A Unified Model for Periodic Non-Linear Dispersive Waves in Intermediate and Shallow Water

Abstract

A numerical model for non-linear dispersive monochromatic wave propagation is developed in this work. The new model has a unified form, being valid in shallow as well as in intermediate water. The approach is based on the expansion of the vertical velocity in power series and on an analytical solution of the Laplace equation. It has a similar form with two types of the Boussinesq equations but instead of the constant coefficient 1/3 (or 1/15) in the momentum equation it is proposed a function of the water depth and the wave period. The continuity equation, which is exact in deep, intermediate and shallow water without any restriction in nonlinearity, remains unchanged. In the momentum equation terms of order up to O(εσ²) - with ε=H/d, σ=d/L  (H=wave height, d=water depth, L=wave length)-are considered. The horizontal and the vertical velocity as well as the pressure distribution are given in relation to the wave period and the instantaneous depth averaged horizontal velocity. The model is validated both in intermediate and shallow water against the non-linear theory and experimental data.