The Theory of Sound 2
The Theory of Sound 2
The Theory of Sound 2
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THE
THEOU Y OF SOUND.
TRK
.1..1..1
.lnJ
THEORY OF SOUND.
DY
JOHN
WILLIAM
BARON
ST.RUTT,
F.R.S.
RAYLEIGII,
M.A.,
FOttMRKLY
FULLOW
0F
TKIXITV
CAMr!):inGE.
COLLEG):,
VOLUME M.
bonbon;
MACMILLAN AND CO.
1878
[~.Z~/t~MMn'cf~
<E'n)nbWt!t:
A
i'~tffT)!t)nY(').(').AY.t!A
~T')'UKttf[YH)tS)TV)'ttKSS.
CONTENTS.
CHAPTER XI.
23G254.
Equations of
Aerial vibrations.
E.pmlity uf prcswuro in n)l directions.
motion. Equation of contimiity. Spcial form for incompressible ihnd.
Motion in two dimouBioua. Stroam function. Symmotry abont nu axis.
inVolocity-potontial.
Lagrango's thcorom. Stokes' proof. l'itysioU
Equfttion of cnn.
ThonMOti's iuYCfiti~tion. Circuhttinn.
tcrprotntiou.
Expression iu po)nr co-ordiu~teH.
tinuity iu torma of vcL.city-poifmti~.
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 no rotation.
In-otn.tiotml motiou bas tho Jeast possible euergy. AtmioHy with thoorioff
Gnerai oqu~iou for
of heat nud ctuctricity.
EquatioM of pressure.
sonorous inotiou. Motion m ouo dimension. I'ositi\-u aud uc~tiYc proHargrcssivo wt~vea. Rolatiou bobveou yclocity aud coudeiMatiou.
monie type. Euorgy propat~tod.
Haf tho ouo~y is potoutia), aud
htdf Junotio. Nowtou'H ealcutatiou of yclucity of Hound. Laplaec~ corroction. Expression of volucity iu tutius of rutio of spcifie hcats.
trou)
Experimout of CIcmout and Dosormcs. liaukine's ualeulation
Stokos' invusti~tio)).
Joute')) equivalout.
Possible cilcct of radiation.
of Itcat ha.s
Rapid HtifHng of tho sound. It appears that communiHation
no Beusibk oSect iu practico. Voloeity dopondunt upon toupcraturo.
Exact
Variation of pitchoforgan-pipos.
VclueityofMuudiuwatb)-.
diCcreutial cquatiou for pluuo WttVM. Aiq)iicatioii to wayes of tlicory
of Htcady motion. Ou]y 01 ouo supposition ns to tho law conucctuiH
without tho assi.stpresHuro aud donsity Otn n. wavomaintain ils fo)-!u
type. Puisson's
ttncsof nu improsscd force. ExpiauatiottofchanHoof
Relation botwoou Yotoeity and condensation in a pro~'ssivo
cquation.
Harn.
wavo of tinito ampUtndo. DifKcu]ty of ultiniatu disccntinuit.y.
Limitud initial dist.urbanc<j.
shaw's intL~rats. Rx!Uiann'Niuvestit;atiun.
Expcrimontal deturmixationH of thu -v'dooity of sound.
t'AOU
1
Y)
CONTENTS.
CIIAPTER XII.
r.\<in
-H.f.
2552Rf)
Vibratiousintnbcf). Goocrnifonn for Hinipbharmo~io type. Nodoannd
loops. Condition for (in opcn ond. lu Ktfttionnryvibrations tboro nn)st
ho nodHnt intct')t)s of
Rottcctionof putsCHnt doned ;md opou cu()s.
rrob]f!)ui]icnmpouudvibMttons. VibratiouiufttuboduotucxtoronI
sourcfs. H(~hc))dsf)poi). ]'rf'K''B3i\'o~vod))todiHt))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 upont)to ~s. Comparifiou of YcloeitiHof Houud iu Vfn'iounRttsoH. Exnminntton of
vi)))'nti))Roolumn of nir by motubrnno and mutd. Dy Kiini~s i))nucfi.
Cm'Ttidpipes. Ih'nnehud pipes. CouditiottH to ho Kntisnod at titt)
junctioi].ofco!)noct<id]iipc!H. Yftn~bIotiGct.ic'n. Approxinmtoc~cutatiottofpitch for pipes ofvfn'inMoMctio)]. I])))nc)tooofYaritttio]Jtof
soctiouon prof.ft'e.ssivo
wavoH. Varitttiou of denffity.
CHAPTER XIII.
65
~2<ji7
chmubo'. Cuhicmbox. Pr.sf~tnnt'oof
Acrin) vibration.') in nrectans"
rooma. liectan~ufar t))bo. CouipoHttionof two e<pt)dtrainH of wnvoM.
I!(;f!octio)i)'y)i)'i~;idp!ftnWft)I.Ct'con'HinvestigatiotiofrcOccUonfmd
Mfrnction of pfnno W)t\'Mnt n [duno surface. Law <~f!'inM. Casuuf ttir
nnd watt;)-. Hot]t ixcdin ~scuus. J''rL'fi))<!l's
cxprcsHion. J!('f!cuti(~nat
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.
273SD.~
Aj'Liti'dryinitial disturbnneo in an uulimited .at.)uobpLL')'o.J'oiHKon'saolution. Verification. Limitod initiul disturbanco. Cnf<oof two dimeu.
aiotts. Doductiou of Bnhttion for n. disturbauco contiiiiially ronwcd.
Sources of Hound. Hnrtttojuotype. Vorifiottinj~ of soJutiou. Sources
diiitriLutcdovcrn.Rurfaeo.
Shoot ofdottbio
hifi!)itp)n)icw)(l).
Hourcef). Wn.vcsin threo dimnjtax~t' symjuttrieni about n point. Harmonie ty]ic. A coudfnsf'd or mre'ffcdwnvo ctuntot exist niono. Cot)Inititdcirc~jnstn))cc.s. ydoeity-potentinlofa. a
tiuuitytbro))(;hpo)o.
givou h'ourco. CtHeubttion of eno~y Mnittcd. SpcaMnt? trompet.
Theory of conicnl tubes. Position of nodcH. Coinpositiou of vibrations
Interfrence ofsoundufrom
fronitwosiuJpiofiurc'csofliiiopitL'b.
cjcetriodiy jmtiutftinc'd tunin~ fot'hH. l'oints of (iikueo. Existoicu
ottu!) to be infcrrcd from eoutiidcrutious of hy)!)mctry. CuBeof bd).
85
CONTENTS.
vu
rAa~
CHAPTER
XV.
~C-302
.13,
Sceondurywnvosduc to a VM-intionin tho modium. Botativo importanco of
Mcondury wnvoa dcpondHnpnn tho w~n-Iength. A rgion of a]terod
comprosaibUityacts lilo [t simple sourco, 0.rogion of n.Jtoroddonsity liko
futoubio sourco. Law of inverse fourth powers infcrred
by method of
dimonsions. Exp])umtion of harmonie cchos. Altration of cimmotor
of compound sound.
Scoondfu-ysourcos duo to excessivo nmpHtudo.
Alteration of pitch hy ro]ntivo motion of source M)d
rcipient. Expori.
mental IHnstrationHof Dopp)er'H
principto. Motioti of a simple source.
Vibrations in a rectanguiar chambor dn to intcmal sources.
Simpio
source situatud in au unHmitcd iubo.
Enorgy ornitted. Comparison
with conicnl tubo. Further discussion of tho motion. Calcuia.tion
of
tho raction of tho air on a
vibrating cireular plate, whoBopiano is corn.
ploted by n. fixed Bango. Equation of motion for tho plato. Caso of
coincideucoof uatural and foi-eodperiods.
CHAPTER XVI.
303322.
.1UU
CONTENTS.
oniyi.ss.t.ciuut.
S.tponorandmfonor!im:t.stothuudnctivityof
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~y
to 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 of dctcrmininR tlio
pitch of resoiintors.
DMcuMton of motion oriHiuntioH within au
op<u pipo. Motion duo to
oxtcrual Boureos. Effoct of cn]a~omcut at a closod
end. Absorption of
Sound byresonators.
Qnmcho'. tnbM. Opomtionofaro.souatoreto.so
to a sourco of sound. Rcitiforeomeut of sound
),y re.4oi-intors. Idea)
resonator.
Oporatiou of a rosonator eJoao to a double source. Savart'H
Two or more rcsona.toM. Qu~tiou of
oxporimcnt.
formation of jets
uunug souorona motion.
UHAPTER XVU.
323-~5
Ai)plieatious of Lapiacu'H functiuns to ncuusticat
prohiems. Cunc.ral .sohition
tho
terni
of
tho
mYotv.ug
order. Expre.s.-jioa for mdLd
Di.
vclocity.
vergent wavcs, Ori~iu at aspJ.L.rieal Hurfaco. TL formation of Honoronti
waYCHrcquu-cs in ~nM-al a eortaiu arca of
movinH Hurfaco; othcrwiso tho
mcchauictd couditions uro HatiKti<dly a )ucat tmnsfercucu of air
withont
appMcinh)ocondc))H,ttiou or rarfaction.
StokcH'discussion ofthoeffect
of 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 i-iHinatoa in
tho vibration of a 8olid
)jody. licaction of thc snrroundinH air on a
Dg.dv.brnt.nHSphoro.
Incroascofotcctivoiuurtia.
W).cnth~p)tcru
'asma)Imcomparisou~iththowaYo-Ienf;th.t))croiabutlittlL.connuu.
nicntion of euorHy. Vibration of an
eHi]Moid. MuMipIo Hourc~. In
cases of symmotry Laplaco's f.mctions reduec to
L~endrG's functions
Caleulat.ou of tho encr~ M.uttcd from a
vibmtinn sphcrical surface
Caso whou tho disturbaneo ia limited to a f,)naH
part of tho 8phcric.U
Burf~o. Numorical rcsu)tH. Effcct of a sma))
f,p]~.ro Mtuatcd doso to a
Bom'co of sound.
Auatyiical tranMformati.
Caso of coutinuity
throngit polo. Aualyiieal MpressiouH for tl.o
-~loeity.potcntial
Expression in torms of Bes~I'H fnnctious of fractionai ordM-.
Particntar
casos. Vibrations of f~s confinf.d within a
rin:d tipherical envciopo
I:ad)a)vi),rnL<i(.s.
Diauh:tral vibrations.
Vibrations (..xpresscdby
a
Lapiaeo's fonction of tho Hccond ordM'. Ta))]o of wavc.i~~th.s.
li~ativo
pitch of varions toiles. Gnral motion oxpressiblu
by Himpio vibrations
Case ofuniform
initia) ve)ocity. Vibrations ofKasiNchtdodbft.wocu
eoncottricsphurica)
surfaces. Spitcricatsitcctof~as.
Investigation of
tho 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
in
a
sourco
at
a
originatinn
nnito distance. Symmetriod oxprossiou
for socondtu'y wavcs. Case of a f~seons obstacle.
E<iual conipreosi.
bitities.
PAOE
PAULP
CONTENTS.
Ix
C'HAPT.-t XVin.
?'~m
?~3
33G343.
l', Expansion cfYtdoeity-potcntifd in
rroUcmofn.Rpbt'rieatiaycroftdt'.
ExFouricr'Hsorif.'s. Din'orcntm.t quation Bfttisficdbyc.ichtorm.
Solution fur thoc~stiof symmetry.
prcHscdintormsof~n.ndof~.
Conditious to ho Rn.tisdcdwhon tho ples arc uot sources. Rduction
tot'egcndro'Hfnnetions. Conjugntopropcrty. Transition fromnphnricnl to p)n.nn hyor. Densel'Hfunctionofzoroordcr.
Sphorioal
tayor boundcd by pnrallola of Itttitndo. Solution for sphorictd layor
bonuded by smnHcirclo. l'nrticular enscs HoluhJoLy Lcgondro'H funetiouH. 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 drivation. Corrospott<lin(;formula in Dcssol'a fnnotions for two
dimensions. Indupcndcnt invcstit;n.tiouof pifmo proLJom. Tranavorsc
vibrations in a cylindricat nvbpo. Cn.8of uniform initia] volocity.
Soctor hounded by mdini w~ts. Application to watc-rw~vcn. Vibntn. circnl~u'eylindur witi) piMio
0
tionn,not ncpcss)u'i)ytraMs\'t~'Hc',witi)in
end~. Cn)np)otoHointion of diff'it'cntial ('(jn~ti~u without restriction
n.Hton.bsoncnnf potarRourco. l''uriunIn.KfdcriYn.tio)). Expression of
vulooity-potentift)by df'scondingffcnn-convorHcntxericH. Cnso of pure').
divorcent wf~vo. Stokcs' n.pp)icationto \i))rntinR Htrinsa. Importn.ueo
of Boundin~-bon.rds.l'rovcntion of latral motion, Volocit-y-potcntitti
rrobloui of
of n lincn.r Houreo. Siguificanef! of l'etardfttiou of
)))M)wavoa impinf.;i))f;upon n. cy)indrica.Iobstacle. Fixcd. nnd ri(;id
nylindor. I\rftthcnmtic[~!yannicH" probicm rolnting to tho trftnavMc
vibrations of nn dMtic soHJ. Application to thoory of light. Tyndfttl's
oxperimonta shewin~ tho sm~IIncHsof tho ohstmction to pound nitorcd
t'y hbricH. whnsoporps f~rr'0))cn.
XIX.
CIIAPTER
280
3443~8
Fluid Friction. Kfituru of viscocity. Cocn'tcientof viscocity. ludcpendont
of tho density of tho gff. ~raxwell'a oxpcrimonts. Cotnpn.rifionof
quations of vincous motion with thoso fipplicnbto to tin cin.stio fiolid.
Assnmption thut ft motiun of ~niform dihtatiou or contrfteHon ianot
opposed hy viacons force. hitoTtcs'
expression for dissipation fonction.
Appheation to theory of ]~ut wnYos. Craduni dceay of harmonie
wavesmaintainctl fit thn ori~i)). To n. first approximation thovolocity
of propagation is nnn.u'c'ctedt'y viscosity. Kumcricn.~cnleuhttion of
coctciontof decity. Tbo ciTeotof viscosity nt rttmosphcric prcHsuroia
scusitilofor vory hiRh notes only. A hiss boooinoa inaudibto nt n,moderato distunec fron) ils h'curcc. lu rnroncd n.ir tho h'ect of viscosity ia
muo!) ineroftscd. Transvorso 'vibrations duo to 'viscosity. Application
to caictunto cffcctHof viscosity on Yibmtious iu nfurow tubea. Holmholtii's nnd Kirehhon''s rcaults. Obson'ntions of Schuoebeli nnd Sccbcck.
Principio of dynamien.1simiifn'ity. Thcory of shipa fmd rnodets. Application of prineipio of Eimi]Mityto dnstic plates.
CONTENTS.
APPENDIX A.
Correction to Opcn
End
Noteto 273
Note on ProgrcssiYc
Wavcs
l'AOP,
9~
~'7
CHARTER XI.
Ai~lUAL VIBRATIONS.
23(!. StNCH t))c atrnosp))erc is thc abnost miivc'rsa] vehicic of
Sound, tbc i)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 Lhc pt-(~):)g:Ltin)tof sumul iu ouc dilnbnstO)), t!t0
]!t:Lt.hc)n:tt,tc.ddinicuitie.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 bc
submit.(('d(.t)t)tctcsbofuxpcrimc))t,indcf:Utkcf
;u;cun).~ nic't.)hj<).sofmL';tMuri)~ thcintcnsityof
vibrations. Iti
woctm dois tu suive thuso
.tncj~rts
oi'thcMubjuctnHUt!~
~'robh'm.s \vh~Memathon~ticu) conditions :n'L!.suf)iui(.-)iLty
sitn)))L-to
:'d))utof solution, :).))([to trust t" thon
a.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(!.sns
prfL'ct, H)at is
to say, wc sh~![ aHsntnc th:tt t!)C mutu:U action bct\VL'cti
any two
au
port)un.ss(jpamtu()by
u)<).)surfa!s);o~i~o~Hi'ce.
itcrc:Lftcr wc Hha)t say souicthing' about Unid
frietio)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.
Thc cqu;)j:ty of prcsfini'L: in a!) diructtons about a
givcn point
is !), ])'jccMsary
cuns(.'<ptcncc of permet Ouidity, ~hutbcr thcrc bc
rcstor]Moho)t,a.sisprovc<}byconside)'in~t)ic<(pu)ibi-iumcfa
smal) tctrahcdron uudct- tbc
opf.'ration uf tbc fbm) pressures, t])c
J:.Jf.
0
EQUATJOX.S 0F
~r~
P~un.
thcir
'r
w I
FLUJD
L-
"y
O. <)
~au.
'f<L.
~o~)Lht
Oc dUIIOLud
Lcdc~.U~, by l,,
JriUIll is
'h.
r.int.
..in
lil/'('{'s
~<
~nc.nt
,~r,
~=~(.Y<
clmtya
~t'i'
t~,
-1"
'r
'r
~o-f-fux.
,).
,~j,,
r~+~)
"<c.
cl, cly, cl.= iu tlte co-ordill:tlL's
uf tlm ]lUill!, ut whielt l:ltc
-p~u.,
,,j t(..nn.s of~.
f.
})fll'ticll', ",ltidl(,l~1' it muy 1..Jl"tli:vt aL
tlie t.illlu t is fouud a tlm
puiut rc,,1/,
1l.f~ura HIJlflll illtcl'd uf
t.
~r'
~s,
un,
))u).f ~,t.
.f.bvc)~ity~.U~<.n,
thu otlicr hand
eXIJI'usscsthe
(i,,t ,j,~
chauge in Il tlw \'I,luei ty of tin;
umjimcl
which
is uut 1ixe itl
EH
L~u jM&p~M,buti.iutcs
slrtcc, Lat IIluVOSwit.h
tll() ilnicl.
w.t).t).<;))tn,j.
..ot..U.
~,dt.i.
;hcd.a,
Tu
'Jutf.is
tjtis
237.]
EQUATION0F CONTINUITY.
;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 th tmjtiott. i.s vcry
sm:).]!, tlic tut'ni.s ,-(f.~ ~c.d)'Ish
7)
utUmatcly
~=~.
in !'L']..ttiv<.i importance,
n.nd
.ST!!t.:A~[-r)rx<'Trnv
~s
~ci.)
r~nnc>
~t~u,
<).
fil
~Ln
..s c.t.iy
p~anututhcphmGuf~
~1''
fll'lJitl'rtl'Y. TIlt! 1'nuctiinl
~i:l.r~
't = ('UlIstnIJt,
ti)
11'
"Lid,f.
is call1,t! the
.strcrmi-l'mnctiml, HiIJl'<:
tlll!
cnrves
~1'Imn tlm mnticul is
Htl'ady, tlmt is, Hlways t,lo;
"~<h~
HII1':llnl5~tic~ully,llre Huh,tit.lltioll of ouc f'm~c.tir~n
t
In ?)h])n.sf,a])
(it~c.
rr
,<)
i-
we s]:dl ll:we fu
in \'iltlll":
'f'))U))mh..n.r
ml,r-1-mlr~_l_.i~,rh III' al. IIIOIW/III"/'
u pel'fect
ditr(~l'eIJtiaI clc~, it, will rurrt:lin
so tur ull
Sldsl.'III/I'Il/'
alld Le tlmn
LAfiHANCE'~
)s THEORE~r.
:23f).]
Hf't in motion
Ly cnn.scrvntive
furcc.s axd
pressures
transnuLt.cd
trL))ntj]n.:c.\t.c't'i<'r,t)K!~)t:Lnt.ittL's
(whiuh Aves)):t]Ldnote, h y
Wt.;ass)nu<jth:Lt.pi.s!).iunctio!t()rF~,andwcH))!)))
hrcvitvy
~.)'h('t.'<[Uatit))tS(')'in"tit'n<)ht:uuedf)'uni(I),(2),~37,tn'e
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
.subtt'acting,
respect tu 7/:UHtL))L'St'r(m() wit,L respect U').nd
i.c)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
s:uiiu furni
\Ytt)i
1 t\vo
two oLhcrsoi'
ullici;s1thc s,wu;
J
furm givin~
~iviy 1~
-L/C JL/C
Jnthuca~
of.'u) incotnp)'cs.sib!cfh)id,wcmaysuL.st.itut(jfnr
<
</ f/o
i,
,!m(tthn.st)t:)n)
~' + < ](..spqmv!).iL't)t <
LACRAXOE'STHEOREM.
<~ c.
~~y~
Tho,
'r'
r~Sf).
c.
"' ~i~
'TT
..c~
~-~"
"~
tlie sulution of the
c(I1l:1tion
its
~u,c).y;[,ut
~cw
'y
,r,
is
.<)<c.tdw.d.~v<sdMt.,S..
~f.t~mc,t
sllcw that il-, aud
"en t.,
.E,
{'v,u,
is v point tlmt is
.v~
.n
tll;tt
~.r~r'
s
,r; Yalljslll.'s
Y.
ia tllc
w,
lcticm
it
'?
limit, nnt to
to P.~
tlo: first orcler, Lut
~t
~r~
'<)on<t).c
c,
body. ~t'.ocf,tio,,
~"<''~n1>ccn 8CF..s,~)aH t),c
Iml
the ~m"n
ditli:1'vlltial cocflicimts of s witll
l'o.;pect
w'illt if' (li(l
and
HO,
tllcn il, miglt hc infClTcd
that
s cOldeIHOn'1'
1\'gitilllateJy
var)' t'1'01l1zc:ro.
By Htlll!orel!l rIl/(' to 8tll/('S,
tllc II1f~111C11t5
of 1iIOIIIcntuIll:t)mat
.~F:
jllfillitesill1aJ sj:,lmric:tl
nf' Ilui~i
hurtiutl
tu
0(til~ti
i;, ~7, r, lI1uJtidil'd hy tllc
II1t)J}W/ltof
"L~='s.r:
tlie torms11t'pl'IJ']"llt
nf'gl"C:1ing
nn iuertin, \0
olitill'il
14j11:Iti~llH
to
111~1~IiC7ll~IU
tIlL.
11)(-tioli
of L'JL'ell'il'ity
=~
;olldlldol'R.
tlwuuglr111Jiflll'Ill
-(~?;7/7']
un. p.(,7.
~.A.I~r,nir~mie.)7.
HOTATOtTY VELOCITIES.
230.] -1
J!yt)tui\un]:m)Lcut:de'p)atiuus
IntoH'n'tIn~
wiLh
th
Unit),
t,his
c'juidiu'i
~!ong
!ny
fmitc
'I1
!U'c
7\
moving'
wch.tvc
CIHC'ULATfO~.
Lr.Q
L")))WO)'(!s,
~7<f
/~)~/t;r,/
,')
nernuins
tinte.
~r~~y;tpt!
con,tunt
;Ll(Ia;ci~
7'UrlIrtG
nll
tlrrurr~hout
is
nppmpl'iatt'Ij' ca.l/l'i! t,llc cinculcction, :ual the prllposill'"
IlIn,)')lu, st.utucl
-morrinr~vcritlr thc ,llrticl ~'e-
'1/1((
II8cumv'tmot.
L.
t).
as~.f~.
'.y~s
").
c.v.<,n.j.j,
.s~ ~hy
.c.
''caco)np)..<G(hn'<rc)!<i;,).
~td.n
c.u
c.
I.h.
Lu
ail)'
'y-t-
.),~
in..j~i.)
