1492-1-F THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory 1492-1-F = RL-2032 DUCT STUDIES Final Report 22 January 1968 - 1 May 1969 John J. Bowman and Vaughan H. Weston By 6 May 1969 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

1492-1-F FOREWORD This report, 1492-1-F, "Duct Studies," 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 report, the Final Report, covers the period 22 January 1968 through 1 May 1969. i

1492-1 -F ABSTRACT Problems in the high frequency regime of electromagnetic scattering by ducts and open waveguides whose walls obey an impedance boundary condition are investigated. In general, the impedance boundary condition means that the walls are absorbing electromagnetic energy. Three structures of particular interest are treated: the semi-infinite parallel plane waveguide, the semi-infinite circular cylinder, and the rectangular duct of finite length. When the surfaces are perfectly conducting, exact solutions for the first two structures are available, and these solutions are expanded asymptotically to yield term-by-term comparisons with the corresponding results of ray diffraction theory. When the surfaces are absorbing, exact solutions are not available, but the ray treatment is applied to advantage. In the case of the rectangular duct, the main problem is to determine the interior waveguide modes. This modal problem is non-separable due to the assumed impedance boundary conditions; nevertheless, information can be obtained by means of an asymptotic analysis applied to the transcendental eigenvalue equation of infinite order. ii

1492-1-F TABLE OF CONTENTS I INTRODUCTION 1 II THE HALF PLANE 4 III THE SEMI-INFINITE PARALLEL PLANE WAVEGUIDE 19 IV THE SEMI-INFINITE CIRCULAR CYLINDER 28 V THE RECTANGULAR DUCT 43 APPENDIX 69 REFERENCES 73 DD FORM 1473 77 iii

1492-1-F I INTRODUCTION Problems of radiation and scattering by open waveguides with ideal boundary conditions have been studied extensively in the literature, and indeed, for some semi-infinite guides, exact solutions have been made available. In sharp distinction, however, comparatively few investigators have directed attention to such problems when the boundaries are not ideal -- for example, when the boundaries are nonconducting or absorbing in the electromagnetic case, or nonrigid in the acoustic case. Nevertheless, many practical applications arise for nonideal boundaries, and it would seem desirable to investigate the effects of these boundaries on the radiation and scattering properties of the structures involved. In this report, therefore, we shall investigate problems of electromagnetic scattering by ducts and open waveguides whose walls may be characterized by an impedance boundary condition, which in general means that the walls are absorbing electromagnetic energy. The wavelength of the incident field is assumed to be small compared to the obstacle dimensions, so that high frequency techniques may be utilized to advantage; however, in many cases the results remain useful even for wavelengths as large as the relevant dimensions of the scatterer. Since we are assuming the validity of the high-frequency regime, it is natural to adopt ray-optical techniques to calculate the effects of diffraction due to the edges forming the mouth of an open waveguide or duct. In this case, the ray treatment is intimately dependent on known solutions for plane wave diffraction by a half plane, and in particular, we desire solutions for a half plane with arbitrary face impedances. For incidence normal to the edge of the half plane, the desired solutions are available from Maliuzhinets (1958, 1960); however, for oblique incidence on the edge, the solution is 1

1492-1 -F available only if the impedances on both sides of the half plane are identical (see, e. g., Bowman and Weston, 1968b). This means that when oblique incidence is involved in the ray treatment, both the interior and exterior surfaces of the open waveguide or duct must display closely similar electrical properties. The half plane solutions are summarized in Chapter II. In the succeeding chapters, three structures of particular interest are treated: the semi-infinite parallel plane waveguide, the semi-infinite circular cylinder, and finally, the rectangular duct of finite length. When the surfaces are perfectly conducting, exact solutions for the first two structures are available. These configurations therefore represent important prototypes for testing the utility of the ray procedure, and also provide insight into the various diffraction mechanisms occurring at the mouth. For both of these canonical bodies we shall provide some new asymptotic representations of the scattered field which yield term-by-term comparisons with the corresponding ray-optical results. It is found that the ray method yields 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 quite accurate results. When the surfaces are absorbing, exact solutions are not available, but the ray treatment is straightforward in view of the perfectly conducting results. For the rectangular duct of finite length, the ray method is applied to obtain the primary diffraction fields generated at the mouth of the duct, although the simple ray expressions must be suitably modified to account for the finite lengths of the diffracting edges. From this information, the energy launched into the waveguide modes within the duct is to be determined. The modal problem, however, is nonseparable due to the assumed impedance 2

1492-1 -F boundary conditions; nevertheless, the modes may be determined by means of an asymptotic analysis applied to the transcendental eigenvalue equation of infinite order. These modes obey a general orthogonality condition that leads to the determination of the conversion coefficients which couple the induced aperture fields to the waveguide modes in the duct. The scattered field due to modes reflected from the closed termination of the duct are obtained by means of a Kirchhoff approximation over the aperture formed by the mouth. 3

1492-1-F II THE HALF PLANE 2. 1 Normal Incidence Before we consider problems of diffraction by ducts and open-ended waveguides we must first examine the pertinent half plane results. To fix our notation, we shall employ natural units with free-space constants Co,,o set equal to unity and suppress the harmonic time dependence exp(-iot) throughout. A plane electromagnetic wave of unit amplitude is assumed incident at an angle 0 to the semi-infinite screen as shown in Fig. 2-1. The screen is assumed to be comprised of material of such a kind as to make the total tengential field components satisfy the following impedance boundary condition on the surface (n is the unit outward normal to the surface) A; E-("nE)n= Hn^H, (2.1) where n = r1 on the upper surface and r = r;2 on the lower surface. The face impedances r1 and 7r2 are complex constants whose real parts, because of energy considerations, must be non-negative. Further, the surface impedances are assumed to account for the presence of thin layers of highly refractive absorbing materials applied as a coating on a perfectly conducting half plane. Although the validity of the impedance boundary condition near an edge of the diffracting structure is then open to question (Weston, 1963), one nevertheless expects some general features of the scattering process to emergewhen the overall effect of an absorber coating is treated in terms of a constant surface impedance (see, e.g., Bowman, 1967). It may be noted that if the surface is to be considered as perfectly conducting, then the corresponding impedance must be set identically equal to zero. In particular, a special case of interest would be that of a perfectly conducting 4

1492-1-F r Impedance r1 Impedance rl2 FIG. 2-1: PLANE WAVE INCIDENCE ON ABSORBING HALF PLANE WITH TWO FACE IMPEDANCES. 5

1492-1-F half plane coated on one side with radar absorbing material. The exact solution for plane-wave scattering by an absorbing half plane with two face impedances is available from Maliuzhinets (1958, 1960), who treated the more general problem of diffraction by a wedge with arbitrary face impedances. For the application of ray-optics techniques, we shall require the asymptotic far-field form of the solution. This result, obtained by means of a steepest descent approximation to the exact contour integral solution, may be written in the form ud ~ e-i74 e k U (O, e) (2.2) 2i (27rkp)1/2 ~ representing a cylindrical wave emanating from the edge of the semi-infinite screen and produced by a plane wave incident at an angle 0 to the screen (see Fig. 2-1, where the geometrical configuration is different from that used by Maliuzhinets). The total exact solution, u, which is a function of two complex quantities al and a2, is derived by imposing boundary conditions of the third kind, Ikr a + ucosa =0 (O=0 27r), (2.3) ikrae 1 osl,2 on the faces of the semi-infinite screen. The quantities a1 and a2 are constants whose real parts lie in the closed interval [0, 7r/2]. For observation angles e bounded away from the geometric optics boundaries (i.e. for 0 f r + eO), the amplitude factor U(0, e ) appearing in (2.2) is given by sin( o/2) __ + k(27-9) U O (-0) sin(0/2)+cos(0o/2) sin(0/2)-cos(e /2)J (2.4) 6

1492-1-F where the auxiliary function /(/3) is expressed in terms of a special meromorphic function 0 (3) by the product O3R) = r(0+ 7r+a)1) (3+7r 1r-) a (/3- - 2) r(- 37r+ 2) (2.5) and r (/) has the representation -r *Cos V p(13J) = exp sinv-2F2' sin(vz2)+2. (2. 6) Maliuzhinets (1960) mentions that the special function i (/3), along with its generalization for the wedge problem, has been tabulated by M. P. Sacharowa, although no reference to the literature is given. The important analytical properties of the functions (/3) and (IP) are given in Appendix A. An examination of the boundary conditions (2. 1) and (2.3) indicates that the scalar function u may be employed to represent the half-plane solution for either of the two fundamental electromagnetic polarizations. Thus, in the case of H polarization, the function u represents the z component of the magnetic field H, provided we make the identifications z H H r), Cos I 2(2.7) t1 -cosa1, 2= cos (2.7) while in the case of E polarization, it represents E under the identifications z E E I =l/cosa, a 1 /1/cosa2. (2.8) This property is a result of the natural duality of Maxwell's equations and the impedance boundary condition (2. 1) under the transformation E -O H, H -- -E, r - 1/r. The affix E or H is attached according as the boundary conditions describe E- or H- polarization states, and in the sequel, all functions will be written with such an affix whenever this polarization distinction is necessary. 7

1492-1-F An alternative representation for U(e,o ) may be obtained by successive use of the identity given in (A. 10); in particular, after some trigonometric reduction we obtain the following expression, valid so long as 0 $ 7+ 0: -o P(O) P(0o) 1 a a u(0, ) = os - o- + -- cos + O cOS0+cos0 { (2 2 2 + 0 o a0 o 1 + co s - 2- o + co os Co } (2.9) with P(0) = -sin( 2) 4 (r/2). (2.10) The quantity qr (r/2) is given in (A. 9). The expression in (2.9) is manifestly symmetrical in the two variables 0, 00, thereby confirming the principle of reciprocity: U(0,0 ) = U(0O, 0). For a1 = a2 = (7r/2), corresponding to a perfect conductor with H polarization, we have upon employing (A. 10) and (A. 15) = 2 [( 2)] sin 2 and therefore (2. 9) reduces to 4cos(0/2) cos(0 /2) U(O0,0) = coO+cos. (2.11) 0 COS0+ Coso 0 On the other hand, for 1 = a2 -+ + i o, corresponding to a perfect conductor with E polarization, one finds asymptotically ( )1 [a,74 a elI/2 (7-o0) ~ 4 ()2 e - 8

1492-1-F so that (2. 9) now reduces to 4 sin(0/2) sin(0 /2) i(Os 0 )- I... (2.12) cos+cos (2.12) 0 These limiting expressions are in accord with known results for a perfectly conducting half plane. In order to explore the effects of multiple interaction, such as the interaction existing between the two edges forming the mouth of a semi-infinite parallel plane waveguide, we shall also require an asymptotic expression for the half-plane solution along the ray-optics reflection boundary 0 = 7r-0 It may be shown from the Maliuzhinets (1958, 1960) contour integral solution that for 0 = 7r-0 the total field behaves as 0 o 1 ik uep c 9S ---1 -- ee + ( 0 ) (2.13) 2 cos a + sin0 r provided 0 < 0 < ir. For incidence from the lower half space Xr < 0 < 2r, the same expression obtains except that al is replaced by a2. The result in (2. 13) is hardly surprising physically and shows that far along the reflection boundary the scattered field is given by the perfectly conducting result multiplied by the infinite flat plane reflection coefficient. However, to apply the ray-optical procedure to the semi-infinite parallel plane waveguide, we need to know the field generated along the reflection boundary by a line source located at a finite distance from the semi-infinite screen. Rather than solve the line source problem for an absorbing half plane, we shall draw upon an analogy (in view of Eq. 2. 13) with the known result for an isotropic line source over a perfectly conducting half plane. Thus, for an incident field given by i P(-'-r02 ik oI-rI 4 u=H (k r|-r kr- e (2. 14) — O 9

