Nonlinear Dynamics of Random Surface Gravity Waves Over Water of Slowly Varying Depth

Abstract

The weakly nonlinear propagation of a random surface wave over a water region of varying depth h is considered. A random, stationary Gaussian process describing perturbations in the volume of the liquid is assumed at infinite depth. The characteristics of the nonlinear random wave are evaluated as the wave propagates into shallow coastal waters. It is found that the mean amplitude of the fundamental harmonic of the wave is proportional to h^7/8, while its variance is proportional to h^7/4 as the wave approaches the coastal zone. The  mean  amplitude  of  the  second  harmonic  is proportional to h^-1/4. Manifestation of nonlinear effects becomes abrupt when the water depth falls below a certain critical value. This behavior occurs for relatively small energy densities of the perturbation at infinite depth, as specified in the study. For larger energy densities, nonlinear effects increase gradually.