The spectrum of one-dimensional eigenoscillations of two-phase layered media with periodic structure

We study the spectrum of one-dimensional eigenoscillations along the $Ox_1$ axis of two-phase layered media with periodic structure occupying the band $0<x_1<L$. The period of the media is a band $0<x_1<\varepsilon$ composed of $2M$ alternating layers of an isotropic elastic or viscoelastic material (the first phase) and a viscous incompressible fluid (the second phase). It is assumed that the number of periods $N=L/ \varepsilon$ is an integer, and the layers are parallel to the $Ox_2 x_3$ plane. The spectrum is denoted by $S_\varepsilon$ and is defined as the set of eigenvalues of a boundary value problem for a homogeneous system of ordinary differential equations with conjugation conditions at the interfaces between the solid and fluid layers. These conditions are derived directly from the initial assumption on the continuity of displacements and normal stresses at the interfaces between the layers. It is shown that the spectrum $S_\varepsilon$ consists of the roots of transcendental equations, the number of which is equal to the number of periods $N$ contained within the band $0<x_1<L$. The roots of these equations can only be found numerically, except for one particular case. In the case of multi-layered media with $N\gg 1$, the finite limits of the sequences $\lambda(\varepsilon)\in S_\varepsilon$ as $\varepsilon\to 0$ are proposed to be used as initial approximations. It is established that the set of all finite limits coincides with the set of roots of rational equations, denoted by $S$. The coefficients of these equations, and hence the points of the set $S$ depend on the volume fraction of the fluid within the layered medium and do not depend on the number $M$ of the fluid layers within the period. It is proved that for any $M\geq 1$ the spectrum $S_\varepsilon$ converges in the sense of Hausdorff to the set $S$ as $\varepsilon\to 0$.