011764-505-M MEMO TO: T. B. A. Senior FROM: Sharad R. Laxpati SUBJECT: Derivation of integral equations for a two dimensional scatterer with impedance boundary condition and partially clad by an absorptive sheet. The possibility of reduction in radar return from a conducting scatterer by means of cladding it with highly absorptive material has generated a considerable amount of both experimental and theoretical literature in this area. Oshiro, et al. (1966, 1971) have developed integral equations for a two dimensional conducting body (fully and partially) clad by a resistive shell of finite and infinitesimally small thickness. They have confined their study to E-polarized incident field. Numerical solutions to the integral equations have been obtained for several different shapes of the scatterers. Knott et al. (1973 a, b) have considered the scattering from a twodimensional scatterer with an impedance boundary condition. This form of the boundary condition has an advantage, that the two polarizations of the incident field do not have to be considered separately. Knott et al. have obtained numerical results for a variety of conducting scatterers clad by an impedance surface; the magnitude of the surface impedance varying as a function of position. Their results indicate that the non-/specular return (for example, from an edge) can be reduced by as much as 13 dB by proper choice of impedance surface cladding. This idealized version of the boundary condition, an impedance boundary condition, has a drawback of not being related to the physical parameters of absorptive materials prevalent in experimental work. It is then of considerable DISTRIBUTION Hiatt/File Liepa Hiatt/File Liepa 11764-505-M = RL-2243 Knott Senior Laxpati

011764-505-M interest to investigate the problem of cladding by absorptive materials whose permittivity and permeability are different from those for free-space. In this memo, the integral equations for a two-dimensional scatterer with impedance boundary condition and partially clad by an absorptive sheet of infinitesimally small thickness are derived. The formulation is a scalar one, and consideres an E-polarized incident wave. The integral equations are derived for the unknowns; the electric surface current on the impedance surface and the electric and magnetic polarization currents in the absorptive material. Geometry and the boundary conditions A n 0 C 22 gum impedance rS Z Absorptive medium 50 of thickness A/2, < parameters E and p. Figure 1. Let a scalar V represent the z-component of the electric field. Figure 1 shows the geometry of the problem. C2 represents the surface of an impedance scatterer and encloses volume V. C is the boundary of the absorptive A 2 material of small thickness -2 and encloses volume V. Note that C is a 0o o closed contour, since this is necessary for the application of scalar Green's theorem. In the next section the case of an open contour C is treated as a 0 limiting case of a closed contour. Let i(a) be the electric field at an arbitrary point p in V, where V is 2

011764-505-M the infinite volume less V and V2. Let Lo (P) be the corresponding scalar o o field in V. We shall derive the integral equation assuming an incident field o inc () in V. The problem will be formulated by the application of the scalar Green's theorem to ib in V and 0i in V. The application of the boundary conditions o 0 and the evaluation of i(pa) for p in C2 and C and o(pa) for p in C provides 2 o the desired integral equations. Since the thickness of the absorptive medium is very small, the volume polarization currents may be approximated as surface currents. These surface currents are defined in terms of b and X on C as an 0o follows: K (ao ) Xik AXe and 0 (1) K*(p ) " o s o 2 an 0 where K and K* are the electric and magnetic surface currents respectively. z s o is in C0, k is the free-space wave number and Z the free-space impedance. s-o o o Xe and Xm are the electric and magnetic susceptibilities of the absorber. Using the above defined equivalent surface currents, the boundary conditions on C are ('ao) - To ~o): K* (.o) and (2) a~(2o) aor(Qo) h —) ~~- 5 ikZ K ( ) an an o z o o o A A Note that the unit vectors n and n2 are always directed into volume V; the 0 observation point p is located in this region. Derivation of the integral equations Application of the scalar Green's theorem to 0(p_) in V, and noting that the contribution of the surface at infinity leads to the incident value of i, we 3

011764-505-M write, for _ in V, i) nc If ( ) -^ AaJ d 0 0 - _ PI )i.]ds' (4) I q(R(,)_ _ aG(p, p ) ( p2) an2' s2 an ( 2 where G(s, ') is the free-space Green's function for the two-dimensional geometry and ei harmonic variation, viz., G(p_,) = i H (1) (k - (5) Application of the scalar Green's theorem to io in V leads to, for p in V 0 0 qoG (&. (a) ds' (6) some algebraic manipulations lead to the two integral expressions for and a o o 0O theorem. The basic property of interest here is the nature of the discontinuity in q in its representation through the Green's theorem. These operations and in their appropriate domains. For p in V, n z22 () (ikZ ')ds'Z K(2 a ds o 0 Ga_,G( p,) - ikZK~ Kz2 ( 2) ds) G (I 2 ds' 2z2 2 an' C2 4

011764-505-M and, for p in V inc K*(^')0 () o (-) + Ks ) ds' + ikZ o Kz(o )G(p, p )ds' C C o o -ikZ Kz2na ) (* d G( P2)9d s' - ZW. (8) 2K z2() an2ds (8) C 2 Equations (7) and (8) are used to derive the integial equations. First, evaluate (7) at p = p This leads to equation (9). If we substitute p =Q2 in equation (7) we obtain equation (11). Third equation is obtained by evaluation of equation (8) at p =Qo This is equation (10) below. In the following equations we have also used the definition of the Green's function from equation (5). kZ f Kinc) ( ') H (kr )ds - 0 =p(.0 0o 00 C 0 o 4 s 0 00 1 00 4 z2 2 (kr2 C C 4 o C2 inc kZ C 0 A (1) kZ (1) + - K*(' ( ~ )2 H(1(kr )ds'+ Kz2Pf)H (kr0)ds'4 so o oo 1 oo 4 z2 2o0 02 C o 5

