lord rayleigh - the theory of sound vol 2
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The theory of sound / byJohn William Strutt,baron Rayleigh,...
Source gallica.bnf.fr / Ecole Polytechnique
Rayleigh, John William Strutt (1842-1919 ; 3rd baron). The theory of sound / by John William Strutt, baron Rayleigh,.... 1877-1878.
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c 3
THE
THEOU Y OF SOUND.
i
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
<E'n)nbWt«!t:
i'~tffT)!t)nY(').(').AY.t!A A~T')'UKttf[YH)tS)TV)'ttKSS.
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.
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).
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
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.
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.
CONTENTS.
Correction to Opcn End 9~
Noteto§ 273
Note on ProgrcssiYc Wavcs ~'7
APPENDIX A.
l'AOP,
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
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,
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
.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
: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
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.
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,
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<)
°
~]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
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
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
<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 \('/1 asl;illlp/('IIIII}('ct(~d
1;1':1('(' if vsd in .ionhy.
tllc wJJilh! 11I:ISScumes to l'cst so 80UII vs tltc IJJt¡tioll ut' thu LUIIJJ-'y<.ua.s~. ~L.fthL.Lu~-
H'Lnm'd'~vI~~ith~Lf.r.).ti~ .H,,
olltsidurcei~ut t.,bc-!iko surjhœof,nr~out.si.~
~s)y su~, t).bec. i.t,
~iD.i. tJ.c.tubccn.nc..s~ <witlliu t1IU taLe comes in l't'st. '1.'llis IlIech:lIlil'alilltc'j>ctati01I,L.
tùlIuLlcl'sLand ) Ilu!'c
~)~.] J.\XAL()(:Y W)'rif H~AT AX)) EH':(")'):r('!TV. 13
(.'))':n']\'w)'!tt ismoanthy a(tuit) havit'gnocircuhttion, axdit
h~~s <t)t <'xh.;))M)')norSt.uhcs''h<t~)~ witht'rspt'< tnxx'if-
cuhtr n'tation. i'or, H ait. thc Hmd ())iovh)j.; suhjt.'ct tu n,
v~h"it.y-p~<)'ti:'t) cnLsidu n.spho'io:)) f':(\'ity(~f :t))y r:n)ms bc-
c<n))L' M)nh!untyso)itt,t]tL! fhud in.sitk' Un.' c:Lvity cnur<jt:nnno
motion. 0' aswu m:ty :so.st:)hj it,a!)y sp)K'rica[ portion oF
:t)) i)T')t:)tionn)ty n~vrn~ Ouid ht'cumin~' suti(!t'!)ty .sotid wou)d
))()SSt.'s.so)ttyatUt)ti())i(jft)':m.i1:~it)n,<'i'<ru<<~<'o~
A Mi)t)i!!0'proposition w)))app)yt')!L(.'ircul!))'t1iHC,f))'cy1in(1~r
~it)ti)at.<'utls,i!).t.ho(.)..s(jofih)i<t ]uo\'iu~it'rot.:ttio))!(Hyintwo
(iitm'n.sit'jisonty.
Thu )))')(!')!) ofn.n i)tComp)'('i.si))tL'fhnd\)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.
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
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
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
~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.
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.
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
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
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.
~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
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
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.
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
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-
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).
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
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
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.
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)
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
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.
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.
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
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.
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~;(.
~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\
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~
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-.
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
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.
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~.
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
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-
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
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-
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
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
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~,
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
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.
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.
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.
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.
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
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
~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
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.
~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
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
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
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.
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.
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
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'.
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
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.
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
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-
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
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&'),
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,
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~
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
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,
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
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.
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
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,
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.(;
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
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
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.
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.
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
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
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.
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.
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
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 )' =«(<-<')
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 (~
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
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.
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
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.
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
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,
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
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
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.
~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)
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,
~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
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.
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.
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
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.
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
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.
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
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
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
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.
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.
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
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~.
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.
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
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.
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,
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
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."
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,
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.
~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'
~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(!.
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
~-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).
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
] 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~
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-
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'
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.
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
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.~
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
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).
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
~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
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.
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.
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.
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 ())
~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
!()
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
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
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-~)
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.
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~?).
~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.
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
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.
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 '~(~).).
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.
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;
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.
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(!
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
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
~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.
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-
~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
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)',
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.
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 /)'.
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
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).
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
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
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
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~.
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..)
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.
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.
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.
~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.
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
~-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
.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.
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<,
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
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.'
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,-
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
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.
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~
~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).
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
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,
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
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
~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.
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
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;
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
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
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?,
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-)
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.
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.
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~.
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.
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.
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.
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.
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.
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.
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
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
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
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.
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
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.
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 <,
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.
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
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 (§ :)
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)
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.
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~
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
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.
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.
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~)
ANALYTICAL EXPRESSIONS. [~20.22G
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
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
~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
230 DESSEL'S FU~CTIOXS.[330.
Now tlie function wlt.!i which wc :u-c at prc.soit conccmu'.),
satisfics (.i.) § 32:3, vi~.
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
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).
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.
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;<
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
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]
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
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'
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)!~
~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
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
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.
~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,
~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,
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
~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,
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
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.
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.
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,
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.
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.
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.
~.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
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).
('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
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
~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.
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
~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.\(')~j~hcd)U'(ht).yt))<ji(.<c,,ft)j(.),hm(.);).,
338.] UMYMMETMCAL MOTrON. 2G1 1
Wc havo ])o-(!t.!)cco)np)(j<c so)))ti')n t'f t)~!prf<)!ein oftLc
vibrations «fa ~)))(.')'ic;)! taycrof~]~))))))!)'') ))V!tsn)!)])circ)c
\vt)'r!u)iusisI(-<,st.)Hmt))~(jU!t(h-))t. ~o)'(':n-!)\)!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
~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
:~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
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;('); §;)!)().
:~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'.
~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~.
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
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,
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~
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
: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
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,~ ~~+~=~ {~~
°
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)~=~,
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
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
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
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),
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
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.
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-
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.
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
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:
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.
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
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)
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.
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.
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 <)
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
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.
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.
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~
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:
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
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.
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
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.
~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
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
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
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
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\.
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