Bloch–Floquet band gaps for water waves over a periodic bottom

Abstract

A central object in the analysis of the water wave problem is the Dirichlet–Neumann operator. This paper is devoted to the study of its spectrum in the context of the water wave system linearized near equilibrium in a domain with a variable bottom, assumed to be a $C^2$ periodic function. We use the analyticity of the Dirichlet–Neumann operator with respect to the bottom variation and combine it with general properties of elliptic systems and spectral theory for self-adjoint operators to develop a Bloch–Floquet theory and describe the structure of its spectrum. We find that, under some conditions on the bottom variations, the spectrum is composed of bands separated by gaps, with explicit formulas for their sizes and locations.