~c.r..u~i.n,
u-.)~iU..h~~t,
..si.
a ,,f,i,
h.
is
tu
vt.uut
<
p..ss.<.
n
n~utu.,fy.cii~
to hc
r~i~jc
..I..n
I.nf.
tr.iU~
uf.st~.r.o.c.pi..)
L,
Ly In.o~tionaHv
i:
;lru s;licl
f.n<h..t
tiunally
~=;
1I1ovillg'flnicl.
'\Vit).in.n ova)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~
c. bc
,drc..).tir.Li
~vc.n..t.uiu,.diy.U.cre
c](I~(~II
C\\l'I'U (1rawlI wiillill it.
HlIC:1tspa,('cs ~re C:LJIed
simplysurliwu d' :1I1;tnclmr
~r~j~r~
rils" a clmmi <!1Jl'ugoillg rmmci tltc
ring is
reclucille tu a point, and tIlcl'cfol'o
t}I()!'ma)' lie
t'n-cn]:)t).)n n!ot)< t cvoi
.i)t~
v.),c
.dn~
jBut Un- c.ircH).n
.tth~vh.!c
everyc~)
c.rve
round thcrin. ' an.
~).p~
~o
'c.~nccun.s~t~iuu~a])<)~<.).at..Io.
24L When
(,
~+~s~c~ct.h-n<~
in a.y .h..c.ctl.n
is .pr~d
hy th c..n.)i
~cL.~y
.)~h i.sc.)M
cha.f~
L<)
~.vd.ci
y-p~f,
~]
YHLOCrrY-rOTEXTIAL.
tlie
It\S''k')t<)tcn.nyc]<.sc(Ls)))'fnc~,t))-at.co('Howf'utw~.sMros.s
f~S', whcrc is thc r~ of va.
ute.nont <? Is expresse by
~nti~ocu.st-ut'
ti<)n()f~inr'c'it)~<'ntwn.]-(1s:).]'gtLun")-n):Ll.
constant d~)).sity,t))ut.jt:d!nss ut'Omdiutttuc~i.sthos
-whc'nitiH'l~irud
towurk
in(.k'})(;ndL'ntv:u'):th)L's.
thc pt:u)C A'y, so t)~t
Thus, if wu ttd poiar ct)-"n1inat.s ia
r~OPERTy
op IKp.
r :,2.1L
~1.
~!nltlir, m,~tln!1~Ilnll
file
~L~.r
"<y'.sby(~
'c~t(,u..uf~
fur t!I(~IJ1'llld('11I
'1~
il!
cunvcniunt
Ilancl.
-y~p~
flllid witllin
allY ,~iIlJpl.r-c()lIlll'dl'd ('Iusl'd
1'11'(' 8 i, i~wnlilc~ful,S~
vlutmwnimnl l,y
r:
~y
~n'j.,)),j,
~cri,d
"j.~ ~.s.s,f..
in ,y.yt.
~.i.i,n.)h.
')
it (';IIJ
'<
l't'st,
,.t
.)'i.J.L~
~JUH.
al'llllil'e III) IIIO/I'cillal' l'otatioll
1111l1(,)'
tlH: ulu,rutiun uf
)~
.f
the spaeu
value
;u~~
'b~~thcc.
in 7'Imnu~oynul
.-P
.S ~<~<~M~y
~t
~<Mtf)('<jrcni.)rr7~-n
i,
1'1'u1.JJVIJl
.L'~J:f.?, jlossilJe.
tlie iut(-~t;t,~
j;
iL.r(-h.j.~
't.
JiU <tll'O l'lIlIelillns,
"c.
s~~ry;
'~J~<
..h.r.
am!
Ju,
'L~r.
L.
S
1
242.]1
MULTIPLY-CONNECTEDSPACES.
11
~Y-Cr,xx,,(.T,
,p,
<0
all<l il tilt'
dYllaIll(';/:1111di(,iltillll('XPI'('S,s(:s
h,Jl'o=~
lllu circolulic,;l l'olilld
i11; l'jll~
''ri''j"i'cc.;t,;ui.,u
<h.
sincc
~o.
())),
0\1'
illn
i11'o
1;11'1'
J\IIIV
;i:r:
th.sL.u.L
Lc.u.cuh~i.n
.iiO-.n.~
,)
of
cd'
if rc \'<I/lisll,
T).
-J
cl~e
~Lc~
be giV011,
ifep aml cj~-1-V~, IJU twa 1'mnclimns
1!~ur,
~T:
snlisfJ'illgLnplaee's l'Cjuation
alld the snrnc l1uI'lIlal
~i=~
t iUIl :lIId ilm cOllditio)1
.LaplaCl!'S l'(IIIILtIlaL tl)(,l'u
.t~
s];J/ I.JUIII.itl,CI'
('il'cula/ioll
~s~cr
.S' p
\1:
~<cu)..<i..n
~-d.~n~
~J.
~sin.p.
(at;.nvp,v!
't,.
J~
~s
n,,r hy
~rf~,
circ.).t.i.,
-ya.
IlItlll
il'ly-cIIIJIJ('dl.d asn.s \('/1 "as
I~c..
f.,
'y-<~cd
l;illlp/('IIIII}('ct(~d 1;1':1('(' if v
sd in .ion
hy.
tllc wJJilh!11I:ISS
cumes to l'cst so 80UII vs
tltc IJJttioll ut' thu
'y<.ua.s~.
~L.fthL.Lu~LUIIJJH'Lnm'd
'~vI~~ith~Lf.r.).ti~
rcei~ut
t.,bc-!iko surjhof,nr
~s)y
su~, t).
witlliu
taLe comes in l't'st.
~iD.i. t1IU
tJ.c.tubccn.nc..s~
'1.'llis
L.
~)~.] J
.\XAL()(:Y
W)'rif
H~AT
AX))
EH':(")'):r('!TV.
13
a(tuit) havit'gnocircuhttion,
axdit
(.'))':n']\'w)'!tt ismoanthy
<'xh.;))M)')norSt.uhcs''h<t~)~
tnxx'ifh~~s <t)t
witht'rspt'<
cuhtr n'tation.
i'or, H ait. thc Hmd ())iovh)j.; suhjt.'ct tu n,
v~h"it.y-p~<)'ti:'t) cnLsidu n.spho'io:)) f':(\'ity(~f :t))y r:n)ms bcc<n))L'M)nh!untyso)itt,t]tL! fhud in.sitk' Un.' c:Lvity cnur<jt:nnno
motion. 0' aswu m:ty :so.st:)hj
it,a!)y sp)K'rica[ portion oF
:t)) i)T')t:)tionn)ty n~vrn~ Ouid ht'cumin~' suti(!t'!)ty .sotid wou)d
))()SSt.'s.so)ttyatUt)ti())i(jft)':m.i1:~it)n,<'i'<ru<<~<'o~
A Mi)t)i!!0'proposition w)))app)yt')!L(.'ircul!))'t1iHC,f))'cy1in(1~r
~it)ti)at.<'utls,i!).t.ho(.)..s(jofih)i<t
]uo\'iu~it'rot.:ttio))!(Hyintwo
(iitm'n.sit'jisonty.
Thu )))')(!')!) ofn.n i)tComp)'('i.si))tL'fhnd\)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 th uncr~'y is <))u icast, possib)c. R'!)ny uthcr
nn'l
motion 1)L' jToposml H:(tistyhi~' thc ('qu:t.ti"nofc(mti;)uity
thchotnx.hu'y cumtitioo.s, itst.'Ot'r~y is ncccssat'ity~rc'ittcr t)):)n
thatofth'jmotio))'\vhichwoutdb~.L;'<)'ci';Ltcdn'o)nrust.
Du; f:).ct that thc irrot:ttiu));~ motion of mcomprt-ssihh'!
2tjt!)ud (tcpcnds upon
V(.-]o'-it.y-potcntir).]Sittisfyi))~ Lap)nc(;'s
foum):<.tion of:). F!U'-i'u:L(.)i))~)))!))o~y butwccn
L'(~n:t.t.io)t,isthc
t)~! motion of'.such~
U~tof
thtid.and
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
wttich dc'pcH(tmat))(j)n!).tical)yo)i
att. thc b[':UiL'))C.sofDiysics
potuntia), i'"r ItoftL')). happuns tl~t thu :i.n:tto~ous thuorcms
:).ru f:n' from cqnidiy uhviou~. For (.'xampic, thc ~na)ytie:d
t)tcui'cni that, if \7~ = 0,
.Vot;), !or.ctY.
EQUATION
2.). t.
!n')t0t)i(.
W)u..n
0F
v..).)city-p.(.nti..d
prus.sm-ct,):!y
1-KESSUHH.
c.xisLs,
Die
f~
Lcquation
`.
!<ut:)'i)L!L.i
t.,
.X.tcr-
},jj.M
'J~-t'\JJ
~~iJ,
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
uc.
t.o. ~d)
siit~c
fur
~pncahon,
t.
if
part
2.N.)
rf.AKE~VA.VEM.
15
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
tinc. Lut tis t)!L-)'L'f~rc!
s:u)tui)t:t!)])):uu'.spt'rpuHt)icu!;n'tu),)t:)t.
(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
~)').
tl.ie w~vc-k'ngt!)
't'ln; condition
tlu,
whatcvcr tbc
vlic~tcver
b~ Tix;
comlil,i~m
Hati.sficd
s~,tisfic~l
vwve-lcn~tl>>na.y
mnybu.
l~ythc
by
pusiLivcWi~ve,:)n<[LhL'rutbrchy th ixitiai disturbimcc ii' n. posiulunu bc gctict'aLud,is
tive Wt).vc
~t5.
Whatcvf.r tlie
init.iat.t)Ht.urb.'U)cuniayhc: (axd Mand.sanibuth
arhjt.rary), it c-an niway.s bc <))\'i(!,jd into two parts, .s~LiHfyi)~
n:-s).ucUvdy(:{) :u.(-t.), which arcj.mpagat.udundistm-hc(t.
In
')'(-)t);)..nuntw:tvcthc
(H )~ct.i.)ni'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
E~EUGY
~5.]
0F
FLANE
WAVES.
17
R. ir.
1873.
18
NEWTON'SINVESTIGATION.
[245.
2-]
LAPLACE'S
CORRECTION.
Toobtmnanumurical
)-csu)t "/c r~'uiretcju~w
~pnir<f
simuftancous v.uc5 of f~d
It i.s found by cxpcru.tent ti)at
fit (~ Cent. undor a pressure of 1033
grammes per squ~-c ccntitnctrc, thc dcn.sityof dry air i.s '001293 grammes pci- cubic ntimetr~. If wc takc tho ccntimctrc,
grannnc, f~d second as tho
fmidamcuta! unit.s t)tc (c.O.s. System), tl~sc
dat:imvc
20
LAPLACE'S
CORRECTION,
[346.
246.]
21
Th cnclosed air is uext. a.!lowcdto absorb heat until it bM reg:Lincd thc a.tmosphcnc temprature 0, and its pressure (jp') is
thcii obscrved. During tlie last c!~nge thc volume is constant,
:uid thereforo tlie relation bctwecu pressure and temprature
c~vos
~2
[2-iG.
s(jU)at,hydhnu)nticnof~:("),
By cxpurnnontsuf
thisnat.)))~
~))tinL:d'y=l~t!bt)t,).JtcnuLhudiso))\'iouHtynots))sc(-ptih)c
of :my grcat iK-cm-i~y. Thc v~hnj uf
'y j'C((uir<jd tu ~ccmcittj
t))L: catcn!atud:n)doL.S(j)-rdvc)uc~iL!.st'.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-nnd
fut- unr knowlud~c of thc !nagnit))()c of 'y. Thu ViLtuc (.F
/<
thc spcifie liuat at constant prcs.sorchn.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 .spcoitc ])cats, Lruu~))t tu Ji~ht Ly t]iu mudum science of Thcrniodynamics.
Iffroiat.hccquatiun.s
2.1 G.]
RANKIN-E'S
C~LCUJjATION.
23
By
of that
<)~rcc
on i t.
24
STOKES' INVESTIOATJON
[247.
bcing a function of
247.']
0F EFFECT 0F RADIATION.
25
lat
:s tthat
IIC1 1:'>
winch
in 1.1
t.hc
~IJlutter,
;n.d JJI
\lUCI',W
l'lnur Cli:'JC
Li11tthc
l\ l'unnur
c~c <tC
~~x~~
v-a, ;LIll
Ncwton's
(sctcct.edby Stokes for :u)a)ytienlinvestigation,
h~v uf radiation bcing ftssumcd a.s a, suficicnt approximation to
Ihc trutli. We i~ve thoi
TII~ AMPL~'UDE
1S ~fOUE
r~.i~.
2t7.~ J
INrLUEXCRD
27
if f/ and
worc cointhc pik-h of t!i<j soum~
wc h.'t.vc no ]'(,'ason to
of observation gocs to
From (10) wc sec that thc faUlng off in thc intcnsity, cstland.
tnatcd pcr wavc-lcn~tl), is a maximum \it)i tan~, or
= ~Y. In this case
is a m~xininm, whcn
by (!))
Calcula.tmg from titesc Jut~ wc fhtd th~t for cach wavclun~h of :).dvnncc,thc innptitudc of the vibration would Le
dn)T.i)iishudinthcrntIu'C172.
To tnkc a.iiumerical cxampic, Ict
38
EFFECT0F CONDUCTION.
[247'.
247.]
VELOCITY DEPENDENT
UPON
TEMPERATURE.
29
30
[248.
~=-0000457,
siucc p = 1.
7'/<xLV.
theory of ~sca,
)).])-<.
t1o voiof.ity
M)!t').y).y,ttndis))roportin)mlt.i,t)R~unmv(.)ocityafthn)))n]<.fu]('f).
]~77.
.)/(5)tn.;)..tU.
1873.
of onun~ is detorminol
)'rc!-t<),
249.]
EXACTDIFFERENTIALEQUATION.
31
If V)
(Ic-nncthc a.ctua! pnsitions n.t timc < of
2/+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
wltic))
f~)
by its np-
coustitutioh
:m(l(3)
bc
may
of
the
thu
relation
mdium,
betwoon
t!tc
cquatiou
n.nd
o~
(k'pcndi))~
(~f motion
is
Ly
32
\VAVE80F PERMANENTTYTH.
[249.
is to be eHminatcd by mcans nf
.f/ u
it~~1j.
tllu
rc~muiut:
therci:(.:ionbc:)VC(;!)~anL),cxpreHS(.(tirt(l~.
1)c";vw;n
cl:c
2.'i0. lu thc prcccding investigations of acria! wf~es wc
that thc air is at t'est cxcfpt in s') f:n' as it i.s
h:ivc HU}'pf)sc<l
distnrbcd by thc vibrations of sound, but we arc of course at
Itbcrty to attribntc to th 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 th wavcs and with the s:unc actuat
velocity, thc wavc form, if permanent, is stationary in spacc,
and th motion is .~eftf~ In th prsent section we will considcr th prol)lcm under this aspect, as it is important to ohtain
a)! possibledcarness in our vicws on t!)e mechanics of wave propagation.
If
p~ dnote respect!vcly the velocity, pressure, and
p bc
density of thc nuid in its nndisturbed state, and if
th currcsponding q~antitics at a. point in the wave, wc !iavc
fur th equation of continuity
250.]
WAVE
0F
PERMANENT
TYPE.
33
34
[251.
pressure
~)
In th ca.scct't.hchuv expn.ssp<1 in (;')) 2.'i(),t)iC'
presen<)y,of/f.
rotation Letw-'L-u M a.n'tpfora,
progressive \vaveis sxch that
A/(;
j+~
is existant,
asnu~h
propa~atit)))()nct<)nugniente(t()cn.si<yas
tion ot'thevetucity~.
So far as t))c constitution
ofth<'
is~'ainc'ttby
superposi-
mdium 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)c foUowing
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~))~
p.)U. ]sos.
~N~t.
vn.
BETWEEN
251.] RELATION
VELOCITY
AND PEXSITY. 35
W)):)t(;vcr
may
bc t!)c
n 1IIl,: j Il,'itlt!.J;III('
I"II
'n.H.!is~uL.
j~.s.(.vf'p)-~r.ssn-uwLvu
d<)!s;Ltioni.s
Luv
ofpre.ssm-e,
!.s bJ",
1.
.'jy~
tlie
vaiccity
of
:1.) ..<('I'WI. 1
tn.=.
'V~
p]-cpa"!i-
,'PII
thcrc)atinnbctwcc..vc.)oeityan.!cnn-
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~,Uch~H
at
YL.!uc.t.y~n<h)..nsit,y:m.v..ry ~nu)u..t) h,,t bc.-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.at
th<.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
tho r't,a.hh.tt,p,~
i~m~].nLci~)~uur)n,ud
t.hc.stator
thu~.s.-Lt~<)..sL-L.K.cfr.nni(,n.twm
t!
bu d~cnmn.j<tby
thu cn).L.n~nnpp)ic.)~.
<.us,na))di.st.rb~~s.
ln ap].)yi,rt).i.s
~rcto~.m~h-r
D.c v..)uciti..saHdeund~t,im.s,
nut
c~nuuwc
h, <).pmvai[ms
~)ut~y,l,trd~.ivdy
h, thcnci~.bnuri,~
f.rt..s oi ti.c m..hui~ .s.jtJuLt tliu furni uf
(1) jn.upL- fur t.).c prsent
pm'po.sujtj
wh!d. )sthc
dation
im~rcs.,ve ~vc.
i'.in'n.sfiaw'.
y~
lion bchv~.u
vciucity ~nd den.sity~ given fir.st, 1 LeHcve,
by
Huhnholt/iij
bc tiic ()c)ts)t.y
co)Tcspon(]ing to = 0.
In this c~c Poissfjn'.s
aHow.s n.s to ~.rm .-L<).fmitc I.)c.-t
intgre
et U.c chang-c oitlie c.-u-Iier st~cs uf t!io
type ~ccompanying
7'/N7.~'nf)).<.l.s~:), p. Hf;.
= J~r~r)~.
(~ y'
,y. p. ioc. 18~2.
n
32
ULTMATE
3G
DISCONTINUITY.
[25L
it finaUyIcads us to n. diMcultywhich
progressofthcwavc,n.nd
has not a~iyct bccn surmonntcd'. 1. If wc dra.w a curvc to rcprescnt
th 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 th positive direction para1)''t
to a;, and of k'n~th cqmd to (f7.+ ?~) or, as wc n.)').;co))cc)')u''dwith
thc shapc of thc on'vc on!y, cqual to Il <.
t. Thu ]ocns of tbc ends ut'
thse linc's is th vulocity on'vc aftur a, time <,
Thc
But thi.s Ia.wofdcrivation caxnot ))o)d gond iudcnoitciy.
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 intgra! htyorni thc point at which thf
velocity becomc's discontinuons, or th Ych'city cnrvc bas n,vertical
In ordcr to nnd wlicn this happons !ct us ta~c two
tangent.
ncighbonring points on any p:u't of thc Ctu'vf which sh)pcs downwards in thc positive direction, and inouirc attc-r what time this
part of thc curvc beconcs vcrtica]. ]f thc diffrence of ahscissa!
bc ~.r, thc hindcr point will ovcrtake thc forward point in thc
Thus thc tnotion,as dctcrmincd by Poisson's
timc ~(f/~).
e<ptation, bccomcs discoitinuons aftcr n, titnc C(p)at to th ruciprocah takcn positivciy, of tl~c grca.tc.st ngative value of ff~
For cxa-mple, lot u.s suppose tha.t
Is
251.]
]
EA.RNSIlAW'S
INVESTIGATION.
37
M~C
7'ruct'~t~u/' </<e
~! ,S'~t\ Jau.C,1850./<t' 2')~. 18CO,
p. 1~;(.
~8
HARNSHAW'S INVESTICATICN.
[252.
bytLc
usuat
as might aiso ha.vc bccn info'rcd from (.).) 2.')1. Thc cnstant (7
othcnvisc
vanisi)CH,if~(o!),viz.<,V!U)i.s)tw!)(j)iet=I,)-jp=~;
itrc{-)renent.iavu!uuityot'thc)n<;(ih[)u:Lsa\Y)tut(j,!)avin~])()L!)in"'
tu do wit.h ti)c w.t.voas such. Fur a~oA~~e
pro~russivu \vayc t)~
)<vcr si~tts in ttie :uubig(ntics :u-c to Le uscd. Titus in
pta <jf
(!{),wc;ha\'e
) t is an arbitra~ function
1 Loulu'h'
Z~t')i<t<;<
A'</t<ff<<(~f.<,
Ch.xt\
253.]
MEMANN'S EtJUATtONS.
39
of &,or of ll.
ebencr
'Uub~dicrurtpfJnnxunH
witc.
(iiittixn~n, /)~/nt~/)n<t,t.vm.
i~the/.<r/~t'<<r)'<XY.p.l~
L"fbv(4)nvnne))d]i(..hcrSrLwin~u,)~1~0.
S<:e)d.un.ucxc(;Hnt
nb.itt'tMt.
40
IJMITED
INITIAL
DISTURBANCE.
[253.
253.]
POISSON'S
INTEGRAL.
41
42
EXPERIMENTAL
[253.
DETERMINATIONS
tlie
con()it,io))s<~n~ss :md motnottum, but it viutatus thc
cututitiou of cnc~y
~) cxpru.sHudby tiic c~xatioa
i'I.fs fu-~mcnt ha.s b~-ca ah-~dygivo.i In anothcr for.n
in 2.~
whicit wuu)d atonejus) ify us in rL.JMt.n~ t!.ca.ss..mod
rnoti.-n.sinco
it a]))ic:u'.s(.[)~tno stcady motion is
po.ssibfccxcoptundcr theiawof
Fmm ~~u.-Ltiun(.S) of t).t .s~ctiun wu
dcn.sity D.crc dctcrxnnc.t.
can fuxt wliat unp)-('ss~) furcc.s woui.t be
nccc.ssary (.) nia:).t:))i t)jo
jnuLio.. (tcHned ).y (7). It f.p].c:)r.sthat 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) !t p!t.s.s.
t.),u y:,),)u
~'h..
whoc moving
!u.d tiji.s cxphun.s i.ow
fu)~,vi~p~v;u)i.s].c-.s,
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~t cxp~ritncnta] .~crnnxation
nf tlte VL-toci~y
of 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.sin thc u]K-.i .nr arc ti~Hc to crror.s f.ou
thc ctt'uct.s of wind, am] frum
nnccrtfunty wi~i rc.spcct to th
exact eon<titio;t of thc ntmo.sphcrcasto
tonpcnt.tureand drync.s.s.
On thc uthci- Jtand wht~i sunnd is
p)-pa~!)tcd throug]i air cc.~
tained in pipes, disturhance ~risos fron fricLimi :md fr.nn
tt-iu).sf.;r
of Iicat; and, aithou~h in) ~t'cat o-rot-s frmu tht-s .sources
arc
tu 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(;ej
sure that tt~c iduat ptanc wavcs of
tt~-ory aru iicarty cno~h
rcalixcd.
T)~c foDuwin, Table' nontain.s a list of the
principe
meutai dutermiim.tiona \iuc)t )mvo bcen madc hithcrto.
Nnmos of Observera.
cxpcri-
v.
of
<J"Cuut.i)iMutruH.
AcadomcdesScicnccs (1738).
