Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Ocean Sciences Supplement
  • Published:

Breaking waves

Why do waves break ? What happens during breaking itself? More than a hundred years of mathematical studies still leave some aspects of these questions unanswered, although we are now understanding much about the processes leading up to breaking and the period after breaking has occurred.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

References

  1. Stokes, G. G., Trans. Camb. Phil. Soc. 8, 441–455 (1847).

    Google Scholar 

  2. Stokes, G. G., Mathematical and Physical Papers 1, 314–326 (Cambridge University Press, Cambridge, 1880).

    Google Scholar 

  3. Wilton, J. R., Phil. Mag. 27, 385–394 (1914).

    Google Scholar 

  4. De, S. C., Proc. Camb. Phil. Soc. 51, 713–736 (1955).

    ADS  Google Scholar 

  5. Boussinesq, J., C. R. Acad. Sci., Paris, 755 (1871).

  6. Rayleigh, Lord, Phil. Mag. 1, 257–279 (1876).

    Google Scholar 

  7. McCowan, J., Phil. Mag. 32, 45–58 (1891).

    Google Scholar 

  8. Korteweg, D. J. & de Vries, G., Phil. Mag. 39, 422–443 (1895).

    Google Scholar 

  9. Weinstein, A., Atti Accad. Lincei, Rend. Cl. Sci. Fiz. Mat. Nat. 3, 463–468 (1926).

    Google Scholar 

  10. Long, R. R., Tellus 8, 460–471 (1956).

    ADS  Google Scholar 

  11. Laitone, E. V., J. Fluid Mech. 9, 430–444 (1960).

    ADS  MathSciNet  Google Scholar 

  12. Grimshaw, R., J. Fluid Mech. 46, 611–622 (1971).

    ADS  Google Scholar 

  13. Stokes, G. G., Mathematical and Physical Papers 1, 225–228 (Cambridge University Press, Cambridge, 1880).

    Google Scholar 

  14. Michell, J. H., Phil. Mag. 36, 430–437 (1893).

    Google Scholar 

  15. McCowan, J., Phil. Mag. 38, 351–358 (1894).

    Google Scholar 

  16. Havelock, T. H., Proc. R. Soc. A 95, 38–51 (1919).

    Google Scholar 

  17. Longuet-Higgins, M. S., Proc. R. Soc. A 331, 445–456 (1973).

    ADS  Google Scholar 

  18. Grant, M. A., J. Fluid Mech. 59, 257–262 (1973).

    ADS  Google Scholar 

  19. Schwartz, L. W., J. Fluid Mech. 62, 553–578 (1974).

    ADS  Google Scholar 

  20. Norman, A. C., J. Fluid Mech. 66, 261–265 (1974).

    ADS  Google Scholar 

  21. Yamada, H., Rep. Res. Inst. appl. Mech. Kyushu Univ. 5, 37–52 (1957).

    Google Scholar 

  22. Yamada, H., Rep. Res. Inst. appl. Mech. Kyushu Univ. 5, 53–67 (1957).

    Google Scholar 

  23. Dean, R. G., J. geophys. Res. 70, 4561–4572 (1965).

    ADS  Google Scholar 

  24. Schwitters, D. J. thesis, Univ. Arizona (1966).

  25. Thomas, J. W., Mathematika 15, 139–148 (1968).

    Google Scholar 

  26. Byatt-Smith, J. G. B., Proc. R. Soc. A 315, 405–418 (1970).

    ADS  Google Scholar 

  27. Strelkoff, J., Proc. 2nd Int. Conf. Num. Meth. Fluid Dyn. (Springer, New York, 1917).

  28. Von Schwind, J. J. and Reid, R. O., J. Geophys. Res. 77, 420–433 (1972).

    ADS  Google Scholar 

