Spatial period-multiplying instabilities of hexagonal Faraday waves

A recent Faraday wave experiment with two-frequency forcing reports two types of ‘superlattice’ patterns that display periodic spatial structures having two separate scales [Physica D 123 (1998) 99]. These patterns both arise as secondary states once the primary hexagonal pattern becomes unstable. In one of these patterns (so-called ‘superlattice-two’) the original hexagonal symmetry is broken in a subharmonic instability to form a striped pattern with a spatial scale increased by a factor of 2√3 from the original scale of the hexagons. In contrast, the time-averaged pattern is periodic on a hexagonal lattice with an intermediate spatial scale (√3 larger than the original scale) and apparently has 60◦ rotation symmetry. We present a symmetry-based approach to the analysis of this bifurcation. Taking as our starting point only the observed instantaneous symmetry of the superlattice-two pattern presented in [Physica D 123 (1998) 99] and the subharmonic nature of the secondary instability, we show: (a) that a pattern with the same instantaneous symmetries as the superlattice-two pattern can bifurcate stably from standing hexagons; (b) that the pattern has a spatio-temporal symmetry not reported in [Physica D 123 (1998) 99]; and (c) that this spatio-temporal symmetry accounts for the intermediate spatial scale and hexagonal periodicity of the time-averaged pattern, but not for the apparent 60◦ rotation symmetry. The approach is based on general techniques that are readily applied to other secondary instabilities of symmetric patterns, and does not rely on the primary pattern having small amplitude.