1492-2-Q THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory 1492-2-Q = RL-2034 REDUCTION OF RADAR CROSS SECTION OF DUCTS Interim Report No. 2 1 April - 30 June 1968 By John J. Bowman and Vaughan H. Weston 6 July 1968 Purchase Order 504-855029 Contract F33615-67-C-1843 Contract With: Northrop Corporation - Norair Division Electronic Systems Research Group Hawthorne, California Administered through: OFFICE OF RESEARCH ADMINISTRATION * ANN ARBOR

FOREWORD This report, 1492-2-Q, "Reduction of Radar Cross Section of Ducts, " was prepared by The University of Michigan Radiation Laboratory, Department of Electrical Engineering, under the direction of Professor Ralph E. Hiatt, Head of the Radiation Laboratory, and was written under Purchase Order 504-855029 to the Northrop Corporation, Norair Division. This Second Interim Report covers the period 1 April through 30 June 1968. i

1492-2-Q ABSTRACT As part of an investigation concerning radar scattering by a rectangular duct lined with absorbing materials, some problems associated with the diffraction of plane electromagnetic waves incident obliquely (with respect to the edge) on an absorbing half plane are considered. The half plane is initially assumed to be governed by an impedance boundary condition wherein the impedances on the upper and lower surfaces of the diffracting screen may differ from one another; however, the investigation then reveals that a rigorous solution appears possible only in the case for which both impedances are identical. This means that in order to apply ray optical techniques to estimate the field generated at the mouth of the duct, it is necessary to consider a duct which is coated on both the interior and exterior surfaces with absorbing materials manifesting closely similar electrical properties. In connection with the modes sustained by a rectangular waveguide whose four walls obey an impedance boundary condition, the boundary conditions are found to be inseparable and no explicit results for the modes are available at present. On the other hand, closed form solutions can be obtained for the modes that exist in a rectangular guide in which two parallel walls are absorbing while the remaining two walls are perfectly conducting. These modes are studied as a preliminary to the more difficult fully-lined duct problem. Also briefly discussed are the modes sustained in a circular waveguide whose wall obeys an impedance boundary condition. ii

1492-2-Q TABLE OF CONTENTS I INTRODUCTION 1 II OBLIQUE INCIDENCE ON AN ABSORBING HALF PLANE 3 III MODES IN RECTANGULAR WAVEGUIDES WITH IMPEDANCE BOUNDARY CONDITIONS 17 IV MODES IN A CIRCULAR WAVEGUIDE WITH IMPEDANCE BOUNDARY CONDITIONS 22 APPENDIX A: PROPERTIES OF THE FUNCTIONS ( 3) AND (). 24 REFERENCES 28 DD 1473 iii

1492-2-Q I INTRODUCTION The purpose of this contract is to investigate the radar cross section of an air intake duct similar to that fund on jet aircraft. In order to reduce the radar cross section, the interior of the duct is supposed to be lined with a layer of radar absorbing material, thereby providing an absorption of the energy launched into waveguide modes which may be reflected by the termination of the duct and reradiated back toward the radar transmitter. The opening of the duct is taken to be rectangular, although the cross sectional shape gradually changes to circular at the rear of the duct. The tasks of the contract are, therefore, divided into two parts; the first being to calculate the energy launched into the duct and its consequent reradiation based upon an unknown reflection coefficient at the termination of the duct, and the second being to calculate the effect of the gradual cross sectional change of shape of the duct on the waveguide modes carrying the energy. From this knowledge, it is hoped to produce a good estimate for the radar cross section due to the interior of the duct. The theoretical approach we adopt is to use ray optical techniques to calculate the field generated at the mouth of the duct and from this information to determine the energy launched into the waveguide modes. As a first approximation to this very complicated problem we have previously studied (Bowman and Weston, 1968) the two-dimensional problem of a plane wave incident upon a semi-infinite parallel plane waveguide wherein the outside surfaces of the guide are perfectly conducting and the interior surfaces of the guide obey an impedance boundary condition. Ray optical techniques were employed to obtain both the field generated at the mouth of the guide and the field scattered into the far zone by the edges forming the waveguide opening. The ray treatment is intimately dependent on the known solution for plane wave diffraction by a half plane with arbitrary face impedances (Maliuzhinets, 1958,1960). The ray optical methods were also applied to the case of a perfectly conducting parallel plane waveguide for which an exact solution is 1

1492-2-Q is available, and the results were compared with the asymptotic expansion of the exact solution. It was found that the two methods yield complete agreement for the primary diffraction, along with the first and second interaction contributions, but that for each successive interaction after the second, the ray optical result underestimates the asymptotic result obtained from the exact solution. The higher order interaction contributions are, however, small in magnitude, and the ray treatment is therefore expected to yield accurate results. For application to a rectangular duct, the ray optical procedure must be extended to include the case of oblique incidence on the edges forming the mouth of the duct, and this aspect of the problem will be treated in the present report. We begin by seeking the three-dimensional generalization of the Maliuzhinets (1958, 1960) scalar solution for diffraction by a half plane with arbitrary face impedances. Unfortunately, it developes that such a generalization does not seem possible (by the method employed here) except in the case where the impedances on both sides of the plane are identical (Senior, 1959; Williams, 1960). This means that in order to apply the ray optical technique it is necessary to consider a duct which is coated on both the interior and exterior surfaces with the same absorbing material, or at least with absorbers manifesting closely Similar electrical properties. Expressions for the diffracted fields due to a plane electromagnetic wave incident at an oblique angle on a half plane governed by a single constant impedance are presented in the following chapters. The investigation of the induced aperture fields for a fully-lined rectangular duct is near completion, and a detailed derivation will be presented in the next interim report. In connection with the modes sustained by a fully-lined rectangular guide, the boundary conditions are inseparable and no explicit results for the modes are available at present. On the other hand, closed form solutions can be obtained when two parallel walls of the guide are perfectly conducting. These modes are studied as a preliminary to the more difficult fully-lined duct problem. Also discussed are the modes supported by circular waveguides with absorbing walls. 2

