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1、FUNDAMENTALSOFACOUSTICS(6)聲學基礎（英文版教學課件）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
