Multiple Waves Propagate in Random Particulate Materials

Abstract

For over 70 years it has been assumed that scalar wave propagation in (ensembleaveraged) random particulate materials can be characterised 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 authors' knowledge, the first of its kind to give such accurate predictions across a broad frequency range and for general particle volume fractions.

Transmission and reflection at the boundary of a random two-component composite

“…Retaining only the monopole term gave an explicit formula for the reflection coefficient, analogous to (4.33) with ρ 0 = ρ 1 . It may be noted that the trial fields (2.9) have a close relationship with the QCA and, in fact, the QCA as employed in [3,4] can be derived as an optimal approximation, given the radial distribution function. The present work offers possible improvement over use of the matrix as comparison, but at the expense of introducing additional polarization fields.…”

“…(2.13) Equations (2.11) have recently been derived in [6]; they were first developed in [8]. 3 Very informally, any particular random medium is one selected at random from a set of possible media. If the set contains only a finite number of possibilities (or is discrete and countably infinite), the probability of any sample medium being characterized by the label α is p α (say), and α p α = 1.…”

How Do Acoustic Materials Work?

“…The relationship between the acoustic impedances does not only have to do with the modification of the amplitude of the incident wave: it is also related to the fraction of the incident energy that will be reflected or transmitted. Most of the acoustic properties of materials can be studied by the application of new laboratory and numerical methods as ultrasonic characterization [4], inverse characterization with basis on the acoustic impedance measurement [5], or ensemble averaged scattering [6].…”

A proof that multiple waves propagate in ensemble-averaged particulate materials

“…Here we prove that there does not exist a unique effective wavenumber but instead there are an infinite number of them. Gower et al [18] first showed that there exist many effective wavenumbers, and provided a technique, the Matching Method , to efficiently calculate the effective wave field. In the present paper and [18], we show that for some parameter regimes, at least two effective wavenumbers are needed to obtain accurate results, when compared with numerical simulations.…”

“…Gower et al [18] first showed that there exist many effective wavenumbers, and provided a technique, the Matching Method , to efficiently calculate the effective wave field. In the present paper and [18], we show that for some parameter regimes, at least two effective wavenumbers are needed to obtain accurate results, when compared with numerical simulations. We also provide examples of how a single effective wave approximation leads to inaccurate results for both transmission and reflection for a half-space filled with particles (figure 1).…”