 Wave-action conservation law

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 $\overline{L}=A^{2}D(\omega ,k)/4$ and simple dispersion relation $D(\omega ,k)=0$, assume $D_{\omega }\ne 0$. Implicit differentiation gives the group velocityHence $W_T+{\partial }_X(c_{g}W)=0$. In several space dimensions this becomes $W_T+{\nabla }_X\cdot ({\mathbf{c}}_{g}W)=0$ 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.