  The effects of a correlated linear energy/velocity chirp in the electron beam
in the FEL, and how to compensate for its effects by using an appropriate taper
(or reverse-taper) of the undulator magnetic field, is well known. The theory,
as described thus far, ignores velocity dispersion from the chirp in the
undulator, taking the limit of a `small' chirp. In the following, the physics
of compensating for chirp in the beam is revisited, including the effects of
velocity dispersion, or beam compression or decompression, in the undulator. It
is found that the limit of negligible velocity dispersion in the undulator is
different from that previously identified as the small chirp limit, and is more
significant than previously considered. The velocity dispersion requires a
taper which is non-linear to properly compensate for the effects of the
detuning, and also results in a varying peak current (end thus a varying gain
length) over the length of the undulator. The results may be especially
significant for plasma driven FELs and low energy linac driven FEL test
facilities.