1492-2-Q II OBLIQUE INCIDENCE ON AN ABSORBING HALF PLANE The exact solution for the problem of a plane electromagnetic wave incident at an oblique angle on an absorbing half plane was first presented by Senior (1959), who solved the coupled Wiener-Hopf integral equations which determine the currents induced on the surface of the sheet. As pointed out by Senior, the standard technique (Clemmow, 1951) used for the derivation of three-dimensional solutions from known two-dimensional solutions in the case of perfectly conducting cylindrical structures fails to yield correct results when the diffracting structures are imperfectly conducting. This is due to the coupling of TE and TM modes in the presence of dielectric or absorbing cylindrical bodies (see, e.g. Wait, 1955). On the basis of Senior's (1959) investigation, however, Williams (1960) subsequently developed a generalized technique by which the oblique xicidente solution may be deduced directly from the known scalar solutions for a half plane with nonzero surface impedance. These treatments of diffraction by a half plane assume that the screen is characterized by a single constant surface impedance ri, whereas for the purposes of our investigation it would be desirable to consider a semiinfinite sheet with two face impedances, say ri = r1, on the upper surface and Y1 = rX2 on the lower surface. A particular case of interest would then be that of a perfectly conducting half plane coated on one side with radar absorbing material. Since the scalar solutions for plane-wave scattering by a half plane with arbitrary face impedances are available from Maliuzhinets (1958,1960), we have attempted to carry out the Williams (1960) procedure to obtain the corresponding threedimensional solutions for oblique incidence. Unfortunately, it has come to light that the technique fails except in the case for which the impedances on both sides of the half plane are identical. The Williams technique appears to be intimately dependent upon the inherent symmetry of the problem with regard to the plane defined by the diffracting screen. Nevertheless, in order to highlight where the 3

1492-2-Q procedure breaks down, we shall assume 1l: r12 until it is no longer feasible to do so, at which point is is necessary to impose the condition r1 = 72 in order to gain a rigorous solution. Future efforts may yield a more general method than the Williams technique. We shall employ natural units with free-space constants, A set equal to unity and suppress the harmonic time dependence exp(-iwt) throughout. The diffraction screen will be taken to occupy the half plane y = 0, x > 0. In cylindrical coordinates (p, 0, z), as well as in spherical coordinates (r,O, ), the upper and lower surfaces of the half plane are prescribed by 0 = 0 and 0 = 27r, respectively. The primary excitation will be due to a plane wave of unit intensity propagating from the direction (eo, 0o): E (-o e s A Aoe ikS = (-x cos 0 sin0-y sin osin0+z cos0e)e (2.1) Hi (-x sin+ cospo)e iks with S =xcos0 cosO +ysin0 cos%0 +zsine =p cos(0-0 )cos +z sin 0. (2.2) The incident field is thus taken to be a pure TM mode (no Hi component); however, the other polarization case (TE incident mode) presents no difficulty in its treatment, and may indeed be obtained upon making use of the natural duality of Maxwell's equations and the impedance boundary conditions. When e0=0, the plane wave is incident normally to the edge of the semi-infinite screen. Since, as we have mentioned earlier, there is a coupling between outgoing TE and TM modes when the half plane is imperfectly conducting, the total electromagnetic field must be expressed as a linear combination of the two basic types of modes, and these may be derived from the electric and magnetic Hertz potentials T and TTas follows: for the TM (or E) modes: 4

1492-2-Q &2 a2 _ { A A a A 2 TT E ax a z +Y az (- +k H " (-ik) (2.3) while for the TE (or H) modes: E=x - y kax k) A a2 A a+ 2 _ x axaz + y a 2 k~)' (2.4) The incident plane wave specified by (2. 1) and (2.2) is derivable from the Hertz potentials -ikS 1;= e r=o, (2.5) k cose o and in view of the z dependence here and the fact that the diffracting structure itself is two-dimensional, it is reasonable to take the total Hertz potentials in the form -ikz sine, -ikz sine = e U(x,y), TT= e U(x,y,), (2.6) where U and U are both solutions of the equation (v2 +2cos20 =. (2.7) xy o The screen is assumed to be comprised of material of such a kind as to make the total tangential field components satisfy the following impedance boundary condition on the surface: E= ( E) 'n = nAnH (2.8) where Yr = r)i on the upper surface, rl = r12 on the lower surface, and n denotes 5

