We consider a transmission problem where a structurally

damped plate equation is coupled with a damped or undamped wave equation

by transmission conditions. We show that exponential stability holds

in the damped-damped situation and polynomial stability (but no exponential

stability) holds in the damped-undamped case. Additionally, we show

that the solutions first defined by the weak formulation, in fact have higher

Sobolev space regularity.