  This paper studies the asymptotic stability of shock profiles and rarefaction
waves under space-periodic perturbations for one-dimensional convex scalar
viscous conservation laws. For the shock profile, we show that the solution
approaches the background shock profile with a constant shift in the $
L^\infty(\mathbb{R}) $ norm at exponential rates. The new phenomena contrasting
to the case of localized perturbations is that the constant shift cannot be
determined by the initial excessive mass in general, which indicates that the
periodic oscillations at infinities make contributions to this shift. And the
vanishing viscosity limit for the shift is also shown. The key elements of the
poof consist of the construction of an ansatz which tends to two periodic
solutions as $ x \rightarrow \pm\infty, $ respectively, and the anti-derivative
variable argument, and an elaborate use of the maximum principle. For the
rarefaction wave, we also show the stability in the $ L^\infty(\mathbb{R}) $
norm.
