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

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.