Resonances and superlattice pattern stabilization in two-frequency forced Faraday waves

We investigate the role weakly damped modes play in the selection of Faraday wave patterns forced with rationally related frequency components mω and nω. We use symmetry considerations to argue for the special importance of the weakly damped modes oscillating with twice the frequency of the critical mode, and those oscillating primarily with the “difference frequency” |n−m|ω and the “sum frequency” (n+m)ω. We then perform a weakly nonlinear analysis using equations of Zhang and Viñals [J. Fluid Mech. 336 (1997) 301] which apply to small-amplitude waves on weakly inviscid, deep fluid layers. For weak damping and forcing and one-dimensional waves, we perform a perturbation expansion through fourth-order which yields analytical expressions for onset parameters and the cubic bifurcation coefficient that determines wave amplitude as a function of forcing. For stronger damping and forcing we numerically compute these same parameters, as well as the cubic cross-coupling coefficient for competing standing waves oriented at an angle θ relative to each other. The resonance effects predicted by symmetry are borne out in the perturbation results for one spatial dimension, and agree with the numerical results for two dimensions. The difference frequency resonance plays a key role in stabilizing superlattice patterns of the SL-I type observed by Kudrolli et al. [Physica D 123 (1–4) (1998) 99].