（精选）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 1 Theequationofmotion 1 theequationofmotion Euler sequation First wewritetherelationbetweensoundpressureandvelocity Considerafluidelement 2 Whenthesoundwavespass thepressureis SotheforceonareaABCDwillbe istheforceofperunitarea TheforceonareaEFGHwillbe ThenetforceexperiencedbythevolumedVinthexdirectionis 3 AccordingtoNewton ssecondlawF ma theaccelerationofsmallvolumeinxdirectionwillbe Forsmallamplitude wecanneglectthesecondordervariableterms 4 When Forsmallamplitude Similarly inthedirectionofyandz wecanobtain 5 Nowletthemotionbethree dimensional sowrite isgradientoperator SinceP0isaconstant andobtain Thisisthelinearinviscidequationofmotion validforacousticprocessesofsmallamplitude 6 2 Theequationofcontinuityrestatementofthelawoftheconservationofmatter Torelatethemotionofthefluidtoitscompressionordilatation weneedafunctionalrelationshipbetweentheparticlevelocityuandtheinstantaneousdensityp 7 Considerasmallrectangular parallelepipedvolumeelementdV dxdydzwhichisfixedinspaceandthroughwhichelementsofthefluidtravel Thenetratewithwhichmassflowsintothevolumethroughitssurfacemustequaltheratewiththemasswithinthevolumeincreases 8 Thatthenetinfluxofmassintothisspatiallyfixedvolume resultingfromflowinthexdirection is Similarexpressionsgivethenetinfluxfortheyandzdirections 9 Sothatthetotalinfluxmustbe Weobtaintheequationofcontinuity 10 Notethattheequationisnonlinear therightterminvolvestheproductofparticlevelocityandinstantaneousdensity bothofwhichareacousticvariables Considerasmallamplitudesoundwave ifwewritep p0 1 s Usethefactthatp0isaconstantinbothspaceandtime andassumethatsisverysmall 11 Weobtain Similarexpressionsgibethenetinfluxfortheyandzdirections 12 Where isthedivergenceoperator 13 3 Theequationofstate WeneedonemorerelationinordertodeterminethethreefunctionsP andu Itisprovidedbytheconditionthatwehaveanadiabatic 绝热的 process thereisinsignificantexchangeofthermalenergyfromoneparticleoffluidtoanother Undertheseconditions itisconvenientlyexpressedbysayingthatthepressurepisuniquelydeterminedasafunctionofthedensity ratherthanadependingseparatelyonboth andT 14 Generallytheadiabaticequationofstateiscomplicated Inthesecasesitispreferabletodetermineexperimentallytheisentropic 等熵 relationshipbetweenpressureanddensityfluctuations WewriteaTaylor sexpansion WhereSisadiabaticprocess thepartialderivativesareconstantsdeterminedforadiabaticcompressionandexpansionofthefluidaboutitsequilibriumdensity 15 Ifthefluctuationsaresmall onlythelowestordertermin Needberetained Thisgivesalinearrelationshipbetweenthepressurefluctuationandthechangeindensity Wesuppose 16 Inthecaseofgasesatsufficientlylowdensity theirbehaviorwillbewellapproximatedbytheidealgaslaw Anadiabaticprocessinanidealgasisgovernedby Hereristheratioofspecificheatatconstantpressuretothatatconstantvolume Air forinstance hasr 1 4atnormalconditions 17 Forideagas Inthesoundfieldofsmallamplitude 18 Speedofsoundinfluids Thisistheequationofstate givestherelationshipbetweenthepressurefluctuationandthechangeindensity Wegetathermodynamicexpressionforthespeedofsound 19 Wherethepartialderivativeisevaluatedatequilibriumconditionsofpressureanddensity Forasoundwavepropagatesthroughaperfectgas thespeedofsoundis Forair at00CandstandardpressureP0 1atm 1 013 105Pa Substitutionoftheappropriatevaluesforairgives 20 Thisisinexcellentagreementwithmeasuredvaluesandtherebysupportsourearlierassumptionthatacousticprocessesinafluidareadiabatic Theoreticalpredictionofthespeedofsoundforliquidsisconsiderablymoredifficultthanforgases Aconvenientexpressionforthespeedofsoundinliquidsis Bsisadiabaticcompressionconstant 21 Thewaveequation Fromtherequirementofconservationofmatterwehaveobtainedtheequationofcontinuity relatingthechangeindensitytothevelocity formthethermodynamiclawswehaveobtainedtheequationofstate relatingthechangeinpressuretothechangeindensity 22 Byusingonemoreequation theequationofmotion thatrelatingthechangeinvelocitytopressure Weshallhaveenoughequationtosolveforallthreequantities 23 Thethreeequationsmustbecombinedtoyieldasingledifferentialequationwithondependentvariable 24 Insmallamplitudesoundfield wecanneglectthesecondordersmallquantity sothat 25 Weobtain Form Equation 3 4 isthelinearized losslesswaveequationforthepropagationofsoundinfluids cisthespeedforacousticwavesinfluids Acousticpressurep x y z t isafunctionofx y z andtimet 26 Where isthethree dimensionalLaplacianoperator Indifferentcoordinatestheoperatortakesondifferentforms Rectangularcoordinates Sphericalcoordinates Cylindricalcoordinates 27 Thevelocitypotentialofsound Fromtheequation 3 1 weget Whererotisrotationoperator 28 Sothevelocitymustbeirrotational 无旋的 Thismeansthatitcanbeexpressedasthegradientofascalar 标量 function 29 where isdefinedasthevelocitypotentialofsound Thephysicalmeaningofthisimportantresultisthattheacousticalexcitationofaninviscidfluidinvolvesnorotationalflow therearenoeffectssuchasboundarylayers shearwaves orturbulence 30 Indifferentcoordinatesittakesondifferentforms Rectangularcoordinates Sphericalcoordinates Cylindricalcoordinates 31 Differentiatingtheequation 3