Skip to main content
SpringerLink
Log in

Waves in a Plane Rectangular Lattice of Thin Elastic Waveguides

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

We study the spectrum of a thin (with the relative width h ≪ 1) rectangular lattice of elastic isotropic (with the Lamé constants ⋋ ≥ 0 and μ > 0) plane waveguides simulating joining seams of a doubly periodic system of identical absolutely rigid tiles. We establish that the low-frequency range of the essential spectrum contains two spectral bands (passing ones for waves) of length O(e−δ/(2h)), δ > 0. Above these bands there is a gap of width O(h−2) (stopping zones for waves) and then, in the mid-frequency range, above the cut-off value μπ2h−2 of the continuous spectrum of the infinite cross-shaped waveguide, there is a family of spectral bands of length O(h); moreover, between some of these bands there are opened up gaps of width O(1). The character of the wave propagation depends on whether the frequencies are below or above the cut-off value. In the first case, the oscillations are strictly concentrated near the lattice nodes and the edges are practically immovable. In the second case, the oscillations are localized on the lattice edges, i.e., the nodes are left at relative rest. We show that single perturbations of nodes or edges can cause the appearance of points of the discrete spectrum under the essential spectrum or inside the gaps; moreover, an infinite collection of identical perturbations of nodes can also change the essential spectrum. Bibliography: 78 titles. Illustrations: 5 figures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. S. Birman and M. Z. Solomyak, Spectral Theory and SelfAdjoint Operators in Hilbert Space, Reidel, Dordrecht (1987).

  2. I. M. Gelfand, “Decomposition into eigenfunctions of an equation with periodic coefficients” [in Russian], Dokl. AN SSSR 73, 1117–1120 (1950).

  3. M. Reed and B. Simon, Methods of Modern Mathematical Physics. III: Scattering Theory, Academic Press, New York etc. (1989).

  4. M. M. Skriganov, “Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators,” Proc. Steklov Inst. Math. 171 (1987).

  5. P. Kuchment, Floquet Theory for Partial Differential Equations, Birchäuser, Basel (1993).

  6. P. A. Kuchment, “Floquet theory for partial differential equations,” Russ. Math. Surv. 37, No. 4, 1–60 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Kuchment, “Graph models for waves in thin structures,” Waves Random Media 12, No. 4, R1–R24 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Bressan, S. Čanić, M. Garavello, M. Herty, and B. Piccoli, “Flows on networks: recent results and perspectives,” EMS Surv. Math. Sci. 1, No. 1, 47–111 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  9. S. A. Nazarov, “General scheme for averaging selfadjoint elliptic systems in multidimensional domains, including thin ones,” St. Petersburg Math. J. 7, No. 5, 681–748 (1996).

    MathSciNet  Google Scholar 

  10. S. A. Nazarov, “The polynomial property of self-adjoint elliptic boundary value problems and the algebraic description of their attributes,” Russ. Math. Surv. 54, No. 5, 947–1014 (1999).

    Article  MATH  Google Scholar 

  11. S. A. Nazarov and A. S. Slutskij, “Arbitrary plane systems of anisotropic beams,” Proc. Steklov Inst. Math. 236, 222–249 (2002).

    MathSciNet  MATH  Google Scholar 

  12. S. A. Nazarov and A. S. Slutskij, “Asymptotic analysis of an arbitrary spatial system of thin rods,” Transl., Ser. 2, Am. Math. Soc. 214, 59–107 (2005).

    MathSciNet  MATH  Google Scholar 

  13. V. V. Zhikov, “Homogenization of elasticity problems on singular structures,” Izv. Math. 66, No. 2, 299–365 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  14. V. V. Zhikov and S. E. Pastukhova, “Averaging of problems in the theory of elasticity on periodic grids of critical thickness,” Sb. Math. 194, No. 5, 697–732 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  15. G. Panassenko, Multi-Scale Modelling for Structures and Composites. Springer, Dordrecht (2005).

