Rayleigh, John William Strutt (1842-1919 ; 3rd baron ). The theory of sound . Volume II. 1995.

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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 .lnJ

VOLUME M.

bonbon;

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

DY

CONTENTS.

CHAPTER XI.t'AOU

23G254. 1

Aerial vibrations. E.pmlity uf prcswuro in n)l directions. Equationsof

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

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

Volocity-potontial. Lagrango's thcorom. Stokes' proof. l'itysioUin-

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

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

Motion of meomprossibio Unid in fiimplyeonncctedHpMM is dotcnninod

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

of u-rotittionally moviug iluid suddeuly soliditiod wouid havo norotation.

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

of heat nud ctuctricity. EquatioM of pressure. Gnerai oqu~ioufor

sonorous inotiou. Motion m ouo dimension. I'ositi\-u aud uc~tiYc pro-

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

monie type. Euorgy propat~tod. Haf 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 spcifie 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 appears that 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)iicatioii to wayesof tlicory

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

presHuro aud donsity Otn n. wavomaintain ils fo)-!u without thoassi.st-

ttncsof nu improsscd force. ExpiauatiottofchanHoof type. 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

Y) CONTENTS.

CIIAPTER XII.r.\273SD.~ 85

Aj'Liti'dryinitial disturbnneo in an uulimited .at.)uobpLL')'o.J'oiHKon'saolu-tion. Verification. Limitod initiul disturbanco. Cnf

CONTENTS. vu

Exprimental motliods. Mayor'a oxporimont. Sound shadowa. AperturorAa~

:np!)-Gcn. Huy~htinri'~o~rs. Ccner.T.:f-.).ud~h~(dow!Obliqua Rcroon. Conditions of approximatoly compictorofloction.DivorginR Wavos. 'Variation of intcnsity. Foei. RoOoction fromcnrvod snrfacoH. Elliptical nnd parabolic reflectors. Pormat's prin-cipto. Whisporing e~ories. Obsorvationa inStranl'aenthodraLProbaMo oxplnnntion. Rosonaneo in buildings, Atmosphorio rfrac-tion of H0)md. Convoctivotiquitibrium of tcmporn.turo. Diilorectialoqnntion to path of ray. Rofractiou of sound by wind. Stokos'cxpiMmtion. Lawof rcfra.ctio)]. Total reflection from wind ocrhenc!.In tho case of rofraotion by wind tito eonrHoof n. Sound ray ia uotrovorsiblo. Observations by RoyjMids. Ty~Il's observations on fogfiRnaIs. Law of diverROt~oof Mund. SpoakinRtmmpot. 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.

CHAPTERXV.~C-302 .13,Sceondurywnvosduc to a VM-intionin tho modium. Botativo importanco of

Mcondury wnvoadcpondHnpnn tho w~n-Iength. A rgion of a]terodcomprosaibUityacts lilo [t simple sourco, 0.rogionof n.Jtoroddonsity likofutoubio sourco. Law of inverse fourth powers infcrred bymethod ofdimonsions. Exp])umtion of harmonie cchos. Altration of cimmotorof compound sound. Scoondfu-ysourcos duo to excessivo nmpHtudo.Alteration of pitch hy ro]ntivomotion of source M)d rcipient. Expori.mental IHnstrationHof Dopp)er'Hprincipto. Motiotiof a simple source.Vibrations in a rectanguiar chambor dn to intcmal sources. Simpiosource situatud in au unHmitcd iubo. Enorgy ornitted. Comparisonwith conicnl tubo. Further discussion of tho motion. Calcuia.tion oftho raction of tho air on a vibrating cireular plate, whoBopiano is corn.ploted by n. fixed Bango. Equation of motion for tho plato. Caso ofcoincideucoof uatural and foi-eodperiods.

CHAPTER XVI.

303322. .1UUThcory of rosouatora. RoMnator composod of n piston and air rGscrvoirrotontml onergy of compression. Poriodiotimo. In a largo dMBof airrosonatorHth compressionis sensiblyuniform thronghout tho rservoir,nnd tho kmetio onergyif.senaibty eonfinedto tho noiHhbourhoodof thoair pasMRca. Expression of kinetic one~y of motion through pasmgosin tcrrns of cloctriealeonductivity. C~)cu)ationof nnturnl pitch. Caseof sovoral eliannols. Snperior and inferior Jimits to conductivity ofchannols. Simple aperturos. Eltiptic apertnro. Comp.LriHonwitli cir.cular aportnro of cqual nroa. In many caseBa catculatiou bascdon arca

1 CONTENTS.

PAOEoniyi.ss.t.ciuut.

S.tponorandmfonor!im:t.stothuudnctivityofPAULP

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

CONTENTS. Ix

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

33G343. ?~3

rroUcmofn.Rpbt'rieatiaycroftdt'. l', Expansion cfYtdoeity-potcntifd inFouricr'Hsorif.'s. Din'orcntm.t quation Bfttisficdbyc.ichtorm. Ex-

prcHscdintormsof~n.ndof~. Solution fur thoc~stiof symmetry.Conditious to ho Rn.tisdcdwhon tho ples arc uot sources. Rductiontot'egcndro'Hfnnetions. Conjugntopropcrty. Transition fromnphn-ricnl to p)n.nn hyor. Densel'Hfunctionofzoroordcr. Sphorioaltayor boundcd by pnrallola of Itttitndo. Solution for sphorictd layorbonudedby smnHcirclo. l'nrticular enscs HoluhJoLyLcgondro'H fune-tiouH. Conomi prohicm for unsynnDctricat motion. Transition totwo dimcnfiiona. Comptoto sointinu for ontirc Hphoro in tcrma ofLftpiftfo'sfunctiins. Expansion of fin nrhitrfn'y function. Fonnuin.of drivation. Corrospott

CONTENTS.

Correction to Opcn End 9~Noteto 273Note on ProgrcssiYcWavcs ~'7

APPENDIX A.

l'AOP,

J:.Jf. J

CHARTER XI.

Ai~lUAL VIBRATIONS.

23(!. StNCH t))c atrnosp))erc is thc abnost miivc'rsa] vehicic ofSound, tbc i)t\'cstig'atio)toft.bc vibrations of a gascm).s mcdiumbas alwuys boc'n con.sidcrcd tbc pcculifu- problon of Physic:~A

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

~r~ 'r 'r -1"P~un. "y O.

237.] EQUATION0F CONTINUITY. 3

position in spacc (dut!'min(jdby thc vn-incsof .T,y, z) is rctn-incd

1.] ] ]). l f 1. 1 1 1inv.u'i:).biL!,w)t!ithu sn-c:)Hc

.ST!!t.:A~[-r)rx

:23f).] LAfiHANCE'~)s THEORE~r. 5Hf't in motion Ly cnn.scrvntive furcc.s axd pressures transnuLt.cd

trL))ntj]n.:c.\t.c't'i

LACRAXOE'STHEOREM. r~Sf).j'. '1' it i; J 1ilie,J IC10, ur oh III :').

'lit II. iiiiti. H!rWISU\VIt1

cxist in IJllids, mul vrc clc-llcndellt o0 the oclolinc \Iocities uf their parts.

~H~

.

230.] -1 HOTATOtTY VELOCITIES. 7

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

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

CIHC'ULATfO~. r.QL-L")))WO)'(!s,

~7.tth~vh.!cv.),c .dn~ jBut Un- c.ircH).n

~oeveryc~) c.rve ~).p~ round thcrin. an.

