Geodesic
On $PC$-ansatz
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 33, Tome 308 (2004), pp. 9-22

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The subject of the paper is detailed consideration of known from seventies ansatz: $$ e^{\operatorname{i}kl(x)}[AD_p(\sqrt{k}e^{-\frac\pi4}m(x))+ k^{-\frac12}e^{\frac\pi4}BD_p^\prime(\sqrt{k}e^{-\frac\pi4}m(x))], $$ where $A$ and $B$ are series: $$ A=\sum_{s=0}^\infty\frac{A_s(x)}{(\operatorname{i}k)^s};\quad B=\sum_{s=0}^\infty\frac{B_s(x)}{(\operatorname{i}k)^s}. $$ Here $D_p$ are parabolic cylinder functions. Analytical expressions in the first approximation for wave field in the penumbra of the wave reflected by impedance or transparent cone were obtained. 
@article{ZNSL_2004_308_a1,
     author = {V. M. Babich},
     title = {On $PC$-ansatz},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {9--22},
     publisher = {mathdoc},
     volume = {308},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_308_a1/}
}
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V. M. Babich. On $PC$-ansatz. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 33, Tome 308 (2004), pp. 9-22. http://geodesic.mathdoc.fr/item/ZNSL_2004_308_a1/