= Modulated-wave first integral
For a periodic travelling profile $U(\theta)$, an autonomous fast-phase <Euler-Lagrange equation> has a <first integral>. For example, the field <Lagrangian> $\mathcal L=\tfrac12u_t^2-\tfrac{c^2}{2}u_x^2-V(u)$ and phase $\theta=kx-\omega t$ give
$$
(\omega^2-c^2k^2)U_{\theta\theta}+V'(U)=0.
$$
Multiplication by $U_\theta$ yields
$$
\frac{d}{d\theta}\left[\tfrac12(\omega^2-c^2k^2)U_\theta^2+V(U)\right]=0.
$$
Thus 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.
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