RL 664 The Boundary Limits of Certain Integrals in Scattering Theory* by S. K. Cho and T.B.A. Senior Radiation Laboratory, The University of Michigan Ann Arbor, Michigan 48109 Abs tract Two integrals that arise in scattering theory are discussed. In contrast to recent statements in the literature it is shown that the correct interpretation is not as a Hadamard finite part, and this has implications conceptually as well as numerically. * This work was supported by the Air Force Office of Scientific Research under Contract F49620-79-C0128 and Grant 77-3188. RL-664 = RL-664

I. Introduction In the application of integral equation methods in scattering theory it is necessary to determine the boundary limits of the integrals involved, and in some cases this requires special care. Factors that can influence the limits are the kernel itself, the smoothness of the boundary geometry, and the smoothness of the density function whose values can be obtained only by solution of the integral equation. For a density function in a particular space, the boundary limit may or may not exist depending on the kernel and the boundary, but since the kernel is given, it is the interplay of the smoothness of the boundary and the density function that we are concerned with. As a general rule, the smoother the boundary, the less stringent the conditions on the density function. Two integrals of interest are O an an G(kR)f(r')dS' (1) r I2(rO) = xJ J(r') - v'vO G(kR )dS' (2) r where the integration is over the closed boundary surface r of a finite body, R = Ir - r'l witn r,' e r, n0 = n(rF) and ni = n ) are unit vector normals to r directed into the exterior region Q, J(r') is a tangential vector function, and G(kR ) = eikRo/(47R ) is the free space Green's function. The first integral was used by Davis and Mittra [1] in the analysis of a thin scatterer problem, and the two-dimensional -1 -

analogue of (2) was considered by Bolomey and Tabbara [2] and Mautz and Harrington [3]. In all cases the integrals were interpreted in the sense of a Hadamard finite part, but this is incorrect practically as well as on strictly formal grounds. The Hadamard finite part integral is a singular integral of a specific type in which the integrand is known and has a fixed singular point. With (1) and (2), however, the singularities are not fixed, but spread over the entire domain of integration, and in addition the integrands are unknown, since they contain the density function whose determination is the objective of the integral equation technique. The purpose of this note is to show the circumstances under which (1) and (2) exist and to give the meaning that the integrals then have. The main results are contained in the theorems of the next two sections, and we then examine the two-dimensional analogues of the integrals for which the 'self cell' contributions are trivially obtainable. II. The Integral Il(r ) The integral is the normal derivative of a (generalized) doublelayer potential in scattering theory and its properties can be deduced from those of the corresponding one in potential theory [4,5,6]. In particular, if the exterior limit of the integral exists, so does the interior one, and both limits are equal. In other words, if the integral exists, it is continuous across r. Schauder [4] has shown that for r E Cl+v(O <v <1) the integral operator in potential theory corresponding to that in (1) maps C1+ into C, 0 < <1. IA1 -2 -

Henceforth we assume that r is closed and in C2. This assumption is more restrictive than Schauder's and implies that the curvature is defined everywhere on r. Let w(Q), Q E r, denote the direct value* of the double-layer potential w(Q) = G(krpQ)f(P)dSp (3) r where the integral is not a principled-valued one as frequently supposed, but one defined in the Riemann sense. Theorem 1: If r E C2, the integral operator M(Q,P;k) such that Ii(Q) = -w(Q) = jJ - n G(kr Q)f(P)dSp M(Q,P;.k)f(P) (4) nQ J p P r Q maps C1 into C1 for all Q e r. Omitting details which are a direct extension of those in [4], an outline of the proof is as follows. Divide r into r' and r where r ~ e is a sufficiently small open surface about a fixed point Q1 e- r For Q2 _ r A double-layer potential defined at r G Q yields an exterior limit which is the sum of l/2f(Q) and w(Q). The integral (3) is called the direct value in potential theory. -3 -

___ w Q - 3 w(Q2) p 'Ur C DfQ ~~G(krQ ) {ff(P) 3P 1Q - f(QN)AS~ -fi" DfQ Dn G(kr PQ2){f(P) - f(Q29}dSp + f(QN) f m n 31n G(kr PQ )dS p - f(Q2) fi an Q2 IL'r )dS Tn-9 KPQ)(zpt P - (5) Since f is in C1, the terms in square brackets tend to zero as Q2 -*> Q1. The integrals over ~ F are IJ(Ql sQ2) -f p a Dn Q 3 G(kr PQ ){f(P) - f(Q1)}dSp -"i c 3 Dn Q ~1P G(krQ){If (P) - f (Q2) }dSp where rD 3 G(kr Q) nQ nP 47rr 3Q n Q 3(pQ np)(r~pQ flnQ)} and since 1' 6 C2, IJ(Ql9Q2) I <ABr QQ (6) -4 -

