PLANE WAVE SPECTRA IN GRATING THEORY: I. SCATTERING BY A FINITE NUMBER OF BODIES

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Abstract
A scalar plane wave is incident on a grating of N cylinders. A solution of the scattering problem, useful when N is large, is the ultimate goal. It is suggested that this may be accomplished by solving first the problem for a semi-infinite grating; this may be solved in terms of the scattering properties of an infinite grating, which, in turn, depend on the properties of an isolated cylinder.In the present paper, attention is confined to certain aspects of scattering by a few (in particular, one or two) cylinders. The field scattered by the cylinders is expressed as a sum of continuous plane wave spectra with amplitudes unknown, the scattering amplitude functions being considered as given for each cylinder in isolation. The field is studied, both outside the grating and between the elements. The convergence of the integral representation for the scattered field is treated in detail. Consequences of analyticity in the spatial variables of the scattered field are discussed; it is shown that the integral representation provides the continuation of the solution into a scatterer up to the convex hull of singularities of the field scattered by an isolated cylinder. Integral equations relating the scattering functions for cylinders in the grating and in isolation are derived. The uniqueness of the solution of these equations under a variety of boundary conditions is discussed by an indirect method; the case of two cylinders is treated in detail. It is inferred that there is at most one solution whose growth in the complex plane of the angle of observation is appropriate to scattering by a physical body.
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