Geodesic
Summatory and integral equations in periodic problems of the diffraction of elastic waves by defects in layered media
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2008), pp. 76-82.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we consider the diffraction problem for an elastic wave at a periodic set of defects located at the interface of stratified media. We reduce the mentioned problem to a paired summatory functional equation with respect to coefficients of the expansion of the desired wave by quasiperiodic waves (the Floquet waves). Using the method of integral identities, we reduce the paired equation to a regular infinite system of linear equations. One can solve this system by the truncation method. We prove that the integral identity is the necessary and sufficient condition for the solvability of the auxiliary overspecified problem for a system of equations in a half-plane in the elasticity theory. We obtain integral equations of the second kind which are equivalent to the initial diffraction problem.
Keywords: elastic waves, stratified media, overspecified boundary value problems, integral equations.
Mots-clés : diffraction problem, Floquet waves 
@article{IVM_2008_9_a8,
     author = {N. B. Pleschinskii and E. A. Osipov},
     title = {Summatory and integral equations in periodic problems of the diffraction of elastic waves by defects in layered media},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {76--82},
     publisher = {mathdoc},
     number = {9},
     year = {2008},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2008_9_a8/}
}
TY  - JOUR
AU  - N. B. Pleschinskii
AU  - E. A. Osipov
TI  - Summatory and integral equations in periodic problems of the diffraction of elastic waves by defects in layered media
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2008
SP  - 76
EP  - 82
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2008_9_a8/
LA  - ru
ID  - IVM_2008_9_a8
ER  - 
%0 Journal Article
%A N. B. Pleschinskii
%A E. A. Osipov
%T Summatory and integral equations in periodic problems of the diffraction of elastic waves by defects in layered media
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2008
%P 76-82
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2008_9_a8/
%G ru
%F IVM_2008_9_a8
N. B. Pleschinskii; E. A. Osipov. Summatory and integral equations in periodic problems of the diffraction of elastic waves by defects in layered media. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2008), pp. 76-82. http://geodesic.mathdoc.fr/item/IVM_2008_9_a8/

[1] Shestopalov V. P., Metod zadachi Rimana-Gilberta v teorii difraktsii i rasprostraneniya elektromagnitnykh voln, Izd-vo Kharkovsk. un-ta, Kharkov, 1971, 400 pp.

[2] Guz A. N., Kubenko V. D., Cherevko M. A., Difraktsiya uprugikh voln, Nauk. dumka, Kiev, 1978, 308 pp. | MR

[3] Israilov M. Sh., Dinamicheskaya teoriya uprugosti i difraktsiya voln, Izd-vo MGU, M., 1992, 208 pp. | Zbl

[4] Pleschinskii N. B., Tumakov D. N., Metod chastichnykh oblastei dlya skalyarnykh koordinatnykh zadach difraktsii elektromagnitnykh voln v klassakh obobschennykh funktsii, Preprint 2000-1, Kazansk. matem. ob-vo, Kazan, 2000, 50 pp.

[5] Pleschinskaya I. E., Pleschinskii N. B., “Pereopredelennye granichnye zadachi dlya ellipticheskikh uravnenii s chastnymi proizvodnymi i ikh primenenie v teorii difraktsii voln”, Kn. 3, Uchenye zapiski Kazanskogo gos. un-ta, 147, 2005, 4–32

[6] Pleschinskii N. B., Otrazhenie, prelomlenie i difraktsiya dvumernykh uprugikh voln. Metod pereopredelennoi zadachi Koshi, Preprint PMF-04-01, Kazansk. matem. ob-vo, Kazan, 2004, 34 pp.