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1、THREEBASICEQUATIONS理想媒质中的三个基本方程1.Theequationofmotion1.theequationofmotion(Euler'sequation)First,wewritetherelationbetweensoundpressureandvelocity,ConsiderafluidelementF1F2F1F2Whenthesoundwavespass,thepressureisSotheforceonareaABCDwillbeistheforceofperunitareaTheforceonareaEFGHwillbeTh
2、enetforceexperiencedbythevolumedVinthexdirectionisAccordingtoNewton’ssecondlawF=ma,theaccelerationofsmallvolumeinxdirectionwillbeForsmallamplitude,wecanneglectthesecondordervariableterms,WhenForsmallamplitudeSimilarly,inthedirectionofyandz,wecanobtainNowletthemotionbethree–dimensional
3、,sowriteisgradientoperatorSinceP0isaconstant,andobtainThisisthelinearinviscidequationofmotion,validforacousticprocessesofsmallamplitude2.TheequationofcontinuityrestatementofthelawoftheconservationofmatterTorelatethemotionofthefluidtoitscompressionordilatation,weneedafunctionalrelation
4、shipbetweentheparticlevelocityuandtheinstantaneousdensityp.Considerasmallrectangular-parallelepipedvolumeelementdV=dxdydzwhichisfixedinspaceandthroughwhichelementsofthefluidtravel.Thenetratewithwhichmassflowsintothevolumethroughitssurfacemustequaltheratewiththemasswithinthevolumeincre
5、ases.Thatthenetinfluxofmassintothisspatiallyfixedvolume,resultingfromflowinthexdirection,isSimilarexpressionsgivethenetinfluxfortheyandzdirections,SothatthetotalinfluxmustbeWeobtaintheequationofcontinuityNotethattheequationisnonlinear;therightterminvolvestheproductofparticlevelocityan
6、dinstantaneousdensity,bothofwhichareacousticvariables.Considerasmallamplitudesoundwave,ifwewritep=p0(1+s).Usethefactthatp0isaconstantinbothspaceandtime,andassumethatsisverysmall,WeobtainSimilarexpressionsgibethenetinfluxfortheyandzdirections,Whereisthedivergenceoperator3.Theequationof
7、stateWeneedonemorerelationinordertodeterminethethreefunctionsP,,andu.Itisprovidedbytheconditionthatwehaveanadiabatic(绝热的)process,(thereisinsignificantexchangeofthermalenergyfromoneparticleoffluidtoanother).Undertheseconditions,itisconvenientlyexpressedbysayingthatthepressurepisunique
8、lydet
