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

Stabilization of a locally transmission problems of two strongly-weakly coupled wave systems