Using Green's function methods, we express the field of a grating of cylinders excited by a plane wave as certain sets of plane waves: a transmitted set, a reflected set, and essentially the sum of the two "inside" the grating. The transmitted set is given by \psi_{o} + 2\SigmaC_{\upsilon}G(\theta_{\upsilon}, \theta_{o})\psi_{\upsilon} , where the \psi 's are the usual infinite number of plane wave (propagating and surface) modes; G(\theta_{\upsilon}, \theta_{o}) is the "multiple scattered amplitude of a cylinder in the grating" for direction of incidence \theta_{o} and observation \theta_{\upsilon} ; and the C's are known constants. (For a propagating mode, C_{\upsilon} is proportional to the number of cylinders in the first Fresnel zone corresponding to the direction of mode v .) We show (for cylinders symmetrical to the plane of the grating) that G(\theta,\theta_{o})= g(\theta,\theta_{o}) +(\Sigma_{\upsilon} - \int dv)C_{\upsilon}[g(\theta,\theta_{\upsilon} + g (\theta,\pi - \theta_{\upsilon})G(\pi-\theta_{\upsilon},\theta_{o})] , where g is the scattering amplitude of an isolated cylinder. This inhomogeneous "sum-integral" equation for G is applied to the "Wood anomalies" of the analogous reflection grating; we derive a simple approximation indicating extrema in the intensity at wavelengths slightly longer than those having a grazing mode. These extrema suggest the use of gratings as microwave filters, polarizers, etc.

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