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Mathematics of the USSR-Sbornik, 1984, Volume 49, Issue 2, Pages 305–323
DOI: https://doi.org/10.1070/SM1984v049n02ABEH002712
(Mi sm2195)
 

This article is cited in 3 scientific papers (total in 3 papers)

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

G. V. Zhdanova
References:
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.
Received: 05.01.1982
Russian version:
Matematicheskii Sbornik. Novaya Seriya, 1983, Volume 121(163), Number 3(7), Pages 310–326
Bibliographic databases:
UDC: 531.262
MSC: 73D25
Language: English
Original paper language: Russian
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/sm2195}
\mathscinet{http://mathscinet.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}
Linking options:
  • https://www.mathnet.ru/eng/sm2195
  • https://doi.org/10.1070/SM1984v049n02ABEH002712
  • https://www.mathnet.ru/eng/sm/v163/i3/p310
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics
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    Abstract page:376
    Russian version PDF:100
    English version PDF:4
    References:51
     
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