Plane wave modes

Considering nondispersive media, for the propagation direction $k_x = k \sqrt{\sin^2 \theta \cos^2 \alpha}$, $k_y = k \sqrt{\sin^2 \theta \sin^2 \alpha}$, and $k_z = k \sqrt{\cos^2 \theta}$, where the angles $\theta$ and $\alpha$ are measured from the vertical axis and the medium's symmetry axis, respectively, as shown in Figure 1, equation (5) produces the following three plane wave modes (results obtained with the symbolic calculation software package Mathematica):

$$\begin{eqnarray} \rho V_{\omega}^{2} = c_{44}\,(1-k_x^2) + c_{55} k_x^2 & {\rm a... ...amma \left( \begin{array} {c} 0 \\ -k_z \\ k_y\end{array}\right)\end{eqnarray}$$ (6)

where $\gamma$ is an arbitrary constant, usually taken as a normalizing factor.

This is a pure shear mode, since its direction of particle motion (0,-k_z,k_y) is always perpendicular to the direction of propagation (k_x,k_y,k_z). This propagation mode is called an SH mode.

The other two solutions are expressed as follows:

$$\begin{eqnarray} \rho V_{\omega}^{2} & = & \frac{1}{2} \left\{ (c_{33} + c_{55}... ... 2 \chi k_x k_y \\ 2 \chi k_x k_z \end{array} \right) \nonumber\end{eqnarray}$$ (7)

The sign of ``$\pm$'' selects one of the two remaining modes. The ``+'' sign selects the ``fast'' wave solution, while the ``-'' sign selects the ``slow'' wave solution. For all angles of direction $\theta$ and $\alpha$, the fast solution is always greater than the slow solution. In the case of isotropy, the fast solution corresponds to the P-wave solution, the slow to the SV-wave. For transverse isotropy and for anisotropy in general, the fast wave-mode may be neither a pure P-wave nor an approximate P-wave, just as the slow wave-mode may be neither a pure SV-wave nor an approximate SV-wave. For this reason and to avoid bias, it is better to call these two solutions the fast and the slow wave solutions, instead of using the conventional notation, qP and qS.

It is important to notice that any plane wave propagating in the y-z plane, will have a constant-value velocity function. This plane is referred to as the isotropy plane (Figure 1).

Even though it may not be obvious from equations (6) and (7), the three propagation modes, fast, slow, and SH, are perpendicular to each other as expected.