Parametrically excited surface waves: two-frequency forcing, normal form symmetries, and pattern selection.

Motivated by experimental observations of exotic free surface standing wave patterns in the two-frequency Faraday experiment, we investigate the role of normal form symmetries in the associated pattern-selection problem. With forcing frequency components in ratio m/n, where m and n are coprime integers that are not both odd, there is the possibility that both harmonic waves and subharmonic waves may lose stability simultaneously, each with a different wave number. We focus on this situation and compare the case where the harmonic waves have a longer wavelength than the subharmonic waves with the case where the harmonic waves have a shorter wavelength. We show that in the former case a normal form transformation can be used to remove all quadratic terms from the amplitude equations governing the relevant resonant triad interactions. Thus the role of resonant triads in the pattern-selection problem is greatly diminished in this situation. We verify our general bifurcation theoretic results within the example of one-dimensional surface wave solutions of the Zhang-Viñals model [J. Fluid Mech. 341, 225 (1997)] of the two-frequency Faraday problem. In one-dimension, a 1:2 spatial resonance takes the place of a resonant triad in our investigation. We find that when the bifurcating modes are in this spatial resonance, it dramatically effects the bifurcation to subharmonic waves in the case that the forcing frequencies are in ratio 1/2; this is consistent with the results of Zhang and Viñals. In sharp contrast, we find that when the forcing frequencies are in a ratio 2/3, the bifurcation to (sub)harmonic waves is insensitive to the presence of another spatially resonant bifurcating mode. This is consistent with the results of our general analysis.