Pattern formation in weakly damped parametric surface waves driven by two frequency components

A quasi-potential approximation to the Navier-Stokes equation for low viscosity fluids is developed to study pattern formation in parametric surface waves driven by a force that has two frequency components. A bicritical line separating regions of instability to either one of the driving frequencies is explicitly obtained, and compared with experiments involving a frequency ratio of 1/2. The procedure for deriving standing wave amplitude equations valid near onset is outlined for an arbitrary frequency ratio following a multiscale asymptotic expansion of the quasi-potential equations. Explicit results are presented for subharmonic response to a driving force of frequency ratio 1/2, and used to study pattern selection. Even though quadratic terms are prohibited in this case, hexagonal or triangular patterns are found to be stable in a relatively large parameter region, a fact that is in qualitative agreement with experimental results.