Growth Rates and Equilibrium Amplitudes of Forced Seiches in Closed Basins

A weakly nonlinear analysis of two-dimensional, standing waves in a rectangular basin of arbitrary depth is presented. The waves are resonated by periodic oscillation along an axis aligned with the wavenumber vector. First, linear analysis is pursued in order to determine the growth rate of the resonated wave as a function of the basin dimensions, mode number, forcing amplitude, and fluid parameters. The effects of viscosity and frequency mismatch (detuning) are considered. It is found that waves in transitional depth have the largest growth rates. Experiments are found to agree extremely well with the theoretical results. Second, cubic nonlinearity is considered in order to describe limits on the maximum amplitude of the resonated wave that can not be accounted for by viscosity or initial detuning. Results indicate that, for the case of perfect resonance, the maximum amplitude can be two orders of magnitude larger than the forcing amplitude. Theoretical predictions of amplitude response curves confirm the nonlinear `frequency reversal' that has been demonstrated previously. Experiments again agree very well with the theory.