We examine the effect of a rapidly varying periodic bottom on the free surface of a fluid. We consider the shallow water scaling regime and derive a model system of equations which consists of the classical shallow water equations describing the effective surface wave dynamics coupled with nonlocal evolution equations for a periodic corrector term. A rigorous justification for this decomposition is given in the form of a consistency analysis. The free surface can however exhibit the effect of resonances with the periodic bottom, which leads to secular growth and can influence the time interval of validity of the theory.

Motivated by the need of analytical tools to address the dynamics of such resonant situations, we consider as a first step, the water wave system with a periodic bottom, linearized near the stationary state, and develop a Bloch theory for the linearized water wave evolution. This analysis takes the form of a spectral problem for the Dirichlet – Neumann operator of the fluid domain with periodic bathymetry. As in the case of Bloch theory for many second order partial differential operators, we find that the presence of the bottom results in the splitting of double eigenvalues near points of multiplicity, creating a spectral gap.