Editing Harmonic oscillator (section)

== Parametric oscillators ==
{{main|Parametric oscillator}}
A [[parametric oscillator]] is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of the oscillator, such as the damping or restoring force.
A familiar example of parametric oscillation is "pumping" on a playground [[swing (seat)|swing]].<ref name=Case>{{cite web |title=Two ways of driving a child's swing |url=http://www.grinnell.edu/academic/physics/faculty/case/swing/ |first=William |last=Case |access-date=27 November 2011 |url-status=dead |archive-url=https://web.archive.org/web/20111209230414/http://www.grinnell.edu/academic/physics/faculty/case/swing |archive-date=9 December 2011 }}</ref><ref name=Case96>{{Cite journal | last1 = Case | first1 = W. B. | title = The pumping of a swing from the standing position | doi = 10.1119/1.18209 | journal = American Journal of Physics | volume = 64 | issue = 3 | pages = 215–220 | year = 1996 |bibcode = 1996AmJPh..64..215C }}</ref><ref name=Roura>{{cite journal |last1=Roura |first1=P. |last2=Gonzalez |first2=J.A. |year=2010 |title=Towards a more realistic description of swing pumping due to the exchange of angular momentum |journal=European Journal of Physics |volume=31 |issue=5 |pages=1195–1207 |doi=10.1088/0143-0807/31/5/020 |bibcode = 2010EJPh...31.1195R |s2cid=122086250 }}</ref>
A person on a moving swing can increase the amplitude of the swing's oscillations without any external drive force (pushes) being applied, by changing the moment of inertia of the swing by rocking back and forth ("pumping") or alternately standing and squatting, in rhythm with the swing's oscillations.  The varying of the parameters drives the system.  Examples of parameters that may be varied are its resonance frequency <math>\omega</math> and damping <math>\beta</math>.

Parametric oscillators are used in many applications.  The classical [[varactor]] parametric oscillator oscillates when the diode's capacitance is varied periodically.  The circuit that varies the diode's capacitance is called the "pump" or "driver".   In microwave electronics, [[waveguide (electromagnetism)|waveguide]]/[[Yttrium aluminum garnet|YAG]] based parametric oscillators operate in the same fashion. The designer varies a parameter periodically to induce oscillations.

Parametric oscillators have been developed as low-noise amplifiers, especially in the radio and microwave frequency range.  Thermal noise is minimal, since a reactance (not a resistance) is varied.  Another common use is frequency conversion, e.g., conversion from audio to radio frequencies.  For example, the [[Optical parametric oscillator]] converts an input [[laser]] wave into two output waves of lower frequency (<math>\omega_s, \omega_i</math>).

Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. Parametric excitation differs from forcing, since the action appears as a time varying modification on a system parameter. This effect is different from regular resonance because it exhibits the [[instability]] phenomenon.