  In this report we present a systematic study of the magnonic modes in the
disordered Fe$_{0.5}$Co$_{0.5}$ alloy based on the Heisenberg Hamiltonian using
two complementary approaches. In order to account for substitutional disorder,
on the one hand we directly average the transverse magnetic susceptibility in
real space over different disorder configurations and on the other hand we use
the coherent potential approximation (CPA). While the method of direct
averaging is numerically exact, it is computationally expensive and limited by
the maximal size of the supercell which can be simulated on a computer. On the
contrary the CPA does not suffer from this drawback and yields a cheap
numerical scheme. Therefore, we additionally compare the results of these two
approaches and show that the CPA gives very good results for most of the
magnetic properties, including the magnon energies and the spatial shape of the
eigenmodes. However, it turns out that while reproducing the general trend, the
CPA systematically underestimates the disorder induced damping of the magnons.
This provides evidence that the physics of impurity scattering in this system
is governed by non-local effects missing in the CPA. Finally, we study the real
space eigenmodes of the system, including their spatial shapes, and analyze
their temperature dependence within the random phase approximation.
