On Faraday waves

The standing waves of frequency ω and wavenumber κ that are induced on the surface of a liquid of depth d that is subjected to the vertical displacement a o cos 2 wt are determined on the assumptions that: the effects of lateral boundaries are negligible; ε = ka 0 tanh kd [Lt ] 1 and 0 < ε−δ = O(δ 3 ), where δ is the linear damping ratio of a free wave of frequency ω; the waves form a square pattern (which follows from observation). This problem, which goes back to Faraday (1831), has recently been treated by Ezerskii et al. (1986) and Milner (1991) in the limit of deep-water capillary waves ( kd, kl * [Gt ] 1, where l * is the capillary length). Ezerskii et al. show that the square pattern is unstable for sufficiently large ε—δ, and Milner shows that nonlinear damping is necessary for equilibration of the square pattern. The present formulation extends those of Ezerskii et al. and Milner to capillary–gravity waves and finite depth and incorporates third-order parametric forcing, which is neglected in these earlier formulations but is comparable with third-order damping. There are quantitative differences in the resulting evolution equations (for kd, kl * [Gt ] 1), which appear to reflect errors in the earlier work. These formulations determine a locus of admissible waves, but they do not select a particular wave. The hypothesis that the selection process maximizes the energy-transfer rate to the Faraday wave selects the maximum of the resonance curve in a frequency-amplitude plane.