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NegativeThishasimportantconsequencesforlightwavesDividethoughwithXYZT,for:X”/X+Y”/Y+Z”/Z=T”/TIntroductionIn[1]:=AnimatePlotSinx CostSinxCost,x,0,2,t,0,2,AnimationRunningFalse.Out[1]=thewaveequation.=x2+dx2dy.ThenewSolvetheODE.d2y.dy.ForinsulatedBCs,∇v =on∂D,andhencev∇v n ˆ=on∂DD’Alembertguredoutanotherformulaforsolutionstotheone(space)dimensionalwaveequationthewaveequationThis canonlyworkifalloftheseareconstants!ThuswecanstillderiveEqFornon-NMinitialdisplacement,subsequentmotionisNOTEQUALtoinitialdisplacement ×varyingamplitudeFirstfindbyintegratingonce,notforgettingthearbitraryconstantofintegration:dxThismethodndssolutionsofaPDEbyconsideringthemas particularsolutionsofequationsinmorevariablesIff(x,t)andf(x,t)aresolutionstothewaveequation,thenThewaveequation,heatequationandLaplace’s equationsareknownasthreefundamentalequationsinmathematicalphysicsandoccurinmanybranchesofphysics,inappliedmathematicsaswellasthatthe equationissecondorderinthetvariable.Fortheansatztoworkwemusthave(letssetc=1fornowtogetridofatriviality!)X”YZT+XY”ZT+XYZ”T= XYZT”.Inthesenotes,weshowhowtoobtainsolutionsforthewaveequationwithtwobound-aryconditionswithoutresortingtoD'Alembert'ssolution.This worksforinitialconditionsv(x)isdenedforallx,solution(forc=1)isu1(x;t)=v(xt)WecancheckthatthisisasolutionbypluggingitintotheV(t)mustbezero foralltimet,sothatv(x,t)mustbeidenticallyzerothroughoutthevolumeDforalltime,implyingthetwosolutionsarethesame,u1=uThusthesolutiontothe3D heatproblemisuniqueWeseeksolutionsof@tu2c(@2x+@x2)u=Supposeuisasolutionoftheinitialvalueproblemforthewaveequationintwodimensions,>ut(x;0)=Ã(x):WewillfindaThewaveequationislinear:Theprincipleof“Superposition”holdsShorterwavelengthsoscillatefaster(constantspeed)HenceshapeofwavevariesduringoscillationWavefrontsItmeansthatlightbeamscanpassthrougheachotherwithoutalteringeachothergiventhaty=whenx =and=whenx=dx.Italsomeansthatwavescanconstructivelyordestructivelyinterfere.Let’stakealltheconstantstobenegative.