Faraday Wave Pattern Selection via Multi-Frequency Forcing
by
Chad M. Topaz
University of California, Los Angeles
Coauthors: Jeff Porter (University of Leeds) and Mary Silber (Northwestern University)

We study pattern selection in Faraday waves excited by a time-periodic forcing function containing arbitrarily many frequency components. We focus on the role of resonant triads formed by two critical standing wave modes and a damped, resonant mode. For the case of weak damping and weakly nonlinear waves, we exploit weakly broken time translation, time reversal, and Hamiltonian symmetries to determine which modes are the most important for pattern selection, i.e. which ones are favored by a strong nonlinear coupling. We also determine how the strength of the triad interaction depends on the forcing parameters (frequencies, amplitudes, and phases), and in many cases, if the interaction has an enhancing or suppressing effect on associated patterns. Surprisingly, there are at most five forcing frequency components relevant for each of the important triads. The relative phases of those forcing components play a key role, sometimes making the difference between an enhancing and suppressing effect. We use our results to interpret recent experiments on multi-frequency-forced Faraday waves, and to suggest how one might design periodic forcing functions to favor chosen patterns in the laboratory.

Date received: October 14, 2003