  We study the behavior of the pion dispersion relation in a pion medium at
finite density and temperature. We introduce a pion chemical potential to
describe the finite pion number density and argue that such description is
valid during the hadronic phase of a relativistic heavy-ion collision between
chemical and thermal freeze-out. 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 $T$. We conclude by exploring some of the
possible consequences for the propagation of pions through the boundary between
the medium and vacuum.
