Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-06-10T23:56:02.892Z Has data issue: false hasContentIssue false
  • English
  • Français

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: yuming@mit.edu
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Becker, J. M. & Miles, J. M. 1991 Standing radial cross-waves. J. Fluid Mech. 222, 471–499.Google Scholar
Becker, J. M. & Miles, J. M. 1992 Progressive radial cross-waves. J. Fluid Mech. 245, 29–46.Google Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.Google Scholar
Bernard, B. J. S. & Prichard, W. G. 1972 Cross-waves. Part 2. Experiments. J. Fluid Mech. 55, 245–255.Google Scholar
Evans, D. V. 1968 The effect of surface tension on the wave produced by a heaving circular cylinder. Proc. Camb. Phil. Soc. 64, 833–847.Google Scholar
Faraday, M. 1831 On a peculiar class of acoustical figures, and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 299–341.Google Scholar
Garrett, C. J. R. 1970 On cross-waves. J. Fluid Mech. 41, 837–849.Google Scholar
Havelock, T. H. 1929 Forced surface waves on water. Phil. Mag. 8, 569–576.Google Scholar
John, F. 1950 On the motion of floating bodies. II. simple harmonic motions. Commun. Pure Appl. Maths 3, 45–101.Google Scholar
Jones, A. F. 1984 The generation of cross-waves in a long deep channel by parametric resonance. J. Fluid Mech. 138, 53–74.Google Scholar
Lin, J. D. & Howard, L. N.1960 Non-linear standing waves in rectangular tank due to forced oscillation. MIT Hydrodynamic Laboratory Tech. Rep. 44, ONR Contracts Nonr-1841-12.Google Scholar
Luke, J. C. 1967 A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395–397.Google Scholar
Mahoney, J. J. 1972 Cross-waves. Part 1. Theory. J. Fluid Mech. 55, 229–244.Google Scholar
Miles, J. M. 1976 Nonlinear surface waves in closed basins. J. Fluid Mech. 75, 419–448.Google Scholar
Miles, J. M. 1984 Nonlinear Faraday resonance. J. Fluid Mech. 146, 285–302.Google Scholar
Miles, J. M. 1988 Parametrically excited, standing cross-waves. J. Fluid Mech. 186, 119–127.Google Scholar
Miles, J. M. & Becker, J. M. 1988 Parametrically excited, progressive cross-waves. J. Fluid Mech. 186, 129–146.Google Scholar
Rajchenbach, J., Clamond, D. & Leroux, A. 2013 Observation of star-shaped surface gravity waves. Phys. Rev. Lett. 110, 094502.Google Scholar
Rhodes-Robinson, P. F. 1971 On the forced surface waves due to a vertical wave-maker in the presence of surface tension. Proc. Camb. Phil. Soc. 70, 323–337.Google Scholar
Taneda, S. 1986 Transfiguration of surface waves around an oscillating sphere. Fluid Dyn. Res. 1, 1–2.Google Scholar
Taneda, S. 1991 Visual observations of the flow around a half-submerged oscillating sphere. J. Fluid Mech. 227, 193–209.Google Scholar
Tatsuno, M., Inoue, S. & Okabe, J. 1969 Transfiguration of surface waves. Rep. Res. Inst. Appl. Mech. Kyushu University 17, 195–215.Google Scholar
Tsai, W. R., Yue, D. K. P. & Yip, K. M. K. 1990 Resonantly excited regular and chaotic motions in a rectangular wave tank. J. Fluid Mech. 216, 343–380.Google Scholar
Wehausen, J. V. & Laitone, E. V. 1960 Surface Waves. Springer.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar