Exploring Reflection and Transmission Coefficients in Elastic Media

Reflection and transmission (scattering) of plane waves at a planar boundary between two elastic half-spaces are important fundamental processes in seismology. Such plane waves may be compressional (P) or shear (S) in an elastic medium. In this article we apply Mathematica to computing the complex algebraic expressions describing the reflection and transmission amplitudes, phases, and angles of propagation. I also illustrate the energy flux quantities. Brewsters angle, which represents where zero reflected energy occurs, is an important variable for SH waves, which are polarized out of the cross section, and is dependent on both the velocity and density ratio of the two half-spaces. For the P-SV system, the P and SV waves are polarized in the plane of the cross section. Poissons ratio (a function of the P and S velocities) affects the energy flux behavior in the case of incident SV but not incident P. Also for the P-SV system, there are four critical angles that affect the energy flux behavior. I also study the limiting P-SV case where the top medium becomes a vacuum, thus creating a "free surface." For the P-SV system, energy flux of the four scattered waves for any incident wave can be partitioned into top and bottom layer contributions and into total P and total SV contributions, providing further insight into the nature of P-SV scattering. The single-boundary formulation can easily be extended to a stack of layers giving the amplitude and phase at receivers offset along the surface from the source.