1492-2-Q the unit outward normal to the surface. The face impedances r1 and r2 are complex constants whose real parts, because of energy considerations, must be nonnegative. For the rectangular components of the total field the boundary condition (2. 8) is such that Ez -l 1H E = 2H EZ=lx X E r=lHz E = -r2H x 2 z on 0 = 0, on 0 = 27r. (2.9) On 0 = 0 these boundary conditions lead to the coupled equations 2 2 8U aU k cos 0 U = ikrL -+ ikrlsin -- o 1 ay 1 o 0x 2 20 aU aU r1k cos U = -iksin0 + ik - - 1 o o 'x ay (2.10) while on 0 = 27r the corresponding equations are obtained from (2. 10) upon replacing ri1 by -r2 Following Williams (1960) we can uncouple the boundary conditions on U and U by considering certain linear combinations of the derivatives of these potentials; in particular, one may show by invoking (2. 7) and (2. 10) that the scalar functions fl and f2 defined by fl =i ai +sinO BA oby' -iU f =-i - +isine - 2 ax o oy (2.11) satisfy the following boundary conditions: af = -ik l1fl. af=ik2f R kA f f2 ik 8 —'- f2 Cy al1 2 on 9 = 0, on p = 27r. ik "2 2 (2.12) 6

1492-2-Q The scalar functions fl and f2 are thus directly related to the solutions of the corresponding two-dimensional diffraction problems for H and E polarization, respectively. The original potentials U and U are related to f and f2 by Poisson-type differential equations. a2U 2 +2s f f1 -i s 2) -a -sine, (2" 2 a2 1 2 i j+sin a )= +sin (2.13) which follow from (2. 11). The original boundary-value problem has now been reduced to two independent boundary value problems for fl and f2' and because of (2.13) four arbitrary constants will be involved in the final expressions for U and U. It turns out (Senior, 1959) that these constants are determined by the condition that the final solution should contain no plane-wave terms other than the incident and reflected waves of geometrical optics. In order to obtain expressions for fl and f2 we now turn to a modification of the Maliuzhlnets (1958,1960) two-dimensional solution and consider the function -ikcos0 (xcos0 -tysino ) 1 ikcos0 (xcosp+|y3sinp) Q.o 0;y) X =e + - d cosdos' sC (2.14) where Q (, 0o;yy) v(u, O) +V(2r-A., ~o) +-gn(y)[v., }o)-V(2,7-A, 0o)- (2.15) with V(p, 00) defined as,(2 )sin 2 sin( al a) ' o: (7_-P) O(70_0) 2 2.. + 0 (co8 +cos-) (c -cos ) +os cos. (2. 16) ^z 2 Os- \ -Co 2 2 2 7

1492-2-Q The auxiliary function O(O8)in (2.16) was introduced by Maliuzhinets (1958) and expressed in terms of a special meromorphic function V'7 (3) by the product L0(1) = 0(B rts)n0 B+tr-a1)B — at2)on7r +a2) v where Q0 (3) has the representation (2.17) *L8=- l 1 ex r sin -,2Z'sin(/2)+ 2 v v cos v (2.18) The important analytical properties of the functions VL(3) and 1(f) are given in Appendix A. Thee quantities c1 and a2 are complex constants whose real parts lie in the interval [, xr/ 2; these constants are related to the impedances r1 and rl2, as we shall see. The contour C in (2.14) is shown in Fig. 2-1 and consists of a path in the complex M plane along which cos p ranges through real values from oo to - o, although it passes above the pole p = r-00. Now the important feature about the function X defined above is that it obeys the boundary conditions ay + ikX cos 80 cos a, 2 = 0 ay - on = 0, 2 r. (2.19) C 0 FIG. 2-1: PATH C IN COMPLEX p PLANE. 8

1492-2-Q By comparison with (2. 12) we see that X satisfies boundary conditions appropriate to fl provided we define a, 2 by the identification H 1,2 o 1,2 '20) r1, 2= Cos0 cos a1, (2.20) whereas conditions appropriate to f2 obtain if we make the identification 1 1,2 E * (2.21) cos 0 cos a 2 o 1,2 The affix E or H is attached according as the boundary conditions describe E- or H-polarization states, and in the sequel, the function X as well as other functions will be written with such an affix whenever this polarization distinction is necessary. In all instances the notation is employed simply to indicate where (2. 20) or (2. 21) should be used to determine the constants a1,2 from the impedances 7,i 2. The function X taken by itself is not sufficient to satisfy all the requirements of our problem, such as the requirements at the edge, although it is sufficient to yi eld the proper incident and reflected fields of geometrical optics. Thus we must seek other solutions to (2. 7), subject to the boundary conditions (2. 19), but which do not contain plane-wave components. Such a solution would be, for example, the function ax+ikX cos 0 cos; (2.22) however, it turns out that yet another independent solution is required. Williams (1960) obtains two independent solutions by taking separately the even and odd parts with respect to y of the function (2. 22). Thus consider the functions (even and odd in y) ikcos (xcosp yjsini) X el_ dAe o~ rO)+v(2r-Ip sn(y) f ikcos O(xcos+ y IsinL) X0 J8 d Ve V(P, O)-V(27r-, 0]. (2. 23) C/ 9

