1492-1-Q THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory REDUCTION OF RADAR CROSS SECTION OF DUCTS Interim Report No. 1 22 January 1968 - 31 March 1968 By John J. Bowman and Vaughan H. Weston 6 April 1968 1492-1-Q = RL-2033 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, 1492-1-Q, "Reduction of Radar Cross Section of Ducts", was prepared by The University of Michigan Radiation Laboratory, Department of Electrical Engineering, ider 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 First Interim, covers the period 22 Jan - uary through 31 March 1968. i

142-1-4 ABSTRACT As a preliminary to an investigation owcerning radar scattering by a rectangular duct lined with absorbing materials, the simplified problem of plane-wave scattering by a semi-infinite parallel plane waveguide Is omsldbrod. The surfaces of the guide are assumed to obey impedance boundary conditions, where the impedance on the interior surface may be different from the impedance on the oter'ior surface. A case of particular interest would be that in which the exterior surface is perfctly conducting. Ray-optical techniques based upon known results for a semi-flinite screen with two face impedances are employed to calculate both the field scattered into the far zone and the field generated at the mouth of the guide. The ray optical procedure is also applied to the case of a perfectly conducting, open-ended parallel plane waveguide for which an exact solution is available, and the ray optical result is compared with the asymptotic expansion of the exact solution. ii

14T2-1-Q TABLE OF CONTENTS I m IV INTRODUCTION THE HALF PLANE THE SEMI-INFINrTE PARALLEL PLANE WAVEGUIDE PERFECTLY CONDUCTING CASE - RAY OPTICS VS EXACT SOLUTION 1 3 8 13 17 20 APPENDIX REFERENCES ID FORM 1473 21 iii

I INTRODUCTION The purpose of this contract is to nvestigate the radar cross section of an air intake duct similar to that found 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 hy the termination of the Ast sad rHradiated back toward the radar transmitter. The opening of the duct is taken to be rectangular, although the cross sectional shape gradually chnes to circular at the rear of the duct. The tasks of the contract were therefore, divided into two parts; the first being to calculate the energy launcked into the duct and its consequent reradlation 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 shall employ 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 studied the two-dimensional problem of a plane wave incident upon a semi-infite 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 are 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_ int mately dependent on the known solution for plane wave diffraction by a half plane with arbitrary face impedances. At the present time, it is n extra complication to dr th tide surfaces of the guide as being governed by a separate constant surface impedance. The ray optical methods are also applied to the case of a perfectly conducting parallel plane waveguide for which an exact solution is available, the motivation being to compare the ray optical result with the asymptotic expansion of the exact solution. 1

142-l-Q For application to a rectangular waveguide, th results nost be ehrtded to include the ase of oblique incidence on the edges ng the waveguide opening. This aspect of the problem is presently under investigo At the same time, the waveguide m es exsting in a rectangular waveguide with an impedance boundary oondition are being investigated. The orthogLoality properties of these modes must be determined in order to match the field in the gide to the field p rated at the mouth. 2

14B2-l — II THE HALF PLANE Before we consider the problem of diffraction by a mi-infinite parallel plane waveguide we must first examine the pertinent half plane results. To fix our notation, we shall employ natural units with ree-space constants o, o set equal to unit and suppress the harmonic time depen ce exp(-iw) throughout. A plane electromagnetic wave of unit amplitude is assumed incident at an angle 0 to the semi-infinite screen as shown in Fig. 1. The screen is assumed to be comprised of material of such a kind as to make the total tangential field components satisf the following impedance boundary condition on the surface ( / is the unit outward normal to the surface) E-(-n E) n = n^A, (1) where 77 = 71 on the upper surface and 7 = 77 on the lower surface. The face impedances n1 and 772 are complex constants whose real parts, because of energy oonsiderations, must be non-negative. Further, the surce impedances are assumed to account for the presence of thin layers of highly refractive absorbing materials applied as a coating on a perfectly oonducting 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 emerge when the overall effedct of an absorber coating is treated in terms of a constant surface impedance (see, &.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 oducting half plane coated on one side with radar absorbing material. The exact solution for plane-wave scattering by an orbing 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 3

