Recent progress on particle trajectories in steady water waves

Mats Ehrnström; Gabriele Villari

doi:10.3934/dcdsb.2009.12.539

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2009, Volume 12, Issue 3: 539-559. Doi: 10.3934/dcdsb.2009.12.539

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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
2.
Dipartimento di Matematica, Viale Morgagni 67/A, 50134 Firenze

Received: February 28, 2009

Revised: May 31, 2009

Published: April 2009

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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.

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Mathematics Subject Classification: 35Q35, 37N10, 76B15, 76F10.

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