Superlattice Patterns in Surface Waves
  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. 
A. Kudrolli, B. Pier, and J.P. Gollub