Editing Resonance

{{Short description|Physical characteristic of oscillating systems}}
{{About|resonance in physics}}
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[[File:Resonance.PNG|thumb|upright=2|Increase of amplitude as damping decreases and frequency approaches resonant frequency of a driven [[Damping ratio|damped]] [[simple harmonic oscillator]].{{sfn|Ogata|2005|p=617}}{{sfn|Ghatak |2005|p=6.10}}]]

'''Resonance''' is a [[phenomenon]] that occurs when an object or [[system]] is subjected to an external force or [[vibration]] whose [[frequency]]  matches a '''resonant frequency''' (or '''resonance frequency''') of the system, defined as a frequency that generates a maximum amplitude response in the system.  When this happens, the object or system absorbs energy from the external force and starts vibrating with a larger [[amplitude]]. Resonance can occur in various systems, such as mechanical, electrical, or acoustic systems, and it is often desirable in certain applications, such as musical instruments or radio receivers. However, resonance can also be detrimental, leading to excessive vibrations or even structural failure in some cases.<ref>{{Cite book |last=Taylor |first=John R. |title=Classical Mechanics |date=22 January 2023 |publisher=University Science Books |publication-date=1 March 2003 |pages=187}}</ref> 

All systems, including molecular systems and particles, tend to vibrate at a natural frequency depending upon their structure; when there is very little damping this frequency is approximately equal to, but slightly above, the resonant frequency. When an [[Oscillation|oscillating]] force, an external vibration, is applied at a resonant frequency of a dynamic system, object, or particle, the outside vibration will cause the system to oscillate at a higher [[amplitude]] (with more force) than when the same force is applied at other, non-resonant frequencies.{{sfn|Halliday|Resnick|Walker|2005|p=324}}

The resonant frequencies of a system can be identified when the response to an external vibration creates an amplitude that is a [[relative maximum]] within the system.{{sfn|Halliday|Resnick|Walker|2005|p=324}} Small periodic forces that are near a resonant frequency of the system have the ability to produce large amplitude [[oscillation]]s in the system due to the storage of [[vibrational energy]].

Resonance phenomena occur with all types of vibrations or [[wave]]s: there is [[mechanical resonance]], [[orbital resonance]], [[acoustic resonance]], [[Electromagnetic radiation|electromagnetic]] resonance, [[nuclear magnetic resonance]] (NMR), [[electron paramagnetic resonance|electron spin resonance]] (ESR) and resonance of quantum [[wave function]]s. Resonant systems can be used to generate vibrations of a specific frequency (e.g., [[musical instrument]]s), or pick out specific frequencies from a complex vibration containing many frequencies (e.g., filters).Some recent theoretical models explore resonance as a unifying structure across quantum, gravitational, and cosmological scales, proposing phase alignment rather than time as the key organizing principle.<ref>Kanda, Yu. "Theory of Resonance – Series Index", Zenodo, 2025. https://doi.org/10.5281/zenodo.15522546</ref>

The term ''resonance'' (from [[Latin]] ''resonantia'', 'echo', from ''resonare'', 'resound') originated from the field of acoustics, particularly the [[sympathetic resonance]] observed in musical instruments, e.g., when one string starts to vibrate and produce sound after a different one is struck.

==Overview==

Resonance occurs when a system is able to store and easily transfer energy between two or more different [[storage mode]]s (such as kinetic energy and potential energy in the case of a simple pendulum). However, there are some losses from cycle to cycle, called [[Damping ratio|damping]]. When damping is small, the resonant frequency is approximately equal to the [[natural frequency]] of the system, which is a frequency of unforced vibrations. Some systems have multiple and distinct resonant frequencies.

==Examples==
{{more citations needed section|date=February 2024}}
[[File:Little girl on swing.jpg|thumb|Pushing a person in a [[swing (seat)|swing]] is a common example of resonance. The loaded swing, a [[pendulum]], has a [[natural frequency]] of oscillation, its resonant frequency, and resists being pushed at a faster or slower rate.]]

A familiar example is a playground [[Swing (seat)|swing]], which acts as a [[pendulum]]. Pushing a person in a swing in time with the natural interval of the swing (its resonant frequency) makes the swing go higher and higher (maximum amplitude), while attempts to push the swing at a faster or slower tempo produce smaller arcs.<ref name="Hüwel">{{cite book
  | last = Hüwel
  | first = Lutz 
  | title = Of Clocks and Time
  | publisher = Morgan and Claypool
  | date = 2018
  | location = 
  | language = 
  | url = https://books.google.com/books?id=Ej9iDwAAQBAJ&pg=SA2-PA26
  | doi = 
  | id = 
  | isbn = 9781681740966
  | mr = 
  | zbl = 
  | jfm =}}</ref>{{rp|p.2–24}} This is because the energy the swing absorbs is maximized when the pushes match the swing's natural oscillations.

