The propagation and scattering of waves on the Timoshenko beam are investigated by using the method of wave propagators. This method is more general than the scattering operators connected to the imbedding and Green function approaches; the wave propagators map the incoming field at an internal position onto the scattering fields at any other internal position of the scattering region. This formalism contains the imbedding method and Green function approach as special cases. Equations for the propagator kernels are derived, as are the conditions for their discontinuities. Symmetry requirements on certain coupling matrices originating from the wave splitting are considered. They are illustrated by two specific examples. The first being an unrestrained beam with a varying cross-section and the other a homogeneous, viscoelastically restrained beam. A numerical algorithm for solving the equations for the propagator kernels is described. The algorithm is tested for the case of a viscoelastically restrained, homogeneous beam. In a limit these results agree with the ones obtained for the reflection kernel by a previously developed algorithm for the imbedding reflection equation.