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