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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. The set of observations consists of 315 high-resolution optical spectra collected with the spectrograph ELODIE at the Haute-Provence Observatory, during 27 observing nights at one-month intervals and spanning two years. 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-1, 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. The LPVs clearly exhibiting line-doubling around maximum light with the K0III mask appear to be the most compact ones, the stellar radius being estimated from their effective temperatures (via the spectral type) and luminosities (via the period-luminosity relationship). It is not entirely clear whether or not this segregation between compact and extended LPVs is an artefact of the use of the K0III mask. Warmer masks (F0V and G2V) applied to the most extended and coolest LPVs yield asymmetric cross-correlation functions which suggest that line doubling is occurring in those stars as well. Although a firm conclusion on this point is hampered by the large correlation noise present in the CCFs of cool LPVs obtained with warm masks, the occurrence of line doubling in those stars is confirmed by the double CO $\Delta v = 3$ lines observed around 1.6 μm by Hinkle et al. (1984, ApJS, 56, 1). Moreover, the Hδ line in emission, which is another signature of the presence of shocks, is observed as well in the most extended stars, although with a somewhat narrower profile. This is an indication that the shock is weaker in extended than in compact LPVs, which may also contribute to the difficulty of detecting line doubling in cool, extended LPVs.