Scattering of plane longitudinal elastic waves by a slender cavity of revolution. The case of axial incidence

G. V. Zhdanova

Sbornik. Mathematics, Tome 49 (1984) no. 2, pp. 305-323

Résumé

The system of equations of elasticity theory $$ A(\partial_x)\overline u+\omega^2\rho\overline u=0,\quad x\in D_\varepsilon;\qquad T\overline u=0,\quad x\in S_\varepsilon, $$ is solved in a homogeneous isotropic medium. Here $A(\partial_x)$ is a matrix differential operator, $T$ is the stress operator, $x\in R^3$, $\varepsilon>0$ is a small parameter, $S_\varepsilon$ is a smooth bounded closed surface of revolution, and $D_\varepsilon$ is the exterior of $S_\varepsilon$. The case where $$ \overline u(x)=A_le^{ik_lz}\overline i_z+\overline u^{(s)}(x),\qquad A_l=\mathrm{const}, $$ is considered. The reflected wave $\overline u^{(s)}(x)$ satisfies the radiation condition. The asymptotics of $\overline u^{(s)}(x)$ is constructed with $O(\varepsilon^{(m)})$ precision as $\varepsilon\to+0$, where $m>0$ is arbitrary. The formulas obtained are useful everywhere near $S_\varepsilon$, including its endpoints, and at a distance. The asymptotics of the scattering amplitudes of the reflected waves is found. Figures: 1. Bibliography: 16 titles.

Bibliographie

[1] Landau L. D., Lifshits E. M., Teoriya uprugosti, Nauka, M., 1965 | MR

[2] Fedoryuk M. V., “Asimptotika resheniya zadachi Dirikhle dlya uravnenii Laplasa i Gelmgoltsa vo vneshnosti tonkogo tsilindra”, Izv. AN SSSR. Seriya matem., 45 (1981), 167–186 | MR | Zbl

[3] Fedoryuk M. V., “Zadacha Dirikhle dlya operatora Laplasa vo vneshnosti tonkogo tela vrascheniya”, Teoriya kubaturnykh formul i prilozheniya funktsionalnogo analiza k zadacham matematicheskoi fiziki (Trudy seminara S. L. Soboleva, 1980, No 1), Novosibirsk, 1980, 113–131 | MR | Zbl

[4] Geer J., “The scattering of a scalar wave by a slender body of revolution”, SIAM, J. Appl. Math., 34:2 (1978), 348–370 | DOI | MR | Zbl

[5] Van-Deik M., Metody vozmuschenii v mekhanike zhidkosti, Mir, M., 1967

[6] Koul Dzh., Metody vozmuschenii v prikladnoi matematike, Mir, M., 1972 | MR

[7] Loitsyanskii L. G., Mekhanika zhidkosti i gaza, Nauka, M., 1978 | MR

[8] Ilin A. M., “Kraevaya zadacha dlya ellipticheskogo uravneniya vtorogo poryadka v oblasti s uzkoi schelyu. I. Dvumernyi sluchai”, Matem. sb., 99 (141) (1976), 514–537 | MR

[9] Ilin A. M., “Kraevaya zadacha dlya ellipticheskogo uravneniya vtorogo poryadka v oblasti s uzkoi schelyu. II. Oblast s malym otverstiem”, Matem. sb., 103 (145) (1977), 265–284 | MR

[10] Geer J., “Electromagnetic scattering by a slender body of revolution: axially incident plane wave”, SIAM, J. Appl. Math., 38:1 (1980), 93–102 | DOI | MR | Zbl

[11] Kupradze V. D., Trekhmernye zadachi matematicheskoi teorii uprugosti i termouprugosti, Nauka, M., 1976 | MR

[12] Uitteker E. T., Vatson Dzh. N., Kurs sovremennogo analiza, 2, Fizmatgiz, M., 1963

[13] Vladimirov V. S., Uravneniya matematicheskoi fiziki, Nauka, M., 1967 | MR

[14] Fedoryuk M. V., Metod perevala, Nauka, M., 1977 | MR

[15] Mazya V. G., Nazarov S. A., Plamenevskii B. A., “Ob asimptotike reshenii zadachi Dirikhle v trekhmernoi oblasti s vyrezannym tonkim telom”, DAN SSSR, 256:1 (1981), 37–39 | MR

[16] Mazya V. G., Nazarov S. A., Plamenevskii B. A., Asimptotika reshenii ellipticheskikh kraevykh zadach pri singulyarnykh vozmuscheniyakh oblasti, Izd-vo TGU, Tbilisi, 1981 | MR