damped oscillation

Damped oscillation is what real oscillations actually do: they gradually die away. A plucked guitar string fades to silence, a child on a swing slows and stops if nobody keeps pushing, and a car settles after bouncing over a bump. It answers the honest question that ideal SHM sets aside: what happens when friction or drag steadily drains the energy out?

Precisely, a resistive force, such as air drag or friction, often roughly proportional to speed, F = -b v, where b is a damping coefficient, removes a bit of energy every cycle. The amplitude then shrinks exponentially over time, roughly A(t) = A_0 e^(-(b/2m) t). There are three regimes: underdamped, where the system oscillates while slowly fading; critically damped, where it returns to rest in the shortest time without overshooting; and overdamped, where it creeps back so sluggishly that it never oscillates at all.

Damping is often exactly what engineers want. Car shock absorbers and self-closing doors are designed near critical damping so they settle fast without bouncing. One subtle but honest detail: damping also lowers the oscillation frequency a little compared with the undamped natural frequency, because the drag slightly slows each cycle.