“Generalizing Shallow Water Simulations with Dispersive Surface Waves” by Jeschke and Wojtan

  • ©Stefan Jeschke and Chris Wojtan




    Generalizing Shallow Water Simulations with Dispersive Surface Waves

Session/Category Title: Pushing the Boundaries




    This paper introduces a novel method for simulating large bodies of water as a height field. At the start of each time step, we partition the waves into a bulk flow (which approximately satisfies the assumptions of the shallow water equations) and surface waves (which approximately satisfy the assumptions of Airy wave theory). We then solve the two wave regimes separately using appropriate state-of-the-art techniques, and re-combine the resulting wave velocities at the end of each step. This strategy leads to the first heightfield wave model capable of simulating complex interactions between both deep and shallow water effects, like the waves from a boat wake sloshing up onto a beach, or a dam break producing wave interference patterns and eddies. We also analyze the numerical dispersion created by our method and derive an exact correction factor for waves at a constant water depth, giving us a numerically perfect re-creation of theoretical water wave dispersion patterns.


    1. George Biddell Airy. 1841. Tides and waves. Encyclopedia Metropolitana, Mixed Sciences 3 (1841).
    2. Omri Azencot, Orestis Vantzos, and Mirela Ben-Chen. 2018. An explicit structure-preserving numerical scheme for EPDiff. In Computer Graphics Forum, Vol. 37. Wiley Online Library, 107–119.
    3. Fa Biesel. 1952. Study of wave propagation in water of gradually varying depth. Gravity waves (1952), 243–253.
    4. Joseph Boussinesq. 1872. Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. Journal de mathématiques pures et appliquées (1872), 55–108.
    5. José A. Canabal, David Miraut, Nils Thuerey, Theodore Kim, Javier Portilla, and Miguel A. Otaduy. 2016. Dispersion Kernels for Water Wave Simulation. ACM Trans. Graph. 35, 6, Article 202 (2016), 10 pages.
    6. Nuttapong Chentanez and Matthias Müller. 2010. Real-Time Simulation of Large Bodies of Water with Small Scale Details. In Proceedings of the 2010 ACM SIGGRAPH/Eurographics Symposium on Computer Animation. Eurographics Association, 197–206.
    7. Nuttapong Chentanez, Matthias Müller, and Tae-Yong Kim. 2015. Coupling 3D Eulerian, Heightfield and Particle Methods for Interactive Simulation of Large Scale Liquid Phenomena. IEEE Transactions on Visualization and Computer Graphics 21, 10 (2015), 1116–1128.
    8. Fang Da, David Hahn, Christopher Batty, Chris Wojtan, and Eitan Grinspun. 2016. Surface-only liquids. ACM Transactions on Graphics (TOG) 35, 4 (2016), 1–12.
    9. B De St Venant. 1871. Theorie du mouvement non-permanent des eaux avec application aux crues des rivers et a l’introduntion des Marees dans leur lit. Academic de Sci. Comptes Redus 73, 99 (1871), 148–154.
    10. Ronald Fedkiw, Jos Stam, and Henrik Wann Jensen. 2001. Visual simulation of smoke. In Proceedings of the 28th annual conference on Computer graphics and interactive techniques. 15–22.
    11. Alexey V Fedorov and W Kendall Melville. 1998. Nonlinear gravity-capillary waves with forcing and dissipation. Journal of Fluid Mechanics 354 (1998), 1–42.
    12. Alain Fournier and William T Reeves. 1986. A simple model of ocean waves. In Proceedings of the 13th annual conference on Computer graphics and interactive techniques. 75–84.
    13. Franz Gerstner. 1809. Theorie der Wellen. Annalen der Physik 32, 8 (1809), 412–445.
    14. Trond Runar Hagen, Jon M Hjelmervik, K-A Lie, Jostein R Natvig, and M Ofstad Henriksen. 2005. Visual simulation of shallow-water waves. Simulation Modelling Practice and Theory 13, 8 (2005), 716–726.
    15. Richard Wesley Hamming. 1998. Digital filters. Courier Corporation.
    16. Damien Hinsinger, Fabrice Neyret, and Marie-Paule Cani. 2002. Interactive animation of ocean waves. In Proceedings of the 2002 ACM SIGGRAPH/Eurographics symposium on Computer animation. 161–166.
    17. Libo Huang, Ziyin Qu, Xun Tan, Xinxin Zhang, Dominik L Michels, and Chenfanfu Jiang. 2021. Ships, splashes, and waves on a vast ocean. ACM Transactions on Graphics (TOG) 40, 6 (2021), 1–15.
    18. S. Jeschke, C. Hafner, N. Chentanez, M. Macklin, M. Müller-Fischer, and C. Wojtan. 2020. Making Procedural Water Waves Boundary-Aware. In Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation. Eurographics Association, Article 5, 8 pages.
