AbstractThe purpose of this paper was to study the propagation of longitudinal waves in non-homogeneous four-parameter viscoelastic rods of arbitrary thickness. The rods were initially supposed to be unstressed and at rest. Apart from a sudden rising traction uniformly applied over the boundary of the opening and parallel to the faces of the plates, which is steadily maintained thereafter, the rods are otherwise free from loading. Methods for treating reflection at the free end of the finite rod and reflection and transmission at an interface between two media in the semi-infinite bi-viscoelastic rod are also presented. Asymptotic techniques are used throughout, and formal asymptotic wave-front expansions of the solution functions are obtained.