Effective waves for random three-dimensional particulate materials

How do you take a reliable measurement of material whose microstructure is random? When using wave scattering, the answer often to an ensemble average (average over time or space). By averaging we can calculate scattered and effective wavenumber. To date, literature has focused on calculating wavenumber for plate filled with particles. One clear unanswered question was how extend this approach any geometry source. For example, does depend only microstructure, also geometry? In work, demonstrate that wavenumbers not geometry, though beyond long wavelength limit there are multiple wavenumbers. We show from random particulate shape, broad frequency ranges. As sphere