'c.~nccun.s~t~iuu~a])

~] YHLOCrrY-rOTEXTIAL.9

It\S''k')t

r~OPERTy op IKp. r :,2.1L

~1.

~L~.r~!nltlir, m,~tln!1~Ilnll file

"

242.]1 MULTIPLY-CONNECTEDSPACES. 11

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

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

an()

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

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

EQUATION 0F 1-KESSUHH. f~L-2.). t. W)u..n

v..).)city-p.(.nti..d c.xisLs, Die cquation t., .X.tcr-!n')t0t)i(. prus.sm-ct,):!y !

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

~)'). T)K!simph;ntkin() uf'\vfi\'c'-)nn

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

Whatcvf.r tlie init.iat.t)Ht.urb.'U)cuniayhc: (axd Mand.sanibutharhjt.rary), it c-an niway.s bc

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

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

Thus the work consumed in gcnerating wavcs of harmonie typeis t)'f!sa.tncf~iwould bc rcquil'cd to givc the maximum vclocity~3to t))cwholemass of air through which th wavcs extend 1,

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

whcro T-(=\

18 NEWTON'SINVESTIGATION. [245.that would bc gained during the expansion of thc con'cspondin~qua-ndtyofga;: from ita .tctualto its normalvolume, thc expansionbcing opposcd t])ronguout by tlic uonnal pt'cfisurc At anystage of thc expansion,whou the condcnsntit)!i is s- the cn'ccdvcpressure ~) is by 2.J4f~~s', wl)icl) pressurebas to bc muItipHcdby thc corresponding incrment of voh)mc fn'c/s'. Thc wholowork ga.Incd dunng thc expansion from ~~to Jr(l+s) isthcrcforc ~p,~F. or ~F. s' Thc gcncra! expressions

1 nfor tho potential and Mnctic nergies arc accordi;)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 th kinetic energy of the whole mass ofair concerned,moving with titc maximum vclocity to bc fouud inth waves,or to tho potcntial cncrgy of tlie same masa of airwl)cnconclensedto the maximum dcnsity of thc wavcs.

246. Tiie first theoretica~ investigation of thc vclocity ofsound 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 vulocityof sound is oxpressedby V/f,or in which t!ic dimensionsof p (= force-area)arc [37] [Z]-' [2']" and thoscof (= mass voJume)arc [.Vj [Z]-Newton Gxprcssedt))c rcsult in terms of thc 'e~ of ~e/iOM:o-~eHeo:~n/M~.f~Aerc,'dcnncdhy the quation

wherc~) and p rcfcr tu thc pt-cssun;a)i(t t)m dc-nslty{tt thc carth'ssurface. Thc velocityof sound is thus or thc vclocitywhichwould be :tC()nirud)~y body i:d[ing frudy under thu action ofgravlty tin'ougt) ludfthc h~ht uft)t(.' homogcncousatmosphore.

2-] LAPLACE'S CORRECTION.

Toobtmnanumurical )-csu)t "/c r~'uiretcju~w ~pnir

20 LAPLACE'SCORRECTION, [346.rec~oncdfromth a.bsohttcxcro'. Tho conittionof thc gn.s(ifuniform)is de~nedby any twoof:tt0thrcc quantitiGs~), a.ndi.h(tUth'dt~tt.ybc -t~)Ld n. td'ins i tiicn!. Thc rc~tioubetweenth simuku-Dcousvarin-tionsof thc tfjrcequantiticsis

In ordcr to cffect tLo change spccificd by f~) and dv, it isin general nccess:n'yto communic!ttc hcat to tlie gn.s. C~IHn"'th nccessM'yqua.utity of hcat ~Q,wcmay writc

246.] EXPERIMENT0F CLEMENTAKD DESORMES. 21

if, as usua!, tlie ratio of thc spcifie Iicats be denoted by 'y.Lapl~cc's vaiuc of titc velocity of Soundis thcrcfore grcater t)ta.nNewton's in thc ratio f~y 1.

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

Maud p, on thc supposition of uo communication of hcat,

whcrc )),, o,; arc two smuuta.neous values. Unuer the samecircumstaaccs tlic rci~tion 'betwecu pressure and temprature is

Ly(3)

Th magnitude of 'ycannot bc dctermined with accuracyby direct

cxpcrhncnt, but an approximate value may be obtained by aiiietliod ofwhich tlie followingis tlie principle. Air is compressedinto a resci'voir capable of being put into communication withthc external atmosphere by opening a wide valve. At first th

tcinperatui'o of tlie compressed air is raised, but after a timetite superiluous hcat passes away and the whole mass assumestlie temprature of thc atmosphre0. Lct thc pressure (measuredby a manometer) be p. Thc valve is now opened for as shorta time as is sufHcieut to permit thc equilibrium of pressure tobe compictcly estabhsbed, that is, until thc internai pressurebas become cqual to that of thc atmosphre P. If th experimentbc properly arrangcd, this opration is so quick tliat tlie air inthc vcssel lias not sufncient time to reoive heat from tlie sides,and thercfore cxpands ucarly aceording to the law expressed in

(9). Its temprature at tlie moment th operation is completeis thcrefore detcrmincd by

V

Th cnclosed air is uext. a.!lowcdto absorb heat until it bM re-

g:Lincd thc a.tmosphcnc temprature 0, and its pressure (jp') isthcii obscrved. During tlie last c!~nge thc volume is constant,:uid thereforo tlie relation bctwecu pressure and tempraturec~vos

It Is horo nssnmod that is const.nut. This quation appears to havo beoueh'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,).JtcnuLhudiso))\'iouHtynots))sc(-ptih)cof :my grcat iK-cm-i~y. Thc v~hnj uf 'y j'C((uir

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

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

By Rcgnault's cxperniicnts thc spcifie hcn-tof air is -2379of that of watcr; and in ordor to raise a gramme of watcr onc

24 STOKES'INVESTIOATJON [247.

presse and density wouid bo th:tt w)uc]t I~~d.sto Ncwton's valueof titc vulocityofsound. On 'hc othcr !~m], Mhnvca.cci-t:unHmit~iqmckm.s, Lhc !,a.-iY.'uufd.~Jt.vc~.jifc~nhcdm.~no)i-L:~n-dxcttng vc.ssci,a.s suppo.scd in L~pt~s Uicory. Nuw fdthoughtito circumstiu~cc.sof thc autoa-t pi-obtf!n anj hctt(.;r rcpi-c.scatcdby titc lattct' t)~n Ly t))c formut' supposition, theru ]n:Lystill(it m:)y bu sn-id)bo n.sensible d(;vi:tt.io)ifnjni t)ic law of pressureand dcnsity invulved in Laplace'.s theory, ent:u)i))g so)newli:).tslower velocity of propag:tt.iunof sound. Thi.s

247.'] 0F EFFECT0F RADIATION.25

not scnsibty absorbcd within distruice of svre wavc-lcngths.

Li thc l'unnur c~c ~~x~~ ;n.d in t.hc lutter, winch :s that11t l\ l'lnur Cli:'JCt lat

sctcct.edby Stokesfor:u)a)ytienlinvestigation, (-Ncwton's

h~v uf radiation bcing ftssumcda.s a, suficicnt approximation to

Ihc trutli. We i~ve thoi

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

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

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

Thc ~rst t))in~ tu bc noticed is thf).t t)io Honm! ca.nnot 'bui't'upa~tttjd tu dis(.;)ncL'untc.snMm Le insensible.

