A model to validate effective waves in random particulate media: spherical symmetry

Abstract

There has not been a satisfying numerical validation of the theory of effective waves in random particulate materials. Validation has been challenging because the theoretical methods for effective waves have been limited to random particulate media in infinite slabs or half-spaces, which require a very large number of particles to perform accurate numerical simulations. This paper offers a solution by providing, from first principles, a method to calculate effective waves for a sphere filled with particles for a spherically symmetric incident wave. We show that this case can excite exactly the same effective wavenumbers, which are the most important feature to validate for effective waves. To check correctness, we also deduce an integral equation method which does not assume the effective wave solution. Our methods are, in principal, valid for any frequency, particle volume fraction and disordered pair-correlation. With the methods we provide, it is now possible to validate, with a heavier Monte Carlo simulation, the predictions from effective wave theory.

The average transmitted wave in random particulate materials

“…Finally, substituting ( 57) into (48), we conclude the extinction theorem w inc (r) = 0 for r that satisfies (35). That is, there is no term in the average transmitted wave that satisfies the background wave equation.…”

“…The distance b 12 appears in (11) and is sometimes called the correlation length. Our analysis shows that when (35) is true, the transmitted field ( 31) is a sum of effective waves, and the incident wave is no more.…”

“…In papers such as [24] that calculate the average field from first principals, it is not clear that w inc = 0. Here we remove any doubt by proving that when sufficiently inside the material, given by condition (35), we have that w inc := 0 for any incident field, any material region R, any frequency, and all types of particles. Using the results from section 4.1, we collect the terms in equation ( 36) that satisfy the background wave equation to obtain:…”

“…As in this case each frequency has only one mode: the plane-wave. An alternate approach to validate the theory of effective waves is to do Monte-Carlo simulations of particles within a sphere [35], or circular particles within a circle [36]. These have the added benefit of requiring a finite computational domain, though the transmitted fields are more complex.…”

“…Using equations (B.13) and (34) in the integral (B.12), and swapping the order of integration leads to: where we used that B(r; a) is completely contained in R 1 for every r 2 due to (35). There are values for r 1 for which the integral in equation (B.15) is zero.…”