Energy flux of a transverse string wave ($P=-T{y}_x{y}_t$)

The small-slope energy density is $\mathcal{E}=(\rho y_t^2+T{y}_x^2)/2$ and power flux is $P=-T{y}_x{y}_t$. The wave equation gives ${\partial }_t\mathcal{E}+{\partial }_{x}P=0$. For a right-going wave, $P=c\mathcal{E}$; a harmonic wave has mean flux $\rho c{A}^2{\omega }^2/2$.