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Linear stability of finite-amplitude capillary waves on water of infinite depth

Published online by Cambridge University Press:  06 March 2012

Roxana Tiron  and
Wooyoung Choi
Roxana Tiron
Affiliation:
Department of Ocean Systems Engineering, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea
Wooyoung Choi*
Affiliation:
Department of Ocean Systems Engineering, Korea Advanced Institute of Science and Technology, Daejeon, 305-701, Korea Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102-1982, USA
*
†Email address for correspondence: wychoi@njit.edu
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Abstract

We study the linear stability of the exact deep-water capillary wave solution of Crapper (J. Fluid Mech., vol. 2, 1957, pp. 532–540) subject to two-dimensional perturbations (both subharmonic and superharmonic). By linearizing a set of exact one-dimensional non-local evolution equations, a stability analysis is performed with the aid of Floquet theory. To validate our results, the exact evolution equations are integrated numerically in time and the numerical solutions are compared with the time evolution of linear normal modes. For superharmonic perturbations, contrary to Hogan (J. Fluid Mech., vol. 190, 1988, pp. 165–177), who detected two bubbles of instability for intermediate amplitudes, our results indicate that Crapper’s capillary waves are linearly stable to superharmonic disturbances for all wave amplitudes. For subharmonic perturbations, it is found that Crapper’s capillary waves are unstable, and our results generalize to the highly nonlinear regime the analysis for small amplitudes presented by Chen & Saffman (Stud. Appl. Maths, vol. 72, 1985, pp. 125–147).

JFM classification

Waves/Free-surface Flows: Capillary waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 696 , 10 April 2012 , pp. 402 - 422
Copyright
Copyright © Cambridge University Press 2012

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