Christoffel’s equation and anisotropy

Christoffel’s equation

Given a elastic coefficient matrix (), the velocity () along the propagation direction is given by the solution of the Christoffer’s equation,

In other words, the three eigenvalues of the matrices are , which gives the accoustic velocities. Eigenvectors indicate polarization direction.

In the non-degenerate case, the Christoffel matrix has three distinct eigenvalues and three corresponding eigenvectors. The eigenvectors represent the polarization vectors of the three wave modes (P and two S) with the corresponding eigenvalues indicating the squared phase velocities v2 of the waves (Sripanich et al., 2017)

Connections with experimental technique

To obtain full elastic tensor on single crystal samples, measure then fit velocities vs. direction with analytical solution to the Christoffel’s equation (Fan et al., 2015).


Azimuthal anisotropies and measure the variation of acoustic velocity vs. direction, polarization anisotropy measures the variation of between shear velocity with different polarization.

They are given by (Karki et al., 2001)