rcprcsenting the magnitudes of

thc sources, (which without luss of genorality may bc supposcd to

hâve thé samesign), and N

rcpreseuts tlie retardation. (considered

as a distance) of the second source reiatively to thé nrst. The two

trams of spherical waves are in agreemcnt at any point P, if

~+ 'x ~'t= ± whcrc 7n is an intcger, that is, if P lie on any

one of a systeni of hypcrboloids of révolution ha.ving foci at

and 0,. At points )yh)g on the intermediate hyperboloids,

represented by ?a + af p-~= + (2/~ + 1) tlie two sets of waves

are opposed in phase, and nentraHze one another as far as thcir

nctualmagnitudes permit. Thé neutratization is complete, if

7\ ?', = ~1 ~C, and then thé density a.t 7~ continues pGt-manentIy

unch~nged. Thé intersections of this sphère with thc system of

hypcrboloids will thus mark out in most cases sevcml circlcs of

absoiate silence. If the distance C\ Oj, between thé sources be gréâtm conparison with thc Jengtil ofa wave, and thé sources tljcmselves

bc not very unequal in powcr, it will bc possible to départ from

t)te sphère ?'j :?'~=~1 Z? for a distance of several wave-Icnn'thswithout

appreciably disturbing thecquatityof intcnsities, and thus

to ohtain over finite surfaces several a!ternations of sound and ofalmost

complète silence.

There is sone diniculty in aetun.I]y rea.!isiug a satisfactory Inter-

férence of two indcpendent sounds. Unicss the unison 'be extra-

ordman]y perfuct, tlie silences are only momcntary and arc

co!)sequcnt)y dinicult to appreciatc. It is thcrefore bcst to employsources whicli are mechanicaHy connected in such a way tijat thé

relative phases of thc soundsissuin~ from them cannot vary. The

situp!cst plan is to rcpcat thc first sound by renection from a HatW!t!I (§§ 2G9, 278), but thé cxperiment tbun Joses

somcthin'r in

dircctncss owing to the fictitious charactcr of the second source.

Pcrhaps tlie most satisfactory furm of t]jc experimeut is that

Page 117: Lord Rayleigh - The Theory of Sound Vol 2

282.] POINTS 0F SILENCE. 105

deseribcd in the Philosophical Magazine for June 1877 by myscif."An intermittent olGctric'un'cnt obtaincd froni a. fork interrupter

making 128 vibrations pur second, cxcitcd by mcans of etectro-

ïnagtiets two othcr forks, whose frequeney was 25G, (§§ G3, 64.).Tficse latter forks were placed at a distance of about ton yardsapart, and were provided with suitably tuncd resonators, by which

their sounds were reinfurccd. Thc pitch of the forks was

ncccss!n-I[y ideutica!, since thé vib~tions werc furcc(t by electro-

in~guetic forces uf absulutely thc s.unG period. With one carclosed it was found possible to define thc ptuccs of silence with

considérable a.ccm-iicy, a motion of about :m incb bcnig sufficientto pro<)uce a markcd rcvival of somid. At a point of silence, fromwhich the line joiniug tlie forks subtended an angle of about GO",the apparent strikiug up of onc fork, wltcn the other was stopped,had a very peculiar eH'uct."

Another method is to duplicate a sound coming along a tube

by means of branch tubes, wiiose open ends act as sources. Butthe experimcnt in this form is uot a very casy one.

It often happens that considérations of symmctry are sufEcientto indicate tlie existence of places of sitence. For exampte, it is

évident that therc can bc no variation of density in the coutinua.-tion of thé plane of a vibrating plate, nor in the equatorial planeof a symmctrical sulid of révolution vlhrating in the direction of

its axis. More gcner:d!y, any plane is a plane of silence, with

respect to which the sources are symtnetrictd in such a mannertLat at any point and at its image in thc piane there are sources

of cqual intcnsities and of opposite phases, or, as it is oftcn moro

convcniently expressed, of the sa)ue phase and of opposite ampli-tudes.

If any numbcr of sources in thé same phase, whose amplitudesare on tbe whole as mucii négative as positive, bc placcd on thé

circumfcrcnce of a circlc, thcy will give rise to no disturhance of

pressure at. points on the straight linc which passes tbrou~h thc

contre of the circle and is directed at rig)it aug)cs to its plane.This is thé case of the symmctrical LcU

(§ 232), which emits uo

sound in tlie direction of its axis*.

Thé aceurate expérimental Investigation of acrla.1 vibrations is

bosct with cousiderable diniculLles, wiuch have been only partiaily

'J"/t~(5),m.p..lCO. 1877.

Page 118: Lord Rayleigh - The Theory of Sound Vol 2

EXPERIMENTAL METIIODS.f282.

Eurmounted hitherto, In order to avold unwished for reflectionsit is generally necessary to work in tbc open aIr.wberRd~ate

~pp-u-atus,sue)i as a sensitive namc, is dimcult of management.

Another impeduncnt arises from the présence of tho experimenterh.msdf, w)mse person is large enough to disturb

materially thestate of tlungs w))ich he wl.shcs to examine. Among indicators ofsonnd may be mentioncd membranes stretched over cups, the agita-tion being made apparat by sand, or by small pendulums re~Inghghtiy aga.nst thcm. If a membrane be simp)y stretched across a

hoop, both its faces arc actcd upon by nearly the same forces, and

consequentlythe motion is muchdiminished, uniess the membrane

be hu-ge euough to cast a sensible sbadow, in which Its hinder faco

may be protected. rrobabiy the best mcthod ofexamining t!ic

intensity of sound at any point in the air is to divert a portion ofit by mcans of a tube ending in a small cono or resonator thésound so diverted being !ed to the car, or to a manometric

capsuic. In this way it is not difncult to détermine places ofsilence witli considerabic précision.

By mcan.s of the same k:nd of apparatns it is possible toexamine cvcn the phase of thé vibration at any point in air, and totrace out the surfaces on which thc phase ducs not vary'. If théînterior of a resonator be connected by flexible tubing with amanomctric capsule, Nyliicli influences a small gas name, thc motionof thc namc is rclated in an invariable mauner (dependincr on theapparatus Itself) to the variation of pressure at thé mouth of thoresonator and in particular thc interval between the Jowest dropof thc name and thc lowcst pressure at thc resonator is Indepcndentof the ahsolute timc at which thèse effects occur. In Mayer'aexpcrimeut two fiâmes were empioycd, placcd close togetber in onevertical Iine, and were examined witb a

rcvotving mirror So longthé assocmted resonators were undisturbed, the serrations ofthetwo Hames occnp.ud a Hxed relative position, and this relativeposition was also maintained when onc resonator was moved aboutsu as to trace out a surface of invariable phase. For furtherdutails thc readcr must bc referred to the original paper,

283. Whcn wavcs of sound inlpinge upon an obstacle, arort.on of the motion i.s thrown back as an écho, and under covcrof tbc obstacle therc is formed a sort of sound shadow. In orderhovever, to produce shadows in

anything like optical perfection,1

Mlyor, P;,t/. ~), sLiv. p. 321. 1672.

Page 119: Lord Rayleigh - The Theory of Sound Vol 2

283.] souND snADows. 107

tho dimensions of théintervening body must be considerable.

Thé standard of comparison proper to tho subject is tho w&ve-

length of the vibration it requires almost as extreme conditions

to produco rays in thé case of sound, as it requires in optics to

avoid producing thcm. Still, sound shadows tlirowu by hills, or

buildings, are often tolerabiy compiute, and must be within thé

expérience ofaU.

For closer examination lot us takc first the case of plane waves

of harmonie type impinging upon an imtnuvable plane screen, of

infmitesimai thichness, in which thûre is an aperture of any form,tbc plane of thé scrccn (.v= U) buing paraDel to tlie fronts of the

waves. The velocity-potential of tlie undisturbed train of waves

may be takcn,

If the value of over the apcrturo be known, formula (6)

and (7) § 278 aHow us to catcuh).te thé value of at any point on

the further sidc. In tlie orduiat'y tlicory of (Ufïra.ction, n.8 givonin works on optics, it is assumod that thc disturbance iu thé planeet' thc apertm'e is t!ie sfunc as if the Bcrœt) were away. This

hypothesis, though it eau acvcr be rigorously exact, will sufHce

when tlie aperture is very large in comparison with the wave-

lungth, as is usually thé case in opties.

For the undisturbed wa.ve we have

thé integration cxtcnding over tlie area of tlie aperture. SInco

~=2-n- we sec by comparison with (1) th:bt iu supposing a.

primary wave brokeu up, with thé vicw of applying Huyghens'

priuciptc, ~<S must be divided by \?', and tlie phase must ba

a.cceleratcid by a qun.rtcr of {), period.

Whcn ?' is large in comparison with the dimensions of tho

n.pcrture, thc composition of tlie Intégral is best studied by the a.Id

ofHuyghens' zones. With thc point 0, for which is to be

cstunated, as centre deRcribc a series of sphères of radii increasing

Page 120: Lord Rayleigh - The Theory of Sound Vol 2

108 nUYGHENS' ZONES.[283.

1

hy thc constant ditFerence ~Â, thc first spticrc of thé series beingof such radius (c) as to touch thé pjanc of t!ic scr~-p']. On t')is

plane are tttus markcd ont a. séries o; circ)e.s, whose radii p aregivcn by ~+c'=(c+~\)', or~=~c~, vcry llcarly so that

therings into witich the plane is divided, heing of approximately

equal area, make contributions to cp which are approximately

equal in nnmericalmagnitude and

a)tcrrmtc)y opposite in Hign.If 0 lie decidcdiy within thc projection of the area, tho first tcrm

of thc scrics rcprcscnting titc Intégral is finite, and the tenns

-\vlnc!t follow are atternately opposite in sign and of numerical

jnagnitudc at first nearly constata, but !d'terw!U-d8 diminishing

gradnally to zéro, as thé parts of thc rings intercupted within thc

aperture become less and less. Tiic case of an aperture, wliose

boundary is cquidistant from is cxceptcd.

In a séries of this description any tcrm after thc first is

neutralizod almost cxact!y hy haïf tlie sum of tliose whicii iinmc-

diatety prccedc and follow it, so that thé sum of the who!e series

is rcprcsentcd approximately by hatf the nrst tcrm, which stands

over uneompcnsated. We sec that, provided a sumcient uumbcr

of zones be ineluded within thé aperture, the value of at tho

point 0 is independent of thé nature of the aperture, and is there-

fore thé same as if there had been no scrcen at ait. Or we maycalculate directly thc effect of thé circle with which thé system of

zones bcgins; a course wliieh will have thé advantagG of bringingout more clearly the significance of thc change of phase which wefound it necessary to introduec when thé primarywave was broken

up. Thus, let us conçoive thc cirele in question divided into in-

nnitcsimal rings of equal ai-ca. Thé parts of <~ due to each of

thèse rings are eqoal in amplitude and of phase ranging uniformlyovur haïf a complète period. The phase of the resultant is therc-

forc midway betwcen tftose of thc extrême éléments, that is to

say, a quarter of a period bchind that due to thé élément at

thc centre of thc circ)e. Thé amplitude of the resultant will bc

less than if aU its componcnts had Leen in thé same phase, in

thé ratio ~sin.<; Tr, or 2 -n-; aud thereforc since thé area

of the circle is TrXr, ha]f the encet of tlie first zone is

thc samc as if tlie primary wave were to pass on undisturbed.

Page 121: Lord Rayleigh - The Theory of Sound Vol 2

283.] HUYGHENS' ZONES. 109

When the point 0 is well away from thé projection of the

n.pc!n-c, tho rp~dt is fjui~ .~ffacut. Thé scr'CHrq'rc'scnLing t))o

intégra,! then converges at botti ends, and by the samc rcasoningas before its sum is sccn to bc approximately zéro. We coneludo

that if thé projection of 0 on the plane a:==0 fait within thé

aperture, and be nearcr to 0 hy a grcat many wavc-Iengths than

the nearest point of tho boundary of t!)e aperture, thon thé

disturbance at 0 is ncarly thc samc as if thcrc were no obstacle at

a)! but, if the projection of 0 fa)l outside the aperture and be

nearer to 0 by a grcat )nany wavc-Icngths thn.)] thé nearcst point of

thcboundary, t!)oi the disttu'bancc at 0 practicaHy vanisjics.

Dus is the thcory ofHonndrays

in itssimplest

form.

Thc argument is not very di~ront if the screen he oblique to

thc phme oftho waves. As hcfo-e, tlie motion on tho further side

of tlle screen may bc rcgarded as due to thé normal motion of the

particles iu the plane of tbe aperture, but this normal motion now

varies in phase from point to point. If the primary waves procccdfrom a source at Q, IIuyghens' zones fur a point 7~ arc thé séries of

citipses represented by += P~ + where and ?-“ are

the distances of any point on thc screen frnm Q and 7~ rcspectively,and M is an integer. On acconnt of tho assumed smallness of in

comparison with?', and ?'~ tho zones are at first of equal area and

make cqual and opposite contributions to thc value of <~ and

thus by t))G samc rcasoning as hefore we may conclude that at any

jx'int decideclly outsidc the gûornetrica! projection of the aperturethe disturbance vani.shcs, while at any point decidcdly within tlie

geometrical projection the disturbance is thc samc as if thc

primary wavc had passcd thc screen unimpedcd. It may be

rernarkGd thut the incrcasc of area of thé Huyghens' zones due to

obliquity is compensated in t!)e calcuiation of the intégral by the

correspondingly dintinisbed value of thé normal veloeity of the

uuid. Tho cnfccblement of thc primary wave between the screen

and thé point .P duc to divcrgency is representcd by a diminution

in thé area of tbe Huygbcns' zones below that correspondinrr to

plane incident waves in the ratio ?', + ?'~ ?'.

TIicre is a simple relation between the transmission of sound

tljrough an aperture lu a screen and its reflection from a planeleficctor of thc same form as the aperture, of which

advantage maysometimoa bc taken in pxpcriment. Let us imagine a source

sirnitar to (~ !Uid in the samc pliase to be placed at (~ thé t~e of

Page 122: Lord Rayleigh - The Theory of Sound Vol 2

110 CONDITIONS 0F COMPLETE REFLEXION.[283.L.

Q in thé plane of thé screen, and Jet ussuppose th~t thé screen is

removed and repfaccd byap!.tc whoseform and position isexactiythat of tlie aperture; then we hnow that tt~e effect at of the twosources is nnu~ucnccd by thé presence of thc plate, so that thévibration from Q renoctcd from the plate and thé ~hration fromOtmnsm.ttcd round tho p]ate togethcr make up the same vibra-

tion as would be rcccived from (? if thcrc werc no obstacle at a!!Now aecordmg to thé

as.su.nption winch we madc at thc b~in-niug of this section, the unimpcded vibration from Q may boregardod as composa ofthe vibration that nnds its way round théplate and of ti.at which ~ou!d pass an aporturc of Die sa.ne formin an infinite screen, and thus thc vibration from Q as tr~mittcdthrou~h the aperture i.9 equal to the vibration from <2' as reriectedfrom thé piato.

In order to obtain a nearly complète reflection it is not noces-sary that the

reneeting p]atc inctude more tl,an a small numhor ofHuyghens zones. In thé case of direct reflection the radius p ofthe first zoue is dctenniued by tlie équation

wherc c aud c, arc thé distances from the rejeter of thc sourceand of the

point of observation. When thc distances conocrnedare grcat, the zones becumc so large that ordin.~ry wdfs areinsufficieut to give a con.ptcte rc.Heetion. but at more moderatodistances écho. arc

cft.u nea.-Iy perfuet. Tf~ areaneccssary for

con~cte reflection depcuds also upon thé~avc-!cngth and thus

it happens that a hoard or plate, which wou!d be quitc inadéquateto reflect a

bravemusical note, may rcficct very fairly a hi~ or

tlie sound of a high ,vhi,t!e. In experiments on reHection byscrccns of moderate size, theprincipal cliflictilty i. to gct ricl

suSc~ent y uf thé d.rcct sound. Thc ~p~t plan is to reflectthe sound from an

eicctric beit, or other fairiy steady source, roundthe corner of a large buildin~

.un

28~ In thépreceding section we have apphed Huy~hens'

principle to thé case where the primary wave is supposed to bebroken up at t!.c surface of an

in~ginary phu.e. If wcreaiivknow what thc normal motion at thc pkne is, we can calculate

1~~)!. 3/<t~. (5) in. p..1C8. 1S77.

Page 123: Lord Rayleigh - The Theory of Sound Vol 2

S84.] DtVERGING WAVES. m

thé distnrbance at any point on thé further side by a ri~oro~process. For surface other ti.an tlie p!ane the problem ].as no<beciisolved genemlly; nevertheless, it is not difBeu!ttoseethatwhen tlie radii of curvature of thc surface are very grcat in com-

parison witli thc w~e-Icngth, the e~et of a normal motion of anGtement of the surface must be very nearly thé same as if thosurface were plane. On this

understanding we may employ thosame mtegral as before to calculate tlie

aggrogate rcsutt As Ilmatter of convenience it is usually bcst to suppose thc wavc to bebroken up at what is calied in opties a

~e-~r/ac< that is, asurface at every point of which the~Me of thé disturbance is 'thosame.

Lot us considcr tho application ofHuyghcns' principle to

calcu!ate the progress of a given divergent wave. With any poiutat which thé disturbance is required, as centre, describe a séries

of spheres ofradii coiltinuaHy increasing by the constant dinfercneetlie first of tlie séries being of such radius (c) as to touch tho

given wave-surface at C. If 2i; be the radius of curvature of thosurface m any plane through 1' and C, the

corresponding radius pof the outer bouudary of the zone is given bv the ennnHnn

If the surface be one of revolution round thé arca ofthc firstn zones is and since p2 is proportional to M, it fullows that thezones are of equal area. If tlie surface be net of révolution théarca of the rirst zones is reprcsentcd ~p'f~, where is théazimuth of the plane in which p is measured, but it still i-eniainstrue tliat tlie zones M-c of equal arca. Since by hypothesis thcnormal motion docs not vary rapidiy over ttie

wave-surface thédtsturbances at P duc to thé various zones are nearly equal inmagnitude and

alteriiatoly opposite in sign, and we conclude that,as in thé case of plane wavcs, thé aggregate effect is the haïf ofthat due to thé first zone. Ti~e phase at Is

according!y retardedbehmd that

prevailing over thé given wave-surface by au amount

corresponding to tlie distance c.

Theintensity of thé disturbance at P depeuda upon the area of

Page 124: Lord Rayleigh - The Theory of Sound Vol 2

112 VARIATION OF INTENSITY.[284.

thc first Huygtiens' zone, ïmd upon thc distance c. In thé case of

svmmetrv. wo havo

'n*~ 7r\ 7i~

'c"+c'

which shows tliat tl)0 disturhancc is less than if R were innuitc in

tho ratio J~-t-c J~. This duninution is thoefïcct of divo'gcticy,

and is the samc as Avould bc obtfuncd on the supposition that the

motion is !unitt;d by a co)nc:d tube wttosc vertex Is a.t the centre of

curvature (§ 2CG). Whcn thé surface is not of révolution, tho

v:due of ~"p~M c may Le expressed in tGi'ms of thc pnnc'ipa.!

radii of curvaturo 7~~ and with which is connectud hythe

relation

so that tlie amplitude is diminished by divcrgcncy in thé ratio

~/(jf~ + c) (7~ + c) ~Jf~, a. rcsutt which might bc anticipated by

supposing tite motion hmitcd to a tube formed by normals dt'awn

through a sn~U coutour ti'ttced on thc wave-surface,

Although we !)ave spoken liitherto of divcrging waves only,

thé preceding expressions )t)~yalso bc a.ppHed to waves converging

in one or in buth of the priacip:).! planes, if wc a.tta.ch suitable

signs to and 7)~. In such a, case tlie arca of the nrst Huyghens'

zone is grea.tcr than if the wavc were plane, aud the intensity of

thé 'vibration is correspondingly increased, If thé point jP

coincide with one of thû principal centres of curvature, the

expression (2) becomcs innnite. The investigation, on which (2)

was fuundcd, is thon insurhciolt; ail tlint we :n'e entitled to afîirm

is that the disturbancc is tmtchgreatct'

at ~'than at othcr points

on thé samc normal, that thc disproportion incrcascs with thc

frequeucy, and that it would becoinc infinité for notes of infinitcly

high pitch, whose wavc-tcngth woutd be negtigible in comparison

with the distances coneemed.

285. Huyghens' principle may also bc applicd to Invcstigato

thé reflection of souud frum cnrvcd surfacctj. If thc materia.1

surface of thc rL'ncctoi' yicidcd so compictely to thé aërial

Page 125: Lord Rayleigh - The Theory of Sound Vol 2

385.] REELECTION FROM CURVED SURFACES. 1]3

pressures that the normal motion at every point were the same as.1 won!.) have been in the absence of thc reHector, then the soundwaves would pass on undisturbed. Thc retlection which actuallyensues when the surface is

unyielding may thcreforc be re~ardedas due to a normal motion of each élément of therenector°erma)and opposa to that of the

primary waves at thé same point, andmay be nwest.gatcd by the formuh. propcr to

p).nc surf~ecs in thomanner of thc

preccding section, and subjcet to a simiL-u. limita-tion to the relative magnitudes of thé

wavc-Icngth aud of thootitcr distances concerncd.

Thé mostintcrcsting caso of reflection occurs wbcn the

surface is so s))apc<! as to cause a concentration of rays upon apart~tar po.nt (P). If t),,

ori~naHy from a simplesource at (2 and thé surface be aneHipsoId of revolution havL

itsfoc~aW

and ~t].e concentration i.s compfete, the vibrationre ected from

cvery eicme.t of thé surface being in the samcPb se ou arnval at <2. If <? be

innnitely distant, so that thénc,dcnt

waves

are piane, t),c surface becomes a paraboloid havin~its f.cus at P and ~ts axis para)!cl to thé incident rays. We mustnot suppose, however, that a

symmetrical wave diverging fromO.s converted hy rencctioa at thé cHipsoi.hd surf.~e into a

spher~calwave

converging symmetrica!Iy upon P, in fact it iscasy to see that thé

ii~tensity of théconvergent wave must he

~erc.tNevertheless, when the wave-

len.this

very small incon~parison .vith t!~ radius, thé dinereut

parts of théconvergent wavc hccotne

approximately indepeudeutof one another, and theirprogress i.s uot materially aneeted bythé faiture of pcrfcct symmetry.

Thé mcrcase of Juudne.ss dne to curvature dépends upon thearea of

rd)cct.ng surface, from which di.sturba.K.cs of unitbnnphase arrive, as

comparud wit). thu area of t!.e hrst Muy<d.ens'.one of a

piane rcH.ctor in thc samoposition. If thc distances of

t).e rencctor from thé source and fro.n thé point of observation becons.dcrahfe, and thé

wave-Jcngth bc nul very sma)!, tj.e nr.stMuygbens zone is

atready ratherlarge, and tbcrefore in thé ca~

of a reOector of moderate dimensions but Httie is muned by

~d~

it concave. On t!.e other i~uaL in laboratory expérimentawhen the d..stances are moderato and thé .sounds en.ptoycd are of..=h p~ch. thé

ticking of a watch or thc cracking of etectricp. L.s concave reHectors are very enicient and give a distinct cun-ccuttatton of somd on particu):~ spots

R. JI.8

Page 126: Lord Rayleigh - The Theory of Sound Vol 2

114 FERMAT'SPRINCIPLE. [28G.

28G. Wc ha.vc secn that if a ray procccding from passes

aftcr rcncction a.t a. planeor curvcd surface throngh 7' thc point

7t'atwhici< it. mcots t]tc; .surface is dcu.:r!nmcd hy thé condition

that ~/?+-K.Pis a minimum (or in sonc cases a m:).xinium).

Thc pointIl I.s thcn thc centre oftitc systcm of Hnyghcns' xonc's;

thc amptitudcuf tho vibmiion at

.<tc'pon)s npon thc arca of tLc

first zone, and It,s pha.sc dc))(;)n)s upon thf di.stmtcc ~~)'+ ~7~. If

thet'e bo uo puinton t)K; mn'facc of Utc runccto)', for which

()/~+ 7~~1s a maxitunm or a nlininunn, thé .systf'm of Huy~tcns'

xoncs bas no centre, fmd tito'c is no ray prucc~din~ from Q w])ich

an'ivcs a.h aftt;r rcHcetion fron thc surface. In )ike tnanno' if

sound bo n'ncctcd ])iorc than oncu, thé course nr a rny is dctcr-

mincd Lytin' condition that its ~)H))u Itj-n~'t!) hctwccn any two

pointsis maxinnnu or a ininitnmn.

Thc same prioci{))c may hc apnik'd toinvcsti~atct.hc r~Y/e~'o~

of sound in a mcdimn, whosu )ncchanic:d pr<~pcrt.ics vary gradua~y

fnnn pointto point. T!)u variation is

supposcdto bc so s)o\v

that no s~nsih)u rcHcotiou occurs, and this is nut incf'nsistcnt

~ith ducided l'tTractionof t))u rays in tra\'L'))it)~ distances winch

im.')udt.! fL vcry ~rL'at, !tnnd)(.'r of wavc-)t-n~t!).s. It is évident

that \vhat wc arc n<'w conf'crnod with i.s not. mcrr'iy t))C )cn~th

of thé r!'Y, hnt a).sn tho yotocity \it.h ')nch thc wave trav(.s

a)on~ it, inasnnu'!) as this vetocity is no )o))~'r constant. Thc

conditinn to hc satisth'd is that tho time occuph'd hy a wa\'c

in tra.v~itin"' :don~ !t. ray )x't\ct.'n nny two points s)):d[ bc

m!txi<nu)H '~r a mini!!]un) so that, if bc th< votooity of propa-

"atiunat any p"int,

:U)d (~' an L'iemunt «f thu Icngth of t)~cr~y,

~hc condition ]nay ~c cxprcsscd, Sj't~'t/==<). Tins is FcDnat's

principtcof tcast tum;.

T))o fnrthor duvelopomentof this part, of thc suhjcct wou.)d

Icad us too far into t]ic domain of ~coniL'trica] optics. T))C fundi).-

mcnta) assu)npti"nof thn s)naUn(;ssoft))cwavL'-h'n~th,onw!)i<')l

thc doctrine of r:'ys is hui)<, havin~ a far wIdcrappHcation tu thc

phcnorncnaof t~'ht than to thosc of .sonnd. th<j task of<h'vc)(~pin~

ils consc'qucnces nmy propt'rtyhe tuft to thc cnitiv.'t.tors of thc

sister sc'h'ncc. In t)~' fo)to\vin~ scctiotis thc m(;t))od.s uf optics a-rcSiHII' seic'uco, Irl tl Il! flJlIIII\'ing sudiolls the nwtJ¡ods uf' optics a!'c

an~hcdtoonr' < tw<~ isoiat~'d qut'sti~ns, \)to.s<; acoustica! intcrcst

issutnri~nt (odcm.'nh! th''irc~nsid.'r:Ltion io <))cprésent

\vork.

Page 127: Lord Rayleigh - The Theory of Sound Vol 2

287.] WnrsPERINOGALLERIES. 115

8

287. Onc of thé rnoststrikir~ of thc phenomena conno~ted

withtbcpr')~tionof sound withi'ido- hui!). isL.h~

prescnted by ".vhispering gai.-ries," of which a ~ood ~nd easityaccess.b!e

cxampie is tu be found in thé cirodar~aiïcryat thé ba~

of the dôme of' St Pau)'s catt.edral. As to thé précise modo ofact.on Mou.stic:d auD.oritics arc not

c-ntirc.Iy ~.rreed. In theopinion of theAstn.M.ncr

Roy:d' thc eft-t is to"bc ~cribcd torcHcct.on fro.n t],c stu-faco of the dôme ovcrh~ut, ftnd is to bnoh.scrvcd at thé poh.t of

t.heg.-dtcrydiametricdty opposKo to t).es.n-ee of .sou..<). Evcry r~y procccding f-om a niduint point fmdrcHcctcd from thc surface of a sphcrica! rcHcctor, will aft(Tre~.enon .t..r.s~t that diamctcr of thc

sphorc which contins tho

'antpomt. This.)iain<.tcr is infect ad~graded fonnofoneof

t))etwocin).sti<'surfaistouchai hy.sy.stemsofray.s in~c-nora),

ht')ngt)~iodofthuœntrcsof principal cnrvaturcoftftosm-fac-ctotlie

my.s a.'u nor.na). Thé concentration of rays on oned.amcter Uu.s cf}~t(.d, doc-.s not

rcquirc thcproxinn'ty ofthe

radtant point tu thcrcHL-ctm~ surface.

J~d~ing fro.n .somc observations that I ])!wc madc in St Pau!'<!

wh.spL.rn~ga)Jury,I an) dispn.scd to think thatthe principal

phunon~nonistobc.p)aint'd.somL-w)tat.tiff..runt)y. Tbca))-

~nnat jo,,dnc..ss withwhicha~hi.sp.ri.sh-.ard i.snotconnn.-d

to thcposition diamctricaify opposite to that

occupicd hy the

w)..spcn.r, and thcrL.Jorc, it won!.)appuar, doo.s not

donendmaturndiy npo.i thc

symmutry of thc dôme. Thcwhisp~.r secins

tocrccp rom~ thc

~aHcry itor~ontatfy, notncce~ari)y ainn~ the

si.ortcrarc. but rath~-aton~ that arc towards w)iich thc

whispcrcrh'(~.s. Thisis

:Lco))s~ptun<-cofthcvcrynnc(p)atau()ihi)ityofft.wh.spcr in front of' and bchind thc speaker a phcno.nci.on '~bich

mayca.sdybcob.scrvcdinthcopcnair'.

Lct ns considcr tho course of thurays divc~i.~ fro.n a radiant

po.nt .situated ncar thc .snrfarc ofarcOcctin~ .spt.crc, and fut lis

dénote thc centre of t)tc spi.crc by and thc dian~.tcrpa.ssin.ï

tt.n.ugb by .ij', so ti.at~ is thopoint on the surface ncares~

<) J) we ux our attention on arny which issues fro.n P at an

au~e ±~v.th t).et~cnt plane at ~,we see

tbatarierauynu.nber of renection.s it continues to touch aconccntric

sp).erc ofrad.u.-j ~~cos~, so that the wi.oie conicat pcncil of

rays ~hich

'Airy(;S',<x;t,2)H!(..]i)i~n.].s7!,p.lJj.

~t/A~.(.)ut.t)..l.').s,)s77.

Page 128: Lord Rayleigh - The Theory of Sound Vol 2

116 WIIISPERING GALLERIES. [287.

originaUy makc angles \vith the tangent plane at ~4 numcricaliy

Icss than is over aftcrwfu'ds inf'tuded betwccn thé reHecting

surface and t!~at of thc conecntric sphère of radius 0~ cos Ti)o

usuat divergence in threc ditncnsions cntailinga diminishing

intensity varying as is rcptaced hy a. divergence in two dimen-

sions, hke that of waves issuing from a source situatcd between

two parallel renecting planes, \vitb an intunsity varying as ?'

T)ie less rapid cnfccbicnicnt of sound by distanco than that usuatty

expcrienccd is tite luading feature iu thc phcjioincaa of whispering

ga.nerics.

T)~ thidtness of thé shcct included betwccn thc two spherea

becomes ]<ss und )css as -/1 approachc.s and in thc Ihniting case

of a radiant point situatcd on tnc surface of tbc rcHcct.or is

cxpresscd hyC~I (1-eos~), or, If h(; sn~)), ~Oj-1 approxi-

]uatc]y. Thc soHd ang)u ofthc p(.'nci),whicb dct.ct'mincs thuwltoio

amount of radiation in thc shcet, is 4-n-~ so that as is

diminishcd wit)iout litnit t)tc intunsitybecontjs inrinitc, ils coni-

pa.rcd wit!i thc intcnsity at a nuitc distance from a shai!ar source

in thé opeu.

It is évident ihat this clinging, sn to spcak, of sonnd to tho

Rurface of a concave waU docs nut dépend upon thc exaciness of

the sphcncal foi'tn. But in thc case ofa. truc sp)icrc, or rathcr of

any surface symmctnca! witit respect to ~1~1', thcre is in addition

thcother kind of concentration spoken ofat thc commencement of

thé présent section which is pccuiiar to t!)c point ~f dianietricaHy

opposite tu thé source. It is pro~ahte that in thc case of a nearly

spherica! dotne like that of St Paut's a part of thc obscrvcd cuect

dépends upon thc symmctry, though perhaps thé grcater part is

rcferable ëimpiy to titc gênerai concavity of tttc walls.

Thé propagation of earthquake disturbances is probably arfected

hy thc curvaturc of tbe surface of thé g)obe acting )Ike a whisper-

ing gaiïery, andperhaps

cven sonorf)US vibrations generated at thé

surface of thé !and or water do not entirely escapc thé same kind

of inHuencc.

In connection with thé aeoustics of public buildings there are

many points which. still remain obscure. It is important to bear

In mind that t))C loss of sound in a singie renection at a smooth

waïï is very smaU, wbethcr tlie wai) be ptane or curved. In order

tu prcvcnt réverbération it may oftcn be necessary to introduce

Page 129: Lord Rayleigh - The Theory of Sound Vol 2

S88.J RESONANCE IN BUILDINGS, 117

i. Jcarpcts or hangi.igs to absorb the sound. In somc CMCs thé

présence of an audience is found surncient to produco Die d~ircde<!ect. Jn t))o absence of a)l dcadening matcriat H)e prolongationof sound may bc very eonsi.tcraldc, of' wfticii perhaps thc most

striking ex:L)nptc is that ~urdc.t by thé Baptistcty at Pis~ ~herothé nuttj.s of thc commoti chord sung consccutivdy !nay bc Lcar<t

ringing on togcthet- formany sccorKis. AcconHng to Henry' it is

iinporL-mt tu prcvcnt thu rcpcatcd rencetion of sound baekwards

~nd forwards along tJK! of a h.-dt Int(.ndL.d for pnbiic speak-n'g, w))ic)i may bc

acconpHshcd by suit~hty piaeud objiquosurfaces. I~ tbi.s way tho munbcr of rcHcctions in a given tinie is

iucreasud, :utd thc unducprutongation ofsouud is checked.

288. Aimost thoon!y instance of acoustical réfraction, which

ha.s apractica! inicrest, is t)ic déviation of soaorous rays from a.rcctihnuar course duc to

bctcrogenuity uf thc atlnosphorc. Thévan:Ltion of prcssm-c at diffurent levcl.s (tocs uut of itsctf givc riseto rufraction, since thu vu]ocity ofsou.td is indcpcndcnt ofdo.sity;but, as was first pointcd eut by Prof. Osbornc. Rcynoids', thc caseJH dtfTcrcnt witb tite variations of

température whicb arcusuaHy

to bc met with. Thé température of t)~cat~nosp)~ere is determined

prineipaHy by the condensation or raréfaction, which any portionof air mustundcrgo in its passage frum onc )cve) to anotbcr, and

its tiormal state is t.neof'convcctivecquDibrium' rathcr thau of

uniformity. According to this view thé rchtio.1 betwecn pressureand dcnsity is ttuit exprcsscd in (U) § 240, and thé velocity of Soundis given by

if r. bc tlie vulocity at thc surface. Tho corrcspottding rcht.tion

'h))f'r.~Mo<t-oc.l.s;n,p.ii{)_

~7'n)<'f-~tH~H/<t.f<y.Yni.xx!i.p.C31. 187t.

T))omnu)), ~t f; r~fn'e~'M c~Mt;t&)'tMM or <emp<M u! the (Kmo~/t~..u<)))e/tM<tT~m())'r.18(jI.–C~.

Page 130: Lord Rayleigh - The Theory of Sound Vol 2

118 ATMosniERic nnr~CTioN.['288.

bctwccn tcmpct'n.turc! :U)d c)cvatiun obtainciJ by mcansof équationriUlS2Ki!s

v'ho'e is thé tonpcratnrc a.<- t))e sm'fncc.

According to (4.) tl)c fa]t of Ictupc-raturc wouH he about

]"C.~))t. in ~;{(H'c(.t,whit;t)<Iuu.S])(~d!Ht'r)nuc)t front t!)~rc.su!t,.sof

(!t:d.s))('['s b.dtoouoh.st'rv~tiuns. W))L'ttt.hL'.skyi.s('JL!:L)-,t])cf:(.))oft~)nj)c)'!du)-edu)'i)~ thedayis mot'c t':)pi([t,))an\v!)(')tthc nkyis

c-h)u'ty,})uLt('w:L)'dssu)ts<tth<'t('ni))~)-at.ut'<jbt'c<)))K's:tpj))'()Xnn:Lt(;)ycu)tst.!))tt'.

l'i'ub:tb!yot)c)uarni~tt.sitisuf):unw:tr;nurabuvc:th:).n

b~uw.

Thc cxplnna<)()n of ncnust!c:)t rcfrnctmn as(~-pcnth'nt, upon a.

va.)'i:ttiun oftt't~pcrnture with

!K;i~))t is :d)n').st(jxactiy

thu .s:unu as

thi~t of'thcoptic'al phcnnniRnon f)f mirage. Thc euryn-ture (o"') uf

a ray, v'husu course is appfuxhnatdy !)nrixont:i), i.scasi)y cstitnatud

by t)ic tnctimd givc'n hy Prof. Jiuucs T))om.son". Not-nud phmcsdmwn at two cunscentivc points a)t))i~ thc ray mc'ct at, L))u ccutre uf

curvature aud a.rc tangcnt.I:),! to t))~ wave-surfaœ in its two con-

Stjcutivc positions. TtK'p')rti()nsufraynatt.c\'ations2:ands-+8~

rcspcetivciy int~rccptcd behvccn thc mx'ntat pianos arc to one

anutt~r in t)K; ratio p p- a))d atsf), sincc tm'y aru dcscribcd

in tliu ~nitj tituc, in Uiu ratio r F+ 8K Huncc in ttiu li)nit

In tbc nonn;).! st:ttc of thc attnrtsphc'ro a r~y, whidi stn.rts

))()n/ont.:L)ty, turns~-mduidty upwn.nLs, !Ut(t :tt a suthcicut ()i.st:t.ncc

[):s~so\'u[' t,huhc:)df't'a)tuI)s'')'c't'w!)osL;st:()))i.satthcsfui)c

ic\'(.-fnst.i)c.suu)-cc. Ji:'L)K!suu)'C(3 bu (;t~;v:i.tc(),t.))C sound isltc!u-(t

n.tt))c.surface utt.)~c.'u-thby!)-n.).HSof:trayw))ichst:u'tswit)i

()ownw:u-() inctitt~tion; hut, h' ))uth thc oh.survcr and thc

suu)'C(jLn()))tht.surf:)cc,<.)tcrcIsnodi)-uct)'ay,:uidt)tcs()u)i(~s

hcin-d, if'ttta)),by]nc:u)sof'<]i<}'ra<"t!on. 'J'hc observer maythcnbc .said to bu sit.uatcd iti a souxd shadow, n)L)iou~)i th~rc

tnay be

no obst:ic)c lu titc direct liuc butwccu biuMuIf a.ud thé source.

Accordm~ to (3)

'A"(ff)tr< Sept. 20, 1877.

'Su(!KY('rttt.,0;;i/tt~ttMn/'j1/)r<~< ~tf<))ï;.v.p~.t~j.218.

Page 131: Lord Rayleigh - The Theory of Sound Vol 2

288.]C'ONVECTIVE EQUILIDRIUM. H 9

or thc ramus of curvaturc of ft hnrixnntid my is n-Lont ton timcs

thc ])(.'i~))Lthn'ug'h witich:), b()<)y)nnstf:dtu)hk'rthn action off

~).vity ni onh'rtn ;).('<[))))'€! :i,\'(.')(jcityc(j))~I to thevc~ocityof

s<'u))().Jt't)tCt'tt;v!iti<)nsut'<))û()!).s(')-v('rnmtnf<hcmn)rccbcz,

n.~d .?.~t)H:)'e:).t.L'st())ntimcuatw)ti(;)i thus~md c:mbe ttc~t'd

cLhei'tYisct.itu.nbyditt'i'it.ctujnI.s

Tt is not to hc suppose') that thc condition ofthc fdmosphcrc

I.sidways.su('hthatthc'r(d:tti('nhL't\vcenvu!oeityandc]nY:).ti')nis

thatcxp)'L'.sscdin(~). '))ui)thtj.su)) i.SM])ini)~,t,])c variation ot'

tt']))))(')'atu!'c upw:u-d.s is ]uurc]-:q)i(); ontLcot.ht't'Itnnd.as.Pt~f'.

]!~yno)ds h:ts rctnm'k~d, whcn j-ai)i is f.;).)jit)g', a niuc!t s)o\vcr varia-

tion istobu t~xpccLud. Jnt)ica)'eticrcg'if)))S,))c.'rc'thc t)ig!tts

nrc]<)))g'a))(I.sti)),ra()iat.i<~nn)ny])avc tuut'c influence titancon-

vcction i)t(]ut(;i')ni)ii)i~ tJjCL'quiiibnum uf tcinpcratum,n.nd ifsothe

propi~itinu nf .soi)))t! in :t. )H))'i/out~t (Ht'cct.io)) wou)d hc f:t,vourcd

by MK't)pp)'oxi)n:i<.t;!y i.sotht;ro):t.I o~diMun ui' t.])c atino.sphcrc.

Tito grnt.'nd dift'm'c)iti:U équation for thc p:).t]f uf :t ra.y, wltûu

t))C surfaœs off.~uat vctt'eityat~ p;u':dt(.'l phLHû.s,i.s readityoljt:)!ned

fron thc ]:t\v of silcs. It'~ bc thu at)g!tj of incidence, ~–sin is

notnttcrcd by:t. r('f[-a<;ti!)~su)-f;)ce, fmdthcrcfhrci)). titccaso

supposcd rcmain.s constaxt .'don~ titu \v)io!c course ni' n. my. If x

Lot.h(jl)0)'!xonta)co-urdu~tc,and thc constant value of ~–siu~bu calicd c, wc ~ct

Page 132: Lord Rayleigh - The Theory of Sound Vol 2

120 PATII OF A RAY.[288.

or, on cfïueting théIntégration,

in which Fmay be cxprcsscd in terms of~by (3).

A si)np!cr rcsult will bc obt:uncd by ti~ing an n.pproximn.tcfur)n ot'(~), which will

beiLCcuratuutiuugh torc-prc.scnt tho cuscs

ofpmctica.1 intùrust..Nu~tucting tiic square aud Itigitcr puwurs of

s, wu m:t.y take

thc ori~!n of .f buing takcn so as to correspond with ~= c, that is

at thu place wi~re thc my is Lurixuutat. ExprusHnj"- ~iu tenua

of~,wc~nd

Thc path of each ray is theruforc a catenary whose vcrtex is

9l/downwards thé liuear parajuctcr

Isaud varies fi-om

~('y-i)c C

ray to ray.

289. Anothor cause of atmospheric rcfmction is to be found

in tlie action uf wind. It Las long bceu known that suunds arc

gencraUy bcttcr Ituardto lucward th:ui tu windward uf thc source-but thc faet rumaincd uncxpjamcd utitit Stokus' pointcd out thuttiie

incrua.si))g vetucity of t.hu wind ovurhead mnst interfcru wit)tthu ructilmcar p)-up:)gation of sound i-ays. Fro)n Fcrtnat's law of!ast tinic it fo)Iuw.s that tlic course of a ray Ifi a movin", but

/~t'<1~. ~'< ma?, 22.

Page 133: Lord Rayleigh - The Theory of Sound Vol 2

289.] REFRACTION DY WIND. 121

otherwisc bomogenoous, médium, is tlic samc as it wou!d be in n

médium, of which a.U tin' pa.rts arc :)t rcst, if th< '/(~ty of

propa.ga.Liou bc inr-rcased at cvcry point by thécomponcnt of

thé wind-velocity in thc direction ai' thé ray. If thé wind bo

borizonta), and do nul vary In the s:uno hurixont:d p)anc, ttie

course ofa ray, wliosc direction is evuryw!icrc b~t slIgtttLy inclined

to t)):i.t of thu wiud, inny be c:dcula,tcd ou t!ie sfune principlus aswo'c app!icd in tlie prcct;ding scctio!i to thû citsc of :), y;t.riabtc

tonpcraL'u'c, thc nonmd vu)ucity ofpropagn.tion at any point being

inct'L'asod, or ditnini.slicd, by thc luc:d wind-vu]ocity, according n.M

thc motion of thc sound is to iccward or to windw~rd. Tbus,

w)icn thcwind ittcrcusesovcrttcad, whicit m~y bclooked uponasthc

uonnal statu of tbings, )torizont:d my tr:LVt.i][ing to windward is

g)~du;d]ybeut upw.n'ds, and at a moderato distance

passesovcr

t!~ hc'ad of an observer; rn.ys tm,vuHing \vittt t)ic wind, on thc

othcr hand, are bcnt downw!U'(.)s,so t)<a.t n.n observur to Iceward of

thc source bcars by a direct my which starts witb asiight upward

mciiuatio)i, !t.t)d ))as tho advantage of buing uut of thc way of

obstructiuMs for thcgrcatcr part.

of ils courue.

Tbc law of tufraction at a horizontal surface, in crossin~ which

tbe velocity of thé windchanges discontinuousiy, is casiiy invcsti-

gatcd. It wiM bcsufîtcient

to consider tbu case in which thé

direction of thé wind and thc ray are in thé samc vertical plane.If0 be thé augic of iucidcnco, which is also tbe angle bctwecn t!ie

pliulc of thé wavc and thé surfilée of séparation, be thc velocityof thc air in that direction wbicb makes tbe smaUer an'de with

tbc ray, and F'be thc commonveiocity of

propagation, thé vulocityof thé trace of thc

planeof tbc wavc on tbe surface of sépara-

tion is

wLich qu~ntity Is unclianged by tho réfraction. If thereforc ~7' bc

t)tu vclocity uf tlie wiud ou ttie second sidc, aud be tlic au"e uf

rctru.ct.ion,

which dirfcrs frorn the oi-dinnry npticnl !a.w. If théwnni-vclopity

v:u'y coutmnou.sty, the course oi' n. my )~~y bL; c;Llculatcd from tlie

condition that thé expression (1) re)na.Ius constant,

Page 134: Lord Rayleigh - The Theory of Sound Vol 2

t22 TOTAL REELECTION I!Y -\vrND. [:289.

If wc suppose that !7'=(), thc ~reatcst ~hmssib)e value of~'i.q

Atastmt.utnwhcrc ~7hasthi.svi).]nc,t!)0(1ircct.mnf)fthumy

whi(')t.s):u'tudt~uia])~]u6))nsh(!('o)nt' pamjiutt.~titcrufracti))~

mu'i'.u~.s, :)))() :).st.)'at)]))i~h(;)-u~'))as agr~~T ViLtuccannotbc

])t;tmt('(tat.!L)). T)tusari)yt)'avc])i)!~))j)\v;))-(].si)).sti)tairata.)i

)tH')Hi:t.<)«n(~7r–~tt)t))(!hn)'i~<))iisr('fi('f'tc<)])y!t.win(tovcr))t'a(l

('t'V(')()C'ity('C('('<iin~~t;t.t.givti)in(~),a)t(] ttti.siodupL'm~'uOyof

~'ttV(.]<)cit)(.s<.t':))<('nnL-()i:t<(j.st.rata.'i'ct.akcatnnncricalcxamph-,

aHray.s w))t).s~- upwant IxctitnUion i.s Ju.ssthan ])", a.ru totaDy

r('nuc[L'()t'y ~i"d~f't!)('s!U)n'a/!in)uthnt<i)!~att!)c moderato

Sj)Q('dofL') !nl)t;.s])ft-])(n)r. 'J'hc (-flirts uf.sm'aa~indonLhc!

]"){)!ati())iof.s«)m(t cnmtotfiLiitohcvt-ryintjxn'tiU)). Ovu)'t.)t(;

nurf.Lceoj'.stii! WiLto- ~nun) )))uvin~ tu )cu\v:trt), ht'in~ con~nc't

~t.;t.\v(;('n paraitt') )'f(!(.'ct.)!)g p!:nh!.s, diverses in t\vo (HxicnsionH

0!~y,n))<!)nay (tn'r~t'cruhc ])c:U(tat (ti.stiUH-s f:Lr~rc:ttc)' U~n

wou)d')t,))(;rwi.suht.;pusst))iL'. Anot)K'i-p<).s.si))]L;t;t]L'cL<)t't!)(-reH(;ctor

ovcrhc.'nt h' )~H()~r .sounds :m(ti))i~ w)nch in HtHt air wontd

bcititcrcujttcd by))i!i.s (~-othuroLst. lotcrvc!)))~ Fur the

pr()(h]c(.i()n'))'t.!ic.s(.;))))).)on)('nait, i.suot ticccssaryt.hatthL'rcbc:d).s<n('c«fwHK) n<,).)h-

.S(')u'cuufs<))tn<t)n(,:).sa))pc:n-s:Lt,(~)ccfr~m tJ.c ff))m of(~), mc-rdy t!)at t.hc(/<ccut' \'L')ucitiu.s U

at,t:)Li)ia,sunit'icnt.v:L)m'.

T))(*()ifrc)'t.'])ti;d 't"t.iuntnt)it.'pat.hufaray,w])(jLithc\nnd-

\'cIocity~i!jconLmuuusIy\u'i:dj)c,is

lucn)np.-n'ing(.'i) ~-ith

~) of (Lep)-(.c~)i))g section, which

tMthcCt)!T<j.spHU(ting (.~nation forordixary rufractio)~, wc must

]-(;nic)nhcT))):~risn(.w c..x.st.at)t.Jf,furt!m.Si~t'ufobtaitiin'ra

(t'imite~su)t,wcsu))jx.su that. U.c huvuf v;u'iatiuu<,f wiud°at

(iitTt.'t'L'nt luvds is titat uxprcssud t)y

Page 135: Lord Rayleigh - The Theory of Sound Vol 2

289 J HEYNOLDS' OBSERVATIONS. ')23

wMchisofthcsnmc formas (II) ofthcp)-HC('d!ngscct;io)i. T)ic

cour.st.'oi'a ravisacc()r()i))if)y!W:)tu))ary"~ti)('p)-f's't('s")d.sn

t"'t.t.)"r(~.sa.most!ntp<)r[:))<Ldist)nc;U()n),~t\t'u)tti)ut\opr~btcms.

Wftunt))C!rcfracti<))ti.softhcor(hnary)u)n),dt;p~ndi))~ upon a

variab)c\'c)()(;it,yofpropa~atiu]),thu()ir(;cti~nufar!)y !nayhcn'vcrscd. lu timcasc

')fat)n«.s]))tL'ri(;(!fr:n;t.it)jt,()ucto:)()i)ni)m-

tinno)'température up\Ya)'(t.s,t.huoo))r.sc ('t'arayisa catcnft.ry,

w))nHCY(;rtuxi.st)uw))wan)s,inw)tic)~crdin.'ctiunt))t'r!y)n:)ybc

prnp;)gatt'<). Whcn thcf'tractionisthtGtuwind.whosc'vcfocity

i)~r(;:).si)))wa)-t)s,a(~r<ti)~tot)K!!a\v cxprcss(;(Ii)t(f!)wit)L/3

po.s!t)\'r-,t~L'))at])<){'ai':)y,\v))().sc(1i!-u('ti<)))i.supwa.)-t),i.sat.s.)a)nng

~catt.)..Nywit)) v~rtGxd-.wnwanIs, buta ray wh~cdin-ction is

duw)<\van[c;u)th.ttra\-L-tai<.ng t))is pat)), h) thutatt~r case thé

Ycrtux ot'thecat~naryalong whicit

Oturaytravci.s is dircctud

npward.s.

~().I"H'(!pap(ThyR..yn()h)sa!rf;a(]yr(;f(-)-rc<Ito,anacco))nt

Ls~i\'t'n')f.S())n(!it)(rn;stin~<Xpurit)tcnt.s<'s))t.;cia!)y()ircct(;t)t()tcs(,thu

theury of rdraction by \vin)!. If wa.s fount) that fn tho

<.)irrcti()nof't)tcwi)u!,w])t;tt itwasstrH)~,t)t(.;sou)H)(~f'anc!c(;tric

bd))cout<))'ch~ardaswL-i)witht)tuh~<]u)tt))c~ronrt(l;).swh~)i

]'ai.s~),(.'v<'nw))(.nina)iuHuw\Yi).htht.ihu)) )u~h~ fr.xn vtewby

t))cs)<)pc()ft))n~roun<);an.) uoa(iva))ta~v!)atu\rw:t.s~m)cd

eith<hyf)s<'(!n()in~tna))(;]~va(in))o!-r:)i.sin~t!tc))<j]). Tiu~,wit)ttht.;

wit)doi.'L'rt!)(j~r:~st)tcso))ndcuu)(.[ hc ))car<) I-t-0 yards, atxl

o\'crH))()W:i!f)<)yiU'()s,<jit))t::r\vitht))L'hcadiift(.'doruntI)c~rou))d;

w)t(TGasatri~))t:m~]<\st('tI)uwi)Klon:d!of'(;!)sionHtitumu<~cwascxtt'n<)cdl.'yraisi)igcit))crt)t(! observer or t)tu bu))."

