This theoretical paper determines the effect of the propagation velocity of heat on the temperature and heat-flux distribution in a semi-infinite body due to a step change in temperature at the surface. The solution yields a maximum but finite heat flux under the conditions of a step change. This a contrary to the infinite value predicted by the error function solution to the Fourier transient conduction equation. In addition, assuming convection is conduction limited, an upper limit for convective heat transfer coefficients is postulated.