  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 much
less than 1, patterns of square symmetry are obtained in the capillary regime
(large frequencies). At lower frequencies (the mixed gravity-capillary regime),
a sequence of six-fold (hexagonal), eight-fold,... 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.