1492-1-F it is physically reasonable to take cos a -sinG0 ik(r +r ) 4i s 1 1 o J 2 o 4(2. 15) -2 cos a1+sinO0 7rk(rr e r( as the scattered field far along the reflection boundary 0 = -r-O, where 0 < 0 < 7rT. For 7r < Q < 2r, replace a1 by a2 in (2.15). 0 01 2. 2 Oblique Incidence The exact solution for the problem of a plane electromagnetic wave incident at an oblique angle to the edge of 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 incidence 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 a, whereas for the purposes of our investigation it would be desirable to consider a semi-infinite sheet with two face impedances, say 77 = ri on the upper surface and rh =r2 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 10

1492-1-F available from Maliuzhinets (1958, 1960), we attempted (Bowman and Weston, 1968b) to carry out the Williams (1960) procedure to obtain the corresponding three-dimensional solutions for oblique incidence. Unfortunately, it came 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. In the following, therefore, it is necessary to assume n1 r2 77. For later application to the rectangular duct, we are interested in the field diffracted into the far zone by the half plane. 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, 0, a), 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 0= (Xr/2) -0, 0= 0: i A A -ikS E = (-xcos0 sine -ysin0 sino +zcosO ) e (2.16) 0 0 0 0 0 i A A-ikS H = (- xsin ycos ) e ikS with S = xcosocos co +ysino cos o+zsin0. (2.17) The incident field is thus taken to be a pure TM mode (no H component); z 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 0 = 0, the plane wave is incident normally to the edge of the semi-infinite screen. 11

1492-1-F Away from the optical boundaries -= r + 00, the diffracted field in the far zone is (Bowman and Weston, 1968b). 7r -1 4 Ed= e - 4cose 0 -ikzsin +ikpcosO 0 e (27rkpcosO 1/2 (27rkpcos0 ) 1/2 0 -ikzsin +ikpcos o o E (0, 00) 2 2 2 ' cos 0 cos 0 +sin 0 0o. 7r -1 4 d -e H = - 4cos0 o e (27rkp cos0 )1/2 H(. 0_o) 2 2 2 cos 0cos 0 +sin 0 0 0 J (2.18) where E (, _) = x [sinOcos 0 G1(0, 0)+ sin 0 G2(, 0)] + + o [sin sin (, 00)- cos 0 G2(. 00)] + zcoso0 G1(0 (2.19) H(0,)=x [sin, O)- sincos G20, )]- 0[coso) + sin0 sin0 G2(, O)]J -z coso G2(p, o ) (2.20) The functions G1 2(0, 0 ) are given by G1(O, A = ikcos0 cos [co CoS0 +oo0 o cos* +cos 00 +AQE(0,o) + + DQE (0, 0 - ik cosO sinO sin0 sinsin + BQo(,0 o)+CQeH(, ) ]' 0 0 eo QO (0' 0)+ Qe(, (0 ) Cos 0 + cos + (2.21) 12

1492 -1-F G2(O.,O)=ikcosO sine sin 2 io 0 0 [Cos 0 Qe( )Q(0 ) AE(~ k cos + Cos 00o +DQE, ~o]+ ik cosO cost snoiO ~(~,~)Q(.~ + BQ H (0.9 ) + CQH O~~) where Q eO=, - P(~ P (p,) Cos a(O)] + (2. 22) (2. 23) Q 0(0j 0 = P (0) P(o0) 2 cos (0/2) cos (00/2) with P(O) = sin (012) 4(7r/ 2) and where the constants A and B are determined by (2. 24) LQE (7T~ +i, QH (I+iv, 0 ) tan & 0 k (- itan&0 +cosP ) Q H(7Ti. 0) tan 2tanO 0sinG0 sinko ik(a20 Co20 ikta &+o Q (-+ iv~o) e-02 '0 +QE(7 - iv,~.1A H(7r iP (2.25) E E(I 7T " Le 2 o sinG 0sinoo 0 0 QE (-r -iv.,O) eo2 0 sinG0 sinko k(i tanG + cosO0)J -2 tanG coso k(tan20 cs20 ) Q ( (- -iv,1 ) e 2 0 Q E(7Z*+.fl0 (2. 26) 13

1492-1-F In (2. 25) and (2. 26) the parameter v is related to the angle of incidence 0 through the relations sinh v = tan 00, cosh v = sec 0. (2.27) Finally, the constants C and D are obtained from (2. 25) and (2. 26) upon E E H H replacing Q by Q and Q by Q, in which instance A -. D and e o 0 e B -* C. In the above, the affix E or H is attached to indicate where the equations 1 E H o) = cos0 cos a (08), = cos co ( ) (2.28) 0 0 should be used to determine the constants a (0 ) or a (0 ) from the impedance 0 0 ri. For normal incidence (0.= 0), the equations in (2.28) are equivalent to those in (2.7) and (2.8) with a1= a2 = -(0). As a check, let us consider the perfectly conducting case which corresH E ponds to a = (r/2) and a -e- + i co. From Appendix A we employ (A. 26) and (A. 27), along with (2. 24) to find Qe (U ) = -8sin sin 2 Q0 0E ) = 0, (2.29) Qo (o) = 8cos2L cos 2 ' Qe (0o) = 0 Only the constants A and B need be evaluated, and with the help of QE2 + iv, -O) i o J o e 2e 0 0 0 -e tan- (2.30) o (-+i 2,) o t 2 2 we obtain from (2.25) and (2.26), after some trigonometric reduction, 14

1492-1-F -sinO sin0 A = 0 B k (l+coS0) (2.31) With these values of the constants we finally have -8 sin (0/2) sin (0 /2) G.( 0) = icosO(oos2 0+8in ein2 0) 0 + (2.32) G2(0 00) = 0, which leads to agreement with the results for a perfectly conducting half plane. Equation (2.18) is fundamentally important for the application of rayoptical techniques to problems involving planar structures that are coated with radar absorbing material, and in particular, the diffraction coefficient derived from (2.18) will play an important role in estimating the fields generated at the mouth of a fully-lined rectangular duct. For this purpose we specialize to the case of edge-on incidence, as illustrated in Fig. 2-2, since only principal-plane incidence will be considered in our treatment of the rectangular duct. Ei z FIG. 2-2: EDGE-ON OBLIQUE INCIDENCE ON ABSORBING HALF PLANE. 15

1492-1-F For edge-on incidence (0o= rT) the incoming field becomes ik(x cos -z sin ) i AoA0 0 E =(xsin +z cos ) e - I 0 0 (2.33) ik(xcos -z sin ) 1 ye H = - y e Now, from (2.24) we see that Q-O0 as0 -, and from (2.25) and (2.26) it is evident that great care must be taken in deriving the correct expressions for E (0, r), H(0, 7r). Further, since we are ultimately interested only in the aperture fields for a rectangular duct, we need only consider the half plane diffraction fields along 0 = 7r/2, 37r/2. There is much algebra involved, although simplification is achieved by noting the following relationships P(2+ iv) = 2 7 iv/ 2 7 2 7t 4cos( 4- 2) (l-iz+2 )r(1 2 4 (r a iv c iv (-) (cos + cos -) (cos -+ sin- ) 7r2 2 2 2 2 (2.34) (- iv)P( -i.v) 2 2 l+cosO cos cosa o P() = 1/2 (1+cos a)12 It turns out that the diffraction fields of interest may be written as: For 0 = (7r/2): -ikz sin0 +ikp cos0 -d =i _it e k4 0 0 An =2 (.. M(02 kpco)-(ysin0 +cose )M(0)] (27r kpcos0 )/ 2 -ikz sin + ikp cos 0 d i -i/4 e 0 0 2 (27rkpcos )1/ 0 (2.35) [-xM(0 )-(sy s + zcos )M(O )O. 0L 0 0 0 16

1492-1-F For 0 = (37r /2): d i E =~ el where i4-ikzsinO +ikpcosO 7r ~ 0 2 (-irkzsinO +ikcos 0 -e e [y M&z(sin6 -zco 2(27r kp cosO0 )1/2 Oil0 0.,the M functions are Cos20 8t (7r /2) PE(7r) M(0) = n7 0 0.0 0.0 (1~(Cos 00+) 1L 2K (e )+e K (- 0 J (2.37) 00 0. i0 = pE(7) Lel 2K(G)..e 2 K (-0) 0 0 0 0 Co 0(r os0 ) 2Ve2 KIO )+e 2K - 0 0+11/ K(-0 with 2 H 7T 2 H?'r 2 E 72 E_ i~'(a -iv/+) (a +iv —)Vi (ae +iv+-H) O(aivK (0 O 7T 2 Ir 2 ir 27r 2 o/aH i H i\ v E a\ E Os +o o sin Os +co O - sin-) (2.38) For a perfect conductor r0 — * 0, in' which case 17

1492-1-F 00 -i 1 8 2 K(O )- 2 (r/2) cos e 2 P ( )- 2 (2n cos0 )1/2 0 0 and M(0 ) = 2 M(O ) = 0. This checks with known results. On the o o other hand, for a perfect magnetic conductor, M ( ) i M(0 ) O0; henoo, 0 0 there is no scattered field in the case of edge-on incidence, as expected. Finally, for normal incidence (0 =0) we have M(0) = UE ( 3r/2, A), M(0) = 0, in agreement with Maliuzhinets' (1958, 1960) solution. 18

1492-1-F THE SEMI-INFINITE PARALLEL PLANE WAVEGUIDE 3.1 Perfectly Conducting Case The problem of diffraotion by two parallel half planes constitutes a valuable prototype for testing the utility of the ray optical method. On the one hand, exact solutions based on the Wiener-Hopf technique are widely known in the literature, while on the other hand, the ray method can be applied systematically and without difficulty. Ray-optical. techniques and their relation to canonical problems with parallel plane geometries involving perfect conductors have been discussed extensively by Felsen and his collaborators (Yee and Felsen, 1967a, b; Felsen and Yee, 1968a, b; Yee, Felsen and Keller, 1968; Yee and Felsen, 1969). Their work has been centered on the study of reflection and radiation of waveguide modes incident on the open end of a waveguide, and they have reported that the ray method yields remarkably accurate results even at small ka values. In this section we shall discuss the plane-wave scattering case. Some new asymptotic developments of the exact solution are presented which yield a term-by-term comparison with the corresponding ray-optical results. We consider an E-polarized plane wave incident from direction 0 on o a pair of perfectly conducting parallel half planes with geometry as illustrated in Fig. 3-1. By combining the results of Vajnshtejn (1954) and Clemmow (1951), we can write the far field as obtained from the exact solution in the following manner: s 1 4 2ika sinO V+Vo + -4ri snr 1 isoO1e l+e +e 0 + (i2ika s ) i sin0 U+U 1 -ika(sinO+sinO0 + -e -e e o (3.1) 19

1492-1-F NIlI N% (r, 0) \ rl \ x 0 0 r1 FIG. 3-1: PLANE WAVE INCIDENT ON A SEMI-INFINITE PARALLEL PLANE WAVEGUIDE WITH TWO SURFACE IMPEDANCES. IN THE PERFECTLY CONDUCTING CASE l7= 172= 0. 20

1492-1-F where we have assumed cos 0 < 0, cosO < 0, and where 0 r-f+ioo V = 1 lg (12ikacosTr) cosrdT V= i log(+e )sin r- cose ' (3.2) U= 21 -- +i0oo /(-e 2ikacos7) cos rd r I *2 log(1-e J sin'7- cos 0 2 The quantities V, U are obtained from (3.2) upon replacing 0 by 00. Now for ka > > 1, a steepest descent approximation yields (Vajnshtejn, 1954): I 9 v-o log (1+ e 2ika- T dt; - eir/4cos t- 2ka e 4 cos 0 (3.3) We expand the logarithm under the assumption that k has a vanishingly small positive imaginary part, t2 mt2 2ika - 2 0 3 m 2imka - 2 m=1 and integrate term by term to obtain 1 O ()m 2imka V,, -2 sgn(cosO) ( e2 G ' w) m=l (3.4) 21

