The behaviour of water waves over periodic beds is considered in a two-dimensional context and using linear theory. Three cases are investigated: the scattering of waves by a finite section of periodic topography; the Bloch problem for infinite periodic topography; and sloshing motions over periodic topography confined between vertical boundaries. Connections are established between these problems.

A transfer matrix method incorporating evanescent modes is developed for the scattering problem, which reduces the computation to that required for a single period, without compromising full linear theory. The problem of the existence of Bloch waves can also be posed on a single period, leading to a close relationship between it and the scattering problem. Sloshing motions over periodic beds, which may be regarded as special cases of the Bloch problem, are also found to have a significant connection with wave scattering.

Integral equations methods allied to the Galerkin approximation are used to resolve the three problems numerically. In particular, the full linear solution for Bragg resonance is presented, allowing the accuracy of existing approximations to this phenomenon to be assessed. The selection of results given illustrates the main features of the work.