damping

Damping is the process by which an oscillating system loses energy to its environment, causing the amplitude of oscillation to decrease over time. In the simplest model, the damping force is proportional to velocity: F = −γv, where γ is the damping coefficient. The amplitude then decays exponentially as e^(−γt/2m).

Three regimes emerge depending on the ratio of damping to stiffness. In the underdamped case, the system oscillates with gradually shrinking amplitude — a struck tuning fork, a plucked guitar string, a pendulum in air. In the critically damped case, the system returns to equilibrium as fast as possible without oscillating — this is what door closers and car shock absorbers aim for. In the overdamped case, the system creeps back to rest sluggishly, without any oscillation at all — like a pendulum immersed in honey.

No real oscillator is perfectly undamped. Air resistance, friction at the pivot, internal flexing of the material — something always drains energy. The question is only how fast.