"KJ(.ation w~ fouudt.<;ta'cctthcra)~cof's()un(]a~amstt))0

wm<)in!t.ntucht)U)rL'tn:u-kc()]n:L)jncrt)):m:d,r!'dtt:U)"'h's."

"Ov~r<))(;~r:tss))(..snn))dc(.)dd))ch(~rd\Yit)tt]tL!)«.~<[ont))c

~oundnt~Oy~-dsfruHi thc !K.)),.t.id :Lt:!()yard.sit.w;ts!ustwit)tthc ))~td 3ic(jt inon t))c gro)md,fmdi(.si))H i))tc))Hit.yw!)s)ostwht-n

standing- cr~t;~:!()yard.s. At7()yi,rds,)~nnta))(ting

crcct.titc.suundw:Ls)nstn.t)ong int~r~d.s,:u)d was

otdyfiuntty

hc'f)rduvcn<I~c));))nt iti)Cf-an)c'c<)]ttmu(jt).sn~inw)~-n thcc'a.r

wnsmi~<tOfL~tfn.rnt)h.~uud,tUtditr~chcditsiu!tiDtu))sityata)tc]t.-va.tiu)iût'12iL'<jt."

Prof. Rcyno)()s t)n)s smnsup thé rcsults of his experimcnts

1. "Wi)oi th(!rc is oowind, soondprocucdit)g' ovcr a rou~h

surface is niorc iutcusc above than hulow."

Page 136: Lord Rayleigh - The Theory of Sound Vol 2

TYNDALL'SOBSERVATIONS

124

[290.2. '~A.s

!nng as tt.evelocity of tbc wind is ~cat~r abovc th.n

LcJ.w, .o..d U~d.)

~c~rd'h" ~ë'~~d, a~d hcnce its

range cxtc,.d.d ut thc surface of t).c gruuud."

At.nosph.ric r.f~tion ),a.s ,u.1in,po,.tant b.ri~ on thc

ud.L.h-.y~j, tlw j,J, ,)'U1lI'S

L.cur.Jtt.c.att.nti.n of twoc..i.cnt

,,hy.sici.ts Pr

~y. ~y

)~

L

it

'T 7~'o.

~~ccul PLcn.nncn. which

T~ ~-rvcr.. wl.ilc~Lfl who.so

.nv.sL.~tions havo b,.encuu~))y e.tcnsivc

~r~?'

n Ly ~ccu~t

ue ~sphcre.smg fr.r.

uncqua! hc.tiug orlatter

~ti.g in~hi.

P o by~"L~d. Ty,M bas

~b~n~ 7~~tric bc!I

~~s d'densitit,s; Ilml, altllOlIgh it lnust hu ndluitted tllltt the al turllatÍ()lIfo!

cOllSl,ll'I'ill)IC illlll IIIO)'(!ahrllht th anean w'c:ll !Je

SlipplJSl'd to occur ill tllc;°l'Ull ail', uxct'pt }JI)l'hap8 in

ot' the suliclgrullnd, SO111C of the

~i" ~i-~My toLULc\j)iai]at)on nj

<)ucsttûn

'rh us it was fcnmcl tlt.vt the ùlast of a sirun lal,vcecl on the

ofbrlulually dillliuiHl1Íng ilitunsity, wllUSC rlurutiuu solllotilrles

ob.sc.rvcd "v-].n t)

mllch~OHH.s. l),).s

phu))u)nc)ton was

SlJ!oothllcss," aud cannot11'h'`~1'clltly IJU o,ttrihutc(1 tu

any otlter cause tluln tlmtasSI~IIC(1 to

Tyn~H. It is ti~ref-Lprob~

acoustic.1opacity arc bot), concorn d i ho

offub-sigl,vls, l1?wiuo we slloulcl

cert;lillly butlisposud to attacll

~=~=~=

suluc of '1`ymi<lll's ovll oùservatiolls mlluit ofexplallation 1111011 t]Jis

~Z:?,~ltJ~l'Jril. L'rmts, l~ï~l. S'uun~l, 8rc1 c<litiuIJ, C'h. YII,

Page 137: Lord Rayleigh - The Theory of Sound Vol 2

290.] ]ON FOG-SIGNALS. 125

principe.A faihn'e in ?'ec~)?'oc:7y can only bc cxplained in

accordance with thcoy Ly t.hc action of wind (§ 111).

According to thc nypothcsis of aconstic c]onds, a difforcncc

mightbc cxpcctcd in the bchaviour ofsounds oflottg and of short

dnration, winch it may hcworth w!iiic to point out hère, as it docs

]tot nppcar to hâve becn notiecd by any préviens writer. Since

cnergy Is not lost in rcficction !).nd rctraction, t]ic intcnsity of

KuHatu'n at agiven

distance from a continuons source of sound (or

Ji~ht) is not a)tcrcd byan cnvcioping c!oud of sphuric'n.) form and of

uniform tlensity, tlle hjss due to t)iuIntcrvL'ning pa.rt.s of thc ciond

hcing compcnsatud by rcncction froin thosc wbich lie bcyond tlie

sonrco. Whcu, howevL'r, thc sound is of short duration, the

intt'nsity at a distance may bu vcry t)U)ch (H)ninishcd hy the ctond

on ac<'o'mt o)' the diH'crcnt dista!icus of its rcfk'ctin~ piu'ts and thc

conHcqncntdrawing out of thc sonnd.ait.hongh thuwitotc intcnsity,

ns mcasnrcd ))y thc tirnc-intc'gra), may ))c thé samc as if thcrc ))ad

ht'cn no c]ond at aH. This is porhaps t!tc cxptanation ofTyudaii'sfthscTvat.ion, t)u(t dincrcnt ]\i)x)s of

signais do not aiways pruscrve

tho sa)nc ontur of ctï'cctivuncss. In sonic statc.s of thé wc-atitC)' a

howitzcr nriog a !))). charge connnandod a ta.rgcr range than tho

witistiûs, trnn))~ts, or syre!)," \v!nh.i on othor days "thc htfo'iority

uf tlte gntt to thc syrcn \vas (~nionstrato) ill thu cicarcst tnanncr."

It shuntd bc noticcd, howcvcr, t))at in thc sanic scrics ofcxpcri-

nn'nts il, \vas fonnd that thc liahi)ity oftin; sonnd of a gnn "to bo

qnonchcd or doncotcd by an opposing wind, so as to bupracticaily

~sctcss at a vcry short distance to windward, i.s vcry ronarkab))!

Thé refraf.'tion propcr mu.stbc t!)c saine for all kinds of sonmts,

lmt for the reason cxptahn.td ahovo, thc diffraction round thc cdgc

ofan obstacle tnay bc h.ss cfï'cctivc for t)ic report of a gun than fur

t)je snstained note ofa sircn.

Another point cxatnincd l)y Tynd:dt was thé inftncnccof fug on

ti~c propagation of sonnd. In spite of isotated assertions to t)i0

eontrary', it was gcnera)!y bciicvud on thc anthorityot Dirham

tbat t])C innncnee of fog was prcjudicia!. TyndaH's observations

prove sati.sfactority ti)at tbis opinion is crroncous, and that tno

passage ofsomtd is favoured by thc homogcncons condition of thc

atmosphère which is tbc usnal concomitant of'foggy weathcr.

Wben thu air is satnrated \vit)i ntoisturc, thé f:dl ot'tonpel'atxrc

with ctuvationaccording

to thc law uf cunvectivceqniHbrimn is

~L'e fur oxtuuplu Dcsor, /«r<it;/trt'«<' f~'r ~«/ xt. p. H17. 1SP5.

Page 138: Lord Rayleigh - The Theory of Sound Vol 2

~26 LAW 0F DIVERGENCE 0F SOUND.[29 1.

mnch ]c.ssraj.id than in tho case nf <!ry air, on accent of thc

Cf'nd.~sati<.nof'vapuur~])i.~t)K.nacc<.tnpanicH(~p:u)si..n. FroniL

~t'<t!atio))hy'r))<))))sf)M'.ta.})p~at.,t)):)t!)., f.~rh. <t.

"f''vap()r;)<it)na))()<~nth'))saHo)two)dd hctodiminish thc fa)) of

t.p<.)-tur(.Ly .,).).n)f: T)u.acu.s<i. n.fr..t<.ti.m(]uetut.n-

p~Ltun.wn.idthus '!f)..ss(.u.),;t)H) in <,t1.r rMp..ot.'j no<)o)tbt

')'n~tLiun<.ft),c:tirwuuh)bc.f~'uur!th)u~thu}u-<.p!~Lion.'fsoutx), pruvid~) nu obstruction WL-ru (.n'd hytito Hus)~n<)cdrartic).st))(.t)).st.v~.

h.afuh)~c!)~)<.cr~.sh:)i)i)tv~ti~.t.-thc

d)st))rba))œofj));mu s..u(,r<u)s wavcshy a .smati u)).stac!u,an(t\VR

.shatI)ir)(tthaL(h..un~-)(t..{,n.!st)))(.nt)t<.n~io..f<.)iC (tiatnctrrof

thuoh.stacit-tuthcwavt.-)~))!)]) of <).< .s.Huni.

lh~)'t'adt'r\\h").s(h..sir<)us(~jn))'sui))~t])is sultjcct tnnycfm-

su)ta]):t)~r)<y]~.yn.,)().s "Ou ti~cKt.tractio)) ni' Sound hy thc

At!n(hsp~.rc'a.wc)t as th..au<)Hn-)t.i<satt~tdyn.~rrudto. Tt

"):tyl~n)(.tin).t.) t)mtR..y)i..)~)~cswith H~nryincf~.sid~

n' rc.f)-acth.))tu!)u()mrc.d)y important cau.s~ofdishtrhanœ, but

fm't)tL'r<jb.~naL[u)).s:tn.'tnuc)tn(;cdu().Suca).so§i!')4.

2f)). 0"IL''assutnptiott ());)(. thudiMturhanf'cat an a~rtorc

n)!inc)'L'('niMt))(~ain(.a.sit.wf)u)<t)tavc:))(;<.natt))('san)rpiaouin

t)K'a~)K'(;ut't))(..s.'rr(.n,w~]nay.su)v..va)-i~u.s))n)t)!~t)i.sr.'sp..t-t.i)~t))~! <)it))-a<').)nn(,f.s<iU))<)

by t)n-.sa)n.'))h')h~d.sa.sarL'cn)pj.)yo()f(.r

t))~currc.spundi,rn).)c.siH},y.sirai «phrs. l-rcxat..p!)i.o

<))~turha)H'(-atadi.sta)~~<)uthct)))'t)tt..rHi.i.- (jfat)in)it)it<-pia)tR

wa~.pK~ed ~itjta<-irr)))ara))urh))-(.on whidtpianu wavusot'

Sounditnpit.~dir~Uy, n.ay hu cafotiah.dasi.i thu a).:d.~ot)s

probk.Utui'Lhc(Htrr.K-ti~)tpath.i~furn)udaLt).cfocu.s()f'acin~dar

object-~a.s.s. Tt.usi..tf.c~.scof'a.syh)mu!.nc;dspL.a)dn~tnu,)})~

t)ics..und Isa maximum a)u)~t))u axis(d't))ciu.sL)-mncnt.v!t(jrc

aH thu(.-)u))h'n)a)'y ')i.sturbaaru.si.s.s))i))~fro)n thc'varions puints

of't!)up]a)tc <t)tL;m.)))t)tarcinon<p))asu. Inohtifpu-tHroc-tions thc

in~n.sit.yi.s ).s.s; ),nt itd~.snotra))m~Tia))ys)t<.rt

of thc rt.n.xi.nnm vahtu unti) tlh;<.)j!i<juity is suc)) that thc

difr.-runcc of distants uf t))c ucar~t fu..) fm-thc.st points of thon~uth :L..tu).nts to about h:df;L

wavc.-tf.n~th. At ns~n~vh~

gn.atcrubiiquity thc'ttout!ttnayh.divid~[intotw<) parts, of

whieh ti~ ncarcr .-ivcs ann~r~atu L-t~ct

cquai in magnitude,

~.V.;)tr/n\<f..).)/)H.,fr.<.1;ir,]-(:

i:jïl~,'A~7'«t;j.<.Yu). !<!< p. :;j. )n7t'

Page 139: Lord Rayleigh - The Theory of Sound Vol 2

~1-] SPEAKIJSTG TRUMPET. 127

butopposite in phftsc, to th~t of t1~ fnrthcr; so th~t the

intensityin t)d.s direction vani.sht.s. In (Hâtions sti!! tu~-coLixjUf. thcs'~nd œvi~.s, !,).)~s tu ..n

ixt~ihitye.jtia! te. ~<7jf f' ~~i,

!L!t.hc:Lxi.s'ain.)i),.i..is].cs to xcro,:Lnd sunn.D.u .r~tions

c..)T<spr,n.ii,~tot).cb.~))tnud<h.rk rmgs w],ichsn.-m).nd tl.e

ce).tndj)atc!t ofJi~M t).<!in~f~ .sh.r. If~'d.-notptho

md.u.sut'<hcniouth,th~u~it

~i.ichti.L.fir.stsik.aœoccur.s is

sin'C!0~. W)K.ntlK-(]i:unctcr.,f<hctnout)t.)oc.St)otcxccc~

~X, t)tcc!<jmu)~t:u-y di.sturhtUK-c.s c..)n))ine witjx.ut any con.sido~bb

anta~.ti.stnnt'pf.aso.nnd thcintc.sityi.su..ar)yu..ifurn) in~)l

'inactions.Itnpp~u'sthit.tconcctttndi~naj-s~mtd :).!ot)~)JK.is

rc.~i~.s t)~t thu ratiu :s).ui.[). ~oo.i'iti..n ncb

u.su.Oy.sati.fiud in <k. ordinary us<jnf.sp~ku~tru)n)K.t.s,wh.se

cHi~~nryd~nd.snLtfh.ruponan in<T~cint).uun~i~d vohunc

<~s(nmd(§~S()). AVh.)..nvcv. tt~vitu~t.ion.sf.n/of~.ry.short

~vo-)..ngt)),fth-mn}.d,ofm«dc)-;)tGsixt.i.sc:)}.:d.h- oft.ctm.rn. v

<Lsidt.ra~~cm.~ntrationa)~~ti,caxi.s,a.s]I~vcmy.sdt'~nficdJ"t)'cc:isu(jf:t.)iis.s.

2f)2. AXhnx~h sn(-I.c;L)cu)atmt).s ns ih~c rd~n~dtoin thc

p~c~di)~(iu)):)rcu.sd)t) ;).s~ivi))~ ~u.di()~ ..fthc

]')),K.)~ .,f diftract.i.m, it tnust nnt h.-~.r~.tt~n t1)at thc

:u)xiii:ny:t.ssnmptiun ..nwi.idt~Hy:n~fmmdcd !shy )«.),)c:m.s

.stn.-Ny:u.d~n..)~)jy <Thn.si))t))(!(-;t.s~«i'i~v:LVLidi)-œt)y

J'x'idd.tup~) a.s<-n.n <hc n~n~d

v..)<xity inthcphmcofthu

ap.'rtumis)~.t.<uthst;mt,:Ls).ash.-cn.supp~s.],hnt mcn~sc-st'rotn

ti)C(~)ttr<jtuw:)n).st!t<'o)~hc.(.<))nmL;-inf!nih; att)tù~i~(;itsc)f.

.i"<.rd~rt.)i.tn..shn~tct))(;(~)tditi..)i.shywhi<-h t)n-tu:J')uf-it.y

~d('~nni))('(),tt-tnsiu)-H)<')n())!it'))ts)JppuHuth!ttt~(j:)p(.rt)))'eis

")'. Ti'e i).(-i<k.).t. w~vu<=eus(~<) i.s th~)

p..rfœtiy

~t)..ctud,nndt)tu vcl«city-putu)itl:d <jn tj)uhc~ttivu.siduut'tfic

sct'L~n(.<;=0)is

= co.s (/<< x~) + cos (/;< + ~.r) (1),

S''vi"g-,whcn.-c=(), <~=2cf.s~. T)tisc.))T<sp.)nd.s t.uthe~mi.sh-

i"g 'jf<))cnor))):dv~tocity m~r thc:))' ~fth~~p~~m~; t))c

cc)npf(jtiuuf)ft)m pr<)b)c)nnv[Hir(.;s tt.s to tk't~rmi)t(! :). \tn:Lb!c

J'u)-m:d v~).K'ity ovur tLu :q.ht~ such that ti)c poUttit~I <Inu to it

(§2~')).s]!:dti))cr(;a.s~-):)y t))uc~n'!t:u)t(p):mt[ty~cus/~iitcn).sni!)"-

'~T~t,7.r~');<t~t~f~t.).)~u(!.

Page 140: Lord Rayleigh - The Theory of Sound Vol 2

128 DIFFRACTION TIIROUOn SMALL APERTURE.[392.

from the négative to the positive Ride; or, sincc the cros-singinvolves simp)y a. citangc of .sign, to détermine :i vfdue of the

n'nialYi-).i)yovcr tlif.irc:a.f{h[)ap.-rtu)'c'whi.f'hsh;~) c~n

tlie positivt. sit)o <p=cos?;i' ovcr thc samc fn-cn.. T)te resn)t of

RUpo-pnsing t)tc two motions thus dL-Hncd salisses aïï tho condi-

tions ot' thc prob)cm, giving thc s:nno vc'tocity imd pressure on t))o

two sidcs oft))c !),po'turc, aud avanishilig norma.lvctocityovcr Uio

rcmaindur of thc serceu.

If T~cos (<~+e) dénote tho value of at tlio varions points

of tho arca. ()S*) of thc apc'rturo, tho condition for detcnuining7' and e is by ((i) § 27M,

whcre ?' dénotes t))G dista.ncc hctwcen t!)c cloncnt fLS* and a~y

n.~cdpoinLhttI~capcrtnrc. Whcn~fH)<I<;arck)-to\vt),t.hecom-

piL't.t't)uuuf<~ fur .mypuint on t]tc positive sit)cuft)tc .sc'rccuis

i''ivcuby

Thé expression ofZ'and e fora fixité i)pertn)'e,e\'cn if of circuler

fot')n, is pn)))!d))y beyond tbLi poWL'r (jf kouwtt mcthod.s; but in tho

cah!Cw))L-)'uthe()in)~t).si()))s:u'uVL-tys)n!).))inc<))np!H'Is()nwith titc

W!t\c-I~n~ththL;.s<))ut.i(j))('ft))tJj))'ub)(.ti m~yLuL'H'tjctcd iurtitc

ct!'c)u amtthu~ilipsc!. If )'bu t.)tc<)ist:mcLibutwu(jtit.wcpoit~s,

bot,h of wbictt :).)'u situ:~tL''t in Uic npci'tm'~ Kr tnny Le ncg)L:ctud,ahd wu tttc'u obtain frutn (~)

shuwin~ that–. p i.sthc dcusityofthc mfi.ttcrwhicit must beo ~7r

distribntcd over ~S'in f'rdRr to producc thcrc thcconst:).nt potcntift.!

uxity. Atn, distance! from thcnpc])ing on thu positive sidewe

j])!)LycotJHidcr?'asco)ist!Lnt,n(!t:)L)\c

Page 141: Lord Rayleigh - The Theory of Sound Vol 2

~-J J HLLU'TXJ APERTUK~. ]29

):. H.

1 :J

1

u'hbrc

~=-~J~'S',dcnotiog

t)to totd<~fUttity

of natter

r¡¡~t.'t.l! :uu~ L-

.u;.}.f.~i~ )~ (ii.iLtKud. ir. wiH oc s)icw)t

1

"n n. future- p)g-ct)nLt fur :utcHip.scufsotmnajnr axis ff,!u.d

ccccntrit'itvc,

j ~Lsr(-su)h!s.juit.cdirfL-rcntfn,.nU)atw).ic]iwcHLou!.)aLt:,i,)

l

'hi:

l'esult,

is fjuite

tlie[ro!l1 tIl1tt which tile

s1101IId

]las 011Z tt"']'y)"jtitosisth:(t,t))on(jr)n:dve]f)(-ity i))thcnpL')'<u)-cha.st.))c

j

\duuprnpcr to t))L-

pri)n:u-y w:tVL-. Li thaL c:Lsc by (:!) § 2~

1T))C(]:m-;t(.ti())]~sm)hd

i.s~.snhj~.t.which i~s nttr.K't.cd!)).).!'tt)' !)t.~))ti(.n <.it,])(.r frum

")nth~)j):)(i(:-i:uiso[-cxpfTi)nL'nt:d!.s<s.

Ai<))0)~htt)c~t)u)-:))c).:u-~(~uft).t-p!K.no.n(.n:Lisw~tI t.ndt.r-

st'd, :md thcr~'orc- no vcry shrHh~ <)).<r<(.ncs :)r<! to b(!

'j['<'ct..< tl)~ cxnct t]H~rcUcit)s<)htth.noraf.of'Utu.sin)j))cr

)"-D)))u).s,w!ii<.)tt))u.suhj.~t.})n..scnts, wo).)JLc i))tc)-~)i)~n.,),< witit tix;

pt-t'.scot impci-frct. )))(.!)ju().s, sr~))(..t))ii).L;'}))'<.haL)y

"ng]tUj(-du))L.inthc\i)y<jt't.p(..ri,,)t_.t)t;Lh'))h):)tiu)).

TLc'va)tH'r)f;Lfn))c)!()))~w]H<-j).sati.stics\7'=<)t))n)U~))-'t titc mh-ri~t- of'~ .si)u)']y-(~)))]<<t(.~] ch~~ sp:n'c ,S' c;))i''h(.

c.\).rcs.s~] ast,).~pot.<~t)i!t!<.i'nj:)t~rdi.sfriL)ttut[nv~r<)tu surfa~

"t~.I"!tccrtai)).sd)scth!s].s:)).~t)'))cof'th(.'d:(s.so['f)))tcti<)j).s

~~L whx'hvc iu-(; now occ))))ic<], w1)!<-)t .s:)ti.sfy \7'(/)+A-=().

i'~uwixg i.sHc)).t)t~)~spr<~r'. Dy(:t~).'stitr<.n.))i, if~

!t!dcn(.)t(-:)nytw'jft)j)<t~)).sf)f;<

77<f'f))')'fy,jr.),/7.<r/;t<')')<H~<');t))~<r''))Mf<f.f'H.'f<7'f~;) r,'(!i.. H.) ,y,.j'.i. ].sf!f).

Page 142: Lord Rayleigh - The Theory of Sound Vol 2

EXTENSION 0F CREEN'S TIIEOREM.[293.

130

whcrcp-~prcsott.st)~distance <.rfn)yp..in<.fr<.)nf).~x<~n)~in~

within At a)I points, cxccpL (~ (1) va)u.s)ic~; :md t)t0 ]~st, tcnn

I't(l)buco)nL's

"i~c))]t~v:m].s)),WtiIia\-t.uipxprL-s.sion <<.rthcv:))ucof~at

anyia~riur point 0 ~tcnnsufth~ snrfaœ va)ucs of-~ andot'

~l~1n the c;isc uf llm conmam potl.'lItial, uu wllÍch We l'ail lr,lrl:-Int]icc.LM(;oi't))tje<)mnt()npotc'nt.i:tI,0)twl)ichw(;f!L]t back

by putting A:= (), w~uld bu ~utc-Dniocd hy thc surface vahtu.s of

<t ~tc, thi.s Jaw ccascs to bc imivursalJy truc.

Fur a givon spacc <S' thcrc is, ns in thu ca.sc iuycstigatcd in § 2(i7,sorics of <)chTtni)tatc vaht~.s oi'

corrcsponding to thc periods ufthc

possible ]no(]c.s <.)'simplu ijannonic vibration, winch may takc

p]a<'('wit))iHa<-]u.sct]ri~i() cnv~)up(;Jtav!n~tL(.f.,rm<)f<S'. AVhh

a)iy<.f'tht'.suva)n(\s<d'/<itiso])\-i.nhstIi!)t-c:m)H)t))C<1('tL'rn)incd

hy its t~nna) variation ovcr <S', n.)xt U~- fact i))at it satisfis

thruu~x.ut <),u (~uation ~+~=0. Bntij'tLc supposa!va]uc ui/t du nut coincidc wit)i (juc of t))u scric-s, t)<L'u thc prob)~;n

Page 143: Lord Rayleigh - The Theory of Sound Vol 2

] IIELMIIOLTZ'S THEOREM.];~ 1

~1_- L. n

~s dctcrn~tc for thc .Mercncc of any two possible ~nti.ns ifhn~c would

sat.sfy ti.e condition ~i~go~ a

cond~ouw!nel~ by hypottic.sLs ~uuot be ..tis~d with

th(.' assutncd value of/

Jf tho ~i.nen.i. ,f ~c sptcc .?be very sm.]I in cc.np.n-son

~7~/7't.t .hf). but !.tt!o fro.n Aniction winch .s~i.fics tia-ou.huuLthc o~t~tton ~7~=0. (),

20.). On hi.s oxf:c~.sion of Grcun's th~rcm (J) Hd.nh.ionn.Ls h..s proofofthc i.nportant ti.~rc.n c<it..Linc<) in t.)m

fotfnwin..s~tcmc.nt:.cc~ ~A M ~r~ &

~M is

r< ~.y point A.

B

~ee~ A, /~<~ E ~ec~ ~e M~rce <6- ~~w~.

If tlie équation i

]~.cdto a sp..co c~j~tdy cncl.s. hy a ri~i.) ~.m.Luy a,.)

-nngany nu,nbcr of-dutac). r~i.) H.~ h.dics, and

If~buvcloc.ty-potcnttafs due to so.u-CL-.s witJ.in <S' w~

Page 144: Lord Rayleigh - The Theory of Sound Vol 2

t32 IIELMIIOLTZ'S TIIEOREM. r20-J.

itfo!!ow.stha.t

~,=<(-).),

which i.s thcsymbuticat statcnu'nt ut' Hcimholtx's thcorcm.

Iftbespa.cc <S'<'xtend to intinity, thc surface Intt'~rai sti)t

vanis)tcs, and thc rcsuit is thc samc Lut it is not~cccssary

togo

intodct!u!!)t;rc,asthisthL-oretnisinchnh:'dinthcvast)yn)on'

gêner:).) prin('ip]ûof'rcciprocityc'stab)ish(.'<) inChaptnrV. T!)o

investigation tiicrc givcn s))cws titnt thcprincipe ronaitts truc in

t)ic présence of (iis.sipativc ft'rcc.s, pruvidcd th~t, thcsc :u'isn from

resist.ancL's \nyi!)gas thu fn'st puwcr ofthcvetucity, that thc

ftuid m;cd tiot bu Ihtinugcneons, nur t)tct~'iglihonritu'' h'xhc.s rigi't

ûr fix~J. Iti t)ic ;).pp)ic:Ltion to infnutc sj):)('c, ~)) obson'ity is

avnitk'd hy suppo.sing thc vihr:tti~')is tu bc Hio\v)y (hssip:).tcd afto-

h:tvl))g (/sc.)po~ to :L distance (j'ont ~t :md 7~, thé sourct.'s undor

('ontooptfdio)].

Thc rcadcr must ca]'cfuHy rentonLcr titat in fhis thcort'm

cqunl.sources of soundaruthnHcpruthtcctIhythL'pL'riodic intro-

duction andft.h.strftctionofcfptat <p):mt.itic.s of finit), or HonK'thin"'

who.sc crï'cct, is thc .s:uno, and th:d c(ptal .sonrocs do not ncccs.stu'ity

uvu!vucqn:t.) amountfjoi'L'nc)'gyittcqu:d ti;ne.s. For instance, n.

source cdosc to thu .surface oi'n. hn'gc ubst~ch.' cmits twiee as much

cnc'rgy as an C(;ua! sonrecsituated in thc opt'n.

As an cxiunpic <'f tlic n.~c ûf this t))corcm wc)nay takc thn

r'asf ot'ahcaring, or.spea]\ing, trtunput L'onsisting of a Ct~ni(.d tubt.

whusccHicic'ncy is thns St'cn tu be thc s:))ne, whcthcra sonnd pro-

(Incud at a point onLsidc i.sobscn'ed at titu V(.'rtcx 01 t]tc cône, ora. source ofcqua) strengt]) situatcd at thc VL'rtcx is obsct'Yc'd at thé

cxterna! point.

It is in)pcrt:mt atso to bprn' in )nind that IH.dmImhx'.s fonn nf

thu rcciprocity theorcm is apptic.tb)L; oj))y to &/t~/e souro'.s ot'sound,

wbic)t in thu absence of obstac]cs wuuld gcncrate synunctricat

wavc.s. As wc sh:d] sec More ch-arly in a.sub.scqnL'ntchaptcr, it is

possibh' to ha.vc sonrccs of sound, \vhich, t))ongh conccntratcd in

anintinitdy

sma)) rf'~ion, do not sa.tisfy t))is condition. It will bo

.snfficicnt hc-n' to considcr tho case of~/o~g sources, for which thc

modined reciprocal thcorcm hn~ a.n intercst of its own.

Lot ussuppose

that is a. si)n])lc source, giving at a. point

thcpotentia! fmd that yl' is an equal a.nd

opposite sonrcc

sitnatcd at a nf'ighbouring point, who.sc potontial :)t 7/ is + A-

Page 145: Lord Rayleigh - The Theory of Sound Vol 2

29-Lj AtTLICATION TO DOUBLE SOURCES. 13.-3

rr) 1 .)botb sources bc in

operation simultaneousiy, the potential at 7,'is

n~. New let us suppose that tbcrc is asimple source at 7~

whoscmtcusity and

p).a.sc arc t).c san.c as ti.ose of t)ic sources at'an.! J'; thé

rc-sulting potcntial at is and at.r + A~It thc .h.stance .Lr bc <)cnotud by and )~

support todmuni.s),~it .ont hnnt, th<-

v<.)~.i(.y <,r t).c fiuid nt .f ill the direction ~L.r..s thu ,nut

ufA~ H.~c, if vu ~.n,, ~),as tf.u imnt oi t~o

~) (,p~ ,)~is

dnnnns).), ~nd who.suink.n.sity i.s incrcascd v'it).o..t

""in. suc), a manner t).~ thc j.rudncb uf ti.cintcn.sity ,u.d

t 'u di.stancc ,s t],c .sa.n. a.s for t~-o unit.sin.ptu sources j.kcud

t''D unit distance npart, wc.nay s.y ti.at t)ic

vulocity of- thc fh.id'Lt. iu .hrcctKm J~f' duc to u..It

simple source 7.' is numc.ri-caify cqua! to the potuntia) at duc to a unit

source ~t Jwf.osc ax.s is i.) t!.o dir~Iou .L. Tiu.s t)icr,rcrn, l;c it observed'Ls truc .n

sp.tc ofauy u]).sta<'tcs or rctiL.ctors that

may cxist in théJ'e~tdjnurhood oithu sourucs.

~iu. ifJJ'aud 7.reprospnt twn u!)itdoub!c sources of t)~

.sa.ncp)..s<tLcvu)of.ityat~i,, direction 7~' duc tnt)iG sourceis tho sainL. a.s t)ic

vcjocit.y at .4 in dir.-ction .L~' duc to thcsource 7~7~. Tj.osc and othcr rcsutt. of IL likc ctt.-n-~c~r

may alsobcobtamcd on an inuncdiate application of the ancrât principte of

IUS. ijtcsccxampiM will hc sufHcicnt to sttew t!~t ill

app)y,n<rthopr.nc.pfeofrcr-iprocityit is

ncccssary to attend to the characte'oi the .sources. A double source, situated h) an

open spar~ is in-audibfc fro.n .-u,y pon.t in its

c.juatonat phu.c, but it doL.s noto'low tl~at a sunjde source In the e.p.atoriat plane is inaudiblefrom the position of tbo duuh!c source. On tllis

priucipte, 1 beiievc'~y bc

c.)]auied a cnrious cxpcri.ucnt by Tyndal)~ in wjdcb'ti'erc was an apparent faifurc of reciprocity'. Thé source of sound

c.nptoycd vas a recd ofvery high pitch-, mo~ntcd i.L a tnbc, aion.r

whuseax.stlieinteusitywasconsidembfygrcatcr than iuoblique

dn'ectio))M.

Tbo kincticcncrgy T of thc motion within a c!oscd

surface iscxpresscd by

e<ul.0" -S~ 3rJ

etlilioll, p, ,105,

't""OnthcApj.)icationofthorri,)cip]cnfJ!MipMcitytnAc~stif.s"A.

.Su< ~~c~r/ Vu), xxv. p. 118, 1876, or r/.< J~. (o) 111. p. 3u'

Page 146: Lord Rayleigh - The Theory of Sound Vol 2

VARIATION 0F TOTAL ENEHGY.f295.

'~wbicbt))cfi!sttcr)nt-(-p)-c.sc-nt.st.bnw«)'ktr:u)Stnit(udacn).sstbc

huundary A', and t))c .second n'prc.sunt.s tbu wurk dune by Intcrn:dsource ut'.sunnd.

If tbu bound:n-y 6' bu :). Hxc(! ri~id cnvch'pc, nnd (hcrc Le no

uttern:d S(un-c(j.s, .rf;t.;m)s its initiai viduc t))ruo~])outt.bc motiou.T))is princip]e !):LS bec!) npp)ied by KircbboiP toj.rovt! tbc (!cto--

tunut~ocss

of tbo motiox)-c.s~)]ting from ~ivun :u'bitr.ny initiât

cunthLiuHs. Sincc every ctcmojt of7~' is positive, tbcre can 1jc nonx.ti.m witbin if bu zuro. Now, if tbcrc werc two motions

possibfc corruspondin~ to tbc sf~mc initi:d. conditions, tbdr diffcr-(.'ncc wuu)t[ bc n, motion for w)neh tbc initi:d Y:duc of ~was zcrubut by what itas jn.st bcen s:ud .sncb a. motion aumot cxist.

')7<ii)<~f'tt<t')'.l/tff/j/(~p.3n.

Page 147: Lord Rayleigh - The Theory of Sound Vol 2

OUAPTER XV.

DJRTHER A1THCATION 0F TUE GENERALEQUATIONS.

-o. Wf!~ n. train of p]anc ~vcs, othc.-wi.sc uni.npcdcd"npn~.s upon a .spac~. ocu.piud hy natter, w].o.sc ~cch~uc~t

pro-i.t.L..sd.ncrfrmn

t).<<,fthusnn-.n.i.!i,~med:un,, .sccun<hu-y~av..s..u.u

iLruwn~~vhk.].~ayLcr~c.das~)i.st,u~,Lncc.)uctu

t)..ch.cn~ ~u.~Hrcoft).nc.)in,u-~ point of~rcc.sp.<iy.,ppn,pr.atc, ~hcn thc

7-~ ~ce,a.swc]I.s thu a!tcr~t,un of ~nccham~.1 pn,)x.tics, i.s snin!). If thc~u.uuu .[ ~)..st.c)e h. nuid, t).e ,nod.n:c.)

partiessp..)~n nf ..u-c twc-thc

~7-7~ and t)i~~y. no

acc..n,.t ,.s h.rc t~kcn of' fridiou orvi.scu.sity. In thc

cil~tcr on.si.hcr.c~) hannonic

ana)y.sis wc .s)udl consi.ter the proh]om i.crcp.paso.! ou t)K..suppositiun t.hat t),c .b.stac]c i.s

.sp).cnca), without~"y rcst..ct.uu tu the s,na)Inc.s.s cf thc

c).u,~ of ~nccl.mica)

!up<.rt..H;Juti.cprc.scntinvc..sti~atiuu t).c furm of t)~ oLst~cJc

is~arb.tr.ry, but a.s.su.nc t)..t i).c

s.juarc.s audhigbcrpowcrs

of t)icchfuigc.s ofjncciia)]ic-al

pt-opcrLius may bc onutted.

If (ienotc thc(U.sphiceincnt.s parniid to tbc .i.xc~ of

co-ord.natcs of thep:u-tic)c, who.sc

c<)ui)i)jriu,n position is dL.fh.cd

~y, and if~ Le t!,e norn~Udcnsity, and tho constant

ofco.npre.s.sib.hty .so HuLt =~ t!.c

équations of motion arc

.m.) ~vo .smuhu.cqu~.on.s ~n ~ud Oa thc

a.s.sn.npCunLi.at titu witoJu inot.un is

proporti.nia) to e'-<, w)iurc as usu:d

=~7r\ anL) (§ 2-l..t) ~=?~o- (1) m~y bc writtcn

Page 148: Lord Rayleigh - The Theory of Sound Vol 2

1~6 SHCO~DARY WAVES!29(J.

Die rctu.Lion Lctwcen th(; condensation s, nnd t!)od!sp!:tce-

ments ??, ubtainud by mtograting (3) § 2:!S witlirespect

tat,hctiin<is

For thuSystem of pmnfu-y w:t,vcs :u!vancing in thc direction

"f -A', ?;:U)() ~vimi.s)); if~ ~hcHn.v:).)u(-.s<-)f~:ut(is,:m(t

~o-~bct))(.' )nQc)):uuc:t! con.sta'tt.s fur thc H)t(HstnrLcd))i(;(Un)i)wc!ta\'(.'asii)f2)

L"t.s')'~n«t.snt,i.sfy(~):Lt,t))cr(~K'nof'di.stu)'h;mc<<jn:K;(~)U!tt,

"i't)nj\H'iatiu)) i)i?~a)~[o-,w]tit;))oœnrsth<jru. f~;t.u.sas.smnu

O'at(.h(-(.~)np)(.Lc! v;i!u~ ;u'u~-)- .~+~ ~~j .su)~t.hnt.u

'"(~). ')'i)(.'nt:).kin~:)~'<~)ntof(~),w(.'gct,

!t. ).s tu bf <.))scrvc<) t)i:i( ~o- vanish, Gxccpt Hn-m~]) :).sh.:dt .sp. w).i.-)L i.s n~k-d as Un. n~-iun d..st.ur)):~(..c i

?;,~.s',I"))t~<))crc.su!t:~t'tttO(!ist.urh:LuccfL)-ct~betrcatcd

us s).)! .[nnntki~rthu <r~w,A~; su that m

onr.i.p-!xiin:dn anajy.sis )))« Viu-i:Ltio).sof~iu. cr intite fir.st~

~Hs.,f(.)..u..t((;)a~h.Lc-)~.t(.d,1,u.~ thfTcmuft.iplicd'sn':t)! .t)):.nti)ics. ~'cthu.s~htainfm.n

(.)a)~((!)by(1itfct--

'tiatiu.u..):u)ditiun,wiLhu.seoi' ~), astl.cdt~i-cnti.-Ucunation'n.~

Page 149: Lord Rayleigh - The Theory of Sound Vol 2

296.] J DUE TO VARIATION 0F MEDIUM. 137

mw)nuh thointention cxtcn.Ls ovcr.t volume

co.npietdy in-

dud.thor..g.«nof<ti.stu.~Lr.ce. Thcmt~!sm(M) mayLc

t~u.sf..rmc~ withtt.caidot'Grcun'sthcorcni."

('aHmr. tlie tw.<

parts j-cspuc~V(j]y aud (), wc );;Lvc°

wt.o ~dcnotu.s tiie.st.rf.Lce ut- thc spaco thro~h whidi thc trit~

i..t.tioa cxtcnd.. Nu.' oa ,S', A~ ..u~~~(A~) v.uus).,

su U.at, buLh t)i& surfaceint~r;t!.s (fisappL-iu-. Morcovcr

~hcru <IeuuLL.s thc cosinc of thu nn~)c ),etwucn and TLcfmcardnncn.sn)n oi-thc i-~iou of di.stnrbaucc M nL.-dcct.cd mc~~mrison with and is nc.~ccted in cornpariso~ witli ?..

If 2' bc thé volume of t!.c spacc throu~h whic)i AM, A<r arc.sc]).-i))jfc,wcmay writo

Page 150: Lord Rayleigh - The Theory of Sound Vol 2

1~8 LAW 0F DEPENDENCE ON WAVE-LENCTII.f29G.

if on Lhc r~ht-hand sidc.s A~, ~<r rcfer to tite mean ofthc\'n.ri:).ho)tHH) question. TJtusfromCS)

o

exprèsm terms of ,vo hâve from (3), = ~d

L'").s,)t the e~~ot.s~i,,)) foi, thcprim~ry wavc.s bc =e'<+.~

= and (12) may Le; put Intu t)iu fonn

~vh.ch~dL.notcs thc condc'n.sation oft).cr"y wa.vc.sat

t''cpiaœufdisturbanccattimu /,and~dL.notus<.).c condcn.sa-

<'onnft).c.su~.n)ary~v~att).L..sanu.time atadi.stan~o~-frn.n

<1.ud..s<urb.-mc.Sinceth.dif!crc,ph..LScr~u.su.~dLythe

~c-~coiTc~nd.s.simp)ytothodi.sta.tc~w~nayc.,nsIdcrt.aLa.snnpiorcvLT.satof phas.ocursatt).. l.taccnfdi.st.u.hancc.J'an.p),Lu.k-ofthc. s~n~uy ~.sisinvc.i~)y i.rupurtion~

<c d..s(anco r, ~ndt"t.)ie~~reuft))cw~vc-)..n.t), of

"'c twotcrn~ (.xp~.sscd in(J3) t).c first i.s.sy.mnct~~tinnH

'chunsr<,un.)Lhup)~ofd~urbancc~vhi~t)~scc.-n.tvar!c.s~sthc

e..s,ur<h.a.hvc.unthupri,nru-y andthcsccondary

r~s.iLus~,h~~t~Lic]t~v;u-!c.sbc.avcsa.s.~ .9~ .source~"d a ],)acc ~Y),ich o- v.u.ius hd.avc.s as a duuble .source (§ 2!).t).

T!'at t).c.sf-cnn.L-uy .ii.sturhanc-o ~u.st

vm-y asmiybc

pr~)..amcdia<.L.)ybytLc)nL.thod uf.)im~.siun.s. A~ a..<)Ao-L''))~n, U.c

atnpHt.td~ i.snc-ccs.s;u'i!y pr~~rti-.nal t. T .utd in

~eurd.mccwiL),thcprincIp)u.,fc..K.r,ry,t ,j,

~yinvcrsdya.No~t).co,.]y<tu~nt.tic.s(dcpundcnt, u).,n s)~ti,nc,a.nd

~ss) u) ~h.cht!.cr~ioofa..npI.LudL..s c;ui bc

afunctio.), arc

"c'ty<'t'sound), an.) <T,uf~hid.thota.stcannot

occurluthûexp.~iun ut'a.sim~fumt:.), asiti.sU.co.dyoneoft'.uhvc ~iehinvutvc.sa rcf~i.ceton~s.s. Oi-ti.ercmah.i,~

f.)uant,tic.s 7', am! thc )~t is t).c on!y<,ncwhi~')v~ a

.~re.tccto ti.nc, and i.s (.h~-cf.).-e oxdndGd Wc arc

'cf~wHi..-u~t of wllicll ().c

n.dy cu.nhmatinnvaryi,as 7' at)(t jodcpL'ndcnt, or thc uni of

Jcn~h, is 7'r'' X'1

~i.it.kTcstit~ application c.f thc rusu?ts of this sec'ti<m

maybemadctu

cxph.i.i what ).av~j b~n ca!)ud /«<</c cc/~e.

'f~nt~i~~i~e~ of

I'{!t'nut))]',trti<.)(..s,"7~J/Jut)t.,lb71..

'f<t<lH7:),v)tt.:)H).

Page 151: Lord Rayleigh - The Theory of Sound Vol 2

ROUNDSALTEREDIN CIIARACTEH.296]]139

Jf the primary sonud bc a co.npound nmsical note, thc variuuscntnponcnttonM.u.- ~tt~d!u«i.}.

~)-t~ Thuocta.'u<")-cxa.up)c, is stxtccn timcs

strongcr rcJativdy to tl.c fund~'.ncntat tonc in t)tc

sccondary t).an itw.~ i.i thcprunaryMund

hcre is thus nodirriculty in

undcr.stan<)i.~ ho~ it may f~apnen'that cd.oe.s rcti.rncd fro.n .snchrcf!.ct:ng bodics as ,o,,p, of ~es

.nay Le nused .~n oct~e. Thu phenu.n.non Las ai.so a conujiu-n~nt~y si.I.. If a nu.nhcr of- smajl bodics lic in thcp.~<,fw~v~s u) .oun<), t],c vibrations wl.idi i.s.suc fro.u U.c.u in dl dirce-t.un.s arc at tlic expcn.so of f].. cncrgy of thc m~in

stro~n, andwhcrc t).o sonnd is co.npuund, t).e oxatt~tion of the hi~her h-u--n~n.c.s H~ thc .seattcrud wavus involvcs a pr~orthmal ~r.L.icncyot thcm iu thé direct w~vu art.cr

passif thc ob.stae!c..s Tins ispL.ri.aps thé cxpfanati~ ~certain cchocs whicit arc .said to .-t..t.,rn

sound g,r t)h.n thé .n~ina], fur it is ~-no~n that thc pitch ofI"o to.ic

.s~ptto he c.stimatcd too )uw. Duttheovid.nccis

cunH.ct.ng, and t!.c w)..de .suhjcct n~nirc.s f.u.th.r carc.fui cxno-'nta)invc.st!,ration; i~nayhuco.n.ncndudtott.o~.ntionof«.se

~).o may hâve thé n..c..ssary upport.uni~.s. W).i)e a.t altère"< m thec~~c~-of aso.md is

ca..siiyi,itu))igib!u, and~nst".deed ~,n.nJ)y j.appcn al:,n:t..d c.xto..t, acha.~m thc]" cft ut ;b ~,n),)c tone wuu)d Lu a viofa~nn .,f t).c iaw'of furcudvib~tto.Ls, and i):u-d)y to bc ruconeifud wiLJi t)~uretie:Ll id~s.

Inobtnini,,g (1.3) wc ),LVo n~ctud thc L.f)-uct uf t)ic variable

"~urcui thc tucdimn ~f~~r~cc. ~Vhc~thc di-sturb-ancc ou th..s

.supposition is th~-o.~b)y !<nown, wc mi~ht appro~i-'~atc a~ t),c .s:unc inanuer. Tha ad.Htio~a! tur.n.s .su uhhuncdw..uhl be

i.cccs.sariiy uf thc second ordcr i.i A~ Ao-, so t).it ourcxpre.s.s.un.s arc ili ail cases c.jrrcct as f.r as thc fir.st powers ofUiu.sc(~)a))titics.

EvL-n wfioi t)io rcgi<j)i of disturbanco i.s not .sma[I in con-panMfm with ti.c .s:uno .~ti.od is appticabic, prcvidcd thc~.arcs nf A~, bc rcai)y nc~i~i)~. Ti.c tu~d cOuct of anyub.stactc

inay t)ic.n hoea!c.datcd by int~rati~i fro.n tho.se of its

P~t.s. hitinsway wc inay trace thc tra.nsltio.i f~.n as.nnit

'-cgion of d)stt)i-bancc whose~<?y~cc docs not corne into cut)sidcr:t-

to a. tbin p)atc of a. fcw or of a grcat many square wavc-'c'~ths in area, ~inc)t wi!! u)timatc!y rcHcet

accord: to thc

r~u!arapLl(..a!Hntift),c.,b.stad~huatai)d.atcdin t)~

'hrccHonofthcprimaryrays, t)ns inethod ofcaiL-utatiuu suon

Page 152: Lord Rayleigh - The Theory of Sound Vol 2

~0 SECONDARY SOURCES. f2DG.

couses to bopracticaDy available, because, cven atdiough tbc

change ofmechauical properties buvery s)n!).)L tl.e iuteraction

of tifc varions parts of t)ic obstac'te c.umot bc )cft out ut' accuunt.

Titis caution is moruespucia)iy ueodet) In duaiin~ witb t)ic case ut'

f!ht,w!ture t)tc wavc-)cngUi is socxcce<!ing!yHmidl i)i

cujnpun.suMwit)i t.hc dinicnsi~tis ot'urdin:u'y ubstacic's.

2!)7. lu sumc(]c~rcc .si)ni)n.r to tbe cftcct prudnccd by n.

ch:u~c iji tliemGc)t:Uiic!d pt-upL'rLtG.s of~ sn):)U r(.-gi(.n ot' t)m tinid,

i.stb:)Lt.w)tichcn.suc.-iwJ!ut) thcsq~u~ ufthutm~hm i'I.-n.\s:u)y-wtiurc to .suc)) i)))port!tUt!c th~t. it c;m bu un iun~crn~uctcd.\7'~+~ Lhun :tC(p)irL'M:t.fi))itn vaiuudcpendcnt upott thc squareof thc jnotiot. Such ptaccs tftct-uiurc act !ikc som-ccs uf sound;

thc pcridds ofthc sourcc.s inc)ndu)~ thusubtnxttfph'.s of thé on-

gnmJ p~riod.Thu.sn.nyjtart of.spit.cc, :ttwhic)t thé intcnsity

nccunndatc'.s to a MufH<;iunt cxt(;)~t, Lccotncs itsc]fas~coodary

source, c)tuttingt])<:h:u-!)~)nic tonus t~t'thép)-im:u-y .suond. If

thc'rebGtwopt-ini:nyso)mdsofs)tf!icicntittt(;nsi<y, thcsucundfu'yvibr:tti(nts hin-c t'rc([UC)tH)(.\s which arc thc .smn.s fmd din'urcnccs of

thcft-L'~Hcuc~s oftttc prinKH-ics (§ GS)'. t

2!)~. Tjic pLtdi of a sound is !!ab)c to modiiication whcn thc

som-L-c aod thc rccipicnt nrc ili !-c:]:t.tive motion. It is c)c;u', for

]').st:(ncc, th:tt an observerapproac)d)ig a (ixed source will tnect

thc wavc.s with afrcqucneycxccodin~ that

propcr to the sound, t)ytlio numhcr of

w:wc-]cn~ths pa.s.scd uvcr in asocond of timc. Thus

if v bc tbc veducity of titc! observer nnd M t]):tt of sonn<), thé

freqnotcy is n.tturcd in tbc ratio M i f<, accor<)in~ f~ tho motion

:.s towardaor fro)n tho Mourcc. SInccthca!tcratio)i ofpitch i.s

constant, a musical pc'rfonnaucc wou]d stiH bc iioard in tune,

:dthough in the second c:LSc', wnoi ft and v arc nc:u-)y cquid, tl(cfid) in pitcb wout(t be so ~reat as to

dcstroy a)l musical c))aracter.

If wc coutd su])posc! to bu greittcr than M, a soundprodocot aftcr

thé motion bad bcgun. wodd ncvcr i-c'ach thc observe')-, but sounds

prcvious)y cxcitcd wotdd bc graduaHy overtakon and itcard in tbc

ruvcrsu of tbc natural ordcr. If u=2(t, titc observer wouht ficur

a musical piccc m correct tinic and tune, but ~<fc/~w~.

Corrcsponding resn)ts cnsuc when tbe source is lu motion and

tbc observer at rest thu altérationdcpoiding only on thc relative

tuotion in tbc Hne of hearing. tlie source and thé observer movcj

\vitb the samovelocity tbcre is nu a!tcration of

frcquoicy, witethcr

Hotmhoitz iibor Combinatioustuuc. Pogg.H. Bd. xcix. H..107. 18.C.

JI10VC

i1 Helmholtz Ühor Combiul1tiollstünc, Pogg, AlI/l, Bd, X<IX,H, ,HJ7,

t

Page 153: Lord Rayleigh - The Theory of Sound Vol 2

2~8.') JDOrpLER'S PR!NCirLE. 14t. t

thc médium be in motion, or not. Witb a. rcJativo motion of

40 rnUcs pur bonrthc i~t.crationofpitcb is 'jry c'iit'ptcnous,

amountin~ to a,bo't)t a. semitonc. Thc whistte of a, loconctivc is

hcard too hi~I; as it.ipproaches.,

and too Io\v as it !'GCC(Jcs from an

observer at a. station, (.'han~'in~rathcr

fiLubteidyat tho tuon~cnt of

pass:

T))L' pritipipicofthc altération ofpitch by relative motiol) was

first <unnciat(.'d ))y Doppler', and is of'tcn caited Duppto-'s prin-

cipe. Str:u~'n]y (jnou~h its Ic~itimacy was <sputud by Put/va)'

whosc ot'jccttun \vaH t!tn rcs~dt of a. cont'usxm bctwcct). tw')

pcrf~ctiy distinct casc.s, that I)t which titere is a rciativc motio))

ut' tt)L: sunrce and t'ucipiunt, aud tha.t in \vhic)i thc medhun is in

tnotiun whiJu thu .suurcc and thé récipient arc !).t rest. In thc

bttc)' ca.SL' thc circmn.stfUK'cs a.rcmechanic:dly

t))C satuc as if thfi

nu'ditnn wcrc at rcst and t)'c source a)id thc récipient had

connnon motion, and thcrcforc by Doppicr'.s principfcno change

ut' piteh is tu bu c'xpcctud.