1492-l-Q Ii I FT, ".:j '... i I ~ t l '., FIG. 1: PLANE WAVE INCIDENCE ON ABSORBING HALF PLANE WITH TWO FACE IMPEDANCES application of ray-optics techniquee, we sball require t asymptotic far-field form of the solution. This result, obtained by means of a steepet decent approximaon to the exact contour integral solution, may be written in the form u acp [i(kr.- |) U(e, 00) (2) representing a cylindrical wave emanoting from the edge of the semi-infinite screen and produced by a plane wave incident at an angle 80 to the screen (see Fig. 1, where the geometrical configuration is different from that used by Maliuhlnets). The total exact solution, u, which is a function of two oomplex quantities a and a, is derived by impos dary o the thrd by imposing boundary conditions of the third kind, 1i - u cOs i a- uco asla =0 (8 = 0, 2), (3) on the faces of the semi-infinite screen. The quantties a and a2 are constants whose 1L 4

real parts lie the closed Interval Eo, /2). Por *bsrtna nes bxu-aded away from the geometric optics boundaries (i e. for 0 r ~ e, the amplitude factor u(e, e9) appearing in (2) is given by sin(U/2J 0) ^ "A U(O. 0,+ 0(2x o (4) (ef g+o)= r~ ( s-n(( 2))+-2 -+) 2 ) (/ -2)} where the auxiliary function 0(b) is expressed in terms of a special meromorphic function () by the product 0(9* (hr+al)o7l (HO -al)I (-ir-a 2)07r((_r_2) (5) and p (0 hs the representation,()=, ep i 1 -i rainv-2J saf(v/2)+2v } (6) W- - ^- (6) Malluzhnets (1960) mentions that the special function '1y(), along with its generalization for the wedge problem, has been tabulated by M.P.haowa, although no reference to the literature is given. The important analytal properties of the functions g(p ) and ) are given in A-ie*A. An examination of the boundary conditions (1) and (3) indicates that the scalar function u may be employed to represent the half-plane solution for either of the two fundamental electromagnetic polarizations. Ths, n the case of H polarization, the function u represents the z component of the gnetic field Hz, provided we make the identifcation 1 - cos a, co o (7) while in the case of E polarization, it represents Ez under the identifications 771 l/cos a1, r2 l/cosa2. (8) This propert is a result of the natural duality of Maxwells equations and the impedance boundary condition (1) under the transformation E - H. H-p, -E, r-f- 1/r7 5

1482-1-Q An alternative representation for U(, o) may be obtained by aaucessive use of the identity given in (A. 10); in particular, ar troin nat cr-c wet>tain the following expression: sin a i0 a2 a1\ e 00o sin join Ar- r M< ---------- COB -+OSm-coo --- —------- r 2 2O (2 ( 2( U( (r) -0e(ir-8o) o e + oos o (9) which is valid so long as o $ r + o0. The quantity, (r/2) is given in (A. 9). The expression in Eq. (9) is manifestly symmetrical in the two variables 0, 00, thereby confirming the principle of reciprocity: U(, GO)=U(Qo, 8). For a1=2=(r/2), corresponding to a perfect conuctor with H polarization, e have upon enmplyng 1 (A.) and (A.15) i-(r-),= (r)]4 in and therefore (9) reduces to U(0e e = 4 cos(/2)co(.8o/2) (10) - tho oOSr os On the other hand, for a1=a2 + i a, corresponding to a perfect conductor with E polarizaton, one finds asymptotically,r 4 1a11/2 ^-e}L[ ( )]e so that (9) now reduces to 4 sln(8/2)sin(o0/2) U(o, )- ocos (11) These limitng expressions are in accord with knwn results for a perfectly conducting half plane. In order to explore the effects of multiple interaction between the two edges forming the mouth of a semi-infinite parallel plane waveguide, we shall also require an asymptotic 6

1482-1-Q expression for tbs half-plane solution alcg the rayoptics fecion boundary 0=7r-00. It may be shown from the Maliuzhlnets 1958,1960) ootour Integral solution that for 0=7-0o the total field behaves as ikrcos2o 20 oosr a -sino \ u-e r 0 - (12) 2 oosa f-tno provided 0 < 00 < r. For incidence from the lower half paoe < < < 2, the same epression obtains except that is replaced by 2. The re t in (12) is hardly surprising physically and shows that far along the reflection dary the scattered field is given by the perfectly onducting result multiplied by the nfinite flat plane reflection ooefficient. However, to apply the ray-optical procedure to the emi-infinite parallel plane waveguide, we need to know the field generated alogg the reflection boundary by a line source located at a finite distance from the semi-infinite screea. Rather than solve the line source problem for an absorbing half plane, we shall draw upon an analogy (in view of Eq. 12 ) with the known result for an iotopic line source over a perfectly conducting half plane. Thus, for an incident field given by i (1) 2 u= H (k Ir-rJ )r= (13) it is physically reasonable to take 1 cos a -sino -- i-ro)-i4r14 2 ooaal+sink rk(r+rd) as the scattered field far along the reflection bomdary 9=r-%, where 0 < 0- < r. For 7r < 00 < 2r, replace a1 by a2 in (14). 7

