Geodesic
Application of topological methods to estimate the number of longitudinal elastic waves in crystals
Teoretičeskaâ i matematičeskaâ fizika, Tome 94 (1993) no. 1, pp. 146-152 
Yu. G. Borisovich; B. M. Darinskii; O. V. Kunakovskaya. Application of topological methods to estimate the number of longitudinal elastic waves in crystals. Teoretičeskaâ i matematičeskaâ fizika, Tome 94 (1993) no. 1, pp. 146-152. http://geodesic.mathdoc.fr/item/TMF_1993_94_1_a10/
@article{TMF_1993_94_1_a10,
     author = {Yu. G. Borisovich and B. M. Darinskii and O. V. Kunakovskaya},
     title = {Application of topological methods to estimate the number of longitudinal elastic waves in crystals},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {146--152},
     year = {1993},
     volume = {94},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1993_94_1_a10/}
}
TY  - JOUR
AU  - Yu. G. Borisovich
AU  - B. M. Darinskii
AU  - O. V. Kunakovskaya
TI  - Application of topological methods to estimate the number of longitudinal elastic waves in crystals
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1993
SP  - 146
EP  - 152
VL  - 94
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_1993_94_1_a10/
LA  - ru
ID  - TMF_1993_94_1_a10
ER  - 
%0 Journal Article
%A Yu. G. Borisovich
%A B. M. Darinskii
%A O. V. Kunakovskaya
%T Application of topological methods to estimate the number of longitudinal elastic waves in crystals
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1993
%P 146-152
%V 94
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_1993_94_1_a10/
%G ru
%F TMF_1993_94_1_a10

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

In the problem of estimating the number of longitudinal normals of elastic waves in crystals, several approaches based on the methods of modern topology and associated with the use of topological invariants are proposed. It is shown that for any symmetry class of the crystals there exists not less than three directions along which purely longitudinal waves can propagate. Under the Morse condition on the fundamental function, connections are obtained between the numbers of directions of longitudinal normals corresponding to different types of critical points of this function. Further prospects are discussed.

[1] Fedorov I., Teoriya uprugikh voln v kristallakh, Nauka, M., 1965 | Zbl

[2] Sirotin Yu. I., Shaskolskaya M. P., Osnovy kristallografii, Nauka, M., 1979 | MR

[3] Relei Dzh., Teoriya zvuka, Gostekhizdat, M., 1940

[4] Kolskii G., Volny napryazheniya v tverdykh telakh, IL, M., 1955

[5] Musgrave M., “Elastic waves in anisotropic media”, Progress in Solid Mechanics, vol. II, North-Holland, Amsterdam, 1961, 61–85 | MR

[6] Aleksandrov P. S., Kombinatornaya topologiya, Gostekhizdat, M., 1947 | MR

[7] Borisovich Yu. G., Bliznyakov N. M., Izrailevich Ya. I., Fomenko T. N., Vvedenie v topologiyu, Vyssh. shkola, M., 1980 | MR | Zbl

[8] Dubrovin B. A., Novikov S. P. Fomenko A. T., Sovremennaya geometriya. Metody teorii gomologii, Nauka, M., 1984 | MR

[9] Krasnoselskii M. A., Topologicheskie metody v teorii nelineinykh i integralnykh uravnenii, Gostekhizdat, M., 1956 | MR

[10] Krasnoselskii M. A., Zabreiko P. P., Geometricheskie metody nelineinogo analiza, Nauka, M., 1975 | MR

[11] Borisovich Yu. G., DAN SSSR, 101:2 (1955), 205 | Zbl

[12] Borisovich Yu. G., DAN SSSR, 104:2 (1955), 165 | Zbl

[13] Shvarts A. S., Tr. Mosk. matem. ob-va, 1961, no. 10, 217–272

[14] Shvarts A. S., Tr. Mosk. matem. ob-va, 1962, no. 11, 99–126 | Zbl

[15] Khirsh M., Differentsialnaya topologiya, Mir, M., 1979 | MR | Zbl

[16] Marzantouriez W., Zesz. nauk. Wydz. Mat. fiz. i chem Mat. UG., 1987, no. 6, 55–60

[17] Arnold V. I., Usp. matem. nauk, 34:2 (1979), 3–38 | MR | Zbl

[18] Krasnoselskii M. A., Vektornye polya na ploskosti, Fizmatgiz, M., 1963 | MR

[19] Kunakovskaya O. V., “Nekotorye zamechaniya k indeksam osobennostei 1-form”, Topologicheskie i geometricheskie metody v matem. fizike, Voronezhsk. un-t, Voronezh, 1983, 118–122 | MR

[20] Kunakovskaya O. V., Kraevye indeksy pary sechenii $n$-mernogo vektornogo rassloeniya nad $n$-mernym mnogoobraziem kraem, Dep. v VINITI 29.08.86, No 6317-V86, Gos. ped. in-t, Tambov, 1986, 23 pp.