Thu vcioc'iLyafpro~ag.Ltion (F) in

i\ow irom (O)w(j suc t)

2t7.~ J INrLUEXCRD TIFAX TIH VELOCITY. 27

tion is no longer con)p)ct(dy rcfnodod during thu corrcspondingrart.'t'action on accunnt of thu ditninishcd tutnporaturo, part ofthc I~'at dcvulopud by t))c compression havin~ in Utc tnciUttime

csc~pL'd. ln f:Lcbt))u pa.s.su~'cof ]x.tt by conduction or nuti~tionfront n-wiu'nn't' to a nnitcty co)dur Lody :d\V!tyHinvoh'cs dissipa-tion, a p)'I)K'ip)'j whiutL occupius a futuhunoit.al positi

38 EFFECT0F CONDUCTION. [247'.

Suppose th~t by a rigid envdopc t.i-anspa.t-cntto raJiiUit hut,thu vohnuc of small ])):),.ssof ga.s wcru n):u))t:uncWe arc thcreforc justihcd in assuming th:tt

247.] VELOCITY DEPENDENT UPON TEMPERATURE. 29

teaving thc velocity of propagation to tbis order uf approximationstill equal to ~/x'y.

From (18) it appears tliat thc nrst cffect of conduction, asof radiation, is on tlie amplitude ratbcr than on tlle velocity of

propagation. In truth the conductmg powcr of g~ses is sofec~c, Md in the case of audible sounds at any rate thc timedurin"' 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 arc supposed to bc

propagatcd in an opcn spacc. Whcn thc air is confined wlthina tube, whosc diamctcr is small in comparison with thc wavc-

Icngth, thc conditions of th problem are aitcrcd, at least inth case of conduction. Wliat we have to say on this hea.dwill, however,comcmore conveniently in n-noUierplace.

24-8. From the expression \/(~y) ~p, wc sec that in thsame gn.s thc velocity of sound is independent of tlie denslty,becausc if the temprature be constant, varies as p (~ =~p0).On th other hand th vclocity of sound is proportional to thesquare root of the n-bsolute temprature, so that if f~ be ilsvalue at 0"Cent.

wherc th temprature is mcasurcd in thc ordinary manncr fromthe freezing point of watcr.

Th most conspicuouseffect of th depondenceof thc velocityof sound on temperature is thc variability of tlie pitch of orga.upipes. We shn-llsec in thc following chapters that th periodof thc note of a flue organ-pipo is th time occupicd by a.pulsein runnin~ over a distance which is a dennitc multiple of th

Icngth of th pipe, and therefore varies inversely as the velocityof propagation. Thc inconvenience arising from this altration

30 YELOCITY0F SOUXI)IN W.ATKH. [248.of pitch is nggravatctt ))y thc tact ti)at t))c rccd pipc.s arc notsnodarly 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'ntutnperattn-c and wa~c-l~ngt,)~th fonn

249.] EXACTDIFFERENTIALEQUATION. 31If V) 2/+- (Ic-nncthc a.ctua! pnsitions n.t timc < of

nt'!Q'bot!n!g i~ym'~of :!)r ~'husou'jUiiibt'iutttpositions :u'c (tt~no)by .f and a;+~, thc dcnsity of t!)CIttchn'cd s)icois givc'n hy

th expansions a.nd condcnsntinnsbci))~ sxpposcd to tn.kc nhicc';iccurdit)g to tlie adiabatic titw. 'l'lic tn:s of uniL cf :u-~ ofJ 1.. l 1 1 1" r ~l!'1tlie slicc i8 and thc corruspondin~]no\'ing force Is fZj;~iving for tlie cqnation of motion

Equation (~) is a.ne.wtc~equ!Ltnj)idctmin~ thc actual absci.ss:iin tci-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 th sni:dt qnn.ntlty by its np-

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

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

quations con-csponding to (~) !tn

32 \VAVE80F PERMANENTTYTH. [249.

from whichp, occurring in is to be eHminatcd by mcans nf

tllu rc~muiut:1)c";vw;n.f/ u it~~1j.therci:(.:ionbc:)VC(;!)~anL),cxpreHS(.(tirt(l~.cl:c

2.'i0. lu thc prcccding investigations of acria! wf~es wch:ivc HU}'pf)sc

250.] WAVE 0F PERMANENT TYPE. 33may bc thcvcJocity?~ ofthc gnera) currcnt, or in othcr wor~s tha.ta w:),vcCfmnot hc jn'n])!~at(;d ru)ativc')y to th undi.sturbcd partsof thc g~s withriut uttdurguitt~ !in :)ltc)-ation of ty[)0. Ncvcrtixjtc.s,wh~n Lho ch:U)~us 01 (jcnsity concenn-tt are sma!], (;')) niny ))Gsatisfied :).ppruxi)natc)y; arxt wc sec fron (-t) th:tt. tho vubcityoi struatu 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.tivdytut)ic(ini(l.

Tins ))i('t))()d of rc~.i,)'

SUPERPOSITION 0F PARTICLE VELOCITY. [251.34

V(')oc!tyand dunsity are insensible fur a. considrante distantea)o))~it,:tnd:tt.:),p)acc w])('rcthcvc!oeity()isnrntf'Jct )).si' ts!)).!t!s~co')'r.t\~ t'h'~)r. TL'v 1with w)tich thc sccondary wavc is pn.pnniUcd

251.] RELATIONBETWEENVELOCITYANDPEXSITY. 35W)):)t(;vcr may bc t!)c Luv ofpre.ssm-e, tlie vaiccity of p]-cpa"!i-

'n.H.!is~uL. !.s .'jy~ ..\.c/ '1

)nt).stbo

THEORY0FOPENENDS. [312.184Thse formu~ -showthat, if thc dissipation be ]oft out of accountthc vcioeity-potejttia) i.s thc sa.nc M if t)iu tube v/cre Jun.~hcncdby of thu radins, and th opcn end tlicii bchaved as a loop.Tho amount of thc cnrrcetlun agres WLthwhat prev:ous investi-ssons wouM Ii:LveJed us to cxpect as H.c rc.sutt of thc Intro-duction of thc pisto!). Wc i.avc sccMrcason tu know that t!.utrue val.ic of a Hcs

bctwccn and ~7~, ~nd U.~t thc prescnof t!~Gpiston dousnot aHcct t)to tenu rcpresonting thu dissipation.But, b~forc discussing our rcsults, it will bc advanta~cous tu ii]-VL-st.~tc thon afrush by a rathcr dincrcnt lucthod,which Lcsidesbcm~ of somcwhat grcater gcncndity, will hdp to tlu-owligtit outhc tncc!):u)icsofthc

3J3.] TMEORY0F OFEN ENDS. 185

Lcb us considur thc bch:n'iour of t.hc nmss of air Inchu~d bu-twnthcp~ncHecL~H~t C:md n,hcnusp))

CORRECTION TO LHNCTH. [313.18G

If/t! bc thu radius uf t,hc tube, wc ni~y rcp!:LCCo- by 7r7~.

Whoi tlie tube is a. simple cylindo-, and thc origin !ic.s at :tdi.st.utcc A7, iruta Litu rnouLh.wuknow that o-c''=AZ,+~, who'ui.s :).n)[)n))o- i-at.!tcrgrc.itcr thau 7r. In such a case (thc oi-i~in

Luing t:d

3)3.') RATU 0F DISSIPATION. 187If tho-c Le no n:U)gcnt J, the valueof c is s)ight)ymodined

bythu)\'niova)ofwhat actsasan obstruction, but th principalcH~ctis o)) thc tcrtn rL'prc-sontingtbc dissipation. Ifwc snpposf-a~ an approxitnation t)iat thuwavcsdivo-gingfrom~ arc sphcricai,wc nmst takc fur t]te current 4-n-r instcad of ~Trr'~ Thu~veiiiiist titl~e1(irtite ettri-ciit

f/ iIlSte~t(IC)r ~y-' 0

u!ti

~83INFLUENCE 0F FLAN(!E.

