Institute for Mathematics and Its Applications
Jerry Gollub, Haverford College
An experimental survey will be presented of both the primary patterns and the secondary instabilities of parametrically forced surface waves (Faraday waves) in the large system limit. The symmetry of the primary pattern (stripes, squares, or hexagons) depends on viscosity n and driving frequency fo. Hexagons are observed at low fo over the whole viscosity range despite the subharmonic symmetry that tends to suppress them. Possible mechanisms for the occurrence of hexagons for single frequency forcing are discussed. Phase defects occur between hexagonal domains differing in temporal phase by p (with respect to the forcing). Patterns of different symmetry coexist in certain parameter ranges. Some of our observations have been explained theoretically by Chen and Vi=F1als using systematically derived amplitude equations. The basic physics governing the relative stability of the different ordered states can be explained qualitatively in terms of nonlinear wave interactions.
The transition to spatiotemporal chaos (STC) depends on the symmetry of the primary patterns. The Hexagonal patterns undergo a order/disorder transition in which the angular anisotropy in Fourier space declines continuously to zero. Striped patterns at high viscosity become unstable via transverse amplitude modulations in regions of high curvature; this instability results in a spatially nonuniform mixed state in which domains of STC coexist with stripes. This phenomenon may be understood in terms of a critical curvature that depends on the acceleration. An oscillatory zig-zag secondary instability of the striped pattern is also observed at intermediate viscosities. At smaller aspect ratios, we find that the onset of spatiotemporal chaos is strongly affected by the pattern's symmetry, with symmetric patterns being more stable.
When forced at two different frequencies, a variety of additional wave patterns are realized, including twelve-fold quasicrystalline patterns composed of two hexagonal lattices aligned at 30 degrees relative to each other. Interactions between hexagonal lattices at other angles gives rise to striking "superlattice" patterns. In some cases the resulting patterns are neither simple standing nor traveling waves.