Skip to main content Accessibility help

Frequency spectra evolution of two-dimensional focusing wave groups in finite depth water

  • Zhigang Tian (a1) (a2), Marc Perlin (a2) and Wooyoung Choi (a1) (a3)


An experimental and numerical study of the evolution of frequency spectra of dispersive focusing wave groups in a two-dimensional wave tank is presented. Investigations of both non-breaking and breaking wave groups are performed. It is found that dispersive focusing is far more than linear superposition, and that it undergoes strongly nonlinear processes. For non-breaking wave groups, as the wave groups propagate spatial evolution of wave frequency spectra, spectral bandwidth, surface elevation skewness, and kurtosis are examined. Nonlinear energy transfer between the above-peak () and the higher-frequency () regions, with being the spectral peak frequency, is demonstrated by tracking the energy level of the components in the focusing and defocusing process. Also shown is the nonlinear energy transfer to the lower-frequency components that cannot be detected easily by direct comparisons of the far upstream and downstream measurements. Energy dissipation in the spectral peak region () and the energy gain in the higher-frequency region () are quantified, and exhibit a dependence on the Benjamin–Feir Index (BFI). In the presence of wave breaking, the spectral bandwidth reduces as much as 40 % immediately following breaking and eventually becomes much smaller than its initial level. Energy levels in different frequency regions are examined. It is found that, before wave breaking onset, a large amount of energy is transferred from the above-peak region () to the higher frequencies (), where energy is dissipated during the breaking events. It is demonstrated that the energy gain in the lower-frequency region is at least partially due to nonlinear energy transfer prior to wave breaking and that wave breaking may not necessarily increase the energy in this region. Complementary numerical studies for breaking waves are conducted using an eddy viscosity model previously developed by the current authors. It is demonstrated that the predicted spectral change after breaking agrees well with the experimental measurements.


Corresponding author

Email address for correspondence:


