Acta Mechanica Solida Sinica

, Volume 21, Issue 6, pp 517–528 | Cite as

Elastic wave localization in two-dimensional phononic crystals with one-dimensional quasi-periodicity and random disorder

  • Ali Chen
  • Yuesheng WangEmail author
  • Guilan Yu
  • Yafang Guo
  • Zhengdao Wang


The band structures of both in-plane and anti-plane elastic waves propagating in two-dimensional ordered and disordered (in one direction) phononic crystals are studied in this paper. The localization of wave propagation due to random disorder is discussed by introducing the concept of the localization factor that is calculated by the plane-wave-based transfer-matrix method. By treating the quasi-periodicity as the deviation from the periodicity in a special way, two kinds of quasi phononic crystal that has quasi-periodicity (Fibonacci sequence) in one direction and translational symmetry in the other direction are considered and the band structures are characterized by using localization factors. The results show that the localization factor is an effective parameter in characterizing the band gaps of two-dimensional perfect, randomly disordered and quasi-periodic phononic crystals. Band structures of the phononic crystals can be tuned by different random disorder or changing quasi-periodic parameters. The quasi phononic crystals exhibit more band gaps with narrower width than the ordered and randomly disordered systems.

Key words

phononic crystal quasi phononic crystal disorder localization factors plane-wave-based transfer-matrix method periodic average structure 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Kushwaha, M.S., Halevi, P., Martinez, G., Dobrzynski, L. and Djafarirouhani, B., Acoustic band structure of periodic elastic composites. Physical Review Letters, 1993, 71: 2022–2025.CrossRefGoogle Scholar
  2. [2]
    Martinez-Sala, R., Sancho, J., Sanchez, J.V., Gomez, V., Llinares, J. and Meseguer, F., Sound-attenuation by sculpture. Nature, 1995, 378: 241–241.CrossRefGoogle Scholar
  3. [3]
    Yablonovitch, E., Inhibited spontaneous emission in solid state physics and electronics. Physical Review Letters, 1987, 58: 2059–2062.CrossRefGoogle Scholar
  4. [4]
    John, S., Strong localization of photons in certain disordered dielectric superlattices. Physical Review Letters, 1987, 58: 2486–2489.CrossRefGoogle Scholar
  5. [5]
    Sigalas, M.M. and Soukoulis, C.M., Elastic-wave propagation through disordered and/or absorptive layered systems. Physical Review B, 1995, 51: 2780–2789.CrossRefGoogle Scholar
  6. [6]
    Vasseur, J.O., Djafari-Rouhani, B., Dobrzynski, L. and Deymier, P.A., Acoustic band gaps in fibre composite materials of boron nitride structure. Journal of Physics: Condense Matter, 1997, 9: 7327–7341.Google Scholar
  7. [7]
    Tanaka, Y. and Tamura, S., Two-dimensional phononic crystals: surface acoustic waves. Physica B, 1999, 263–264: 77–80.CrossRefGoogle Scholar
  8. [8]
    Wu, T.T., Huang, Z.G., and Lin, S., Surface and bulk acoustic waves in two-dimensional phononic crystal consisting of materials with general anisotropy. Physical Review B, 2004, 69: 094301.CrossRefGoogle Scholar
  9. [9]
    Vasseur, J.O., Deymier, P.A., Khelif, A., Lambin, Ph., Djafari-Rouhani, B., Akjouj, A., Dobrzynski, L., Fettouhi, N. and Zemmouri, J., Phononic crystal with low filling fraction and absolute acoustic band gap in the audible frequency range: A theoretical and experimental study. Physical Review E, 2002, 65: 056608.CrossRefGoogle Scholar
  10. [10]
    Wu, F.G., Liu, Z.Y. and Liu, Y.Y., Splitting and tuning characteristics of the point defect modes in two-dimensional phononic crystals. Physical Review E, 2004, 69: 066609.CrossRefGoogle Scholar
