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Experiments on ripple instabilities. Part 3. Resonant quartets of the Benjamin–Feir type

Published online by Cambridge University Press:  26 April 2006

Marc Perlin  and
Joe Hammack
Marc Perlin
Affiliation:
Department of Aerospace Engineering, Mechanics and Engineering Sciences, University of Florida, Gainesville, FL 32611, USA Present address: Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA.
Joe Hammack
Affiliation:
Department of Aerospace Engineering, Mechanics and Engineering Sciences, University of Florida, Gainesville, FL 32611, USA
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Abstract

Instabilities and long-time evolution of gravity-capillary wavetrains (ripples) with moderate steepnesses (ε < 0.3) are studied experimentally and analytically. Wave-trains with frequencies of 8 ≤ f ≤ 25 Hz are generated mechanically in a channel containing clean, deep water; no artificial perturbations are introduced. Frequency spectra are obtained from in situ measurements; two-dimensional wavenumber spectra are obtained from remote sensing of the water surface using a high-speed imaging system. The analytical models are in viscid, uncoupled NLS (nonlinear Schrödinger) equations: one that describes the temporal evolution of longitudinal modulations and one that describes the spatial evolution of transverse modulations.

The experiments show that the evolution of wavetrains with sensible amplitudes and frequencies exceeding 9.8 Hz is dominated by modulational instabilities, i.e. resonant quartet interactions of the Benjamin–Feir type. These quartet interactions remain dominant even for wavetrains that are unstable to resonant triad interactions (f > 19.6 Hz) – if selective amplification does not occur (see Parts 1 and 2). The experiments further show that oblique perturbations with the same frequency as the underlying wavetrain, i.e. rhombus-quartet instabilities, amplify more rapidly and dominate all other modulational instabilities. The inviscid, uncoupled NLS equations predict the existence of modulational instabilities for wavetrains with frequencies exceeding 9.8 Hz, typically underpredict the bandwidth of unstable transverse modulations, typically overpredict the bandwidth of unstable longitudinal modulations, and do not predict the dominance of the rhombus-quartet instability. When the effects of weak viscosity are incorporated into the NLS models, the predicted bandwidths of unstable modulations are reduced, which is consistent with our measurements for longitudinal modulations, but not with our measurements for transverse modulations.

Both the experiments and NLS equations indicate that wavetrains in the frequency range 6.4–9.8 Hz are stable to modulational instabilities. However, in these experiments, wavetrains with sensible amplitudes excite one of the members of the Wilton ripples family. When second-harmonic resonance occurs, both the first-and second-harmonic wavetrains undergo rhombus-quartet instabilities. When third-harmonic resonance occurs, only the third-harmonic wavetrain undergoes rhombus-quartet instabilities.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 229 , August 1991 , pp. 229 - 268
Copyright
© 1991 Cambridge University Press

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