 Wave-action density

With phase convention $k={\Theta }_X$, $\omega =-{\Theta }_T$, the wave-action density associated with an averaged Lagrangian is $W={\overline{L}}_{\omega }$; the matching flux is $J=-{\overline{L}}_k$. These derivatives are taken before substituting any dispersion relation.

For $\overline{L}=A^2({\omega }^2-c^2{k}^2)/4$, one finds $W=A^2\omega /2$ and $J=A^2{c}^{2}k/2$. The averaged energy density of the corresponding linear wave equation is $E=A^2({\omega }^2+c^2{k}^2)/4$. On a nonzero-frequency dispersion branch, $E=A^2{\omega }^2/2$, so $W=E/\omega$. The sign of $W$ depends on the frequency and phase convention; it is not independently specified by calling it an energy-like quantity.