Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 333 / 3 / ii / Solution

At $\mu ={\mu }_c$, the real undamped part $F$ vanishes. PutThe neutral equation implies $d=1/4-1/{\mu }_c^2>0$. The radicand is thereforeIf $\sqrt{z}=u+iv$ is chosen with $v>0$, thenThus $v>ϵ/2$, and the plus branch hasFor $k\Lambda >0$ this is exponential growth. Lower-boundary damping therefore destabilizes the formerly neutral cutoff mode, an example of dissipation-induced instability.