14-1I-Q &THEI7SEMI-INFINITE PARALLEL PLANE WAVEGUIDE The semi-infinite waveguide to be considered here consists of to parallel half planes specified by y = + a, x > 0, as shown in Fig& 2, where polar coordinates r, 9 measured from~ the center of the guide are also illustrated. The upper half plane is referred to as (T) and the lower as (), and th distanes from edges ( and ) to the field point (r, 9) are respectively denoted as ri and r2. The Interior surfaces of the waveguide are governed by a constant surface impedance AI, while the exterior surfaces are governed by an impedance i77 which in future applications will generally be taken as zero. Finally, a plane wave is assumed incident upon the structure from the direction Oo, where (r/2) < 0 < r. We shall employ the ray optical procedure to calculate first the field scattered into the far zone by the edges forming the mouth of the wavegulde, and then the field generated at the mouth of the guide. The latter field will be required in the future when we consider the case of a rectanglar waveguide. In particular, an estimate of the field generated at the mouth will play an important role in obtaining the energy launched into the waveguide modes. For waveguides of finite length - such as air intake ducts of aircraft - a portion of the energy carried by the waveguide modes will be reflected back toward the mouth and reradated into the far zone, thereby making a contribution to the radar cross section of the duct. The primary diffraction due to the open-ended parallel plane waveguide is obtined 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 rl, r2 are approximately r + a sin 9 as r - oo, and takng 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: 8

!1492-1-Q (r, 0) IG. 2: PLANE WAVE INCDENT ON A SEMI-INFITE PARALLEL PLANE WAVEGUIDE WITH TWO SURFACE IMPEDANCES. \ ~ na FIG. 2: PLANE WAVE INCIDENT ON A SEMI-INFINITE PARALLEL PLANE WAVEGUIDE WITH TWO SURFACE IMPEDANCES. 9

1402-1-Q Ti l (kr-4V) 1 -ika(sn+sina0) ka(instid U 2 U8 4 g e e U(O8, + U(2r-, 2.-o) J (15) where (r/2) < (8, 0)< r. It may be noted that the Emction U(2xr-, 2r-8d properly accounts for the fact that the lower plate Q is illuminated on the side rather than on the ri7 side as is the case for the upper plate Q. As a special instance of (15), the backscattered field (0=00) is BS 1 (kr-) -2ikasinO 2ikasin ) u B4 r! e e U(8, 0)+e U(2,-8, 2vr-0) (16) The immediately preceding approximations are valid so long as ka~ 1>, so that the interaction between the edges is weak. However, for more closely spaced waveguides, it may become 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 ths 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, on the basis of Eq. (2), to provide the correct diffraction field in the direction toward the other edge. Kow 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 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 hlf plahe (. The field 10

1492-4-Q scattered back toward the line source- along the reflection boundary of plate ( - is then (in view of Eq. 14) given by coa2-1 ik(r2+2a)-i "u '^'2 cosa2+1 k(r2+2a) e ' and this cylindrical wave appears to emanate from an image line source located at a distance 2a behind the half plane ~. From this erample, 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 multiple interactions. Forsaking-farther details, we shall merely present our final result for the field scattered into the far zone by the parallel plane waveguide: Ne U(6,o)+e:U (27r-,2r-Qo) - i r rE 2m i(2m+1)2ka ika(i-sino) e4 e R2ml )u(, 22o + ika(sin0-sin0o) 3) 3U +e U(2r-O, ()U( j, +. OD 4 R2m-1 i(2m)2ka e-ika(aino )n ) +, 3)IM 2m e2 lka(sin+sin0 0) 3w +e U(2r-O, )U(-, 2,-o) + 0 (l) (18) where cos a2-1 R = -. (19) cos 2+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 to zero. 11

