Application of time-dependent wave equations to random waves over ripple patch

In a linear dispersive system, the combined effect of water wave frnnsformations such as refraction, diffraction, shoaling, and reflection can be predicted by the mild-slope equation which was developed by Berkhoff (1972) using the Galerkin-eigenfunction method. In the derivation of the equation, he assumed a mild slope of the bottom $\nabla$h/kh << 1 (where $\nabla$ is the horizontal gradient operator, k is the wavenumber, and h is the water depth) and thus neglected second-order bottom effect terms proportional to O($\nabla$h)$^2$ and O($\nabla$$^2$h). (omitted)