Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 331 / 3 / a / i / Solution

Under the convention $e^{i(kx-\omega t)}$, a mode's amplitude grows as $e^{{\omega }_{i}t}$. The linear stability analysis of real-wavenumber disturbances is therefore summarized byAt ${\omega }_{i,\max }=0$ there is marginal or neutral hydrodynamic stability: no normal mode grows exponentially, but some may fail to decay. If “linearly stable” includes neutral modes, its condition is ${\omega }_{i,\max }\le 0$. These modal statements do not rule out transient growth from non-normal modes or algebraic growth.