Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 336 / 1

The chain rule andgiveApplying the same rule to $L_1$ and $L_2$ in the Euler-Lagrange field equation yields the exact modulated equation

Multiply this equation by ${\Phi }_{\theta }$. Since $k$ and $\omega$ are independent of the fast phase,The product rule then rearranges the field equation into the exact modulated-wave first integralAverage this identity over one $2\pi$ period in $\theta$. Periodicity kills the first term, while differentiation of the averaged Lagrangian givesConsequently

The same two equations follow directly from the modulated variational principle. Varying $\Phi$ giveswhich is the exact field equation above. For a variation of $\Theta$, use $\delta k={\partial }_X\delta \Theta$ and $\delta \omega =-{\partial }_T\delta \Theta$. Integration by parts in $X$ and $T$ gives the averaged equation. Thus variation with respect to the periodic profile reproduces the local wave equation, whereas variation with respect to its slow phase gives the Whitham modulation equation for wave action.

Hereso the Euler-Lagrange field equation is the variable-coefficient wave equationAt leading order, $\Phi =A(X,T)\cos \theta$ hasand henceThe leading part of the modulated-wave first integral isA nonconstant periodic wave therefore requires the local dispersion relationThis is usually called the eikonal equation, or equivalently the leading geometric-optics dispersion relation.

Since ${\Phi }_{\theta }=-A\sin \theta$ and the average of ${\sin }^2\theta$ over one period is $1/2$, the leading averaged Lagrangian isIts Euler-Lagrange equation for the slowly varying amplitude is algebraic:For a nonzero wave this again gives the dispersion relation. It does not determine $A$ because the original wave equation is linear and homogeneous: the leading amplitude is fixed only by the next-order transport equation and by initial or boundary data.

Define the wave-action densityThe local dispersion relation implieswhich is the group velocity. Moreover,Substitution into the phase-variation equation gives the wave-action conservation lawOn the two nondispersive branches $\omega =\pm ck$, the group velocities are $c_g=\pm c$.