Localization properties of acoustic waves in the random-dimer media (original) (raw)
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Physical Review B, 2008
Localization of elastic waves in two-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 − Ω c ) −ν , where Ω c 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.
1993
-DIMENSIONAL RANDOM MEDIA WITH LARGE-SCALE INHOMOGENEITIES V. S. Filinov UDC 535.36 Path integrals and a complex Monte Carlo method are used to describe wave propagation and localization in 1D, 2D, and 3D media with small-scale random and large-scale regular inhomogeneities. The probability of wave passage over distance I r --r ' ] averaged over an ensemble of scatterers is calculated. A comparison with analytical results is performed. Unexpected sharp peaks interrupting the exponential decay of the mean square absolute value of Green's function are detected numerically together with the possibility of wave delocalization in the presence of regular inhomogeneities.
Physical Review Letters, 2005
Using the Martin-Siggia-Rose method, we study propagation of acoustic waves in strongly heterogeneous media which are characterized by a broad distribution of the elastic constants. Gaussian-white distributed elastic constants, as well as those with long-range correlations with non-decaying powerlaw correlation functions, are considered. The study is motivated in part by a recent discovery that the elastic moduli of rock at large length scales may be characterized by long-range power-law correlation functions. Depending on the disorder, the renormalization group (RG) flows exhibit a transition to localized regime in any dimension. We have numerically checked the RG results using the transfer-matrix method and direct numerical simulations for one-and two-dimensional systems, respectively.
Localization of nonlinear waves in randomly inhomogeneous media
It is shown that localization of the interface in randomly inhomogeneous bistable media is possible. The equation describing slow dynamics of the interface is obtained. Conditions of the localization are considered. It is shown that the localization may lead to formation of stochastic dissipative structures.
Acta Mechanica, 2009
We describe and discuss the recent progress in the study of propagation and localization of acoustic and elastic waves in heterogeneous media. The heterogeneity is represented by a spatial distribution of the local elastic moduli. Both randomly distributed elastic moduli as well as those with long-range correlations with a nondecaying power-law correlation function, are considered. The motivation for the study is twofold. One is that recent analysis of experimental data for the spatial distribution of the elastic moduli of rock indicated that the distribution is characterized by the type of long-range correlations that we consider in this study. The second motivation for the problem is to understand whether localization of electrons (which, in quantum mechanics, are described by wave functions) has any analogy in the propagation of classical waves in disordered media. The problem is studied by two approaches. One of them is based on developing a dynamic renormalization group (RG) approach to analytical analysis of the governing equations for wave propagation. The RG analysis indicates that, depending on the type of the disorder (correlated vs. uncorrelated), one may have a transition between localized and extended regimes in any spatial dimension. The second approach utilizes numerical simulations of the governing equations in two-and three-dimensional media. The results obtained by the two approaches are in agreement with each other. Using numerical simulations, we also describe how the characteristics of a propagating wave may be used for probing the differences between heterogeneous media with short-and long-range correlations. To do so, we study the evolution of several distinct characteristics of the waves, such as the amplitude of the coherent wave front, its width, the spectral densities, the scalogram (wavelet transformation of the waves' amplitudes at different scales and times), and the dispersion relation. It is demonstrated that such properties have completely different characteristics in uncorrelated and correlated media. Finally, it is shown how wave propagation may be used for establishing a link between the static and dynamical properties of heterogeneous media.
Theory of vibration propagation in disordered media
Wave Motion, 2007
Irregularity can have a significant impact on the vibrational behavior of elastic systems and can effect a broad range of physical properties, ranging from the acoustic scattering cross section of marine structures to the thermal conductivity of semi-conductors. In many instances, the spatial behavior of the modes of the system is fundamentally altered by the irregularity and decays exponentially with distance. This behavior is known as Anderson localization and dynamically generates a new length scale, the localization length n. The modeling of such systems can be quite challenging, with numerical simulations often being misleading owing to finite size effects, and analytical methods being highly specialized and inaccessible to the non-expert in many body theory. We present here an approach which has been highly successful in recent years, the self-consistent diagrammatic theory. Published by Elsevier B.V.
Physical review. E, Statistical, nonlinear, and soft matter physics, 2003
The localization length of classical waves in two-dimensional random media is calculated exactly, and is compared with the theoretical prediction from the current analytic theory. Significant discrepancies are observed. It is also shown that as the frequency varies, critical changes in the localization behavior can occur. However, by a rescaling of parameters the two results tend to match each other for weak scattering. Possible reasons for the discrepancies are discussed.
On elastic waves in a medium with randomly distributed cylinders
International Journal of Mathematics and Mathematical Sciences, 1981
A study is made of the problem of propagation of elastic waves in a medium with a random distribution of cylinders of another material. Neglecting 'back scattering', the scattered field is expanded in a series of 'orders of scattering'. With a further assumption that the n (n > 2) point correlation function of the positions of the cylinders could be factored into two point correlation functions, the average field in the composite medium is found to be resummable, yielding the average velocity of propagation and damping due to 2 scattering. The calculations are presented to the order of (ka) for the scalar case of axial shear waves in the composite material. Several limiting cases of interest are recovered.
Statistical and dynamic localization of plane waves in randomly layered media
Soviet Physics Uspekhi, 1992
This article presents a detailed discussion of the problem of the localization and various methods of describing it on the basis of plane waves multiply scattered in randomly layered media. It is noted that the field of localized waves has a complicated structure, with sharp peaks and extended "dark" regions, where the intensity of the wave is small. It is shown that because of this complicated structure the wave field in a randomly layered medium, the dynamic and statistical characteristics of the wave behave in fundamentally different ways. For example, the statistical moments of the intensity of the wave increase exponentially into the interior of the medium, while the energy of the wave penetrating into randomly inhomogeneous medium can be finite with unity probability. The concept of a majorant curve and of an isoprobability curve, helpful for understanding the phenomenon of localization, are introduced. Also taken into account is the effect of a small regular absorption on the statistical and dynamic properties of the wave, and the localization of space-time pulses in a randomly layered medium is also studied.