THE PROPAGATION OP ELECTROMAGNETIC WAVES IN PARABOLIC PIPES Thesis for the Dogma of M. 5. MICHIGAN STATE COLLEGE Robert Dean Spence I942 [1: S. _U 4L- :5.) IMH‘ . IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 31293 01774 9643 .. . LIBRARY I Michigan State University PLACE IN RETURN BOXto remove this chedtout from your record To AVOID FINE retum on or before date due. MAY BE RECALLED with earlier due date if requested. TJE PROPAGATION OF ELECTROEAGNETIC WAVES H PARABOLIC PIP“ by Robert Dean Spence ”.7 A THEE 3 Submitted to the Graduate echool of Kichigan State College of Agriculture and Applied Science in partial fulfilment of the requirements for the degree of DIAS; .43 CF Jul—Ii? E Department of Physics 1942 ACKNO'I'I‘L EDGLIAICT The writer 'I'Ii shes to express his sincere thanks to Dr. C. P. Wells for his aid and counsel on this problem and to those members of the staff of the Department of Physics who dontributed numerous helpful discussions. I. II. III. IV. 8 :3 TENTS Introduction Solution of the Wave Equation Perfectly Conducting Pipes Imperfectly Conducting Pipes Appendix 149§78 /j. /% I. INTRODICTION The purpose of this work is to discuss and formulate mathematically the physical properties of a particular system for the transmission of guided electromagnetic waves. The system consists of a straight, hollow, con- ducting sheath or pipe of parabolic cross section as shown in figure 1. The electromagnetic waves are assumed to be confined to the interior of the pipe and are prop- agated along its axis. The discussion 3 limited to the problem of determining the waves that may exist in such a system and to the attenuation that they experience as they move down the pi e. The question of reflection and radia- tion that may occur at discontinuities in the interior dielectric or the conducting sheath will not be discussed here. V The first diseussion of such systems for guiding electro- magnetic waves appears in a paper by Lord Rayleigh, pub- lished in the "Philosophical hagazine"l in 1897. In this er he showed that for perfectly conducting pipes of rap ‘ C re tangular and circular cross section all waves greater than a certain critical wave length were completely attenuated, while wave lengths shorter than the critical wave length were freely transmitted. Futhermore, he was able to separate the types of waves that occur in such See references 1-5 in bibliography at the end of this paper. systems into two types which he called "waves of the first and second kind" which are identical with the E and H waves to be defined later in this paper. In 1931 South- wortnl began an experimental investigation of wave guides of circular cross section. He reported the results of his work in 1936 and at the same time Carson, Mead, and Schel- kunoffg published the first theoretical study of wave guides whose sheath was of finite conductivity. Ba row and anus reported theoretical work on pipes of rectangular and elliptical cross section in two papers that appeared in lads. 1n the case of the elliptical pipe Chu found that there existed two kinds of E and H waves which he des- ignated as odd and even and both of which degenerated into waves of the circular pipe for the case of zero eccentric- ity. The work on rectangular pipes by Barrow and Chu is interesting because it includes a discussion of the radia- tion fronrthe Open and of the pipe; The radiation from a circular pipe was discussed by Uhu in a paper published in 1940. Ibid. 2 Ibid. Ibid. -J II. SOLUTION OF THr HAVE EQUATION I: According to Eisenhart the scalar wave equation is sep- arable in only four distinct systems of cylindrical coor- dinates. On the basis of Eisenhart's criterion1 the only types of pipes for which the scalar wave equation may be separated are those whose cross sections are either rectangular, circular, elliptical, or parabolic. The first three have been well investigated theoretically, but the fourth has received little if any attention.2 The parabolic coordinate system used to describe the parabolic pipe is defined by the transformation: [I = é'CP‘Tfé/ i=7; The Z axis is taken to be the axis of the pipe and the surface of the pipe itself is formed by the intersection of the two parabolic cylinders;=;=constant. Plane waves, nonhomogeneous in f’ and fr are assumed propagated down the pipe in a nositive direction. This and the assumption of a simple periodic time variation is equivalent to postulating an electromagnetic field of the form: A? :_ /—'-; / I j ,1. -A2 (chat /—7 / 7f (1) Eisenhart, L. P., Annals of Mathematics 35, pp. 284—304. 