  We identify the characteristic times of the evolution of a quantum wave
generated by a point source with a sharp onset in an absorbing medium. The
"traversal'' or "B\"uttiker-Landauer'' time (which grows linearly with the
distance to the source) for the Hermitian, non-absorbing case is substituted by
three different characteristic quantities. One of them describes the arrival of
a maximum of the density calculated with respect to position, but the maximum
with respect to time for a given position becomes independent of the distance
to the source and is given by the particle's ``survival time'' in the medium.
This later effect, unlike the Hartman effect, occurs for injection frequencies
under or above the cut-off, and for arbitrarily large distances. A possible
physical realization is proposed by illuminating a two-level atom with a
detuned laser.
