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 Mat. Sb. (N.S.), 1983, Volume 121(163), Number 3(7), Pages 310–326 (Mi msb2195)

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

G. V. Zhdanova

Abstract: 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.

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English version:
Mathematics of the USSR-Sbornik, 1984, 49:2, 305–323

Bibliographic databases:

UDC: 531.262
MSC: 73D25

Citation: G. V. Zhdanova, “Scattering of plane longitudinal elastic waves by a slender cavity of revolution. The case of axial incidence”, Mat. Sb. (N.S.), 121(163):3(7) (1983), 310–326; Math. USSR-Sb., 49:2 (1984), 305–323

Citation in format AMSBIB
\Bibitem{Zhd83} \by G.~V.~Zhdanova \paper Scattering of plane longitudinal elastic waves by a~slender cavity of revolution. The case of axial incidence \jour Mat. Sb. (N.S.) \yr 1983 \vol 121(163) \issue 3(7) \pages 310--326 \mathnet{http://mi.mathnet.ru/msb2195} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=707999} \zmath{https://zbmath.org/?q=an:0571.73023|0533.73031} \transl \jour Math. USSR-Sb. \yr 1984 \vol 49 \issue 2 \pages 305--323 \crossref{https://doi.org/10.1070/SM1984v049n02ABEH002712} 

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This publication is cited in the following articles:
1. G. V. Zhdanova, “Asymptotics of the solution of the Dirichlet problem for the system of elasticity theory in the exterior of a thin body of revolution”, Math. USSR-Sb., 62:1 (1989), 1–27
2. Zhdanova G., “Asymptotics of the Solution of Dirichlet Problem for the System of the Elasticity Theory Equations Outside a Slender Body of Revolution”, 296, no. 5, 1987, 1041–1045
3. Ming-Chun Cai, Hui-Ji Shi, Xian-Feng Ma, “Yield initiation of compressible material with a central void under dynamic load”, Comptes Rendus Mécanique, 338:4 (2010), 207
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