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

Put $A=\gamma -i\omega$ and $B=\alpha -i\omega$. For $\hat{v}=0$,Continuity gives the damped Equatorial Kelvin wave dispersion relationOnly roots satisfying $\mathrm{Re}(i\beta k/A)<0$ are trapped; for weak damping this is the eastward branch $k\mathrm{Re}\omega >0$. The other algebraic root makes the Gaussian grow with $|y|$ and must be rejected.