This paper presents a time domain algorithm for solving the direct wave scattering problem for a Timoshenko beam. The beam is assumed to be made of a homogeneous material and to be restrained by a viscoelastic suspension of finite length. In the Timoshenko beam, the characteristic wave-fronts propagate with two distinct velocities. This implies that the equation satisfied by the reflection operator contains three separate families of characteristic curves. Furthermore, the equation is both temporally and spatially dependent due to the finite extent of the suspension. The mathematical problem this paper addresses is the construction of an algorithm to solve an associated partial integro-differential equation with three distinct wave-front velocities. The mathematical techniques that are developed are applicable to any set of matrix-valued, functional, first order equations appropriate for reflection kernels. Some numerical examples are presented in order to validate the algorithm.