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Weakly nonlinear analysis of impulsively-forced Faraday waves.

Anne Catllá1, Jeff Porter, Mary Silber

  • 1Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA. acatlla@math.duke.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 31, 2005
PubMed
Summary

Parametrically-excited surface waves exhibit amplitude suppression due to a 1:2 resonance, tunable by impulse spacing. This finding offers insights into nonlinear wave dynamics under impulsive forcing.

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Area of Science:

  • Fluid Dynamics
  • Nonlinear Dynamics
  • Wave Phenomena

Background:

  • Parametrically-excited surface waves are modeled using the Zhang-Viñals framework.
  • Linear stability analysis of impulsive forcing reveals subharmonic instabilities.
  • Asymmetric impulse spacing is crucial for harmonic resonance tongues.

Purpose of the Study:

  • Extend linear analysis to the weakly nonlinear regime for N=2 impulses.
  • Determine nonlinear saturation of standing waves as a function of forcing strength.
  • Derive an analytic expression for the cubic Landau coefficient.

Main Methods:

  • Exact linear stability analysis for impulsive forcing.
  • Weakly nonlinear analysis to determine saturation.
  • Derivation of the cubic Landau coefficient as a function of spacing and fluid parameters.

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Main Results:

  • An analytic expression for the cubic Landau coefficient is derived.
  • A parameter regime of wave amplitude suppression is identified.
  • This suppression arises from a 1:2 spatiotemporal resonance between subharmonic and damped harmonic modes.

Conclusions:

  • The 1:2 resonance occurs even without harmonic resonance tongues in neutral stability curves.
  • The resonance strength is tunable by varying the spacing between impulses.
  • Findings align with symmetry-based analyses of multifrequency forced Faraday waves.