  We consider a coupled wave system with partial Kelvin-Voigt damping in the
interval (-1,1), where one wave is dissipative and the other does not. When the
damping is effective in the whole domain (-1,1) it was proven in H.Portillo
Oquendo and P.Sanez Pacheco, optimal decay for coupled waves with Kelvin-voigt
damping, Applied Mathematics Letters 67 (2017), 16-20. That the energy is
decreasing over the time with a rate equal to $t^{-\frac{1}{2}}$. In this
paper, using the frequency domain method we show the effect of the coupling and
the non smoothness of the damping coefficient on the energy decay. Actually, as
expected we show the lack of exponential stability, that the semigroup loses
speed and it decays polynomially with a slower rate then given in, H.Portillo
Oquendo and P.Sanez Pacheco, optimal decay for coupled waves with Kelvin-voigt
damping, Applied Mathematics Letters 67 (2017), 16-20, down to zero at least as
$t^{-\frac{1}{12}}$.
