Introduction to Acoustics of Phononic Crystals. Homogenization at Low Frequencies

Abstract

A short introduction to propagation of sound in phononic crystals is given. Special emphasis is put to the description of the properties of phononic crystals in the long-wavelength limit when periodic inhomogeneous medium can be replaced by a homogeneous one with effective parameters (speed of sound, elastic modulus, and mass density). Two approaches to calculate these effective parameters are given: the plane-wave method and the multiple-scattering method. Metafluid with anisotropic mass density is discussed.

Keywords

Microwave Anisotropy Mercury Attenuation Hexagonal 

Notes

Acknowledgements

JSD acknowledges useful discussions with D. Torrent and the support from the ONR (USA) grant N00014-12-1-0216, and the MINECO (Spain) grants #TEC2010-19751 and #CSD2008-66 (CONSOLIDER program). AAK acknowledges support from the DOE grant # DE-FG02-06ER46312.

References

  1. 1.
    E. Yablonovitch, Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58, 2059–2062 (1987)CrossRefGoogle Scholar
  2. 2.
    S. John, Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett. 58, 2486–2489 (1987)CrossRefGoogle Scholar
  3. 3.
    E. Yablonovitch, T.G. Gmitter, K.M. Leung, Photonic band structure: the face-centered-cubic case employing nonspherical atoms. Phys. Rev. Lett. 67, 2295–2298 (1991)CrossRefGoogle Scholar
  4. 4.
    M. Sigalas, E.N. Economou, Band structure of elastic waves in two dimensional systems. Solid State Commun. 86, 141–143 (1993)CrossRefGoogle Scholar
  5. 5.
    M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 71, 2022–2025 (1993)CrossRefGoogle Scholar
  6. 6.
    R. Martínez-Sala, J. Sancho, J.V. Sánchez, V. Gómez, J. Llinarez, F. Meseguer, Sound attenuation by sculpture. Nature 378, 241 (1995)CrossRefGoogle Scholar
  7. 7.
    M.S. Kushwaha, Stop-bands for periodic metallic rods: sculptures that can filter the noise. Appl. Phys. Lett. 70, 3218–3220 (1997)CrossRefGoogle Scholar
  8. 8.
    M.F. de Espinosa, E. Jiménez, M. Torres, Ultrasonic band gap in a periodic two-dimensional composite. Phys. Rev. Lett. 80, 1208 (1998)CrossRefGoogle Scholar
  9. 9.
    J.V. Sánchez-Pérez, D. Caballero, R. Mártinez-Sala, C. Rubio, J. Sánchez-Dehesa, F. Meseguer, J. Llinares, F. Gálvez, Sound attenuation by a two-dimensional array of rigid cylinders. Phys. Rev. Lett. 80, 5325–5328 (1998)CrossRefGoogle Scholar
  10. 10.
    M.S. Kushwaha, Classical band structure of periodic elastic composites. Int. J. Mod. Phys. B 10, 977–1094 (1996)CrossRefGoogle Scholar
  11. 11.
    R.H. Olsson III, I. El-Kady, Microfabricated phononic crystal devices and applications. Meas Sci Technol. 20, 012002–012015 (2009)CrossRefGoogle Scholar
  12. 12.
    L.D. Landau, E.M. Lifshitz, A.M. Kosevich, L.P. Pitaevskii, Theory of Elasticity (Pergamon Press, Oxford, 1986)Google Scholar
  13. 13.
    C.G. Poulton, A.B. Movchan, R.C. McPhedran, N.A. Nocorovici, Y.A. Antipov, Eigenvalue problems for doubly periodic structures and phononic band gaps. Proc. R. Soc. A 457, 2561–2568 (2000)Google Scholar
  14. 14.