1492-2-Q These, however, satisfy the boundary conditions (2.19) only in the case a1=a2, and it is at this point that the Williams procedure breaks down for the general case of arbitrary face impedances. Before continuing with the case a, =a2 let us seek a solution to (2. 7) and (2. 19) in the form ik cos e(xcos - yls inP) dP e ~ S) (2.24) C where S(p) is independent of y. The boundary conditions (2. 19) demand that ikx cos08cosp dpe (sin0+cosa1c 2) =, (2.25) JC and in order to satisfy (2. 25) one may show, with certain analyticity assumptions concerning SG.), that the integrands in (2. 25) must be odd functions of p. Therefore, we must have S(u) sing-cosal sinfi-cosa2 (2.26) S( —A) sinp+cosal sinw+cosa2 which is possibly only if a1=2. From this point on we assume that c1=a2 in which case Xe and Xo in (2. 23) take the form ikcose (xcos+i ylsinp) Xe 87ri cde Qe ( o) gy) ikcosO (xcosp+-ylsinp) = J djue Q (o) (2.27) C where sin(u/2)sin( 0/2) 8 o Qe P, o)= - 7( —) (' ( k) cos a sinp/2)sin(0o/2) 8 A (2.28) Q W (- (O) 2 cos 7cos. Qo~,~o= (~~.)~lr,~)..... 10

1492-2-Q The function Qu(, 0;y) in (2. 15) may now be written as QA, o0;y)=Qe(0, 0)+sgn(y)Q0(., 0). (2.29) The relationships between the Q functions here defined and the "split" functions K +(kcos/) and L +(kcosp) of Senior (1952) may be determpined from Appendix A. The scalar functions fl and f2 may be expressed as sine sin H H H f k X +CXe +BX — i E E E f x+AXe +DX (2.30) 2 2 e 0 k cose o where the form of fl and f2 have been chosen to ensure the appropriate incident field behavior sin s8in0 -ikcos0 (xcos0o ysin0o) 1 k e V -kcos (0 -ikcos0 (xcospo+ysinn o 0e (2.31) 2 k and where the constants A,..., D are closely related to those of Senor (1959). In integral form fl and f2 are given by + 1 d ikcose (xcosA+(yI sinP) H 1 1 8iri 0 1 ikcos0c(xcos+1y| siW) E with S insainp QH.o0;y) H y)- coscos0 +Q u 0o)+B QO yo)gn) E sO( C'o;:Y) F EE S, o;Y)= C k cos +AQe(p 0)+DQ(O, o)sgn(y) * (2.33) It follows from (2.13) and (2.32) that U and U may be written as 11

1492-2-Q sec 20 ikcose (cosp+Y sing) Gl(, o;y) U=U --- -Ie 2 0 0 87rk C cos 2++sin 0 sin p sec 2eo ikcose (xcosP+IyI, sinp) G2(0,0;y) 8rk C cos p+sin 9 sin p 0 where G (,, y)=ikcos0 cospS (, 0;y)-ikcoso in0 sinS,;y)sgn(y), 1 0 0 0 0 0 0 G2(, o; y)=ikcose cospS H(, 0;y)+ikcos0 sin6 sinpS E(, o;y)sgn(y). 2 o o (0 0 (2.35) But the integrands in (2. 34) have spurious poles in the p plane at the zeros of 2 2 2 2 2 2 cos 2+sin 20 sin2= =cos 2cos 0 +sin 2 =0, (2.36) 0 0 0 that is, at P +iv, (2.37) where v satisfies the equations sinh v=tan 0, coshv = sec 0. (2. 38) In order that U and U be free of plane-wave terms other than the incident and refelcted waves of geometrical optics, we must have G1( i v,;y) =, G2( 2i v, 0;y) =0 (2.39) for all y. Thus (2.39) must hold separately for both the even and the odd parts in y. If we write G1, 2' 00;Y)=G1 2( 0)+G1, 2 ( o)sgn(y) (2.40) where, for example G e,,0 )=ik cos0 Qi PQ ) E(Pcosp z+cAP - oe E k(Focosp+ o0 ) H sinO sino - -ikcos0 sin0 sinpQ Y ) ~ +o B (2.41) o o 0~^ 0o L(oo^cosM40 J 12

1492-2 —Q G 0(p,00) =:ikcosO0 cosp Q E 00s p 1 Ik(cosp+coso 0)+ -i rsine sine, - ikcos&9 sine sinpQs He~roIk0cos cos -11 then A and B are determined by Ge (f jiv'O0)= 0 or explicitly, (2.42) (2. 43) QE~(Z~-ilO ), Q1 (! -iv, 0) r i tanG9 LkMitan~O0+cop0 1Fsin O)0sin 0 + A -k(i tanG +coso.0+Bj 0, *QE (f+iv, 0) Q' H(7r+iv, 00) F -i tanG e sine sine L k(-itaneG+coso0)+jS Lk-i tanG0h- oa 00) +B (2.44) Solving for A and B, we obtain Q (:+iv, ) -QH(Z+iv~o) tan e 0 k(-i tanG +,cos ~) Q H (7r*i, o) 2tanG 0sine sinoo ik(tane 0 Ic os20) 0 0 +i. Q - ~iv, 0 )I e 2 0 tanG0 kMi tanG 0+cosO j) o 2 o k(i tane0 +cosp0 Q (-7r+iv, 0 ) s ine s ino o2 '(~ 0 0 0 Q E(~iO )k(-i tanG e+cos0( e 2 0o - 2 tanG coso k(tan29 +Cos 20) o 0 H v 41v, 0 ) QO(5 0 QE(I+iVo ) e 2 0 + (jiv ~0j B. (2.46) The constants C and D are similarly determined by (2.47) 13

