Amplitude equation and pattern selection in Faraday waves.

A nonlinear theory of pattern selection in parametric surface waves (Faraday waves) is presented that is not restricted to small viscous dissipation. By using a multiple scale asymptotic expansion near threshold, a standing wave amplitude equation is derived from the governing equations. The amplitude equation is of gradient form, and the coefficients of the associated Lyapunov function are computed for regular patterns of various symmetries as a function of a viscous damping parameter gamma. For gamma approximately 1, the selected wave pattern comprises a single standing wave (stripe pattern). For gamma<<1, patterns of square symmetry are obtained in the capillary regime (large frequencies). At lower frequencies (the mixed gravity-capillary regime), a sequence of sixfold (hexagonal), eightfold, ...patterns are predicted. For even lower frequencies (gravity waves) a stripe pattern is again selected. Our predictions of the stability regions of the various patterns are in quantitative agreement with recent experiments conducted in large aspect ratio systems.