Abstract: Two approaches for analyzing the viscoelastic stress wave propagations in a bar are provided. The first approach applies the finite difference scheme to the characteristic differential equations derived from the governing wave propagation equations. The second one uses the Laplace transform and the numerical inverse technique to solve the equations directly. It is shown that the characteristic finite difference scheme effectively handles strong discontinuities on a wave front as well as the material’s non-elasticity, while the Laplace transform approach is concise and neat in a mathematical form, and can be used effectively in combination with mathematical software.