[314.

f~vnr't'imfntnUt)' ~t\.r~ i i

L.

cxpcnmcnta.Hy'. Two organ-pipcs ncariyenougb in unison.withonuanot!)L-r to givc countablobcatswerc biownfrom an or'~anbdiows; tho cifect of t)~ nangc was (icdnccd fro!n titc dift'cr~ccin thc frcqucucics of t).c be;Lts according as ono of tho pipes wasnitugcd or not. Ti)c correction dnc tu th Oangc was aboit -2/t'.A (prohaDy more tmsLw.rttfy) ~}K-t,itio)i of' this cxpuritnt.nt ).yAir Jj

314.] EXPERIMENTALMETHODE. 183

Varions mcthods hve bccnu.scdto dtermine thc pitc!)nfre.sooators cx])cri)n~i)ta1)y. Most frc;()ncnt)y,perhaps, tl)c rcsonatorshavu b'jcn madu t.o N/)e~' after t)ic manncr of organ-pipcs by aKt,t'(?:)ni.ofnir bh)\vn obhfjnc'Jy across thci)' months. AIthoufh gnodrL's~it.s hve bcun obtaincd lu this way, onr ignorance as to thomodt~ of action oFthc wiod rc:)](tL')'nLhombthod uns~tisfitetory. InBusnoquct's Mxpt'rimeiits thc pipes wcrc not acLu~Hy mn.dc tospc:).k, but sitort discontinuons jets of air wurc btowu a.cross t])e~))t;n end, thc piteh bcing usti)nntL-dfrum thc frcc vibrations asthc sound di(.-dawa.y. A )n(.'thud,simi!at'in principte, that 1 Iiave.sonn;titncs cmpioycd wit!t a.dvn.tita~'cconsists I)i uxcitingfroei'vihra-

DISCUSSIONOPMOTION [314.~0i.s known, it is atmost, impossible to an'ivc at an Indcpcndcntopinion hv'xp('itiir'nt.

31.'). Wcwi)) no\v, fu])uwingHu]mhohx,examine more c!ose)y~hc nature of thc motion within t)ic pipe, rcprcscntcd by thcfunnn~(H)3)3. Wchave

whcrc M is a.posttivc jntcgcr.T!)C(1ist:mcchctwccu conscutive m~xim~is thns a~)(~thc

v:duc cf thc m~xnnu))!is sec~a. Thc miniinnmva)uu.sci' Z/' occurnpproxitnatcfy wlieu (a; a) = M

315.] ORIGJNAT1NG WTHIN AN OPHN l'IPE. 191tTh fipproxin~te tuagnitudc of tLc maximum is ~sec~x, a.nd

1 QO AIOU'IO.DUETOMOTIONDUETO f;~Il. Iv W oJf

~nc.j EXTERNALSOURCES. 193t)ic elosing plate may be rcmovcdwithont any altration in thmotion. Now, ijowcvcr, tficre is in gnral a finite velocity at.T=- and Htcrefure \vc cannnt suppose the pipe to be t!icrcstop])cd. But \vhen therc liappois to bc a nodo at a; == that ist" say wiicn is Huefit))at eus/

I!)4 EMARGEMENT AT A CLOSED END. 1317.

whero a is thc correction fur thu op(;n (-n

318.J ABSORPTION 0F SOUND 13Y HE.~ONATO!!S. 1S5

good dcat of confusion among acoustical writcrs, thc autbor notexcepted.

Titcrc are cases wbcrc a rcsonator absorb.s Sound, as it wcrcattracting thc vibrations to itself aad so (tiverting them fronrL'gions whero otherwisc thcy would bc fe!t. For cxampte,suppose that thcre is a simple source of sound sitnated in anarrow tube at a distance (or any odd muttip)c thercof) from acfnscd end, and not too ncar tho mouti): thoi at any distantoxicmal point its cfuct is ni). This i.s an inuncdiatc consu-'ptenco of thc principtc of recipt-ocity, bccanso if ~1werc thsource, t])Ct'e cuuld bu no variation of potcutial at A Threstriction, precludin~' too grcat a proxinuty to thc mouth, maybc dispcnscd with, if wc suppose th source to be din'usedunifornily (jvcr t))Ccross section, instcad of conccntratcd in oncpoint. Thcn, wltatcvcr may bc thc Hizcand shapc of thc section,thcrc is nbsoiutuiy ne disturbanco on thc furthc-r si(h'. This isc)car frum thu thuory of vibrations in onc ditm.-n.sion thc reci-procal form of th propositionthat wbatcvcr sources of'distmb-ancc ;nay cxi.st bcyond thc section, jy~-rZo-= 0tuay bc provcdft'om Hc))nh()ltx'sfurmuia (2) 2'): hy inking for t]te vclocitypotcntia! 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 shaHbc no variation of pressure at th source, for thatin thc case of a simple source Is impossible, but that th variablepart of th pressure sbaU bave exact)y th phase of thc aeccicr!tion, and no componcnt with tho phase ofthc vclocity.

Othcr cxanip!es uf th absorption of sonnd by resonators arean'orded hy certain modincations of Hcrschers interfrence tubeuscd by Quinekc' to stop tones of de~nite pitcb froni reacitin'tlie car.

In th combinations of pipes rcprcscntct) in Fig. (!3,th soun()c'ntcrs frcciy at at it nnds itscif at th mouth of a resn-nator of pitch identical with its own. Under thse circumstanccsit is absorbcd, and thcre is no vibration propagatcd a]on~ 7?~.H is cJear that th cylindrica! tube ~C' may bu rcp)aced by anyothur rcsonator of th same piteh (7), without pn~ndicc tu t)tu

'{'{;f~.cxx\'))i.!77.tn(;f!.

L;

l~G QUTNCKE'S TUBES. [31g.1 mf' 1actionof thc apparatu.s. Thc ordinary cxplanatiou by intcrfcrcnc

(soea-Hcd)of direct t~udreffcctcttwavcsis thun lussapplicablu.

Thcsc cases ~hcrc t]te source Is at thc mouth of a i-csonatorinnst net bc confuscdwith othcr.swbcru thu source i.sin thu intc-nur. If be :i source at t)i huLtotnuf a. stuppcd tubn w])osorcductjd 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 thcpotential ~t duc tu t)ie sour :Lt is thu s:unc as it would Let).t wct'cthu Muurecat J..

31~). For a do.soruxatninatiuti of thc mcc)h'micsor rsonancewo shaH oblain thc proDcm iu a !'L.rnidisutnbarrasscd of unue~ccssary dif!icutt!cs by suppusing t)ic rcsuj.ator to consist of asmati eircular ptatc, bae!.c-

319.] RKSONATOR CLOSE TO SOURCE. 197

Lct ua now suppose t]iat F is duo to a.)i externe source ot'sound, giving whui Dmplate i.s rcst a poto)ti:d which ~illbc nc~riyconsta,!i.tovci-th fu'ca(jfthc plate. Thus

so that 2-n-A:i.stfie 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.kcstliu i'ortu

from winch wc may infer us bcforc that if /

REINFOnCEMENTOT SOUND [319.198

of tlie phLtcs. T]tc intensity of 7~may be suppo.scdto be suchtim.titsnutc)iti:t.[is

Th relation of phases )nay bc reprcs~ntcd hy rcgarding thJnduc~d vibration as proce~hog frojn by way of J, and asbt-ing suhject to au additionat retaniation of su that D.e whoieretar(!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]cgreater, nnd thc tuta] soun() is much hn~cr tnau if wcrc nutpermittc.! to po-ate. lu this case thc phase is rct:u-dcd by aquarter of a pcriod.

