Asymptotic solutions of the problem of steep capillary-gravitational Faraday waves on the interface between fluids

Abstract

We construct asymptotic solutions of the 2D problem on weakly nonlinear capillary-gravitational Faraday waves and the interface between two fluids (of distinct density) subjected to vertical oscillations. The problem is formulated in the Lagrange variables, and the solutions are sought in the form of expansions in powers of the small parameter characterizing the steepness of the waves, and it is assumed that the ratio of the amplitude of the vertical acceleration of the fluids to the acceleration of gravity is of the order of some power of this small parameter. We construct solutions in the domains of parameters of the problem that correspond to both ordinary regimes (in which one can restrict oneself to two terms of the expansion) and critical and near-critical regimes (in which one must consider fourth-order terms). The resonance curves and wave profiles of the waves of maximal steepness are represented graphically. The results for ordinary regimes are compared with the known asymptotic solutions for free standing waves and with data of laboratory experiments concerning Faraday waves.