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The system of equations for water waves, when linearized about equilibrium ofa fluid body with a varying bottom boundary, is described by a spectral problemfor the Dirichlet -- Neumann operator of the unperturbed free surface. Thisspectral problem is fundamental in questions of stability, as well as to theperturbation theory of evolution of the free surface in such settings. Inaddition, the Dirichlet -- Neumann operator is self-adjoint when given anappropriate definition and domain, and it is a novel but very natural spectralproblem for a nonlocal operator. In the case in which the bottom boundaryvaries periodically, $\{y = -h + b(x)\}$ where $b(x+\gamma) = b(x)$, $\gamma\in \Gamma$ a lattice, this spectral problem admits a Bloch decomposition interms of spectral band functions and their associated band-parametrizedeigenfunctions. In this article we describe this analytic construction in thecase of a spatially periodic bottom variation from constant depth in two spacedimensional water waves problem, giving a construction of the Blocheigenfunctions and eigenvalues as a function of the band parameters and adescription of the Dirichlet -- Neumann operator in terms of the bathymetry$b(x)$. One of the consequences of this description is that the spectrumconsists of a series of bands separated by spectral gaps which are zones offorbidden energies. For a given generic periodic bottom profile$b(x)=\varepsilon \beta(x)$, every gap opens for a sufficiently small value ofthe perturbation parameter $\varepsilon$.

Bloch theory and spectral gaps for linearized water waves