Resonance occurs widely in nature, and is exploited in many devices. It is the mechanism by which virtually all [[sinusoidal]] waves and vibrations are generated. For example, when hard objects like [[metal]], [[glass]], or [[wood]] are struck, there are brief resonant vibrations in the object.<ref name="Hüwel"/>{{rp|p.2–24}} Light and other short wavelength [[electromagnetic radiation]] is produced by resonance on an [[Atomic spacing|atomic scale]], such as [[electron]]s in atoms.  Other examples of resonance include:

* Timekeeping mechanisms of modern clocks and watches, e.g., the [[balance wheel]] in a mechanical [[watch]] and the [[quartz crystal]] in a [[quartz watch]]
* [[Tidal resonance]] of the [[Bay of Fundy]]
* [[Acoustic resonance]]s of [[musical instruments]] and the human [[vocal tract]]
* Shattering of a crystal wineglass when exposed to a musical tone of the right pitch (its resonant frequency)
* [[Friction idiophone]]s, such as making a glass object (glass, bottle, vase) [[vibration|vibrate]] by rubbing around its rim with a fingertip
* [[Electrical resonance]] of [[tuned circuit]]s in [[radio]]s and [[TV]]s that allow radio frequencies to be selectively received
* Creation of [[Coherence (physics)|coherent]] light by [[optical resonance]] in a [[laser]] [[Optical cavity|cavity]]
* [[Orbital resonance]] as exemplified by some [[natural satellite|moons]] of the [[Solar System]]'s [[giant planet]]s and resonant groups such as the [[plutino]]s
* Material resonances in atomic scale are the basis of several [[spectroscopy|spectroscopic]] techniques that are used in [[condensed matter physics]]
**[[Electron spin resonance]]
**[[Mössbauer effect]]
**[[Nuclear magnetic resonance]]

==Linear systems==
Resonance manifests itself in many [[Linear system|linear]] and [[nonlinear system]]s as oscillations around an equilibrium point. When the system is driven by a sinusoidal external input, a measured output of the system may oscillate in response. The ratio of the amplitude of the output's steady-state oscillations to the input's oscillations is called the gain, and the gain can be a function of the frequency of the sinusoidal external input. Peaks in the gain at certain frequencies correspond to resonances, where the amplitude of the measured output's oscillations are disproportionately large.

Since many linear and nonlinear systems that oscillate are modeled as [[harmonic oscillator]]s near their equilibria, a derivation of the resonant frequency for a driven, damped harmonic oscillator is shown. An [[RLC circuit]] is used to illustrate connections between resonance and a system's transfer function, frequency response, poles, and zeroes. Building off the RLC circuit example, these connections for higher-order linear systems with multiple inputs and outputs are generalized.

===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]]]]
<!--<ref>{{cite book|last=Ogata|first=Katsuhiko|title=System dynamics|year=2004|publisher=Pearson Education|location=Upper Saddle River, NJ|isbn=9780131247147|edition=4th}}</ref><ref>
{{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>

=== RLC series circuits ===
{{summarize section|date=January 2021}}
{{See also|RLC Circuit#Series circuit}}
[[File:Rajz RLC soros.svg|thumb|An RLC series circuit]]
Consider a [[electrical network|circuit]] consisting of a [[resistor]] with resistance ''R'', an [[inductor]] with inductance ''L'', and a [[capacitor]] with capacitance ''C'' connected in series with current ''i''(''t'') and driven by a [[voltage]] source with voltage ''v''<sub>''in''</sub>(''t''). The voltage drop around the circuit is
{{NumBlk|:|<math>L \frac{di(t)}{dt} + Ri(t) + V(0)+\frac{1}{C} \int_{0}^t i(\tau)d\tau = v_\text{in}(t).</math>|{{EquationRef|4}}}}

Rather than analyzing a candidate solution to this equation like in the mass on a spring example above, this section will analyze the frequency response of this circuit. Taking the [[Laplace transform]] of Equation ({{EquationNote|4}}),
<math display="block">sLI(s) + RI(s) + \frac{1}{sC}I(s) = V_\text{in}(s),</math>
where ''I''(''s'') and ''V''<sub>''in''</sub>(''s'') are the Laplace transform of the current and input voltage, respectively, and ''s'' is a [[complex number|complex]] frequency parameter in the Laplace domain. Rearranging terms,
<math display="block">I(s) = \frac{s}{s^2L + Rs + \frac{1}{C}} V_\text{in}(s).</math>

