On scattering of waves by the infinite grating of circular cylinders

The boundary value problem of the infinite grating of circular cylinders is treated by specializing the new functional equation obtained previously for arbitrary elements. This specifies the problem in terms of a set of algebraic equations which involves only the known scattering coefficients of an isolated cylinder, and certain series of elementary functions. The special results for normal incidence are identically those obtained originally by Ignatowsky who worked with the separation-of-variables solution; as a check we also extend his procedure to arbitrary angles and show that the results can be transformed to those we obtain by the Green's function approach. The equations are used to construct series and closed form approximations for both polarizations, for conducting and dielectric cylinders, for arbitrary angles of incidence. The results are applied to consider multiple scattering or coupling effects for a mode near grazing (Wood anomalies), and for closely spaced scatterers (packing effects). For example, we show that to a first approximation the packing effects for E perpendicular to the axes merely increase the dipole moment of the isolated cylinder; in this range, the circular cylinder within the grating is equivalent to an isolated elliptic cylinder whose size and shape are independent of angle of incidence. We also obtain simple closed forms for low frequencies which take explicit account of coupling effects up to multipoles of order 2^5, etc.