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In this paper, we obtain very general decay rate estimates associated to a wave–wave transmission problem subject to a nonlinear damping locally distributed employing arguments firstly introduced in Lasiecka and Tataru (1993) and we shall present explicit decay rate estimates as considered in Alabau-Boussouira (2005) and Cavalcanti et al. (2007). In addition, we implement a precise and efficient code to study the behavior of the transmission problem when k 1 ≠ k 2 and when one has a nonlinear frictional dissipation g ( u t ) . More precisely, we aim to numerically check the general decay rate estimates of the energy associated to the problem established in first part of the paper. [ABSTRACT FROM AUTHOR]