Energy properties of regular water waves over horizontal bottom with increasing nonlinearity

Abstract

The energy of surface water waves is a very important concept. The most common method for calculating the wave energy is based on the linear or first-order Stokes wave theory. This study investigated the energy density (i.e., the potential, kinetic, and total energy) of very nonlinear waves for a finite water depth and deep water over horizontal bottoms. Analytical expressions for the wave energy density were derived based on the nonlinear Stokes wave theories using the perturbation expansion method and Taylor expansion. The energy density was accurate to the fourth order and sixth order for the second- and third-order Stokes wave theories, respectively. As for the fifth-order Stokes wave theory, higher-order energy density expressions were proposed, which were accurate to the tenth order. Compared with previous findings, the present fifth-order Stokes wave energy density was more consistent with the numerical integration result. Then, the homotopy analysis method (HAM) was applied to obtain the energy density. According to the expressions for the nonlinear wave energy density, the energy density was no longer a single-variable function of wave height and varied with wave parameters such as the wave height and wave number. In addition, the kinetic energy and potential energy were not equal, and the former was larger than the latter.