Boussinesq Model for Weakly Nonlinear Fully Dispersive Water Waves

Abstract

In the present work a new Boussinesq dispersive wave propagation model is proposed. The model is based on a system of equations expressed in terms of the free-surface elevation and the depth-averaged horizontal velocities. The approach is developed for fully dispersive and weakly nonlinear irregular waves propagating over any constant water depth in two horizontal dimensions, but it can also be applied in mildly sloping beaches with considerable accuracy. The model in its two-dimensional formulation involves in total five terms in each momentum equation, including the classical shallow water terms and only one frequency dispersion term. The latter is expressed through convolution integrals, which are estimated using appropriate impulse functions. The formulation is fully explicit in space and thus no inversion is required for the numerical solution. The model is applied to simulate the propagation of regular and irregular waves using a simple explicit scheme of finite differences. Numerical integration of a convolution integral is also required. The results of the simulations are compared with experimental data, as well as with linear and nonlinear wave theory. The comparisons show that the method is capable of simulating weakly nonlinear dispersive wave propagation over finite constant or slowly diminishing water depth in a satisfactory way.