Elastic wave-vector decomposition in heterogeneous anisotropic media |

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where denotes the Christoffel matrix , in which is the stiffness tensor, and and are the normalized wave-vector components in the and directions: . denotes the phase velocity of a given wave mode for the given phase direction ( ), and denotes the corresponding polarization vector (Cervený, 2001).

In the non-degenerate case, the Christoffel matrix
has three distinct eigenvalues and three corresponding eigenvectors. The eigenvectors represent the polarization vectors of the three wave modes (P and two S) with the corresponding eigenvalues indicating the squared phase velocities
of the waves. In the degenerate case, any two or all three eigenvalues become equal and the corresponding phase direction is referred to as the *singular* direction (Vavrycuk, 2001). Note that if two eigenvalues for S waves coincide for all phase directions, the problem reduces to isotropy, which is a special case of Christoffel degeneracy.

Wave propagation in low-symmetry anisotropic media involves at most twenty-one independent stiffness tensor coefficients in the case of triclinic media. The elements of general Christoffel matrix
can be defined as follows:

Equation A-2 reduces to the case of orthorhombic media when , and further to the case of TI media when, additionally, , , and .

Elastic wave-vector decomposition in heterogeneous anisotropic media |

2017-04-18