  The origin of long secondary periods (LSPs) in red giant variables is
unknown. We investigate whether stellar pulsations in red giants can explain
the properties of the LSP variability. VIJHKs light curves obtained by OGLE and
the IRSF/SIRIUS survey in the Small Magellanic Cloud are examined. The sample
of oxygen-rich LSP stars show evidence of a phase lag between the light curves
of optical and near- IR band. The change in radius contributes the bolometric
change roughly half as much as the change in temperature, implying that the
change in effective temperature plays an important role in the luminosity
change associated with the LSPs. We have created numerical models based on the
spherical harmonics to calculate the light amplitudes of dipole mode
variability and have found that the models can roughly reproduce the amplitude
- amplitude relations (e.g. ($\Delta I$, $\Delta H$)). The LSP variability can
be reproduced by the dipole mode oscillations with temperature amplitude of
$\lesssim$ 100 K and $\lesssim$ 150 K for oxygen-rich stars and most carbon
stars, respectively. Radial pulsation models are also examined and can
reproduce the observed colour change of the LSPs. However, there is still an
inconsistency in length between the LSP and periods of radial fundamental mode.
On the other hand, theoretical PL relations of the dipole mode corresponding to
so-called oscillatory convective mode were roughly consistent with observation.
Hence our result suggests that the observations can be consistent with stellar
pulsations corresponding to oscillatory convective modes.
