= Whitham modulation theory
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<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>.<A general approach to linear and non-linear dispersive waves using a Lagrangian by Gerald Beresford Whitham>[abstract, nonuniform wave trains and variational formulation]{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 law
$$
k_T+\omega_X=0.
$$
Variation 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.
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