When a crystal layer grows from the cooling boundary, the solid-liquid interface moves and the concentration profile evolves with time. In this Stefan-type solidification problem, convective instabilities of the melt were analysed numerically by using the propagation theory. It was found that the onset time of compositional convection increased with increasing the non-dimensional growth rateλwhich is an important factor governing the instabilities of the melt during solidification. The local critical Rayleigh number R_c having the length scale of the concentration boundary-layer thickness decreased with increasing λ. For small values of λ, R_c, was strongly dependent upon λ, while for large values of λ, its effect was not significant. The present theoretical results were compared favorably with existing experimental results of aqueous ammonium chloride solution.