Stochastically forced water waves in a circular basin

An analysis of wave motion under stochastic excitation is presented. The starting point is a Hamiltonian model of surface wave motion. This model is augmented with linear damping and stochastic forcing terms. An asymptotic scaling parameter is then introduced to show that the problem has three time scales. Integrability of the system is established for the case of two wave modes near 1: 1 resonance. Stochastic averaging is used to reduce the dimensions of the system from four to two and the evolution of surface waves is now described, over long time scales and small amplitude stochastic forcing, by the evolution of the integrals of motion, as a random process. The domain of the generator of this random process is characterized and it includes a 'gluing' boundary condition. The adjoint of the generator yields the Fokker-Planck partial-differential equation (FPE). This equation governs the time evolution of the joint probability density of the two integrals of motion. The coefficients of the FPE are calculated numerically and the steady-state solution is found using the finite-element method. Results of the analysis show a distinct peak in the probability density along one edge of the reduced domain.