Zeitschrift für angewandte Mathematik und Physik

, Volume 65, Issue 5, pp 977–1002 | Cite as

Criteria for trapped modes in a cranked channel with fixed and freely floating bodies

Article

Abstract

Trapped modes in the linearized water wave problem are localized free oscillations in an unbounded fluid with a free surface. For sometime, it has been known that certain structures, fixed or freely floating, can support such modes. In this paper, we consider the problem on a channel, which consists of a finite part and two cylindrical outlets into infinity. The finite (bounded) part may contain some submerged and/or surface-piercing bodies. Since the ordinary scattering matrix can by no means contribute any information on trapped modes, we introduce the fictitious scattering operator and present a criterion for the existence of trapped modes. The criterion states that the number of trapped modes is the difference of the multiplicities of the eigenvalue 1 of the fictitious scattering operator and the eigenvalue −i of the scattering matrix.

Mathematics Subject Classification (2010)

Primary 35P25 Secondary 76B15 

Keywords

Trapped modes Spectral problem Water waves Fictitious scattering operator Steklov–Poincaré operator Artificial boundary condition 

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References

  1. 1.
    Agoshkov, V.I.: Poincare–Steklov operators and domain decomposition methods in finite dimensional spaces. In: SIAM Proceedings of the First International Symposium on Domain Decomposition Methods, Paris (1987)Google Scholar
  2. 2.
    Aslanyan A., Parnovski L., Vassiliev D.: Complex Resonances in Acoustic Waveguides. Q. J. Mech. Appl. Math. 53, 429–447 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bonnet-Ben Dhia A.-S., Joly P.: Mathematical analysis of guided water waves. SIAM J. Appl. Math. 53, 1507–1550 (1993). doi:10.1137/0153071 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bonnet-Bendhia A.-S., Starling F.: Guided waves by electromagnetic gratings and non-uniqueness examples for the diffraction problem. Math. Meth. Appl. Sci. 17, 305–338 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Birman M.S., Solomyak M.Z.: Spectral Theory of Self-Adjoint Operators in Hilbert Space. Reidel Publishing Company, Dordrecht (1986)CrossRefGoogle Scholar
  6. 6.
    Euler L.: Principia motus fluidorum. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tom VI, 271–311 (1761)Google Scholar
  7. 7.
    Evans D.V., Levitin M., Vassiliev D.: Existence theorems for trapped modes. J. Fluid Mech. 261, 21–31 (1994)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Garipov R.M.: On the linear theory of gravity waves: the theorem of existence and uniqueness. Arch. Rat. Mech. Anal. 24, 352–362 (1967). doi:10.1007/BF00253152 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gohberg, I.C., Krein, M.G.: Introduction to the theory of linear non-selfadjoint operators. Transl. Math. Monogr. 18, Am. Math. Soc. (1969) (Translated from Russian)Google Scholar
  10. 10.
    John F.: On the motion of floating bodies I. Comm. Pure Appl. Math. 2(1), 13–57 (1949)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    John F.: On the motion of floating bodies II, simple harmonic oscillations. Comm. Pure Appl. Math. 3, 45–101 (1950)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Jones D.S.: The eigenvalues of ∇2 u +  λ u =  0 when the boundary conditions are given on semi-infinite domains. Proc. Camb. Phil. Soc. 49, 668–684 (1953)CrossRefMATHGoogle Scholar
  13. 13.
    Kamotskii, I.V., Nazarov, S.A.: An augmented scattering matrix and exponentially decreasing solutions of an elliptic problem in a cylindrical domain, Zap. Nauchn. Sem. St.-Petersburg Otdel. Mat. Inst. Steklov 264 (2002), 66–82 [English transl.: Journal of Math. Sci., Vol. 111(4), 3657–3666 (2002)]Google Scholar
  14. 14.
    Kuznetsov N.I., Mazya V.G., Vainberg B.R.: Linear Water Waves. Cambridge University Press, Cambridge (2002)CrossRefMATHGoogle Scholar
  15. 15.
    Leis R.: Initial Boundary Value Problems of Mathematical Physics. B.G. Teubner, Stuttgart (1986)CrossRefGoogle Scholar
  16. 16.
    Linton C.M., McIver P.: Embedded trapped modes in water waves and acoustics. Wave Motion 45, 16–29 (2007). doi:10.1016/j.wavemoti.2007.04.009 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Maniar H.D., Newman J.R.: Wave diffraction by a long array of cylinders. J. Fluid Mech. 339, 309–330 (1997)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mei C.C., Stiassnie M., Yue D.K.-P.: Theory and Applications of Ocean Surface Waves. Part 1: Linear Aspects. World Scientific Publishing Co., Singapore (2005)Google Scholar