  11. [11]
    Khelif, A., Deymier, P.A., Djafari-Rouhani, B., Vasseur, J.O. and Dobrzynski, L., Two-dimensional phononic crystal with tunable narrow pass band: Application to a waveguide with selective frequency. Journal of Applied Physics, 2003, 94: 1308–1311.CrossRefGoogle Scholar
  12. [12]
    Psarobas, I.E. and Sigalas, M.M., Elastic band gaps in a fcc lattice of mercury spheres in aluminum. Physical Review B, 2002, 66: 052302.CrossRefGoogle Scholar
  13. [13]
    Yan, Z.Z. and Wang, Y.S., Wavelet-based method for calculating elastic band gaps of two-dimensional phononic crystals. Physical Review B, 2006, 74: 1.Google Scholar
  14. [14]
    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 and Astronomy, 2007, 50: 622–630.CrossRefGoogle Scholar
  15. [15]
    Yan, Z.Z., Wang, Y.S. and Zhang, C.Z., Wavelet method for calculating the defect states of two-dimensional phononic crystals. Acta Mechanica Solida Sinica. 2008, 21: 105–109.Google Scholar
  16. [16]
    Torres, M., Montero de Espinosa, F.R., García-Pablos, D. and García, N., Sonic band gaps in finite elastic media: surface states and localization phenomena in linear and point defects. Physical Review Letters, 1999, 82: 3054–3057.CrossRefGoogle Scholar
  17. [17]
    Sigalas, M.M., Defect states of acoustic waves in a two-dimensional lattice of solid cylinders. Journal of Applied Mechanics, 1998, 84: 3026–3030.Google Scholar
  18. [18]
    Kafesaki, M., Sigalas, M.M. and García, N., Frequency modulation in the transmittivity of wave guides in elastic-wave band-gap materials. Physical Review Letters, 2000, 85: 4044–4047.CrossRefGoogle Scholar
  19. [19]
    Anderson, P.W., Absence of diffusion in certain random lattices. Physical Review, 1958, 109: 1492–1505.CrossRefGoogle Scholar
  20. [20]
    Li, F.M., Wang, Y.S. and Chen, A.L., Wave localization in randomly disordered periodic piezoelectric rods. Acta Mechanica Solida Sinica, 2006, 19: 50–57.CrossRefGoogle Scholar
  21. [21]
    Zhang, Y.P., Yao, J.Q., Zhang, H.Y., Zheng, Y. and Wang, P., Bandgap extension of disordered 1D ternary photonic crystals. Acta Photonica Sinica, 2005, 34: 1094–1098 (in Chinese).Google Scholar
  22. [22]
    Zhao, Z., Gao, F., Peng, R.W., Cao, L.S., Li, D., Wang, Z., Hao, X.P., Wang, M. and Ferrari, C., Localizationdelocalization transition of photons in one-dimensional random n-mer dielectric systems. Physical Review B, 2007, 75: 165117.CrossRefGoogle Scholar
  23. [23]
    Bendiksen, O.O., Localization phenomena in structural dynamics. Chaos, Solitons and Fractals, 2000, 11: 1621–1660.MathSciNetCrossRefGoogle Scholar
  24. [24]
    Ariaratnam, S.T. and Xie, W.C., Wave localization in randomly disordered nearly periodic long continuous beams. Journal of Sound and Vibration, 1995, 181: 7–22.MathSciNetCrossRefGoogle Scholar
  25. [25]
    Asatryan, A.A., Robinson, P.A., Botten, L.C., McPhedran, R.C., Nicorovici, N.A. and Martijn de Sterke, C., Effects of disorder on wave propagation in two-dimensional photonic crystals. Physical Review E, 1999, 60: 006118.CrossRefGoogle Scholar
  26. [26]
    Vinogradov, A.P. and Merzlikin, A.M., Band theory of light localization in one-dimensional disordered systems. Physical Review E, 2004, 70: 026610.CrossRefGoogle Scholar
  27. [27]
    Zhang, D.Z., Hu, W., Zhang, Y.L., Li, Z.L., Cheng, B.Y. and Yang, G.Z., Experimental verification of light localization for disordered multilayers in the visible-infrared spectrum. Physical Review B, 1994, 50: 9810–9814.CrossRefGoogle Scholar
  28. [28]
    Chen, A.L. and Wang, Y.S., Study on band gaps of elastic waves propagating in one dimensional disordered phononic crystals. Physica B, 2007, 392: 369–378.CrossRefGoogle Scholar
