  We consider the perturbations of a relativistic star as an initial-value
problem. Having discussed the formulation of the problem (the perturbation
equations and the appropriate boundary conditions at the centre and the surface
of the star) in detail we evolve the equations numerically from several
different sets of initial data. In all the considered cases we find that the
resulting gravitational waves carry the signature of several of the star's
pulsation modes. Typically, the fluid $f$-mode, the first two $p$-modes and the
slowest damped gravitational $w$-mode are present in the signal. This indicates
that the pulsation modes may be an interesting source for detectable
gravitational waves from colliding neutron stars or supernovae. We also survey
the literature and find several indications of mode presence in numerical
simulations of rotating core collapse and coalescing neutron stars. If such
mode-signals can be detected by future gravitational-wave antennae one can hope
to infer detailed information about neutron stars. Since a perturbation
evolution should adequately describe the late time behaviour of a dynamically
excited neutron star, the present work can also be used as a bench-mark test
for future fully nonlinear simulations.
