Numerical simulations are performed to investigate the linear stability of a two-dimensional incompressible free convection flow induced by a vertical semi-infinite heated flat plate. A small-amplitude local temperature disturbance with a slowly increasing frequency is introduced on the surface near to the leading edge in order to generate disturbance waves within the boundary layer. The aim is to compare the response of the thermal boundary layer with that obtained by selecting discrete disturbance frequencies. In the present study, air is considered to be the working fluid for which the value of the Prandtl number is taken to be . The computational results show that the disturbance decays initially until it reaches a critical distance, which depends on the current frequency of the disturbance. Thereafter the disturbance grows, but the growth rate also depends on the effective frequency of the disturbance. Comparisons with previous work using constant disturbance frequencies are given, and it is shown that the sine-sweep technique is an effective method for analyzing the instability of convectively unstable boundary layers.