  We investigate the spectral properties of the time-dependent linear transport
equation with bounce-back boundary conditions. A fine analysis of the spectrum
of the streaming operator is given and the explicit expression of the strongly
continuous streaming semigroup is derived. Next, making use of a recent result
from M. Sbihi \cite{Sbihi}, we prove, via a compactness argument, that the
essential spectrum of the transport semigroup and that of the streaming
semigroup coincide on all $L^p$-spaces with $1<p<\infty$.
