An attempt to find the analogue of the Schwarzschild criterion in radially pulsating stars is described. The procedure is to transform to pulsating coordinates to make the problem resemble that of ordinary convective stability theory, but with fictitious forces acting. Attention is confined to layers in a star's envelope that are so thin that the pulsation is homologous within them. This also permits the use of the Boussinesq and the plane-parallel approximations. It is found that pulsation is destabilizing and that it promotes monotonic convective instability for large horizontal scales and overstability for small horizontal scales.