Editing Harmonic oscillator (section)

===Sinusoidal driving force===
[[File:Mplwp resonance zeta envelope.svg|thumb|300px|Steady-state variation of amplitude with relative frequency <math>\omega/\omega_0</math> and damping <math>\zeta</math> of a driven harmonic oscillator. This plot is also called the harmonic oscillator spectrum or motional spectrum.]]
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{{cite book
 | title = Optics, 3E
 | author = Ajoy Ghatak
 | author-link = Ajoy Ghatak
 | edition = 3rd
 | publisher = Tata McGraw-Hill
 | year = 2005
 | isbn = 978-0-07-058583-6
 | page = 6.10
 | url = https://books.google.com/books?id=jStDc2LmU5IC&pg=PT97 
 }}</ref>
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In the case of a sinusoidal driving force:
<math display="block"> \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + 2\zeta\omega_0\frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = \frac{1}{m} F_0 \sin(\omega t),</math>
where <math>F_0</math> is the driving amplitude, and <math>\omega</math> is the driving [[frequency]] for a sinusoidal driving mechanism. This type of system appears in [[alternating current|AC]]-driven [[RLC circuit]]s ([[Electrical resistance|resistor]]–[[inductor]]–[[capacitor]]) and driven spring systems having internal mechanical resistance or external [[air resistance]].

The general solution is a sum of a [[Transient (oscillation)|transient]] solution that depends on initial conditions, and a [[steady state]] that is independent of initial conditions and depends only on the driving amplitude <math>F_0</math>, driving frequency <math>\omega</math>, undamped angular frequency <math>\omega_0</math>, and the damping ratio <math>\zeta</math>.

The steady-state solution is proportional to the driving force with an induced phase change <math>\varphi</math>:
<math display="block"> x(t) = \frac{F_0}{m Z_m \omega} \sin(\omega t + \varphi),</math>
where
<math display="block" qid=Q6421317> Z_m = \sqrt{\left(2\omega_0\zeta\right)^2 + \frac{1}{\omega^2} (\omega_0^2 - \omega^2)^2}</math>
is the absolute value of the [[Mechanical impedance|impedance]] or [[linear response function]], and
<math display="block"> \varphi = \arctan\left(\frac{2\omega \omega_0\zeta}{\omega^2 - \omega_0^2} \right) + n\pi</math>

is the [[phase (waves)|phase]] of the oscillation relative to the driving force. The phase value is usually taken to be between −180° and 0 (that is, it represents a phase lag, for both positive and negative values of the arctan argument).

For a particular driving frequency called the [[resonance]], or resonant frequency <math display="inline">\omega_r = \omega_0 \sqrt{1 - 2\zeta^2}</math>, the amplitude (for a given <math>F_0</math>) is maximal. This resonance effect only occurs when <math>\zeta < 1 / \sqrt{2}</math>, i.e. for significantly underdamped systems. For strongly underdamped systems the value of the amplitude can become quite large near the resonant frequency.

The transient solutions are the same as the unforced (<math>F_0 = 0</math>) damped harmonic oscillator and represent the system's response to other events that occurred previously.