  We study the behavior of the pion dispersion relation in a pion medium at
finite density and temperature, introducing a chemical potential to describe
the finite pion number density. Such description is particularly important
during the hadronic phase of a relativistic heavy-ion collision, between
chemical and thermal freeze-out, where the pion number changing processes,
driven by the strong interaction, can be considered to be frozen. We make use
of an effective Lagrangian that explicitly respects chiral symmetry through the
enforcement of the chiral Ward identities. The pion dispersion relation is
computed through the computation of the pion self-energy in a non-perturbative
fashion by giving an approximate solution to the Schwinger-Dyson equation for
this self-energy. The dispersion relation is described in terms of a density
and temperature dependent mass and an index of refraction which is also
temperature, density as well as momentum dependent. The index of refraction is
larger than unity for all values of the momentum for finite \mu and T. Given
the strong coupling between \rho vectors and pions, we argue that the
modification of the pion mass due to finite pion density effects has to be
taken into account self-consistently for the description of the in-medium
modifications of \rho's.