Lcnxcubcrg(lMlJ)
(33~3
f~33'7
s~i
3~o.('
3~')")
(~l()ing-ham(182l)
~ur~u des Longitudes (1822)
MuHandv~nBbdk
'Lusanrjnet,U~.Apri),
1877.
0);' TilK
254.j
VELOCITY 0F
KaincsufObsorvM~.
St.'U))j'J'c)'andA!yt'b;K'k.
;)h-!LV:Lisan<[AhtrL)ns(lS4-).).
W<jrt.hcnn
~tunu (IM71)
Lui~ux.
R~-)t:mJtt
SOUND.
4:3
VctocityofSnuudu.t
U"C(jut.u)Mct.run.
M2~
H~l'C
:4
:j~()'7
3!:3()'7
hasbcoi
proposcd by BusHc)):rf"t- d(jteniu!)in~
thc vctocity ut'sound wit)K~tt t!<c nsu oi'gruat distances. It
car is!).b)c to dcide
dpends ~pu))thc!prccisiotiwiU)w)nc]tt)iu
~')~(jt.))ernh()rttiekHar('sinndt:U)eouH,ornot.
InKonig''s"f()rni<jf
thc cx])cnincnt, two sn)!LHc)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.sof 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.)',th two scrics of tic)<s
f;d!.!Lsuud<r. AVficnt))u dii!'cruncc of distances is ~bout 3-)<jnctres 1
coincidc-ncc a~ain takcs 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.
VIBRATIONSIN' TUi~ES.
255. Wn hve 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 no ctm.ugcwoutd be ciUt.scdby tlie introduction of a.ny
munber of iixed cylituh'icit.Isurfaces, w)ioso geucrating lincs a.re
parallul to thc eo-ordumt.clu questiou for evcn whmi t)ie Hurfa.cca
arc absent thu ihud ba-sno toidcncy to move across tbeni. Ifons
surfu.CLj.s
of t)ic cyHtKh'ica.1
bc ctosed (in rc'.spcctto its transverse
section), wc ftavc tbu impot-t:uitprubicm of tlic axitd motion of air
wltinn :).cylindricat pipe, whici), wncn once tho tnechanical conditions 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 (~
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. Th const:).nts
and m tlic syinLiiciL!sotution
may bc co]np!px, a.nd thus the
<in:d expression in ternis of rcfd
qufmtit.ics wi)I invoive/arbltrary cousta.nts. If wo wish to use re.d ()H!UttHicsthroughout, we
must takc
4G
[255.
Thcothci'ton's
pns.sib)c foradoubiyclonu()pi])c
i~vcporiods
whic)t :u'< sub)))))tti))iL's of t]):Ltof thu gr:LVcst t'~tc, :Utd thc whotc
sy.st,('nifo]'msah:n'tnu)ticsc:t)c.
toinLutttsnnwHUpjto.sc, witLout, htnppm~forthcnn~ncnt.
f[nn't.;howsn(;)t:t. condition ()fthin~sc':U)~t''sc'f'u)'(.t)):)tt))C)'ciH
1. )'u:t<i<n)((i)~ives
:).It)opinstt.K)ot':).])<)()c!tttnej"~mt~=/.
cn.s/< = 0, whc'nco = -)-~ (2/~ + J), whf'rc ?)!is zero <')':), pusit~iv'
t~his c:).se thc ~'t'nvc.st to))(; hit.s w:n'c-tu))~L)t oq'):)l
iutc'gcr..)n
thu nodctu
lof<'n]'ti)ncst))c]t')in't)tr)fthcpipci'u('k')))''(It't'<))<)
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!.
25G.] 1
END.
47
proccss thc conditions to bc satisficd a.t thc en(). For our imtnctiiato purpnse it will 1w su~cicut to kuow, w)):)t is imIcL-d
t<)!c)-;i.b)yobvi.jus,t))!).ttlle ope;) cndofft.pijtCjn~y
i~ L'uat.d~
:).f""p,if thc diiunutcruf thc pipe bc n<~)<ctcd iKcotnpitrison
W)th<.))c~tvc-ju~t,]t,p)'uvi(tedthucxtct')]:Ltp)-c.sm'i~t.)t(.!]tc!r'-hLu).)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
Th
!t\ny.
in)pC()i)))C))tto.secun))~t))nfnHn))t(;!)tot').))(_! conditio)i
i"r a )~<)))~t:),ny(]('sircd point fies in th
inertie 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'dhy
a 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~')].)wsif thercbcatanypuiut
a )<)op; hutit
)nay])L-rf'L'('tiywc]l
thattho-o
arc
n~ititcr Utiles norioop.s, as
h:t,pcn
jt)rt..x:mii))c in
thL! case whcutitu motion i'(j()))~s to a positive or
~cg-ativu prowavc.
gressive
In.st.atiu)):u-yvi))rati(jnthcrci.s)]ntra)).sfc)-(;nccof
('nL!r~y atun~ t])c tube in ei[,)tcr direction, for enurgy 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 aurnny ~so
l'c invostigato] hy it.Jtowi])~ t])c inotioa nf n,
~?/7. by ~1,ie]) is
umkTstuod a wavc c<~)(i)tc<!within n;UTow )nnits and
compo.scd
of uniformiy cun<I<;))su<)
nr r!U'cfi<j<!
finie). Jn ]<x)hiu~ at th ])):),t,t~r
i'rom this point of vi<j\vit i.s nccc.ssfu-y to takc intoncunt e:u'ci'nily t]io cil-curn.sta.xccsnndcr 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. Sincc
thc CDO-gycouta.mct) in th \va.vc ca)ni"t
cscapc h'om t!tc t)tbc
thcrc must bc n. rcficctcd wave, ami thiT.t this rcftcctcd wave is
a).so n. w:i.vcof 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 cfuet of thc hu'ricr may bc inutatcd by thc introduc-
48
REELECTION
AT AN OPEN
END.
[257.
258.]
PRODLEM.
49
othe)-,whichisM)(Mcn]yhrougLttorcstn,tthctimc
=0,-<('t(jr
Lcing for sonc tune I]) motion vi).h a uniform velocitypand)~! te
its ieu~t.)). Th Initia! statc of t)te cont:uncd air is tiK-ti onc 0!'
unif'ornivducity p:u-nHu]to .-K,
{mdoffrocdom from compression
:ui(! r:u'<jfactic)i. If wc suppose t)):Ltthc
ori~iu i.s at tl~e clu.sett
un<),Du!gf;ne]-it.I
sofutiuuis by (7) 25;'j,
con.st:).nt,.s.
thccocfnS!ncc~!st<)LcKcroirnbi;).])yfor:tnv;L)u(-Sfjf~
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:tLioucxtcnds
vaincs of?' frotu
tc~fintc~-al
to . Thc ~cte)'mi))!itinn of thc cocfficiotts jd fron
(2) is
(-fbctc<) int).c usu:dway.
Muitipfying hy sin~d
intgra t)ng fron 0 tn we gct
Jn U)c
cascofatubestnp])(.<]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) .scmrofso.nxL
Dcf.onnit.i.~ r and 6' ili
<)!ahon(7)25~, \vff!)t.)
50
FORCEDVIBRATION.
[259.
259.]
DOTH ENDS
OPEN.
51
th variation of pressure
(proportion~ to vanishcs
~inr,that
yy +A =(), tj~t j~ ;(di.stm.b:tncMn,t thc ohi.s bc cqu~l un.!
if
"i
phases. U.i)css this condition bc .s~Is~d, t)~
~o.s~c
expression bccomcs infini(,o, wiicu
2/ = (2~ + 1)
At a point disant
from tlie middte of tho tube t!)o
express) on for is
~t
P~arc
w.H ~e
onc
duc
end
of an
to an
cxternal
i.s expre~d
Ly
un]i,uitcd
tube
thcre
Le a variation
of
a train
of progrc.s.sivc
source
wavu.s
inward.s
pror.~atc<)
from that
end.
Thns, if th Jcn~th
t)ic tube Mcasnrcd
froiu t!,e
t).e velocityopun end bo
Potential
~=co.s(~).
~"J'
corrcsponding
tu
of
(Phil.
co.np.nn~ the ~cn.itic.
of .ourcc.s of .onnd of the ,~nu
of tho
iH
th .~u,.ce. to bc
v
the
nntil
-'1"
"ibr.t.0..
't~
"?'
~
'
"tric
c.su)c
a(.c 'of
~y'"R
point of J""cti.n tho dist.n.h.
this al,
tho
test it nppcars tL.t
""M
nsbmnptton is 110tthooretic)t])y cornet.
I.itcb
hitcli.
E~
43
l-'ORCHD VIBRATION
52
0F
PISTON.
[25~.
~)=cos7~at~=0;sut)):)t,ii'thcc!tusRoftLL'di.stm1j:))]cc\YitJtiH
thc tubu bc t))c p:ts.s:(~(jof :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.cuthfLt.th di~tneto' of thu
tnL(iIs.s))ppos(;(t to bI)tfihitc)ysm~niucu)ttu:t.riHunwit))t)tL:
k'))gt))of:Lwavc.
Lct us ncxt suppose thnt thc sonrec of t))Cmotion is within thc
tnl)uitsc)f,'h)u ~(.-xinnpteto thc inexorabletnotiunof;). piston
~t the origin'. Tho constants in (~) ~5.'):).)'cto hc detenninctl
thc COI1<
couditions
ttIons tt!ia.tw!)uu
lat Wlen ~,c=
cos ?lt (s~y),
1.)ytlie
by
,t;=0,0, ~'=cos?)<
th:Lt,
.111(l and th:lt,
.IJ
il:lJ
T)ms ~=-t:m~,
-hcn ~=~, ~=0.
~7~=1, :nid thc ext))'<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 odd mu)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)tst]iccncr~y frum accumulatmg beyond a ccrt~m puint-;
but no account [-:ui Le tiL~oi of <)ns so !on~ as UtC opoi end is
trca.tud ri~orousiyasfLioop.
Wcshidt
rc'sutne t))u question n)'
rsonance at'Lct-wc hve 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 satisfactoj-ijy.
j!i )i)<(-mnnnc')- if th point ~'== bc
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))Int)tis e:Lsc~ecor()it)gto(~)thc
tiotisot't.hc'.jucstiun.
motion
Luc()H)C.sin()nit.c,w!ncIt)t)c:)n.stt)atinthc!d)S(;'ucoofdis.sipativc
furcus thu vibmtiuti wou)d Incrc:~o witituut limit.
'TitMcpruUc)tLsarucotMiJct'cdbyl'ui.ssu)),.VJw.<)).<t.n.t'.M)j.
2GO.J
KUNDT'S
EXPERIMENTS.
5:3
54
KUNDT'S EXPERLMENTS.
FSGO.
2GOJ
KUNDT'SEXPERIMENTS.
55
5G
EXPERIMENTS
0F
SAVART AND
KUNm.
[2G1.
midwvy
'tw'thcnrst.a)'dth~s(o'~rd(')-d~fth~p:p.
ru thc case of th <.pcn
or~an-pipo both end.s are )nops, and
thL-rc .nnst bc at luast cnc interne n~.dc. Tj.c
wave-fcr~~H. of tho
th icngthof the pipe, ~uch is dividcd
r'nc,pattonc~twice
into two sniular
parts by a noduI)i thu iniddic. Fnj!n tt.i.s wc sec
<~ foundatiun ut' thu
ordi.~uy r.ttc t)uit Lhu pitch of an open pipe
is
For rasons
to 'csatnc~th~ufa.pp~[pip.d')Laifit.s]o,~th.
bc ~norcf.,))y
in a sub.s.[Uunt
e.xpi.i.d
co.n.ectc.d
ch~pt.
with onr prc.sL.,it i.npc.rfL.cLfreinent
uf t),c opcn cm), thc ru)c is
.y
appruxinmtdy eon-.ct. Th opc.n pipe, ditrurin~ in th:.s ruirmn U,c stuppud pipe, i.s
~t
cnp.)c of soundinq thu whu]c .sc.rics
ut ton~h.nnin~
thc )~r.jiunicsc~fuuuded
upun 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).u frequcncy of thc vibration in a
pipe Is proportionrd
to tlte vdoc.ty of
propagation of .sound in thc ga.s with which tl.c
f")"~ is r.iicd, thc compari.son of thc pitd.cs of thc nutes
ohtaincd
thosarno pipe in
dinbruutgasc.s i.s auobviou.sn.ethodof
'n
dGtenn.nn., thu vclucity r~fpn.pa~ati.n, in c~scs whcrc th.
impos~L'lLy of ..hta.uiDg a .snHicicntiy long co)nrnn of thc
~as prccludcs
tho
<,ithc dn-cct moD.od. In this
appiicntion C'fdadnl ~it], his
"sna ~.sc
H.o way. T).c suhjcct ~.s rc.snnK.d at
iatcr
i~
d.~c by.s~city
D~on~~ and by Werthein~, w)~ obtainud
fair)y satisfactoryrcsnit.s.
2<'2. Thc condition of tlle air in the
inicrior of an or~n-pinc
,,Sa..rt~ho
I~dIntJ
-t~~
~i
.strctd.cd m.n.bmncon
~Uch a htHo sand .vas
1"~
~ttcrud.
In ~o nc-ighbou.i.uud u)anode thesandrc.maincd
und.sturb.d, but, a.s a fo.p wa.sapproac!.cd, It danccd .vi~
~~biy
.c
and muru v,go..r. But
by far th. n..st striki,~ funn of tin.
:.s that
invcntcd hy K.ini~. In Dus nK-th~ thc v:bra~
~r~.n.nt
"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~tric
capsule.
i..L~c~
1.xr,c.p, 11a.
t,
'-f'~<.<~(~tw.ii.'l.]~,t.xx)l.p..t;(f.
'O.f/fC/ftM., t. xxtY.),);
~j
~C7~
2G2.]
CURVED riPE.
t)~ih"I:)g:nr
:)oN.
A:ifhc~i~nbr;inc~ih)'itt.i-c:idcri))gthc
57
shcwing that
dpends upon .r in tlie samc wnyas if t))0 pipe
wo'c strai~ht.. By means of uquation (1) t.))e vibrations of itir hi
~S
8
DRANCIIEDPIPES.
f3G3.
~of'n'h.\nsa).pn~tn~.tonninot).o
refluer
nn~~frncteJwaYcs
thc junc~-n of ~o tKL.s uf MC'Juns 7.
+ r, and
rc.spc.ctivcfy, arc b'iv~ Ly
2G4.]
It appcars tha.t/, ~nl/
tion. if
DRANCHEDriPES.
5!)
~0
BRA~CHEDPIPES.
L
r-?(;
~ntnuwundcr
considration v.asinvcutcdhy
Rc~dK~andh~s
~'uun cmp)..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..s
n~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;
diverti
~i..)K.r.;yi.s
fro.,)'
""cp!itn.)pj[,car.si)tanut)KT.
I"t!h.})n,st.t(.asuthcp.).sit!vu
~vcin~
n't.).)-is
c.m~ys(..H~ywithi<.
nu wavea)u..rr~'
O~rc. ~hvo],s.s.),)u
alternative.
-it!r.ya(.cnnm~t.s
'n Dbranches, or dscit passes
in't.),c f-onant-aa
La<ka)<t
In.~urtos~
n~at.vewav.
what r<.uiy]~pj~.s,
lut u.s trace
t))e pro~ress of thu wavt'.s rcfkcttt back at 7'
Thse wav(.s arc-~qua] !na~tittxic
a)u) .start fn.tn 7;:n
oppnsitc p).a.s~;i.)<
pa.s.sa~ihjin7;tu~<)~~
~rav..]
.)i.sta.. ti.anthuoth~-Ly
!atLT
au (,<)
and
,uu)tip)uof'
thc.rdurc ou arriva! at
bo iuc.anp!u(u ac.-ur.i'ancp
t).y~ij)
whirh
Ut.(h-rth(.sc<.irc)..n.stanc~t)~.ycun.b.ncInt~.si.)L;Icw:u'(.
tr.~(.su~ati~.)y
a]o~.t,an.itl.c.rc
i.snn rcf)ccti.,n. W).~t),c
ne~atn-c ~avc ruac)K..str,c c.nd of thc tu),G J, or i.s .,t.hurw..sc
<!ist.trbc.d ni its euur.se, t).c ~hoie or a
part may !~crdJcctud, and then
tl.c procs .s rupoatcd. But h.,wcvcr <t.c~
this ~ay haj<pcn thcrc
will bc i)o wavu
a)u,~ F, un!c..s.sl,y ac.uunda.tam in oon.sc.qucnceof
.'L
ur pcri.~s, thc. vibration ia thc
hranehc.s bccon.G so
c.an~)cn
g-rL-t that a .sma)I fraction ofit eau no tonner bc
u~k-ctcd.
Fig.CG.
Or wc n.un
2G4.J
BRANCHED
PIPES.
Gl
L
57.
VARIABLE
SECTION.
f2G5.
<~
section ntat x, ~?.
rcckoned f.-om thc
G()ui)ibnum condition thcn
reprosents the total vclocity of tlle currcnt, .nd .Yrcprc.scnts
actu~}.ty
.f.
thu kiucuc
~i,~t!~
of
t)ic
motion
Avithinthc tube is
enorgy
cxprcs.sc-dhv
Rr~ntir\ti
2G5.]
VARIABLE
SECTION.
63
G-~
VARIABLE
DENSITY.
F 2 GG.
CHAPTER
XIII.
~'hure
AEMAL
VIBRATIONS
Lf2G7.
Taking three edgcs which meet as axes of rcctungular co-ordi~ates, .~d supposing th~b tho lengths of the
edges are respeetivelv
wo
a, p, 7,
kuow ( 255) t)):~
whcre a..sbcforc
7'are intcgcrs.
T!~
ail particule~ncml soh.tinn, cl.tained by compounding
267.]
INA RECTANGULAR
CIIAMBER.
67
NOTES OF NARROW
PASSAGES.
[~f.
f2G7.
268.]
RECTA.NGULA.R TUBE.
G9
70
MMCTANUL'LAR TUBE.
)-~U.
FSCS.
2 G9.]
Tl
72
[2G9.
270.]
]
REFRACTFO~
OP PLANEWAVES.
7:3
thcnndisturhcd
GREEN'S
1
INVESTIGATION
'1
Fs~Q.V.
respect:vc!y
Th
coemc.cnt of < is
ncccs.sariy tlic same in ait thrcc waves
on ~ccountof tlie
periodicity, and ti~ecoemcicut of y nu.st be ti.c
samc, .~ncct).c tracus of a)l th waves on thc
p!.nc ,f section
must n.ovc togctl~
With regard to ti.c coefHcicntof if
appc~ by substitution in thc diHcrentud
that It.s si~n
quations
in p~Ing i-ro~ thc lucidcnt to
th rcncctcd wavc'. In
~h~ged
fact
Now
&- V(..+ ~) ,s th sine of tlie
angle Included between the
axis of x and tlie norn~! tu thc
plane of t]~ w~vcs-in optic.1
t)~ sine of the
an~u,
a~]c of incidence,~d & ~(. "+ is in
T7~
of'
1~
angles
be c.I!cd
(~ asserts th.t sin~: sin~ is
cqual to the constant rat.o
.cU-J.ncwn law of sincs. TI.c )~ of re= ~-the ~!cw
f.act~n .nd raction
simply from tlie fact that the vc]oof
city propag~n normal to tlie wave-fronts is
constant in cacli
n~dunn that
to say, indcpcndcnt of the
~c~
of thc wavet.ken
in
front,
conucetiou with thc equ.! velocities of tlie
traces of
aH thc waves on the
phtnc of sparation (
sin = F sin )
It renoms to satisfy th
boundary conditions (7) and (8).'
Thse mvo
(auj
bc rca!, wc
'v~
270. jJ
OF BEFLECTION
AND REFRACTION.
75
FRESNEL'SEXPRESSIONS.
[270.
comcidmg with that givcn by Fre.snc! for light polarized pcrpcn'hcufarly to t].e plane of incidence. In nccordimcowitli Brcw.stcr's
huv 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
= 0,
370.]
AND MOISTURE.
77
~=
Suppose, for cxamp]c, that aftor p(.rppndicu)ar incidence rc~'cdott takcs place at a surface
scp~-atin~ air au() ]iydro~ou. Wu
iiavu
TYNDALL'S
EXPERIMENTS.
r'270.
from whieh by
discarr!ing th in~ginary parts, wc ohtain
'yo~/)(f,3rddition,p.282.
270.]
TOTALREELECTION.
79
shcwing t!mt ti~c distm-bance does not penetrate into the second
mcdtum more thfui a. few
wa.ve-Ienn'ths.
Thc difFercnccof phase bctweeli the
rcHc-ctcdand th inddcnt
waves is 2e, wlicrc
L~W
0F ENERGY
VERIFIED.
f~O.
-~n
u.e .n.rgy cun~~ou, ~d
agres with the rcsult of nu.itipfy.ng togethcr tbu two bonudary quations
(13),
WJien tho vcloeity of
propagation is grever ia th lowort!~u
th uppcr mdium, aud the
angle of incidence excecd. thc
critical ~g!c no enorgy i.s tr~mitteJ
into the second inediu.n.
othor words thc reficctiou is total.
Tlie method of tho present
invcsti~tiou is substantia))y ~c
a.ne as th.t~pjoyed
by Grccn :n p.per on the ReHectJand
Icract.onof Sound
T). case ofpcrpcndicu]. incidence
~.t u.vc.st.g.ted byPoi.s.~ who
cbtained
(3) and (2.t). ~).eh I~d i.wcvcr bc.n fonnui~corrc.sp.n,);n.
airc.dy givcu tj Y.u
tho rcf)cct.cn of
Ligi.t. lu a sub~~cnt ~oi/Poi~
c.n.s.dcrcdt!.e gnera! c.sc
ofobtiquc incident H.nitinghimscif
.owcvc, to g.cous n..di. for ~ich
Boyie-s law hoids~od,
d
a
.y
.cryccmpi.c.tc. ana]ysis an-ived at a rcsult
cqui~icnt to
Hc a!so vor.hod th~t t),c
-').
nergies of the rcftected aud ro~-acted wavcs make up that of Htc
hicidunt wavo.
271. If
cxtcnded do~i'to]y
v~d. w,th comp],.to
.nlfon.ity in its ~ch.
do
parties
tr.ns.n, ted wave is prop.g~d onw.rd.
'B~jf
at
eonti,I)y.
c'gc i~ thc co.sihi)ity,
o both p.r of th w.vo wHIbc
density
throwll back, .nd ou .riv~t
tlie b~
~.=0; will hc divid.) iuto t.o parts, o e
11~ ~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
'c.
~"th~
is
tlie .~hod
/r''i"
In
u.s..diy ad.pt.d
()p,ics for thc c~poud
'i~
;?
.s~
-s
huu~tiy c.p)anK.d but it ducs not
appc.r to hve any ad~nt..m
.a
~cr. straightf~rd
auaJy.si.s. r. t.f.c f.Uo~. ri
.Ld
cu~nc ou~dvc. te thu a.hcre
th J
~~n
."cdu.,n is .u.njar ill its
thu
p,
~<
= ~rM;.
rlr l'In,tilrrt,
<~
~r'~i"
7'fu<.~))).~
]~jg
t. Jt. p. iJOg
~7,
1. X'l',
;i17,
1,-j;Jl,
IQjf)
271.]
PLATE
OF FINIT
TIIICKNESS.
g).
