Anomalous polarization of elastic waves in transversely isotropic media

The orientation of the displacement vector U of a plane wave in a homogeneous anisotropic elastic medium is the polarization of that plane wave. For transversely isotropic media, U of the fastest plane wave propagating in a given direction need not be in or near the direction of propagation, i.e., the direction of the slowness vector s. Moreover, with the x3 axis the transverse axis of the medium, when c13+c440, there are directions of propagation for which U can be perpendicular to s which means that the first arrival in that direction is a purely transverse wave. The two plane waves whose displacement vectors lie in the plane of the direction of propagation and the x3 axis—the faster usually called ‘‘quasi-P’’ and the slower usually called ‘‘quasi-SV’’—have mutually orthogonal displacement vectors. The polarization of these waves as a function of propagation direction depends strongly on the sign of c13+c44 whereas the magnitudes of the slowness vectors, which give the shape of the slowness surface, do not. In the x3 direction and perpendicular to the x3 direction the waves are purely longitudinal and purely transverse. When c13+c44 is positive, which is the usual case, the particle displacement vectors rotate in the same sense as the slowness vector. When c13+c44 is negative, which is the ‘‘anomalous’’ case, the sense of rotation of the particle displacement vector is opposite to that of the slowness vector and there always exists a direction of propagation in the medium for which U of the fast (or inner or quasi-P) sheet of the slowness surface is perpendicular to that direction and U of the slow (or outer or quasi-SV) sheet is parallel to that direction. The stability conditions on the transversely isotropic moduli even admit media that are (almost) kinematically isotropic, i.e., characterized by spherical wave fronts emanating from point sources, but which are characterized by anomalous polarization. A layered medium can be equivalent, in the long-wavelength limit, to an anomalous transversely isotropic medium. For a layered medium composed of two alternating constituent substrates, this can occur when the larger volume fraction constituent has a negative Poisson’s ratio, the smaller volume fraction constituent has a positive Poisson’s ratio, and the ratio of the shear modulus of the negative Poisson’s ratio constituent to the shear modulus of the positive Poisson’s ratio constituent is very large.