Gap detection in the spectrum of an elastic periodic waveguide with a free surface

  • S. A. NazarovEmail author


A three-dimensional periodic elastic waveguide is constructed whose continuous spectrum (the frequencies that admit propagating waves) contains a gap, i.e., an interval that has its ends in the continuous spectrum but contains at most a discrete spectrum. The waveguide consists of an infinite chain of massive bodies connected by short thin links, and its surface is assumed to be free. The method for detecting a gap also applies to plane problems, including scalar ones. Periodic elastic waveguides with different shapes or contrasting properties are indicated in which a gap can also be detected.


three-dimensional periodic waveguides gap in an eigenvalue spectrum Floquet waves elasticity problems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. I. Vorovich and V. A. Babeshko, Mixed Dynamic Problems of Elasticity Theory for Nonclassical Domains (Nauka, Moscow, 1979) [in Russian].zbMATHGoogle Scholar
  2. 2.
    M. M. Skriganov, /ldGeometric and Arithmetic Methods in the Spectral Theory Multidimensional Periodic Operators,” (Nauka, Leningrad, 1985; Am. Math. Soc., Providence, 1987).Google Scholar
  3. 3.
    P. Kuchment, Floquet Theory for Partial Differential Equations (Birkhäuser, Basel, 1993).zbMATHGoogle Scholar
  4. 4.
    A. Figotin and P. Kuchment, /ldBand-Gap Structure of Spectra of Periodic Dielectric and Acoustic Media: I. Scalar Model,” SIAM J. Appl. Math. 56, 68–88 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    E. L. Green, /ldSpectral Theory of Laplace-Beltrami Operators with Periodic Metrics,” J. Differ. Equations 133, 15–29 (1997).zbMATHCrossRefGoogle Scholar
  6. 6.
    V. V. Zhikov, /ldGaps in the Spectrum of Some Elliptic Operators in Divergent Form with Periodic Coefficients,” Algebra Anal. 6(5), 34–58 (2004).Google Scholar
  7. 7.
    N. Filonov, /ldGaps in the Spectrum of the Maxwell Operator with Periodic Coefficients,” Commun. Math. Phys. 240(1/2), 161–170.Google Scholar
  8. 8.
    S. A. Nazarov, /ldThe Rayleigh Waves in an Elastic Semi-Layer with Periodic Partly Clamped Boundary,” 423(1) (2008) Dokl. Akad. Nauk [Dokl. Math. 423, No. 1 (2008)].Google Scholar
  9. 9.
    I. M. Gel’fand, /ldEigenfunction Expansion for an Equation with Periodic Coefficients,” Dokl. Akad. Nauk SSSR 73, 1117–1120 (1950).zbMATHGoogle Scholar
  10. 10.
    S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries (Nauka, Moscow, 1991; Walter de Gruyter, Berlin, 1994).Google Scholar
  11. 11.
    S. A. Nazarov, /ldElliptic Boundary Value Problems with Periodic Coefficients in a Cylinder,” Izv. Akad. Nauk SSSR, Ser. Mat. 45(1), 101–112 (1981).zbMATHMathSciNetGoogle Scholar
  12. 12.
    P. A. Kuchment, /ldFloquet Theory for Partial Differential Equations,” Usp. Mat. Nauk 37(4), 3–52 (1982).MathSciNetGoogle Scholar
  13. 13.
    O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973; Springer-Verlag, New York, 1985).Google Scholar
  14. 14.
    M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space (Leningrad. Gos. Univ., Leningrad, 1980; Reidel, New York, 1986).Google Scholar
  15. 15.
    T. Kato, Perturbation Theory for Linear Operators (Springer-Verlag, Berlin, 1966; Mir, Moscow, 1972).zbMATHGoogle Scholar
  16. 16.
    S. A. Nazarov, /ldKorn’s Inequalities for Elastic Junctions of Massive Bodies and Thin Plates and Rods,” Usp. Mat. Nauk 63(1), 143–217 (2009).Google Scholar
  17. 17.
    V. A. Kondrat’ev and O. A. Oleinik, /ldBoundary Value Problems for the Elasticity System in Unbounded Domains: Korn’s Inequality,” Usp. Mat. Nauk 43(5), 55–98 (1988).zbMATHMathSciNetGoogle Scholar
  18. 18.
    S. A. Nazarov, Asymptotic Theory of Plates and Rods: Dimension Reduction and Integral Estimates (Nauchn. Kniga, Novosibirsk, 2002) [in Russian].Google Scholar
  19. 19.
    W. G. Mazja, S. A. Nazarov, and B. A. Plamenewski, Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten (Akademie-Verlag, Berlin, 1991), Vol. 1.Google Scholar
  20. 20.
    D. V. Evans, M. Levitin, and D. Vasil’ev, /ldExistence Theorems for Trapped Modes,” J. Fluid Mech. 261, 21–31 (1994).zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    S. A. Nazarov, /ldConcentration of the Point Spectrum on the Continuous One in Problems of Linear Water-Wave Theory,” Zap. Nauchn. Semin. POMI Ross. Akad. Nauk 348, 98–126 (2007).Google Scholar
  22. 22.
    I. Roitberg, D. Vassiliev, and T. Weidl, /ldEdge Resonance in an Elastic Semi-Strip,” Quart. J. Appl. Math. 51(1), 1–13 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    S. A. Nazarov, /ldTrapped Modes for a Cylindrical Elastic Waveguide with a Damping Gasket,” Zh. Vychisl. Mat. Mat. Fiz. 48, 132–150 (2008) [Comput. Math. Math. Phys. 48, 816–833 (2008)].Google Scholar
  24. 24.
    M. S. Agranovich and M. I. Vishik, /ldElliptic Problems with a Parameter and Parabolic Problems of General Type,” Usp. Mat. Nauk 19(3), 53–160 (1999).Google Scholar
  25. 25.
    F. Ursell, /ldMathematical Aspects of Trapping Modes in the Theory of Surface Waves,” J. Fluid Mech. 18, 495–503 (1988).Google Scholar
  26. 26.
    A.-S. Bonnet-Bendhia, J. Duterte, and P. Joly, /ldMathematical Analysis of Elastic Surface Waves in Topographic Waveguides,” Math. Models Methods Appl. Sci. 9, 755–798 (1999).CrossRefMathSciNetGoogle Scholar
  27. 27.
    C. M. Linton, and P. McIver, /ldEmbedded Trapped Modes in Water Waves and Acoustics,” Wave Motion 45, 16–29 (2007).CrossRefMathSciNetGoogle Scholar
  28. 28.
    S. A. Nazarov, /ldArtificial Boundary Conditions for Finding Surface Waves in the Problem of Diffraction by a Periodic Boundary,” Zh. Vychisl. Mat. Mat. Fiz. 46, 2267–2278 (2006) [Comput. Math. Math. Phys. 46, 2164–2175 (2006)].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Institute of Mechanical Engineering ProblemsRussian Academy of Sciences, Vasil’evskii OstrovSt. PetersburgRussia

Personalised recommendations