Hecei\'cd 3~Iarch t9n2 llnd in rc\-ised form 21 September 1992) 671 \ I The standing waves of frequency wand wavenumber k that are induced on the surface of a liquid of depth d that is subjected to the vertical displacement a, cos 2wt are determined on the assumptions t1mt: the effects of lateral boundaries are negligible: £ = ka, tanh kd~1 and 0 < £-0 = 0(0'), where 0 is the linear damping ratio of a free wave of frequency w; the ""wes form a square pattern (which follows from observation).

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... s problem. which goes back to Faraday (1831) , has recently been treated by Ezerskii el al. (1986) and Milner (1991) in the limit of deepwater capillary waves (kd, kl.~I, where I. is the capillary length). Ezerskii et al. show that the square pattern is unstable fOI' sufficiently large £-0, and Mil.nel· shows that nonlinear damping is necessary for equilibration of the square pattern. The present fOl'nlliiation 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-Ql'der damping. There are quantitative ditl'el'ences in the resulting evolution equations (for kd, kl.~I), \\'hich appeal' 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 energytransfer rate to the Faraday wave selects the ma.ximum of the resonance curve in a. frequency-amplitude plane.