Integral Equations for Electromagnetic Scattering by Perfect Conductors with Two-Dimensional Geometry

In this paper, we consider the scattering of time-harmonic electromagnetic fields by perfect conductors with 2-dimensional geometry, in which the boundaries are independent of the 2-coordinate. The zdependence of the fields is assumed to be of the form exp(t'A sin a z), where k is the free space wave number and | a | tt/2, so that the scattering of obliquely incident plane waves may be investigated. It is known1 that the electromagnetic fields may be expressed in terms of the longitudinal components, Ez and Hz, and that each of these two quantities satisfies the scalar wave equation, with wave number k cos a. Moreover, since the boundary conditions on a perfectly conducting surface imply that both Ez and the normal derivative of Hz are zero, there is no coupling between Ez and Hz, and we refer to E waves and H waves, respectively. Integral equations for scattering problems have been considered by numerous authors. A relatively recent treatment of this topic is that of 409 Poggio and Miller,2 but they give only a brief discussion of the 2dimensional case. A useful discussion of integral equations for the scalar problem is given by Noble.3 Poggio and Miller state that the integral equation which they derive for the surface current in the case of H waves is useless when the scatterer is infinitely thin. Noble points out that the corresponding integral equation, when applied to the problem of scattering by an elliptic cylinder, degenerates as the eccentricity tends to unity, so that the scatterer becomes an infinitesimally thin strip.