    J.O. Vasseur, P.A. Deymier, B. Chenni, B. Djafari-Rouhani, L. Dobrzynski, D. Prevost, Experimental and theoretical evidence for the existence of absolute acoustic band gaps in two-dimensional solid phononic crystals. Phys. Rev. Lett. 86, 3012–3015 (2001)CrossRefGoogle Scholar
  15. 15.
    S.M. Rytov, Acoustic properties of a thinly laminated medium. Sov. Phys. Acoust. 2, 68 (1956)Google Scholar
  16. 16.
    S. Nemat-Nasser, M. Yamada, Harmonic waves in layered transversely isotropic composites. J. Sound Vibrat. 79, 161 (1981)CrossRefMATHGoogle Scholar
  17. 17.
    R.E. Camley, B. Djafari-Rouhani, L. Dobrzynski, A.A. Maradudin, Transverse elastic waves in periodically layered infinite and semi-infinite media. Phys. Rev. B 27, 7329 (1983)Google Scholar
  18. 18.
    D. Djafari-Rouhani, L. Dobrzynski, Simple excitations in N-layered superlattices. Solid State Commun. 62, 609 (1987)CrossRefGoogle Scholar
  19. 19.
    M. Grimsditch, Effective elastic constants of superlattices. Phys. Rev. B 31, 6818 (1985)CrossRefGoogle Scholar
  20. 20.
    S. Nemat-Nasser, J.R. Willis, A. Srivastava, A.V. Amirkhizi, Homogenization of periodic composites and locally resonant sonic materials. Phys. Rev. B 83, 104103 (2011)CrossRefGoogle Scholar
  21. 21.
    A.A. Krokhin, J. Arriaga, L.N. Gumen, Speed of sound in periodic elastic composites. Phys. Rev. Lett. 91, 264302 (2003)CrossRefGoogle Scholar
  22. 22.
    Q. Ni, J. Cheng, Anisotropy of effective velocity for elastic wave propagation in two-dimensional phononic crystals at low frequencies. Phys. Rev. B 72, 014305 (2005)CrossRefGoogle Scholar
  23. 23.
    A.W. Wood, Textbook of Sound (Macmillan, New York, 1941)Google Scholar
  24. 24.
    M. Kafesaki, R.S. Penciu, E.N. Economou, Air bubbles in water: a strongly multiple scattering medium for acoustic waves. Phys. Rev. Lett. 84, 6050 (2000)CrossRefGoogle Scholar
  25. 25.
    M. Kafesaki, E.N. Economou, Multiple-scattering theory for three-dimensional periodic acoustic composites. Phys. Rev. B 60, 11993 (1999)CrossRefGoogle Scholar
  26. 26.
    A.A. Ruffa, Acoustic wave propagation through periodic bubbly liquids. J. Acoust. Soc. Am. 91, 1 (1992)CrossRefGoogle Scholar
  27. 27.
    D. Bai, J.B. Keller, Sound waves in a periodic medium containing rigid spheres. J. Acoust. Soc. Am. 82, 1436 (1987)CrossRefGoogle Scholar
  28. 28.
    F. Cervera, L. Sanchis, J.V. Sánchez-Pérez, R. Martínez-Sala, C. Rubio, F. Meseguer, C. López, D. Caballero, J. Sánchez-Dehesa, Refractive acoustic devices for airborne sound. Phys. Rev. Lett. 88, 023902 (2002)CrossRefGoogle Scholar
  29. 29.
    B.C. Gupta, Z. Ye, Theoretical analysis of the focusing of acoustic waves by two-dimensional sonic crystals. Phys. Rev. E 67, 036603 (2003)CrossRefGoogle Scholar
  30. 30.
    E. Meyer, E.G. Neumann, Physical and Applied Acoustics (Academic Press, New York, 1972)Google Scholar
  31. 31.
    A. Bensoussan, J.-L. Lions, G. Papanicolau, Asymptotic Analysis for Periodic Structures (North-Holland, Amsterdam, 1978)MATHGoogle Scholar
  32. 32.
    N.S. Bakhvalov, G.P. Panasenko, Homogenization. Averaging Process in Periodic Media. Mathematical Problems in the Mechanics of Composite Materials (Kluwer, New York, 1989)Google Scholar
  33. 33.
