Archive for Rational Mechanics and Analysis

, Volume 171, Issue 1, pp 129–150 | Cite as

Analysis of Elastic Band Structures for Oblique Incidence

Article

Abstract.

We propose a canonical basis that is used in the expansion of eigensolutions for a problem of oblique incidence of elastic waves on a doubly periodic array of cylindrical channels. We apply a multipole method to study the spectral properties of waves in such a structure. Dispersion diagrams constructed on the basis of an analytical solution show the presence of a full phononic band gap when the angle of oblique incidence exceeds certain critical value. Explicit asymptotic formulae are presented for the effective refractive index associated with shear waves in oblique incidence.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Achenbach, J.D., Fang, S.J.: Asymptotic analysis of the modes of wave propagation in a solid cylinder. J. Acoustical Soc. Am. 47, 1282–1289 (1970)Google Scholar
  2. 2.
    Armenàkas, A.E., Gazis, D.C., Herrmann, G.: Free vibrations of circular cylindrical shells. Pergamon Press, New York, 1969Google Scholar
  3. 3.
    Chin, S.K., Nicorovici, N.A., McPhedran, R.C.: Green’s function and lattice sums for an electromagnetic scattering by a square array of cylinders. Phys. Rev. E 49, 4590– 4602 (1994)Google Scholar
  4. 4.
    Dowling, J.P., Everitt, H., Yablonovitch, E.: Photonic and sonic band-gap bibliography. http://home.earthlink.net/∼jpdowling/pbgbib.html 2003Google Scholar
  5. 5.
    Gazis, D.C.: Three-dimensional investigation of the propagation of waves in hollow circular cylinders I: Analytical foundation. J. Acoustical Soc. Am. 31, 568–573 (1959)Google Scholar
  6. 6.
    Guenneau, S., Poulton, C.G., Movchan, A.B.: A spectral problem for conically propagating elastic waves through an array of cylindrical channels. C. R. Acad. Sci. Paris, Sér. IIb 330, 491–497 (2002)Google Scholar
  7. 7.
    Guenneau, S., Poulton, C.G., Movchan, A.B.: Oblique propagation of electromagnetic and elastic waves for an array of cylindrical fibres. Proc. Roy. Soc. Lond. A 459, 2215–1163 (2003)Google Scholar
  8. 8.
    Knight, J.C. Broeng, J., Birks, T.A., Russell, P. St. J.: Photonic Band Gap Guidance in Optical Fibers. Science 282, 1476–1478 (1998)Google Scholar
  9. 9.
    Kushwaha, M.S., Halevi, P., Dobrzynski, L., Djafari-Rouhani, B.: Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 71, 2022–2025 (1993)Google Scholar
  10. 10.
    Maradudin, A.A., McGurn, A.R.: Out of plane propagation of electromagnetic waves in a two-dimensional periodic dielectric medium. J. Mod. Opt. 41, 275–284 (1994)Google Scholar
  11. 11.
    Martinez-Sala, R., Sancho, J., Sánchez, J.V., Gómez, V., Llinares, J., Meseguer, F.: Sound attenuation by sculpture. Nature 378, 241 (1995)Google Scholar
  12. 12.
    McPhedran, R.C., Dawes, D.H.: Lattice sums for a dynamic scattering problem. J. Electromagn. Waves Appl. 6, 1327–1340 (1992)Google Scholar
  13. 13.
    Movchan, A.B., Movchan, N.V., Poulton, C.G.: Asymptotic models of fields in dilute and densely packed composites. Imperial College Press, London, 2002Google Scholar
  14. 14.
    Movchan, A.B., Nicorovici, N.A., McPhedran, R.C.: Green’s tensors and lattice sums for elastostatics and elastodynamics. Proc. Roy. Soc. Lond. A 453, 643–662 (1995)Google Scholar
  15. 15.
    Nicorovici, N.A., McPhedran, R.C., Botten, L.C.: Photonic band gaps for arrays of perfectly conducting spheres. Phys. Rev. E 52, 1135–1145 (1995)Google Scholar
  16. 16.
    Pendry, J.B.: Photonic band structures. J. Mod. Opt. 41, 209–229 (1994)Google Scholar
  17. 17.
    Poulton, C.G., Movchan, A.B., McPhedran, R.C., Nicorovici, N.A., Antipov, Y.A.: Eigenvalue problems for doubly periodic elastic structures and phononic band gaps. Proc. Roy. Soc. Lond. A 456, 2543–2559 (2000)Google Scholar
  18. 18.
    Lord Rayleigh: On the influence of obstacles arranged in rectangular order upon the properties of a medium. Phil. Mag. 34, 481–502 (1892)Google Scholar
  19. 19.
    Servant, J., Guenneau, S., Movchan, A.B., Poulton, C.G.: Vibrations of a circular cylinder in oblique incidence revisited. In: Asymptotics, singularities and homogenisation in problems of mechanics. Proceedings of the IUTAM Symposium, (Edited by A.B. Movchan), Kluwer Academic Publishers, 2003, pp. 95–104Google Scholar
  20. 20.
    Sigalas, M.M., Economou, E.N.: Elastic and acoustic wave band structure. J. Vib. 158, 377–382 (1992)Google Scholar
  21. 21.
    Watson, G.N.: A treatise on the theory of Bessel functions. Cambridge University Press 1944Google Scholar
  22. 22.
    Zalipaev, V.V., Movchan, A.B., Poulton, C.G., McPhedran, R.C.: Elastic waves and homogenzation in oblique periodic structures. Proc. Roy. Soc. Lond. A 458, 1887–1912 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of LiverpoolLiverpoolUK
  2. 2.Department of PhysicsImperial CollegeLondonUK

Personalised recommendations