  29. Sasaki, K. and Murakami, T., J. Ocean. Soc. Japan 29, 94–105 (1973).

    Google Scholar 

  30. Thomas, J. W., Q. app. Math. 32, 403–410 (1975).

    Google Scholar 

  31. Fenton, J., J. Fluid Mech. 53, 257–271 (1972).

    ADS  Google Scholar 

  32. Longuet-Higgins, M. S. and Fenton, J. D., Proc. R. Soc. A 340, 471–493 (1974).

    ADS  Google Scholar 

  33. Longuet-Higgins, M. S., Proc. R. Soc. A 342, 157–174 (1975).

    ADS  Google Scholar 

  34. Cokelet, E. D., Phil. Trans. R. Soc. A 286, 183–230 (1977).

    ADS  MathSciNet  Google Scholar 

  35. Byatt-Smith, J. G. B. and Longuet-Higgins, M. S., Proc. R. Soc. A 350, 175–189 (1976).

    ADS  Google Scholar 

  36. Longuet-Higgins, M. S. and Fox, M. J. H., J. Fluid Mech. 80, 721–742 (1977).

    ADS  MathSciNet  Google Scholar 

  37. Daily, J. W. and Stephen, S. C., Proc.—Separate No. 107 (American Society of Civil Engineers, 1951).

  38. Daily, J. W. and Stephen, S. C., Proc. 3rd Conf. cst. Engng., 13–30 (1953).

  39. Danel, P., Gravity Waves, U.S. National Bureau of Standards, Circular 521, 35–38 (US Government Printing Office, Washington D.C., 1952).

    Google Scholar 

  40. Wiegel, R. L., Oceanographical Engineering (Prentice-Hall, Englewood Cliffs, 1964).

    Google Scholar 

  41. Van Dorn, W. G. and Pazan, S. P., Scripps Inst. of Ocean., Adv. Ocean Engng. Lab., Rep. AOEl. 71 (1975).

  42. Morison, J. R. and Crooke, R. C., Tech. Mem. No. 40 Beach Erosion Board, US Army Corps Engineers (1953).

  43. Le Méhauté, B., Divoky, D. and Lin, A., Proc. 11th Conf. cst. Engng. 1, 86–107 (1968).

    Google Scholar 

  44. Iwagaki, Y. and Sakai, T., Proc. 12th Conf. cst. Engng. 1, 309–325 (1970).

    Google Scholar 

  45. Dean, R. G., Proc. Symp. Long Waves, 129–152 (University of Delaware, Newark, Delaware, 1972).

  46. Cox, C. S., J. mar. Res. 16, 199–225 (1958).

    Google Scholar 

  47. Longuet-Higgins, M. S., J. Fluid Mech. 16, 138–159 (1963).

    ADS  MathSciNet  Google Scholar 

  48. Crapper, G. D., J. Fluid Mech. 2, 532–540 (1957).

    ADS  MathSciNet  Google Scholar 

  49. Kinnersley, W., J. Fluid Mech. 77, 229–241 (1976).

    ADS  MathSciNet  Google Scholar 

  50. Iversen, H. W., Gravity Waves, US National Bureau of Standards, Circular 521, 9–32 (US Government Printing Office, Washington, DC, 1952).

    Google Scholar 

  51. Mason, M. A., Gravity Waves, US National Bureau of Standards, Circular 521, 215–220 (US Government Printing Office, Washington, DC, 1952).

    Google Scholar 

  52. Iversen, H. W., Proc. 3rd Conf. cst. Engng. 1–12 (1953).

  53. Galvin, C. J., J. geophys. Res. 73, 3651–3659 (1968).

    ADS  Google Scholar 

  54. Galvin, C. J., Waves on Beaches (ed. Meyer, R. E.), 413–456 (Academic, New York, 1972).

    Google Scholar 

  55. Kjeldsen, S. P. and Olsen, G. B., Breaking Waves (16 mm film) Coastal Engineering Laboratory, Technical University of Denmark (1968).