(;
S2
REELECTIONFROM A PLATE
s).cwingt)~L cxccptf.thc
.hcr~i.n r,f p].e,
!ncd)U)nm~ht as wut! ijavc Leen unifonn.
r-27l
If bc small, wc h~-c
apprcxHnatcfy for L),crcf)ccto.]
wavc
~-mu]aapp)yu.g~hcnthop!at.cist)uuin
eom~n.son ~ith
tlie w~c-!cngt).
cos~. it appears t)~t for a givc.
Sinco
=~
ang!c uf incidence th a)np!itud~ varies
or as
im-c.-sclyas
Jn any case t!.c rcHection
vanishe.s,if cot~</ =
t)~t is, if
~bc.n~n
Let
us
nc.v
suppt
t!.it
th
.ecuud
mdium
i.s
ii.eu~prcssibic,
.so
OF FIXITE
371.]
t}):Lt
oui'
cxprcssKtM
TlifCKNESS.
83
bceomc.s
a-
0j
81
~0 LOSS 0F ENEUGY.
H'S.X
transl11ittcd(Lecotilit
f~
encrgies of
flll"tllc wholo
eller~y of tlle incident
front
aro c(lual fur ail threc
=~F"
it
~h" (12), (B),
tiulls
cil'
waiL:-
1~ destroyed
I)y ail)'
IlIlJnber
of
i,oflectioll,4
S~
tiull ulwa~~s
iu HllOthcr,
l'l.:al'l)(!al'ing
011 aCC01llltof tlm
;~rc.,tt dit}'ercl1ccuf' (1C11SlLICS
l'efluction is
liclnitll11attl!r, ~oululs 1)1'o(IticL-(l
iu ;lir arc )lot
cvsily coml11l1lliI:WUI)(ls,
il>1Iudel'water,
IYIIUSI:
U1'1bj11
wit.c.c<. );
(liflictilt,yiu air. ~L~"iofwooL),ora.
mctaJiic
distances with very littlo loss.
g
CHAPTER XIV.
GENERALEQUATIONS.
273. 1~ conncetion with 1)~0 gnrt probicm of aurial
vibrations in thrcc dImcusiujiHone of thc first
questions, whic]~
natu!U)y of~rs itsdf, is tit dctern)i;iaLio)iof thc motion in an
uniimitud atmosphre consquent upon
arbitt-ary initia.1 disturbances. It will be assumcd t))at thc disturbancc is
small, so
that thc ordinat-y~pproxiniatuquations arc
applicable, aud furt'hcr
that the initial vu!oclticsare snch as cnn bc dcrived from a
vclocity])otcntial, or ( 240) that tllere is no CM-c/
If th Jatter condition bc violntell,the probtcMiis onc ofvortex
motion, on whieh
wc do not enter. \Ve s]iaH idso
suppose in the m-st place that no
cxtcrna! f..rccs act upon the uuid, so t)iat tlie
motion to bc
i.s
duc
invcstigatcd
soldy to a disturbanco actuaDy cxistin~ at
a titnc (<=0), prcviou.s to which wc do jiot
push our inrp~-ius
Thc mct.hod that wcs!.a!l c.npjoy is not
very dinTo-entfromthat
of Poisson hy whom thc proHon was first
succcssfn!!yattackcd.
If M.,
bc tlie initial velocities at the point a-,
z, and 80
by which
ditrereutlal
t!.G ituLi~l
coefficient
values
with
of thc
volocity-potcntial
to tunG
respect
and
arc
of
its
Jetermined.
1 Sur
l'int~~tin..
quc.]qncs <juation~ lindairos aux di~rcnccs
pM-ti~M
et p,u'hcul~remcnt
do l'quation ~n.rato
du luouvcmeut de. fluides
6iaBtiq~
~~i. <!t; <Y)~ft<x~t. m. p. 121. 1820.
8G
ARBITRARY
INITfAL
DISTURBANCE.
.jiujmjM~tjjt,.
f273.
)Z/,),
Whcn is ]<nown,
it.s dcrivativcs~ivc
ZD t)tc ccrnponcnt;vetocitics at.
:nyponit.
Th~ symhuHca)sohnion of
(:3),,my hc ~-ritten
~hcrc~and
which
thc question of the
Into~rctation of od.) po~.s
in
equ~on
Hlly"
syllibulic
\lol]y evcl.
In the
was a faction of .r
c~y, .vo s.w ( 245)
c.~hc.-c
t!.t its ~uc
fur .nyp.int..t
on thc nit
tin.c~.pe,.dcd
valllcs .f
at
thc p.i.ts ..h~
and
cc-n~
we~
at
-I.
+~, and .s w.Iiy i.
~J
a)I uthcr points, In thc
pru.sentca.s. Lho.si.np)~ supposition
is
point 0 clepenclson
t..n
acs of nu.t at points .situatcd on f)~ ,f,~
of t~
spf.crc .-I.o.scc.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'
be
writtell
lay
273.'J
1AR13ITRAP.Y INITIAL
DISTURBANCE.
87
+ +
VERIFICATION
0F'SOLUTION.
~3
whichi.sPoissoii'sresuIt'.
Ncw~~
satisHcd.
i. t~
~J~
~S-~
mutiacAc l'h~~ik, 1, 517,
in
1876.
~rchh~
273.]
LIMITED
INITIAL
DISTURBANCE.
89
of which the first term boeomcsin the limit 7~(0). Whcn < = 0,
wavc
Dn-oughout
thc
who)c
of
f'2~.
Its
subs~uent
course
but
as Prof.Stores bas rcniarked, such
a conclusion wouhtbe erron'eous'
For vah.es of th tune )ess than
r, -a t].c poientia! at is ~roit then becoiacs ngative
(~ being positive), aud continues nc.~at.ve unt.I ttv.uu.shesagain wheu <=?-,
after wl.ie)i it ajw~ys
ren~n.s equ.'d to xero. W)u!c is
t!.c mdian at
is in a .statoof cmKk.nsatIou,but di.nini.shiug,
a~ incroases a~u.i to .en, th~
statc. of t]ic med.um at is onc of nu-cfactioa. Thc
wavc prom
gatcd .-utwants cnnsi.st.sthurcforc of two parts at )cast, of ~-hich
thc first is cundc-nscdand thc. !ast nu-efi~.
~~),atcvcr ,nay hc t).c
charactcr of thc. ~mai di.sturhanec.wit)un
t),c <i,d y;
of6
tlt anyextc.ja) poillt (J is t,),c.sa.nu .LSt)h.
Initit Ya]uc,an<[therciurc, sutcu a~=t)ic n.c.aucon~n.sati.)).
thu pas.sa.reof
dun)~thc wavc.,d~cndiu~ on U.u
is ,ero. Undcr"'thc
int~r~
hcad of sp)~.nca! wavcs wc shali h~vu occasion tu
rcturu to this
suhjcet ( 27!)).
Thc gnral solution cmbodied in
(8) 273 must of course
embracc the part.cuiar c~su of p)anc ~vc.s, but a few
words on
this application may not bc supurfiuous, for it
nn~ht appear at
first si~ht that the cUcct at a ~iveu point ofn
di.sturbancc i)iiti:U)y
connnL.dto a sficc of thc tncdiunt oyclosed between two
paraHet
planes woutd not pa.s.soit- in any tinitc titnc, as wc know it ourrj.t
to do. Let us suppose for
is zero throug-In~t
shnpHcity t!.at
and U.at wit)mi thc. slice in
<p~atiou thu initial value -A is
constant. Fron. the thcory of p)anu waves we ]n.ow that
at any
~rbitrary point the di.sturb.-inccwi)) fioaDycca.se aftcr thc
iapse of
a time .such that <!<i.s c<{u:dtu thc distance
(~ of thc point
undcr considration froni th furtller
boundary of the initially
disturbcd rgion ~diiie on thc oti.er hand, sincc thc
sphcrc of
radius ~< continues to eut the rgion, it woutd
appear D-omtbc
formuk
t))at thc di.sturbancc continues. It is truc indced
gnera)
remains tillite, but this i.s nut incon.sistentwith rest.
that
It
will in fact appear on cxatnination that t]~c me:m value
of
t)~
mdius
of
t!.c
multipiicd by
sphre is th .same whatever may
bc t)te position and sixe of tl.e spbcre,
provided on]y tjiat it
eut con.plcte)y through th
n.gion of original disturbance. If
a0f/, cpis thus constant v-ith respect both to
space and timo,
and accordingjy tbe jnedium is at rc.st.
275.]
TWO DIMENSIONS.
91
where thc intgration 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 Jaycr
of gns
bctwccn two piu'a.lld -phuics, or to th:it of fm unUoutcd stretchc~
mutnbmnc, wtuctt dpends upon thc s:unc fundftmcuta.t quation.
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 disturhancc. Let us suppose tliat throughout the spa.cc
(w)tich
will uttirnatc~y be !i)ade to vanish), a, uniform disturba.ucc
T!)0 rcsultiu~vatuc
cqua) tu (~')~, is communicn.tcd at tmic
of 6 at timc t is
SOURCES0F SOUND.
j~~Jt'J.
f-S/C.
2 7 G.]
IIARMONIC
TYPE.
through fLsma)t space Including thc point at which <Pis ultinm.tL'iyconcuutrat.cd,-c Und in th limit
VERIFICATION0F SOLUTION,
f'277
thmnghwtnchthcy~
t!.c
s~n,
~c-]L.n~e-n,aybcro.nov~fn,,n~dcr
and
at n. sufhcicnt.
distance
+ ~=0.
with
buvuty.s)na])i,.co.np!U-i.son
ihuint~d
wc
may
<),u
U.c.samcph.and
ta~c
-satires
~+~=~
~j,
course is to cxprc.s.s
in p.]ar co-ordi.i~es
.s.m~st
thc c!cmcnt itsuffas
pulc, whcn it appels tha.t
referrcd to
278.]
SURFACE
DISTRIBUTIONS.
95
la
i.sthcsn-niou
ihctwos[<]cscf~butthcrc
Thcvatocff~
(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.
L-,
[278.
DOUBLE SHEETS.
_u-.
"E,
m. 1
'1..
~7
~t~~T
source
concentrated in a point
close h~u
a corresponding
e norm~ ~n
'J.tJ3o~ui.Hccof th plane itself,
Th operation of the
plane is to doubte th effective pressures
which oppose th
expansion and contraction at th
.so~.ce
'ed~
double
total energy
and since this energy
is diffused r'~
only
space,
~y~
~~(~
~1~
amplitude,
or potential (~
2.I~),
Wc will now
.upposc that instead
the prcscribcd
of~=0,
condition at tho infinite
pL.ne is th.t ~=0. In this case the
fictitious distribution
on the second side of
of
the plane
rnust s If fT"" of
on
first side, so
the sum of
values at two
corresponding points is always zero. Tins .ocurcs
that on th plane
ofsymmctry itself shall vanish throughont.
Lot us next suppose th.t there
arc two parallel
surfaces
~h~r
small ~~1
and ~'at th
o~n~
second
is equal
opposite to the value
of
on the first. In
thcre is by (2) finite
crossing
change
in the value of
to thc amount of
but in
d7&
'e th
crossing
same finite change occurs in the reverse
direction. When
is
reduced without Iimit, and
replaced by
will
th
dit
same on the two sides of the double
shcet, but there will be
diseontinulty in th value of < to thc amount
At tho
of
same time (1) becomes
po~or~
""T"' sign
Positive on the one side and
native
R.II.
~"rfacc-potentiaL
on th other, due to the
7
98
SPIIERICALWAVES.
[278.
279.]
CONTINUITY
THROUGH POLE.
99
If a divergent distance
bc conflucd to a
sphencal she
w.t~r.:
~iLhcur. ~L
t!
udthur condens~ion nor
veioctt~ tho chamctcr of the w~e Mlinutcd by remarkable relation, first poiuted out hy Stokcs'.l, From
quations (4) wc have
shewing th~, th v.~ue of/(~)
I, the s.-unc, viz. zero, both
in.s.deand outside tbc she]] to which t])e w~ve is
IImited. Henec
if
M
bo radii less aud grcater tliau tlie
by (~
and
cxtremo
radn f tho shel!,
100
I~ITEAL CIRCUMSTANCES.
[279.
wo obttun
280 J
ENERGY
EMITTED
101
~02
SPEAKING TRUMPET.
[280.
281.]
103
~04
[281.
whcre A and
arc coc~eicnts rcprcsenting the magnitudes of
thc sources, (which without luss of genorality may bc supposcd to
hve th same sign), and N rcpreseuts tlie retardation. (considered
as a distance) of the second source reiatively to th nrst. The two
trams of spherical waves are in agreemcnt at any point P, if
whcrc 7n is an intcger, that is, if P lie on any
~+ 'x ~'t=
one of a systeni of hypcrboloids of rvolution 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
nctual magnitudes permit. Th neutratization is
complete, if
7\ ?',= ~1 ~C,and then th density a.t7~continues pGt-manentIy
unch~nged. Th intersections of this sphre with thc system of
hypcrboloids will thus mark out in most cases sevcml circlcs of
absoiate silence. If the distance C\Oj,between th sources be grt
m conparison with thc Jengtil ofa wave,and th sourcestljcmselves
bc not very unequal in powcr, it will bc possible to
dpart from
t)te sphre ?'j :?'~=~1 Z?for a distance of several
wave-Icnn'ths
without appreciably disturbing thecquatityof intcnsities, and thus
to ohtain over finite surfaces several a!ternations of sound and of
almost complte silence.
There is sone diniculty in aetun.I]yrea.!isiuga satisfactoryInterfrence of two indcpendent sounds. Unicss the unison 'be extraordman]y perfuct, tlie silences are only momcntary and arc
co!)sequcnt)ydinicult to appreciatc. It is thcrefore bcst to employ
sources whicli are mechanicaHyconnected in such a
way tijat th
relative phases of thc sounds issuin~ from them cannot
vary. The
situp!cst plan is to rcpcat thc first sound by renection from a Hat
W!t!I( 2G9, 278), but th 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.]
POINTS0F SILENCE.
105
1877.
EXPERIMENTALMETIIODS.
f282.
283.]
souND snADows.
107
108
nUYGHENS'
ZONES.
[283.
283.]
HUYGHENS'
ZONES.
109
110
L.
[283.
S84.]
DtVERGINGWAVES.
112
VARIATION
OF INTENSITY.
[284.
'n*~ 7r\7i~
'c"+c'
385.]
REELECTIONFROM CURVEDSURFACES.
1]3
114
FERMAT'S
PRINCIPLE.
[28G.
287.]
WnrsPERINO
GALLERIES.
115
116
WIIISPERINGGALLERIES.
[287.
S88.J
RESONANCE
IN BUILDINGS,
117
J
carpcts or i.hangi.igs to
absorb the sound. In somc CMCsth
prsence of an audience is found surncient to produco Die d~ircd
e<!ect. Jn t))o absence of a)l
dcadening matcriat H)e prolongation
of sound may bc very eonsi.tcraldc, of' wfticii
perhaps thc most
striking ex:L)nptcis that ~urdc.t by th Baptistcty at Pis~ ~hero
th nuttj.s of thc commoti chord
sung consccutivdy !nay bc Lcar<t
ringing on togcthet- for many 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.dfor pnbiic speakn'g, w))ic)i may bc acconpHshcd by suit~hty piaeud objiquo
surfaces. I~ tbi.s way tho munbcr of rcHcctionsin a
given tinie is
iucreasud, :utd thc unduc prutongation ofsouud is checked.
288. Aimost tho on!y instance of
acousticalrfraction, which
ha.s apractica! inicrest, is t)ic dviation of soaorous
rays from a.
rcctihnuar course duc to bctcrogenuity uf thc
atlnosphorc. Th
van:Ltionof prcssm-c at diffurent levcl.s(tocs uut of itsctf
rise
givc
to rufraction, since thu
vu]ocityofsou.td is indcpcndcnt ofdo.sity;
but, as was first pointcd eut by Prof. Osbornc.
Rcynoids', thc case
JHdtfTcrcnt witb tite variations of
temprature whicb arc usuaHy
to bc met with. Th temprature of
is determined
t)~cat~nosp)~ere
prineipaHy by the condensation or rarfaction, which any portion
of air must undcrgo in its
passage frum onc )cve) to anotbcr, and
its tiormal state is t.ne
of'convcctivecquDibrium' rathcr thau of
uniformity. According to this view th rchtio.1 betwecn pressure
and dcnsity is ttuit exprcsscd in
(U) 240, and th velocity of Sound
is given by
118
ATMosniERic nnr~CTioN.
['288.
v'ho'e
is th tonpcratnrc
a.<-t))e sm'fncc.
288.]
C'ONVECTIVE EQUILIDRIUM.
H 9
120
PATII OF A RAY.
[288.
~'<
ma?,
22.
289.]
REFRACTIONDY WIND.
121
t22
TOTAL REELECTION
I!Y -\vrND.
[:289.
If wcsuppose
that !7'=(),thc ~reatcst~hmssib)e
valueof
~'i.q
Atastmt.utnwhcrc
~7 hasthi.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
ViLtuccannotbc
agr~~T
])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-,
aH ray.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
Ovu)'t.)t(;
cnmtotfiLiitohcvt-ryintjxn'tiU)).
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.
Fur the
lotcrvc!)))~
pr()(h]c(.i()n'))'t.!ic.s(.;))))).)on)('nait, i.suot ticccssaryt.hatthL'rcbc
:d).s<n('cfwHK) n<,).)h.S(')u'cuufs<))tn<t)n(,:).sa))pc:n-s:Lt,(~)cc
fr~m tJ.c ff))m of(~), mc-rdy t!)at
\'L')ucitiu.s
U
t.hc(/<ccut'
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
289 J
wMchisofthcsnmc
cour.st.'oi'a ravis
HEYNOLDS'OBSERVATIONS.
')23
formas
T)ic
(II) ofthcp)-HC('d!ngscct;io)i.
acc()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
!nayhc
n'vcrscd. lu timcasc ')fat)n.s]))tL'ri(;(!fr:n;t.it)jt,()ucto:)()i)ni)mtinno)'temprature
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'traction
isthtGtuwind.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.t tra\-L-tai<.ng t))is pat)), h) thutatt~r case th
Ycrtux ot'the cat~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)t
tht.; 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<~cwas
observer or t)tu bu))."
cxtt'n<)cdl.'yraisi)igcit))crt)t(!
"KJ(.ation w~ fouud
t.<;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
thc !K.)),.t.id :Lt:!()yard.sit.w;ts!ustwit)t
~oundnt~Oy~-dsfruHi
thc ))~td 3ic(jt inon t))c gro)md,fmdi(.si))H
i))tc))Hit.yw!)s)ost
wht-n standing- cr~t;~
:!()yard.s. At7()yi,rds,)~nnta))(ting
crcct.titc.suund
int~r~d.s,:u)d was otdyfiuntty
w:Ls)nstn.t)ong
hc'f)rduvcn<I~c));))nt
thcc'a.r
iti)Cf-an)c'c<)]ttmu(jt).sn~inw)~-n
wnsmi~<tOfL~tfn.rnt)h.~uud,tUtditr~chcditsiu!tiDtu))sity
ata)tc]t.-va.tiu)it'12iL'<jt."
Prof. Rcyno)()s t)n)s smns up th rcsults of his
experimcnts
1. "Wi)oi th(!rc is oowind, soond
procucdit)g' ovcr a rou~h
surface is niorc iutcusc above than hulow."
124
TYNDALL'S
OBSERVATIONS
2.
[290.
it
L
~~ccul
T~
~Lfl
'T
7~
who.so .nv.sL.~tions
~r~?'
n
ue
havo
.smg
~sphcre
latter
b,.en
'o.
PLcn.nncn. which
wl.ilc
~-rvcr..
cuu~))y e.tcnsivc
Ly ~ccu~t
fr.r. uncqua!
hc.tiug or
in~hi.
~ti.g
~"L~d.
bas
Ty,M
~~tric bc!I
P o
by
~b~n~
7
~~s
d' Ilml, altllOlIgh it lnust hu
densitit,s;
ndluitted tllltt the al turllat()lIfo!
cOllSl,ll'I'ill)IC
illlll
IIIO)'(!
th an
ean w'c:ll !Je
ahrllht
SlipplJSl'd to occur ill tllc; l'Ull ail',
uxct'pt }JI)l'hap8in
ot' the sulicl
grullnd, SO111C
of the
LULc\j)iai]at)on
~i-~My to
~i"
nj <)ucsttn
'rh us it was fcnmcl tlt.vt
the last of a sirun
lal,vceclon the
of brlulually
dillliuiHl1ng ilitunsity, wllUSC rlurutiuu
solllotilrles
mllch
ob.sc.rvcd
"v-].n
~OHH.s.
t)
l),).s
phu))u)nc)ton
was
aud
SlJ!oothllcss,"
cannot
11'h'`~1'clltlyIJUo,ttrihutc(1 tu any otlter cause tluln
tlmt asSI~IIC(1to
Tyn~H.
It is ti~ref-L
prob~
acoustic.1 opacity arc bot),
concorn d i ho
of
fub-sigl,vls,
l1?wiuo we slloulcl cert;lillly bu
tlisposud to attacll
suluc of '1`ymi<lll'sovll
oservatiolls mlluit of
explallation 1111011
~=~=~=
t]Jis
l'Jril. L'rmts,l~~l. S'uun~l,
8rc1c<litiuIJ,
C'h.YII,
~Z:?,~ltJ~
290.]]
ON FOG-SIGNALS.
125
~26
LAW
0F
DIVERGENCE
0F SOUND.
[29 1.
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
~itjt a<-irr)))ara))urh))-(.on
wa~.pK~ed
wavusot'
whidtpianu
Sound itnpit.~dir~Uy,
thu a).:d.~ot)s
n.ay hu cafotiah.dasi.i
probk.Utui'Lhc(Htrr.K-ti~)tpath.i~furn)udaLt).cfocu.s()f'acin~dar
object-~a.s.s. Tt.usi..tf.c~.sc
of'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-tHroctions 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 tho
n~uth :L..tu).nts to about h:df;L
At ns~n~vh~
wavc.-tf.n~th.
thc 'ttout!ttnayh.divid~[intotw<)
gn.atcrubiiquity
parts, of
whieh ti~ ncarcr .-ivcs an
L-t~ct cquai in magnitude,
n~r~atu
~.V.;)tr/n\<f..).)/)H.,fr.<.1;ir,]-(:
'A~7't;j.<.Yu).
!<!< p. :;j.
)n7t'
i:jl~,
SPEAKIJSTG TRUMPET.
~1-]
127
sn(-I.c;L)cu)atmt).s
AXhnx~h
;).s
p~c~di)~(iu)):)rcu.sd)t)
.,f
]')),K.)~
it
diftract.i.m,
:u)xiii:ny:t.ssnmptiun
..nwi.idt
.stn.-Ny:u.d~n..)~)jy
J'x'idd.t
up~)
<
ap.'rtum
a.s<-n.n
<hc
ns
ih~c
~ivi))~
tnust
rd~n~dtoin
..fthc
~u.di()~
nnt
h.-
~.r~.tt~n
~Hy:n~fmmdcd
!shy
t1)at
thc
).),)c:m.s
Thn.si))t))(!(-;t.s~i'i~v:LVLidi)-t)y
n~n~d
in thcphmcofthu
v..)<xity
mcn~sc-st'rotn
is)~.t.<uthst;mt,:Ls).ash.-cn.supp~s.],hnt
ti)C(~)ttr<jtuw:)n).st!t<'o)~hc.(.<))nmL;-inf!nih;
at
.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(~<)
~t)..ctud,nndt)tu
thc
vclcity-putu)itl:d
<jn tj)u
t)t~i~(;itsc)f.
i.s th~)
p..rftiy
hc~ttivu.siduut'tfic
sct'L~n(.<;=0)is
128
DIFFRACTION
TIIROUOn
SMALL APERTURE.
