Journal of Applied Mathematics

Waves Trapped by Submerged Obstacles at High Frequencies

A. M. Marín, R. D. Ortíz, and P. Zhevandrov

Full-text: Open access

Abstract

As is well known, submerged horizontal cylinders can serve as wavegorderes for surface water waves. For large values of the wavenumber k in the direction of the cylinders, there is only one trapped wave. We construct asymptotics of these trapped modes and their frequencies as k in the case of one or two submerged cylinders by means of reducing the initial problem to a system of integral equations on the boundaries and then solving them using a technique suggested by Zhevandrov and Merzon (2003).

Article information

Source
J. Appl. Math., Volume 2007 (2007), Article ID 80205, 17 pages.

Dates
First available in Project Euclid: 27 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.jam/1204126687

Digital Object Identifier
doi:10.1155/2007/80205

Mathematical Reviews number (MathSciNet)
MR2342134

Zentralblatt MATH identifier
1180.76009

Citation

Marín, A. M.; Ortíz, R. D.; Zhevandrov, P. Waves Trapped by Submerged Obstacles at High Frequencies. J. Appl. Math. 2007 (2007), Article ID 80205, 17 pages. doi:10.1155/2007/80205. https://projecteuclid.org/euclid.jam/1204126687


Export citation

References

  • F. Ursell, ``Trapping modes in the theory of surface waves,'' Mathematical Proceedings of the Cambridge Philosophical Society, vol. 47, pp. 347--358, 1951.
  • A.-S. Bonnet-Ben Dhia and P. Joly, ``Mathematical analysis of guided water waves,'' SIAM Journal on Applied Mathematics, vol. 53, no. 6, pp. 1507--1550, 1993.
  • N. Kuznetsov, V. Maz'ya, and B. Vainberg, Linear Water Waves, Cambridge University Press, Cambridge, UK, 2002.
  • A. M. Marín, R. D. Ortíz, and P. Zhevandrov, ``High-frequency asymptotics of waves trapped by underwater ridges and submerged cylinders,'' Journal of Computational and Applied Mathematics, vol. 204, no. 2, pp. 356--362, 2007.
  • P. Zhevandrov and A. Merzon, ``Asymptotics of eigenfunctions in shallow potential wells and related problems,'' in Asymptotic Methods for Wave and Quantum Problems, vol. 208 of Amer. Math. Soc. Transl. Ser. 2, pp. 235--284, American Mathematical Society, Providence, RI, USA, 2003.
  • L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory. Course of Theoretical Physics, Vol. 3, Addison-Wesley Series in Advanced Physics, Pergamon, London, UK, 1958.
  • B. Simon, ``The bound state of weakly coupled Schrödinger operators in one and two dimensions,'' Annals of Physics, vol. 97, no. 2, pp. 279--288, 1976.
  • R. R. Gadyl'shin, ``On local perturbations of the Schrödinger operator on the axis,'' Theoretical and Mathematical Physics, vol. 132, no. 1, pp. 976--982, 2002.
  • D. S. Kuznetsov, ``A spectrum perturbation problem and its application to waves above an underwater ridge,'' Siberian Mathematical Journal, vol. 42, no. 4, pp. 668--684, 2001.
  • M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, USA, 1970.