Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Recent progress on particle trajectories in steady water waves

Pages: 539 - 559, Volume 12, Issue 3, October 2009      doi:10.3934/dcdsb.2009.12.539

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Mats Ehrnström - Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany (email)
Gabriele Villari - Dipartimento di Matematica, Viale Morgagni 67/A, 50134 Firenze, Italy (email)

Abstract: 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, 76F10.

Received: March 2009;      Revised: June 2009;      Available Online: July 2009.