1492 -1-F where w = lca/7IcosOI and G()=2 -i22 00 c -A2 e~ dp. (3. 5) (1 -i) g For values of ka and 0 such that 'Z aI~2Icoso >j~ 1, we have. 7 1 - 4 (3. 6) and consequently * 7r V e4 e Ph j- 47rka co so m= 1 (-m2imka m 3/2e + 1 ka (3.7) Similarly,. 7T u _ _ _ _ e '^' 1 -%Tc m= 1 1 -m3/2.2imka o(1 ka (3. 8) When (3. 7) and (3. 8) are employed in (3. 1), we obtain E 2 E w 7Tk ikr+i -T 4 2sin(9/2)sin(00/2) co[asiOin cos0+cos0 0 L i 4 +T f47kacose 1 cose 0 OD ei(2m+1).2ka X-ecCos ka~sineO-sie)1 -mO(2m+1) 3/2 - )D ei(2m)2ka e 32- cos[1 (sinO+sin90)]+O( i 7T 47a cos0 cos0o In L= I (3. 9) 22

1492-1-F which is the desired expansion of the scattered field. In order to interpret (3. 9) physically, we now direct attention to the ray-optical calculation. The primary diffraction due to the open-ended parallel plane waveguide is obtained in the ray method by neglecting the mutual interaction process that takes place between the two straight edges forming the waveguide mouth. In this first approximation, then, the two edges behave as independent semi-infinite screens, each in the absence of the other and each excited by the incident field alone. Noting that the distances r1, r2 are approximately r.; asin0 as r -- co, and taking into account the phase of the incident plane wave at the two edges, we may at once write the scattered field from the parallel plane waveguide as a superposition of the two primary edge waves: f ikr+i4 2sin(0/2) sin (0 /2) EAW -- e C ----COSO cos |ka(sin0+sin00)J (3.10) I rrcos0+cos0 coLa~i0 siG) provided (7r/2) < (0,0 ) <yr. This approximation is valid so long as ka > > 1, so that the interaction between the edges is weak. However, for more closely spaced waveguides, it becomes necessary to include mutual interaction effects in order to improve the accuracy of the ray optical procedure. To calculate the secondary diffraction contribution, for example, each edge is considered to undergo an excitation due to a cylindrical wave emanating from the other edge in addition to the excitation provided by the incident plane-wave field. The cylindrical wave from the edge is assumed to emanate from an equivalent isotropic line source whose strength is chosen to provide the correct diffraction field in the direction toward the other edge. Knowing how the field of such a line source is diffracted by the half plane representing the other edge, one may write down the first interaction contribution to the far scattered field. The physical situation becomes somewhat more complicated for the successive interactions past the first. This is due essentially to the fact that 23

1492-1-F each edge lies precisely along the ray optical reflection boundary of the half plane corresponding to the other edge. Consider, for example, an isotropic line source of unit strength located at the position of edge ( in the presence of the half plane (. The field scattered back toward the line source -- along the reflection boundary of plate (~ -- is then given by E -, - e (3.11) EBS 1 2 ik(r2+2a)-i4 (3.11) 2 wk(r +2a) 2 and this cylindrical wave appears to emanate from an image line source located at a distance 2a behind the half plane (. From this example, it may be seen that each successive interaction between the edges of the waveguide gives rise to a new image source and that an infinite number of image sources is required to account for the mulitple interactions. By adding the contributions due to the image sources, retaining in the process only terms through O(1/j1), we obtain the following ray-optical result: ikr + i 2 sin (0/2)sin (O /2) F E v- e. - + - os ka(sin 0 + sin 0 + e ~kr cosO+cosO [k 0 ~ L.7T 1 4 oo i(2m+1)2ka + eT +s c 22m- - 2 cos ka(sine-sin0o) - eos A 2m 1'2m+l' m=0 * 7T e4 1 1 i(2m)2ka r 1 ~-4kc (~-s +;- cos rka(sin0+sin0 +O( f4Tr cos0 cos 0 2 2m- ka (3.12) which is to be compared with (3. 9). The two results in (3. 9) and (3. 12) differ because of the numerical factors in the denominators of the summands. It is seen, however, that the primary 24

1492-1-F diffraction, along with the first and second interaction contributions, are in compllete agreement. For each successive interaction after the second, the ray-ol)tical result (3. 12) underestimates the asymptotic result (3. 9). In order to obtain a more accurate result from the ray method, which is based on diffraction of isotropic line sources, modifications are required in order to take into account the fact that the equivalent line sources are not isotropic. This is not necessary, however, because the exact edge interaction functions tell us immediately how to modify the ray-optical interaction functions, at least in the case of perfectly conducting surfaces. 3. 2 Absorbing Case Armed with the results of Section 3.1, we can now deal with the problem of scattering by an absorbing- parallel plane waveguide, for which, unfortunately, an exact solution is not available. The geometry of the problem is illustrated in Fig. 3-1 where, as shown, the interior surfaces of the waveguide are governed by a constant surface impedance ri2, while on the exterior surfaces an impedance rl is prescribed. A plane wave is assumed incident upon the structure from the direction 0, where (7/2) < 0 < r. We shall employ the ray-optical procedure to calculate the field scattered into the far zone by the edges forming the mouth of the waveguide. We make use of (2. 2) to calculate the primary diffraction, which takes the form ur1 i(kr- ) -ika(sinO+sinO ) ika(sin0+sin ) U i- -e <e U(,o0)+e U(27r-, 2-O (3.13) where (7/2) < (0,0 ) < r. It may be noted that the function U(27-0, 2a-Qo) properly accounts for the fact that the lower plate 0 is illuminated on the r2 side rather than on the nl side as is the case for the upper plate (. The 25

1492-1-F edge interaction contributions are calculated just as in Section 3. 1 except that in place of (3. 11) the following equation (in view of Eq. 2. 15) is employed: BS 1 cOS 22 ik(r+2a) -i u -2 cosa2+1 7 k(r +2a) e (3.14) 2 2 The final ray-optical expression for the far scattered field is 1 i(kr-) F-ika(sin+sinika(insin+si ) U r ee U(QO,O)+e 0U(2 r-O,27r) - 4 O R2m i(2m+1)2ka -ika(sin-sin 3r U37 2 + __ R e2mika(sin - o 1 22m+l 22m+1 L m=0 2 ika(sinO-sinO ) + e U(27r-0, -) U( 2 )+ 2 2 0 i1 r 4 Oo 2m-1 i(2m)2ka -ika(sin0+sinO ) 3 3 + e ~ U(O.' U(' o m:01 2 m, 2 2 102m+ m=1 2e 2 2 2 ka where cos a2-1 R = 2-. (3. 16) cos2 0+1 It may be noted that each successive interaction is reduced in magnitude not only by the numerical factors appearing in the denominators of the summands, but also by the reflection coefficient R which for good absorbers is very close 26

1492-1-F to zero. The higher interaction terms are thus negligible in the absorbing case, although in the perfectly conducting case they may play a more important role. 27

1492-1-F IV THE SEMI-INFINITE CIRCULAR CYLINDER 4.1 Perfectly Conducting Case The Wiener-Hopf technique has also been applied to yield exact Solutions to thb imponrtant problomi of oauttating tn4 radiation from a hollow semi-infinite circular cylinder. This configuration therefore constitutes another important prototype for testing the application of the ray method. Levine and Schwinger (1948) (see also Jones, 1952, 1964; Morse and Feshbach, 1953; Noble, 1958) investigated the radiation and reflection of sound waves in an open-ended cylindrical tube, while Vajnshtejn (1954) treated the problem in more detail for both acoustic and electromagnetic radiation. The reflection and transmission properties of electromagnetic waves (H11 mode) in an open-ended circular pipe has also been studied intensively by Iijima (1952) although his report does not seem to be readily available. The problem of scattering of electromagnetic plane waves by a semi-infinite circular tube was treated by Pearson (1953) for axial incidence and by Bowman (1963) for general incidence; these results may be found in the exhaustive review of Einarsson et al (1966). Ray-optical analyses of modal reflection in an open-ended circular waveguide have been carried out for sound waves by Felsen and Yee (1968c) and for electromagnetic waves by Yee and Felsen (1969). These authors have also provided numerical comparisons of the ray-optical results with data computed from the exact solutions. Again, the ray method is shown to be capable of high accuracy even at the lower frequencies including the dominant mode regime. We shall direct attention here to the plane-wave scattering problem and present some new asymptotic developments. As in Section 3.1, these results provide a term-by-term comparison with the ray-optical expressions. The physical configuration is illustrated in Fig. 4-1 where, in terms of cylindrical coordinates (p, 0, z), the semi-infinite tube is located at p=a, 28

1492-1-F FIG. 4-1: PLANE WAVE INCIDENCE ON SEMI-INFINITE CIRCULAR CYLINDER. z > O. The incoming plane wave is propagating along the z axis with spatial variation prescribed by E =xe z(4,1) Part of this wave will propagate along the axis as a set of modes within the tube and part will be scattered into the space surrounding the tube. In terms of spherical coordinates (r, 0, 0), the field scattered into the far zone can be written in the form (Bowman, 1963) ikr E H = - f (0,) & r (4.2) ikr E = H r- g(ii ) & r where the angular distribution functions are given by e cn os 0 Ji(ka sin 0) L (ka) f(Q, 0) =, (4.3) 2kcos(0/2) M (ka) L (-kacos0) (1- A ) 29

1492-1-F g(0.0) = asin 0 J1 (kasin0) M (ka) sin (0/2) M (ka) M (-ka cos 0) (1- A ) 1 L(ka) 2ka M(ka) (4.4) with (4.5) The functions L(w), M () are factorization functions analytic and non zero in the upper complex w plane, and are defined by L(w)L(-w) L = 7T J1(X) H( )(A) (4.6) M()M (-w) = r X J' (A) H(1) (A) where where X = (ka)2-2 7' ImX> for Imk>. (4.7) The imaginary part of k is assumed to vanish in the final formulae. Explicit expressions for the factorization functions have been given by Vajnshtejn (1954), who also derived their asymptotic approximations. In particular, for ka >> 1 and cos 0 < 0, we have (-kacos0) = e L (-ka cos 0) = e (4.8) where U is given asymptotically by (Vajnshtejn, 1954) 00 1 U~ 2i log - -D \ dt t l i cos/4 t + ^2ka e 'cos0e (4.9) with 7k 3 q =ka- -+ + 4 8ka (4.10). 30

1492-1-F The Method of Section 3.1 may now be employed to obtain the following alternative asymptotic representation:.7r 00 i /4.i)m -3/2 2imka 1 L(-kacos)N1l + (-mi)m e + O( ) (4.11) 14Jka'cos m=l provided ka/2 cose >> 1. In similar fashion we obtain M(-kacos0)-l1+ e m3/22imka). (4.12),i /4. i - 3/2 2 imka 1 M (-ka cos ) e 1 +O( ) (4.12) 47rka coss m=1 The assumption cos 0 < 0 means that we are confining our attention to observation points in the backward half-space z < 0. When the immediately preceding results are employed in (4.2) through (4.4), the field scattered into the backward half-space is found to take the form eikr J (ka sin ) cos i /4 3/2 2i se e e2imka X 0 2kr cos (0/2) m=1 kam=l ) (4.13) X C- ()m sec cos2(0/2)] ikr J'(kasine)sin0 i /4 2 _ES. _ a- e 1 ) 0 e sin (0/2).m -3/2 2imka r sin(0/2) \ a cos m e (4.14) For kasin >> 1 the scattered field is thus 31

1492-1-F cos(kasin — ) si] Es _ e ( 2a 4 -- k ' insin (0/2) 2 sin (0/2).m -3/2 2imka os _ i m e m=l }o e l/4 7a ka (4.15) whereas, for backscattering (0=7r) ik r r iA /4 BS A ae, 1.m m-3/2 e2imka E -~x 2l1 m e [ ^?rka m=l (4.16) It is interesting to note that the high-frequency backscattering cross section is aBS r a2+ 1 (4.17) which is just the area of the mouth. The result obtained by Ross (1967) for backscattering along the axis is apparently in error. It is a relatively straightforward task to provide a ray-optics description of the scattering process for axial incidence. For observation at (0, 0) bounded away from the backscattering direction, the points of diffraction (or scattering centers) onthe rim of the open tube are located at 0 and 0 + r. This means that we can begin with the parallel plane result in (3.12) specialized to the case 0 = v, o E.l 2 ikr+iR/4 7kr ika sin O -ika sin 0 e +e0 2 sin (0/2) ei /4 sin2(0/2) ei(2m) 2ka1 cos O m=l 22m-l(2m)1/2 k (4.18) ei(2m+l) 2ka 1 22m (2m+1)1/2 j 9 32