1954 2 See references 1-5 in bibliography at end of this paper. Sigure 1. Parabolic cross-section with parabolic coordinates inserted. in which a) is equal to 277 times the frequency of the source, and h is the prOpa"ation canstant in the positive 2 direction. The propaga.ion constant is itself complex and equal to a +A3i where cr is the attenuation constant and K3 is the phase constant. The scalar components of the electromagnetic field are obtained from the wave equation and from Kaxwell's equations. In parabolic coordinates these are: 2.. ,4 £2: ,1 a’fll/ér’r/ry/fi/fy = 0 _ 32‘ afz (2) rL/Tl- fl :_/ L94 1, {c'wévwr/ £41 7 f r a}, 3? (5a) 3?, 3:; (3b) r’I/T/‘j A” 2-,! 24:6 - ca.) fl/flz 7/ r f A 27 I...) (5d) 3' where/-cwgw.tqflrand_:wuzz‘;g“ . We shall call 7- the wave guide constant. "It will depend only on the physical size of the pipe and the mode of the wave excited. The practical system of units is used, where: e :dielectric constant ‘éepf farads per cm. (in air) /U = permeability .xi 1/0 7 henrys per cm. (in air) ”'= conductivity in mhos per cm. /9 I magnetic field in amoeres per sq. cm. 1? electric field in volts per cm. Separating variabilos in the wave equation leads to the two ordinary differential equations: éfib"w flsy”:*nw/é7 =c9 ( 4a) df‘ .5525 i /r7‘-/m/// : 0 ( 4b) where m is the separation constant, restricted here to pusitive integral values. Both of these are forms of the i l - . l confluent hyperseOLe.ric ecuation or as they are some- times called, parabolic cylinder equations. Solutions in 2 the form of definite intesrals were given by Weber for the problem of the parabolic membrane. Later all solutions 5,4 were classified by Epstein in connection with the problem of diffraction of lirht by a parabolic cylinder. Both even and odd solutions of order m exist for these eouations. we shall desisnate the even solutions of (4a) and (4b) as eUm’ V and the odd solutions as U , V e m o m o m l Whittaker and natson, Modern Analysis, p.341,2nd ed.(1915) 2 aeber, Die Eartiellen Qifferentialslsichuqen der Kath. Phy81k,“Bd. a, nnfl. S. 238 (1912 Epstein, miss. Kunich. 4 See appendix for list of solutions. The solution of the wave equation can be written: 5 gunman V// l . waxy/1‘13”” as) where bm and cm are complex constants and depend upon the str noth of excitation. III. PERFEU TLY CONDUCTING PIPES The boundary conditions depend upon the assum mptions made regarding the conductivity 0' . In this section 0 we consider 0/ to be infinite in the walls of the pipes and zero in the dielectric inside the pipe. under this condition the tanaential conponents of the electric field vanish at the walls and the prOpagation constant h reduces to a}! , since the attenuation onstant must be zero. Boundary conditions of this type can he sat- isfied by considering two partial fields to exist in the pipe. The first, called an E wave is defined by Hz 3 O everywhere inside the pipe, and the second, called an H wave is defined by EZ - O everywhere inside the pipe. Thus for the two partial fields the boundary conditions are: 5=olf=7=a [We [7 :o,f :a— (08) £120, 7 :a, Hove/m: [zzo/fgq (6b) :a J 11W: I3 "’~ 1:}:0 where f =1 srjf'8a cat-ininaa l)“ lggndnry of the pipe. The boundary conditigns are exceptional in that the walls of the pipes are formed by two cylindrical surfaces rather than just one as in the case of the circular or elliptical pipe.1 The above boundary conditions can be satisfied by using only one term of the series siven by (5). For the E waves the components of the field are: E. ~—= easy/14W , C Jar/meal. ”vs-W: 2; a) where the primes indicate the differentiation with respect to either f or z' . In order that the tangential components of the electric field vanish at the boundary, it is necessary that U and V satisfy the equations: eUnda) = 0 eVm(a) - 0 (8a) See appendix for list of solutions. OUm(a) - o on(a) = 0 (8b) The roots of these equations serve to determine the crit- ical wave lennth. corresponding to a root anm where n indicates the number of root and m the order, the critical wave length is: )Lm” = 37' 2;,” For m = O the functions of U and V are identical. In this case it is not hard to verify that they can be written in terms of Bessel functions of order i and -:. The even solution is: .44ng = c,/j7'ez .