1492-1-Q By a procedure similar to that above, we have also calculated the field generated at the mouth of the waveguide; in particular, for x = O, ly I < a the diffracted field is -ikasine2 ia-y)+2ika(2m) 4xU(2, 20o)...m.l.. ika sin -i r 2 2m-1 ik(a —)+2ika(2m-l) 0-4 U 2r- R e + 4rimm m=o 22m /k(ay)+2ka(2m)' ikasine- 2m- ik(a+y)+2ika(2m-l) 3.ikaini r R eia o) 20) m 22m k(a.y)+2ka(2m-1) maanitude, as expected. to calculate the mutual interaction effects is based upon the simplifying assumption that the equivalent RfSe sources are isotropic. in order to modify the ray optical method described here, we would require the solution to the complicated boundary value problem of a non-isotropic line source over an absorbing half plane with arbitrary face impedances. ince the higher interaction terms are still expected to be very small, especially in the case of highly efficient absorbers, the slight improvement in accuracy for these terms is most likly not worth the labor involved. 12

I -QL IV PERFECTLY CONDUCTING CASE - BAY OPTICS VS EXACT SOLUTION It wouldibe interesting to compare the ray optical description of scattering by two parallel half planes with the exact solution to the problem; unfortunately, a solution for the case where the parallel plane waveguide is absorbing does not seem to be available. Nevertheless, we can make a oompariso in the case where the waveguide is perfectly conducting since in this case exact solutions, based on the Wiener-Hopf technique, are widely known in the literature. Ray optical techniques and their relation to canonical problems with parallel plane geometries involving perfect conductors have been discussed by Yee and Felsen(1967a, b), by Felsen and Yee (1968a, b) and by Yee, Felsen and Keller (1968). 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 optical method yields remarkably accurate results even at small ka values. In this section we shall discuss briefly the results that are obtained in the plane-wave scattering case. No numerical calculations have been carried out, so we shall confine our comments to analytical results. We consider an E-polarized plane wave incident from direction 0 on a pair of perfectly conducting parallel half planes. By combining the results of Vajnshtejn (1948) and Clemmow (1951), we can write the far field as obtained from the exact solution in the following manner: s 1 _ g / 1i0kr-r) 2sin(9/2)8in(90/2)| 2 () V - 2i+e2asinO) +22ikasinao eV+VO + +(ie2iksnG) (12iksi 6o eU+UO } -ika(sinO inO0),(21) 13

I~t-I.. where we have assumed cog 6 < 0OB 80 < O,* aid whr -+ico 2ikcos 'r) c-r dT v Aj' log (+eT 2(22) - O - 2Jkacosm coar dr 2i1i oselr-cos6 ic The quantities VO, U0 are obtindfrom (22) upon replacing 6 by %. Now for ka~ > 1,p a steepest descent approximation Yields (Vajnshtejn, 1948): C2 t2 V.,i lbg(l+e 2iWA d (23) We epand the logarithm under the assmk a vanishingly small positive imaginary part, t2 2ika-Tr H 2imk log -e 1+e -- and integrate term by term to obtain VO- ugn(cosO) e2i G(4m w) (24) where w = ika/12 I coselad zOD G(g) = e 2e e' Cik (25) (1-i)g 14

Forvialues of kca an 9suchthtka/:ffaOBGI~1>, wehbave 4 6 adconsequently (26) (27) 7r U e4ra 'coso 3 2imka CD 1 2imka e +0 m (3k2 (28) When (27) ad(28) are employed in (2 1), we obta — in El'-IV,- 7rir lkr 4 2sin(O 12)sin(901j2) co jka(sine+sino 5 + e cos9+cosGo.ir 4 O + s- ( 1 + 1 cos90 cosO0. 4 ODji e 1 1 0 4rk Coss cos90 & 1 I% e 32r /i2k Cos [ka(sine-sino0](2m)23ka Cos [ka(sinO in%~J + 0 (29) On the other hd, the ray optical result in (18) reduces to ES /'eir142sin(9/2)sin(9~2 o k~lh~~e5 + 1 ~0 ( 1 CD, i(2m-i)2k +~- (.i 080f1) e 2m Cos kgstinO-sin~O)J - cosO cos8a m=o 2 V1 i F ~~CD i2)k - ( ---- SOO 2 — Cos asniB +0 cos9 cos90 m=1 2 m --- icsn9snO" (30) 15