011764-505-M ikZ - t U ) KZ n2 r )H (kr (10) 4 j 2 z2 2 r02 02, C2 ie2- ~(2 oKz52)+ 0, ^)H(1)(kr20)ds,o2 4 Kz r)H (kr )ds' +(')H (kr)ds 2- Q H2 01 S 20 oz2 2 0 22 o 2 ikZ (1) o nt) (0')(i~ e )H (kr )d s (11) 2 where = i-; i, j = 0, 2. Equations (9) through (11) are the required integral equations. They can be readily transformed into the usual form by means of the boundary condition equations (2) and (3) along with the definition of the currents from equation (1). For numerical purposes, it will be found advantageous to simplify equation (10) by use of equation (9). This explicit form of the integral equations is not shown. Integral equations for a partially clad impedance surface The previously derived integral equations, equations (9) through (11) are valid only for a closed boundary C. Since the practical cladding is mostly partial, it is necessary to develop the integral equations for this case. The technique employed here is to start with the closed boundary C and consider the limiting case of the region enclosed by C approaching 0. Figure 2 (a) shows the contour C along with the identification of the sections of C. O O 6

011764-505-M 2 2 c IOR COL n1 C1 C1 Figure 2 (a) Figure 2 (b) Figure 2 (b) shows the result of the limit 6 -e 0 on the geometry of Figure 2 (a). The contour C consists of C, C1 C and COL Note that as o 1 1 OR OL 6 — 0, the length of contours COR and COL - 0. We rewrite our definition of the currents and the boundary conditions as follows: +K ik Xe + z Z 2 o o /+ (12) K*+ am a ) o 50 2 + 0 0( 0)-^ aK^ + (13) a 0(fi) a0 ( )0 + -= +=- ikZ K (1)O an- ar o o Note that the superscripts + are associated with the contours CL. Under the limit 6 — 0, we observe that 7

011764-505-M (1) Contour C-> C C 1: Cl 'C1 (2) +o(Qo) —,o(~-) (3) Contributions from the integral over COR and COL will both approach zero since: (a) the length of the contours are proportional to 6 and (b) for infinitesimally small thickness A, the currents tangential to COR and COL must approach zero. In order to arrive at the necessary integral equations, first evaluate (9) at j 4 and and add the two. We also replace integrals over C by C1 with appropriate changes in the integrand. Furthermore, we define Kz( 1) = KZ ce ) + KZ(P1) (14) Ks(ei ) K*1 +) + K (1 ) and use the definition of currents through equation (12). We obtain *K(w K2)H (k'r1)dsfikZ fk An analogous manipulation of equation (10) and use of the boundary condition e C1 - )(n1- 1)H 1(krll)ds'+ kZ | (2H (krl)ds C1 C2 ikZ, 4 p (2)Kz2 )(n2' r2) 1 1 12 2 (13) to eliminate leads to the following integral equation: k ax1^ Kz( l)2K (Q1)+ H((1) (kr*dst + C 8

011764-505-M ikf kZ0 )1 H1(1) 1(kr d )d sf sK*(' 2)(. ~Kll)H((kr )ds' +(kr2 12 C1 C2 Ci 1C1 ) Jr H (kr2)ds ) ik (1), (1) eas to te ti intea e 0C2 Scattered Fields The expression for the scattered component of r can be readily obtained from equation (9). The C epesofotesCatee2opnn f^ca eraiyotie fromequaikon() 9

011764-505-M (kZ~ 1) ik (1) ( () - |K (H (krll)ds' + 4 I K(e n r. H krt ds - 2 2 (18) Remarks: The three integral equations in this work have been derived based on scalar Green's theorem. It does have one shortcoming. In this derivation, although boundary condition on the tangential component of the magnetic field (via the normal derivation of b) has been used explicitly, no integral equation for the magnetic field is developed. To do so would require a vector formulation and consequently second derivatives of the Green's functions would appear in the integrand. This would cause difficulties in numerical solution of these equations. The numerical solution may convince us that it is necessary to use an integral equation for the magnetic field. At that time, the necessary equations can be readily developed. The numerical difficulty anticipated in such a case is the evaluation of the contributontribution of the self cell (-). However, I believe, that this can b can be circumvented by the use of the theory of distribution functions, and in particular, the Hadamard's principal value technique. 10

References E. F. Knott and T. B. A. Senior, "Non-Specular Radar Cross Section Study," University of Michigan Radiation Laboratory Technical Report AFAL-TR73-2, February 1973. E. F. Knott, V. V. Liepa and T. B. A. Senior, "Non-Specular Radar Cross Section Study," University of Michigan Radiation Laboratory, Technical Report AFAL-TR-73-70, April 1973. F. K. Oshiro and R. C. Cross, "A Source Distribution Technique for the Solution of Two Dimensional Scattering Problems," Phase III Report, Northrop Corporation Report No. NOR-66-74, March 1966. F. K. Oshiro, K. M. Mitzner, et al., "Calculation of Radar Cross Section," Northrop Corporation Report No. NOR-71-14, 1971. 11