An evolution equation for nonlinear wave transformation

Nonlinear energy transfer through resonant interactions among gravity waves is an important phenomenon especially in the shoaling region. In this study, a new evolution equation is derived from the mild-slope equation that includes the nonlinear quadratic interaction terms. The resulting equation is used to effectively model the two-dimensional wave transformation (reflection, diffraction, refraction, dissipation, etc.) and the nonlinear interaction between frequency components. The evolution equation has linear characteristics of fully-dispersive waves and is applicable to a spectrum of waves over arbitrary water depths. The Alternating Direction Implicit (ADI) scheme is employed to solve the equation with appropriate boundary conditions. The computations are compared with the laboratory data and numerical models from literature. In general, an improved agreement is observed in data-model comparisons and in the applicability of the present equation.