  We investigate the time evolution of waves in evanescent media generated by a
source within this medium and observed at some distance away from the location
of the source. The aim is to find a velocity which describes a causal process
and is thus, for a medium with relativistic dispersion, limited by the velocity
of light. The wave function consists of a broad frequency forerunner generated
by the onset of the source, and of a monochromatic front which carries the
oscillation frequency of the source. For a medium with Schr\"{o}dinger-like
dispersion the monochromatic front propagates with a velocity which is in
agreement with the traversal time, and in the relativistic case the velocity of
the fronts is limited by the velocity of light. For sources with a sharp onset,
the forerunners are not attenuated and in magnitude far exceed the
monochromatic front. In contrast, for sources which are frequency-band limited,
the forerunners are also attenuated and become comparable to the monochromatic
front: like in the propagating case, there exists a time at which a broad
frequency forerunner is augmented by a monochromatic wave.
