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Mathieu Progressive Waves

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Mathieu Progressive Waves

    Mathieu Progressive Waves

Commun.Theor.Phys.56(2011)733739Vo1.56,No.4,October15,2011

    ;MathieuProgressiveWaves

    ;AndreiB.Utkin

    ;INOVInescInovaqgo,RuaAlvesRedol9,Lisboni000029,Portugal

    ;ICEMS,InstitutoSuperiorT6cnico,TecnicalUniversityofLisbon,Av.RoviscoPais1,Lisbon10490

    01,Portugal

    ;(ReceivedMay30,2011;revisedmanuscriptreceivedJuly11,2011) ;AbstractAnew~milyofexactsolutionstothewaveequationrepresentingrelativelyundjstortedprogres

    sivewaves

    ;isconstructedusingseparationofvariablesintheellipticcylindricalcoordinaresandoneoftheBatemant

    ransforms.

    ;ThegeneralformofthisBatemantransforminanorthogonalcurvilinear~vlindricalcoordinatesystemis

    discussed

    ;andaspec

    cproblemofphysicalfeasibilityoftheobtainedsolutions.connectedwiththeirdependenceonthecyclic

    ;coordinate,isaddressed.Thelimitingcaseofzeroeccentricjty‟inwhichtheellipticcylindricalcoordinat

    esturninto

    ;theircircularcylindricalcounterparts,isshowntocorrespondtothefocusedwavemodesoftheBessel-G

    ausstype.

    ;PACSnumbers:43.20.Bi,03.50.De,41.20.Jb

    ;Keywords:waveequation,Batemantransform,progressivewave,Mathieufunction ;1Introduction

    ;Nowadaysintcrestintherepresentationofsomeprob

    ;lemsoftheoreticalandcomputationalphysicsinterms

    ;oftheellipticcylindricalcoordinatesisstipulatedbytwo

    ;majorreasons:0ntheonehanditcanprovidesimpli.

    ;fledmodelsofsomePhenomenathat,duetothepres

    ;enceofanadditionalparameter.areclosertorealitythan

    ;thoseharnessingthecylindricalsymmetryapproximation

    ;fcircularcylindricalcoordinates).IJThisisespeciallythe

    ;casefordescriptionofstressedanddeformedaswellas

    ;imperfectcylindricalstructures.0ntheotherhand.con.

    ;temporarytechnologymakesfabricatingsurfacesofthe

    ;ellipticalandhyperbolicprofileseachtimemoreafrord

    ;able.Thisenablesagammaofnewgadgetssuchasel

    ;lipticalwaveguides[andconcentrators.

    ;.]whichusespe

    ;cificpropertiesofthesesurfaces,tobeproducedonever

    ;growingscale.

;Sincetheanalyticalrepresentationofthebasicoper

    ;atorsofmathematicalphysicsintheellipticcylindrical ;coordinatesiswellknown.inmanycasesthedesiredso

    ;lutionscanbeobtainedbygeneralizingthemethodsde

    ;velopedfortheordinarycircularcylindricalcoordinates. ;However,thepresenceofthecyclicvariablev,describing ;thepositionalongtheellipticcoordinateline,putalimi

    ;tationonthemanifoIdofphysicallyadmissibleresults:if ;specificboundaryconditionsarenotimposed,oneshould ;expectperiodicityofthesolutionwithrespecttov.Here ;theapplicationoftraditionaltechniqueslcadstosolutions ;involvingtheperiodicangularfunctions--——Mathieusines

    ;andcosines.[41

    ;Forthepractica1reasonsdiscussedabove,thebegin- ;ningofthe21stcenturywitnessedarebirthofthein

    ;teresttotheellipticwaves.Theearlyfindings[5were

    ;Email:andrei.utkin@inov.pt

    ;@2011ChinesePhysicalSocietyandIOPPublishingLtd ;rediscoveredandadaptedtonewapplications.Doing ;localizedwaveresearch.Guti6rrezVegaeta1.carriedout ;modeling[6andexperimentalinvestigation[7ofpropaga.

