Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 71 / 2 / d

The internal-wave ray tracing equations preserve the horizontal wavenumber $k$ and laboratory frequency. For the increasing wind, the upward branch satisfiesThe group-velocity ratio therefore givesIntegrating from $(0,0)$ yieldsThis is the ascending branch of the circular mountain-wave ray in linear shear:It reaches $z_t=H(q_0^{-1}-1)$ at $x=x_t$, where the energy direction is horizontal. After reflection, $m<0$ and the ray continues downwind and downward along the other branchreturning to $z=0$ at $x=2x_t$. The last panel of the figure in part (b) shows this path. These are WKB energy rays, with the short neighborhood of the turning height supplied by its transition solution.