Breakdown of Self-Similarity at the Crests of Large-Amplitude Standing Water Waves

Jon Wilkening
Phys. Rev. Lett. 107, 184501 – Published 25 October 2011

Abstract

We study the limiting behavior of large-amplitude standing waves on deep water using high-resolution numerical simulations in double and quadruple precision. While periodic traveling waves approach Stokes’s sharply crested extreme wave in an asymptotically self-similar manner, we find that standing waves behave differently. Instead of sharpening to a corner or cusp as previously conjectured, the crest tip develops a variety of oscillatory structures. This causes the bifurcation curve that parametrizes these waves to fragment into disjoint branches corresponding to the different oscillation patterns that occur. In many cases, a vertical jet of fluid pushes these structures upward, leading to wave profiles commonly seen in wave tank experiments. Thus, we observe a rich array of dynamic behavior at small length scales in a regime previously thought to be self-similar.

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  • Received 10 November 2010

DOI:https://doi.org/10.1103/PhysRevLett.107.184501

© 2011 American Physical Society

Authors & Affiliations

Jon Wilkening*

  • Department of Mathematics, University of California, Berkeley, California 94720, USA

  • *wilken@math.berkeley.edu

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Issue

Vol. 107, Iss. 18 — 28 October 2011

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