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| Within the setting of periodic rotational water waves, we present a new intrinsic characterization of the symmetric gravity waves. Namely, we show that the symmetry property of the free water surface can be characterized in terms of the flow supporting the wave. To be more precise, we prove that a gravity wave - which may possess a priori arbitrary many crests and trough per period - is symmetric and has only one crest and trough per period if and only if there exists a vertical line within the fluid domain such that all the fluid particles located on that line minimize there simultaneously their distance to the fluid bed. This characterization is new even for Stokes waves. |
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