1492-2.-Q however, they may be obtained from (2.45) and (2. 46) upon replacing Q Eby QE H QH e 0 and Q 0by Q J in which instance A -op D and B -*w C. Finally, it is easy to verify that the vanishing of G2(fj iv. O;y) yields the same constants A,.., D. The solution for oblique incidence is now completely determined. Upon combining results, we can write the total electromagnetic fields in the form sec9 ikde ics0(CSP jsn)E(, y E=i -i0 osj iko9(cs+y sn)E0,;y E=E ---e O pe 8ir cos 2.scos2& +sin 2e 0 0 sec90 -ikzcosO koO0xopysru H~- 8wre H (p, 0_; y) 2 2 2 J cos pcos 9 +s in 9 (2. 48) where E*Uo;Thc —xin0cosijG (41 0.;y)+.iS,.G2(14,,o;y)sgn(y] + +-Lsine sinuG 1(P, 00;y)sgn~y)-cossG 2(P, 0;y) 4 cos e G1 (As, 0Y (2.49) -Y SMG(p, 00;y)+s inO s inp~G2(p, 0 0;y)sgn(Y}-ZcosO oG2(p, 0;y) (2. 50) Away from the optical boundaries V irt 0, -i74 -ikzsMin9+ikpcos9 _de e 0 E 4cose V o (2irkpcose0 ) - i V4 -ikzsintl+ikp'cos90 '~ -e e =4cos9 (2rpos 12 0 ( 7rk~oS0) the diffracted field in the far zone is E(O, 0) 2 2 2 Cos ocos e +sin e 0 0 H (O., _) 2 CO2 2 ' coal cs +sin9 0 0 (2. 51) where E(O, 00 ) =[intl0cosoGI(O., 0 )+sinoG 2(0, 100]+ (2. 52) 14

1492-2-Q H(p, po)=x[sinOG1((0, O)-sino0cosOG2(, o )] - y [cosG1,(4, 0o)+sino sin0G2(, o)]- -Z cos0 G2(0, 0), (2.53) iith G1 2c, o)G1 2, o)+G, 2(' (2.54) Equations (2. 51) through (2.54) are valid for all 0 and 0o except r ~ +. As a check, let us cansider the perfectly conducting case which corresponds H E to a = (r/2) and a - + i oo. Employing from Appendix A (A. 26) and (A. 27), 1 - along with (2. 28), we find Qe, o) = -8sin 2 sin Q,) = 0, H j o H QHou, 0) = 8 cos Co 2 Q, =0. (2. 55) Only the constants A and B need be evaluated, and with the help of Q! e(+ iv, 0o) Qe(2- 0 e ootan- 0 (2.56) QH( + iv 2 0 2- '0 we obtain from (2.45) and- (2.46), after some trigonometric reduction, -sine sino A = 0, B k(l= +S (2. 57) With these values of the constants we finally have 2 2 2 -8sin(M/2)sin(0o/2) Gl(, o;y)= ikcose0(cos /u+sin8 sin p) cosp+cos (2.58) G,;y)=0 G 2@ p;Y) = ~ s which leads to agreement with the results for a perfectly conducting half plane. Equation (2.51) is fundamentally important for the application of ray-opttical techniques to problems involving planar structures that are coated with radar absorbing material, and in particular, the diffraction coefficient derived from (2. 51) 15

1492-2-Q will play an important role in estimating the fields generated at the mouth of a fully-lined rectangular duct. As pointed out in the Introduction, however, the condition, =r/2 restricts our consideration to ducts whose walls are coated on both the interior and exterior surfaces with absorbers of closely similar electrical properties. A detailed derivation of the aperture fields - some aspects of which are not complete at present - will be given in the next interim report. 16

1492-2-Q III MODES IN RECTANGULAR WAVEGUIDES WITH IMPEDANCE BOUNDARY CONDITIONS The problem of determining the modal structure of a rectangular waveguide whose four walls are absorbing is complicated by the fact that the boundary conditions are inseparable, and no simple representation for the eigenfunctions is presently available. Contrarywise, closed form solutions can be obtained for the modes that exist in a rectangular waveguide in which two parallel walls obey an impedance boundary condition while the remaining two walls are perfectly conducting. The geometry for the rectangular waveguide is displayed in Fig. 3-1, where the origin of the Cartesian coordinates is taken at the center of the guide cross section. An impedance boundary condition of the type in Eq. (2.8) will be imposed on the walls of the waveguide, with the top and bottom face characterized by an impedance tll and the side faces by an impedance r2. In terms of the field y 2b z x FIG. 3-1: COORDINATES AND DIMENSIONS OF RECTANGULAR WAVEGUIDE. 17

