Modeling Water Waves Beyond Perturbations

  • Didier ClamondEmail author
  • Denys Dutykh
Part of the Lecture Notes in Physics book series (LNP, volume 908)


In this chapter, we illustrate the advantage of variational principles for modeling water waves from an elementary practical viewpoint. The method is based on a ‘relaxed’ variational principle, i.e., on a Lagrangian involving as many variables as possible, and imposing some suitable subordinate constraints. This approach allows the construction of approximations without necessarily relying on a small parameter. This is illustrated via simple examples, namely the Serre equations in shallow water, a generalization of the Klein–Gordon equation in deep water and how to unify these equations in arbitrary depth. The chapter ends with a discussion and caution on how this approach should be used in practice.


Velocity Field Variational Principle Water Wave Lagrangian Density Gordon Equation 
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  1. 1.
    Boussinesq, J.V.: Théorie de l’intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. C. R. Acad. Sci. Paris Sér. A-B 72, 755–759 (1871)zbMATHGoogle Scholar
  2. 2.
    Broer, L.J.F.: On the Hamiltonian theory of surface waves. Appl. Sci. Res. 29(6), 430–446 (1974)MathSciNetCrossRefADSzbMATHGoogle Scholar
  3. 3.
    Clamond, D.: Variational principles for water waves beyond perturbations. (2014)
  4. 4.
    Clamond, D., Dutykh, D.: Practical use of variational principles for modeling water waves. Phys. D 241(1), 25–36 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Craik, A.D.D.: The origins of water wave theory. Ann. Rev. Fluid Mech. 36, 1–28 (2004)MathSciNetCrossRefADSGoogle Scholar
  6. 6.
    Dutykh, D., Clamond, D.: Shallow water equations for large bathymetry variations. J. Phys. A Math. Theor. 44(33), 332001 (2011)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dutykh, D., Clamond, D., Chhay, M.: Numerical study of the generalised Klein-Gordon equations. Phys. D, 304–305, 23–33 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Dutykh, D., Clamond, D., Milewski, P., Mitsotakis, D.: Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations. Eur. J. Appl. Math. 24(05), 761–787 (2013). Google Scholar
  9. 9.
    Dysthe, K.B.: Note on a modification to the nonlinear Schrödinger equation for application to deep water. Proc. R. Soc. Lond. A 369, 105–114 (1979)CrossRefADSzbMATHGoogle Scholar
  10. 10.
    Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics. Addison-Wesley, San Francisco (1964)Google Scholar
  11. 11.
    Goldstein, H., Poole, C.P., Safko, J.L.: Classical Mechanics, 3rd edn. Addison–Wesley, San Francisco (2001)Google Scholar
  12. 12.
    Green, A.E., Naghdi, P.M.: A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237–246 (1976)CrossRefADSzbMATHGoogle Scholar
  13. 13.
    Green, A.E., Laws, N., Naghdi, P.M.: On the theory of water waves. Proc. R. Soc. Lond. A 338, 43–55 (1974)MathSciNetCrossRefADSzbMATHGoogle Scholar
  14. 14.
    Grue, J., Clamond, D., Huseby, M., Jensen, A.: Kinematics of extreme waves in deep water. Appl. Ocean Res.25, 355–366 (2003)CrossRefGoogle Scholar
  15. 15.
    Jensen, A., Clamond, D., Huseby, M., Grue, J.: On local and convective accelerations in steep wave events. Ocean Eng. 34, 426–435 (2007)CrossRefGoogle Scholar
  16. 16.
    Johnson, R.S.: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge (2004)Google Scholar
  17. 17.