1492-1-Q The two results in (29) and (30) differ because of the numerical factors in the denominators of th summands. It is seen, however, that the primary diffraction, along with the first and second interaction contributions, are in complete agreement. For each successive interaction after the second, the ray optical result (30) uderestimates the asymptotic result in (29) obtained from the ct solution. In order to obtain more accurate results the simple ray optical method employed above would have to be modified. 16

1492-1-Q APPENDIX A PROPERTIES OF THE FUNCTIONS rL(3) AND (.) The meromorphic function 07( B) is definedin Eq. (6) as - { 1 r sin v - 2V rsin (v12) + 2d (A.1) ~.(t~ = exp -c from which it will be observed that ir(I) is an even function of - whose logarithmic derivative is given by '() _!sin + V2sin(0/2) 1 B. (A. 2) 8*( 8) 8cos 1 4 cos t 4r cos P ' By means of the elementary integrals Isin v., sin v dv = -In (cos 8), 0 COSV 2 fsin(v/2) dv=-In /2 cos(P/2)-1 V2 + 1 o os V /2Vcos(8/2) + 1 /2 - 1 we obtain the following alternative representations for 07(O): ( -) F/2Vcos(f/2)+ 1 pIfvdv, (A.3) Tm exp r(A. 3) LV2 + 1 (cos) 4 -.os,-,- - -.. F V( [ cos (/2) + 1 1 I sin v' - 2v (A. 4) 2 + 1 ex r Cos When 1 < (rr/2), the integral in (A..3) can be expanded as I (vdv _ 1 5 +8s 61 8 + (-)"E. +2 5) J.cosv 2 2 4 24 6 720 8+ (2n)! 2n+ 2 ( where -E2n are the Euler numbers. When p = ioo, we have (Grobner and Hofreiter 1949)? fsvd. ' f' xx d -2 0 (-+-1)"): b, (A. 6) ',o cos, ': Jo coshx - "'.o (2n + 1) 17

1492-1-Q where b = (2/7r)K with K = 0. 9159656... (Catalan's constant). On the other hand, when j3 = (r/2), we employ (A. 4) with a change of integration variable v =Or/2)-u to find LVj2+ I 8J sinu du '4Jo sin"uj" The first integral is elementary and the second integral Is (Grobner and Hofreiter 1949) 1 r'/2u d 2 (-1)" 1 Jo siu - X o (2n + I)2 = b. (A. 8) We obtain then (7r/2) = e] (A. 9) It is easy to verify the following fundamental identity (Maliuzhinets 1958) 4'(i + ~,r)',.(# - ~r) = [,.(r/2)]2 cos(f/4), (A. 10) and by successive application of (A.- 10) one obtains 0 + 7r) - 7r) = [T/2)]4 [cos(8/2) + cos(7r/4)], (A. 11) 3 1+ -)\-) = (/2)]2C(. (A. 12) From this last equation we observe that the zeros of 7.(3) which are closest to the point 3 = 0 and the corresponding poles are the points 13 = ~ (5ir/2) and 3 = t (74-/2), respectively. From Eq. (A. 10) one also derives *,.(3 + 7r) cos(0j + + )(A13) -1 -) cos(i- ) - \ () 41(O + 27r) ^( 2) = cot(~j + i). A.14 2)(A. 14) The function ~(t is expressed in terms of the function T&(3) by the product in Eq. (5): - q (5 ) = ^(0 + r + al)(0 + r - a)*.( - -a2)(f -T + a2), (A. 15) from which, by means of (A. 14), we derive 18

l492-1-Q 0(7?r + )9 ) I( + al + 27r),( a, * 0(7r8) h( + al *~( al - 2r) (A. 16) cot(jf + Iaij+ ir) cot(j# al + IT) Coeal- Sin m# COB af + sin ' and Similarly t(- r-) _Cos a2 - sinf (A. 17) V(-+ P) COS a2+sinl 19

