We report novel superlattice wave patterns at the interface of a fluid layer
driven vertically. These patterns are described most naturally in terms of two
interacting hexagonal sublattices. Two frequency forcing at very large aspect
ratio is utilized in this work. A superlattice pattern ("superlattice-I")
consisting of two hexagonal lattices oriented at a relative angle of 22^o is
obtained with a 6:7 ratio of forcing frequencies. Several theoretical
approaches that may be useful in understanding this pattern have been proposed.
In another example, the waves are fully described by two superimposed hexagonal
lattices with a wavelength ratio of sqrt(3), oriented at a relative angle of
30^o. The time dependence of this "superlattice-II" wave pattern is unusual.
The instantaneous patterns reveal a time-periodic stripe modulation that breaks
the 6-fold symmetry at any instant, but the stripes are absent in the time
average. The instantaneous patterns are not simply amplitude modulations of the
primary standing wave. A transition from the superlattice-II state to a 12-fold
quasi-crystalline pattern is observed by changing the relative phase of the two
forcing frequencies. Phase diagrams of the observed patterns (including
superlattices, quasicrystalline patterns, ordinary hexagons, and squares) are
obtained as a function of the amplitudes and relative phases of the driving
accelerations.Comment: 15 pages, 14 figures (gif), to appear in Physica