We examine the motion of the free surface of a body of fluid with a periodically varying bottom. We consider the water wave system linearized near a stationary state and develop a Bloch theory. The analysis takes the form of a spectral problem for the Dirichlet– Neumann operator in a fluid domain with a periodic bottom and a flat surface elevation. We find that, generically, the presence of the bottom results in the splitting of double eigenvalues creating a spectral gap. The analysis is uniform in the spectral parameter and provides the gap asymptotics.