Unsteady and nonlinear effects near the cusp lines of the Kelvin ship-wave pattern

Abstract

According to the linearized water-wave theory, a localized pressure source travelling at constant speed on the surface of deep water generates the classical Kelvin ship-wave pattern, which follows behind the source and is confined within a sector of half-angle equal to 19.5°. In this paper, an asymptotic theory is developed which takes into account finite-amplitude and unsteady effects near the boundaries of the Kelvin sector, the so-called cusp lines, where the far-field wave disturbance takes the form of a modulated wavepacket. A nonlinear equation governing the spatial and temporal evolution of the wavepacket envelope is derived. It is shown that, for a pressure source turned on impulsively, a nonlinear steady state is reached. All unsteady effects are found in a region of finite extent which moves away from the source. Numerical calculations indicate that the steady-state nonlinear response is very similar to the steady-state linear response.