Mathematics of transverse isotropy with a horizontal axis of symmetry

The elastic modulus matrix of an HTI medium with a symmetry axis along the x direction (which I will denote as ${\rm HTI}_{x}$) has the following form (in compressed notation):

 

$$C_{ \alpha \beta }= \left( { \matrix{c_{11}&c_{13}&c_{13}&0... ..._{44}&0&0\cr 0&0&0&0&c_{55}&0\cr 0&0&0&0&0&c_{55}\cr }} \right)$$ (3)

This representation can easily be obtained by a $\frac{\pi}{2}$rotation of the VTI's elastic matrix along the y axis. Incorporating this expression into equation (1) and identifying $\omega/k$as the velocity $V_{\omega}$ of the plane wave with propagation direction (k_x,k_y,k_z) gives us

$$\begin{eqnarray} \left( \begin{array} {ccc} c_{11} k_x^2 + c_{55} (k_y^2 +k_z^2)... ... \begin{array} {c} v_x \\ v_y \\ v_z\end{array}\right) \nonumber\end{eqnarray}$$ (4)

From here on, to simplify notation, I use the variables and , introduced by Dellinger 1991. Thus we rewrite equation (4) as

$$\begin{eqnarray} \left( \begin{array} {ccc} c_{11} k_x^2 + c_{55} (k_y^2 +k_z^2)... ... \begin{array} {c} v_x \\ v_y \\ v_z\end{array}\right) \nonumber\end{eqnarray}$$ (5)

 

HTIprop Figure 1 Definition of propagation angles. The azimuthal angle $\alpha$ is measured with respect to the symmetry axis, and the incidence angle $\theta$ with respect to the vertical axis.