It is important to hve a c~ar i

319.] BY REHONATOUS. 1~9

by modifymg' t))c othcrwisc unfavourabic relation betwccn t)tophasesut' tbu pressure aud of t]In tllu prsent .statc of our knowtcdgc of thc incchanics ofvibrating Huids, whiic titc difHcultics of dduction arc for thomo.st part still to bu ovurcutne, any simplification of cotKJItionnwhie!)afiow.sprogruss to bumade, wltitout ~Lo]]y dcstroying t!topr:).cticalcharactur of thc qucstiuu, tnay bc a stcp of "-reatimportaocc. Sucit, for cxamptc-,was thc introductioti by Hchn-hu[tz of th idca ofa source concuhtratcd in onc point, rcpruscntedanulyticaHy by thc violation at tliat point of th eqnatiou ofeoutinuity. 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 moruimpossibleto construct than a simple source.

320. Wo havo sccn that tburc is a grca.t augmentation ofsnund, 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 doublesource is (294~,324-)

RESONATOR AND DOUBLE SOURCE. [320200and thcrcforc thc potcutial duc to the resonator at distance is

If~, Vtuush, thc rcsonator is without effcct; hut whcn = + 1that :s, wi~n thc r~on~or Iles on thc axis oi- th double source,wchnvc

Thus wo mny consur that th potentiat duc to ti)c rcsonatorisgreater than that duc to thc double .sourcein thc ratio 1th anguiar variation bcing disregarded.

A vibrating rigid .spiicrcgives thc same kind of motion to thcsurroundmg au. a. a double .sourcesituatcd at its centre; Lut thsubstitution suggcsted by this fact is on)y pennissibic wl.cn thradius of th sphre is .smaHIn coinparison wit!, c: oLhcnvisetlic presse of tlic .sphre modines th action of thc resonatorNcvertitdes.s th prcceding Investigation, shews how powcrfuiin gcuM-d th 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 cxampicsof tins use of a rcsonator is an-ordedby a vibrating bar of g)ass, or mtal, heid at thc nodcs A stripof plate glass about a fout ]ong and an inch broad, of mdiumthiekness (say inch), supported at about 3 inches from th endsby mcans of .string twisted round it, an.swerst!.o purpcsc verywc)i. WI.en struek by a hanter it gives but Iitt)e .soun

320.] TWOOR MORERESONATORS. 201sidcrabic force and purity may bc obtnincd hy a. wcU managcdbio\v. lu this way a.n irnprovcd )j:u'tnuiuc

FORMATION0FJETS [322.202322. 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 th ]cngth ofor~an pipes. Wfienitnid iiows in a.stca'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.dt, ot'pitch an oct,a\'(!jowcr(~. (:[oscr examinatioa npvufdcdthc fact that at t))~ .sidcs of t.]te )npp!c tI)G outward nowiu-Tstrcam \vas rcpiaccd ))yonc in thc opposite direction, so that n,tondue of n:nno frotn

C'HAPTER XVII.

APPLICATIONS0F LAPLACE'SFUNCTIONS.

323. Tim gcncral cqn~tion of n, vcioeity potcntial, w!)cnrcfcrrcd to polar co-ordin~tos,takcs thc foi'tn ( 241)

If /cvanish, wc hve thc quation of tho ordinary poicnUa.1,which, as woknow, is sa.tisHc

333.] SOLUTION IN LAPLACE'S FUNCTIONS. 205

In order to solvc this quation, we may observe th~t when ri.n-~).u, i!,c ).,i(!ji(; tcnuisrelativcjy ncgligiblc,andthntLhcuLhcsolution is

.nu h:ut~Mnn nmy Deassumer ta itoid good for tho compltecquatujn (4-),it' we look upuu !Utd uo lungcr as constants, but,as funcLit.n.sof ?-,who.so!i:Lturcis to bc dctcrmined. Substitutin~in (-~),wc

20G EXPRESSION FOR RADIAL VELOCITY. [323.

Thc forms of thc functions F, n.sfar as 11= 7, n.rcexhUjttud inth n.ccompa.nyingt!t.L]c

Jo m'dur to intd thc ]!u)ingturtus iu 7'~(~)') whcuMris srn:t,[),wc It:u on rtjvct'sm~'thu surius in (!))

32t-. An important case of our gnera.! formulai occurswhenrcprcsents n.(iisturbance which is propagft.tcdw)tol]yoM

324.'} DIVERGENT WAVES. 207

T))c m'igin of tlie (hsturbfinco mny bc in n.prcsci-iLed normalmotion of tho surf~cof n, sphre ofmdiu.s p. Lf't. us suppose

FORMATION0FSONOROUSWAVES.[324.208portion of th cxccss of ihud in front going to supply th dc-ncicncy behind. Now conoive t))c po'iodic time of th motionto bc contmuany diminishcd. Gradu:dtytho altcrnation of movG-mcnt beco;ncs too nLpidto permit of th(.!fu!IGsta.Dishmcntof thcmct'c-)yIoc:Urcciprccn.tin~f)ow;th air is .scnsibiycotuprcs.sedandrarcRcd, a-ndi),sensible suunft wa\'c (or wnvcof t))e samu nature,in c:t.set))c po'iodic tima bu bcyoudt)ic )i)nits suit~bicto hcarin"')ispro}~giitu()to f),()ist;u)ce. TLc s:unu t:(.kcsp)!).ccin a.nygas,an

324.J EFFECT OF LATERAL MOTION. 20~)9

Le reduccd to its Hrst tcrm and seeing tha.t (~r) = 1,weshou)d i)ftvcfrom (.~),

Thi.sexpressionwi)! apply tn a.nyp:.rticu!ar cana) if we t~c todoiotc Lhunormal vctocityat t.hcspl.crc's sm-fafur t,hn.tpart.icJtarcanal and thcrufure to uLtiun tm expression appHcabic at H))ccto aU tlie e!Lna)s,w~ )):tvumcrdy to writc ~'fo!- 'i\) faci)ita.toa- compari.sonwith (~ and ((!),I sha)t, howevcr, writc for U.Wchve 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 takesplace whollyalollg thc radius vector.and accordin~tyt!je expres-sion is not to be differcntiated wltit respect to or Mwith th viewof fmding tlie trans versovelocities.

Oncomp~nng (7) with the expressionfor the function inth fK;tu:d motionat agrt distance from th sphre ((!), wc secth:it tlie two arc idcntictd with the exception that is dividudby two different constants, na.mc!y~(~c) in tlie former case and7'(!

210 EFFECT0F LATERALMOTION. [:24.

grt distance from thc centre wi!)no h)nger vary from one direc-tion to another according to th saine Ia\v as thc normal velocityof thc surface of th sphre, sincc t))C moduh~s ami hkcwisethc amp]itu

324J STOKES' INVESTIGATION. 211

the table bas bccn extendcd further in the direction of valuesgreater than than it lias in tlie opposite direction. FIve signi-ficant figuresat least are retained.

