Skip to main content
SpringerLink
Log in

Analysis of Elastic Band Structures for Oblique Incidence

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

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.

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. 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. Armenàkas, A.E., Gazis, D.C., Herrmann, G.: Free vibrations of circular cylindrical shells. Pergamon Press, New York, 1969

  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. Dowling, J.P., Everitt, H., Yablonovitch, E.: Photonic and sonic band-gap bibliography. http://home.earthlink.net/∼jpdowling/pbgbib.html 2003

  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. 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. 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. 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. 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. 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. 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. McPhedran, R.C., Dawes, D.H.: Lattice sums for a dynamic scattering problem. J. Electromagn. Waves Appl. 6, 1327–1340 (1992)

    Google Scholar 

  13. Movchan, A.B., Movchan, N.V., Poulton, C.G.: Asymptotic models of fields in dilute and densely packed composites. Imperial College Press, London, 2002

  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. 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. Pendry, J.B.: Photonic band structures. J. Mod. Opt. 41, 209–229 (1994)

    Google Scholar 

  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. 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. 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. 95–104

  20. Sigalas, M.M., Economou, E.N.: Elastic and acoustic wave band structure. J. Vib. 158, 377–382 (1992)

    Google Scholar 

  21. Watson, G.N.: A treatise on the theory of Bessel functions. Cambridge University Press 1944

  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 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, UK

    A.B. Movchan

  2. Department of Physics, Imperial College, London, SW7 2AZ, UK

    S. Guenneau

Authors
  1. S. Guenneau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A.B. Movchan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A.B. Movchan.

Additional information

Communicated by G. Milton

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guenneau, S., Movchan, A. Analysis of Elastic Band Structures for Oblique Incidence. Arch. Rational Mech. Anal. 171, 129–150 (2004). https://doi.org/10.1007/s00205-003-0288-z

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-003-0288-z

Keywords

Search

Navigation