  Two-dimensional free-surface potential flows of an ideal fluid over a
strongly inhomogeneous bottom are investigated with the help of conformal
mappings. Weakly-nonlinear and exact nonlinear equations of motion are derived
by the variational method for arbitrary seabed shape parameterized by an
analytical function. As applications of this theory, band structure of linear
waves over periodic bottoms is calculated and evolution of a strong solitary
wave running from a deep region to a shallow region is numerically simulated.
