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BASICAOUSTICAS(6)TransverseMotion-TheVibratingStringBASICAOUSTICAS(6)Transver1VibrationsofextendedsystemsInthepreviouschapteritwasassumedthatthemassmovesasarigidbodysothatitcouldbeconsideredconcentratedatasinglepoint.However,mostvibratingbodiesarenotsosimple.Aloudspeakerhasitsmassdistributedoveritssurfacesothattheconedoesmoveasaunit.Apianosting.Vibrationsofextendedsystems2Aflexiblestringundertensionprovidestheeasiestexampleforvisualizinghowwavesworkanddevelopingphysicalconceptsandtechniquesfortheirstudy.Thevibratingstringisinterestingbothforitsownsake(asasourceofsoundonaguitarorviolin)andasamodelforthemotionofothersystems.Westudyfreemotionofastring.Theproceduresweusewillapplyinourlaterstudyofotherkindsofwaves.Aflexiblestringundertensio3FUNDAMENTALSOFACOUSTICS6聲學(xué)基礎(chǔ)(英文版教學(xué)課件)4FUNDAMENTALSOFACOUSTICS6聲學(xué)基礎(chǔ)(英文版教學(xué)課件)5Initialdisturbanceatt=0

Separatedisturbanceatt1>>0

Separatedisturbanceatt2>t1PropagationofatransversedisturbancealongastretchedstringInitialdisturbanceatt=0Sep6ItisobservedthatthespeedofpropagationofallsmalldisplacementsisindependentoftheshapeandamplitudeoftheinitialdisplacementanddependsonlyonthemassperunitlengthofthestringanditstensionExperimentandtheoryshowthatthisseedisgivenbyWherecisinm/s,TisthetensioninNandplisthemassperunitlengthofthestringinkg/m.Itisobservedthatthespeed7TheequationofmotionAssumeastringofuniformlineardensityplandnegligiblestiffness,stretchedtoatensionTgreatenoughthattheeffectsofgravitycanbeneglected.Alsoassumethattherearenodissipativeforces(suchasthoseassociatedwithfrictionorwiththeradiationofacousticenergy)TheequationofmotionAssumea8Fig.Aisolatesaninfinitesimalelementofthestringwithequilibriumpositionxandequilibriumlengthdx.Whenthestringisatrest,thetensionsatxandatx+dxarepreciselyequalinmagnitudeandoppositeindirection,makingzerototalforce.

Fig.AFig.Aisolatesaninfinitesi9If

(thetransversedisplacementofthiselementfromitsequilibriumposition)issmall,thetensionTremainsconstantalongthestringandthedifferencebetweentheComponentofthetensionatthetwoendsoftheelementis

If(thetransversedispl10Ifissmall,

WegetApplyingtheTaylor’sseriesexpansionIfissmall,WegetApplyingth11Sincethemassoftheelementispldxanditsaccelerationinthe

directionis

Newton’slawgivesThenyieldstheequationofmotionwheretheconstantc2isdefinedbySincethemassoftheelement12GENERALSOLUTIONOFTHEEQUATIONGOFMOTIONEquation(2-1)isasecond-order,partialdifferentialequation.Itscompletesolutioncontainstwoarbitraryfunctions.Themostgeneralsolutionis

arecompletelyarbitraryfunctionsofarguments(ct-x)and(ct+x),respectively.Possibleexamplesofsucharbitraryfunctionsincludelog(ct+x),(ct+x)2,sin[w(t+x/c)],etal.GENERALSOLUTIONOFTHEEQUATI13Wecanprovethatanyfunctionofargument(ct-x)isasolutionofthewaveequation(2-1).Similarly,itcanbeshownthatf2(ct+x)isalsoasolution.Thesumofthesetwofunctionsisthecompletegeneralsolutionoftheequationofmotion.Wecanprovethatanyfunction14Considerthesolutionf1(ct-x).Attimet1thetransversedisplacementofthestringisgivenbyf1(ct-x).AssuggestedbyFig.B

x1x2Atalatertimet2theshapeofthestringwillbegivenbyf1(ct2-x2)

Considerthesolutionf1(ct-x)15Theparticulartransversedisplacementf1(ct1-x1)ofthestringthatwasfoundatx1whent=t1mustbefoundatapositionx2whent=t2wherect1-x1=ct2-x2Thus,thisparticulardisplacementhasmovedadistancex2-x1=c(t2-t1)totheright.Theparticulartransversedisp16Sincetheparticulardisplacementchosenwasarbitrary,anytransversedisplacementmustmovetotherightwiththesamespeed.Thismeansthattheshapeofthedisturbanceremainsunchangedandtravelsalongthestringtotherightataconstantspeedc.Thefunctionf1(ct-x)representsawavetravelinginthe+xdirection,calledwavefunction.Sincetheparticulardisplacem17STANDINGWAVESConsidernowastringoffinitelengthL.Describingallmotionsofthisstringintermsoftravelingwavesremainspossibleinprinciple.Becauseofrepeatedreflectionsbetweenthetwoends,thatisusuallynotthemosthelpfuldescription.Wefinditmoreconvenienttostudystandingwaves.STANDINGWAVESConsidernowas18FUNDAMENTALSOFACOUSTICS6聲學(xué)基礎(chǔ)(英文版教學(xué)課件)19Welimittosolutionsthatmeettheproperboundaryconditions.Supposespecificallythatbothendsofthestringarefixed,thatis:Substitutetheinitialconditions,andobtainWelimittosolutionsthatmee20TheonlywaytomeetthefirstconditionistosetA=0.ThenthesecondconditionallowstheamplitudeBtobeanythingaslongaswerequirethatul/cbeanintegralmultipleof.Theonlywaytomeetthefirst21Weuse