> any as was mpmanompfim .mpspm snoopt may mo macaw m - ha. 6 1M. 96. .:.E pmvpo N; BN. .0 o madmwm .1 RN. 3mg. y, / / : //- l -| l I .E... . ' .12.. I Y ”AV" _ O 1 U! o. .4 . \. — _.. _ I... slid-I“ If. ‘ ‘irure 9. Electric field configurations for the ground state modes shown for quarter section:- of the pipe- The subscripts are written in the order m,n. “—4- ———. an-———~. “—— AZ. /5 As in the case of the E wave there appears to be no simple method for calculating roots of higher orders. IV. IMPERPECTLY CON UC IKG PIPE 0) In this case we consider the conductivity to be finite thouch large. This required modification of the boundary conditions which are now the continuity of the tangential electric and tanrential mannetic fields. The actual fields that exist will be a superposition of the two partial fields which we previaisly designated as E waves, or as H waves. However, if the conductivity is large, it will be assumed tlat one of these partial field will predominate and we shall still speak of E and H *aves. ror each case we shall define a set of associated impedances which will help clarify the physical statement of the probleml. A superscript (l) on the impedance will designate an E wave, 22’ z 1;; :5; _-_ ifs. _—_ f5 (13a) where Z:=f%“'is the Characteristic impedance. The imped— ance for the coordinates f’ and f7 are defined by: m, [z =‘Zf (a, Z} : Zg/ff (13b) Likewise for the H wave: Zia) : é; :- Q T: .Ié £31: 7% 47 '4? ’ (lda) .. ~’ 1 ": "fl -' Stratton, alectromarmatic Theory, pp. 554-o5. 1941. /7 Because of the similarity of the two boundaries]'- a andZV a a we shall limit the remainder of the discussion to the boundaryjr : a. Loreover we shall no longer dis- tinguish be ween even and odd U and V as t.e det ils are the same for both. for the field outside the boundary of the pipe I = 8 we shall take a... p ’11 JI'WZ x" =2: V; 7.172242% X (15) z? ...._.o whereIZflfi/represnnts an asymptotic solution of the conflu- ent hyperncometric equation. The primes will hereafter desionate the value of the functions outside the pipe. I As an approximation forZV/we shall use: _ ('7; 2' cI/ZWJ/jg— 7::‘gr2 m .4: 2 1 (16) zrjz which is valid for values ofJCU~1. Then (15) can be written: / I p, , (r? 2' .. Zz flu/Z ‘5; = 5 AZ?J.4L_.£;_E .g (17) 27"] where 77/7]: [51,; V 7/.."/Z”””/;’Z /JT and b' is a new constant. The boundary conditions require 3? =§g at'f - 8 where: A? = ”if"? .4”? ” , W : a: WW- ii? 24¢171 2f / rlzyz-Jfl af. We have assumed for the second equation thatofi)cwc and / . ’ f ' q , 7c’n/M! 4/ Vua¢m- nquatine h/ and né anc setting 7fi%4%7/ equal to unity since it is independent of)” , sites a relation for the constant b' in terms of b: ,9“ z - écoc’i’nz ,1 if" gl/Q/ (is-’1) 0' 7'1 if In this expression we have assumed 17%2’))’ . Jalculation of the attenuation constant is the prin- ciple objective in the imperfectly conducting pipe. This can be done by means of the definition: a :3 {4 _é‘/ (19) z ”,3 where W]! is tte power lost throuWh the wall, f': a and wZ is the power transmitted down the pipe. In order to evaluate w? it is necessary to intecrate the real part of the Poyntino vector in the z direction over the cross section of the ripe. for tie a wave this yields: _ m this“ o/o/afl/V/jff ””727; ”(r-:0) 2?M The value of wZ for the R wave can be obtained by re— placins 6 by /4 . /6- The power lost throu“h the wall of the pipe is: —/ 1v -- . /r ~— , V; .