Nonlinear Forced Water Waves in a Shallow Channel near a Cut-off Frequency

Abstract

A theoretical study is presented of the free-surface flow induced by a wavemaker performing torsional oscillations about a vertical axis in a shallow channel near a cut-off frequency. It is shown, through a perturbation analysis using characteristic variables, that the nonlinear response is governed by a forced Kadomtsev-Petviashvili (KP) equation with periodic boundary conditions across the channel; this nonlinear initial-boundary-value problem is investigated analytically and numerically. When surface-tension effects are negligible, so that dispersion in the KP is negative, it is found that the nonlinear response reaches a steady state and exhibits jump phenomena, typical of nonlinear forced oscillations. On the other hand, in the high-surface-tension regime, when dispersion in the KP is positive, there is no sign of a steady state and the response undergoes a recurrence phenomenon.