The Classification of Linearly Polarized Transverse Electric Waves

Transcription

1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 5, ARTICLE NO. AY97515 The Classificaion of Linearly Polarized Transverse Elecric Waves P. R. Baldwin Deparmen of Physics, The Uniersiy of Akron, Akron, Ohio Submied by Rober L. Anderson Received June, 1995 Recenly he auhor has underaken he classificaion of coninuous soluions o some common vecor PDEs. These include he simples of Belrami soluions o hydrodynamic flows and elecromagneic wave equaions Žwhich are, in fac, closely relaed.. In his paper, we consider linearly polarized ransverse elecric wave soluions o he elecromagneic wave equaion: E ih; H ie. Using Clebsch funcions and differenial geomeric echniques he auhor is able o give a discussion of he mos general possible forms of propagaion of linearly polarized waves. A zero curvaure condiion Žwhich assers ha he Clebsch funcions necessary o represen he soluions are expressible in erms of Caresian variables. reduces he sudy of vecor PDEs o nonlinear PDEs, which may be solved in special cases. The benefi is ha he Clebsch represenaion is he one ha is he mos useful for analyzing he srucure of he flow. This mehod of analysis is now a feasible soluion mehod due o he remendous advances in mahemaical sofware allowing one o compue curvaures, given meric funcions. For he special case of linearly polarized ransverse elecric waves he analysis may be applied o a grea exen Academic Press 1. INTRODUCTION Recenly Baldwin 1, has underaken he classificaion of cerain soluions o sandard parial differenial equaions of mahemaical physics. These include special cases of soluions o he Belrami equaion and linearly polarized ransverse elecromagneic Ž TEM. soluions o he elecromagneic wave equaions. In his work we consider he more difficul problem of aemping o wrie down general soluions o he wave equaion. This becomes much easier when one assumes a direcion of propagaion for he wave; his is a sandard procedure in elecromagneism exs. In his aricle we presume X97 $5. Copyrigh 1997 by Academic Press All righs of reproducion in any form reserved.

2 314 P. R. BALDWIN o deermine wha possible propagaion direcions for ransverse elecric Ž TE. waves are allowed. The mehod ha is employed is represening he fields in erms of Clebsch funcions and using a Lame analysis o solve for he Clebsch funcions. An arbirary soluion o he wave equaion may be expressed by Eq. 5., where he funcions P, s, u saisfy Eq. 7. and are such ha he Ricci curvaure compued from he inverse meric ensor Žwhere,, are he coordinaes. given by Eq. 9. vanishes. This curvaure sor of analysis owes is heriage o Lame 5. The novel feaure in he analysis is he use of he Clebsch poenials o describe he vecor fields. In paricular, we consider linearly polarized ransverse elecromagneic wave soluions. We find he following resuls: we are able o reproduce familiar soluions o he elecromagneic wave equaions, including spherical waves and waves propagaing down circular and recangular waveguides. The auhor is only able o consruc soluions where he propagaion direcion is in he radial spherical direcion or in a Caresian direcion. This is reminiscen of he linearly polarized TEM case.. SOLUTION TO THE WAVE EQUATION In order o sudy elecromagneic waves propagaing in a cerain direcion, we should firs ask how we should define a wave o be propagaing in a cerain direcion? Clearly his needs some inerpreaion. We feel ha he following is he mos naural inerpreaion. We need o have he direcion of propagaion be described by he gradien of a real coordinae funcion Ž., so ha we may wrie wih E e i E, He i H, E H real, real. 1.. Tha is, he componens of he fields resriced o he planes of consan phase are all in phase wih one anoher. This is consisen a leas wih Rund. 3 Moreover, he E and H fields mus saisfy he elecromagneic wave equaions, which in appropriae variables, read E ih, HiE. 3.