1492-2-Q components, the boundary conditions are as follows: (i) Top face y=b: (ii) Bottom face y=-b: (iii) Side face x=a: (iv) Side face x=-a: x 1 z E z=-711Hx, E =r1Hx E =- 2H J Ey=-12Hz' E =72H. (3.1) (3.2) (3.3) (3.4) The fields will be expressed in terms of the electric and magnetic Hertz potentials T', T as in Eqs. (2.3) and (2.4), and in addition, these potentials will be taken in the form ihz i; (hz " = e X(X, Y), T =e (x, y), (3.5) where h is an unknown parameter and where both X and I satisfy Lv^y+^ h ] x-o. (3.6) yv +(k-2h2 X=~ (3.6) Consider first the boundary conditions on the top and bottom faces. In terms of the potentials X and i, these boundary conditions become, for y = + b; +i O-. +k W. rll 7, +i l (h k a) =a2, - 77 = k X, (3.7) - 1 iX ay with (3. 8) 2 2 2 a =k -h. 18

1492-2-Q We define two classes of modes: (I). 3 = sin 3 y F(x), X = cos y F2(x) (II). w = cos y Gl(x), X = sin y G2x), (3. 9) where F1 2 and G1, 2 satisfy the differential equation 2 d F+(2-2)F =0. (3.10) dx For these modes it is immediately seen that the boundary conditions at y=b are identical to those at y=-b, and in particular, the modes of class (I) satisfy hdF 2 -i hx + kI FL = 71 Ftan 3b, / dF1 2 in1 Kh dx +kF2) a F2cot 3 b (3.11) whereas the modes of class (II) satisfy / (hdG1 2) 2 i1 - -kG = a G2tan3b, -i dx) = r1 a G cot 3 b (3.12) It should be noted that (3. 12) may be obtained from (3. 11) upon replacing F1(x) by -G2(x), F2(x) by G1(x), and then replacing rY by (1/X71). The modes of class (I) may be further distinguished into two classes depending on the parameter,; in particular Eqs. (3. 10) and (3. 11) are satisfied in the following two cases: k dF2if (Ia) Fi=- h dx for tanbX-k a k k dFl ik (lb) F-h- dfor tan b=- — 1 19

1492-2-Q similar results follow for the modes of class (II): k dG1 ig (IIa) G2= dx for tan bS- k T 1 k dG2 ik (nb) G1 d-xr for tanbb - t 1 (3.14) Now we consider the boundary conditions on the side walls. These become, for x +a x =2 - a: ax h aay 2 I2 in2(kxx + h )=a2x. (3.15) First consider modes of class (Ia); the boundary conditions (3.15) are, for x= t a: 2 2a F1(X) = 0 dF (x) dF1(x) 2 2 k x - i n(k2-3 )F(x) = 0. (3.16) Since, by virtue of (3. 10), Fl(x) has the general form F (x)=A sin (/ x )+Bcos ( /a x ), (3.17) it follows that a non-trivial solution; exists only if r12=0, and this is also true for the other modes. When r2 = 0, in which case the side walls are perfectly conducting, the modes of class (Ia) or (Ha) may be represented in the form (n=0, 1, 2,...) Fle or G1,3 sin n [(n a with_ (n+) _ {1(-) F or Gn cos ( n) with =3-, (3.18) 1,,e la' a where F1 or G1 is chosen according to the specification of B. Similarly, the modes of class (Ib) or (IIb) may be written as 20

1492-2-Q n nfx nir f or G = sin - with:f - 2,0 2,o a a IF e or Ge I co (n+ ) - with a22= (n+) (3.19 2,e G2,e a 2 a The modes are now completely determined, and the eigenvalues h are given by h = k22-.] or h = k2-2- (n+ )2 (3.20) 2depending on the mode classification. La a I depending on the mode classification. 21

1492-2-Q IV MODES IN A CIRCULAR WAVEGUIDE WITH IMPEDANCE BOUNDARY CONDITIONS In this chapter we devote attention to the modal structure sustained by a circular waveguide whose wall is lined with absorbing material. The electromagnetic field may again be represented in terms of electric and magnetic Hertz potentials FT and T. respectively, and in cylindrical coordinates (p, 0, z) the relationships of the fields to the potentials are: a2 j2 + 2 2 2 n -- a ) H= - (-ik)T1 (4.1) for TM (or E ) modes, and E= P r - (ik)T, H: p 8-p' + + Z +k k2 (4.2) apaz p t(z 2) for TE (or H) modes. Boundary conditions of the impedance type (2.8) will be imposed; in particular, for p = a we demand that Ez = -r H E = rH. (4.3) Because of these conditions, only hybrid EH or HE modes will exist in general. The Hertz potentials will be represented in the form rl(p,, z)=e X (Pp, ), i(P,, n z) =e (p,, z), (4.4) where h is an unknown parameter. With this assumption, the boundary conditions 22

