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On the steady-state fully resonant progressive waves in water of finite depth

Published online by Cambridge University Press:  07 September 2012

Dali Xu ,
Zhiliang Lin ,
Shijun Liao  and
Michael Stiassnie
Dali Xu
Affiliation:
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
Zhiliang Lin
Affiliation:
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
Shijun Liao*
Affiliation:
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
Michael Stiassnie
Affiliation:
Faculty of Civil and Environmental Engineering, Technion IIT, Haifa 32000, Israel
*
†Email address for correspondence: [email protected]
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Abstract

The steady-state fully resonant wave system, consisting of two progressive primary waves in finite water depth and all components due to nonlinear interaction, is investigated in detail by means of analytically solving the fully nonlinear wave equations as a nonlinear boundary-value problem. It is found that multiple steady-state fully resonant waves exist in some cases which have no exchange of wave energy at all, so that the energy spectrum is time-independent. Further, the steady-state resonant wave component may contain only a small proportion of the wave energy. However, even in these cases, there usually exist time-dependent periodic exchanges of wave energy around the time-independent energy spectrum corresponding to such a steady-state fully resonant wave, since it is hard to be exactly in such a balanced state in practice. This view serves to deepen and enrich our understanding of the resonance of gravity waves.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 710 , 10 November 2012 , pp. 379 - 418
Copyright
Copyright © Cambridge University Press 2012

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References

1. Benney, D. J. 1962 Non-linear gravity wave interactions. J. Fluid Mech. 14 (4), 577–584.CrossRefGoogle Scholar
2. Bretherton, F. P. 1964 Resonant interactions between waves: the case of discrete oscillations. J. Fluid Mech. 20 (3), 457–479.CrossRefGoogle Scholar
3. Chen, Y. Y. 1990 Nonlinear interaction between two gravity wave trains in water of uniform depth. J. Harbour Technol. 5, 1–46.Google Scholar
4. Craik, A. D. D. 1988 Wave Interactions and Fluid Flows. Cambridge University Press.Google Scholar
5. Davey, A. & Stewartson, K. 1974 On three-dimensional packets of surface waves. Proc. R. Soc. Lond. A 338 (1613), 101–110.Google Scholar
6. Katsardi, V. & Swan, C. 2011 The evolution of large non-breaking waves in intermediate and shallow water. Part 1. Numerical calculations of uni-directional seas. Proc. R. Soc. A 467 (2127), 778–805.CrossRefGoogle Scholar
7. Liao, S. J. 1992 Proposed homotopy analysis techniques for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University.Google Scholar
8. Liao, S. J. 1999 A uniformly valid analytic solution of two-dimensional viscous flow over a semi-infinite flat plate. J. Fluid Mech. 385, 101–128.CrossRefGoogle Scholar
9. Liao, S. J. 2003 Beyond Perturbation: Introduction to the Homotopy Analysis Method. CRC.CrossRefGoogle Scholar
10. Liao, S. J. 2010 An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 15 (8), 2003–2016.CrossRefGoogle Scholar
11. 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
12. Liao, S. J. 2012 Homotopy Analysis Method in Nonlinear Differential Equations. Springer & Higher Education Press.CrossRefGoogle Scholar
13. Longuet-Higgins, M. S. 1962 Resonant interactions between two trains of gravity waves. J. Fluid Mech. 12 (3), 321–332.CrossRefGoogle Scholar
14. Longuet-Higgins, M. S & Smith, N. D. 1966 An experiment on third-order resonant wave interactions. J. Fluid Mech. 25 (3), 417–435.CrossRefGoogle Scholar
15. Madsen, P. A. & Fuhrman, D. R. 2012 Third-order theory for multi-directional irregular waves. J. Fluid Mech. 698, 304–334.CrossRefGoogle Scholar
16. 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.CrossRefGoogle Scholar
17. Mei, C. C., Stiassnie, M. & Yue, D. K. P. 2005 Theory and Applications of Ocean Surface Waves: Nonlinear Aspects, vol. 2. World Scientific.Google Scholar
18. Onorato, M., Osborne, A. R., Janssen, P. A. E. M. & Resio, D. 2009 Four-wave resonant interactions in the classical quadratic Boussinesq equations. J. Fluid Mech. 618, 263–277.CrossRefGoogle Scholar
19. Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9 (2), 193–217.CrossRefGoogle Scholar
20. Phillips, O. M. 1981 Wave interactions: the evolution of an idea. J. Fluid Mech. 106, 215–227.CrossRefGoogle Scholar
21. Stiassnie, M. & Gramstad, O. 2009 On Zakharov’s kernel and the interaction of non-collinear wavetrains in finite water depth. J. Fluid Mech. 639 (1), 433–442.CrossRefGoogle Scholar
22. Stiassnie, M. & Shemer, L. 1984 On modifications of the Zakharov equation for surface gravity waves. J. Fluid Mech. 143, 47–67.CrossRefGoogle Scholar
23. Whitham, G. B. 1962 Mass, momentum and energy flux in water waves. J. Fluid Mech. 12, 135–147.CrossRefGoogle Scholar
24. 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