  We study the behavior of shallow water waves over periodically-varying
bathymetry, based on the first-order hyperbolic Saint-Venant equations.
Although solutions of this system are known to generally exhibit wave breaking,
numerical experiments suggest a different behavior in the presence of periodic
bathymetry. Starting from the first-order variable-coefficient hyperbolic
system, we apply a multiple-scale perturbation approach in order to derive a
system of constant-coefficient high-order partial differential equations whose
solution approximates that of the original system. The high-order system turns
out to be dispersive and exhibits solitary-wave formation, in close agreement
with direct numerical simulations of the original system. We show that the
constant-coefficient homogenized system can be used to study the properties of
solitary waves and to conduct efficient numerical simulations.