.i)opp)crs principicbas bccn cxpcriinc~taHy vcrinc'd by Unijs

f!at)ut" and Scott Ku.sHcH, who cxantincd thc .'Llturati"n.s of pitch

of ]nusica[ instrnmcnts carricd ou. fucomotivos. A iahoratory in-

strument for proving thc cbiu~'o of pitcli due to nmtio)i Ijas becn

invcnted hy Mach~. It consists of a tube six icub in icngth,

capabtu ofturni]]L;'a.houtanaxisn.tits coitru. Atone ond is

p)ac-t;d a.s)na]lw)nsth; orrced, chichis bhjwn bywindforccd

ah)))"- thc axis of t))C tnbe. An o))Mt'rvcr situatcd in tt)G p!ancof

rutatiou !)c;u's a note of~uct)U).tin~ pitch,but if )tc p)accs ]n)nscff

in thc proion~atinnof thé a.\is of rotation, titc sonnd becomes

.stuady. P(-'r))ap.s tlic si;nplL'st cxpc'ritncnt i.s that dcscrihcd by

Koni" Two c" tuning' furks mountcd on rL'soxancc casc.s are

prcparcd togivc '\v)t!)(-:u-hcL)tcr four bcatspcr second, rfthc

~)'avfjrofth(-'forkshc)na()L!t('app)-oac]tthccarwhi)etbcothr'r

ru!nain.s at rust, one bcat is ~.s'< fur cacb two feet of approach if,

howcvcr, it bc tbe more acute of the two forks ~'bich approachf'H

)hc ca.r, onc béat is~t:~ in thc samc distance. A modification

ThnnriG dcf! ffU'tji~'] I~ichtof) (h'r D<ipp(.')stcrnp. rmK, 18)2. Sec l'isko, /)/<'

;t«')'<'M.-(~ftrff<<<'r~ \Viun,18C~.

H'/fx. vm. Ut. M52. ~'or~f~t- ~f)' /< vnr. lf!7.

~'r~g.< t.xvt. p..T~l.

'P<tf.;g.cxn. p. nf!,lMt, and cxvi.)).):M.

"]~'ni):sf'(;~)yf)y;«f </<< ..t/«)'~f/Jr~tt.<f)')~. rnt-is,1RC.

Page 154: Lord Rayleigh - The Theory of Sound Vol 2

DOPPLER'S PRINCIPLE.['39 g.

of this cxpcrhnent duc to Maycr' may atso bu notiecd. In thiscase onc fork excites thé vibrions of a second -in unison wit),

Jtscif, tho cxcitationbcing madc apparent by a sn.aH p~dui~m,

whosc bob rcsts against t))c extrc.nity af une of thc pro.s. If thc

cxcitmg fork be at rest, the etfcct isapparent np to a di.stn.cc of

('~ct, but .t ccascs ~.).cn thc cxciting f. j, ,novcd rapidiy too'- iro m the dm.cti(jn cf tjta Jinc joining t]ju twu furks.

Ti~cre is somedi~cutty lu troating inatbcmat:c~!y the p.-obic.nof a movu~ source, .-u-ising fro.n thc fact th..L any practic.-d smn-cc

~ts a so as an ob.stacie. Thus in tbc case of a bc)I cm-ricd

t'.rougb tho an, we s!.ou)d rcquire to solve a probk..n dif)1cu)t

ouougb withoutincfuding H.e vibmtion.s at a)]. But thc so)utio,i

of .such aprob]cm, cvcn if it couJd bc oLtainod, woutd throw noparticuJar i.ght on

DoppJcr-s law, and wc ,nny tbc.rcforc advan-

ta~cou.sly snnp)ify the question by idca~n.g thc bc!! into a snnpiosource of sound.

Tn § l.t7 v-c considcrc-d thé prob]cm of a moving .source ofd..st..rbance Lu t),u case of a strc-tchcd string. Thc thcory foraenal .vavc.s m o,,o di.ncnsion is

prcci.sdy simDar, but for tlleancrai case of thrcc di.ncnsiun.s son.c extension i.s

ncec.~ry inontcr to takc account of the

po.s.sibiiity uf a motion acro.ss tt.cd~ct.ou of the sound r~ys. From §§ 273, 27(! it appuars th.t t!.cefrcet at any point 0 of a .sourœ of sound is thc samc, ~hcth.r t),csource he at rcst, or ~hethcr it. ,novc in any ~nncr on thc surfaceof a. spi.crc dL.scn))cd aLont as centre. Jf thc source inovc in.sucb a manncr as to

change its distance(~) fr.un its e~et i.s

aKcrcd in hvo way.s. Not o.dy is thé ~~c of tho distnrbancc onarr.val at ~a~cted hy t).c variation of distance, Lut H.c~a).so

undergoes achange. Thc L.tter co.nphcation Lowcver mayhe put out of account, if wc limit oursc.Ivc.s to the case in ~hich

tlie source is sun.cicntly distant. On thi.sunderstanding we may

assert that thc enuct at 0 of adisturhance gcncratc.I at time nnd

at d.stancc ?-LS thc sa)nc as t!tat ofa .siniilar distnrhancc ~ncratedat the time t + and fLt t])c distance )-- a~. In thé case of a

penodic disturba..cc avelocity of

approach (r) is cquiva~ent to anmcruase offrcqucncy in thc ratio ft f<+r.

20f). AVc ~-i)[ nowinvcstigatc tlie forccd vibrations of t]jc

an- conta.ncd within arcctang.dar chambcr, duc to internat sonrccs

of sound. By § 2(!7 itnppcars ti.at thc rcsutt at timc < of an

()) xt.nt. p. 27~. 1R72.

Page 155: Lord Rayleigh - The Theory of Sound Vol 2

2M.'] RECTANOULAR CHAJMDER. 143

B

g initia condensation confincd to tticucig!.bour!)ood of t!tû

point

g

'3

f.-n.n wltich tLc c.Hcct of .tni)n)~rcsscd force

m~y bo(Jc.)uc(~

S a.s ..) § ~7(!. T).cdi.sturb:u.ccjj'j' co.umu.ne;LtL..[ at'

B Un.c Lun~dcnotcJbyJJJ')~.< or

~,(~ thé

g ru.suhant di.sturbancc at tituc is

1T)'c

.symmctryof thi.s cxpt-CMion with respect to y, a.td

?/, '.s an cxamp]c of thc pnuciptc ofrcciprocity (§ 107).

.In fJtG ca.c «f h.-u-mntnc force, for which (0 =~ cos?~wc hav'u to considcr thc vn)uc of

StncUy spc.ing, thi.smt~al bas !ia <tcf.).;tc

va)uc- L..t ifwc ~s). for thu

c.xpre.s.si.m of (hu f.n.ccd vit~ionson)~ wu nmst

~t t).cu)tc~tcdf.mr.ii.,natt).c )~.cr ii.n.t. ,n.yb~cc.a

.y supposm~ t),u ii,trud.,ctiou ofvury .sn~)I

<)i.ssip~ivc forcesWct,h)t.suut:ti)i

As m.~ht !c bccn prc~ictctL thccxp.-cssions b~eomo infinitc

in

c~eof a comci~~cc LeUvce.i tho pc.ria.t uf t)~ snurcc an.! one

oftho.tnratpc.nrKf.s ofthc c).mLcr. A.yparticuf.-u. normal

Y.b,t.n will i~ot bc cxci~<), if tl.e source bo situatcd on oncof its luop.s.

Thé cffuct cf anu.p)icity of .sources n~y r~dify bc infcrrud

''ysumnnattonori)it(.raLi(jn.

Page 156: Lord Rayleigh - The Theory of Sound Vol 2

H.i U~LIMJTHD Tt'UK. t/

300. When sound iscxcitcdwithma.cylindne~pipc,the

si)np)cstMnd of excitation t.h~t we cun suppose

isby

thé forced

vibration of n piston.Jn ttu.scase thc w:u'(js f~'c ph~c

irmn

thé hcgimung.But it is Imp~rLnnt

:dso toirtqnirc

w1ta.b h~ppcn.s

\Yt)cn the source, inst(:'n.d ot'hcinL; uniiorndy'Hftuscdover thc

section, is conc~ttr~tuditiotn.'pointofit. it. Ifwc:~sumG(wh:tt,

howcvoi-, Isnfjtu)n't;r\'<jdty tn~) t]):itat:L sufficient distance

from thc source thu wn.vcs Lccomc plane, thc law oi' rccipt-uclty

issuHitcicntto~uidcu.stothcdusircttinf'orxKLtion.

L(.'t~). bc asitnph'source in nn un~n~cd <uh(~/)', two

pointsoi'tttcs~Tnc normal St.;ct.ion iu thHrc~ionot'phincwavcs.

~.<- A~fC.thc potuntiaisut/~and

7/~h~ tn thc .source J

arcthusHmc, ~nd accordin~y hythch~vof rcciprn(;i(yenu:d

sourcos :<.t 7~' funi 7~' wou)d ~ivc thf .s:unc potontia!nt ~). From

this it fo!!uws tlutt the cn'<'ct of :my source is t))c sn.)nc n,t a.

distance, as if thc source wcrc uniforndy(hifusL-d ovci' tho section

whidt passes throu~h it. Forex:).m])ic, if~:U)d7~werce<)U:).)

sources in opposite phases, thc di.sturb:u)ce at /1 wou)d hc ni).

T)te cner~y etnittt'd Ly a, simpit' .source situated within a,

tuhc ]na.ynow be cah'nhdcd. If thc section of'ti'c tuhc he cr,

:md thé source sueh thatin theopcn the potentieldue to itt

woutdhc

thé \'u)ocH.y-p()t.c-))ti:Uat :L (Hshtm-c withio thc tuhc 'i)l bc

U)e s:n))e:iLS if thé C!U).sc ofthc dist-m'hanec wct'c thc mnt.iou

of:), piston !ttt)tenri~it),~ivin~ t))cs:i.mc total displacemcnt,

!U)d thé cncr~y C]nittcd will nt.so be ttic MmnL'. New i'rum ())

Page 157: Lord Rayleigh - The Theory of Sound Vol 2

~1 I~ERGY EMITTnn. 145

I!cncc,~sjn§2.{.i, <)'cc!)p)-gy(~r)Rmitt~o)!C(7c/~t'(~t)~ .source

isgivc'nhy

If thc tube Le stoppcd hy an innnovit.bte {ti.stun p):).cc() dose tnthc source, tho \vho!e cncr~y is onittcd in one direction; bnt

tins )s !i(-)t a)). In con.scqnc-ncc of thé (ionblcd prossurc, twiccas n)uch

cno-gy f).s bcforc is ()cvc)nped, an.1 t))us in this c:tse

Thn)i;))-r<-)wpr thé tubc,Lhogr~tcri.stiicono-~yissuinrffroni

?L ~)\'L;)t suut'cf;. It i.sinto-c.sting

to compa.rc tjtc cfHcicncy of

a,sou)-cc;)Ltt))cstopp(-;(!en(tofn. cy)in<)ncn.t tubowiththatot':") ef)))~) source situatcd at t))e vcrtex of a, cone. From § 280

wujtavcinthclattf-rcnso,

Thc Rncr~iM omittcd in thc two casps are t))C sfune whcn m= ~'o-,t))at is, whoi thc section of thn cy!int!(.;t- is

cqu:d to tho an;it

eut otT hy thc cône: frnni a sphère of )'a(]jus x'

301. '\Vc ))nvc now to examine hnw far it is truc! that vibra-t)f)ns w)t.I)in a. cyimdrical tube bccomo

approxhnatdy p)anc at n.

suuicicnt ()ist:mce from tbcir source.Taking <,hc nxis of~ pa)'a!)c)

to thugcnurati))~ Jincs of thé cyiitxicr, lut. us

invc-.stigatc t))t:

)notinn, whosc potcntial varies as e" nu thh {.ositivc ~dc of a.

-soxrcf-, sitnatcd at ~=(). If bc t))C potcntial and stamt furr/'

.+ j'tlie

cqnatton oftho ]not.inn is

If' bt!I))()upct)dcnt, uf it t-cprcscnt.s \')))!-atKm.sw))o])~

tmnsvt.t-sGtnttiCfLxi.sofUiccytindur. li'tite potentiel bcHto.

prr-pnrtion~dtde'it ntnstsatisiy

!()

Page 158: Lord Rayleigh - The Theory of Sound Vol 2

14 G VIBRATIONS IN UNLIMITED TUBES. [301.

as well as thc conditioti that over tlie boundary of tho sectioM

In ordcr that thnso équations n~y be compatible, is rcst.rictod

to certum dufinite vêtues con'usponding to thé pcriods of thc

D~tu)':dvi!)r:Ltiuns. A xcro value of~) gives ~)= constant, wh~'h

sotution, t)tougii it is of no significancu in thé two dinicusion p)'f)-

bk'm, we nha)l prusentty bave to c<'usi(!(.'r. rur oach athnissi~G

v.duc of thcrc is f).dcfinite norma.1 fnuction M of and y (§ 0~),

such that a sutution iH

in which corrcspondin~ tn ~= 0, is constant.

[n thu actun] prnb)cin mity stiU bc expandcd in thc samc

séries, provide~ t)):~t J~, ~1,, &(. bc t~gardeJ as fnnctions of

Hy substitution in (t) wc gct, liaving regard to (~),

Thc solution of the gcnc'rai cq~ation in ~t nssunn-s a di~crcnt

f'urin, aecordixg as is punitive or ncg'~tivp. K'thc furccd

Page 159: Lord Rayleigh - The Theory of Sound Vol 2

DISCRIMINATION 0F CASES, t~

vibrion be graver in pitch than thc gr~vest of thé purdy trans-vcr.sc n~ura)

vibration.s. cvery fimtc valueof;~ i..

g,-(;aterth~

Ls<m.Hi,(j~at!vL-. Pm.(,n)g

~w un.). thé L.ireu.n.sL.u~.ssuppôt, it is C'vidc'I1t thatthp

.notjun ducs not b.co.nc infinie with so that a)! t).o c.ciïicicnts

.sh.I'su,ncwi.atdim..rc,.troason thc.~ncistn.cof. u,..s t).c can bc. no wavc in t),c

négative dirccLiun. W~,navt.Jt'jrctut'eta.ko

<~=~e')+~+~+,,(~~

111e

Q +.(1_),

a..r.p,-ossi.n which rc.ducc.s Le its n.st tc.n. wi.en i.s

suf~iu-UySe~. A\ c c.ndu.)c that. in ..]! ca.sus U.c ~vc..s

ukin.d.c.)y bccunK.

!o,~c~

Jrctr~est cf tlie?<<?-«/ ~-(~~c~e ï;<M).s-.

v.~cg.u-c.st trans-

to 2.1.S.H =3..H.~33J) If ~ca thc

wavc-Jc.ngth nf the forccd vibrion e..c~

i' t? !T! ~'o~-

~u

hat

i.cwav~uh.n.atdy

bc.cun..p)..n.,n)t.].ou~. tho w.vc-~.1. iaH .s .t of tl.c aL~.c limit. F.

exa,.p)e. if thc suurccot v.bmt.on ho .sy.nn~t.ca) with

respect to thc axi.s cf t),c tubc<7. a

,unp), source .situai) on the Hxis it.c.!f, t),c ~avc..st trans~vci-.SG ~hrat.o.j ~.ith whieh wc .shouhi ).ave to d.at wouid bc .nor.

~n

an octave

J.i~cr

than in th.g.ncra! ca.sc. an.) H.

'c .)t uf thc forcedv.brat.on n,i.r).t ),avc Jc.s.s than i~f t),c. abovu

valne.

inasmncha. r/<r, A.c., ~) vani.s)~.

It appcar.s accr,n!mg)y t).at t)~ p]anc wavos a<, a dist.anco a..'esamcaswonM bcpro.juce.] hya rigir) pi.ston at,

thcm~m,

K)–2

Page 160: Lord Rayleigh - The Theory of Sound Vol 2

t~8 REACTION OF AIR [!

~iving thé samc mcan normal vdocity actuaUy cxists. Any

"orm~ motion of w~ich t.)'f- ~-t~tivc ~n~ positive p.~tsarc equaL

produccs uitimateiy no crï'cct.

Wilcn thcrc is no restriction on thc eharactcr of tLe source, :ujd

whcn some ot' thc transverso natnrat vibrations are gravert1':m

the !tCtn!U onc, sotne of thc vah'cs of are positiveand thon

tcnns enter of thé form

Indicating that ttic pcculiantic.sof tlie source arc tiropagatc-d

to

au in~n'tc distance.

Thcprubt~m

hère consido-c~ may hc rog~'d'~ as n. gcncraH~-

tionnfth!),tof§26M. For thc cfiLSuoi'n.eirc-nhu-cyHn'io-Itmay

t,<- wcn-l.L.d out complutdy with Htt. aid or BL-sseFs functiun.s, but

this tnust be !uft tu tlic rcudu)'.

302. In § 27H wc )):u'e fuUy dutentuncd thû !noti<'n of th'~

au- duo tu thc normal perdue motion of a bnunding p)anc plate of

ii~nitc cxtci)t. If bc thc givcn ~on~al yc-Ioeity nt tlie dc-

clo

mcut~.

gives thc vdocity-potcntia) at nny point 7' distant ?' from f~ Tho

~maimter of this cha.ptcr is devot'~ to thé cxfuninatum of thc

pjirticulai-case of thi.s

probicmwhioh crises whcn the nnrn~l

Yulocity Itas a giv~constant vaine nvM- rL (-ircuhu- :u'ca of ra-hns

7~, whiln over t)tc rcmai)xk.r of thc- p!anc it is zéro. In particnhtr

wc shan invcstig:itc wh~t forces <1uc to thc réaction of thé air will

act on a rigid circ~n- p)ate, vn.raiing with a snnp~ harmonie

motion in an (-qua)cirodar arcrturc eut out of a rigid ptanc p)atc

extcndin~ to intmity.

For tlie wh.~e variation of pressure acting nu thé platewn

l.avc (§ 2-~)

Page 161: Lord Rayleigh - The Theory of Sound Vol 2

3~.j UN A VmKATfNG CIKCLJLAJ~ PLATE. ~f)

wherc o- is Ui<j naLurat dunsity, !u.d varies as <<"<. D~us by (1)

~.ch we hâve .,ow to évaluée, cac)~pair of éléments is to bu

takeu once o.dy, and ti.c product is tu be sun.med after n.ut~pi.-c~unbythc~tor.«.ti~ ti.d~nutua! distancu.

1 he bc.s mcthud ~.ts~cstcd by Prof. A[~xwe[i f..r thé co.n.non

potcntiat Ihcqu.ntit.y (.) i. rc~rded a.s thcwork that wou)d bu

con.su.nc.! t).ccomj.~c di.ssuci~tion of thc ni~tcr

composn,O.c dLsc, t)~ is to .s.y. iu tlie rc.n.vd ofcv.,y c.icmcnt from ti~

mfh.enco ot.v.ry othcr, oa thé

.supp.sit:on tlmt the putu.iti~ oftwo clement.s is proportio~d to r-'e-- » Tj~ a.nou.it et- workrc.[.urcd ,v)nch

dépends o.dyo~ t).c initia ~d ~dstate.s,,naybu c~eui~tod

by .suppc.sin~ theoperatiou pcrfbnned in any wayL.t may Le must convc.ni.nt. Fur this

purpo.so ~e suppose thatthé d.sc is divided ,ut.

elu.ncntary nn~, ,~d th..t caeli rin.r~~d

away toin~ity bcfuro any et- the Intenor

ring.s arc ~i.s-tlll'ued.

firststop is thé adcnhLt.on of thc

potcntlal (~) at théc~c of dise of radius c.

Takmg potar co-ordinatc-s(p~ .yith

any pomt of thé circmnfuroico for pôle, wc hâve

Dus

quan~tymust bc

nu,]t:Iic.! hy 2~~ a.~d aftcrw~d.sntte~ate(t with respect to c bctwcun thc Ihuits 0 aad Butit will bc convenieut ih'.st tu cn'cct transfonuatioD Wc trn-c

whcrc is writtca for 2.c. (.) is thc B~c!~ onction of .cro

Thcory of Jtesouanco. J7t)7. yMfx. 1870.

Page 162: Lord Rayleigh - The Theory of Sound Vol 2

150 REACTION 0F AIR [303.

order (§ 200), a.nd 7~(~) is a. function deHncd by thé cquaticm

Frox) this thc <ot:t! pressure is do'ivud by introdnctton of thc

!<'f/0-f/~)sofactur

7rj.sothat

TT K~

Thc re!K;ti"n of thc air nu thc (fisc may tho.s hc divulud into

two p:u'ts, nfwhict) Hœ îu-~t is proportional to tlie vclocityof tlie

dise, and thc sccun't to thc accolcr~tion. If dénote thé dis-

pt!LCC<ne))t of the dise, sothat =

wc hâve ~= :t =

and thercforc in thc cquation of motion ofthe dise, thé réaction of

thc air is rcprospnted by a frictional force ao-. 'n'-R".( 1 tj"~ )

retarding thc motion, and by an accession to the mcrtia c~ual to

'7TO'r~ r)\

~~(2~?).

Page 163: Lord Rayleigh - The Theory of Sound Vol 2

~02.] ON A VIBRATING CIRCULAH PLATE. 151

When is stual), we have from thc ascendinc. séries for ,7(~§2()().

iz

From thc nature of the case thc cocMcicnt of must bepositive, otherwisc thc réaction of tlie air would tend to au~mcnti't.stc~J uf to retard, thc motion. That (~) is in f~ct dways Ic.ss

t))an x may he vcrincd as Miows. If lie betwcen 0 and pr, andbc positive, sin (.: sin 0) sin is negaLive, and theruforc also

Is négative. But this intcgm! is J, (.:) z, winch is aceordinglyncgahvc for a!!

positive values ofz.

When is gréât, ,7, (2~) tends to v~nish, and then t))Cfnct.cnat tenu bccomes

si.npiy a<r.7r~ This i-esu)t mighthâve becn expeeted; for w!ien is very large, t))c wave motion

theueighbonrhood of the dise becotnes

~pproxi.nate)yp)ane.h~e then by (G) ~nd (8) § 2~5, ~=~ in ~hich is the

density (o-); so that thé retardmg force is 7r7);=

a<r.7r~

We h~vc now to considcr the term rcprcsGnting !ni itération

ofmertia, and among otiter titings to prnvc t)~t this altemtton Isan increase, or t)tat (~ is positive. By direct Intégration of the

asccndiug séries (5) for ~(whieh Is aiways convergent)

This part of thé reaction of thé ah- is tlierefore representfd bysnpposing tlie

vibmting plate to carry with it n. mass of air cquatto that cont:uned in a cylindcr wliose base is the plate, ~nd whoscQ 7~

hcight is equal to so tliat, whcn tlic p)~tc is s~ciently sn~H,

the mass to be added is independent of tlie period of vibration.

Page 164: Lord Rayleigh - The Theory of Sound Vol 2

152 RHACTKJX UF AHi[~U~.

an Intcgrn.1of which cvo'y douent is positive. When s is very

l!).rgc, cos(3sin6) nuctn~tcswith ~ruct.t nLpit.ht,y,:t)td tims 7~(~)

tends to thc i'orm

When is grcat, tho aacemhng séries fur 7~ aud 7~, thougb atways

n)ti)n:~te)y convcrguut.bccomG uscicss furpract,ic:).t c:t.)cuh).t,ion,a.nd itt

Is nccu~sary tu rcsort to othcr pruccs.s(;M. It will bc ohservcd thui

thc diMcrcnti:).l cquaLion (1C) sa.tisHt.'d by 7~ is thé saine as th:Lt

be!onging to thé Bcsscl's function < with tho cxccptimi of thc

terni on tho right-ha.nJ sidc, viz. Thc funetiou A'is thereforuTT~

included iu thc form obt~nned hy iutding to thé genenti sulution of

Hcsscl's c<(U:T.t[<)n contiUtnngtwo :n'hitr:u'y constauts :U)y particular

soiutiun of (!()). Snch a particuhn'sulution is

~7r.A''(~)=~+l'3'3"l'C'=.~+l'.3=.5~.7'(21),

as may bc rcadi)y Ycntled on substituttun. Dtû scrics on thc

right uf (~t) notwithMtandin~ its utthnato divcrgc'ncy, tnay bc

usc't) succu.ssfuHy forcomputatum

whcn is g'n'at.. It is in fact

Page 165: Lord Rayleigh - The Theory of Sound Vol 2

302.) ]UN A YlBRA'nNC! CIRCL'LAH PLATE. 153

thc anatyticd c<iulv~]e!tt uf~e'(s''+/3'')- and we nii~ht. taku

dcteDnining thc t\v<j fu'bitt'iu'y cnn.stiUlts by !ui cxa.nunation of thu

furms a~su!U(;d w!)cn .? is very grott. But it is pct-hups simpicr tu

foilow thé method uscd hy Lipschitz' for Bcssefs fnuctions.

By ('i') wc hâve

fe"f~w

Considcr tlic intcgral ) .-=– wltcrc ï~ i.s a complex vfu'mLIc of~vl+~'

thé fn)'m ?<+! Rcprc.scuLing,as

u.sua!, simultn.neouap:u)'s of

values of a)i(t u Ly t,hc co-ordinatûs of a point, wo sec that tt)c

value of Die intc~r:t.) will be xero, if thc intt.'gra.tion w!th respect

to M r~UT~~roundthé rcct!mg)c,hosc angutar points are ruspcc-

tivcty 0, A+~ ï, w!ier(; A is any rcat positive quautity. Thus

Thcri!sttcrmonthjri~htiu(24.) Iscntirelyunnginiu'y;M

therctorc fuDows by (22) that ~7rJ.(~) is thc rc:U part of titc

.second terni. By expanding tho binumial nndur t)ie late~r.d si~n,

and aftcrwit.rds iutcgmting by thé fui'tuu)a.

Page 166: Lord Rayleigh - The Theory of Sound Vol 2

REACTION 0F AIH [302.154

By stopping thc expfUisiut) n.ftcr any (tcsircd number of tcrms,

and forniutg thc expression <u)' thc l'cmaindt;)', it inny ~)c .~vcd

th;~ titC c-rt-ut' c~tunuLtud by Hc~icL'ijtn~ LiiC rcma.mdci' 'cU.aoL

cxcccd thc i:~t tci'ni rctainct) (§ 200).

In )ikc m:t.nncr thoItUi~inary part

of thc nght-hand mcmbcr

uf (24') Is thc e~uivak'ut of –~t'7r/~(2), su tha.t

It nppears then th~t 7~ docs not. vanish when is grea.t, but

approximates to J.z. But ntthough tlie accession to thé mcrtin,

As wn.f!to ho cxpccted, tho ficrieH wititin brac~ets are tho samo as thoso thatoccur in tho expression of thofuuctiou '~(~).).

Page 167: Lord Rayleigh - The Theory of Sound Vol 2

303.] ] ON A VIBRATINO CIRCULAR PLATE. 155

wh:ch ispmpnrtio~al to

7~, bccomcs !nH)]ite with it vn.nishcs

')!thnat(;iywhcnc()rnparo()wit!)th''a)-(':),r,t't))(i~isr;)n<)wlt.!)t.!)n

nt.h<jrtcrm whicii rc'prcsotts t))u ttisnip~Lion. And t!tls ~t-ucswit!t whiLt \vc shouht nnticipitte f'rom tlic

thuory ofptane \vavcs.

If, Im)cpcHdently of thc rc:tction of Lhc nii-, tlie m~s uf tito

p]atc bu jV, :U)d t))u furce of restitution bc thé cquatiott of

jnotiu!) of t!fu pJatu whun nctcd u!t by miimprcsscd furcc pro-

port,iot)!t.) to e" will bu

Two p:u-t!cu]:u' cases of' this problem (k'servc notice. First let

.Vand v:mi.s)), so t)mt t.hc plate, itscifdcvox) uf !nass, issubject

Lo no ut.hcr forces tha.n F a.m! t!iosearisin~ from acrifj

pressurc-s.

Smce ~=~(~,thc friction:~ tcrm isreta.tivctyncg)igiblc,andwc

~et wh(;n is very smail,

Ncxt !ct aud bc such (.hat t!te n:ttund penod of tlie plate,wltcn subject tu t!)e react.Kjn oft)ie !ur,is thc same us that imposcd

'tpoa it. Undur tljcse circumsta.nccs

Comparing with (31), we sec tha.t tlie amplitude of vibration is

grcater in thc case whcn tlie incrtia of thé air is bahtnecd, in thc

ratio of J C 3~ shcwing !a.)'~o increasc whcn i.s mnalt. In

thu Hrst case tho phase of Lhc motion, is such that compa-rativ~ly

A'cry )itt)c work is <!one by the force w])i]c in thé second, thc

incrtia of thc air is compcnsatol by thé spring, and t)tcn bL-in'ïof tim same phase as the vcbcity, docs thu maximum a.mount (~'

work.

Page 168: Lord Rayleigh - The Theory of Sound Vol 2

C'UAPTEU XVI.

'nmuRYUL'R).JSUNATUR.S.

;ill,`3. Wth, pipe If, (),le end alld

open nt tllc ot]Il~I' wc luul

vilrratin~ln cl'l.taill clafillitelmriruls to to itsco/f il IJI(JI'(~ ur Il'ss COIl1-Jdl'tl~ i(JUI)L-11(lellc(" of tho extl'rll;d

atlliusphere. If tlll! airI¡cyolld

thu nuutiul witllin tlmpil: \l'ollld Ilavu 110

to tuusca)Jo, :tilt! the cuutuiuecl coIlIll1n

system 110t sII1dect todi¡.;sipatiou" lit aetllal

tilt! illertiu üf tllC C'xturnaI air

~t?~?~ uf tlw pipe

.?.r~~ builJsiglli/ièilnt, :lI1d tllull viùmtions ou ce caciteti itl tliepipe have Ilul' }Jl'I'¡.;i:tulll'C, '1.'llu Ilarl'O\or tho ch:lIInd of co«1-

'~di.

,),,"f

~.)M,.I ti,a extc.n.J

'< 'c. U~'c. L~n, s.wvvities cOllstitllte

l'usonators; ill tlle lresellce uf <tu ex tu l'Il a1ofsound, tllu

colltviuuul «,ic vibratus in ullisou, alld witll ail

~EE=~and fOl'ced

llcriml,, l'iliing ts .gl'l!atintcllsity iu tlie of

appl'Oxi-

~S~~rI 'SUlla/uryiclds tliu vil)l.~Ltioli.4

111) as it wel'e witllil it,

th~rvIJel'UlIling

~t.m,~s.e..n<i,u-y .source T),c~t.tu~

i,sll10œt.

.s'yH~

-Ic case of

c. <' T~< <~

i"c.~i.s.

t

~i<! of8o tlutt tlm 11l'I'SSllr!! is

absulut.dy COIIstallt. If nosv the

F: is cli:«r thatt.lm CUI¡[aillt'd air will lm aL:111)' tillle

vury nuarly ¡Il tlm (~t¡lIi-~<

dc:nsity)concH!wlldillg to tllr;

Page 169: Lord Rayleigh - The Theory of Sound Vol 2

303.]rOTnXTIAL EXKRCY 0F COMPRKSSIOX. 157

momentary positon uf thé piston. If thé tna.s.s of the piston bc

vcry considérable ni coniparison with that of the induded air, thé

natural vibrations resu)ting from a displacemc'nt wiH occur ncarfy

~s if the air Lad no inerti:).; and in dcriving thc poriod frotn thc;

kinctic and putcntifd cnergics, tLe former nmy bc ca!cu)ated with-

out aHowancc for thé incrtia of thc ni)', fmd tho taticr as if t)ic

ra~factio)i and condensation Wt;re nnifonn. Undc'r thc c'ircnm-

stances cnntumptatcd the air actsmeruiy as a sprin~ in virtuc of

its résistance tn compression or dilatation; thcform ofthc.contain-

ing vessel ia thcrcforc Immateria!, and t)ic p(;ru:)d of vibration

rcmains tiic samc, providcd t)te capacity bo not varicd.

W))cn a gas is comprcsscdor rarL'ncd, th<; mccitanical value of

the rcsuiting displaccmcnt is found hy rnu[t!p!ying cach infinitési-

mal Incrcmcnt of vo]n)nc by the corrcsponding' prcssurG and

intcgrating over thc range rco~uircd. In tho present case It is of

course only the différence of pressure on thc two sides of t!)C

piston -winch is rcatiy operative, aud this for a smaU cha~~e is

proportionat to the altération, of volume. 'Thewhoicmechit.nical

vaJuc of the sma)l change is thé samc as if thc expansion wcro

opposcd thronghout by thc ??;<?f!;?, that is )):i]f tlie nna), pressure

thuscorresponding

to a chitng'! of vobfnn; froin ,S to /S'+8)S',

since ?) = f<

Lut us now Imagine a vcs.se't containing air, ~vliose intcrior

connnnuicat.cs witli thc cxtûrnattittnospLcro by a nan'ow

apurturo

or nc'c~. It is not t!if)icu)t to sce that this system is capablu of

vihmtiuns smiilartot.lto.sc j)tsbc()nH)')u)'cd,tt)e air in thé nei"'h-hnm'hood of titc apcrturc suj~dying the p)acf! of thé piston. By

sufMcicntly increasin~thé pc-riod of thc vibration may be madc

ns lon~ aH wc ptc~se, a)td wc obtiun rinaUy a state of thin'rs in

(!nmrfu-(! (1'~) § 3iH.

Page 170: Lord Rayleigh - The Theory of Sound Vol 2

KtNETIC ENEHGY OF MOTION158

1-

whichthc~nct.cc.c~y.fthc nation

n~ybo n~cctc.) c.~ptt'.c

~hourh..f U.c apeuré, and the potcntial c,.o~yle c.ic.fatc.d as if th.

the

T.~-0

c~~ou t).c two si~.s, or in virtuc .f'it.s own

<L~ aftc, .suc). p,-<ss.n.. ).,s~asc.]. thc. air ,vcs

npproxin~.tefy

;~d'r r" ~s~"v d that t.).. sj.< D.ruug). w),id. t),c )<i,ti,.~T.y i

ill

w).c.L arc ..h.ut t.pr.cc.cd n.t of

..st.)y, t.)arcmwd

r~i"?~

< s < <"J!su~<t)y accm-ntc ca)r.,)ati.~ of tj.c

r,iLch ~'Jn-v ),

=:

isilldufinjt('!ygrullt io

eUlI1pal'iSOIl witll fllo dilJH'lJsions of tlm is

,`~()~1. 'l'lie l:il'ticcll('rg'j' tlie motion uf ail

111(Y71171)!'CSSI))1C

nl;ly lle uXpl'l'ssl'd il terms tliedellsity p~ tlie rnte or

tI'aIlHfel', orcnrrcnt, l', fur mulcr the cil'-

=''=:-?.~ tllc illotioli isal \nj'stll(: S:UIIl'. Hi 1)('('

l' 11('('I'SS:ll'i]y varies as p and as .2, 11'l~ Illll.y put

~~T'~y on U.c ..t.<f,nlt;mm~l, i, a Ji'll'nl'

I~Il;ltltir,y, ;1~lllay 1)(t iufurrcnl t'l'III 11tllCfilet flmt:3 in SP1l<'l' alJd -1 1 in timc. 111

il'Obe tlie\'('!c)('it,y-pntvlItial,

l'y GrL'Oll'S1111'01'('11), w]¡(.re tlie

Ílltl'gmtion i; to hu l'xtf'lllll'¡} 0\'(11'

~F=~

i,sensihle. l1t a sld1il'il'nf clistanr~c~ 01} l.illll'r sicll' uf tlmapl'I'IIII'I'. rpbecOIIIOS

C'ollstnllt, :lIlt! if t1~ constant. \"allll's lm ¡)PIIClt(.cIlu- ~y1 nncl

:EE~

tllat Ilaif nt' fCl\Y<lrd.wlJieh the' {J'liC]flo\s, wc' ¡I,W(!

Page 171: Lord Rayleigh - The Theory of Sound Vol 2

304.] TIIROUGHNARROWPASSAGES. 159

Now, sincewithin is detcrmincd littearly by its surface

vah.cs, or is proportiona) to(~). If wc put.

.Y == c (~~ ~), wo gût as bcforc

Thc nature of thé constantcwitibnhnttcrundcrstondbyco))-

si(b;)-ing t)ie (dectricid probicm, w))osc conditions arcmathonatK-a!)~

ideatica! with thoso of that undcr (Hucossio)~ Let us .suppose thattheHuid i.s replaccd by unift)nn]ycon(]uctingmatcria], an.! t)iat thc

Lun)](]!nyof titu dtanne! or aporhu-c is rcp]aœd h~ insulators. Wc

know that ifhy batk'ry powcr nr utho-wiso, a di~rcnco ofc.Icctnc

tx'tcntial bc maintaincd on thc two sicles, stoa()y currcnt tlirou~htho

apo-turu of propottiona) jna~nitudc will bu gt-nf~-atcd. Thc

ratio of the total currcnt to t)n. ul~ctrontotivc forcu i.s caik-d thc

C(~!</M~/< of the (-hiulnc), and tLus wc sec that our coxstantc rcprusunts si).)p)yt))is cnn<h)cti\'ity, on 1)~'

supposition that thé

sp~Hiccoid.tcti)~ powcr of thc hyputhctira! sub.stancc isunity.

Thc sa.tnct!)~ may bu «tbcnvisccxprc~ud by sa-yin~ tii:Lt c is thc

sicle of tbu cube, w)K)sc rc.sist.ancc bcit\vu<;nopposite f:)c<~ is t)to

.satncast))atoft)icc)unmu[.Lttbcs~~tut wushidioftcnavaii

m))-.sL'lvesof't))cc)cct,ric!t,)ana]ugy.

~Vhf-n cisknuwt), theprnpo-tnnc of tb~ rc.sonatorcanhc

c'asi)y (tuduecd. Since

Page 172: Lord Rayleigh - The Theory of Sound Vol 2

NATURAL riTOt OFJ~OXATOt{S. f'~o-t.1CO

nnd

~ncsd.rc~iyasthchncardhnc.n.sio.. The~vc-k.n~h h.

w.H ho observe. is functiun ufth~sizc. and.shape ofth~

'natoron)y,whi)ot!,efreqncncyd.pcnds al.soupon thc natur.

<.i U.c ga.s a.id it ..simportant <.o rc.nark that it i.s on t).c nature of

g.s n. and ncar t).c ci.a.nc.I that D.c pitch deponds and not onthat

oecupymg the inturior of th. vc.ssd, for thu incrtia of tlle airin the latter situation dr.cs not cône int~ p)ay,whilc the eom-

pres.siblilty cf aiïga.scs is vcry approxunatciy thc .sa.uc.. Th..s In

the cascofapipe, thc substitution

ofhydrr~.n f.)r mrin t).<~

nc.,g)t],ourhood oi- a a~jc wo.dd ,nakc 1)ut )itt)c di~.n-nco but its~rtcct in <ho neighhourhuod ofa !nop wuu).! hc c<.nsidL.rah)e.

Hithc.~wchavcsp~f-u r.fth.hann.)ofcnnunun!ca)!onas

sn~h',hmdt!K.rehcm.~than..n(..d,ann..),t).c pn.Demis nnt

os.scnt.attyaltcn.d. Thc.sam<.for.m.)a~rt).c~.pK-ncyi.sst!)t

npp).cah)G, ifa.s hcforc~ u.)d~-s(a.,d hy c <hc whoie cnnduc-

t'vlty hctwccn the intL.rior aud cxturh.r of the vessc). Whcu thochannct.s are .s~uatc-d

sumcicnt.fy faraj~-t

tn actindcpcndonDvonc ofanothcr, thc

rcsuXant conductivity i.s thcsimple .su.n of

thoscbcinng.ng to t].e

~.paratc cha.u,c.]s; othurwi.sc t)ic résultant!.sJcsstha)ithatca]cn)atedhymcrcad()ition.

If tho-e be twoprccisciy simitar c!)a!)nc].s, which do ])<.<

iiiterfc~, nnd ~vh.~conductivity takcn

.sc.pa.-ateiy is p, wc ].ave

Page 173: Lord Rayleigh - The Theory of Sound Vol 2

~Oi.J StJrEfUOR A~D I~FERIOR L~n'rs. ICI

shcw.ng that thé note is ).igher than Ifthcre wcreonly o,

channct in thc ratio V:~ = ], or by rat.h<.r Jc.ss t).an a f.ftl~-a. ]~Jobser~cd

by Sondj.au.ss and pro~.d tj.eo~.ticaiiy by Hdmho!t/ int''e case, whero thc

d.anndsofconunnnicationconsistofsimntcLofes

u)t]tcntnnitc)ythi;)sidc.soft))<.ro.s<.rvoi)-.

305. Théinvcsti~t; of thc

condn<.tivity fur varions MndsofchanndsLs an

important part oi-theth.cryofrcson.tor.s; b.,t

m a!Ic-.xccj.t very icw c.~ thc ~ecur~tc so)ution of thc nroLi~

bcyo, t).epo~. of

cxi.stin~ ,,mthc,ties. So.nc gcnem[r.nc.pic.s throwin~ )ig).t on thc ~.e.sthu. jn.~y howcver hc Jai.)<!own ~d ,n in.-u.y ca.se.s of inturc.st an ~pp..oxi,n..Ltc solution, .suHi-cicnt for

pt~cttca! purposes, may hc obt~in~

Wc know (~ 2~) th~ thcc.nc~y oF fh.ij fio.vi..

tin-o.~h channel aumot bc grcat.r fhan th..t of.~ny ~ctitiou~motion

g-vm~th~nc total cnn-cnt.Hcnco, if thé channct Lo

r'arrowed inany way, or ~ny r.b.s~ruetion hc

intn,.)uccd, thc con-

<ct.v,<y,.stt.crchy <)unini.sh~,)~use tho attention is ofth.nature oi an additions) con.strai..t. Hetor. thc

change t).cilm.!~freeto adopt thc distribution ofnow

finat!ya.s.su,ncd rncases whcrc a rigorons .sutution cannot be chtainod we

may use t),emuninnm

property to e.sti.na~ an inf.rior limit to thocon.iu<-tivitv.i.c

cncrgy ca)cu!atcd fn,.n a hypr,t)H.tica! law of <)<nv eau ncv.r hoc.ss than th.truO. aud nu.st cxc..c.d it un!cs.s )hc

hynothctica!a.ndt))eactua!niot)ou coïncide.

Anothcr~cncra! principh, Avtuch is of

frcqncnt u.so mav !.r.more

convc.nientiy .statcd in .k.ctrica)iauguago. T).c

qua~ityw.t). wh.ch wc arc concc.-ncd is thcconduc.tivity of a certain con-

<ctorcon.po.scd ofmattcr ofunihspécifie con~.n.tivity. Thc

rrn.c,pic ,.s that if t).cconductivity <,f

any part of t).c condnctor'Je jncrca.scd that of thc who]c is incrca.scd, and if t}.c

conductivi~ofany part hc (hmini.sf.cd that of thc who)e is

dindnishcdMccption bcmg ma.]e..f certain

vcryparticn)ar cases, v-hcrc n.'.-dterat.oncn.suc.s. In its

pas.sa~ Hn-o,~hacondnctorclectricity

'L.strd~tc.s itscif. se t],at t).cencr.~y di.ssipatc<! i.s for a givcn total

cnrrcntthc]ca.stpo.ssib!u(§.S.'). -[fnowD.c.specinc résistanceofany

P~-t bedumn.8t.cd, thé total

(fissipation would bc !cs.s t!.an heforceven if t),c di.strib.ttion ofc..rrent.s rcmaincd

unci.an~dwill this he thc case, w),c.n <he currents redistribua the.n-

.sctves so a.s to makc thcdissipation a minimum. If an

innnite)y~.ir.

Page 174: Lord Rayleigh - The Theory of Sound Vol 2

SIMPLE APERTURES. [305.162

thin lamina ofmathTstrc-tc-hiog across thc channd bcmade

perfcctiy conductin~. tin; rt.-sist.tncu of f])e wholu will bu diminishcd,nn]css t)tc iarnina coincitk) with onc ofthu undistorb~)

cqui~n-ttal surfais. In thé

excc-ptcd case MO cH'cct will Le produccd.

HOG.A))]f)i)~ (1iffu'(;t)t k)n(f.s of cl):u)nc)s !m Important place

!nu.st))Ca.s.s)g))C()tot))0.su cunsiHt.in~ofsim~tcapcrturusinun-

!m)it;c(tpi!uuj\v:).!)sofin<tnitcsin):dti)te!<))L'ss.Inpractica.):q)]))t-

catiuns it is HufHciunt thitt :). wa)! bu vury thin inproporHon tu thu

'-hn)(.-nsi<)t)s of t.)tu a;)urt.urc, aud:t.p[)t-uxi))):ttc~y piano wiUutt a

(hst.Lncu frum thuapurturu lar~u in propurt.iun tu thc i~mc

quant.it.y.

On account of (.hc;symmct,i-y un t)t<j two sides of thc wa!), tho

tH()ti())t()t't!icf)nidint!~p):m(j()ft)tcapcr<,))!-emu.stbc!tonHa),aud tfturef'otc t)~

vu)(jcit.y-}tf)t.cntia.I must bc c~st.imt; over thc

n-)n:undcr<jft.)t~ p)a.!)u thu mution mosL buexc!))M[\'c)j tfn~ntia),

so that. tudtjturtninc on one sidc of thc

piaoc ~e h:tvu thc

coudrions (a) = const:mt ovcr tl.c ~pertuj-e, (/9)= 0 over

H~

thc rcst uft))c phmc cft!.c waH, ~) ~== constant at ititinity.

Since wc arc conccrncdon)y with thc di ~renées

of \ve may-st'ppf'.so tt.at at

itifinity vani.s].c.s. It will b~ .sccn Htat condit.i..u.s

(/3) and (y) arc satist.ud hy supposing to bu t)tc potoitfat of

attractif natter diHtnhutcdov(;rt))capc!-turc-; t!te rcmaindcr of

thc prubjon cunsists i)idcterïni.ung thé distribution of mattcr so

that its putcntial may bc constant over thé .satnc fu-ua. Tho

proi.icm ism:Lt]t~tn!Ltic.i)ty thc saine as that of

determining t].cdf.stnbntiou of

cluctricity on achargcd eonducting pjatc situated

"ianopcnsp~cc, w))osc fcnti is that of t))c

~pcrtm-c nndcr con-

stdLration, .nd t)iucooductivity (,f thc apt-rturc tnay be cxprc-sscd

'n ~i-ms oftj.ee~x~ of tiju piate of thc statical

prohicm. Ifdc-untc tho constant, potcntiat in thé a.pL-rtnre, thc ulcetricai

résistance (fur onc side oniv) will bu

thé intégration cxtcndmg ov-cr thc arca of tlie opcning.

AI ff~!

jj~~=27rx(whu)c qu~ntity of mattcr Jistributcd),

and thus, ifj~be t).ccaj~eity, or charge corn-spon()i,,gtounit-

putc.nti.d, thc total rcsi~ncc is (7rJ/)- Aceordiugly fur thé coa-

Page 175: Lord Rayleigh - The Theory of Sound Vol 2

~1 J ELLrr'riC APERTURE. T~~

ductivity, whieh is tlie rcciproc..J of t).G résistance,

So hu. as 1aw.ro, tho

eitipse I.s t)ioo.]y f.,nn of apertm-cf.r winch cor Veau be dcter.ni.ed

thcorctijy', in w!n~'.c rc-.s.k is ~ch.d.1 i. t,

1~.?'e.Iip.conduct.r.

l'.u n th. h et th.~ a shujt Loundcdby two

conccntric, si.nihu..uds..n!arly

~d.H.p.suids exert. uo f.rcc on .n i.tcru.)

p.t..ie~.sc..sy to sce t)..t t).u sup~ci.I dcn.sity at

.uypoiu~f. ne.?-'

~dnece.ss.uytu

.i~ ~j., j~P

Pc peud.cui.r ~) let <)~~J tallgentLe pc.nt in

.sti.u. Tj.us if bo thédunsi~, ;=whole

qu~L.ty ofnu~cr is givcn by

.iwenowsuppose

t~tci.sinfinitdysma)!, wco~unthcpar-t.cuhu.c..se of' an

citiptic pi.to, and if uolungcr disth~ui.~

Lutwccu tlic t\osurfaces, wu gct

Wu !,avc nc.xt to find thé va!ue of t]ic const.-mt potcnt.al (P)

~con~dcring

thc value of at tUc ccutre of thc p!~ wc sectliat

(c~IsnTT"LyHoi~h.itz

~ruiL, j~t. ~7, l8),0), whnHo rusutt is cquivn)t;nt tu(.S)

b~ë fur the momeut tho thirj principe axis of thu sHipsoid.

11-2

Page 176: Lord Rayleigh - The Theory of Sound Vol 2

1C4 ELLIPTIC APERTUHK.['30G.

nsthcf)na] expression f()rt.))cc:)pf)c!tyofnne)]ip.se,wh(~cscn)i-

t))!)j')raxi.si.sf!nth)cpc~)~ricityi.se. Jnthcpnrt.icolarcnsbofihc

circle, e =0, ~'(e)

=~7r, and Ums for circ)Li ~t' radius 7)',

Page 177: Lord Rayleigh - The Theory of Sound Vol 2

30G.] COMPARISON WITH CIRCULAR APERTURE. 1G5

From this rcsult we scn that, if its ecccntricity bc smd), the

oonductiv~y of au eiïiptic apcrturc is very ncar]y thc .samc astitfLt of a cirotiiu- aperture o/' e</?<~ ~-e< Among various furms

"t'apcrturc ofgtvcu m-cn. titcre inust bc ouc whic)t has a nnnimutu

conduetivity, a)i(], t]mugh A format proof ]night bc <)ifHcu)t, it. is

c~sy torcœ~ni.sc t)iat this eau ho no ot~cr thati thc circle, An

itifurtor limit to thé value of c is thus a)wa,ys ailui-ttcd hy tho cou-

'Livlty of ~e circle of c.~al arca, that is 2 and ~hcnV \7!

t)ic truc furm is ncarty cu-cnt!u-, t]ti.s H.mb may bc takcu M a closeapproximation tu thé rcat v~nc.

Ti'c vainc of thon givcn ))y

In ordcr to shuw iiûw sti~ht)y a moderatoccccntnci~y~cc~

t).c value of c, 1 hâve c~cuktcd thé foHowing sl.ort tab)c wit)i thc~d of L~en.h-c's va]ucs of 7-). Putti~- e=si.i~ wc hâveeus as tho mho of axes, tuul fur the condnctivity

e==shi~. ~eoH~. 7r-27''(<')(l-e~.

o" -ooooo i-ooooo i-oooo

-~204 -!)39<i!) l.ooon30" -50000 .8GG03 1-0013

-C~79 .7~04 1-OO.it5~ -rcGOt -C427!)0 1-0122

~0" 'SGG03 ~0000 1-0301 l

70" -939M!) -3.~03 r07i~~0" -98481 -173G5 l-l'J5-t

90" 1-00000 -00000 co

Thc vainc of t])e last factor ~-ivoi in t!ic fourtii column is thoratio of thu

eonJucLivity of thc e))ip.sc to o/' ci')-c/e q/' ~)<f~f<e~. It appears that cvcn whcn tho cl) ipso is so ccccntric tha.tthc ratio of t])e axes is 2:1, thc conductivity is incrcascd byonty about 3 per cent, which wou!<) correspotut to an attentionof littic more th~n a comma (§ 18) m tho pitcli of a reson~or.

Page 178: Lord Rayleigh - The Theory of Sound Vol 2

CALCULATION BASED ON AREA.166

[306.

Thcrc Hccms uo rcason to suppose that this approximate in<!c-

pendence of shape is a property pcculiar to thé cHipsc, and we

may condudc wit)t soinc conHdcnce that itt thé case of fmy mode-

r~t,c)y c!o))~tcd oval aperturc, théconducth'ity may be calculatod

from thc arca alouc wittt a co))si(tcrab!u dcgrce of accuracy.

If thc arca bc givcn, therc is no snperlor lunit to c. For sup-pose the arca o- to be distnbutcd over M cqual circtcs

su~ciuntfyfarnparttoact iudcpGndcnDy. Titc arca of cach circle is M-and its comh.cttvity is 2 (~) Thc whole

ccuductivity is )~titues as grcat, and thercturc incrcasc-s

indefinitcjy with 7;. As a

gc-]iGt-:d ruic, thc more thc opcning is c]ungatcd 01- Lrokf;u up, tho

grca.ter will bu thccunductivity for a givcu arca.

To find a supcrif.r linut to thcconductivity of agivcn apcrtm-e

wc may av~il oursctves of t!.cprincip)c that any addition to thé

apcrttu-c must be attunded hy an incrcaso in thé vaine of c. Thisin thc c:MC ot'a

square, wuinay be s..rc that c is ie.ss th.ui for t)te

c.rcutnscnbcd circle, and wo ),aveah-cady scen that it is <r,.e:iter

than fur thc circlc of equal arca. If be thc side of thé s<, ~u-c

fhc tones of a rcsonator with a .square aperture calcntatcd fromthcsc two )nnits wo.dd difïcr t.y abont a. wi.otc tone; thc ~-avor oft))cm would <toubt)e.s.s be muett t!ic ncarcr to thc truth This

exampic sl.cws t).at cvcu w)icnanajysi.s fai)s to give soutien in

t).c niathe.nat.c~ sensc, we ncpd not bc a)tugct),cr in thé <hu-k asto thc magnitudes of-thc quantiLics with whicii wc are deaHug.

