Non-linear wave equations for free surface flow over a bump

Abstract

ABSTRACT This study aims to develop a new wave equation model by modifying the Fully-nonlinear and strongly-Dispersive Surface wave (FDS) equations. The modification was performed by applying a new expansion in a series of the vertical coordinate, zμ , to the velocity potential while a simple expansion in a series of z was applied to the FDS equations. Verification of the model was conducted by comparing with the theoretical solutions of surface solitary waves. We applied the modified FDS equations to wave fields over a bump under conditions with and without currents, which agreed very well with the time series of wave heights and velocity obtained from laboratory experiments. The dispersion relationship computed using the normalized modified FDS equations also agreed very well with the theoretical solution when we gave the number of expansion terms as 3 with μ = 2.5. Additionally, the profile of surface waves computed with the modified FDS equation was shown to have a larger width ridge, a bulbous-type wave, by comparing with a Trochoidal wave under the condition of waves against a current.