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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

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 and Continuous Dynamical Systems - B, 2009, 12 (3) : 539-559. doi: 10.3934/dcdsb.2009.12.539
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