In t!ic case of si.mhu- orifices, or sy.stcms of oriHccs, c varies asthé hucar dtmcnsiou.

307. Most rcscnatnrs uscd in p~cticc hâve nccks of ~-GaterorIc.ss length, cvcn w),cn thc.rc is

nothing that woutd be ca!)edneck, thc t)ne].ncss of thc sidc uf t).c réservoir cannot a!ways hcnc~ccted. Wc .s].~ t!)erefore examine tf.c

conductivity ofchanne! formcd by a cytindrieal horing tfn-ough a.i

obstruetin.plate b.~undcd hy para))d planes, and, thun~h wc fait to solvc t)~prob)cm r.gorousfy, we sh..dj ubtain information .sufHcicnt for mostpract.ca! p.n-pn.se.s. Ti.u thick~ss of thc p)ate wc sha!! eall Z andthc radtus oft!(c cytindricat ch:n)!)d /)'.

Page 179: Lord Rayleigh - The Theory of Sound Vol 2

307.] CONDUCTIVITY 0F NECKS.IG~

Wh~tcver tho rc.si.s~nce of thé c~nnc! mn.y bcwill bc lessened by tlic intr.xtuetiun of Iufinitc)y

Dnn d)sc.sofperfucb co~tnctivity m n.n,) ~)

TI.e ctï-cct of tlie dises i.s toproch.ce constant

potJntudovcr thon- are~, ~d t].e pruDcm thu.s inodiflud is

su.sccpt.b)c cf rigDrou.s sointiun. Outsidc J .~Ktthc motion is thc ~ne as tL~t

prcviou~y invcsti-

~tcd,wh.n thc

obstn,cting plate is infim-tciy t.hin.hctwucn ~ud thc ftow is unifunn. Thc rc.~t-'a-nco jti t))ercfurc ou thc whoic

Th.s correction is in générât undcr thc mark, but, whcn Z is

~ysn~i,ri cuntp..u-isonwith~ thc a.ss.nncd motion coincid~

n~-e

an<hnurc n.arjy ~.ith tho .ctual nation, ~nd t)n,s t!,o va!ueoi a: m (2) tonds to buuotne correct.

A snperior limit to the résistance mny be c.-dcu!.tc<! from

I.ypoth~c..tmotiun of thé Huu). For t).is pm-pose wc will suppose

n~n.tetyt!nn p.stnns introduecd at Yl and t!.c cr~ct uf which

will be to inakc t)~ normat vdocity coa.st.nt nt those p].sWithu, t).e tube thé How will t,c. unifor.n a.s befo~ but fur tlieexterne space wc Ih~'c a ncw prcb)em tocon.si.tcr–To .L-ter.ninctbe mot.o.1 of a fh.id bounded byan i~nitc phu.c, thenor.n~vclocity over a cireur, ar~ of t).e ptanc having a givcn constantvaiuc, and over tbc rc!n;undur of thé p)anc being ~cro.

Thc potential may sti)t bcrogardcdasd~tomattcl-distribntcd 1

ovcr tlie di.sc, but it is no longer coûtant over thc arca; tt.e~6~

of thc mattcr, I.owcvcr, being proportional to is constant.

T)ie kinctic cncrgy oftttc mûtiun

Page 180: Lord Rayleigh - The Theory of Sound Vol 2

I<jS CONDUCTIVfTY 0F NECKS.[:~7.

If thu dcnstty uf th(; tnatt<r Le takcn as unity, ~=27r, a)ul

thc ruqxir't! rath)t.s~xprc.s.hy

-t,wta'rc ~dcnote's t)tC

p')tc))t!:ttt)))it.st']t'<'ft).('h'(.'u!;)t');)y(j['L'fj)):t<tei'(jfuniLdL'nsitynndcf')'!h)i)[s7t'.

T))csi)))[)k'.st)nL't])<)(l<)f(.dun].L(,it)~<)o))(')KLsup")it)icc~n-

si~crnt.iuit <,)).')t itruprt.'MCtitMthewo'k i'c'([U)t'(jdt()Lruak un thc

(Hscintoiufmit.L'HunaIc'iL'mcnt.s a)i(t toi'L'movcthL'tnfruni cncit

othcr'.sinOucnct' Ifw(.!t:d<:cp')]:n'co-r)t'(]it):itc.s (/?,~),thHp")c

L)L'i))~).tt))u(;~Q"t'<tL'tU.scwh'jne)'a<n))sisH,L! )tavc for t))C

putent.ia.t atthu put(. )''=;)'~6'< t]K: !i)))its<ji'p bum~Oand

H(tc<js~tnd<)tus<of~bc'in~7rand-t-.t7r.

T'sr=~(:).

Now )ub us eut ()<)':).sh'ip uf brcadt.)t f)-o]n titc cdgo cf t))c dise.

Thé work j'c<[uin'd tu rmuovu titls to an inHuite di.stanco i.s

~7r«~t.4H. If we gr;u)u!tl)y parc thé discduwutouot])ingfu)d

canyidi thcpiu'ing.s

toinfixity, wc fhid fur thc tot:d work hy

intc~r~tij)~ \Itft rc.spL'ct tu from 0 to /<

A part of !i<)'J is r<'jtcntcd hnre for thc nnhc of thoso who niay wish to avutti

tht'(im!cutU(.'softhM)t)ur(.'cf))nplt;tuin\-(.'i,t)Ht~H)n.

TJiiti tncthoj uf calcult~tiug was st~ëcstej to thc outhor by Professer(.'tf'rkMnxwci).

Page 181: Lord Rayleigh - The Theory of Sound Vol 2

307.j CORRKCTION TO LENGTH. 1~9

It must bc observai th:~ a hore dcnotcs thc con-cctiun fur oneend. T)te who)c rL-sistiUtce cotTfjspottds to a. !cn~th Z+2a oftube Iia-vin~ the section 77 7~.

Wftcu Z is vary grcat m rci.Lt! to 7~ ~'c may tf~œ slinpiy

Tbe correction for an open ond (~ is a fonction of eo.ncidin~-WithU.e

!ow<.r!imIt,vi~w))unZvani.s!h-s. AsZincrca~mcrc.a.seswithit; butdo~r.ot,evenwhcnZis infinitc.attain

thc snpci-ior limit 7~. For consi.lcr thc motion going on in any

n'iddie piccc of thc tube. Thc kinctiecncrgy is grcatcr than

cf.rrcspuitd.s incrciy to thc lo~tli of t).c picc-c. If therefore thc

piucc be ronoved, and thé frce cnd.s brougttt togcttier, the motion<jthcrwise conthmmg as bcforc, thé kinetic energ-y will hc dimin-isited more th:ui con-e.spond.s to tt)c fength of thc pièce subtmcted.

~~?'i't'M~ will this bc true oftite real motion which would exist in

tbe sitortcned tube. Th~t, wlion Z = ce, a docs not becomo is

evident, bccausc thé normal vc-tuclty at thé end, far from beingconstant, as was nssumcd in thc calculation of titis i-esn]t mu.~increaso from tiie coitrc out~-ards and becomc Innnite at the cd'~e.

A furthcr approximation to tlic value of a may be obtained byassuming a variable vclocity at thc plane of tbe mouth. Théc-alculation will be found in Appeudix A. It appears that in théc;tse of an Innnitdy long tuLoeccannot bc so gréât as '82422~.T!tC real value ofa is probab!y not far from -82

308. Eesidcs thé cyHndcr there are very few forms ofchannci -whosc conductivity can be dctcrmined

mathcmatica!)y.

W))cn howevcr tlie fonn is approximatbiy cylindricfd we mayobtain limits, which arc usefut as altowiug us to cstimatc thc cHect

Page 182: Lord Rayleigh - The Theory of Sound Vol 2

TUBES 0F REVOLUTION'. [308.1~0

of snch dcpartures from mathema.tica.l accumcy M must occur I[i

practicc.

Au infurior Hmit tothorcsistanccofn.tiyc)nngatRdn.n<t~pproxi-

matc~y Htt-:u~))t c(jn()uct,(jrtnay be obtait]~)

ini!)ic()in.tc)y by the

i))ia~itj!uyintroduction of ait mftnite numbcr of piane po'fccDy

co)i(h)cti))g' inycrs pc-rj)nn<ti(j))iar to thé axis. Ifo- dototc tLe arc;).

of thu ficction at nny point y, tho rcsist:u~c LctwL'ca t\vo layursdi.stunt (~ wiH bc o-(~and thuruforc thé whotc actu:).! résistance

isccrtain!y g)'cat(jr

thaii

un]ess indccd tho conductur betruly cynndricn.1.

l'i or<]er to find asupcrior Innit wc

n~y c~cu)atc tlie kinctic

cncrgy of t)tc cun'cnt on t))c hyp~thcsis tii~t t)tc vutocity pM-~Oclto titu axis iH unifurm over c~c)i .st.-(.-t.!on. Thc

!)ypot))eticaJ motiuji

tH thi).t whi<;)t wou)(t fuliow frotn thé intrnttuction of a.n inHnito

munbcr of n~id pistons tnuving frecty, :),nd dtc calot~tcd rcsu]t i.s

ncce-ssiu'ityin exccss of thé truth, untess thc suctiun he :d).so)utc!y-

const;u)t. Wc shaH suppose for thc sakc ofsimplicity that t)tc

channol issynnnutricfd nhuut H)) axis, in which case oi' course tho

motion of'thc nnid is synnnctricf).! :d.so.

If (IcnotR the total cun-cnt, wc hâve M for tho

n.xial vc)ocit.y a.t any poitjf rc

Page 183: Lord Rayleigh - The Theory of Sound Vol 2

308.] SUPERIOR MMIT.1~1

This is tho qnMtlty which gives a superior limit to tlie resist-~nco. Thc first to-m, which corrc.sponds to thc componcnt velodty?<, i.s thc s:m)c a.s t)tat

prcvious)y obtamcd for the lowcr Junit as!~ht ttavc b~-cn forus~n. Thé difÏcrcncc butwecn t)ie two, wh'ichgivcs t).c utmost en-or invutvcd in

tddng eit)ier of titGH/:ts thotruc vatuc, is

Ina ncariy cyHndrical d.annd is a sma]!

qnantity and so

thc rusult found in this manucr isclosely approximatc. It is not

ncœs.s:ny U~t the section .simnid benca.r)y consent, but

only thatit shouid

vary s)uw)y. 'l'lie success of the appi-oxi.nati..n in thisiu.d snni! cases dupends upon thc fact ti)at, tfto

quantity to Le~tHuat~d is at a ttnnitnuin. Any rc!tsonab!e

appmxim.ttion to thcr~) motion will ~ivc a ru.suft vcry uear thc truth

accordinrr to thc

p)'mcip)cs of t)m din'urentia! calcuhts.°

By mcans of thcpropertics of thc potcntial and strc.tm

ftmctions thé présent probJou adtnits of actualapproxitnatc

suhttion. If find dénote thé values of titcsc ftitictioiis at anypoint 7'; M, dénote tlie axi:d and transverse

vclocitics

Page 184: Lord Rayleigh - The Theory of Sound Vol 2

173 APPROXIMATE CALCULATION. [308.

If 7'\Jcuote tho vn.tuc of as a function of x, v'hcn = 0, thc

général values of <~ a))<! may bc expi'CM.scd in ternis of by

m('n.nsof(7)a!id(H))nthc,(.'ric.

i.s thc équation conncct.mg y and 7~ lu tlic prc.soit proUon yis

gtVtjn, au<t wc ]iavc tu express hy ])tc:m.s of tt. By succcsijivo

approxnn:itiu)iwcubtainiro]n(l())

Jhc expression for thé resisLmcc n.dniits of considcrn-bic si)np!I-iic~tiou by intcgra.tinn by pru-t.s in thc case whcti tho channct is

truly cylindrical iu Utû nui~hbotu-hooLl oft!)u !iniits of intcgrat.ion.I)i tUs way we Und fur tlic fi)i~ rusu)t,

dcnoting thc ()iir~rcni.i:L) eoc~n'iolts ofy wiLh respect to a'.

It thus nppc:u-s th:it thc supcriL'r ]umt of thc prcccd!n~

mvestign.tion is in f~et tho cun-ecL i-L-su)t to L)tc second ordci- of

r~'OCCCfh')~f'f~/f; /.<))t)yott~/Mf/)<Htf;<;<:()/.S'OC't'/y,Vo). Yfl. Ne. f~.

Page 185: Lord Rayleigh - The Theory of Sound Vol 2

308.] COMPAmsON WITII EXPERIMEKT. 1~3

npproxun~ion.Jfwc

regard 2/as a fonction of~herc ~isas.n:dt

'tu~tity.n~)IsMrr<.cmsf.i.sf(Tu~cont~,n,

30.'). Our]<n<~v)..)gc c,f t)ic j.vs on wf.ieh t).o pitch of

resunators dcponds, 1. duc to tl.c i.hour.s of scvcndcxpcr ncntor.s

:mdm:).thcnt:t.t)ci:ms.

T).e oh.surv~iun <h~ fur.nouthpieec H.c pitch of.t

rosun~or dupcmLs~uuiy upun thc vu)mn. is duc to

Li.scuvius..1.0 f..und th.t t],c pitch ofM part)y ~)cd wit]. watc.r

MtaUcrcdwhcnt).ci)n.sk w.-LS iuciincd. Thi.s r.suk~sconf!nncd hy Sond).aus.s\ Ti.o )~~r observer f.und iur~cr, th..t i,.thc case o< r..son~or.s witf.out neck.~ thé influence of thé aperture

d~pcndcd .nandyupon its ..u-<t, ,dt).ou~i ~hcn thé ~po vc..y

chjng~tcd, a certain ri.s~ cf i.hc]~ u.s.ted. Ko g. t].c formu!~

thc unit uficn~th bcmg t)œ mii]imctrc.

Thc ~Mry of Uns kind of rcsonator wc owc to Hdmholt~whorieiornuu~ts

ln pmct.cc it doc.s not oftu.i h.~pj.un dt).crt).at thé ucek).s so long that thc currc~iun fur t).L. opui ends ean bc )K-]cctcdns

(4) .supposes, m, on thé othcr hand, so short U~t~b c.uitsdf bc ncg!cctc.), us

suppose.! i~(~. \Vcrt)R.i,n< ~s t).c first

7-T.

'r cubi~hon I.feif~.l'npll. :1 rrn, r.xxxr.

=Crc))c, Bd. n'ii. 1–72. 1S(!0.

U.ber dk.Seha)I.~hwinK,n ~r Luft in c.rhit~.n Glasri.Lrcu u~ in .edecktpn l'f~fun von n~ichcr Wuitn. 7'),t.y~ix. l.S~f)

Mcn.uire Hur )c.s vibrations sonorM I',tir..)

Page 186: Lord Rayleigh - The Theory of Sound Vol 2

HELMIIOLTZ'S INVESTIGATION.[309.

174

to shc\v that titc cffc'ct of an open end codd bc reprcsoited byan addition (~) to tim leugt,h, indcpundt.'nt, ~arh.' .o, of

aitd\.

Thé approximate thcnrctica! dutcHnination of N is due to

Hu)t)dK')tx, who gavu 7r7~ as thc correction fur an opcn end

<itt.ed wiLh an infiititu Hangc. His mcinud consisted In invcntingi'ormH of tntjt! for which thc prubfon was so)nb)G, and scluctin~tt).t.b onc w])ic)i agrcud most m!:u-)y wit)t a cy)indcr. Tite cor-

t-ccti~)) } 77- i.s ri~orunsty applicable to a tnbc whosc radins at thé

opc'n end and at a ~ruat (Ustancc ft'otu it is 7/, but whic!t in tho

HL'i~'hbu)))'])um) ofUte opun end bulbes .slightiy.

From t))c tact that thc true cy)indcr !nay bc dcrivcd by in-

t)'tj()ucin~ an obstruction, wo u)ay infur tbat t)ie re.sult thus obtanicd

is too smal).

It is curions that tbc proccss foUowcd in this work, which was

Ht'titgivcn

in thé menioir oa rc'snnancc, leads tocxact)y

thc same

r<sult, thon~h it would be dif~cult to couccive two mcthuds more

uniiko c'ach othcr.

Thc correction to tbc Icngtb will dcpcnd to some extcnt upoo

wbeLhcr t!iC itow of air front thc opun end is obstructcd, or not.

Whcu thc ncck projccts into opcn spacc, thcrc will bu less ob-

strnctiun than whcn a backward Ho\v is prcvcnted by a nange as

Sttppo.d in onr approxinmte caicn!:<.tions. Howcvcr, thé un-

c<rt,ainty in<rodnccd in this way is not very important, and we

may gou;)'a))y take a=~'7r~ as a sufncicnt approximation. In

practicu, wht,'ti thc nccks arc short, Die hypothesis of thc Han~e

a~rccs prc'tt.y wc)I witti tact, and whcn thc uccks arc Jong, tlio

curnjcLion is itSL'tfofsuhordinatc itnportaucG.

Thc gênerai formula will thuu run

whure <r Is tlie M'en uf thc section of thé ncck, or in numbers

A formula. not dtffuring mneit from this was given, as thé em-

Loduncnt of tlic rcsults of I)!s )neitsuru)ncnts, by Son<n~uss' who

'?..i"cxL.53,219. 1870.

Page 187: Lord Rayleigh - The Theory of Sound Vol 2

MULTIPLERESONANCE. 17530D.]

at tlie .samc timecxprc.s.s.d a eunviction t).at it was no mère

".np.n~)~mu)aofinturp.)atiM,L~t!,o.u..n, ,,atn,

Incti,uury pf r~sonators wib). t.ucks wa.s

give-a abo~u thé~.nc tnnc lu a mc.noir un Rc.s~nance pub~hud in ti.c

fur1871, fru.u w].icit inust uf thu JasL icw

p~cs )s durivcd.

:310. Tho.simph .nethod of

c.~c.uh.tingth. pitch ..frcsouatorsw.di ~.ch ),~c bccn uc<picd is

.j.),)iu.bfc to thu ~vc.st.~cof v,br.tion ou)y, ti.o ch.r .,f

~.ch i.s.juitc distinct.'ho ov.Ttu.cs ofrc.s<.n..Lur.s witll eontrac~d ncch.s ~.o

n.Iativdv~y h~h and tlic

c.,rrc.spon.]i~ un.des of vibr.~iun arc1)y no

'nc.a,.s.ndupcndcnt of t).c inertie of the ..ur in tho intcrior of the

r~rvoir.

l),c ci.a~ctcr of thc.sc mo<).s wi)) be more ovidentK.n wc comc to considcr H.c vibrat.-ons of air within a cu.n-'

]. c y c!d v~d..such as asphorc. but it will

nu.dy h.ppcntllat thop)tch can be ca!cu)atcd

tJ.eurcti(;:diy.Ti.crc arc, howcvcr, cases uf

,n,dtip)c rcson~cc to which ourt'.cory is

app),cabJe. Thcse occur wf.un two or ~orc vcs.sd.s c..n-"atc

by channcfs ~th each ot).cr and wit), thc externat air..d .u.crcaddy trc.atcd

Ly L.rangc. n.cth.d, pro.idcd «f courseti.at thé

wav.tc.~th af thc vihrati.n issuf!icic.ntjy I.rge iu com-

p.-tn.sonwitht)~di.Hc.iu,).suft)tuvc..s.sd.s.

Suppo~ ti.at thc.rc arc tworc.s.rv.irs.

con~nunicatin~~it!. cach othcr aud with tl.c c.Lcrnat airby ~arro~

passage, o~

ncc~.If wo wero to con.si~r a. a sin~c réservoir an.! ~pp]y

~f prenonsfurmuf~

wc shouM bc !.d to cn-.n.ou.src.snit fur

:~tluLt formula is fOlllldcd an tlleaS~lImptiou tll1Lt witlia the rescrvuir

,~hatiormu)~sf.,unded may thé ~u.nption t).atw~h:n tho rc.sc.'vcir isthé u.crti.. of thc air j.~ bc Jcft eut of ac.our~, ~hcr~ it Iscvnicnt that thé

cne~y of the motion t)n-o~h théconncctin~

pa~gc iuay hc a.s ~rcat a.s through thé two others.Ho~cv. a~

~')-uc~t' (~ </te~(~(~ .S-~i(. Koy. 2.i, 1870.

Page 188: Lord Rayleigh - The Theory of Sound Vol 2

17G DOUBLE RESONATOR. [~~0.

invcsLig~ionon thé Hfune K'c~l P~"

~'°

perfuctiy. Dcuoti)~ by .Y,, A\ t])C totat tr:u).s~.rs of Hnid 1

tlirou~h thc Un-ce pa-s~gcs,wc Lave as m (2) § 304. fur thc kiu(jtic

cncrgyt))C cxpi't's-iion

as thc cqu:ttirm to détermine thc nutund toncs. IfJVbe thc

frcqucucyof vibt-~tio)), ~=- ~e two values of~" bcing of

course rca.1 and ucgativc. Thc fonnu]:). simptifles considcr~Myif

~=c,, ~'=<S'; but it will be inore inst-ructivc to wurk eut this

case from tlie bp~inning. Let = = ?"~= ~'c.

Page 189: Lord Rayleigh - The Theory of Sound Vol 2

~J noUBLE RESONATOR. ~y

~'hichrc.juirc.s th~

= 0. Tho motion is thei-cforc thé .sa.nc

'night t.:Lkc p)acc wcro thé Ct))u)nun!c:tt:on bctwccn ~'an<~S"cutofT, aud bas its

fi'Gfjuuncy ~ivcn by

~1 ).c vibmtion.s ~ro tl.us opposcd m phase. T!.e ratio of frcqucncicsis ~vcn Ly~=~+2: s~cwin~ th.tt thé second modeh~ thc .shortcr penod. In this mo.Ic of vibration t))c

conncctinrrpassage acts m .somc mcasurc as a.second opcning tn hoth ve~c]~:md t!)us nuscs t])e pi~-).. If thc pesage hc contractcd. thc intervatof pitc)i bctwecn thé t\vo notes is stna)).

A pfu-t!cu)M- easc of thc~cncriL) fonnuia

woi-t))y of notice isohtfuned

byputting~=(), ~Lich amounts tosupprcssins one of

t))c commumcatiuns with thc cxtcrna) air. Wc ttms obtain

n. ir.

Page 190: Lord Rayleigh - The Theory of Sound Vol 2

178 PAUTICULAR CASE.~310.

It appcars that thu into-v~ from JV, to A~ I.s tnc samc as from

~to~,namc)y,(2'(JlS)=l'(i1.s, or ratbcr more th.in a fifth.)t will be fonnd that wh~~vcr thé vainc of w ]nay hc, <Lc httcrv:d

hetwool thc two toncs caunot bc 1~-ss th:in 2'4.).i., winch IsaLout

:m octave !uid a minor titini. J'hucon-c.spunding vaiuc of?/t is 2.

A sunilar mcthod is applicable to any combination, howGvcr

f'otnp)icat.u(!, of rcscrvuirs audcunnecting puisages uado- thc

~in~c rcstricLioH as tu thu comparative magnitudes of t)tc rpser-voir.s aud

wn-c-Icngth.s; Lut Hieexamp]c just ~ivcn is sumci~nt

to iliustratc titcthuory of']nu)tip)c résonance. A few mensure-

)UL-nts of thépitch of duubtc rcsonators arc dctailcd in

]ny rnc-moiron rcsottanc-c, atrcady refon'L'd to.

:]. Thcéquations winch wc ))ave cmploycd Intticrto t:~c

no account of thc (-scapc of encrgy from a resonator. If' tho-cwcn-

rcnHyno transfcr of

cnurgy ~ctwcon a rcsonator and thc

cxt'jrna)attno.sph~ru, t)tc motion won)d bo isoiat.cd and of IItUc

jn'acticat iuterc.st; nuvcrthcic.s.s 1)10 characteri.sticcfa rcsonatorru.i.sists In it.s vi))rations b~in~ in

~rcat mca.surcindcpendent.

Vibrations, once c.xci~d, wii! c-onti.mc for consiuL-riLbic number of

pt-riods \vit!tout nmoh ]oss of rncrgy, and HK.ir frc.mcDcy will bcat.nost

cntiœfy indcpc.ndcnt of t).L. rate ufdissipation. Tbe rate

ofdissipation is, howhvf-r.an important fcatm'c in tLc chamcter

Page 191: Lord Rayleigh - The Theory of Sound Vol 2

~-1 COMMUNICATION 0F ENHH(.Y. l~:)

~f rcsonator, on whic). it.s bcL~io.u- undcr certain circ.tm.stanc~.~tcnaHy <!e,x.nd.s. It ~j)!bc n~crst~!t).~t t).c<]i.ssip~ion'.cro spoken uf n.e~ns c..)y tii.. cscape «f cner~y fn.m thu vc.Iaud its

nc~),bour)tood, and Its ditl-us~on in thc.s.n-roundm~

~n~hmn,and ~ot. t).c tr.sfu.-nmtion of

o,-di,.a,.y cncrgy into ),~Oi such tran.sforn~tion (nu- c.t..atio.).s tako no accuunt, uniussspcc.at tcnns bc imroduccd for t)u- purposo of

roprcscnLi, t].cfïucts

ofvi.cosity, and of t!,e cond~ctb.i and r;u)iatiou of hcat.

Cl.

In ~rcvicus chaptor (§ 278) wc .s.w ).ow L. cxp.-o.ss thé motion<'n thc r.~t ,f Lh. i.~nitc H.ngc (1~. (n). in tenns oi-ti.c ~rn.dYcjoetty of the <huJ over tlie di.sc We foun.), § 278 C:~

~cre~iHp)'op(H-tion!(I<oc'

If r hc t),o distant bctween any two p.,ints of thc .)i.sc, ~<. is.s'n-tU

qna~.ty, an<} 6--=1appruxi.n.~civ.

T~. nrst term <)op.n.).s npon thc <!istribut:on of thé c.n-cnt. Jf

wc suppose th.~ isconstant, wc obtain u]ti,natc!y terni rcprc-

~nting anincrcasc <~f I.c.-tia, or a correction to'thc Icrgth.

oqua! to Thi.s ~-c j,avcn)rc.~)y considorcd, undcr t],c

s"p!.os,tion of pi.f, ,t r~~

'"s.s.pat.on ~.pcnd.s, i.s ind.-pc.ndcnt of thc distributiun of current,

')~_2

Page 192: Lord Rayleigh - The Theory of Sound Vol 2

.RATH 0F DJSSÏl'ATrON. t':nt.180

hc'in~ a fonction of<.hctct,a) cxn-cnL(.Y)nt))y. Coxfitiin~our

at-(.t.'ut.i()H~)<.)n.st.t.')')n,W('i)ave

Thé ('f))')'(;sp"n()Ii)~work ()om; (hn-ingit.LriULsj'ur of ()))!<) ë~Vis

(J7I~ j'. ~Lllll siacu, ;~s in :31)~ tlie exlressions fnr t~üe Imtent,i;~l~V; fm~ sincc, as It) § 30-t, thé cxprc'sston.s ftO- t.hc potentat

nn(ikinetict;)iC)'gics:n'c

in phtcc of (3) § 30~. In thc valuntiun of c nu aDowancc nut.st hc

i))C-t!)(i(~fu)'t))uin(.ti:).(~t't,h(' <!ni(tont))cright-I):U)(I sittoof-~t,

<'o)'rc.s])f)))(!in~ <,o tim tcr)n oxtittcd m th~ expresMon fur~).

-K<)na<,io)i(;')) is of thc stamLtntfoDnfurt.hcfrccvihmttons

f)f' (1i.ssip!tti\'c sy.st~ms ofnnu dc~rcc afi't-cc~om (§ 't5). T~e

M'

:m')phtm!cv:uiL'sa.sc't'l)~i))g(Hmmished itithu r~tic e:l 1

af'tcr:). tnoc&quatto Ift]tcpit.L-h((]ctcrminc<! by?<)bc

~ivcn, )))(' vihratiotLS hâve thc ~re:ttcs<,p<')'.si.s<onccwhc)i c in

mn!~fL'st.,<.h!).t.i.s,w))L')t Lhc )tcck isjnoutcontractL'tt.

If )S' t'c ~i\'ct), wc !i:tV(.t unsubsLitnting fur c its vahtc in to'ms

ui'~and;

shc\vi))gt))a[.tm(L'rthnnccircuni.st.;uiCt'Ht.])U(h[['~tiu))('fL))L')t)<~ion

it)C)'tsc's)':q)i'Hy:t.s/t(HtninihihcH.

Jnthcc:).sc()fsin)i!art'e.sot)at('r.sex7<):UKnLcn

'K.)Uftti(.n(.)it<<~)]ynpprf.xin)tt~nrLsmuc]tnHthodissi[)ati\-c force iscfttcu-

Ltt('dont)jt;H))p])(~iti~nt)t)ttthavi))rati(~)isj)ermnuentiLntthiswi)Ucndtono!n)ttcritdo'rur\lK'nt.I)() dissipation iiiHmftU.

Page 193: Lord Rayleigh - The Theory of Sound Vol 2

NUMERICAL EXAMPLE. 1813U.]

which shows that lu fhis case t)te samcproportional loss ot'

n!np)it.udc atways oecut-H :tftu)- thc)apse uf thc satnc numbcr of

]'criuds. T))i.s rcsu)Lmay bc obtidncd Ly thc mcthu<t of d!-

tncnsions, as neonscqncncc

of thcpriocipic of dyn:unica.)

.si))ii):).)-ity.

As anex:unj))û of

(.), I may rdc!- to thc c~G of a g!ubc wit)t

ncck, mtu))(]c<) furhurtti))~ pho.sphorus itt

oxygol gas, wltosu

capaeity is -251 cuhic fuct. It wa.s f..u,x! by (-xpcri.ncnt t).at thunote of tnaxhnu))). rusntianGu )nath 120 vibrations pur sucom),so that 7t=12()x27r.

Taldn~ thcvch.city ofsuuud (<;) at )i.20

f(.'ut pur sccon!, \Ye mn) fron th(.'sc (]ata

Jm)~i)ig from tllC Soundpro.htccd wLcn tlie g~bo is

struck,] t))ink t))at this cstimn.tc m)t.sf, bc too !ow; but it .shoutd buohs.j)-vc<) U): tho ab~ncu of t.hu iatinite fhmgc itssorncd ia t,)tc

th~ury nmst infjucnccvcry njaLcriaUy thc rato of dissipation.

Wc will oow ~xanur~e tlie f~rced vibrations duc to n, sourceof somu) externat tu tb<; t-eson~tn)-. If the pressure 8;) at thcjnout)) of )))(' r(;sQi)!t<,or duc t,) tlie source, i.c. c;t.!cuktc<I on thé

supposition t,)):).t the mouth is cioscd, hc 7''c""<, t)ic équation ofmotion

corrcspon<)it)~ to(: but ~pplicabic to thc forcod vibra.-

tiononly, is

w))!chagrccs w:(.]i thc cqo~tinn ob~inc<! by Hutm])o!f,z for tho

case whcru thu commnnicatiott with thu cxtGr!)fit fur i.s hy n.

simple !ipcrLuro(§S()(;). ThcprL-sc)itproh)cm i.sne:u')y,butnu<,

Page 194: Lord Rayleigh - The Theory of Sound Vol 2

1S~ r<JH(JKD \')I!i{.ATIO\.S.j~L).

(putL', :). c~so uf t.])fit tru~tcd iji § .K!, t)tu différence dcpcndin~

upou thc fact that, i))(ic')'-)H<io))L(.)t'()is.sip;).tiu))in (7)i;!it.sr!f r

:Lfu'ict.~)t~ft))Lipc)'in(f,a)Hl!)ut!t.n:L))S()!ut.(.')ycnn.-it.:u)t([um)tit,y.tf'tiu'

p'')-h)L),()(!t('nttit)t'dby /c,and~'hn ~i\'(-n,('))s))c\v.sU)at

th<)in~Tii:([\'n.ri:diunofp~.s.sur<~(<)!s!Lnt~in])nnw!)cuc=/<d)aL i.s,\vh<'n t))un:)t.m-;d

n"tc()f't.))uro.son:tt,~(f~1cu~(:(;<)wit)t-

<.)nLfU)u~it!ci()t-tIi.s)j)at.iun)i.st)t(;s:))))L':)st)t!~ot'thu"-('t)c!).tu)"'.sutm'). Tt'c maximum

\i))r:)ti()!),A\'))()nth)ic<'inc)()('))cu()t'p(,'['if)ds

i.sjx.-r~ct, variasinvt.r.~ty as <S;

hnt,if'A'bcHm:L]),!tvcrys)i~ht,

int'')')atityi)tL))c pcn~i.si.s.s)tf!!ci(..nt tn cause a )narl<L'(tfa)nn~ufl'inth~

mtcxsity uf'ti)u~so)):Lnce (§-)'!)'). h) t))<!p)'~cticatus<-<d' rcsonat~r.s

iti.sn()t.at!v:utta~'uustocrn-)yt))C!'cdnct.[u~

t)hS':n)(t<cryfar,pr''ba))iyh('c:Lu.st!thc;u-ra))~(;)nc))~)](TL'ss!u'y

t'orronm'cti))~ thc inttïior w:~ t!)cci)roroLhtjr.~nsidv'ia.))-

)):n':tt.usi))\-uh'ca.t]rj)a)'t)n-cfn~n tttDHUppo.sit.ixnso)) w)nc))thn

cahi)hth()))s :).)'(-fu)))h)(;(],w)Nchb()t-())nL'sn~)ntant) !n<)rc impurtanL:L.sti~c()inte))si.))ts:u-t' rr()uc~). W)~at!m scusitivc nppfLmt.n.si.s itct.ineotmuuti~n~ith

Lhemtc-rior.it.sInthccxpcmnL-nLut'

r(;i)tihrci!)~thc n<)un(i()t'~tUHin~-i'nrkhyn)c:U).s()f!L résonant)'(-t))ur ck'ntoits Gnt.cr Intu thc (tuuiiLiuu, :md adist-inct

mvcsti~Uon

i.s nt'ccs.Siuy (§;!)')).

)n\irLnuuft)tupt'it)cip!u()frc(;]p)-()cityt.cinvc-sti~th)nofthn

prec<din~pamgr:)p)t )nnyb(;a])pijc-dt<jc.d(;ut:ttc thu ci)'cctof:L

soun'uof sound mt.uat.t'd in t))cintu)-ior(!t'iL)-('.so)i!tt.or.

3]~. 'now p:)s.s on tu thci'm-thcr discussion()ft]tcpnd)]rtn

ofthu.~pL-upipt'. \Vusl)!))[m)pj)f)sc t))!t.t t.hc opcncndofthc

])ipc ispr«\'i(K'd wiL)):m intinit.c ();u)~c,:t))d th:).t.its dmmc~r

i.s sm;dt in C(j)np:u-is(j)~vit.)t t))u\v;Lvcie)~t.)i ot'thu vibration

um)o'cot).sidûr:Ltit)n.

As :m introduction tut))C())K.stion,wc winftn-thct-suppnst;

thatthemoutiiofthc pipe i.shtt.<-()with~fr<dyi)h)ving pi.st.un

wiLhuut thic)<n(.'ss inu) )n:t.ss. Th~p)'ccc<)in~ pn-)!)tons, froin

w!)if)) thL'pt-L~ntdiH'cr.si"rc:)titybutiitt)u,)):LY<j:d)'<u!ygiv(;n

usn~son tuthink t))at Du; j'n-s~t)C(i.)t't))(j piston wiH~u)s.;

nuitopurtitttt )nudinc;((ion.Witiii)ith~t)))jc~csup])u.sc(§2.5.'))

t)t:t.tthL;VL')u(.'i(y-pot.u))ti.di.s

Page 195: Lord Rayleigh - The Theory of Sound Vol 2

31~.] 1 oi'E~ rjpE. 183

On thé right of thupistou thé rchition bctwccn and

~)t on~i \njs by 302

v

buing thc radius of thé pipe. Froni this t)tû solution of thc

]"'oLh'm tnn,y hc obtrunod without :H)y restriction as to thc

.smuHness ui' sinco, Imwcvur, it is ody whcn /c~ is smid)

tf~.Lt t))Li présence uf thé piston wan)d nutnt!).tcri:d)y mudify

thc([«estion, wu nmy as weU )):Lve thc hoiefit of thc sitnpiification

aL unccby taking as in (1) §3)1

N~w,sincc thcpist.onoccuntcsnospa.œ, tlie vaincs ot'Nuw, sinc:e the pistoll occllpil's no RpaCl', the valllos of

Ol.c>\«.c/ '1

)nt).stbo<))Cs:u)tC()nbothsi()(;i()fit,n.n<lsinect)[C)-cisnont:Ms,thc Itku must bu truc oi't)tc values ofj~f~o-. Thu.s

In thisexprcssum tlie te-rmcuntfunin~Hin~ dépends upouthc

dissipation, :ut(! is thé sunic as if thcrc wcre no piston, whitc tha.t,

1.

`~"Il

1 cfti:ct f']' f. 1 caterual 1involviu~ rcprcsent.sttto cft'L-ct uf t)tc inertie of thé cxtcnmi

air in thc ncigtdjuurhood of thc mouth. In ordcr to comp:u-c with

pruvious rcsult.s, !ct a be sucti th~t

~7.'

Page 196: Lord Rayleigh - The Theory of Sound Vol 2

THEORY0F OPENENDS. [312.184

Thèse formu~ -show that, if thc dissipation be ]oft out of accountthc

vcioeity-potejttia) i.s thc sa.nc M if t)iu tube v/cre Jun.~hcncd

by of thuradins, and thé opcn end tlicii bchaved as a loop.

Tho amount of thc cnrrcetlun agrées WLth what prev:ous investi-

ssons wouM Ii:Lve Jed us to cxpect as H.c rc.sutt of thc Intro-duction of thc pisto!). Wc i.avc sccM rcason tu know that t!.u

true val.ic of a Hcsbctwccn

and~7~,

~nd U.~t thc prescnœ

of t!~G piston dous not aHcct t)to tenurcpresonting thu dissipation.

But, b~forc discussing our rcsults, it will bcadvanta~cous tu ii]-

VL-st.~tc thon afrush by a rathcr dincrcnt lucthod, which Lcsides

bcm~ of somcwhat grcater gcncndity, will hdp to tlu-ow ligtit outhc tncc!):u)ics ofthc<)ucstio)).

313. Fur tins purposc it ~iit bu convcniont to tiltift thc ori~tain thc négative direction to such distance from t)ie jnonth t!at,thé wavcs arc thcrc

~pproximatcly plane, a disp~eonent which

aecording to oursuppositions nccd not :unount to more than a

sm~H action of thc wave-Icngth. ThcdifHctdty of thc question

consists in finding thc connecta betwccu thé wavcs in thc pipu,whic-h at n sunicicnt dist:uice from tt~ mouth are p)ane, and thu

divcrgin~ wavc.s ontsido, wi.Ich at a modumtc distance may bo

treatcd as sphcrica). If t!)c transition tako ptacc within spacc.sinaH comparcd witb thc

wavc-Ic~th, whicit it must evidcntiydo,tt tho

dmn)ctorbc.s.na)Icnough, thé prubtumadn.itsof solution,wh~tcvci- .nay bu tiic for)n of thc pipe in thé ncighbourhood ufthu niouti).

.L puint, 7~, wiiosc distnncc from ~1 is n.~k-mtc, t)iu vducit.y-put,cu(.j:dis(§27U)

tliu l'elucitj,-

Page 197: Lord Rayleigh - The Theory of Sound Vol 2

3J3.] TMEORY 0F OFEN ENDS. 185

Lcb us considur thc bch:n'iour of t.hc nmss of air Inchu~d bu-

twœnthcp~ncHecL~H~t C:mdn,hcnusp))<ric:dmtr{~ccw!tosu

centre is~l,iuid radius ?',)'bcin~I:n-g-tj inconipari.so)iwit.)t t)tu

d):U)tct.L-r ot'thopipc,b)tt..sma)linc(')))p:).risonwith<L)i<j \t\'L'-

)';))~tL. Wit)tD) tins sp:L(:ct))u airnu~tniuveitpproxitni~cty~s :).n.

incun)}))-c.ssib)c Ouid -\vuuid du. Nuw t.hc uun'cnt ncru.s.s thc hcmi-

sphcric:dsm'i'n.c<j

This is tlic first condition; tlie second is to bc fuuud from t))e

cunsidoratton that thc total currcnt (wliose two v;ducs hn.vc justbuen c(tun.ted) is proportionfd to thu di~crcncc of p(jtcuti:d at tiiu

tun)H!t:).!H. Thus, if c dcuotc thé conductivity of thc pussa~c hc-

t\vccn ttiu tertninal surfaces,

fn tins expression thc sccotid tcrm is ncg]i~!))]c in compariso)~ with

thu first, i'ur c is at niu.st. 'tunat.ity ot' t!~ s:unc ordor as thc radius

Page 198: Lord Rayleigh - The Theory of Sound Vol 2

CORRECTION TO LHNCTH. [313.18G

If/t! bc thu radius uf t,hc tube, wcni~y rcp!:LCC o- by 7r7~.

Whoi tlie tube is a.simple cylindo-, and thc

origin !ic.s at :t

di.st.utcc A7, iruta Litu rnouLh.wu know thato-c''=AZ,+~, who'u

i.s :). n)[)n))o- i-at.!tcr grc.itcr thau 7r. In such a case (thc oi-i~in

Luing t:d<~tisxtHcicuDy uc:u- titu inouth) ycK is :). mnaU qu:mt~y,

tmd titL-rcturc fron. (10)

At tllc .~m~ Litnc cu.s~ nmy bu iduntiticd whh unity.Th~ principe) tcnn itL

~invotvu~ co.s?~, )n~y t.)ten heealot-

latud, if t.)iu tuhu wct-cprolon~cd, and i))ct-c wcru :t, luop :tt ,).

i~mL.situat.udaL:n)Istancc~ huyont the actual posidun of ti.u

hiunL)j, iu nccord:Uicu witit w).at wc t'ound bufurL-. Th(jsu rusult.

appruxitnatuiat-urdimn-y tubes, bucomuri~-uruus w)iun thu diatnutut'

i.-j rud~ced \vit.Luut iimit, fi-ict.Ioii bcin~ nc~cctud.

Page 199: Lord Rayleigh - The Theory of Sound Vol 2

3)3.') RATU 0F DISSIPATION. 187

If tho-c Le no n:U)gc nt J, the value of c is s)ight)y modined

bythu)\'niova)ofwhat actsasan obstruction, but thé principalcH~ct is o)) thc tcrtn

rL'prc-sonting tbcdissipation. Ifwc snpposf-

a~ an approxitnation t)iat thuwavcs divo-gingfrom~ arc sphcricai,

wc nmst takc fur t]te current 4-n-r instcad of~Trr'~

Thu~ve iiiiist titl~e 1(ir tite ettri-ciit

f/i IlSte~t(I C)r

~y-'0

u!ti<nate€<ït.ctoft))<t-at,i<jn wiH )j2 to hnjvc thc cxprcsHion forH'c V(.-)<)city-p()tcntifLt uutsi()(; tho muuth, tis wcU ILS thc corrc-

spun<)ing second tunn in (invoivi))~ .sinM~). T)tu :uuount of

')tMs)]):Lti<.))jLis t!ms .scctL todupcnd ))):ttcri:).t)y ontitcdcgrecittwhich t)m \v:Lvc's arn fruc t<j divo-gc, atK) onr a)i:L)yttc~ cxprf.'s.~iuusmust nut bu r~mk'd :L.s tnurc than run~h c.sthnit.t~.

Tiiu c~rrt.'ct thcory <'f thc open org.'ni-pipc, including cqu~tioxs

(H) :nid (i2), wfm di.scovcrcd hy Hut)n)toltx', w)K)SG method,

f~nvt-vur, di'rct-.s cot)sideï-:).b)y from t)mt hcru adoptc'd. Th~

c:n'!ic.st suintions uf thc probicni by L~t-fin~L-, ]). BcruunU), and

Etdc-r, weru foundcd on tho a,sst))nptio)i that~ta~ opcn end

thc pt-c.s.sut'o cou)d not, vary ft-om thi).t of tho~urrutmding atmo-

Mphcru, a.principic w))ich )nay pcr!)aps cvu)i now be consi()crc()

apphcahic to an ond wliosc opcnm-ss is ideaUy pcrfcct. TIiu tact

ttiat iu au ordiaary casus cncr~y cscapus is a, proof that tttcrc is

nut anywfturu in thc pipe au absolu te )oop, nnd it might hâve bccn

('xpectud t))at the ino-ti~ oi'thc air just ontside the tnouth would

ha.vu the eUccI of an iocreasG in thL: Iu)gth. Thc positions of tho

nudcs in a soun()in~ pipe \cre invc.sti~at(;() cxpcrimcntauy hy

!S<).vai-t"andI[o])!dns",wit)t thc r~uit that thc intervai bctwecn

thu moût)) a)td thu ncarustnoduisa.I\vays )c.ss titim t)tu h:dfof that

s~pa.t'ating consccutivc nod~'s.

31~ Expo-imcnta) d~'tDi-nnnationsof t!.c correction for an

opun on) )iavc gcn(.'ra))y beun )na.dc wittiout t)ic usu of a riangc,a)))) it t)tL'ruforcbcconics itnpurtant tofortna.tanyra.tGarough

u.stitnatcofitsctrect. No ttK'orctiea.tsotution ofthuproDt~nof:m unOangt'd opcn cn<) ))as )nthurto hccn givc'n, but it is casy to

scu t)i:tt t))o rcmovalof tho ffangc will ruducu thc correction

tn:Ltcria)!yhc'!uwthcv:dnc -~i27t' (Appcndix A). In the abscncu

ul' tticory I hâve attcmptcd to dutcrtninc Hic innuoice of a naxgu

'Cn'))t',]3~7,r.l. 18(!t).

~HcchL'r(.-)tL'isnr)MYibn)ti<'))St))jt')ti)'t~f/o'xt.t.xxn'.l.S'

-'Acnftt vihrittiuns incyliudncnit.ubt; c'ftHt&ro/~f j~Y~

)')1. 1~

Page 200: Lord Rayleigh - The Theory of Sound Vol 2

~83INFLUENCE 0F FLAN(!E.

[314.

f~vnr't'imfntnUt)' ~t\.r~ i i

L.

cxpcnmcnta.Hy'. Twoorgan-pipcs ncariyenougb in unison.with

onuanot!)L-r to givc countablobcatswerc biownfrom anor'~an

bdiows; tho cifect of t)~ nangc was (icdnccd fro!n titc dift'cr~ccin thc

frcqucucics of t).c be;Ltsaccording as ono of tho

pipes was

nitugcd or not. Ti)c correction dnc tu thé Oangc was aboit -2/t'.A (prohaDy more

tmsLw.rttfy) ~}K-t,itio)i of' this cxpuritnt.nt ).yAir

Jj<.s:uu)uct ~:Lvc -2.')~. Jf wc Huhtt-act -2~7t; frntn -S27~ wuct'tain -8/

w''ic)tninybcrc-g:u~(~asabntttt)hJprubah!(-v;dnL-ofthu conwt.ion f<m

unft!.)~) npt.ncn.), ..))t])e.s))pp)siLi.)t).at

thcwavc-IcngtitisgrcaL iticunip;u'i.-iunwi(.h thc diani~Lcr <jf thc

pipc.

Attcmpts t~ (~to-muic thé cnrrcctioncntirdy from cxpo-inx-.nt

Lavu not )cd !.iL))crto to vcry prccisc rc.sutt.s. ALua.suronent.s LyWurthcita' on

doubfy opoa ptpcs gave as a. muan (fur cachen.)')

-~i: winiu ~r pip(j.s opcu at o).c end onjy t))c muan resutt was-7-tUA'. In two carctut

uxpoi.ncnt.s Ly Bo.sa!X)u~' onduubiy

('pt'n pipes thc correction fur onc end was -C~7~, whcn \= 12 7/an.) -5.):i w),en =:~U.

Bosan.,nct Jay.s it duwn as a gcn~-ati-)t)c tLatt))ccorr~iun

(uxpr~.dasafr;Mtiunof7.') i.t'crcn.scswit]i t).c ratio of diatnctcr tu

wave-fcngth; part of this I.icrpa.sc

inay Ilowuvur bc d~c to thu .nutunt rcaction of thc- ends, w].ichcauscs thc p!anc of

.symmetry to behave likc a rigid wa)!. Whc.nt).c pipe is oniy modcratdy long in

proportioli to itsdia.netcr, a

statc ofthings is

appro;Lchcd w!ac)irnay bc more

ncarly rcpro-Hcntc-d

by tho prc.scnec t!)a)i by thc absence ofa ftangc. TIjc com-

pan.sun ofHicory and obscrvatiort on this .subjcct is n mattcr of

-soincdimctdty, because whcn tlic correction is sma)), its va)u~ as

calcutatud frumobservation, i.s aH'uctud

by uncertaintics as'toab.s.dntc pitch and t).c

vclocity of sound, wtnie for thc ca..so, wbcnt))C correction is relativdy iarger, w!tic)t

expcrimcnt is more co.n-

pctctit to duat wit)), titcre is at présent uo thcory. rrobab)y a. moreaccumte v~luc of thé correction cou!d be obtaincd from a re.sonatorof tbc kind considcrcd m § 3<)(i, wh~-ro thc communication witb

t)ic outstdu air is by a simple aperture; thé "k-ngth" is in tbatcase ~ro,and thc correction is

cverytiung. Somc mca.suroncnt.scfthis hmd, in whic]), ))n~-cvc.r, no

grr..at accurapy wasattemptcd,

will bu fonnditimytncmuir on résonance'

'r/;<7..v~(.))!iS(;. ]S77.

=~t;t).(;/NM.(:t)t.x.\xt.p.;);)~

~~t/1A;f/.(n))v.p.~)u. is77.''7'/<t/.7'nu<.<.lH7].

S~a]sn.S<~t])i)tuss,r,)~t.ltn,21U(lH7<'),(ind[~uutc rcjuarki, tiiermjx~ ],y myself (/u, Hrj.t. 1870).

Page 201: Lord Rayleigh - The Theory of Sound Vol 2

314.] EXPERIMENTAL METHODE. 183

Varions mcthods hâve bccnu.scdto détermine thc pitc!)nf

re.sooators cx])cri)n~i)ta1)y. Most frc;()ncnt)y, perhaps, tl)c rcsonators

havu b'jcn madu t.o N/)e~' after t)ic manncr of organ-pipcs by a

Kt,t'(?:)ni. ofnir bh)\vn obhfjnc'Jy across thci)' months.AIthoufh gnod

rL's~it.s hâve bcun obtaincd lu this way, onr ignorance as to tho

modt~ of action oFthc wiod rc:)](tL')'n Lho mbthoduns~tisfitetory. In

Busnoquct's Mxpt'rimeiits thc pipes wcrc notacLu~Hy mn.dc to

spc:).k, but sitort discontinuons jets of air wurc btowu a.cross t])e

~))t;n end, thc piteh bcing usti)nntL-d frum thc frcc vibrations as

thc sound di(.-d awa.y. A )n(.'thud,simi!at'in principte, that 1 Iiave

.sonn;titncs cmpioycd wit!t a.dvn.tita~'c consists I)i uxcitingfroei'vihra-

<io)tS hy !ncans uf a Uuw. In order to obtain as we)] dufincd n. note

as pnssibic, it, is of importance te accomtncdate t))c hardnc.ss of thé

substance with w))ich thc rcsonator coincs into contact to thé pitch,a loAv pitch rcqninng a soft, b)ow. Thus thc pitch nf a tcst-tnbc

may bG dutcrmincd in a mumoit by striking it against t)~o bout

!\u(;c.

I)i nsin~ this mcthod \vc onght not cntirdy to ovorloo~ tho

fact that thc natnra) pitch of a vibrating hody is attcrcd hy a

<(iDn dépend Ing npon t))C sqnarn of thc dissipation. With thc

nntfLtion of § 45, tho frcqucncy is diminishcJ from ?t to

?'() –c~r''), or if A' bc thc nurnbcr of vibrations t'xccntcd whi!e

tt)C amplitude faHs m thc ratio e 1, from M to

Thc correction, howcvcr, -\von)d mrc]y be wortit tahin"' illto

itccnuut:.

