= Damped wave equation
{title2=$u_{tt}+\alpha u_t=c^2u_{xx}$}
A <wave equation> with a linear <damping> term and positive <wave speed> $c$ separates into spatial <normal modes> and <damped harmonic oscillators>. For a spatial <eigenvalue> $k^2$, the time factor obeys $T''+\alpha T'+c^2k^2T=0$. Its characteristic roots are $-\alpha/2\pm\sqrt{\alpha^2/4-c^2k^2}$. Thus <underdamping> requires $|\alpha|<2c|k|$, <critical damping> gives a repeated root and the second solution $te^{-\alpha t/2}$, and <overdamping> gives two real exponential rates. Negative $\alpha$ amplifies rather than damps the motion.
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