Plane wave solutions

For a plane wave with horizontal slownesses p_x and p_y and frequency $\omega$, the total displacement vector $\vec{\bf u}$ at some point in space $\vec{\bf x}$ is given by

$$\vec{\bf u} = \sum_{\lambda=1}^{6} S_{\lambda} \vec{\bf v}_{... ...} e^{i \omega ( t - \vec{\bf x} \cdot \vec{\bf p}^{\lambda} )}$$ (9)

where $\vec{\bf p}^{\lambda} = ( p_x, p_y, p_z^{\lambda} )^T$. The summation is over the six wave-types in the medium, which have slowness vectors $\vec{\bf p}^{\lambda}$, particle-motion direction $\vec{\bf v}_{\lambda}$, and amplitudes $S_{\lambda}$.

The strains in the medium are given by $e_{kl} = \frac{1}{2} ( u_{k,l} + u_{l,k} )$,the stresses by $\tau_{ij}=C_{ijkl} e_{kl}$.