Math @ Duke

Publications [#43577] of Anne Catlla
Papers Published
 A. Catlla, J. Porter, M. Silber, Weakly nonlinear analysis of impulsivelyforced Faraday waves,
Physical Review E, vol. 72 no. 5
(November 17, 2005),
pp. 056212
(last updated on 2005/12/12)
Abstract: Parametricallyexcited surface waves, forced
by a repeating sequence of N deltafunction
impulses, are considered within the framework
of the ZhangViñals model [W. Zhang and J.
Viñals, J. Fluid Mech. 336, 301 (1997)]. With
impulsive forcing, the linear stability
analysis can be carried out exactly and leads
to an implicit equation for the neutral
stability curves. As noted previously [J.
Bechhoefer and B. Johnson, Am. J. Phys. 64,
1482 (1996)], in the simplest case of N=2
equallyspaced impulses per period (which
alternate up and down) there are only
subharmonic modes of instability. The
familiar situation of alternating subharmonic
and harmonic resonance tongues emerges only
if an asymmetry in the spacing between the
impulses is introduced. We extend the linear
analysis for N=2 impulses per period to the
weakly nonlinear regime, where we determine
the leading order nonlinear saturation of
onedimensional standing waves as a function
of forcing strength. Specifically, an
analytic expression for the cubic Landau
coefficient in the bifurcation equation is
derived as a function of the dimensionless
spacing between the two impulses and the
fluid parameters that appear in the
ZhangViñals model. As the capillary
parameter is varied, one finds a parameter
regime of wave amplitude suppression, which
is due to a familiar 1:2 spatiotemporal
resonance between the subharmonic mode of
instability and a damped harmonic mode. This
resonance occurs for impulsive forcing even
when harmonic resonance tongues are absent
from the neutral stability curves. The
strength of this resonance feature can be
tuned by varying the spacing between the
impulses. This finding is interpreted in
terms of a recent symmetrybased analysis of
multifrequency forced Faraday waves [J.
Porter, C. M. Topaz, and M. Silber, Phys.
Lett. 93, 034502 (2004); C. M. Topaz, J.
Porter, and M. Silber, Phys. Rev. E 70,
066206 (2004)].


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