Cross-waves. Part 1. Theory

Abstract

An instability mechanism, leading to the generation of cross-waves in a closed channel, was examined recently by Garrett (1970). His theory is not applicable to long channels where the wavemaker produces a primary field which is a progressive wave train. In such cases, the heaving of the mean surface, of considerable significance in the instability mechanism, is confined to the nonpropagating field near the wavemaker. Here the theory of resonant interactions is extended to describe the energy transfer from this forced localized field to the cross-wave field. There are close analogies between the present results and Garrett's, although the resonant bandwidth estimated here is an order of magnitude smaller. The theory indicates that nonlinear effects may control the decay of cross-waves down the channel.