Acta Mechanica Solida Sinica

, Volume 21, Issue 2, pp 104–109 | Cite as

Wavelet Method for Calculating the Defect States of Two-Dimensional Phononic Crystals

  • Zhizhong Yan
  • Yuesheng Wang
  • Chuanzeng Zhang


Based on the variational theory, a wavelet-based numerical method is developed to calculate the defect states of acoustic waves in two-dimensional phononic crystals with point and line defects. The supercell technique is applied. By expanding the displacement field and the material constants (mass density and elastic stiffness) in periodic wavelets, the explicit formulations of an eigenvalue problem for the plane harmonic bulk waves in such a phononic structure are derived. The point and line defect states in solid-liquid and solid-solid systems are calculated. Comparisons of the present results with those measured experimentally or those from the plane wave expansion method show that the present method can yield accurate results with faster convergence and less computing time.

Key words

acoustic wave phononic crystal defect state wavelet band structure 


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  1. [1]
    Joannopoulos, J.D., Meade, R.D., and Winn, J.N., Photonic Crystals: Molding the Flow of Light. Princeton: Princeton University Press, 1995.zbMATHGoogle Scholar
  2. [2]
    Bayindir, M., Temelkuran, B. and Ozbay, E., Tight-binding description of the coupled defect modes in three-dimensional photonic crystals. Physics Review Letter, 2000, 84: 2140–2143.CrossRefGoogle Scholar
  3. [3]
    Yablonovitch, E., Gmitter, T.J., Meade, R.D., Rappe, A.M., Brommer, K.D. and Joannopoulos, J.D., Donor and acceptor modes in photonic band-structure. Physics Review Letter, 1991, 67: 3380–3383.CrossRefGoogle Scholar
  4. [4]
    Sigalas, M.M., Elastic wave band gaps and defect states in two-dimensional composites. The Journal of the Acoustical Society of America, 1997, 101: 1256–1261.CrossRefGoogle Scholar
  5. [5]
    Khelif, A., Choujaa, A., Djafari-Rouhani, B., Wilm, W., Ballandras, S. and Laude, V., Trapping and guiding of acoustic waves by defect modes in a full-band-gap ultrasionic crystal. Physical Review B, 2003, 68: 214301.CrossRefGoogle Scholar
  6. [6]
    Kafesaki, M., Sigalas, M.M. and Garcia, N., Frequency modulation in the transmittivity of wave guides in elastic-wave band-gap materials. Physics Review Letter, 2000, 85: 4044–4047.CrossRefGoogle Scholar
  7. [7]
    Khelif, A., Choujaa, A., Benchabane, S., Djafari-Rouhani, B. and Laude, V., Guiding and bending of acoustic waves in highly confined phononic crystal waveguides. Applied Physics Letters, 2004, 84: 4400–4402.CrossRefGoogle Scholar
  8. [8]
    Wu, F.G., Hou, Z.L., Liu, Z.Y. and Liu, Y.Y., Point defect states in two-dimensional phononic crystals. Physics Letters A, 2001, 292: 198–202.CrossRefGoogle Scholar
  9. [9]
    Torres, M., Montero de Espinosa, F.R., Garcia-Pablos, D. and Garcia, N.: Sonic band gaps in finite elastic media: surface states and localization phenomena in linear and point defects. Physics Review Letter, 1999, 82: 3054–3057.CrossRefGoogle Scholar
  10. [10]
    Xiang, J.W., He, Z.J. and Chen, X.F., The construction of wavelet-based truncated conical shell element using B-spline wavelet on the interval. Acta Mechanica Solida Sinica, 2006, 19: 316–326.CrossRefGoogle Scholar
  11. [11]
    Zhang, C. and Zhong, Z., Three-dimensional analysis of functionally graded plate based on the Haar wavelet method. Acta Mechanica Solida Sinica, 2007, 20: 95–102.CrossRefGoogle Scholar
  12. [12]
    Yan, Z.Z. and Wang, Y.S., Wavelet-based method for computing elastic band gaps of one-dimensional phononic crystals. Science in China Series G: Physics, Mechanics, Astronomy, 2007, 50: 622–630.CrossRefGoogle Scholar
  13. [13]
    Yan, Z.Z. and Wang, Y.S., Wavelet-based method for calculating elastic band gaps of two-dimensional phononic crystals. Physics Review B, 2006, 74: 064303.Google Scholar
  14. [14]
    Kittel, C., Introduction to Solid State Physics. John Wiley & Sons, Inc, 7th edition, 1996.Google Scholar
  15. [15]
    David, S.W., Fundamentals of Matrix Computations. New York: John Wiley&Sons, Inc. 2002.zbMATHGoogle Scholar
  16. [16]
    Cohen, A. and Masson, R., Wavelet methods for second-order elliptic problems, preconditioning, and adaptivity. SIAM Journal on Scientific Computing, 1999, 21: 1006–1026.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2008

Authors and Affiliations

  1. 1.Institute of Engineering MechanicsBeijing Jiaotong UniversityBeijingChina
  2. 2.Department of Civil EngineeringUniversity of SiegenSiegenGermany

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