  In this paper, we study the motion of the free surface of a body of fluid
over a variable bottom, in a long wave asymptotic regime. We assume that the
bottom of the fluid region can be described by a stationary random process
$\beta(x, \omega)$ whose variations take place on short length scales and which
are decorrelated on the length scale of the long waves. This is a question of
homogenization theory in the scaling regime for the Boussinesq and KdV
equations. The analysis is performed from the point of view of perturbation
theory for Hamiltonian PDEs with a small parameter, in the context of which we
perform a careful analysis of the distributional convergence of stationary
mixing random processes. We show in particular that the problem does not fully
homogenize, and that the random effects are as important as dispersive and
nonlinear phenomena in the scaling regime that is studied. Our principal result
is the derivation of effective equations for surface water waves in the long
wave small amplitude regime, and a consistency analysis of these equations,
which are not necessarily Hamiltonian PDEs. In this analysis we compute the
effects of random modulation of solutions, and give an explicit expression for
the scattered component of the solution due to waves interacting with the
random bottom. We show that the resulting influence of the random topography is
expressed in terms of a canonical process, which is equivalent to a white noise
through Donsker's invariance principle, with one free parameter being the
variance of the random process $\beta$. This work is a reappraisal of the paper
by Rosales & Papanicolaou \cite{RP83} and its extension to general stationary
mixing processes.
