Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 345 / 1 / b / Solution

A localized monochromatic cylinder emits four narrow beams, one in each quadrant, forming a St Andrew's cross. If $\alpha$ is the beam angle to the horizontal,equivalently, its angle $\theta =\pi /2-\alpha$ to the vertical obeys $\cos \theta =\omega /N$. As $\omega$ rises from zero to $N$, the beams rotate from horizontal toward vertical. For $\omega >N$ no freely propagating internal wave exists and the response is evanescent.

The wavevector and phase velocity are parallel. The dispersion relation is homogeneous of degree zero in $(k,m)$, so Euler's theorem gives $\mathbf{k}\cdot {\mathbf{c}}_g=0$: the group velocity is perpendicular to both the wavevector and phase velocity. Energy travels along the beams in the group-velocity direction.