where A and B are some constants independent of Q1 and Q2, but dependent on the smoothness of r and f. The integrals over the open surface r' are similarly bounded, and from these estimates it follows that II(Q) is in C1. It is easily shown that if the density function is merely continuous on a boundary of class Cl+v, 0 < v < 1, I(Q) is unbounded and hence does not exist. In particular, the theorem fails if r has a surface singularity, but ifr is in C2, II(Q) is in C1, at variance with a Hadamard finite part interpretation. III. The Integral I2(r) 0 Theorem 2: If r e C2 the integral operator N(Q,P;k) defined by T2(Q) = nQ x f/ J(P) vpvQG(krpQ)dSp = sQN(Q,P;k) * J(P) (7) r maps C1+, into C, 0 < <1, where sQ is a unit tangent vector at Q e r. The proof is trivial. For Q G Q and R = IQPI J/ J(P). vpVQ G(kR)dSp = - / VQG(kR)Vp * J(P)dSp r r provided, of course, J e Cl(r). Hence, for Q e r, I2(Q) = -nQ x f vQG(krpQ)vp * J(P)dSp (8) r which is simply the tangential derivative of a single layer potential with density function SQVp * J(P). From the fact well known in potential -5 -

theory that this is continuous across a boundary r c C2 and is in C on the boundary if the density function is in C, 0 < < 1, the theorem follows. Here again, therefore, the interpretation as a Hadamard finite part is incorrect. In the numerical solution of an integral equation containing either of the integrals (1) or (2), the error affects the self cell contribution and, as a result, the diagonal terms in the matrix. The determination of the self cell contributions is most easily considered in two dimensions where (1) is directly related to an integral similar in form to (2). IV. Two-Dimensional Case The two-dimensional analogue of (8) and, hence, (2) is - 4nQ x VQH (krpQ) Ds J(sp)dsp (9) Y where s is arclength along a smooth closed boundary y, H(1) is the 0 Hankel function of the first kind of zero order, and the density function J(sp) = spJ(sp). If (nqs^Qz) is a right-handed rectangular coordinate system at Q s y, the scalar product of (9) with z2 yields 4 spJ() Hi )(kr)dp. (10) Y By a tedious but straightforward analysis, it can also be shown that -6 -

i an H()(krp)ds 4 J(p) Q 3np pQ p Y k2 J(s ) (sp * )H()(kr )ds 4 P o PQ P Y + J(s ) H(1)(krp)ds (11 ) p P Q Y and this not only relates the two-dimensional versions of (1) and (2) explicitly, but can serve to give a proper interpretation of the integral on the left. In particular, in any numerical evaluation of the integrals on the two sides of (11), the self cell contributions must be equal. The first integral on the right-hand side poses no problem and its contribution is aJ(sQ) where a tends to zero with the cell size A, assumed small compared with the wavelength. The second integral produces (-A/27)( a/ Cs)J(sQ) which also tends to zero with A, but when, in the moment method, the second derivative is expressed in terms of J via finite differencing, we obtain a contribution proportional to (l/7A)J(sQ). This is therefore the dominant term in the expressions for the self cell contributions of the integrals on both the left- and right-hand sides of (11). It is a term which would be omitted were the first and third integrals treated as Hadamard finite parts, but one whose retention is implied by our stricter interpretation. It must be admitted that the retention of these terms does not seem to have a major impact on a numerical solution of an integral equation in which either of the integrals appears. Mathematically the self cell -7 -

contributions do tend to zero with decreasing cell size, and the solution does remain stable as A decreases. At most, therefore, the omission of the terms 0(1/A) would produce some loss of efficiency in the program by forcing a smaller cell size than otherwise necessary to achieve the same accuracy of solution. References [1] W. A. Davis and R. Mittra, "A new approach to the thin scatterer problem using the hybrid equations," IEEE Trans. Antennas Propagat., vol. AP-25, pp. 402-406, May 1977. [2] J-C Bolomey and W. Tabbara, "Numerical aspects on coupling between complementary boundary value problems," IEEE Trans. Antennas Propagat., vol. AP-21, pp. 356-363, May 1973. [3] J. R. Mautz and R. F. Harrington, "A combined-source solution for radiation and scattering from a perfectly conducting body," IEEE Trans. Antennas Propagat., vol. AP-27, pp. 445-456, July 1979. [4] J. Schauder, "Potentialtheoretische Untersuchungen," Math. Zeit., vol. 33, pp. 602-640, 1932. [5] N. M. Gunter, Potential Theory, Frederick Ungar Pub. Co., New York, N. Y., 1967. [6] N. S. Landkof, Foundations of Modern Potential Theory, Springer Verlag, New York, N. Y., 1972. -8 -

References [1] J-C Bolomey and W. Tabbara, "Numerical aspects on coupling between complementary boundary value problems," IEEE Trans. Antennas Propagat., vol. AP-21, pp. 356-363, May 1973. [2] J. R. Mautz and R. F. Harrington, "A combined-source solution for radiation and scattering from a perfectly conducting body," IEEE Trans. Antennas Propagat., vol. AP-27, pp. 445-456, July 1979. [3] W. A. Davis and R. Mittra, "A new approach to the thin scatterer problem using the hybrid equations," IEEE Trans. Antennas Propagat.., vol. AP-25, pp. 402-406, May 1977. [4] J. Schauder, "Potential theoretische Untersuchungen," Math. Zeit., vol. 33, pp. 602-640, 1932. [5] N. M. Gunter, Potential Theory, Frederick Ungar Pub. Co., New York, N. Y., 1967. [6] N. S. Landkof, Foundations of Modern Potential Theory, Springer Verlag, New York, N. Y., 1972.