A theoretical analysis was carried out studying the formation and propagation of a weak shock wave through a gas with axial temperature gradients. The focus of the study was the shock wave - glow discharge plasma interaction problem. In particular, the thermal gradient effects on the shock structure and speed, caused by the discharge were investigated. First, the equation governing the formation and propagation of a weak shock wave through a non-uniform medium in the absence of energy dissipation has been derived. Next, an approximate analytical solution to the one-dimensional wave propagation equation is established. With this, the effects of axial (longitudinal) temperature gradients on shock wave speed and strength have been investigated. The present results indicate that axial temperature gradients affect both the shock wave speed and its strength. The present analysis was used to reanalyze the experiments of [1] and [2] who studied the effect of glow discharge plasma on shock wave parameters.