The z-transform Method and its Applications in Layered Media

The z-transform method is a transform technique developed in the early fifties, applied to analyze discrete systems. It represents the counterpart of the Laplace transform, which applies to continuous systems. In the field of geophysics, z-transform is used to analyze the seismic wave propagation in the earth’s layered media, and it relates with communication or filter theory.

In this project, we use the z-transform method to find closed form solutions of the stress wave propagation in one-dimensional finite elastic layered media. First, we state the definition of the z-transform and its inverse, and show how the z-transform method relates with the Laplace transform. Then, we apply the z-transform method to solve the problem of stress wave propagation in a three-layered elastic strip of Goupillaud-type, subjected to given boundary and initial conditions. We state/solve the problem in two different ways: by solving a difference equation of second order, and by solving a system of three difference equations of first order. In addition we demonstrate how to obtain/validate some of our results using Maple software. Finally, we generalize the results and state a conjecture for the m-layered media.