In connection with the development of a technique for determining shear mechanical properties of liquids and deformable solids at high frequencies, the basic equations are presented for a layered waveguide geometry. The model is an infinite plate surrounded by an infinite viscoelastic material. The propagation of shear waves in such a system is described and the appropriate equations are developed to relate explicitly the shear mechanical properties of the surrounding material to the velocity and attenuation of the wave in the plate. For the case in which the characteristic shear mechanical impedance of the surrounding material is less than 2% of that of the plate, a simplified equation is obtained. This equation is useful for a wide range of liquids. The equations for the layered waveguide technique are compared to the corresponding equations for other techniques and the similarities are discussed.