Mach reflection and KP solitons in shallow water

Article

Abstract.

Reflection of an obliquely incident solitary wave onto a vertical wall is studied analytically and experimentally. We use the Kadomtsev-Petviashivili (KP) equation to analyze the evolution and its asymptotic state. Laboratory experiments are performed using the laser induced fluorescent (LIF) technique, and detailed features and amplifications at the wall are measured. Due to the lack of physical interpretation of the theory, the numerical results were previously thought not in good agreement with the theory. With proper treatment, we demonstrate that the KP theory provides an excellent model to predict the present laboratory results as well as the previous numerical results. The KP theory also indicates that the present laboratory apparatus is too short to achieve the asymptotic state. The laboratory and numerical results suggest that the maximum of the predicted four-fold amplification would be difficult to be realized in the real-fluid environment. The reality of this amplification remains obscure.

Keywords

Soliton Solitary Wave European Physical Journal Special Topic Soliton Solution Laser Induce Fluorescent 

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References

  1. 1.
    V.B. Barakhnin, G.S. Khakimzyanov, J. Appl. Mech. Tech. Phys. 40Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 1008 (1999)CrossRefADSGoogle Scholar
  2. 2.
    R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves (Intersciences Pub. New York, 1948), p. 464Google Scholar
  3. 3.
    S. Chakravarty, Y. Kodama, J. Phys. A: Math. Theor. 41Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 275209 (2008)CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    S. Chakravarty, Y. Kodama, Stud. Appl. Math. 123Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 83 (2009)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J.D. Diorio, X. Liu, J.H. Duncan, J. Fluid Mech. 633Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 271 (2009)MATHCrossRefADSGoogle Scholar
  6. 6.
    J.H. Duncan, V. Philomin, M. Behres, J. Kimmel, Phys. Fluids 6Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 2558 (1994)CrossRefADSGoogle Scholar
  7. 7.
    J.H. Duncan, H. Qiao, V. Philomin, A. Wenz, J. Fluid Mech. 379Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 191 (1999)MATHCrossRefADSGoogle Scholar
  8. 8.
    M. Funakoshi, J. Phys. Soc. Jpn. 49Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 2371 (1980)CrossRefADSGoogle Scholar
  9. 9.
    S.M. Gardarsson, H. Yeh, J. Engrg. Mech., ASCE, 133Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 1093 (2007)Google Scholar
  10. 10.
    D.G. Goring, Tsunami–the propagation of long waves onto a shelf, Ph.D. thesis, Calif Inst. Tech. (1979)Google Scholar
  11. 11.
    K. Guizien, R. Barthlemy, J. Hydraulic Research 40Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 321 (2002)CrossRefGoogle Scholar
  12. 12.
    R. Hirota, The Direct Method in Soliton Theory (Cambridge University Press, Cambridge, 2004)Google Scholar
  13. 13.
    B.B. Kadomtsev, V.I. Petviashvili, Sov. Phys. - Dokl. 15Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 539 (1970)MATHADSGoogle Scholar
  14. 14.
    C.-Y. Kao, Y. Kodama, Math. Computers in Simulation (2010) (submitted)Google Scholar
  15. 15.
    S. Kato, T. Takagi, M. Kawahara, Int. J. Numer. Meth. Fluids 28Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 617 (1998)MATHCrossRefGoogle Scholar
  16. 16.
    Y. Kodama, J. Phys. A 37Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 11169 (2004)MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Y. Kodama, M. Oikawa, H. Tsuji, J. Phys. A: Math. Theor. 42Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 1 (2009)CrossRefMathSciNetGoogle Scholar
  18. 18.
    X. Liu, J.H. Duncan, J. Fluid Mech. 567Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 433 (2006)MATHCrossRefADSGoogle Scholar
  19. 19.
    M.S. Longuet-Higgins, J.D. Fenton, II. Proc. R. Soc. Lond. Ser. A 340Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 471 (1974)MATHCrossRefMathSciNetADSGoogle Scholar
  20. 20.
    W.K. Melville, J. Fluid Mech. 98Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 285 (1980)CrossRefADSGoogle Scholar
  21. 21.
    J.W. Miles, J. Fluid Mech. 79Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 157 (1977)MATHCrossRefMathSciNetADSGoogle Scholar
  22. 22.
    J.W. Miles, J. Fluid Mech. 79Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 171 (1977)MATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    J.V. Neumann, Oblique Reflection of Shocks (Dept. Comm. Off. of Tech. Serv., Washigton, D.C., 1943), PB-37079Google Scholar
  24. 24.
    E. Pelinovsky, T. Talipova, C. Kharif, Physica D 147Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 83 (2000)MATHCrossRefADSGoogle Scholar
  25. 25.
    P.H. Perroud, The solitary wave reflection along a straight vertical wall at oblique incidence, Institute of Engineering Research, Wave Research Laboratory, Tech. Rep. 99/3 (University of California, Berkeley, 1957), p. 93Google Scholar
  26. 26.
    J.D. Ramsden, Tsunamis: Forces on a Vertical Wall Caused by Long Waves, Bores,, Surges on a Dry Bed, Ph.D. thesis, Calif Inst. Tech. (1993)Google Scholar
  27. 27.
    M. Tanaka, J. Fluid Mech. 248Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 637 (1993)MATHCrossRefADSGoogle Scholar
  28. 28.
    H. Tsuji, M. Oikawa, J. Phys. Soc. Japan 76Discussion & Debate : Rogue Waves - Towards a Unifying Concept?, 84401 (2007)CrossRefGoogle Scholar
  29. 29.
    J. Von Neumann, Oblique reflection of shocks, Explosives Research Rep. No. 12, Navy Dept., Bureau of Ordnance, Washington, D.C. Also in John von Neumann Collected Works, edited by A.H. Taub, Vol. 6Discussion & Debate : Rogue Waves - Towards a Unifying Concept? (MacMillan, New York, 1963), p. 238Google Scholar
  30. 30.
    G.B. Whitham, Linear and Nonlinear Waves (A Wiley-Interscience Pub., New York, 1974), p. 636Google Scholar
  31. 31.
    R.L. Wiegel, Oceanographical Engineering (Prentice-Hall, Englewood cliffs, N.J. 1964), p. 532Google Scholar
  32. 32.
    H. Yeh, A. Ghazali, A Bore on a Uniformly Sloping Beach (Proceedings 20th International Conference on Coastal Engineering) (1986), p. 877Google Scholar

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© EDP Sciences and Springer 2010

Authors and Affiliations

  1. 1.School of Civil & Construction Engineering, Oregon State UniversityCorvallisUSA
  2. 2.Department of MathematicsOhio State UniversityColumbusUSA

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