An approximation to the dispersion relation for wave propagation through a doubly-periodic array of scatterers may be obtained by "homogenization". However, this technique yields only an approximation for waves that are long relative to the periodicity, and does not describe the phenomena such as band gaps that are associated with the periodicity. (A band gap is a range of frequencies for which wave propagation through the array is not possible.)

In the present work a technique is described that yields approximations to the dispersion relation for waves that have lengths of the same order of magnitude as the periodicity, and hence that recover the band-gap structure. The approximations are obtained from perturbations of those plane-wave solutions that exist in the absence of the scatterers and have the same spacial periodicity as the array. This is illustrated below for a doubly-periodic array of rigid circular scatterers arranged in a square lattice with cell-size L. Here the wave-number vector has components q₁ and q₂ and k=ω/c, where ω is the radian frequency of the waves and c is the wave speed for the medium. The left- and right-hand figures show the approximate dispersion relation for the array in regions of the parameter space for which there are respectively three and four plane-wave solutions in the absence of the scatterers.