Analysis and Mathematical Physics

, Volume 2, Issue 4, pp 347–366 | Cite as

A dispersive model for undular bores



In this article, consideration is given to weak bores in free-surface flows. The energy loss in the shallow-water theory for an undular bore is thought to be due to upstream oscillations that carry away the energy lost at the front of the bore. Using a higher-order dispersive model equation, this expectation is confirmed through a quantitative study which shows that there is no energy loss if dispersion is accounted for.


Undular bore Energy loss Dispersion 


  1. 1.
    Alazman, A.A., Albert, J.P., Bona, J.L., Chen, M., Wu, J.: Comparison between the BBM equation and a Boussinesq system. Adv. Differ. Equ. 11, 121–166 (2006)MathSciNetMATHGoogle Scholar
  2. 2.
    Albert, J.P.: Concentration compactness and the stability of solitary-wave solutions to nonlocal equations. In: Applied Analysis (Baton Rouge, LA, 1996), pp. 1–29. Contemp. Math. vol. 221. American Mathematical Society, Providence (1996)Google Scholar
  3. 3.
    Ali, A., Kalisch, H.: Energy balance for undular bores. C. R. Mecanique 338, 67–70 (2010)MATHCrossRefGoogle Scholar
  4. 4.
    Ali, A., Kalisch, H.: Mechanical balance laws for Boussinesq models of surface water waves. J. Nonlinear Sci. 22, 371–398 (2012)Google Scholar
  5. 5.
    Benjamin, T.B., Lighthill, M.J.: On cnoidal waves and bores. Proc. R. Soc. Lond. A 224, 448–460 (1954)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Binnie, A.M., Orkney, J.C.: Experiments on the flow of water from a reservoir through an open channel II. The formation of hydraulic jumps. Proc. R. Soc. Lond. A 230:237–246 (1955).Google Scholar
  7. 7.
    Bjørkavåg, M., Kalisch, H.: Wave breaking in Boussinesq models for undular bores. Phys. Lett. A 375, 157–1578 (2011)CrossRefGoogle Scholar
  8. 8.
    Bona, J.L., Chen, M., Saut, J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory. J. Nonlinear Sci. 12, 283–318 (2002)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Bona, J.L., Colin, T., Lannes, D.: Long wave approximations for water waves. Arch. Ration. Mech. Anal. 178, 373–410 (2005)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Bona, J.L., Dougalis, V.A., Mitsotakis, D.E.: Numerical solution of KdV-KdV systems of Boussinesq equations. I. The numerical scheme and generalized solitary waves. Math. Comput. Simul. 74, 214–228 (2007)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bona, J.L., Grujić, Z., Kalisch, H.: A KdV-type Boussinesq system: From the energy level to analytic spaces. Discrete Contin. Dyn. Syst. 26, 1121–1139 (2010)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Bona, J.L., Pritchard, W.G., Scott, L.R.: An evaluation of a model equation for water waves. Phil. Trans. Roy. Soc. Lond. A 302, 457–510 (1981)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bona, J.L., Varlamov, V.V.: Wave generation by a moving boundary. In: Nonlinear partial differential equations and related analysis, pp. 41–71. Contemp. Math., vol. 371, American Mathematical Society, Providence (2005)Google Scholar
  14. 14.
    Borluk, H., Kalisch, H.: Particle dynamics in the KdV approximation. Wave Motion 49, 691–709 (2012)CrossRefGoogle Scholar
  15. 15.
    Byatt-Smith, J.G.B.: The effect of laminar viscosity on the solution of the undular bore. J. Fluid Mech. 48, 33–40 (1971)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Chanson, H.: The Hydraulics of Open Channel Flow. Arnold, London (1999)Google Scholar
  17. 17.
    Chanson, H.: Current knowledge in hydraulic jumps and related phenomena. A survey of experimental results. Eur. J. Mech. B Fluids 28, 191–210 (2009)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Chanson, H.: Undular tidal bores: basic theory and free-surface characteristics. J. Hydraul. Eng. ASCE 136, 940–944 (2010)CrossRefGoogle Scholar
  19. 19.
    Chen, M.: Exact solution of various Boussinesq systems. Appl. Math. Lett. 11, 45–49 (1998)MATHCrossRefGoogle Scholar
