  The effect of Alfv\'en waves on the thermal instability of the Interstellar
Medium (ISM) is investigated both analytically and numerically. A stability
analysis of a finite amplitude circularly polarized Alfv\'en wave propagating
parallel to an ambient magnetic field in a thermally unstable gas at thermal
equilibrium is performed, leading to a dispersion relation which depends on 3
parameters, namely the square ratio of the sonic and Alfv\'en velocities
($\beta$), the wave amplitude and the ratio between the wave temporal period
and the cooling time. Depending on the values of these 3 parameters, the
Alfv\'en waves can stabilize the large-scale perturbations, destabilize those
whose wavelength is a few times the Alfv\'en wavelength $\lambda_{AW}$ or leave
the growth rate of the short scales unchanged. In order to investigate the
non-linear regime, two different numerical experiments are performed in a slab
geometry. (TBC)
