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Singularities in the complex physical plane for deep water waves

Published online by Cambridge University Press:  22 September 2011

Gregory R. Baker  and
Chao Xie
Gregory R. Baker*
Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210, USA
Chao Xie
Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210, USA
*
†Email address for correspondence: baker@math.osu.edu
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Abstract

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Deep water waves in two-dimensional flow can have curvature singularities on the surface profile; for example, the limiting Stokes wave has a corner of radians and the limiting standing wave momentarily forms a corner of radians. Much less is known about the possible formation of curvature singularities in general. A novel way of exploring this possibility is to consider the curvature as a complex function of the complex arclength variable and to seek the existence and nature of any singularities in the complex arclength plane. Highly accurate boundary integral methods produce a Fourier spectrum of the curvature that allows the identification of the nearest singularity to the real axis of the complex arclength plane. This singularity is in general a pole singularity that moves about the complex arclength plane. It approaches the real axis very closely when waves break and is associated with the high curvature at the tip of the breaking wave. The behaviour of these singularities is more complex for standing waves, where two singularities can be identified that may collide and separate. One of them approaches the real axis very closely when a standing wave forms a very narrow collapsing column of water almost under free fall. In studies so far, no singularity reaches the real axis in finite time. On the other hand, the surface elevation has square-root singularities in the complex plane that do reach the real axis in finite time, the moment when a wave first starts to break. These singularities have a profound effect on the wave spectra.

JFM classification

Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 685 , 25 October 2011 , pp. 83 - 116
Copyright
Copyright © Cambridge University Press 2011 The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution-NonCommercial-ShareAlike licence <http://creativecommons.org/licenses/by-nc-sa/2.5/>. The written permission of Cambridge University Press must be obtained for commercial re-use.

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