Thé mc.isuroncnt.s givoi m jnytncmoh'nn rc.sDna.nce wcrc

co)uh)ct.f;<l npotin.dii'f'L'rt-'nt, princi~tchyc'stimii.t.n)~ the note of

tnft.xImnniL rt.'son.'utcc'. TItcearwaspLLCctIincommonicn.tinnwith

thL'[)jteri()rofthccaYit.y,Y))i)ct])ccI)ron)!'ttie8c:t.)cw~.ssoun()c!d.

tu ~)us way it was found pns.stbtc with !i HtUe practicc to o.stiinn.t.e

t)K'pitcl) of a gnod rcsor):Ltf)r to about tt qufu'tcr ofa scnnt.onc. In

the citsc of'.sman n~s~s with Jong noc~.s, to wL!c')) thc n.bovc mct,hod

W()ut<] not bu ap])!ic:d))c, it wfts funod sumocub nu'rcly to ho)d thc

0; nea.r thc vibmting wires of n. pianofoj'te. Tito resonant note

:)~))~o~)ncu(~ itsutfby a.'nuvu)']))~ ofthn body ofthc f)as]e, e:~i]y npr-

<'cptil)tu hy titcnngcrs. ]nunH)~t)ns)nc't))odit,i.si))ip(.)rtantt)):t.t

thcïnittd .s])ou]d bcfrccfrombiasinsnb-dividing'

the intcrv:).!

bchvcoi two consécutive .sonitone.s. Whcn thc thcorctic~t rcsu)t

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DISCUSSIONOP MOTION [314.~0

i.s known, it is atmost, impossible to an'ivc at an Indcpcndcnt

opinion hv'xp('itiir'nt.

31.'). Wcwi)) no\v, fu])uwing Hu]mhohx, examine more c!ose)y~hc nature of thc motion within t)ic pipe, rcprcscntcd by thc

funnn~(H)§3)3. Wchave

whcrc M is a.posttivc jntcgcr.

T!)C (1ist:mcc hctwccu consécutive m~xim~ is thns a~)(~ thc

v:duc cf thc m~xnnu))! is sec~a. Thc miniinnm va)uu.s ci' Z/' occur

npproxitnatcfy wlieu (a; a) = M<7r,

Page 203: Lord Rayleigh - The Theory of Sound Vol 2

315.] ORIGJNAT1NG WÏTHIN AN OPHN l'IPE. 191t

Thé fipproxin~te tuagnitudc of tLc maximum is ~sec~x, a.nd

<hnt ofthe mminuuu A:o-coH'47r". It appcars that ti)C

tnaxima. ofvelocity oceur in the s:unc parts of thc tuhe as thc

nnnimit of condcnsit.tio)! (:md rtn'cfact.ion), and the nuninm of

yclocity in thc samc places as <.I)e nmxi)))~ ofcondun.sfitiot). Thc

scries ofloops nnd nodcs :u-G :uTa.)tgc() if thc in-st loop werc at a.

(listance a bcyond thé month.

WIH) rcg'fn-d in thé phnscs, wû sec t!~t bnth and arc in

~encrât s)naH and thcreftjre with t1)C exception of thc ph~ccswhcre 7,' and ,7' arc ])ofn' tileir rninima. the w]io!c motion is

.synchronous, as if there werc no dissipation.

Hit!)crto we hâve considercd thc prohtcm of th(; passade ofp):ui0

wavcs fdong thc pipe and t)~Ir gradu:d dinnslon frojn tho

month, -\Ylthcntregard to thc ori~in of thé plane -avcs thcm-

sctvcs. A)l tliat wc hâve assumcd is that t1ic origi)t ofthc motion

is somowhcre within thc pipe. \Vc will nowsuppose that tiic

mution is duc to the known vibration of a. piston, situated

at A'=- tlie origin of co-ordinaLcs bcin~ at thc moût]!. Thus,

whcn ==

n.nd tins imist hc !n:~c to corj'cspom) with titcexpression for thc

p):inew!).vcs,~cncmlixc<) bythc it)tro(h)ction ofarbitniry ampjitude

:uidp)ia.sc.

AVcmayta~c

by which tind e arc detcnnincd.

In :tccoi-d!uicc wiL)i (12) § 313, thé corrcsponding divergentwa.vc is rcprcsc'iited by

Tt

Page 204: Lord Rayleigh - The Theory of Sound Vol 2

1 QOAIOU'IO. DUE TOMOTION DUE TO

f;~Il. Iv W o

Jf<?)M givcn, ,s ~.catmt,whcneo.,K(<+.).n t)~t h

~.en. isIn

°71of t).c

~t.i~ .Ibr~i.n v.y.t., th.n~h n.,

n~mtc. sincccos~can~t~is),. Wi.cn

hun~uU..muchcont.rac~,

oos.y bcco..c .sn..)., h.t tin t!s c~ isncccssa.y that t)~

a.)ju.st.,ncnt of ncrio hovery c. in onicr thut thc Hr.st te. cf ( i'.) ,n.y b. n~ le:~r: tlie ¡.;ecolH1.

~p~c.i:l'quaI to unity.il CC)SAC2i.

lie.-ll-]Y

Tho ininhnum of vibration occur.s whcn snch t)~t

tLc pi.~n is ~cd at r~ptlmt Case

T.cv.br.ionout.si.jc tbc tube i.s tben, accorda te tbo value ofc.,ua)to.r.s~a]fc,- than thé vibrion ~.ich there wouU be't!~

"S P~~tlie hlane,

316. Onrcqu~ions may ..J.so bc ~ppUed to the

investigation.f tl.c motion cxc~cd in tube by cxtcrna! sources of~nndLet uyuppo.sc in the first place tliat tho ~out). of tbc tube i.c).soJ by a ~.cd pL.tc fo~ing part of the y. p]ane, .nd tl.at the

p~cnha]

<h.c to t).c cxt.n~I .sources(approximatoly constat

ovur t!.c plate) .s undcr tbc.sc c;rcum.sta)icc.~

~-he.-c Iscomposcd of thc potcntini due to each .source and its

.mngc .n thé pi~no, asc.xpt.h~d in § 27H. Inside t!~ tube lut

t)~ potcutiat be

so that <~ and ils difrcrcntia! c~mciont are eoutinuons ae.-oss tho).:UT.cr. Th<y.sical nK~in~ofthi.s Is.simp)~. Wci.n~me~OnntLctuhc suc), a ~r-t.ion as is extermine.)

hy thé conditions

ti.atthcvdoc.ty at thc n~nth is zéro, and that thc condensationat thc moutli is thé same as that duc to thc sources ~f sound who)ithc.nouth inc~sed. h. i.s obvions that undcrthc.scn.-f-u.nstancos

Page 205: Lord Rayleigh - The Theory of Sound Vol 2

~nc.j EXTERNAL SOURCES. 193

t)icelosing plate may be rcmovcd withont any altération in thé

motion. Now, ijowcvcr, tficre is in général a finitevelocity at

.T=- and Htcrefure \vc cannnt suppose the pipe to be t!icrc

stop])cd. But \vhen therc liappois to bc a nodo at a; == that ist" say wiicn is Huefi t))at eus/<-(/+a) =0, a)l thc contiitions :u-e

sattsficd, a))(t t)tc actual mution withiu thc pipe is that cxprcsscd

~y (2). T)iis tnot.ion is cvi'K'ntiy thc same as might obLain, iftttc

p)pc wcrc c]osQ(l at LoUt ends; atn! in cxtcrnalspacu thc potcutiat

is thc sa.mc ~s if thc mout-h uf thé pipe wcro ctusud wit,]i tho ri~idp)n.te.

lu tljc gcncra! case in ordcr to rcftucc thc air at = to restwe must superpose on tho motion rcprcsentcd hy (2) another oftho kind invcstigatcd in § 313, so dctcnnincd as to givc a.t a; = ia

vclocityequat and opposite to that of thé first. Thus, if thésecond motion bc ~ivcn by

It nppcars, as might ]~vc bccn expcctcd, tha.t tito résonance is

grc~tcst wiieu thc t-cduccd lûngth is an odd multiple of

317. From thc principtc that in t~to neighbourhood of a nodethc mortm of thu air docs not conic much into play, wc sce thatin snc)i p]accs thc form of a, tube is of little conscquencc, and that

cnly thé capacity need be attcndcd to. T))is considération ~lowsus tu calcu~tc tlic pitch of a pipe which is cylindrica.1 througb'nost of its leng-th (~ bttt ncfn- the closcd end cxpands into aL"]b of small

c~paeity (~. Thc rcducc<t Ict)gth is thcn cvi-

Jcut]y

'Hd)n))ott~,C)-18<!0.

R. H.

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I!)4 EMARGEMENT AT A CLOSED END. 1317.

whero a is thc correction fur thu op(;n (-n<], nn<) o- is thc Mrcfi of

the tr:).nsve)'.se s(-c),io)) of thc cy)in!))'ic:d piu't. 'i'his fm'muta is

-c).u~L~~y!ph~i~h~~ih~dc~aU~uf~~)~~

cy)[n(h-ic;d f«)-)u <)ucs not take thé .shapc ot'onenitu-~emunt.

AVhGn thc oikrgGmcnt rcprcsunted by <S' is too )a)'~o to a)]o\v

of tlic abovu trcatment, wc )n~y procccd as fuitows. T))c dissipa-tion buing ucglectc~, Die vuincity potential lu the tube !n:).y bc

takcn tu bo

is thecqufttion dctermining thc pitch. Numcrical cxamphs nf

thcapplication of (3) are

given injny mcmoir on résonance

(P/i~. 7'm7i.9. 1871, p. H 7).

Simil:u-rca.sonit)~ provcs tl~t iti any cn.se of

sta.tion!i)-y vibra-

tions, for ~'hich tlie w~vc-Ien~t!) is Hcvcra.t thncs as gréât as the

duunctor of t)ic bulb, thé end of t)ic tubeadjnining thc I)u)b

bchitvcsapproxitna.tcty as an opcn end if ~,9 bc inuch grcatcr

Huui o-, and :ts :istnppud oïd if Le mncb luss than o-.

3~8. Thc actio)) of a rcsonator whcn under thu innucncc of :t

source of sound in unisoi with itsctf is a point uf considérable

duUcacy andi:npor~ncu, aud onc on w])ic)t t))crc ])as b(~n

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318.JABSORPTION 0F SOUND 13Y HE.~ONATO!!S. 1S5

good dcat of confusion among acoustical writcrs, thc autbor not

excepted.

Titcrc are cases wbcrc a rcsonator absorb.s Sound, as it wcrc

attracting thc vibrations to itself aad so(tiverting them fron

rL'gions whero otherwisc thcy would bc fe!t. For cxampte,

suppose that thcre is asimple source of sound sitnated in a

narrow tube at a distance (or any odd muttip)c thercof) from a

cfnscd end, and not too ncar tho mouti): thoi at any distant

oxicmal point its cfïuct is ni). This i.s an inuncdiatc consu-

'ptenco of thc principtc of recipt-ocity, bccanso if ~1 werc thé

source, t])Ct'e cuuld bu no variation of potcutial at A Thé

restriction, precludin~' too grcat a proxinuty to thc mouth, maybc dispcnscd with, if wc

suppose thé source to be din'used

unifornily (jvcr t))C crosssection, instcad of conccntratcd in onc

point. Thcn, wltatcvcr may bc thc Hizc and shapc of thc section,thcrc is nbsoiutuiy ne disturbanco on thc furthc-r si(h'. This is

c)car frum thu thuory of vibrations in onc ditm.-n.sion thc reci-

procal form of thé proposition–that wbatcvcr sources of' distmb-

ancc ;nay cxi.st bcyond thc section, jy~-rZo- =0–tuay bc provcd

ft'om Hc))nh()ltx's furmuia (2) § 2'): hy inking for t]te vclocity

potcntia! of thc purcly axial vibration ofthc saine period.

It is scarcc)y ncccssary to say that, whcncver no cnen'-yis cinittcd, tho source does no work; and this reqnircs, notthat thcre shaH bc no variation of

pressure at thé source, for that

in thc case of a simple source Is impossible, but that thé variable

part of thé pressure sbaU bave exact)y thé phase of thc aeccicr!

tion, and no componcnt with tho phase ofthc vclocity.

Othcr cxanip!es uf théabsorption of sonnd by resonators are

an'orded hy certain modincations of Hcrschers interférence tube

uscd by Quinekc' to stop tones of de~nite pitcb froni reacitin'tlie car.

In thé combinations of pipes rcprcscntct) in Fig. (!3, thé soun()

c'ntcrs frcciy at at it nnds itscif at thé mouth of a resn-

nator of pitch identical with its own. Under thèse circumstanccsit is absorbcd, and thcre is no vibration propagatcd a]on~ 7?~.H is cJear that thé cylindrica! tube ~C' may bu rcp)aced by anyothur rcsonator of thé same piteh (7), without pn~ndicc tu t)tu

'{'{;f~.cxx\'))i.!77.tn(;f!.

L;

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l~GQUTNCKE'S TUBES.

[31g.

1 mf' 1

action of thc apparatu.s. Thc ordinary cxplanatiou by intcrfcrcncû

(so ea-Hcd) of direct t~ud reffcctctt wavcs is thun luss applicablu.

Thcsc cases ~hcrc t]te source Is at thc mouth of a i-csonatorinnst net bc confuscd with othcr.s wbcru thu source i.s in thu intc-

nur. If be :i source at t)iû huLtotn uf a.stuppcd tubn w])oso

rcductjd Icngth is t])e iatcnsity at fm cxtet~id point m~yLu vast)y grL-atcr t)i:ui if th~u h:td bccn ])o tube. lu fact thc

potential ~t duc tu t)ie sourœ :Lt is thu s:unc as it would Let).t wct'c thu Muurec at J..

31~). For a do.sor uxatninatiuti of thc mcc)h'mics orrésonance

wo shaH oblain thc proDcm iu a !'L.rni disutnbarrasscd of unue~

ccssary dif!icutt!cs by suppusing t)ic rcsuj.ator to consist of asmati eircular ptatc, bae!.c-<I hy a spritig, and hnbeddcd in a!iinde~nttc ri~id p)anc. It was provcd iu a proviens chaptcr (:30)

§.tl.at if J/ bc thé ma.s.s uf t)tc

ptatu, itsdisplaccmc.t,

titc furcc of restitution, 7i' thc radius, aud o- thc dcnsity of thcair, thc équation of vibration is

where Fand arc propoi-HonaI to c"

If tl.c natund pcriod of vibration (tlie rcaction of cxtcrnal airindudcd) cojncide wit), t!.at i.nposed, t!.c cquatiou rcduces to

Page 209: Lord Rayleigh - The Theory of Sound Vol 2

319.] RKSONATOR CLOSE TO SOURCE. 197

Lct ua nowsuppose t]iat F is duo to a.)i externe source ot'

sound, giving whui Dm plate i.s rcst a poto)ti:d which ~ill

bc nc~riy consta,!i.t ovci- thé fu'ca (jf thc plate. Thus

so that 2-n-A: i.s tfie w.ivc-icngth of thc )in.tum! noto of thc rcso-

n~tor. If ~bu writtcli for ~/+!jo-7~ t.)iccquation con'cspond-

in~ to (5) ta.kcs tliu i'ortu

from winch wc may infer us bcforc that if /<=/<: thc cfDcicncy of

thc rcsonator as a..source is iii<)cpe)x)cnt of ./< W)tcn t!tc ndjttst-mcnt is impc'rCcct, tho ]a.w of f~Hing oil' deponis upu)). J/T))U.s ifj/' be grcat fm<tTt; MmaH, atthou~h thc mnximum cfHcicncyof the rcso!]n.tor is no le.ss, a, grc~tcr ficcuracy of adjustnicnt is

rcquircd in ordcr toapproac]) t)io maxunun) (§ 4')). lu t!ic case

of rc.sonators wit)t suup)c npcrturcs J/'= 'o- so t)iat ~)/varies as jf)' Accon]iii~)y j-esonators with s)n:LH apcrtoi-c.s rc-

'p)irc thc grcatcst pt-ccision of ttUiing, but thcdi~-rcnco is not

t'nport.-mt. Front acomp:n-i.son of Hic prcscnt investi~tiou with

th~t uf § :ni it appcnrs tl~t thc conditions of cOicifjncy arc dif-iuront

a.ccordin~ as inturnal or externat cffucts a.ru considurcd.

Wo wift no\v rut.m-n to the casu of isucfn-omsm fmd supposef'n'ther tftat ttic extornal source of Sound to which thc rcsonator

yl rusponds, is thc motion of a simifar plate w]jose distance

c from ~1 is a quantity I:u-c iti comparisou wilii thc dimcusious

Page 210: Lord Rayleigh - The Theory of Sound Vol 2

REINFOnCEMENT OT SOUND[319.

198

of tlie phLtcs. T]tc intensity of 7~ may be suppo.scd to be such

tim.titsnutc)iti:t.[is

Thé relation of phases )nay bcreprcs~ntcd hy rcgarding thé

Jnduc~dvibration as

proce~hog frojn by way of J, and asbt-ing suhject to au additionat retaniation of su that D.e whoie

retar(!atiun betwccn 7~ axd is c + In respect of amptitudc13 ~i-catcr t))au in thc ratio of 1 ~c.

Tf~ts when ~e is .sj))n!), H.c Induccd vibration is much H]c

greater, nnd thc tuta] soun() is much hn~cr tnau if wcrc nut

permittc.! to «po-ate. lu this case thc phase is rct:u-dcd by aquarter of a pcriod.

It isimportant to hâve a c~ar i<)e!t of thé cause of this

aug-mcntation of sount). I)t a préviens ch.-tptcr (§ 2.SO) wc saw

that, ~heit ~1 is ~xed, gives ont much ics.s sound tkui n.ightat ilr.st hâve been cxpcctcd from thc pressure duvclopcd. Thc

expiu.nat.ou was tJ.at t))e of thcprésure was unfavourah)c

thc hn~cr pfu-t of it is conccrncdouiy in

ovcrconung thc incrti:t.of t]ic

surruunding :ur, and Is incHectivc towards thuperfonuanee

of work. Now thc pressure which sets m inutlun Is t)~ who)e

pressure, aud uot inuruiy thcinsignifiant part that would of itscif

do work. T)ic motiuu of is duturnu.tcd by thc condition thatthat

cu.nponcnt of thé whoto prcs.surc upo.i it, which i.as t))c phaseof thc

vulocity, shai) vanish. But uf' t!m pressure that is due tothc mution of~, thc

largcr part bas thc phase of thcaccélération;

aud thcrcforc thc prcscri))ed conditiuu requires a)i cfptaUt.yhetwccn thé stu:dl componoit of' thc pressure duc to ~t's niotion,~ud a pressure comparable with t)te large conponeut of thc

pressure due to ~'s motion. Thc rL-.s~h is that .i becoines a

much tnorc powerfui source titan Of course no work is donf

by the piston ~1 its efTect is toaugmcut thu work dune at 7?,

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319.]BY REHONATOUS. 1~9

by modifymg' t))c othcrwisc unfavourabic relation betwccn t)to

phases ut' tbu pressure aud of t]<e vciocity.

Thc inHnitc ptane in thc preccding (hscussion is otdyrcquircdm ordcr tfiat wu may nnd roon) bchind it fur oui' toachincry of

sprints. If -wc arc content with still ïnorc hig)~)y idcabzcd

sources it.u<! t-~unatu)-s, wc ni~y dispoisc \viL)t it. Tu utn'h pistonniustl)ca(t<.)cd a. dupUcit.tc, vibrating Inas!n)i)ar tnfmncr, but in

thc cppo.sito <tir~ctiun, t))e ufluct oi' which A\'i)l bc to tttalœ tho

nonuat vulocity of thé nuid vanish ovur t)tc pl:tnc JA Under

thusc circmn.st:utC(js tho plium is without in~ootcc fmd may Le

runovcd. If thu sizc of thu pintes bc rcduccd witLuut limit tlicybcc'oxH-

uh.imatcty ('(jnivajcuttoshnpie sourcps of <]uid; aud wo

co)ic)udc tlif).t a simple source will Lccumc more efficient than

bc'furc in thc ratio of 1 /cc, w))cn at a sma)l distiuico e from

It thuru is a!)uwcd tu oporatc a. simple rcsonator (as wc may cali

it) of I)kc pitch, t!)at is, a sourco in whicit t))c inci-tia of t))c

iinmcdiatuiy surroutiding H nid iscompcn.sated Ly sfxne adéquate

m:).chine)-y, imd which is set in motion by cxtcrna) causes ordy.

In tllu présent .statc of ourknowtcdgc of thc incchanics of

vibrating Huids, whiic titc difHcultics of déduction arc for tho

mo.st part still to bu ovurcutne, any simplification of cotKJItionnwhie!) afiow.s progruss to bumade, wltitout ~Lo]]y dcstroying t!to

pr:).ctical charactur of thc qucstiuu, tnay bc a stcp of "-reat

importaocc. Sucit, forcxamptc-, was thc introductioti by Hchn-

hu[tz of thé idca ofa source concuhtratcd in onc point, rcpruscnted

anulyticaHy by thc violation at tliat point of thé eqnatiou of

eoutinuity. Pcrhaps in Hkc manncr t))e i(!ea. of a sin)p)c rcso-

nator may bc uscfut, althou~h thc thin~ would bu stiil moru

impossible to construct than a simple source.

320. Wo havo sccn that tburc is a grca.t augmentation of

snund, when a suitabty tuned rcsonator is close to a sinip)osource. Much more is this thc casu, whcu thc source of sound i~

cumpound. Tiio potential duc to a double source is (§§ 294~, 324-)

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RESONATOR AND DOUBLE SOURCE. [320200

and thcrcforc thc potcutial duc to the resonator at distance is

If~, Vtuush, thc rcsonator is without effcct; hutwhcn = + 1

that :s, wi~n thc r~on~or Iles on thc axis oi- thé double source,wchnvc

Thus womny consœur that thé potentiat duc to ti)c rcsonator

isgreater than that duc to thc double .source in thc ratio 1thé

anguiar variation bcing disregarded.A vibrating rigid .spiicrc gives thc same kind of motion to thc

surroundmg au. a. a double .source situatcd at its centre; Lut thésubstitution

suggcsted by this fact is on)y pennissibic wl.cn théradius of thé

sphère is .smaH In coinparison wit!, c: oLhcnvisetlic pressée of tlic .sphère modines thé action of thc resonatorNcvertitdes.s thé

prcceding Investigation, shews how powcrfuiin gcuM-d thé action of a resonator is wt~'i placcd in a suitableposition e)ose to a

compound source of sound, whosc characteris suc)t that it would of itself produce but little efruct at adistance.

One of tlie best cxampics of tins use of a rcsonator is an-ordedby a vibrating bar of g)ass, or métal, heid at thc nodcs A stripof plate glass about a fout ]ong and an inch broad, of médiumthiekness (say inch), supported at about 3 inches from thé endsby mcans of .string twisted round it, an.swers t!.o

purpcsc verywc)i. WI.en struek by a hanter it gives but Iitt)e .soun<t exceptovcrtones; and e.cn t)K'se mayaimo.stbe got rid of by choosi~a hammer of suitable .soft.ic.ss. This

denciency of' sound is aconséquence of thé sn~U di.nunsious of t).c bar in

comnarison

~tht).e

~ye-Jength, wiucfj a!tows of thé ca.sy tran.sfercncc of airfrom onc s.dc to ti.c other. If now t!.c mouth ofa resonator ofthc nght pitch' be I~dd ovci- one of thé free ends, a .sonad of con-

ToKet tho bo.st efïeet. t!,c mouth of tho rc.s.nnt~. c.nH'.t io hopr~tv e!o~ tohc .r and thcu thc pitch ifs d~MIy t.~u it ..o.d b. 1~

fmai adju.tmcut n~y bo m,.l. by v~yi~. tho amouut of cb~ructiuu T ~usc ofreBotiators )g of grcat nutiquity.

Page 213: Lord Rayleigh - The Theory of Sound Vol 2

320.] TWO OR MORE RESONATORS. 201

sidcrabic force and purity may bc obtnincd hy a. wcU managcdbio\v. lu this way a.n irnprovcd )j:u'tnuiuc<jn may bo consLructcd,with toncs much !owcr thau wouRI Le praeticab)o wi~tout reso-

na.tors. In tlic ordina-ry instrument t!ic wavc-IcngLhs a-ro snfH-

eicntly short to pcrnut ttic bar to communic~Lc -vibrations to tlica.ir indepundently.

Tho rcinforcemGn), of thé soutid of a LcU in n, weti-known

oxperiment due to Snvart~ is an exemple of t)ic same ~odc of

action, but peritaps tlie most striking mstancc is in thc M'-

rangomcnt ndopted by Hchn!)oltx in hiscxpcrimcnts rcqnirixg

pure tonc.s, which a.re obtaiued Ly Itolding tuning-forks over thc

menthe of reson~tors.

321. Wlicn two snnpic rcson~toi-ssepartitely in tune

wit)i tho source, arc close togcther, the eMbct is Icss th:m if Lhcrowure ouly onc. If thc potcutials duc rcspcctivcly to J~ ~(~ be

wu nn).ytn.ko

Lot i-cprescnt tlie distance aud tlie potentiaLsth:u, would uxist at ~t~, if t))cro wct-c no rcso~toi-s; thon the

couditiuns to détermine arc by (5) § 31:)

bincc is Hmd), thc cfTuct is much !css than if there wcro

r'n!y onc rcsonn-tor. It mu.st bc obscrved howevcr t!mt tlio

dnninishcd cUbctivcncss is due to titc rcson:).torsputting

ono

anothcr out of tune, tuid if thi.s tcodoicy be conipcn.sn,tc(! 'by a)iftftcration in t!œ .spriog-, any numbcr of rcsonn.tors ])c~t- to'thcr)mvc just t]te cÛcct of one. This point is iHnstratctt hy § :{()~whcrc it win bc SGcn (32) t!)!it titougit thc rcsuuiLnce docs not

depond upuu thc sixc of t))C ptatc, still thc incrtia of thc air, whichbas to Le couipcnsated by a spi-in~ docs dcpuud upou it.

J~t. ff. C/t!'m. t. xxiv. 1823.

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FORMATION0F JETS [322.202

322. It wiH be propcr to say a. fcw words in th!s place on

an objection, w!:if-h ha.s bccn brougbt forwa.rd hy Busam~tct' .t.

possibiy inva!idnting thensual caicutationsof thu pitch of re-

sonators and ci' thc correction to thé]cngth ofor~an pipes. Wfien

itnid iiows in a.stca<Iy strc'atn thn)))~h :(, hoie in a t]un phtc, thc

motion onthcfowpn.s.suœsidcisby no ]nca))softhcc)):L!-actci-

invc.sti~Ltcd in ~OU. ]nstc:ut ofdivo-~in~ :Jtcr

p~sing Htc hu!c

su us tu futiuw t)ic .surfaco oft))c p)atc, L).c ftnid sh~pc.s itself into

fmappn~i)nat(.')ycy)in')rK'a)jt.'t,whn..cfonn forthuca.scot't~o

dimensions canbccahutatcd ironiformuki~ivcnbyKirchhnfï'

On Htc high prc.ssurc .si()c thn motion dues not dc-viatu sowiduiy

fro)nt))atdctunnincdbyt).cdcctrica!Ia\ InlikumantK-i-nnid

pas.sin~fn)t.\v:u-dsfrom pipe continues toinovc in :icy)indricalstremn. If thc extcrnal prcssnru hc thu ~rc~tei-, thu ci):u'actui- of

t))u tnotion is di~urent. In tftiH case tilL- .strcani Hnt.s convoiefrom a)) directions to thcmunth of t.])C

pip(., aft.c-rward.s~tin.ri~t)tumsolv~ into n. pnr.diui ))nnd)c, whosu .s~tiou is

considcrabtyIcsH than t))at of thc

pipe. It is ck-ar that, if tite formatlun ofjetstook

pla.ce tu aoy considuraDu uxtcntdnring t!ic

passage of air

thrung-h H)C niontits of rusonator~ our c~IcuLtions of pitclt woutd

Itave to buscriousty modinud.

Thc précise conditions nnd~- winch Jets arc formod is a. snhjcct

ofgreatd.dicacy. It!nay(iV.jnh.()onbt~Iwhct]K't-t]toyw,,n)doccurat aU in irictiuldcss ituid

movin~ with vdocitios so sntal) that thc

cor)'ospondin~prL-.ssun's,whie]t nrcpn~portional to tho squares of

thc vc)ocitk's, are incon.sidcraoh'. Hut wittt air, as \vuactuaity

!)a.vcIt,moYingurH!cr<!)u action nfthp pressures ta Le fonnd in

n'sonators, it must bo admitted that JL-ts n):t.y sonnjtimcs occnr.

\Vhitccxptjritnuntin~ about twu yuars ago witit onc of Kuni~'H

brass i-csanatur.s (jf pitclt c', 1 noticct! that whcn thcnorrusp<.n()iu~

fork, stnjngiy cxcitcd, was ]ietd to thu moutt), a witid uf cunsid~

~bic force issncd froni thé nippie at t))c opposite side. T!iis ~ft'uctjnay rise to snch

intcnsity as to L]ow out a catidio npon whosc

wick thc strcam is diructud. It docs not dupcnd npon any pL'cniia.rtnotion of thé air nL-ar thc ends of thc fork, as is proved hy

inounting thc forkupon its rcsojiance-box an()

prcscntin~ thc opunend of thc hox, instcad of t))C fork itscif, to t)tc mout)i of thc

rcsonator, whcu thc cilect is ohtaincd with but sligtitjy dinunishcd

'J"~f;VM;Aun.lH77.r.l2S.

l'hil. Dec, 1S7~.

Page 215: Lord Rayleigh - The Theory of Sound Vol 2

322.]OURI~G SONOROUS MOTION. 203

Intcnsity. A simitar rcsult wnsobtaiijudwith a forkand rc-

bou.dôt, ot'pitch an oct,a\'(! jowcr (~. (:[oscr examinatioa npvufdcd

thc fact that at t))~ .sidcs of t.]te )npp!c tI)G outward nowiu-Tstrcam \vas rcpiaccd ))y onc in thc

opposite direction, so that n,

tondue of n:nno frotn <). suitabtyp)~ccd c:m)HuftppO!a-C(t to enter

t))c nipp]n thc sanic Urne thn.t annt))cr c~hHc situatcd

imm<-([i~tc')yin front was )))o\v)) nw.~y. Ti.o two efFocts n.rc of

cour.su in rc:L!itya!t(.!rnatit)g,:u)d onJy :tppe:u- to bc sitmdtanerms

in conscqucncc of thé in:d)i)ity of thc oyc tn foUow snch rn.pid

changus. Thc fortnation ofjct.s mxst ma.kc .1.scrious draft on thc

Otur~y of thc tmjtion, an() thi.s is no donbt t)tc rcasnu w))y it is

nncc.s.s:).ry todosc thc: nipptc in ot'dut- to ohtaitt ;). po\vcrf)n Sound

fru)n :). t-c.son&tor uf this furm, witen suit:ddy tunud f'ork is prc-scnted to it.

At thc same fiinc it docs not nppca.r prob;tb]c tt)~t jet fonnft-

tion occurs to any appreciabie extcnt !).t t)tc )nont)ts of t-csonators

as ordinariiy uscd. Thc ncar agrcenK'nt bctwccn t))C obHGrvud and

thc c:dcu!atu(t pitcit is ahnost a sntneinnt pnx'f of this. Anutl~r

iLr~umcnt tonling to thc samc conclustonninybu dt'awM frum thé

pcrsistenccofthu frcc vtbmtions ofresoun.tors (§ :ni), whosc dura-

t)on scoms to excludo nny hnpurt.fmt Ct).usc of dissipa-tiou bcyuadthe commuuicatiu)i of niotion to thé surrouuding air.

lu thc case of organ pipes, wt)crc thc vibrations are vcry powcr-

fu), ttjcscarguments

arc Jcss cogent, but 1 ncc no reason for t!unk-

ingt!<at thé motion nt the uppcr&pt'n ouhHO'crs greatlyfrom thn.t

supposcd ni irchnhottx'tj cfdcuhttion. No conclusion to t))u con-

tnu'y c:u), 1 thhik, safc)y bc dra.n from the phenomena. of .stcady

tnotion. la thc opposite extrême case of impulsive motion jets

certainjy ca.nrtct bc fonnod, as fot)o\v.s from Thmuson's pri~icipic

of least cncrgy (§ 7!)), and it is doubtfu) to wilich extroue tho

case of ponodic tuotiou )nay with gruatcst plausihihty bu assitni-

Jatcd. Observation by thu mcthod of intermittent illumination

(§ 42) mightlcu.d to further Infortna.t.icn upou this subjuct.

Page 216: Lord Rayleigh - The Theory of Sound Vol 2

C'HAPTER XVII.

APPLICATIONS 0F LAPLACE'S FUNCTIONS.

323. Tim gcncral cqn~tion of n, vcioeity potcntial, w!)cn

rcfcrrcd to polar co-ordin~tos, takcs thc foi'tn (§ 241)

If /c vanish, wc hâve thc équation of tho ordinary poicnUa.1,

which, as wo know, is sa.tisHc<I, if '=~ wttcrd dcuotes thc

sphcric:U surface harmonie' of ci-dur ?;. On substitutiuli itappc:u's

t)mt thé équation sati.sficd hy is

whcrc will satisfy a.!i.cqun.ti<m

sueh ns (2).

Comp~ring (1) n.nd (2) wc sec that to détermine as a

function of ?', wu hâve

1 On tho tttcory of thoso fnnctinns the If~tost Enf{)ish \vnr]is nrc ToJhnntor'sï' ~'M)t<-<;uM Laplace, JL~t/tc, fuif~ ~t'M< aud Fei'rcfs' ~/terf'cat TYan/tottt'ctf.

Page 217: Lord Rayleigh - The Theory of Sound Vol 2

333.] SOLUTION IN LAPLACE'S FUNCTIONS. 205

In order to solvc thiséquation, we may observe th~t when r

i.n-~).u, i!,c ).,i(!ji(;tcnuisrelativcjy ncgligiblc,andthnt

LhcuLhc solution is

.nu h:ut~ Mnn nmy De assumer ta itoid good for tho complètecquatujn (4-), it' we look upuu !Utd uo lungcr as constants, but,as funcLit.n.s of ?-, who.so !i:Lturc is to bc dctcrmined.

Substitutin~in (-~), wc <iud fur 7)',

°

.Lhcsymbois ~1, and~, though indcpcndeiit of?-, M-e functions

of thé angular co-ordin~tes in thc most gGiiei-~ case, they arc

any two spherical surfMo harmonies of order M. Equation (a)maythcrcforc bo Avrittcn

On thc Communication of Vibration ~om a Vibratiug Dody to a surrouudiDc<i. ~<t!.rr<;))s.l8C8.

Page 218: Lord Rayleigh - The Theory of Sound Vol 2

20G EXPRESSION FOR RADIAL VELOCITY.[323.

Thc forms of thc functions F, n.s far as 11= 7, n.rc exhUjttud in

thén.ccompa.nying

t!t.L]c

Jo m'dur to intd thc ]û!u)ing turtus iu 7'~(~)') whcuMris srn:t,[),

wc It:u on rtjvct'sm~' thu surius in (!))

32t-. An important case of our gênera.! formulai occurs when

rcprcsents n. (iisturbance which is propagft.tcd w)tol]y oM<~(?'

At:tgrc:).t distance i'rom tlie ong:)),(w)=~(-~)-)=l, a.nd

thus, ifwc rcstorc tlie time factor (e~), wuhnvc

of which thc second part represcnts a. disturhn.ttcc travcDittg

iuwnnis. Under the circumstanccs contemp)a.tcd wc are thcrc-

fm'u to Utkc =0, and thus

which rcprcscnt~ in t.hc must gêner:).! manncr the ?)"' hiu'numic

c'))u])t'n~ntcfndiMturb:uT.cc oft)ic~ivcnpcriod<.]ifru~i!igit,sc)f

out.a)'()s]ntoi~f))))t'jsp.icc.

Page 219: Lord Rayleigh - The Theory of Sound Vol 2

324.'} DIVERGENT WAVES. 207

T))c m'igin of tlie (hsturbfinco mny bc in n. prcsci-iLed normal

motion of tho surf~cof n, sphère ofmdiu.s p. Lf't. us suppose<.ha.t :)t !').ny point on thc

spitere thé outward vchjcity is reprc-suntcd hy ~'e' bcing in ancrai n,i'uuction of t))e ROHition of

thc puitit consi()e]'c().

Lt' bc cxpfmded i)i tlie sphcric.'L) hiu-mnnic scries

whcrc the sunimation is to bc extcndcd to all (nitcgra)) values of

M. Tlie rca.! part of thise({uation will give thu vulucity potcntiid

duc to thc normal vuloeity f/cos/<1

at thu surface of thc

sphère )'=c.

Prof. Stukcs bas apphcd this solution to the cxp~natioti of a

remarl<ab)c cxpcrinicnt Ly Ij(\stic, ncc-onhng to which it n.ppea.rudthat t!ic sound of:L bu)! vibrn.ting In f). part~Hy cxh~ustcd reccivcr

is (bmini.shud by the i)it)-oductiou of )iydrogcn. This p:u-a(tuxical

phcnomcnoji ha.s its o'i~in in t!tc n.))gmentc'd w:ivc-!cngth duc to

the addition of hydrogcu, inconséquence of whiuh tlic hc)). loscs

its hold (so to spc:d~) ou thesurrounding gas.

T!)c gc'ncnd exp!a-nation Cimnot be Lutter givcn than ni titu words uf Pruf. Stukus

Suppose :). pet-son ~o jnovc his han() to and fro through n. sma.U

spaco. Ttic motion w]neh is ucc~sionud in t))c air is ;dmost cxacttythc stLmu as it would Itave huun if thu air ])!n) hL'cn nu

incompres-

sible fhud. Thcrc is mcrc iocrt.! reciprocaLing motion, in which

thc air unmcdiatdy in iront is push'jd forw:u-d, :md that Immc-

diittuly bchind itnpcDûd aftur t)tc inoving hody, w!n!c in thc

antcriursp:Lcc gcucr;d)y thu :ur rccudus from the eno-oachmcnt of

t]ie moving body, and itt Die postcrior spacc gcnoraDy f!n\vs in

from :in sides to supp!y tlie vacuum which tends to bc crcatcd so

tha.t in latéral directions thc now of the fluid is backwards, a.

Thc assuniptiou of fi real vulue for U is cquiya]out tu limitinH tho normal

Yc)~ciLy t~ Lu ht thé Htunn phitsc ai) f)v<'[' t!)n f<]))u')-o !'=r. To inc)~ tho nio.t t

t;cncM! uuriut iaotion wouid hfLYt' t') hn tt'c:it,f'~ fts co)i)j.)cx.

Page 220: Lord Rayleigh - The Theory of Sound Vol 2

FORMATION0F SONOROUSWAVES. [324.208

portion of thé cxccss of ihud in front going to supply thé dc-

ncicncy behind. Now conçoive t))c po'iodic time of thé motion

to bc contmuany diminishcd. Gradu:dtytho altcrnation of movG-

mcnt beco;ncs too nLpid to permit of th(.! fu!I Gsta.Dishmcnt of thc

mct'c-)y Ioc:U rcciprccn.tin~ f)ow; thé air is .scnsibiy cotuprcs.sed and

rarcRcd, a-nd i), sensible suunft wa\'c (or wnvc of t))e samu nature,

in c:t.se t))c po'iodic tima bu bcyoud t)ic )i)nits suit~bic to hcarin"')

ispro}~giitu()to f), ()ist;u)ce. TLc s:unu t:(.kcs p)!).cc in a.nygas,

an<) thc niure rapi<) bu tho pro])ng:Ltiun ofeondunsatKjns :md rfu'e-

factions i)i t)tc gas, tbu niurc ncarly will it approacb, in rctation to

t)ic motions wc ]t:LVu undur considcra.tion, tu thc condition of a.n

incompressible nnid thc more ncariy will thc conditions of thé

dispincmnunt of t!)Ggas

n.t thc surface oftiie solid bc satisncd by a

murcly local rcciprocating itow."

In discussing the solution (.), Prof. Stukcs gocs on to say,

"At a gréât distance from thc sphère thé function~(~?-)' bc-

coincs ulthnatulycquat to 1, and wc hâve

"It nppcars (from t)ie va.Iuo of ) that thc couipcmcutof the

velocity a]ong thc radius vcctor is of tlie ordcr ?' and that in any

direction pcrpcndieular to thc radins Ycctor of the order f" so

that tho latcra! motion may bo disrc'gardcd cxcept in tlie neigh-

bourhood ofthc sphère.

In order to examine the influence of thc Interd motion in thc

ncighbourhood of thé sphère, !ct uscompare

tlic nctual <.1isturb-

a.ncc at a grcat distance with what itwcuht hâve bccn if a-Il latéral

motion had bcen preventcd, suppose hy inHnitdy thin conica)

partitions dividing tlic ftuld into donontary canats, each boundcd

by a couical surface having its vcrtcx at thé centre.

"On this supposition thc motion in any canal would cvidentlyhc thc samc as it would bc in a!l directions if thc sp))crs vibratcd

by contraction and expansion of the surface, thc same aïï round,and such that thé normal velocity of thé surface was thc samc as

it is at t)ic pa.rticuhu' point at which thc canal in question abuts

on tlie surface. I~uw if~werc constant thc expansion of ~wou!d

1 hn've mane aomc sliHttt chMRca m rrof. Stores' notation.

Page 221: Lord Rayleigh - The Theory of Sound Vol 2

324.J EFFECT OF LATERAL MOTION. 20~) 9

Le reduccd to its Hrst tcrm and seeing tha.t(~r)

= 1, weshou)d i)ftvc from (.~),

Thi.s expression wi)! apply tn a.ny p:.rticu!ar cana) if we t~c to

doiotc Lhu normal vctocity at t.hc spl.crc's sm-faœ fur t,hn.t part.icJtarcanal and thcrufure to uLtiun tm

expression appHcabic at H))ccto aU tlie e!Lna)s, w~ )):tvu mcrdy to writc ~'fo!- 'i\) faci)ita.toa- compari.son with (~ and ((!), I sha)t, howevcr, writc for U.

Wc hâve then,

Tt must be rcmcmhcrcd that this is merciy an expression appli-cabto at once to ail thc canais, t)ic motion in each of wincft takes

place wholly alollg thc radius vector.and accordin~tyt!je expres-sion is not to be differcntiated wltit respect to or M with thé viewof fmding tlie trans verso velocities.

Oncomp~nng (7) with the expression for the

function inthé fK;tu:d motion at agréât distance from thé sphère ((!), wc secth:it tlie two arc idcntictd with the exception that is dividud

by two different constants, na.mc!y ~(~c) in tlie former case and

7'(!<-c) In the latter. Tlie sa.me will he true of thé Icading terms

(or those of the order r'') in the expressions for the condensation

and velocity. Hcnce if the mode of vibration of thé sphère besuch t)tat the normal velocity of its surface is expressed by a

Lap~ce's function of any one order, the disturbancc at a grea.tdistance from t!iG spiiere will vary from one direction to another

according to thé same law as if latéral motions had been prc-vented, thé amplitude of excursion at a givcn distance from tliecentre varying In bot)) cases as the amptitude of excursion, in a.

Jtonnal direction, of thé surface of the sphère itself. Thé' on)ydinerence is that cxpresscd by ttie symbohe ratio ~(~e) 7~ (<).If we suppose (t'/te) reducecl to thé form (cos a,, + i sin a ),tlie amplitude of vibration in the actual case will be to that in the

stlpposed case as to and thé phases in thé two cases willdiffer by o~-a,

"If thé normal velocity of thé surface of thé sphère be not

expressible by a sing!e Lapkce's Function, but only by a series,imite or Innaltc, of sucli functions, the disturbancc at a given

R.IL 14

Page 222: Lord Rayleigh - The Theory of Sound Vol 2

210 EFFECT 0F LATERAL MOTION. [:24.

gréât distance from thc centre wi!) no h)nger vary from one direc-

tion to another according to thé saine Ia\v as thc normal velocity

of thc surface of thé sphère, sincc t))C moduh~s ami hkcwise

thc amp]itu<tc ofthc imaginary quantity 7~(~c) vary with thc

ordcr of thu fttnctiou.

Lct us now supposethc disturhunce cxprcsscd hy a La.pIaCH's

'fonction ofsomo ono ordcr, and scck thc numcnciLl v:duc of thc

ahcration of intcnsity a.t a. distance, produccd by thé latcrfd

motion winch actutdly cxists.

"T!)e intensity will bc measurcd hy tlic vis ~< produecd in a

.~ivoi timc, :mdconsc(jucnt!y

will vary as thé density muttiptictt

hy t]ic velocity of propagation mtdtipHcd by thé s<)uaro of thé

amplitude of vihratton. It is thu )ast factor atone that is diifercut

h'otu what itwouhl hâve bccn if titerc had bcen no latéral motion.

T]te amplitude is ahered in tlic proportiou of/~ to so tliat if

/~u''== thc quantity by winch thc intcnsity that would

ttave existcd if tho nuid had bcea hindcrcd fTom latéral motion

lias to bc dividcd.

"If be the Icngth of thc sound-wavc corrcsponding to thé

period of thc vibration,/c=27r\so that ~c is t))C ratio of thc

cn'eumfo'cncc of thc .sphcrc to thc lengtb of a wttve. If we sup-

pose thé gas to be air a))d to bc 2 fcct, which '\vou)d correspondto about 550 vibrations iu a. second, and tlie circurnfL'rcnce 27rc to

be 1 foot (a sizc and pitch which v'ou)d correspond with thc case

ofa connnon houfic-bc))), wc sh:dl hâve A'c=. Tho fuHowin"'

tahtc givcs thc values of thc squares of tho modtnus and of thé

xc ;t=0

l ;;=! M=2

?;=!}

--l.

t;=t

_1.

4 17 1(!'25 l.l'87n l!8-t8 20-1775 S <)';)lL!5 M l.i!);8 ?

1 2 5 89 3i)' 30()):t7 S(~ l-ax 1(! l:):i<)'2 2:«~91 720H)::)71 S.

0'5 1'2:' 1B'2i¡ W:\O'2

12:!IJl!H

720Hlj:J71 lo~S-

0~5 l'()<~5 M'()t!2 ~()87H ll.Sii789t) 181f;()xl0"

4 1 0'M58~ 0'87.T~ 0-81.15:) l-tuf::)2 1 1 ]'8<J 16 2!)U-1C F

1 1 'J'C 41~ 1M~~ 1;-it)<w:n S

()-~ 1 1:< JDfii~ 18Si).'<:i 57':<)')!)7 2.

0" 1 (:U-2'.)t J'JHM muCS~l)~ 1?(J!JJ:<]Û"

ratio 7~ for tlie ftmctions .7~(t\'c) of thc first ih'c on]t;)'s, for e:Lch

ofthc vaincs~, 2, l,[md~of~-c. JtwiDprescntlyappcar~y

Page 223: Lord Rayleigh - The Theory of Sound Vol 2

324J STOKES' INVESTIGATION. 211

the table bas bccn extendcd further in the direction of values

greater than than it lias in tlie opposite direction. FIve signi-ficant figures at least are retained.

"Whcn ~c=cc wo gct from thé ana!ytica! expressions 7' =1.Wc sec from thé table tluLt when ~c is somewhat larg-o f,, is liableto bu little !css titan 1, and

consequentty the sound to Le a little

more Intense thf).n if IntGrn.1 motion }i~d becn prcvcnted. T)tc

possihility of that is cxpin.incd byconsidci-ing that tlie wavcs of

condensation spreading ft-on thoso compartmcnts of thé sptto-cw!uch fit a givun moment arc vibt'ating positively, ~.e. outwards~ftcr the hpsc of a hatf pcriod n)ay Iut.vc sp)-ca.d over the nci'di-

bonring cottipiu-ttncnts, wnich arc now in t)icir turn vibrating

posittvcly, so that Htcsc latter compartments in theh- outward

motion work ag:u)t.st a somewhat grcatcr pressure than if snch

comparttnont )iad opposite to it only ttie vibration of tlie baswinch It Lad itself occasioncd; and thé sajnc cxp]anailuu applicsMtM~~t&- -);t!<i~~ to the waves of raréfaction. Howevcr, thé in-

crca.sc ofsound thus occasionecl by the existence of Jatm'at motionis but snitdl in any case, whereas when /cc is somewhat small Initrcreases enormous)y, a.nd tho sunnd hecomes a mcro

uothing

comparcd with what it would ])a,ve been had lateral utatiua heen

pre-vcnted.

TIiG higher be the order of thé function, the grenter will he the

numbcr of compartments, alternately positive and négative as to

their mode of vibration at a given moment, into which the surface

of tho sphère will hû dividcd. We sec from t)ie table that for a

g)ven periodic timc as well as radius the value of 7,, becomcs con-

sidérable whcn ?t is somewhat high. Hovever practically vibra-

ttons of this kind are produecd when thc elastic sphère exécutes,not its principal, but one of Its subordinate vibrations, thé pitch

corrcsponding to which rises with the ordur of vibration, so that /cincreascs witti that order. It was fur this reason that the tab)c

was extended from /fc=0'5 further in tho direction of high pitchthan low pitch, namely, to three octaves higher and only oue octave

lower.

"WIien the sphere vibrates symmetricaHy about the centre, i. e.

so titat any two opposite points of thé surface are at a givenmoment

moving with cqual velocities in opposite directions, or

more generally when the mode of vibration is such tha.t there is

no change of position of thé centre of gravity of the volume, there

14–3

Page 224: Lord Rayleigh - The Theory of Sound Vol 2

LESLIE'S EXPERIMENT.[324.

212

is no term of order 1. For a sphère vibrating in thc manner of a

bcU t!tc principa.1 vibration is that exprcsscd by a terni of thc

order 2, to which 1 sha.11 now more particu)a.r!y attend.

ruttinc. for shortness. ~c" <7. wc hayo

so that the utmost incrcasc of sound produccd by latéral motion

ïunounts to about 15 pcr cent.

"I now come more particuhu-Iy to Lcsiie's expcrimcnts. Nothingis stated as to thé fonn, sizc, or pitch of lus bcll and evcn if thèse

had been accuratdy describcd, there would have bccn n. good dea.1

of guess-work in fixing on the sizc of thé sphorc which should be

considercd thé bcst représentative of the bcU. Hencc a)t we cn.n

do is to choose such values for and c as are comparée with the

probable couditions of thc cxperimcnt.

"I posscss a bcll, belonging to an o!d boU-In-air appn.ratus,winch may probabjy be somewhat sMnihtr to that used by LcsIIc.

It is ncarly hcmi.spheric:d, t!t0 diamctcr is I-9G inch, and Hic pitchan octave above t)te middtc c of a piano. Taking titc numbur of

vibrations 105C per second, and thé vcloelty of sound in air 1100

feet pcr second, we hâve \= 12-5 inches. To reprcsent thc be)l bya. sphère ofthe samc radius wonid be vcrygreatly to undcrra-tc t!ic

influence of local Ctrcu)ation,slnce ncar the mouth the gas bas but

a little way to gct round from thc outside to thé Insidc or the

reverse. To rcprescnt it by a sphcre of hait thé radius would stiU

apparentiy be to underratc thc cnect. Neverttte)ess for tho saké

of rathcrundcr-cstimating than

cxnggcrating thc innuence of titc

cause Itèreinvestigated, 1 will make thèse two suppositions suc-

cessively, giving respectivcty c = -98 aud c = -4.9, ~-c= ~D2(~ und

~c = -G3 for air.

Page 225: Lord Rayleigh - The Theory of Sound Vol 2

324.] NUMER1CAL RESULTS. 213

"If it wore not for latéral motion tlie intensitywould vary from

gas to gas in t))o proportion of tlie density into tho velocity of

propagattot~ and thci-efore as tliepressure iuto tlie square root of

the density under a standard pressure, if we take tlie factor de-

pcuding on tho devoiopment of heat as sensibly t)ie sa.me for tlie

gasus andgascons mixtures wit!i winch we have to deal. lu the

ibHowJug Table the rirst column gives thé gas, the second tho

o m

O* ~-j co Mo w o t- op o e5

o 'pc-<

i-)

$000000

~o°~So~c~oCID

sgg~ssp:~n

'<' co

t~-00 f- <f: [~

g~ts~cacafC

~Q 'p!~

Ô

«

Ô Ô G~~1

O

oo

Q

o

UQ^ pIp 9 p

OD.O~

B ~f m '-<

0~ '-t

rl

<-< eo M

0 f< 0 r- 0

<p p <po

<p ~-<

r-<

tg<g<DO<C<OMmBfoBmMM

)) t_" t~- W r-) *-< M'-<-}<~0f-<9<

u ob e~c~

t~~<

he?

L~ tr f~·r- t~ t- Q r- t~ f–&< M M 0~ 0 Ot

M 0 M o e<) c) r-<

[~ CO M .-<t. o

.-io~ m <-<

c<<?

o) o 'o

d M MOt 00 CO

f~ co w t" r~ Mo o o <?

S?< W t~

rd r·~P

.·r

0 t,n.

