Whitham modulation equation

For local wavenumber $k={\Theta }_X$, frequency $\omega =-{\Theta }_T$, and averaged Lagrangian $\overline{L}(k,\omega ,\dots )$, slow-phase variation gives

For an averaged Lagrangian, the wave-action density is $W=\partial \overline{L}/\partial \omega$, up to the sign convention used for the fast phase.

In a slowly varying conservative medium, wave action obeys a continuity equation with flux equal to wave-action density times group velocity.