  We discuss, in the perturbative regime, the scattering of Gaussian pulses of
odd-parity gravitational radiation off a non-rotating relativistic star and a
Schwarzschild Black Hole. We focus on the excitation of the $w$-modes of the
star as a function of the width $b$ of the pulse and we contrast it with the
outcome of a Schwarzschild Black Hole of the same mass. For sufficiently narrow
values of $b$, the waveforms are dominated by characteristic space-time modes.
On the other hand, for sufficiently large values of $b$ the backscattered
signal is dominated by the tail of the Regge-Wheeler potential, the
quasi-normal modes are not excited and the nature of the central object cannot
be established. We view this work as a useful contribution to the comparison
between perturbative results and forthcoming $w$-mode 3D-nonlinear numerical
simulation.
