  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 half-space of
multi-species cylindrical inclusions is also formally derived.
