Skip to main content
SpringerLink
Log in

The effect of a fixed barrier on an incident progressive wave in shallow water

  • Published:
Il Nuovo Cimento B (1971-1996)

Summary

The effect of a fixed barrier on the propagation of an incident wave inside an ideal fluid is investigated within the frame of the shallow-water theory in two dimensions. The fluid is supposed to occupy an infinite channel of constant depth. Further, the horizontal extent of the barrier is assumed to be small. An asymptotic double series expansion for the solution is used. This procedure enables us to obtain analytic expressions for the local perturbations. The results of the first-order approximation indicate that no reflections exist. The second-order approximation of the solution is found to be the superposition of a progressive wave and local perturbations. For approximations of order higher than two, a secular term which increases monotonically with time appears in the expressions for the progressive wave. This unacceptable result is due to certain aspects in the mathematical procedure used. For this reason, the procedure is modified by using a suitable transformation of variables which reduces the determination of the transmitted wave to the solution of the K.d.V. equation. As an illustration, the special case of the incident uniform flow is considered and the stream lines of the resulting flow are drawn.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. V. Wehausen andE. V. Laitone:Surface Waves, Handbuch der Physik, Vol.9 (Springer-Verlag, Berlin, 1960).

    Google Scholar 

  2. W. R. Dean:Proc. Cambridge Philos. Soc.,41, 231 (1945).

    Article  MathSciNet  ADS  Google Scholar 

  3. F. Ursell:Proc. Cambridge Philos. Soc.,43(3), 374 (1947).

    Article  MathSciNet  ADS  Google Scholar 

  4. T. F. Ogilvie:Propagation of waves over an obstacle in water of finite depth, Ph. D. Thesis, University of California, 1960.

  5. C. Takano:Houille Blanche,3, 247 (1960).

    Article  MathSciNet  Google Scholar 

  6. R. L. Wiegel:Transmission of waves past a rigid thin barrier, J. Water Harbors, Div. Proc. ASCE, 86, WW.1, 1960, p. 2413.

    Google Scholar 

  7. D. V. Evans:J. Fluid Mech.,40(3), 433 (1970).

    Article  ADS  Google Scholar 

  8. D. J. Struik:Math. Ann.,95(4), 595 (1926).

    Article  MathSciNet  Google Scholar 

  9. R. Gouyon:Contribution à la théorie des houles, Thèse de doctorat d’état, Faculté des Sciences, Université de Grenoble, 1958.

  10. R. W. Yeung:Anu. Rev. Fluid Mech.,14, 395 (1982).

    Article  MathSciNet  ADS  Google Scholar 

  11. J. P. Germain andL. Guli:Ann. Hydrog., Ser. 5,5, 7 (1977).

    Google Scholar 

  12. S. Badawy:Etude théorique de la réflexion des ondes de gravité de grandes longueurs d’onde relatives sur les plages, Thése de Ph. D., Faculté des Sciences, Université du Caire, 1981.

  13. A. Kabbaj:Contribution à l’étude du passage des ondes de gravité sur le talus continental et à la génération des ondes internes, Thèse de doctorat d’état, I.M.G. Université de Grenoble, 1985.

  14. S. L. Cole:Wave Motion,7, 579 (1985).

    Article  MathSciNet  Google Scholar 

  15. A. F. Ghaleb andI. A. Z. Hefni:J. Mec. Théor. Appl.,6(4), 463 (1987).

    Google Scholar 

  16. J. P. Germain:C.R. Acad. Sci. (Paris), Ser. A,273, 1171 (1971).

    MathSciNet  Google Scholar 

  17. J. P. Germain:C.R. Acad. Sci. (Paris), Ser. A,274, 997 (1972).

    MathSciNet  Google Scholar 

  18. N. I. Muskhelishvilli:Singular Integral Equations (Noordhoff Pub., 1946).

  19. I. N. Sneddon:Mixed Boundary Value Problems in Potential Theory (North-Holland Pub. Co., Amsterdam, 1966).

    Google Scholar 

  20. T. Omar:Solution of certain trigonometric series inConference on Applied Mathematics in the Honour of Professor A. A. Ashour, 3–6 January 1987, Cairo (1987).

  21. S. J. Russel:Report on waves, Report toThe 14th Meeting (1844) of the British Ass. for Adv. of Science, London (1845), p. 311.

  22. A. temperville:Contribution à la théorie des ondes de gravité en eau peu profonde, Thèse de doctorat d’état, I.M.G. Université de Grenoble, 1985.

  23. R. K. Dodd, J. C. Eilbeck, J. D. Gibbon andH. C. Morris:Solitons and Nonlinear Wave Equations (Academic Press, 1982).

  24. C. S. Gardner, J. M. Greene, M. D. Kruskal andR. M. Miura:Phys. Rev. Lett.,19, 1095 (1967).

    Article  ADS  Google Scholar 

  25. M. Wadaty andM. Toda:J. Phys. Soc. Jpn.,32, 1403 (1972).

    Article  ADS  Google Scholar 

  26. A. Kanji andI. Osamu:J. Comp. Phys.,34, 202 (1980).

    Article  Google Scholar 

  27. N. J. Zabusky andM. D. Kruskal:Phys. Rev. Lett.,15, 240 (1965).

    Article  ADS  Google Scholar 

  28. N. J. Zabusky andC. J. Galvin:J. Fluid Mech.,47, 811 (1971).

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

    M. S. Abou-Dina & M. A. Helal

Authors
  1. M. S. Abou-Dina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. Helal
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Author’s names are alphabetically ordered.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abou-Dina, M.S., Helal, M.A. The effect of a fixed barrier on an incident progressive wave in shallow water. Nuov Cim B 107, 331–344 (1992). https://doi.org/10.1007/BF02728494

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02728494

Keywords

Search

Navigation