WAVE EQUATION AND PLANE WAVE SOLUTIONS

When considering plane waves propagating in a homogeneous anisotropic medium, it is convenient to use the Christoffel equation. This equation is simply the elastodynamic wave equation Fourier transformed over space and time. It specifies the propagation velocity and particle-motion (also called polarization) direction for each plane-wave component in the Fourier domain. The Christoffel equation takes the form of a simple eigenvalue-eigenvector problem, as follows:

$$\begin{eqnarray} {\bf D C} {\bf D}^{T} \vec{\bf v} & = & \rho \left(\frac{\omega... ... 0 & 0 & k_{z} & k_{y} & k_{x} & 0 \end{array} \right) \nonumber\end{eqnarray}$$ (1)

where $\rho$ and the $6 \times 6$ symmetric matrix $\bf C$ define the homogeneous medium, $\rho$ is the density, and $\bf C$ (in compressed notation) gives the elastic constants. The derivative matrix $\bf D$defines the direction of plane-wave propagation. The leading $\frac{1}{k}$ term normalizes the wavenumber vector ($k=\vert\vec{\bf k}\vert=\sqrt{k_x^2+k_y^2+k_z^2}$). The terms $\omega$ and $\vec{\bf v}$define the resulting wave modes; $\vec{\bf v}$ is the particle-motion direction, and $\omega$ is the associated phase velocity.

Solving the eigensystem in equation (1) is straightforward. For any wavenumber vector $\vec{\bf k}$, the symmetry of the matrix ${\bf D C}{\bf D}^{T}$ ensures that the underlying eigenvalue-eigenvector problem is well-behaved: we can always find three distinct modes associated with three orthogonal directions of particle motion. This eigensystem has solutions only when its determinant vanishes:

$$\det\vert {\bf D C} {\bf D}^{T} - \rho \left(\frac{\omega}{k}\right)^{2} {\bf I} \vert = 0$$

For a general anisotropic medium, this equation requires the solution of a third-order polynomial in $\rho \left(\frac{\omega}{k}\right)^{2}$, where we can identify the term $\omega/k$ as the velocity $V_{\omega}$ of the plane wave with direction $\vec{\bf k} / k$, which results in  

$$\det\vert {\bf D C} {\bf D}^{T} - \rho V_{\omega} ^{2} {\bf I} \vert = 0$$ (2)

The three roots, which correspond to the eigenvalues of equation (1) are the velocities of the three fundamental plane waves. The eigenvectors of equation (1) are, respectively, the three directions of particle motion. In general, none of these fundamental waves will be purely longitudinal or purely transverse.