[392.
from the ngative to the positive Ride; or, sincc the cros-sing
involves simp)y a. citangc of .sign, to dtermine :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-pnsingt)tc two motions thus dL-Hncdsalisses a tho conditions ot'thc prob)cm, giving thc s:nno vc'tocityimd pressure on t))o
two sidcs oft))c !),po'turc,aud avanishilig norma.lvctocityovcr Uio
rcmaindur of thc serceu.
If T~cos(<~+e) dnote tho value of
fLS*and a~y
Whcn~fH)<I<;arck)-to\vt),t.hecomon t]tc positive sit)cuft)tc .sc'rccuis
mfi.ttcrwhicit must be
p i.sthc dcusityofthc
shuwin~
o that. ~7r
distribntcd over ~S'inf'rdRrto 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
HLLU'TXJ APERTUK~.
u'hbrc
1
r
~t.'t.l!
"n
:uu~
n.
future-
t)to
dcnotiog
~=-~J~'S',
L-
)~
.u;.}.f.~i~
p)g-ct)nLt
fur
:ut
totd
(ii.iLtKud.
]29
1 :J
<~fUttity
ir.
wiH
of
natter
oc
s)icw)t
axis
cHip.scufsotmnajnr
ff,!u.d
ccccntrit'itvc,
l'esult,
s1101IId
~Lsr(-su)h!s.juit.cdirfL-rcntfn,.nU)atw).ic]iwcHLou!.)aLt:,i,)
'hi:
is fjuite
tile
]las 011
[ro!l1 tIl1tt which
Z tt"']'y)"jtitosisth:(t,t))on(jr)n:dve]f)(-ity
i))thcnpL')'<u)-cha.st.))c
tlie
\duu prnpcr to t))L-pri)n:u-y w:tVL-.Li thaL c:Lsc
by (:!) 2~
jl
j
T))C(]:m-;t(.ti())]~sm)hd i.s~.snhj~.t.which
i~s nttr.K't.cd!)).).
!'tt)' !)t.~))ti(.n <.it,])(.rfrum
")nth~)j):)(i(:-i:uiso[-cxpfTi)nL'nt:d!.s<s.
t.ndt.rAi<))0)~htt)c~t)u)-:))c).:u-~(~uft).t-p!K.no.n(.n:Lisw~tI
:md thcr~'orc- no vcry
st'd,
<)).<r<(.ncs :)r<! to b(!
shrHh~
'j['<'ct..< tl)~ cxnct t]H~rcUcit)s<)htth.nora
f.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~))titc mh-ri~t- of'~ .si)u)']y-(~)))]<<t(.~]ch~~
sp:n'c ,S' c;))i''h(.
c.\).rcs.s~] as t,).~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.s
i'~uwixg
Hc)).t)t~)~spr<~r'.
Dy(:t~).'stitr<.n.))i,
if~
!t!dcn(.)t(-:)nytw'jft)j)<t~)).sf)f;<
't
77<f'f))')'fy,jr.),/7.<r/;t<')')<H~<');t))~<r''))Mf<f.f'H.'f<7'f~;)
j'.i. ].sf!f).
):. H.
r,'(!i..
H.) ,y,.
130
EXTENSION
0F
CREEN'S
TIIEOREM.
[293.
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~
snrfa va)ucs of-~ andot'
~l~
1n
the c;isc uf llm conmam potl.'lItial, uu wllch We l'ail back
lr,lrl:
-Int]icc.LM(;oi't))tje<)mnt()npotc'nt.i:tI,0)twl)ichw(;f!L]t
]
~1_- L.
IIELMIIOLTZ'S
THEOREM.
];~ 1
to a sp..co
cncl.s. hy a ri~i.) ~.m.Luy a,.)
c~j~tdy
any nu,nbcr of-dutac). r~i.) H.~ h.dics, and
-nng
If~
bu
vcloc.ty-potcnttafsdue to so.u-CL-.s
witJ.in <S'w~
]~.cd
t32
IIELMIIOLTZ'S
TIIEOREM.
r20-J.
itfo!!ow.stha.t
~,=<(-).),
which i.sthc symbuticat statcnu'nt ut' Hcimholtx's thcorcm.
<S'<'xtend to intinity, thc surface Intt'~rai sti)t
Iftbespa.cc
vanis)tcs, and thc rcsuit is thc samc Lut it is not ~cccssary to go
intodct!u!!)t;rc,asthisthL-oretnisinchnh:'dinthcvast)yn)on'
T!)o
gner:).) prin('ip]of'rcciprocityc'stab)ish(.'<)
inChaptnrV.
investigation tiicrc givcn s))cws titnt thc principe ronaitts truc in
t)ic prsence of (iis.sipativc ft'rcc.s, pruvidcd th~t, thcsc :u'isn from
resist.ancL's \nyi!)gas
thu fn'st puwcr ofthc vetucity, that thc
ftuid m;cdtiot bu Ihtinugcneons, nur t)tc t~'iglihonritu'' h'xhc.s rigi't
r 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 aftoh:tvl))g (/sc.)po~ to :L distance (j'ont ~t :md 7~, th sourct.'s undor
('ontooptfdio)].
Thc rcadcr must ca]'cfuHy rentonLcr titat in fhis thcort'm
introcqunl.sources of soundaru thnHcpruthtcctIhythL'pL'riodic
duction andft.h.strftctionofcfptat
<p):mt.itic.s of finit), or HonK'thin"'
who.sc cr'cct, is thc .s:uno, and th:d c(ptal .sonrocs do not ncccs.stu'ity
ti;ne.s. For instance, n.
uvu!vucqn:t.) amountfjoi'L'nc)'gyittcqu:d
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 f this t))corcm wc )nay takc thn
r'asf ot'a hcaring, or.spea]\ing, trtunput L'onsisting of a Ct~ni(.d tubt.
whusc cHicic'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 cne, or
a. source ofcqua) strengt]) situatcd at thc VL'rtcx is obsct'Yc'dat th
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
an intinitdy 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 us suppose that
is a. si)n])lc source, giving at a. point
thc potentia!
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 a
simple source at 7~
whosc mtcusity and p).a.sc arc t).c san.c as ti.ose of
t)ic sources at'
an.! J'; th
is
rc-sulting potcntial at
and at .r
+ A~
It thc .h.stance .Lr bc <)cnotud
and )~ support todmuni.s),
by
~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.su ink.n.sity i.s incrcascd v'it).o..t
""in.
suc), a manner t).~ thc
j.rudncb uf ti.c intcn.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 .nays.y ti.at t)ic vulocity of- thc fh.id
'Lt. iu .hrcctKm J~f' duc to u..It
7.' is numc.risimple source
caify cqua! to the potuntia) at duc to a unit
source ~t J
wf.osc ax.s is i.) t!.o dir~Iou .L.
Tiu.s t)icr,rcrn, l;c it observed'
Ls truc .n sp.tc of auy u]).sta<'tcsor rctiL.ctors
that may cxist in th
J'e~tdjnurhood oithu sourucs.
~iu.
ifJJ'aud
7.
dn'ectio))M.
surface
Tbo kinctic
cncrgy T of thc motion within a c!oscd
is cxpresscd by
e<ul.
etlilioll, p, ,105,
A.
.Su<
0" -S~
't""OnthcApj.)icationofthorri,)cip]cnfJ!MipMcitytnAc~stif.s"
Vu), xxv. p. 118, 1876, or r/.<
~~c~r/
J~. (o) 111.p. 3u'
3rJ
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 .secondn'prc.sunt.stbu wurk dune by Intcrn:d
source 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-of tbo motiox )-c.s~)]tingfrom ~ivun
tunut~ocss
:u'bitr.ny initit
cunthLiuHs. Sincc every ctcmojt of7~'is positive, tbcre can 1jc no
nx.ti.m witbin
if
bu zuro. Now, if tbcrc werc two motions
possibfc corruspondin~ to tbc sf~mciniti:d. conditions, tbdr diffcr(.'nccwuu)t[bc n, motion for w)neh tbc initi:d Y:ducof ~was
zcru
but by what itas jn.st bcen s:ud .sncba. motion aumot cxist.
')7<ii)<~f'tt<t')'.l/tff/j/(~p.3n.
OUAPTER XV.
DJRTHER
A1THCATION
0F
TUE GENERAL
EQUATIONS.
-o.
1~6
SHCO~DARY WAVES
!29(J.
L"t.s')'~nt.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;))onrsth<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,
296.] J
DUE TO VARIATION
0F MEDIUM.
137
mw)nuh
tho intention
cxtcn.Ls ovcr.t volume
co.npietdy indud.thor..g.nof<ti.stu.~Lr.ce.
Thc mt~!sm(M)
mayLc
t~u.sf..rmc~ withtt.c
aidot'Grcun'sthcorcni."
('aHmr. tlie tw.<
parts j-cspuc~V(j]y aud (), wc );;Lvc
wt.o
<IeuuLL.s
thc cosinc of thu nn~)c ),etwucn
~hcru
and
TLc
fmcardnncn.sn)n oi-thc i-~iou of di.stnrbaucc M
nL.-dcct.cdm
and
is nc.~ccted in cornpariso~witli ?..
c~~mrison with
If 2' bc th volume of t!.c
spacc throu~h whic)i AM, A<rarc
writo
.sc]).-i))jfc,wcmay
1~8
LAW 0F
DEPENDENCE
ON WAVE-LENCTII.
f29G.
o
m terms of ,vo hve from
(3), =
exprs
L'").s,)t the e~~ot.s~i,,)) foi, thc
prim~ry wavc.s bc
and (12) may Le;put Intu t)iu fonn
=
of
~d
=e'<+.~
of
296]
]
ROUNDS
ALTERED
INCIIARACTEH.139
~0
SECONDARY
SOURCES.
f2DG.
j
i
t
2~8.')J
DOrpLER'S
PR!NCirLE.
14t.t
</<<..t/)'~f/Jr~tt.<f)')~.
rnt-is,1RC.
DOPPLER'S
PRINCIPLE.
['39 g.
2M.']
B
g
g
S
B
g
initia
RECTANOULAR
condensation
'3
confincd
CHAJMDER.
to ttic ucig!.bour!)ood
143
of t!t point
thi.s mt~al
StncUy spc.ing,
bas !ia <tcf.).;tc va)uc- L..t if
wc ~s). for thu c.xpre.s.si.mof (hu f.n.ccd
vit~ions on)~ wu nmst
~t
t).c u)tc~tcdf.mr.ii.,natt).c
)~.cr ii.n.t.
,n.yb~cc.a
.y supposm~ t),u ii,trud.,ctiou of vury .sn~)I
<)i.ssip~ivc forces
Wct,h)t.suut:ti)i
As m.~ht !c bccn
prc~ictctLthc cxp.-cssionsb~eomoinfinitc
in
of a comci~~cc LeUvce.itho
pc.ria.tuf t)~ snurcc an.! one
c~e
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 onc
of its luop.s.
Th cffuct cf a
nu.p)icity
''ysumnnattonori)it(.raLi(jn.
U~LIMJTHD
H.i
Tt'UK.
t/
the
When sound iscxcitcdwithma.cylindne~pipc,
is by th forced
si)np)cst Mnd of excitation t.h~t we cun suppose
vibration of n piston. Jn ttu.scase thc w:u'(js f~'c ph~c irmn
th hcgimung. But it is Imp~rLnnt :dso to irtqnirc w1ta.bh~ppcn.s
over thc
\Yt)cn the source, inst(:'n.d ot'hcinL; uniiorndy'Hftuscd
ofit.it. Ifwc:~sumG(wh:tt,
section, is conc~ttr~tuditiotn.'point
sufficient distance
tn~) t]):itat:L
howcvoi-, Isnfjtu)n't;r\'<jdty
from thc source thu wn.vcs Lccomc plane, thc law oi' rccipt-uclty
inf'orxKLtion.
issuHitcicntto~uidcu.stothcdusirctt
300.
two
L(.'t~). bc asitnph' source in nn un~n~cd <uh(~/)',
points oi'tttcs~Tnc normal St.;ct.ioniu thHrc~ionot'phincwavcs.
7/~h~ tn thc .source J
thc potuntiaisut/~and
~.<- A~fC.
~nd accordin~y
arcthusHmc,
hythch~v of 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-dovci' 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,
the potentiel due to itt
:md th source sueh thatin
theopcn
woutdhc
~1
I!cncc,~sjn2.{.i,
t)~ .sourceisgivc'nhy
I~ERGYEMITTnn.
145
<)'cc!)p)-gy(~r)Rmitt~o)!C(7c/~t'(~
Thn)i;))-r<-)wpr
th tubc,Lho
gr~tcri.stiicono-~yissuinrffroni
?L~)\'L;)tsuut'cf;. It i.s
into-c.sting to compa.rc tjtc cfHcicncy of
a,sou)-cc;)Ltt))cstopp(-;(!en(tofn. cy)in<)ncn.ttubowiththatot'
:") 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 cne: frnni a sphre of )'a(]jus x'
301. '\Vc ))nvc now to examine hnw far it is truc! that vibrat)f)ns w)t.I)in a. cyimdrical tube bccomo
approxhnatdy p)anc at n.
suuicicnt ()ist:mce from tbcir source.
Taking <,hcnxis of~ pa)'a!)c)
to thu gcnurati))~ Jincs of th
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 fur
r/'
tlie cqnatton oftho ]not.inn is
.+
j'
If'
bt! I))()upct)dcnt, uf
it t-cprcscnt.s
\')))!-atKm.sw))o])~
li'tite potentiel bcHto.
tmnsvt.t-sGtnttiCfLxi.sofUiccytindur.
prr-pnrtion~dtde'it
ntnstsatisiy
!()
14 G
VIBRATIONS
IN UNLIMITED
TUBES.
[301.
in which
corrcspondin~ tn ~= 0, is constant.
DISCRIMINATION
0F CASES,
t~
~w un.). th L.ireu.n.sL.u~.s
suppt, it is C'vidc'I1tthatthp
.notjun ducs not b.co.nc infinie
so that a)! t).o c.ciicicnts
with
I'su,ncwi.atdim..rc,.troason
thc.~ncistn.cof.
u,
.sh. t).c
..s
can bc. no wavc in t),c
ngative dirccLiun. W~ ,nav
t.Jt'jrctut'eta.ko
e
111
+
.(1_),
(~~
a.. r.p,-ossi.n which rc.ducc.s Le its
n.st tc.n. wi.en i.s
suf~iu-Uy
Se~.
A\ c c.ndu.)c that. in ..]! ca.sus U.c ~vc..s
ukin.d.c.)y bccunK.
~c~
!o,
Jrctr~est cf tlie
?<<?-/ ~-(~~c~e ;<M).s-.
<~=~e')+~+~+,,
v.
inasmnch
a.
r/<r,
A.c.,~) vani.s)~.
It appcar.s
accr,n!mg)yt).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
[!
cxists. Any
normal
mcan
actuaUy
th
samc
vdocity
~iving
"orm~ motion of w~ich t.)'f-~-t~tivc ~n~ positive p.~ts arc equaL
produccs uitimateiy no cr'cct.
Wilcn thcrc is no restriction on thc eharactcr of tLe source,:ujd
are graver t1':m
whcn some ot' thc transverso natnrat vibrations
are positive and thon
the !tCtn!Uonc, sotne of thc vah'cs of
tcnns enter of th form
3~.j
UN A VmKATfNG
CIKCLJLAJ~ PLATE.
wherc o- is Ui<jnaLurat
dunsity, !u.d
~f)
Dus
must bc nu,]t:Iic.! hy
a.~d aftcrw~d.s
2~~
quan~ty
with
ntte~ate(t
respect to c bctwcun thc Ihuits 0 aad
But
it will bc convenieut ih'.sttu cn'cct
transfonuatioD Wc trn-c
whcrc
150
REACTION0F AIR
[303.
Thc re!K;ti"nof thc air nu thc (fisc may tho.s hc divulud into
two p:u'ts, nfwhict) H u-~t is proportional to tlie vclocityof tlie
dise, and thc sccun't to thc accolcr~tion. If dnote th disof the dise, so
wc hve ~= :t
=
pt!LCC<ne))t
that
=
and thercforc in thc cquation of motion ofthe dise, th raction of
thc air is rcprospnted by a frictional force ao-. 'n'-R".
tj"~ )
( 1
retarding thc motion, and by an accession to the mcrtia c~ual to
'7TO'
r~
r)\
~~(2~?).
~02.]
When
(~2()().
ON A VIBRATINGCIRCULAHPLATE.
151
z, winch is aceordingly
When
is grt, ,7,(2~) tends to v~nish, and then t))C
fnct.cnat tenu bccomes
This i-esu)t might
si.npiy a<r.7r~
hve becn expeeted; for w!ien
is very large, t))c wave motion
the ueighbonrhood of the dise becotnes
~pproxi.nate)yp)ane.
h~e then by (G)~nd
in ~hich
(8) 2~5, ~=~
is the
density (o-); so that th retardmg force is 7r7); = a<r.7r~
We h~vc now to considcr the term
rcprcsGnting !ni itration
ofmertia, and among otiter titings to prnvc t)~t this altemtton Is
an increase, or t)tat
(~ is positive. By direct Intgration of the
asccndiug sries (5) for ~(whieh Is aiways convergent)
152
RHACTKJX UF AHi
[~U~.
302.) ]
UN A YlBRA'nNC!
CIRCL'LAH
PLATE.
153
and we nii~ht.taku
fe"f~w
Thcri!sttcrmonthjri~htiu(24.)
Iscntirelyunnginiu'y; M
therctorc fuDowsby (22) that ~7rJ.(~) is thc rc:U part of titc
.secondterni. By expanding tho binumial nndur t)ie late~r.d si~n,
and aftcrwit.rdsiutcgmting by th fui'tuu)a.
154
0F AIH
REACTION
[302.
mcmbcr
303.]]
155
wh:ch is pmpnrtio~al
Ncxt !ct
aud bc such (.hatt!te n:ttund penod of tlie plate,
wltcn subject tu t!)e react.Kjnoft)ie !ur,is thc same us that
imposcd
it.
Undur
tljcse
circumsta.nccs
'tpoa
C'UAPTEU XVI.
'nmuRY
UL'R).JSUNATUR.S.
ln cl'l.taill clafillite
vilrratin~
lmriruls to
to itsco/f il IJI(JI'(~ur Il'ss
COIl1Jdl'tl~ i(JUI)L-11(lellc("
of tho extl'rll;d
If tlll! air Icyolld
atlliusphere.
thu nuutiul witllin tlm
pil: \l'ollld Ilavu 110
tu usca)Jo, :tilt! the
to
cuutuiuecl coIlIll1n
system 110t sII1dect to
lit aetllal
di.;sipatiou"
tilt!
illertiu
tllC
C'xturnaI
air
~t?~?~
uf tlw pipe
.?.r~~
bu
ilJsiglli/iilnt,
:lI1d tllull vimtions ou ce
caciteti itl tlie pipe have Il
ul' }Jl'I'.;i:tulll'C, '1.'llu
Ilarl'O\or tho ch:lIInd of co1,),,
"f
~.)M,.I
ti,a
extc.n.J
'~di.
'c.
L~n,
s.
'<
wvvities
'c.
U~
cOllstitllte
ill
tlle
l'usonators;
lresellce uf <tu extu l'Il a1
of
sound, tllu colltviuuul ,ic vibratus in
ullisou, alld witll ail
and
fOl'ced llcriml,,
~EE=~
l'iliing ts .gl'l!at intcllsity iu tlie of
appl'Oxi~S~~
rI 'SUlla/uryiclds tliu
vil)l.~Ltioli.4
111)
IJel'UlIling
~t.m,~s.e..n<i,u-y
th~rv
~t.tu~
i,
sll10t.
.s'yH~
case of
<
<~
t c.
<' T~
~i<!
of
i"c.~i.s.
8o
tlutt tlm 11l'I'SSllr!!is
absulut.dy COIIstallt. If nosv the
is cli:r that
F:
t.lm CUI[aillt'd air
will lm aL :111)'tillle
vury nuarly Il tlm (~tlIi~<
dc:nsity) concH!wlldillg to tllr;
-Ic
303.]
rOTnXTIAL
EXKRCY 0F
COMPRKSSIOX.
157
158
KtNETIC
ENEHGY OF MOTION
1-
which
thc~nct.cc.c~y.fthc
nation n~ybo n~cctc.)
c.~pt
t'.c
.f U.c apeur, and the
potcntial c,.o~y
le ~hourh.
c.ic.fatc.d as if th.
the
~-0 c~~
T.
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
"v ;~d'r
d
..st.)y,
mwd
~s~
that t.).. sj.<
r"
D.ruug). w),id. t),c )<i,ti,.
i
~T.y
ill
w).c.L
arc ..h.ut t.
pr.cc.cd
n.t of
t.)
arc
r~i"?~
<
"J!
s < < accm-ntc
su~<t)y
ca)r.,)ati.~ of tj.c r,iLch
~'Jn-v ),
is
illdufinjt('!y
grullt io eUlI1pal'iSOIl
witll fllo dilJH'lJsions of tlm
=:
is
,`~()~1. 'l'lie l:il'tic
cll('rg'j' tlie
motion uf ail
111(Y71171)!'CSSI))1C
nl;ly lle uXpl'l'ssl'd il terms tlie
dellsity p~ tlie
rnte or tI'aIlHfel',or
cnrrcnt, l', fur mulcr the cil'=''=:-?.~
tllc illotioli is
tll(: S:UIIl'.
al \nj's
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)(tiufurrcnl t'l'III 11tllCfilet flmt
:3 in SP1l<'l'alJd-1 1 in
timc. 111
be tlie
il'O
\'('!c)('it,y-pntvlItial,
l'y GrL'Oll'S1111'01'('11),
w](.re tlie lltl'gmtion i; to
hu l'xtf'lllll'} 0\'(11'
i,
sensihle. l1t a sld1il'il'nf clistanr~c~
01}l.illll'r sicll' uf tlm
~F=~
becOIIIOSC'ollstnllt, :lIlt! if t1~
apl'I'IIII'I'. rp
constant. \"allll's lm )PIIClt(.cI
lu- ~y1 nncl
tllat Ilaif nt' fCl\Y<lrd.
wlJieh
the'
{J'liC]
wc'
:EE~
I,W(!
flo\s,
304.]
TIIROUGH
NARROW
PASSAGES. 159
is proportiona) to (~).
.Y ==c (~~ ~), wogt as bcforc
If wc put.
of tb~ rc.sonatorcanhc
1CO
NATURAL
riTOt
nnd
w.H ~ncsd.rc~iyasthchncar
ho observe.
is functiun
OFJ~OXATOt{S.
f'~o-t.
dhnc.n.sio..
ufth~sizc.
The~vc-k.n~h
h.
and .shape ofth~
al.so upon thc natur.
