This paper is a study of the problem of nonlinear wave motion of the free
surface of a body of fluid with a periodically varying bottom. The object
is to describe the character of wave propagation in a long wave asymptotic
regime, extending the results of R. Rosales & G. Papanicolaou (Rosales &
Papanicolaou, 1983). We take the point of view of perturbation of a
Hamiltonian system dependent on a small scaling parameter, with the
starting point being V.E. Zakharov's Hamiltonian (Zakharov, 1968) for the
Euler equations for water waves. We consider bottom topography which is
periodic in horizontal variables on a short length scale, with the
amplitude of variation being of the same order as the fluid depth. The
bottom may also exhibit slow variations at the same length scale as, or
longer than, the order of the wavelength of the surface waves.
In the two dimensional case of waves in a channel, we give an alternate
derivation of the effective KdV equation that is obtained in (Rosales &
Papanicolaou, 1983). In addition, we obtain effective Boussinesq
equations that describe the motion of bidirectional long waves, in cases
in which the bottom possesses both short and long scale variations. In
certain cases we also obtain unidirectional equations that are similar to
the KdV equation. In three dimensions we obtain effective three
dimensional long wave equations in a Boussinesq scaling regime, and again
in certain cases an effective KP system in the appropriate unidirectional
limit.
The computations for these results are performed in the framework of an
asymptotic analysis of multiple scale operators. In the present case this
involves the Dirichlet-Neumann operator for the fluid domain which takes
into account the variations in bottom topography as well as the
deformations of the free surface from equilibrium.