Non--linear Long Waves Generated by a Moving Pressure Disturbance

The evolution of long waves generated by a pressure disturbance acting on an initially unperturbed free surface in a channel of finite depth is analysed. Both off-critical and transcritical conditions are considered in the context of the fully nonlinear inviscid problem. The solution is achieved by using an accurate boundary integral approach and a time-stepping procedure for the free-surface dynamics. The discussion emphasizes the comparison between the present results and those provided by both the Boussinesq and the related Korteweg-de Vries model. For small amplitudes of the forcing, the predictions of the asymptotic theories are essentially confirmed. However, for finite intensities of the disturbance, several new features significantly affect the physical results. In particular, the interaction among different wave components, neglected in the Korteweg-de Vries approximation, is crucial in determining the evolution of the wave system. A substantial difference is indeed observed between the solutions of the Korteweg-de Vries equation and those of both the fully nonlinear and the Boussinesq model. For increasing dispersion and fixed nonlinearity the agreement between the Boussinesq and fully nonlinear description is lost, indicating a regime where dispersion becomes dominant. Consistently with the long-wave modelling, the transcritical regime is characterized by an unsteady flow and a periodic emission of forward-running waves. However, also in this case, quantitative differences are observed between the three models. For larger amplitudes, wave steepening is almost invariably observed as a precursor of the localized breaking commonly detected in the experiments. The process occurs at the crests of either the trailing or the upstream-emitted wave system for Froude numbers slightly sub- and super-critical respectively.