'natoron)y,whi)ot!,efreqncncyd.pcnds
<.i U.c ga.s a.id it ..s
important <.orc.nark that it i.son t).c nature of
g.s n. and ncar t).c ci.a.nc.I that D.c
pitch deponds and not on
that oecupymg the inturior of th.
vc.ssd, for thu incrtia of tlle air
in the latter situation dr.cs not
cne int~ p)ay,whilc the eompres.siblilty cf ai ga.scs is vcry approxunatciy thc .sa.uc.. Th..s In
the cascofa
pipe, 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
ifa.s
npp).cah)G,
hcforc~
u.)d~-s(a.,d hy c <hc whoie cnnduct'vlty hctwccn the intL.rior aud cxturh.r of the vessc). Whcu tho
channct.s
are
.s~uatc-d
sumcicnt.fy
far
aj~-t
tn
act
indcpcndonDv
onc ofanothcr, thc
i.s
thc
rcsuXant conductivity
simple .su.n of
thosc bcinng.ng to t].e
~.paratc cha.u,c.]s; othurwi.sc t)ic rsultant
!.sJcsstha)ithatca]cn)atedhymcrcad()ition.
If tho-e be two
prccisciy simitar c!)a!)nc].s, which do ])<.<
iiiterfc~, nnd ~vh.~ conductivity takcn
.sc.pa.-ateiyis p, wc ].ave
~Oi.J
StJrEfUOR
A~D
I~FERIOR
L~n'rs.
ICI
162
SIMPLEAPERTURES.
[305.
thin
lamina ofmathT
strc-tc-hiog across thc channd bcmade
perfcctiy conductin~. tin; rt.-sist.tncuof f])e wholu will bu diminishcd,
nn]css t)tc iarnina coincitk) with onc ofthu undistorb~)
cqui~nttal surfais.
In th excc-ptcd case MOcH'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))C a.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)]))tcatiuns it is HufHciunt thitt :). wa)! bu
vury thin in proporHon tu thu
'-hn)(.-nsi<)t)sof t.)tu a;)urt.urc,
aud:t.p[)t-uxi))):ttc~y piano wiUutt a
(hst.Lncu frum thu apurturu lar~u in
propurt.iun tu thc i~mc
quant.it.y.
On account of (.hc;symmct,i-y un t)t<jtwo 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.Imust bc
c~st.imt; over thc
n-)n:undcr<jft.)t~ p)a.!)uthu mution mosL bu
exc!))M[\'c)j tfn~ntia),
so that. tu
on one sidc of thc piaoc ~e h:tvu thc
dtjturtninc
coudrions (a)
= 0 over
= const:mt ovcr tl.c ~pertuj-e, (/9)
H~
thc rcst uft))c phmc cft!.c waH,
~) ~== constant at ititinity.
Since wc arc conccrncd on)y with thc di ~renes
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
natter diHtnhutcd ov(;rt))capc!-turc-; t!te rcmaindcr of
attractif
thc prubjon cunsists i)i
dcterni.ung th distribution of mattcr so
that its putcntial
may bc constant over th .satnc fu-ua. Tho
proi.icm is m:Lt]t~tn!Ltic.i)tythc saine as that of
determining t].c
df.stnbntiou of cluctricity on a
chargcd eonducting pjatc situated
"ian opcnsp~cc, w))osc fcnti is that of t))c
~pcrtm-c nndcr constdLration, .nd t)iu cooductivity (,f thc apt-rturc
tnay be cxprc-sscd
'n ~i-ms oftj.e e~x~
of tiju piate of thc statical
prohicm. If
dc-untc tho constant, potcntiat in th
a.pL-rtnre, thc ulcetricai
rsistance (fur onc side oniv) will bu
~1 J
ELLrr'riCAPERTURE.
T~~
.iwenow
suppose t~tci.sinfinitdysma)!,
t.cuhu.c..se of' an citiptic pi.to, and if
Lutwccu tlic t\o surfaces, wu
gct
(c~IsnTT"
LyHoi~h.itz
~ruiL, j~t. ~7, l8),0), whnHo rusutt is
cquivn)t;nt tu (.S)
b~ fur the momeut tho thirj principe axis of thu
sHipsoid.
11-2
1C4
nsthcf)na]
ELLIPTIC APERTUHK.
expression f()rt.))cc:)pf)c!tyofnne)]ip.se,wh(~cscn)it))!)j')raxi.si.sf!nth)cpc~)~ricityi.se.
Jnthcpnrt.icolarcnsbofihc
=
=
e
circle,
0, ~'(e) ~7r, and Ums for circ)Li ~t' radius 7)',
['30G.
COMPARISON
30G.]
WITH
CIRCULAR
APERTURE.
1G5
e==shi~.
o"
30"
5~
~0"
70"
~0"
90"
-ooooo
-~204
-50000
-C~79
-rcGOt
'SGG03
-939M!)
-98481
1-00000
~eoH~.
i-ooooo
-!)39<i!)
.8GG03
.7~04
-C427!)
0
~0000
-3.~03
-173G5
-00000
7r-27''(<')(l-e~.
i-oooo
l.ooon
1-0013
1-OO.it
1-0122
1-0301
l
r07i~
l-l'J5-t
co
166
CALCULATION
BASED ON AREA.
[306.
307.]
CONDUCTIVITY
0F
NECKS.
IG~
I<jS
CONDUCTIVfTY 0F NECKS.
[:~7.
rath) t.s~xprc.s.hy
-t,
wta'rc ~dcnote's
a)ul
t)tC
p')tc))t!:ttt)))it.st']t'<'ft).('h'(.'u!;)t');)y(j['L'fj)):t<tei'(jfuniLdL'nsitynnd
cf')'!h)i)[s7t'.
T))csi)))[)k'.st)nL't])<)(l<)f(.dun].L(,it)~<)o))(')KLsup")it)icc~nsi~crnt.iuit <,)).')titruprt.'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
)tavc for t))C
L)L'i))~).tt))u(;~Q"t'<tL'tU.scwh'jne)'a<n))sisH,L!
t]K: !i)))its<ji'p bum~Oand
putent.ia.t atthu
put(. )''=;)'~6'<
H(tc<js~tnd<)tus<of~bc'in~7rand-t-.t7r.
T's
r=~(:).
Now )ub us eut ()<)':).sh'ip uf brcadt.)t
f)-o]ntitc cdgo cf t))c dise.
Th 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 th discduwu touot])ingfu)d
canyidi thc piu'ing.s to infixity, wc fhid fur thc tot:d work hy
intc~r~tij)~ \Itft rc.spL'ct tu from 0 to /<
307.j
CORRKCTION TO LENGTH.
1~9
1~0
TUBES 0F REVOLUTION'.
[308.
for tho
308.]
SUPERIORMMIT.
1~1
173
APPROXIMATECALCULATION.
[308.
Jhc expression for th resisLmccn.dniits of considcrn-bicsi)np!Iiic~tiou by intcgra.tinn by pru-t.sin thc case whcti tho channct is
oft!)u !iniits of intcgrat.ion.
truly cylindrical iu Utnui~hbotu-hooLl
I)i tUs way we Und fur tlic fi)i~ rusu)t,
COMPAmsON
308.]
WITII
EXPERIMEKT.
1~3
:mdm:).thcnt:t.t)ci:ms.
ln
pmct.cc
).s so long that
ns
itsdf
(4)
it
doc.s
thc
.supposes,
bc ncg!cctc.),
m,
not
oftu.i
currc~iun
on
th
us suppose.!
dt).crt).at
h.~pj.un
fur t).L. opui
ends
othcr
so
hand,
i~ (~.
th
ucek
ean bc )K-]cctcd
short
U~t~b
c.u
\Vcrt)R.i,n<
~s
t).c
first
174
HELMIIOLTZ'S
INVESTIGATION.
[309.
A formula.not dtffuring mneit from this was given, as th emLoduncnt of tlic rcsults of I)!s )neitsuru)ncnts,by Son<n~uss' who
'?..i"cxL.53,219.
1870.
30D.]
MULTIPLE
RESONANCE.
175
~mu)aofinturp.)atiM,L~t!,o.u..n,
pf r~sonators
ti,uury
Inc
,,atn,
wib).
~.nc tnnc
t.ucks
wa.s
give-a
abo~u
th
lu a mc.noir un Rc.s~nance
pub~hud in ti.c
fur 1871, fru.u w].icit inust uf thu JasL
icw
durivcd.
p~cs )s
:310. Tho .simph .nethod of
c.~c.uh.tingth. pitch ..frcsouators
w.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 arc
1)y no
'nc.a,.s .ndupcndcnt of t).c inertie of the
..ur in tho intcrior of the
l),c ci.a~ctcr of thc.sc mo<).s wi)) be
more ovident
r~rvoir.
K.n wc comc to considcr H.c
vibrat.-ons of air within a cu.n-'
]. c y c!d
as a sphorc. but it will
v~d..such
nu.dy h.ppcn
tllat tho p)tch can be ca!cu)atcd
tJ.eurcti(;:diy.
Ti.crc arc, howcvcr, cases uf
,n,dtip)c rcson~cc to which our
is
t'.cory
app),cabJe. Thcse occur wf.un two or ~orc vcs.sd.s
c..n"atc
channcfs
~th each ot).cr and wit), thc externat
by
air.
.d .u.c rcaddy trc.atcd
Ly L.rangc.
n.cth.d, pro.idcd f course
ti.at th wav.tc.~th
af thc vihrati.n is
suf!icic.ntjy I.rge iu comp.-tn.sonwitht)~di.Hc.iu,).suft)tuvc..s.sd.s.
Suppo~ ti.at thc.rc arc two rc.s.rv.irs.
con~nunicatin~
~it!. cach othcr aud with tl.c c.Lcrnat
air by ~arro~
passage, o~
If wo wero to con.si~r
a. a sin~c rservoir an.!
ncc~.
~pp]y
wc shouM bc !.d to
~f prenons
cn-.n.ou.s rc.snit fur
furmuf~
formula is fOlllldcdan tlle
:~tluLt
witlia the rescrvuir
aS~lImptioutll1Lt
th
u.crti.. of thc air j.~
,~hatiormu)~sf.,unded
bc~u.nption
Jcft eut of
mayth
t).atw~h:n
ac.our~, ~hcr~
tho rc.sc.'vcir
it Is
is
cvnicnt
pa~gc
that
iuay
th
hc
of
cne~y
a.s ~rcat
~')-uc~t'
a.s
the
through
motion
th
t)n-o~h
two others.
th
conncctin~
Ho~cv.
a~
17G
DOUBLE
RESONATOR.
[~~0.
~'
P~"
invcsLig~ion on th Hfune K'c~l
Hnid1
.Y,, A\ t])C totat tr:u).s~.rs of
perfuctiy. Dcuoti)~ by
Lave as m (2) 304. fur thc kiu(jtic
wc
thc
Un-ce
pa-s~gcs,
tlirou~h
cncrgy t))C cxpi't's-iion
~J
noUBLE RESONATOR.
~y
~'hich rc.juirc.s
= 0. Tho motion is thei-cforc th
th~
.sa.nc
t.:Lkc
'night
p)acc wcro th Ct))u)nun!c:tt:on bctwccn ~'an<~S"cut
ofT,aud bas its fi'Gfjuuncy~ivcn by
n. ir.
178
PAUTICULAR
CASE.
~310.
dissipation
is,
howhvf-r.an
important
fcatm'c
in
tLc
chamcter
~-1
COMMUNICATION
0F ENHH(.Y.
l~:)
~f
In ~rcvicus chaptor (
278) wc .s.w).owL. cxp.-o.ssth motion
<'n thc r.~t ,f Lh. i.~nitc
H.ngc (1~. (n). in tenns oi-ti.c ~rn.d
Ycjoetty 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--=1
appruxi.n.~civ.
180
t':nt.
.RATH 0F DJSSl'ATrON.
Coxfitiin~our
Ift]tcpit.L-h((]ctcrminc<!
by?<)bc
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
forceiscfttcu'K.)Uftti(.n(.)it<<~)]ynpprf.xin)tt~nrLsmuc]tnHthodissi[)ati\-c
Ltt('dont)jt;H))p])(~iti~nt)t)ttthavi))rati(~)isj)ermnuentiLntthiswi)Ucndtono
!n)ttcritdo'rur\lK'nt.I)()
dissipationiiiHmftU.
3U.]
NUMERICAL
EXAMPLE.
181
1S~
r<JH(JKD \')I!i{.ATIO\.S.
j~L).
(putL', :). c~so uf t.])fit tru~tcd iji .K!, t)tu diffrence dcpcndin~
upou thc fact that, i))(i c')'-)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, varias invt.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)'~cticat
us<-<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 inttior w:~ t!)cci)r oroLhtjr.~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.s
i.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
rsonant)'
(-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
thu ci)'cctof:L
prec<din~pamgr:)p)t )nnyb(;a])pijc-dt<jc.d(;ut:ttc
soun'uof sound mt.uat.t'd in t))cintu)-ior(!t'iL)-('.so)i!tt.or.
3]~. 'now
p:)s.son tu thci'm-thcr discussion ()ft]tcpnd)]rtn
\Vu sl)!))[m)pj)f)sc t))!t.t t.hc opcncndofthc
ofthu.~pL-upipt'.
])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).sidr:Ltit)n.
As :m introduction tu t))C())K.stion,wc
winftn-thct-suppnst;
thatthemoutiiofthc
pipe i.s htt.<-()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.s
i"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
oi'E~ rjpE.
31~.]1
183
and
~)
\n
v
buing thc radius of th pipe. Froni this t)t 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 prsence uf th piston wan)d nut
nt!).tcri:d)y mudify
thc ([estion, wu nmy as weU )):Lvethc hoiefit of thc
sitnpiification
aL uncc by taking as in (1) 3)1
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
184
THEORY
0FOPEN
ENDS.
[312.
3J3.]
185
Lcb us considur thc bch:n'iour of t.hc nmss of air Inchu~d buC:md n,hcnusp))<ric:dmtr{~ccw!tosu
twnthcp~ncHecL~H~t
centre is~l,iuid
radius ?',)'bcin~I:n-g-tj in conipari.so)iwit.)t t)tu
d):U)tct.L-rot'tho pipc,b)tt..sma)linc(')))p:).risonwith<L)i<j
\t\'L')';))~tL. Wit)tD) tins sp:L(:ct))u air nu~tniuveitpproxitni~cty~s
:).n.
incun)}))-c.ssib)cOuid -\vuuiddu. Nuw t.hc uun'cnt ncru.s.sthc hcmisphcric:dsm'i'n.c<j
18G
CORRECTION
TO LHNCTH.
[313.
3)3.')
RATU 0F
DISSIPATION.
187
instcad C)r
iIlSte~t(I
of
Thu0
f/
~Trr'~~y-'
u!ti<nate<t.ctoft))<t-at,i<jn wiH )j2to hnjvc thc cxprcsHionfor
H'c V(.-)<)city-p()tcntifLt
uutsi()(; tho muuth, tis wcU ILSthc corrcspun<)ing second tunn in
(invoivi))~.sinM~). T)tu :uuount of
is t!ms .scctLtodupcnd ))):ttcri:).t)y
')tMs)]):Lti<.))jL
ontitcdcgrecitt
which t)m \v:Lvc's
arn fruc t<j divo-gc, atK)onr a)i:L)yttc~cxprf.'s.~iuus
must nut bu r~mk'd :L.stnurcthan run~h c.sthnit.t~.
Tiiu c~rrt.'ctthcory <'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)side-:).b)yfrom 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)ithat~ta~
opcn end
thc pt-c.s.sut'o
cou)d not,vary ft-om thi).t of tho ~urrutmding atmoMphcru,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 anywfturuin thc pipe au absolute
)oop,nnd it might hve bccn
('xpectud t))at the ino-ti~ oi'thc air just ontside the tnouth would
ha.vu the eUccIof an iocreasGin 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)tthc r~uit that thc intervai bctwecn
thu mot))a)td thu ncarust
noduisa.I\vays)c.sstitim t)tu h:dfof that
s~pa.t'atingconsccutivcnod~'s.
31~ Expo-imcnta) d~'tDi-nnnationsof t!.c correction for an
opun on) )iavc gcn(.'ra))ybeun )na.dcwittiout 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 hve 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
)')1.
vihrittiuns
1~
incyliudncnit.ubt;
c'ftHt&ro/~f
j~Y~
~83
INFLUENCE
0F
FLAN(!E.
[314.
L.
~t\.r~
314.]
EXPERIMENTAL
METHODE.
183
Thc correction,
itccnuut:.
howcvcr, -\von)d
mrc]y be wortit
tahin"'
illto
~0
DISCUSSION
OPMOTION
[314.
ORIGJNAT1NG
315.]
WTHIN
AN OPHN l'IPE.
191
t
==
1 QO
Il.
MOTIONDUE
AIOU'IO.
DUE TO
Iv W
f;~
Jf<?)M givcn,
is
~.en.
,s ~.catmt,
whcneo.,K(<+.).n
In
t)~t h
of t).c ~t.i~
.Ibr~i.n
v.y
71
.t.,
n.,
th.n~h
n~mtc. sincc cos~can~t~is),.
Wi.cn
hun~uU..much
cont.rac~, oos.y
bcco..c .sn..)., h.tt
in t!s c~
is ncccssa.y that t)~
a.)ju.st.,ncnt of ncrio ho
c.
in
onicr thut thc Hr.st te. cf
very
( i'.) ,n.y b. n~
le
tlie
:~r:
.;ecolH1.
CC)SAC2
i. lie.-ll-]Y
~p~c.i: il
l'quaI to unity.
Tho ininhnum of vibration occur.s
whcn
snch t)~t
tLc pi.~n is ~cd
at r~p
tlmt Case
T.cv.br.ion
out.si.jc tbc tubei.s tben,
accorda te tbo value of
c.,ua)to.r.s~a]fc,than th vibrion ~.ich there wouU
be
't!~
"S
P~~
tlie
hlane,
316. Onr cqu~ions
may ..J.sobc ~ppUedto the investigation
.f tl.c motion cxc~cd in
tube by cxtcrna! sources of~nnd
Let 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
<h.c to t).c cxt.n~I .sources
(approximatoly constat
p~cnha]
ovur
t!.c plate) .s undcr tbc.sc c;rcum.sta)icc.~
~-he.-c
Is composcd of thc
potcntini due to each .source and its
.mngc .n th
pi~no, as c.xpt.h~d in 27H. Inside t!~ tube lut
t)~ potcutiat be
~nc.j
EXTERNALSOURCES.
193
'Hd)n))ott~,C)-18<!0.
R. H.
I!)4
EMARGEMENT
AT A CLOSED END.
1317.
318.J
ABSORPTION
1S5
L;
l~G
QUTNCKE'S TUBES.
[31g.
1
action1 off'mthc
apparatu.s. Thc ordinary cxplanatiou by intcrfcrcnc
(so ea-Hcd)of direct t~udreffcctcttwavcs is thun luss applicablu.
where Fand
319.]
197
so that 2-n-A:i.s tfie w.ivc-icngth of thc )in.tum!noto of thc rcson~tor. If ~bu writtcli for ~/+!jo-7~ t.)ic cquation con'cspondin~ to (5) ta.kcstliu i'ortu
198
REINFOnCEMENT
OT SOUND
[319.
Th relation of phases
)nay bc reprcs~ntcd hy rcgarding th
Jnduc~d vibration
as proce~hog frojn
by way of J, and as
bt-ing suhject to au additionat retaniation of
su that D.e whoie
retar(!atiun betwccn 7~ axd
is c +
In respect of amptitudc
13 ~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 a
quarter of a pcriod.
It is important to hve a c~ar i<)e!t of th cause of
this
aug-mcntation of sount). I)t a prviens ch.-tptcr ( 2.SO) wc saw
that, ~heit ~1 is ~xed,
gives ont much ics.s sound tkui n.ight
at ilr.st hve been cxpcctcd from thc
pressure duvclopcd. Thc
expiu.nat.ou was tJ.at t))e
of thc prsure was
unfavourah)c
thc hn~cr pfu-t of it is conccrncd
ouiy in ovcrconung thc incrti:t.
of t]ic surruunding :ur, and Is incHectivc towards
thu perfonuanee
of work. Now thc pressure which sets
m inutlun Is t)~ who)e
pressure, aud uot inuruiy thc insignifiant part that would of itscif
do work. T)ic motiuu of
is duturnu.tcd by thc condition that
that cu.nponcnt of th whoto
prcs.surc upo.i it, which i.as t))c phase
of thc vulocity, shai) vanish. But uf' t!m
pressure that is due to
thc mution of~, thc largcr
part bas thc phase of thc acclration;
aud thcrcforc thc prcscri))ed conditiuu
requires a)i cfptaUt.y
hetwccn th 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 to augmcut thu work dune at 7?,
319.]
BY REHONATOUS.
1~9
200
RESONATOR
AND DOUBLE
SOURCE.
[320
distance
is
320.]
TWO OR MORERESONATORS.
201
Lot
aud
i-cprescnt tlie distance
tlie potentiaLs
th:u, would uxist at ~t~,
if t))crowct-c no rcso~toi-s; thon the
couditiuns to dtermine
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:).tors
putting ono
anothcr out of tune, tuid if thi.s tcodoicy be
conipcn.sn,tc(!'bya)i
ftftcration in t! .spriog-,any numbcr of rcsonn.tors])c~tto'thcr
)mvc just t]te ccct of one. This point is iHnstratctt
hy :{()~
whcrc it win bc SGcn(32) t!)!it titougit thc rcsuuiLncedocs not
depond upuu thc sixc of t))Cptatc, still thc incrtia of thc air, which
bas to Le couipcnsated by a spi-in~ docs
dcpuud upou it.
J~t. ff.C/t!'m.
t.xxiv.1823.
202
FORMATION
0FJETS
[322.
322.
322.]
OURI~G
SONOROUS MOTION.
203
Intcnsity. A simitar rcsult wnsobtaiijudwith a forkand rcbou.dt, ot'pitch an oct,a\'(!jowcr (~. (:[oscr examinatioa npvufdcd
thc fact that at t))~ .sidcs of t.]te )npp!c tI)G outward
nowiu-T
strcam \vas rcpiaccd ))y onc in thc opposite direction, so that n,
to enter
tondue of n:nno frotn <).suitabtyp)~ccd c:m)HuftppO!a-C(t
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 efFoctsn.rc of
cour.suin rc:L!itya!t(.!rnatit)g,:u)donJy :tppe:u-to bc sitmdtanerms
in conscqucncc of th in:d)i)ity of thc oyc tn foUowsnch
rn.pid
Thc
fortnation of jct.s mxst ma.kc.1.scrious draft on thc
changus.
Otur~y of thc tmjtion, an() thi.s is no donbt t)tc rcasnu w))y it is
todosc thc: nipptc in ot'dut-to ohtaitt ;). po\vcrf)n Sound
nncc.s.s:).ry
fru)n :).t-c.son&toruf this furm, witen suit:ddy tunud f'ork is
prcscnted to it.
At thc same fiinc it docs not nppca.r prob;tb]c tt)~t jet fonnfttion occurs to any appreciabie extcnt !).t t)tc )nont)ts of t-csonators
as ordinariiy uscd. Thc ncar agrcenK'ntbctwccn t))CobHGrvudand
thc c:dcu!atu(t pitcit is ahnost a sntneinnt pnx'f of this. Anutl~r
iLr~umcnttonling to thc samc concluston ninybu dt'awMfrum th
pcrsistenccofthu frcc vtbmtions ofresoun.tors ( :ni), whoscdurat)on scoms to excludo nny hnpurt.fmtCt).uscof dissipa-tiou bcyuad
the commuuicatiu)iof niotion to th surrouuding air.
lu thc case of organ pipes, wt)crcthc vibrations are vcry powcrfu), ttjcsc arguments arc Jcss cogent, but 1 ncc no reason for t!unkingt!<at th motion nt the uppcr&pt'nouhHO'crs greatlyfrom thn.t
supposcd ni irchnhottx'tj cfdcuhttion. No conclusion to t))u contnu'y c:u), 1 thhik, safc)y bc dra.n from the phenomena.of .stcady
tnotion. la thc opposite extrme case of impulsive motion jets
certainjy ca.nrtct bc fonnod, as fot)o\v.sfrom 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 assitniJatcd. Observation by thu mcthod of intermittent illumination
( 42) might lcu.dto further Infortna.t.icnupou this subjuct.
