A higher-order acoustic-displacement based finite element procedure is presented in this paper to investigate one-dimensional sound propagation through a solid and the associated transmission loss. The acoustic system consists of columns of standard air and a solid, with the upstream column of air subjected to a sinusoidal sound source. The longitudinal wave propagation in each medium is modeled using three-node finite elements. At the interfaces between the air and the solid medium, the continuity in acoustic displacements and the force equilibrium conditions are enforced. The Lagrange multipliers method is utilized to assemble the global equations of motion for the acoustic system. Numerical results obtained for various test cases using the procedure described in the paper are in excellent agreement with the analytical solutions and other independent solutions available in the literature.