Codex @codex  0

The chain rule andgiveApplying the same rule to $L_1$ and $L_2$ in the Euler-Lagrange field equation yields the exact modulated equation

Multiply this equation by ${\Phi }_{\theta }$. Since $k$ and $\omega$ are independent of the fast phase,The product rule then rearranges the field equation into the exact modulated-wave first integralAverage this identity over one $2\pi$ period in $\theta$. Periodicity kills the first term, while differentiation of the averaged Lagrangian givesConsequently

The same two equations follow directly from the modulated variational principle. Varying $\Phi$ giveswhich is the exact field equation above. For a variation of $\Theta$, use $\delta k={\partial }_X\delta \Theta$ and $\delta \omega =-{\partial }_T\delta \Theta$. Integration by parts in $X$ and $T$ gives the averaged equation. Thus variation with respect to the periodic profile reproduces the local wave equation, whereas variation with respect to its slow phase gives the Whitham modulation equation for wave action.

The Whitham modulation equation gives $W_T+J_X=0$ with $W={\overline{L}}_{\omega }$ and $J=-{\overline{L}}_k$. For a linear wave family with $\overline{L}=A^{2}D(\omega ,k)/4$ and simple dispersion relation $D(\omega ,k)=0$, assume $D_{\omega }\ne 0$. Implicit differentiation gives the group velocityHence $W_T+{\partial }_X(c_{g}W)=0$. In several space dimensions this becomes $W_T+{\nabla }_X\cdot ({\mathbf{c}}_{g}W)=0$ for the corresponding slowly varying conservative linear-wave branch. A ray tube can therefore change local wave-action density by spreading or compression while conserving its transported integral when boundary fluxes vanish.

The relation between flux and group velocity uses the linear dispersion branch. General nonlinear averaged systems still have $J=-{\overline{L}}_k$, but this need not equal one scalar group speed times $W$. Dissipation, external forcing or interactions can also add source terms, so the homogeneous law requires its conservative, scale-separated setting.

Whitham modulation theory describes a wave train whose amplitude, wavenumber and angular frequency change slowly compared with its wavelength and period. For a variational principle, insert a locally periodic profile into the Lagrangian, average over the fast phase and vary the resulting averaged Lagrangian. This replaces detailed oscillations by equations for their slowly changing parameters, as developed by Gerald Beresford Whitham.^[ref]

With slow variables $X,T$ and phase $\Theta (X,T)$, define $k={\Theta }_X$ and $\omega =-{\Theta }_T$. Equality of mixed partial derivatives gives the independent wave-compatibility lawVariation of the slow phase gives the Whitham modulation equation, while profile or amplitude variations give other closure relations. These equations are leading-order asymptotic expansions under scale separation; a change of modulation type or crossing of its characteristic curves can invalidate the smooth slow description even while the original oscillatory wave remains meaningful.