Perturbation theory of parametrically driven capillary waves at low viscosity

We present a critical review of the Hamiltonian and the Lagrangian theories 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 waves. Our presentation is detailed, and we carefully point out the differences between our results for the nonlinear coefficients and the results obtained by others. From our standing wave analysis we find that the square pattern is subcritical. Among the supercritical standing wave patterns, we find that the eightfold quasi-crystalline pattern, observed by Christiansen et al. (1992) and by Bosch (1995), is more stable than both rolls and hexagons. We outline the high-aspect-ratio experimental results obtained so far, and discuss them in the light of the theory.