Non-Gaussian Properties of Shallow Water Waves in Crossing Seas

  • A. Toffoli
  • M. Onorato
  • A. R. Osborne
  • J. Monbaliu


The Kadomtsev–Petviashvili equation, an extension of the Korteweg–de Vries equation in two horizontal dimensions, is here used to study the statistical properties of random shallow water waves in constant depth for crossing sea states. Numerical simulations indicate that the interaction of two crossing wave trains generates steep and high amplitude peaks, thus enhancing the deviation of the surface elevation from the Gaussian statistics. The analysis of the skewness and the kurtosis shows that the statistical properties depend on the angle between the two wave trains.


Wave Height Wave Train Shallow Water Wave Unimodal Condition Petviashvili Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Battjes(1974).
    Battjes JA (1974) Surf similarity. In: Proceedings of the 14th international conference on coastal engineering, Copenhagen, DenmarkGoogle Scholar
  2. Bitner-Gregersen and Eknes(2001).
    Bitner-Gregersen E, Eknes M (2001) Ship accidents due to bad weather. Technical Report 2001-1330, Det Norske VeritasGoogle Scholar
  3. Demirbilek and Vincent(2002).
    Demirbilek Z, Vincent L (2002) Water wave mechanics. In: Demirbilek Z, Vincent L (eds) Coastal engineering manual, Part II. Hydrodynamics, chap. II-1, Engineer manual 1110-2-1100. US Army Corps of Engineers, Washington, DCGoogle Scholar
  4. Donelan and Magnusson(2005).
    Donelan M, Magnusson AK (2005) The role of meteorological focusing in generating rogue wave conditions. In: Proceedings of the 14th Aha Huliko a Hawaiian winter workshop, University of Hawaii at Manoa, USA, 24–28 January 2005Google Scholar
  5. Fornberg and Whitham(1978).
    Fornberg B, Whitham GB (1978) A numerical and theoretical study of certain nonlinear wave phenomena. Philos Trans R Soc Lond Ser A 289:373–404CrossRefGoogle Scholar
  6. Forristall(2000).
    Forristall GZ (2000) Wave crests distributions: Observations and second-order theory. J Phys Ocean 30:1931–1943CrossRefGoogle Scholar
  7. Goda(2000).
    Goda Y (2000) Random seas and design on marine structures. Advanced Series on Ocean engineering, vol. 15. World Scientific, SingaporeGoogle Scholar
  8. Greenslade(2001).
    Greenslade DJM (2001) A wave modelling study of the 1998 sydney to hobart yacht race. Aust Met Mag 50:53–63Google Scholar
  9. Hauser et al.(2005)
    Hauser D, Kahma KK, Krogstad HE, Lehner S, Monbaliu J, Wyatt LW (eds) (2005) Measuring and analysing the directional spectrum of ocean waves. Cost Office, BrusselsGoogle Scholar
  10. Herbers(2003).
    Herbers THC, Orzech M, Elgar S, Guza RT (2007) Shoaling transformation of wave frequency-directional spectra. J Geophys Res 108(C1):doi:10.1029/2001JC001304Google Scholar
  11. Janssen et al.(2006)
    Janssen TT, Herbers THC, Battjes JA (2006) Generalized evolution equations for nonlinear surface gravity waves over two-dimensional topography. J Fluid Mech 552:393–418CrossRefGoogle Scholar
  12. Johnson(1997).
    Johnson RS (1997) A modern introduction to the mathematical theory of water waves. Cambridge University, CambridgeGoogle Scholar
  13. Kadomtsev and Petviashvili(1970).
    Kadomtsev BB, Petviashvili VI (1970) On the stability of solitary waves in weakly dispersive media. Sov Phys Doklady 15:539–541Google Scholar
  14. Komen et al.(1994)
    Komen GJ, Cavaleri L, Donelan M, Hasselmann K, Hasselmann H, Janssen PAEM (1994) Dynamics and modeling of ocean waves. Cambridge University, CambridgeGoogle Scholar
