Difference between revisions 111804241 and 111804244 on dewiki

A '''parametric oscillator''' is a simple harmonic oscillator whose parameters
(its resonant frequency <math>\omega</math> and damping <math>\beta</math>)
vare variabley in time in a defined way

:<math>
\frac{d^{2}x}{dt^{2}} + \beta(t) \frac{dx}{dt} + \omega^{2}(t) x = 0
</math>

This equation is linear in <math>x(t)</math>.  TBy assumption, the parameters 
<math>\omega^{2}</math> and <math>\beta</math> depend only ⏎
⏎
on time and notdo ''not'' depend on the state of the oscillator.  In general, ⏎
⏎
<math>\beta(t)</math> and/or <math>\omega^{2}(t)</math> ⏎
⏎
are assumed to vary periodically with the same period <math>T</math>.

⏎
⏎
Remarkably, if the parameters vary at roughly ''twice'' the natural ⏎
⏎
frequency of the oscillator (defined below), the oscillator phase-locks⏎
⏎
  to the parametric variation and absorbs energy at a rate proportional to ⏎
⏎
the energy it already has.  Without a compensating energy-loss mechanism, ⏎
⏎
the oscillation amplitude grows exponentially. (This phenomenon is called ⏎
⏎
'''parametric excitation''', '''parametric resonance''' or '''parametric pumping'''.)  ⏎
⏎
However, if the initial amplitude is zero, it will remain so; this ⏎
⏎
distinguishes it from the non-parametric resonance of driven simple ⏎
⏎
[[harmonic oscillator]]s, in which the amplitude grows linearly in time ⏎
⏎
regardless of the initial state.  

⏎
⏎
A familiar experience of parametric oscillation is playing on a swing. ⏎
⏎
By alternately raising and lowering their center of mass (changing their⏎
⏎
  moment of inertia and, thus, the resonant frequency) at key points⏎
⏎
  in the swing, children can quickly reach large amplitudes provided⏎
⏎
  that they have some amplitude to start with (e.g., get a push).  Doing⏎
⏎
  so at rest, however, goes nowhere.

⏎
⏎
==Transformation of the equation==

We begin by making a change of variables

:<math>
q(t) \equiv e^{D(t)} x(t)
</math>

(contracted; show full)
* [[Optical parametric oscillator]]
* [[Optical parametric amplifier]]

[[Category:Oscillators]]
[[Category:Amplifiers]]
[[Category:Dynamical systems]]
[[Category:Ordinary differential equations]]