C'HAPTER
XVII.
APPLICATIONS0F LAPLACE'SFUNCTIONS.
323. Tim gcncral cqn~tion of n, vcioeity potcntial, w!)cn
rcfcrrcd to polar co-ordin~tos,takcs thc foi'tn ( 241)
whcrc
as a
333.]
SOLUTION
IN LAPLACE'S
FUNCTIONS.
205
<i.
20G
EXPRESSION
FOR RADIAL
VELOCITY.
[323.
324.'}
DIVERGENT WAVES.
207
208
0FSONOROUS
FORMATION
WAVES.[324.
portion of th cxccss of ihud in front going to supply th dcncicncy behind. Now conoive t))c po'iodic time of th motion
to bc contmuany diminishcd. Gradu:dtytho altcrnation of movGmcnt beco;ncs too nLpidto permit of th(.!fu!I Gsta.Dishmcntof thc
mct'c-)yIoc:Urcciprccn.tin~f)ow; th air is .scnsibiycotuprcs.sedand
rarcRcd, a-ndi),sensible suunft wa\'c (or wnvcof t))e samu nature,
in c:t.set))c po'iodic tima bu bcyoud t)ic )i)nits suit~bicto hcarin"')
ispro}~giitu()to f),()ist;u)ce. TLc s:unu t:(.kcs p)!).ccin a.nygas,
an<)thc niure rapi<)bu tho pro])ng:LtiunofeondunsatKjns:md rfu'efactions 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 th
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 grt distance from thc sphre th function~(~?-)' bccoincs ulthnatulycquat to 1, and wc hve
324.J
EFFECT
OF LATERAL
MOTION.
20~)9
(~r) = 1, we
EFFECT0F LATERALMOTION.
210
[:24.
grt distance from thc centre wi!)no h)nger vary from one direction to another according to th saine Ia\v as thc normal velocity
of thc surface of th sphre, 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 suppose thc 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 th latcrfd
motion winch actutdly cxists.
"T!)e intensity will bc measurcd hy tlic vis ~< produecd in a
.~ivoi timc, :md consc(jucnt!y will vary as th density muttiptictt
hy t]ic velocity of propagation mtdtipHcd by th s<)uaro of th
amplitude of vihratton. It is thu )ast factor atone that is diifercut
h'otu what itwouhl hve bccn if titerc had bcen no latral motion.
T]te amplitude is ahered in tlic proportiou of/~ to
so tliat if
thc quantity by winch thc intcnsity that would
/~u''==
ttave existcd if tho nuid had bcea hindcrcd fTomlatral motion
lias to bc dividcd.
"If
be the Icngth of thc sound-wavc corrcsponding to th
that ~c is t))C ratio of thc
period of thc vibration,/c=27r\so
cn'eumfo'cncc of thc .sphcrc to thc lengtb of a wttve. If we suppose th gas to be air a))d to bc 2 fcct, which '\vou)d correspond
to about 550 vibrations iu a. second, and tlie circurnfL'rcnce27rc to
be 1 foot (a sizc and pitch which v'ou)d correspond with thc case
ofa connnon houfic-bc))),wc sh:dl hve A'c=.
Tho fuHowin"'
tahtc givcs thc values of thc squares of tho modtnus and of th
xc
;t=0
?;=!}
l
4
1
(~
17
5
2
l-ax
;;=!
1(!'25
S
5
1(!
M=2
l.l'87n
<)';)lL!5
89
l:):i<)'2
0'5 l'()<~5
0~5
1'2:' M'()t!2
1B'2i ~()87H
W:\O'2
4
2
1
()-~
0"
1
1
1
1
1
0'M58~
1
'J'C
1:<
(:U-2'.)t
0'87.T~
]'8<J
41~
JDfii~
J'JHM
t;=t
--l.
l!8-t8
M
3i)'
2:~91
ll.Sii789t)
12:!IJl!H
0-81.15:)
16
1M~~
18Si).'<:i
muCS~l)~
_1.
20-177
l.i!);8
30()):t7
720H)::)71
720Hlj:J71
181f;()xl0"
lo~
l-tuf::)
2!)U-1C
1;-it)<w:n
57':<)')!)7
1?(J!JJ:<]"
?
S
SS.
F
S
2.
324J
STOKES'
INVESTIGATION.
211
212
LESLIE'S
EXPERIMENT.
[324.
324.]
NUMER1CAL
RESULTS.
213
~-j
o
p
i-)
w
o
co
to
o
e5
M
o
c-<
'p
$000000
~o~So~c~o
CID
~n
sgg~ssp:
'<'
co
00
t~g~ts~cacaf
~Q
'p!~
O
f-
<f:
Ip
Q
oo
OD
B
'-t
0
<p
0~
[~
o
.O~
~f
<-<
0
<p
f<
rl
p
r-<
G~~1
UQ^
pp
<p
'-<
M
0
~-<
m
eo
ro
tg<g<DO<C<OM
))
t_"
&<
mBfoBmMM
t~W
'-<-}<~0f-<9<
ob
e~
c~
~<
t~
rt~
M
0
[~
o
t.
c<
L~
r-
t~
tr
0
f~
f
Ot
c)
r-<
h
tM
e?
Q
0~
o e<)
.-i
<?
d
Ot
f~
co w
o
o
?<
CO
o~
M
m
o)
M
00
M
CO
t"
o
r~
.-<
<-<
'o
r~
<?
S
t~
P
rd
r-) *-<
t,n.
.r
11
s
.ri
g
u
g
:a
.h
E-'
g
S
'o
<i
214
DEFI.CJKNCY
0FTERM
0FXEHO
ORDER.
[324.
325.] ]
REACTION
ON RIGID VIBRATINCt
SPHERE.
215
and ~is proportiona! to thc cosineof t]ie angle bctwccu tho direction considcrcd a.nd somo iixcd axis. Tliis expression is of thc
same fonn as thc potcutial of n,(loitblesource ( 2D4-),situated a.t
thc centre, a.ud coniposcd of two cqun).i~ndopposite simple sources
]yin~ on thc axis in question, wtiose distance npai't is innnitcly
S)na.t),andintensi.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 hutwcenx and r bc it is vident that tlie potential of tlie
duubtc source is proportional to
21G
[325.
is proportionat to
~.hcre there
335.]
MULTIPLESOURCES.
317
hve bccn cxpectcd that a simitar taw wou)dhohi for tho velocity
for ?'
Tins howcver
potcntia) with thc substitution of r"
is not thc case; it tuay bc .shcwtitliat thc potentia.1of a quadruple
(~ e
dcnotcd
in ccncnU uot to thc
source,
by
corresponds
?'
M/<
io'm of thc second ordcr simpiy, vix., e" ~(!'x)-), but to a
cu)nhin:diunof this with a. term of zuru ordcr. Thc ana.h)gytherei'urc hutds uniy in thc sing)c iastim uf tlie ~6
point or source,
fd'tcrany nuniber of dii'crtliough of eoursu titu function )-e"
cnUations continues to satisfy tlic fuudfuncntal quation
218
ENERCYEMITTED
[32G.
327.]
FROM A VIBRATJNQBPnERICALSURFACE.
219
220
SOURCE
SITUATED
[~27.
328.J
2211
0
1
S
3
4
C
+
+
2~
2
C4
~(:(i
+I.t')()~
+175CU2
1
7
35
8M
+
8M1
.)-:}~1~H)
+
(M+~a~.(~+~)
+'4
+'J8td.)
--()6()J:i!)l
-'()();i-lM7
+-n<)(~()5!)
+'UUUUlii
(,t.~)~-(~+~)
+-22
-)2307n8
--0:M88M
+-()OC:M()1
-)--n<)<)~~
-'UUUU2)i~
222
NUMERICAL
RESULTA.
[328.
/<-c=l.
r'
a
(N+t)ft-(a.).j8")
();)~)~-T(n')
~J
i
2
n j
i
2
-t.
+
i)
't
n:)
tBi)(!
d~t
.)
.)())!):)
-U:}(::i.i()
3
n
7
-<-
+
+ -I
i
i
H
3l
:il7')
f'h~it
+'!<)1~17
f/
a
0
1
i<
4
.'i
G
7
1
2
!.(2
1
+
S!'5
1'75
8
4
IC-IM73 + 35'm.~
-)-]8<(~i;
~t~ 8;')'4:;7.ri
.t
+!i:!H'H()
-U77-H~iG'8
-8M1-7
-)+
+
~~a
+'<:
-J.tD.ttO
--f)tr.7H.t
f'()(~t.t:tH
.f.'()')')7.s7
-'0()<~))7
--ooouOf!
-)-~5
-1
-'a
-t7]0
.)'o:t0t))~ a
-)'()()(i!))~
-'()<))).();
-'<'()))t)7:!
-t'ooooo.~
2.
'(;).~)a-(a.)-)
-)-I
+-<:
+-tn'.)80
-(.') i
--t))H70
-)-f'l:<tf!
.).'()'U
--(JUU7'3
(t))-)~(ft=+~)
+-3
+':(
-'(.7114
--173
-t'IOM?
.)'()1)!~
--tHtt.f!
--oou;)~
NUMHIUCA.L RHSULTS.
328.]
323
7'f'
J~hC=
I._p
1
-1 1
1
-1 1
0
-).():!)-'L):H17'
'J.Lt!)-)~))t!);
'iH~ii-~K:i:!i)t
-(!)i7H.8) '2:i.-<(T.!)<
-t<)(~)(~);():)f
-)-'H'!li)Uii-'i)()JH)7.i;
-7')'!)-)-2:i)21t
o0
'lij:HH
--15:)M1-~7(!<J;
'2')tH')l
-H;);)7~)
-}1U!~)
-(M1(;1 1
-s.')~j'o
-CM.S
-(!M!)8
-;i.(;~
'ajlj~
'[j7(jIW
224
[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(.'))tapproximatiu))
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
Titus
v:t.)uc'.sthL-nn,ismnt:J!)y<t(j fonctions/
wmcn
f~rcc
prctty
ctosciy
witti
thc
rcsuiLs
of
thc
more
complote
catculit.tlun.
0F SOUND.
ONA SOURCE
328.]
]
9'~
-j*7
t1)~:L)t''r;(ti<~t tjfpim.se
wch:LVcfo]'a,s)n:dl
.s))!teru
C'unsidcr))f)W
th
sytnMicopc')'))-
niny bc expresscd in
t
P,,(!),fmd)ctit.
(/Y
f'pcratf'nn~
'Thnmsnn)U)dT~it't,.Yf~.7~7R~(~u~t<'[UnnM~fnr~~yt.
H. 11.
)5~)
22G
ANALYTICAL
EXPRESSIONS.
[~20.
330.]]
227
\vu )):t.vo
2288
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~
thccrr('spu)n!i)~p)W(;i-<.t'
!tt-u)i)!U)tst.jcx)):H)'[t)nj~i))~.s~cha\u u
Miun()nt))eti~))Linn.sccndi))~pu\vL-r.s<jf'7'.
~0.]
FOR VELOCITY-POTENTIAL.
32!)
230
DESSEL'SFU~CTIOXS.
[330.
3:~0.'j
PARTICULARCASES.
231
tha.t
cnvclupc. T!tus in the case of tlie symnictrical vibrations, wc
havetodutenninc~
ta.n~'=A:r.(1),
{m cquntion which wc hve n.h'e~t~yconsidcrcd in thc ehaptcr
on mctnhnuics, 2()7. Thc first nnitc i-oot (/<= l'-t3037r) eon-csponds to the symmetrica.!vibratio)i of lowcst pitch. lu the ca.so
of hi~hcr root, Uio vibt'i.Ltionin quc.stiun !ms .sphcrical node.<i,
wh.scmdli correspond to t)tc infurior roots.
Any cne, w)tosc vortex is at the ori~'in, m~y bc nuulc rigid
without ~n'ccting thc conditions of thc question.
Th loop.s, or places of no pressure v;u-i:).tion,aro giveil by
or /<-)'=W7r,w])crc M is any intcgct-, except,
(~)''sin/o'=U,
zro.
T)tG case of )t=l, whcn tito vibnttion.s may bo ca.Dcd<U;T.tnct)':).],is pcrha.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'))oco~ieal)iutfc.-i' ~vithvL'rt.cxn.tt.'iiccentre.
Any jne]-i(H:ui!dpiiLnc,!~o\vcvo-,is nod:d, n.nd tnay be supfoscd
]'!gi(L AJong nny spccificd ra.dins vectur,
fim)
vanish, iuxl
vix. wLon i;u) /<-r=/<-r.T))c
d~ngc sign, Avithcos
C
(~)' sin
\it)) thu nod:).)surfaccH
loops in thc' prc'.sutttcase t)ierctut-ucoinci<tu
ofthe r:u)ia) vibraLiun.s.
Tof!)x~))csph(.'ri('a]))H:)c.s,\v(;]);tvc
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 sphre titan for !).ciosc(l
pipe ofoqua! Jung't)), hecausc thc spho-c may bc (turivcd f'rotn thc
cylin<)(.-rwith 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.sthtoMlt:)rpc~thc
t))C))i:)s.stohc mo\')'(I rondins but.iit.Uc c));H~(.'d.
.v])i)c
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)"vibrt
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
tli;Lt
nn
~>,i
t)~tA~=~4()~or
or
Jo~
: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'r!H\\hi(;i)Uterc
i)-!)o)nutt~t).
VIBRATIONS
331.] ]
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(')'
'Hcrc;
is proportiunal
to s!n20,
tlie
.sccLon:d
H'crc again
nodat.
But
I~n'tnonic
plane is
Inthiscasc~~v[H)t.shcsindcpcn<Icnt)yofMwithcos2~t))at
is, Avitun0=J7r, or .7r, \v]nch givcs a nodal cne of i'cv<j!utlun
mitose
1
vcrtic:d1 a))g]u
1.. i.s n rig-ht
1t :mg)u.
] rl~ v.u'ic.sa.s sinM, und1
M
<hust))Ci'cisu)K'n')f')'idi:)n:)tno(hdp'):mc',nn'1))))tonu
234
AVAVE LEXTHS 0F
VU3RATIONM
['331.
~
c '3r
ICI
gt~
l~:i
-81:).~
-f;7(i~
-C8~51
~~U8
'0'*8' -c~.i8
-.il<:7i)
-SOM:i
-.15:~0
-:if!
..10;);~)
'n<)8:M
3
-;);)C2;<
1~77
-80!).~
-7;~<)
4
].ll:t
.c;).~
C
.o:ioo
C
.g~
wf'niiN
33L]
sniEnicAL
ENVELorE.
2:35
TA!<M;n.
I-H.chofcaeh
tuoo, ruturruM
tuno.r.ferrud
toh'raveat.
_`_
l'OOOO
Ordc.r
Or.h.r
~ritchofc.~
1 ~')
N.UI~lher
Il
of of
f.l1Itlll)(H'
1
tnoe
of
CllC\
01',1(,1'
of
"t..ue,~fcn-cd
"f'"t~
HMinuine.
IlM'manic.
tuH~vcst.
"tlIUI CH..
1101OH
III
1
1
2'85.10
J-C056
3~~8
~'l'~S
:<)21
2-KiU
!)
!711t
3-713
!i'773
Ii
il
II
!2.
If wc drop mmcccs.sfu-yconstants, t))o p:u'tictd:n-solution for the vibrations ot'gns\vit]nn~sphcric:d e:)LSCofradius
uuit.yi.src'prc.scntedby
In gc'ncra.ii.singthi.s, \vc must rcnicmbcr t.])n.t mily bGcomposcd 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 hve
23GC)
CASE OF UNIFORM
[:}32.
which!snnimni~)iatcc<mscfpK'not'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))(;(!initit cij'cmn.staaces.
L]0)-d<'rt()i)i)).st.t-;dcthissuLj('ctwcwiHt:).kcthcc;tS(\AvIicro
in its position ()f'('()uiiih)'i))mh))t.is))')<)vin~wi(h
nntiaDytheg'asis
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)iuniftn'm motion, w~r~.sud~chly stopper.
Sincctnct'C! isno
Inititnc'ondcn.sntionot'nu'ofactiun,
(jnn.))titiL'.s~v:u)i.s)). ]f ;~)'u
~'hicit nh(~H tlhitth('.S!))ntion
o)'duri)isp)K')'ic-at Itar~tonics.
forn]
nHthc
i)uti;diyu)tity,wch:i\'('~==.c=~,
ff~tt.'titisoxiy tc-nxs of t))c fir.st
'J'hcsotution is t))<jru)'urcof U~
33~] 1
iNrrrAL
fl
Thc (.'ViUttation
of )'<
YHLoci'rv.
237
233
Sl'HERICALSHKLL.
[332.
is by fiu-t,!m most
H33.]
When thcdittbroice
l'LANE WAVES.
239
\v)iorc J.
:u'c functions of~ but not uf/t. Frorn w))at bas bccn
t!~t ~t, conHidcreda.sa func:drc.tdy provcd wc mn.ytmti(.-ip:Lt.c
t.i<uiof)',m.ustv~ryas
M)))tip!yi)!~
~0
SI'IIERJCAL
C~) ''Y
M, i~.d intcgrating
~=+I,wc~hu)
OBSTACLE
wit]i ro.spM-t.io
f~
fron)
~=
1 to
I" thcpn,L]<niin).an<!
th ~vh<.)umotion
nut.si.L.O.c.spl.cre
Le d.v.)u.) into <wo
parts; th. f,that,
'Y
n.pn.su.h.d ),y<&
to un.li.sturL..) phm.v..s,
.-u.Xhc.s~
c.o,)i,
~s<ha..rc.hn,t~h<
prince
~thusp!.<.rc~ndmdi..Lti,~ct,tofU.. ]aU.r ,.u-L
h~, ~havo
~I~H.i..d
on
(-~
rq))acmg t])c ~oto-a! )t:u')no))ic ,S' bv
~ovc!<~t.ypotcnt~ofthow)m!c
.notion isf.n)n.t Lya,)it,on
r~a,.i
~,t~
c-nsta.ts
hci~do~ni~by
c.n~it.ons. ~)H,scfonn .Icp.nds ..pnn t).c ct.u-actcr of<).c'h<-n.y
t).c ob~-uc
t.r.nh.dbyth.s).hcro.
Tbc.sHnp)cstca.sei.st)~of~"ri.ri.l
an.
hxcd .sphc.rc,anj t),cn Ll.ucundition
to bc sati.silcd whcn 7. lec
i.sthnt
334.]
rnc![D
spHH)m'L
ousAC'~H.
241
As cxa.mp)c.swe may writc down th ternis in [~], invoiving hannonic. of ontcrs 0, 2. TIie futiuwing tab!e of the
futictinns 7~ (~) wi)) bc uscfu).
1C
L:.tl.
DJSTUBBANCE
DUE TO
f334.
PIOID
~d
SPUKRCAL
OLiSTACLK.
343
of~"
s<bccon')
~44
I\TE~8ITY
OP SMCON))ARV AVAVES.
'i')ieYc]ocit,y-pot('n)iaIof<))cdisturb:u)ce
f'.n.ni:\e(!n"~).'<tit~i\-t~r.j.})r..Aj,,)ith'!v,
[~34.
duct~asm~tngid
334.] ]
FURTIIHR
APPROXIMATION.
2455
thc
Byn)C:msof
(;!J) wc nmy vo'ifytitc rn-st two t~nnsin
In ()7),(t.S). ). T.t))cc:tsc<.t'~=-.(),(:)
(~pressions ir,r[~],j~],
<k)~.sn(jt;(pp)y.
th<:
Ag!un,hy (3)),
~C
i'R.SS(JR.S
ON OUSTAt.'LH.
{'334.
334.]
SOURCE AT FINITE
DISTANCE.
247
SYMMETRICAL
KXI'UH.s~ION
r~S~.
~'ichisthesfuneasifthc
source h~Lconoi
th spito-c.fmd
th point at w!)ic]i t!).' putcnti.d is requij-cd ~t gr~at (hstancc
i.s;n)cx;unp)c<jf-t)K.gnera! rr:neip)c
(.3~8),!U]d
ofReciprocity. J!y
n~m)iing tlic prineip!c,and n~kin~ )t.sc oft)tere.suJt(~) cf'~ :~)S,\vc
Hect).at if thu s..))rccuf tlic
pri)n!uy w:ivu.s bc a fi..itc.ti.sta'ticc
li, thc va)nc of~.e tot.al potcntiat:Lt nny p.,int on
t~.sphcrc i.s
which~iv~tl.ispartufthc
uthct')S!)unitso)))w.
334.j ]
FORSECONDA.RY
DI.STUHBANCE.
2-H)
f)'omwhidi wc pnss to tlie ca.sc of n. imite Ti*by t)tc sit'uptc inLroductiu)) ofthu i':tci<jr/, (t~~).
T)ms tlie poteot):).)at fi)tit,L'ty(KsttUitp~jint uf :Lmut.sotu'cc
at ~1 is
is of
Exterior to the Hphcrc, <~ is thc sanie cxac~y, a.)td
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,
UASEOUS
OBSTACLE.
r33~,
335.]]
CASEOUSOBSTACLE.
25 L
as thc expression for th rno.st important part of thc disturbaucc, corrcspondixg to (~1) 334 for a nxed rigid sphre. Ibt
as inight hve bccn expecto), that thc term of zero order
:ipp(;n.)'s,
is ducto thc variation ofcomprcssibility, and tt)at of ordcr one to
th variation, ofdettsity.
From (13) wc tnay faUhach 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 th same time
rem:t.i))cdhnitc, ~e'cwoutd not bc smaH, as thc investigation bas
assumed.
Wlicu
KQL'AL C'OMPRESSIUILITIES.
t):~5.
Inactu;dg!LSG.s~'=~,an.tthct(-r.nof~)-oon)crdi~ppcars.
If tiic
~as occupyin~ t)tc sphcric:d .sp;tcobG inca)np:D-ab)ylimiter
tti<mt!tC!ut,)iL'rgits,er'=(),an(t
XVIH.
CIIAPTER
Kt'HHRICALSHUETS0F AIR.
~.J-t
(:H\E);AL
DJFJ.'KJf.EXTJALJ.Tjcx.
r.3:;G.
L.
