Effective mass density for wave propagation in layered media: a study of the elastic/acoustic transition.

It is well-known that the effective mass density of a layered elastic medium is isotropic whereas its acoustic analogue is anisotropic. Given that the acoustics regime is recoverable from elasticity by taking the limit as the shear modulus µ tends to zero, one should recover this result for the effective density in this limit. However, the effective mass density does not explicitly depend on shear. Thus, how the effective density transitions between the (isotropic) elastodynamic and (anisotropic) acoustics regimes remains unclear. To understand the dependence of the effective density on shear, a dynamic self-consistent method is employed. Reflection and transmission coefficients of an elastodynamic system comprising two layers with different densities, placed between
two half-spaces of identical, but unknown, anisotropic mass densities, are calculated. The effective density is deduced by imposing zero reflection and complete transmission. The zero shear (acoustic) limit is taken, resulting in the anisotropic effective acoustic mass density. Therefore, for the first time, the nature of the transition to this case from the isotropic effective elastodynamic mass density is illustrated. Additionally, it is shown that a parameter space exists for small but non-zero µ where the density of a layered elastodynamic material is anisotropic.