"Whcn ~c=cc wo gct from th ana!ytica! expressions 7' =1.Wc sec from th table tluLtwhen~c is somewhatlarg-o f,, is liableto bu little !css titan 1, and consequentty the sound to Le a littlemore Intensethf).n if IntGrn.1motion }i~d becn prcvcnted. T)tcpossihility of that is cxpin.incdbyconsidci-ing that tlie wavcs ofcondensation spreading ft-on thoso compartmcnts of th sptto-cw!uch fit a givun moment arc vibt'ating positively, ~.e. outwards~ftcr the hpsc of a hatf pcriod n)ay Iut.vcsp)-ca.dover the nci'di-bonring cottipiu-ttncnts,wnich arc now in t)icir turn vibratingposittvcly, so that Htcsc latter compartments in theh- outwardmotion work ag:u)t.sta somewhat grcatcr pressure than if snchcomparttnont )iad opposite to it only ttie vibration of tlie baswinch It Lad itself occasioncd; and th sajnc cxp]anailuu applicsMtM~~t&--);t!

LESLIE'S EXPERIMENT. [324.212

is no term of order 1. For a sphre vibrating in thc manner of abcU t!tc principa.1 vibration is that exprcsscd by a terni of thcorder 2, to which 1 sha.11nowmore particu)a.r!yattend.

ruttinc. for shortness. ~c"

324.] NUMER1CAL RESULTS. 213

"If it worenot for latral motion tlie intensitywould vary fromgas to gas in t))o proportion of tlie density into tho velocity ofpropagattot~ and thci-efore as tlie pressure iuto tlie square root ofthe density under a standard pressure, if we take tlie factor de-pcuding on tho devoiopment of heat as sensibly t)ie sa.mefor tliegasus and gascons mixtures wit!iwinch we have to deal. lu theibHowJugTable the rirst column gives th gas, the second tho

o mO* ~-j co Mo w o t- o

p o e5 o 'p c- .~s- 1, it. nppc:u's U)at thc so'tcs iu thc expressionfor

2G4 FORMULA. OF DERIVATION, f'3~8.

which is Laptacc's expansion in .sphoical surf.LCC )):u')no!)ics.

Froin thc difTurcnt.iid cquation (;~), or its g~n(.')-:t) solution

(':),itis c;)..sytop)-ovGth:)t~i!-i nfthcsii.muforHia.s

:~38.]1 VIBRATION 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

~GG HIGID CIRCULER BOU~DARY. [339.

Thc !owc;r vahu\s of /< satisfyin~ (.) a)'(; ~ivcn in thc f'uH~witigt:d))u', whidt w:)..sc:t)fn)a,L-tliront iL'm.smt'.st~tjlc.soi'thu ruocti.tu.st/ by tnn;u).s uf

339.] CASEOF COMPOUND VIBRATIONS. 2G7

an C()U!tt.ionwhich may bc vurificd m)tncric~))y,or hyan a.n:))y-tic:d pt'uct'.ss siniit: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'regardto Liiu i'undameut~l diit'L'rott.iaIcqua-tiuu s~tisticd by whichshuwstii~t

IIitherto Avchavc supposcdthc cyl!ndcrcomptetc,so thatrucm's ~ft.cc cach rvolution, witich l'cf~Hrcs that bu mtL'gnd

but ii'instc.ut ot'titu conphtc cy)indur wo ta]\c thc soctor iuciududtjut.wcoi 0=0 ~uiJ ~=/3, ft'i).ctiun:Llvtues 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))

2G8 ANALOOOUS PHOHLEMS rOR WATER WAV). [339.

st)-f;tc))i))g f)o))i thc-o('ntt't't.()t])(-circni))fum))(c

3K).]1 VnmATOXSIN A CLOSRDCYfJNDHR. 2G9

.'m'! c:n'h tcr)n)~n)st sati.sfy thc difrt.')'c))ti:dc(p)nt.innHCpar!itc;!y.Ti.n.s

winchis

270 C!HRAL SOLUTION. ['341.

rt'tjnit'cd.solution )nny,])nwt'V(')',hc'

:i.]] HXt'HH.SSION BY DESCEND)~; S)':I{')KS. 271Tho fu)-m))):L

DIVHiKiHXTWAVE. f:~n.L..Whcn H is intc.L;-)'a),(~c.so sries !~m infimtcand u!ti.natc]y

341.] ] DIVERGENT WAVE. 273

whcre y is Euler's constant ('5772.); and, as we m~y casilytiatisiyourscivcs by Intgration hy parts, th other Intgrais do notcontnbute anythmg to the Ic-a.dingtemM. TIius, \vltcnz is verysmall,

Reptacing by and compa.rmgwith the form assumcd by (4.),when ?'is sma! wc sec tliat in order to makc thc sones identicatwe must tako

so that a.sries of w;),vesdtvo-gmg from tlic ple, wLuscexpression1in desccNdingsns is

In applying th formula of drivation (11) to tlie dcsccndiugseries, the parts coutaininge- and e+'~ as fa.ctorswill evidentlyromain distinct, and tlic complete intgrt for th gnral valueof ?, subject to th condition that the part containing e+" sha.11not appear, will be got by diferentiittion from tlie completeintgral for M= 0 subject to the same condition. Thus, sinco

by(5)~=~,

SOUNDINGBOARDS. [341.274

or, in tcrms of tlie asccndingseries,

Thcso expressions arc apphed by rrof: Stukcs tosbewbowfccblythe vibrations of a string, (corrcsponding to th term of orderonc), arc cotmnunicatcd to thc surt-onndinggas. For titis purposoite makcs a comparison bctwccu th actuiUsound, .md wh:~ wonidh.ivc been onittc-d in thb ~mc direction, wcre thc latral Motionof thc gas m tlie ncighbourhood of thc stnng prcveutcd. For a.piano string corrc.spondiogto tho middic C, thc radius of t)iewirc may be abont-02 u.ch, and is about 25 inchcs; and itappcars that t)tc sonnd is ncarty 4(),()0()timcswcaker than it wou!dh.ivc bcctt if t))omotion of thc partic)e.sofair )md taln placemp)anL-spassittg tbt-ough thc a.s of titu strittg. "Thi.s shews thovit:d nuportance uf .so)t))(]ing-horn-d.si)t strmged instruments.AIthough thc amp)itudc of vibration of thc pin-tidusof th sound~n)g-board is extrcnK.Jysma]I compa.rcdwith tha.t of t))c partic)csof thc string, yet as it prGMcntsa broad surface tu tho air it is ableto excite loud sonorous vibrations, whcreas wcrc the stringsupported in an absotnteJyrigi(t manncr, the vibrations whici)cou)d excite directiy in t)tc air w~u)dbc so .s.nal)as to be abnust oraltogcther Inaudibic."

l''it!'!t.

"Thc incrcase of soun.! pro

341.] SYMMETRICALDIVERGENT WAVES. 275

ma.dc to vibrato, place a. sbpct of paper, or th bladc of a. broadknifn, witb its cd~e par~tiu) L~Lhu:t,xis of {.hcfork, and as ncfn' totho fork ft.sconvc'DKjnUymnybcwithout toucjnng. If thc plane; ofth bstac)c coincide witit eit.))Ht'of thc p)ancs of symmetry of thcfurk, a-s t'cpt'csL'titcd in section at ~1 or j~, no effect is producedbut if it bc p)accd in au into'mcdia.tc position, such as (7, tticSound becomos mucb strougor

~42. Tbo rcfd expression for the velocity-potential of sym-mctricn.1 wavu.s divo'ging in two dimensions is obt~incd from (1M) ~-H aftcr introduction of tbc time factor e' by rujccting th

imaginary pa.rt it is

in which,as usu;d, twn :n'hit)'aryconstants may hc InscTtcd,one asa.mu!tip)ier of thu w)))c:cxprcs.siouand th othcr as an addttionto thc thnc.

