Advertisement

Journal of Elasticity

, Volume 124, Issue 2, pp 193–223 | Cite as

Trapped Modes in Piezoelectric and Elastic Waveguides

  • Sergei A. Nazarov
  • Keijo M. RuotsalainenEmail author
  • Minna Silvola
Article
  • 190 Downloads

Abstract

We derive a sufficient condition for the existence of trapped modes in a cylindrical piezoelectric waveguide \(\varOmega\) with a compact void \(\varTheta\). An infinite part \(\varGamma_{D}\) of the exterior boundary \(\partial\varOmega\) is clamped along an electric conductor and the remaining part \(\varGamma_{N}=(\partial\varOmega\setminus \varGamma_{D})\cup\partial\varTheta\) is traction-free and is in contact with an insulator, e.g., a vacuum. The condition permits the limit passage either to an elastic or a compound waveguide but it crucially differs from the pure elastic case due to the involved electric enthalpy instead of the energy functional. Examples of concrete waveguides and voids supporting trapped modes are given, and open questions are formulated. In particular, in contrast to a pure elastic waveguide where “almost” any crack traps a wave, no example of a crack trapping a wave in a piezoelectric waveguide is known yet.

Keywords

Trapped mode Piezoelectricity Waveguide 

Mathematics Subject Classification

35P15 35Q74 74F15 74F20 

Notes

Acknowledgements

The first author is supported by the Russian Foundation of Basic Research grant 15-01-02175.

