Square Patterns and Quasi-patterns in Weakly Damped Faraday Waves

Details Square Patterns and Quasi-patterns in Weakly Damped Faraday Waves Square Patterns and Quasi-patterns in Weakly Damped Faraday Waves

1996-03-19

arxiv.org/abs/patt-sol/9603005v1

Phys. Rev. E 53, R4283 (1996)

Nonlinear Sciences

Pattern Formation and Solitons

RevTeX, 11 pages, 8 figures

Scientific paper

10.1103/PhysRevE.53.R4283

Pattern formation in parametric surface waves is studied in the limit of weak viscous dissipation. A set of quasi-potential equations (QPEs) is introduced that admits a closed representation in terms of surface variables alone. A multiscale expansion of the QPEs reveals the importance of triad resonant interactions, and the saturating effect of the driving force leading to a gradient amplitude equation. Minimization of the associated Lyapunov function yields standing wave patterns of square symmetry for capillary waves, and hexagonal patterns and a sequence of quasi-patterns for mixed capillary-gravity waves. Numerical integration of the QPEs reveals a quasi-pattern of eight-fold symmetry in the range of parameters predicted by the multiscale expansion.

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