= Wave-action density
{title2=$W$}
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 $\overline L=A^2(\omega^2-c^2k^2)/4$, one finds $W=A^2\omega/2$ and $J=A^2c^2k/2$. The averaged <energy density> of the corresponding linear <wave equation> is $E=A^2(\omega^2+c^2k^2)/4$. 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.
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