This paper is concerned with the dynamic treatment of a transient thermoelastic problem for a semi-infinite medium which is constrained against transverse displacements and is exposed to a uniform time-dependent heating (or cooling) of its entire plane boundary. The stress distribution appropriate to this problem, in the event that the surface temperature is a step-function of time, was previously established by Danilovskaya [1] and by Mura [2]. In the present investigation the accompanying displacements are determined in closed form. In addition, an exact closed solution, in terms of error functions, is obtained for the case in which the time-dependence of the given surface temperature is of the ramp-type. The ensuing field of thermal stress is compared with the corresponding quasi-static stress distribution, with a view toward a quantitative assessment of the accompanying inertia effects as influenced by the rate at which the temperature of the boundary is altered. The results indicate that the conclusions reached in [1] and [2] are in need of essential modification once the assumption of an instantaneous change of the surface temperature is abandoned.