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Parametrically excited surface waves: two-frequency forcing, normal form symmetries, and pattern selection.

M Silber1, A C Skeldon

  • 1Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|April 24, 2002
PubMed
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Normal form symmetries simplify pattern selection in the two-frequency Faraday experiment. When harmonic and subharmonic waves emerge together, specific frequency ratios diminish the role of resonant triads, impacting wave pattern formation.

Area of Science:

  • Fluid dynamics
  • Nonlinear dynamics
  • Pattern formation

Background:

  • The two-frequency Faraday experiment exhibits complex free surface standing wave patterns.
  • Understanding pattern selection requires analyzing instabilities and wave interactions.

Purpose of the Study:

  • Investigate the influence of normal form symmetries on pattern selection.
  • Analyze scenarios where harmonic and subharmonic waves destabilize simultaneously.
  • Compare cases with different relative wavelengths of harmonic and subharmonic waves.

Main Methods:

  • Employing normal form transformations to simplify amplitude equations.
  • Analyzing resonant triad interactions in pattern selection.
  • Utilizing bifurcation theory to study wave stability.

Related Experiment Videos

  • Examining the Zhang-Viñals model for one-dimensional surface waves.

    • Main Results:

    • Normal form symmetries can eliminate quadratic terms in amplitude equations, reducing the impact of resonant triads.
    • A 1:2 spatial resonance significantly affects subharmonic wave bifurcation when forcing frequencies are in a 1:2 ratio.
    • When forcing frequencies are in a 2:3 ratio, bifurcations are insensitive to spatially resonant modes.

    Conclusions:

    • Normal form symmetries play a crucial role in simplifying pattern selection problems.
    • The relationship between forcing frequencies and spatial resonances dictates the influence on wave pattern formation.
    • Findings are consistent with experimental observations and theoretical models.