Journal of Engineering Mathematics

, Volume 77, Issue 1, pp 1–18 | Cite as

Numerical use of exterior singularities for computation of gravity waves in shallow water

Article

Abstract

This paper describes fully nonlinear, fully dispersive computation for two-dimensional motion of periodic gravity waves of permanent form in an ideal fluid of arbitrary uniform depth. The rate of convergence of a series solution of this motion becomes slow with decrease of the water depth-to-wavelength ratio, and a considerably large number of terms in series are required to obtain accurate numerical solutions for long waves in shallow water, even if the wave amplitude is small. Domb–Sykes plots for coefficients of the computed series solutions show that the slow rate of convergence is due to a singularity outside the flow domain in a conformally mapped complex plane. This exterior singularity can be theoretically estimated using analytic continuation of an ordinary differential equation given by the free surface condition. It is shown that a conformal mapping regularizes this singularity and enlarges the radius of convergence of a series solution. In addition, iterative use of this mapping for regularization further improves convergence. This work proposes a new method to compute long waves in shallow water using regularization of the exterior singularity. Numerical examples demonstrate that the proposed method can produce accurate enough solutions for a wide range of water depth-to-wavelength ratios. Validity of cnoidal wave solutions is also discussed using comparison with the computed results.

Keywords

Analytic continuation Conformal mapping Numerical computation Singularity Water waves 

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References

  1. 1.
    Schwartz LW, Fenton JD (1982) Strongly nonlinear waves. Ann Rev Fluid Mech 14: 39–60MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Ursell F (1953) The long-wave paradox in the theory of gravity waves. Math Proc Camb Philos Soc 49: 685–694MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    Fenton JD (1999) Numerical methods for nonlinear waves. Adv Coast Ocean Eng 5: 241–324CrossRefGoogle Scholar
  4. 4.
    Yamada H (1958) Permanent gravity waves on water of uniform depth. Rep Res Inst Appl Mech 6(23): 127–139Google Scholar
  5. 5.
    Yamada H, Shiotani T (1968) On the highest water waves of permanent type. Bull Disas Prev Res Inst Kyoto Univ 18 part 2 no. 135: 1–22Google Scholar
  6. 6.
    Schwartz LW (1974) Computer extension and analytic continuation of Stokes’ expansion for gravity waves. J Fluid Mech 62: 553–578ADSMATHCrossRefGoogle Scholar
  7. 7.
    Cokelet ED (1977) Steep gravity waves in water of arbitrary uniform depth. Philos Trans R Soc Lond A 286(1335): 183–230MathSciNetADSMATHCrossRefGoogle Scholar
  8. 8.
    Vanden-Broeck J-M, Schwartz LW (1979) Numerical computation of steep gravity waves in shallow water. Phys Fluids 22: 1868–1871ADSMATHCrossRefGoogle Scholar
  9. 9.
    Vanden-Broeck J-M, Miloh T (1995) Computations of steep gravity waves by a refinement of Davies-Tulin’s approximation. SIAM J Appl Math 55: 892–903MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Williams JM (1981) Limiting gravity waves in water of finite depth. Philos Trans R Soc Lond A 302: 139–188ADSMATHCrossRefGoogle Scholar
  11. 11.
    Rienecker MM, Fenton JD (1981) A Fourier approximation method for steady water waves. J Fluid Mech 104: 119–137ADSMATHCrossRefGoogle Scholar
  12. 12.
    Wu TY, Kao J, Zhang JE (2005) A unified intrinsic functional expansion theory for solitary waves. Acta Mech Sinica 21: 1–15MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Murashige S, Wu TY (2010) Dwarf solitary waves and low tsunamis. J Hydrodyn 22(5, supplement 1):960–968Google Scholar
  14. 14.
    Wu TY, Murashige S (2011) On tsunami and the regularized solitary-wave theory. J Eng Math 70: 137–146MathSciNetCrossRefGoogle Scholar
  15. 15.
    Grant MA (1973) The singularity at the crest of a finite amplitude progressive Stokes wave. J Fluid Mech 59: 257–262ADSMATHCrossRefGoogle Scholar
  16. 16.
    Tanveer S (1991) Singularities in water waves and Rayleigh–Taylor instability. Proc R Soc Lond A 435: 137–158MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    Havelock TH (1919) Periodic irrotational waves of finite height. Proc R Soc Lond A 95: 38–51ADSGoogle Scholar
  18. 18.
    Vanden-Broeck J-M (1986) Steep gravity waves: Havelock’s method revisited. Phys Fluids 29: 3084–3085MathSciNetADSMATHCrossRefGoogle Scholar
  19. 19.
    Fenton JD, Gardiner-Garden RS (1982) Rapidly-convergent methods for evaluating elliptic integrals and theta and elliptic functions. J Austral Math Soc (Ser B) 24: 47–58MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Van Dyke M (1974) Analysis and improvement of perturbation series. Q J Mech Appl Math 27: 423–450MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Littman W (1957) On the existence of periodic waves near critical speed. Commun Pure Appl Math 10: 241–269MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Bulirsch R (1965) Numerical calculation of elliptic integrals and elliptic functions. Numerische Mathematik 7: 78–90MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Longuet-Higgins MS (1975) Integral properties of periodic gravity waves of finite amplitude. Proc R Soc Lond A 342: 157–174MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.School of Systems Information ScienceFuture University HakodateHakodateJapan

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