The onset of buoyancy-driven convection in an initially isothermal, quiescent fluid layer heated from below with time-dependent manner is analyzed by using propagation theory. Here the dimensionless critical time τ_c to mark the onset of convective instability is presented as a function of the Rayleigh number Ra_φ and the Prandtl number Pr. The present stability analysis predicts that τ_c decreases with increasing Pr for a given Ra_φ. The present predictions compare reasonably well with existing experimental results. It is found that in deep-pool systems the deviation of temperature profiles from conduction state occurs starting from a certain time τ=(2~4)τ_c.