One-dimensional Random Medium Research Articles

Influence of the Parameters of a Random Medium on the Characteristics of Transient Reflections

Based on the invariant imbedding method, we study numerically the statistical characteristics of the kernel of the backscattering operator in the case of normal incidence of a plane wave on a one-dimensional random medium with strong fluctuation intensities and various correlation radii of the irregularities. The local reflection coefficient of the medium is modelled by a centered Gaussian process with an exponential correlation function. The first eight one-point cumulants and the correlation functions of delta-pulse reflection are considered and the fluctuation phenomena are analyzed. The transition to the diffusion scattering regime is studded, and the numerical results are compared with the known analytical solutions.

Nonlinear hyperbolic wave propagation in a one-dimensional random medium

We use an asymptotic expansion introduced by Benilov and Pelinovskiĭ to study the propagation of a weakly nonlinear hyperbolic wave pulse through a stationary random medium in one space dimension. We also study the scattering of such a wave by a background scattering wave. The leading-order solution is non-random with respect to a realization-dependent reference frame, as in the linear theory of O’Doherty and Anstey. The wave profile satisfies an inviscid Burgers equation with a nonlocal, lower-order dissipative and dispersive term that describes the effects of double scattering of waves on the pulse. We apply the asymptotic expansion to gas dynamics, nonlinear elasticity, and magnetohydrodynamics.

Comparison of the Maxwell and Boltzmann theory for multilayered dielectric random media.

Based on the transfer matrix theory for electromagnetic fields, we develop a corresponding theory for intensity transport in a one-dimensional random medium. We show the conditions for which the Maxwell equations, the intensity transport theory, and the Boltzmann equation can lead to similar predictions. We generalize the transfer matrix theory to study intensity modulated waves and compare its predictions with those obtained from the Maxwell and Boltzmann equations.

Propagation of wave packets in randomly stratified media.

The propagation of a narrow-band signal radiated by a point source in a randomly layered absorbing medium is studied asymptotically in the weak-scattering limit. It is shown that in a disordered stratified medium that is homogeneous on average, a pulse is channelled along the layers in a narrow strip in the vicinity of the source. The space-time distribution of the pulse energy is calculated. Far from the source, the shape of wave packets is universal and independent of the frequency spectrum of the radiated signal. Strong localization effects manifest themselves also as a low-decaying tail of the pulse and a strong time delay in the direction of stratification. The frequency-momentum correlation function in a one-dimensional random medium is calculated.

Acoustic wave propagation in one-dimensional random media: the wave localization approach

Multiple wave scattering in strongly heterogeneous media is a very complicated phenomenon. Although a statistical approach may yield a considerable simplification of the mathematics, no guarantee exists that the theoretically predicted and the observed quantities coincide. The solution of this problem is to use self averaging quantities only. A multiple scattering theory that makes use of such self averaging quantities is the so called wave localization theory. This theory allows one to study both numerically and theoretically the influence of the presence of heterogeneities on the frequency dependent dispersion and apparent attenuation of a pulse traversing a random medium. I calculate the localization length (penetration depth), the inverse quality factor and both the group and phase velocities for several chaotic media described by different autocorrelation functions. Calculations are limited to 1 D acoustic media with constant density. However, media studied range from very smooth to fractal like and incidence is not limited to be vertical. I then compare the theoretical results with estimates of the same quantities obtained from numerical simulations. The following can be concluded. (1) Theoretical predictions and numerical simulations agree in nearly the whole frequency domain for angles of incidence ≤30° and relative standard deviations of the fluctuations of the incompressibility ≤30 per cent. (2) An inspection of the inverse quality factor confirms that the apparent attenuation is strongest in the domain of Mie scattering except for fractal like media. In such media, no particular ratio of the wavelength to the typical scale length of heterogeneities is preferred since no such typical scale length exists. Hence, the inverse quality factor is constant over a large frequency band. (3) The group and phase velocities obtained agree with the effective medium theory and the Kramers–Krönig relations. That is, both converge to the effective medium velocity and the geometric velocity in the low and high frequency domains respectively. However, for intermediate frequencies, the exact behaviour strongly depends on the type of medium. Differences are related mainly to the number of extrema and Airy phases.

