The propagation of waves through a doubly-periodic array of identical rigid scatterers is
considered in the case that the field equation is the two-dimensional Helmholtz equation.
The method of matched asymptotic expansions is used to obtain the dispersion relation
corresponding to wave propagation through an array of scatterers of arbitrary shape
that are each small relative to both the wave length and the array periodicity. The
results obtained differ from those obtained from homogenization in that there is no
requirement that the wave length be much smaller than the array periodicity, and
hence it is possible to examine phenomena, such as band gaps, that are associated with
the array periodicity.