11

s s

.rig

ug

g:a

gS 'o S

$.h

J jM E-' <i H

Page 226: Lord Rayleigh - The Theory of Sound Vol 2

DEFI.CJKNCY0F TERM0F XEHOORDER. [324.214

pressure ~), in atmosphères, the third tlie density Z) undcr tho

pressure referred tu thc dunsity of thc air nt thé atmospht.'nc

pressure as unity, t))e fnurth, w])a.t would hâve been tlie intot-

sity h:ut t))0 motion beeu \vhol)y radial, referred to tho iutcnsity

in iur :).t attnosphcric pressure a.s unity, or, in othcr word.s, a.

quantity varying as x (the dcosity at pressure 1)\ Thon fol-

in\v thc values of (/ a.nd thé Jast Lciu~ thc aetual iutensity

ruferred to air as htjfurc.

An it~pcction ofthc numbers eontaincd in the-columns headed

wl!! shcw that thc cause hère investiga.tcd is amply sufRcIcnt to

aceount for tho fa.cts mcntioncd hy Lcsiic."

Thc importance of the suhjcct, and thc master!y mander in

which it lias hecn tre~ted hy Prof. Stokes, \viH prubably bc thought

sufHcicnt tojustify tins long quotation. Thé simpiieity of thc truc

cxphufation contrastsrcinarhabiy

with conjectures that had prc-

Tiousiy bccn advanced. Sir J. Hc'rsehe), for cxampic, thought

tbat thc mixture uf two gascs tcndingto

propa~atcsnund wit.h

din't'rcnt velocities might producc a. confusion rcsuttingia a. rapid

stining of t))C suund.

~25. Thé to'm of zero order

where is a, complex constant, corresponds to the pntential of a

~'w~ M!t?'ee of arbitrary intensity and phase, situated. at the

centre of thc sphcrc (§ 27!)). If, as often Imppens in practice, tlic

source of sound be a solid body vibrating without much changeof

vulume, this tcrm is relatively dcHcicnt. In the case of a rigid

spliere vibrating about a position of cquMihrium, the deficiciiey is

absotutc', inasmucli as thé whole motion will then be rcpresented

by a tcrm of order 1 and whenever thc body is very smn,)l in

comparison with titc wave-length, thé term of zero order must

be insignificaut. For if wc intc'grate thé equation of motion,

\7'+~=(), ovcr the sma,U volume Inciuded between thé body

and a sphère closely surrounding it, we sce that the whole quan-

tity of nuid which enters and leaves this space is small, and that

therefore thcre is but little total flow across thé surface of the

sphère.

1 Thc centre of tho spitore being the origin of coordicateB.

Page 227: Lord Rayleigh - The Theory of Sound Vol 2

325.] ]REACTION ON RIGID VIBRATINCt SPHERE. 215

and ~is proportiona! to thc cosine of t]ie angle bctwccu tho direc-

tion considcrcd a.nd somo iixcd axis. Tliis expression is of thc

same fonn as thc potcutial of n, (loitble source (§ 2D4-), situated a.t

thc centre, a.ud coniposcd of two cqun). i~nd opposite simple sources

]yin~ on thc axis in question, wtiose distance npai't is innnitcly

S)na.t), and intensi.ties Such t)iat tho product of the intensities aud

distance i.s fiuite. For, if ? bo tlie axis, and thc cosinc of the

angle hutwcen x and r bc it is évident that tlie potential of tlie

duubtc source is proportional to

It appca.rs thcn that thc disturbfmeo duc to the vjbrutioM of a

sphcro as a. rigid body is thc same as tha.t corresponding to n,

douh)e source a.t tho ccutrc whosc n.xis coiucidcs with titû Hue of

thc snhcrc's vibratinn.

Tho réaction of thc air on a small sphci'c vibrating as a, rignl

body with a. harmonie motion, may bo l'cadily catculatcd from prc-

ceding formula. If dénote thc vcbcity of the sphcrc a.b time <,

Page 228: Lord Rayleigh - The Theory of Sound Vol 2

21G INCREASE 0F EFFECTIVE INERTIA. [325.

Thc opération of thc a.ir is thcrcforc to incrcase tho effective

incrti. of thé spho-c by~) timesthc inorthlof t))cairdisp!nccd,and to ret:n'd t))G motion by a. force proportiond to t.))C vctocity,:u)d cqu:t! to ~-n-pc" thèse

eifcetsbeing in gcucm] fmicuon.

of thc f)-c<}ucney of vibration. By introduction of thé vatucs of/:md 7~ we nnd

Whon xc is sma)), we hâveapproximatu)y ~=~, <y=~V.

Hcncc thé cucctivc incrtia of a sma)! spbcrc is incrc:iscd by onc-

ba]f of that of tho air displaecd–a (p)antity indcpcndcnt of thc

frc(]uoncy and thc same as if tlio nuid wcre mcomprcssiDc. T))c

di.ssip~tivc term, which corresponds to thc cncr~y cmittcd, is of

high ordcr in ~c, and thcrcforG (the cfïccts of viseosity bcing

disrc~rdcd) thc vibrations of a smait sphère arc but slowly

datnpcd.

Thé motion ofan etHpsoid through an incompressible fluid bas

bccn investigatcd by Grccn', and his rcsnit is applicabto to thc

c:).icu)ation ufthe inercascofufFt.'ctivc int-rti~duc to a compressiblenuid, providcd tbo dimensions of thc body bc smiJt Ht compariso!!with

titcwavc-JcngtIi ofthc vibration, rur a snndt circulai' dise

vibrating at rigbt an~es to its p]anc, thé increasc of cneetivc

incrtia is to tbo mass of a sphère uf nnid, wbosc radins is optai tu

tbat of thé dise, as 2 te 7r. Tbc rcsntt for tbe case of a sphumgivoi abovc was ohtaincd by Poisson", a sbort tinic bufore tbu

publication of Grucn's papcr.

Jt bas bccn provcd by Maxwell' that thc various tcrms oftbc harmonie expansion of thé connnou

potcntiat may bc rc-

gardcd as duc to p~if~t/~e~o~~ ofcorre.spoading dpgrces of com-

plexity. Thus is proportionat to ~.hcre there

arc t differentiations nfr'' with respect to'thc axes 7~ &c., a)iy

numberofwbicfnnay in particutar cases coincittc. Itmigbtpo-haps

7~ ~H.~tctt-oM~, Dec. ]R, 183H. AJso Grccu's ~ti/«.w<tffc«t J'~x.cthted by Furrers. ~LtoniDat) & Co., lti71.

.1/<?w.«'n'.< (/c /f(~f~)~ ~< .S'c'«'))cc<, Ton. xi. p. 521.

M)t)LWt;H'sA'~c~-«;)'<y n)M/ .)j<if<t!M<, Ch. !x.

Page 229: Lord Rayleigh - The Theory of Sound Vol 2

335.] MULTIPLE SOURCES. 317

hâve bccn cxpectcd that a simitar taw wou)d hohi for tho velocity

potcntia) with thc substitution of r" for ?' Tins howcver

is not thc case; it tuay bc .shcwti tliat thc potentia.1 of a quadruple

(~ esource, dcnotcd by corresponds in ccncnU uot to thc

M/< ?' °

e"io'm of thc second ordcr simpiy, vix., ~(!'x)-), but to a

cu)nhin:diun of this with a. term of zuru ordcr. Thc ana.h)gy there-

i'urc hutds uniy in thc sing)c iastimœ uf tlie ~6 point or source,

tliough of eoursu titu function )-e" fd'tcr any nuniber of diiÏ'cr-

cnUations continues to satisfy tlic fuudfuncntal équation

It is pcrhaps wort,h notice that t!ic disturbancc outsidc any

imaginary sphère wltich comp]ete)y encloses thc origin of sound

tnay Le reprcsentcd as duc to thc normal motion of the surface of

auy smaHcr concontric sphcre, or, as a p!U'ticn!ar case whcn t!ic

ra(hus of thc sphère is innnit.uly sina!), as due to a source concen-

tratud la one pohit at thc centre. T!ns source will m gênerai bc

composcd of a combina.tion of multiple sources of ail ordcrs of

complexity.

32G. WIien thc origin of thc distm'hancc is the vibration of a

r!g)d body p:t!'aUel to its axis of révolution, the varlous sphcrica.1Itarmonics rcduce to simpte multiples of thé zonal fia-rmonic

(~). which may Le durincd as thc coemcicnt of c" in thc cxpan-

mon of {1 2e~+e~~ in rising powcrs of c. And whencvcr thc

sohd, busides bcing symmetric:d about an axis, is a.Iso symmetricalwith respect to an equatoriat plane (whosc intersection with thc

axis is takcn asorigin of co-ordinatcs), the expansion of thc

rcsultingdisturbance in spho-ical harmonies wlllcontaintcrmsof

odd order ouly. Fur exampic, if thc vibrating body wcre a circulai'

dise moving pcrpcndieutarty to its plane, tho expansion of -t~would contai n tenus proportiona! to (~), 7~ (/<.), (/u.), &c. In

thc case of thé sphcre, as we hâve S(.tcn, thc séries reduccs

absn!utc)y to its nrst tcrm, amt titis tei'))i wiH gencrally be prépon-dérant.

On thc othcr hand womay hâve a vibrating System symmetri-

cal about an axis and Avith respect to a.n cquatorial plane, but in

such a mannur that thé motions of thé parts on thc two sides of

thc plane arc opposed. Undur Uns !ica.d cornes thc idéal tuning

Page 230: Lord Rayleigh - The Theory of Sound Vol 2

218 ENERCY EMITTED [32G.

furk, cumposcdof cqual spitcrcs or para!t<jl circuhu' dises, wbo.sc

di.stattCG apart varies pcriodiea!]y. t"!ym)nc;try shcws titat thc

vutocity-potcutiaL bcit~g thc satue :).b any pointant} :).L itsimii.ge

in

thc pLnu of nytntnuLt'y, must bc an <;v<j)] function of ~u.,and ttœrc-

foru L'x)))'essihic by a, scries c<jnt:uriin~ ouly thc cven funct.Ion.s

7~(~),(~), ~c. Titc second fhnc-tion ~(~) wou!d u.sn.dfy

prcji~ndcra.tc, though in particidar ca.scs, a.s fur cxn.mp!c if thc

Lody ~o'c cutnposcd of two (Uses very cioRC tngctitcr in conpfu'isonwit)) thci)' dia.mctcr, thé symmctric:d tcrm of' zcro ordcr nu~htbceome important. A conm~-ison with thé k)iown sotutioti for tho

Hphcrcwhosc surface vibrâtes

a.cconh)~to any ]aw, will in most

c:t.sc.s fttD~sit matcrmi fur an c.stima.t<: a.s to thc relative i)nport:mœ

of thc various tcrms.

327. Thé tôt:).! émission of cucrgy hy a vibrating spitcrc is

round by )nu)tip)ying thé variable part of thc pressure (proportioualtu ~-) by thé nonnal velocity and. integrating over thc Hurfacc

(§ 2-t.). In virtue of tbe conjugatc propcrty thé varions sphcriealharmonie terms tnaybc takeu scparatclywitttout lossofgeucrality.

Wu bave (§ :)

Page 231: Lord Rayleigh - The Theory of Sound Vol 2

327.] FROM A VIBRATJNQ BPnERICAL SURFACE. 219

Now, since titero eau Le on tho whol.c no accumulation of

cno'gy in thc space mciudcd bctwccu two concentric spherical

surfhcL's, t))C rates of tr:uis)~is.siou of cner~y n.cross titesc surfaces

inust bu t))c same, that is to say ?- (:['/3–/3'ct) must bc mdepcndcnt

of?'. It~ order to dctunninc t))C constant value, wemaytnkctlK;

p!).rticu!ar case of 7' mdcfinit.dy gi'cat, whcn

It may bc obscrvcd that the !eft-hand membcr of (5) when

multipticd by t is thcimaginary part of (x+z/3) (a'–t'/S') or of

(~r)~ (-?'), so tha.t our resuit may be cxprc.sscd by sayingtha.t thé iniaginary part of j~, (~r)~, (- ï'/o') is ï'/cr, or

In this form we s!~l! hâve occasionprcscntly to mako use of it.

Thc samo conclusion may be arrived at somcwhat ]norcdircct!y

by a.)i application of Heimhoitz's thcorcm (§ 2!)-t), i.e. that if two

functions M and satisfv tlu'ouirh a. closcd sDa-ce S the Gnua.tmr)

Page 232: Lord Rayleigh - The Theory of Sound Vol 2

220 SOURCE SITUATED[~27.

It will bc more instructive tocxhibit ~a.safut]ctionof tho

no)')n:).l motion at thc suri'ace ofi), sphurc of ra<)ius c. From (2)

Tins fur)nu!a )Mybc vcnfied for tho partieular cases ~=0and

M =1, tt-catcd in §§ 280, 32;') respectively.

~28. If thé source of disturb:uice bc a normal motion of a

~)n:U[ p:u't of thc surface of tlie Hphurc (?'=c) iti Lhe mnne'Imtc

)K'i~)tbour))Ood of titu point /t=l, wc must takc in thû gcncr:d

Hulution appHc:t.btc to divergent waves, viz.

Page 233: Lord Rayleigh - The Theory of Sound Vol 2

328.J ON THE SURFACE 0F A SPHERE. 221 1

We will nûw cx~tnino t)tcprobluni wjtun /<-c i.s not vcry smn.)),

ta~ing fct'simplicitythc cascwhr'is !'?<))'<t n ~)'c'distance Ot~iy, su th:).t/,(~)-)

= 1. Tj:f! ihctor on windi tite rela-

tive iutun.sitic.s in various dirucLiuns (tcpcttd is

Thc foHowing table givcs tho mcans of calculating J~ and

fur any value of whcn /<:c=~, 1, or 2. lu thc last c~se it in

ncecfjs~ry to ~o as far as ?: = 7 to gct a tolumbly accurate rcsult, a.nd

for !arger values of /vc tho calculation would scon bceome veryJit-bonons. I)i a.)]

prubtcm.s of Uu.s sort. tbc harmonie analysis sccms

to lose its powcr whcn thc wavcs are vcry small ni comparisonwith thc dimensions ofbudic.s.

/<c==~.

2ft 2~ (M+~a~.(~+~) (,t.~)~-(~+~)

0 + 2 + 1 +'4 +-2 21 + 7 +'J8td.) -)2307n8S C4 35 --()6()J:i!)l --0:M88M3 ~(:(i + 8M -'()();i-lM7 +-()OC:M()14 +I.t')()~ .)- 8M1 +-n<)(~()5!) -)--n<)<)~~C +175CU2 -:}~1~H) +'UUUUlii -'UUUU2)i~

Page 234: Lord Rayleigh - The Theory of Sound Vol 2

222 NUMERICAL RESULTA.[328.

/<-c=l.

a (N+t)ft-(a.).j8") ();)~)~-T(n')

r'

~J

-1n j -t. i -<- i ~~a -)-~5

i + 2 i +'<: -'a

2 H -J.tD.ttO -t7]0

i) n:) + 3l --f)tr.7H.t .)'o:t0t))~ a

't t- Bi)(! + -I f'()(~t.t:tH -)'()()(i!))~

3 .) d~t :il7') .f.'()')')7.s7 -'()<))).();

n.)())!):) f'h~it -'0()<~))7 -'<'()))t)7:!

7-U:}(::i.i() +'!<)1~17 --ooouOf! -t'ooooo.~

f/ 2.

a j'(;).~)a-(a.)-) (t))-)~(ft=+~)

0 -)- 1 !.(- 2 -)-I +-3

1 + 2 + 1 +-<: +':(

+ 1'75 S!'5 +-tn'.)80 -'(.7114i< 8 4 -(.') i --173

4 IC-IM73 + 35'm.~ --t))H70 -t'IOM?.'i -)-]8<(~i; .t~t~ 8;')'4:;7. ri -)-f'l:<tf! .)'()1)!~G +!i:!H'H() -U77- .).'()'U --tHtt.f!

7 -8M1-7 -H~iG'8 --(JUU7' 3 --oou;)~

Thctnost interesting question on which thisanaly.sis informs

usisthe influence which a rigid sphère, situatcdcfosc tot))C

source, lias on tho i)itcns!ty of Sound in dif['c')'cnt directions.

Ï3y ti)c principlo of reciprocit-y (§ 29'j-) thc som-CG and thu p]~cc of

"bscrvation may be into'chnngcd. AVhcn Uterefore wc know t)m

relative intcn.sitics nt twn distantpoints 7~, 7?', dnc to a. source ~t

en thc surface of the sphère, wc hâve tdso thé relative intcnsitics

(tncasm'cd by potcntial) at thc point ~1, (h]0 to distant sources a.t

:md 7~ On this account thc problem lias a. doub)' Intcrcst.

Asa. nnmerical cxatnptc 1 hâve c~dculated t)ic values ofj~-}- ï'<7

aud J~+ C'~ fur thé a.hovc vtdues of ~c, whcn ~.=~=–J,~=0,that is, louking from thc centre of thé sphère, in the direction ~fthé source, in thc opposite direction, and IatcraHy.

Whcn ~c If! zero, thc value of F"+ <v~ !s '25, which therctorc

rcprc~'nts on thc samc sca)e as in thé t:d))e thc intensity due to

an unubKtructcd source ofc'fpia] magnitude. Wc may intcrprct ~c

Page 235: Lord Rayleigh - The Theory of Sound Vol 2

328.] NUMHIUCA.L RHSULTS. 323

as thc ratio of tlic circtnuibreuCL- of t)tc sj'hcrc to thé \vavc-)ci~thoi'thc sound.

0

M

I._p

7'f' J~hC=

1 -).():!)-'L):H17' '2')tH')l

-1 1 'J.Lt!)-)~))t!); -H;);)7~)

'iH~ii-~K:i:!i)t -}1U!~)

1 -(!)i7H.8) '2:i.-<(T.!)< -(M1(;1 11 -1 1 -t<)(~)(~);():)f -s.')~j'o

0 -)-'H'!li)Uii-'i)()JH)7.i; -CM.S

1 -7')'!)-)-2:i)21t -(!M!)8

2

0 --15:)M1-~7(!<J; -;i.(;~o 'lij:HH

'[j7(jIW

'a¡jlj~

In lookingat thèse figues tllo nrst point \v]nch attracts

attottion is t)tc conparativeiy shgttt déviation from unifunnityi)i thc intensiticsindifTerent directions. Evenwt~n thccircmn-

fercncc of t!)C sphère amounts to twice thu wavc-It-o~t)), tito-c is

sca)-(;e!y anyLhht~ to bo cancd n. sound shadow. But w)).tt is

pcr))~ps still more uncxpcetcd is t])at in tl)c fh-st two cases tlic

iatensity behmd tlic sphère cxcccds t))at in a transvcrso diroction.

Tias rcsult dcpcuds !n:LHi!y on tl)c prcpondcra.ncc of t!)c tcrm of

thc first order, whic)i vfnushcs with /n. Thc ortler of the more

importfuit tcrm.s Ino-c'ascs with ~c; whc-)i A-c is 2, the pruictpu.lt(.'rtn is that of UtC second ordcr.

Up to a certain point the augmentation of thc sphère wH!

incrcase tho total cnorgy cnnttef), bccausc a simple source cmits

twiec as muehellcrgy whcn close to M rigid plane as -whcn entirely

lu thc open. Within the limits of thc table this effect masks tlle

obstruction due to an ino-casing sphère, so that when u.=–l,the ihtensity is greato- whcn t))C circumfo'cncc is twicc thc wavc-

Jcngth than whon it is hatf thc wavc-Iength, thc source itscif

rcmaining constant.

If the source hc not simple harmotlic with respect tn timc, thc

rebdiveproportions

of thc varions const.ituent.s wih vary tosonic

cxtcntboth wititthcsixc of the sphère, andwith thc direction

<jf t))C point of ohservati~!], illustrating t)ie faudatnc'nta! cbaractcr

of thcanalysisinto simple harmonies.

Page 236: Lord Rayleigh - The Theory of Sound Vol 2

224 KFFECT 0F SMALL SPHERE[328.

WhcnA-~i.S(]('('i(]r)))y!(;.ssth:u)onc-ha)f,t))cc!dru]nti<'n o~y

Le con'iuct.cd with .s))f)ici(.'))t approximatiu)) a)~uLr;uc;!))y. Thu

!su)t,[.s

It:ipj)(':).)-s<))f).t.st)f:)r:tst)K;tL']'tr]in~t))cint(;nsityis;ut

('vc))f)))ict.iu))<)fjU,,vix.t)tc.s:L))K;at:tHytwn points ()ia)nc(t'ic:d!y

")'))".s(!<L F"rt.)tupti))dj):)) directions ~=+~ur(),thc))U)))cri(-:).)

c'!L)cu):).t.h)t)ut'L))Li('<)cf)i(.'iL'nt.<)f'/f'c;'isc!tsy<)))!tcc())mtufU)Csin)p)u

v:t.)uc'.sthL-nn,ismnt:J!)y<t(j fonctions/ Titus

\Yhcn /<:c" catTL bcncgicctcd, tlieItitcnsityi.s ]c.s.s iu a h~c~!

direction t.hn.nnnmcdi:).tc]y i)i frunt, uf or buhind thc sphère. 0)-,

by théreciprocalproperty, a Kourcu at a (H.st.:mcc

wingivca.grctd.cr

inten.sity 011 t)ic sm-~cc of a small Hphere ab titc: point fm-t.hcst

frum the source titan in a hLtcra.1 pusitiuu.

If we apply t))c.so formu).c to tho case of ~c =t, wc "'ct

wmcn f~rcc prctty ctosciy witti thc rcsuiLs of thc morecomplote

catculit.tlun.

For othcr v:ducs of thé coc~cient of ~< In (~0) nil~ht bc

CiJcuIn.tcd with t!ic aid of tables ofLegcndre's functions, or fi-oni

t)tc fullowm~ aigebraic cxprc.ssion in tcnns of~

Thc f~o'~ïce of iutcnsities in thé directions /~=+1 1 at)d

= 1 inay bc very simply cxprcsscd. Thus

1 For tLc forma of thé funetious 7~, Hoc § 3M.

Page 237: Lord Rayleigh - The Theory of Sound Vol 2

328.]] ON A SOURCE0F SOUND. 9'~-j*7

At thu .sanic timu thc totat v:L)uc of J~~+ f< nppruxiin:t.t~s tu

'wh(ji)~<)SH!un)).

~'ttCsct)))h'1"sh:).(.'at)mt,L!rc's(i))~))u:u'h~('Ht))Ccx~]a:)ati"n

()f't.hep:trt.p):)ymt)'yt)~t:\vou!u'.sinthc pcrc(.'pLtun<)t'tiiu<[t)!U'tcr

f)'U))TL~')Hc)t:L.S()))~dp)'('CL'C()H.

jft shouht ))uohHurvct1 that. thc Viu'i~tions (jfiotcnslty in (Hffui'f.'nt

diruct.iotts :Lbuutwhic)i \u itavcbucn spL'!tkit)~:u'u <)uc totitc

prc.scncu «t't.hespho'c !L.s:t))<)Lst:Lc!c,n.n<l)n'ttothcfactt)):it

Lhc source i.s «n titc circtu~fo'cucc ot'titCHphuroiusLcfut ofat

thu c(.'])trc. At a ~)'cn.t (tistfuicc :i small (i)sp):).cc'mc))t cf

finurcuof.su)m')wHt:Ln'L'(.'t Lhc ~~<-sc but )h)t,t))(ju~c;i.uia.))y

direction.

In onh')'tf'fhxl t1)~:L)t''r;(ti<~t tjfpim.se wch:LVcfo]'a,s)n:dl

.s))!teru

iront which ~'c tnay infcr <.)~t, thc pitasc ut (1!.st:uicc i.s thé .satnc

{t.s if thc .sotn'cc Lad hccu sit.ua.tcd at thc puint /t=l, )'==:jc c

(in.stc:).d(.)t'~=(.'),a)idthc't'c)tadbcunno obstacle.

32!). Thc fm~ctiona! Hy)n))ols ~nd niny bc expresscd in

tcrms of 7\ It is !\nown' that

C'unsidcr))f)W thé sytnMicopc')'))-P,,(!),fmd)ctit.

t

f'pcratf'nn~

(/Y

'Thnmsnn)U)dT~it't,.Yf~.7~§7R~(~u~t<'[UnnM~fnr~~yt.

H. 11. )5~)

Page 238: Lord Rayleigh - The Theory of Sound Vol 2

ANALYTICAL EXPRESSIONS. [~20.22G

Page 239: Lord Rayleigh - The Theory of Sound Vol 2

330.] ] MOTION CONTIN UOUS THROUGII POLE. 227

MO. Wc havo ah-cady co]).sidcred m sonie détail t!tc form

iLSsumcd Ly om- gC)~r:Ll exprensions whcn thcre is no source at

infmity. Au ci]u:L)!y important eta.ss of cases is dcnncJ Ly tiic

condition that Lttût'c be no soarcc at t.hc origin. Wc 8h:d! now

investi~'atc wluit l'c.stt-iction is t!i(;rcby itnposcd on. our gcncra!

cxprc.ssion.s.

Rcvcr.singt.hc scrics for~, \vu )):t.vo

Sincc thc fmictiou P,, is cithcr \vho!)y odd or who)!y evc']), thc

expression for is whuDy rcid cr whoDy hn~ii);u'y.

Itt.o'dcrto provcthatthc~'n.toc of~in(.))'cma.!nsfinitc

\hcn t- vani.si~s, wc bogin by obscrving that

~5–3

Page 240: Lord Rayleigh - The Theory of Sound Vol 2

228 8 ANALYTICAL EXPRESSION![330.

asis()Lvi()n.sw~c)iiti~consn)crcdtI):ttt))ccfT~ct()f()itTt'L')tti:)t)nL;

c")))yitun)))<~t)rtin)(!.s\ith!-('.s~~ttow ist'))n))itit)!yi<.h~

thcc«rr('spu)n!i)~p)W(;i-<.t' !tt-u)i)!U)tst.jcx)):H)'[t)nj~i))~.s-

Miun()nt))eti~))Linn.sccndi))~pu\vL-r.s<jf'7'. ~cha\u u

Nuw any positive intégra! po\vt.r of such ns c-nn hc

cxpnmtcd in ater)tnn:~ingHc)'K-soft))C fonctions 7', H)cfnncticn

uf itigh~st. ontcr buin~ 7~, Jt ful!o\~ th:it, if~ < n,

by knowu pn'po'tic.s of Utcsc fonctions; so t,I):tt thc lov'c.st powcr/-n

of~-inj~(/<.)c' i.s(/~)". K~ainingon)y Lhc )cadmg

tenn, wcmay writc

Page 241: Lord Rayleigh - The Theory of Sound Vol 2

~0.] FOR VELOCITY-POTENTIAL. 32!)

~'hich shcws that ViUti-ihcs with ?', cxccpt when ?! = 0.

Thu cctttplutc séries fo)' whcn t))crc is ho Sûnrcc at thc

p()t~,i.s)u'))'e cot)vc))K'nt.!y obt:um-dLyt.ht':iLi<t oftttcttt~oryoi'iiL'.s.sL-1'.s fnuct.ion.s.

Ti)C(Hn'ut-CtiU~eqm).tion.s (-)§ ~00, sat~Hud

byt))c;.seinnct.ioMS,vix.

i.s thu Bcs.sul's fmtction ofordcr ~i.

Wt)0) ?«. is mtc~rd, r (~~ + I)= 1 2 ?~ but hcrc wc hn.ve

tu do with )?t fractiona). :n)d of' thc fonn ): + :'i, being' mt.cgra!.

ih t.]tls CiLSC

Page 242: Lord Rayleigh - The Theory of Sound Vol 2

230 DESSEL'S FU~CTIOXS.[330.

Now tlie function wlt.!i which wc :u-c at prc.soit conccmu'.),

satisfics (.i.) § 32:3, vi~.

Page 243: Lord Rayleigh - The Theory of Sound Vol 2

3:~0.'j PARTICULAR CASES. 231

It wHI bc convctucnt to write duwn fur rcfcrcticc tlic forms of

-Jr~nd ,f"r<.h'Htt.t~cor<I')'

33!. 0))c of the most intorcsting n.pplica.tions of thèse rcsulLs

is to t))e investigation of thé motion of a, g:is within a, rigid

sphuricid cnvulopc. Tu detcuninc t)tc frcL: pcriod.s wc Imve on]y

to suppose tha.tva.nisttcs, when )' is cqua.1 to thc radms of t!tc

cnvclupc. T!tus in the case of tlie symnictrical vibrations, wc

havetodutenninc~ta.n~'=A:r.(1),

{m cquntion which wc hâve n.h'e~t~y considcrcd in thc ehaptcr

on mctnhnuics, § 2()7. Thc first nnitc i-oot (/<= l'-t3037r) eon-c-

sponds to the symmetrica.! vibratio)i of lowcst pitch. lu the ca.so

of hi~hcr root, Uio vibt'i.Ltion in quc.stiun !ms .sphcrical node.<i,

whû.sc mdlicorrespond to t)tc infurior roots.

Any cône, w)tosc vortex is at the ori~'in, m~ybc nuulc

rigid

without ~n'ccting thc conditions of thc question.

Thé loop.s, or places of no pressure v;u-i:).tion, aro giveil by

(~)''sin/o'=U, or /<-)'=W7r, w])crc M is any intcgct-, except,

zéro.

T)tG case of )t=l, whcn tito vibnttion.s may bo ca.Dcd <U;T.-

tnct)':).], ispcrha.ps thc most intcrcsting. ~S' bcing

a harmonie

ofordcr 1, is proportion:d to cos 0 whcrc is the ang)e bctwecn r

Page 244: Lord Rayleigh - The Theory of Sound Vol 2

L~2 DIAMETRAL VIBRATIONS. [_L

an<tsomc~xcd()h'cct.ion ofrcfcrcncc.Si)ice~va.uis))fson!y

it.tthcpi)lu.-i,t!)('():u'û))oco~ieal)iutfc.-i' ~vithvL'rt.cxn.tt.'iic centre.

Any jne]-i(H:ui!d piiLnc, !~o\vcvo-, is nod:d, n.nd tnay be supfoscd

]'!gi(L AJong nny spccificd ra.dins vectur, fim) vanish, iuxl

d~ngc sign, Avith cos(~)' sin vix. wLon i;u) /<-r=/<-r. T))c C

loops in thc' prc'.suttt case t)ierctut-ucoinci<tu \it)) thu nod:).) surfaccH

ofthe r:u)ia) vibraLiun.s.

Tof!)x~))csph(.'ri('a]))H:)c.s,\v(;]);tvc

Thc fh'nt root is ~'=0. Cidcutatin~ from Tri~nomct.ric:~T:')')(.-s Ly t.riat :m() c'rmr, 1 fi)idfurt))u ))t'\t,)'(~)(,w)(K')tco)'-

rc~punt.Lstu tlieYibr~thm orm'j.st.intpo-Latiœwitttiti f)..sp)tcrc,

~-=ll!)'2Gx-s<jtlt:tt. )-: \='3.313.

1 ~tl

T))cairs\vny.sfrun)sidut<'si(k'in]tU)ch <.))C.samcmnnncras

n)a.(1ou))!yc)<).sc(t pipe. Withunt-an~ty.si.swu nug'httmticip~tc

tt)att.)iL;pit(.tWonhtLo h~~u' fur thu sphère titan for !).ciosc(l

pipe ofoqua! Jung't)), hecausc thc spho-c may bc (turivcd f'rotn thc

cylin<)(.-r with c]osc<!C!t(ts,hyf)))i))~t)pp:irt'.f'tt)u]:itturwit.)i

u1)St)-)K-tin~)natcr)!)],thccnL'ct,of'w))ic))]nu.sthûtoMlt:)rpc~thc

.v])i)c t))C))i:)s.stohc mo\')'(I rondins but.iit.Ucc));H~(.'d.

lu f:K;t, fur a. c)').sc<)pipe of'lungDt 2/

Tf'csphcrci.sthn.s])i~))crinj)it.eht))ant)tecy)i))(IerLvab(.)ut,aFourth.

T)"vibrât ion!)ow)i))d<;rcons!(Ic')'f)t!onist])C~ra\'cstofw]ti('))

<)'<sp))(;r(-i.sca))a.b)<iti.sn)()rut.)ta))a)t<)('ta\-c~rav(;r<.))ant))c

~vc.st, radia) vibration.Tf)un('xt\')))r:diuHuf'i)u!jty})e i.ssuch

t)~tA~=~4()~ortli;Lt nn ~>,iJo~

or

:HKttst,J)L'ruforch)~hc)-lh:mU)u)ir.sLra()icL).

'A)h~).!i~asut'~cp\\i!it'hn)i;;ht))CHt)pp()~Jnj.th),v.x.('m.aL'rœ!H\\hi(;i)Uterc

i)-!)o)nutt~t).

Page 245: Lord Rayleigh - The Theory of Sound Vol 2

331.] ] VIBRATIONS 0F SECOND ORUER. 233

Whcn is gre:<.t, t!n; rocts of (2) mny be convcnicutly CtUcu-

);'t''dbv!"f'f'~sof'n.Hn)'n:s. It'/<'r=)/t7r–?/.t)t(')'

fn'tn which wc mny .SL'tcct fur spccud con.sidL'ratu~n L)iu fuHo\vh)~nutabtu cases:

(a) thuxonai])fL)'tnonic,

'Hcrc; is proportiunal to s!n20, and thcrcfui~ YtUii.shes

whcn ~=~7T. Titis.shcwsU~tthcc~hd p1a))f isn.!)f)(1:~)

surface, .so t))nt thc s:mic niotiou might tak~ pt~cu within a closcd

)tu)t)).sptx;rc. Ai.~u .sinec duu.s not invulvc N, :my muridiatud pIcUte

n):ty Le rcg:u'()cd :m rigid.

(~3) tlie .sccLon:d I~n'tnonic

H'crc again varies as sin 2~, and thc cquaturial plane is

nodat. But varies as sin~M, and tito'cfcrû ducs nut vanish

it)dcp(.:ndcnt]y ui' 0, cxcupt, whcn sin 2M = 0. IL appcar.s accord i))'dythat h\'o, aod hut.two, inuridiana! planes an;' uoda), aud th:tt thusu

are at ri~'ht angles to onc anut))cr.

('y) thc tct-scral )iai')nonic,

Inthiscasc~~v[H)t.shcsindcpcn<Icnt)yofMwithcos2~t))at

is, Avitun 0=J7r, or .7r, \v]nch givcs a nodal cône of i'cv<j!utlun

1 1 1.. 1 t ] rl~ 1mitose vcrtic:d a))g]u i.s n rig-ht :mg)u. v.u'ic.s a.s sinM, und«M

<hust))Ci'cisu)K'n')f')'idi:)n:)tno(hdp'):mc',nn'1))))tonu.

Page 246: Lord Rayleigh - The Theory of Sound Vol 2

234 AVAVE LEXÛTHS 0F VU3RATIONM ['331.

gtvi)]g a Lonc g)-a.vct- t]i:).n any of tlic radi:t) gr~op.

Li t)tc ca.su of thc gcn~ral hannonie, thc cqu~tioti givit~r tLc

toncspo.s.sibtuwidun:), sphcrc of radius ;'jf):tybc ~'riLtL'n~))§~(J

corrcspnmfH.g Lu n.einuruimportant .uod~ofvibr.i..u. In~iJ

cxhihitu.] thuf'rc(j))(.-ncy of thu varions vibrations rcfL-rn~ to tho

~-avcst,of thc whoïc My.stchi. Thc 'i~bic i.s extcndcd iar

cnou~h t«includc two octaves.

°

TAni.n A,

Civu~ tho Yalnes of for a at'bcro of unit radius.

Order of II~naonic.

0 1 2 3 4 C C

0 l~:i S'OJSf; l.MOO l.:)!)3 ].ll:t .o:ioo .g~

3~1 -81:).~ 1~77 -80!).~ -7;~<) .c;).~

~§Â 2 -f;7(i~ -C8~51 ~~U8 '0'*8' -c~.i8

c '3

ICI

r3 -.il<:7i) -SOM:i -.15:~0

gt~

4 -:if! ..10;);~)

C 'n<)8:M3 -;);)C2;<

Page 247: Lord Rayleigh - The Theory of Sound Vol 2

33L]wf'niiN A sniEnicAL ENVELorE. 2:35

TA!<M;n.

I-H.chofcaeh Ordc.r~ritchofc.~ Or.h.r ~°')

tuoo, ruturruM of f.l1Itlll)(H' 1tnoe of CllC\¡ 01',1(,1' 1 ÍN.UI~lher Iltuno.r.ferrud of

"t..ue,~fcn-cd of "f'"t~

toh'raveat. HMinuine."t

III

tuH~vcst. IlM'manic.lIUI CH.. 1101 OH

_`_

11

l'OOOO 1 o 2'85.10 1 1

J-C056 2 0 3~~8 S 0

~'l'~S 0 0Ii

:<)21 2 1

2-KiU !) 0

il

!711t 0 1

3-713 4 0II

!i'773 G O

!2. If wc drop mmcccs.sfu-y constants, t))o p:u'tictd:n- solu-

tion for the vibrations ot'gns\vit]nn~sphcric:d e:)LSCof radius

uuit.yi.src'prc.scntedby

In gc'ncra.ii.sing thi.s, \vc must rcnicmbcr t.])n.t mily bG com-

poscd ut'scvcml terms, corrcspondin~ tu cach of winch there may

c'xi.st :), vibration oi'm'bitnn'y ampHtndc :utd pha.se. Furtiicr, each

tcrm in ~S' may bc associatcd witil any, or a)!, of thc vaincs of

(tctcrnnncd by (2). For example, nndcr t)iu Iicad of M= 2, wc

might hâve

A]iy two of thc constitncnts of arc co))jugn.tc, i.e. will vanihitt,

w))L'u mu!ti])iicd togctiio', and intcgrat.cd ovcr thé Yolmne of thé

sphère. This fo)!uws fron thc propurty of t)icsphurical harmonies,

whcrcvcrthctwotGt'ms considcrcd co'rcsputidto diffcroitvtducs of

or totwo din'ut'enL constituent.s ofA' T)~c ouly ca,sc rcmamin~

for considura.tion rct~uires us to .shuw titat

Page 248: Lord Rayleigh - The Theory of Sound Vol 2

23GC) CASE OF UNIFORM[:}32.

which!snnimni~)iatcc<mscfpK'nœot'a.fn))(huncntaipn)p('rty<)f

<)K-S(;functit)n.s(§~0~). Ttn't'cisthcn-forcno(Hfticuityhtadapt-

!tI)<jgu))ut';UM())))ti()n tu pr<\scri))(;(! initiât cij'cmn.staaces.

L]0)-d<'rt()i)i)).st.t-;dcthissuLj('ctwcwiHt:).kcthcc;tS(\AvIicro

nntiaDytheg'asis in its position ()f'('()uiiih)'i))mh))t.is))')<)vin~wi(h

constant.\'c)ocit.ypa)';t]!t;)t');r. Thi.s condition of'thin~-s wuul't bc

:)ppn)xi)n:)Lt<iyrca!i.s<),ift))t;casc,h:u'in~ )~c])pruvi«)[.s)yi)iuni-ftn'm motion, w~r~.sud~chly stopper.

Sincctnct'C! isno Inititnc'ondcn.sntionot'nu'ofactiun, nHthc

(jnn.))titiL'.s~v:u)i.s)). ]f;~)'u i)uti;diyu)tity,wch:i\'('~==.c=~,

~'hicit nh(~H tlhitth('.S!))ntion ff~tt.'titisoxiy tc-nxs of t))c fir.st

o)'duri)isp)K')'ic-at Itar~tonics. 'J'hcsotution is t))<jru)'urcof U~

forn]

Page 249: Lord Rayleigh - The Theory of Sound Vol 2

33~]1

iNrrrAL YHLoci'rv. 237

fl

Thc (.'ViUttation of)'< (/<r)~ )n:Ly Le cH'cct.a! hy t.hc !ud of

:)~cuur:).tt.hcurc))i )'c)atingi.othc'sc fonctions. HythcfutKhuncnt.iLl

ttinL't'untiaiL'quat.toB

Page 250: Lord Rayleigh - The Theory of Sound Vol 2

233 Sl'HERICAL SHKLL.[332.

shcwin.~ t)~t Lho Hi-st tcrm m théscrics fur is by fiu- t,!m most

ijnpurt.nmt.

Jft may )jo wcit to j-ccait hpt'u t)t;tt

333. In n. sunihu- manncr ~cm.~y trcat t))C

proDcm of thé

vibrattons of air inchtdcd bc-twœn ri~-id conc~ntric stj)uj)-ic;d

surfitccs, whosc mdli arc?-,

nnd For Ly (J3) § ~3, if

vani.s)) for thc'.sc vatucs of)',

dl'

Page 251: Lord Rayleigh - The Theory of Sound Vol 2

H33.] l'LANE WAVES. 239

When thcdittbroice botwcct~ )-, :md is vcry snudi comparcd with

citttcr.Lhcpnjbtum "tctittftL-siLscH'witttth~Lot'tJx.iviItrationcff~ i

.sphct'ic~tHhuut,uf:ut',n.nd isbustnuh'<diu(k'pt.'udL'Ht.iy. ln(l)

§:!2~f-~bc]ndcpu))duttuf; as itisévident, tim.tiLntu.st:).!)-

proxim~t.ciy bu iu t)tc c~ML;snpposcd, -\vc )):ivu

Thu jntcrv.d bctwecn ttmgravest tonu (/< =i) !Uid Um ncxt i.s sncli

thitt t,wo of t,hcni wou!d nuLkc a hvuti'Lh (octave +<ift,h). Thc pro-btum of Die .sphcnc:d .sliect of g!M will bc f\athcr con.sidcred inL)iu i'uUuwing chapter.

~t. TI)û next ~ppticn.tio)t t)i~ wc st):t.!i n~kcof Litu .sphcnctU

hat'niuuicantJyMLs is tuinvu.st.i~tc tLe (Hsturb~ncc whidi unsues

w])(;)). pttUtu wn.ves of sound hnpingc on aiLûLstructinf spitere.

Tnkin~ th(.' centre of thé sphcre as origi.n of po)<u' co-o)-din:t.tes, and

thé direction frum -\vtiic)i tijc wavcs corne as thu :ixis of tel 6

hetttcpotcntktofthuunobstructcd plane waves. Thun !e:u'in"'ont !t.n

nnn(!Ct's.s:uy co)np)ex coeincicut, wc havo

nnd (:hc solution cftiicproHon rc<juh'cs L!tû c.\p:msiou of c" in

sphcrica! hat'moruc.s. Ua !Lcc(jnnt of thosytHi~ch-y tlic tnn-monics

roducc thonscivcn to Lpgonh'c's i'uncttous 7~ (~), so t)iat wc maytahc

\v)iorc J. :u'c functions of~ but not uf/t. Frorn w))at bas bccn

:drc.tdy provcd wc mn.ytmti(.-ip:Lt.c t!~t ~t, conHidcred a.s a func-t.i<uiof)',m.ustv~ryas

but Lhc; s:imc rcsu]t may f~sHy hc oht;unt'ddircctiy. M)))tip!yi)!~

Page 252: Lord Rayleigh - The Theory of Sound Vol 2

~0SI'IIERJCAL OBSTACLE

f~

C~) ''Y M, i~.dintcgrating wit]i

ro.spM-t. io fron) ~= 1 to~=+I,wc~hu)

I"thcpn,L]<niin).an<! thé ~vh<.)u motion

nut.si.L.O.c.spl.cre

'Y

Le d.v.)u.) into <wo parts; th. f,that, n.pn.su.h.d ),y<&

c.o,)i,to un.li.sturL..) phm.v..s, .-u.Xhc.s~

~s<ha..rc.hn,t~h< prince~thusp!.<.rc~ndmdi..Lti,~ct,t-

~I~H.i..dofU.. ]aU.r ,.u-L h~, ~havo

(-~ onrq))acmg t])c

~oto-a! )t:u')no))ic,S' bv

~ovc!<~t.ypotcnt~ofthow)m!c .notion isf.n)n.t Lya,)it,onr~a,.i ~,t~ c-nsta.ts hci~do~ni~by <).c'h<-n.yc.n~it.ons. ~)H,sc fonn .Icp.nds ..pnn t).c ct.u-actcr of t).c ob~-uc

t.r.nh.dbyth.s).hcro. Tbc.sHnp)cstca.sei.st)~of~"ri.ri.lan. hxcd .sphc.rc, anj t),cn Ll.u cundition to bc sati.silcd whcn 7. leci.sthnt

Page 253: Lord Rayleigh - The Theory of Sound Vol 2

334.]rnc![D spHH)m'L ousïAC'~H. 241

At a. su~dott distance from thu source of ()is<,U)'')):mcc wc m!iy

h)k~ ~(~r)=l. Ta o)-t)t-)- t.o to thé mitution of i-L~)

probton, wc m:tys(.'par!tt,o thc ruid au<t imn.~in:L)'y pa.rt.s, and

t))nnv!).w:tyth(.'):).tt.('r. (Jti this .sn))posit!o)~ thu p!:).nc w~vus arc

)'cpr(.sc))tc'!hy

Cunfhnng our.sch'cs for simpiicit.y's sa.ko tn p~rts cf spa.œ a.t ft

~)'(j:Ltdi.st!).ncci'n))n <ho sphère, w)u'rc/~ (;<)=], we pt'ocpcd to

<xtrt).ct thc rca.t pa.rb of (8). Sincc thc functiona 7~ arc w)io)]y

cvcn or wIioUy f)d<1,

Ascxa.mp)c.s we may writc down thé ternis in [~], in-

voiving hannonic. of ontcrs 0, 2. TIie futiuwing tab!e of the

futictinns 7~ (~) wi)) bc uscfu).

L:.tl. 1C

Page 254: Lord Rayleigh - The Theory of Sound Vol 2

DJSTUBBANCE DUE TO f334.

Ihc solution oft~e probicm hcre obtained, though ann.!ytica))yquito gênerai, is ]iard!y of pmctical use cxccpL wltcn ~c is a sm~

'{uautity. la this case we may aJvatit~eousIy expa.)id our resultsJU rising powo's ofM.

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~d PIOID SPUKRïCAL OLiSTACLK. 343

Itnppcfu-.st!i:tLwhi~ [~,] and [~,] aro of t)iG s~mc orJcrh)

t!.p.srnfJIquant.ity/ce,[~Ji.shvom-dur.s in~ier. Wcsha)IHn<i

prcscutty t)t:it thehighur )):u'mn))ic cronponouts in [~-] dépend uponstii) nxn-c e]cvatcd powc-rs of A-L'. For n first approximation, thon,wc may confine cur~tvc.s to tho ctcmunts of ordcr 0 aud J.

AIthou~h [-J conta.in.s a cosino, n.nd [~-J t). sine, they never-

thetcs.s (.liH'm- in ph:LSc hy a. sm:Ut fptit.ntity on)y. Comparlug two

ofthe valuesof~"

in (2)) § 330 wc s~c that.

Whcn M is at fi)) high, tftc expressions tan /<:)'nnd /3 s< bccon')

very ncady idcnt.ica.) for mojcmtn vaincs «f~r.

'Whcn M is û(M, wc gct in !i ncar)y similar mnnncr,

Page 256: Lord Rayleigh - The Theory of Sound Vol 2

~44 I\TE~8ITY OP SMCON))ARV AVAVES.[~34.

'i')ieYc]ocit,y-pot('n)iaIof<))cdisturb:u)ce duct~asm~tngid

f'.n.ni:\e(!n"~).'<tit~i\-t~r.j.})r..Aj,,)ith'!v,

For a givcu obst:M-]c aud a gi\)i (Hst~)ice thc ratio of H)C

n.mp]it~d(js of t)n; sc~to-cd and thé ttircL-t wavcs i.s in ~cnen~ p)-o-portiorjft! to thc ioverso

s~uiu-c uftLc wavc-Lngt.h,and Lhe n~iu ofintenshicH is proportiona! to t)tc inverse i'ourt)) pûwcr (§ SOC).

In order to compare thé intensitics of thc pruniny nndscatto-cd Rounds, we may suppose t!ic former to ori~nH.te in a

simple source, providcd it, bc surMcient]y distant (/<') from 7'.

Thus, if

It must bc wc)t undcr.stood tL~t In ordor that ihis n..su!t in!tynppty must bc grcut compiu-cd with thé iinear dunc~ion ofaud must bc grcat compared witb X.

To find thé leadhig term in thé expression for when issma!], wc !)avp in tlic ~rst place,

Page 257: Lord Rayleigh - The Theory of Sound Vol 2

334.] ] FURTIIHR APPROXIMATION. 245 5

wLiie if M bc o<td, we hâve mcrc)y to replace t" hy thc

t-esnkbch~thcustiltre:).

Byn)C:msof (;!J) wc nmy vo'ifytitc rn-st two t~nnsin th<:

(~pressions ir,r[~],j~],In

()7),(t.S). ). T.t))cc:tsc<.t'~=-.(),(:)

<k)~.sn(jt;(pp)y.

Ag!un,hy (3)),

C~nbihin~ (17), (18), (~3), (:~t),wc !ubvcthf!v:dnoof[~-j

comptute as fur a~ t!ic ternis which arc uf thc ordur x"c" cuinparud

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~C i'RË.SS(JRË.S ON OUSTAt.'LH.{'334.

witli thé two leadiu~ terms givcn ui (21). la conpounding thc-

pfu-tial expressions, it is a.sncœssary

to Le exact with respect tutLu [jhascs oi' thc cutuponcm.s us with respect, to their amptitmtcs-but

forpurposes rc()uir)')!g ody nue !)n)-)nouic clément at ft titu~thé phase is uften (jf suburdinatc importance. In sueh cases wc

jnay takc

Ft'om (:H) (,r (~!2) if appoar.s ~hat t)tc tuadin~tcrtu iti riscs

two ontur.s in wlt)) cac]t st~p in the ordcr of the Larmoxic; andthat is itsuU'

cxprcsscd by ;). serins co)it.:uning on!y evct], or (jnfyudd, powers of/cc. But bcsidcs huin~ uf hi~hur ordcr in A-c, Utc

!mt)n~ ict-m hecùtucs)'apid)y smitHur as ?! incrL'i~c.s, ou account of

thu uthur factors wh!ch it contnm.s. Tins I.sc-vident, becausc fut-

:Ui values uf ?t andjf\ (ju.) < 1, thu .sa)nc i.s truc uf + ï

\vhiK- t" oniy aiïuct.s thc phase.

lu particuiar cases any one of the harmonie cléments of[~ l

may vains).. Frum (11), (12) sinco ~+~ cannot vanish, wc

havu in such a case

tLc f~mo crjuatinn as th.~t which givus L))c pcriud.s of thc Ylbra-

tious ufort.k'r?! in n.c)o.su.) s~hcœof )'i).dius c. Aiitticco))-

sutL'f-ationwtttshuwthatt))isrc.s)))tmig])thn.(3bccn (..xpcctcd.

Tho taI))L: «('§ ~:U is app)ic~b)e t.n <.)ns(~<stiu)t a!)(t sl~ws, :unon"-

cUtcrt)!H)~,t)j:tL\))cnA:cissni;t)t,no)iarmotiie(dcn~cntiu[~] Jcati\i).))i.~)t.

Inconscqucnec of tlic ncrial

pj-essurcs thésphnro is f~f;tfi(~ on

by a force p:)ra)tcl to tlie axis of w])ose tcndency is to set tLc

Mphci-c into vibration. Titom~gnimdc of Dus furcc, if o- bc thc

dcnslty of the uuid, is givc'n by

iii which, by thé conjngate propcrty of Legcudrc'.s functions, oii]ythc tenn of tlie first ordcr aH'ccts t)tc rcsuit uf tlie intégration.~ow,when~=c,

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334.]SOURCE AT FINITE DISTANCE. 247

which cannot be sattshcd by any rc&l value of xo. We conclude

that, if thû sphère be frcc to movc, it will alw&ys bo set into

vibration.

If InstG~d of bcing absohtt.cly p):me, the primary waves have

their orig'itiiu a. unit source at a grea.t, though dnitc, distance J~

from tlie centre of tlie sphère, wc hâve

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SYMMETRICAL KXI'UH.s~ION r~S~.

~'ichisthesfuneasifthc source h~Lconoi thé spito-c.fmdthé point at w!)ic]i t!).' putcnti.d is requij-cd ~t gr~at (hstancc

(§.3~8),!U]d i.s ;n) cx;unp)c <jf- t)K.gênera! rr:neip)c ofReciprocity. J!yn~m)iing tlic prineip!c, and n~kin~ )t.sc oft)tere.suJt (~) cf'~ :~)S, \vcHec t).at if thu s..))rcc uf tlic pri)n!uy w:ivu.s bc a fi..itc.ti.sta'ticcli, thc va)nc of~.e tot.al potcntiat :Lt nny p.,int on t~.sphcrc i.s

li .4 and 7~ Lo any two points extcntal to thc spi.crc, a unitsource at ~1 will give thé sanic total putcutial at as a unit t.source at would ~ive at J. In (.iH.cr case thé total potentiat is

madcup or two parts, ofwi.ieh tf~fir.stis thé same as ifthcrowprono ob.stac)c to t]tc fn.c propagation of thé wavcs, aint tlic secondrcprcsGnts tlic <li.sturbance <))tc to tlie <J)Kt!tc)c. Of thèse t\voparts thé nrst i.s obviou.siy Die sanu-, whictiever of thc two pointsLu ~~u'ded as source, and thercfore thc other parts mnst a)so be

cquaL ti.at is tl.e v:.htc of~- at 7~ whcn i.s a .source ise~ua) to

thc value of~at .1 when is an cqua) source. Nuw when thésourcu is at a gréât distance t).e vatue of at a pointwhose anguiar .Hstanc.. fro.n is cos- and iinuar distanceh-ota thé centre iM ?', is (30)

andaccordnigly On.s 1~ aiso tLc value of~~ta greM di.stancc

w!icn t!~ .source i.s 7~. Hutsince Is disturbancc

radiati,~outwnrds from the sphcrc, it.s value at

ai)y finitc distance 7);n,Le intcn-cd frum t).at at an limite distance hy intro.hci. in't.,

cach i~r.non.c tcnn tho factur(~. Wu ti.u.s obtain t~ fut-

luwtng- symmctricn) oxpre.ssiû]i

which~iv~tl.ispartufthc potentia! ~tcith~r point, whenthputhct')S!)unitso)))w.

