Slow evolution of nonlinear deep water waves in two horizontal directions: A numerical study
type: Item
instance of: scholarly article
title: Slow evolution of nonlinear deep water waves in two horizontal directions: A numerical study
zbMATH Open document ID: 0613.76015
DOI: 10.1016/0165-2125(87)90014-X
author: Edmond Y. Lo
author: Chiang C. Mei
published in: Wave Motion
publication date: 1987-01-01T00:00:00Z
review text: The two-dimensional evolution of Stokes waves over a long fetch has been studied only by means of the cubic Schrödinger equation which is accurate up to the third order in wave slope. For uniform Stokes waves unstable sidebands are known to induce modulation and demodulation at the beginning but energy leakage to higher frequencies leads to chaos eventually and invalidates the theory. For one-dimensional evolution, the fourth-order extension by \textit{K. B. Dysthe} [Proc. R. Soc. Lond., Ser. A 369, 105-114 (1979; Zbl 0429.76014)] has been shown to corroborate better with experiments than the third-order. In this paper we pursue further the implications of the fourth-order theory on two-dimensional evolutions. For the instability of uniform wave trains due to oblique sidebands, we find that the narrower instability region and the presence of a higher-order dispersion term tend to delay the trend toward chaos. We also study the evolution of a single three- dimensional envelope with various aspect ratios. Comparisons between the third- and fourth-order theories are made. Specifically, if the initial envelope is elongated in the direction of the crests, new groups are first born near the center, with ridges parallel to the crests, then everything flattens out. If the initial envelope is elongated in the direction of wave propagation, then the tendency of group-splitting is reduced. Furthermore, the envelope elongates crest-wise to form a ridge before eventual flattening.
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 76M99
Mathematics Subject Classification ID: 86A05
zbMATH DE Number: 3993114
zbMATH Keywords: two-dimensional evolution
zbMATH Keywords: Stokes waves
zbMATH Keywords: long fetch
zbMATH Keywords: cubic Schrödinger equation
zbMATH Keywords: wave slope
zbMATH Keywords: uniform Stokes waves
zbMATH Keywords: unstable sidebands
zbMATH Keywords: modulation
zbMATH Keywords: demodulation
zbMATH Keywords: energy leakage
zbMATH Keywords: one-dimensional evolution
zbMATH Keywords: fourth-order extension
zbMATH Keywords: instability
zbMATH Keywords: uniform wave trains
zbMATH Keywords: oblique sidebands
zbMATH Keywords: instability region
zbMATH Keywords: higher-order dispersion term
zbMATH Keywords: three- dimensional envelope
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3993114