Steady state and time-dependent energy equilibration in two-dimensional random elastic slabs

The static and dynamic transport properties of elastic wave propagation through two-dimensional random slabs without internal reflection were studied at two different scattering parameters: one for Rayleigh scattering and the other for Rayleigh-Gans scattering. The spatial distribution and temporal evolution of shear (s-) and compressional (p-) wave energy densities inside the slabs were calculated by solving the radiative transfer equation and the generalized diffusion equation (GDE). The comparison of their results can determine the region of validity of the GDE. The process of energy equilibration between the two wave modes was demonstrated explicitly as well as the process of diffusion. The depth inside a slab that is needed to reach energy equilibration or diffusive behavior is found to be dependent on source polarization. The results also show that the bulk equilibration ratio can be found inside a sample only when the sample is sufficiently thick. Deviations of the equilibration ratio from its bulk value are found near the output surface due to the absence of in-flow energy flux. The behavior of the deviations is sensitive to the scattering parameter but independent of source polarization.