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Long-time behaviour of a random inhomogeneous field of weakly nonlinear surface gravity waves

Published online by Cambridge University Press:  20 April 2006

Peter A. E. M. Janssen
Peter A. E. M. Janssen
Affiliation:
Department of Mechanical Engineering, University of Rochester, Rochester, NY 14627 Permanent address: Department of Oceanography, Royal Netherlands Meteorological Institute, De Bilt, Holland.
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Abstract

In this paper we investigate nonlinear interactions of narrowband, Gaussian-random, inhomogeneous wavetrains. Alber studied the stability of a homogeneous wave spectrum as a function of the width σ of the spectrum. For vanishing bandwidth the deterministic results of Benjamin & Feir on the instability of a uniform wavetrain were rediscovered whereas a homogeneous wave spectrum was found to be stable if the bandwidth is sufficiently large. Clearly, a threshold for instability is present, and in this paper we intend to study the long-time behaviour of a slightly unstable modulation by means of a multiple-timescale technique. Two interesting cases are found. For small but finite bandwidth – the amplitude of the unstable modulation shows initially an overshoot, followed by an oscillation around the time-asymptotic value of the amplitude. This oscillation damps owing to phase mixing except for vanishing bandwidth because then the well-known Fermi–Pasta–Ulam recurrence is found. For large bandwidth, however, no overshoot is found since the damping is overwhelming. In both cases the instability is quenched because of a broadening of the spectrum.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 133 , August 1983 , pp. 113 - 132
Copyright
© 1983 Cambridge University Press

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