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334.j ]FORSECONDA.RY DI.STUHBANCE. 2-H)

It shodd be ohscrved tha.t thé général paît of thc argument

ducs nut dépend upon thu obsta.c!u bunig eiti~er sphurical or ri~td.

From thc expansion of e" ut sphet'ica.1 Itarniooics, 'WG may

(tc'htce t)t:tt of thc pot.eutia.l of w~ves i.s.sumg i'rora a )nut simple

som'cc -'f. fhiitL~y distant (/') frnm thc origin of co-ordnm.tcs. T!te

potcnti.d at :). p<j!nt 7~ :).t a)i itifinitc distance -K from thé origin,

:md iu a dh'ectiun making au auglc cos't with )', will bc

f)'om whidi wc pnss to tlie ca.sc of n. imite Ti* by t)tc sit'uptc inLro-

ductiu)) ofthu i':tci<jr/, (t~~).

T)ms tlie poteot):).) at fi)tit,L'ty (KsttUit p~jint uf :L mut. sotu'cc

at ~1 is

!). Ha.ving considcrcd at some icngth the case of a, rigid

spherica! obstacle, wc will now skctcit briuf!y tho course of t))c

investigation ~vlieu thé obst:).c)e isgascons. A!t.hough In ail

hatura.1gascs

thc eotnprcssibility is nearJy thcsa.mc, wc will sup-

pose for thé sakc of' gcncndityth:Lt t))c m:Lttcr occupying thc sp))urc

ditTcrsin comprcssihiHty.as\ve)l as in dcnsity.froin tttu médium in

which thé phmc waves advanf'c.

Exterior to the Hphcrc, <~ is thc saniecxac~y,

a.)td is of

t))L'. sfuno furm as bcf'x'c. Fur thc motion inside thc sphcrc, if

~=27r-X' bc thc internai Wfn'c-k-ngth, (2) § 330,

satisfying Lhe ecudition ofcontimnt.y throug)i thc œnti'c.

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0

UASEOUS OBSTACLE.r33~,

cxpressu~ re.spectivdy thé equalitic-.s of thé normal motions and'of thé pressures on thé two sidc.s of thé

bounding surface. Fromthcsc Gquat.ons the coni~Gtc soh.tion may be wor~d eut; butwe will hère confino our~.fves to

rinding tite Vfdoe of thcIcadinc

torms, w))o) /<-c, A:'c arcvo-y sina!

lu this case, when ?'= c,

Page 263: Lord Rayleigh - The Theory of Sound Vol 2

335.] ] CASEOUS OBSTACLE. 25 L

as thc expression for thé rno.st important part of thc disturb-

aucc, corrcspondixgto (~1) § 334 for a nxed rigid sphère. Ibt

:ipp(;n.)'s, as inighthâve bccn expecto), that thc term of zero order

is ducto thc variation ofcomprcssibility, and tt)at of ordcr one to

thé variation, ofdettsity.

From (13) wc tnay faU hach on thc case of a rigid nxcd sphcrc,

by maMug both o-' aad Mt' inrinitc. It is not surucicnt to makc o-'

by it.sdf infinitc, apparently bccausc, if ); at thé same time

rem:t.i))cd hnitc, ~e'c woutd not bc smaH, as thc investigationbas

assumed.

Wlicu -w, o-' o- a.ro smal!, (13) bccomcs equivalent to

correspoudmg to <~=c<')S<~ at the centre of thé sphère. This

agrées with thé resuk (13) cf§ 296, in which thc obstacle may bc

ofnnyform.

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KQL'AL C'OMPRESSIUILITIES.):~5.t-

Inactu;dg!LSG.s~'=~,an.tthct(-r.nof~)-oon)crdi~ppcars.If tiic ~as occupyin~ t)tc sphcric:d .sp;tco bG

inca)np:D-ab)y limitertti<mt!tC!ut,)iL'rgits,er'=(),an(t

so th:Lt in thc term uf onh.r onc, t)te effect is twicc th~t of a n.ri()Lody, a)td )in..s t)te ruYursc si~n.

Tin.- grcfd.L-t- ]~:u-t of Uns c))!~ptG)- is takcn froa two paper.s bythé author "Un thc vibmtiuns uf~ ~.s contimicd within ri~<!.sphcricfd cnvuinpc," funi fui "Jnvesti~~ion ofthe disturbancu pro-duecd by ~spi.urica! ohst.adc on thc waves uf sound' aud fromtho pa.pcr by Profussor Sioke.s atrcafiy refut-rcd to,

'S'uc~~2'~w<MarehH, 1872; Nov. 1.1, 1872.

Page 265: Lord Rayleigh - The Theory of Sound Vol 2

CIIAPTER XVIH.

Kt'HHRICAL SHUETS 0F AIR. ~OTLON IN TWO DIMENSIONS.

~30. IN former cbnpter (§ 135), we s:w tlu~t a proof of

Fnuncr's tlicorummight bcobtaiucd hyconHulering the mccha~k's

(~t' n, -vibmting strmg.A sunilar trcn.tmcnb of thé probicm of

:). spl]erical shect of air witl. Icad us to n. proof of Luplace's

t-xpunsionfor n. function which is nrbitrary ~t evcry point of

a spbenca.1 surface.

As in § 333, if is thc vclocity-potentia], thc equation of

contiuuity, rcfcrred to the ordinary polar co-ordinates M, t~kcs

the form,

Whatcvcr may bc tho chamctcr of thé frec motion, it cnu

Le iina.lyscd into n. séries of simple harmonie vibrations, thc

nature of which ia detennined by thé eorrcspouding functions

considorcd M dépendent ou sp!K'e. Thua, if -<xe" thc

équation to dotenuilie as a function of and N is

Again, wh~tcivcr fonction n~y be, it can bc cxpanded by

Fourier's theorem' iu a séries of sincs and cosincs of thc multiples

of~. Thus

Wc hère iutroducc thé condition tho.t recuis af<cr onc révolution round thé

Bphnro.

Page 266: Lord Rayleigh - The Theory of Sound Vol 2

~.J-t (:H\E);ALDJFJ.'KJf.EXTJALJ.Tjcx. r.3:;G.

L.

.v~-c

<).. c.~c.c.nt.sf,ti, ,f i

an.) hy titoœnju~atcpn.pc.-t.y of t),e <ircu)ar ft)nct.ioi]s, cnch<crm of tlie .scnc..s must

.sati.~y thcc~u~~u indcpcndcut)y

Accot'dn~ty,

is thc équation frf.m wt.ich thcc).a)-ncter of r.)- is to bp

<!<-te.-mincd. Tin.s cqu~bn n~y bc vmtten m vanuu.s way.s~

Jn tcrms of (= cos B),

\\hen thé on~na! onction is symmctncal with respectto the pole, that is, dépends upon Jatitu.Ic on!y, vanishos, andthe equations simplify. Thi.s case we .n.y conveniently takefirst. In tcrtns of

t hc sofnt.cn of thi.s cqnation In~)vc.s t~-oarbitrary constants

mn!t.p!y.ng t.vo dcHnitc fnnction.s of and rnny be ch<ainedin thé o.-d.n.ry w.iy hy as.snmin~ ~n

asccndin~ seric.s anc) dc-t~rnmnng t)if. cxpnnonts an.) cne~cient.s hy .substitution ~rhu~

in which .1 and H arctu-bitrnry eo))stantH.

Lct us now f.nt.Lcr.suppose t)..t t~.siJcs

bc.i~synunct,ie.]round thcpôle a~

.sy,nn.e(.-ic.!r.sp.ct t:

tL a(~<.h is~.n).,g!y

th;it

Page 267: Lord Rayleigh - The Theory of Sound Vol 2

33C-] coxDiïiox ïo nj.; sATL-sf.'m;) AT po!s. 255

ere~ function of thc sine of thc )atitudc (~). Undcr thcse circun)-

stances it is cicar that 7~ mnst vanish, and thc vrduc of Le

cxpresscd si)np)y by thé nrst seriez, multiplicd by t))c arbitraryconstant ~1. This va]uo of tho vctucity-potûnLia! is thc It~icnJ

conséquence of t)tc onginal (Uifc)\;tit.i:d c-quation {uid of thù°two

restrictions as tosymn~try. T))c vainc of A' might appcar

to be arbitrary, but fron what we know of tho mcchanics of t!ie

pru'bicm, it is certain befurehand that /r is reaHy iinnted tn a

séries of particular values. T])e condition, which yet remains

to 'bc introduced and by which /i, is dctcrmincd, is that thc

original équation is satisned at thc po)c itsc!f, or in othcr wordsthat thé pole is not a. source and this rcquires us to considcr

t)ie value of thé scrics when ~.=1. SIncc .thc scries is aneveu function of if thc pole ~=+1 bo not a source, ncithc-r

will bu thé pôle /~=- 1. It is (-.vidcut at once that if ~bo of

thc fonn ?t(M+l), whcrc is an even integer, tho scrics termi-

nâtes, and therefore romains nnitc whcn ~=1; but what wc

no\v want to provc is tliat, if thé séries renmin unité forjM.=I,is neeessarily of thé a.bovc-mentioned form. By thé ordinary

rule it appears at once that, whatevcr be the vahto of /t"thé ratio of successive terms tonds to the limit and there-fore thc scries is convergent for ail values of;u. less than unity.But for thé extreme va)uc /<-=]f, a highcr method of discrimi-

nation isneccssary.

It is known' tliat t!tc infinité hyporgeometrical séries

is convergent, if c+~& be grcatur t)mn .1, and c1ivc)'r''cnf.if c+~-H–~ bc (~(~u~I tn, or )css t)tan ]. Jn thu Ia.ttcr case

thé va)nc of c+~ itttbrds a critcrion of t)jcdc~rce of

(Uvcrgcncy. Of two divo-~ent scrics of thc abovc form, for

w))ic]t thc v:)hu'sof c+~a.)-c(tiffcrcnt.,t))atot)cis?-c~~c~

mfinitc for which titc value of c +~- a- is t])C smaDcr.

Our présent scries (7) may bc rcduc~) to thc standard fonn

Ly taking ~=7! (7;)-]~ whcrc )t is not nssutocd to bc intpf-nJThu.s

nn<i)c''f ~'t')))' /n')t''f.<, ]). 7f).

Page 268: Lord Rayleigh - The Theory of Sound Vol 2

('RtTHRtox OF D!VEnr:CV.f'33G

Accnrdn~y, sincc c+~l, tho serins is divergent for

~= 1, ~x/CM ~-)~i';)f~c; and it to-minatus oniy wtien is aneven intcgcr. AVo ~)-c tf.u.s ]cd to thc conclusion that whentho pote is not n. source, an<! is an (.ven fnnction

of~u. mustbc of thé form ?;. (); -}-1), whci-û )t is an cvcn intcgcr.

In !ihc .nanncr, ~-c juay prove t!~t w])cn is an odd function

ff~, !Utd thé poles :u-c notsourcc.s, =(), attJ mu.st bc of thc

ibrm ?:(?t + ~), bcing au o~(/ intogo-.

If H bc fraction~, both sc-rics arc divergent for ~=~ and

although a combundion of (hon may bc ibnnd which remfun.s~nite at eue or othcr po]c., t).crc c.-ui bc no cojnbination wbichrein~ns finitc at p(,!e.s. If thcreforc iL be a condition tbat"o point on thc .sm-face of

thc..sphcre is a source, wc hâve noatten.~t.ve but to makc

int~-al, aud cvcn thcn wc do not.sccut-c hnitenosa at tho poh.s ui.Icss we

furthcr.suppo.se ~=0whcn js odd, and ~=(), wh~n 7~ i.s cvun. Wc concludc that<or a.

compote .sphcrical Jaycr, t).c only admissible vahtes ofwhich are function.s ofj~titudc on!y, and proportional to !.arnionic

fmicttons oft))c ti)nc. arc inc]tn)r.d .m.~n..

w.K-rc(~ is

J~cndrc's funcLion, and isany ndd or even

'n~e. Thépn.ssibility of cxpanding an arbiLrary functio.i of

I~.tnde in n, .séries of Legcndrc's onctions is anccc.ssary con-

.sc.jnence of w)~t I.~s now b.-cn p.-ovcd. Any pus.sH~ motionof thc laycr of~.s is rc-pt-escnted hy t)tc so-ics

Page 269: Lord Rayleigh - The Theory of Sound Vol 2

3:} G.]1TRANSITION TO TWO DIMENSIONS. 257'

;ind tho vatuu of~r who) ~==0 is au ~?7~<tr// function of latitude.

')'))<nct])()<tt))at\(!]):)V(']m)'(ifot)(j\vud)ta.s!t)s<)tI)(.ulv:uita.gcofpt'uvm~thcet~tjx'atcpr~ncrLy,

who-n Hand~t!U-c<1H~;)-('))ti))tt\n\'r.s. Furthcfunctions -P(~)

arc thu ))<t(~functious(§')4)for thu vibt-!).ti))g System nudm-

('onsitturation, fuutacct))'()in~)y thc expression for titc ]unct,ic

('no~'y c:m ot))y lHVu)\-).; thu A'<<c.s ut' thc ~Oto'n.lixcd vctociLics.

tf(!~) ttu nut. hotd ~uud, t))C~(jf~<c~'a).su of thé \'c!ocities mustL'nter.

Thu vaiuc of i~- npprf~x-iit.te to n, ~/«)!e )aycr of vibmting g:).s

cnn ofcom'.sc! bu ()u<)ucc(l !Ls :). pa.t'ticuiar c'n.sc of Die gcnend .sotu-

tion :L))])Iic:)b!c to a npherical I.~yor. Cottfmixg oursutves to U)o

c.'LSc witcrc t)H'rc i.s nu source at ttiu pote (~.= )), wc; h:(,vc to m-

VL'.sti~iLte t))C )i)nitin~ fumi nf -~= C'(~), whcru (~+ 1)=/<whcn c'' a)t(t /t' an.: infinitc. At thé s:L!nc tune 1 and a.rc

infmitcsitïta), attd c~ passas into thé piano poiïu' radiu.i (/'), sn

that ?u<=M'. Fur thi.s purposc die jnosteonvcuient funn ofj~(~)

is that uf Murphy'

shcwing that thc BcssuJ'.s function of zcro'ordcr is an extrême case

of Lcg'cndrc's functions.

W))0i thc sphcncal IfLycr is not complète, thc probicm rc-

rjun'cs :h dif~'rcnt tt-fntmcnt. Thus, If thc g:ts he bouxtied bywa.))s

.strctctting' :dong two paral)c)s of.t:t.titudc, thc compote intégra)

tuvûlving two :u'bi.trary constants witi ingc'ncral

benecessary.

Thom.'i<mnu<t'nut.'s~<t<.P/f!§783. [/-Bin=.uotf!iitt~P.] Todhuntcr's

~~)y«f;t' 2''NN('<))i!, §Ii).

n. t). ~7

Page 270: Lord Rayleigh - The Theory of Sound Vol 2

~58 VIBRATIONS 0F A SPIIKRICAL .SIIEET[33G.

Tt'c ratio nfthc constants and thc admi.ssihh. v.~uc-.s of/~ nrc tn ho~tcnnuicd

LyU.L.,um!ary~.n)itinn.S(.xpn~h~'],;)<. :dtLc

p:)LmlIu).s in fjucst,i<.iit)tc motioni.sw])(.!)y in

]<it,u~ Th~v~u~

of~ bdns'thrm.g].out 'ncricd~!c.s.s'tnanu~<y,t)ic.suncsarc:mvay.s convergent.

If ~c portion of)]ic.surfaceocL-pi~])y~s Le thaLindu.]~

hchvcont~o parafK.I.s <.f )atitu.)c ai.c.,ua) distants from ~hc

~nator, thcqu~iun],cc(,)np.ssi.n)~r,si.jCf D.cnf.ncarothur.-ft'.c constants and 7~ in (7~ vani.si.u.s in t).c case of c-adi nor.uallunctiot).

337.~)'L'nthc.sphcrica)arcac<.]t~int)]ato(tindndMap..]c.

we hâve, as I.i thu caseuft).u.up)utcsp),L.n.,tointr..ducct}.u

cond.tionthatthcpulei.suot a.source.F~th~puq~s~sdu-

tioniMtcr)nsof~i.c.sin<?,wi))Lcnior<convc))iunt.

If wc restrict oursetvcs for Otc prc-suut to Hic caseofsynunctry

wc liavc, putting = 0 in (~) § ~3G,

Onc solution of tinscqu~ionisr~diiyobtaiHcdinU.cnrdinnry

way by as.sn.ning ;,n asccndit.g seri~ andsu)j.sLit)tti).~ in <).c

.hfturcntiai cq..aLion to détermine L).c cxpunpnt.s and co~i~nts

Wc~ct'J

Dus valueof~, is thcmn.st ancrât so)u~,n

of(1), suhjccttotl.e condition of (hntcnc.ss wh~.u ~=(). Thc

c.,np)<tc .suiutJ(,n

".voh'n~ twoarbitrary co.,stant.s prnvi.tc.s f<,r a

snurccofa.tntra.-yintens.ty at U.c p.,Jc, in which case H.e value

nf~ is innnit~v~n~=0.

-A.t)y.soiuti<.nw]tic)trc)namsfiHitc-wfion~=()!,n(]mvo!vo<onc

arLitrarycunstant, is t)..i-L.f~-u t),. ,nost gc.ucra! pos.s:)<)o uudcrtitc ru.sL.-)chun t!,at thc pulo bc n~L a sou.rc.

Aecnn]inrr]y it

uunc.œs.sary f.,r o,,r pu.-po.sc toœ.npjc.tc thé suh.tiun. T)~ ..aturc

of Lhc second funcLion(inv<.)ving a )o~arit].ni of~) wiJI bf. iih.s-

tndcd in théparticu!ar case of a

pianc Jnycr tu be con.si.tc.)

prcscntly.

'n('in).'sA';f';f~.h)M(~/f~),'?t,{;28.

Page 271: Lord Rayleigh - The Theory of Sound Vol 2

337.] HOUNDED BY A SMALL CFRCLE. 259

By writin~ )) ("+1) for/~ t.ho so-ius wiH'in ~rac~cts bccomcn

Sioco c + f7– <v= j,)h' so'ics convo'n'cs for al) vo.h)cs of t'

from 0 tn 1 i)K'h)si\'p. To values of <?(= sin'' ~) grca.tcr than ~r

<hcso)ut.i(~nisi!t;)]~))ic:d))c.

'h('t) is nn int<)', thc scrics bocomcs i~oitica.! wit))

Lc~cn()n''s functinn 7~ (~.). If t])(; iotc~nr bc cven, thé scrinH

<'oi'ninn:)h"but,()t))C)'wi.su)'cmaiit.s)))nnit(?. T)tus,w)iC)iM=l,t))c'

HrriL'.s i.s i([u))t.ic'al \vit.)) thc <j\j):u)sion uf~, viz. \/(1 in powurs

of~.

'J'hc cxp)'f'ssi<i)) fur in to'tns nf~ mnybcconvononttyapplicd

to thc invc.stigat.iot) <~f t,h(;; froc synuncLncat vihratioo.s of n. sp))cri-

cal inyo' uf !m', buuttdcd Ly n. small cit'c)e,vhoHc radius is Juss titan

the quadrant. TtK; condition tubcsatisficd i.s simply .~=0,an(IV

uquatinn hy which thé possible va)ucs of or A:V, :u'c connectcd

with thc givcn boundary Viduc of f.

Certain p:n'ticn!:Lr cases of thisproLIem may

bc trcatcd hy

mcaus of Lc~'ndrc's f'unctions. Suppose, furcxa)np)e, t))at 7t =

G, Hf)

that /r=/c'~=42. T)tccon'cspunding suhttion is

-~=ytjf~(/~).

Thc ~rcairs!. \'a)))o of /n for ~hic)) '=() is ~.=-.S3()2, corrc-f jl,

.spnnding to <9= 3:)"): = ')0f 37 radians'.

If wo takc r~=?'so tha.t,r is thc mdius of the smfdl ch'c!c

)'nef).surcdaiongt)tcsp)tcr(\wc~'ct

which is tht- f'quft.Lion conncctin!~ thc v:duc "f'/c (=27r\) with thf

eurvp<t radius ?', In thé case of :). suiatt ci)'c)< \v)x'.s(; a~gulfn' rn,(ii))s

is :;3" 5;}'. If thc; )ayci- wcrc pl~rtc (§ 339), thé \-a)uc of K)- wou)()

hc 3'~3')7; so tha.t it makes nopci'CL'))t.ii)!u

<]ifrcrcuc< in thc pit.d)

of tlie gravest tono wl)cthcr thc radins ()') of given tengtil h~

T))~ r)).<t)ftnin thc nnit. uf cu'cnhu' mca.snrG.

)7–3

Page 272: Lord Rayleigh - The Theory of Sound Vol 2

~GOUNMVMMHTRICAL T. MOTION.

['337.

strai~ht, crhccun-<.d to an arc nf M". Thcn~tofM~c~.n-

par)sonw<nt!<tjHnv~Y<.rJicniah..ri.)ny (tinrent, !fwcwu)-(..t.otak(.t.hc ~)]gt!i ot' thc circutnfo-cnce ns t)m s:nuc :;) L).u hv.~ cases, Otat

is, rcp)acc c~ = ?' hy c~ = ?'.

Inord.;rto(]L-ducoUH.sym.not.-i<-aI.s<.)uti.,nr<n-~p]:c ],,ycr

.Us<u,)y~.ssaryt~cinfini)<w),i).c.,nain.s~.itc On.~untnfthc infinie va)uc~r,t),u

.s.ut.i.,nas.su.uc.si)K..simple)~)!1a

An")<iq~)).h.ht!))V..sti~t!<nta)~.so]))ti<.hfot-tI)c

phtnc.praLbm

wi)ihugi\'('np)-cs(.)tt)y.

~{S. ~< ,sis<]i~.n.nt r~n.~<).)incnnti!dcquati<t

s!ittMi)c'(H)yt!t<L'()('nicK')tts()f.sii).s'f<),c()H.~u,i.s

'.c~utinnn.n.vh.').yt~)it!~nufa~.o~.Wun..tinn~.riv.'J

~n.~)~<n~n~u~~(~n~y~h~

.s

n. ..)~

~itiY.i,,t~r. Th.nu.th~of procure will ).c

r.\(')~j~hcd)U'(ht).yt))<ji(.<c,,ft)j(.),hm(.);).,

Page 273: Lord Rayleigh - The Theory of Sound Vol 2

338.] UMYMMETMCAL MOTrON. 2G1 1

Wc havo ])o-(!t.!)cco)np)(j<c so)))ti')n t'f t)~!prf<)!ein oftLc

vibrations «fa ~)))(.')'ic;)! taycrof~]~))))))!)'') ))V!tsn)!)])circ)c

\vt)'r!u)iusisI(-<,st.)Hmt))~(jU!t(h-))t. ~o)'(':n-!)\)!H('()t'.s',t.)n'r(j

arc a St't'ten of'p~JHsiUe\L)n(j~uf ~,t)ct.ci'<))i)!~d Lyt)tUt.~)tdi-

~=<w'Hy()ft.)tcscvduesof~thL;t'nnctmn un Oif:

)-i_L;ht-)):n~.si<)rof(2),\vI)~nn)u)tipn(.-d t)ycr).s.?M or.siu~M, is~

n~rnmtnxH.~iun ())'<.))< .syst.utn. '.i'!hj:)~r~!tt.(.~)t':dtth<-n<)nn:tt

fu]j('(i())is<'orr<'spn))<]H)g).(jt-vcryfn!))ti.s.si))!uvatncof s :)!)(! ;vith

;m a)'))itr:))'y cocf)k'ic)it pn~xud tu L'acii, ~ivL-s :mcxprcssio:)

capable (.)t'))(!i))jL;'it)t-t)tific(twit)) t,hui)titi:L! vainc of-i.u.vit.hn

i'mtctiungivu)):u-))itrat-i!yuvL'rt.hc:u-u;).()fUnjM)U!dici)'c)u.

W)[uut.h(;ra(1i)~()ft.hc.sph(!rcci.sini)))it,ctygrc:).t,isiufi))it.(j.

Jt'C~=; ~=A~~ .').!)() ~)))L'L'U)ttL"j

:). funcLtun of )- prupnrtiomd to </ (A:~).

In tunns of'/t, thc di)ï'e)-(jnt..ial cquatu~tt sa.ti~fiet! by tLc co-

cfHcicuL ofcus&'M, (jr siu&'ct), id

a.ud su)).st.it,utL- in (~). Thu corfticicnt~ftht' Inwcstpo~rof

isa(x-1); so tliat :(=(), or ?=). T)t(.'r~)ati<j)) but.wct.'a

:tf~

f'~undby c~)):ttn)g U) xui'o t]~! Ct)'j(tic'iunt.

ut'iH

Page 274: Lord Rayleigh - The Theory of Sound Vol 2

~2 COXDITIOXS TO BE SATFSFIKD[338.

Wuha\~]i.)\vtu})ruvc th:ttt.).ucut)dit.iunt.)t!Lb)~.it.)H.Tpu]cisa.source

ru<)uirc.st)i;~ M- hua positive i)it~r,m w)u'c)i~scune uruL).r ut' thu .suriu.s m

titcuxpr~.s.siu.t fur~t~ninatu.s.

Ft'rthi.s}n)rj)(..s~ it~iii nuLhc cnou~!i Lo.sftcw Lhi~thusurn.~

(u)iI~sturnun:~in~)arL-in)i.ntuw)tun~=il;it~i)jh~s..uytu pruvu U~L L),y ronnitt div~r~-nt :d't.t.r nmitipiicfLtifnt by

(1-~)~ura.swc

J'):Lyj)ntiL))H)rL;c.)nvutuunUy,t)i:(tt))'y:u'u

innn[~whL'n~=ilï'cf~~<<r~?;A(l- ItwHibusuf)iciunt to cousider iit détail t)iu casu uf t.hu fli-tit scrics.

Wchavc

Page 275: Lord Rayleigh - The Theory of Sound Vol 2

:~8.jWIIEN THE POLES ARE NOT SOURCES. 2G3

On thé otho' h:md, tlic bmomnU thcorem gives for tlie ex-

n:u)siunut'(l–)"*

Sincc s 1 > .~s- 1, it.nppc:u's U)at thc so'tcs iu thc expression

for<~aru inHnitiL's of ]ti~ht'r ordur t.)):m (l–~)" :).ud Lhurc-

f()rurc)naiui)ifmit.u:Lft<t'ntuttipi[(.'i).t.U)tt))y (!)~. Accordingty

canitut bu finitu at Luthpotc.s utituss onc ur ut.itcr of thu .scries

).~)')nin~tu,\v)H':hc:mù)))yii:tp))c)t\vI[cn~s'isxL;ro,c).'a.))o.sitivû

h)(t.~cr. If Lhuint.)'bu L'vc<),wch:Lvestillt.t) suppose/~=0;

:unt it' thc intu~L'r be odd, ~i=(), in order to sccm'c linitcuess at

UtUpufL'.S.

Jn ciLhcr c:)sc t))û value of for t!t0 conpietc sphcre may bc

put inLo L)tC iur))l

whcrc thc constant mi)I(ip!icr is omittcd. Tho complete cxpres-

Hiun l'or t)):Lt part of wtuch contains cos~'M or sniSù) as :), factor

li-ithurcfot'c

For most purposes, ]]owcvcr, it is more convcnicnt to groupthe tcrms for w)uc)t M ts thc s~mc, ratticr tliau thoso for whicit s

isthc!:i:).!nc. Titus for any vainc ot'~

who'e cvcry cocfHcicnt ~1,, 7?, ma.y bc rcgn.rdcd ascoutaining a

Lune factor of tfte funn (10).

ItiLtiaDy is atifu-bitr.n-y function of ju. n.n(~ M, and tLcrcforo

any suc)~ funct.ion is cupalde' of)jeu)~ roprosente') i)) tt)c ff)r)u

Page 276: Lord Rayleigh - The Theory of Sound Vol 2

2G4 FORMULA. OF DERIVATION,f'3~8.

which isLaptacc's expansion in

.sphoical surf.LCC )):u')no!)ics.

Froin thc difTurcnt.iidcquation (;~),

or itsg~n(.')-:t) solution

(':),itis c;)..sytop)-ovGth:)t~i!-i nfthcsii.muforHia.s<)!),so

t)i:).t.\vctn:)v writc

Equation (13) i.s agcncralixaticn of tho

propcrty of Laptacc'.s

iunctiunsu.S(j(iin(~!).

Thccon-(-H))un(ti))g]-~]at.in))s furthc-ph).nc probfcmmnyhû

()c(h!f'cd,!)s ))tjfo)-c, byat.tachin~an inHnito va)nc! to ?<, wh~h

in(1~), (14.) is :u-bitra)-y, am!

writing ?~=Smce + =

1,

~n bcing rpgnnh~ as a functitm of n. In Ujo ],it. (~cn

()t'))]~h .s))))JL-et ta')ii'['cuti;Lt.iu!t)]nayl)eidu)ttitiL-dwit.)tt))tity,

n)idt!mswujnaYt:(.ke

\\hen ihc poic is not a source, isproportiona) to

./(~-).T)'u constant c<)c)!)cit.'nt,, )cft. utxtctcrmincd

hy ()5), )n;)'v ho

ru:uj)fy foundby aco]))p:u'isun ot' t)n:i<utin~ ter)n.s. iLthu.'j

:U)peartith:Lt

a wcil-knownpropcrt.y ofUcssc! .s functiuns'. 1.

Tho vibrations of a plane iaycrofgns iu'cofcou~c more

casilydcaitwit)j,than t))usuofaJay(.T<)i'ih)itceurvatu)-c,Lut,

T(~hn))tt.')'<~f~~r<<t;('); §;)!)().

Page 277: Lord Rayleigh - The Theory of Sound Vol 2

:~38.] 1VIBRATION IN TWO DIMENSIONS. 2G5

1 h:tvf prcfcD'cd te cx))iLit thc indirect a.s wc)! as thc'. du'cet

!netht)(1nfit)vc.sti~!(,tim),h<)t.hfo)'t.})CS!)1\C()t't.LcHphut'ic:))pn)b)cni

it.sutfwiUtthccorn.'spontii))~ Liiphtcc's explosion', :ux) bucause

thc cotmcction ))(;t,wucn I!c.sHct's:t.))d L:tp)iL(;c's functioosappeiu's

nut to bc ~oto'n.Hy undL-rst.out]. '\V<j tnn.y !)o\v~ howuvur, procucd

to t))C in~(.!f')i<)L!nt trc:).ttnLiuL of t))C phuic }u'ol))u))).

33f). Ifi't. thc gcncml cqna.ti~nof simple aermi vibraduns

This équation isofthcsamcformasthat.'withwhichwchadto

d~d in troating ot' circulât'tnoabraocs (~ ~00); thc principal

]na.t)icmatic!Lt dit~crutTUC bctwcun t)i0 t\vo questions IIus iti tlic

tact tliat w)ntu m thc case of jnonbram.is thc cuudttitja to bc

sati.sncd at tlie botun)ary is '==0, i't H'c prusunt case intct'cst

attaches itsuif ratticr to the boundary coudltio)i< '=0.

cùi'rc-

spondmgtotitc couinement ofthcg~sby ft.rigtdcylmdnca.I

cnvclopc.

Tlie polo not bcing a source, tLc solution of (3) is

1 1 )m\'o hccn t))U(;h assista L.y Hcinu's /~«M~tt«;/t ~<T 7~),(M~ Derl!n,

1801, und by Sir W. Thonison'H papurH ou L~ptaeo's Thuury of tltu Tides, ~<t/.

~Yot.)v.m75.

2 1 itère rcou- to tho HS)m) natation, but tho rG~lnr will undot-stand that n cnr-

rf"~pnndH to the uf prcccdin~ Hoctio)~. Thé )t of Laj)tacL''f)fnitctious isnnwmfthitt'.

Page 278: Lord Rayleigh - The Theory of Sound Vol 2

~GG HIGID CIRCULER BOU~DARY.[339.

Thc !owc;r vahu\s of /<satisfyin~ (.) a)'(; ~ivcn in thc f'uH~witig

t:d))u', whidt w:)..s c:t)fn)a,L-tl iront iL'm.smt'.s t~tjlc.soi'thu ruocti.tu.st/ by tnn;u).s uf <.)tc ruht.t.lun~

:d)<j~i)tg tu bu ~)['cSHcd in LurutS

of ~and

fsntnhrrofm-

h'rnn) circu- tt-0 p -/t 1 M-~2 i't-.3 3)!H'))~(h'S.

0 :}-.S;!2 t-.S!l :)~.t .j.~oi

1 7' ~i 5-:i~~ C-7"~ 8'()L~

3 I()')7~ S-)C u.U!;5 ll~).i3 l:t H-7<)<!

4 I(:')7[( 1 ]t-.S);j

3 H'-Ci': ].s'u!(;

Thup:n'ticuhu'm)htLiuH)i):~yLuwnt.t.ut

whcrc ~1, 7?, 6', D :u'c:u-1)itnu-y j.,r cvcty !utn)issib]c value of

M. a)id/< Asi'iLL)n'um'.spum}In~pru])]c)n.siurtItosphut-en.))<t

uIrcni:u-n)L-inbranc,Lhu.suiti.)t':tHthcp!u'tic))int-Hu)utK)tis )nust.

bc ~ctio-id enou~it tu rcprcsoit, w)icn <==U, arbitmry v.ducs of

and

As an exampJc of conpound vihr~tio~s wcm.iy suppose, as

iu § ~32, that thu initiât coudit.ioi ci' thu gas is t)):~ dchtiudby

'X..trs~t)tc.s~r.sL'nuc~u:/))')/,< Kuv.l~.

Page 279: Lord Rayleigh - The Theory of Sound Vol 2

339.]CASE OF COMPOUND VIBRATIONS. 2G7

an C()U!tt.ionwhich may

bc vurificd m)tncric~))y,or hyan a.n:))y-

tic:d pt'uct'.sssiniit:n'to tit~tapptiu~ i)t t))u casuof()'~ §:{:j.2.

~Vu)nny}'t'uvuL))aL

Frnm this (12) i.s d(;rivnd by pntt.in~ ~=1~ and ha.ving' regard

to Liiu i'undameut~l diit'L'rott.iaI cqua-tiuu s~tisticdby

which

shuwstii~t

IIitherto Avchavc supposcdthc cyl!ndcrcomptetc,so that

rucm's ~ft.cc cach révolution, witich l'cf~Hrcs that bu mtL'gnd

but ii'instc.ut ot'titu conphtc cy)indur wo ta]\c thc soctor iuciudud

tjut.wcoi 0=0 ~uiJ ~=/3, ft'i).ctiun:Ll vêtues of~ witi lu gcuul'a.i pre-

sent t)iemsc!ves. Siuco vanishcs at both IInuts of must

beofthefonu

whcrc !=!7T~f beinguttcgnu. Itpbu~nahtptot partot'

7r (oi'Tr itscif), t))<j com~!utc so~utiou mvulvcs on)y mt~nd values

tjf astni~ht

)i:).vc Lucu iurcsL'ot but, iu gênera)) fnuct.tuns oi'

tractional urdc).' must bc mtroduccd.

Au ititct'estuig cxampicoccurs \Y!)cn ~=27r, wfuch corrc-

spftuds tu t.))c casu of a. cytindcr, travct'sc() by a rigi't watt

Page 280: Lord Rayleigh - The Theory of Sound Vol 2

2G8 ANALOOOUS PHOHLEMS rOR WATER WAV).[339.

st)-f;tc))i))g f)o))ithc-o('ntt't't.()t])(-circni))fum))œ(c<))tip.u't'§2n7).

T))L-L-ih'<'t<)ft))c~-atiis<.<)n;)).)~)-j)('ssi))!c:t()ifr(-)'(;))L;(~)f}))-t'ss)))-(i

un]tst.~omdns;h)tt,w]t(_'u no.such (Hff't'~tjccoccurs, thcwa))

!n:)y ))C ]~nYovu(), ~nd t)<c vibrations itruinchnh~) undL-r tLn

<.)K'.)ry of :Lcu)u]))(.h; cyHndt..)'. This .st:t<c

oftLu~s oconr.s

~L~tt~i.scvc!). But wttcn~is (.ni.), ~is<)rt.h<'f.))-n)(int~(.t-+~),

anttt)i(.!])r~su)~s (.))<.))(; Lwoni(tt'.s ut t)tt'wa!):u'<<)i)1c)~nt. ht

())''):)Lt.Lct'('iLSc~ is cxprrs.sihh- in finimtunn.s. T)n; gr:).vc.st

(.uHuisubta!)K!dbyt.:Lkin~~=J,orM==A,whL!ti

""<)t])cadniisHih)o\ah)t'S()f'~a)-uthn)-outs(jf<:)n~ Tho

fu-.st. rout,(;dtur /<-=()) i.s K=.i-l(! curn~jx~htin~ (n a <«))(!

~L-ci()c(I)y~nn-crt]):))ia))y<)n(-,of\\)ticht)n;c..)Utp)ct.ur\)Ht(tt.ri.-j

C!)p!Lh)c.

Ttie prcecding an:))y.si.s bas anintût'cstit)~ application to

thc~n!Lt,))e)n:LticatIy:m:).)ugut)n])rub)uniuft.hu vibrations ut wa.Lcr

in HL cyHndricat vcssc! of mn~nn (L-pth. T)tc r~dut- mayconsulta

])apcr on wavc.s hy t)tc- authur in t])c7V«7~)/<r</7

~/f~c furApri),l87<J,an()papGrsby Prof. (it)())ricto\vhi.t

rcfcrcnccisthL.rc !na<)c.Thc()1).s.;rvat.iu))ot'thcpcri~]icti)n(j

isv'')-ycasy,!mtIint)nsway)naybc()htai))L! ai) expérimentât

so)))t.iu)iut'pn)b!e)n.s,wl)u.su thuoruticnl trcatincntisfarbL-yuntt

tlie power uf knuwn tnuthud.s.

340. Rct.)n-ning to Utc compL'to cytind~r, Jc-t vts sopposc it.

closcd by rigid tran.svcr.se waHs at ~=0, and ~=/, and rctnnvu

thL')-u.sti-ict.i()ntIjatthun)ut,I),ni.stu))(jt])c.s:unuina!i transvo'.su

suctiuns. Titogtjnerat diiî'crcntia) c<)ua.tlon (§ H) is

whf'rctitccocfïifu.nts7~ mayhc fonctions of?-:ut(~. Tiusfonn

su(-ur<jst]tcf)[tfi!)nc))tr.f't))L:1)u))))d:n-yc<jm!iti())).s,w])(.'ns=0,~=/, I,

Page 281: Lord Rayleigh - The Theory of Sound Vol 2

3K).] 1 VnmATÏOXS IN A CLOSRD CYfJNDHR. 2G9

.'m'! c:n'h tcr)n)~n)st sati.sfy thc difrt.')'c))ti:dc(p)nt.innHCpar!itc;!y.

Ti.n.s

winch is<'f'<]h'.S!mt(!fo)'tn as \hc)t).h); motion isuxit'pùntiuntof

s, ~)H'i))~')'t.')'i!n'r'thy/<7: 'i'))~p!U'ticuhn'MnhtLit))t ))'ay

t))urt.'t'(j)'cbcwrittcn

w)u<'h)))u.st,l~<'t)c']':))ix<'(thy:),t.)-i}))os)t)n)t):tt)<)t),W)th respect to

:< intc~'r:)) v:)hn's uf'~ ;t))(.) )),;))«) id.so \ith rL'.spccttoati U~

V!).h)('.sut/f,(.tL;t(.')')uiHL'(thyt))CL'')uati<))),

Thé pun')y ftxi:d vibrationscurt't.'spond

to n. zcro v:).[uc of 7t",

nf)tinch)')cditit))ctabtc.

3~L Die contplutc intcgnd of thc cqun.tio)!

w))('nth<')'(!isno)i)nit:)<.i~))!t.s to thé .thscuccof:),.source.'t.tt!x'

))~)(',mvo)v(,'s!),sccotit1 fonction ()f?',whichm~y))o<)c!)ot('(1 by

'(~')- T)'"s, omitting unuccL;ss:u'y con.stiUtt multipiicrs, we may

takG(§2()0)

bntt))uH('c')ru)s(.'ric-s]'c()uii'c's)n<)dific:tt)0)),if))bc!i)ttc'gr:d. Whc'n

/<=(), <hc hvo séries bccomû ith-'utica), nnd tims thc imnic~ia.tu

)'~utt~t'.s))))p()si))g~=()in(2)t:Lch.sthûncccss!U'ygcner:Uity.Th~

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270 C!HRAL SOLUTION.['341.

rt'tjnit'cd.solution )nny,])nwt'V(')',hc'<'Lt:)i))(''t1Ly{hcnn1!)):t l'y )'utc

ap))]icah)'su<'hf':L~s. Dt'))~tn)~t])t.: couOici~nts ot'~tfu~i

in (~j by .(/'), /'(- '<), wu ]):L\'c

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:i.] ] HXt'HH.SSION BY DESCEND)~; S)':I{')KS. 271

Tho fu)-m))):L<)f(]t'rivat.ir))t(.~tnnyhc oLCuxcddirr'o~yfrom

()tcditrm'cnti:tt<([)t:)tiun()). ~r!t)))~))'/<')':Uhf;)Ht.ti)~

which is equivaicnt.to (.')), .sincc thc constant.s iniu-earbitrary

mb~t)t<(~ta)i('n.s.

Tin' scri:Lt (.'x)')'ûssujns for thns obiaincd arc cf'nvc)'<nt fnr

:d) v:duc.s of titc argument, Lut arcpractic:J)y u.s<)css whcn th(.'

ar~mount i.sgruat. Jn suc]) cases wc tnxst. hâve r~course to .suuu-

convur~cut scncscort'c.spondu)~ to <)):tt of

(10) § 200.

Equation (1) may he put into titc fonn

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DIVHiKiHXT WAVE.f:~n.L..

Whcn H isintc.L;-)'a),(~c.so séries !~m infimtcand

u!ti.natc]y<).ve,ut, hut(~~)(). ;~)thi.scircu.nstaucc dœsnot interfère

Wtt])t)K'H-pr;t.ic;d)tti!i)y.

Thc)))u.sti)np.j)t:u)t,)p)ic:ionnft])ccu!)tpl(.tcl))tp~ra]nf(~J.s to

rq~.nt.h.sturt,anœ.!ivcr~n~frn.nthHp«]c, ~prnh)c.

w'ch)n,.sbt.ntn.at.n,ySL.,).sin).i.s.n..moironth..(.on.muni-canon

~vth.-ati.s<.0!s. T~.cu.~iti..nL).;L<,t).cdi.sturhanœ

n.)~s..ntc.) by (1;~) s),)) Lu..xdu.sivt.jy divergent is

.sin.piy

'~J'i~a~orCiKnl l Y yll

1I1

In5111~r 9' tO l)C 1'l',1'3 ,~t'c~ut;m tlu; 1)rilmil~;)1 1

1. fI' l ufe

Ly.s..ppn.si.~?- hevu.-y ~t; t).upnn<.ip~<Hfncnity<,j-'t.c q.st.un (.sist.s H,

.]i.s(.vcri.~w)~tr..)atiun b.jtw~. théf-o<fhc.~nts.-i't)~

as~jh~ .sc.riL-.sc.)rn.s)H,,).Lst.)t],is~n.].ti..n

~.r~nc).purp..scSt.sc.n,p)..y.st)n'.su!.t(i.muf'())i,.tL~m-.uofa. <.h.f,n,t. !nt~ra]. Wc.shaDattaiu

t!.c.sa.nuobjcct,purInLp.s

t~oro.S)).)p]y,1,ynsH)~t))cr~))]t.S.jf§~0~.

-md thustho qucsti.,nn.dncc.s itsdft.uthc détermination ofthcf.'nu of tlie r.~).t-ha,n) H.u.nber .,f (1.~ wi.c.i is ~n~H. By (:)§;~and (.)§ 200 wchuvc

~) +~.(~)} =~+~'7r+ higher termsin~(l.-j),

s~U~taHth~rcma~sistufmdthc~nnofthcddn~cintc~~tni (l- ~hen ..s s.na)i.P~ti,~ ~~+~=~ {~~

°

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341.] ] DIVERGENT WAVE. 273

whcre y is Euler's constant ('5772.); and, as we m~y casily

tiatisiy ourscivcs by Intégration hy parts, thé other Intégrais do not

contnbute anythmg to theIc-a.ding temM. TIius, \vltcn z is very

small,

Reptacing by and compa.rmg with the form assumcd by (4.),when ?' is sma! wc sec tliat in order to makc thc sones identicat

we must tako

so that a. séries of w;),ves dtvo-gmg from tlic pôle, wLusc expression 1

in desccNding sénés is

In applying thé formula of dérivation (11) to tlie dcsccndiug

series, the parts coutaining e- and e+'~ as fa.ctors will evidentlyromain distinct, and tlic complete intégrât for thé général value

of ?, subject to thé condition that the part containing e+" sha.11

not appear, will be got by diferentiittion from tlie complete

intégral for M = 0 subject to the same condition. Thus, sinco

by(5)~=~,

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SOUNDINGBOARDS.[341.

274

or, in tcrms of tlie asccnding series,

Thcso expressions arc apphed by rrof: Stukcstosbewbowfccbly

the vibrations of a string, (corrcsponding to thé term of order

onc), arc cotmnunicatcd to thc surt-onnding gas. For titis purposoite makcs a

comparison bctwccu thé actuiU sound, .md wh:~ wonid

h.ivc been onittc-d in thb ~mc direction, wcre thc latéral Motion

of thc gas m tlie ncighbourhood of thc stnng prcveutcd. For a.

piano string corrc.spondiog to tho middic C, thc radius of t)iewirc may be abont-02 u.ch, and is about 25 inchcs; and it

appcars that t)tc sonnd is ncarty 4(),()0() timcswcaker than it wou!dh.ivc bcctt if t))o motion of thc partic)e.s ofair )md talœn place m

p)anL-s passittg tbt-ough thc a.s of titu strittg. "Thi.s shews thovit:d

nuportance uf.so)t))(]ing-horn-d.s i)t

strmged instruments.

AIthough thc amp)itudc of vibration of thc pin-tidus of thé sound~

n)g-board isextrcnK.Jy sma]I compa.rcd with tha.t of t))c partic)cs

of thc string, yet as itprGMcnts a broad surface tu tho air it is able

to excite loud sonorous vibrations, whcreas wcrc the stringsupported in an

absotnteJy rigi(t manncr, the vibrations whici)

cou)d excite directiy in t)tc air w~u)d bc so .s.nal) as to be abnust or

altogcther Inaudibic."

l''it! '!t.

"Thc incrcase of soun.! pro<!ucc<I hy thc stoppao-c of tatcr~motion ,~y Le prcttilycxLihitct] hya vory .silnpie cxpen-.ncnti~ke a

tuuing-fork, and holdmg it in thc fit.gcrs aftcr it bas bccn

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341.]SYMMETRICAL DIVERGENT WAVES. 275

ma.dc to vibrato, place a. sbpct of paper, or thé bladc of a. broad

knifn, witb its cd~e par~tiu) L~ Lhu :t,xis of {.hc fork, and as ncfn' to

tho fork ft.s convc'DKjnUy mnybcwithout toucjnng. If thc plane; of

thé «bstac)c coincide witit eit.))Ht' of thc p)ancs of symmetry of thc

furk, a-s t'cpt'csL'titcd in section at ~1 or j~, no effect is produced

but if it bc p)accd in au into'mcdia.tcposition,

such as (7, ttic

Sound becomos mucb strougor

~42. Tbo rcfd expression for the velocity-potential of sym-

mctricn.1 wavu.s divo'gingin two dimensions is obt~incd from (1M)

§ ~-H aftcr introduction of tbc time factor e' by rujccting thû

imaginary pa.rt it is

in which, as usu;d, twn :n'hit)'ary constantsmay

hc InscTtcd, one as

a. mu!tip)ier of thu w))û)c: cxprcs.siou and thé othcr as an addttion

to thc thnc.

TitG prnDcm of a. lincfu' source of uniforni Intcnsity may f).1so

hc tt'e:itGd by the g'cncnd )nut!)')d ~ppliciL~tc In thrcc dl;ncnsions.

Thus by (3) § 277, if p bu thc distance oi'any ctcmott ~.<; from 0,

thc point at which t)te potoitin.! is to hc cstimatcd, and r hc thc

smaHest value ofp, so thut =)'' +?' we ]nay tal<e

from which thc vfiriouscxjx'cssions

fu!!ow a.s in (14) § 341. When

~r is grcat, an nppn'ximit.t~ ViUue of t!ic mtu~r:).l may ue obtaiucd

by nc~ecttng titc vm'iatioa cr \/(2r+y), sinco on [(.ccount of the

rapid Hnctuntiou uf sign caused by thé factur e' wu uecd attend

r/(t!. yra' 18C8.

18–3

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276 LINEAR SOURCE.[342.

as tlie value of thc vclocity-potcntial at a grcat distance. A

.shnilar argument is appHcabIe to shcw that (1) is aiso the expres-sion fur tlie velocity-potential on one sidc of an Innnite p!anc

(§ 273) due to thc unifurm normal motion of an Inmutcshnal stripboundcd by paraUel Unes.

In Ii!~ mn.nncr wn )n:ty regard thc tcrm of t)ic first or<!er

(20) § :~1 as the expression of the vclocity-potentinl due to double

sources uniform)y distributcd aloug an infinité strai~ht linc.

FrMn tlie point of view of tlie présent section we sec the

si~ificancs of thc rctardation of winch f~ppcars in (1) and in

the results of thc foUowing section (l(i), (17). In tlieordinary

intégration for .surface distributions by Huyghcns' zones (§ 283)thé who)c effuct is tlie lmlf of tliat of tlie nr.st zone, and the phaseof thc cnect of the first zone is midway between the ph~ea due

to its extreme parts, i.c. behind thc phase duc to thc central

point. In thc présent case tho retardation of thé résultant rcla.-

tivcly to the central clement is less, on account of thc prepon-derance of the central parts.

343. In illustration of tlie formutœ of § 341 wc may take

the prohicm of tho disturbancc of plane waves of sound by a

cyhndrical obstacle, whose radius is small in comparison with

the Icngth of the wavcs, and whose axis is paraUel to tlicir

plane. (Compare § 335.)

Let tlie plane waves be represented by

Thc général expansion of in Fonncr's séries may be readily

uSeetod, the coefficients of thc various tcrms being, as might

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343.]CYLINDRrCAL OBSTACLE. 277

bc antieipa.ted, simply thc Bcs.scI'H fmictions of corresponding

ordcrs; but, as wc confine onrscivcs ho-e to the case wLere c

t])C mdms ûf thc cyliudcrIs sma.li, Ave will at once cxpa.ud 1)1

powurs of 1'.

Thus, when )'=c, if e" be omitted,

Thé amount and cvoi thc la.w of thé disturbance dépends upon

t]tc c])ara.ctet' of thc obstacle. We will bc~m by supposing thé

mn.tcrial of thc cylinder to bu a. gas of dcnsity o-' and comprcssi-

bility ?/6' thu solution of tbc probtem for a, rigtd obstacle may

finaUy be durivod by suitu.btc suppositions with respect to o- ?/t'.

If K' bc tliû mtern~I vïduc of K, wû have inside thé cyliuder by

thc condition that thc axis is not a source (§ 3~0),

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CYLINDRICA.L OBSTACLE.[343.

278

Thé tact that varius invcr.sc)y .as \"S might Ijave been

anticipatcd by thc motitod uf duncn.sions as in thu con-espondmgproDum fur the sphcrc (§33.-)). As in that case, thc

synunctric:3

pfn-t of tlie divcr~nt w:L\-c dopcnd.s upon t)tc van~tiuu of com-

pœ.ssibiJity, and wouht di.s:tppc;n- in thu application to an actua.1

gas, and tlie turm of the first ordet- deponis npon tlic variation of

dL'nsity.

