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The small-angle approximation replaces $\sin \theta$ by $\theta$, giving the damped harmonic oscillator equation $\ddot{\theta }+c\dot{\theta }+\theta =0$. Its characteristic roots satisfy $r^2+cr+1=0$.

For $0<c<2$, put $\nu =\sqrt{1-c^2/4}$; the motion is underdamped:It oscillates with decaying amplitude. For $c=2$, the repeated root gives critical damping:For $c>2$, the motion is overdamped:Both real exponents are negative, so there is no oscillatory factor. The two constants are arbitrary in each case; these approximations require the displacement to remain small.

With gravity omitted, the initial data contain only the first fixed-end normal mode, so writeThenBecause $k=\pi c/l$, this oscillator is critically damped, andTherefore

Gas in hydrostatic balance has zero vertical velocity. A dust grain subject to linear drag with aerodynamic stopping time $\tau$ therefore obeysThis is a damped harmonic oscillator. For ${\Omega }_z\tau >1/2$, the motion is underdamped, with angular frequencyand envelope $e^{-t/(2\tau )}$. Critical damping occurs at ${\Omega }_z\tau =1/2$. For ${\Omega }_z\tau <1/2$, the motion is overdamped. In the strong-drag limit ${\Omega }_z\tau \ll 1$, a rapid transient on timescale $\tau$ leaves slow dust settling in an astrophysical disk at rate ${\Omega }_z^2\tau$; in the weak-drag limit the grain makes many damped vertical oscillations.