    Google Scholar 

  16. P. Exner and O. Post, “Convergence of spectra of graph-like thin manifolds,” J. Geom. Phys. 54, No. 1, 77–115 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Kuchment and O. Post, “On the spectra of carbon nano-structures,” Commun. Math. Phys. 275, No. 3, 805–826 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  18. O. Post, Spectral Analysis on Graph-Like Spaces, Springer, Berlin (2012).

    Book  MATH  Google Scholar 

  19. P. Exner and H. Kovařík, Quantum Waveguides, Springer, Cham (2015).

  20. R. Mittra and S. W. Lee, Analytical Techniques in the Theory of Guided Waves, Macmillan, New York (1971).

    MATH  Google Scholar 

  21. L. Pauling, “The diamagnetic anisotropy of aromatic molecules,” J. Chem. Phys. 4, 673 (1936).

    Article  Google Scholar 

  22. P. Kuchment and N. Zeng, “Asymptotics of spectra of Neumann Laplacians in thin domains,” In: Advances in Differential Equations and Mathematical Physics, pp. 199–213, Am. Math. Soc., Providence, RI (2003).

  23. T. Petrosky and S. Subbiah, “Electron waveguide as a model of a giant atom with a dressing field,” Phys. E 19, No. 1, 230–235 (2003).

    Article  Google Scholar 

  24. D. Grieser, “Spectra of graph neighborhoods and scattering,” Proc. Lond. Math. Soc. 97, No. 3, 718–752 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  25. S. Molchanov and B. Vainberg, “Scattering solutions in networks of thin fibers: small diameter asymptotics,” Commun. Math. Phys. 273, No. 2, 533–559 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  26. S. A. Nazarov, “Bounded solutions in a T-shaped waveguide and the spectral properties of the Dirichlet ladder,” Comput. Math. Math. Phys. 54, No. 8, 1261–1279 (2014).

    Article  MathSciNet  Google Scholar 

  27. S. A. Nazarov, K. Ruotsalainen, and P. Uusitalo, “The Y-junction of quantum waveguides;;, Z. Angew. Math. Mech. 94, No. 6, 477–486 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  28. S. A. Nazarov, “The spectra of rectangular lattices of quantum waveguides,” Izv. Math. 81, No. 1, 29–90 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  29. K. Pankrashkin, “Eigenvalue inequalities and absence of threshold resonances for waveguide junctions,” J. Math. Anal. Appl. 449, No. 1, 907–925 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  30. P. G. Ciarlet, Plates and Junctions in Elastic Multi-Structures. An Asymptotic Analysis, Masson, Paris (1990).

    MATH  Google Scholar 

  31. D. Cioranescu, S. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer, New York, NY (1999).

    Book  MATH  Google Scholar 

  32. V. Maz’ya, S. Nazarov, and B. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vols. 1, 2, Birkhäuser, Basel (2000).

  33. V. A. Kozlov, V. G. Maz’ya, and A. V. Movchan, Asymptotic Analysis of Fields in Multi-Structures, Clarendon Press, Oxford (1999).

    MATH  Google Scholar 

  34. S. A. Nazarov, “Junctions of singularly degenerating domains with different limit dimensions,” J. Math. Sci., New York 80, No. 5, 1989–2034 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  35. S. A. Nazarov, “Junctions of singularly degenerating domains with different limit dimensions. II,” J. Math. Sci., New York 97, No. 3, 4085–4108 (1999).

    Article  MathSciNet  Google Scholar 

  36. S. A. Nazarov, “Asymptotic analysis and modeling of the jointing of a massive body with thin rods,” J. Math. Sci., New York 127, No. 5, 2192–2262 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  37. S. A. Nazarov, “Korn inequalities for elastic junctions of massive bodies, thin plates, and rods,” Russ. Math. Surv. 63, No. 1, 35–107 (2008).

    Article  MATH  Google Scholar 

  38. R. L. Shult, D. G. Ravenhall, H. D. Wyld, “Quantum bound states in a classically unbounded system of crossed wires,” Phys. Rev. B. 39, No. 8, 5476–5479 (1989).

    Article  Google Scholar 

  39. Y. Avishai, D. Bessis, B. C. Giraud, G. Mantica, “Quantum bound states in open geometries,” Phys. Rev. B 44, No. 15, 8028–8034 (1991).

