Reflection and refraction of antiplane shear waves at a plane boundary between viscoelastic anisotropic media

Abstract

We consider two monoclinic viscoelastic media in contact, with the incidence and refraction planes coincident with the respective planes of symmetry. Then, an incident homogeneous antiplane shear wave generates pure reflected and refracted antiplane waves, whose slowness and Umov–Poynting vectors lie in the planes of symmetry. The simplicity of the problem permits a detailed investigation of the phenomena caused by the combined anisotropic––anelastic properties of the media and waves. A general approach and the analysis of a numerical example provide a complete picture of the physics. In general, the reflected and refracted waves are inhomogeneous, i.e. equiphase planes do not coincide with equiamplitude planes. The reflected wave is homogeneous only when the incidence medium is transversely isotropic, i.e. its symmetry axis is perpendicular to the interface. If the refraction medium is elastic, the refracted wave is inhomogeneous of the elastic type, i.e. the attenuation vector is perpendicular to the Umov–Poynting vector (energy direction). The angle between the attenuation and the real slowness vectors may exceed 90 degrees, but the angle between the attenuation and the Umov–Poynting vector is always less than 90 degrees. If the incidence medium is elastic, the attenuation of the refracted wave is perpendicular to the interface. As in the anisotropic elastic case, energy flow parallel to the interface is the criterion for obtaining a critical angle. As in the isotropic viscoelastic case, critical angles exist only in rare instances. Indeed, they do not exist if one of the media is elastic. The existence of Brewster angles (related to a zero reflection coefficient) is also severely restricted by anelasticity. To balance the energy flux at the boundary, it is necessary to consider the interference flux between the incident and reflected waves (this flux vanishes in the elastic case). For the particular example, the refracted flux is always greater than zero and there is transmission for all the incident angles. This phenomenon is related to the absence of critical angles. For a transversely isotropic incidence medium, attenuations, quality factors and phase and energy velocities of the incident and reflected waves coincide for all the incidence angles. It is important to point out that the relevant physical phenomena are related to the energy flow direction (Umov–Poynting vector) rather than to the propagation direction (real slowness vector). For instance, the characteristics of the elastic type inhomogeneous waves, the existence of critical angles, and the fact that the amplitudes of the reflected and refracted waves decay in the direction of energy flow despite the fact that they grow in the direction of phase propagation.