Plane-wave reflection coefficients for anisotropic media: Practical implementation
Charles P. Ursenbach, Arnim B. Haase
A step-by-step procedure is described for calculating reflection coefficients between media of monoclinic or higher symmetry, which possess a mirror plane parallel to the horizontal reflecting plane between the two media. This is based on a theoretical description available in the literature, and attempts to make these valuable results available to a wider audience.
In general outline, one begins with an incident ray in the upper medium, specified by a polar and azimuthal angle. One form of the Christoffel equations can be used to obtain the slowness vector of the incident wave. Next, using another form of the Christoffel equations, and the constancy of the horizontal slowness components, one obtains a unique vertical slowness for each reflected and transmitted wave, six in all. Using the Christoffel equations once more, but in their original form, one can then obtain the polarization vectors of each reflected and transmitted wave. One then has sufficient information to construct impedance matrices, and these may be combined with matrix algebra to obtain two new matrices, one with all of the reflection coefficients and one with all of the transmission coefficients.
A key aspect of programming a non-trivial code such as this is testing. This paper includes benchmark numbers which can be used to begin rigorous testing of an implementation of the above theory.