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On the local uniqueness and the profile of the extreme Stokes wave

Published online by Cambridge University Press:  25 February 2010

L. E. FRAENKEL  and
P. J. HARWIN
L. E. FRAENKEL
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK email: [email protected], [email protected]
P. J. HARWIN
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK email: [email protected], [email protected]
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Abstract

Proceeding from a recent construction (Fraenkel 2007, Arch. Rational Mech. Anal. 183, 187–214), this paper contains three contributions to the theory of the extreme gravity wave on water (or “wave of greatest height”) initiated by Stokes in 1880. The first is a solution, of the extreme form of the integral equation due to Nekrasov, that consists of an explicit function plus a (rigorously) estimated remainder; the remainder vanishes at the crest and at the trough of the wave and in between has a maximum value less than 0.0026 times the maximum value of the explicit part. The second contribution establishes local uniqueness for a variant of the Nekrasov equation; we prove that this equation has only one solution between upper and lower bounding functions that are moderately far apart. The third contribution consists of a table and graph of strict upper and lower bounds for the interface between air and water in the physical plane.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 21 , Issue 2 , April 2010 , pp. 137 - 163
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Abramowitz, M. & Stegun, I. A. (1972) Handbook of mathematical functions. New York, NY: Dover Publications.Google Scholar
[2]Amick, C.J. & Fraenkel, L.E. (1987) On the behavior near the crest of waves of extreme form. Trans. Amer. Math. Soc. 299, 273–298.CrossRefGoogle Scholar
[3]Amick, C.J., Fraenkel, L.E. & Toland, J.F. (1982) On the Stokes conjecture for the wave of extreme form. Acta Math. 148, 193–214.CrossRefGoogle Scholar
[4]Cokelet, E.D. (1977) Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183–230.Google Scholar
[5]Fraenkel, L.E. (2007) A constructive existence proof for the extreme Stokes wave. Arch. Rational Mech. Anal. 183, 187–214.CrossRefGoogle Scholar
[6]Fraenkel, L.E. (2010) The behaviour near the crest of an extreme Stokes wave. Euro. Jnl of Appl. Math. 21, 165–180.CrossRefGoogle Scholar
[7]Gandzha, I.S. & Lukomsky, V.P. (2007) On water waves with a corner at the crest. Proc. R. Soc. Lond. A 463, 1597–1614.Google Scholar
[8]Grant, M.A. (1973) The singularity at the crest of a finite amplitude progressive Stokes wave. J. Fluid Mech. 59, 257–262.CrossRefGoogle Scholar
[9]Kobayashi, K. (2004) Numerical verification of the global uniqueness of a positive solution for Nekrasov's equation. Japan J. Indust. Appl. Math. 21, 181–218.CrossRefGoogle Scholar
[10]Kobayashi, K. & Okamoto, H. (2004) Uniqueness issues on permanent progressive water-waves, J. Nonlinear Math. Phys. 11, 472–479.CrossRefGoogle Scholar
[11]Lewy, H. (1952) A note on harmonic functions and a hydrodynamical application. Proc. Amer. Math. Soc. 3, 111–113.CrossRefGoogle Scholar
[12]Longuet-Higgins, M.S. (1975) Integral properties of periodic gravity waves of finite amplitude. Proc. R. Soc. Lond. A 342, 157–174.Google Scholar
[13]Longuet-Higgins, M.S. & Fox, M.J.H. (1978) Theory of the almost-highest wave. Part 2. Matching and analytic extension. J. Fluid Mech. 85, 769–786.CrossRefGoogle Scholar
[14]McLeod, J.B. (1987) The asymptotic behavior near the crest of waves of extreme form. Trans. Amer. Math. Soc. 299, 299–302.CrossRefGoogle Scholar
[15]Rainey, R.C.T. & Longuet-Higgins, M.S. (2006) A close one-term approximation to the highest Stokes wave on deep water. Ocean Eng. 33, 2012–2024.CrossRefGoogle Scholar
[16]Toland, J.F. (1996) Stokes waves. Topological Meth. in Nonlinear Anal. 7, 1–48.CrossRefGoogle Scholar
[17]Williams, J.M. (1981) Limiting gravity waves in water of finite depth. Phil. Trans. R. Soc. Lond. A 302, 139–188.Google Scholar