[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

On the Particle Paths and the Stagnation Points in Small-Amplitude Deep-Water Waves

  • Published:
Journal of Mathematical Fluid Mechanics Aims and scope Submit manuscript

Abstract

In order to obtain quite precise information about the shape of the particle paths below small-amplitude gravity waves travelling on irrotational deep water, analytic solutions of the nonlinear differential equation system describing the particle motion are provided. All these solutions are not closed curves. Some particle trajectories are peakon-like, others can be expressed with the aid of the Jacobi elliptic functions or with the aid of the hyperelliptic functions. Remarks on the stagnation points of the small-amplitude irrotational deep-water waves are also made.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amick C.J., Fraenkel L.E., Toland J.F.: On the Stokes conjecture for the wave of extreme form. Acta Math. 148, 193–214 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  2. Banner M.L., Peregrine D.H.: Wave breaking in deep water. Annu. Rev. Fluid Mech. 25, 373–397 (1993)

    Article  ADS  Google Scholar 

  3. Byrd P.F., Friedman M.D.: Handbook of Elliptic Integrals for engineers and scientists. Springer-Verlag, Berlin, Heidelberg, New York (1971)

    Book  MATH  Google Scholar 

  4. Constantin A.: On the deep water wave motion. J. Phys. A 34, 1405–1417 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Constantin A.: Comment on “Steep Sharp-Crested Gravity Waves on Deep Water”. Phys. Rev. Lett. 93, 069402 (2004)

    Article  ADS  Google Scholar 

  6. Constantin A.: The trajectories of particles in Stokes waves. Invent. Math. 166, 523–535 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Constantin, A.: Nonlinear water waves with applications to wave-current interactions and tsunamis. CBMS-NSF Conference Series in Applied Mathematics, vol. 81. SIAM, Philadelphia (2011)

  8. Constantin A., Ehrnström M., Villari G.: Particle trajectories in linear deep-water waves. Nonlinear Anal. Real World Appl. 9, 1336–1344 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Constantin A., Escher J.: Particle trajectories in solitary water waves. Bull. Am. Math. Soc. 44, 423–431 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Constantin A., Escher J.: Analyticity of periodic traveling free surface water waves with vorticity. Ann. Math. 173, 559–568 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Constantin A., Varvaruca E.: Steady periodic water waves with constant vorticity: regularity and local bifurcation. Arch. Ration. Mech. Anal. 199, 33–67 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Constantin A., Villari G.: Particle trajectories in linear water waves. J. Math. Fluid Mech. 10, 1–18 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Constantin A., Strauss W.: Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57, 481–527 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Constantin A., Strauss W.: Pressure beneath a Stokes wave. Comm. Pure Appl. Math. 63, 533–557 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Constantin A., Strauss W.: Periodic traveling gravity water waves with discontinuous vorticity. Arch. Ration. Mech. Anal. 202, 133–175 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Debnath L.: Nonlinear Water Waves. Academic Press Inc., Boston (1994)

    MATH  Google Scholar 

  17. Dubreil-Jacotin M.-L.: Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie. J. Math. Pures Appl. 13, 217–291 (1934)

    MATH  Google Scholar 

  18. Ehrnström M., Villari G.: Linear water waves with vorticity: Rotational features and particle paths. J. Diff. Equ. 244, 1888–1909 (2008)

    Article  MATH  Google Scholar 

  19. Gerstner F.: Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Ann. Phys. 2, 412–445 (1809)

    Article  Google Scholar 

  20. Goyon R.: Contribution à à la théorie des houles. Ann. Fac. Sci. Univ. Toulouse 22, 1–55 (1958)

    Article  MathSciNet  Google Scholar 

  21. Henry, D.: The trajectories of particles in deep-water Stokes waves. Int. Math. Res. Not., Art. ID 23405, 13 pp. (2006)

