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Trapped modes in a T-shaped waveguide

  • Classical Problems of Linear Acoustics and Wave Theory
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Abstract

We study the discrete spectrum of the Laplace operator with the Dirichlet boundary conditions in a plane waveguide made of a band of unit width and half-band of width h perpendicular to it. The existence of critical width h * ∈ (1,2] is established: in the case of h ∈ (0, h*), the discrete spectrum σ h d of the problem contains only one eigenvalur λ h1 , but σ h d = ⊘ at h ≥ h *. The asymptotics of the eigenvalue λ h1 at h → + 0 is found.

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References

  1. M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-sopryazhennykh Operators in Hilbert Space (Leningr. Univ., Leningrad, 1980) [in Russian].

    Google Scholar 

  2. M. Sh. Birman and G. E. Skvortsov, Izv. Vyssh. Uchebn. Zaved., Matem. 5, 11 (1962).

    Google Scholar 

  3. S. A. Nazarov and B. A. Plamenevskiĭ, Elliptical Problems in Areas with Piecewise Smooth Boundary (Nauka, Moscow, 1991; Walter de Gruyter, Berlin, 1994).

    Google Scholar 

  4. F. Ursell, Proc. Camb. Phil. Soc. 47, 347 (1951).

    Article  MATH  MathSciNet  Google Scholar 

  5. D. V. Evans, IMA J. Apl. Math. 49, 45 (1992).

    Article  Google Scholar 

  6. C. M. Linton and P. McIver, Wave Motion 45, 16 (2007).

    Article  MathSciNet  Google Scholar 

  7. E. L. Shenderov, Akust. Zh. 45, 661 (1999) [Acoust. Phys. 45, 589 (1999)].

    Google Scholar 

  8. E. L. Shenderov, Akust. Zh. 46, 816 (2000) [Acoust. Phys. 46, 716 (2000)].

    Google Scholar 

  9. A. D. Lapin, Akust. Zh. 52, 716 (2006) [Acoust. Phys. 52, 425 (2006)].

    Google Scholar 

  10. R. L. Shult, D. G. Ravenhall, and H. D. Wyld, Phys. Rev. B 39, 5476 (1989).

    Article  ADS  Google Scholar 

  11. Y. Avishai, D. Bessis, B. G. Giraud, and G. Mantica, Phys. Rev. B 44, 8028 (1991).

    Article  ADS  Google Scholar 

  12. P. Duclos and P. Exner, Rev. Math. Phys. 7, 73 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  13. D. Borisov, P. Exner, R. Gagyl’shin, and D. Krejcirik, Ann. H. Poincare 2, 553 (2001).

    Article  MATH  Google Scholar 

  14. D. Borisov, P. Exner, and R. Gagyl’shin, J. Math. Phys. 43, 6265 (2002).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. D. V. Evans, M. Levitin, and D. Vasil’ev, J. Fluid Mech. 261, 21 (1994).

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. A.-S. Bonnet-Bendhia and F. Starling, Math. Meth. Appl. Sci. 77, 305 (1994).

    Article  MathSciNet  Google Scholar 

  17. I. V. Kamotskiĭ and S. A. Nazarov, Mat. Sb. 190(1), 109 (1999).

    MathSciNet  Google Scholar 

  18. I. V. Kamotskiĭ and S. A. Nazarov, Mat. Sb. 190(2), 43 (1999).

    Google Scholar 

  19. I. V. Kamotskiĭ and S. A. Nazarov, Zap. Nauchn. Semin. Peterb. Otd. Mat. Inst. RAN 264, 66 (2000).

    Google Scholar 

  20. S. A. Nazarov, Funktsion. Analiz Prilozh. 40(2), 20 (2006).

    Google Scholar 

  21. I. V. Kamotskiĭ and S. A. Nazarov, Mat. Zametki 73, 138 (2003).

    Google Scholar 

  22. S. A. Nazarov, Zh. Vychisl. Mat. Mat. Fiz. 46, 2265 (2006) [Comput. Math. Math. Phys. 46, 2164 (2006)].

    Google Scholar 

  23. S. A. Nazarov, in Proc. of the 18 Seminar of I.G. Petrovskii (Mosk. Gos. Univ., Moscow, 1995), pp. 3–78.

    Google Scholar 

  24. S. A. Nazarov, in Proc. of the 20 Seminar of I.G. Petrovskii (Mosk. Gos. Univ., Moscow, 1997), pp. 155–195.

    Google Scholar 

  25. S. A. Nazarov, in Proc. of the 24 Seminar of I.G. Petrovskii (Mosk. Gos. Univ., Moscow, 2004), pp. 95–214.

    Google Scholar 

  26. V. A. Kondrat’ev, Trudy Mosk. Matem. Obshch. 16, 219 (1963).

    Google Scholar 

  27. V. G. Maz’ya and B. A. Plamenevskiĭ, Math. Nashr. 76, 29 (1977).

    Article  MATH  Google Scholar 

  28. V. G. Maz’ya and B. A. Plamenevskiĭ, Math. Nashr. 77, 25 (1977).

    Google Scholar 

  29. S. A. Nazarov, Usp. Mat. Nauk 54(5), 77 (1999).

    Google Scholar 

  30. V. A. Kozlov, V. G. Maz’ya, and J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities (Amer. Math. Soc., Providence, 1997).

    MATH  Google Scholar 

  31. W. G. Mazja, S. A. Nasarow, and B. A. Plamenewski, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains (Akademie-Verlag, Berlin, 1991; Birkhauser, 2000), Vol. 1.

    Google Scholar 

  32. O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973; Springer, New York, 1985).

    Google Scholar 

  33. S. A. Nazarov, in Sobolev Spaces in Mathematics, Vol. 2, Ed. by V. Maz’ya, Int. Math. Ser., Vol. 9 (2008), pp. 261–309.

  34. S. A. Nazarov and Yu. A. Romashev, Izv. AN ArmSSR, Mekhan., No. 4, 30 (1982).

  35. S. A. Nazarov, Trudy SPb. Matem. Obshch. 5, 112 (1996).

    Google Scholar 

  36. V. G. Maz’ya, S. A. Nazarov, and B. A. Plamenevskiĭ, in Problems of Math. Analysis, No. 8 (Len. Gos. Univ., Leningrad, 1981), pp. 72–153 [in Russian].

    Google Scholar 

  37. A. M. Il’in, Matching of Asymptotic Expansions of Boundary Problems Solution (Nauka, Moscow, 1989) [in Russian].

    Google Scholar 

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Authors and Affiliations

  1. Institute of Problems of Machine Manufacture, Russian Academy of Sciences, Vasilevskiĭ Ostrov, Bol’shoĭ pr. 61, St. Petersburg, 199178, Russia

    S. A. Nazarov

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  1. S. A. Nazarov
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Correspondence to S. A. Nazarov.

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Original Russian Text © S.A. Nazarov, 2010, published in Akusticheskiĭ Zhurnal, 2010, Vol. 56, No. 6, pp. 747–758.

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Nazarov, S.A. Trapped modes in a T-shaped waveguide. Acoust. Phys. 56, 1004–1015 (2010). https://doi.org/10.1134/S1063771010060254

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