We present a linear analysis of a minimal model of moist convection under a variety of atmospheric conditions. The stationary solutions that we analyse include both fully saturated and partially unsaturated atmospheres in both unconditionally and conditionally unstable cases. We find that all of the solutions we consider are linearly unstable via exchange of stability when sufficiently driven. The critical Rayleigh numbers vary by over an order of magnitude between unconditionally unstable and conditionally unstable atmospheres. The unsaturated atmospheres are notable for the presence of linear gravity wave-like oscillations even in unstable conditions. We study their eigenfunction structure and find that the buoyancy and moisture perturbations are anticorrelated in
$z$
, such that regions of negative buoyancy have positive moisture content. We suggest that these features in unsaturated atmospheres may explain the phenomenon of gravity wave shedding by moist convective plumes.
