  In this paper, we embark on a captivating exploration of the stabilization of
locally transmitted problems within the realm of two interconnected wave
systems. To begin, we wield the formidable Arendt-Batty criteria\cite{AW} to
affirm the resolute stability of our system. Then, with an artful fusion of a
frequency domain approach and the multiplier method, we unveil the exquisite
phenomenon of exponential stability, a phenomenon that manifests when the waves
of the second system synchronize their propagation speeds. In cases where these
speeds diverge, our investigation reveals a graceful decay of our system's
energy, elegantly characterized by a polynomial decline at a rate of $t^{-1}$.
