FUNDAMENTALS OF ACOUSTICS(7)

BASICACOUSTICS(7)Acousticwavetravelinginperfectlyinviscidfluidmedium•Manyphysicalphenomenahaveapropertycalledlinearity.Thismeansthattheforcesactingonthemediumproducemotionsinexactproportion;doublingthecausewilldoubletheeffect.•Althoughnonlinearitydoesoccurforsoundwavesofverylargeamplitude,theapproximationthatsoundwavesbehavelinearlyisgenerallyinexcellentoneformanyuses.•Wheneverlinearityholds,weareentitledtousetheextremelypowerfulconceptofsuperposition.•Thismeansthatwhenmanywavespassthroughthesameregion,theireffectsaresimplyadditive,eachcontinuestobehavejustasiftheotherswerenotpresent.Thenatureofsound•Soundisawavephenomenon•Inagasorliquid,soundwavesareclassifiedaslongitudinalwaves—thevibrationisalwaysparalleltothedirectionofwavetravel•Thespeedofsoundinairisroughly340meterpersecond;inwaterisroughly1500meterpersecondSoundwavewithcompressionatAandrarefactionatBSoundPressure•WemostoftendescribethestrengthofasoundwaveintermsofitsPressureamplitude.•Pressureamplitudemeaningthesizeofthepressurefluctuationsproducedbythewave.•IffluidalreadyhasasteadyequilibriumpressureP0,Thesoundwaveisrepresentedbythesmalldisturbancepinthetotalpressure•P=instantaneouspressureatanypoint•P0=constantequilibriumpressureinthefluid.•Definition:thedifferencebetweentheinstantaneouspressureatapointinasoundfieldandtheaveragepressureatthatpoint.•Note:pispositiveornegative,anditisthefunctionofx,y,z,t.p=p(x,y,z,t)00PpPpPP•Sincepressuremeansforceperunitarea,itismeasuredinNewtonpersquaremeter.•IthasbecomeincreasinglycommontousethedesignationPascalforthiscombinationofunits:•1Pa=1N/m2=10dynes/cm2=10ubarEffectivesoundpressure•Root-mean-squarevalueoftheinstantaneoussoundpressureatapoint,duringagiventimeintervaliscalledeffectivesoundpressureTedtPTp021Forharmonicvibrationcos,,2mmeppptsopThefluidvelocityinordinarysoundwavesisalwaysmuchlessthanthesoundspeedc•Underthesoundpressure,thedisplacementofanyparticularbloboffluidfromitsundisturbedpositionisutItsvelocityis•Particledisplacementfromtheequilibriumposition•Particlevelocityinstantaneousdensityatanypoint•Constantequilibriumdensityofthefluid•condensationatanypointu),,,(tzyx),,(0zyx),,(),,(),,,(),,,(00zyxzyxtzyxtzyxsSomeassumptions•Thefluidmaybethoughtofasacontinuousmedium•Allacousticvariablesmaybeconsiderednearlyconstantthroughoutthesmallvolumeelementandmaybetreatedasanunchangingunit•Thefluidisalsoassumedtobehomogeneous,isotropic,andperfectlyelastic,nodissipativeeffects•Finally,theanalysiswillbelimitedtowavesofrelativelysmallamplitudesothatchangesindensityofmediumwillbesmallcomparedwithitsequilibriumvalue.•Theseassumptionarenecessary,toarriveatthesimplesttheoryforsoundinfluids.•Weareindeedfortunatethatexperimentshaveshownthatthissimplesttheoryadequatelydescribesmostcommonacousticalphenomena.Threebasicequations•1.theequationofmotion(Euler'sequation)•First,wewritetherelationbetweensoundpressureandvelocity,•ConsiderafluidelementzyxVABCDEFGHyxzzxyF1F2ABCDEFGHyxzzxyF1F2Whenthesoundwavespass,thepressureis0(,)Ppxy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,validforacousticprocessesofsmallamplitudeHOMEWORK•1.Provethatthefollowingexpressionsarecorrectsolutionsfortheone-dimensionalwaveequation:(seethetextbookP2763-1)