  The trapped $w$-modes of stars with a first order phase transition (a density
discontinuity) are computed and the excitation of some of the modes of these
stars by a perturbing shell is investigated. Attention is restricted to odd
parity (``axial'') $w$-modes. With $R$ the radius of the star, $M$ its mass,
$R_{i}$ the radius of the inner core and $M_{i}$ the mass of such core, it is
shown that stars with $R/M\geq 5$ can have several trapped $w$-modes, as long
as $R_{i}/M_{i}<2.6$. Excitation of the least damped $w$-mode is confirmed for
a few models. All of these stars can only exist however, for values of the
ratio between the densities of the two phases, greater than $\sim 46$. We also
show that stars with a phase transition and a given value of $R/M$ can have far
more trapped modes than a homogeneous single density star with the same value
of $R/M$, provided both $R/M$ and $R_{i}/M_{i}$ are smaller than 3. If the
phase transition is very fast, most of the stars with trapped modes are
unstable to radial oscillations. We compute the time of instability, and find
it to be comparable to the damping of the $w$-mode excited in most cases where
$w$-mode excitation is likely. If on the other hand the phase transition is
slow, all the stars are stable to radial oscillations.
