Surface gravity waves over a two-dimensional random seabed.

We extend homogenization theory to study the two-dimensional evolution of weakly nonlinear waves in a sea where the bathymetry is random over a large area. A deterministic nonlinear Schrödinger equation is derived for the envelope of a nearly sinusoidal progressive wave train. Randomness is shown to yield a linear term with a complex coefficient depending on a certain statistical average of the bathymetry. Numerical solutions are discussed for the diffraction of a Stokes wave in head-sea incidence towards a bathymetry of given plan form. Effects of the height and plan form of the randomness, as well as wave nonlinearity are examined analytically and numerically.