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

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 $c_g=\pm c$.