Explicit transfer matrix for an incompressible orthotropic elastic layer and applications

In this paper, we establish transfer matrix for an incompressible orthotropic elastic layer. It is explicit and expressed compactly in terms of square brackets. This transfer matrix is a very convenient tool for solving various problems of wave propagation in layered elastic media including incompressible orthotropic layers. To prove this point, we apply it to investigate the reflection of SV-waves from an incompressible orthotropic layer overlying an incompressible orthotropic half-spaces and the propagation of Lamb waves in a composite plate consisting of two incompressible orthotropic layers. By using the obtained transfer matrix along with the effective boundary condition technique, we reduce the reflection of SV-waves from the layer to the reflection of SV-waves from the surface of half-space. The necessary and sufficient conditions for one or two reflected waves to exist have been established, and formulas for the reflection coefficients have been derived. Unlike the previously obtained formulas, the formulas derived in the present paper for the reflection coefficients are totally explicit. Employing the obtained transfer matrix, we arrive immediately at explicit dispersion equation of Lamb waves. Based on the obtained dispersion equation, it is shown numerically that for a two-layered plate with high-contrast material properties of the layers, the cutoff frequency of the first harmonic is close to zero. That means the low-frequency vibration spectrum of strongly inhomogeneous two-layered plates involves not only the fundamental bending mode, but also the first harmonic.