In this study, the propagation of SH waves in a coupled plate consisting of a piezoelectric layer and an elastic layer is analytically investigated. The piezoelectric material is polarized in the z-axis direction and perfectly bonded to an elastic layer. The mathematical model of the SH wave propagation in this plate is based on the type of surface wave solution. Dispersion relations with respect to phase velocity are obtained for electrically open and mechanically free. Numerical examples are provided to illustrate graphically and compare the variations of the phase and group velocities versus the wave number for the different layers. The thickness ratio and the properties of the two layers have a significant effect on the propagation of SH waves. The conclusions are meaningful both theoretically and practically for the design of high-performance surface acoustic wave (SAW) devices.