Physica D
**146** (2000) 367-387.
doi:10.1016/S0167-2789(00)00124-X
## Spatial period-multiplying instabilities of hexagonal Faraday waves

D.P. Tse(1),
A.M.Rucklidge(1),
R.B. Hoyle(1)
and
M. Silber(2).

(1) Department of Applied Mathematics and Theoretical Physics,

University of Cambridge, Cambridge, CB3 9EW, UK

(2) Department of Engineering Sciences and Applied Mathematics,

Northwestern University, Evanston, IL 60208, USA

**Abstract.**
A recent Faraday wave experiment with two-frequency forcing reports two types
of `superlattice' patterns that display periodic spatial structures having two
separate scales [1]. These patterns both arise as secondary states once the
primary hexagonal pattern becomes unstable. In one of these patterns (so-called
`superlattice-II') the original hexagonal symmetry is broken in a subharmonic
instability to form a striped pattern with a spatial scale increased by a
factor of 2sqrt{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 (sqrt{3} larger than the original scale) and apparently has 60
degree 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-II pattern presented in [1] and the
subharmonic nature of the secondary instability, we show (a) that the
superlattice-II pattern can bifurcate stably from standing hexagons; (b) that
the pattern has a spatio-temporal symmetry not reported in [1]; 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
degree 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.

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