Whitham modulation theory

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.^[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 lawVariation 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.

The averaged Lagrangian is the mean over one fast phase of a Lagrangian evaluated on a locally periodic wave profile. Holding its slow parameters fixed during this average yields a function of those parameters, the wavenumber and the angular frequency. A simple explicit example starts fromSince the average of ${\sin }^2$ is $1/2$, one obtainsFor $A\ne 0$, variation with respect to $A$ gives the dispersion relation ${\omega }^2=c^2{k}^2$. Variation of the slow phase instead gives the wave-action conservation law. The distinction matters: substituting the dispersion relation into $\overline{\mathcal{L}}$ before differentiating would set it to zero and erase the nonzero derivatives encoding the conserved wave quantity. General nonlinear waves require averaging their full periodic profile and retaining its shape parameters.

For a periodic travelling profile $U(\theta )$, an autonomous fast-phase Euler-Lagrange equation has a first integral. For example, the field Lagrangian $\mathcal{L}=\frac{1}{2}{u}_t^2-\frac{c^2}{2}{u}_x^2-V(u)$ and phase $\theta =kx-\omega t$ giveMultiplication by $U_{\theta }$ yieldsThus 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.

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.