Water waves over a strongly undulating bottom

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 an arbitrary seabed shape parametrized by an analytical function. As applications of this theory, the band structure of linear waves over periodic bottoms is calculated and the evolution of strong solitary waves running from a deep region to a shallow region is numerically simulated.