Reflection and transmission of qP-qS plane waves at a plane boundary between viscoelastic transversely isotropic media

We consider the problem of reflection and transmission in two viscoelastic transversely isotropic (VTI) media in contact, with the symmetry axis of each medium perpendicular to the interface. The problem is investigated by means of a plane-wave analysis and a numerical simulation experiment. For an incident homogeneous wave, the reflected wave is of the same type and is also homogeneous, while the other waves are inhomogeneous, that is, equiphase planes do not coincide with equiamplitude planes. If the transmission medium is elastic, the refracted waves are inhomogeneous of the elastic type, that is, the attenuation vectors are perpendicular to the respective Umov-Poynting vectors (energy direction). On the other hand, if the incidence medium is elastic and the transmission medium anelastic, the attenuation vectors of the transmitted waves are perpendicular to the interface. The angle between the attenuation and the real slowness vectors may exceed 90 degrees, but the angle between the attenuation and the Umov-Poynting vectors is always less than 90 degrees. As in the anisotropic case, energy flow parallel to the interface is the criterion for obtaining a critical angle, which exists only in rare instances in viscoelastic media. In fact, for this particular example, the transmitted flux of the quasi-compressional wave is always greater than zero. To balance energy flux it is necessary to consider the interference fluxes between the different waves (these fluxes vanish in the elastic case). The relevant physical phenomena are related to the energy flow direction (Umov-Poynting vector) rather than to the propagation direction (real slowness vector). The simulation experiment gives the particle velocity fields caused by a mean stress source. The results are in good agreement with the plane-wave analysis, despite the fact that only a qualitative comparison can be performed. The presence of the conical wave, which cannot be explained with a plane analysis, indicates that, in spite of the absence of a critical angle, some of the refracted energy disturbs the interface.