On the thermally excited stress waves in a rod resting on elastic foundation

A semi-infinite rod x>0 is attached to an elastic foundation reacting to axial displacements of the rod with forces proportional to the displacements. Temperature field T(x) is applied to a portion of the rod at time t=0. The stress disturbance produced by that temperature is divided into two waves which propagate along the rod, one of them being reflected from its end according to the support conditions. The solution is written in an integral form, and it is analyzed in approximate manner using both the analytical methods (integral transforms) and numerical integration.