Elliptic and hyperelliptic functions describing the particlemotion beneath small-amplitude water waves with constant vorticity

Delia Ionescu-Kruse

doi:10.3934/cpaa.2012.11.1475

Article Contents

2012, Volume 11, Issue 4: 1475-1496. Doi: 10.3934/cpaa.2012.11.1475

This issue Previous Article On the formation of singularities for surface water waves Next Article A selection of nonlinear problems in water waves, analysed by perturbation-parameter techniques

Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity

Delia Ionescu-Kruse^1,

1.
Institute of Mathematics Simion Stoilow of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest

Received: April 2011

Revised: July 2011

Published: July 2012

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Abstract

We provide analytic solutions of the nonlinear differential equation system describing the particle paths below small-amplitude periodic gravity waves travelling on a constant vorticity current. We show that these paths are not closed curves. Some solutions can be expressed in terms of Jacobi elliptic functions, others in terms of hyperelliptic functions. We obtain new kinds of particle paths. We make some remarks on the stagnation points which could appear in the fluid due to the vorticity.

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Mathematics Subject Classification: 76B15, 76F10, 33E05, 33E20.

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