  The propagation of the scalar ($\sigma$) and vector ($\omega$) mesons in
nuclear matter is studied in detail using the Walecka model over a wide range
of densities and including the effects of a finite $\sigma$ width through the
inclusion of a two-pion loop. We calculate the dispersion relation and spectral
functions of the $\sigma$ and (transverse and longitudinal) $\omega$ mesons,
including the effect of $\sigma$-$\omega$ mixing in matter. It is shown that
the mixing effect is quite important in the propagation of the (longitudinal)
$\omega$ and $\sigma$ mesons above normal nuclear matter density. We find that
there is a two-peak structure in the spectral function of the $\sigma$ channel,
caused by $\sigma$-$\omega$ mixing.
