Geodesic
On $PC$-ansatz
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 33, Tome 308 (2004), pp. 9-22 
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/
@article{ZNSL_2004_308_a1,
     author = {V. M. Babich},
     title = {On $PC$-ansatz},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {9--22},
     year = {2004},
     volume = {308},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_308_a1/}
}
TY  - JOUR
AU  - V. M. Babich
TI  - On $PC$-ansatz
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2004
SP  - 9
EP  - 22
VL  - 308
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2004_308_a1/
LA  - ru
ID  - ZNSL_2004_308_a1
ER  - 
%0 Journal Article
%A V. M. Babich
%T On $PC$-ansatz
%J Zapiski Nauchnykh Seminarov POMI
%D 2004
%P 9-22
%V 308
%U http://geodesic.mathdoc.fr/item/ZNSL_2004_308_a1/
%G ru
%F ZNSL_2004_308_a1

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

[1] A. L. Brodskaya, A. V. Popov, S. A. Khozioskii, “Asimptotika otrazhennoi ot konusa volny v oblasti poluteni”, 6-oi Vsesoyuznyi Simpozium po Difraktsii i Rasprostraneniyu Voln (Moskva–Erevan), 1976, 227–231

[2] F. Olver, Asimptotika i spetsialnye funktsii, Nauka, M., 1960 | MR

[3] Yu. A. Kravtsov, “Ob odnoi modifikatsii metoda geometricheskoi optiki”, Izv. vuzov. Radiofizika, 7:4 (1964), 659–665

[4] N. V. Tsepelev, “O nekotorykh spetsialnykh resheniyakh uravneniya Gelmgoltsa”, Zap. nauchn. semin. LOMI, 51, 1975, 197–202 | Zbl

[5] I. M. Ryzhik, I. S. Gradshtein, Tablitsy integralov, summ, ryadov i proizvedenii, GITTL, M.–L., 1951

[6] V. A. Borovikov, B. E. Kinber, Geometricheskaya teoriya difraktsii, Svyaz, M., 1978 | MR

[7] V. A. Fok, Problemy difraktsii i rasprostraneniya voln, Sovetskoe Radio, M., 1970

[8] V. P. Smyshlyaev, On the diffraction by cone at high frequencies, LOMI Preprint E-9-89. Leningrad, 1989 | MR

[9] N. Bleistein, “Uniform asymptotic expansion of integrals with stationary points near algebraic singularities”, Comm. on Pure and Applied Math., 19:4 (1966), 353–370 | DOI | MR | Zbl

[10] V. A. Borovikov, Difraktsiya na mnogougolnikakh i mnogogrannikakh, Nauka, M., 1966 | MR

[11] V. M. Babich, D. B. Dementev, B. A. Samokish, V. P. Smyshlyaev, “O rasseyanii ploskoi volny vershinoi proizvolnogo konusa. (Singulyarnye napravleniya)”, Zap.nauchn.semin. POMI, 264, 2000, 7–21 | MR | Zbl