Recent progress on particle trajectories in steady water waves

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October 2009, 12(3): 539-559. doi: 10.3934/dcdsb.2009.12.539

Recent progress on particle trajectories in steady water waves

Mats Ehrnström ^1, and Gabriele Villari ^2,

1.	Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany
2.	Dipartimento di Matematica, Viale Morgagni 67/A, 50134 Firenze, Italy

Received March 2009 Revised June 2009 Published July 2009

Abstract
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We survey recent results on particle trajectories within steady two-dimensional water waves. Particular emphasis is placed on the linear and exact mathematical theory of periodic and symmetric waves, and the effects of a (possibly rotational) background current. The different results vindicate and detail the classical Stokes drift, and also show the transition of orbits when waves propagate into running water. The classical approximation, depicting the trajectories as closed ellipses, is shown to be a mathematical rarity.

Keywords: Steady water waves; Stokes waves, Particle trajectories, Vorticity, Background current..

Mathematics Subject Classification: 35Q35, 37N10, 76B15, 76F1.

Citation: Mats Ehrnström, Gabriele Villari. Recent progress on particle trajectories in steady water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 539-559. doi: 10.3934/dcdsb.2009.12.539

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