= Solution
A mode has dimensionless damping ratio $k/\alpha_n$, and characteristic roots
$$
s_\pm=-k\pm\sqrt{k^2-\alpha_n^2}.
$$
If $k/\alpha_n\ll1$, the <normal mode> oscillates at $\omega_n=\alpha_n+O(k^2/\alpha_n)$ with exponentially decaying envelope $e^{-kt}$. As $k\to0$, the solution approaches the undamped <wave equation> response.
If $k/\alpha_n\gg1$, the roots are real and negative, with
$$
s_+=-\frac{\alpha_n^2}{2k}+O(\alpha_n^4/k^3),\qquad s_-=-2k+\frac{\alpha_n^2}{2k}+O(\alpha_n^4/k^3).
$$
The mode has a fast transient and a slow nonoscillatory relaxation. After the fast transient, neglecting $y_{tt}$ gives the <diffusion equation> $y_t\simeq c^2y_{xx}/(2k)$ for these long-wavelength modes. At $k=\alpha_n$, the repeated root produces $te^{-kt}$: this is <critical damping>.
Thus \b[the critical parameter is $k/\alpha_n=1$ for each mode], and the fundamental threshold is
$$
\boxed{\frac{kl}{\pi c}=1.}
$$
When $k<\pi c/l$ every mode is oscillatory. For any fixed, finite $k$, sufficiently large $n$ still gives oscillatory modes, so large damping of the lowest modes does not mean that every mode of the full string is overdamped.
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