Editing Harmonic oscillator (section)

{{Short description|Physical system that responds to a restoring force inversely proportional to displacement}}
{{About|the harmonic oscillator in classical mechanics|its uses in [[quantum mechanics]]|quantum harmonic oscillator}}
{{Use American English|date = February 2019}}
{{Classical mechanics|Core}}

In [[classical mechanics]], a '''harmonic oscillator''' is a system that, when displaced from its [[Mechanical equilibrium|equilibrium]] position, experiences a [[restoring force]] ''F'' [[Proportionality (mathematics)|proportional]] to the [[Displacement (geometry)|displacement]] ''x'':
<math display="block" qid=Q170282> \vec F = -k \vec x, </math>
where ''k'' is a [[positive number|positive]] [[coefficient|constant]].

The harmonic oscillator model is important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as [[Clock|clocks]] and radio circuits.

If ''F'' is the only force acting on the system, the system is called a '''simple harmonic oscillator''', and it undergoes [[simple harmonic motion]]: [[sinusoidal]] [[oscillation]]s about the [[equilibrium point]], with a constant [[amplitude]] and a constant [[frequency]] (which does not depend on the amplitude).

If a frictional force ([[Damping ratio|damping]]) proportional to the [[velocity]] is also present, the harmonic oscillator is described as a '''damped oscillator'''. Depending on the friction coefficient, the system can:
* Oscillate with a frequency lower than in the [[Damping ratio|undamped]] case, and an [[amplitude]] decreasing with time ([[Damping ratio|underdamped]] oscillator).
* Decay to the equilibrium position, without oscillations ([[Damping ratio|overdamped]] oscillator).

The [[Boundary value problem|boundary solution]] between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called [[critically damped]].

If an external time-dependent force is present, the harmonic oscillator is described as a ''driven oscillator''.

Mechanical examples include [[pendulum]]s (with [[Small-angle approximation#Motion of a pendulum|small angles of displacement]]), masses connected to [[spring (device)|springs]], and [[acoustics|acoustical systems]]. Other [[#Equivalent systems|analogous systems]] include electrical harmonic oscillators such as [[RLC circuit]]s. They are the source of virtually all sinusoidal vibrations and waves.