<).. c.~c.c.nt.s
f,ti,
,f
i
.v~-c
an.) hy
of t),e <ircu)ar ft)nct.ioi]s, cnch
titonju~atcpn.pc.-t.y
<crm of tlie .scnc..s must
.sati.~y thc c~u~~u
indcpcndcut)y
Accot'dn~ty,
t~.siJcs bc.i~synunct,ie.]
r.sp.ct t: tL
a
th;it
33C-]
coxDiiox
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 th nrst seriez, multiplicd by t))c arbitrary
constant ~1. This va]uo of tho vctucity-potnLia! is thc
It~icnJ
of
t)tc
consquence
onginal (Uifc)\;tit.i:dc-quation {uid of thtwo
restrictions as to symn~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
sries of particular values. T])e condition, which yet remains
to 'bc introduced and by which /i, is dctcrmincd, is that thc
original quation is satisned at thc po)c itsc!f, or in othcr words
that th pole is not a. source and this rcquires us to considcr
t)ie value of th scrics when ~.=1. SIncc .thc scries is an
eveu function of
if thc pole ~=+1 bo not a source, ncithc-r
will bu th ple /~=- 1. It is (-.vidcutat once that if ~bo of
thc fonn ?t(M+l), whcrc is an even integer, tho scrics termintes, and therefore romains nnitc whcn ~=1; but what wc
no\v want to provc is tliat, if th sries renmin unit
forjM.=I,
is neeessarily of th a.bovc-mentionedform. By th
ordinary
rule it appears at once that, whatevcr be the vahto of /t"
th ratio of successive terms tonds to the limit
and therefore thc scries is convergent for ail values of;u. less than
unity.
But for th extreme va)uc /<-=]f, a highcr method of discrimination is neccssary.
It is known' tliat t!tc infinit hyporgeometrical sries
is convergent, if c+~&
be grcatur t)mn .1, and c1ivc)'r''cnf.
if c+~-H~
bc (~(~u~Itn, or )css t)tan ]. Jn thu Ia.ttcr case
th va)nc of c+~
itttbrds a critcrion of t)jc dc~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 prsent scries (7)
may bc rcduc~) to thc standard fonn
Ly taking ~=7! (7;)-]~ whcrc )t is not nssutocd to bc intpf-nJ
Thu.s
nn<i)c''f~'t')))'/n')t''f.<, ]).7f).
('RtTHRtox
OF D!VEnr:CV.
f'33G
3:} G.]1
TRANSITION
TO TWO DIMENSIONS.
257'
Todhuntcr's
~7
~58
VIBRATIONS
0F
A SPIIKRICAL
.SIIEET
[33G.
convergent.
If ~c portion of)]ic.surface
Le thaLindu.]~
ocL-pi~])y~s
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.-f
t'.c constants
and 7~in (7~ vani.si.u.s in t).c case of c-adi
nor.ual
lunctiot).
337.
~)'L'nthc.sphcrica)arcac<.]t~int)]ato(tindndMap..]c.
we hve, as
I.i thu case
uft).u.up)utcsp),L.n.,tointr..ducct}.u
a.source. F~th~puq~s~sducond.tionthatthcpulei.suot
tioniMtcr)nsof~i.c.sin<?,wi))Lcnior<convc))iunt.
If wc restrict oursetvcs for Otc
prc-suut to Hic case ofsynunctry
wc liavc, putting = 0 in
(~) ~3G,
337.]
By writin~ )) ("+1)
259
~GO
UNMVMMHTRICALT. MOTION.
strai~ht, crhccun-<.d
['337.
to an arc nf M".
An ")<iq~)).h.ht!))V..sti~t!<nta)~.so]))ti<.hfot-tI)c
wi)ihugi\'('np)-cs(.)tt)y.
~{S.
~<
,sis<]i~.n.nt
phtnc.praLbm
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
~itiY.i,,t~r.
Th.nu.th~of
n.
..)~
r.\(')~j~hcd)U'(ht).yt))<ji(.<c,,ft)j(.),hm(.);).,
procure
will ).c
UMYMMETMCAL
338.]
MOTrON.
2G11
un Oif:
f~
Thu
corfticicnt~ftht'
Inwcstpo~rof
T)t(.'r~)ati<j)) but.wct.'a
:(=(), or ?=).
iH
~2
COXDITIOXS
TO BE SATFSFIKD
[338.
Wuha\~]i.)\vtu})ruvc
th:ttt.).ucut)dit.iunt.)t!Lb)~.it.)H.Tpu]cis
a.source ru<)uirc.st)i;~ M- hua
positive i)it~r,m
w)u'c)i~sc
une 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..uy
tu pruvu U~L L),y ronnitt div~r~-nt :d't.t.r
nmitipiicfLtifnt by
ura.swc
(1-~)~
J'):Lyj)ntiL))H)rL;c.)nvutuunUy,t)i:(tt))'y:u'u
ItwHibu
innn[~whL'n~=il'cf~~<<r~?;A(lsuf)iciunt to cousider iit dtail t)iu casu uf t.hu fli-tit scrics.
Wchavc
:~8.j
2G3
Sincc s 1 > .~s- 1, it. nppc:u's U)at thc so'tcs iu thc expression
:).ud Lhurcfor<~ aru inHnitiL's of ]ti~ht'r ordur t.)):m (l~)"
f()rurc)naiui)ifmit.u:Lft<t'ntuttipi[(.'i).t.U)tt))y (!)~.
Accordingty
canitut bu finitu at Luth potc.s utituss onc ur ut.itcr of thu .scries
).~)')nin~tu,\v)H':hc:m)))yii:tp))c)t\vI[cn~s'isxL;ro,c).'a.))o.sitiv
L'vc<),wch:Lvestillt.t) suppose/~=0;
h)(t.~cr. If Lhuint.)'bu
:unt it' thc intu~L'r be odd, ~i=(), in order to sccm'c linitcuess at
UtUpufL'.S.
Jn ciLhcr c:)sc t)) value of
put inLo L)tCiur))l
sphcre may bc
whcrc thc constant mi)I(ip!icr is omittcd. Tho complete cxpresHiun l'or t)):Lt part of wtuch contains cos~'Mor sniS) as :),factor
li-ithurcfot'c
2G4
FORMULA. OF DERIVATION,
which is Laptacc's
Froin
(':),itis
expansion
in .sphoical
f'3~8.
surf.LCC )):u')no!)ics.
(;~), or its
c;)..sytop)-ovGth:)t~i!-i
nfthcsii.muforHia.s
t)i:).t.\vctn:)v
g~n(.')-:t) solution
<)!),so
writc
.s functiuns'.
1.
of a plane iaycrofgns
iu'cofcou~c
t))usuofaJay(.T<)i'ih)itceurvatu)-c,Lut,
T(~hn))tt.')'<~f~~r<<t;(');
;)!)().
more
VIBRATION
:~38.]1
IN TWO DIMENSIONS.
2G5
spondmgtotitc
cnvclopc.
couinement
coudltio)i
ofthcg~sby
<
'=0.
ci'rc-
ft.rigtdcylmdnca.I
~GG
HIGID CIRCULER
BOU~DARY.
[339.
h'rnn) circu)!H'))~(h'S.
tt-0 p
-/t 1
0
1
:}-.S;!2
7' ~i
t-.S!l
5-:i~~
I()')7~
S-)C
3
4
3
l:t
I(:')7[
(1
H'-Ci':
H-7<)<!
]t-.S);j
].s'u!(;
M-~2
:)~.t
C-7"~
i't-.33
.j.~oi
8'()L~
u.U!;5 ll~).i
Thup:n'ticuhu'm)htLiuH)i):~yLuwnt.t.ut
'X..trs~t)tc.s~r.sL'nuc~u:
/))')/,<
Kuv.l~.
339.]
CASE
OF
COMPOUND
VIBRATIONS.
2G7
Siuco
must
beofthefonu
whcrc !=!7T~f
beinguttcgnu.
Itpbu~nahtptot
partot'
7r (oi'Tr itscif), t))<jcom~!utc so~utiou mvulvcs on)y mt~nd values
tjf
as tni~ht )i:).vc Lucu iurcsL'ot but, iu gnera)) fnuct.tuns oi'
tractional urdc).'must bc mtroduccd.
Au ititct'estuig cxampic occurs \Y!)cn ~=27r, wfuch corrcspftuds tu t.))c casu of a. cytindcr, travct'sc() by a rigi't watt
2G8
ANALOOOUS
PHOHLEMS rOR
WATER WAV).
[339.
Tho
""<)t])cadniisHih)o\ah)t'S()f'~a)-uthn)-outs(jf<:)n~
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
whf'rctitccocfifu.nts
Tiusfonn
7~ mayhc fonctions of?-:ut(~.
su(-ur<jst]tcf)[tfi!)nc))tr.f't))L:1)u))))d:n-yc<jm!iti())).s,w])(.'ns=0,~=/, I,
3K).]1
VnmATOXSIN A CLOSRDCYfJNDHR.
2G9
winchis<'f'<]h'.S!mt(!fo)'tn
s, ~)H'i))~')'t.')'i!n'r'thy/<7:
t))urt.'t'(j)'cbcwrittcn
as \hc)t).h);
motion isuxit'pntiuntof
'i'))~p!U'ticuhn'MnhtLit))t ))'ay
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<))),
w))('nth<')'(!isno)i)nit:)<.i~))!t.s
to th .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)
Whc'n
bntt))uH('c')ru)s(.'ric-s]'c()uii'c's)n<)dific:tt)0)),if))bc!i)ttc'gr:d.
/<=(), <hc hvo sries bccom ith-'utica), nnd tims thc imnic~ia.tu
Th~
)'~utt~t'.s))))p()si))g~=()in(2)t:Lch.sthncccss!U'ygcner:Uity.
270
C!HRAL SOLUTION.
['341.
:i.]]
HXt'HH.SSION BY DESCEND)~;
S)':I{')KS.
271
which is equivaicnt.to
mb~t)t<(~ta)i('n.s.
iu-earbitrary
DIVHiKiHXT
WAVE.
f:~n.
L..
+~.(~)}
s~U~taHth~rcma~sistufmdthc~nnofthcddn~cintc~~t
ni (l- ~hen ..s s.na)i.
P~ti,~ ~~+~=~
{~~
341.] ]
DIVERGENT
WAVE.
273
274
SOUNDINGBOARDS.
[341.
341.]
275
from which thc vfirious cxjx'cssions fu!!ow a.s in (14) 341. When
~r is grcat, an nppn'ximit.t~ ViUueof t!ic mtu~r:).l may ue obtaiucd
sinco on [(.ccount of the
by nc~ecttng titc vm'iatioa cr \/(2r+y),
wu uecd attend
rapid Hnctuntiou uf sign caused by th factur e'
r/(t!. yra'
18C8.
183
276
LINEAR SOURCE.
[342.
343.]
CYLINDRrCAL
OBSTACLE.
277
278
CYLINDRICA.L
OBSTACLE.
[343.
Th tact that
varius invcr.sc)y .as \"S might Ijave been
anticipatcd by thc motitod uf duncn.sionsas in thu con-espondmg
proDum fur the sphcrc (33.-)). As in that case, thc synunctric:3
pfn-t of tlie divcr~nt w:L\-cdopcnd.s upon t)tc van~tiuu of comp.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 corresponding to a rigid and iM)novcab]cobstacle,
343.]
PASSACiH0F SOUNDTUROLTGIIFABRICS.
279
C1IAPTER XIX.
FLUID FRICTION. PRINCIPLE 0F DYNAMICAL8IMFLARITY.
344. TiE quations of Chapter XI.aud the consquencesthat
wehavcdcducett fromthctn arc b~scdupon thc assumption (230),
that the mututd action betwccn anytwo portions of fluid separatcd
by au imagina.rysurface is normal to that surface. Actua.1Sui<tn
howcvcr do not corne np to thi.s idal iu many phcnoinona thc
dcfcct of Huu)it.y,usually callcd viscosityor ftuid friction, plays an
important a,)idevoi a prcponderating part. It will therefore Lo
proper to inquire whctijcr the laws ofacri:d vibrations are sensibly
iunuenced by tlie viscosity of air, and if so in what mauncr.
In order to understand clearly the nature of viscosity, let us
conceivea nuid dividcd into parallel strata. in such a manner that
wItHecach stratum moves in its own plane with uniform velocity,
a change of velocityoccurs 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 incrcasc uniformly
ht magnitude as wo pass along a linc perpendicular to tlie planes
of stratification. Under thse ch'cumstances a tangential force
betwccn contiguo~tsstrata is caHnd into play, in the direction of
the relative motion,and of magnitude proportional to th rate at
which the velocitycitangcs, and to a coeflicicut of viscosity,commonly dcnotcd by the lutter
Thus, if the strata bo paraUcI to
a'~ and t)ie directionof thch' motion bc pamUulto titc tangcntia!
force, reckoned (iikc a pressure) pcr nuit of area, is
345.1
~j
FLUID FRICTION.
281
cordance
with tlie kinctic theory of gascs M resulting from mterof
change of molcules between thc strata, giviug riso to diffusion
mon~ntum. Both by theory and experiment th rcmarka-ble
conclusion haa been esta.bliahed that within widc limits th forco
is mdepcndent of tlie density of th gas. For air at Centigrade
Maxwell' found
tlie centimtre, gramme, and second being units.
345. Th investiga-tion of th equations of 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 th more
geMera.llyknown tlieory of solid cla~ticity.
Thc potential encrgy of unit of volume of uniformly stra.incd
2
bc
mattcr
exprcssed"
may
isotropic
1 On tho Viscosityor Interne Friction of Air nnd other Gases. Phil. 2'rM.
18CC.
T'oxo~/ty. AppondixC.
Thomsou and Tait's ~<f<xr<~
S 82
EQUATIONS0F MOTION.
f345.
345.~
rr.ANn WAVES.
283
cotisidered
Wcmn.y observe ttm.tthcdis.sip~tivcfurccsiturc
correspond tu :). dissipation fnnctiuri, whosc furnt is thc .santCwit)i
with rospect to a, /3, y, i)i ttie ttiuut'y
u,?u vs tlt:).t of
rL'spcct tu
ui'i:jotropic soiids. Thus puttin~ A:= 0, wc Itavc frotu (1)
284
EFFECTS
OF FRICTION.
[346.
lu tho application to air at ordinary pressures ma.y bc conHidcred to bo a vury smaM qu:mtity and its square may Le
ue~lected. Thus
Observations,
1877.
34G.]
TRANSVERSE
VIBRATIONS.
285
28G
PROPAGATION
0F
SOUND
[347.
would imply nn
nppa,rent!y snfHcicnt g)'f)nnd thiittLcc'mtnn'y
itif-mitc'y ~-ren.tersjnc'ithofss of <n (htid w;th "(..sp~f-t t~ t.hi! soiit'
ti):m wit.)t respect tu itscU'. On t.)ns
supposition (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).
Thc tangcnti:d force pet- unit a)'c:i
n.cting on th 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")'cn G<~)iv:Jc-ntto an
ino-caHOin thc ino-ti.i of t.])c vibrating body. Thc m~mtudc uf
both .t'urccs(h;pcmds upon thc fn-f~tuncy ofthc vibration.
Wc wi)i app!y th)src.sulttoca!cu!atcappt-ox!)n:t,t(j!y th VL-iocity
of sound in tubes so Ufu'row tbat t)K' vi.sco.sitynf air (.'x~rciscsa.
scnsiDu mfiucncc. As in 2(!), !ct JV dnote 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 th sticc bctwcun a; and
.e +
i.s, as usua),
347.]
1~ NARROW TUBES.
287
whore /<is Newtons va)ue of thc vcioeity of sound, and t/ is a cocfHciunt of conductiun, equal according to th kinctic thuory of
gasusto~p" `.
Thu dintinntiou of thc vcloclty of sound in nn.n'ow tubes, aa
iudiciitcd by t.hc w: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 -variesas?' in nccordancewitb (12), but, wbcn n varies,it
is proportionat rat]ter to K'~ than to
Since is indcpcnduut
of thc dunsity (~)),t]ic effoctwou!d bc incrcasodin rarcficd air,
34'8. In the course of this work wc Lave )iad frquent occasion
to notice the importance of the conclusionsthat may be arrived at
by the mcthod of dimensions. Now that we are in :t. position to
draw Hhtstrations from a grcatcr varicty of acoustical phcnomcn:),
re!a.t.ing-to th vibrations of both so)ids and iiuids, it will be convenicnt to rcsume th subjcet, and to dcvefopcsonewhat in dtail
the principes upon -\v]tic]ithe mcthod rcsts.
In th case of Systems,such as heHs or tt)ning-fo)\ks,formed of
uniform isotropic mntcna!, and vibmting in virtuc of cJasticity, th
'7'n~)t).t.cxxx!7.177. 1RM.
~7'f~t)f.t.cxxx!X.l(tL 1870.
''ro~in);.t.cxxxvi.29(!. 1SC9.
288
DYNAMICAL
8IMILARITY.
[348.
DYNAMICAL
348.]
SIMILARITE.
289
THE END.
R. !.
1<)
APPENDIX
A.
( 307).
?' dcnotmg thc distance of tlio point cojisido'ed ft'ovntho centre of tho
Mul so as to tunko t)tc whole eucrgy tt
mouth, and thcn dctcnuino
minuuuin. Ttte cncrgy so ca-lcult~cd,tho~tghucccssurily in cxccss, must
bo a very goodupproximn.tionto t!tu tt'uth.
In can'yingout this phui.'wohft.votwo distinct pro~lems to de;d wit.)),
l,
tho dotenninatioli of tho motion (I) ontsidc, tuid (2) it~ido t))o eylindcr.
Th former, bcing t!io e~Iei', wo will takc (Irst.
TIte conditions are tha.t ~) Y{mis!ia.t uiduty, n.ndthat wlien = 0,
MtC
whcre
van.iBh,except over tho area,of thc circle r=
l')2
2S2
CORRECTION
Is to bo estiinated
Thn vahin hf 7' is tu l'f c~cuhtt.f~ hy t.hc ~K'Lhod onpioyud in Lho t,(.t
(~ 307) for t).unifortu dt'n.sity. At, thu edgc uf tho dise, witeji eut du\)t
tu radius f<,wo)t!L\'(:L)t(;puLL'ntiai
Th tut:d cm'n'ut,
293
~= 7-015,
p,i=l(W71,
~=10'17~.
~=19-C1C.
294
KothiLtt.hoscccmdtot'mis 7!7(1+~+~')".
Iti catculating tho ~rst tcnn, wc must rcnx'mhcr thitt if
two difTuL'cnt
values of~,
nnd~ tx:
FOROFENENDS.
CORRECTION
2955
and 0', A', arc dm'tvcd from 0 and A by nu.et'changing thc acccntcd and
unacccntcd Ictturs.
lu t)io prsent case, si)ice<S"is a product of liucar factors, A'=0,
A 0 sunpiy.
0' =- 0, so that
iLud aincc th two factors arc thc MiLUtt-,
tho tmuu'ricul v:dm.s, !md <-f)'uctingLhu c:).!c(dado!is, wc
Substituting
(iud = = '~!898G~,whiu)i M tho nm.xiuumLY.duo uf Lho fracdi). uonsistcuL
wibh i'< values of !(.ttd
Tho con-cHpoudmg Y:duo of a is -821:227. LhtUi winch Lhc tnn:
corrcctiti ciumob bc ~ruatL'r.
If wo assumu
-0, tliu ~rua(,(.'st Ytduc of.: titcn possible is '021~63,
whn:!i fi\us
whh:h
givus
a .'8281.iG7~.
On th 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 parL f'f thu
norinid vclocity at thc nnjuth is buttai' t-~pruscatud t'y a tct-m vm'yio~ as
tliau by une vnryihg as
Jf thu
TI~ value M '82 12~ is probably p)-(.y clo.sc to thf! truth.
ifofLitnji-m 1
normal vclocity bu assmncd constant, a-'8~S2GA',
aud wi~'u Uh;
isHuitabtydctcrntined;
w)f'n
1+u.?'~ a-'828157~,
iuiutbcr arbitrary constant, is !na<h:
forni l+/jL!~+~
e'jntami)~
tho fomulation of thc ca!cn)atiou, w" gt;t a --82i2A'.
Tlic truc Ya]uc ofa is pt-ubab)y about '82/t*.
In thc case of
l'U.):
so tliat
On thia snpposition thc 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;
in
!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
wciitul on tnu)ni'<j)')n!).Lut.w('t'nH~)t(-rica)H)n-f)te('i()i'ra(1ii)'aud
!'+<
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'ioUsurr~cf;of nuiiu.s )'. Eqnatiun (n) nuiy
Lo i'(~)H-(tcdas 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-ordinatcs ?', M.
TJic gcm'nd soluttou of (fjt)iH
wp
H!)7
7'oc<'c~ty.'i
/~r. ~'u.ll').
</; Zo;t(~
a:),
2:~8
PROGRESSIVE
WAVHS.
whichiHuquidt-u
If K'-K,
])o Hinft)), wc htivc n. tmin of W!).vcswhosc amptitudu
vanus slowly from ono point to Miotlier hctwcen tho limits 0 tmd 2,
formiug <).scrics of groups separatcd from ono anoUtor hy l'cgiona cotnpM-attVc!y -ec froni di~turhanco. TLo position at tuno t of tho middiu
of Umt group, which w~s imti:L])y at th origm, iHgivcu by
UmveiocityoftItC!
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 worth notice, a.ud :u'o hcro tabulatcd fur convctiicuco of comparison
Fcc,
fee~,
ree.
)' M .
6'
r =c ",
6'
~-0,
~)
r,
r,
2 r,
]!('yno!ds'di.sconncetedpcndut)))ns.
Dt'<'p-wat<jr~ravit,yw:Lvcs.
A(.')'ia)-a\-('M,Ac.
(.')t])iHary watur wavcs.
Figura) waY.;s.
191).
PROGRESSIVE WAVES.
299
Tins vainc of
Yclci~y-~zuro~-hen
r.+~(~),
7/ dcnoting thc !)iax!i~mn c!cv:Lt.i(jn.
rruf. HcyMida eo!)HukMUtGtroeh.~M Wftvoof n~nkhtc nud Frotulc, which
mvoh'es tnakcuittr rottitifin.
~00
PROGRESSIVE
WAVJ.:S.
PROGRESSIVE WAYES.
301
Ly
<
./<
T))C
sohttion
t!m3
bccojucs
,.ff)fdl<
part ia rcjcctcd, e
A:
e
,1'
'eo8(~<-<.T),
(;'0'<r),
or,
wl(cn
80 t]<at
T))(''mtioofthf'('ncr~vtra))sn)ittf'dinth<'
thc
~<
int.~gina.ry
<
and
unirtinK't"
J'KOCKES.StVH
WAVES.
302
11\!pI'OY(,(1.
was to hc
113wns
as
provc-d.
CAMftIUDO!
t'K)!)TKD
i)Y C. J. CLAY,
H.A.
AT THE
UStVKtta)TT
MESS
TUE
THEORY OF SOUND.
VOL. I.
CO. LONDO~.
In. Crowii
8vo. pricc
S~. G~.
AND
MUSIC.
Two Lectures
By Dr W. n. SON).
dc!tVcrcdat.Snut)tK(;)t.si))~tun. muhtt'~tcd. Cruw)tM\-o. (!<
LECTURES
ON SO~rE
RECENT
ADVANCES
JN PHYM!CAL SC'rKXC!
~y I'r.,fc.s.~)rP. C:. TAfT, M.A. H)n.stratud. Sucund
Edition, ('nhu~ud. L'ruwn.iv~ !).<.
THE
APPLTCATrON.S
OF PHYSICAL
FORC'ES.
Trans~trd hy Mt-.s L.~yt'r, aodcditcdwith
1iyA.(:Unj,EM)X.
Additions a))dXut.('().yJ.I~.ckyu)',F.J;.S.
\VithCut<)n)\.))P!a)<-sat)d
inuori'~u.sinustr.ditU~.
Hny.d~v~. :}).<.(!
~).\('fLLA~
AX)) ~a
3
L~Xno\.