= Wave-action conservation law
{title2=$W_T+\nabla\cdot(\mathbf c_gW)=0$}
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 $\overline L=A^2D(\omega,k)/4$ and simple <dispersion relation> $D(\omega,k)=0$, assume $D_\omega\ne0$. Implicit <differentiation> gives the <group velocity>
$$
c_g=\frac{d\omega}{dk}=-\frac{D_k}{D_\omega},\qquad J=c_gW.
$$
Hence $W_T+\partial_X(c_gW)=0$. In several space dimensions this becomes $W_T+\nabla_X\cdot(\mathbf c_gW)=0$ 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.
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