    ;tionofthetruncatedzeroorderMathieubeamobtained

    ;viadifiractiononaspeciallymodulatedslitoftheDurnin‟s

    ;setup.Higher.orderMathieubeamsobtainedinsimilar ;conditionswereinvestigatedashorttimelaterbyDar

    ;toraeta1.[8-9]Thecommonpointsofthesestudiesare ;(i)thetheoreticaldescriptionofthewavepropagationus

    ;ingtheFourier(and,insomecases,FourierBesse1)rep

    ;resentationofthegeneralsolutioninthedomainofspa

    ;tial/temporalfrequenciesand(ii)constructionofapartic

    ;ularsolutionchoosingaspecificspectralfunction(calcula- ;tionofthedifiractionintegral1.Intheellipticcylinderco

    ;ordinatesv,.thetransversestructureofthemthmode

    ;ofthesolutions[69]canberepresentedbyageneralrela- ;tion

    ;w(u,v)=Je(,q)ce(u,q)(1)

    ;whereJem(,q)issomeradialMathieufunction,Cem(v,q) ;theMathieucosine,andqtheparameterresultedfrom ;separationofvariables.

    ;Doesgeneralexpression(1),matchingtheelliptic ;cylindergeometry,exhaustallpossibletransferstructures ;oftheMathieuwaves?Thisisatopicalissueformany ;practicalapplications,includingthewavepropagation ;innoncylindricalellipticcrosssectionwaveguides(horns,

    ;couplers,etc.).Inthepresentresearchtheauthorgivesan ;exampleofmoresophisticatedspace--timestructurecorre-- ;spondingtophysicallyadmissible,exactsolutionsofthe ;waveequationwithlocalizationpropertiessimilartothose ;ofBrittingham‟sfocuswavemodesandBateman-Hillion

    ;relativelyundistortedprogressivewaves.Derivationofthe ;http://www.iop.org/EJ/journal/ctphttp://ctp.itp.ac.cn ;734CommunicationsinTheoreticalPhysicsVb1. ;56

    ;wavefunctionisbasedonanalternativetechniqueofgen

    ;eratingnewsolutionstothewaveequationbasedonone ;oftheBatemantransforms,successfullyadaptedtothe ;ellipticcylindricalcoordinatesystem.

    ;2GeneralSolutionsforCurvilinearOrthog.

    ;onalCylindricalCoordinates

    ;Inthissectionwewil1discussgeneratingsolutionsto ;thewaveequationusingtheBatemantransformofwave

    ;functionsobtainedbyseparationofvariablesinageneral ;orthogonalcylindrica1coordinatesystem.

    ;2.1BatemanTransform

    ;TheBatemantransform[10

    ;e(,7-)一西G(,)=1c(

    ;,f)(2)

    ;enablesauewsolutionofthehomogeneouswaveequation ;Gtobegeneratedonthebasisofsomeknownsolution ;G.Here,Y,zandstandfortheCartesiancoordinates

    ;andtimet,representedinadimensionlessform ;x/lx,y/ly,z/l,ctfl7_(3)

    ;usingsomecharacteristiclengthlandthewavefrontve. ;1ocityc,whilethetransformedargumentsaredefinedby ;relationships

    ;z

    ;T

    ;XY——

    ;,——,

    ;,7

    ;X2+Y2+Z27-2l

    ;2(Z7_1‟

    ;X2+Y2+Z2一下2+1

    ;2(Zr1‟(4)

    ;OnecanemployEq.(2)forgeneratingCourantHilbert

    ;relativelyundistortedprogressivewavesolutionsofthe ;form[1l12]

    ;(,Y,z,7-)=g(x,Y,z,),((,Y,,7-)),

;withthephasefunction

    ;(z,Y,z,7-)=+7_+x2Y2

    ;

    ;7_

    ;z7-

    ;(5)

    ;(6)

    ;characteristicfortheBatemansolutions[10,121thatwerein ;theoriginofmanyinterestingwavefunctions,inparticu

    ;lar.BrittinghamfocuswavemodesandBatemanHillion

    ;relativelyundistortedprogressivewaves.113J ;InRef.141,theinitialsolutionswereconstructed ;viaseparationofvariablesinthehomogeneouswaveequa- ;tionwithzeroseparationconstant.Severalorthogonal ;coordinatesystemsofthecylindricaltypewereconsid

    ;ered:Cartesian(rectangular),circularcylindrical,elliptic

    ;cylindrical,paraboliccylindricalandbipolar.Borisov[15] ;extendedthistechniquetothecaseofanonzerosepa-

    ;rationconstanta,obtainingnewpropagationinvariant

    ;wavestructuresinthecircularcylindricalP,,zand ;spherical,,0coordinatesystems,wheretransform(2) ;isinvariantwithrespecttothepolarangle ;=

    ;.

