Abstract

It will be shown how solutions to problems involving scattering of waves by a finite periodic array of scatterers can be reduced to a pair of scalar equations. These arise from matching solutions in the two semi-infinite domains either side of the array at appropriately chosen interfaces at the two ends of the array to a general solution within the array. The form of the scalar equations that result depends upon how the matching at these interfaces is performed. The multiple wave scattering that takes place between elements in the finite array can be encoded in the general solution within a single period of the array using an expansion in the eigenfunctions of the infinite periodic Bloch-Floquet problem. The key is the construction of an orthogonality relation for these eigenfunctions which allows the information in one period of the array to be propagated, without mode coupling, to any other period of the array. The absence of mode coupling sets it apart from transfer or scattering matrix approaches. Two examples will be presented to illustrate its application in two different settings. The connection to homogenisation will also be touched upon.