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Unstable Surface Waves in Running Water

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An Erratum to this article was published on 06 February 2013

Abstract

We consider the stability of periodic gravity free-surface water waves traveling downstream at a constant speed over a shear flow of finite depth. In case the free surface is flat, a sharp criterion of linear instability is established for a general class of shear flows with inflection points and the maximal unstable wave number is found. Comparison to the rigid-wall setting testifies that the free surface has a destabilizing effect. For a class of unstable shear flows, the bifurcation of nontrivial periodic traveling waves is demonstrated at all wave numbers. We show the linear instability of small nontrivial waves that appear after bifurcation at an unstable wave number of the background shear flow. The proof uses a new formulation of the linearized water-wave problem and a perturbation argument. An example of the background shear flow of unstable small-amplitude periodic traveling waves is constructed for an arbitrary vorticity strength and for an arbitrary depth, illustrating that vorticity has a subtle influence on the stability of free-surface water waves.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Vera Mikyoung Hur

  2. Department of Mathematics, University of Missouri-Columbia, Columbia, MO, 65203-4100, USA

    Zhiwu Lin

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  1. Vera Mikyoung Hur
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  2. Zhiwu Lin
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Correspondence to Vera Mikyoung Hur.

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Communicated by P. Constantin

An erratum to this article can be found online at http://dx.doi.org/10.1007/s00220-013-1660-y.

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Hur, V.M., Lin, Z. Unstable Surface Waves in Running Water. Commun. Math. Phys. 282, 733–796 (2008). https://doi.org/10.1007/s00220-008-0505-6

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