    ;(7)

    ;Herewewillapplythisapproachtothesolutionsofthe ;waveequationsintheellipticcylindricalcoordinates. ;Fora(genera1)curvilinearorthogonalcylindricalcoor

    ;dinatesystem72,v,zdefinedviarelations ;=

    ;X(u,v),Y=Y(u,v),(8)

    ;Batemantransform(2)takestheform

    ;(,v,z,7_)__+(,v,z,7-),(9)

    ;where

    ;(u,,z,)((,”,),y(u,u),z,7-),

    ;(u,,,7-)=G(,,,:)27_

    ;:=

    ;1_((,),y(,),,)

    ;7-

    ;(,,,)?

    ;So,notion(9)becomes

    ;(10)

    ;(,,z,7_)(u,,z,7.)=(,,,),(12)

    ;where,asininitialnotion(2),thevariablesandare

;definedexplicitly,

    ;;::(!!?:(!12?:二二:

    ;2(z一下1

    ;=2(z,(12)

    ;1‟\

    ;whileandobeytheimplicitrelationsthatfollowfrom

    ;Eq.(4)

    ;(,)=1(

    ;,

    ;),Y(gt,)=1y(

    ;,).(14)

    ;Inprinciple,Eqs.(14)mayresultinmorethanonesolu-

    ;tion,givingrisetoseveraldifferentcurvilinearcoordinate ;representationsofthesameCartesian-coordinateBate

    ;mantransform(2).

    ;2.2InitialWavefunction

    ;Firstletusrepresentintheform ;(,,z,)=(,7-)(,v)(15)

    ;SubstitutingEq.(15)intothehomogeneouswaveequa-

    ;tion,whichintheparticularcaseofcylindricalcoordinates

    ;(8)takestheform

    ;1

    ;hh((hOu)+(hOv)au/‟/

    ;+(m))一一o,(16)

    ;u,(,)beingthemetriccoefficients,onehas ;1

    ;hh((hOu)+Ov(h))\a\/‟\/,,

    ;No.4CommunicationsinTheoreticalPhysics ;/0

    ;+(00

    ;Separationofvariableswiththeseparationconstant0.

    ;+n.:0

    ;.02r02z.‟‟

    ;splitsEq.(17)intotheonedimensionalKlein-Gordon ;equation

    ;+..:0

    ;.02T02z…‟‟(19)

    ;(,,,7-)=1(,,,

    ;f)=1(

    ;,)(,)

    ;1,,

    ;A

    ;andthetwodimensionalHelmholtzequation ;1(()+())+n2=..c2.

    ;2.3RepresentationolRelativelyUndistorted ;ProgressiveWaves

    ;Havingappliedconcretesolutionschemesforaparticu

    ;larorthogonalcylindricalcoordinatesystemandobtained ;(,),(u,),(z,7-),and(u,v)assomesolutionsof

    ;Eqs.(14),(19),and(20),onecaneasilygenerateadesired ;relativelyundistortedprogressivewave:

    ;.(“,)+y(u,)+z.一丁.1(,v)+Y(,)+.7_+1

    ;2(z一丁1

    ;Fromhereonletusconsiderthemethodofcom

    ;pleteseparationofvariablesthatinthecaseofKlein——

    ;Gordonequation(19)yieldsanexactanalyticalparticular ;solution[15]

    ;?k(z,7_)=exp(i(kT-+)),(22)

    ;whichdependsontwoseparationconstantsaand.The ;Helmholtzdifferentialequation(20)admitssolutionby ;separationofvariables

    ;(,v)=()(),(23)

    ;(wherethefunctions()andv)acquireapara-

    ;metricdependenceonanadditionalconstantresulting ;fromthe?2-vseparation1inelevenclassicalorthogonalco

    ;ordinatesystems.Il6Jfromwhichthreecoordinatesystems ;areofthecylindricaltype:circular(ordinary),parabolic ;andellipticcylindrica1.SolutionsEqs.(22)and(23)in ;thecircularcylindrica1coordinateswereinvestigatedby ;Borisov.l5Jandthosefortheparaboliccylindricalcoordi

    ;nateswillbediscussedelsewhere.Subsequentstudywill ;foCUSonsolutionsfortheellipticcylindricalcoordinate ;system.

