Overdamped approximation ($b\dot{x}+kx=F$)

The overdamped approximation neglects inertia when friction relaxes velocity much faster than position changes. For a damped harmonic oscillatorwith $m,k,b>0$ and $mk/b^2\ll 1$, the velocity-relaxation time $m/b$ is much shorter than the position-relaxation time $b/k$. On the slow time scale, and for sufficiently slowly varying drive, the reduced equation is $b\dot{x}+kx=F(t)$.

For zero drive, the exact characteristic roots approach $-b/m$ and $-k/b$. The first gives a fast initial transient; the second is the decay retained by the approximation. A first-order reduced equation cannot independently fit both initial position and initial velocity, so an initial layer may be needed. Merely being in the overdamping regime $b^2>4mk$ does not ensure a large separation of these time scales.