Resonant effects in scattering by periodic arrays

The scattering of plane acoustic waves by an infinite periodic array of circles is considered. Attention is focused on parameters (frequency, incident angle, array spacing) that lead to resonance; that is, when one or more of the waves that is diffracted by the array propagates along the array. By considering the unknowns in the solution as functions of the resonant mode scattering angle, we are able to determine the precise nature of the behaviour of the solution at resonance and thereby to accurately compute the resonant state. Both single resonance, when a single mode propagates along the array, and double resonance, when there are two resonant modes propagating in opposite directions along the array, are considered. Numerical results are presented, with particular emphasis on computations of the scattered field at resonance. Comparisons are also made with scattering by a long finite array.