On the excitation of long nonlinear water waves by a moving pressure distribution

A study is made of the wave disturbance generated by a localized steady pressure distribution travelling at a speed close to the long-water-wave phase speed on water of finite depth. The linearized equations of motion are first used to obtain the large-time asymptotic behaviour of the disturbance in the far field; the linear response consists of long waves with temporally growing amplitude, so that the linear approximation eventually breaks down owing to finite-amplitude effects. A nonlinear theory is developed which shows that the generated waves are actually of bounded amplitude, and are governed by a forced Korteweg-de Vries equation subject to appropriate asymptotic initial conditions. A numerical study of the forced Korteweg-de Vries equation reveals that a series of solitons are generated in front of the pressure distribution.