Hide All
1. Baldock, T. E., Swan, C. & Taylor, P. H. 1996 A laboratory study of nonlinear surface waves on water. Phil. Trans. R. Soc. Lond. A 354, 649676.
2. Banner, M. L. & Peirson, W. L. 2007 Wave breaking onset and strength for two-dimensional deep-water wave groups. J. Fluid Mech. 585, 93115.
3. Benjamin, T. B & Feir, J. E. 1967 Disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417430.
4. Choi, W. 1995 Nonlinear evolution equations for two-dimensional surface waves in a fluid of finite depth. J. Fluid Mech. 295, 381394.
5. Choi, W., Kent, C. P. & Schillinger, C. 2005 Numerical modelling of nonlinear surface waves and its validation. Adv. Engng Mech. 94110.
6. Drazen, D. A., Melville, W. K. & Lenain, L. 2008 Inertial scaling of dissipation in unsteady breaking waves. J. Fluid Mech. 611, 307332.
7. Duncan, J. H. 1981 An experimental investigation of breaking waves produced by a towed hydrofoil. Proc. R. Soc. Lond. Ser. A 377, 331348.
8. Duncan, J. H. 1983 The breaking and non-breaking wave resistance of a two-dimensional hydrofoil. J. Fluid Mech. 126, 507520.
9. Gemmrich, J. R., Banner, M. L. & Garrett, C. 2008 Spectrally resolved energy dissipation rate and momentum flux of breaking waves. J. Phys. Oceanogr. 38, 12961312.
10. Janssen, P. A. E. M. 2003 Nonlinear four-wave interaction and freak waves. J. Phys. Oceanogr. 33 (4), 863884.
11. Jiang, L., Perlin, M. & Schultz, W. W. 2004 Contact-line dynamics and damping for oscillating free surface flows. Phys. Fluids 16 (3), 748758.
12. Kway, J. H. L., Loh, Y. S. & Chan, E. S. 1998 Laboratory study of deep water breaking waves. Ocean Engng 25, 657676.
13. Lake, B. M., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. 1977 Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83, 4974.
14. Longuet-Higgins, M. S. 1983 On the joint distribution of wave periods and amplitudes in a random wave field. Proc. R. Soc. Lond. A. 389, 241258.
15. Mei, C. C. 1983 The Applied Dynamics of Ocean Surface Waves. Wiley-Interscience.
16. Melville, W. K. 1982 The instability and breaking of deep-water waves. J. Fluid Mech. 115, 165185.
17. Melville, W. K. 1994 Energy-dissipation by breaking waves. J. Phys. Oceanogr. 24, 20412049.
18. Melville, W. K. 1996 The role of surface-wave breaking in air–sea interaction. Annu. Rev. Fluid Mech. 28, 279321.
19. Melville, W. K., Veron, F. & White, C. J. 2002 The velocity field under breaking waves: coherent structures and turbulence. J. Fluid Mech. 454, 203233.
20. Meza, E., Zhang, J. & Seymour, R. J. 2000 Free-wave energy dissipation in experimental breaking waves. J. Phys. Oceanogr. 30, 24042418.
21. Nepf, H. M., Wu, C. H. & Chan, E. S. 1998 A comparison of two- and three-dimensional wave breaking. J. Phys. Oceanogr. 28, 14961510.
22. Ochi, M. K. 1998 Ocean waves: The stochastic approach.. Cambridge University Press.
23. Onorato, M., Cavaleri, L., Fouques, S., Gramstad, O., Janssen, P. A. E. M., Monbaliu, J., Osborne, A. R., Pakozdi, C., Serio, M., Stansberg, C. T., Toffoli, A. & Trulsen, K. 2009 Statistical properties of mechanically generated surface gravity waves: a laboratory experiment in a three-dimensional wave basin. J. Fluid Mech. 627, 235257.
24. Perlin, M., He, J. H. & Bernal, L. P. 1996 An experimental study of deep water plunging breakers. Phys. Fluids 8 (9), 23652374.
25. Perlin, M. & Schultz, W. W. 2000 Capillary effects on surface waves. Annu. Rev. Fluid Mech. 32, 241274.
26. Rapp, R. J. & Melville, W. K. 1990 Laboratory measurements of deep-water breaking waves. Phil. Trans. R. Soc. Lond. A. 331, 735800.
27. Shemer, L., Goulitski, K. & Kit, E. 2007 Evolution of wide-spectrum unidirectional wave groups in a tank: an experimental and numerical study. Eur. J. Mech. (B/Fluids) 26, 193219.
28. Shemer, L. & Sergeeva, A. 2009 An experimental study of spatial evolution of statistical parameters in a unidirectional narrow-banded random wavefield. J. Geophys. Res. 114, C01015.
29. Song, J. B. & Banner, M. L. 2002 On determining the onset and strength of breaking for deep water waves. Part I: Unforced irrotational wave groups. J. Phys. Oceanogr. 32, 25412558.
30. Tian, Z. G., Perlin, M. & Choi, W. 2008 Evaluation of a deep-water wave breaking criterion. Phys. Fluids 20, 066604.
31. Tian, Z. G., Perlin, M. & Choi, W. 2010 Energy dissipation in two-dimensional unsteady plunging breakers and an eddy viscosity model. J. Fluid Mech. 655, 217257.
32. Tulin, M. P. & Waseda, T. 1999 Laboratory observations of wave group evolution, including breaking effects. J. Fluid Mech. 378, 197232.
33. West, B. J., Brueckner, K. A., Janda, R. S., Milder, D. M. & Milton, R. L. 1987 A new numerical method for surface hydrodynamics. J. Geophys. Res. 92 (11), 803-11824.
34. Yao, A. F. & Wu, C. H. 2004 Energy dissipation of unsteady wave breaking on currents. J. Phys. Oceanogr. 34, 22882304.
35. Young, I. R. & Babanin, A. V. 2006 Spectral distribution of energy dissipation of wind-generated waves due to dominant wave breaking. J. Phys. Oceanogr. 36, 376394.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Frequency spectra evolution of two-dimensional focusing wave groups in finite depth water

  • Zhigang Tian (a1) (a2), Marc Perlin (a2) and Wooyoung Choi (a1) (a3)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.