  This paper presents statistics of the line-doubling phenomenon in a sample of
81 long-period variable (LPV) stars of various periods, spectral types and
brightness ranges. When correlated with a mask mimicking a K0III spectrum, 54%
of the sample stars clearly showed a double-peaked cross-correlation profile
around maximum light, reflecting double absorption lines. Several pieces of
evidence are presented that point towards the double absorption lines as being
caused by the propagation of a shock wave through the photosphere. The
observation of the Balmer lines appearing in emission around maximum light in
these stars corroborates the presence of a shock wave. The observed velocity
discontinuities, ranging between 10 and 25 km/s, are not correlated with the
brightness ranges. A comparison with the center-of-mass (COM) velocity obtained
from submm CO lines originating in the circumstellar envelope reveals that the
median velocity between the red and blue peaks is blueshifted with respect to
the COM velocity, as expected if the shock moves upwards.