- clef/Lea ’ wax/yard»; (.211 for a unit length of the pipe. nere R indicates the real part of the expression and * indicates the complex con- juqateo It is necessary to en phas sine tLat for an inperfsctlv P conductins pipe the so-called a wave actzally has a small HZ component present. in computina the power lost through the pipe the se cannot be neslected. Both a macr etic and an electric impedance will exist for either wave. for values of é’)) (we the approximate values of the im- ped nce are: / w ’ W1 4'7} Z}! *' ' *7 '7’;— “’ ma) Za/://;T“Ii7lé;:«1(aab) By means of equations (15) and {14) we find h x, {N , 77' ’ 5f W's]; ’ 272‘ n. [2; : :4»? 4}”! A? I / é/:;’/;l//l/jrz a]? (4,5) These viva for the 3 wave: -"’ z 5/2 Jfl/fi/ /z/d_ ’2 1/ Vf= V2 V312 w“ 57;?” ”7/ (:24) 7 /7- The attenuation for the n wave is: 41/22 ~- WWW/M ” 7““ ////z 777/77 where_//ii; and fémm is the frequency correspondine to the have lensth/Jmlm. . Followine the saye method, the vs lue of WJ, for the E have is: g/: J/z/’Z/z7/7//‘/ /7/’~-,11,’//// 1:25) ”I" The attermu tion for t1:e ; wave is: 1/7 1 11., 7f 171/177 //7_Z*—:" /////J!//*/‘/1p// 1/}, \fi The values or the attenuation for both war 3 are in 17$ aereenent with th se fOJnd by Ghu for the pi e of ellip- tica1 cross section. Both expressions for the attenuation have one term that Varies asJ/éi thus increasine with frequency. Thus it of interest to note that the behavior of electronasnetic w;ves, insofar as the above theory applies, is the same f0: all cylindrical pipes whose /5 whose cross sections are defined by coordinates Miich separate the scalar wave equation. jxperinental work is now in progress to det-rmine ways in which the various partial fields may be excited and to check the values 0? tne critical frequencies and attenua- tion constants for VariOJS modes. /7 We now consider the nethod of obtaining the solution of fi( /r' ’ tan/[C : 0 (2C) 1% ’1 / Letting gz=rj_%1F‘/72z/ we obtain 55% ,1 1/7411? f (m «(r/fl :0 . (29) Agri 49’ Setting 1 = 5734' yields .. 2/‘7 a — ‘7 (3“ ~- 21 4/sz 145/ JV 5/4:- .. 6-5 -317 /7 .. a . (50) This is the same form as Krummcr's first confluent hyper- peometric equation 2 £1.17 + éezj/fl ., (2/7 I (bl) '1’” 22: where c = g and a -/§%-;§?' . Nov let a ‘ P is determined Irom the indicial equation -flW~/¢V=v and since c = % P = c or i. The coefficients for the series are given by Cnrl (nfl)(nfc) - Cn (n-a) = o If CO = l the series is 2: C 1"" : / — a- z ,, gfig-d z‘ , g/x-d/a’iLf’..-; (55) ,M if. <43fl7 3/ (fivflk4d 1’ 20 Since p is doutle valued I nay we either odd or even in respect toJf . (/7 —— 475,, 0,74" (an a /‘7 = ,6ng flay/‘1)" (ISM-‘77) If a or (c—a) is an interser the series in finite and w ohtain Eernetian or Sonine polynonials. However this condition can be satisfied only if the separation constant is permitted to take on inacinary values and this is im- possible if we wish to satisfy real boundary conditions. The final solution of our orisinal differential equation must be real and so we take teéQ’=' an'é}f e -Qn[ v‘,awa‘:que/’.afij¢ (353) (55b) 0 6/ - (r—a'déf} (1/71/4/ 7‘ _,¢é'zi1~g2? /7 (”,7 7 These are given in the integral form by Eater: I t a : fwd/Ma/w/y/J-zwx- :5: 4/5 '22:.) \ / 2-; -- ,. J- , H- a :j’\o/ J 6/. / (‘»/-\),/;— / 63:) /}y*/j_ 3///?}% g? [if (2)93) 0 For the cise wte'e m - o the differential equation :5 a special form of the bessel equation: 4’ / "&,c”/0 =0 which has the solutions 7.- i z/Lr L/_: /::‘é:"j / r g? r v62; Dig/(ffgdz'fl/ Thus 6 4/0 -. 4?" J; ("97 7 0420 - 6"- 4; (27,57 2/ BIBLIOCYC Lord Raylei2h, On PassL Tubes or tte Vi hrationsg _—r—- -—_ _—_ ———‘ -——__ _ Philosophical nag; ine, ZZ. .721- lectric ¢avcs Throufh "lectric Jylinders. 50 (1:39 7) Southworth, 8.0.,11 oer-fi'rc eC1ency Jove Guides. The Bell System gechnica Jounr :1 15; 284. (1956) Jarson, L.R., Lead, 3.P., and ’chelkunoff, S.A., Pvper-rrocuencl Iave Guides- ‘ attem' tical ” sorz. lhe Lell vate. Techr ical dour rnal. 15; blO (1956) Chu, L.J. , lECtrOerq*th Javeg in illiptic gollow Y res of et :1 Journal of Applied rhysics, 9; 535, k 958) Chu, L.J., and Barrow, 7.L., Jle ec in mellow rises of Rectan2n ar r ‘ 7 tro notic Laves oss _'_ .Lection. Proceedings of the lnsti 26; 1550 (1930) Pa2e, L.I., and Adams, N Chu, L.J., Calculations of hollow Sines and fiOTU tute of Li di 10 sn2incers, _) .I., ulCthO““”thlC caves in Condictinq fubes. Physical neview 52; _647 {193 7) of the mediation Prooerties s. uournal of noplied Physics; 11, 603, {1940) 4} ) 3 MICHIGAN STQTE UNIV. LIBRQRIES 017749643