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On nonlinear wave groups and crest statistics

Published online by Cambridge University Press:  10 February 2009

FRANCESCO FEDELE  and
M. AZIZ TAYFUN
FRANCESCO FEDELE*
Affiliation:
School of Civil and Environmental Engineering, Georgia Institute of Technology, Savannah, Georgia, USA Civil Engineering Department, College of Engineering and Petroleum, Kuwait University, Kuwait
M. AZIZ TAYFUN
Affiliation:
School of Civil and Environmental Engineering, Georgia Institute of Technology, Savannah, Georgia, USA Civil Engineering Department, College of Engineering and Petroleum, Kuwait University, Kuwait
*
†Email address for correspondence: ffedele3@gtsav.gatech.edu
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Abstract

We present a second-order stochastic model of weakly nonlinear waves and develop theoretical expressions for the expected shape of large surface displacements. The model also leads to an exact theoretical expression for the statistical distribution of large wave crests in a form that generalizes the Tayfun distribution (Tayfun, J. Geophys. Res., vol. 85, 1980, p. 1548). The generalized distribution depends on a steepness parameter given by μ = λ3/3, where λ3 represents the skewness coefficient of surface displacements. It converges to the Tayfun distribution in narrowband waves, where both distributions describe the crests of all waves well. In broadband waves, the generalized distribution represents the crests of large waves just as well whereas the Tayfun distribution appears as an upper bound and tends to overestimate them. However, the theoretical nature of the generalized distribution presents practical difficulties in oceanic applications. We circumvent these by adopting an appropriate approximation for the steepness parameter. Comparisons with wind-wave measurements from the North Sea suggest that this approximation allows both distributions to assume an identical form with which we can describe the distribution of large wave crests fairly accurately. The same comparisons also show that third-order nonlinear effects do not appear to have any discernable effect on the statistics of large surface displacements or wave crests.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 620 , 10 February 2009 , pp. 221 - 239
Copyright
Copyright © Cambridge University Press 2009

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