References

  1. 1.
    Aslanyan, A., Parnovski, L., Vassiliev, D.: Complex resonances in acoustic waveguides. Q. J. Mech. Appl. Math. 53(3), 429–447 (2000) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Birman, M.S., Solomyak, M.Z.: Spectral Theory of Self-Adjoint Operators in Hilbert Space. Reidel, Dordrecht (1986) CrossRefzbMATHGoogle Scholar
  3. 3.
    Bonnet-Bendhia, A.-S., Duterte, J., Joly, P.: Mathematical analysis of elastic surface waves in topographic waveguides. Math. Model. Mech. Appl. Sci. 9(5), 755–798 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borisov, D., Ekholm, T., Kovařík, H.: Spectrum of the magnetic Schrödinger operator in a waveguide with combined boundary conditions. Ann. Henri Poincaré 6(2), 327–342 (2005) ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Borisov, D., Exner, P., Gadylshin, R., Krejc̆iřík, D.: Bound states in weakly deformed strips and layers. Ann. Henri Poincaré 2, 553–572 (2001) ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Bulla, W., Gesztesy, F., Renger, W., Simon, B.: Weakly coupled bound states in quantum waveguides. Proc. Am. Math. Soc. 125, 1487–1495 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cardone, G., Nazarov, S.A., Ruotsalainen, K.M.: Bound states of a converging quantum waveguide. Math. Model. Numer. Anal. 47, 305–315 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cardone, G., Nazarov, S.A., Taskinen, J.: A criterion for the existence of the essential spectrum for beak-shaped elastic bodies. J. Math. Pures Appl. 92(6), 628–650 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Duclos, P., Exner, P.: Curvature-induced bound states in quantum waveguides in two and three dimensions. Rev. Math. Phys. 7(1), 73–102 (1995) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Evans, D.V., Levitin, M., Vasil’ev, D.: Existence theorems for trapped modes. J. Fluid Mech. 261, 21–31 (1994) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Exner, P., Vugalter, S.A.: Bound states in a locally deformed waveguide: the critical case. Lett. Math. Phys. 39, 59–68 (1997) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gadylshin, R.R.: On local perturbations of quantum waveguides. Teor. Mat. Fiz. 145(3), 358–371 (2005). English transl.: Theor. Math. Phys., 145(3), 1678–1690 (2005) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kamotskii, I.V., Nazarov, S.A.: Elastic waves localized near periodic sets of flaws. Dokl. Akad. Nauk, Ross. Akad. Nauk 368(6), 771–773 (1999). English transl.: Dokl. Phys. 44(10), 715–717 (1999) MathSciNetGoogle Scholar
  14. 14.
    Kamotskii, I.V., Nazarov, S.A.: Exponentially decreasing solutions of the problem of diffraction by a rigid periodic boundary. Mat. Zametki 73(1), 138–140 (2003). English transl.: Math. Notes 73(1), 129–131 (2003) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Kato, T.: Perturbation Theory of Linear Operators. Springer, Berlin (1966) CrossRefzbMATHGoogle Scholar
  16. 16.
    Kondratiev, V.A.: Boundary value problems for elliptic equations for the systems of elasticity theory in domains with conical or angular points. Trans. Mosc. Math. Soc. 10, 227–313 (1967) Google Scholar
  17. 17.
    Ladyshenskaya, O.A.: The Boundary Value Problems of Mathematical Physics. Springer, New York (1985) CrossRefGoogle Scholar
  18. 18.
    Linton, C.M., McIver, P.: Embedded trapped modes in water waves and acoustics. Wave Motion 45(1), 16–29 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications. Springer, Berlin (1972) CrossRefzbMATHGoogle Scholar
  20. 20.
    Nazarov, S.A.: The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes. Usp. Mat. Nauk 54(5), 77–142 (1999). English transl.: Russ. Math. Surv. 54(5), 947–1014 (1999) CrossRefGoogle Scholar
  21. 21.
    Nazarov, S.A.: Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigen-oscillations of a piezoelectric plate. Probl. Mat. Anal. 25, 99–188 (2003). English transl.: J. Math. Sci. 114(5), 1657–1725 (2003) zbMATHGoogle Scholar
  22. 22.
    Nazarov, S.A.: Trapped modes for a cylindrical elastic waveguide with a damping gasket. Ž. Vyčisl. Mat. Mat. Fiz. 48(5), 863–881 (2008). English transl.: Comput. Math. Math. Phys. 48(5) (2008) MathSciNetzbMATHGoogle Scholar
  23. 23.
    Nazarov, S.A.: The spectrum of the elasticity problem for a spiked body. Sib. Mat. Zh. 49(5), 1105–1127 (2008). English transl.: Sib. Math. J., 49(5), 874–893 (2008) CrossRefzbMATHGoogle Scholar
  24. 24.
    Nazarov, S.A.: Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains. In: Maz’ya, V. (ed.) Sobolev Space in Mathematics, vol. II. International Mathematics Series, vol. 9, pp. 261–309. Springer, New York (2008) CrossRefGoogle Scholar
  25. 25.
    Nazarov, S.A.: Sufficient conditions for the existence of trapped modes in problems of the linear theory of surface waves. Zap. Nauč. Semin. POMI 369, 202–223 (2009) (in Russian). English transl.: J. Math. Sci. 167(5), 713–725 (2010) Google Scholar
  26. 26.
    Nazarov, S.A.: Variational and asymptotic methods for finding eigenvalues below the continuous spectrum threshold. Sib. Mat. Zh. 51(5), 1086–1101 (2010). English transl.: Sib. Math. J. 51(5), 866–878 (2010) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Nazarov, S.A.: Localized elastic fields in periodic waveguides with defects. Prikl. Mekh. Tekhn. Fiz. 52(2), 183–194 (2011). English transl.: J. Appl. Mech. Tech. Phys. 52(2) (2011) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Nazarov, S.A.: Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide. Theor. Math. Phys. 167(2), 239–262 (2011). English transl.: Theor. Math. Phys. 167(2), 606–627 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Nazarov, S.A.: Enforced stability of an eigenvalue in the continuous spectrum of a waveguide with an obstacle. Ž. Vyčisl. Mat. Mat. Fiz. 52(3), 521–538 (2012). English transl.: Comput. Math. Math. Phys. 52(3), 448–464 (2012) MathSciNetzbMATHGoogle Scholar
  30. 30.
    Nazarov, S.A.: The asymptotics of frequencies of elastic waves trapped by a small crack in an anisotropic waveguide. Mekh. Tverd. Tela 45(6), 112–122 (2010). English transl.: Mech. Solids 45, 856–864 (2010) Google Scholar
  31. 31.
    Nazarov, S.A., Plamenevsky, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. de Gruyter Expositions in Mathematics, vol. 13. de Gruyter, Berlin (1994) CrossRefzbMATHGoogle Scholar
  32. 32.
    Parton, V.Z., Kudryavtsev, B.A.: Electromagnetoelasticity, Piezoelectrics and Electrically Conductive Solids. Gordon & Breach, New York (1988) Google Scholar
  33. 33.
    Roitberg, I., Vassiliev, D., Weidl, T.: Edge resonance in an elastic semi strip. Q. J. Mech. Appl. Math. 51(1), 1–13 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. 1. Functional Analysis. Academic Press, San Diego (1980) zbMATHGoogle Scholar
  35. 35.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. IV. Analysis of Operators. Academic Press, San Diego (1978) zbMATHGoogle Scholar
  36. 36.
    Suo, Z., Kuo, C.-M., Barnett, D.M., Willis, J.R.: Fracture mechanics for piezoelectric ceramics. J. Mech. Phys. Solids 40(4), 739–765 (1992) ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (1977) zbMATHGoogle Scholar
  38. 38.
    Tiersten, H.F.: Linear Piezoelectric Plate Vibrations. Plenum Press, New York (1964) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia
  2. 2.Peter the Great Saint-Petersburg State Polytechnical UniversitySt. PetersburgRussia
  3. 3.Institute of Problems Mechanical Engineering RASSt. PetersburgRussia
  4. 4.Applied and Computational Mathematics Research GroupUniversity of OuluOuluFinland

Personalised recommendations