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Subharmonic resonant interaction of a gravity–capillary progressive axially symmetric wave with a radial cross-wave

Published online by Cambridge University Press:  25 April 2019

Meng Shen  and
Yuming Liu Open the ORCID record for Yuming Liu [Opens in a new window]
Meng Shen
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Yuming Liu*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: [email protected]
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Abstract

We theoretically investigate the problem of subharmonic resonant interaction of a progressive (axially symmetric) ring wave with a radial cross-wave in the context of the potential-flow formulation for gravity-capillary waves. The objective is to understand the nonlinear mechanism governing energy transfer from a progressive ring wave to its subharmonic cross-waves through triadic resonant interactions. We first show that for an arbitrary three-dimensional body floating in an unbounded free surface, there exists a set of homogeneous solutions at any frequency in the gravity-capillary wave context. The homogeneous solution depends solely on the mean free-surface slope at the waterline of the body and physically represents a progressive radial cross-wave. Unlike standing cross-waves, a progressive cross-wave loses energy during propagation by overcoming the work done by surface tension at the waterline and through wave radiation. We then consider the subharmonic interaction of a progressive ring wave, which is forced by a radial swelling–contraction deformation of a vertical circular cylinder, with subharmonic cross-waves. We derive the nonlinear spatial–temporal evolution equation governing the motion of the cross-wave by use of the average Lagrangian method. In addition to energy-input terms from the interaction with the forced ring wave, the evolution equation contains a damping term associated with energy loss in cross-wave propagation. We show that the presence of the damping term leads to a non-trivial threshold value of the ring wave steepness (or amplitude) beyond which the cross-wave becomes unstable and grows with time by taking energy from the ring wave. Finally, we extend this analysis to the experimental case of Tatsuno et al. (Rep. Res. Inst. Appl. Mech. Kyushu University, vol. 17, 1969, pp. 195–215) in which asymmetric wave patterns are observed during high-frequency vertical oscillations of a surface-piercing sphere. The theoretical prediction of the threshold value of oscillation amplitude and characteristic features of generated radial cross-waves agrees reasonably well with experimental observations.

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave-structure interactions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 869 , 25 June 2019 , pp. 439 - 467
Copyright
© 2019 Cambridge University Press 

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