We present a Hamiltonian, potential-flow formulation for nonlinear surface water waves in the presence of a variable bottom. This formulation is based on a reduction of the problem to a lower-dimensional system involving boundary variables alone. To accomplish this, we express the Dirichlet-Neumann operator as a Taylor series in terms of the surface and bottom variations. This expansion is convenient for both asymptotic calculations and direct numerical simulations. First, we apply this formulation to the asymptotic description of long waves over random topography. We show that the principal component of the solution can be described as a solution of a Korteweg-de Vries-type equation, plus random phase corrections. We also derive an asymptotic expression for the scattered component. Finally, we propose a pseudospectral method, using the fast Fourier transform, to numerically solve the full equations. Several applications are presented.
Because of its relevance to coastal engineering, surface water wave propagation in the presence of an uneven bottom has been studied for many years. The character of coastal wave dynamics can be very complex; waves are strongly affected by the bottom through shoaling and the resulting variations in local linear wave speed, with the subsequent effects of refraction, diffraction and reflection. Nonlinear effects, which influence waves of appreciable steepness even in the simplest of cases, have additional components due to wave-bottom as well as nonlinear wavewave interactions, as seen, e.g., in depth-induced breaking. The presence of bottom topography in the fluid domain introduces additional space and time scales to the classical perturbation problem. The resulting nonlinear waves can have a great influence on sediment transport and the formation of shoals and sandbars in nearshore regions. It is therefore of central importance to understand the basic mechanisms that govern the dynamics of such waves.
