Geodesic
Wave propagation in, layered, transversally isotropic, elastic media
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 9, Tome 78 (1978), pp. 149-173

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Wave propagation in a transversally isotropic, elastic medium consisting of plane-parallel layers and half spaces is considered. A generalized matrix method is used to derive the dispersion equation of this medium and to find the coefficients of reflection and refraction. This method makes it possible to consider dispersion curves and the coeffients of reflection and refraction in a broader domain than with Haskell's method. The results obtained generalize to layers in which the elastic characteristics vary with depth according to an arbitrary law. For such layers it is possible to find matrices in the form of series which converge rapidly for low and high frequencies. Moreover, a rule is formulated which makes it possible on the basis of a known field in an isotropic medium to find the field in the corresponding transversally isotropic medium. 
@article{ZNSL_1978_78_a11,
     author = {L. A. Molotkov and U. Baimagambetov},
     title = {Wave propagation in, layered, transversally isotropic, elastic media},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {149--173},
     publisher = {mathdoc},
     volume = {78},
     year = {1978},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1978_78_a11/}
}
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L. A. Molotkov; U. Baimagambetov. Wave propagation in, layered, transversally isotropic, elastic media. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 9, Tome 78 (1978), pp. 149-173. http://geodesic.mathdoc.fr/item/ZNSL_1978_78_a11/