Slowly-varying-envelope-approximation(慢变包络近似)

SlowlyvaryingenvelopeapproximationFromWikipedia,thefreeencyclopediaInphysics,theslowlyvaryingenvelopeapproximation(SVEA)istheassumptionthattheenvelopeofaforward-travellingwavepulsevariesslowlyintimeandspacecomparedtoaperiodorwavelength.Thisrequiresthespectrumofthesignaltobenarrow-banded—henceitalsoreferredtoasthenarrow-bandapproximation.Theslowlyvaryingenvelopeapproximationisoftenusedbecausetheresultingequationsareinmanycaseseasiertosolvethantheoriginalequations,reducingtheorderof—allorsomeof—thehighest-orderpartialderivatives.Butthevalidityoftheassumptionswhicharemadeneedtobejustified.ExampleForexample,considertheelectromagneticwaveequation:220020EEtµε∂∇−=∂.Ifk0andω0arethewavenumberandangularfrequencyofthe(characteristic)carrierwaveforthesignalE(r,t),thefollowingrepresentationisuseful:0()0(,){(,)}itEtEteω⋅−=ℜ0krrr,Where{}ℜ⋅denotestherealpartofthequantitybetweenbrackets.Intheslowlyvaryingenvelopeapproximation(SVEA)itisassumedthatthecomplex-valuedamplitudeE0(r,t)onlyvariesslowlywithrandt.ThisinherentlyimpliesthatE0(r,t)representswavespropagatingforward,predominantlyinthek0direction.AsaresultoftheslowvariationofE0(r,t),whentakingderivatives,thehighest-orderderivativesmaybeneglected:2000EkE∇∇0and20002EEttω∂∂∂∂0,with0k=0k.FullapproximationConsequently,thewaveequationisapproximatedintheSVEAas:22000000000022()0EiEikEtωµεωµε∂⋅∇+−−=∂0k.Itisconvenienttochoosek0andω0suchthattheysatisfythedispersionrelation:2200000kωµε−=.Thisgivesthefollowingapproximationtothewaveequation,asaresultoftheslowlyvaryingenvelopeapproximation:000000EEtωµε∂⋅∇+=∂0k.Whichisahyperbolicpartialdifferentialequation,liketheoriginalwaveequation,butnowoffirst-orderinsteadofsecond-order.Andvalidforcoherentforward-propagatingwavesindirectionsnearthek0-direction.ParabolicapproximationAssumewavepropagationisdominantlyinthez-direction,andk0istakeninthisdirection.TheSVEAisonlyappliedtothesecond-orderspatialderivativesinthez-directionandtime.If2222xy⊥∂∂∆=+∂∂istheLaplaceoperatorinthex–yplane,theresultis:0000000102EEkiEztωµε⊥∂∂+−∆=∂∂.Whichisaparabolicpartialdifferentialequation.ThisequationhasenhancedvalidityascomparedtothefullSVEA:itcanrepresentswavespropagatingindirectionssignificantlydifferentfromthez-direction.