  Considering the space-periodic perturbations, we prove the time-asymptotic
stability of the composite wave of a viscous contact wave and two rarefaction
waves for the Cauchy problem of 1-D compressible Navier-Stokes equations in
this paper. This kind of perturbations keep oscillating at the far field and
are not integrable. The key is to construct a suitable ansatz carrying the same
oscillation %eliminating the oscillation of the solution as in
\cite{HuangXuYuan2020,HuangYuan2021}, but due to the degeneration of contact
discontinuity, the construction is more subtle. We find a way to use the same
weight function for different variables and wave patterns, which still ensure
the errors be controllable. Thus, this construction can be applied to contact
discontinuity and composite waves. Finally, by the energy method, we prove that
the Cauchy problem admits a unique global-in-time solution and the composite
wave is still stable under the space-periodic perturbations.
