Explicit transfer matrices of pre-stressed elastic layers

We establish transfer matrices in explicit form of pre-stressed elastic layers for compressible and incompressible materials. Because these transfer matrices are totally explicit, they are a powerful tool for solving various problems related to the wave propagation in layered pre-stressed elastic media. To prove this point, we investigate the reflection of a qP wave from a pre-stressed compressible elastic layer overlying a pre-stressed compressible elastic half-space. By using the transfer matrix of a pre-stressed compressible elastic layer, we derive the formulas in closed form for the reflection coefficients. Based on the obtained formulas, we carry out some numerical examples to investigate the dependence of the reflection coefficients on the incident angle, the wave frequency, the prestresses and the ratio of mass densities.