    ;3SolutionsforEllipticCylindrical

    ;Coordinates

    ;Intheellipticcylindricalcoordinatesystemequations ;(8)areconcretizedin

    ;=

    ;(u,v)=hcoshuCOSV,

    ;Y=y(,v)=hsinhusinv,u0,(24)

    ;2(27-)

    ;1((,),(,)).(21)

    ;tricities1/cosy.Equations(14)taketheform ;c.shc.s:c.shc.s.

    ;Z一丁

    ;1

    ;sinhsin=sinhsinv.(25)

    ;z7

    ;Turningtothecomplexvariablevibymultiplyingthe ;latterequationbyiandsummingup,onegets

    ;c.s(i)c.s(),

    ;whichleadstotwosolutions

    ;f=Im(u,v,,),

    ;【西=Re(,,z,7-),

    ;f=Im((72,v,z,),

    ;=Re(u,v,z,7-)+,

    ;(26)

    ;(27)

    ;wherethefunctionscanbeformallyrepresentedvia ;themiltivaluedcomplexarecosinefunction: ;=arccos

    ;(c.s(v-iu)),

    ;=arcc.s

    ;(c.s(“一钆)).(28)

    ;ThusintheellipticcylindricalcoordinatestheBateman

    ;transformtakesthefollowingtwoforms ;(,,,7-)}(+)(,,,7-)

    ;=

    ;1(

    ;ImG,Re,,f),

    ;(,,,7-)西()(,”,z,7_)

    ;=

    ;1(

    ;Im,Re+,,),

    ;wherehisarealpositiveparametercharacterizingthewhere

    ;commonsemifocaldistanceofconfocalellipseswithec一,

    ;p2+z27-21P2+z7-2+1

    ;centricities1/coshuandconfocalhyperbolaswitheccen-—‟

    ;(29)

    ;(30)

    ;,(31)

    ;*Moreconventionalrepresentationcosh(~+i)=(7-)cosh(u+iv)impliesintroductionofthecomplexvariableu+iv.However,

    ;aswewillstudythecyclicbehaviourofthevariablesvand,itispreferabletodealwiththetrigonometricfunctionnotation.

    ;736CommunicationsinTheoreticalPhysicsVo1.56 ;d

    ;

    ;ef

    ;v

    ;/~2r

    ;coshitcos2u+sinh2sin2(32)

;Inthecaseinquestionseparationofvariables(23)

    ;splitstheHelmholtzequation(20)intothesystem

    ;d2Ua~

    ;

    ;(#-h2a2cosh2u):.,

    ;d2Va~,

    ;+(#--h2a2cos2v):.

    ;Beingrepresentedintheform

    ;d.

    ;du2((

    ;d.

    ;.

    ;dv2.((

    ;)2(haJ2cosh2)

    ;)2()cos2):.

    ;(33)

    ;(34)

    ;theab

    ;

    ;oveyieldsUa,intermsofthegeneralmod

    ;ifiedM(Q,q,)andordinaryM(,q,v)Mathieufunc

    ;tions——solutionsofthemodifiedandordinaryMathieu

    ;equations[1r~

    ;d2.,

    ;

    ;(Q2qcosh2u)M=0,

    ;d2M

    ;+-2qcos2v)M:0+

    ;(,Th2a2,),

    ;M(,譬,Th2a2,”).

