We consider the movement of a free surface of a two-dimensional fluid over a variable bottom. We assume that the bottom has a periodic profile and we study the water wave system linearized near a stationary state. The latter reduces to a spectral problem for the Dirichlet--Neumann operator in a fluid domain with a periodic bottom and a flat surface elevation. Bloch spectral decomposition is a classical tool to address problems in periodic geometries or equivalently differential operators with periodic coefficients. We show that the spectral problem admits a Bloch decomposition in terms of spectral band functions and their associated band-parametrized eigenfunctions. We find that, generically, the spectrum consists of a series of bands separated by spectral gaps which are zones of forbidden energies.

Bloch theory and spectral gaps for linearized water waves