Multiple scattering of waves by the doubly periodic planar array of obstacles

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Abstract
We analyze the three−dimensional problem of multiple scattering by a doubly periodic planar array of bounded obstacles, and compare the results with those for the grating of parallel cylinders in two dimensions and for the periodic line in three. Plane wave integral forms of the scattered field and of the multiple scattered amplitude lead directly to the array−mode functional representation in terms of the single scattered amplitude and to simple approximations corresponding to array resonances for near−grazing evanescent modes. For the grating and for the periodic line, a grazing mode corresponds to reinforcement of the excitations an obstacle receives from the waves scattered by its neighbors; for the doubly periodic array, to reinforcement of the waves of the lattice lines perpendicular to the mode’s direction. We also derive spherical wave (and conical−cylindrical wave) representations of the solution, and exhibit the results for spherically symmetric scatterers as a special case.

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