1492-1-F and suitably modify this result to account for the axial caustic of the circular tube and for the divergence of rays due to the curvature of the rim. Each time a ray traverses the axial caustic, its phase must be retarded by the amount 7r/2; therefore, the summand in the first summation in (4.18) requires a factor (-i), while that in the second summation takes a factor (-i) 2m. Furthermore, the term e 0 which is due to the diffraction point at 0 + 7r must be multiplied by (-i) since rays diffracted at 0 + 7r pass through the caustic to arrive at (0, 0). The curvature of the rim is taken into account by incorporating an overall divergence factor (a/r sin) 1/2 which is the same for both diffraction points under consideration (see e.g. Felsen and Yee, 1968c). Finally, the incident polarization is A A A x - 0 sin0 + p cos 0 A and only the 0 component contributes since it is the component tangential to the edge at 0 and 0 + r. (The field scattered by the normal component is identically zero for edge-on incidence.) When these modifications are applied to (4.18), the ray-optical result for the semi-infinite circular cylinder becomes (for kasine > 1) 1 7" i4 A ikr 2a /2 cos(kasinO- )sin sin 4 (/2) X E ~0 v - I. X - r 7rksin sin (0/2) cos 0 (4.19).m 2imka X l e1 \ m-1 2m- J in agreement with (4.15) up to and including the m=2 term. Complete agreement would have been obtained if we had started with (3.9) rather than (3.12). In the backscattering case (0=7) a slightly modified ray method is required because all points on the rim contribute to the backscattered field. It is simplest to integrate the contributions around the rim a la Siegel (1959). 33

1492-1-F The final ray-optical result is then BS ikr 1 i /4 im e2imka BS (4.20) - 2r 7rka 2m-1 (4.20) in agreement with (4.16) up to and including the m=2 term. Again, complete agreement would have been obtained by starting with the exact parallel plane result. In the case of off-axis plane wave excitation of the semi-infinite hollow cylinder, both the exact solution and the ray-optics description are considerably more complicated. Let the incident plane wave be 4 A A ik(xsinl3- zcosl3) (x cos a cos 3 + y sina+zcosasinj) e, (4.21) so that the incident wave is propagating parallel to the (x, z) plane from the direction 0 = f3, 0 = 7r. The angle a determines the state of polarization of the wave. If the observation angle is outside the domain of reflected cylindrical waves, that is 0> 7T - then the far-zone scattered field is comprised solely of the diffraction contribution and is given by (4.2) with (Bowman, 1963) a oe J (kasin0)J (kasin/3) a cos a n n ( '0) cos(0/2)cos(B/2) E L (-kacos0)L (-kacosB) X n= -00 n a X 2cos (e/2)cos (3/2)+ 2 (4.22) cos 0+ cosj + A2 n ise in 0 J (ka sin9)JT(kasing3) A la sina n n n n cos(0/2) sin(1/2) L (-kacos0)M (-kacosn) 2 12 n= -oo n n i-A n 34

1492-1-F ~ ein J (kasin O)J' (kasin ) =. a sin a n n g(, 0) sin(0/2) sin( /2) in M(-kacosn )M (kacos 3) M (M-kacos0)M(-kacosj3) n=-co n n 22 2 A 2 sin2(0/2)ssin (B/2) n + (423) ccos + cos/3 n i a cos a n n n sin (0/2)cos (3/2) M (-kaos)L(-kacos)L) 1 ) 2 n=-co n A n where L (ka) n 2ka M (ka) (4.24) n The factorization functions Ln(w), M n() are analytic in the upper complex w plane and are defined by L (~)L ( —) = 7r X J ( H)(), n n n n (4.25) M () M (-) = r XJ'(X)H (X), n n n n with X as in(4.7) For simplification, we shall first restrict our consideration to observation points in the plane of symmetry - the (x, z) plane - and further, we shall consider only source and observation angles such that cos 3 < 0 and cos0 < 0. In this case the factorization functions approach unity as ka - coo, and the leading terms for the angular distribution functions are 2 cos (0/2)cos (3/2) -1)' J in)Jkasin) —, ar cos a {'f uos {(O,, ) acosa C + Cos ) 1 J (ka sin 0)J (ka sing1) (4.cos+ n-oo6) (4.26) 35

1492-1-F J(. 2sin(o/2)sin(/ g 2 a sina c si Z -1 J'(kasinO)J(ka sinl) (0, 0) cos 0 + cosA n n (4.27) The summations immediately above can be carried out explicitly by means of the addition theorems for Bessel functions. We obtain (Oa T2 cos (o/2) cos (p/2) { f(O ) a cosa 2cos(o cs1 [ka(sin + sin)] (4.28) g(O, 7r) 2 sin(0/2)sin(1/2) + ) asina si(/c2) sin J [ka(sin + sin)]. (4.29) g (/, 0) cosO + cos1 1 In the case of backscattering (0 = 3, 0 = 7r) the outgoing field is ikr EBS a e A 2 E BS ae [y sina sin (//2) Ji (2ka sin 3) - rcos1 1 A 2 - (x cos3 + z sin ) cosacos (1/2) J (2kasin 3). (4.30) For axial incidence (3 = 7r) agreement with the leading term in (4.16) is obtained (now the incident field is polarized in the A direction). Moreover, for kasin >> 1 ikr E BS rcs ( ) cos(2kasing - ') [ sinasin2(1/2) - r cos sinT 4 (4.31) - ( cos 3 + zsin3) cosa cos(13/2) which is in agreement with the ray-optical calculation (see e.g. Ross, 1967). Further insight is gained by examining the ray structure for general observation locations. If cos < 0, cos < 0 then at least two diffracted rays pass through each observation point in the far zone. These rays correspond to the existence of at least two points of diffraction (or scattering centers) on the rim forming the mouth of the open waveguide. Let us consider 36

1492-1-F the set of diffracted rays arising from an induced source element at 0= 0 on the curved rim, which may be regarded as locally straight s because of the large waveguide assumption ka > 1. In general the incident ray strikes the locally straight edge at an oblique angle so that the family of rays emanating from the point of diffraction s forms a s right circular cone. Now, the equation rsino sin(0-0) ).1 =. = + cos, 0<V<< (4.32) [r2+ a2 - 2rasin0 cos (O-0 /2] 2 describes a right circular semi-cone of central angle 2 / with apex on the A A rim at s and with interior axis in the 0s or -0 direction depending, respectively, on whether the upper or lower sign is chosen. The angle i is determined by ray-optics considerations; in particular, b is the angle formed by the incident ray and the tangent to the edge at 0. Since A A A A A k = x sin -zsin3, 0 =-xsin0 +ycos0 o (4.33) s s ' and since we want O< V < (7r/2), we can take A A cosI = k sin = sin sin0 (4.34) From our geometry it is seen that the diffracted rays must lie on semicones with axes in the directions + 0 for sin0 < 0. In both cases, the s S "ray cone" emanating from the point of diffraction at 0 is described by the s equation rsine sin(0- 0s) [r.1 =- sin 3 sin0. (4.35) r +a 2- 2rasin0cos(0-0 )] /2 In the far zone (r- ao ) the cone equation (4.35) becomes sin 0 sin (0 - -sinm sin0. (4.36) 37

1492-1-F For a fixed observation location (0, 0), Eq.(4. 36) can be solved for Os I and since both S and 7r + 0s will appear as solutions, it is clear that two diametrically opposite points of diffraction will yield a diffracted ray passing through the point (0, 0). Thus, without loss of generality, we may assume that 0 < ~ < 7r provided the two rays from 0 and 7r + are taken into account. It may be noted that (4.36) is identically satisfied for all 0 if 0 = 13 = =r or if 0 = 3, 0 = 0. The former case corresponds to backscattering with axial incidence; the latter case is described by (4.28) and (4.29) in which the angular functions f(j3, 0), g (3, 0) are governed by Bessel functions whose arguments vanish. In both cases all points on the rim contribute to the diffracted field. Provided the two cases cited immediately above are excluded, the field scattered into the far zone is of the form ikr a sin2 ika [(k-r. * P )] + i7/4 E k) --- DA e + L' s (4.37) asin2 ika [(k-r)- p(7r+s)] + i/4 j(r-K) *(7 +0 e where D is an appropriate vector diffraction coefficient (with a factor i -r/4 e excluded) obtained from the half plane solutions and where k is given in (4.33) while A A A r =x sin cos + y sin 0 sin + z cos 0, A A(4.38) p() = x cos + sin. Combining (4.33), (4.36) and (4.38) we obtain 38

1492-1-F (r-k) P(0) = sine cos (0- 0s) - sinj3cos0 (4.39) = sin0 sin0 = sgn(sino)(sin20+sin2_-2sin-sins cos0)/2 sin 0 s thus the scattered field (4.37) behaves as s eikr a sin2, 1/2 E - D cos (2ka2 -) (4.40) r 4 where 22 = (sin2e+sin2 -2sin singcos )/2, O<2<1. (4.41) It may be noted that n vanishes only if 0 = r or if 0 =, 0 = 0, which are the cases we have excluded above. In the case of backscattering (0 = 3, 0 = 7r) the field in (4.40) will reduce to that in (4.31). 4.2 Absorbing Case In view of the results of Sections 3.2 and 4.1, the problem of axial incidence upon an absorbing semi-infinite cylinder may now be treated without difficulty. The geometry of the configuration remains the same as in Fig. 4-1, except that the interior surface of the cylinder is assumed to be governed by a surface impedance r2, while the exterior surface is governed by an impedance rL. As pointed out in Section 4.1, the parallel plane result may be suitably modified by ray-optical considerations to yield the circular cylinder result, and indeed, in the case of perfectly conducting surfaces the exact parallel plane result can be used to yield the exact cylinder result. When the surfaces are absorbing, we begin with the parallel plane result in (3.15) with 0 = 7, and then incorporate the appropriate ray-optical modifications that are necessary to account for the axial caustic of the circular tube and for the curvature of the rim forming the mouth of the tube. For axial incidence given by (4.1) and for kasin0 >> 1, the far field scattered by the absorbing semi-infinite cylinder is found to take the form 39

1492-1-F E 0-u sin0 + u cos 0, (4.42) where ikr 1/2 -U e 2a (-ikasinO+i14 [ + us ' e (O, 7r) + 4r ksinL i /4.mRm-l1 2imka + U(, 37r/2)U(37r/2,7r)o 1 e) + iYka' ml 2m+1 J - + eika sinO - i /4 L(27r - ) + U(27r-O 37r/2) i /4 imRm-1 2imka 1 x U(37r/2, 7r) >i4 mRl 2m (4.43) r ka m=1 2 In (4.42) the affix H or E is attached to indicate where (2.7) or (2.8) H E should be used to determine the constants al, 2 l 2 that are implicit 19 2, 1,2 in the functions above. For a perfect conductor, H 2 U (O,) U(O7r) sin(/2) - E 8ssin (0/2) UE (o, 3/2) UE (3r/2, r) = - 8sn(O ) cos 0 and agreement with (4.19) is obtained. To obtain the backscattered field, we begin with (3.15) specialized to the case 0 =0 = 7, and incorporate only the ray-optical caustic corrections. O The outgoing electric field from a diffraction point on the rim at 0 then has the form i(kr+7r/4) (Ei SE i H),5, 2 E S S (4 44) 27rkr ~2nkrI 40

