Asymptotic behavior of the spectrum of one-dimensional vibrations in a layered medium consisting of elastic and Kelvin–Voigt viscoelastic materials

The work is devoted to the analysis of the spectral properties of a boundary value problem describing one-dimensional vibrations along the axis $Ox_1$ of periodically alternating $M$ elastic and $M$ viscoelastic layers parallel to the plane $Ox_2x_3$. It is shown that the spectrum of the boundary value problem is the union of roots of $M$ equations. The asymptotic behavior of the spectrum of the problem as $M\to\infty$ is analyzed; in particular, it is proved that not all sequences of eigenvalues of the original (prelimit) problem converge to eigenvalues of the corresponding homogenized (limit) problem.