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.     

Causality and theories of multiple scattering in random media

It is shown that the causal or noncausal behavior of theories of multiple scattering of waves in random media is merely a consequence of the representation of the single scattering process. One can expect causality, and possibly therefore Kramers- Kronig relations, to be satisfied by a theory which either (a) correctly and fully represents the scattering operator of an individual scatterer or (b) though using an on-mass-shell representation such as the Waterman T matrix, excludes scatterer overlap.     

The weak damped oscillation in a random medium

The paper gives a solution of the random differential equation with random coefficient, that is , where K (t) is a random process and is a damp coefficient, it is a little parameter.     

Reflection and transmission by a random medium

From ultrasonic inspections of heterogeneous materials the ultrasonic velocity in a test specimen and the ultrasonic reflection and transmission coefficients can be found over a certain range of frequencies. This paper presents a theoretical analysis of the frequency dependence of the reflection and transmission coefficients caused by random fluctuations of the material parameters.The procedure of using the effective frequency-dependent material parameters of an unbounded random medium to determine the wave propagation in a bounded random medium is not applicable at a distance of the boundary of the order of a correlation length. From this interface effect a contribution to the frequency dependence of the reflection and transmission coefficients arises.A previous consideration assuming the correlation length of the fluctuations to be small compared with the wavelength, is extended to shorter waves. Moreover, the wave propagation in a random half-space whose average material parameter differs from that of the exterior is considered. The consequences of the fluctuations of the elastic constants are of special interest, because they do not go with ω² in the governing wave equation as the density does.The interface effect is also discussed in the static limit of ‘homogeneous deformation’.     

Nonlinear hyperbolic wave propagation in a one-dimensional random medium

We use an asymptotic expansion introduced by Benilov and Pelinovski 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.     

Percolation with a limiting gradient in a medium with random inhomogeneities

The coverage of a medium by percolation and the effective permeability of a medium with stagnant zones are determined. It is shown that effective permeability is a function of external conditions, particularly the average pressure gradient. Three-, two-, and one-dimensional flows are discussed. The theory of overshoots of random functions and fields beyond a prescribed level [1, 2] is used for the investigation. Overshoots of elements of the percolation field in media with random inhomogeneities are studied. Overshoots of energy being dissipated in a volume are discussed in particular; this permits an approximate determination of the coverage of an inhomogeneous porous medium by migration during percolation with a limiting gradient, i.e., in the case of formation of stagnant zones chaotically disseminated in the flow region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 159–165, September–October, 1970.The authors thank V. M. Entov for discussing the article and useful comments.     

Radiation and attenuation of low- and high-frequency waves in a random medium

The wave field radiated by a point source in a weakly inhomogeneous, weakly dissipative, one-dimensional random medium is considered. Approximate expressions for the mean intensity and mean energy flux as a function of propagation range, based on general results obtained previously, are derived for the limiting cases of low and high frequencies. These expressions show that, in both cases, the mean intensity and mean energy flux decrease more rapidly with range than would be the case in the absence of randomness. This more rapid rate of decrease with range can be interpreted as an excess attenuation. These results are similar to those obtained previously for the intermediate-frequency case. The excess-attenuation results are found to be consistent with observations.     

Radiative transfer in a scattering rectangular medium with diffusely reflecting boundaries

This work presents an acurate solution to the integral formulation for radiative transfer in a medium with diffusely reflecting boundaries. The formulation consists of integral equations for the intensity at boundaries and for the source function. Comparisons to existing results for the simplified case with non-reflecting boundaries show excellent agreement. The effects of scattering albedos, boundary reflectivities and optical sizes on the radiative transfer are examined.

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Strahlungsübertragung in einem streuenden rechtwinkligen Medium mit diffus-reflektierenden Grenzen

Zusammenfassung Diese Untersuchung gibt eine genaue Lösung für die integrale Formulierung von Strahlungsübertragung in einem Medium mit diffus-reflektierenden Grenzen. Die mathematische Beschreibung enthält Integralgleichungen für die Strahlungsintensität an den Grenzen des Mediums und für die Quellenfunktion. Vergleiche zu bereits bestehenden Lösungen für den vereinfachenden Fall von nicht-reflektierenden Grenzen zeigen ausgezeichnete Übereinstimmungen. Die Einflüsse von Streu-Albedos, Grenzreflektionen und optischen Größen auf die Strahlungsübertragung werden untersucht.