1492-1-F where land E are the components of the incident field parallel and perpendicular, respectively, to the rim at the point 0: El= sin2 -y sin cos, (4. 45) E= x cos2 0 +y sin 0 cos The quantity S is given by i7r/4 m mn-l 2imka 2 37re iRi 2me( 2S=U(7r, 7r)+U2(r, 37T) e _ R 2 e. (4. 46) k m=l 2 / To convert the field (4. 44) to a three dimensional field, we multiply (4. 44) by Siegel's (1959) factor e i/4(k/27rr) 1 a d and to add up the contributions from all points on the rim, we integrate over d from 0 to 7r (rather than 27r since we began with the parallel plane result). The backscattered field is then ikr E BS ae (sE H E ^.* -S (S^SH). (4. 47) For a perfect conductor S =0, and S becomes sE i e/4 O im 2iika S =1+ m rka m=l 2m Vi' so that agreement with (4. 20) is established. On the other hand, for a perfect absorber (r 1=772=1 ) we have S =S in agreement with Weston's (1963) general absorber theorem. Weston's theorem states that any body whose shape is invariant under a rotation of 900 about an axis will yield a vanishing backscattered field if a plane wave is incident along the symmetry axis and if the surface impedance of the body is unity. The circular cylinder under consideration here is a special case governed by Weston's theorem. In the case of off-axis incidence with the incoming field given by (4. 21), the backscattered field (for which 0= f3, 0=7r) may be derived in the form, 41

1492-1-F provided kasin3 >> 1 and cos3<O, BS A EH E =yu sina +(xcos+B4sin3)u cosa (4. 48) where -ilr a 1/2 -2ikasin4+i7 4r( &n ) e lu(j, ) + 2ika sine-i 1 + e U(27r- f, 2r-3). (4. 49) Equation (4. 48) represents the appropriate generalization of (4.31), and indeed, if the surfaces are perfectly conducting, then Ewc - 2sin2(y/2) UHae, =) 2cos2(..3/2) U (4)= c-Jos ' ( = ost which yields agreement with the result in (4. 31). 42

1492-1-F V THE RECTANGULAR DUCT We now direct attention to the problem of scattering by a rectangular duct of uniform cross section and finite length. The walls of the duct are assumed to be absorbing and such that an impedance boundary condition may be applied. Rather than determine the field scattered directly by the edges forming the mouth of the duct -- a task which is easily accomplished in view of the preceding sections -- we are interested in determining the contribution to the radar cross section due to energy launched into waveguide modes which may be reflected by the closed termination of the duct and reradiated back toward the transmitter. The first step is to calculate, by means of ray optics, the diffraction fields generated at the mouth of the duct by the incident plane wave. This information must then be used to calculate the conversion coefficients which couple the induced aperture fields to the waveguide modes in the duct. In the process, the modal structure for a rectangular waveguide whose four walls obey an impedance boundary condition must be examined. Because of the boundary conditions, the modal problem is not separable; however, the modes may be determined approximately by means of an asymptotic analysis applied to the transcendental eigenvalue equation of infinite order. Orthogonality relations for the modes may be derived by using the Lorentz reciprocity theorem in conjunction with the boundary conditions and symmetry conditions. Finally, the scattered field due to modes reflected from the duct termination are obtained by means of a Kirchhoff approximation over the aperture formed by the mouth. 5.1 Aperture Fields The geometrical configuration for the rectangular duct is displayed in Fig. 5-1, where the origin of the Cartesian coordinates is taken at the center of the guide cross section and in the plane of the open mouth. The duct is assumed 43

1492-1-F to be terminated at a distance L from the mouth by a perfectly conducting plate transverse to the z axis; the waveguide is thus short circuited at the termination and the modes suffer no decrease in amplitude at the point of reflection. The incident plane wave is taken as A -ik(ysine +zcos0 ) O O E =(ycosO -zsin& ) e _O O (5.1) i e -ik(ysino +zcosO ) H =x e and is therefore propagating in a principal plane of the waveguide structure. Finally, an impedance boundary condition of the type (2. 1) will be imposed on the walls of the duct. For simplicity, it will be assumed that both the interior and exterior surfaces of the duct are governed by a single constant impedance r1. This assumption is made because the plane wave in (5.1) is incident obliquely to edges Q and ((see Fig. 5-1), and as pointed out in Section 2. 2, the diffraction coefficient for oblique incidence on an absorbing half plane is available only if the impedances on both sides of the half plane are identical. The primary diffraction fields generated at the mouth of the duct by the field (5. 1) will be determined from ray diffraction theory with suitable modifications to account for the finite lengths of the diffracting edges. We begin with the fields due to edges (),, and employ the half plane results (2.35) and (2. 36). When the change in coordinate systems is accounted for, the half plane diffraction fields take the following form. For edge (: -ikysinO +ikp cosO o 3 o F d i -i r/4 1/2 (xsin0 -ycos0Z )M(0 zM(0 2 (27rkp3cos0) )(5) ir/4e i 1/2 Lx 0sin 0 0 y (27rkp3cosO )1/2 3 0 (27rkp3c~S eo) 1/ 44

1492-1-F Ei k H I % \ FIG. 5-1: PLANE WAVE INCIDENCE ON RECTANGULAR DUCT. 45

1492-1-F For edge ': -iky sinO +ikp 4cos0 Ed i _-i//4 o 4 o F-(xsinO+Ycoso )M(0 )-zM( O) 2 1/2 Lo 1 (27rkp4cos 0 ) (5.3) -iky sinO +ikp 4cosO0 d = i 2-i7r/4 e + CiO/ So )( (27rkp4cos 0 where p3= (a-x), p4= (a+x), and the M functions are given in (2.37). In these E H functions the quantities a (0 ), a (0 ) appear implicitly and are determined by (2.28). To account for the finite lengths of the edges, we consider the integral 1 -ikysin o eik [p2+(y-y )2] 1/2 1-b [P-iy12 +- dy,-tkysinO -iky sin0 +ikp cos0 4 e ~ H (kp cos/ ) e- e - ' H o (27rkp cosO )1/2 0 (5.4) In view of (5.4), better results for the aperture fields are expected if the quantity -i 7r/4 -1/2 -ikysin0 -ikp cosO0 e (2rkpcosO e (5.5) appearing in (5.2) and (5.3) is replaced by the finite integral __.b -iyi eiky.'sin ik +(y-y) 2]1/2 47r e ~i2 [2]1/2 dy (5.6) [b 2+y-y,) 2/"y'. 46

1492-1-F The integral (5. 6) physically corresponds to a finite line source with traveling wave current. Let us denote this integral by Ib(p, y, O0), then the aperture fields due to edges ) and may be taken as E Ib(a-x, y; ao) (R sin - y cosO ) M(0 ) - M()] - -~ 0 o o o o - I(a+x, y; 0 ) [(x sini + coso ) M(o) + z M(o)]. I = Ib(a-x,y;0 ) [(x sinO0 - y cos0) M(O0) + z M(0)] + + Ib(a+x, y; 0 ) [(xsin0 +ycos0) M(0 )-^ M(0 )] (5.7) (5.8) -r The quantity M(0 ) is due to an electric line source, while the quantity M(0 ) is due to a magnetic line source. In the case of the remaining two edges, Q and ~, the half plane diffraction fields are determined from (2. 2) with due account given to the phase of the incident field at the points of diffraction. For edge: d ^ i -ir/4 E z- e 2 -ikbsin0 +ikp o 1 uH( 37 ) (27rkp )1/2 2 1 (5.9) d A i -ir/4 H =-x e 2 -ikb sinG +ikp e (2?rkp )1/2 1 H 3r U (-, -e0 ). 2' o For edge 0: d A i -i7r/4 E =-z e 2 d A i -i7/4 H -x2 e 2 ikb sin0 +ikp2 e H U (-,+e), (27rkp )1/2 2 2 ikb sin0 +ikp2 e - (27rkp )l/2 2. ) 2 (5. 10) 47

1492-1-F where p = (b-y) and p2= (b+y). Here the impedances on the two sides need not be equal; however, we shall still adhere to our assumption that; the impedances are identical. We also note that multiple interaction between edges O and Q can be taken into account by incorporating the results of Section 3. 2. Finally, since co ik[p2+(x-x') 2 1/2 1 e 1 H() e-ir/4 ekp 27n 21 2 o'2e 1/27 oo [p 2+(x-x') J 2 (2rkp) 1/ (5.11) it is clear that better results for the aperture fields can be obtained upon replacing e -i/4 (2rkp)1/2 eikp (5.12) by the finite integral a ik[p2+(x-x)2] 1/2 " CXT ( 5. 13) 27ri a 2+ 2]1/2 dx'. (5.13 The total aperture fields are obtained by combining the above results for all the edgesO,., Q. We define = b -iky'sinO ik[p+ (y- y')2]1/2 Ib(P'; o) b 2 2]1/2 4Y e (5. 14 a ikp 2+ (x-x)2] 21/2 I(.x) =dx' d(5. 15) -a [p2+(x-x1)2] /2 then the aperture fields may be written in the form 48

1492-1-F E= Ib(a-x,y; 00) [(x sin0 -ycosO0) M()-^ M(0o)]-Ib(a+ x, y;0 O) [(x sinO y cos0 ) M(0o)+ M( + -ikb sin0 ikb sinO + z I(b-yx) U (0)e I (b+y,x)U e 0 - 0 Zla(ByOx)uH(-0 ) (5.16) H=I (a-xy;0 )[(xsin0 -ycosO ) M(i0 ) + M(0 )+ +Ib(a+ x, y; O) [( 0sin0 + coso0 0 () - M(A o )]-ikb sin0 ikb sin0 -xI (b-yx)U e xI(b+y,xU (-) e, (5.17) where the M functions are given by (2. 37) with (2. 28) and (2. 38), and where H4(r/2) cos(0 /2) U (0) + cos, ( [n+ fi sin(0 /2)] (5.18) (l+r)1/2 H(0 )cos0 0 0 H E with Yr= cosa. For 0 = 0: M(0) = U (0), MI(O) = 0. O 5. 2 Boundary Conditions in Rectangular Waveguide The problem of determining the modal structure in a rectangular waveguide whose four walls are absorbing is greatly complicated by the fact that the boundary conditions are inseparable. Investigations of alternative techniques for obtaining the fields inside the duct have revealed that the boundary conditions are most convenient when expressed in terms of the electric fields tangential to the normal cross section. of the duct. In this section we shall display a modified form of the standard impedance boundary conditions. The geometry for the rectangular waveguide is as illustrated in Fig. 5-1, although now we are considering a waveguide of infinite length. An impedance boundary condition (2. 1) will be imposed on the four interior walls of the waveguide, 49

1492-1-F with the walls characterized by a single constant impedance r). In terms of the field components the boundary conditions are as follows: On y = +b: E =+ rH, x z E =+ oH, Z - X E =+ r H y z E =+H. z y (5.19) On x = + a: (5.20) We select the transverse components Ex, E as the two fundamental field quantities. In terms of these components, Maxwell's equations allow us to rewrite the boundary conditions in the form On y = + b: aE aE aE y=x ikE = ikyE y -T - E + ay - y ay - r7 x ax (5.21) On x =-+ a aE x = + ik r E ax - x aE aE = +- k E+ ax - r y ay (5.22) Two of these boundary conditions are uncoupled, whereas the remaining coupled boundary conditions appear to have the simplest form -- indeed, other choices of basic field quantities, aside from H and H, lead to much more complicated expressions involving second derivatives. In terms of the transverse magnetic field, the boundary conditions may be expressed as On y=+ b: On x = +a: H +ik aH x ik ax -- x aH aH x- =+ikrHH + ay - x ax aH aH Y= +ikrlH + x ax - y ay (5.23) (5.24) 50

1492-1-F and are, of course, derivable from (5. 21), (5. 22) by invoking the natural duality of Maxwell's equations and the impedance boundary condition. 5.3 Scalar Eigenfunctions The boundary conditions in (5. 21) and (5. 22) strongly suggest that we examine first the eigenfunctions satisfying the uncoupled relations. Such eigenfunctions would be appropriate for a rectangular acoustic waveguide with non-rigid walls. The electromagnetic eigenfunctions will be represented as expansions with respect to the acoustic eigenfunctions. The scalar functions of interest are determined by the differential equations " 2 2 m (X) m = () + n Xn (y)= 0 (5. 25) and the boundary conditions ' (a) = ik r I (a), X (b) = ikrXn (b). (5.26) For convenience, let us consider explicitly only the set of functions{ (x) }; all relationships for the set X { (y)} follow by simple notational changes. The functions m (x) will be defined as follows: m (x) cos A x I ( m for m = 0,2,4,... m(a) cosl a m m (5.27) Zm(x) sin m x fm(a). sin:a for m= 1, 3, 5,... m m where the boundary conditions are met by adjusting jm so that j3 tanf a = -ik r (m even) m m (5.28) 3 cot3 a = ikr) (m odd) m m 51

