Time-reversed refocusing of surface water waves

TIME-REVERSEDREFOCUSINGOFSURFACEWATERWAVESJEAN-PIERREFOUQUEANDANDRENACHBINyAbstract.Atime-reversalmirroris,roughlyspeaking,adevicewhichiscapableofreceivingasignalintime,keepingitinmemoryandsendingitbackintothemediuminthereverseddirectionoftime.Abriefmathematicalreviewofthetime-reversaltheoryispresentedinthecontextofthelinearshallowwaterequations.Inparticularanexplicitexpressionisgivenfortherefocusedpulseinthesimplesttime-reversalcase.Theexplicitexpressionforthepowerspectraldensityofthereectionprocessisusedtoconstructthehighpasslterwhichcontrolstherefocusingprocess.Time-reversalnumericalexperimentsinthe(eectively)linearregimeareusedtovalidatethenonlinearshallowwatercode.Thenumericallyrefocusedpulseiscomparedwiththetheoreticalpredictedshape.Furthernumericalexperimentsillustratetherobustnessofthetheory.Inparticularthetime-reversalrefocusingwithsmallercutowindows,theself-averagingpropertyandnallyrefocusingwhenthenonlineartermissmallbutnotnegligible.Keywords.waterwaves,inhomogeneousmedia,asymptotictheoryAMSsubjectclassications.76B07,76B15,35Q1.Introduction.Atime-reversalmirroris,roughlyspeaking,adevicewhichiscapableofreceivinganacousticsignalintime,keepingitinmemoryandsendingitbackintothemediuminthereverseddirectionoftime.Inthecontextofultrasoundstime-reversalmirrorshavebeendevelopedandtheireectstudiedexperimentallybyMathiasFinkandhiscollaboratorsattheLaboratoireOndesetAcoustique(ESPCI-Paris).Themaineectistherefocusingofthescatteredsignalaftertime-reversalinarandommedium:anacousticpulseissentinadisorderedmediumgeneratinga\noisyreectedsignalwhichistime-reversedandsentbackintothemedium.Thenewreectedsignalisapurepulsewithashapesimilartotheinitialpulse.Amazinglyits\refocusingtakesplaceintimeandspaceandseemstobeindependentoftherealizationofthemedium,intheregimewhereitscorrelationlengthissmallerthanthetypicalwavelengthofthepulse.ExperimentsinthecontextofunderwateracousticshavebeenconductedbyKupermanetal.andshowthesamephenomenon.Werefertoreference[12]foradescriptionoftheseexperimentsandfurtherreferences.Fromthetheoreticalpointofview,arstproofofthisrefocusingeecthasbeenobtainedbyClouetandFouque[10]inthecontextofaone-dimensionalrandommediumforwhichonlythetimerefocusingisrelevant.Therefocusingisobtainedbyusingasymptoticsintheregimewheretherearethreewellseparatedscales:correlationlengthofthemediumwavelengthofthepulsewavelengthofthepulsedistanceofpropagation1:Theuctuationsofthemediumarenotassumedtobesmallbutratheroftheorderofseveraltensofpercent.Thisresulthasbeenextendedtothecaseof3DlayeredmediabyFouqueandNdzie[16]whereitisshownthatrefocusingtakesplaceintimeaswellasinspace.MorerecentlyspacerefocusinghasbeenanalysedinthecontextofsmalluctuationandtheparabolicapproximationbyBlomgren,PapanicolaouandZhao[3]DepartmentofMathematics,NorthCarolinaStateUniversity,Raleigh,NC27695-8205,USA([e-mail]).ThisworkwassupportedbyONRunderGrantN00014-02-1-0089.yInstitutodeMatematicaPuraeAplicada,Est.D.Castorina110,JardimBot^anico,RiodeJaneiro,RJ22460-320,Brazil([e-mail]).ThisworkwassupportedbyCNPq/BrazilunderGrant300368/96-8andinpartbyONRunderGrantN00014-02-1-0089throughthisauthor'svisittoNCSU.12J.P.FOUQUEANDA.NACHBINandPapanicolaou,RyhzikandSlna[23].Theyshowthattime-reversalandrandom-nesshelpsin\beatingthediractionlimit:theycallthiseectsuper-resolution.Acompleteproofofsuper-resolutionforthree-dimensionalrandomlylayeredmediahasbeenobtainedinFouqueandSlna[15].Potentialapplicationsoftheseeectsarenumerousinimagingandwirelesscommunication.Thegoalofthispaperistopresentthetime-reversalandrefocusingofwaterwavesand,ofparticularinterest,toinitiateastudyoftheeectofnonlinearityontherefocusingeect.Fordoingsoweusethecontextofgravitydrivenwaterwaves,propagatinginshallowone-dimensionalchannelsintheregimewheretheuidisconsideredtobeinviscidandincompressible.Therandomnesscomesfromthetopographyatthebottomoftheshallowchannel.Notethattheshallowwaterregimeisdictatedbyhavingasmalldepthtowavelengthratio.Thisisthesameasthelongwaveregime.VelocityandelevationatthefreesurfacearedescribedthroughtheshallowwaterequationswhichcontainaBurgers'typenonlinearitycontrolledbytheamplitudeofthewave.Forsmallamplitude-to-depthratiosthenonlinearityisnegligibleandwearebacktotheone-dimensionalcasestudiedfromClouetandFouque[10].Moreoveraveryecientnumericalcodehasbeendevelopedforsolvingthesenonlinearequations.Itprovidesanextremelyaccuratetooltoconductnumericalexperimentsontheeectofnonlinearityontime-reversalrefocusing.Werstvalidatethistoolinthelinearregimeandshow,aspredictedbythetheory,thatrefocusingindeedtakesplace:arighttravellingpulseshapedwaveissentfromtheleftsideoftherandomportionofthemedium.Itinteractswiththeuctuatingbottomandproducesareected\noisywavepropagatingtowardstheleftofthedisorderedchannel.Attheleftendofthechannelthiswaveistime-reversedbysimplychangingthesignofitsvelocityproleandsentbackintotherandommedium.Itthenproducesanewreectedwavewhichappearstobea\cleanpulsewithashapeindependentoftherealizationofthemedium.Thepulseshapeisofadete