A study of bright Southern Long Period Variables
  In this paper we present radial velocity curves of AGB variables that exhibit various kinds of anomalies: Semiregular variables (SRVs) with typical mira periods, SRVs exceeding the mira 2.5 mag amplitude limit, miras with secondary maxima in their light curves, and a SRV with a long secondary period. The stars with reliable Hipparcos parallaxes from this and from previous studies are plotted in a logP-$M_{K}$-diagram. Our objects nicely follow the logP-$M_{K}$-relations determined for the LMC. This allows the pulsation mode to be identified. While all miras fall on the fundamental mode sequence, the SRVs fall on both the first overtone and fundamental mode sequences. The SRVs on the fundamental mode sequence occur at both high and low luminosities, some of them being more luminous than larger amplitude miras. This demonstrates observationally that some parameter other than luminosity affects the stability of long period variables, probably mass. First overtone pulsators all show velocity amplitudes around 4 km/s. For the fundamental mode pulsators, the velocity amplitude shows a correlation with light amplitude. The two miras R Cen and R Nor, known for their double-peaked light curves, have velocity curves that are quite different. The R Nor velocity curve shows no evidence of the double peaks, meaning that the true pulsation period is the time between alternate minima or maxima. There is slight evidence for a double bump in the R Cen velocity curve. It is suggested that these stars are relatively massive (3-5 Msun). 
T. Lebzelter, K.H. Hinkle, P.R. Wood, R.R. Joyce, F.C. Fekel