FUNDAMENTALS OF ACOUSTICS(8)

BASICACOUSTICS(8)THREEBASICEQUATIONSTheequationofmotion•1.theequationofmotion(Euler'sequation)•First,wewritetherelationbetweensoundpressureandvelocity,•ConsiderafluidelementzyxVABCDEFGHyxzzxyF1F2ABCDEFGHyxzzxyF1F2Whenthesoundwavespass,thepressureis0(,,,)Ppxyzt10()FPpyz0Pp20()FPpPyzSotheforceonareaABCDwillbeistheforceofperunitareaTheforceonareaEFGHwillbeThenetforceexperiencedbythevolumedVinthexdirectionis•AccordingtoNewton’ssecondlawF=ma,theaccelerationofsmallvolumeinxdirectionwillbe12xFFFPyztudtduxxdtdzzudtdyyudtdxxutudtduxxxxxForsmallamplitude,wecanneglectthesecondordervariableterms,zyxVm•WhenxuPyzxyztxuPxt0xxPxPx0limtuxPx00ForsmallamplitudeSimilarly,inthedirectionofyandz,wecanobtaintuyPy0tuzPz0•Nowletthemotionbethree–dimensional,sowrite0uPtPp0dupdtijkxyzppppijkxyzisgradientoperatorSinceP0isaconstant,andobtainThisisthelinearinviscidequationofmotion,validforacousticprocessesofsmallamplitudeTheequationofcontinuityrestatementofthelawoftheconservationofmatterTorelatethemotionofthefluidtoitscompressionordilatation,weneedafunctionalrelationshipbetweentheparticlevelocityuandtheinstantaneousdensityp.•Considerasmallrectangular-parallelepipedvolumeelementdV=dxdydzwhichisfixedinspaceandthroughwhichelementsofthefluidtravel.•Thenetratewithwhichmassflowsintothevolumethroughitssurfacemustequaltheratewiththemasswithinthevolumeincreases.tzyxuxuxx])([Thatthenetinfluxofmassintothisspatiallyfixedvolume,resultingfromflowinthexdirection,istzyxuxx)(Similarexpressionsgivethenetinfluxfortheyandzdirections,tzyxuyy)(tzyxuzz)(Sothatthetotalinfluxmustbe[()()()]xyzmxyzuuuxyztxyz)]()()([zyxuzuyuxt0tttt0lim)]()()([zyxuzuyuxtWeobtaintheequationofcontinuity•Notethattheequationisnonlinear;therightterminvolvestheproductofparticlevelocityandinstantaneousdensity,bothofwhichareacousticvariables.•Considerasmallamplitudesoundwave,ifwewritep=p0(1+s).Usethefactthatp0isaconstantinbothspaceandtime,andassumethatsisverysmall,0•Weobtainxuuxuxxxx00)()(yuuyyy0)(zuuzzz0)()(0zuyuxutzyx0utSimilarexpressionsgibethenetinfluxfortheyandzdirections,()yxzaaaaxyzWhereisthedivergenceoperator0utTheequationofstate•Weneedonemorerelationinordertodeterminethethreefunctionsp,p,andu.•Itisprovidedbytheconditionthatwehaveanadiabaticprocess,(thereisinsignificantexchangeofthermalenergyfromoneparticleoffluidtoanother).Undertheseconditions,itisconvenientlyexpressedbysayingthatthepressurepisuniquelydeterminedasafunctionofthedensityp(ratherthanadependingseparatelyonbothpandT)()PP＝Generallytheadiabaticequationofstateiscomplicated.Inthesecasesitispreferabletodetermineexperimentallytheisentropicrelationshipbetweenpressureanddensityfluctuations.•WewriteaTaylor’sexpansion0220S,00d1()()()()()d2!PdPPPd０Ｓ,2＝WhereSisadiabaticprocess,thepartialderivativesareconstantsdeterminedforadiabaticcompressionandexpansionofthefluidaboutitsequilibriumdensity.0,()SdPdPdd00S,0d()()()dPPP＝Ifthefluctuationsaresmall,onlythelowestordertermin0Needberetained.ThisgivesalinearrelationshipbetweenthepressurefluctuationandthechangeindensityWesuppose•Inthecaseofgasesatsufficientlylowdensity,theirbehaviorwillbewellapproximatedbytheidealgaslaw.Anadiabaticprocessinanidealgasisgovernedby02,()SdPcd00VPPVHereristheratioofspecificheatatconstantpressuretothatatconstantvolume.Air,forinstance,hasr=1.4atnormalconditions•Forideagas,00VV00VPPV00000000dPPPPPdInthesoundfieldofsmallamplitude01d2000PpdPPPdcd0PPp0'＜＜Speedofsoundinfluids2'dpdcdtdtThisistheequationofstate,givestherelationshipbetweenthepressurefluctuationandthechangeindensity.Wegetathermodynamicexpressionforthespeedofsound0,sddPc•Wherethepartialderivativeisevaluatedatequilibriumconditionsofpressureanddensity.•Forasoundwavepropagatesthroughaperfectgas,thespeedofsoundis:002PcForair,at00CandstandardpressureP0=1atm=1.013*105Pa.Substitutionoftheappropriatevaluesforairgives•Thisisinexcellentagreementwithmeasuredvaluesandtherebysupportsourearlierassumptionthatacousticprocessesinafluidareadiabatic.•Theoreticalpredictionofthespeedofsoundforliquidsisconsiderablymoredifficultthanforgases.Aconvenientexpressionforthespeedofsoundinliquidsis51.4021.10310331.6/1.293cms01ScBsisisothermalcompressionconstantThewaveequationFromtherequirementofconservationofmatterwehaveobtainedtheequationofcontinuity,relatingthechangeindensitytothevelocity;formthethermodynamiclawswehaveobtainedtheequationofstate,relatingthechangeinpressuretothechangeindensity•Byusingonemoreequation(theequationofmotion),thatrelatingthechangeinvelocitytopressure.•Weshallhaveenoughequationtosolveforallthreequantities.Thethreeequationsmustbecombinedtoyieldasingledifferentialequationwithondependen