P. Lyngshansen et P. Alstrom, PERTURBATION-THEORY OF PARAMETRICALLY DRIVEN CAPILLARY WAVES AT LOW-VISCOSITY, Journal of Fluid Mechanics, 351, 1997, pp. 301-344
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.