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Amplitude equation and pattern selection in Faraday waves
1Supercomputer Computations Research Institute, Florida State University, Tallahassee, Florida 32306-4130, USA.
Summary
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.
- 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.