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

Document Type

Article

Publication Date

4-10-2012

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 Crappers capillary waves are linearly stable to superharmonic disturbances for all wave amplitudes. For subharmonic perturbations, it is found that Crappers 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). © 2012 Cambridge University Press.

Identifier

84859171773 (Scopus)

Publication Title

Journal of Fluid Mechanics

External Full Text Location

https://doi.org/10.1017/jfm.2012.56

e-ISSN

14697645

ISSN

00221120

First Page

402

Last Page

422

Volume

696

Grant

R31-2008-000-10045-0

Fund Ref

National Research Foundation of Korea

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