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

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.