When an ice layer is melting from below, buoyancy-driven convection often appears in a thermally-unstable water layer. In this study, the onset of convection during time-dependent melting is investigated by using similarly transformed disturbance equations under the propagation theory. The critical Rayleigh numbers based on the water layer thickness are obtained for various conditions and compared with previous experimental and theoretical results. For a slowly melting system, the present prediction is quite close to that under the quasi-static assumption. However, for a rapidly melting system the critical condition deviates from the quasi-static one. With increasing the ratio of the depth of the unstable layer to the whole depth of the liquid layer the system becomes more unstable. But with increasing the phase change rate the system becomes more stable. With decreasing the phase change rate the present results approach the available critical conditions from the quasi-static model. The double cell pattern is predicted at the critical condition and the present results agree reasonably well with existing experimental data. (C) 2008 Elsevier Ltd. All rights reserved.