A New Model for Three-Dimensional Nonlinear Dispersive Long Waves

Abstract

A new formulation of the pair of Boussinesq-class equations for modelling the propagation of three-dimensional nonlinear dispersive long water waves is presented. This set of model equations permits spatial and temporal variations of the bottom topography. Further, the two resultant equations may be combined into a single equation through the introduction of an irrotational layer-mean velocity. An exact permanent-form solution is derived for the combined equation, which is still of the Boussinesq-class and includes reflection. This solution for the surface height is found to describe a slightly wider wave than the permanent form solution to the uni-directional Korteweg-deVries Equation. A numerical scheme using an implicit finite-difference method is developed to solve the combined equation for propagation over fixed sloping bottom topography. The scheme is tested for various grid sizes using the permanent-form solution, and an oscillatory tail is seen to develop as a result of insufficient mesh refinement. Several cases of wave propagation over a straight sloping ramp onto a shelf are solved using the permanent-form solution as initial conditions and the results are found to be in good agreement with previous results obtained by using either the Boussinesq dual-equation set or the single Korteweg-deVries equation. The combined equation is used to solve the related problem in two horizontal dimensions of a wave propagating in a channel having a curved-ramp bottom topography. Depending on the specific topography, focussing or defocussing occurs and the crest is selectively amplified. Indications of cross-channel oscillation are presented. Linear, nondispersive theory is used to solve a case with identical topographical features and initial condition. The solutions using the simplified theory are found to be considerably different from the results for nonlinear, dispersive theory with respect to the overall three-dimensional wave shape as well as in the areas of crest amplification, soliton formation and cross-channel effects.