最新fundamentals of acoustics(6) 声学基础（英文版教学课件课件ppt.ppt

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1、FUNDAMENTALSOFACOUSTICS(6)声学基础（英文版教学课件）VibrationsofextendedsystemsInthepreviouschapteritwasassumedthatthemassmovesasarigidbodysothatitcouldbeconsideredconcentratedatasinglepoint.However,mostvibratingbodiesarenotsosimple.Aloudspeakerhasitsmassdistributedoveritssurfac

2、esothattheconedoesmoveasaunit.Apianosting.Aflexiblestringundertensionprovidestheeasiestexampleforvisualizinghowwavesworkanddevelopingphysicalconceptsandtechniquesfortheirstudy.Thevibratingstringisinterestingbothforitsownsake(asasourceofsoundonaguitarorviolin)andasam

3、odelforthemotionofothersystems.Westudyfreemotionofastring.Theproceduresweusewillapplyinourlaterstudyofotherkindsofwaves.Itisobservedthatthespeedofpropagationofallsmalldisplacementsisindependentoftheshapeandamplitudeoftheinitialdisplacementanddependsonlyonthemassperu

4、nitlengthofthestringanditstensionExperimentandtheoryshowthatthisseedisgivenbyWherecisinm/s,TisthetensioninNandplisthemassperunitlengthofthestringinkg/m.TheequationofmotionAssumeastringofuniformlineardensityplandnegligiblestiffness,stretchedtoatensionTgreatenoughthat

5、theeffectsofgravitycanbeneglected.Alsoassumethattherearenodissipativeforces(suchasthoseassociatedwithfrictionorwiththeradiationofacousticenergy)Fig.Aisolatesaninfinitesimalelementofthestringwithequilibriumpositionxandequilibriumlengthdx.Whenthestringisatrest,thetens

6、ionsatxandatx+dxarepreciselyequalinmagnitudeandoppositeindirection,makingzerototalforce.Fig.AIf(thetransversedisplacementofthiselementfromitsequilibriumposition)issmall,thetensionTremainsconstantalongthestringandthedifferencebetweentheComponentofthetensionatthetwoen

7、dsoftheelementisIfissmall,WegetApplyingtheTaylor’sseriesexpansionSincethemassoftheelementispldxanditsaccelerationinthedirectionisNewton’slawgivesThenyieldstheequationofmotionwheretheconstantc2isdefinedbyGENERALSOLUTIONOFTHEEQUATIONGOFMOTIONEquation(2-1)isasecond-ord

8、er,partialdifferentialequation.Itscompletesolutioncontainstwoarbitraryfunctions.Themostgeneralsolutionisarecompletelyarbitraryfunctionsofa