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 *C*_{ML} is the stiffness
matrix in compressed notation and *k*_{iM}, *k*_{Lj} , denote
the Fourier representations of spatial and temporal derivatives.
is the density tensor, *u*_{j} is the particle displacement.
If we write the derivative matrices as functions of slowness rather
than wavenumber we have,

For the general anisotropic case *C*_{ML} has 21 independent
elements. In the monoclinic case *C*_{ML} has 13 independent
elements.

In equation we can write *p*_{iM} *C*_{ML} *p*_{Lj} as a product of matrices:

The result of the product is a 3x3 matrix *a*_{ij}. 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
*p*_{x},*p*_{y},*p*_{z}. To find the vertical slowness for a given pair of
horizontal slownesses *p*_{x},*p*_{y} I must find the roots of a sextic in
*p*_{z}. If the system has monoclinic symmetry the equations are
somewhat simpler. The matrix elements in equation only
have terms in *p*_{z}^{2} so the roots may be found by solving a cubic
equation in *p*_{z}^{2}.

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.

12/18/1997