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JOURNALS // Pis'ma v Zhurnal Èksperimental'noi i Teoreticheskoi Fiziki // Archive

Pis'ma v Zh. Èksper. Teoret. Fiz., 2013 Volume 98, Issue 11, Pages 767–771 (Mi jetpl3589)

This article is cited in 28 papers

PLASMA, HYDRO- AND GAS DYNAMICS

The complex singularity of a Stokes wave

S. A. Dyachenkoa, P. M. Lushnikovab, A. O. Korotkevichab

a Department of Mathematics and Statistics, University of New Mexico
b L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: Two-dimensional potential flow of the ideal incompressible fluid with free surface and infinite depth can be described by a conformal map of the fluid domain into the complex lower half-plane. Stokes wave is the fully nonlinear gravity wave propagating with the constant velocity. The increase of the scaled wave height $H/\lambda$ from the linear limit $H/\lambda=0$ to the critical value $H_{\max}/\lambda$ marks the transition from the limit of almost linear wave to a strongly nonlinear limiting Stokes wave. Here $H$ is the wave height and $\lambda$ is the wavelength. We simulated fully nonlinear Euler equations, reformulated in terms of conformal variables, to find Stokes waves for different wave heights. Analyzing spectra of these solutions we found in conformal variables, at each Stokes wave height, the distance $v_c$ from the lowest singularity in the upper half-plane to the real line which corresponds to the fluid free surface. We also identified that this singularity is the square-root branch point. The limiting Stokes wave emerges as the singularity reaches the fluid surface. From the analysis of data for $v_c\to 0$ we suggest a new power law scaling $v_c\propto (H_{\max}-H)^{3/2}$ as well as new estimate $H_{\max}/\lambda \simeq 0.1410633$.

Received: 07.11.2013

Language: English

DOI: 10.7868/S0370274X13230070


 English version:
Journal of Experimental and Theoretical Physics Letters, 2013, 98:11, 675–679

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© Steklov Math. Inst. of RAS, 2024