1492-1-F and where i (a) is given by 12 '2 m (a) = 2(5.29) (132 k2 r2) a-ik ri m With these definitions, {m (x)} forms a normal orthogonal set of eigenfunctions such that -a where 6P = 0 (m p p) and 6 = 1. The set of derivatives \ (x) forms m m m a set of functions with the interesting property Ja m (x); (x)dx-2ikr jm(a)I (a)= P, (5.31) m p m p m m a provided m, p are either both even or both odd. This follows easily from integration by parts: _a [ax '(x),(x)dx= [ -(x) (x)] j (x) (x) dx -a -a 2 a - 2ik 7im(a) p(a)+ m T (x)lp(x) dx m;p m (X)m p -a = 2ik r m(a) Zp(a) + 4m2 6P = m p m m Because of the orthogonality properties (5. 30) and (5. 31), expansions of a function f(x) with respect to the systems {m(x)} and i{ (x)} take the form 52

1492-1-F a f(x) = m (X f (x) dx, (5.32) m - X J(x) = a 2 f(x) = m f(x) j I (x) dx-f(a) m(a)+ (-a) (-a). r 2 1 j M m m m a (5.33) For our later application, some particular expansions will be most useful; for example, C X m(a)m (x) cos" u x 2 m -m p sing a+ ik rlcos/s a 2 2 2 m m (5.34) sing xo = (a) (x) sinx __x -_ = -2 m -m p cos, a - ik 1 sin/ a 2 32 m m where the affix e or o indicates whether the sum is over even m or odd m. The summations in (5. 34) are unformly convergent in x < a; however, the differentiated (term-by-term) series are uniformly convergent only in the interval Ix < a- E, 0 < < a. For this reason, it may be important to have expansions whose derivatives are also uniformly convergent at the boundary x= a; in paricular, the series o _ (a)JT (x) cos x _ -2 L'^~ "m -m (a (u cosu a-ikr sin/ a) 2 2 2 m m m (5.35) sin, x e Im(a)j m (x) s =-2 m (a (' sin, a+ ik rcos/ a - 2 2 32 m m n and their derivatives are uniformly convergent in Ix j < a. 53

1492-1-F Equations (5. 34) and (5. 35) lead to the following relations: I-' Cx) m TM (a) =2 (2k2 72) m 2 2 2 = 2(A3 -k ri ) m = 2( 2_2O2) = -2(g 2 -k2 7) m o v WaI~ (x) m p e Iv (a)7 Cx) 132 132 p m p Cm even) (5. 36) Cm odd) TjmWX I (a) In 0 xr~m p p I(a)l? ICx) p C m even) 92(2 2) p m p I Ca)l' ICx) p 1 Cm odd) 92(22. (5. 37) all valid for Jx 1~ a. We also obtain 1 1cs sing acosgI.a -Msing'a cosga -2-,2 (pip~koo )-lipaikiop p 2(a p p (5. 38) 1 2 2 1 AS gz cosg~a sing la-ps'cosm la singa (p. cosApa- ik r7 sinpa) (u. lcosps 'a- ik77 sinAS 'a) p l 2(a) ip p p 54

1492 -1-F Other sets of scalar eigeenftu-nctions are obtained upon replacing rj by 1/a7. These will be denoted as Ii(x) and x n(y), where (x) I + [311 ~( x) = 0 -m m m n n n (5. 39) with boundary conditions ji (a) - il (a) m Ti m n fl n (5. 40) The set{"' (X)1 is I'hiLLted to{ (x)} through the expansions, valid for lx < a.,.*-m =2 ik (i- - ) ~(a) 11 m - 2 ik (r)-1) Tj ) (a)Tj (x) -2 2 mp Tj (a)~ (x)p 0%0 2 [3 - A m p (m even) (5. 41) ~ p (m odd) Also, we have for lxi <a., *,m* (a) = 2 (~2 1? ) m p p TP(a)l (x 'v2 2 mp lp (a)5 (x),v2 2 [3 M-[ mp (m even) (5.42) = 2(I3 -k) m (m odd) Finally we note 55

1492-1-F 2 1 2 2 g'a m' 4k (r- I 2 (a) y (2 = 6 (m,ml both even) 2 (5.43) 2 1 22 9 ma) 4k(-) ( ) 2 p 2 2 -2 6 am (mm' both odd). p) m ' -2/3 2(j m ) p (Pm p) Wppm1) Many other expansions can be derived. The eigenfunctions v (x), X (y) are m n particularly appropriate for representations of the magnetic field [compare the boundary conditions (5. 40) with those in (5. 23), (5. 24)]. 5. 4 Approximate Eigenvalues for Scalar Functions In general the tangent-cotangent equations (see, for example, eqs. 5. 28) that determine the (complex) eigenvalues /3, /3, Y P must be solved by m nn numerical or graphical techniques. In some cases, however, they may be solved apprc Imately. Let us consider the equations (5. 28) j3mtan ma = - K (m even) (5.44) /3 cot/3 a = K (m odd) where K = ikr. For IKa >> 1 ( or for m7r < IKa ) an approximate solution is given by a~ 2- (m+1) a (5.45) m 2 Ka- 1 On the other hand, for IKal << 1 (or for mr~> >Kaj ) we have m 2m [ 22] (m4Ka 0) mL m 2 (5.46) J3ma E~ Ka (m = 0) 56

1492-1-F from which it is clear that the eigenvalues and eigenfunctions of high order (m large) approach those of ordinary Fourier analysis. This very important property is characteristic of Sturm-Liouville problems, and strongly suggests that an asymptotic approach may be applied in the investigation of the electromagnetic modes in absorbing rectangular waveguides. 5. 5 Modes in PIct:;'.i.Lr Waveguides with Absorbing Walls We are now in a position to investigate the electromagnetic modes, which will be obtained by representing the transverse electric field as Fourier series with respect to the systems { (x)} and {X (y). The first step is to decompose the solutions into four independent parts E = ee + Eeo + EOe + EOO Y Y y Y Y (5.47) E = E~~ +E + Ee +Eee E.E4 x x x x x where the superscripts o, e refer to odd and even parity, and the first refers to this property with respect to the variable x, and the second with regard to the variable y. The independent solutions are formed by the pairs (E ~ Ee) (Eoe Ee), (Ee~, E e) (E, E~ ) (5.48) x y x y x y x y Considering the first pair, for example, we can write the fields as ee -ihz ee Xn(b)Xn(Y) cosnx E =e An y n ik 3n An sinp a+ — coslA a n n n r] n (5.49) E= -z B m(a)m (x) sinvmy y Eoo -ihz 0 B MO m. x m v cosv b ---sinv b m m m r; m 57

1492-1-F 2 2 2 2 2 2 2 22 2 where h = k- 7, and V m= a -j3. The uncoupled n r mn boundary conditions in (5. 21), (5. 22) are thereby automatically satisfied. An alternative representation is ee -ihz 2 n 0 m c e A a Xn(y) mi(a)2m' (x) n nm Xn m m 2 2m X (b) B b0 I (x) n(b) X(y) rn "mn n 2 Yn im(a) (5.50) Eo -ihz 2 x n m n where 2 e 2 (b) / (/ncos n a -ik r sin/ a) e n n n n n a nm 2 2 ik (u - Pj )(A sinls a+ — cost a) n m n n / n 2 2Z2 (a)v (v sinv b+ikrcosv b) o m m m mn m b mn 2 2 ik ( -'y) (v cosv b- -sinv b) m m m r mr Yet another representation is (5. 51) e o ee -ihz ~ z n m #V V e X(b) Xn(y) m(a) Im(x) n 2 -2 n- m (5.52) E = -2e-z xm m B~O m n Tm(a)m(x) n(b) Xn(y) 2 ~2 /v - y m n The eigenvalues h are determined by applying the coupled boundary conditions in (5.21), (5.22). To this end, we employ the first two rc,-resentations. On y = b the relation 58

1492-1-F a - ik E = \y 7 x aE ax ax (5.53) becomes o o e E B m I)Im(x) = ) m m n so that e B = A a m n nm n A e (a)f (x), n nm m (5.54) (m odd) (5.55) Similarly, the coupled boundary condition on x = a yields A = I B~ b~ n/ m mn m (n even) (5.56) We thus obtain two sets of homogeneous equations: e o t AeO s m 0 e B s n a b -6 = 0 (n even) sm mn s b a - 6 ~ = (m odd). sn nm s J (5.57) The eigenvalues h (or a) are determined by the requirement that the Ldeterminant vanishes, that is 0(en) Q a b - 6 = 0 sm mn s m (n, s even). (5.58) Thus the eigenvalues are specified by zeros of a transcendental determinant of infinite order. 59

1492-1-F Approximate expressions for the high order modes can be derived by recognizing that only a few terms contribute S;ignificantly to the determinant. To illustrate, we assume M,N to be large integers and set 2 2 2 2 a2 = M y + 7N + 6 (M odd, N even) (5.59) where 3 >> 62 7N >> 6 (5.60) Then the only significant contribution to the determinant arises for s = n = N, m = M, and the eigenvalue equation (5. 58) reduces to ae b0 1 (5.61) NM MN 2 Now, to a first approximation in 6 we have, upon expanding (5.51), XN (b) [1+ 2 2a2M(a) + } e _ M (a) [( ) + ika (n(5.62) M(a) [1+ 2 2bXN (b)+} 2 N, - 2, { thus, in view of (5.61) and (5.62) we see that 6 is specified approximately by a simple algebraic equation. For I >~ k, > ~ k and ka> 1, kb> whichimplies (k/a), (k/b) we kb > 1 which implies ' > > (k/a), 7N > > (k/b) we have 60

1492-1-F 2 -4k (y +j ) 42 2 1' 2 2 (NM + M ) - 2i (Nr- r ) (kaN + kb M ) TN M r) Y (5.63) 2 2 When = 1 note that 6 -4k. Also, when ka = kb = 1 2 -4k 6 ' - 1 1-2i(r)- ) lrl (5.64) and one may expect the technique to be valid for the low order modes in the case of resonance frequencies ka ~ 0(1), kb O(1). Increased accuracy can be achieved by allowing more terms in the determinant to enter. The remaining pairs of modes are treated similarly: eo -ihz~ Ae Xn(b)Xn(y) cos/ x E=e A n Yn ik Y n ~sin a+ r1 cosp, a n n r~ n oe -ihz e Tm(a)m( x)cosvY E=e B( v sinv b+ ikosv m m m r1 m oe -ihz o Xn(b)Xn(y)s in x E =e A Y n ik e n nCos na- - sin/r a Eeo -ihz j Bo m(a)m()sinv y x m, ik m cosV b-isinv k m m m rl m Ae e y o o AnanmnY)~ (a)l I (x) _ -ihzn m emn( ) -b z! 2 ' (a) n m n - =e-o Be mbemnm(X)Xn(b)yX(y) ) Y 2 (a) m n nI(m (5.65) -ihz AnanmX n (Y) n 2 (b) n m mXn -ihz e e Bb0mb0 (x>X(b)x(y) e -i'hz m Bm mnl rn'x 2 rn m 7n m(a) (5.66) 61

