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Finding phase velocities and polarization vectors

When considering plane waves propagating an a homogeneous anisotropic medium it is convenient to use the Christoffel equations for an anisotropic medium. These equations:

are the equations of motion of an elastic anisotropic medium, represented in the Fourier domain. Here CML is the stiffness matrix in compressed notation and kiM, kLj , denote the Fourier representations of spatial and temporal derivatives. is the density tensor, uj is the particle displacement. If we write the derivative matrices as functions of slowness rather than wavenumber we have,

For the general anisotropic case CML has 21 independent elements. In the monoclinic case CML has 13 independent elements.

In equation  we can write piM CML pLj as a product of matrices:

The result of the product is a 3x3 matrix aij. The Christoffel equation can be written now in the form:

This system of equations has solutions when it's determinant vanishes. In general this requires solution of a sixth order polynomial in px,py,pz. To find the vertical slowness for a given pair of horizontal slownesses px,py I must find the roots of a sextic in pz. If the system has monoclinic symmetry the equations are somewhat simpler. The matrix elements in equation  only have terms in pz2 so the roots may be found by solving a cubic equation in pz2.

Solving the system equation  using the previously calculated roots gives the corresponding polarization vectors . I separate the wavetypes into up- and downgoing solutions by examining the energy flow direction, the Poynting vector.

Next: Expressing waves in terms Up: APPENDIX: WAVE PROPAGATION IN Previous: APPENDIX: WAVE PROPAGATION IN
Stanford Exploration Project
12/18/1997