  We consider a one dimensional porous media equation with a spatially periodic
reaction. We first construct a periodic traveling wave whose profile tends to a
positive steady state at left infinity and takes zero on the right half line,
with a free boundary satisfying the Darcy's law. Our method is to take the
limit for a sequence of normalized solutions starting at Heaviside type of
initial data. The main difficulty is to give a uniform positive lower bound for
the instantaneous speed of the free boundary.
