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Pattern formation in two-frequency forced Faraday waves
  • Chad M. Topaz, Macalester College

This dissertation investigates pattern selection in two-frequency forced Faraday waves. In this system, a fluid layer is subjected to a periodic vertical acceleration $g_z[\cos(\chi)\cos (m\omega t) + \sin(\chi) (n \omega t + \phi)]$, where $m$ and $n$ are co-prime integers. For sufficiently large $g_z$, standing waves form on the free surface. Experiments have produced exotic patterns, including spatially-periodic “superlattice” (SL) patterns (Kudrolli, Pier and Gollub, Physica D, 1998) which contain two length scales. This dissertation determines the selection mechanism for the length scale ratio for the SL-I superlattice pattern.

A 12-dimensional $\group{D}_6 \dot{+} \group{T}^2$ equivariant bifurcation theoretic framework (Dionne, Silber and Skeldon, Nonlinearity, 1997) is used to study the competition of stripes, hexagons, rhombs, and SL-I superlattice patterns. Symmetry considerations are used to demonstrate how weakly damped harmonic modes may affect the stability of SL-I patterns through spatiotemporally-resonant triad interactions, which produce resonant contributions to coefficients in the bifurcation equations. To demonstrate this effect explicitly, the coefficients are numerically calculated via a perturbation calculation on partial differential equations of Zhang and Vinals (J. Fluid Mech., 1997) which describe Faraday waves on a deep layer of weakly viscous fluid. A bifurcation analysis reveals that a weakly damped harmonic mode may help stabilize an SL-I superlattice pattern.

For weak damping and forcing, symmetry considerations also determine which particular damped harmonic modes have the most significant effect. These are: (i) modes oscillating with twice the frequency of the pattern modes, (ii) "difference frequency" modes oscillating with dominant frequency $|m-n|\omega$ and (iii) "sum frequency" modes oscillating with dominant frequency $(m+n)\omega$. For weak damping and forcing and one dimensional waves, an analytical expression is derived for the cubic self-interaction term. For stronger damping and forcing and two-dimensional waves, the remaining coefficients are computed numerically. Both calculations yield results in good agreement with those obtained from symmetry arguments.

A bifurcation analysis demonstrates that the difference frequency modes help stabilize the SL-I pattern and determines the length scale ratio, which is well-predicted by the gravity-capillary wave dispersion relation. The SL-I stabilization effect may be enhanced by appropriate choice of periodic forcing functions with more than two frequency components.

Publication Date
Citation Information
Chad M. Topaz. "Pattern formation in two-frequency forced Faraday waves" (2002)
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