Multiple waves propagate in random particulate materials

© 2019 Society for Industrial and Applied Mathematics Publications. All rights reserved. For over 70 years it has been assumed that scalar wave propagation in (ensembleaveraged) random particulate materials can be characterized by a single effective wavenumber. Here, however, we show that there exist many effective wavenumbers, each contributing to the effective transmitted wave field. Most of these contributions rapidly attenuate away from boundaries, but they make a significant contribution to the reflected and total transmitted field beyond the low-frequency regime. In some cases at least two effective wavenumbers have the same order of attenuation. In these cases a single effective wavenumber does not accurately describe wave propagation even far away from boundaries. We develop an efficient method to calculate all of the contributions to the wave field for the scalar wave equation in two spatial dimensions, and then compare results with numerical finite-difference calculations. This new method is, to the best of the authors' knowledge, the first of its kind to give such accurate predictions across a broad frequency range and for general particle volume fractions

Time reversal of ultrasound in granular media

Time reversal (TR) focusing of ultrasound in granular packings is experimentally investigated. Pulsed elastic waves transmitted from a compressional or shear transducer source are measured by a TR mirror, reversed in time and back-propagated. We find that TR of ballistic coherent waves onto the source position is very robust regardless driving amplitude but provides poor spatial resolution. By contrast, the multiply scattered coda waves offer a finer TR focusing at small amplitude by a lens effect. However, at large amplitude, these TR focusing signals decrease significantly due to the vibration-induced rearrangement of the contact networks, leading to the breakdown of TR invariance. Our observations reveal that granular acoustics is in between particle motion and wave propagation in terms of sensitivity to perturbations. These laboratory experiments are supported by numerical simulations of elastic wave propagation in disordered 2D percolation networks of masses and springs, and should be helpful for source location problems in natural processes.Comment: 15 pages, 6 figure

The average transmitted wave in random particulate materials

Microwave remote sensing is significantly altered when passing through clouds or dense ice. This phenomenon isn't unique to microwaves; for instance, ultrasound is also disrupted when traversing through heterogeneous tissues. Understanding the average transmission in particle-filled environments is central to improve data extraction or even to create materials that can selectively block or absorb certain wave frequencies. Most methods that calculate the average transmitted field assume that it satisfies a wave equation with a complex effective wavenumber. However, recent theoretical work has predicted more than one effective wave propagating even in a material which is statistical isotropic and for scalar waves. In this work we provide the first clear evidence of these predicted multiple effective waves by using high fidelity Monte-Carlo simulations that do not make any statistical assumptions. To achieve this, we also had to fill in a missing link in the theory for particulate materials: we prove that the incident wave does not propagate within the material, which is usually taken as an assumption called the Ewald-Oseen extinction theorem. By proving this we conclude that the extinction length - the distance it takes for the incident wave to be extinct - is equal to the correlation length between the particles

Effective wavenumbers and reflection coefficients for an elastic medium containing random configurations of cylindrical scatterers

Propagation of P and SV waves in an elastic solid containing randomly distributed inclusions in a half-space is investigated. The approach is based on a multiple scattering analysis similar to the one proposed by Fikioris and Waterman for scalar waves. The characteristic equation, the solution of which yields the effective wave numbers of coherent elastic waves, is obtained in an explicit form without the use of any renormalisation methods. Two approximations are considered. First, formulae are derived for the effective wave numbers in a dilute random distribution of identical scatterers. These equations generalize the formula obtained by Linton and Martin for scalar coherent waves. Second, the high frequency approximation is compared with the Waterman and Truell approach derived here for elastic waves. The Fikioris and Waterman approach, in contrast with Waterman and Truell's method, shows that P and SV waves are coupled even at relatively low concentration of scatterers. Simple expressions for the reflected coefficients of P and SV waves incident on the interface of the half space containing randomly distributed inclusions are also derived. These expressions depend on frequency, concentration of scatterers, and the two effective wave numbers of the coherent waves propagating in the elastic multiple scattering medium.Comment: 24 page

Multiple scattering by a collection of randomly located obstacles Part IV: The effect of the pair correlation function

The effect of two different pair correlation functions, used to model multiple scattering in a slab filled with randomly located spherical particles, is investigated. Specifically, the Percus-Yevick approximation is employed and a comparison with the simple hole correction is made. The kernel entries of the hole correction have an analytic solution, which makes the numerical solution of the integral equations possible. The kernel entries of Percus-Yevick approximation are integrated numerically after a subtraction of the slowly converging part in the integrand. Several numerical examples illustrate the effect of the two pair correlation functions, and we also make a comparison with the predictions Bouguer-Beer law gives

Multiple scattering theory for heterogeneous elastic continua with strong property fluctuation: theoretical fundamentals and applications

In this work, the author developed a multiple scattering model for heterogeneous elastic continua with strong property fluctuation and obtained the exact solution to the dispersion equation derived from the Dyson equation under the first-order smoothing approximation. The model establishes accurate quantitative relation between the microstructural properties and the coherent wave propagation parameters and can be used for characterization or inversion of microstructures. As applications of the new model, dispersion and attenuation curves for coherent waves in the Earth lithosphere, the porous and two-phase alloys, and human cortical bone are calculated. Detailed analysis shows the model can capture the major dispersion and attenuation characteristics, such as the longitudinal and transverse wave Q-factors and their ratios, existence of two propagation modes, anomalous negative dispersion, nonlinear attenuation-frequency relation, and even the disappearance of coherent waves. Additionally, it helps gain new insights into a series of longstanding problems, such as the dominant mechanism of seismic attenuation and the existence of the Mohorovicic discontinuity. This work provides a general and accurate theoretical framework for quantitative characterization of microstructures in a broad spectrum of heterogeneous materials and it is anticipated to have vital applications in seismology, ultrasonic nondestructive evaluation and biomedical ultrasound.Comment: 70 pages, 42 figure

Ioffe-Regel criterion of Anderson localization in the model of resonant point scatterers

We establish a phase diagram of a model in which scalar waves are scattered by resonant point scatterers pinned at random positions in the free three-dimensional (3D) space. A transition to Anderson localization takes place in a narrow frequency band near the resonance frequency provided that the number density of scatterers $\rho$ exceeds a critical value $\rho_c \simeq 0.08 k_0^{3}$, where $k_0$ is the wave number in the free space. The localization condition $\rho > \rho_c$ can be rewritten as $k_0 \ell_0 < 1$, where $\ell_0$ is the on-resonance mean free path in the independent-scattering approximation. At mobility edges, the decay of the average amplitude of a monochromatic plane wave is not purely exponential and the growth of its phase is nonlinear with the propagation distance. This makes it impossible to define the mean free path $\ell$ and the effective wave number $k$ in a usual way. If the latter are defined as an effective decay length of the intensity and an effective growth rate of the phase of the average wave field, the Ioffe-Regel parameter $(k\ell)_c$ at the mobility edges can be calculated and takes values from 0.3 to 1.2 depending on $\rho$. Thus, the Ioffe-Regel criterion of localization $k\ell < (k\ell)_c = \mathrm{const} \sim 1$ is valid only qualitatively and cannot be used as a quantitative condition of Anderson localization in 3D.Comment: Revised and extended version. 9 pages, 6 figure