  29. [29]
    Chen, A.L., Wang, Y.S., Guo, Y.F. and Wang, Z.D., Band structures of Fibonacci phononic quasicrystals. Solid State Communications, 2008, 145: 103–108.CrossRefGoogle Scholar
  30. [30]
    Xie, W.C., Chaos, buckling mode localization in nonhomogeneous beams on elastic foundations. Chaos, Solitons and Fractals, 1997, 8: 411–431.MathSciNetCrossRefGoogle Scholar
  31. [31]
    Li, F.M. and Wang, Y.S., Study on wave localization in disordered periodic layered piezoelectric composite structures. International Journal of Solids and Structures, 2005, 42: 6457–6474.CrossRefGoogle Scholar
  32. [32]
    Li, F.M., Wang, Y.S., Hu, C. and Huang, W.H., Wave localization in randomly disordered periodic layered piezoelectric structures. Acta Mechanica Sinica, 2006, 22: 559–567.MathSciNetCrossRefGoogle Scholar
  33. [33]
    Aynaou, H., Boudouti, E.H.EI., Djafari-Rouhani, B., Akjouj, A. and Velasco, V.R., Propagation and localization of acoustic waves in Fibonacci phononic circuits. Journal of Physics: Condensed Matter, 2005, 17: 4245–4262.Google Scholar
  34. [34]
    Peng, R.W., Wang, M., Hu, A., Jiang, S.S., Jin, G.J. and Feng, D., Characterization of the diffraction spectra of one-dimensional k-component Fibonacci structures. Physical Review B, 1995, 52: 13310–13316.CrossRefGoogle Scholar
  35. [35]
    King, P.D.C. and Cox, T.J., Acoustic band gaps in periodically and quasiperiodically modulated waveguides. Journal of Applied Physics, 2007, 102: 014902.CrossRefGoogle Scholar
  36. [36]
    Steurer, W. and Sutter-Widmer, D., Photonic and phononic quasicrystals. Journal of Physics D: Applied Physics, 2007, 40: R229–R247.CrossRefGoogle Scholar
  37. [37]
    Velasco, V.R., Perez-Alvarez, R. and Garcia-Moliner, F., Some properties of the elastic waves in quasiregular heterostructures. Journal of Physics: Condensed Matter, 2002, 14: 5933–5957.Google Scholar
  38. [38]
    Sesion Jr, P.D., Albuquerque, E.L., Chesman, C. and Freire, V.N., Acoustic phonon transmission spectra in piezoelectric AlN/GaN Fibonacci phononic crystals. The European Physical Journal B, 2007, 58: 379–387.CrossRefGoogle Scholar
  39. [39]
    Li, Z.Y. and Lin, L.L., Photonic band structures solved by a plane-wave-based transfer-matrix method. Physical Review E, 2003, 67: 046607.CrossRefGoogle Scholar
  40. [40]
    Hou, Z.L., Kuang, W.M. and Liu, Y.Y., Transmission property anslysis of two-dimensional phononic crystal. Physics Letters A, 2004, 333: 172–180.CrossRefGoogle Scholar
  41. [41]
    Wolf, A., Swift, J.B., Swinney, H.L. and Vastano, J.A., Determining Lyapunov exponents from a time series. Physica D, 1985, 16: 285–317.MathSciNetCrossRefGoogle Scholar
  42. [42]
    Castanier, M.P. and Pierre, C., Lyapunov exponents and localization phenomena in multi-coupled nearly periodic systems. Journal of Sound and Vibration, 1995, 183: 493–515.MathSciNetCrossRefGoogle Scholar
  43. [43]
    Xie, W.C., Buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners. Computers and Structures, 1998, 67: 175–189.CrossRefGoogle Scholar
  44. [44]
    Sutter-Widmer, D., Deloudi, S. and Steurer, W., Periodic average structures in phononic quasicrystals. Philosophical Magazine, 2007, 87: 3095–3102.CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Ali Chen
    • 1
  • Yuesheng Wang
    • 1
    Email author
  • Guilan Yu
    • 2
  • Yafang Guo
    • 1
  • Zhengdao Wang
    • 1
  1. 1.Institute of Engineering MechanicsBeijing Jiaotong UniversityBeijingChina
  2. 2.School of Civil EngineeringBeijing Jiaotong UniversityBeijingChina

Personalised recommendations