Weakly nonlinear resonant waves generated by an oscillatory pressure distribution in a channel

This paper analyzes the problem of wave generation in a deep, long channel by an oscillating pressure patch. Special attention is paid to the free-surface response in the neighborhood of the resonance where the frequency of the forced oscillation is equal to one of the natural frequencies of the channel. The linearized solution for an antisymmetric wave generation, which yields an infinite response at the cut-off frequency, is here modified to include contributions from the most-singular third-order terms. The resulting generalized solution is uniform and renders a finite response at critical conditions. The modified dispersion relationship, which incorporates a phase shift in the wave frequency, yields a finite value for the group velocity, hence providing the means for carrying energy away from the disturbance. The case of symmetric wave generation, resulting in a progressive wave moving down the channel, and the related subharmonic resonance are also briefly discussed. A comparison between theory and some laboratory experiments is presented.