  We construct an approximate solution which describes the gravitational
emission from a naked singularity formed by the gravitational collapse of a
cylindrical thick shell composed of dust. The assumed situation is that the
collapsing speed of the dust is very large. In this situation, the metric
variables are obtained approximately by a kind of linear perturbation analysis
in the background Morgan solution which describes the motion of cylindrical
null dust. The most important problem in this study is what boundary conditions
for metric and matter variables should be imposed at the naked singularity. We
find a boundary condition that all the metric and matter variables are
everywhere finite at least up to the first order approximation. This implies
that the spacetime singularity formed by this high-speed dust collapse is very
similar to that formed by the null dust and thus the gravitational emission
from a naked singularity formed by the cylindrical dust collapse can be gentle.
