In this study, an improved mathematical model is presented to investigate the stress wave propagation in two circular cylinders bonded with a functionally graded adhesive layer. In the proposed model, the spatial derivatives of mechanical properties are included in the governing equations of the wave propagation. Also, the finite-difference method was used for the solution of the governing equations and boundary conditions. The Mori-Tanaka homogenization scheme was employed to evaluate the through-thickness mechanical properties of the adhesive layer. The effects of the spatial derivatives of the local mechanical properties and the through-thickness material composition variation in the adhesive layer were examined in detail. The presence of the material spatial derivatives in the governing equations mitigated the stress and displacement levels as well as axial and radial wave speeds.