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Nomenclature b optical dimension iny direction - c optical dimension inz direction - d ₁,d ₂,d ₃ optical distance defined in Eqs. (8), (9) and (14) - I radiation intensity - q _y y component of radiative flux - q _z z component of radiative flux - S source function - S _n generalized exponential integral function defined in Eq. (10) - y,z optical variables Greek symbols emissivity of the bottom - polar angle - reflectivity of the bottom - azimuthal angle - scattering albedo     

Extinction of elastic wave energy due to scattering in a viscoelastic medium

Forward scattering theorem for elastic longitudinal and shear wave scatterings by an arbitrary-shaped three-dimensional object embedded in a viscoelastic medium is derived. It is shown that the formulae for extinction cross-sections of an object in an energy-absorbing medium are formally the same with those of the object in the lossless elastic medium. Numerical calculations are executed for the longitudinal wave scattering in an epoxy matrix by a spherical inclusion with different material properties. The condition of negative extinction is examined with the causality constraint on the viscoelastic medium taken into account. It is found that the negative extinction occurs in the Rayleigh limit when the attenuation of the medium is sufficiently high and, more restrictedly, the wave speed in the object is larger than that in the medium, while it occurs less likely in the high frequency range considered in this paper (0<ka<100).     

Analysis of multiple axisymmetric annular cracks in a piezoelectric medium

An axisymmetric annular electric dislocation is defined. The solution of axisymmetric electric and Volterra climb and glide dislocations in an infinite transversely isotropic piezoelectric domain is obtained by means of Hankel transforms. The distributed dislocation technique is used to construct integral equations for a system of co-axial annular cracks with so-called permeable and impermeable electric boundary conditions on the crack faces where the domain is under axisymmetric electromechanical loading. These equations are solved numerically to obtain dislocation densities on the crack surfaces. The dislocation densities are employed to determine field intensity factors for a system of interacting annular and/or penny-shaped cracks.     

Statistical characteristics of the motion of a fluid particle in a medium with random porosity

In the study of flow of a neutral admixture in a porous medium, it is most often assumed in the stochastic formulation that the porosity is constant and a determinate quantity, and the velocity is a random function [1–4]. The velocity distribution is usually regarded as known. Flow in a porous medium with random porosity has been studied to a far lesser extent. We note [5], which studies the averaged equations obtained within the framework of the correlation approximation. We consider the model problem of one-dimensional motion of a fluid particle (position of the front for flow of a neutral admixture in a porous medium) in a medium with random porosity. For a particular form of random porosity field, expressions are obtained for the one- and two-point densities of the distribution of the position of the particle. A study is made of the dependences of the first four moments and the correlation function of the position of the particle as functions of the time. It is shown that for large values of the time the motion of the particle is asymptotically similar to Brownian motion. It is shown by means of numerical modeling that the results obtained transfer to the case of an arbitrary random porosity field. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 59–65, November–December, 1986.     

Reduction of dimensions in random,elastic soil medium

The paper deals with an application of the plane strain analysis in a stochastic three-dimensional soil medium. In a framework of random elasticity theory, the geostatical state of stresses and the problem of a unit force acting in a statistically homogeneous half-space are considered. Only the modulus of elasticity is considered to be random and is modelled as a three-dimensional (3-D) homogeneous random field. As the result of imposed constrains due to the plane strain assumption the additional body and surface forces are induced. In order to determine them, additional equations must be introduced. The equations in a form of constrain relations are proposed in this paper. These equations are also valid for a case of uniformly distributed external loading.First, the two-dimensional (2-D) problem and its reduction to the uni-axial strain state, for the gravity forces and uniform, unlimited surface loading is considered. Then, it is generalised into a 2-D schematization of the 3-D state. Next, the problem of a unit force acting in a statistically homogeneous half-space is considered. For a 3-D state of stress and strain the resulting stresses are compared with those for a 2-D state. These stresses for the multidimensional state of strain and stress are presented as a sum of two components. The first one reflects plane strain state stresses and is given in a form of a 3-D random field. This term allows for incorporating a spatial, 3-D soil variability into a two-dimensional analysis. The second component can be treated as a correction term and it represents the longitudinal influence of a 3-D analysis.Some numerical results are presented in this paper. The proposed method can be regarded as a framework for further research aiming at application to a variety of geotechnical problems, for which the plane strain state is assumed.