Extreme wave formation past a submerged circular sill: The role of linear spatial focusing and resonant trapping

Abstract

We investigate dispersive wave amplification past a submerged circular sill on an otherwise flat seabed. This phenomenon is important because it can generate large-amplitude waves near the sill, due to spatial focusing and resonant trapping in linear dispersive regime, endangering navigation. Based on the potential flow theory, the velocity potential is solved separately in the ocean region and in the sill region. Matching is then achieved by means of integral equations involving Galerkin expansion of the unknown velocity field at the border between the two regions. The model is successfully validated against known analytical expressions for long waves and a smoothed particle hydrodynamics numerical solution. Our results advance existing theories valid either for non-dispersive waves or for shallow submergence of the sill. We show that, for relatively short waves as compared to the ocean depth, the sill acts as a wave lens focusing energy behind it. Increasing the wavelength of the incident wave promotes transition from wave focusing behind the sill to partial trapping atop the sill. In intermediate water depth, the concurrence of focusing and partial trapping favors the emergence of extreme wave amplitudes that can exceed up to 6 times the amplitude of incident waves. We hypothesize that this phenomenon is the main cause of local peaks in skewness and kurtosis near a submerged circular shoal obtained in recent numerical simulations. Indications for further studies in the nonlinear regime are finally provided.