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 pf normal form symmetries in the associated pattern-selection problem. With forcing frequency components in ratio min, where m and n are coprime integ ers that are not both odd, there is the possibility that both harmonic wave s and subharmonic waves may lose stability simultaneously, each with a diff erent wave number. We focus on this situation and compare the case where th e harmonic waves have a longer wavelength than the subharmonic waves with t he 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 qua dratic terms from the amplitude equations governing the relevant resonant t riad interactions. Thus the role of resonant triads in the pattern-selectio n problem is greatly diminished in this situation. We verify our general bi furcation theoretic results within the example of one-dimensional surface w ave solutions of the Zhang-Vinals model [J. Fluid Mech. 341, 225 (1997)] of the two-frequency Faraday problem. In one-dimension, a 1:2 spatil resonanc e takes the place of a resonant triad in our investigation. We find that wh en the bifurcating modes are in this spatial resonance, it dramatically eff ects the bifurcation to subharmonic waves in the case that the forcing freq uencies are in ratio 1/2; this is consistent with the results of Zhang and Vinals. 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 made. This is consistent with the results of our general analysis. [S1063-651X(99)01505-6].