3 POLARIZED TRANSVERSE ELECTRIC WAVES 315 Now subsiuing.1 ino.3, we have i E E ih, ih H ie. 4. Nex we can ake he do produc of each equaion wih and noe ha he LHS mus be real Ždue o... A once i follows ha E and H mus boh be complex. Therefore we can wrie E iq e, H iw h, 5. where e, h, Q, W are all real and e h. Now we subsiue 5. ino 4. and equae he real and imaginary pars o find e Q h, 6. ew, 7. hwe, 8. hq, 9. where we claimed e and h should have no componen along. Noice hen from 7. and 9. ha e Ž e. h Ž h.. Therefore we may represen e and h wih jus wo funcions. 4 In summary, we have derived he sysem wih e i E E iq s, e i HH iw u, 1. s Q u, 11. sw, 1. u Ws, 13. u Q. 14. These compleely code he wave equaion and condiion.. The nex objecive of he Clebsch mehod is o encode he vecor PDEs appearing in as relaions among he componens of a meric ensor. Using, and as coordinaes Ž 1,, and 3, respecively., we ake he inner produc of each of hese equaions wih he gradiens of each of he coordinae funcions. Defining D, 15.

4 316 P. R. BALDWIN here follow ha Q Duh Ž sq. Duh s DWh sdwh Ž uw Dsh.. WDsh 3 1. udqh 1. udqh 11, 3. as well as h 1 h Recasing in erms of relaions of he inverse meric ensor, we find ha we may wrie for 1. e i E E ip Ž,,. u sž,., e i HH ip Ž,,. s už,., 5. where P Qu Ws and he las equaliy follows from 18. and 3.. Also, DP u Ž Ps. Ž Ps. ij D D h s s 6. Ž Pu. s Ž Pu. D D u u and Ž Pu. Ž Ps.. 7. u s Considerable simplificaions ake place in he case of TE or TM Žrans- verse magneic. waves, where už,. or sž,. will be zero. Noice ha generically for linearly polarized elecromagneic waves he componen of he E field orhogonal o he direcion of propagaion is no necessarily Ž 3 orhogonal o he H field h..

5 POLARIZED TRANSVERSE ELECTRIC WAVES 317 As is well known, he deerminan of he marix given by he conravari- an meric ensor mus be equal o D, where D is given by.15. This implies 1 1 D Ž Pus. Žus sž Ps. už Pu. Ž Ps. Ž Pu. Ž Ps. Ž Pu.. 8. Thus he conravarian meric for he linearly polarized elecromagneic waves may be expressed by Ž. ij h us s Ps u Pu Ps Pu Ps Pu us Ž u Ž Ps.. up Ž Ps. up, 9. Ž Pu. sp s Ž Pu. sp where P PŽ,,., u u,,ss,, he relaion 7. holds, and he above meric yields zero curvaure. The fields are hen given by 5.. We nex resric he se of soluions o o he case of TE waves SOLUTIONS TO THE PLANE-POLARIZED TE PROBLEM The vanishing of he funcion Q Ž equivalenly u. is precisely he case of TE waves Žsee 1.; W for TM waves.. Then 1. and simplify o wih and e i E E sž,., Ž 3.1. e i HHiWŽ,., Ž 3.. Ž 1 W Dsh. Ž 3.3. s DWh 11 Ž 3.4. sd uh 33, Ž 3.5. h 3 h 1 h 13. Ž 3.6.