    19. Stefan Jeschke, Tomáš Skřivan, Matthias Müller-Fischer, Nuttapong Chentanez, Miles Macklin, and Chris Wojtan. 2018. Water Surface Wavelets. ACM Trans. Graph. 37, 4, Article 94 (2018), 13 pages.
    20. Stefan Jeschke and Chris Wojtan. 2015. Water Wave Animation via Wavefront Parameter Interpolation. ACM Trans. Graph. 34, 3, Article 27 (2015), 14 pages.
    21. Stefan Jeschke and Chris Wojtan. 2017. Water Wave Packets. ACM Trans. Graph. 36, 4, Article 103 (2017), 12 pages.
    22. Robin Stanley Johnson. 1997. A modern introduction to the mathematical theory of water waves. Number 19. Cambridge university press.
    23. Steven G Johnson. 2011. Notes on FFT-based differentiation. MIT Applied Mathematics, Tech. Rep. (2011).
    24. Michael Kass and Gavin Miller. 1990. Rapid, stable fluid dynamics for computer graphics. In Proceedings of the 17th annual conference on Computer graphics and interactive techniques. 49–57.
    25. Todd Keeler and Robert Bridson. 2014. Ocean waves animation using boundary integral equations and explicit mesh tracking. In ACM SIGGRAPH 2014 Posters. 1–1.
    26. Theodore Kim, Jerry Tessendorf, and Nils Thuerey. 2013. Closest point turbulence for liquid surfaces. ACM Transactions on Graphics (TOG) 32, 2 (2013), 1–13.
    27. Anita T Layton and Michiel van de Panne. 2002. A numerically efficient and stable algorithm for animating water waves. The Visual Computer 18, 1 (2002), 41–53.
    28. Jörn Loviscach. 2002. A Convolution-Based Algorithm for Animated Water Waves. In Eurographics short papers.
    29. Jörn Loviscach. 2003. Complex Water Effects at Interactive Frame Rates. Journal of WSCG 11 (2003), 2003.
    30. John Marshall, Kerry Emanuel, and Alistair Adcroft. 2004. Atmospheric And Oceanic Modeling Course 12.950, Lecture 2. Massachusetts Institute of Technology: MIT OpenCouseWare, https://ocw.mit.edu/. License: Creative Commons BY-NC-SA. (Spring 2004).
    31. Gary A Mastin, Peter A Watterberg, and John F Mareda. 1987. Fourier synthesis of ocean scenes. IEEE Computer Graphics and Applications 7, 3 (1987), 16–23.
    32. Zherong Pan, Jin Huang, Yiying Tong, and Hujun Bao. 2012. Wake synthesis for shallow water equation. In Computer Graphics Forum, Vol. 31. Wiley Online Library, 2029–2036.
    33. Darwyn R Peachey. 1986. Modeling waves and surf. ACM Siggraph Computer Graphics 20, 4 (1986), 65–74.
    34. William John Macquorn Rankine. 1863. VI. On the exact form of waves near the surface of deep water. Philosophical transactions of the Royal society of London 153 (1863), 127–138.
    35. Camille Schreck, Christian Hafner, and Chris Wojtan. 2019. Fundamental solutions for water wave animation. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1–14.
    36. Camille Schreck and Chris Wojtan. 2022. Coupling 3D Liquid Simulation with 2D Wave Propagation for Large Scale Water Surface Animation Using the Equivalent Sources Method. In Computer Graphics Forum, Vol. 41. 343–353.
    37. Tomas Skrivan, Andreas Soderstrom, John Johansson, Christoph Sprenger, Ken Museth, and Chris Wojtan. 2020. Wave Curves: Simulating Lagrangian Water Waves on Dynamically Deforming Surfaces. ACM Trans. Graph. 39, 4, Article 65 (2020), 12 pages.
    38. G. S. Stelling and S. P. A. Duinmeijer. 2003. A staggered conservative scheme for every Froude number in rapidly varied shallow water flows. International Journal for Numerical Methods in Fluids 43, 12 (2003), 1329–1354.
    39. Jerry Tessendorf. 2004a. Interactive water surfaces. Game Programming Gems 4 (2004), 265–274.
    40. Jerry Tessendorf. 2004b. Simulating Ocean Water. In ACM SIGGRAPH 2004 course notes.
    41. Jerry Tessendorf. 2014. eWave: Using an Exponential Solver on the iWave Problem. Technical Note.
    42. Jerry Tessendorf. 2017. Gilligan: A Prototype Framework for Simulating and Rendering Maritime Environments. Technical Note.
    43. William Thomson. 1887. On ship waves. Proceedings of the institution of mechanical engineers 38, 1 (1887), 409–434.
    44. Nils Thürey, Ulrich Rüde, and Marc Stamminger. 2006. Animation of open water phenomena with coupled shallow water and free surface simulations. In Proceedings of the 2006 ACM SIGGRAPH/Eurographics Symposium on Computer Animation. 157–164.
    45. Huamin Wang, Gavin Miller, and Greg Turk. 2007. Solving General Shallow Wave Equations on Surfaces. In Proceedings of the 2007 ACM SIGGRAPH/Eurographics Symposium on Computer Animation. 229–238.
    46. Qizhi Yu, Fabrice Neyret, and Anthony Steed. 2011. Feature-based vector simulation of water waves. Computer Animation and Virtual Worlds 22, 2–3 (2011), 91–98.

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