1492-1-F E~~0 -ihz Ao Xn(b)Xn(y)sinsmx E =e A y n ik n n cOa~na- - sin.na Eee -ihz e e m(a)Im(x)cosv vy e B - mv sin b+i cos k m m m 'r m -ihz n nmXn(Y) ( m ( n m mXn n(b) i e o Beb mnm(Xn)v (b)xY(y) =- e 2 nz Y (a) m n n m (5.67) where a, b are as defined in (5.51) and nm mn 2 2X (b)Pn(P sinpn a+ik-cosm a) o n n n nnn a nm 2 2 ik n ( an )(uncos/na- sinz, a). m rn (5.68) 212 (a) v (v cosv b-iknsinv b) be m m m rn b _= mn (y2 2 ik m (v2-72)(V sinv b+i cosv b) m n m m qO m The uncoupled boundary concitions are again automatically satisfied, whereas the coupled relations demand 0 Be - ~ Aeae (m odd) rn n nm n (5.69) O e Q e e A =- Be be n m mn m e B =- A a m m n nm n e A= - BO bO a m m mn m (n odd) (m even) (5.70) (n even) 62

1492-1-F O B= a (m even) m n nm n (5.71) e A B be (n odd) m so the eigenvalue equations become (ae be - 6n = 0 (n, s odd) sm mn s b (n b0 - 6 = 0 (n, s even) (5.72) sm mn s rn L (aO be - 6n) =0 (n,s odd) $sm mn s m 0 e 0 be In the asymptotic approximation a a, b b so that the approximate 2m m nm mn mn expressions for 6 have identical forms for all mode pairs. The result of the above analysis is that for each high order mode characterized by a pair of large integers M, N, the unknown eigenvalue h can MN be determined approximately and is given by 2, 2 2 2 h " -^k _ - -62 ImhM < (53) MN M N MN(5.73) 2 where 6 appears in (5.63). In the case of resonance frequencies such that ka 0(1), kb 0(1), the asymptotic expressions are expected to yield good results even for the low order modes. For other frequencies, increased accuracy can be achieved by retaining higher orders in 6 and by including more terms in the determinant close to s = n = N, m = M. The technique is basically a truncation procedure used in conjunction with asymptotic analysis. 63

1492-1-F As such, iteration methods can be used to advantage in order to gain more accurate evaluations of the eigenvalues. Once the eigenvalues are known, then the field expressions for the modes are determined to within an arbitrary multiplicative constant. 5. Orthogonality of Vector Modes In order to determine the fields excited in the duct by the incident plane wave, we need to know o know the orthogonaliy properties of the electromagnetic modes sustained by the waveguide structure. We begin with the reciprocity statement V ~ (EI ^ I' - E'^ II) = 0 (5.74) which we write in the form aA V- (EAH' - E H) + -.(EAHI -E'AH) = 0 (5.75) where v =V -z (zV) ihz ih'z Now assume (E, H), (E', TI') have z-dependences e, e, respectively, then V. (EAH' - E'H) + i(h+h') Z(EAH' - EA H) =0. (5.76) t Integrate over the guide cross section and use the two-dimensional divergence theorem fV A ds = (AzA).(AA) dc = - j n Adc, (5.77) o fC C to find 64

1492-1-F AI Vt (EAH' -E'AH) dS t - -_ [Z^(EAH' -E )] (nA z) dc (5.78) [(z. H') E-(z. H) E'+ (. E' ) H-(.) H'].(^A) dc. However, the boundary conditions on C are (za.E) = -r H-(nAz) *E.(nAA) = r1 (I. II), E n z,,., (5.79) and thus (A I')[E ( )] =(. II) [E'. Az)], (5.80) ( E') [ (nA z)] = (E) [H '. (nA z)] Therefore V, (EAH' - E'AH) dS - 0, and when (5. 81) is used in conjunction with (5.76), we find F. (5.8 ) J z.EAH' -E'AH) dS= 0 if -h/ h' -A Associated with the mode (El ' E'; H' H' ) e ther ' Z t' Z (E' -E'; -H' ie-ih'z; therefore, we have j.(EAH'T+E'AH)dS = 0 if h h'. A (5.82) e is also a mode (5. 83) Adding the two orthogonality relations (5. 82), (5. 83) gives 65

1492-1-F.(EAII1) dS = 0 ifh2 2 (5.84) This relation can be used to determine the modes generated in the duct. Note also that the orghogonality discussed here is valid for uniform waveguides of arbitrary cross sectional shape. For other treatments of orthogonality see Marcuvitz (1954), Kino (1955), Bresler and Marcuvitz (1956, 1957), and Bresler, Joshi and Marcuvitz (1958). 5.7 Determination of the Scattered Field To obtain the scattered field due to modes reflected from the termination of the duct, we shall employ a Kirchhoff approximation. We begin with 7P E(x) = [(AE)AV' G +(-E) V'G+ik(nAH) G] dS', (5.85) H(x) = f [(^HA'G+(nH)' G-ik^_E) G] dS' A where the normal vector n points into the region of interest and ikR G(x, x 7) = R =x- -X. (5.86) Take S = A+S1 where A is the aperture formed by the mouth land S is the outside surface of the duct. We now assume that the currents on S1 due to modes reflected from the rear of the duct are zero to a first approximation. The surface integrals thereby reduce to integrals over the aperture A. In the far zone we then '-.ave Es8f) - e~ kA r(n^AH)A\k+kA(E)] e -- dS1, rikr A (5.87) ikr -. HS(x) 4ekr k AE Ak -k- AH) e1 -- dS' 47nikr JL" EAkk A 66

1492-1-F where k = k(sin0 cosO, sinO sin0, cos0) is the wave vector of the radiated wave and = z. To complete the Kirchhoff approximation, we take the field on A to be comprised of the modes reflected from the termination of the duct. Physically, the incident field excites modes within the duct which progress toward the termination of the duct. These modes are then reflected from the closed end, return to be reflected from the open end, are again reflected from the closed end, and so on. As a result,oscillations are set up inside the duct, although for each reflection at the open end, energy is radiated into the surrounding medium. In general the modes are damped exponentially as they progress along the axis of the duct, and since we are considering the length L to be large, it will be sufficient as a first approximation to take into account only the first reflection from the termination. The amount of energy launched into each mode can be obtained by matching the normal mode expansion at the mouth to the aperture fields derived in Section 5.1. To accomplish this we use the orthogonality conditions derived in Section 5.6. For simpliication of notation, we shall attach a subscript o to denote a normal mode, where in general a may be a multiple index that accounts for the parity of the mode in the x, y variations and also tabulates the doubly infinite set of integers M,N which order the eigenvalues h. In addition, the z dependence is separated out by representing the transverse electric and magnetic fields for a mode as -ih z Et=E (x,y) e t -- ih z (5.88) H =H (x,y)e t -(0 where H (x,y) may be obtained from E (x,y) through Maxwell's equations: -a -a 2 H 1 h2) E + x1 Hy= hk 2 + axay 67

1492-1-F Alternative representations for E (x, y) may be obtained from Section 5. 5. 1 — A A It may be noted from (5. 87) where n = z that the z components of the electromagnetic fields in the integrals are of no consequence in calculating the scattered fields. The transverse fields ( A E) and (Z A H) are evaluated at the aperture formed by the open end of the duct and are represented by normal mode expansions: -2ih L e U (ZAE) A ~(E AAI) dS A — — A -A' zAE - A (5.90) -2ih L A UA e factor exp(-2ih L) in (E fact dS ZAH Z. (E A ) dS modes have traversed the length of the duct and returned to the mouth. Since Imh < 01 this factor means that the summations in (5. 90) may be effectively truncated, for exa, -pple, at some a for which Reh = 0. The minus sign in A the expression for (zA E) accounts for the fact tbat the electric field is reflected by the termi.a.ting plate with reflection coefficient (-1). The scattered field due to the reflected interior modes is obtained upon employing (5. 90) for the integrands in (5. 87). 68

1492-1-F APPENDIX A PROPERTIES OF THE FUNCTIONS QR() AND I(3) The meromorphic function /r( 3) is defi:- d as () = exp - f - (A. 1) 1 ', r sin v-2/ V,rsin (v/2) + 2v dv (A 1) C(, p= x ST Jo S cosv dv from which it will be observed that 07(3) is an even function of 3 whose logarithmic derivative is given by __ I sin 0 + V2sin(8/2) _ 1 (A.2). ( S) c8 os 4 cos 3 47r cos By means of the elementary integrals ' si ydv = -In (cos 3), o0 COS V J sin(v/2) dv — In/2 cos(0/2) - 1 / + 1I Jo cosv - /2cos(3/2) + 1 N/2 - -1 we obtain the following alternative representations for 0(P):.( f /2-cos(3/2)+ l_ 1 dv 1W -ri -- | - (A. 3) N/2 + 1 (cos ) exp 4rJ o '.(3 = [ Vcos(0/2) + 1 ]x Jr sin v 2v (A. 4) V2+1 exp ro cos V When 1 < (7r/2), the integral in (A. 3) can be expanded as Jo cos vy 2 2 + '4 + 46 720 8 (2n)! 2n + 2+ ( where E2n are the Euler numbers. When = i oo, we have (Orobner and Hofreiter 1949) 1 wc vdv 1 rCdx f2 d (A. 6) osv Jo C. (2n + 6) r Co v cosh x -_ (2, + " -. 69

1492-1-F where b = (2/7r)K with K = 0. 9159656... (Catalan's constant). On the other hand, vhen = (7r /2), we employ (A. 4) with a change of lintegration variable v:(t/2) -u to find 2,(7r/2) = (A. __ /+1 e 8 siln u 4 ro sin;The first integral is elementary and the second integral is (Grobner and Hofreiter 1949) 1 r/2 (-d 9 o sin T 2+ ) = b. (A. 8) 7i *o sin u 7r nE.O (2n + 1) We obtain then ',(7r/2) = Cb/2 (A. 9) 2 I[v22 1 It is easy to verify the following fundamental identity (Maliuzhinets 1958) '.(a3 + (r),a - r) = [ [,(r/2)]2 cos (/4), (A. 10) and by successive application of (A. 10) one obtains, (I + ) -. - ( [i( [7 cosG(/2) + cos(7r/4)], (A. 11) ( +32= [O(/] /) (A. 12) cos(3/4) From this last equation we observe that the zeros of -7r(3) which are closest to the point 3 = 0 and the corresponding poles are the points (= + (57/2) and 3 = + (7?r/2), respectively. From Eq. (A. 10) one also derives, r) -COS + ) (A. 13) 4P, ( 73+4r) _cos (-I+Aor) (A.1s).(3 - ir) - cos(? - r) ) + ) cot(-.l + r). (A. 14) q,.( - 2ir) The function O(3) is expresses in terms of the function 7r.(I) by the product: (,W) = (0 + it + ai',).( + T - ai),( - T - a2)X( -r + a2). / (A. 15) from which, by means o; (A. 14), wc derive 70

1492 -1-F 7(7) W+aI+27r)Viir(I3-aI+27r) ~-~8- 7r ILIBa - 4-27r) B-r W ---2r) 1 C 1 1 1 i Cos oA-si.16) Lcot 2a ~1+Vr~COt(Zg-2 al 47n) = a +Bing 1 and similarly rL(-~-P )Cos a 22s in (A. 17) ~-~-iP)cos a2+s inS Another identity of interest may be derived through the application of (A. 13); in particular, ( )-C[cOs (. Q;o (-) [c2 ')+Coso + ) ~,&(7r-I3 2 - 24(A, 18) osi)+cos(+)] Lcos( + )os(2U 2 4 2 2f 4 2 from which it follows that V(7r-13 -L(-7r-/) sinl+cosa1 sinl3-cos a2 ~7-J - sein3-os ar1 sin(+os a (A.19) in agreement with the results in (A. 16) and (A. 17). If a1=a2 (in which case the two surface impedances are equal), Eq. (A. 18) reduces to IL~nB) IlL(n-tS.(A. 20) In this instance rt'4) is related to the "split" functions 1(4k cos S) and L4+k coo of Senior 1952) as follows: Cos3/2) i 4 1 L(kcosp =( Ycos a,' wL'cos. (A. 21) L+kcosP=T h2 F0 8( T]4, with cos a, r? where the constant rn represents the complex impedance of the Balf-plane. Senior's functions may be do. d by the equations 71

