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 Schrödinger-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.