Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-26T20:25:25.423Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady-state resonance of multiple wave interactions in deep water

Published online by Cambridge University Press:  24 February 2014

Zeng Liu  and
Shi-Jun Liao
Zeng Liu
Affiliation:
State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China
Shi-Jun Liao*
Affiliation:
State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China Key Laboratory of Education-Ministry in Scientific Computing, Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, PR China Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University (KAU), Jeddah, Saudi Arabia
*
†Email address for correspondence: sjliao@sjtu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

The steady-state resonance of multiple surface gravity waves in deep water was investigated in detail to extend the existing results due to Liao (Commun. Nonlinear Sci. Numer. Simul., vol. 16, 2011, pp. 1274–1303) and Xu et al. (J. Fluid Mech., vol. 710, 2012, pp. 379–418) on steady-state resonance from a quartet to more general and coupled resonant quartets, together with higher-order resonant interactions. The exact nonlinear wave equations are solved without assumptions on the existence of small physical parameters. Multiple steady-state resonant waves are obtained for all the considered cases, and it is found that the number of multiple solutions tends to increase when more wave components are involved in the resonance sets. The topology of wave energy distribution in the parameter space is analysed, and it is found that the steady-state resonant waves indeed form a continuum in the parameter space. The significant roles of the near-resonance and nonlinearity were also revealed. It is found that all of the near-resonant components as a whole contain more and more wave energy, as the wave patterns tend from two dimensions to one dimension, or as the nonlinearity of the steady-state resonant wave system increases. In addition, the linear stability of the steady-state resonant waves is analysed. It is found that the steady-state resonant waves are stable, as long as the disturbance does not resonate with any components of the basic wave. All of these findings are helpful to enrich and deepen our understanding about resonant gravity waves.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 742 , 10 March 2014 , pp. 664 - 700
Copyright
© 2014 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

