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

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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.

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References

  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)

  2. Aslanyan A., Parnovski L., Vassiliev D.: Complex Resonances in Acoustic Waveguides. Q. J. Mech. Appl. Math. 53, 429–447 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Birman M.S., Solomyak M.Z.: Spectral Theory of Self-Adjoint Operators in Hilbert Space. Reidel Publishing Company, Dordrecht (1986)

    Book  Google Scholar 

  6. Euler L.: Principia motus fluidorum. Novi Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tom VI, 271–311 (1761)

    Google Scholar 

  7. Evans D.V., Levitin M., Vassiliev D.: Existence theorems for trapped modes. J. Fluid Mech. 261, 21–31 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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)

  10. John F.: On the motion of floating bodies I. Comm. Pure Appl. Math. 2(1), 13–57 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  11. John F.: On the motion of floating bodies II, simple harmonic oscillations. Comm. Pure Appl. Math. 3, 45–101 (1950)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)]

  14. Kuznetsov N.I., Mazya V.G., Vainberg B.R.: Linear Water Waves. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  15. Leis R.: Initial Boundary Value Problems of Mathematical Physics. B.G. Teubner, Stuttgart (1986)

    Book  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  17. Maniar H.D., Newman J.R.: Wave diffraction by a long array of cylinders. J. Fluid Mech. 339, 309–330 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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. Motygin O.V.: Trapped modes in a linear problem of theory of surface water waves. J. Math. Sci. 173(6), 717–736 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)]

  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)]

  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)]

  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. 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)]

  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)]

  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. 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)]

  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.)

  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); Russian

  30. Nazarov S.A., Plamenevskii B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walter de Gruyter, Berlin (1994)

    Book  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)]

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  35. Ursell F.: Trapping modes in the theory of surface waves. Proc. Camb. Phil. Soc. 47(2), 347–358 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ursell F.: Mathematical aspects of trapping modes in the theory of surface waves. J. Fluid Mech. 183, 421–437 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  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. Wilcox C.H.: Scattering Theory for Diffraction Gratings. Springer, New York (1979)

    Google Scholar 

  39. Wood R.V.: On a remarkable case of uneven distribution of light in a diffraction grating spectrum. Phil. Mag. 4, 396–402 (1902)

    Article  Google Scholar 

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Authors and Affiliations

  1. Mathematics and Mechanics Faculty, St. Petersburg State University, 198504, Universitetsky pr. 28, Stary Peterhof, Saint Petersburg, Russia

    S. A. Nazarov

  2. Mathematics Division, University of Oulu, P.O. Box 4500, 90014, Oulu, Finland

    K. M. Ruotsalainen

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  1. S. A. Nazarov
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  2. K. M. Ruotsalainen
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Correspondence to K. M. Ruotsalainen.

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Nazarov, S.A., Ruotsalainen, K.M. Criteria for trapped modes in a cranked channel with fixed and freely floating bodies. Z. Angew. Math. Phys. 65, 977–1002 (2014). https://doi.org/10.1007/s00033-013-0386-1

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  • DOI: https://doi.org/10.1007/s00033-013-0386-1

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