We study localization of elastic waves in two-dimensional heterogeneous solids with randomly distributed Lamé coefficients, as well as those with long-range correlations with a power-law correlation function. The Matin-Siggia-Rose method is used, and the one-loop renormalization group (RG) equations for the the coupling constants are derived in the limit of long wavelengths. The various phases of the coupling constants space, which depend on the value ρ, the exponent that characterizes the power-law correlation function, are determined and described. Qualitatively different behaviors emerge for ρ < 1 and ρ > 1. The Gaussian fixed point (FP) is stable (unstable) for ρ < 1 (ρ > 1). For ρ < 1 there is a region of the coupling constants space in which the RG flows are toward the Gaussian FP, implying that the disorder is irrelevant and the waves are delocalized. In the rest of the disorder space the elastic waves are localized. We compare the results with those obtained previously for acoustic wave propagation in the same type of heterogeneous media, and describe the similarities and differences between the two phenomena.
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 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 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 simulation yields, ν ≃ 1.89 ± 0.17, 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.
We investigate the explicit renormalization group for fermionic field theoretic representation of two-dimensional random bond Ising model with long-range correlated disorder. We show that a new fixed point appears by introducing a long-range correlated disorder. Such as the one has been observed in previous works for the bosonic (ϕ 4 ) description. We have calculated the correlation length exponent and the anomalous scaling dimension of fermionic fields at this fixed point. Our results are in agreement with the extended Harris criterion derived by Weinrib and Halperin. *
We revisit the problem of one-dimensional Anderson localization, by providing perturbative expression for Lyapunov exponent of Anderson model with next-nearest-neighbor (nnn) hopping. By comparison with exact numerical results, we discuss the range of validity of the naive perturbation theory. The stability of band center anomaly is examined against the introduction of nnn hopping. New anomalies of Kappus-Wegner type emerge at nonuniversal values of wavelength when hopping to second neighbor is allowed. It is shown that covariances in the first order of perturbation theory, develop singularities at these resonant energies which enable us to locate them.Comment: 5 pages, 3 figure
Extensive computer simulations have been carried out to study propagation of acoustic waves in a two-dimensional disordered fractured porous medium, as a prelude to studying elastic wave propagation in such media. The fracture network is represented by randomly distributed channels of finite width and length, the contrast in the properties of the porous matrix and the fractures is taken into account, and the propagation of the waves is studied over broad ranges of the fracture number density ρ and width b. The most significant result of the study is that, at short distances from the wave source, the waves' amplitude, as well as their energy, decays exponentially with the distance from the source, which is similar to the classical problem of electron localization in disordered solids, whereas the amplitude decays as a stretched exponential function of the distance x that corresponds to sublocalization, exp(-x(α)) with α < 1. Moreover, the exponent α depends on both ρ and b. This is analogous to electron localization in percolation systems at the percolation threshold. Similar results are obtained for the decay of the waves' amplitude with the porosity of the fracture network. Moreover, the amplitude decays faster with distance from the source x in a fractured porous medium than in one without fractures. The mean speed of wave propagation decreases linearly with the fractures' number density.
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