Geodesic
The plane wave refraction on convex and concave obtuse angles in geometric acoustics approximation
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 3, pp. 251-271

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The strict analytical solution to the eikonal equation for the plane wave refracted on convex and concave obtuse angles has been built. It has a shock line for the ray vector field and the first arrival times at the convex angle and a rarefaction cone with diffracted waves at the concave angle. This cone corresponds to the Keller diffraction cone in the geometric diffraction theory. The comparison of the first arrival times, the Hamilton-Jacoby equation times for downward waves and the conservation ray parameter equation times was made. It is shown that these times are equal only for pre-critical incident angles and are different for sub-critical angles. It is shown that the most energetic wave arrival times, which have dominant practical importance, are equal to the times calculated for the conservation ray parameter equation. The numerical algorithm proposed for these times calculation may be used for arbitrary velocity models. 
@article{SJVM_2017_20_3_a2,
     author = {A. N. Kremlev},
     title = {The plane wave refraction on convex and concave obtuse angles in geometric acoustics approximation},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {251--271},
     publisher = {mathdoc},
     volume = {20},
     number = {3},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2017_20_3_a2/}
}
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A. N. Kremlev. The plane wave refraction on convex and concave obtuse angles in geometric acoustics approximation. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 3, pp. 251-271. http://geodesic.mathdoc.fr/item/SJVM_2017_20_3_a2/