Two-frequency forced Faraday waves: Weakly damped modes and pattern selection

Mary Silber*, Chad M. Topaz, Anne C. Skeldon

*Corresponding author for this work

Research output: Contribution to journalArticle

48 Scopus citations

Abstract

Recent experiments [A. Kudrolli, B. Pier, J.P. Gollub, Physica D 123 (1998) 99-111] on two-frequency parametrically excited surface waves produced an intriguing "superlattice" wave pattern near a codimension-two bifurcation point where both subharmonic and harmonic waves onset simultaneously, but with different spatial wave numbers. The superlattice pattern is synchronous with the forcing, spatially periodic on a large hexagonal lattice, and exhibits small-scale triangular structure. Similar patterns have been shown to exist as primary solution branches of a generic 12-dimensional D6+̇T2-equivariant bifurcation problem, and may be stable if the nonlinear coefficients of the bifurcation problem satisfy certain inequalities [M. Silber, M.R.E. Proctor, Phys. Rev. Lett. 81 (1998) 2450-2453]. Here we use the spatial and temporal symmetries of the problem to argue that weakly damped harmonic waves may be critical to understanding the stabilization of this pattern in the Faraday system. We illustrate this mechanism by considering the equations developed by Zhang and Viñals [J. Fluid Mech. 336 (1997) 301-330] for small amplitude, weakly damped surface waves on a semi-infinite fluid layer. We compute the relevant nonlinear coefficients in the bifurcation equations describing the onset of patterns for excitation frequency ratios of 23 and 67. For the 23 case, we show that there is a fundamental difference in the pattern selection problems for subharmonic and harmonic instabilities near the codimension-two point. Also, we find that the 67 case is significantly different from the 23 case due to the presence of additional weakly damped harmonic modes. These additional harmonic modes can result in a stabilization of the superpatterns.

Original languageEnglish (US)
Pages (from-to)205-225
Number of pages21
JournalPhysica D: Nonlinear Phenomena
Volume143
Issue number1-4
DOIs
StatePublished - Sep 1 2000

Keywords

  • Faraday waves
  • Pattern selection
  • Resonance triads
  • Superlattice patterns

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

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