The Multiple Scattering of Waves. I. General Theory of Isotropic Scattering by Randomly Distributed Scatterers
TL;DR: In this article, the problem of multiple scattering of scalar waves by a random distribution of isotropic scatterers is considered in detail on the basis of a consistent wave treatment.
Abstract: While the problem of the multiple scattering of particles by a random distribution of scatterers has been treated classically through the use of the Boltzmann integro-differential equation, the corresponding problem of the multiple scattering of waves seems to have received scant attention. All previous treatments have considered the problem in the "geometrical optics" limit, where the rays are regarded as trajectories of particles and the treatment for particles is then applied, so that the interference phenomena in wave scattering are neglected. In this paper the problem of the multiple scattering of scalar waves by a random distribution of isotropic scatterers is considered in detail on the basis of a consistent wave treatment. The introduction of the concept of "randomness" requires averages to be taken over a statistical ensemble of scatterer configurations. Equations are derived for the average value of the wave function, the average value of the square of its absolute value, and the average flux carried by the wave. The second of these quantities satisfies an integral equation which has some similarities to the corresponding equation for particle scattering. The physical interpretation of the results is discussed in some detail and possible generalizations of the theory are outlined.
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TL;DR: A review of the methods for determining the behavior of solids whose properties vary randomly at the microscopic level, with principal attention to systems having composition variation on a well-defined structure (random "alloys") can be found in this paper.
Abstract: We review the methods which have been developed over the past several years to determine the behavior of solids whose properties vary randomly at the microscopic level, with principal attention to systems having composition variation on a well-defined structure (random "alloys"). We begin with a survey of the various elementary excitations and put the dynamics of electrons, phonons, magnons, and excitons into one common descriptive Hamiltonian; we then review the use of double-time thermodynamic Green's functions to determine the experimental properties of systems. Next we discuss these aspects of the problem which derive from the statistical specification of the microscopic parameters; we examine what information can and cannot be obtained from averaged Green's functions. The central portion of the review concerns methods for calculating the averaged Green's function to successively better approximation, including various self-consistent methods, and higher-order cluster effects. The last part of the review presents a comparison of theory with the experimental results of a variety of properties---optical, electronic, magnetic, and neutron scattering. An epilogue calls attention to the similarity between these problems and those of other fields where random material heterogeneity has played an essential role.
1,213 citations
TL;DR: In this article, a criterion for the validity of approximate integral equations describing the various field quantities of interest was obtained by employing a ''configurational averaging'' procedure, and the extinction theorem was obtained and given rise to the forward-amplitude theorem of multiple scattering.
Abstract: Multiple scattering effects due to a random array of obstacles are considered. Employing a ``configurational averaging'' procedure, a criterion is obtained for the validity of approximate integral equations describing the various field quantities of interest. The extinction theorem is obtained and shown to give rise to the forward‐amplitude theorem of multiple scattering. In the limit of vanishing correlations in position, the complex propagation constant κ of the scattering medium is obtained. Under appropriate restrictions, the expression for κ is shown to include both the square‐root law of isotropic scatterers and the additive rule for cross sections valid for sufficiently low densities of anisotropic obstacles. Some specific examples from acoustics and electromagnetic theory then indicate that at least in the simplest cases the results remain valid for physically allowable densities of obstacles.
929 citations
01 Sep 1980
TL;DR: In this paper, the authors derived the differential-integral equation of motion for the mean wave in a solid material containing embedded cavities or inclusions, which consists of a series of terms of ascending powers of the scattering operator, and is here truncated after the third term.
Abstract: The differential-integral equation of motion for the mean wave in a solid material containing embedded cavities or inclusions is derived. It consists of a series of terms of ascending powers of the scattering operator, and is here truncated after the third term. This implies the second-order interactions between scatterers are included but those of the third order are not.The formulae are specialized to the case of thin cracks, either aligned in a single direction or randomly oriented. Expressions for the overall elastic constants are derived for the case of long wavelengths. These expressions are accurate to the second order in the number density of scatterers.
699 citations
TL;DR: In this article, the authors present a rigorous model for the propagation of pressure waves in bubbly liquids and show that the model works well up to volume fractions of 1% to 2% provided that bubble resonances play a negligible role.
Abstract: Recent work has rendered possible the formulation of a rigorous model for the propagation of pressure waves in bubbly liquids. The derivation of this model is reviewed heuristically, and the predictions for the small‐amplitude case are compared with the data sets of several investigators. The data concern the phase speed, attenuation, and transmission coefficient through a layer of bubbly liquid. It is found that the model works very well up to volume fractions of 1%–2% provided that bubble resonances play a negligible role. Such is the case in a mixture of many bubble sizes or, when only one or a few sizes are present, away from the resonant frequency regions for these sizes. In the presence of resonance effects, the accuracy of the model is severely impaired. Possible reasons for the failure of the model in this case are discussed.
649 citations