The object of this article is to study the numerical methods for solving two-dimensional transient wave propagation problems. The first method proposed by Clifton involves the characteristic method for two-space variables, and the second method proposed by Mengi involves the use of the combination of. a Fourier transform for one space variable and the characteristic method for one space variable. An example problem is presented which involves transient waves in an elastic layer. generated by out-of-plane shear stress acting on its top surface.