This work deals with the extension of the partition of unity finite element method (PUFEM) "(Comput. Meth. Appl. Mech. Eng. 139 (1996) pp. 289-314; Int. J. Numer. Math. Eng. 40 (1997) 727-758)" to solve wave problems involving propagation, transmission and reflection in layered elastic media. The proposed method consists of applying the plane wave basis decomposition to the elastic wave equation in each layer of the elastic medium and then enforce necessary continuity conditions at the interfaces through the use of Lagrange multipliers. The accuracy and effectiveness of the proposed technique is determined by comparing results for selected problems with known analytical solutions. Complementary results dealing with the modeling of pure Rayleigh waves are also presented where the PUFEM model incorporates information about the pressure and shear waves rather than the Rayleigh wave itself.