  22. Henry D.: Particle trajectories in linear periodic capillary and capillary-gravity water waves. Phil. Trans. R. Soc. A 365, 2241–2251 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Henry D.: On Gerstner’s water wave. J. Nonlinear Math. Phys. 15, 87–95 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  24. Ionescu-Kruse D.: Particle trajectories in linearized irrotational shallow water flows. J. Nonlinear Math. Phys. 15, 13–27 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  25. Ionescu-Kruse D.: Particle trajectories beneath small amplitude shallow water waves in constant vorticity flows. Nonlinear Anal-Theor 71, 3779–3793 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ionescu-Kruse D.: Exact solutions for small-amplitude capillary-gravity water waves. Wave Motion 46, 379–388 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ionescu-Kruse D.: Small-amplitude capillary-gravity water waves: exact solutions and particle motion beneath such waves. Nonlinear Anal. Real World Appl. 11, 2989–3000 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ionescu-Kruse D.: Peakons arising as particle paths beneath small-amplitude water waves in cosntant vorticity flows. J. Nonlinear Math. Phys. 17, 415–422 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Ionescu-Kruse D.: Elliptic and hyperelliptic functions describing the particle motion beneath small-amplitude water waves with constant vorticity. Comm. Pure Appl. Anal. 11, 1475–1496 (2012)

    Article  MathSciNet  Google Scholar 

  30. Johnson R.S.: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge Univeristy Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  31. Kalisch H.: Periodic traveling water waves with isobaric streamlines. J. Nonlinear Math. Phys. 11, 461–471 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Lamb H.: Hydrodynamics, 6th edn. Cambridge University Press, Cambridge (1953)

    Google Scholar 

  33. Lewy H.: A note on harmonic functions and a hydrodynamical application. Proc. Am. Math. Soc. 3, 111–113 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lighthill J.: Waves in Fluids. Cambridge University Press, Cambridge (2001)

    MATH  Google Scholar 

  35. Lukomsky V., Gandzha I., Lukomsky D.: Steep sharp-crested gravity waves on deep water. Phys. Rev. Lett. 89, 164502 (2002)

    Article  ADS  Google Scholar 

  36. Matioc, A.-V.: On particle trajectories in linear deep-water waves. arXiv:1111.2490, pp. 1–12

  37. Matioc, B.V.: On the regularity of deep-water waves with general vorticity distributions. Quart. Appl. Math. (2012, to appear)

  38. Plotnikov, P.I.: Proof of the Stokes conjecture in the theory of surface waves. Dinamika Sploshn. Sredy, 57 (1982), 41–76 (in Russian); English transl.: Stud. Appl. Math, 108 (2002), 217–244

  39. Rankine W.J.M.: On the exact form of waves near the surface of deep water. Phil. Trans. R. Soc. A 153, 127–138 (1863)

    Article  Google Scholar 

  40. Smirnov, V.: Cours de Mathématiques supérieures, Tome III, deuxième partie. Mir, Moscou (1972)

  41. Stokes, G.G.: Considerations relative to the greatest height of oscillatory irrotational waves which can be propagated without change of form. Mathematical Physiscs Papers, vol. I, pp. 225–228. Cambridge University Press, Cambridge (1880)

  42. Toland J.F.: Stokes waves. Topol. Methods Nonlinear Anal. 7, 1–48 (1996)

    MathSciNet  MATH  Google Scholar 

  43. Varvaruca E.: On the existence of extreme waves and the Stokes conjecture with vorticity. J. Diff. Equ. 246, 4043–4076 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wahlen E.: Steady water waves with a critical layer. J. Diff. Eq. 246, 2468–2483 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zeidler E.: Existenzbeweis für permanente Kapillar-Schwerewellen mit allgemeinen Wirbelverteilungen. Arch. Rational Mech. Anal. 50, 34–72 (1973)

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Simion Stoilow Institute of Mathematics, Research Unit No. 6, Romanian Academy, P.O. Box 1-764, 014700, Bucharest, Romania

    Delia Ionescu-Kruse

Authors
  1. Delia Ionescu-Kruse
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Delia Ionescu-Kruse.

Additional information

Communicated by A. Constantin

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ionescu-Kruse, D. On the Particle Paths and the Stagnation Points in Small-Amplitude Deep-Water Waves. J. Math. Fluid Mech. 15, 41–54 (2013). https://doi.org/10.1007/s00021-012-0102-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00021-012-0102-5

Keywords

Search

Navigation