Principle of Superposion of wavesSound waves are longitudinal mechanical waves that can travel...
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16-5 Interference of Waves Principle of Superposi3on of waves Let y 1 (x, t) and y 2 (x, t) be the displacements that the string would experience if each wave traveled alone. The displacement of the string when the waves overlap is then the algebraic sum This summa<on of displacements along the string means that © 2014 John Wiley & Sons, Inc. All rights reserved. 16-5 Interference of Waves
Transcript of Principle of Superposion of wavesSound waves are longitudinal mechanical waves that can travel...
16-5InterferenceofWaves
PrincipleofSuperposi3onofwaves
Lety1(x,t)andy2(x,t)bethedisplacementsthatthestringwouldexperienceifeachwavetraveledalone.Thedisplacementofthestringwhenthewavesoverlapisthenthealgebraicsum
Thissumma<onofdisplacementsalongthestringmeansthat
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16-5InterferenceofWaves
16-5InterferenceofWaves
Construc3veandDestruc3veInterference
Twoiden<calsinusoidalwaves,y1(x,t)andy2(x,t),travelalongastringintheposi<vedirec<onofanxaxis.Theyinterferetogivearesultantwavey’(x,t).Theresultantwaveiswhatisactuallyseenonthestring.ThephasedifferenceΦbetweenthetwointerferingwavesis(a)0rador0o,(b)πrador180o,and(c)2/3πrador120o.Thecorrespondingresultantwavesareshownin(d),(e),and(f).©2014JohnWiley&Sons,Inc.Allrights
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16-5InterferenceofWaves
©2014JohnWiley&Sons,Inc.Allrightsreserved.
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Nodes:sin(kx)=0An<nodes:sin(kx)=+/-1
Reflec<onataboundary
16-7StandingWavesandResonance
StandingWaves• Theinterferenceoftwoiden<calsinusoidalwavesmovinginoppositedirec<onsproducesstandingwaves.Forastringwithfixedends,thestandingwaveisgivenby
Stroboscopicphotographsreveal(imperfect)standingwavepaVernsonastringbeingmadetooscillatebyanoscillatorattheleWend.ThepaVernsoccuratcertainfrequenciesofoscilla<on.©2014JohnWiley&Sons,Inc.Allrights
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16-7StandingWavesandResonance
Standingwaves
16-7StandingWavesandResonance
Harmonics• Standingwavesonastringcanbesetupbyreflec<onoftravelingwavesfromtheendsofthestring.Ifanendisfixed,itmustbetheposi<onofanode.Thislimitsthefrequenciesatwhichstandingwaveswilloccuronagivenstring.Eachpossiblefrequencyisaresonantfrequency,andthecorrespondingstandingwavepaVernisanoscilla<onmode.ForastretchedstringoflengthLwithfixedends,theresonantfrequenciesare
16-7StandingWavesandResonance
Wavesl TransverseWaves
l LongitudinalWaves
SinusoidalWavesl Wavemovinginposi<ve
direc<on(vector)Eq.(16-2)
16Summary
TravelingWavesl Afunc<onalformfortraveling
wavesEq.(16-17)
WaveSpeedl Angularvelocity/Angularwave
number
Eq.(16-13)
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Powersl AveragePowerisgivenby
InterferenceofWavesl Twosinusoidalwavesonthe
samestringexhibitinterference
Eq.(16-51)
16Summary
Resonancel ForastretchedstringoflengthL
withfixedends,theresonantfrequenciesare
Eq.(16-66)
StandingWavesl Theinterferenceoftwoiden<cal
sinusoidalwavesmovinginoppositedirec<onsproducesstandingwaves.
Eq.(16-60)
Eq.(16-33)
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Waves-II
Chapter17
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17-1SpeedofSound
Soundwavesarelongitudinalmechanicalwavesthatcantravelthroughsolids,liquids,orgases.PointSrepresentsa<nysoundsource,calledapointsource,thatemitssoundwavesinalldirec<ons.AsoundwavetravelsfromapointsourceSthroughathree-dimensionalmedium.Thewavefronts(surfacesoverwhichtheoscilla<onsduetothesoundwavehavethesamevalue)formspherescenteredonS;theraysareradialtoS.Theshort,double-headedarrowsindicatethatelementsofthemediumoscillateparalleltotherays.
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17-1SpeedofSound
ThespeedνofasoundwaveinamediumhavingbulkmodulusBanddensityρis
AnelementofairofwidthΔxmovestowardthepulsewithspeedv.
Theleadingfaceoftheelemententersthepulse.Theforcesac<ngontheleadingandtrailingfaces(duetoairpressure)areshown.
Throughdirectapplica<onofNewton’sSecondlaw.
17-2TravelingSoundWaves
(a)Asoundwave,travelingthroughalongair-filledtubewithspeedv,consistsofamoving,periodicpaVernofexpansionsandcompressionsoftheair.Thewaveisshownatanarbitraryinstant.
(b)Ahorizontallyexpandedviewofashortpieceofthetube.Asthewavepasses,anairelementofthicknessΔxoscillatesleWandrightinsimpleharmonicmo<onaboutitsequilibriumposi<on.Attheinstantshownin(b),theelementhappenstobedisplacedadistancestotherightofitsequilibriumposi<on.Itsmaximumdisplacement,eitherrightorleW,issm.
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17-2TravelingSoundWaves
Displacement:Asoundwavecausesalongitudinaldisplacementsofamasselementinamediumasgivenby
wheresmisthedisplacementamplitude(maximumdisplacement)fromequilibrium,k=2π/λ,andω=2πƒ,λandƒbeingthewavelengthandfrequency,respec<vely,ofthesoundwave.
Pressure:ThesoundwavealsocausesapressurechangeΔpofthemediumfromtheequilibriumpressure:
wherethepressureamplitudeis
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©2014JohnWiley&Sons,Inc.Allrightsreserved.
©2014JohnWiley&Sons,Inc.Allrightsreserved.
17-3Interference
TwopointsourcesS1andS2emitsphericalsoundwavesinphase.TheraysindicatethatthewavespassthroughacommonpointP.Thewaves(representedwithtransversewaves)arriveatP.
FullyConstruc3veInterference
(exactlyinphase)
FullyDestruc3veInterference(exactlyoutofphase)
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17-3Interference
• Theinterferenceoftwosoundwaveswithiden<calwavelengthspassingthroughacommonpointdependsontheirphasedifferencethereϕ.IfthesoundwaveswereemiVedinphaseandaretravelinginapproximatelythesamedirec<on,ϕisgivenby
whereΔListheirpathlengthdifference.
PathLengthDifference
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17-3Interference
and,equivalently,whenΔLisrelatedtowavelengthλby
• Fullydestruc3veinterferenceoccurswhenϕ isanoddmul<pleofπ,
and,equivalently,whenΔLisrelatedtowavelengthλby
• Fullyconstruc3veinterferenceoccurswhenϕ isanintegerandmul<pleof2π,
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