TitG prnDcm of a. lincfu'source of uniforni Intcnsity may f).1sohc tt'e:itGdby the g'cncnd )nut!)')d~ppliciL~tcIn thrcc dl;ncnsions.Thus by (3) 277, if p bu thc distance oi'any ctcmott ~.

276 LINEAR SOURCE. [342.

as tlie value of thc vclocity-potcntial at a grcat distance. A.shnilar argument is appHcabIeto 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 orFrMn tlie point of view of tlie prsent section we sec thesi~ificancs of thc rctardation of winchf~ppcarsin (1) and inthe results of thc foUowingsection (l(i), (17). In tlie ordinaryintgration for .surfacedistributions by Huyghcns' zones ( 283)th who)ceffuct is tlie lmlf of tliat of tlie nr.st zone,and the phaseof thc cnect of the first zone is midwaybetween the ph~ea dueto its extreme parts, i.c. behind thc phase duc to thc centralpoint. In thc prsent case tho retardation of th rsultant rcla.-tivcly to the central clement is less, on account of thc prepon-derance of the central parts.

343. In illustration of tlie formut of 341 wc may takethe prohicm of tho disturbancc of plane waves of sound by acyhndrical obstacle, whose radius is small in comparison withthe Icngth of the wavcs, and whose axis is paraUel to tlicirplane. (Compare 335.)

Let tlie plane wavesbe represented by

Thc gnral expansion of in Fonncr's sries may be readilyuSeetod, the coefficients of thc various tcrms being, as might

343.] CYLINDRrCAL OBSTACLE. 277

bc antieipa.ted, simply thc Bcs.scI'Hfmictions of correspondingordcrs; but, as wc confine onrscivcs ho-e to the case wLere ct])Cmdms f thc cyliudcr Is sma.li,Avewill at once cxpa.ud 1)1powursof 1'.

Thus, when )'=c, if e" be omitted,

Th amount and cvoi thc la.wof th disturbance dpends upont]tc c])ara.ctet' of thc obstacle. We will bc~m by supposing thmn.tcrial of thc cylinder to bu a.gas of dcnsity o-'and comprcssi-bility ?/6' thu solution of tbc probtem for a, rigtd obstacle mayfinaUybe durivod by suitu.btcsuppositions with respect to o- ?/t'.If K' bc tli mtern~I vduc of K, w have inside th cyliuder bythc condition that thc axis is not a source ( 3~0),

CYLINDRICA.LOBSTACLE. [343.278

Th tact that varius invcr.sc)y .as \"S might Ijave beenanticipatcd by thc motitod uf duncn.sionsas in thu con-espondmgproDum fur the sphcrc (33.-)). As in that case, thc synunctric:3pfn-t of tlie divcr~nt w:L\-cdopcnd.s upon t)tc van~tiuu of com-p.ssibiJity,and wouht di.s:tppc;n-in thu application to an actua.1gas, and tlie turm of the first ordet-deponis npon tlic variation ofdL'nsity.

By snpposin~o-' and 7~' to becomc innnitc, in sud) a mannerthat their ratio rcmaius fiuite, we obtain tlie solution corre-sponding to a rigid and iM)novcab]cobstacle,

Thc aualysis of this section is appUcaDcto th matlicm~ticaDyanaingous problum of H)tdingt!ic ctt'ect of a cytindrica! obstacle

343.] PASSACiH0F SOUNDTUROLTGIIFABRICS. 279

on plane wavcs of transvo'sc vibration ni an clastic solid, thcdirectioti of vibration bcing p:u'aHclto tlie :txis of th cyHndcr.If tlie dcusities be o-,o-' and thc ri~iditics be M, and Ydnotethc tmnsvcrs disptacctncnt, thc bound.u'y conditions arc

Fur an application to the thcory of light thc roader is rcferrcdto a papcr by thc author, 'On tl)c manufacture aud theory ofdiffracLiungra.ting~

T)tc cxcceding smaUness of th obstrnction nncrcd by finewh'cs or nbl'es to th passage of sonud i.s stt'ikin~Iy iHustra.tcdi)i some of Tyndatl's cxpcrnnonts. A pice of stUFfcit hait aniuch in thickness allows mucit more sou)id to pass than a we~erZpockct-handkcrchicf, which in consquence of tho ctosinf ofits porcs behaves rather as a thin lamina. For the same rcasonfogs, aud even raiti aud snow, interfre but little with th freopropagation of sounds of moderato wave-Iength. In tlie caseof a hiss, or other very acute sound, the cilcct would perbapsbe apparent.

P~t

C1IAPTER XIX.

FLUID FRICTION. PRINCIPLE 0F DYNAMICAL8IMFLARITY.

344. TiE quationsof Chapter XI.aud the consquencesthatwehavcdcducett fromthctn arc b~scdupon thc assumption (230),that the mututd action betwccn anytwo portionsof fluidseparatcdby au imagina.rysurfaceis normal to that surface. Actua.1Sui

345.1 FLUID FRICTION. 281~j

cordancewith tlie kinctic theory of gascsM resulting from mter-

change of molculesbetween thc strata, giviug riso to diffusionofmon~ntum. Both by theory and experiment th rcmarka-bleconclusion haa been esta.bliahedthat within widc limits th forcois mdepcndent of tlie density of th gas. For air at CentigradeMaxwell' found

tlie centimtre, gramme,and second beingunits.

345. Th investiga-tion of th equations of nuid motion inwhich regard is pfdd to viscous forces c~u sca.rcclybe considcrcdto belong to thc subject of tins work, but it may bc of serviceto some readers to point out its close conneetion with th more

geMera.llyknown 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, couucetedwith tlic Mtua,!displaccmeutsa,AY

by tlie quations

of which Mmcasurcs thc n'yi'~y, or i-Gnist~ccto &7~(t?- a.ndKmcapurcsth rsistance to change of ~o~t?~. T!)C componeuts ofstress P, r, corresponduig rcspectively to e,~ a, &,C,are f'jund from by simple ditiercutiation with respect to those

S 82 EQUATIONS0F MOTION. f345.If ~Y,F, Z bc thc component.sof tlic applied force reckoned pcrunit of voknnc,tlie equ~ti~nsofequilibrium arc ot' thc form

from \v1)ichthc cqu~ion.s of motion arc immedia-telyobtfuna.biohymens oi'D'Aionburt.sprincipIu. In tenus uf tlie di.sphtcc-tncut.-ja, thcsc c

345.~ rr.ANn WAVES. 283

or, if thcrc bc no n.pplicdforces :ui.d tlie squa.rc of thc motion bonc~ectud.,

Wcmn.y observe ttm.tthcdis.sip~tivcfurccsiturc cotisidered

correspond tu :).dissipation fnnctiuri, whosc furnt is thc .santCwit)i

rL'spcct tu u,?uvs tlt:).t of with rospect to a, /3, y, i)i ttie ttiuut'yui'i:jotropic soiids. Thus puttin~ A:= 0, wc Itavc frotu (1)

in forcement with Prof. St.okcn' ca,!cu!a,tion'. T!n.:theoryuf frictionib)' thc c~sc of n compi'cssibio Huid was first given by Fuissur~.