    Article  Google Scholar 

  40. M. Dauge and N. Raymond, “Plane waveguides with corners in the small angle limit,” J. Math. Phys. 53 (2012) DOI: https://doi.org/10.1063/1.4769993

  41. S. A. Nazarov and A. V. Shanin, “Trapped modes in angular joints of 2D waveguides,” Appl. Anal. 93, No. 3, 572–582 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  42. S. A. Nazarov, “Asymptotics of eigenvalues of the Dirichlet problem in a skewed T –shaped waveguide,” Comput. Math. Math. Phys. 54, No. 5, 811–830 (2014).

    Article  MathSciNet  Google Scholar 

  43. S. A. Nazarov, “Discrete spectrum of cranked quantum and elastic waveguides,” Comput. Math. Math. Phys. 56, No. 5, 864–880 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  44. S. A. Nazarov, “Localization of elastic oscillations in cross-shaped planar orthotropic waveguides,” Dokl. Phys. 59, No. 9, 411–415 (2014).

    Article  Google Scholar 

  45. S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walter de Gruyter, Berlin etc. (1994).

  46. S. A. Nazarov, “Properties of spectra of boundary value problems in cylindrical and quasicylindrical domain” In: Sobolev Spaces in Mathematics. Vol. II, pp. 261–309, Springer, New York etc. (2008)

  47. S. A. Nazarov, “Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide,” Theor. Math. Phys. 167, No. 2, 606–627 (2011). 167, No. 2, 239–263 (2011)

  48. S. A. Nazarov, “Nonreflecting distortions of an isotropic strip clamped between rigid punches,” Comput. Math. Math. Phys. 53, No. 10, 1512–1522 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  49. S. A. Nazarov, “Structure of the spectrum of a net of quantum waveguides and bounded solutions of a model problem at the threshold,” Dokl. Math. 90, No. 2, 637–641 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  50. F. L. Bakharev, S. G. Matveenko, and S. A. Nazarov, “Spectra of three-dimensional cruciform and lattice quantum waveguides,” Dokl. Math. 92, No. 1, 514–518 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  51. M. L. Williams, “Stress singularities resulting from various boundary conditions in angular corners of plate in extension,” J. Appl. Mech. 19, No. 4, 526–528 (1952).

    Google Scholar 

  52. V. A. Kondrat’ev, “Boundary value problems for elliptic equations in domains with conical and angular points” [in Russian], Tr. Mosk. Mat. O-va 16, 219–292 (1963).

    Google Scholar 

  53. M. S. Agranovich and M. I. Vishik, “Elliptic problems with a parameter and parabolic problems of general type,” Russ. Math. Surv. 19, No. 3, 53–157 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  54. J. L. Lions and E. Magenes Non-Homogeneous Boundary Value Problems and Applications, Springer, Berlin (1972).

  55. I. V. Kamotskii and S. A. Nazarov, “Exponentially decreasing solutions of diffraction problems on a rigid periodic boundary,” Math. Notes 73, No. 1, 129–131 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  56. S. A. Nazarov, “Variational and asymptotic methods for finding eigenvalues below the continuous spectrum threshold,” Sib. Math. J. 51, No. 5, 866–878 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  57. M. Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, CA (1975). 58. A. M. Il’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Am. Math. Soc., Providence, RI (1992).

  58. M. M. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Nonlinear Equations [in Russian], Nauka, Moscow (1969).

  59. S. A. Nazarov, “Asymptotics of the frequencies of elastic waves trapped by a small crack in a cylindrical waveguide,” Mech. Solids 45, No. 6, 856–864 (2010).

    Article  Google Scholar 

  60. V. G. Maz’ya and B. A. Plamenevskij, “Estimates in Lp and in Hölder classes and the Miranda–Agmon principle for solutions of elliptic boundary value problems in domains with singular points on the boundary” [in Russian] Math. Nachr. 81, 25–82 (1978).

  61. M. I. Vishik and L. A. Lusternik, “Regular degeneration and boundary layer for linear differential equations with small parameter” [in Russian], Usp. Mat. Nauk 12, No. 5, 3–122 (1957).

