Editing Resonance (section)

===The driven, damped harmonic oscillator===
{{main|Harmonic oscillator#Driven harmonic oscillators}}

Consider a damped mass on a spring driven by a sinusoidal, externally applied force. [[Newton's second law]] takes the form
{{NumBlk||<math display="block">m\frac{\mathrm{d}^2x}{\mathrm{d}t^2} = F_0 \sin(\omega t)-kx-c\frac{\mathrm{d}x}{\mathrm{d}t}, </math>|{{EquationRef|1}}}}
where ''m'' is the mass, ''x'' is the displacement of the mass from the equilibrium point, ''F''<sub>0</sub> is the driving amplitude, ''ω'' is the driving angular frequency, ''k'' is the spring constant, and ''c'' is the viscous damping coefficient. This can be rewritten in the form
{{NumBlk||<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{F_0}{m} \sin(\omega t),</math>|{{EquationRef|2}}}}
where
* <math display="inline">\omega_0 = \sqrt{k /m}</math> is called the ''undamped [[angular frequency]] of the oscillator'' or the ''natural frequency'',
* <math>\zeta = \frac{c}{2\sqrt{mk}}</math> is called the ''damping ratio''.

Many sources also refer to ''ω''<sub>0</sub> as the ''resonant frequency''. However, as shown below, when analyzing oscillations of the displacement ''x''(''t''), the resonant frequency is close to but not the same as ''ω''<sub>0</sub>. In general the resonant frequency is close to but not necessarily the same as the natural frequency.{{sfn|Hardt|2004}} The RLC circuit example in the next section gives examples of different resonant frequencies for the same system.

The general solution of Equation ({{EquationNote|2}})  is the sum of a [[Transient (oscillation)|transient]] solution that depends on initial conditions and a [[steady state]] solution that is independent of initial conditions and depends only on the driving amplitude ''F''<sub>0</sub>, driving frequency ''ω'', undamped angular frequency ''ω''<sub>0</sub>, and the damping ratio ''ζ''. The transient solution decays in a relatively short amount of time, so to study resonance it is sufficient to consider the steady state solution.

It is possible to write the steady-state solution for ''x''(''t'') as a function proportional to the driving force with an induced [[phase (waves)|phase]] change, ''φ''.
{{NumBlk||<math display="block">x(t) = \frac{F_0}{m \sqrt{\left(2\omega\omega_0\zeta\right)^2 + (\omega_0^2 - \omega^2)^2}} \sin(\omega t + \varphi),</math>|{{EquationRef|3}}}}
where <math> \varphi = \arctan\left(\frac{2\omega \omega_0\zeta}{\omega^2 - \omega_0^2} \right) + n\pi.</math>

The phase value is usually taken to be between −180° and 0 so it represents a phase lag for both positive and negative values of the arctan argument.

[[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 [[simple harmonic oscillator]]]]
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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>
-->

Resonance occurs when, at certain driving frequencies, the steady-state amplitude of ''x''(''t'') is large compared to its amplitude at other driving frequencies. For the mass on a spring, resonance corresponds physically to the mass's oscillations having large displacements from the spring's equilibrium position at certain driving frequencies.  Looking at the amplitude of ''x''(''t'') as a function of the driving frequency ''ω'', the amplitude is maximal at the driving frequency
<math display="block">\omega_r = \omega_0 \sqrt{1 - 2\zeta^2}.</math>

''ω''<sub>''r''</sub> is the '''resonant frequency''' for this system. Again, the resonant frequency does not equal the undamped angular frequency ''ω''<sub>0</sub> of the oscillator. They are proportional, and if the damping ratio goes to zero they are the same, but for non-zero damping they are not the same frequency. As shown in the figure, resonance may also occur at other frequencies near the resonant frequency, including ''ω''<sub>0</sub>, but the maximum response is at the resonant frequency.

Also, ''ω''<sub>''r''</sub> is only real and non-zero if <math display="inline">\zeta < 1 / \sqrt{2}</math>, so this system can only resonate when the harmonic oscillator is significantly underdamped. For systems with a very small damping ratio and a driving frequency near the resonant frequency, the steady state oscillations can become very large.

==== The pendulum ====
For other driven, damped harmonic oscillators whose equations of motion do not look exactly like the mass on a spring example, the resonant frequency remains
<math display="block">\omega_r = \omega_0 \sqrt{1 - 2\zeta^2},</math>
but the definitions of ''ω''<sub>0</sub> and ''ζ'' change based on the physics of the system. For a pendulum of length ''ℓ'' and small displacement angle ''θ'', Equation ({{EquationNote|1}}) becomes
<math display="block"> m\ell\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} = F_0 \sin(\omega t)-mg\theta-c\ell\frac{\mathrm{d}\theta}{\mathrm{d}t}</math>

and therefore
* <math>\omega_0 = \sqrt{\frac{g}{\ell}},</math>
* <math>\zeta = \frac{c}{2m}\sqrt{\frac{\ell}{g}}.</math>