Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 333 / 4 / ii / Solution

Elimination of $\hat{u}$ and $\hat{\varphi }$ givesChoose ${\sigma }^2=(\beta /c)(B/A{)}^{1/2}$ with $\mathrm{Re}{\sigma }^2>0$ and set $Y=\sigma y$. The Hermite eigenvalue condition gives, for $n=1,2,\dots$,Squaring the eigenvalue condition introduces an extraneous branch, so one must also impose $ikc/\sqrt{AB}=2n+1$ and $\mathrm{Re}{\sigma }^2>0$. For real $k$, trapped propagating roots exist whenand the accepted root propagates westward, $k\mathrm{Re}\omega <0$. The other root has an outward-growing meridional structure.