= Solution
The <angular frequency> of an underdamped mode is
$$
\boxed{\omega_n=\sqrt{\frac{c^2n^2\pi^2}{4L^2}-\frac{\alpha^2}{4}}.}
$$
For ordinary <damping>, $\alpha\geq0$, mode $n$ oscillates precisely when $\alpha<cn\pi/L$. In particular, \b[all modes, including the fundamental, oscillate when $\alpha<c\pi/L$]. There are some oscillatory modes for every finite $\alpha$, since the allowed $n$ are unbounded; no finite damping coefficient overdamps the entire ideal string spectrum.
If $\alpha<0$, the same mode has negative <damping>: oscillatory amplitudes grow as $e^{|\alpha|t/2}$. For modes with real characteristic roots, both roots are positive, since $|\alpha|/2>\sqrt{\alpha^2/4-c^2k_n^2}$. Hence those modes also grow. Negative damping makes the equilibrium unstable.
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