Christoffel equation for anisotropic waves

Remarks

Substituting the plane wave gives $\rho c^2 d_i F'' = \sum_{j,k,l} C_{ijkl} n_j n_l d_k F''$, which is the Christoffel equation after dividing by $F''$ at a point where it is not zero. Symmetry of $\Gamma(n)$ follows from $C_{ijkl} = C_{klij}$, and strong ellipticity with $b = n$ gives $d^T \Gamma(n) d > 0$ for $d \ne 0$. So the squared speeds $\rho c^2$ are the eigenvalues of $\Gamma(n)$ and the polarizations are its eigenvectors.

Example: for an isotropic medium with Lamé parameters $\lambda$ and $\mu$, $\Gamma(n) = (\lambda + \mu) \, n n^T + \mu I$. Its eigenvalue $\lambda + 2\mu$ has eigenvector $n$, giving the P-wave, and its double eigenvalue $\mu$ has the plane $n^\perp$ as eigenspace, giving the S-waves.

Without strong ellipticity some eigenvalue of $\Gamma(n)$ can be zero or negative, so some plane waves have no real speed, and the equations are no longer hyperbolic.

Going deeper

For a student

In an anisotropic medium the three waves for a given direction are called quasi-longitudinal and quasi-shear waves. The reason is that their polarizations are in general neither exactly along nor exactly across $n$. Plotting $1/c$ over all directions $n$ gives the slowness surface, which determines refraction at interfaces and the directions in which energy travels. Measuring the speeds in several directions and inverting the Christoffel equation is a standard way to determine the elastic constants of a crystal.