Two-frequency forced Faraday waves: weakly damped modes and pattern selection

Recent experiments [A. Kudrolli, B. Pier, J.P Gollub, Physica D 123 (1998) 49-111] on two-frequency parametrically excited surface waves produced an i ntriguing "superlattice" wave pattern near a codimension-two bifurcation po int where both subharmonic and harmonic waves onset simultaneously, but wit h different spatial wave numbers. The superlattice pattern is synchronous w ith the forcing, spatially periodic on a large hexagonal lattice, and exhib its small-scale triangular structure. Similar patterns have been shown to e xist as primary solution branches of a generic 12-dimensional D-6+T-2-equiv ariant bifurcation problem, and may be stable if the nonlinear coefficients of the bifurcation problem satisfy certain inequalities [M. Silber, M.R.E. Proctor, Phys. Rev. Lett. 81 (1998) 2450-2453]. Here we use the spatial an d temporal symmetries of the problem to argue that weakly damped harmonic w aves may be critical to understanding the stabilization of this pattern in the Faraday system. We illustrate this mechanism by considering the equatio ns developed by Zhang and Vinals [J. Fluid Mech. 336 (1997) 301-330] for sm all amplitude, weakly damped surface waves on a semi-infinite fluid layer. We compute the relevant nonlinear coefficients in the bifurcation equations describing the onset of patterns for excitation frequency ratios of 2/3 an d 6/7. For the 2/3 case, we show that there is a fundamental difference in the pattern selection problems for subharmonic and harmonic instabilities n ear the codimension-two point. Also, we find that the 6/7 case is significa ntly different from the 2/3 case due to the presence of additional weakly d amped harmonic modes. These additional harmonic modes can result in a stabi lization of the superpatterns. (C) 2000 Elsevier Science B.V. All rights re served.