    Google Scholar 

  56. Patrick, D. A. and Wiegel, R. L., Proc. 1st Conf. Ships Waves, 397–422 (1955).

  57. Penney, W. G. and Price, A. T., Phil. Trans. R. Soc. A 244, 254–284 (1952).

    ADS  Google Scholar 

  58. Taylor, G. I., Proc. R. Soc. A 218, 44–59 (1953).

    ADS  Google Scholar 

  59. Weggel, J. R., J. WatWays. Harb. Div. Am. Soc. civ. Engrs. 98, 529–548 (1972).

    Google Scholar 

  60. Ippen, A. T. and Kullen, G., MIT Hydrodynamics Lab., Tech. Rep. No. 15 (1955).

  61. Camfield, F. E. and Street, R. L., J. WatWays. Harb. Div. Am. Soc. civ. Engrs 95, 1–22 (1969).

    Google Scholar 

  62. Divoky, D., Le Méhauté, B. and Lin, A., J. Geophys. Res. 75, 1681–1692 (1970).

    ADS  Google Scholar 

  63. Iwagaki, Y. and Sakai, T., Mem. Faculty Engng Kyoto Univ. 38, 11–20 (1976).

    Google Scholar 

  64. Miller, R. L. and Zeigler, J. M., Proc. 9th Conf. cst. Engng 103–122 (1964).

  65. Airy, G. B., Encyclopedia Metropolitan London (1845).

    Google Scholar 

  66. Biesel, F., Gravity Waves, US National Bureau of Standards, Circular 521, 243–253 (US Government Printing Office, Washingtun, D.C., 1952).

    Google Scholar 

  67. Price, R. K., J. geophys. Res. 76, 1576–1581 (1971).

    ADS  Google Scholar 

  68. Harlow, F. H., Shannon, J. P. and Welch, J. E., Science 149, 1092–1093 (1965).

    ADS  PubMed  CAS  Google Scholar 

  69. Chan, R. K.-C. and Street, R. L., J. comp. Phys. 6, 68–94 (1970).

    ADS  Google Scholar 

  70. Chan, R. K.-C. and Street, R. L., Proc. 12th Conf. cst. Engng 1, 345–361 (1970).

    Google Scholar 

  71. Brennen, C. and Whitney, A. K., Proc. 8th Symp. Naval Hydro. 117–145 (Office of Naval Research, Washington, DC, 1970).

  72. Chan, R. K.-C., J. comp. Phys. 17, 311–331 (1975).

    ADS  Google Scholar 

  73. Longuet-Higgins, M. S. and Cokelet, E. D., Proc. R. Soc. A 350, 1–26 (1976).

    ADS  Google Scholar 

  74. Banner, M. L. and Phillips, O. M., J. Fluid Mech. 65, 647–656 (1974).

    ADS  Google Scholar 

  75. Barnett, T. P. and Kenyon, K. E., Rep. Prog. Phys. 38, 667–729 (1975).

    ADS  Google Scholar 

  76. Banner, M. L. and Melville, W. K., J. Fluid Mech. 77, 825–842 (1976).

    ADS  Google Scholar 

  77. Longuet-Higgins, M. S. and Turner, J. S., J. Fluid Mech. 63, 1–20 (1974).

    ADS  Google Scholar 

  78. Rajaratnam, N., Adv. Hydrosci. 4, 197–280 (1967).

    Google Scholar 

  79. Witting, J., CRC Critical Reviews in Solid State Sciences, 2, 555–581 (1972).

    Google Scholar 

  80. Hibberd, S. & Peregrine, D. H., Proc. IUTAM symp. on waves on water of varying depth, (20–23 July 1976). (Australian Acad. Sci., Canberra, 1977).

  81. Meyer, R. E. and Taylor, A. D., Waves on Beaches (ed. Meyer, R. E.), 357–411 (Academic, New York, 1972).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cokelet, E. Breaking waves. Nature 267, 769–774 (1977). https://doi.org/10.1038/267769a0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/267769a0

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing