Skip to main content Accessibility help

On dispersion of directional surface gravity waves

  • Tore Magnus A. Taklo (a1), Karsten Trulsen (a1), Harald E. Krogstad (a2) and José Carlos Nieto Borge (a3)


Using a nonlinear evolution equation we examine the dependence of the dispersion of directional surface gravity waves on the Benjamin–Feir index (BFI) and crest length. A parameter for describing the deviation between the dispersion of simulated waves and the theoretical linear dispersion relation is proposed. We find that for short crests the magnitude of the deviation parameter is low while for long crests the magnitude is high and depends on the BFI. In the present paper we also consider laboratory data of directional waves from the Marine Research Institute of the Netherlands (MARIN). The MARIN data confirm the simulations for three cases of BFI and crest length.


Corresponding author

Email address for correspondence:


Hide All
Alber, I. A. 1978 The effects of randomness on the stability of two-dimensional surface wavetrains. Proc. R. Soc. Lond. A 363, 525546.
Alber, I. E. & Saffman, P. G. 1978 Stability of random nonlinear deep water waves with finite bandwidth. In TWR Defense and Spacesystems Rep., 31326–6035–RU–00, 89.
Bateman, W. J. D., Swan, C. & Taylor, P. H. 2001 On the efficient numerical simulation of directionally-spread surface water waves. J. Comput. Phys. 174, 277305.
Crawford, D. R., Saffman, P. G. & Yuen, H. C. 1980 Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves. Wave Motion 2, 116.
Dysthe, K. B. 1979 Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Phil. Trans. R. Soc. Lond. A 369, 105114.
Fessler, J. A. & Sutton, B. P. 2003 Nonuniform fast Fourier transform using min-max interpolation. IEEE Trans. Signal Process. 51, 560574.
Gibson, R. S. & Swan, C. 2006 The evolution of large ocean waves: the role of local and rapid spectral changes. Proc. R. Soc. Lond. A 463, 2148.
Goda, Y. 2000 Random Seas and Design of Maritime Structures. World Scientific.
Gramstad, O. & Trulsen, K. 2007 Influence of crest and group length on the occurence of freak waves. J. Fluid Mech. 582, 463472.
Hasselmann, K., Barnett, T. P., Bouws, E., Carlson, H., Cartwright, D. E., Enke, K., Ewing, J. A., Gienapp, H., Hasselmann, D. E., Kruseman, P. et al. 1973 Measurements of wind-wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Erg. zur Deutsch. Hydrograph. Z. A 8 (12), 95.
Houtani, H., Waseda, T., Fujimoto, W., Kiyomatsu, K. & Tanizawa, K. 2015 Freak wave generation in a wave basin with HOSM-WG method. In ASME 2015 34th International Conference on Ocean, Offshore and Artic Engng, Paper no. OMAE2015-42284.
Janssen, P. A. E. M. 2003 Nonlinear four-wave interactions and freak waves. J. Phys. Oceanogr. 33, 863884.
Krogstad, H. E. & Trulsen, K. 2010 Interpretations and observations of ocean wave spectra. Ocean Dyn. 60, 973991.
Lo, E. & Mei, C. C. 1985 A numerical study of water-wave modulations based on a higher-order nonlinear Schrödinger equation. J. Fluid Mech. 150, 395415.
Lo, E. & Mei, C. C. 1987 Slow evolution of nonlinear deep water waves in two horizontal directions: A numerical study. Wave Motion 9, 245259.
Mori, N., Onorato, M. & Janssen, P. A. E. M. 2011 On the estimation of the kurtosis in directional sea states for freak wave forecasting. J. Phys. Oceanogr. 41, 14841497.
Naaijen, P., van Dijk, R., Huijsmans, R. H. M., El-Mouhandiz, A. A. & Danneberg, J. 2009 Real time estimation of ship motions in short crested seas. In ASME 2009 28th International Conference on Ocean, Offshore and Arctic Engng, Paper No. OMAE2009-79366, pp. 243255.
Nieto Borge, J. C., Rodríguez, G., Hessner, K. & Izquierdo, P. 2004 Inversion of marine radar images for surface wave analysis. J. Atmos. Ocean Technol. 21, 12911300.
Onorato, M., Osborne, A. R., Serio, M. & Bertone, S. 2001 Freak waves in random oceanic sea states. Phys. Rev. Lett. 86, 58315834.
Simanesew, A., Krogstad, H. E., Trulsen, K. & Nieto Borge, J. C. 2016 Development of frequency-dependent ocean wave directional distributions. Appl. Ocean Res. 59, 304312.
Socquet-Juglard, H., Dysthe, K., Trulsen, K., Krogstad, H. E. & Liu, J. D. 2005 Probability distributions of surface gravity waves during spectral changes. J. Fluid Mech. 542, 195216.
Taklo, T. M. A., Trulsen, K., Gramstad, O., Krogstad, H. E. & Jensen, A. 2015 Measurement of the dispersion relation for random surface gravity waves. J. Fluid Mech. 766, 326336.
Toffoli, A., Gramstad, O., Trulsen, K., Monbaliu, J., Bitner-Gregersen, E. & Onorato, M. 2010 Evolution of weakly nonlinear random directional waves: laboratory experiments and numerical simulations. J. Fluid Mech. 664, 313336.
Trulsen, K., Kliakhandler, I., Dysthe, K. B. & Velarde, M. G. 2000 On weakly nonlinear modulation of waves on deep water. Phys. Fluids 12, 24322437.
Tucker, M. J. & Pitt, E. G. 2001 Waves in Ocean Engineering. Elsevier.
Waseda, T., Kinoshita, T. & Tamura, H. 2009 Evolution of a random directional wave and freak wave occurence. J. Phys. Oceanogr. 39, 621639.
Xiao, W., Liu, Y., Wu, G. & Yue, D. K. P. 2013 Rouge wave occurence and dynamics by direct simulations of nonlinear wave-field evolution. J. Fluid Mech. 720, 357392.
Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190194.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

On dispersion of directional surface gravity waves

  • Tore Magnus A. Taklo (a1), Karsten Trulsen (a1), Harald E. Krogstad (a2) and José Carlos Nieto Borge (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.