:~fi. Wc will )iow app)y tlie diiei'e;ntl!U quations to thc in-vcsUgft.tiunof pt~nc wa.VL'sof sonnJ. Suppo.sin~ t.h:t.tv a)id in'exo'u a.nd tha-t n, &c. arc functions of ou)y, wc obtit.in from~13) ~

whichis thc cquatlo~~gtvcn by Stokc.s~.Lct us nowinquu'e how a, traiu of harmonie waves of wavc-

Jcugth which are inanitcuncd at tlie ongni (a;=0), Me a.way

Com~rf'f7'M;);!ffc

284 EFFECTS OF FRICTION. [346.

as .c iucrcascs. Assuming tliat Mvaries M e' we find as ia14.8.

lu tho application to air at ordinary pressures ma.y bc con-Hidcred to bo a vury smaM qu:mtity and its square may Leue~lected. Thus

It appca-rs th~t to tilis ordcr of a.pproxima.tiontlie vclocity ofsound is unnH'cctcdLy Huidfriction. If we rcptuceMby 27ra\thc expression fur the cocfHcicntof d(jc:t.ybccomcs

s)icwM)gthat tue inimcncc of viscosity is greatest on the wavcs ofshort wavc-)L'j)gth. Tlie :unplitudc is ditnhus!tud iu thu ratioC 1, wlicu x =f" In c.O.S.mca.surcwumay take

Thus the amplitude of wavcs of one centimtre wavc-Icngth isdiminishod in the ratio e 1 after travc)IIng a, distance of 88jnctres. A wave-lcngth of 10 centimtres would correspondncarlyto for this case a;= 8800 mtres. It a.ppe:u'stherefore thu.t atatmospheric pressures t))e influenceof fricLionis not Hkdy to busensible to ordiuaryobservation,cxcept nc:).rtite upperII)nit of themusicalsca)e. 'Diemellowingof soonds by distance,as obscrved Itimountainous countrics, is pcrhaps to bc attribnted to friction, byth opration of which the higher and Iuu's))crcomponcnts arcgradually climinated. It must oftcn have bccn noticeclthat thesuund s is scareciy,if at al], rctnrncd by echos, and I hve fuund~that at a,distance of 200 nictrcs a powcrfui hiss losesits charactcr,even whcnUicrc is no refiection. Proba.b!yUns enect aiso is ducto viscosity.

AcofitictU Observations, P/t~. ~/

34G.] TRANSVERSE VIBRATIONS. 285

lu highiy rarcned air th value of a as givcn in (8) is much

incrcased, being constant. Sounds even of grave pitch may thonbc affectedwithiu niodcrate distances.

From the observations of CoHadon in thc Iakc of Gcncva,itwould appcar that in water grave sonndsare morerapidiy da.mpedthan acute sounds. At a moderato distance from a bcH, struckundcr water, he found the sound short and sharp, without musicalcharactcr.

347. Th effect of viscosityin modifyingthc motion of air incontact with vlbratingsolldswill be best uudcrstood from th solu-tion of tho probicmfor a verysimple casegivcn by Stokes. Lct us

suppose thn.tan innnito plane (~) excutesharmonie vibrations Ina direction (y) parallel to itsclf. TI~c motion bcing in parallelstrata, u and M vanish, and thc variable quantitics are fune-tions of a; otdy. Tho nrst of quations (13) 345 shews that thepressure is constant; th correspondiug quation in v takes thofurm

sun)]n,r to thc quation for tho tincn.r conduction of hcat. If wcnow suppose tlm.t v is proportional to e' tlie resulting quationin a; Is

If th gas bc on the positive si.doof tlie vlbrating plane the motionis to vanish when a:=+cc. Hence J9=0, and thc value of vbccomca onrejection of the imaginary part

a.t a;==0. Th velocity of thc fnnd in contact with thc plane isusually assu)aedto be tlie sn.mea~ tha.t of tlie plane itscif on the

28G PROPAGATION 0F SOUND [347.

nppa,rent!y snfHcicnt g)'f)nnd thiittLcc'mtnn'y would imply nnitif-mitc'y ~-ren.tersjnc'ithofss of

347.] 1~ NARROW TUBES. 287

Thc rc.sult cxpresscd in (12) wn.s first oht.aincd by Hchnhoitz.An

DYNAMICAL 8IMILARITY. [348.288

acoustical clments are th shape, th Hnear dimension c, thcconstants of clastieity q and ( 149), and th density p. Hcnec-,by thc method of dimensions, tho periodic time varies cfe

348.] DYNAMICAL SIMILARITE. 289

Thc same comparison that we bave c'mpJoycdabovc for clasticsobds app)ics a)soto acrial vibrations. T)

APPENDIX A. ( 307).

Tho probtcm of dctcrmming tho correction for th opcu cud of a

tubei8oneofconaIdct'a.b)cdif)icuKy,cvcnwhenHu-rci~au Innnitc

HtLnge. Itisproved in t]

2S2 CORRECTION FOR OPEN ENDS.

wLero p dcnoies tho (]iu

CORRECTION FOR OPEN ENDS. 293

Now

294 CORRECTION FOR OFE~ ENDS.

KothiLtt.hoscccmdtot'mis 7!7(1+~+~')".Iti catculating tho ~rst tcnn, wc must rcnx'mhcr thitt if nnd~ tx:

two difTuL'cntvalues of~,

To t.)usmust hc :ult)

CORRECTIONFOROFENENDS. 2955

)Utdour uljcct is to detei-tninc its mnxhnuta Ytdno. la geneDt] if:

29G

NOTE TC)~ 27;

A nK'Lhndnf o))t..unm~ L'uisson's soluLio!! (8) ~ivoi l'y Mouvitlu' inwru'U)yfjfn()

wpH!)7

jVo~

2:~8 PROGRESSIVE WAVHS.

whichiHuquidt-u

If K'-K, ])o Hinft)),wc htivc n. tmin of W!).vcswhosc amptituduvanus slowly from ono point to Miotlier hctwcen tho limits 0 tmd 2,formiug

PROGRESSIVE WAVES. 299

of Ut)t.tcxL~ing in n. Icngth F, or U timcs tlmt cxis(,ing in tim UltiL

Icngtb. Accorttmgly

Energy pt'opagittcdin unit timo Etx'rgy contn.hicd(on au ~vcragc')inunihiungth =~(!"):(/'

~00 PROGRESSIVE WAVJ.:S.

Wu I):n-c ucxL to cai'uh~t' U)e enci~y p)-o]

PROGRESSIVE WAYES. 301

a variation ofwa.vc-lcngth. Tho fullowlng investigation, in. whichthoinci'cincjttiof wave-IcngHi is ~~Ky

302 J'KOCKES.StVHWAVES.

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.'/

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)toatudyphysics 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 Yhunc. Tho Aut))('r bas routtcred it

po.sHi))!o,)'y th yery cnnvcuicot .sy.stutn~Licarrangutncut ofthc whulc, fur tho

)nost diMeuit pn'hions of acou.stics to 'jo now ntudiudwiHtfi.n' grcator eMe

Umn hitIicrto.Pt'of. IIchtiho~tx in Nature.'

Wolook forward wit.ti tttc grcatc.sb inhcrest tu t!io [tj'peamnco of tt)Csub-

squent volumes, for whiuh t)[is proparcs th way. Thc highor study of

acoustics will bc n diit'crent titiag attoguthcr wilcu LhoyM'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'dS