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The averaged Lagrangian is the mean over one fast phase of a Lagrangian evaluated on a locally periodic wave profile. Holding its slow parameters fixed during this average yields a function of those parameters, the wavenumber and the angular frequency. A simple explicit example starts fromSince the average of ${\sin }^2$ is $1/2$, one obtainsFor $A\ne 0$, variation with respect to $A$ gives the dispersion relation ${\omega }^2=c^2{k}^2$. Variation of the slow phase instead gives the wave-action conservation law. The distinction matters: substituting the dispersion relation into $\overline{\mathcal{L}}$ before differentiating would set it to zero and erase the nonzero derivatives encoding the conserved wave quantity. General nonlinear waves require averaging their full periodic profile and retaining its shape parameters.

For a periodic travelling profile $U(\theta )$, an autonomous fast-phase Euler-Lagrange equation has a first integral. For example, the field Lagrangian $\mathcal{L}=\frac{1}{2}{u}_t^2-\frac{c^2}{2}{u}_x^2-V(u)$ and phase $\theta =kx-\omega t$ giveMultiplication by $U_{\theta }$ yieldsThus the bracketed profile energy is constant during one fast cycle. In a wave described by Whitham modulation theory, its parameters also vary on slow scales. The fast identity alone does not determine that slow evolution: one must retain the next-order terms and impose their averaged solvability condition, or vary the averaged Lagrangian. A periodic total phase derivative averages to zero, which is the useful simplification in either route.

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.

Since ${\Phi }_{\theta }=-A\sin \theta$ and the average of ${\sin }^2\theta$ over one period is $1/2$, the leading averaged Lagrangian isIts Euler-Lagrange equation for the slowly varying amplitude is algebraic:For a nonzero wave this again gives the dispersion relation. It does not determine $A$ because the original wave equation is linear and homogeneous: the leading amplitude is fixed only by the next-order transport equation and by initial or boundary data.

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.

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.