A class of exact nonlinear traveling wave solutions for shallow water with a non-stationary bottom surface

Abstract

Abstract The GN nonlinear shallow water wave equations developed by Green and Naghdi (1976) [5,6] are valid for a non-stationary, non-uniform bottom surface and a non-uniform pressure on the top surface. In contrast, the S nonlinear shallow water wave equations developed by Serre (1953) [12,13] for uniform depth and later generalized by Seabra-Santos et al. (1987) for non-uniform depth are limited to a stationary bottom surface and a uniform pressure applied to the top surface. This paper develops a class of exact nonlinear traveling wave solutions of the GN equations for a non-stationary, non-uniform bottom surface. Also, the explicit expressions for the pressure acting on the bottom surface and the average pressure through the depth in the GN equations are used to place physical restrictions on the motion which ensure that these pressures remain non-negative preventing cavitation.