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