    W.S. Ament, Sound propagation in gross mixtures. J. Acoust. Soc. Am. 25, 638–641 (1953)CrossRefGoogle Scholar
  34. 34.
    J.G. Berryman, Long-wavelength propagation in composite elastic media I. Spherical inclusions. J. Acoust. Soc. Am. 68, 1809–1819 (1980)CrossRefMATHGoogle Scholar
  35. 35.
    J. Mei, Z. Liu, W. Wen, P. Sheng, Effective mass density of fluid-solid composites. Phys. Rev. Lett. 96, 024301 (2006)CrossRefGoogle Scholar
  36. 36.
    D. Torrent, A. Hakansson, F. Cervera, J. Sánchez-Dehesa, Homogenization of two-dimensional clusters of rigid rods in air. Phys. Rev. Lett. 96, 204302 (2006)CrossRefGoogle Scholar
  37. 37.
    D. Torrent, J. Sánchez-Dehesa, Effective parameters of clusters of cylinders embedded in a non-viscous fluid or gas. Phys. Rev. B 74, 224305 (2006)CrossRefGoogle Scholar
  38. 38.
    D. Torrent, J. Sánchez-Dehesa, F. Cervera, Evidence of two-dimensional magic clusters in the scattering of sound. Phys. Rev. B (RC) 75, 241404 (2006)CrossRefGoogle Scholar
  39. 39.
    D. Torrent, J. Sánchez-Dehesa, Anisotropic mass density by two-dimensional acoustic metamaterials. New J. Phys. 10, 023004 (2008)CrossRefGoogle Scholar
  40. 40.
    D. Torrent, J. Sánchez-Dehesa, Acoustic metamaterial for new two-dimensional sonic devices. New J. Phys. 9, 323 (2007)CrossRefMATHGoogle Scholar
  41. 41.
    A. Climente, D. Torrent, J. Sánchez-Dehesa, Sound focusing by gradient index sonic lenses. Appl. Phys. Lett. 97, 104103 (2010)CrossRefGoogle Scholar
  42. 42.
    T.P. Martin, M. Nicholas, G. Orris, L.W. Cai, D. Torrent, J. Sánchez-Dehesa, Sonic gradient index lens for aqueous applications. Appl. Phys. Lett. 97, 113503 (2010)CrossRefGoogle Scholar
  43. 43.
    L. Zigoneanu, B.-I. Popa, S.A. Cummer, Design and measurements of a broadband two-dimensional acoustic lens. Phys. Rev. B 84, 024305 (2011)CrossRefGoogle Scholar
  44. 44.
    L. Zigoneanu, B.-I. Popa, A.F. Starr, S.A. Cummer, Design and measurements of a broadband two-dimensional acoustic metamaterial with anisotropic effective mass density. J. Appl. Phys. 109, 054906 (2011)CrossRefGoogle Scholar
  45. 45.
    L.N. Gumen, J. Arriaga, A.A. Krokhin, Metafluid with anisotropic dynamic mass. Low Temp. Phys. 37, 1221–1224 (2011)CrossRefGoogle Scholar
  46. 46.
    J. Li, L. Fok, X. Yin, G. Bartal, X. Zhang, Experimental demonstration of an acoustic magnifying hyperlens. Nat. Mater. 8, 931–934 (2009)CrossRefGoogle Scholar
  47. 47.
    S.A. Cummer, D. Schurig, One path to acoustic cloaking. New J. Phys. 9, 45 (2007)CrossRefGoogle Scholar
  48. 48.
    D. Torrent, J. Sánchez-Dehesa, Radial wave crystals: Radially periodic structures from metamaterials for engineering acoustic or electromagnetic waves. Phys. Rev. Lett. 103, 064301 (2009).Google Scholar
  49. 49.
    D. Torrent, J. Sánchez-Dehesa, Multiple scattering formulation of two-dimensional acoustic and electromagnetic metamaterials. New J. Phys. 13, 093018 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Grupo de Fenómenos Ondulatorios, Departamento de Ingeniería ElectrónicaUniversitat Politècnica de ValènciaValenciaSpain
  2. 2.Department of PhysicsUniversity of North TexasDentonUSA

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