Transient elastic waves propagating in a multi–layered medium subjected to in–plane dynamic loadings. I. Theory

The propagation of elastic transient waves in a multi–layered medium subjected to in–plane loadings is investigated in this study. One of the objectives of this study is to develop an effective analytical method for determining transient full–field solutions in the layered medium. A matrix method is developed by expanding the matrix solution obtained directly from the boundary–value problem in the integral transform domain into a power series of the phase–related reflection and transmission matrix which characterizes the multiple reflections and transmissions of all waves in every layer. The transient response of the multi–layered medium is decomposed into infinite wave groups in which the waves are either reflected by, or transmitted through, the interfaces. The connection between the proposed matrix method and the generalized ray method is established for the layered medium in the transform domain. The matrix representation of the solution enables us to calculate the transient response of the layered medium without tracing the ray path manually. The obtained analytical solution can easily be applied to numerical calculations. The double inverse transform is performed based on Cagniard9s method and the theoretical transient solution for a layered half–space subjected to an in–plane dynamic force will be presented in part II. An experimental set–up that simulates the plane stress condition for a layered half–space was established to obtain the dynamic displacement response. The experimental result agrees well with the theoretical solution. The proposed methodology in this study can be extended to solve more complicated problems such as waves propagating in three–dimensional space.

Mean survival probability in a one-dimensional random medium with traps

We study the mean survival probability (n) at time n on a random one-dimensional chain with perfect absorbers at 0 and L. The transition probabilities gi at the lattice sites i, are independent identically distributed random variables having the distribution p(gi) = 1 for 0 gi1. We prove the asymptotic inequality, C1(n)n2/(logn)L-3 C2 where C1 and C2 are finite positive constants which depend on the lattice size L, but not on n. We confirm this result by simulations for lattice sizes up to L = 17.

Reflected pulses from a refractive random medium at grazing incidence

A problem of acoustic pulse reflection by a one-dimensional refractive random medium is considered in the case of grazing angle incidence. The material parameters of the medium are assumed to vary with a random microscale and a deterministic macroscale. A system of stochastic equations for random scattering variables is derived based upon the random modelling of three separate scales of variations. The statistical properties of the reflected pulses are characterized by an asymptotic diffusion limit theorem of stochastic differential equations with multiple scales. The transport equations governing the limiting stochastic distributions of the random reflection coefficient are obtained in the propagating regime, which leads to the power spectral densities of the reflected pressure and particle velocity fields.

Reflection coefficient and localization length of waves in one-dimensional random media

We develop a powerful method of exactly calculating various transport characteristics of waves in one-dimensional random media with (or without) coherent absorption or amplification. Using the method, we compute the probability densities of the reflectance and of the phase of the reflection coefficient, together with the localization length, of electromagnetic waves in sufficiently long random dielectric media. We find substantial differences between our exact results and the previous results obtained using the random phase approximation (RPA). The probability density of the phase of the reflection coefficient is highly nonuniform when either disorder or absorption (or amplification) is strong. The probability density of the reflectance when the absorption or amplification parameter is large is also quite different from the RPA result. We prove that the probability densities in the amplifying case are related to those in the absorbing case with the same magnitude of the imaginary part of the dielectric permeability by exact dual relationships. From the analysis of the average reflectance that shows a nonmonotonic dependence on the absorption or amplification parameter, we obtain a useful criterion for the applicability of the RPA. In the parameter regime where the RPA is invalid, we find that the exact localization length is substantially larger than the RPA localization length.

Distribution of Wigner delay time from single channel disordered systems

We consider the scattering of an electron from a semi-infinite one-dimensional random medium. The random medium is characterized by force, −∂V ∂L being the basic random variable. We obtain an analytical expression for the stationary delay time (τ) distribution P s ( τ) within a random phase approximation. Our result agrees with earlier analytical expressions, where the random potential is taken to be of different kind, indicating universality of the delay time distribution, i.e. delay time distribution is independent of the nature of disorder.

Uniaxial waves in randomly heterogeneous elastic media

Propagation of uniaxial sound waves in heterogeneous linear elastic one-dimensional media is considered. The scalar wave equation is transformed by the Liouville substitution and the Dyson integral equation is applied for a statistically homogeneous field of heterogeneities that results in an integral spatial representation for the mean wave field. The mean field is analysed in detail for the following three correlation functions: (i) an exponential one; (ii) a mean-square differentiable correlation function with a hidden periodicity; and (iii) a non-differentiable correlation function with a hidden periodicity. Despite the variety of the stochastic properties of the media, the equation for the mean field takes formally the same form for the three cases. For this reason, the general case of the random elastic medium with an arbitrary heterogeneity of small scale is considered and simple closed form expressions for the mean field and attenuation are derived. Applicability of the modelling of extended complex engineering structures by one-dimensional random media is discussed. The overall mechanical parameters of the primary structure and the secondary systems determine average rigidity, average mass density and average wave speed, while the secondary systems determine the attenuation. The latter is shown to depend upon the size and range of the secondary systems.

Estimation of effective conductivity of random heterogeneous media by diffusion processes

A new Monte Carlo (MC) simulation method is developed for estimating the effective conductivity of random heterogeneous media. The method is based on the Itô formula for differentiable functions of diffusion processes, the similarity of the functional form of the generators of these processes and the operators of the steady state diffusion equation, smooth approximations of the conductivity field of multiphase heterogeneous media, and MC simulation. The article establishes the theoretical framework of the proposed method and gives estimates of the effective conductivity for a one-dimensional random heterogeneous medium with two phases.

