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Numerical simulation of the localization of elastic waves in two- and three-dimensional heterogeneous media

Sepehrinia, R ; Sharif University of Technology | 2008

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  1. Type of Document: Article
  2. DOI: 10.1103/PhysRevB.78.024207
  3. Publisher: 2008
  4. Abstract:
  5. Localization of elastic waves in two-dimensional (2D) and three-dimensional (3D) media with random distributions of the Lamé coefficients (the shear and bulk moduli) is studied, using extensive numerical simulations. We compute the frequency dependence of the minimum positive Lyapunov exponent γ (the inverse of the localization length) using the transfer-matrix method, the density of states utilizing the force oscillator method, and the energy-level statistics of the media. The results indicate that all the states may be localized in the 2D media, up to the disorder width and the smallest frequencies considered, although the numerical results also hint at the possibility that there might be a small range of the allowed frequencies over which a mobility edge might exist. In the 3D media, however, most of the states are extended (with only a small part of the spectrum in the upper band tail that contains localized states) even if the Lamé coefficients are randomly distributed. Thus, the 3D heterogeneous media still possess a mobility edge. If both the Lamé coefficients vary spatially in the 3D medium, the localization length Λ follows a power law near the mobility edge, Λ∼ (Ω- Ωc) -ν, where Ωc is the critical frequency. The numerical estimate, ν 1.89±0.17, is significantly larger than the numerical estimate, ν 1.57±0.01, and ν=3/2 (which was recently derived by a semiclassical theory for the 3D Anderson model of electron localization). If the shear modulus is constant but the bulk modulus varies spatially, the plane waves with transverse polarization propagate without any scattering-leading to a band of completely extended states, even in the 2D media. At the mobility edge of such media the localization length follows the same type of power law as Λ but with an exponent νT 1/2 for both 2D and 3D media. © 2008 The American Physical Society
  6. Keywords:
  8. Source: Physical Review B - Condensed Matter and Materials Physics ; Volume 78, Issue 2 , 2008 ; 10980121 (ISSN)
  9. URL: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.78.024207