 Elastic energy velocity

For a real branch polarization $\mathbf{a}$ and complex displacement amplitude $U$, the cycle averages are $⟨F_j⟩=\omega |U{|}^2{c}_{ijkl}{a}_i{a}_k{k}_l/2$ and $⟨W⟩=\rho {\omega }^2|U{|}^2|\mathbf{a}{|}^2/2$. Differentiating the acoustic tensor eigenvalue equation with respect to $k_j$ gives their ratio as the group velocity. This identifies the energy flux direction with the normal to the elastic slowness surface in a lossless nondispersive elastic medium.