Wave scattering by platonic grating stacks

We address the problem of scattering of flexural waves obeying the biharmonic equation by a stack of a finite number of gratings. We express the solution of the scattering problem for a single grating in terms of reflection and transmission matrices, incorporating the effects of both propagating and evanescent incident waves. The plane wave expansion coefficients above and below the grating are linked to multipole coefficients within the grating using the grating sums and the Rayleigh identities. We derive the recurrence procedure giving the reflection and transmission matrices of the stack in terms of those of individual layers. Trapped waves between a pair of gratings are investigated.