6 318 P. R. BALDWIN Equaions Ž 3.3. Ž 3.6. correspond in urn o 16., 19.,., and 4.. Since u už. from. and 3., hen u in urn may be aken o be simply uniy, as may be seen from 5. and Ž 3... Therefore he conravarian meric ensor reads sdw ij h Ž 1W. Ds. Ž 3.7. sd ij Ž. However, he deerminan of his marix mus iself be D see.15. Therefore, we have and he conravarian meric ensor mus read D W s Ž 1 W. Ž Ž 1 W. ij h WŽ ss. sw Ž 1 W. s ij, Ž 3.9. where W WŽ,., s s,, and,, are he coordinaes. The E and H fields of linearly polarized TE waves are given from he above funcions, as in Ž 3.1. and Ž 3... I remains o calculae he curvaure of his marix and o se i equal o zero. The 1 and 3 componens of he Ricci ensor formed from h ij are paricularly simple. They become 1 ss ss R1 W, 4 ss Ž W 1. 1 ss ss R3 W. 4 ss Ž W 1. Ž 3.1. Case 1. If we assume ha s, may no be decomposed ino he produc of a funcion of imes a funcion of hen i mus necessarily follow from Ž 3.1. ha WŽ,. a consan, so W Ž,. W Ž. L. Ž Case. Oherwise s, may be aken o be simply a funcion of : s, s. This may be shown in he following fashion: one possible soluion o Eqs. Ž 3.1. implies ha s, s fž.. By examining Ž 3.1., however, we see ha fž. may be absorbed in such a way ha fž. 1, as has been done above. We discuss Case a greaer lengh below.

7 POLARIZED TRANSVERSE ELECTRIC WAVES Case 1 If s, may no be decomposed as a produc of a funcion of and a funcion of, hen i necessarily follows ha W W Ž. L, in order ha he 1 and 3 componens of he Ricci ensor vanish. The 13 componen of he Ricci ensor now reads 1 W L R13. Ž 3.1. W Ž. L Thus Case 1 decomposes ino wo cases, each of which ensures he vanishing of he 13 componen of he curvaure ensor. Case 1.1 Ž L.. We shall see in a momen ha his is someimes relaed o he case of spherical coordinaes. The siuaion L implies W W Ž.. Ž Under his assumpion he 11 componen of he Ricci ensor reads Seing he R 11 / 1 W 3 W W R11. Ž 3.14 ž. W componen of he curvaure ensor o zero implies ha Ž 1. W 1 k k The and 33 componens of he Ricci ensor are hen proporional Žhey will vanish ogeher.: and R33 Rs Ž R s 3 s s s 4 s s s s ss s s ss s 1 ss sk s s ss ss ss Ž If s sž., see Case.1.1. Case 1.. In his case, W is a consan which may be aken o be zero, due o a shif in. Tha is, W L. Again R and R33 will vanish ogeher: R 1 L R s

8 3 P. R. BALDWIN and ½ s 3 R s s s s s s s 1L 1L s Ž s. ss s ss ss Ž s. Ž s. sž s. s. s Any soluion s, o Ž hen yields a se of elecromagneic fields, e i E E sž,., Ž 3.. e i HHiL, Ž 3.1. ha represen linearly polarized elecromagneic waves ravelling in he direcion given by. Moreover, he inrinsic coordinaes,, are expressible in erms of he usual Caresian variables. One of he sandard cases ha belongs o Case 1. is he propagaion of TEM waves in a recangular waveguide. One may well ask wha is he advanage of he mehod we develop here in discussing TE waves, as opposed o a sandard discussion say as given in Landau and Lifschiz for example. In he laer, a direcion of propagaion is assumed. Here no such assumpion is made, we aemp o derive all he possible forms of he propagaion direcions. 3.. Case If s sž. he 1 and 3 componens of he Ricci ensor are zero; however, he res of he elemens of he Ricci curvaure ensor are sill quie complicaed. The auhor has been unable o find general soluions from he parial differenial equaions obained by seing o zero he 13 componen of he Ricci curvaure ensor. For compleeness we name hose soluions ha do no fall ino he laer classificaion as exoic soluions. Case. Ž Exoic soluions.. We lis he 11 and 13 componens of he curvaure ensor below for which he zero ses should be he simples o solve: 1 11 ž R 1 R s RR 3R Žs R Ž 1R. Ž 1R R. Ž srr srr sr. R 3 s R R sr R, 3. 1 ss R R R R R RŽ R. RR R 4 RssŽ 1R. 13 /. Ž 3.3.