1409a4-q REFERENCES Bowman, J. J. (1967), "High Frequency Backatting From an Absorbing Infinite Strip with Arbitrary Face Impedances," Ca.J.Phy., 45 2409-2430. Clemmow, P. C. (1951), "A Method for the Exact Solution of a Class of Two-Dimensional Diffraction Problems, Proc. Roy. Soc., A2 286-308. Felsen, L.B. and H. Y. Yee (1968a), "Ray-Optical Technique for Waveguide Discontinuities, " IEEE Trans. AP-16, 268-269. Felsen, L. B. and H. Y. Yee (1968b), "Multiple Interaction by Bay Optics - Some Observations and an Example, " to be published. Grobner, W. and N. Hofreiter (1949), Integratafel. Springer-Verlag, Wien and Innsbruck, Austria. Maliuzhets, G. D. (1958), "Excitation, Reflection and Emislion of Surface Waves from a Wedge with Given Face Impedance, " Soviet Phyic.s-Doklad4, 3 752-755. Maliuzhinets, G. D. (1960), '1Da Sommerfeldcahe Integral und die L8sung von Begungsaunfgaben in kelgebleten," Am d. Phaik. 6y 107-112. Vajnshtejn, L A. (1948), "On the Theory of Diffraction by Two Parallel Half-Planes, " Izv. Akad. Nauk., Ser. Fiz., 12, 166-180. English translation by J. Shmoys, New York University Report No. EM-63 (1954). Weston, V. H. (1963), "Theory of Absorbers in Scattering, " IEEE Trans., AP-11, 578-584. Yee, H. Y. and L. B. Felsen, (1967a), "Ray-Optical Calculation of Reflection from an Open-ended Waveguide, " Spring URSI Meeting, Ottawa,.Ca.nda (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, H. Y., L. B. Felsen and J. B. Keller (1968), "Ray Theory of Reflection from the Open-end of a Waveguide, " SIAM J. on Appl. Math, 16 268-300. 20

Security Classification N DOCUMENT CONTROL DATA. R & D (Security classification of title, Iodly of abstract nnd tideflntId annotation rmut be entered when ithe overall report Is leelilied) 1. ORIGINATING ACTIVITY (Corporate author)POR The University of Michigan Radiation Laboratory, Dept. of UNCLASSIFIED Electrical Engineering, 201.Catherine Street, ab. GROUP N/ Ann Arbor, Michigan 48108 3. REPORT TITLE REDUCTION OF RADAR CROSS SECTION OF DUCTS 4 OESCRIPTIVE NOTES (Type of report and Inclulsve date.) INTERIM REPORT NO 1 22 January 1968 - 31 March 1968 8. AU THOR(S (Firet name, middle Initial, liel ame) John J. Bowman and Vaughan H. Weston 6. REPORT DATE. 7t. TOTAL NO. OF PAGES 7b. NO. OF REFS April 1968 2011 Oa. CONTRAT OR GRANT NO. 9I. ORIGINATOR'S REPORT NUMBERISI P 504 855029 149-1 1492 -la b. PROJECT NO. c. \ h. OTHER REPORT NO(SI (Any other numbers hfat may he assigned thi report) d. 10. DISTRIBUTION STATEMENT II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Northrop Corporation - Norair Division Hawthorne, California 13. ABSTRACT As a preliminary to an investigation concerning radar scattering by a rectangular duct lined with absorbing materials, the simplified problem of plane-wave scattering by a semi-infinite parallel plane waveguide is considered. The surfaces of the guide are assumed to obey impedance boundary conditions, where the impedance on the interior surface may be different from the impedance on the exterior surface.. A case of particular interest would be that in which the exterior surface is perfectly conducting. Ray-optical techniques based upon known results for a semi-infinite screen with two face impedances are employed to calculate both the field scattered into the far zone and the field generated at the mouth of the guide.. The ray-optical procedure is also applied to the case of a perfectly conducting, open-ended parallel plane waveguide for which an exact solution is available, and the ray optical result is compared with the asymptotic expansion of the exact solution. Also briefly discussed are the modes sustained in a circular waveguide whose wall obeys an impedance boundary condition. r %FORM A —I 0 IJ Li i'm0V Gel4 / 3 UNCLASSIFIED S -~luril v (C l.issi i 'ill i.i,

.I 14. LINK A LINK 0 LINK C K EY WORDS. - 1. ROLE WT ROL E WT ROLE WT I 11I r I Ducts Radar Scattering Radar Cross Section Reduction Ray Optics Absorbers i a. am -I I'mm I t I UNCLASSIFIED Sct.rlly ('l slf: i~:I^.rol