By snpposin~ o-' and 7~' to becomc innnitc, in sud) a manner

that their ratio rcmaius fiuite, we obtain tlie solution corre-

sponding to a rigid and iM)novcab]c obstacle,

Thc aualysis of this section is appUcaDc to thématlicm~ticaDy

anaingous problum of H)tding t!ic ctt'ect of a cytindrica! obstacle

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343.] PASSACiH 0F SOUND TUROLTGII FABRICS. 279

on plane wavcs of transvo'sc vibration ni an clastic solid, thc

directioti of vibration bcing p:u'aHcl to tlie :txis of thé cyHndcr.If tlie dcusities be o-, o-' and thc ri~iditics be M, and Y dénote

thc tmnsvcrsû disptacctncnt, thc bound.u'y conditions arc

Fur an application to the thcory of light thc roader is rcferrcd

to a papcr by thc author, 'On tl)c manufacture aud theory of

diffracLiun gra.ting~

T)tc cxcceding smaUness of thû obstrnction nncrcd by fine

wh'cs or nbl'es to thépassage of sonud i.s

stt'ikin~Iy iHustra.tcd

i)i some of Tyndatl's cxpcrnnonts. A pièce of stUF fcit hait an

iuch in thickness allows mucit more sou)id to pass than a we~erZ

pockct-handkcrchicf, which in conséquence of tho ctosinf of

its porcs behaves rather as a thin lamina. For the same rcason

fogs, aud even raiti aud snow, interfère but little with thé freo

propagation of sounds of moderato wave-Iength. In tlie case

of a hiss, or other very acute sound, the cilcct would perbapsbe apparent.

P~t<1/«y. Vul. xn-n. 187.1.

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C1IAPTER XIX.

FLUID FRICTION. PRINCIPLE 0F DYNAMICAL 8IMFLARITY.

344. TiïE équations of Chapter XI. aud the conséquences that

wehavcdcducett fromthctn arc b~scdupon thc assumption (§230),that the mututd action betwccn anytwo portions of fluid separatcd

by au imagina.ry surface is normal to that surface. Actua.1 Sui<tn

howcvcr do not corne np to thi.s idéal iu many phcnoinona thcdcfcct of Huu)it.y, usually callcd viscosity or ftuid friction, plays an

important a,)id evoi a prcponderating part. It will therefore Lo

proper to inquire whctijcr the laws ofacri:d vibrations are sensiblyiunuenced by tlie viscosity of air, and if so in what mauncr.

In order to understand clearly the nature of viscosity, let us

conceive a nuid dividcd into parallel strata. in such a manner that

wItHe cach stratum moves in its own plane with uniformvelocity,

a change of velocity occurs in passing irom one stratmn to anothcr.

TIie simplest supposition which we ca)i makc is that thc vclocitics

ofa)l thc strata are in tho sanjc direction, but incrcascuniformly

ht magnitude as wo pass along a linc perpendicular to tlie planesof stratification. Under thèse ch'cumstances a tangential force

betwccn contiguo~ts strata is caHnd into play, in the direction of

the relative motion, and of magnitude proportional to thé rate at

which the velocity citangcs, and to a coeflicicut of viscosity, com-

monly dcnotcd by the lutter Thus, if the strata bo paraUcI to

a'~ and t)ie direction of thch' motion bc pamUul to titc tangcntia!

force, reckoned (iikc a pressure) pcr nuit of area, is

Thc dimensions of~M arc [J7Z''7~'j.

Thc exa.minatioti of t!)c origit) of thé tangenti~l force Lclon~sto motecutar science. It )ias bee]i exp!:uncd by Maxwell in ac-

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345.1 FLUID FRICTION. 281~j

cordancewith tlie kinctic theory of gascs M resulting from mter-

change of molécules between thc strata, giviug riso to diffusion of

mon~ntum. Both by theory and experiment thé rcmarka-ble

conclusion haa been esta.bliahed that within widc limits thé forco

is mdepcndent of tlie density of thé gas. For air at Centigrade

Maxwell' found

tlie centimètre, gramme, and second being units.

345. Thé investiga-tion of thé equationsof nuid motion in

which regard is pfdd to viscous forces c~u sca.rccly be considcrcd

to belong to thc subject of tins work, but it may bc of service

to some readers to point out its close conneetion with thé more

geMera.lly known tlieory of solid cla~ticity.

Thc potential encrgy of unit of volume of uniformly stra.incd

isotropicmattcr may bc exprcssed"

2

in which 8(= e +/+~) is tlie dihitn.tion, e,~ < a, c arc tlie six

componcntsof stmin, couuceted with tlic Mtua,! displaccmeuts a,AY

by tlie équations

of which M mcasurcs thc n'yi'~y, or i-Gnist~cc to &7~(t?- a.nd K

mcapurcs thé résistance to change of ~o~t?~. T!)C componeuts of

stress P, r, corresponduig rcspectivelyto

e,~ a, &, C,

are f'jund from by simple ditiercutiation with respect to those

<l)innh)t.)f'H thtIS

1 On tho Viscosity or Interne Friction of Air nnd other Gases. Phil. 2'r«M.

18CC.

Thomsou and Tait's ~<f<xr<~ T'oxo~/ty. Appondix C.

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S 82 EQUATIONS 0F MOTION.f345.

If ~Y, F, Z bc thc component.s of tlic applied force reckoned pcrunit of voknnc, tlie equ~ti~ns ofequilibrium arc ot' thc form

from \v1)ich thc cqu~ion.s of motion arcimmedia-tely obtfuna.bio

hymens oi'D'Aionburt.sprincipIu. In tenus uf tlie di.sphtcc-tncut.-j a, thcsc

c<{U!tti'jus bccutnc

1)1 thé ordin.n-y thcory of ihtid friction no forces of restitution

!n'C!lnch)ded,b)ttont)motiiCt-!]a.)idwe)ta.vetoconsiderviscousforces w))osc rc~tion to thu vu)ucities («,M) of'thc nuid cicmentsis of prucisctythu s~mu c)mracter as t)t:).t of thé forces of restitution

to thc <)iHp)accincnts (a, ~3,~) of :misotropic sulid. TiiUH if S' bc

tlie vetocity of dittt.ta.tion, Ho that

Sofaj-x~)d?!f).rcn.)-bitr:uy constants; Lut

Ititasbccnar~ncdwit)i ~-C!tt force byPruf. Stukus, th.-tt there is no rc!t.son

wt~y a,motion ot'(ti):).tatioau)ntunn ni al! (.)i)-cetio)i.ssLou)<t givcrisctovi.scuns forœ, 0- c:u)su tho prc.ssut-c tu diffcr from thc s~ticat pres-sure-

correspond)!)~ to t!tc actu~ density. In a.ccordnnce wit)i this

iu'gmncntwc :u-c to put /t=0; and.as ~ppcars from (C),~ eoincideswitit thé ()u:uitity prcviou.sly denoted by Thé frictions! termsa.re thcreforc

Page 295: Lord Rayleigh - The Theory of Sound Vol 2

345.~ rr.ANn WAVES. 283

or, if thcrc bc no n.pplicd forces :ui.d tlie squa.rc of thc motion bo

nc~ectud.,

Wcmn.y observe ttm.tthcdis.sip~tivcfurccsiturc cotisidered

correspond tu :). dissipation fnnctiuri, whosc furnt is thc .santC wit)i

rL'spcct tu u,?u vs tlt:).t of with rospect to a, /3, y, i)i ttie ttiuut'y

ui'i:jotropicsoiids. Thus puttin~ A:= 0, wc Itavc frotu

(1)

in forcement with Prof. St.okcn' ca,!cu!a,tion'. T!n.: theoryuf friction

ib)' thc c~sc of n compi'cssibio Huid was first given by Fuissur~.

:~fi. Wc will )iow app)y tlie diiïei'e;ntl!U équations to thc in-

vcsUgft.tiun ofpt~nc

wa.VL's of sonnJ. Suppo.sin~ t.h:t.t v a)id in'e

xo'u a.nd tha-t n, &c. arc functions of ou)y, wc obtit.in from

~13) §~

which is thc cquatlo~~ gtvcn by Stokc.s~.

Lct us now inquu'e how a, traiu of harmonie waves of wavc-

Jcugth which are inanitcuncd at tlie ongni (a; = 0), Me a.way

Com~rf'f 7'M;);!ffc<tnH.<, 18:')1. g ~9.

JoNntff/ </e <'A'co/<' ~/)/f<'c'7t)tt'~)«', t. xni. cah. 20, p. 139.

C<tMi')')'f~<' 2'<'«)t.i<fc<<('x. 184:

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284 EFFECTS OF FRICTION.[346.

as .c iucrcascs. Assuming tliat M varies M e' we find as ia

§14.8.

lu tho application to air at ordinary pressures ma.y bc con-

Hidcred to bo a vury smaM qu:mtity and its square may Le

ue~lected. Thus

It appca-rs th~t to tilis ordcr of a.pproxima.tion tlie vclocity of

sound is unnH'cctcd Ly Huid friction. If we rcptuce M by 27ra\

thc expression fur the cocfHcicnt of d(jc:t.y bccomcs

s)icwM)g that tue inimcncc of viscosity is greatest on the wavcs of

short wavc-)L'j)gth. Tlie :unplitudc is ditnhus!tud iu thu ratio

C 1, wlicu x =fï" In c. O.S. mca.surc wu may take

Thus the amplitude of wavcs of one centimètre wavc-Icngth is

diminishod in the ratio e 1 after travc)IIng a, distance of 88

jnctres. A wave-lcngth of 10 centimètres wouldcorrespond ncarly

to for this case a; = 8800 mètres. It a.ppe:u's therefore thu.t at

atmospheric pressures t))e influence of fricLion is not Hkdy to bu

sensible to ordiuary observation, cxcept nc:).r tite upper II)nit of the

musical sca)e. 'Die mellowing of soonds by distance, as obscrved Iti

mountainous countrics, is pcrhaps to bc attribnted to friction, bythé opération of which the higher and Iuu's))cr componcnts arc

gradually climinated. It must oftcn have bccn noticecl that the

suund s is scareciy, if at al], rctnrncd by echos, and I hâve fuund~

that at a, distance of 200 nictrcs a powcrfui hiss loses its charactcr,even whcn Uicrc is no refiection. Proba.b!y Uns enect aiso is duc

to viscosity.

AcofitictU Observations, P/t~. ~/<t.'7., Junc, 1877.

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34G.]TRANSVERSE VIBRATIONS. 285

lu highiy rarcned air thé value of a as givcn in (8) is much

incrcased, being constant. Sounds even of grave pitch may thon

bc affected withiu niodcrate distances.

From the observations of CoHadon in thc Iakc of Gcncva, it

would appcar that in water grave sonnds are more rapidiy da.mped

than acute sounds. At a moderato distance from a bcH, struck

undcr water, he found the sound short and sharp, without musical

charactcr.

347. Thé effect of viscosity in modifying thc motion of air in

contact with vlbratingsolldswill be best uudcrstood from thé solu-

tion of tho probicm for a very simple case givcn by Stokes. Lct us

suppose thn.t an innnito plane (~) exécutes harmonie vibrations In

a direction (y) parallel to itsclf. TI~c motion bcing in parallel

strata, u and M vanish, and thc variable quantitics are fune-

tions of a; otdy. Tho nrst of équations (13) § 345 shews that the

pressureis constant; thé correspondiug équation in v takes tho

furm

sun)]n,r to thc équation for tho tincn.r conduction of hcat. If wc

now suppose tlm.t v is proportional to e' tlie resulting équation

in a; Is

If thé gas bc on the positive si.do of tlie vlbrating plane the motion

is to vanish when a:=+cc. Hence J9=0, and thc value of v

bccomca on rejection of the imaginary part

a.t a;==0. Thé velocity of thc fnnd in contact with thc plane is

usually assu)aed to be tlie sn.me a~ tha.t of tlie plane itscif on the

Page 298: Lord Rayleigh - The Theory of Sound Vol 2

28G PROPAGATION 0F SOUND[347.

nppa,rent!y snfHcicntg)'f)nnd thiittLcc'mtnn'y would imply nn

itif-mitc'y ~-ren.ter sjnc'ithofss of <n (htid w;th "(..sp~f-t t~ t.hi! soiit'

ti):m wit.)t respect tu itscU'. On t.)nssupposition (5) expresses t!)u

nn'tion uf t)ic finid on tlic positive sidc due to a motion of tliu

p!:uic biven by (G).

Thctangcnti:d force pet- unit a)'c:i

n.ctingon thé plane is

'f~t.=l. Thc fir.st tcrm t'cp)-(jscnts a dissipativo forcci holding to

stop thc motion thu second rGprcscnt.s a f")'cnG<~)iv:Jc-nt to an

ino-caHO in thc ino-ti.i of t.])c vibrating body. Thc m~mtudc uf

both .t'urccs (h;pcmds upon thcfn-f~tuncy ofthc vibration.

Wc wi)i app!y th)src.sulttoca!cu!atcappt-ox!)n:t,t(j!y thé VL-iocityof sound in tubes so Ufu'row tbat t)K' vi.sco.sity nf air (.'x~rciscs a.

scnsiDu mfiucncc. As in § 2(!), !ct JV dénote thc total transfur of

<i))Ki across t)tc section of t)m tube at thc point .f. T))c fo~'c,

duo to hydrostatic pi'cssurt. actih~on thé sticc bctwcun a; and

.e + i.s, as usua),

Thc force uuu tu vi.sco.sity may bc infcn-cd fro)n t.hcinvcstigatinn

H)r :Lvibi-ftting' p):u)u, pt-ovidL-d Ui~ thu t))iduiuss of thc fayur of

:ur atUt(j)-i)~ to thu W!i)).s of the tulxj bc sn~]! in co]np!m.s'))i with

t))u (tliDn~t~-r. Thus, if 7~ bc t)tc puntm.'t.ur cf thc tube, :n)<) rbc

thc vutocity of thc auTent at a distance in'm 0~; w:i.!Ls xf thu

tube, thc tangcntiid force on t.hc s!ico, w))usc volume is is

Ly (7)

Page 299: Lord Rayleigh - The Theory of Sound Vol 2

347.]1~ NARROW TUBES. 287

Thc rc.sult cxpresscd in (12) wn.s first oht.aincd by Hchnhoitz.

An <'):dx'mtu invustig:)tiu)t of this pruUon ])as :)!.so b(.-cn givcn by

Kn'chhoff, who inchKicd iti las c~cuht.t.ion aot only tlic dU'ct, of

i'rictiou but atsn th:)i ufthe couduction ofin.-at. Kit'chhofr.s n.'su]h

is d' thc samc form as (12), but ~/(~.p'') i.si'cplucud hy t](c q)):uitity

(c~Hcd ~)

whore /< is Newtons va)ue of thc vcioeity of sound, and t/ is a co-

cfHciunt of conductiun, equal according to thé kinctic thuory of

gasusto~p" `.

Thu dintinntiou of thc vcloclty of sound in nn.n'ow tubes, aa

iudiciitcd by t.hcw:t.VL!-t(.'ngt)Kjfst:Lticna.ry vibrations, was cbscrvcJ

by Kundt (§ 2CO), and bas becu speci:d)y invcstigatcd by

Scimeebuli' and A. Sucbeck~ It appcars thn.t thc ditninution of

vclocity -varies as?' in nccordance witb (12), but, wbcn n varies, it

is proportionat rat]ter to K'~ than to Since is indcpcnduut

of thc dunsity (~)), t]ic effoct wou!d bc incrcasod in rarcficd air,

34'8. In the course of this work wc Lave )iad fréquent occasion

to notice the importance of the conclusions that may be arrived at

by the mcthod of dimensions. Now that we are in :t. position to

draw Hhtstrations from a grcatcr varicty of acousticalphcnomcn:),

re!a.t.ing- to thé vibrations of both so)ids and iiuids, it will be con-

venicnt to rcsume thé subjcet, and to dcvefopc sonewhat in détail

the principes upon -\v]tic]i the mcthod rcsts.

In thé case ofSystems, such as heHs or tt)ning-fo)\ks, formed of

uniform isotropic mntcna!, and vibmting in virtuc of cJasticity, thé

'7'n~)t).t.cxxx!7.177. 1RM. ''ro~in);.t.cxxxvi.29(!. 1SC9.

~7'f~t)f.t.cxxx!X.l(tL 1870.

Page 300: Lord Rayleigh - The Theory of Sound Vol 2

DYNAMICAL 8IMILARITY. [348.288

acoustical cléments are thé shape, thé Hnear dimension c, thc

constants of clastieity q and (§ 149), and thé density p. Hcnec-,

by thc method of dimensions, tho periodic time varies cfe<e?'M

~rM' as thé lincar dimension, at lea.st if thé amplitude of vibra-

tion be in thc same proportion; aud, if thc ia\v of Isochronism

be assamcd, tI)G !a.st-named restriction may be dispcnsed witL. Jn

fact, since thc dimensions o.f q and p arc respectively [Jt7'Z'' 2')and [~Z'"], wliile is a mère number, the only combination

capable ofrepresenting a thue is y'~ ./3~ c.

Thé argument which undcrUes this mathcmatical shorthand is

ofthc following nature. Conçoive two gnomctricallysimitar bodics,

whose mccitanicai constitotion at corrcsponding points is the

same, to exécute similar muvcmc'nts in such a manncr that t))o

corrcspondingctiangcsoccnpytimcs' which are proportional to tho

linear dimensions–in thc ratio, say, of 1 ?;. Then, if the ono

movement be possible as a conséquence of the elastic forces, thé

other will bc also. For thé nasses to bc movecl arc as 1 ?", thé accé-

lerations as 1 and tlicrefore the necessary forces arc as 1 M';

and, sincc the strains are thc same, tins is in fact thé ratio of tlic

clastic forces duc to them when rcferrcd to corrcsponding a-reas.

If thé elastic forces are competent to producc the supposcd motion

in the first case, they arc atso competent to produce thc supposcdmotion in the second case.

Thc dynamical similarity is disturbed by thé opération of a

force like gravity, proportions! tothc cubes, and not to thc squares,

of eorresponding lines; but in cases whcre gravity is thé sole

motive power, dynamical similarity may bc sccured by a different

relation between con-csponding spaccs and corrcsponding times.

Titus if thc ratio of corresponding spaces bc 1 ?), and that of

corresponding times bc 1 )r, thé accélérations are in both cases

thc same, and may bc thc effects of forces in thc ratio 1 m" actingon masses whieh are in thé same ratio. As examples comingundert!us head may be mentionc<) the common

pendufum, sca-waves,

whose velocity varies as thé s()uarc root of the wave-]ength, and thé

whole theory of thé comparison of sliips and thcir models bywhich Mr Froudo prcdicts thé behaviour of ships from experi-ments made on models of moderate dimensions.

1 Thé conception of an altération of sealo in spacc has ~ccn mado familiar bythé nuiverfifd use of mapo and n)0(tc)s, but tho

con-ospondiu~ conception for timo

is often less distinct. Référence to tho caoo nf musieai composition performed atdiflerout spcods may nssist thé imnginntion of tbo student.

Page 301: Lord Rayleigh - The Theory of Sound Vol 2

348.]DYNAMICAL SIMILARITE. 289

Thc same comparison that we bave c'mpJoycd abovc for clastic

sobds app)ics a)so to acrial vibrations. T)<L- pressurer, in thé cases to

bc compared arc thé same, aud thercforc wlicn acth)~ ovcr areasin.

thc ratio 1 ?r, givc forces in tbc sarnc ratio. Thèse forces operato

on masses in thé ratio 1 ?~,and thûreforu pro(h)cc accélérations in

thc ratio 1 /f, winch is thé ratio of th(; acLuat accek'rations wJten

both spaccs and times arc as 1 :7:. Accordix~lythe pcr!odicti)ncs

of simiJar rosonant cavitius, H))t.'d ~'ith thc sa.rnc i~as, arc dirc'ctiy as

thc lincar dimcnsiun–a Yc'ry important )aw fh'st f<n'niu)atcd hy

t-i:LYart.

Sincc thc s:m)c mut.hod of conpaD.son :)pp1i<s both to clastic

.so)id.s and to cta.stic nuids, an cxtcnsiua mayhc nmdc to Systems

into winch both ~ind.s of vibration c'ntcr. For cx:nuph?, tho sc:do

of asystcm compoundcd ofa hining-fork and of nn nir rcsonnfor

maybe supposed to bc altcrcd wit))out change in tbc motion ot.hc'r

~han that i:)vo!vcd in takit~ thc timcs in thc .samc ratio as thé

hncar dl~ncnsions.

JIitItcrto thc altération of sca!c bas b<;un snpposcd to bc

nnifonn m iiU dimensions, but Hierc arc cnscs, not c'oniiog undcr

this hcad, to whieh thc principeof dynannMd simihn'ity maybc

most usefuDy applicd. Let us considcr, for c'xamptc, titc i!cxural

vibrations of a sy.st.em conposedoi' a tbin clastic hunin~, phtncor

cnrved. Dy §§ 2L-i-,2L5 wu sec that thc thicknc.ss of thu Iann<t:t

and thc mechanica) const.'u~ts y and p, will occur onty in tbe cotn-

binations and and thns acontp:n'ison may

bc made cvcn

although tbc attcradon of tinckncss bc not in. t]tc Hamo proportion

as for thc ot-!)or dimensions. If c bc thu ]Inear ditncnsion who~

thc tilickncss is disrcgardcd, t)ic timc.s must vary c~o'/ô' ~)fu't'~<s

as ~3~.c\ yor a givcn uiatcrial, ttnckncss, and shapc, thc

titnes arc thcrcforc as tbe.s~«~' of tbc )inoar dnnc'nsion. Jt must

t)ot bc forgottcn, Itowcvci') t)tat rc'snits such as thcsc, which invo)vc

a )aw whosc truth is oïdy approximatc, stand on a dinercnt Jcvct

from thc more Immédiate conséquences of t))u principic of shni-

Im'ity.

THE END.

R. !ï. 1 <)

Page 302: Lord Rayleigh - The Theory of Sound Vol 2
Page 303: Lord Rayleigh - The Theory of Sound Vol 2

APPENDIX A.(§ 307).

Thoprobtcm

of dctcrmming tho correction for thé opcu cud of a

tubei8oneofconaIdct'a.b)cdif)icuKy,cvcnwhenHu-rci~au Innnitc

HtLnge. Itisprovedin t]<c text (§ 307)

that thé co)')-uction a in grcnter t]mu

Q

~7r7?,n.ud tcsa tha:

7~.Tho latter vahio is obtuuied

l'y c:duuhiting

thé encrgyof tho !not!o)t on tho suppositiou t)mt thc

vn]oc!ty p:u'a))c)

to thc axis is constant ovcr tho planeof thc mouth, and

con)])arit)g this

encrgy with thé square of tho total ctu't'ent. TJtC actnatvolocity, no

doubt, mercasca froni tho ccntroontw:u'ds, bnconnng i))<inito at thc

sl)~)'p

tidgea.ud tho assumption

of tL constant value ia a sotucwhat violent onc.

NovcrthcJoss tho value of a so Cidculatcd turns eut to bc notgrfatly in

cxeess of tho truth. Tt M évident t]tat wo should hf: justifier in cx-

pectingi).

Ycry good rcsult, if we assunic n.n axdal velocity of tito for]u

?' dcnotmg thc distance of tlio point cojisido'ed ft'ovn tho centre of tho

mouth, and thcn dctcnuino Mul so as to tunko t)tc whole eucrgy tt

minuuuin. Ttte cncrgy so ca-lcult~cd, tho~tgh ucccssurilyin cxccss, must

bo a very good upproximn.tion to t!tu tt'uth.

In can'yingout this phui. 'wo hft.vo two distinct pro~lems to de;d wit.)),l,

tho dotenninatioli of tho motion (I) ontsidc, tuid (2) it~ido t))o eylindcr.

Thé former, bcing t!io e~Iei', wo will takc (Irst.

TIte conditions are tha.t ~) Y{mis!i a.t uiduty, n.nd that wlien = 0,MtC

van.iBh, except over tho area, of thc circle r= whcre

Uadcr thcso circumstanccs we know (§ 27'8) that

l')–2

Page 304: Lord Rayleigh - The Theory of Sound Vol 2

2S2 CORRECTION FOR OPEN ENDS.

wLero pdcnoies tho (]iu<.ti!ico of t)(û pumt wlicro Is to bo estiinated

ft'mnLhec'Ioncntofn.rca.f/u-. 'Nuw

Thn vahin hf 7' is tu l'f c~cuhtt.f~ hy t.hc ~K'Lhodonpioyud in Lho t,(.t

(~ 307)for t). unifortu dt'n.sity. At, thu

edgcuf tho

dise, witeji eut du\)t

tu radius f<,wo)t!L\'(:L)t(;puLL'ntiai

on c<ct,Ii)~ thc Integmtinn. This qu)Utt,ity <U\'idcd Ly givcs twice ijto

kitiût,Ic cncrgy ofthe uiotiou JcfmcJ Ly (1).

Thé tut:d cm'n'ut,

Woha.vcnoxtt.ocons!<!c)' thcpi-oLIt'jnofdctcrnunit]~ thc motion of nn

incotupt'pssiLie ~uidwithin)-i~i([ cy)im)frundcr thc coudit.ions t])at thc

!txi:d vt.']ucityshitH bc unifut'nt w)n'~ .'<; M and w)t(;n a: 0 sItaU ho of

t))C furm

if for thc sft!:e of brcvify wc })ut 7~ 1,

'Thp~f')))i)tyofthcnunlisH))pposcdtf))r')n))ty.

Page 305: Lord Rayleigh - The Theory of Sound Vol 2

CORRECTION FOR OPEN ENDS. 293

Now <~ ma.y bc cxpauduj in thc ncr!cs

E)t.ch tenu ofUtis séries sitLi.sfic.sUic conditmnofgivin~

no l'tutitd

vu!octt.y,v))('u)'-lj innlno!))<)tio)ioE!myki)n!,w][cun;co. ]t t

rctninns to dtitcrnuao tlio coc(!iclcnt.n so as tosaLiijfy (G),

Avhun œ-O.

Fron )'- 0 to r- I, wo nmst hin'o

thc sunnrtiLtton cxtcndu~ to :dt Hic admissible values of 1;. Wc ]m.\c

uow to iin'l t))0 cnorgy of jnotioti of so nmch of tlie flilid as Is mcitutcd

bctwccn x = 0, and = l, witoru is so grca.t that tho volocity la tliere

sensibly constant.

By Grccu's tlicoroni

Thc uumcrieal vn.Iuca cf thc roots aro appi'OMmatc)y

~t= 3-83170~ ~= 7-015, ~=10'17~.

~=13-321, p,i=l(W71, ~=19-C1C.

Page 306: Lord Rayleigh - The Theory of Sound Vol 2

294 CORRECTION FOR OFE~ ENDS.

KothiLtt.hoscccmdtot'mis 7!7(1+~+~')".

Iticatculating tho ~rst tcnn, wc must rcnx'mhcr thitt

if nnd~tx:

two difTuL'cnt values of~,

To t.)us must hc :ult)<;d tho cno~y of ttic rnoUon on thc poHitIvc Hido

nf~~O. Ou(.))owho~

Page 307: Lord Rayleigh - The Theory of Sound Vol 2

CORRECTION FOR OFEN ENDS. 2955

)Utd our uljcctis to detei-tninc its mnxhnuta Ytdno. la geneDt]

if:

<S':uid Le two qu~dmtic functionH, thé mnxmuun and mminnnn vaincs

of x= <S'S" tu'o gh'cn by tbo cubiccqn~tioti

and 0', A', arc dm'tvcd from 0 and A by nu.et'changingthc acccntcd and

unacccntcd Ictturs.

lu t)ioprésent case, si)ice<S"is a product of liucar factors, A'=0,

iLud aincc thé two factors arc thc MiLUtt-, 0' =- 0, so that A 0 sunpiy.

Substituting tho tmuu'ricul v:dm.s, !md <-f)'ucting Lhu c:).!c(dado!is, wc

(iud = = 'Û~!898G~, whiu)i M tho nm.xiuumLY.duo uf Lho fracdûi). uonsistcuL

wibh i'< values of !(.ttd

Tho con-cHpoudmg Y:duo of a is -821:227. LhtUi winch Lhc tnn:

corrcctiûti ciumob bc ~ruatL'r.

If wo assumu -0, tliu ~rua(,(.'stYtduc of.: titcn possible is '021~63,

whn:!i fi\uswhh:h givus

a .'8281.iG7~.

On thé othor Iiand if wc put = 0, tho maximum value of s conics

out '027G53, winjucc

a-82 535 3 A'.

It wouhl nppca)' from this n'unit th~t Un) variable parLf'f thu

norinid vclocity at thc nnjuth is buttai' t-~pruscatud t'y a tct-m vm'yio~as

tliau by une vnryihgas

TI~ value M '82 12~ isprobably p)-(.y

clo.sc to thf! truth. Jf thu

normal vclocitybu assmncd constant, a-'8~S2GA', ifofLitûnji-m 1

1+u.?'~ a-'828157~, w)f'n isHuitabtydctcrntined;aud wi~'u Uh;

forni l+/jL!~+~ e'jntami)~ iuiutbcr arbitrary constant, is !na<h:

tho fomulation of thc ca!cn)atiou, w"gt;t

a --82i2A'.

Tlic truc Ya]uc ofa is pt-ubab)yabout '82/t*.

In thc case of thu nnnhnmn uno-gy con'csponds to l'U.):

so tliat

On thia snppositionthc nornud yclucity

of thu cdgf; (<' -/<') wou!d 1~~

about double of that ncar t)tc (.'t-htrc'.

'X~tcsfnUc~t.'l'fifunctious. J'/tf/<Xov.187:

Page 308: Lord Rayleigh - The Theory of Sound Vol 2

29G

NOTE TC)~ 27;

A nK'Lhnd nfo))t..unm~ L'uisson's soluLio!!

(8) ~ivoi l'y Mouvitlu' in

wru'U)yfjfn()<iu(.\

!f)'L('thr'p()]~]')'at!t)ts\T'ct,()r]nMsur('(tf)'o)na!)ypnintO,n.n<lt))<'

g<')u')':t!d)f)i')'(')tLiid<qu!).t,iutt))('htt(~)'at('()()V(;)'t)n!V(~))un(iin(:I)utud

Lu t.w('t'nH~)t(-rica)H)n-f)te('i()i'ra(1ii)'aud !'+< wciitul on tnu)ni'<j)')n!).-

ii')itoft)t(!.s(;(-'o))dint.('~)'!d~y(h'c(')t'.st.]n'or<'m

mw)nch<\ ~\`~~<~r,thaLi.st.os!)yi.sj!n)pnrt.!u))aItothH]aca)tva!m'

of </)rcckotu'd ovcr t])c sphf.'t'ioU surr~cf; of nuiiu.s )'. Eqnatiun (n) nuiyLo i'(~)H-(tcd as a)i cx~'asion of ()) § 27iJ; it may a).so )'o pruved from

thc cxto'f.ssioti (.5) 241 fur ~<~ I)i teru)s of Lhc ordmary polar co-ordi-

natcs ?', M.

TJic gcm'nd soluttou of (fjt) iH

Page 309: Lord Rayleigh - The Theory of Sound Vol 2

wpH!)7

jVo~ <))? 7~)'<.s'tM!)~(u~(~<t 7'oc<'c~t«y.'i </«; Zo;t(~

JA(~eMf<<t'<<s'()f«7y, )~ /~r. ~'u.ll').

Jt hiY.s oftcn ~ceu rcnuu'~cd thaf, whcn.~ro~pof Win'os a~vanccs

iuLo sti!) ~'att')', Oa: Y<(;iLy of tlio ~t'oup is Ic.ss thati th~t of t!)ti indi-

viduitt w~vcsoi'whichitis conpum'd; t.]tCwaY(.'snpp('!ti'toa'h'anu(.'

t)))'ou~))t)!u~ruuj', ')j'")~ !Lwny!'s t!t('yn]~)t-o:(c))it.s :)n)r!)'ior!i)ntt..

'l'hisjtjn'nottu'nunwu.s,

t ))t'.)[u'<iu'Htcxp)!L)m!d))yHt())<cs,w]torc-

~ardcd tho ~r~upns ïonncd t'y

t-))osopo'pusition ot' two infinittttmins

of wttvcs, of C(~]!t) tUttrittudusimd of ])c:D')y cqun) waYG-tc'ngUiS,

ad-

vancingintit(i.S)U~odi)'ccti.on. Myat,t,c))tion\as(:'a))c<ltoth(!sn~uct

ahout two yc'ars Mince Ly ~n' Fronde, !U)d (fx; siuuocxplanatiou

t]ten

occun-cd tu me i)LdcpcndcntIy'. ]u inyboo];; on t)io "Thuoryof

Suund" (§ 191),1 La\'c eonsidcrcd tho

questionmore

gaiendty,and

lu~'H shcwn Hott, if r Lo tito 'ctocity ci'pt'opng~tiou of tmy kind of

w:LVCs \viiose w.LVc-L'ngt]).i.s and K 27r\ tl)cn thc vclocity of

!).gronp compo.scd of n. gi'cat

tnnnbcr cf wn.Ycs, atldmo'ving into (m un-

disturbed p.n't of Utc ntedium, ia cxpresscd )'y

In fttct, if tlio two itiiinito tmius borcprc.seutc.d !jy eon«()"):)

and cos «' ( t~-x), tlicu' resulta.nt is roprcseutcd Ly

cos K( F< œ)

+ cos x'( a:),

1 Another phouomcuou, aJ.-iO mcutiotictl to lue hy ~[[ Froujc!, ndmits of a

similar txpittun.tiou. A steam In.uueti moving quickty thron};!) tho water is nc-

comp&uied by a pcculinr Hy.stcul of Jiver~i])~ Wftvcs, of which tho most stri]:i))g

fon.turo ia tlt0 obliquity of tito Iiuo coutn.iuiHg tho grcatcst devn.tious of suecossivo

waycs to tho wttvo-fronts. Thia ~mvo tmtturii rnny bo oxpiaincd by tho snpor-

positiou of two (or more) infinito trains of wn,Yes, of slightiy diiïering wft.Ye-Icngths,

whoso du'cctionH and velocities of propngn.tiou aro so related in ench easû th~t thero

is no eLfiago of position rcla.tivcly to tlio boat. Tho modo of composition will bo

bcst, uudcrstootl by Jrawiog on papcr two sots of parnUtil and cquidistaut lincs,

eubjeet to tho abovc condition, to rcprcaeut tho crcsts of tho componcut trains. lu

tho cnso of two trains of slightiy diiïercut wavc-Ioneths, it may Lo pro~cd that tho

tangent cf tho angle bctwcou tlio liuo of maxima and tho wavo-fronts ia hait tho

tangent of tho angle betwccu tho waye-fronta and tbe boat's courso.

Page 310: Lord Rayleigh - The Theory of Sound Vol 2

2:~8 PROGRESSIVE WAVHS.

whichiHuquidt-u

If K'-K, ])o Hinft)), wc htivc n. tmin of W!).vcswhosc amptitudu

vanus slowly from onopoint to Miotlier hctwcen tho limits 0 tmd 2,

formiug <). scrics ofgroups separatcd

from ono anoUtorhy l'cgiona cotn-

pM-attVc!y û-ec froni di~turhanco. TLo position at tuno t of tho middiu

of Umt group, which w~simti:L])y

at théorigm, iH givcu by

winch. shews thatUmveiocityoftItC! group is(/<«r)-(K').

InthcHu)it,wi)('uthc]U))nLfrufw!LVusnieaehgroupisindoTinItt']y

grent, titia rcsult eoincidus witit(t).

T))0 fo!)owu)g particutar ca.scs arc worthnotice, a.ud :u'o hcro tabu-

latcd fur convctiicuco ofcomparison

FccÂ, ~-0, ]!('yno!ds'di.sconncetedpcndut)))ns.

fee~, ~) Dt'<'p-wat<jr~ravit,yw:Lvcs.

ree. 6' r, A(.')'ia)-a\-('M,Ac.)' M

Â. r, (.')t])iHary watur wavcs.

r =c À", 6' 2 r, Figura) waY.;s.

T]t0 ci~fiihu-y watcrwavcs arc tLcHû whosc\)n'c-]t.-ngUt i.s su sinidi

thiLt tho furcc of restitution duc toCtq'mfU-it.y I'n-~c)y oxeceds t)tat duu

to gn~-ity. TJtcirtheo-y JtaH hccu givu Ly TitOtnson

(/< J/f~

Nov. 1871). Thc j)cxund wav(.'s, for w])ic)t ~=.2 F, arc t)toso cor-

ruapohdingto thu

Pouding uf an clastiu rud or jdatc ("TJtcory of

Sound," § 191).

In '<- paper rcml at <.itorfynumt.)) )ncct.ing of t))(! British Association

(a.ftc'rwardM printcd in .Nature," ~t)g. ~:), ]877), Prof. Osbornc

l!.(.'yn<j]ds gi'.vun

dyuiunicid L'xp).t!i)ttio)t of thé fact that ft group of

dccp-waturwavcs ad\U)cuswit)ion)yIta]f therapidity of titoindi-

vulual v'avcs. ]taj'pt-fu'ri t))!tt thc cnc'rgy p)-opagat';d in-ross

any point.

whun a tral)i of wavcM i.s pa.s.sing, is oniy onc-h:df of t))oouurgy ncces-

nary t~supp)y

tiiû w.n-c.s w)m.tpass

in thc samc ti)m', so t)tat, if thu

train of wavt's hL- limitL'd, IL i.s i)))j'ossi)j]c th:it its front can Loprop:t-

~atcd wit.L ttic fuHvctocity

of tin! wavcs, Lccat~sc this -ou)dimply tue

ac~uisitmnof morf

('ttf'rgy thnti (-:m m faet hcsu))))!icd. Prof. Rcyno)ds

did not contonpiatc t)i(i cases whurc ?/!e'rt;t'])c'rgy is

propagatcd thau

correspondutu th(; wavcs

passing m tho saine tinu' but Lisargument,

applied converst')yto tho rt'suits a)ruady givcn, s)icws tftat suu)t cnso3

must cxjst. Tito ratio of thccuo-~y pro]'ag!ited to titat of tho

passin'~

wavcsis r; t))ust))n<'ncrgypr()pagatcd in tho unit thncis F

Page 311: Lord Rayleigh - The Theory of Sound Vol 2

PROGRESSIVE WAVES. 299

of Ut)t.t cxL~ing in n. Icngth F, or U timcs tlmt cxis(,ing in tim UltiL

Icngtb. Accorttmgly

Energy pt'opagittcd in unit timo Etx'rgy contn.hicd(on

au~vcragc')

inunihiungth =~(«!"):(/'<, ~y(~).

As tili cx)unp!c,1 wH) tiJœ tho c:)sc of s!n:dl in'otjttional wnvcs i)i

Wittct' of fhuto dcpt!i If x t'c încasut'cd downwai'ds from the Hurfacr,

!md thc cluvtt.Lion (A) of tho waye bo douotcd hy

Tins vainc of fiu.t!afLcs tho goicral dKrcrctitlid Cf~mtmn for m'ot~

tionid motion (\7~=0), makcs tho verticalYclûci~y-~zuro~-hen

and-wf'cn ~==0. Thc vclocity of propngniion is giveJt t'yJ<

Wc inaynow cftieuhtto t,!)C enor~y contiuuRd in n. hngth :c, which is

HUpxoscdto includu no grcut

:<. Jiumber of wayes t!mt fmctio:ml ptu-b

nin.ybc lui't ont of account.

For tho potentud cncrgywc I~c

hy (1)funi

(G). If, m nccordanco wiUt thf ar~mncnt advn.nccd ut U)n

und of thiM p~rM',t!)C cqua!ity

ofF,

nnd 7' bo asanmed, tho Ytduc of

tho vclocityci' propngattou

f'ottow.s ft'oxt t)icprfsoit exprossions. Thc

wholo encrgyitt tlio w.n'csoeeuj'yin~ )i Ibn~Lh is thp!'cfn)-c (fur cncit

umtofbrcadLh) r.+~(~),

7/ dcnoting thc !)iax!i~mn c!cv:Lt.i(jn.

rruf. HcyMida eo!)HukM UtG troeh.~M Wftvo of n~nkhtc nud Frotulc, which

mvoh'es tnakcuittr rottitifin.

Page 312: Lord Rayleigh - The Theory of Sound Vol 2

~00 PROGRESSIVE WAVJ.:S.

Wu I):n-c ucxL to cai'uh~t' U)eenci~y p)-o]<:)~att'(t i~ thue < ncross :t

pLuMfo- which~i.sconstant, or, i

<)t)tcrwo)-(!.s,thcw(M-k(!~)t.h:)tt

mustL('<1ûnniuo:'tïot'tuHU.st!ti)tt!)(;]))<)tinaofthopLm(-((;o!t!ji()t~'cd

an!Lf!t'ib!c))nni)ta)i)it!)('f)tccoi't!mf]ttidpresHU)'c.s acLtn~unouDit'i')-o)tL of it. T))H Y:u'iaMo

pitrt of Hic prc.ssuro (~<), atdr)!t,h

in

~i\'(.t~y

As nn cxfunph' of t!ic direct crdcuLttion of we !nay tllke titc ca.sc

of waves tnovjng und<;r thé joint mHnencc of ~'n.vity and cohésion.

Itis{)roved)'yTi)ontsunt))at

Whcn « is snmU, t]t0 surface tension i.s nngligi)')c, and th~n ~'= r;

butwhcjt, on tbo contrary, « is Ifu'go, ~r, iLS lias alt-eady hccn

s~tcd. Witcn ?"y, ~'=r. This con-cspoiuls to tho nunhmntt

vclocity ofpropngitttou i)ivcst!gatcd ~y

Thomson.

Althongh t]ionrgtimcnt from mtcrfo-cncc

groupascons

s~tisfactory,

fui mdcpoidunt invcstigatio]i H dcsit-a)')c of thc ruititi.ou Letwcen

cncrgy cxisting :md enct-~y propitgatcd. Fu)' sotnc tmio 1 Anm at ft !os8

fur Mcthod applicable to ail khul.s of wavcs, not socing in pfu'ticular

why tho comparison of énergies should hitroducc Uie consLderat,Io!i of

Page 313: Lord Rayleigh - The Theory of Sound Vol 2

PROGRESSIVE WAYES. 301

a variation ofwa.vc-lcngth. Tho fullowlng investigation, in. which tho

inci'cincjtti of wave-IcngHi is ~~Ky<f<r~, inay porhaps Le considcred to

]nect thc want,

Lct ussuppose

that tht! motion of cvcry part of tho jncdium Is

re.sistcdt'ya.forcoofYcrysmaUmaguituduproportionaltothcmaHS

and to thc yc)ocity of' tho part, thc cn'cct ûf winch will bo t))at wavcs

~cnct'atcd at DiC origin graduaUy die :Lway as a; inovascs. Ti)cmotion,

whieti m thca~sonœof frictujn would )j('rcp)'(;HC!tt(.'d by co8(~),

under ttto iniincuco of friction is rcprescjttcd hy e'~eos(;<<-«.'<'),

who'c is a StuaHposiLn-o

cocÛicicnt:1~ stnctncss t!)(; vainc of K is

aLso altcrcd l'y Dte friction; but tho adoration is of titc second ontcr as

r'~ards thé ft'ictional forccH, and ]nay Lu omittcd ~tidor thf circum-

stitnccs hci'c suppo.s<;d. Thc cucr~y of tho waA'cs pcr unit IcngHt nt

anysta~oofdc~radatiott in proportiotiaitothf! square of thc a]np)itudc,

and thus thc w)t(j]c L'nprgy on titc positive sidc nf tho ori~in is to tho

cncrgy of' HO jnuch of tho wa.vcs at their ~reatest Yiduo, i.e., at thc

origin, as wooid hecoutainod lu tho unit of lun~H), ns~e'~f/.E ],

or as (~)' Thocncr~y trnnstnittcd throug)it!tc engin in thé

unit time is t!tC satnc as thc cncrgy dissipatcd and, if t))C frictional

force ncting on thc cloncnt of mass Mt bo/t7~M, whurc'uistitcvclucity

of thc c)cmcnt and is constant, thé onnrgy dissijtatcd in unit thno is

/t~?HU" or 2/ Luing tho kmctic oiergy. Thus, on tho assmnptio~

t)mt tho kinctic cncr~y Is ])a]f tho wholo cucrgy, wc .rntd that tho

cncrgy transuntted in tho unit tuao is to tho grcatcst encrgy cxisting

in tho unitlungth

ns A 3~. It rcmains to tiud thc conncction Lo-

twccn /t and

For this purposc it will bf conYcniunt to regard cos (M< -M') as tho

rca~ part ofc"~c') !t.ndtoinqnirohow«isa.fruct('d,w)x'n~isgi\'('n,

hy thc mtroductiou of friction. New t)tn c'n'oct of friction is rcju'L'scnt.cd

in thc thft'crentiai équations of motiojt hy tho substitutionof j+/t-

in place of.j,

or, mnco titc wl)o!o motion is proportional to e' Ly

snhstituting -?~+t7~ for –?~. Hcncf thc introduction of friction

correspondsto an a!tt;ruLion of M from M to M-Ai/t (t~c Bqnam of

hcing ung)cctcd), and n.ccordingjy « is aht-rcd from « tu K-<

./<

T))C sohttion t!m3 bccojucs e(;'0'<r), or, wl(cn thc

int.~gina.ry

,.ff)f dl<,1'

<

part ia rcjcctcd, e 'eo8(~<-<.T), 80 t]<at~<

and

A: T))(''mtioofthf'('ncr~vtra))sn)ittf'dinth<' unirtinK't"K

Page 314: Lord Rayleigh - The Theory of Sound Vol 2

302 J'KOCKES.StVH WAVES.

U)C cnc'rgy cxiKting in t))R unit h'ngth is tl)('cfoi'(- c'xpro'.sr'd hy

or as was to hc provc-d.'/« <f«

113wns to 11\! pI'OY(,(1.

It )iaa ofton h<'cu noticed, inparticuh~r

cases ofprogressive wavcs,

t))at tho pobentialnnd kiupLic énergies arc cqual Lut 1 do not cfdl to

]nind any gcncra! trcatmcnt of Dm question. Tito tileorem ia not

usutdly truc for Lhc jndividuat j'arts of t)'c tnndhnn', Lut must Le

imdcrstood to rcfct' cithcr to au intcgral tnm~K'r of wave-tcugths, or to

:t spacc so considcrftLIc that t)t0 outstandin~ fra(;t,ion:d parts of waves

may hc k'ft out of nccouut. As an cxtunp)e wt'Il adaptcd to givo in-

Hr'ht into thc question, 1 will tn.k<i thc case of a uniform. stretched

f'ireular jncjn~rano ("Tiieory of Sound," § 200) viLmting with a. g)ve;i

tmmLcr of nodal eirck'a nnd diaumtcrs. Thc fundamcnta! jnodcH arc

not quitc dt'tcrminatc in couijcqucucc of thc synunctry, fur any dia-

ntùtcr may bo mado nodaL In ordcr io get rid of this in()cternunatc-

ncss, wc tn~y Hupposo tito mcinLrauc to carry n, sniaU Joad attachcd to

it anywhcrn exempt on a Jioda! circic. Thcre arc t]~'n two dt'nnito

fundamontal modt's, in oue of which titc Jo~d lies ou a uodal. ditunctcr,

thus ]~roduumgno cfrcct, n.ad in thc othcr jnidway bctwccn nodal dia-

nictcrs, wh~)'c it ])roducoa n. maximum eftoct ("Tix;o)'y of Sound,"

§ 208). If vibrations ofboth modes arc going on. siuuntaneousiy, thc

notcuti~and kitietic cnorgi.cs of tho vhotc motion

nmyne calculatcd

by St'y~e <t~to~ of thosc of tho componcnts. Lcb us now, aupposing

thc load to di)ninisl) wIDtout htnit., haaginc that tlic vibrations arc of

cqual ampHtudoaud diu't'i' in phase hy a quartor of a pcriod. T!ie

rcstdt is a y~'o~ressti'c wavc, whos~ potcutml nn.d kinctic cnergics arc

tho suma of tliosc of thc stationary wavcs of whictt it is composcd.

For tho Ërat componcntwo Iiavo r,==~'cos*?~, ~==J?sin"~<; a!id

for tho second eomponf-nt, ~siu'?~, ~=~'eos'~j so that

+ P' ==7' + 7~ 2', or t)tc potcnti~t and kinctic énergies of tite

pro'n'cssivo'\vavo aro

equa), hcing tho samo as t)io w!io!o encrgy of

cither of thé compojicnts. TIio nifthod of proof hcro entployed appcara

to bo suSIeIentIy gênera!, tiLOUgIi it is rathcr diilicult to cxprcaa it in

Innguago which is a-ppropriate to aU kindH of wa.vcs.

AMal warea Mo an important exception.

CAMftIUDO! t'K)!)TKD i)Y C. J.CLAY, H.A. AT THE UStVKtta)TT MESS

Page 315: Lord Rayleigh - The Theory of Sound Vol 2

THEORY OF SOUND.

VOL. I.

8\-o. dotl), pricc 12$. G~.

Thc Authnr wi)l mo'it in thc highest dcgrcc tho thimk.s of ;dl w)to atudy

physics and inttUtMn~tics if ho c~htinuus Uto wot-k ia thc Munc tf):n)nor in

w]tiuh l)c has ]~nn it in thc first Y«hunc. Tho Aut))('r bas routtcred it

po.sHi))!o, )'y thû yery cnnvcuicot .sy.stutn~Lic arrangutncut ofthc whulc, fur tho

)nost diMeuit pn'hionsof acou.stics to 'jo now ntudiudwiHtfi.n' grcator eMe

Umn hitIicrto.Pt'of. IIchtiho~tx in Nature.'

Wo look forward wit.ti tttc grcatc.sb inhcrest tu t!io [tj'peamnco of tt)C sub-

séquent volumes, for whiuh t)[is proparcs thé way. Thc highor study of

acoustics will bc n diit'crent titiag attoguthcr wilcu Lhoy M'c in onr tutnds.

J ('«t~)y.

IttACMILLAN AN!) CO. LONDO~.

TUE

Page 316: Lord Rayleigh - The Theory of Sound Vol 2

In. Crowii 8vo. pricc S~. G~.

SOUND AND i~rusrc.

A Non-Mathcmaticfd Trcn.tiric Oti t))0 Physica) C~OHtituLh'n of Mn.sK'd

S<)H)n).'jatHUIann<)))y,i))L')ndh~t])cchiufA(.'nu.st,ic:dDi.sc~YcnL!snf]'ro-

Hi.ssor [Idn)h.)it/. Liy Si'JIJijf' TAVLOU, ~I.A., latc ycUuw of Trinity

CoHcgo, (Ja)uhrittgc.

"In no prcviou~ .sciuntiftc treatitiodn wc remcinhcr so cxhimst-ivc ami

Ho l'ichly i)tu.stt'atcd a. <Iuscript!(Ut of form.s uf vibration iUtd uf w:n'c-

ttlut.imtianuidtj.J/tM/(.'<:<S'~</n~<n<

ON SOUND AND ATMOSPHUHIC VIBRATIONS. Wit!. thc

Mat)tctnatic:d Eturnuntu of ~fn.sic. Dy Sir G. K AIHY, Astronon~r

l{nyfd. Sccund cdit~n], ruvi.scd .md cnt.n'gcd. Ct'own Hvo. Oif.

AN ELEMENTARY TREATtSE ON MUSICAL INTERVALS

AND TE~t'EXA~fHXT. With an accouxt of an Enhfu-innmc Uar-

ntoniutn (.-xttit'itcd in tho Lo.~H Cn))t;ct.i~n <~ Scic-nti~c ïnstnunotts, Soutit

Kunsit~ton, Ls7(!; i also df:ut Hnimnnonic 0)-K:m <x))ihitcd tn Lhc ~tn~ical

Associ.ttit'n of'Lando)), May, 1S7.'). ]!y R. 11. M. BOSANQUET, FcHuw

<)fStJ(jltt)'~C'un(;gc,Oxfnrd. Hvo. (if).

SOUND AND MUSIC.By Dr W. n. SïON). Two Lectures

dc!tVcrcdat.Snut)tK(;)t.si))~tun. muhtt'~tcd. Cruw)tM\-o. (!<

LECTURES ON SO~rE RECENT ADVANCES JN PHY-

M!CAL SC'rKXC!~y I'r.,fc.s.~)r P. C:. TAfT, M.A. H)n.stratud. Sucund

Edition, ('nhu~ud. L'ruwn.iv~ !).<.

THE APPLTCATrON.S OF PHYSICAL FORC'ES.

1iyA.(:Unj,EM)X. Trans~trd hy Mt-.s L.~yt'r, aodcditcdwith

Additionsa))dXut.('().yJ.I~.ckyu)',F.J;.S.

\VithCut<)n)\.))P!a)<-sat)d

inuori'~u.sinustr.ditU~. Hny.d~v~. :}).<.(!

3

~).\('fLLA~ AX)) ~a L~Xno\.

Page 317: Lord Rayleigh - The Theory of Sound Vol 2