    Google Scholar 

  62. S. A. Nazarov, Asymptotic Theory of Thin Plates and Rods. Vol. 1. Dimension Reduction and Integral Estimates [in Russian], Nauchnaya Kniga (IDMI), Novosibirsk (2002).

  63. M. Lobo, S. A. Nazarov, and E. Perez, “Eigen-oscillations of contrasting non-homogeneous bodies: asymptotic and uniform estimates for eigenvalues,” IMA J. Appl. Math. 70, 419–458 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  64. S. A. Nazarov and B. A. Plamenevskii, “On radiation conditions for selfadjoint elliptic problems,” Sov. Math., Dokl. 41, No. 2, 274–277 (1990).

    MathSciNet  MATH  Google Scholar 

  65. S. A. Nazarov and B. A. Plamenevskii, “Radiation principles for selfadjoint elliptic problems” [in Russian], Probl. Mat. Phys. 13, 192–244 (1991).

  66. S. A. Nazarov, “The asymptotic analysis of gaps in the spectrum of a waveguide perturbed with a periodic family of small voids,” J. Math Sci., New York 186, No. 2, 247–301 (2012).

  67. L. I. Mandelstam, Lectures on Optics. Theory of Relativity and Quantum Mechanics [in Russian], Collected Works, Vol. 2. Izd-vo AN SSSR, Moscow (1947).

  68. I. I. Vorovich and V. A. Babeshko, Dynamic Mixed Problems of the Theory of Elasticity for Nonclassical Domains [in Russian], Nauka, Moscow (1979).

  69. N. A. Umov, The Equations of Motion for the Energy in Bodies [in Russian], Ulrich–Schulz tipogr., Odessa (1874).

  70. J. H. Poynting, “On the transfer of energy in the electromagnetic field,” Phil. Trans. Royal Soc. London 175, 343–361 (1884).

    Article  MATH  Google Scholar 

  71. S. A. Nazarov, “Justification of the asymptotic theory of thin rods. Integral and pointwise estimates,” J. Math Sci., New York 97, No. 4, 4245–4279 (1999).

    Article  MathSciNet  Google Scholar 

  72. S. A. Nazarov, “Estimates for second order derivatives of eigenvectors in thin anisotropic plates with variable thickness,” J. Math. Sci., New York 132, No. 1, 91–102 (2006).

    Article  MathSciNet  Google Scholar 

  73. S. A. Nazarov, “Various manifestations of Wood anomalies in locally distorted quantum waveguides,” Comput. Math. Math. Phys. 58, No. 11, 1838–1855 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  74. S. Molchanov and B. Vainberg, “Scattering on the system of sparse bumps multidimensional case,” Appl. Anal. 71, No. 1–4, 167–185 (1999).

    MathSciNet  MATH  Google Scholar 

  75. D. Hundertmark and W. Kirsch, “Spectral theory of sparse potentials,” In: Stochastic Processes, Physics and Geometry: New Interplays, pp. 213–238, Am. Math. Soc., Providence, RI (2000).

  76. V. Jakšić and P. Poulin, “Scattering from sparse potentials: a deterministic approach,” In: Analysis and Mathematical Physics, pp. 205–210, Birkhäuser, Basel (2009).

  77. S. A. Nazarov and J. Taskinen, “Essential spectrum of a periodic waveguide with nonperiodic perturbation,” J. Math. Anal. Appl. 463, 922–933 (2018).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St.-Petersburg State University, 7-9, Universitetskaya nab., St. Petersburg, 199034, Russia

    S. A. Nazarov

Authors
  1. S. A. Nazarov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. A. Nazarov.

Additional information

Dedicated to Nina Nikolaevna Uraltseva who taught the author a lot in mathematics

Translated from Problemy Matematicheskogo Analiza99, 2019, pp. 47-88.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nazarov, S.A. Waves in a Plane Rectangular Lattice of Thin Elastic Waveguides. J Math Sci 242, 227–279 (2019). https://doi.org/10.1007/s10958-019-04476-7

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-019-04476-7

Search

Navigation