1492-1-F K+ (k cos )K (-k cos,)= (A. 23) L+(kcos) L+(-kcos 3) (A. 24) whereas I3) with a =a2 satisfies the relation mploy (A. 10) and (. 12i1 2 ( 2 8 3) ( o+cos ). (A.25) It is easily verified that (A. 25) is consistent with (A.21) through (A. 24). For a1ma2-(ir/2). corresponding to a perfect conductor with H polarization, we have 0)-1 ' r (2) cos (0/2); (A.26) on the other hand, for 1=a2 -> t 1 o, corresponding to a perfect conductor with E polarization, one finds asymptotically (7 4 12 /2 (A.27) These limiting expressions are valuable when checking the results of calculations against known perfectly conducting results. 72

1492-1-F REFERENCES Bowman, J. J. (1963) "Scattering of Plane Electromagnetic Waves from a Semi-Infinite Hollow Circular Pipe (Axial Incidence)" Memnorandum D0620-125-M41-10; "Scattering of Plane Electromagnetic Waves from a Semi-Infinite Hollow Circular Pipe (Off-Axis Incidnlcetj" Memorandum D0620-127-M41-13; "Scattering from a Semi-Infinite QuarterSector Duct Bounded by Two Infinite Planes," Memorandum D0620-129 -M41-16; "On the Vajnshtejn Factorization Functions" Memorandum D0620-135-M41-20, Conductron Corporation, Ann Arbor, Michigan. Bowman, J. J. (1967) "High-Frequency Backscattering from an Absorbing Infinite Strip with Arbitrary Face Impedances," Canad. J. Phys. 45 (2409-2430). Bowman, J. J. and V. H, Weston (1968a) "Reduction of Radar Cross Section of Ducts, " The University of Michigan Radiation Laboratory Report 1492-1-Q. Bowman, J. J. and V. H. Weston (1968b) "Reduction of Radar Cross Section of Ducts, " The University of Michigan Radiation Laboratory Report 1492-2-Q. Bresler, A. D. and N. Marcuvitz (1956) "Operator Methods in Electromagnetic Field Theory, " Research Report R-495-56, PIB-425, Polytechnic Institute of Brooklyn. Bresler, A.D. and T. Marcuvitz (1957) "Operator Methods in Electromagnetic Field Theory," Research Report R-565-57, PIB-493, Polytechnic Institute of Brooklyn. Bresler, A. D., G. H. Joshi and N. Marcuvitz (1958) "Orthogonality Properties for Modes in Passive and Active Uniform Wave Guides," J. Appl. Phys. 29 (794-799). Clemmow, P. C. (1951) "A Method for the Exact Solution of a Class of TwoDimensional Diffraction Problems," Proc. Roy. Soc. A205 (286-308). Einarsson, 0., R.E. Kleinman, P. Laurin andP.L.E. Uslenghi (1966) "Studies in L:ar Cross Section L-Diffraction and Scattering by Regular Bodies IV: The Circular Cylinder," The University of Michigan Radiation Laboratory Report No. 7133-3-T. Felsen, L. B. and H. Y. Yee (1968a) "Ray-Optical Techniques for WVaveguide Discontinuities," IEEE Trans. AP-16 (268-269). 73

1492-1-F Felsen, L. B. and II. Y. Yee (1968b) "Multiple Inlraction by Ray Optics -- Some Observations and an Example, " IEEE Trans. AP-16 (360-362). Fclsen, L. B. and H.Y. Yee (1968c) "Ray Method for Sound Wave Reflection in an Open-Ended Circular Pipe, " J. Acoust. Soc. Amer. 44 (1028-1039). Grobner, WV. and N. Hofreiter (1949), Integraltafel Springer-Verlag, vVien and Innsbruck, Austria. Iijima, T. (1952) "On the Tlici llii:i lic Fields in Case of Existence of a Semi-Infinite Hollow Conductive Circular Cylinder II, " Electrotechnical Laboratory, Agency of Industrial Science and Technology, Tokyo, Report No. 531 (in.iT, L.l I.: i). Jones, D. S. (1952) "A Simplifying Technique in the Solution of a C:Lass of Diffraction Problems," Quart. J. Math. 3 (189-196). Jones, D. S. (1964) The Theory of Electromagnetism, The Macmillan Co., New York. Kino, G. S. (1955) "Normal Mode Theory in Perturbed Transmission Systems," Technical Report No. 84, Electronics Research Laboratory, Stanford University, Stanford, California. Levine, H. and J. Schwinger (1948a) "On the Radiation of Sound from an Unflanged Circular Pipe," Phys. Rev. 73 (383-406). Maliuzhinets, G. D. (1958) "Excitation, Reflection and Emission of Surface Waves from a Wedge with Given Face Impedances, Soviet Physics - Doldady 3 (752-755). Translation of Dokl.Akad. Nauk. SSSR 121 (436-439). 1958. Maliuzhinets, G. D. (1960) "Das Sommerfeldsche Integral und die Lsung von Beugungsaufgaben in Winkelgebieten, " An.;. Physik 6 (107-112). Marcuvitz, N. (1954) "Field Representations in General Cylindrical Regions. I. " Research Report No. EM-69, New York University. Morse, P.M. and H. Feshbach (1953) Methods of Theoretical Physics, Vol. 1 and 2, McGraw-Hill Book Co., Inc., New York. Noble, B. (1958) Methods Based on the Wiener-Hopf Technique, Pergamon Press, London. Pearson, J. D. (1953) "Diffraction of Electromagnetic Waves by a Semi-Infinite Circular \Waveguide," Proc. Cambridge Philos. Soc. 49 (659-667). 74

1492-1-F Ross, R.A. (1967) "Investigation of Scattering Center Theory," Technical Report No. UA-2462-E-2, Cornell Aeronautical Laboratory, Inc., Buffalo, New York 14221. See also R.A. Ross and H.R. Witt (1967) Interim Technical Report No. UA-2462-E-1, Cornell Aeronautical Laboratory. Senior, T. B. A. (1952) "Diffraction by a Semi-Infinite Metallic Sheet," Proc. Roy. Soc. A213 (436-458). Senior, T. B. A. (1959) "Diffraction by an Imperfectly Conducting Half-Plane at Oblique Incidence, " Appl. Sci. Res. B8 (35-61). Siegel, K. M. (1959) "Far Field Scattering from Bodies of Revolution, " Appl. Sci. Res. B7 (293-328). Vajnshtejn, L.A. (1954) "Propagation in Semi-Infinite Waveguides, " NYU Research Report EM-63, Institute of Mathematical Sciences, New York University (six papers translated from Russian by J. Shmoys). The six original papers are: (1) 'Rigorous Solution of the Problem of an Open-Ended Parallel-Plate Waveguide," Izv. Akad. Nauk. SSSR Ser. Fiz. 12 (144-165) 1948. (2) "On the Theory of Diffraction by Two Parallel Half-Planes," Izv. Akad. Nauk. SSSR Ser. Fiz. 12 (166-180) 1948. (3) "Theory of Symmetric Waves in a Cylindrical Waveguide with an Open End," Z. Tehn. Fiz. 18 (1543-1564) 1948. (4) "The Theory of Sound Waves in Open Tubes," Z. Tehn. Fiz. 19 (911-930) 1949. (5) "Radiation of Asymmetric Electromagnetic Waves from the Open End of a Circular Waveguide," Dokl. Akad. Nauk. SSSi 74 (485-488) 1950. (6) "Diffraction at the Open End of a Circularly Cylindrical Waveguide whose Diameter is much Greater than Wavelength," Dokl. Akad. Nauk. SSSR 74 (909-912) 1950. Wait, J.R. (1955), "Scattering of a Plane W: -e from a Circular Dielectric Cylinder at Oblique Incidence," Can. J. Phys. 33 (189-195). Weston, V.H. (1963) "Theory ox Absorbers in Scattering," IEEE Trans. AP-11 (578-584). Williams, W. E. (1960) "Diffraction of an Electromagnetic Plane Vrave by a Metallic Sheet," Proc. Roy. Soc. A257 (413-419). 75

1492-1-F Yee, N.Y. and L. B. Felsen, (1967a) "Ray-Optical Calculation of Reflection from an Open-Ended Waveguide, 1 Spring URSI Meeting, Ottawa, Canada, (May). Yee, H. Y. and L. B. Felsen (1967b) "Ray-Optical Calculation of Scattering by Waveguide Discontinuities," IEEE G-AP International Symposium, Ann Arbor (October). Yee, II. Y. and L. B. Felsen (1969) "Ray-Optical Analysis of Electromagnetic Sca:tterhing in Waveguides," IEEE Trans. MTT (.o be published). Yee, H. Y., L. B. Felsen and J. B. Keller (1968) 'TRay Theory of Reflection from the Open End of a Waveguide, " J. Soc. Indust. Appl. Math. 16 (268-300). 76

I L i I I I TUNCLA SSI PI"'] Ii) S<,i(rly C( 'I it < I I - II Ii(on DOCUMENT CONTROL DATA. R & D ( lafiy l.; 4 '/ /Ie,,,1,i 1 II.;1. n l l. l y...1 1.. I guiy I. * d *.. n. fNl I i.I -W it whl n n l ia 'rnll rf. tnr In. a li.) j (i,,irLA TINO AC rIVITY (t,)oerpotelf ieflhirl) a, H4IPOR T,CU nl Y C L A 5 CA TION The University of Michigan Radiation Laboratory, Dept. of NCLASSIFIErD Electrical Enginccring, 201 Cathorino Street,. OUP Ann Arbor, Michigan 48108 -P I HLP* OP T TIT L DUCT STUDIES -f *. O lCICPTIVC NOTIIe (7TYp of rporl nnd Inclialvo ldnlo) Final Rel)ort (22 January 1968 - 1.May 1969) ei. AU TtOqt*ll (-"lrfi name, miUddlo lefllal, lost1 iuamn) John J. Bowman and Vaughan H-. Weston '- HLPORT OATL 71,, TOTAL NO. OF PAOQI. 7b. NO, Ot rCFls 6 May 1969 76 36 "a. CONTIRACT 014 GRANT NO. 'u. O'IOINATOR'S II.POIT NUMO.;II.ll P.O. 504 855029 1492-1-F b. POJIKC T NO. F336 15-67-C-1843 e.;lh. OTHER tiPORT NOIS) (Any o#lle t tor mtmhb Ihol mily bte I*lilnol / IlEla rvporl) d. eo. 0,1T onuTlON STATCMcNT II.-.UPLEMENTARY NOTES 12. SPONSORING MILI TARY ACTIVITY Northrop Corporation - Norair Division Hawthorne California I). ADuTtACT I Problems in the high frequency regime of electromagnetic scattering by ducts and open waveguides whose walls obey an impedance boundary condition are investigated. In general, the impedance boundary condition means that the walls are absorbing electromagnetic energy. Three structures of particular interest are treated: the semi-infinite parallel plane waveguide, the semi-infinite circular cylinder, and the rectangular duct of finite length. When the surfaces are perfectly conducting, exact solutions for the first two structures are available, and these solutions are expanded asymptotically to yield term-byterm comparisons with the corresponding results of ray diffraction theory. When the surfaces are absorbing, exact solutions are not available, but the ray treatment is applied to advantage. In the case of the rectangular duct, the main problem is to determine taie interior waveguide modes. This modal problem is non-separable due to the assumed impedance boundary conditions; nevertheless, information can be obtained by means of an asymptotic analysis applied to the transcendental eigenvalue equation of infinite order. I nn lFOPM FA7'R - - i NOV 6aI 'l / J UNCLASSIFIED St'curiry Ci l Hi'lii'llonI

UN CLi- \SSI FI E'-)_________ S -,rli, 1 4 LIN1K A LINI'. I) L INK C K F9? WO1HDS 0LL IIROL T w FI 0 L T- i 6T _ _ _ _ _ _ _ _ _ _ _ _IIIt k Ducts MIode s Radar Scatteringr Ray Optics Albsoi-bers I I I I I i 11.1 d i I I K L- I. — - I — II UNCLASSIFIED ___ w iS atiir i t% - _____ I