### Abstract

A nonlinear theory of pattern selection in parametric surface waves (Faraday waves) is presented that is not restricted to small viscous dissipation. By using a multiple scale asymptotic expansion near threshold, a standing wave amplitude equation is derived from the governing equations. The amplitude equation is of gradient form, and the coefficients of the associated Lyapunov function are computed for regular patterns of various symmetries as a function of a viscous damping parameter [Formula Presented]. For [Formula Presented], the selected wave pattern comprises a single standing wave (stripe pattern). For [Formula Presented], patterns of square symmetry are obtained in the capillary regime (large frequencies). At lower frequencies (the mixed gravity-capillary regime), a sequence of sixfold (hexagonal), eightfold, [Formula Presented] patterns are predicted. For even lower frequencies (gravity waves) a stripe pattern is again selected. Our predictions of the stability regions of the various patterns are in quantitative agreement with recent experiments conducted in large aspect ratio systems.

Original language | English (US) |
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Pages (from-to) | 559-570 |

Number of pages | 12 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 60 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1999 |

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**Amplitude equation and pattern selection in Faraday waves.** / Chen, Peilong; Viñals, Jorge.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Amplitude equation and pattern selection in Faraday waves

AU - Chen, Peilong

AU - Viñals, Jorge

PY - 1999/1/1

Y1 - 1999/1/1

N2 - A nonlinear theory of pattern selection in parametric surface waves (Faraday waves) is presented that is not restricted to small viscous dissipation. By using a multiple scale asymptotic expansion near threshold, a standing wave amplitude equation is derived from the governing equations. The amplitude equation is of gradient form, and the coefficients of the associated Lyapunov function are computed for regular patterns of various symmetries as a function of a viscous damping parameter [Formula Presented]. For [Formula Presented], the selected wave pattern comprises a single standing wave (stripe pattern). For [Formula Presented], patterns of square symmetry are obtained in the capillary regime (large frequencies). At lower frequencies (the mixed gravity-capillary regime), a sequence of sixfold (hexagonal), eightfold, [Formula Presented] patterns are predicted. For even lower frequencies (gravity waves) a stripe pattern is again selected. Our predictions of the stability regions of the various patterns are in quantitative agreement with recent experiments conducted in large aspect ratio systems.

AB - A nonlinear theory of pattern selection in parametric surface waves (Faraday waves) is presented that is not restricted to small viscous dissipation. By using a multiple scale asymptotic expansion near threshold, a standing wave amplitude equation is derived from the governing equations. The amplitude equation is of gradient form, and the coefficients of the associated Lyapunov function are computed for regular patterns of various symmetries as a function of a viscous damping parameter [Formula Presented]. For [Formula Presented], the selected wave pattern comprises a single standing wave (stripe pattern). For [Formula Presented], patterns of square symmetry are obtained in the capillary regime (large frequencies). At lower frequencies (the mixed gravity-capillary regime), a sequence of sixfold (hexagonal), eightfold, [Formula Presented] patterns are predicted. For even lower frequencies (gravity waves) a stripe pattern is again selected. Our predictions of the stability regions of the various patterns are in quantitative agreement with recent experiments conducted in large aspect ratio systems.

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U2 - 10.1103/PhysRevE.60.559

DO - 10.1103/PhysRevE.60.559

M3 - Article

C2 - 11969795

AN - SCOPUS:18144379623

VL - 60

SP - 559

EP - 570

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

SN - 1539-3755

IS - 1

ER -