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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.

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.