Annenkov, S. Y. & Shrira, V. I. 2006 Role of non-resonant interactions in the evolution of nonlinear random water wave fields. J. Fluid Mech. 561, 181–208.Google Scholar
Badulin, S. I., Shrira, V., Kharif, C. & Ioualalen, M. 1995 On two approaches to the problem of instability of short-crested water waves. J. Fluid Mech. 303 (1), 297–326.CrossRefGoogle Scholar
Benjamin, T. B. & Brooke, F. J. 1967 The disintegration of wave trains on deep water. J. Fluid Mech. 27 (3), 417–430.CrossRefGoogle Scholar
Benney, D. J. 1962 Non-linear gravity wave interactions. J. Fluid Mech. 14 (4), 577–584.Google Scholar
Bretherton, F. P. 1964 Resonant interactions between waves. The case of discrete oscillations. J. Fluid Mech. 20 (3), 457–479.CrossRefGoogle Scholar
Chen, B & Saffman, P. G.1978 Steady gravity–capillary waves on deep water. I. Weakly nonlinear waves. Tech. Rep. DTIC Document.Google Scholar
Craik, A. D. D. 1985 Wave interactions and fluid flows. Cambridge University Press.Google Scholar
Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary–gravity waves. Annu. Rev. Fluid Mech. 31 (1), 301–346.Google Scholar
Francius, M. & Kharif, C. 2006 Three-dimensional instabilities of periodic gravity waves in shallow water. J. Fluid Mech. 561 (1), 417–437.Google Scholar
Fructus, D., Kharif, C., Francius, M., Kristiansen, O., Clamond, D. & Grue, J. 2005 Dynamics of crescent water wave patterns. J. Fluid Mech. 537, 155–186.Google Scholar
Grimshaw, R. 2005 Nonlinear Waves in Fluids: Recent Advances and Modern Applications. Springer.CrossRefGoogle Scholar
Hammack, J. L. & Henderson, D. M. 1993 Resonant interactions among surface water waves. Annu. Rev. Fluid Mech. 25 (1), 55–97.CrossRefGoogle Scholar
Hasselmann, K. 1962 On the non-linear energy transfer in a gravity-wave spectrum. J. Fluid Mech. 12, 481–500.CrossRefGoogle Scholar
Hasselmann, K. 1963a On the non-linear energy transfer in a gravity-wave spectrum. Part 2. Conservation theorems; wave–particle analogy; irreversibility. J. Fluid Mech. 15 (2), 273–281.Google Scholar
Hasselmann, K. 1963b On the non-linear energy transfer in a gravity-wave spectrum. Part 3. Evaluation of the energy flux and swell–sea interaction for a Neumann spectrum. J. Fluid Mech. 15 (3), 385–398.Google Scholar
Ioualalen, M. & Kharif, C. 1993 Stability of three-dimensional progressive gravity waves on deep water to superharmonic disturbances. Eur. J. Mech. Fluids B 12 (3), 401–414.Google Scholar
Ioualalen, M. & Kharif, C. 1994 On the subharmonic instabilities of steady three-dimensional deep water waves. J. Fluid Mech. 262, 265–265.Google Scholar
Janssen, P. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33 (4), 863–884.Google Scholar
Kartashova, E. 2010 Nonlinear resonance analysis: theory, computation, applications. Cambridge University Press.Google Scholar
Komen, G. J., Cavaleri, L., Donelan, M., Hasselmann, K., Hasselmann, S. & Janssen, P. A. E. M. 1996 Dynamics and modelling of ocean waves. Cambridge University Press.Google Scholar
Liao, S. J.1992 Proposed homotopy analysis techniques for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University.Google Scholar
Liao, S. J. 2003 Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman & Hall/CRC Press.CrossRefGoogle Scholar
Liao, S. J. 2011 On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves. Commun. Nonlinear Sci. Numer. Simul. 16 (3), 1274–1303.CrossRefGoogle Scholar
Liao, S. J. 2012 Homotopy Analysis Method in Nonlinear Differential Equations. Springer & Higher Education Press.Google Scholar
Liao, S. J. 2013 Advances in Homotopy Analysis Method. World Scientific Press.Google Scholar
Longuet-Higgins, M. S. 1962 Resonant interactions between two trains of gravity waves. J. Fluid Mech. 12 (3), 321–332.Google Scholar
Longuet-Higgins, M. S. & Smith, N. D. 1966 An experiment on third-order resonant wave interactions. J. Fluid Mech. 25 (3), 417–435.Google Scholar
MacKay, R. S. & Saffman, P. G. 1986 Stability of water waves. Proc. R. Soc. Lond. A 406 (1830), 115–125.Google Scholar
Madsen, P. A. & Fuhrman, D. R. 2006 Third-order theory for bichromatic bi-directional water waves. J. Fluid Mech. 557, 369–397.Google Scholar
McGoldrick, L. F., Phillips, O. M., Huang, N. E. & Hodgson, T. H. 1966 Measurements of third-order resonant wave interactions. J. Fluid Mech. 25 (3), 437–456.Google Scholar
McLean, J. W. 1982 Instabilities of finite-amplitude water waves. J. Fluid Mech. 114 (1), 315–330.CrossRefGoogle Scholar
Okamura, M. 1996 Notes on short-crested waves in deep water. J. Phys. Soc. Japan 65 (9), 2841–2845.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. J. Fluid Mech. 9, 193–217.Google Scholar
Phillips, O. M. 1977 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
Phillips, O. M. 1981 Wave interactions – the evolution of an idea. J. Fluid Mech. 106 (1), 215–227.Google Scholar
Roberts, A. J. 1983 Highly nonlinear short-crested water waves. J. Fluid Mech. 135, 301–321.CrossRefGoogle Scholar
Shrira, V. I., Badulin, S. I. & Kharif, C. 1996 A model of water wave ‘horse-shoe’ patterns. J. Fluid Mech. 318, 375–405.Google Scholar
Xu, D., Lin, Z., Liao, S. & Stiassnie, M. 2012 On the steady-state fully resonant progressive waves in water of finite depth. J. Fluid Mech. 710, 379–418.CrossRefGoogle Scholar
Young, I. R. 1999 Wind Generated Ocean Waves. vol. 2. Elsevier.Google Scholar
Yuen, H. C. & Lake, B. M. 1982 Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech. 22, 67–229.CrossRefGoogle Scholar
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 190–194.CrossRefGoogle Scholar

Liu supplementary movie

Movie1: Perspective view of surface elevation of group 1 when $k_{2,x}=0.9$ and $varepsilon=1.001$. With angles closed to collinear limit, the wave pattern behaviors like long-crested waves, which is warped by the wave group node.

Download Liu supplementary movie(Video)
Video 5 MB

Liu supplementary movie

Movie 2: Perspective view of surface elevation of group 2 when $k_{2,x}=0.9$ and $varepsilon=1.001$. With angles closed to collinear limit, the wave pattern behaviors like long-crested waves, which is warped by the wave group node.

Download Liu supplementary movie(Video)
Video 4.9 MB

Liu supplementary movie

Movie 3: Perspective view of surface elevation of group 3 when $k_{2,x}=0.9$ and $varepsilon=1.001$. With angles closed to collinear limit, the wave pattern behaviors like long-crested waves, which is warped by the wave group node.

Download Liu supplementary movie(Video)
Video 5 MB