  19. 19.
    Motygin O.V.: Trapped modes in a linear problem of theory of surface water waves. J. Math. Sci. 173(6), 717–736 (2011)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Nazarov, S.A.: A criterion for the existence of decaying solutions in the problem on a resonator with a cylindrical waveguide. Funkt. Anal. i Prilozhen, 40(2), 20–32 (2006) [English transl.: Funct. Anal. Appl., 40(2), 97–107 (2006)]Google Scholar
  21. 21.
    Nazarov, S.A.: Artificial boundary conditions for finding surface waves in the problem of diffraction by a periodic boundary. Zh. Vychisl. Mat. i Mat. Fiz. 46(12), 2265–2276 (2006). [English transl.: Comput. Math. and Math. Phys. 46(12), 2164–2175 (2006)]Google Scholar
  22. 22.
    Nazarov, S.A.: A simple method for finding trapped modes in problems of the linear theory of surface waves. Dokl. Ross. Akad. Nauk. 429(6), 746–749 (2009) [English transl.: Doklady Mathematics, 80(3), 914–917 (2009)]Google Scholar
  23. 23.
    Nazarov S.A.: Sufficient conditions for the existence of trapped modes in problems of the linear theory of surface waves. Zap. Nauchn. Sem. St.-Petersburg Otdel. Mat. Inst. Steklov 369, 202–223 (2009)Google Scholar
  24. 24.
    Nazarov, S.A.: Eigenvalues of the Laplace operator with the Neumann conditions at regular perturbed walls of a waveguide. Probl. Mat. Analiz. No. 53. Novosibirsk, 2011, 104–119. [English transl.: Journal of Math. Sci., 172, 555–588 (2011)]Google Scholar
  25. 25.
    Nazarov, S.A.: Incomplete comparison principle in problems about surface waves trapped by fixed and freely floating bodies, Probl. mat. analiz. No. 56, Novosibirsk, 2011, 83–115 [English transl.: Journal of Math. Sci., 175, 309–348 (2011)]Google Scholar
  26. 26.
    Nazarov, S.A.: Trapped waves in a cranked waveguide with hard walls. Acoust. J. 57, 746–754 (2011) [English transl.: Acoustical Physics 57, 764–771 (2011)]Google Scholar
  27. 27.
    Nazarov, S.A.: Enforced stability of an eigenvalue in the continuous spectrum of a waveguide with an obstacle. Zh. Vychisl. Mat. i Mat. Fiz. 52(3), 521–538 (2012) [English transl.: Comput. Math. and Math. Physics 52, 448–464 (2012)]Google Scholar
  28. 28.
    Nazarov, S.A.: The enforced stability of a simple eigenvalue in the continuous spectrum of a waveguide. to appear in Funkt. Anal. i Prilozhen (English transl.: Funct. Anal. Appl.)Google Scholar
  29. 29.
    Nazarov, S.A., Plamenevskii, B.A.: Radiation principles for self-adjoint elliptic problems. Probl. Mat. Fiz., No. 13, pp. 192–244. Leningrad Univ., Leningrad (1991); RussianGoogle Scholar
  30. 30.
    Nazarov S.A., Plamenevskii B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walter de Gruyter, Berlin (1994)CrossRefMATHGoogle Scholar
  31. 31.
    Nazarov S.A., Taskinen J.: On essential and continuous spectra of the linearized water-wave problem in a finite pond. Math. Scand. 106, 1–20 (2009)MathSciNetGoogle Scholar
  32. 32.
    Nazarov, S.A., Taskinen, J.: Double-sided estimates for eigenfrequencies in the John problem for freely floating body. Zap. Nauchn. Sem. St.-Petersburg Otdel. Mat. Inst. Steklov 397, 89–114 (2011) [English transl.: Journal of Math. Sci. 397 (2011)]Google Scholar
  33. 33.
    Nazarov S.A., Videman J.H.: Trapping of water waves by freely floating structures in a channel. Proc. R. Soc. A 467, 3613–3632 (2011)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Pagneux V., Maurel A.: Scattering matrix properties with evanescent modes for waveguides in fluids and solids. J. Acoust. Soc. Am. 116(4), 1913–1920 (2004)CrossRefGoogle Scholar
  35. 35.
    Ursell F.: Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47(2), 347–358 (1951)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Ursell F.: Mathematical aspects of trapping modes in the theory of surface waves. J. Fluid Mech. 183, 421–437 (1987)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Vainberg B.R., Maz’ja V.G.: On the plane problem of the motion of a body immersed in a fluid. Trans. Moscow Math. Soc. 28, 33–55 (1973)Google Scholar
  38. 38.
    Wilcox C.H.: Scattering Theory for Diffraction Gratings. Springer, New York (1979)Google Scholar
  39. 39.
    Wood R.V.: On a remarkable case of uneven distribution of light in a diffraction grating spectrum. Phil. Mag. 4, 396–402 (1902)CrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Mathematics and Mechanics FacultySt. Petersburg State UniversityStary PeterhofRussia
  2. 2.Mathematics DivisionUniversity of OuluOuluFinland

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