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.