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Amplitude equation and pattern selection in Faraday waves.

P Chen1, J Viñals

  • 1Supercomputer Computations Research Institute, Florida State University, Tallahassee, Florida 32306-4130, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|April 24, 2002
PubMed
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This study presents a nonlinear theory for pattern selection in Faraday waves, revealing how viscous damping influences wave patterns. Different damping levels predict stripe, square, and hexagonal patterns, matching experimental results.

Area of Science:

  • Fluid dynamics
  • Nonlinear dynamics
  • Pattern formation

Background:

  • Parametric surface waves, or Faraday waves, exhibit complex pattern selection.
  • Previous theories often assumed small viscous dissipation, limiting their applicability.
  • Understanding pattern selection is crucial for fluid dynamics and nonlinear systems.

Purpose of the Study:

  • To develop a nonlinear theory for Faraday wave pattern selection beyond small viscous dissipation.
  • To derive and analyze an amplitude equation for standing wave patterns.
  • To predict pattern selection as a function of viscous damping and frequency regimes.

Main Methods:

  • Utilized multiple scale asymptotic expansion near the pattern formation threshold.
  • Derived a gradient form amplitude equation from governing fluid equations.

Related Experiment Videos

  • Computed Lyapunov function coefficients for various pattern symmetries and viscous damping parameters (gamma).

    • Main Results:

    • Identified stripe patterns for gamma approximately 1 (high viscosity).
    • Predicted square patterns in the capillary regime (gamma << 1, high frequency).
    • Showcased hexagonal and other symmetries in the mixed gravity-capillary regime, and stripe patterns in the gravity regime.

    Conclusions:

    • The derived amplitude equation accurately predicts pattern selection across different viscous damping and frequency regimes.
    • Theoretical predictions show quantitative agreement with experimental observations in large aspect ratio systems.
    • The theory provides a comprehensive framework for understanding Faraday wave pattern selection.