In this paper, we will solve for the exact solutions for reflection coefficients of the elastic wave equation in 1D, 2D, and 3D. The velocities in which we are most interested have a transition zone, or a portion of the velocity which is non-constant. As such, we will discuss what occurs at the start and end of the transition zone. In particular, we will find that certain continuity conditions are required. We will also discuss the case when the plane wave is orthogonal to the transition zone as well as the non-normal incidence case in 2D and 3D. This work is a extension of a paper by Hardeman and Lamoureux written in 2016. Finally, using the general solution for the reflection coefficients we find in the 1D and 2D cases, we will compare the results of the exact solution to numerical solutions of the elastic wave equation in 1D and 2D.
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