  We consider the Cauchy problem for the damped wave equation under the initial
state that the sum of an initial position and an initial velocity vanishes.
When the initial position is non-zero, non-negative and compactly supported, we
study the large time behavior of the spatial null, critical, maximum and
minimum sets of the solution. The spatial null set becomes a smooth
hyper-surface homeomorphic to a sphere after a large enough time. The spatial
critical set has at least three points after a large enough time. The set of
spatial maximum points escapes from the convex hull of the support of the
initial position. The set of spatial minimum points consists of one point after
a large time, and the unique spatial minimum point converges to the centroid of
the initial position at time infinity.
