  Effective medium theory aims to describe a complex inhomogeneous material in
terms of a few important macroscopic parameters. To characterise wave
propagation through an inhomogeneous material, the most crucial parameter is
the effective wavenumber. For this reason, there are many published studies on
how to calculate a single effective wavenumber. Here we present a proof that
there does not exist a unique effective wavenumber; instead, there are an
infinite number of such (complex) wavenumbers. We show that in most parameter
regimes only a small number of these effective wavenumbers make a significant
contribution to the wave field. However, to accurately calculate the reflection
and transmission coefficients, a large number of the (highly attenuating)
effective waves is required. For clarity, we present results for scalar
(acoustic) waves for a two-dimensional material filled (over a half space) with
randomly distributed circular cylindrical inclusions. We calculate the
effective medium by ensemble averaging over all possible inhomogeneities. The
proof is based on the application of the Wiener-Hopf technique and makes no
assumption on the wavelength, particle boundary conditions/size, or volume
fraction. This technique provides a simple formula for the reflection
coefficient, which can be explicitly evaluated for monopole scatterers. We
compare results with an alternative numerical matching method.
