An Accurate Model of Wave Refraction Over Shallow Water

  • Manuel N. Gamito
  • F. Kenton Musgrave
Part of the Eurographics book series (EUROGRAPH)


A computer model of wave refraction is desirable, in the context of landscape modeling, to generate the familiar wave patterns seen near coastlines. In this article, we present a new method for the calculation of shallow water wave refraction. The method is more accurate than previously existing methods and provides realistic wave refraction effects. We resort to Fermat’s principle of the shortest path and compute the propagation of wavefronts over an arbitrary inhomogeneous medium. The propagation of wavefronts produces a phase map for each terrain. This phase map is then coupled with a geometric model of waves to generate a heightfield representation of the sea surface.


Computer Graphic Wave Train Wave Breaking Wave Refraction Wave Path 
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. 1.
    Arthur, R., Munk, W., and Isaacs, J. The direct construction of wave rays. Trans. Am. Geophys. Un. 33, 6 (1952), 855–865.Google Scholar
  2. 2.
    Benamou, J.-D. Multivalued solution and viscosity solutions of the eikonal equation. Tech. Rep. 3281, INRIA Rocquencourt, October 1997. available at http: // Scholar
  3. 3.
    Biesel, F. Study of wave propagation in water of gradually varying depth. In Gravity Waves. 1952, pp. 243-253. U.S. National Bureau of Standards Circular.Google Scholar
  4. 4.
    Born, M., and Wolf, E. Principles of Optics: electromagnetic theory of propagation, interference and diffraction of light. Cambridge University Press, Cambridge, UK, 1997.Google Scholar
  5. 5.
    Campion, D., and Brewer, A. The Book of Waves: Form and Beauty on the Ocean, third ed. Roberts Rinehart Pub., December 1997. ISBN 157098168X.Google Scholar
  6. 6.
    Chen, J. X., da Vitoria Lobo, N., Hughes, C. E., and Moshell, J. M. Real-Time fluid simulation in a dynamic virtual environment. IEEE Computer Graphics & Applications 17, 3 (May-June 1997), 52–61. ISSN 0272-1716.CrossRefGoogle Scholar
  7. 7.
    Coxeter, H. S. M. Introduction to Geometry. Wiley, New York, 1961.Google Scholar
  8. 8.
    Foster, N., and Metaxas, D. Realistic animation of liquids. Graphical Models and Image Processing 58, 5 (1996), 471–483.CrossRefGoogle Scholar
  9. 9.
    Fournier, A., and Reeves, W. T. A simple model of ocean waves. In Computer Graphics (SIGGRAPH’ 86 Proceedings) (Aug. 1986), D. C. Evans and R. J. Athay, Eds., vol. 20, pp. 75–84.CrossRefGoogle Scholar
  10. 10.
    Gamito, M. N., and Musgrave, F. K. Non-height field rendering, available at http: // Scholar
  11. 11.
    Gelfand, I. M., and Fomin, S. V. Calculus of Variations, rev. english ed. Prentice-Hall, Englewood Cliffs, N.J., 1963.Google Scholar
  12. 12.
    Gonzato, J.-C., and le Saec, B. A phenomenological model of coastal scenes based on physical considerations. In Computer Animation and Simulation’ 97 (1997), D. Thalmann and M. van de Panne, Eds., Eurographics Association, Springer Computer Science, pp. 137-148. ISBN 3-211-83048-0.Google Scholar
  13. 13.
    Hindmarsh, A. C. Odepack: A systematized collection of ode solvers. In Scientific Computing, R. S. Stepleman, Ed. North-Holland, Amsterdam, 1983, pp. 55-64. Package available at Scholar
  14. 14.
    Kass, M., and Miller, G. Rapid, stable fluid dynamics for computer graphics. In Computer Graphics (SIGGRAPH’ 90 Proceedings) (Aug. 1990), F. Baskett, Ed., vol. 24, pp. 49–57.CrossRefGoogle Scholar
  15. 15.
    Kinsman, B. Wind Waves. Dover, 1984.Google Scholar
  16. 16.
    LE Méhauté, B. An Introduction to Hydrodynamics and Water Waves. Springer-Verlag, New York, 1976.Google Scholar
  17. 17.
    Lewis, J.-P. Algorithms for solid noise synthesis. In Computer Graphics (SIGGRAPH’ 89 Proceedings) (July 1989), J. Lane, Ed., vol. 23, pp. 263–270.CrossRefGoogle Scholar
  18. 18.
    Mastin, G. A., Watterberg, P. A., and Mareda, J. F. Fourier synthesis of ocean scenes. IEEE Computer Graphics and Applications 7, 3 (Mar. 1987), 16–23.CrossRefGoogle Scholar
  19. 19.
    Max, N. L. Vectorized procedural models for natural terrain: Waves and islands in the sunset. In Computer Graphics (SIGGRAPH’ 81 Proceedings) (Aug. 1981), vol. 15, pp. 317–324.CrossRefGoogle Scholar
  20. 20.
    Peachey, D. R. Modeling waves and surf. In Computer Graphics (SIGGRAPH’ 86 Proceedings) (Aug. 1986), D. C. Evans and R. J. Athay, Eds., vol. 20, pp. 65–74.CrossRefGoogle Scholar
  21. 21.
    Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. Numerical Recipes in C: The Art of Scientific Computing (2nd ed.). Cambridge University Press, Cambridge, 1992. ISBN 0-521-4310Google Scholar
  22. 22.
    Schachter, B. J. Long crested wave models. Computer Graphics and Image Processing 12 (1980), 187–201.CrossRefGoogle Scholar
  23. 23.
    Tessendorf, J. Simulating ocean water. In Simulating Nature: From Theory to Applications (1999), D. S. Ebert, Ed., no. 26 in SIGGRAPH 99 Course Notes.Google Scholar
  24. 24.
    TS’O, P. Y., and Barsky, B. A. Modeling and rendering waves: Wave-tracing using beta-splines and reflective and refractive texture mapping. ACM Transactions on Graphics 6, 3 (1987), 191–214.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2000

Authors and Affiliations

  • Manuel N. Gamito
    • 1
  • F. Kenton Musgrave
    • 2
  1. 1.ADETTIEdificio ISCTELisboaPortugal
  2. 2.FractalWorlds.comWaterfordUSA

Personalised recommendations