Editing Harmonic oscillator (section)

===Step input===
{{See also|Step response}}

In the case {{math|''ζ'' < 1}} and a unit step input with&nbsp;{{math|1=''x''(0) = 0}}:
<math display="block"> \frac{F(t)}{m} = \begin{cases} \omega _0^2 & t \geq 0 \\ 0 & t < 0 \end{cases}</math>
the solution is
<math display="block"> x(t) = 1 - e^{-\zeta \omega_0 t} \frac{\sin \left( \sqrt{1 - \zeta^2} \omega_0 t + \varphi \right)}{\sin(\varphi)},</math>

with phase ''φ'' given by

<math display="block">\cos \varphi = \zeta.</math>

The time an oscillator needs to adapt to changed external conditions is of the order {{math|1=''τ'' = 1/(''ζω''<sub>0</sub>)}}. In physics, the adaptation is called [[relaxation (physics)|relaxation]], and ''τ'' is called the relaxation time.

In electrical engineering, a multiple of ''τ'' is called the ''settling time'', i.e. the time necessary to ensure the signal is within a fixed departure from final value, typically within 10%. The term ''overshoot'' refers to the extent the response maximum exceeds final value, and ''undershoot'' refers to the extent the response falls below final value for times following the response maximum.