Seaquake waves: Standing wave dynamics with Faraday excitation and radiative loss

When a body of deep water is subjected to vertical oscillations with frequency $w0$, linear theory predicts that a standing wave field with frequency $w=w02$ forms on the surface. The wave number $k*=w2g$ is determined by the deep water dispersion relation. These waves are called Faraday waves after Michael Faraday [16] who first documented them in 1831. Since then, much work has been done to try and understand them. However, all the existing results are for high frequency capillary waves subjected to viscous damping. What is observed is a variety of standing wave patterns that may be stabilized for various forcing strengths and viscosities. We analyze a system in which the dominant mechanism for energy dispersal is radiation rather than viscosity. We suggest this as a first approximation to the transient motions associated with earthquakes at sea and find that the results give insight into seaquake dynamics and the complex interactions involved with water wave formation. These assumptions introduce a number of challenges along with introducing interesting dynamics. In particular, phase dynamics arise that lead to remarkable quasi-stationary states that evolve on a slow time scale.