Application to characteristics of equations of motion

Rauch (1991) says that a smooth hypersurface $\Sigma$ is characteristic only if the principal symbol $L^P(\mathbf{x},i\mathbf{\xi})$ is zero for $\mathbf{\xi}\in N_\mathbf{x}^*(\Sigma)\backslash0$, where $N_\mathbf{x}^*(\Sigma)$ is a 1-D subspace of $T_\mathbf{x}^*(\Sigma)$.I have used this principal-symbol method of determining characteristics to check against the method of characteristics used in the previous subsection. But in the case of the system of PDEs (5) the system is formulated in terms of a matrix operator

$$L_{mi}(\mathbf{x},D)u_i=\sum_{\vert\alpha\vert\leq 2} \frac{a_{\a... ...ha^1}\partial x_2^{\alpha_2}\partial x_3^{\alpha_3} \partial x_4^{\alpha_4}} .$$ (14)

In that equation $m$ and $i$ range from one to three and $x_4=t$. The procedure of replacing $D^\alpha$ by $(i\xi)^\alpha$ and considering only the highest order terms, $\vert\alpha\vert=2$, results in a principal-symbol matrix $\mathbf{L}_P(\mathbf{x},i\xi)$. (I have not included this matrix here; it is available should readers wish to see it.) Setting the determinant, given in appendix C, of this matrix to zero gives a characteristic or eikonal equation. Again, as in the previous section, it is a nonlinear first-order equation in the $\xi_i$ but is one-fifth as long as the one in the previous section. However, when the isotropic elasticity matrix is inserted it reduces to the eikonal equations for $P$ and $S$ waves as before. Due to the differences in length of appendix B compared to appendix C I suspect there is an error in my Mathematica formulation of the first case, which I will attempt to trace, despite it reducing to the correct equations for the isotropic case.