Reflection from a multi-species material and its transmitted effective wavenumber

We formally deduce closed-form expressions for the transmitted effective wavenumber of a material comprising multiple types of inclusions or particles (multi-species), dispersed in a uniform background medium. The expressions, derived here for the first time, are valid for moderate volume fractions and without restriction on the frequency. We show that the multi-species effective wavenumber is not a straightforward extension of expressions for a single species. Comparisons are drawn with state-of-the-art models in acoustics by presenting numerical results for a concrete and a water–oil emulsion in two dimensions. The limit of when one species is much smaller than the other is also discussed and we determine the background medium felt by the larger species in this limit. Surprisingly, we show that the answer is not the intuitive result predicted by self-consistent multiple scattering theories. The derivation presented here applies to the scalar wave equation with cylindrical or spherical inclusions, with any distribution of sizes, densities and wave speeds. The reflection coefficient associated with a halfspace of multi-species cylindrical inclusions is also formally derived.


Effective waves for uniformly distributed species
We consider a halfspace x > 0 filled with S types of inclusions (species) that are uniformly distributed. The fields are governed by the scalar wave equation: ∇ 2 u + k 2 u = 0, (in the background material) (1) ∇ 2 u + k 2 j u = 0, (inside the j-th scatterer), The background and species material properties are summarised in Table 1. The goal is to find an effective homogeneous medium with wavenumber k * , where waves propagate, in an ensemble average sense, with the same speed and attenuation as they would in a material filled with scatterers. See Gower (2017) for the code that implements the formulas below.
Below we present the effective wavenumber, for any incident wavenumber and moderate number fraction, when the species are either all cylinders or spheres * . For cylindrical inclusions we also present the reflection of a plane wave from this multi-species material.
Background properties: wavenumber k density ρ sound speed c Specie properties: number density n j density ρ j sound speed c j radius a j total number density n effective wavenumber k * species min. distance a j > a j + a Table 1: Summary of material properties and notation. The index j refers to properties of the j-th species. Note a typical choice for a j is a j = c(a j + a ), where c = 1.01.

Cylindrical species
We consider an incident wave and angle of incidence θ in from the x-axis, exciting a material occupying the halfspace x > 0. Then, assuming low number density n (or low volume fraction πa 2 n ), the effective transmitted wavenumber k * becomes with f • and f •• given by (8). The above reduces to Linton and Martin (2005) equation (81) for a single species in the low frequency limit; This equation (81)  The ensemble-average reflected wave measured at x < 0 is given by where which reduces to Martin (2011) equations (40-41) for a single species, which they show agrees with other known results for small k.
The ensemble-average far-field pattern and multiple-scattering pattern are † , the J m are Bessel functions, the H m are Hankel functions of the first kind and a j > a + a j is some fixed distance. The Z m j describe the type of scatterer: with q = (ρ j c j )/(ρc). For instance, taking the limits q → 0 or q → ∞, recovers Dirichlet or Neumann boundary conditions, respectively.

Effective wavenumber for multi-species spheres
The results here are derived by applying the theory in our paper to the results in Linton and Martin (2006). We omit the details as the result follows by direct analogy.
See Caleap et al. (2012) for details on reflection from a single species, although, to our knowledge, a formula for reflection from a single species valid for moderate number fraction and any wavenumber has not yet been deduced.