  20. 20.
    Constantin, A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Constantin, A., Strauss, W.: Pressure beneath a Stokes wave. Comm. Pure Appl. Math. 63, 533–557 (2010)MathSciNetMATHGoogle Scholar
  22. 22.
    Duruk, N., Erkip, A., Erbay, H.A.: A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity. IMA J. Appl. Math. 74, 97–106 (2009)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Dutykh, D., Dias, F.: Energy of tsunami waves generated by bottom motion. Proc. R. Soc. Lond. A 465, 725–744 (2009)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Ehrnström, M.: On the streamlines and particle paths of gravitational water waves. Nonlinearity 21, 1141–1154 (2008)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Ehrnström, M., Escher, J., Wahlen, E.: Steady water waves with multiple critical layers. SIAM J. Math. Anal. 43, 1436–1456 (2011)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    El, G.A., Grimshaw, R.H.J., Smyth, N.F.: Unsteady undular bores in fully nonlinear shallow-water theory. Phys. Fluids 18, 027104 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Favre, H.: Ondes de Translation. Dunod, Paris (1935)Google Scholar
  28. 28.
    Fokas, A.S., Pelloni, B.: Boundary value problems for Boussinesq type systems. Math. Phys. Anal. Geom. 8, 59–96 (2005)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Henry, D.: Analyticity of the streamlines for periodic travelling free surface capillary-gravity water waves with vorticity. SIAM J. Math. Anal. 42: 103–3111 (2010)Google Scholar
  30. 30.
    Kalisch, H.: A uniqueness result for periodic traveling waves in water of finite depth. Nonlinear Anal. 58, 779–785 (2004)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Kalisch, H., Bjørkavåg, M.: Energy budget in a dispersive model for undular bores. Proc. Est. Acad. Sci. 59, 172–181 (2010)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Keulegan, G.H., Patterson, G.W.: Mathematical theory of irrotational translation waves. Nat. Bur. Standards J. Res. 24, 47–101 (1940)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Koch, C., Chanson, H.: Unsteady Turbulence Characteristics in an Undular Bore in River Flow 2006, pp. 79–88. Taylor& Francis Group, London (2006)Google Scholar
  34. 34.
    Koch, C., Chanson, H.: Turbulence measurements in positive surges and bores. J. Hydraul. Res. 47, 29–40 (2009)CrossRefGoogle Scholar
  35. 35.
    Lemoine, R.: Sur les ondes positives de translation dans les canaux et sur le ressaut ondule de faible amplitude. Jl. La Houille Blanche. 183–185 (1948)Google Scholar
  36. 36.
    Madsen, P.A., Schäffer, H.A.: Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis. Phil. Trans. Roy. Soc. Lond. A 356, 3123–3184 (1998)MATHCrossRefGoogle Scholar
  37. 37.
    Peregrine, P.G.: Calculations of the development of an undular bore. J. Fluid Mech. 25, 321–330 (1966)CrossRefGoogle Scholar
  38. 38.
    Rayleigh, Lord: On the Theory of Long Waves and Bores. Proc. R. Soc. Lond. A A90, 324–328 (1914)Google Scholar
  39. 39.
    Sturtevant, B.: Implications of experiments on the weak undular bore. Phys. Fluids 6, 1052–1055 (1965)CrossRefGoogle Scholar
  40. 40.
    Svendsen, I.A.: Introduction to Nearshore Hydrodynamics. World Scientific, Singapore (2006)MATHGoogle Scholar
  41. 41.
    Varlamov, V.: On the initial-boundary value problem for the damped Boussinesq equation. Discrete Contin. Dynam. Syst. 4, 431–444 (1998)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Varlamov, V.: On the spatially two-dimensional Boussinesq equation in a circular domain. Nonlinear Anal. 699–725 (2001)Google Scholar
  43. 43.
    Wei, G., Kirby, J.T., Grilli, S.T., Subramanya, R.: A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 71–92 (1995)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Whitham, G.B.: Linear and Nonlinear Waves. McGraw-Hill, New York (1975)Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BergenBergenNorway

Personalised recommendations