✓ Sources checked · 10 September 2026 · View verification
An underdamped linear oscillator has 0 < ζ < 1. It oscillates around equilibrium with an exponentially decreasing amplitude.
By Matan · Updated
Decay and frequency
For mx″ + bx′ + kx = 0, ζ = b/(2√mk), ω₀ = √(k/m) and ωd = ω₀√(1 − ζ²). The motion is x = A e^(−bt/2m) cos(ωdt + φ), where A and φ depend on initial conditions. Zero damping is the undamped limit: its ideal oscillations do not eventually die away. [1]
Worked oscillator
Take m = 1 kg, k = 25 N/m and b = 6 N·s/m. Then ζ = 0.60, ωd = 4 rad/s and the period is 2π/4 = 1.57 s. The amplitude envelope after one second is e⁻³ = 0.0498 times its initial envelope value.
Underdamping is not automatically a defect
Some resonators are intended to sustain detectable oscillations; other systems need rapid settling. The desired damping depends on the task. A mechanical analogy does not establish the cause of a neurological condition or justify treatment claims.
Use the stated model
The exponential form assumes linear restoring force, velocity-proportional damping and no ongoing drive. A forced oscillator can retain a steady oscillation even though its initial transient decays.
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How this article was checked
Sources checked; calculations checked where shown. The definitions, model assumptions and main explanatory claims were cross-checked against the sources below. Worked-example arithmetic and units were checked separately. These examples are illustrative calculations, not experimental measurements.
Source cross-check and editing assisted by AI. No independent human scientific review is recorded for this revision. The check applies to the version dated 10 September 2026; it is not a guarantee that every possible interpretation is correct.
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Sources and further reading
- OpenStax: Underdamped oscillations — Underdamped oscillations.
Cite this article
Matan. “Underdamping.” Your Physicist. Originally published 2023-03-17; updated 2026-09-10. https://your-physicist.com/underdamping/
Revision history
: Removed unsupported medical claims; corrected the zero-damping limit and added a checked decay example. Added linked sources and a transparent verification record.
