Gravity-Capillary Surface Waves

Abstract

We consider in this chapter the weakly nonlinear dynamics of a wave train propagating at the surface of a liquid, a question also referred to as the water-wave problem. When the wave is modulated in the direction of propagation only, the long-time large-scale dynamics of the envelope is governed by the cubic Schrödinger equation in one space dimension (see Zakharov 1968b for the case of infinitely deep water, Hasimoto and Ono 1972 for the extension to a finite depth, and Kawahara 1975 for the inclusion of the surface tension). We review here the more general case where perturbations in the transverse directions are also permitted. In this case, a mean flow and a mean elevation of the free surface are induced by the amplitude modulation of the wave. The original derivations of the envelope equations in the case of pure gravity waves are due to Benney and Roskes (1969), who retained the dynamics of the mean fields, and Davey and Stewartson (1974), who consider these fields on the long-time scale where they are slaved to the amplitude modulation (see also Johnson 1997). It is noticeable that the former system is identical with that arising in situations where the carrying wave modulation drives low-frequency acoustic waves (Zakharov and Rubenchik 1972, Zakharov and Schulman 1991), thus providing another example of the canonical character of the amplitude equations. The effect of surface tension was included by Djordjevic and Redekopp (1977) and Ablowitz and Segur (1979).