Following the first principle procedure outlined by Buchen (1971a) and Borcherdt (1973), we describe the derivation of SH wave propagation in a homogeneous transversely isotropic linear viscoelastic (HTILV) solid. Plane SH wave propagates with frequency-dependent complex phase velocity
β 2 (ω) = β h 2 (ω) sin 2 b +β v 2 (ω) cos 2 b
where β h and β v are complex shear wave velocities perpendicular and parallel to the axis of symmetry of the medium and b is a complex angle that the complex wavevector makes with the axis. The energy flows in a direction governed by the propagation vector, attenuation vector and the rigidities. The attenuation angle between the propagation vector and the attenuation vector can be uniquely determined by the complex ray parameter at the saddle point of the complex traveltime function. Complex ray can be traced between source and receiver locations with intermediate coordinates being complex. By means of the method of steepest descent, the wavenumber integral representing the exact SH wave field generated by a line source for layered-case problem can be approximated to give complex ray amplitudes for reflected and transmitted body waves. The factor accounting for cylindrical divergence is similar in form to that of the isotropic case. However, the similarity is not so obvious without going through the mathematics.
For a simple two half-spaces model, the complex ray result agrees well with the ω-k solution in regions away from the critical area. For pure SH mode propagation through a planar HTILV structure with 20% anisotropy, the reflected amplitudes in both cases (transversely isotropic and isotropic) look similar. However, the most significant is the kinematic difference.
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