Elastic-wave energy flux
= Elastic-wave energy flux
For displacement velocity $\dot u$ and stress $\sigma$, the instantaneous elastic-energy flux is
$$
P_i=-\sigma_{ij}\dot u_j.
$$
For complex harmonic amplitudes, its average over one temporal period is
$$
\langle P_i\rangle=-\frac12\operatorname{Re}(\widehat\sigma_{ij}\widehat{\dot u}_j^*).
$$