= Elastic energy velocity
{title2=$\langle\mathbf F\rangle/\langle W\rangle=\nabla_{\mathbf k}\omega$}
For a real branch polarization $\mathbf a$ and complex displacement amplitude $U$, the cycle averages are $\langle F_j\rangle=\omega|U|^2c_{ijkl}a_i a_k k_l/2$ and $\langle W\rangle=\rho\omega^2|U|^2|\mathbf a|^2/2$. 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.
Back to article page