Weuse22Normal-modefrequenciesthatformaharmonicseriesasaboveareaveryspecialfeatureofone-dimensionalsystemswhosepropertiesareuniformeverywherealongtheirlength.Normal-modefrequenciesthatf23Anysumofthesesinusoidalstandingwavesisalsoasolutionofboththeequationofmotionandtheboundaryconditions.SowecanrepresentverygeneralmotionsofastringfixedatbothendsbyAnysumofthesesinusoidalst24HomeworkHowtodeterminetheamplitudesandphasewheninitialconditionshavebeenspecified.(textbookP71-72)HomeworkHowtodeterminethea25

BASICAOUSTICAS(6)TransverseMotion-TheVibratingStringBASICAOUSTICAS(6)Transver26VibrationsofextendedsystemsInthepreviouschapteritwasassumedthatthemassmovesasarigidbodysothatitcouldbeconsideredconcentratedatasinglepoint.However,mostvibratingbodiesarenotsosimple.Aloudspeakerhasitsmassdistributedoveritssurfacesothattheconedoesmoveasaunit.Apianosting.Vibrationsofextendedsystems27Aflexiblestringundertensionprovidestheeasiestexampleforvisualizinghowwavesworkanddevelopingphysicalconceptsandtechniquesfortheirstudy.Thevibratingstringisinterestingbothforitsownsake(asasourceofsoundonaguitarorviolin)andasamodelforthemotionofothersystems.Westudyfreemotionofastring.Theproceduresweusewillapplyinourlaterstudyofotherkindsofwaves.Aflexiblestringundertensio28FUNDAMENTALSOFACOUSTICS6聲學(xué)基礎(chǔ)(英文版教學(xué)課件)29FUNDAMENTALSOFACOUSTICS6聲學(xué)基礎(chǔ)(英文版教學(xué)課件)30Initialdisturbanceatt=0

Separatedisturbanceatt1>>0

Separatedisturbanceatt2>t1PropagationofatransversedisturbancealongastretchedstringInitialdisturbanceatt=0Sep31ItisobservedthatthespeedofpropagationofallsmalldisplacementsisindependentoftheshapeandamplitudeoftheinitialdisplacementanddependsonlyonthemassperunitlengthofthestringanditstensionExperimentandtheoryshowthatthisseedisgivenbyWherecisinm/s,TisthetensioninNandplisthemassperunitlengthofthestringinkg/m.Itisobservedthatthespeed32TheequationofmotionAssumeastringofuniformlineardensityplandnegligiblestiffness,stretchedtoatensionTgreatenoughthattheeffectsofgravitycanbeneglected.Alsoassumethattherearenodissipativeforces(suchasthoseassociatedwithfrictionorwiththeradiationofacousticenergy)TheequationofmotionAssumea33Fig.Aisolatesaninfinitesimalelementofthestringwithequilibriumpositionxandequilibriumlengthdx.Whenthestringisatrest,thetensionsatxandatx+dxarepreciselyequalinmagnitudeandoppositeindirection,makingzerototalforce.

Fig.AFig.Aisolatesaninfinitesi34If

(thetransversedisplacementofthiselementfromitsequilibriumposition)issmall,thetensionTremainsconstantalongthestringandthedifferencebetweentheComponentofthetensionatthetwoendsoftheelementis

If(thetransversedispl35Ifissmall,

WegetApplyingtheTaylor’sseriesexpansionIfissmall,WegetApplyingth36Sincethemassoftheelementispldxanditsaccelerationinthe

directionis

Newton’slawgivesThenyieldstheequationofmotionwheretheconstantc2isdefinedbySincethemassoftheelement37GENERALSOLUTIONOFTHEEQUATIONGOFMOTIONEquation(2-1)isasecond-order,partialdifferentialequation.Itscompletesolutioncontainstwoarbitraryfunctions.Themostgeneralsolutionis

arecompletelyarbitraryfunctionsofarguments(ct-x)and(ct+x),respectively.Possibleexamplesofsucharbitraryfunctionsincludelog(ct+x),(ct+x)2,sin[w(t+x/c)],etal.GENERALSOLUTIONOFTHEEQUATI38Wecanprovethatanyfunctionofargument(ct-x)isasolutionofthewaveequation(2-1).Similarly,itcanbeshownthatf2(ct+x)isalsoasolution.Thesumofthesetwofunctionsisthecompletegeneralsolutionoftheequationofmotion.Wecanprovethatanyfunction39Considerthesolutionf1(ct-x).Attimet1thetransversedisplacementofthestringisgivenbyf1(ct-x).AssuggestedbyFig.B

x1x2Atalatertimet2theshapeofthestringwillbegivenbyf1(ct2-x2)

Considerthesolutionf1(ct-x)40Theparticulartransversedisplacementf1(ct1-x1)ofthestringthatwasfoundatx1whent=t1mustbefoundatapositionx2whent=t2wherect1-x1=ct2-x2Thus,thisparticulardisplacementhasmovedadistancex2-x1=c(t2-t1)totheright.Theparticulartransversedisp41Sincetheparticulardisplacementchosenwasarbitrary,anytransversedisplacementmustmovetotherightwiththesamespeed.Thismeansthattheshapeofthedisturbanceremainsunchangedandtravelsalongthestringtotherightataconstantspeedc.Thefunctionf1(ct-x)representsawavetravelinginthe+xdirection,calledwavefunction.Sincetheparticulardisplacem42STANDINGWAVESConsidernowastringoffinitelengthL.Describingallmotionsofthisstringintermsoftravelingwavesremainspossibleinprinciple.Becauseofrepeatedreflectionsbetweenthetwoends,thatisusuallynotthemosthelpfuld

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