声学基础课件(许肖梅)fundamentalsofacou

Transmissionfromonefluidtoanother:obliqueincidence,Applyingtheconditionofcontinuityofpressureattheboundaryz=0yields,Applyingtheconditionofcontinuityofnormalvelocity,(3-7-11),(3-7-10),Lawofreflection:,theangleofincidenceisequaltotheangleofreflection,andSnellslaw:,Snellslaw:,Fromtheequation(3-7-10),weobtain:,Fromtheequation(3-7-11),weobtain:,when,Givestheangleofincidenceforwhichthereisnoreflectionandthereforecompletetransmission.Thisangleknownastheangleofintromission:,Itwillbeexistonlyif,Thetotallyreflected,Fromthereflectioncoefficientequation,wesee,Theangleoftransmissionisrealandlessthantheangleofincident.Atransmittedbeamexitsinthesecondfluid.,(1),Theangleoftransmissionisagainrealbutgreaterthantheangleofincidence,(2)when,Whereisthecriticalangledefinedby,(3)When,Inthiscase,thetransmittedwaveassumeaverypeculiarform,thereflectioncoefficientisacomplex,Consider,560,1,0.5,2a,1800,00,0.4,00,900Angleofincidence,IRI,IRI,Reflectionofcoefficient,Transmissionthroughalayer:Normalincidence,AssumethatalayerofuniformthicknessLliesbetweentwodissimilarfluidsandthataplanewaveisnormallyincidentonitsboundary,assuggestedinfollowingFig.,LetthecharacteristicimpedancesofthefluidsbeZ1,Z2andZ3respectively,Fig.,Fig.Reflectionandtransmissionofplanewavesnormallyincidentonalayer,WhenanincidentsignalinfluidIarrivesattheboundarybetweenfluidsIandII,someoftheenergyisreflectedandsomeistransmittedintothesecondfluid.TheportionofthewavetransmittedwillproceedthroughfluidIItointeractwiththeboundarybetweenfluidsIIandIII,whereagainsomeoftheenergyisreflectedandsometransmitted.ThereflectedwaveproceedsbacktotheboundarybetweenfluidsIandII,andthewholeprocessisrepeated.,Fromthewaveequationandtheequationofmotion,weobtain(neglectthetimefactor):,InfluidI,InfluidII,InfluidIII,Atx=0,twoboundaryconditions,Atx=l,twoboundaryconditions,(3-7-28),Definethereflectioncoefficientofthelayer:,Similarly,definetherefractionandintensitytransmissioncoefficientsofthelayer:,Fromtheequationof(3-7-28),WhereZ21istheinputimpedanceofthelayer.TheboundarybetweenfluidIandfluidIIcorrespondstoanimpedance.Thentheformofthepressurereflectioncoefficient,Note:Z21hasbothrealandimaginarycomponents,From(3-7-28),Nowthesoundintensitytransmissioncoefficientis,Thereexistanumberofspecialformsofthetransmissioncoefficientwhichareofparticularinterest.Inthefollowingcaseswecanfurthersimplifythesoundintensitytransmissioncoefficient.,Fromtherelationshipofsoundenergy:,Whenmedium3isthesameasmedium1,wehaveZ3=Z1,and,If,inaddition,(1)Ifthelengthofmiddlemedium,(2)If,(3)If,T=1,Maximumtransmissionofplaneacousticwavesfromwaterintosteelisrequired.Whatshouldbetheoptimumcharacteristicimpedanceofthematerialtobeplacedbetweenthewaterandthesteel?Ifthethicknessofthelayerofmaterialtobeusedis0.02mandthefrequencyofsoundtransmittedis1000Hz,findthespeedofsoundinthematerialandthedensityofthematerial.,TextbookP2773-15,Thesoundintensitytransmissioncoefficientfortransmissionthroughthreemediaatnormalincidenceisgivenby:,WhereZ1,Z2,Z3arethecharacteristicimpedancesofthemedia,k2isthewavenumberofmedium2,andlisthethicknessofmedium2.,Formaximumtransmissionofacousticintensity,If,Thenandthetransmissioncoefficientbecomes,WhereZ1=1.48*106MKSraylsisthecharacteristicimpedanceofwater,andZ3=47*106MKSraylsisthecharacteristicimpedanceofsteel,Hence,Z2=8.35*106MKSrayls,Therefore100%transmissionofsou