9 POLARIZED TRANSVERSE ELECTRIC WAVES 31 Case.1. Ž Special cases.. Le us assume ha he crieria of Case 1 is also saisfied: WŽ,. W Ž. L Žsee Ž As before, he analyses of Case 1 mus hold rue. Therefore, i mus follow ha eiher WŽ,. W Ž Case.1.1. or WŽ,. L Ž Case.1... Case.1.1 Ž See also Case A furher case may be reduced o spherical coordinaes. Under he assumpions of Case.1 wih WŽ,. W we see, by comparing o Case 1.1 ŽEq. Ž , ha he vanishing of he 13 componen of he curvaure ensor mus yield up o a shif and rescaling. Thus, Seing he R A paricular soluion is Then subsiuing implies 1 W Ž 3.4. h11 1, Ž 3.5. ds h sž., Ž 3.6. d h33 gž s.. Ž 3.7. componen of he curvaure o zero implies Case.1.. Now we ake 3 s s ss s s s s. 3.8 ' s 1 e. 3.9, Ž 3.3. ln sin, Ž r Ž 3.3. cos i e E E, ˆ Ž r sin ˆr cos i e H H i ˆ. Ž r r sin W L ; sž,. sž.. Ž All he componens of he Ricci curvaure ensor are found o vanish excep for he and 33 componens. These will vanish when 3 s s s s s sss s s 3.36

10 3 P. R. BALDWIN Žcompare o Ž 3.17.; Eq. Ž may be obained from Ž by seing s, ec., where denoes a derivaive wih respec o. This may be seen o be relaed o a Bessel equaion by a change of variables from o by means of he ransformaion d d. Ž s This leads o he covarian meric ensor, where,, and are he variables, whose nonvanishing componens are g11 1 L, Ž Ž. g, Ž L g33 Ž L 1. g. Ž 3.4. The dos refer o derivaives wih respec o. I is now clear, ha upon subsiuing Ž. JŽ., Ž we find g11 1 L, Ž 3.4. g L, Ž g33 Ž 1 L. L, Ž wih all oher componens of he covarian meric ensor vanishing. This is immediaely recognized as he meric of cylindrical coordinaes. We hen may idenify z' 1 L, Ž ' L, Ž ' LŽ 1L. r, Ž where r,, z are he usual cylindrical coordinaes. FIG. 1. A schemaic of he soluions o he linearly polarized TE waves via he Clebsch mehod. The E and H fields are given by Eqs. Ž 3.1. and Ž 3... The conravarian meric ensor for he coordinaes,, and is given by Eq. Ž The parial differenial equaions ha are solved in his work, are hose resuling from seing he Ricci curvaure compued from he meric ensor o zero. Transformaions beween,, and and sandard orhogonal coordinae sysems are hen sough.

11 POLARIZED TRANSVERSE ELECTRIC WAVES 33

12 34 P. R. BALDWIN Combining everyhing, we find ha ž ' / dj r L 1 L E ' L, ˆ Ž dr L djž r' LŽ 1 L. ' / ž / ˆ H i J r LŽ 1L. z ˆr. Ž ' 1 L dr To make his soluion look like a sandard soluion for TE waves ravelling down a cylindrical waveguide as in Landau and Lifschiz, one may subsiue L k z. Alhough here are exoic soluions Ž Case.., all he soluions which he auhor has consruced belong o eiher Case 1 or Case.1. The auhor hanks R. M. Kiehn of he Universiy of Houson for his commens and encouragemen and G. M. Townsend of he Universiy of Akron for consrucing he figure. REFERENCES 1. P. R. Baldwin and G. M. Townsend, Complex Trkalian fields and soluions o Euler s equaions for he ideal fluid, Phys. Re. E 51 Ž 1995., 59.. P. R. Baldwin and R. M. Kiehn, A classificaion resul for linearly polarized principal elecromagneic waves, Phys. Le. A 189 Ž 1994., H. Rund, The Hamilon Jacobi Theory in he Calculus of Variaions, Van Nosrand, London, H. Flanders, Differenial Forms wih Applicaions o he Physical Sciences, Dover, New York, P. Moon and D. Spencer, Field Theory for Engineers, Van Nosrand, Princeon, NJ, 1961.