PERTURBATION-THEORY OF PARAMETRICALLY DRIVEN CAPILLARY WAVES AT LOW-VISCOSITY

We present a critical review of the Hamiltonian and the Lagrangian the ories of pattern formation in driven capillary waves at low viscosity and high aspect ratio. We construct a Hamiltonian perturbation theory in the spirit of Milner's (1991) formulation, and derive the amplitude equations and their coefficients relevant at the onset of surface wav es. Our presentation is detailed, and we carefully point out the diffe rences between our results for the nonlinear coefficients and the resu lts obtained by others. From our standing wave analysis we find that t he square pattern is subcritical. Among the supercritical standing wav e patterns, we find that the eightfold quasi-crystalline pattern, obse rved by Christiansen et al. (1992) and by Bosch (1995), is more stable than both rolls and hexagons. We outline the high-aspect-ratio experi mental results obtained so far, and discuss them in the light of the t heory.