= Solution
At the <absolute wavenumber>, the <group velocity> $d\omega/dk$ is zero. The corresponding accessible saddle therefore governs the disturbance seen at a fixed position. Its exponential <growth rate> is
$$
\operatorname{Im}\omega_0=\mu-\frac{U^2}{4(1+c_d^2)}.
$$
Saddle accessibility can be checked directly here. With $D=1+ic_d$, whose <real part> is positive, <Fourier inversion> gives the <Green function of the linear complex Ginzburg-Landau equation>,
$$
G(x,t)=\frac1{\sqrt{4\pi Dt}}
\exp\left[\mu t-\frac{(x-Ut)^2}{4Dt}\right],\qquad t>0.
$$
At fixed $x$ its exponential rate is $\operatorname{Im}\omega_0$, while along the packet centre $x=Ut$ it is $\mu$. For real <wavenumbers> of <Fourier modes>, the <temporal growth rate> is $\operatorname{Im}\omega(k)=\mu-k^2$, so the flow is temporally unstable exactly when $\mu>0$.
Consequently the classifications for a localized disturbance in this laboratory frame are
$$
\boxed{\begin{aligned}
0<\mu<\frac{U^2}{4(1+c_d^2)}&:\ \text{convective instability},\\
\mu>\frac{U^2}{4(1+c_d^2)}&:\ \text{absolute instability}.
\end{aligned}}
$$
For <convective hydrodynamic instability>, a travelling <wave packet> amplifies but the response at each fixed position decays. For <absolute hydrodynamic instability>, that fixed-position response amplifies. The equality is the marginal absolute threshold: its exponential rate is zero and the impulse response has a $t^{-1/2}$ prefactor. For $\mu<0$ there is no temporal growth; $\mu=0$ is temporally marginal. When $U=0$ the convective window is empty. These conclusions use the infinite-line impulse problem; a general <dispersion relation> requires its own spatial-branch or saddle selection.
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