= Solution
The transient is the homogeneous part of a <damped harmonic oscillator>, $m\ddot x+b\dot x+kx=0$. Define $\alpha=b/(2m)$ and $\omega_0=\sqrt{k/m}$. Its characteristic roots are $-\alpha\pm\sqrt{\alpha^2-\omega_0^2}$.
* For $\alpha<\omega_0$, the <underdamped> response is $x=e^{-\alpha t}(A\cos\omega_dt+B\sin\omega_dt)$, with $\omega_d=\sqrt{\omega_0^2-\alpha^2}$. The displacement oscillates with a decaying envelope.
* At critical damping, $x=(A+Bt)e^{-\omega_0t}$. It has a repeated root, so two identical exponentials are not an independent pair of solutions.
* For $\alpha>\omega_0$, two real negative exponentials produce an <overdamped> return without sustained oscillation. In the strong-damping limit the slow decay rate is approximately $k/b$, not $b/m$.
Initial position and velocity determine A,B. For an imposed force or support motion, the full solution is a particular response plus this transient, which decays for b>0 and k>0. The mechanical energy obeys $d(K+U)/dt=-b\dot x^2$. In light damping its cycle-averaged envelope decays as $e^{-2\alpha t}$, with $Q\simeq\omega_0/(2\alpha)$. Initial conditions can still produce an overshoot in a nonoscillatory regime; the damping category describes the roots rather than every possible trajectory.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2010/paper-1-pa2010-oscillator-transients.png]
{title=Oscillator transients}
{description=Dimensionless release from unit displacement and zero velocity for three damping regimes.}
{source=paper-1-pa2010-oscillator-transients.py}
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