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Stokes waves in water with a non-flat bed

Published online by Cambridge University Press:  08 January 2014

A. Constantin
A. Constantin*
Affiliation:
Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
*
†Email address for correspondence: adrian.constantin@kcl.ac.uk
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Abstract

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We investigate the relevance of Stokes waves for the propagation of ocean swell in the absence of strong currents. By providing estimates for the depth of the near-surface layer to which the main effects of a Stokes flow are confined, we show that wind-generated uniform wave trains can be modelled as Stokes waves over a fictitious flat bed, immersed in the water. Throughout the lower parts of this layer the deviations of the flow from a pure current are negligible.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 740 , 10 February 2014 , pp. 17 - 27
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
©2014 Cambridge University Press.

References

Amick, C. J. 1987 Bounds for water waves. Arch. Rat. Mech. Anal. 99, 91–114.Google Scholar
Buffoni, B. & Toland, J. F. 2003 Analytic Theory of Global Bifurcation. An Introduction. Princeton University Press.CrossRefGoogle Scholar
Clamond, D. 1999 Steady finite amplitude waves on a horizontal seabed of arbitrary depth. J. Fluid Mech. 398, 45–60.CrossRefGoogle Scholar
Clamond, D. 2012 Note on the velocity and related fields of steady irrotational two-dimensional surface gravity waves. Phil. Trans. R. Soc. Lond. A 370, 1572–1586.Google Scholar
Constantin, A. 2006 The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535.CrossRefGoogle Scholar
Constantin, A. 2011 Nonlinear Water Waves with Applications to Wave-current Interactions and Tsunamis, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81, SIAM.Google Scholar
Constantin, A. 2013 Mean velocities in a Stokes wave. Arch. Rat. Mech. Anal. 207, 907–917.Google Scholar
Constantin, A. & Johnson, R. S. 2008 On the non-dimensionalisation, scaling and resulting interpretation of the classical governing equations for water waves. J. Nonlinear Math. Phys. 63, 58–73.Google Scholar
Constantin, A. & Strauss, W. 2010 Pressure beneath a Stokes wave. Commun. Pure Appl. Maths 63, 533–557.Google Scholar
Drennan, W. M., Hui, W. H. & Tenti, G. 1992 Accurate calculations of Stokes water waves of large amplitude. Z. Angew. Math. Phys. 43, 367–384.Google Scholar
Fenton, J. D. 1985 A fifth-order Stokes theory for steady waves. J. Waterway Port Coastal Ocean Engng 111, 216–234.Google Scholar
Henry, D. 2008 On the deep-water Stokes wave flow. Int. Math. Res. Not. IMRN 2008, Art. ID rnn 071, 7 pp.Google Scholar
Plotnikov, P. I. & Toland, J. F. 2002 The Fourier coefficients of Stokes waves. In Nonlinear Problems in Mathematical Physics and Related Topics, vol. I, pp. 303–315. Kluwer/Plenum.Google Scholar
Stokes, G. G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455.Google Scholar
Toland, J. F. 1996 Stokes waves. Topol. Meth. Nonlinear Anal. 7, 1–48.CrossRefGoogle Scholar
Umeyama, M. 2012 Eulerian–Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry. Phil. Trans. R. Soc. Lond. A 370, 1687–1702.Google Scholar