Hyperbolic partial differential equations with one space variable are used to investigate the longitudinal wave propagation through an elastic composite medium. A high order Lagrangian finite element is used to model the wave propagation and the weak-form Galerkin weighted residual method is adopted for solving the governing differential equations, viz., the one-dimensional wave equation which is extended to include damping and strain-rate effects. The numerical solutions are compared to analytical solutions (where they exist) and excellent temporal and spatial correlation is achieved, within 90-95% accuracy. It is found that damping leads to a decrease in peak stresses and strains by up to 11% for 5% of critical damping, even during the direct loading phase. It is shown that the inclusion of strain-rate did not have an effect on strains but led to an increase in stresses by almost 95%. The inclusion of both damping and strain-rate effects together increased stress values by up to 70% compared to the non-viscous cases.