PLANE WAVE SPECTRA IN GRATING THEORYL III. SCATTERING BY A SEMIINFINITE GRATING OF IDENTICAL CYLINDERS

Abstract

A time-harmonic, plane wave is incident on a semiinfinite grating of identical, but otherwise arbitrary, cylinders. The scattered field is expressed as an angular spectrum of plane waves with an unknown scattering amplitude function. By decomposing the grating into two "bodies" (namely the first N elements and the remaining infinity of elements), integral equations are derived for the unknown. The case N = 1 leads to two integral equations which relate the scattering functions of the grating and its end element to the scattering function of the end element in isolation. The case N → ∞ provides an equation relating the scattering function of the grating to that of an element in an infinite grating. The scattering function of the grating is shown to possess branch points in the complex planes of the angles of incidence (α) and observation (θ), and poles whose positions depend on both. The scattering function is examined in the neighborhood of the branch points. For all relative configurations of poles, θ, and branch points, the field far from the end element is calculated by asymptotic methods. Contributions arise from poles, branch-cut integrals, and a steepest-descent integral. Poles give rise to plane waves existing in wedge-shaped regions bounded on one side by the grating. Branch-cut integrals also yield waves in wedge-shaped regions; their contribution is usually negligible in comparison with that of the steepest-descent integral.