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.

Parametrically Forced Surface Waves

“…, We have already noticed that progressive plane waves in the neighborhood of one of the cutoff frequencies of a long channel may induce cross waves that exhibit a slow spatial modulation in the direction of propagation. The corresponding analytical problem has been considered by Mahony (1972), Jones (1984) The somewhat simpler problem of spatially modulated cross waves excited through vertical oscillation of a channel has been studied by Larraza & Putterman (1984) and Miles (l984b), who obtain results in quantitative agreement with the experiments of Wu et a1. (1984).…”

The mode number dependence of neutral stability of cross-waves

“…Kit and Shemer (1989), as well as some other works dealing with the long-time evolution of nonlinear resonant standing waves (Lichter and Chen 1987;Shemer andKit 1988, 1989) have shown that both dissipation along the tank and at the wavemaker have to be taken into account in order to obtain agreement between the theoretical and the experimental results. Lichter and Bernoff (1988) have indicated that at sufficiently strong forcing, the neutral stability curve will asymptote to the lines of constant 2, while Kit and Shemer (1989) have demonstrated that the asymptotic value of 2 can be considerably different from the one predicted by Mahony (1972) due to dissipation at the wavemaker and along the tank.…”

“…Mahony (1972) has shown that in the absence of dissipative effects, crosswaves are linearly unstable for I AI < 1. Kit and Shemer (1989), as well as some other works dealing with the long-time evolution of nonlinear resonant standing waves (Lichter and Chen 1987;Shemer andKit 1988, 1989) have shown that both dissipation along the tank and at the wavemaker have to be taken into account in order to obtain agreement between the theoretical and the experimental results.…”

Methods for Boundary-Value Problems in Free-Surface Flows: The Third David W. Taylor Lecture, 27 August through 19 September 1974,

“…It is not the same as a similar phenomenon known as "cross-waves," recently studied by Garrett (1970), Mahony (1972), and others (Garrett gives a history of this phenomenon).…”