1492-2-Q become i {k a + h 2X - a+hp ax, -ik ak -h r 2 (4. 5) 2 2 2 with a = k -h as in (3. 8). Now it is sufficient to consider modes of the form X = A ein J(ap), = BeinJ(p), (4.6) n n in which case the conditions (4. 5) yield ka i nha AQ J'(u)+ Jn(U) +iBi Jn(u) =0, u n ] n u2 -iAi Jn(u)) +B1 J (u)+irJ(u) =0, (4.7) U2 u n la where u = (k2-h2) 2a. A non-trivial solution of the homogeneous equations in (4.7) exists provided the determinate vanishes; thus we obtain the eigenvalue equation J (u) Jn (U) 2 aka c) - C (. - i2 r= n ka (4. 8) n ) (4.8) which determines the eigenvalues u. In the symmetric case (n=O) the eigenvalue equation uncouples into two equations, one for the Eo mode and one for the Ho mode: ka J1(u) 1 'a J (U) = for E mode, uJ (u) i o ka 1(U) a J ( for H mode. (4.9) u j (u) o 0 The modes corresponding to n > 1 are coupled or hybrid EH modes. In general, the roots of u of the eigenvalue equation will be complex, so that the axial propagation h given by 2 2 h = k +()2 (4.10) will also be complex. The mode will thus attenuate as it proceeds down the guide, that is, provided the restriction Im h > 0 is imposed. 23

1492-2-Q APPENDIX A PROPERTIES OF THE FUNCTIONS fr(3) AND L(j3) The meromorphic function ql( (3) is defined as { 1 1iC sin v- 2 /2r sin (v/2) + 2vdv (A. 1) exp dv (A. 1) from which it will be observed that 07(p) is an even function of 3B whose logarithmic derivative is given by '() _ IsinB /2sin(t/2) _1 B (A. 2),(8) 8 cos B 4 cos 3 4r cos ' By means of the elementary integrals sn — dv = -In (cos 8), ocosv sin(v/2) dv = -n - cos( /2) - 1 /2 + 1 cos V L/2cos(,/2) + 1 /2- 1 we obtain the following alternative representations for qr(p): ' [ ]{- yA03 V = cos(/2 )~+ 1 { 1f 1 1 rvd (A.3) F /2 + 1 (co0S /) /8 exp 4 J- c v (A. 3) N/2 =1[ (0/2)1.Ij i sin v - 2 dv5. (A. 4),(0) = / exp 8r Cos( + When I31 < (Ir/2), the integral in (A. 3) can be expanded as I vdv _ 12 1 5361 + (-)" E2x \ + J cos v 2 2 4 24 6 720 8 (2n)! 2n+2* -\ (A.5 where E2n are the Euler numbers. When 3 = i o, we have (Grabner and Hofreiter 1949) 1 f'vdv C 1f "xdx - 2 (-1), (A. 6) I cos r Jo cosh x o (2 '

1492- 2-Q where b = (2/7r)K with K = 0. 9159656... (Catalan's constant). On the other hand, when,B = (r/2), we employ (A. 4) with a change of/ integration variable v (=r/2)-u to find 1 ]x /'cos u 1 du I T/2 u du) (A. 7),. (7r/2) = ['2 + 1 1 (A. 7) /2 + 1 8 o suin 4ro sin u' The first Integral is elementary and the second Integral is (Grlbner and Hofreiter 1949) 1 du 2 _ (-1)" ~ o sinu T.._(2n+l)=b We obtain then ^[,(/2) '- (A.9) It is easy to verify the following fundamental identity (Maliuzhinets 1958) + I,( + Ir),()f - jir) = [,(.r/2)]2 cos(/9/4), (A. 10) and by successive application of (A. 10) one obtains 2. + 7r) -) [1(2) [cos(,B/2) + cos(7r/4)], (A. 11) ) - ) = I[4(7r/2)]2COS(./ (A. 12) From this last equation we observe that the zeros of -0,3) which are closest to the point 13 = 0 and the corresponding poles are the points 3 = + (5ir/2) and 1 = t (72r/2), respectively. From Eq. (A. 10) one also derives #(, '+7r) - cos(3 + ir) (A 13),,(t- -) - cos(i )' - + 2r) = cot(~, + ir). (A. 14) - 2.r) The function (j3) is expressed in terms of the function,(,3) by the product: ( 1) =.(3 + T + ai)~.(P + Tr -. ai)/ - r (2) -T + a), / (A. 15) from which, by means of (A. 14), we derive 25

1492-2-Q V(73 = (_ +2r=)+ -al+2r_) (r4)_ 1^ (3 2) ^C/(n-SB) ^ (S+'a, -2f 03- (0 -27r) iT i i ii 1 1 1 1 1 1 cos al-sin/ =cot (2 l+4Xr)cot(2-2 l+4X)= o (A. 16) 2 4+4i2 ) cosa +sin(A 16) and similarly (-r-/c) os a2-sin / i(-ir) -cosa2- *(A. 17) Another identity of interest may be derived through the application of (A. 13); in particular,, co.cs(o+ )-c -,o.s ] -)+C 1 (A. 18) 0(-T- [o) Cos(.... 2 4 2 ] 4 '2 from which it follows that,(-7rfl) (-r-fs) sin+-cos a1 sin+-cos a2 (A. 19) in agreement with the results in (A. 16) and (A. 17). If al=a2 (in which case the two surface impedances are equal), Eq. (A. 18) reduces to (r-9) = /(-+ ). (A. 20) In this instance (B) is related to the "split" functions K+(k cos 3) and L+(k cos 3) of Senior 1952) as follows: K+(k os ^ (I)= as4 ', with cos c- (A.21) KC(kcosl jo(L/2) cr o4o L+(kcoss )= " r( 4cosn with w coschi = r where the constant rl represents the complex impedance of the half-plane. Senior's functions may be defined by the equations 26

