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Progressive radial cross-waves

Published online by Cambridge University Press:  26 April 2006

Janet M. Becker  and
John W. Miles
Janet M. Becker
Affiliation:
School of Mathematics, University of New South Wales, Kensington, NSW 2033, Australia
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093-0225
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Abstract

Progressive radial cross-waves in a deep, periphractic wavetank are investigated on the assumption that the vertical component of the capillary force vanishes at the wavemaker. For a cylindrical wavemaker, the envelope of the radial cross-wave is shown to obey an evolution equation that differs from the cubic Schrödinger equation only in the presence of a factor 1/R in the cubic term, where R is a slow radial variable. Weak, linear damping is incorporated, and the transition conditions at which the directly forced concentric wave loses stability to a parametrically forced cross-wave are obtained. The cylindrical problem is used to develop an asymptotic approximation to the corresponding problem for a spherical wavemaker. The theory is compared with the experiments of Tatsuno, Inoue & Okabe (1969). The theoretical predictions of resonant wavenumbers are consistent with their data, but the corresponding predictions of wavemaker amplitudes, on the assumption of linear damping that is confined to an inextensible (fully contaminated) free-surface boundary layer, are an order of magnitude smaller than those observed by Tatsuno et al. (1969). This underprediction of the transition amplitudes may be due to nonlinear phenomena — in particular, nonlinear effects at the contact line and ‘undersurface flows’ (Taneda 1991) – that are not comprehended by the theoretical model.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 245 , December 1992 , pp. 29 - 46
Copyright
© 1992 Cambridge University Press

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