Reflection source term for the wave action equation

The wave action equation is a widely used governing equation in wave forecasting models. Whilst it accounts for wave propagation, its analytical derivation neglects various other effects, such as the sea bottom reflection. The present work derives an analytical source term for the bottom reflection of oblique incident waves to be used in numerical forecasting models. This is done by means of a coupled oblique parabolic approximation of the mild-slope equation, which is then decoupled by introducing a perturbation solution. The resulting first two orders produce the wave action equation itself for an on-going wave in the first order and a wave action equation with a reflection source term in the second order for the reflected wave. A method to implement this source term in two-dimensional wave action forecasting models is discussed. Numerical simulations show this new source term to be in excellent agreement with the mild-slope equation for different slopes, wave periods and attack angles.