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The trapped $w$-modes of stars with a first order phase transition (a densitydiscontinuity) are computed and the excitation of some of the modes of thesestars by a perturbing shell is investigated. Attention is restricted to oddparity (``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 isshown that stars with $R/M\geq 5$ can have several trapped $w$-modes, as longas $R_{i}/M_{i}<2.6$. Excitation of the least damped $w$-mode is confirmed fora few models. All of these stars can only exist however, for values of theratio between the densities of the two phases, greater than $\sim 46$. We alsoshow that stars with a phase transition and a given value of $R/M$ can have farmore trapped modes than a homogeneous single density star with the same valueof $R/M$, provided both $R/M$ and $R_{i}/M_{i}$ are smaller than 3. If thephase transition is very fast, most of the stars with trapped modes areunstable to radial oscillations. We compute the time of instability, and findit 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 isslow, all the stars are stable to radial oscillations.Comment: To appear in Physical Review

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

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Trapped and excitedwmodes of stars with a phase transition andR>~5M