Nonlinear evolution of surface gravity waves over an uneven bottom

Abstract

A unified theory is developed which describes nonlinear evolution of surface gravity waves propagating over an uneven bottom in the case of two-dimensional incompressible and inviscid fluid of arbitrary depth. Under the assumptions that the bottom of the fluid has a slowly varying profile and the wave steepness is small, a system of approximate nonlinear evolution equations (NEEs) for the surface elevation and the horizontal component of surface velocity is derived on the basis of a systematic perturbation method with respect to the steepness parameter. A single NEE for the surface elevation is also presented. These equations are expressed in terms of original coordinate variables and therefore they have a direct relevance to physical systems. Since the formalism does not rely on the often used assumptions of shallow water and long waves, the NEEs obtained are uniformly valid from shallow water to deep water and have wide applications in various wave phenomena of physical and engineering importance. The shallow- and deep-water limits of the equations are discussed and the results are compared with existing theories. It is found that our theory includes as specific cases almost all approximate theories known at present.