New nonlinear evolution equations are derived that generalize the system by Matsuno
[16] and a terrain-following Boussinesq system by Nachbin [23]. The regime considers finite-amplitude
surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable,
finite depth. The asymptotic simplification of the nonlinear potential theory equations is performed
through a perturbation anaylsis of the Dirichlet-to-Neumann operator on a highly corrugated strip.
This is achieved through the use of a curvilinear coordinate system. Rather than doing a long
wave expansion for the velocity potential, a Fourier-type operator is expanded in a wave steepness
parameter. The novelty is that the topography can vary on a broad range of scales. It can also have
a complex profile including that of a multiply-valued function. The resulting evolution equations are
variable coefficient Boussinesq-type equations. These equations represent a fully dispersive system in
the sense that the original (hyperbolic tangent) dispersion relation is not truncated. The formulation
is done over a periodically extended domain so that, as an application, it produces efficient Fourier
(FFT) solvers. A preliminary communication of this work has been published in the Physical Review
Letters [1].