Asymptotics of eigen-oscillations of a massive elastic body with a thin baffle

We construct asymptotics of eigenvalues and eigenvectors in the elasticity problem for an anisotropic body joined to a thin plate-baffle (of variable thickness $O(h)$, $h\ll 1$). The spectrum contains two series of eigenvalues with stable asymptotic behaviour. The first is formed by eigenvalues $O(h^2)$ corresponding to the transversal vibrations of the plate with rigidly clamped lateral surface, and the second contains eigenvalues $O(1)$ generated by the longitudinal vibrations of the plate as well as eigen-oscillations of the body without baffle. We verify the convergence theorem for the first series, estimate the errors for both series, and discuss the asymptotic correction terms and boundary layers. Similar but simpler results are obtained in the scalar problem.