  We study the asymptotic stability of a planar rarefaction wave (in the $ x_1
$- direction) for the 3-d isentropic Navier-Stokes equations, where the initial
perturbation is periodic on the torus $ \mathbb{T}^3 $ with zero average. To
solve this Cauchy problem in which the initial data is periodic with respect to
only $ x_2 $ and $ x_3 $ but not to $ x_1, $ we construct a suitable ansatz
carrying the oscillations of the solution in the $ x_1 $- direction, but
remaining to be periodic in the transverse $ x_2 $- and $ x_3 $- directions. In
such a way, the difference between the ansatz and the solution can be
integrable on the region $ \mathbb{R}\times\mathbb{T}^2, $ which allows us to
utilize the energy method with the aid of a Gagliardo-Nirenberg type inequality
on $ \mathbb{R}\times\mathbb{T}^2 $ to prove the result.
