理想媒质中的声波方程.ppt

THREEBASICEQUATIONS 理想媒质中的三个基本方程 1 Theequationofmotion 1 theequationofmotion Euler sequation First wewritetherelationbetweensoundpressureandvelocity Considerafluidelement Whenthesoundwavespass thepressureis SotheforceonareaABCDwillbe istheforceofperunitarea TheforceonareaEFGHwillbe ThenetforceexperiencedbythevolumedVinthexdirectionis AccordingtoNewton ssecondlawF ma theaccelerationofsmallvolumeinxdirectionwillbe Forsmallamplitude wecanneglectthesecondordervariableterms When Forsmallamplitude Similarly inthedirectionofyandz wecanobtain Nowletthemotionbethree dimensional sowrite isgradientoperator SinceP0isaconstant andobtain Thisisthelinearinviscidequationofmotion validforacousticprocessesofsmallamplitude 2 Theequationofcontinuityrestatementofthelawoftheconservationofmatter Torelatethemotionofthefluidtoitscompressionordilatation weneedafunctionalrelationshipbetweentheparticlevelocityuandtheinstantaneousdensityp Considerasmallrectangular parallelepipedvolumeelementdV dxdydzwhichisfixedinspaceandthroughwhichelementsofthefluidtravel Thenetratewithwhichmassflowsintothevolumethroughitssurfacemustequaltheratewiththemasswithinthevolumeincreases Thatthenetinfluxofmassintothisspatiallyfixedvolume resultingfromflowinthexdirection is Similarexpressionsgivethenetinfluxfortheyandzdirections Sothatthetotalinfluxmustbe Weobtaintheequationofcontinuity Notethattheequationisnonlinear therightterminvolvestheproductofparticlevelocityandinstantaneousdensity bothofwhichareacousticvariables Considerasmallamplitudesoundwave ifwewritep p0 1 s Usethefactthatp0isaconstantinbothspaceandtime andassumethatsisverysmall Weobtain Similarexpressionsgibethenetinfluxfortheyandzdirections Where isthedivergenceoperator 3 Theequationofstate WeneedonemorerelationinordertodeterminethethreefunctionsP andu Itisprovidedbytheconditionthatwehaveanadiabatic 绝热的 process thereisinsignificantexchangeofthermalenergyfromoneparticleoffluidtoanother Undertheseconditions itisconvenientlyexpressedbysayingthatthepressurepisuniquelydeterminedasafunctionofthedensity ratherthanadependingseparatelyonboth andT Generallytheadiabaticequationofstateiscomplicated Inthesecasesitispreferabletodetermineexperimentallytheisentropic 等熵 relationshipbetweenpressureanddensityfluctuations WewriteaTaylor sexpansion WhereSisadiabaticprocess thepartialderivativesareconstantsdeterminedforadiabaticcompressionandexpansionofthefluidaboutitsequilibriumdensity Ifthefluctuationsaresmall onlythelowestordertermin Needberetained Thisgivesalinearrelationshipbetweenthepressurefluctuationandthechangeindensity Wesuppose Inthecaseofgasesatsufficientlylowdensity theirbehaviorwillbewellapproximatedbytheidealgaslaw Anadiabaticprocessinanidealgasisgovernedby Hereristheratioofspecificheatatconstantpressuretothatatconstantvolume Air forinstance hasr 1 4atnormalconditions Forideagas Inthesoundfieldofsmallamplitude Speedofsoundinfluids Thisistheequationofstate givestherelationshipbetweenthepressurefluctuationandthechangeindensity Wegetathermodynamicexpressionforthespeedofsound Wherethepartialderivativeisevaluatedatequilibriumconditionsofpressureanddensity Forasoundwavepropagatesthroughaperfectgas thespeedofsoundis Forair at00CandstandardpressureP0 1atm 1 013 105Pa Substitutionoftheappropriatevaluesforairgives Thisisinexcellentagreementwithmeasuredvaluesandtherebysupportsourearlierassumptionthatacousticprocessesinafluidareadiabatic Theoreticalpredictionofthespeedofsoundforliquidsisconsiderablymoredifficultthanforgases Aconvenientexpressionforthespeedofsoundinliquidsis Bsisadiabaticcompressionconstant Thewaveequation Fromtherequirementofconservationofmatterwehaveobtainedtheequationofcontinuity relatingthechangeindensitytothevelocity formthethermodynamiclawswehaveobtainedtheequationofstate relatingthechangeinpressuretothechangeindensity Byusingonemoreequation theequationofmotion thatrelatingthechangeinvelocitytopressure Weshallhaveenoughequationtosolveforallthreequantities Thethreeequationsmustbecombinedtoyieldasingledifferentialequationwithondependentvariable Insmallamplitudesoundfield wecanneglectthesecondordersmallquantity sothat Weobtain Form Equation 3 4 isthelinearized losslesswaveequationforthepropagationofsoundinfluids cisthespeedforacousticwavesinfluids Acousticpressurep x y z t isafunctionofx y z andtimet Where isthethree dimensionalLaplacianoperator Indifferentcoordinatestheoperatortakesondifferentforms Rectangularcoordinates Sphericalcoordinates Cylindricalcoordinates Thevelocitypotentialofsound Fromtheequation 3 1 weget Whererotisrotationoperator Sothevelocitymustbeirrotational 无旋的 Thismeansthatitcanbeexpressedasthegradientofascalar 标量 function where isdefinedasthevelocitypotentialofsound Thephysicalmeaningofthisimportantresultisthattheacousticalexcitationofaninviscidfluidinvolvesnorotationalflow therearenoeffectssuchasboundarylayers shearwaves orturbulence Indifferentcoordinatesittakesondifferentforms Rectangularcoordinates Sphericalcoordinates Cylindricalcoordinates Differentiatingtheequation