Asymptotics of the Spectrum of One-Dimensional Natural Vibrations in a Layered Medium Consisting of Viscoelastic Material and Viscous Fluid

Abstract

The spectrum of one-dimensional natural vibrations propagating through a two-phase layered medium in the direction of the normal to the layers is investigated. The medium considered consists of many periodically alternating layers of isotropic viscoelastic material and viscous compressible fluid. It is found that the spectrum mentioned above consists of the roots of transcendental equations whose number is proportional to the number of the layers of original medium. As the initial approximations of the roots for solving these equations numerically, it is proposed to use the points of the spectrum of one-dimensional natural vibrations of the corresponding homogenized medium. These points represent the roots of linear fractional equations. It is shown that the points at which the denominators of fractions in the linear fractional equations vanish should be also taken as the initial approximations. The accuracy of the initial approximations is proved to increase when the number of layers of the original medium increases and the layer thickness decreases simultaneously.