Transient and steady-state amplitudes of forced waves in rectangular basins

A weakly-nonlinear analysis of the transient evolution of two-dimensional, standing waves in a rectangular basin is presented. The waves are resonated by periodic oscillation along an axis aligned with the wavenumber vector. The amplitude of oscillation is assumed to be small with respect to the basin dimensions. The effects of detuning, viscous damping, and cubic nonlinearity are all simultaneously considered. Moreover, the analysis is formulated in water of general depth. Multiple-scales analysis is used in order to derive an evolution equation for the complex amplitude of the resonated wave. From this equation, the maximum transient and steady-state amplitudes of the wave are determined. It is shown that steady-state analysis will underestimate the maximum response of a basin set into motion from rest. Amplitude response diagrams demonstrate good agreement with previous experimental investigations. The analysis is invalid in the vicinity of the “critical depth” and in the shallow-water limit. A separate analysis, which incorporates weak dispersion, is presented in order to provide satisfactory results in shallow water.