Whitham modulation theory

Whitham modulation theory averages a variational wave equation over a fast periodic phase and derives slow equations for its local wavenumber, frequency, and amplitude.

The averaged Lagrangian is the phase average of a Lagrangian evaluated on a periodic wave profile. Variation with respect to profile parameters gives the local dispersion relation, while variation of the slow phase gives a modulation conservation law.

Multiplying a one-dimensional Euler--Lagrange wave equation by the derivative of its periodic profile with respect to fast phase produces a phase first integral plus slow divergence terms. Phase averaging removes the periodic derivative and yields the slow phase 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.