    Korteweg, D.J., de Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag. 39(5), 422–443 (1895)CrossRefzbMATHGoogle Scholar
  18. 18.
    Lanczos, C.: The Variational Principles of Mechanics. Dover Publications, New York (1970)zbMATHGoogle Scholar
  19. 19.
    Laughlin, R.B.: Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations. Phys. Rev. Lett. 50(18), 1395–1398 (1983)CrossRefADSGoogle Scholar
  20. 20.
    Li, Y.A.: Hamiltonian structure and linear stability of solitary waves of the Green-Naghdi equations. J. Nonlinear Math. Phys. 9(1), 99–105 (2002)CrossRefADSGoogle Scholar
  21. 21.
    Luke, J.C.: A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 375–397 (1967)MathSciNetCrossRefADSGoogle Scholar
  22. 22.
    Mei, C.C.: The Applied Dynamics of Water Waves. World Scientific, Singapore (1989)Google Scholar
  23. 23.
    Murayama, H.: Berkley’s 221A Lecture Notes: Variational Method. (2006)
  24. 24.
    Petrov, A.A.: Variational statement of the problem of liquid motion in a container of finite dimensions. Prikl. Math. Mekh. 28(4), 917–922 (1964)zbMATHGoogle Scholar
  25. 25.
    Radder, A.C.: Hamiltonian dynamics of water waves. Adv. Coast. Ocean Eng. 4, 21–59 (1999)CrossRefGoogle Scholar
  26. 26.
    Rajchenbach, J., Leroux, A., Clamond, D.: New standing solitary waves in water. Phys. Rev. Lett. 107(2), 024502 (2011)CrossRefADSGoogle Scholar
  27. 27.
    Rajchenbach, J., Clamond, D., Leroux, A.: Observation of Star-Shaped Surface Gravity Waves. Phys. Rev. Lett. 110(9), 094502 (2013)CrossRefADSGoogle Scholar
  28. 28.
    Salmon, R.: Hamiltonian fluid mechanics. Ann. Rev. Fluid Mech. 20, 225–256 (1988)CrossRefADSGoogle Scholar
  29. 29.
    Serre, F.: Contribution à l’étude des écoulements permanents et variables dans les canaux. La Houille blanche 8, 374–388 (1953)CrossRefGoogle Scholar
  30. 30.
    Serre, F.: Contribution à l’étude des écoulements permanents et variables dans les canaux. La Houille blanche 8, 830–872 (1953)CrossRefGoogle Scholar
  31. 31.
    Stoker, J.J.: Water Waves: The Mathematical Theory with Applications. Interscience, New York (1957)zbMATHGoogle Scholar
  32. 32.
    Stoker, J.J.: Water waves, the Mathematical Theory with Applications. Wiley, New York (1958)Google Scholar
  33. 33.
    Su, C.H., Gardner, C.S.: KdV equation and generalizations. Part III. Derivation of Korteweg-de Vries equation and Burgers equation. J. Math. Phys. 10, 536–539 (1969)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Su, C.H., Mirie, R.M.: On head-on collisions between two solitary waves. J. Fluid Mech. 98, 509–525 (1980)MathSciNetCrossRefADSzbMATHGoogle Scholar
  35. 35.
    Wehausen, J.V., Laitone, E.V.: Surface waves. Handbuch der Physik 9, 446–778 (1960)MathSciNetCrossRefADSGoogle Scholar
  36. 36.
    Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1999)CrossRefzbMATHGoogle Scholar
  37. 37.
    Wu, T.Y.: A unified theory for modeling water waves. Adv. Appl. Mech. 37, 1–88 (2001)CrossRefGoogle Scholar
  38. 38.
    Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194 (1968)CrossRefADSGoogle Scholar
  39. 39.
    Zakharov, V.E., Kuznetsov, E.A.: Hamiltonian formalism for nonlinear waves. Usp. Fiz. Nauk 167, 1137–1168 (1997)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Laboratoire J. A. DieudonnéUniversité de Nice – Sophia AntipolisNice Cedex 2France
  2. 2.Université Savoie Mont Blanc, LAMA, UMR 5127 CNRSLe Bourget-du-Lac CedexFrance

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