    ;(35)

    ;(36)

    ;ApplyingtransformsEqs.(29)and(3o),wefinallycon

    ;structthefollowingprogressivewaves ;)=1(一譬,,m)M(,

    ;)=1(一譬,,m)(一譬,

    ;Here/kakisdefinedbyrelation(22),andbyEq

    ;(?byEq.(2s),whileMandMsatisfyEq.f35). ;(31),

    ;4ConstructionofPhysicallyFeasibleRela- ;tivelyUndistortedProgressiveWaves ;Thecomplexnatureoftheobtainedwavefunctions

    ;doesnotprecludetheirphysicalfeasibility:asfaras

    ;=D(Re+iIm):(Re)+iVl(Im):0,(39)

    ;(isthewaveoperator),itjustindicatesthatboththe ;realandcomplexpartsofthewavefunctionseparatelysat ;isfythehomogeneouswaveequation,yieldingtworea1. ;valuedsolutions.

    ;Theconventionalapproachtoconstructingperiodicso. ;1utions,widelydiscussedformorethanacentury,, ;[a16]im

    ;pliesimposingtheperiodicityrequirementuponunlimited ;rangeofthecoordinatevariation,?<<?.Asfok

    ;lowsfromEq.(31),thetransformedlongitudina1temporal ;componentoftheconstructedwavefunctionn(,)de

    ;pends(viaP)onlyonsinandCOSV.Thus

    ;.((“,+2不礼,,7_),(,+27r,z,7-))

    ;=

    ;Aak(Z(it,72,,),:(,,,7-)),(40)

    ;resultingonlytotherequirementuponthetransversal ;component

    ;((,v+2n))((,v+27rn))

    ;=

    ;((,u))((,)),=0,4-1,4-2,…(41)

    ;Consequently,theproblemisreducedtofindingspecific ;representationsofthefunctions,andM,Msatisfying ;Eq.(41).

    ;),

    ;,Re(+1

    ;(37)

    ;(38)

    ;UsingthenotationA:f

    ;21.237byKornandKorn[8]

    ;andtheformula

    ;arccosZ=ilog(Z+i=)

    ;=

    ;ilog(Z+),

    ;weobtainasimpleasymptoticrelation

    ;=areeos(A~COS(Vi))ilog(A+e”+‟,

    ;=+7r+27rn-il.A?l,

    ;(42)

    ;uo(3,(43)

    ;wheretheintegerndependsonhowwefixthevalueof ;thecomplexlogarithmfunction:forexamplethechoice ;oftheprincipalvalue,Log(?),definesninsuchaman- ;nerthatRealwaysresideswithinthesegment0,27r).

    ;WewilluserelationEq.(43)forfixingthevalueofin ;Eq.(28)insuchawaythat

;limfit,v1=lim

    ;“—†..”—?..

    ;Re

    ;

    ;case(z)>0,

    ;+7rcase(7_)<0,

    ;forand

    ;.

    ;!im.(u,)=liraRe+U,..U_oo

    ;rv+7rcase(7_z)>0,

    ;vcase(7_)<0,

    ;for,whichcanbecombinedinto

    ;lim

    ;..

    ;(,)=+:aaseez-r<>.0,

    ;,

    ;(44)

    ;(45)

    ;(46)

    ;forbothwavefunctions.Followingthisprocedure, ;forthe

    ;observationpointslocatedsufficientlyfar(withrespectto ;No.4CommunicationsinTheoreticalPhysics737 ;thedistancescaleh)fromtheoriginofcoordinates ;havethetransformconditionssimilartoEq.(7)for ;wethetransformedellipse

    ;the

    ;circularcylindercoordinates.

    ;As(i)thetrigonometricfunctionscomposingEq.(28) ;areanalyticand(ii)eachofrelationsEq.(28)conditioned ;byEq.(46)mapsalmostsmoothlytheRiemannsurface ;?({i))intotheRiemannsurface?({)),analyticity

    ;ofthedependence=(,v)holdsforallvaluesofv,pro

    ;videdthatbothandarenon-zero(i.e.,thetransform,

    ;v_.doesnotinvolvethesingularpointsofitsparent ;transformZ(vi)-_†?(),whicharelocatedbetween

    ;thefociofthefundamentalellipse{-hxh,Y=0}). ;Moreover,onecaneasilycheckbydirectcalculationsthat ;coincidewith”orv+7rforallvaluesof.correspond—

    ;ingtothemultipleofdirectionsofsymmetryofthe

    ;ellipticandhyperboliccoordinatelines ;c,”,”=+

    ;

    ;for

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