  An understanding of the oscillations of differentially rotating systems is
key to many areas of astrophysics. It is of particular relevance to the
emission of gravitational waves from oscillating neutron stars, which are
expected to possess significant differential rotation immediately after birth
or binary merger. In a previous paper we analysed the normal modes of a simple
system exhibiting differential rotation. In this complementary paper we address
the initial value problem for the same simple model using both analytical
methods and numerical time evolutions. We derive a necessary and sufficient
condition for dynamical shear instability. We discuss the dynamical behaviour
of the continuous spectrum in response to an initial perturbation, and show
that certain singular solutions within the continuous spectrum appear
physically indistinguishable from the discrete modes outside the continuous
spectrum.
