Whitham modulation equation

Fix the phase convention $k={\Theta }_X$, $\omega =-{\Theta }_T$. If the averaged action is $\int \overline{L}(k,\omega ,\dots )dXdT$, a compactly supported phase variation gives $\delta k={\partial }_X\delta \Theta$, $\delta \omega =-{\partial }_T\delta \Theta$. Integration by parts then yieldsDefining the wave-action density $W={\overline{L}}_{\omega }$ and its flux $J=-{\overline{L}}_k$ puts the same equation in the conservative form $W_T+J_X=0$. This sign follows from the declared negative time derivative of phase. It must be combined with $k_T+{\omega }_X=0$ and any profile-parameter equations to close a modulation problem.

With phase convention $k={\Theta }_X$, $\omega =-{\Theta }_T$, the wave-action density associated with an averaged Lagrangian is $W={\overline{L}}_{\omega }$; the matching flux is $J=-{\overline{L}}_k$. These derivatives are taken before substituting any dispersion relation.

For $\overline{L}=A^2({\omega }^2-c^2{k}^2)/4$, one finds $W=A^2\omega /2$ and $J=A^2{c}^{2}k/2$. The averaged energy density of the corresponding linear wave equation is $E=A^2({\omega }^2+c^2{k}^2)/4$. On a nonzero-frequency dispersion branch, $E=A^2{\omega }^2/2$, so $W=E/\omega$. The sign of $W$ depends on the frequency and phase convention; it is not independently specified by calling it an energy-like quantity.

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.