1492-2-Q K+(k cos 3)K+(-k cos )3= sin ' (A. 23) + + 1+)rsinO L+(kcos 3)L+(-kcos )= rl sin (A. 24) whereas ()3) with al=a2 satisfies the relation Employ (A. 10) and (. 12I) 2' 2- 3) (+) = (2 (cos 3+cos a ) (A. 25) It is easily verified that (A. 25) is consistent with (A.21) through (A. 24). For a1=a2=(ir/2), corresponding to a perfect conductor with H polarization, we have (3): L ()]4 cos (3/2); (A. 26) on the other hand, for a1=a2 -* + i o, corresponding to a perfect conductor with E polarization, one finds asymptotically ~ > ir (2)]4 e 2 (A.27) These limiting expressions are valuable when checking the results of calculations against known perfectly conducting results. 27

1492-2-Q REFERENCES Bowman, J. J. and V. H.Weston (1968), "Reduction of Radar Cross Section of Ducts," The University of Michigan Radiation Laboratory Report 1492-1-Q. Clemmow, P. C. (1951), "A Method for the Exact Solution of a Class of TwoDimensional Diffraction Problems, " Proc. Roy. Soc., A205, pp. 286-308. Grobner, W. and N. Hofreiter(1949), Integraltafel, Springer-Verlag, Wien and Innsbruck, Austria. Maliuzhinets, G. D. (1958), "Excitation, Reflection and Emission of Surface Waves from a Wedge with Given Face Impedances," Sov. Phys-Doklady, 3, pp. 752-755. Maliuzhinets, G. D. (1960), "Das Sommerfeldsche Integral und die Losung von Beugungsaufgaben in Winkelgebieten, " Ann. d. Physik, 6, pp. 107-112. Senior, T. B.A. (1952), "Diffraction by a Semi-infinite Metallic Sheet, " Proc.Roy. Soc., A213, pp.436-458. Senior, T. B.A. (1959), "Diffraction by an Imperfectly Conducting Half-Plane at Oblique Incidence," Appl. Sci.Res., B8, pp. 35-61. Wait, J. R. (1955), "Scattering of a Plane Wave from a Circular Dielectric Cylinder at Oblique Incidence," Can. J. Phys., 33, pp. 189-195. Williams, W. E. (1960), "Diffraction of an Electromagnetic Plane Wave by a Metallic Sheet, " Proc.Roy. Soc., A257, pp. 413-419. 28

UNCLASSIFIED Security Classification ~~~~~~~~~~~~~~~~~.... [-l[...~ I - I I I I I l l I I III l I - I ~ l JI l I Im [ [ I DOCUMENT CONTROL DATA - R & D (Security classification of title, hody of nabstract iind intlx'/ilm.uiotwninn nsiat be entered when the overall report Is clasSlfied) I -- - - --- - - 9 1. ORIGINATING ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION The University of Michigan Radiation Laboratory, Dept. of UNCLASSIFIED Electrical Engineering, 201 Catherine Street, 2b. GROUP Ann Arbor, Michigan 48108 3. REPORT TITLE REDUCTION OF RADAR CROSS SECTION OF DUCTS 4. DESCRIPTIVE NOTES (Type of report and inclusive dates) Second Interim (1 April - 30 June 1968) ] 5. AUTHOR(S) (First name, middle iniltial, last name) John J. Bowman and Vaughan H. Weston 6. REPORT DATE 7ii. TOTAL NO. OF PAGES 7b. NO. OF REFS l 6 July 1968 28 9 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) PO 504 855029 b. PROJECT NO. 1492-2-Q F33615-67-C- 1843 c. )h. OTHER REPORT NOIS) (Any other tnumcbr.s that miay be ja..ssrnt, this report) d. 10. DISTRIBUTION STATEMENT II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Northrop Corporation-Norair Division Hawthorne California 113. ABS T R A C T As part of an investigation concerning radar scattering by a rectangular duct lined with absorbing materials, some problems associated with the diffraction of plane electromagnetic waves incident obliquely (with respect to the edge) on an absorbing half plane are considered. The half plane is initially assumed to be governed by an impedance boundary condition wherein the impedances on the upper and lower surfaces of the diffracting screen may differ from one another; however, the investigation then reveals that a rigorous solution appears possible only in the case for which both impedances are identical. This means that in order to apply ray optical techniques to estimate the field generated at the mouth of the duct, it is necessary to consider a duct which is coated on both the interior and exterior surfaces with absorbing materials manifesting closely similar electrical properties. In connection with the modes sustained by a rectangular waveguide whose four walls obey an impedance boundary condition, the boundary conditions are found to be inseparable and no explicit results for the modes are available at present. On the other hand, closed form solutions can be obtained for the modes that exist in a rectangular guide in which two parallel walls are absorbing while the remaining two walls are perfectly conducting. These modes are studied as a preliminary to the more difficult fully-lined duct problem. I ir r FORM I A "7' LJ i NOV 65 1 4 / j UNCLASSIFIED St uriv ('I.s;i I ci'. I. itl