Evolution of wave packets in quasi-one-dimensional and one-dimensional random media: Diffusion versus localization

We study numerically the evolution of wavepackets in quasi one-dimensional random systems described by a tight-binding Hamiltonian with long-range random interactions. Results are presented for the scaling properties of the width of packets in three time regimes: ballistic, diffusive and localized. Particular attention is given to the fluctuations of packet widths in both the diffusive and localized regime. Scaling properties of the steady-state distribution are also analyzed and compared with theoretical expression borrowed from one-dimensional Anderson theory. Analogies and differences with the kicked rotator model and the one-dimensional localization are discussed.

Diffusion in a one-dimensional random medium and hyperbolic Brownian motion

Classical diffusion in a random medium involves an exponential functional of Brownian motion. This functional also appears in the study of Brownian diffusion on a Riemann surface of constant negative curvature. We analyse in detail this relationship and study various distributions using stochastic calculus and functional integration.

Fragmentation and coalescence in simulations of migration in a one-dimensional random medium

A simple one-dimensional model of fluid migration through a disordered medium is presented. The model is based on invasion percolation and is motivated by two-phase flow experiments in porous media. A uniform pressure gradient g drives fluid clusters through a random medium. The clusters may both coalesce and fragment during migration. The leading fragment advances stepwise. The pressure gradient g is increased continuously. The evolution of the system is characterized by stagnation periods. Simulation results are described and analyzed using probability theory. The fragment length distribution is characterized by a crossover length s ∗ (g) ∼ g − 1 2 and the length of the leading fragment scales as s p ( g) ∼ g −1. The mean fragment length is found to scale with the initial cluster length s 0 and g as 〈s〉 = s 0 1 2 f(gs 0 3 4 ) .

Fine structure of transient waves in a random medium: The half N wave

The propagation of a transient scalar wave in a one-dimensional random medium is considered. The initial disturbance is assumed to have a bilinear form characterized by a relatively short compression phase followed by a relatively long expansion phase. A wave having such a form is referred to here as a half N wave. Based on a general formula derived previously, an expression for the variance, which is a measure of the fine structure, of the half N wave is obtained. In a region encompassing the compression phase and the initial portion of the expansion phase of the wave, the behavior of the fine structure, as determined by this expression, is found to be consistent with observations of actual transient waves, such as sonic booms. Further into the expansion phase, however, discrepancies between theory and observations appear. These findings are described in detail. [Research supported by NASA.]

First-passage time and the fluctuation of the quenched disorder in biased media.

We consider the exit-time problem for a particle diffusing in a one-dimensional random medium with a global nonrandom bias field. It is found that, for small bias, in addition to the usual renormalization of the diffusion coefficient the fluctuations in the hopping rates have the effect of increasing the mean exit-time, independently of the direction of the bias and the initial position of the particle. We also introduce an effective-medium approximation that becomes exact in the limit of small bias.

Propagation and attenuation of a Ricker pulse in one-dimensional randomly heterogeneous media: a numerical study

Numerical simulations are extensively used to study some major features of acoustic wave propagation in one-dimensional random media. Gaussian and exponential stationary media, in which a Ricker pulse is applied, are considered. The properties numerically shown in this paper deal with the irregular progression of the wave front and the changes in its energy distribution with respect to time. Particular emphasis is given to the study of the apparent attenuation factor. Not only are the results shown to be in good agreement with existing analytical studies, but they also provide a more thorough insight into propagation phenomena, especially when the corresponding analytical predictions no longer hold true.

Mean wave propagation in a slab of one-dimensional discrete random medium

A study has been made of the propagation of time harmonic waves through a one-dimensional medium of discrete scatterers randomly positioned over a finite interval L. The random medium is modeled by a Poisson impulse process with density λ. The invariant imbedding procedure is employed to obtain a set of initial value stochastic differential equations for the field inside the medium and the reflection coefficient of the layer. By using the Markov properties of the Poisson impulse process. exact integro-differential equations of the Kolmogorov-Feller type are derived for the probability density function of the reflection coefficient and the field. When the concentration of the scatterers is low, a two variable perturbation method in small λ is used to obtain an approximate solution for the mean field. It is shown that this solution, which varies exponentially with respect to λL, agrees exactly with the mean field obtained by Feldy's approximate method.

Hopping transport in one-dimensional random media: A master equation approach

Abstract Low-temperature drift, diffusion and energy relaxation of non-equilibrium carriers in a one-dimensional hopping system with built-in intrinsic energy disorder are treated analytically employing a master equation approach. The analytical results for energy relaxation are compared with computer simulation and very good agreement is found.