Numerical study of pattern formation in weakly damped parametric surface waves

We present a numerical study of parametrically driven surface waves in fluids of low viscosity. The equations governing fluid motion in the bulk and the appropriate boundary conditions at the free surface are approximated by a nonlocal set of equations involving surface coordinates alone, thus allowing the numerical analysis of the large aspect ratio limit (the aspect ratio being the ratio between the size of the system and the wavelength of the waves). The numerical calculation supports the predictions of a previously given multi-scale asymptotic analysis, yielding stable patterns of square symmetry above onset in the capillary dominated regime. In the case of mixed capillary-gravity waves, patterns of eight-fold symmetry are obtained, again in agreement with the predictions of the asymptotic expansion. These calculations provide indirect evidence about the stability of such patterns, analysis that is too involved to be carried out analytically.