  15. Lehner et al.(2005)
    Lehner S, Günther H, Rosenthal W (2005) Extreme wave observations from radar data sets. In: Ocean waves measurements and analysis, Fifth international symposium WAVES 2005, Madrid, Spain, 3–7 July 2005, paper 69Google Scholar
  16. Longuet-Higgins(1952).
    Longuet-Higgins MS (1952) On the statistical distribution of the heights of sea waves. J Mar Res 11:1245–1266Google Scholar
  17. Miles(1977a).
    Miles JW (1977a) Diffraction of solitary waves. Z Ang Math 28:889–902CrossRefGoogle Scholar
  18. Miles(1977b).
    Miles JW (1977b) Note on solitary wave on a slowly varying channel. J Fluid Mech 80:149–152CrossRefGoogle Scholar
  19. Onorato et al.(2006)
    Onorato M, Osborne AR, Serio M (2006) Modulation instability in crossing sea states: A possible mechanism for the formation of freak waves. Phys Rev Lett 96:014503CrossRefGoogle Scholar
  20. Osborne and Petti(1994).
    Osborne AR, Petti M (1994) Laboratory-generated, shallow-water surface waves: analysis using the periodic, inverse scattering transform. Phys Fluids 6(5):1727–1744CrossRefGoogle Scholar
  21. Pelinovsky and Sergeeva(2006).
    Pelinovsky E, Sergeeva A (2006) Numerical modeling of the kdv random wave field. Eur J Mech B Fluids 25:425–434CrossRefGoogle Scholar
  22. Peterson et al.(2003)
    Peterson P, Soomere T, Engelbrecht J, van Groesen E (2003) Soliton interaction as a possible model for extreme waves in shallow water. Nonlinear Proc Geophys 10:503–510CrossRefGoogle Scholar
  23. Segur and Finkel(1985).
    Segur H, Finkel A (1985) An analytical model of periodic waves in shallow water. Stud Appl Math 73:183–220Google Scholar
  24. Shukla et al.(2006)
    Shukla PK, Kaurakis I, Eliasson B, Marklund M, Stenflo L (2006) Instability and evolution of nonlinearly interacting water waves. Phys Rev Lett 97:094501CrossRefGoogle Scholar
  25. Soomere and Engelbrecht(2005).
    Soomere T, Engelbrecht J (2005) Extreme elevation and slopes of interacting solitons in shallow water. Wave Motion 41:179–192CrossRefGoogle Scholar
  26. Soomere and Engelbrecht(2006).
    Soomere T, Engelbrecht J (2006) Weakly two-dimensional interaction of solitons in shallow water. Eur J Mech B Fluids 25:636–648CrossRefGoogle Scholar
  27. Tanaka(2001).
    Tanaka M (2001) A method of studying nonlinear random field of surface gravity waves by direct numerical simulations. Fluid Dyn Res 28:41–60CrossRefGoogle Scholar
  28. Tayfun(1981).
    Tayfun AM (1981) Distribution of crest-to-trough wave heights. J Waterw Port C Ocean Eng 107(3):149–158Google Scholar
  29. Tayfun(1980).
    Tayfun MA (1980) Narrow-band nonlinear sea waves. J Geophys Res 85(C3):1548–1552CrossRefGoogle Scholar
  30. Toffoli et al.(2006a)
    Toffoli A, Lefèvre JM, Bitner-Gregersen E, Monbaliu J (2006a) Towards the identification of warning criteria: Analysis of a ship accident database. Appl Ocean Res 27:281–291CrossRefGoogle Scholar
  31. Toffoli et al.(2006b)
    Toffoli A, Onorato M, Monbaliu J (2006b) Wave statistics in unimodal and bimodal seas from a second-order model. Eur J Mech B Fluids 25:649–661CrossRefGoogle Scholar
  32. Ursell(1953).
    Ursell F (1953) The long wave paradox in the theory of gravity waves. Proc Camb Philos Soc 49:685–694CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2008

Authors and Affiliations

  • A. Toffoli
    • 1
  • M. Onorato
    • 2
  • A. R. Osborne
    • 2
  • J. Monbaliu
    • 1
  1. 1.Katholieke Universiteit LeuvenLeuvenBelgium
  2. 2.Universitá di TorinoTorinoItaly

Personalised recommendations