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A generalized stability criterion for resonant triad interactions

Published online by Cambridge University Press:  26 April 2006

Carson C. Chow ,
Diane Henderson  and
Harvey Segur
Carson C. Chow
Affiliation:
NeuroMuscular Research Center, Boston University, Boston, MA 02215, USA
Diane Henderson
Affiliation:
Department of Mathematics, Penn State University, University Park, PA 16802, USA
Harvey Segur
Affiliation:
Program in Applied Mathematics, University of Colorado, Boulder, CO 80309–0526, USA
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Abstract

It is well known that in any conservative system that admits resonant triad interactions, a uniform (test) wavetrain that participates in a single triad is unstable if it has the highest frequency in the triad, and neutrally stable otherwise. We show that this result changes significantly in the presence of coupled triads: with coupling, the test wave can be unstable to a high-frequency perturbation. The coupling sends energy from the (weak) high-frequency source into particular low-frequency waves that grow even though they had zero amplitudes initially. This mechanism thereby selects these low-frequency waves from the spectrum of low-frequency waves available for triad interactions. Moreover, the instability persists in the presence of weak damping, provided the wave amplitudes exceed two thresholds. First, the initial amplitude of the test wavetrain must be large enough for the instability to dominate the damping. Secondly, the (small) initial amplitudes of the high-frequency perturbations must exceed a threshold in order for the low-frequency waves to grow to a prescribed amplitude.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 319 , 25 July 1996 , pp. 67 - 76
Copyright
© 1996 Cambridge University Press

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