= 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 by
$$
\boxed{\omega_{i,\max}<0:\ \text{strictly stable},\qquad
\omega_{i,\max}>0:\ \text{linearly unstable}.}
$$
At $\omega_{i,\max}=0$ 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 $\omega_{i,\max}\leq0$. These modal statements do not rule out <transient growth from non-normal modes> or algebraic growth.
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