====Voltage across the capacitor====
An RLC circuit in series presents several options for where to measure an output voltage. Suppose the output voltage of interest is the voltage drop across the capacitor. As shown above, in the Laplace domain this voltage is
<math display="block">V_\text{out}(s) = \frac{1}{sC}I(s)</math>
or
<math display="block">V_\text{out}= \frac{1}{LC(s^2 + \frac{R}{L}s + \frac{1}{LC})} V_\text{in}(s).</math>

Define for this circuit a natural frequency and a damping ratio,
<math display="block"> \omega_0 = \frac{1}{\sqrt{LC}},</math>
<math display="block"> \zeta = \frac{R}{2}\sqrt{\frac{C}{L}}.</math>

The ratio of the output voltage to the input voltage becomes
<math display="block">H(s) \triangleq \frac{V_\text{out}(s)}{V_\text{in}(s)} = \frac{\omega_0^2}{s^2 + 2\zeta\omega_0s + \omega_0^2}</math>

''H''(''s'') is the [[transfer function]] between the input voltage and the output voltage. This transfer function has two [[zeros and poles|poles]]–roots of the polynomial in the transfer function's denominator–at
{{NumBlk||<math display="block">s = -\zeta\omega_0 \pm i\omega_0\sqrt{1-\zeta^2}</math>|{{EquationRef|5}}}}

and no zeros–roots of the polynomial in the transfer function's numerator. Moreover, for {{math|''ζ'' ≤ 1}}, the magnitude of these poles is the natural frequency ''ω''<sub>0</sub> and that for {{math|''ζ'' < 1/<math>\sqrt{2}</math>}}, our condition for resonance in the harmonic oscillator example, the poles are closer to the imaginary axis than to the real axis.

Evaluating ''H''(''s'') along the imaginary axis {{math|''s'' {{=}} ''iω''}}, the transfer function describes the frequency response of this circuit. Equivalently, the frequency response can be analyzed by taking the [[Fourier transform]] of Equation ({{EquationNote|4}}) instead of the Laplace transform.  The transfer function, which is also complex, can be written as a gain and phase,
<math display="block"> H(i\omega) = G(\omega)e^{i\Phi(\omega)}.</math>

[[File:RLC Series Circuit Bode Magnitude Plot.svg|thumb|upright=1.35|Bode magnitude plot for the voltage across the elements of an RLC series circuit. Natural frequency {{math|''ω''<sub>0</sub> {{=}} 1 rad/s}}, damping ratio {{math|''ζ'' {{=}} 0.4}}. The capacitor voltage peaks below the circuit's natural frequency, the inductor voltage peaks above the natural frequency, and the resistor voltage peaks at the natural frequency with a peak gain of one. The gain for the voltage across the capacitor and inductor combined in series shows antiresonance, with gain going to zero at the natural frequency.]]

A sinusoidal input voltage at frequency ''ω'' results in an output voltage at the same frequency that has been scaled by ''G''(''ω'') and has a phase shift ''Φ''(''ω''). The gain and phase can be plotted versus frequency on a [[Bode plot]]. For the RLC circuit's capacitor voltage, the gain of the transfer function ''H''(''iω'') is
{{NumBlk||<math display="block"> G(\omega) = \frac{\omega_0^2}{\sqrt{\left(2\omega\omega_0\zeta\right)^2 + (\omega_0^2 - \omega^2)^2}}.</math>|{{EquationRef|6}}}}

Note the similarity between the gain here and the amplitude in Equation ({{EquationNote|3}}). Once again, the gain is maximized at the '''resonant frequency'''
<math display="block">\omega_r = \omega_0 \sqrt{1 - 2\zeta^2}.</math>

Here, the resonance corresponds physically to having a relatively large amplitude for the steady state oscillations of the voltage across the capacitor compared to its amplitude at other driving frequencies.

==== Voltage across the inductor ====
The resonant frequency need not always take the form given in the examples above. For the RLC circuit, suppose instead that the output voltage of interest is the voltage across the inductor. As shown above, in the Laplace domain the voltage across the inductor is
<math display="block">V_\text{out}(s) = sLI(s),</math>
<math display="block">V_\text{out}(s) = \frac{s^2}{s^2 + \frac{R}{L}s + \frac{1}{LC}} V_\text{in}(s),</math>
<math display="block">V_\text{out}(s) = \frac{s^2}{s^2 + 2\zeta\omega_0s + \omega_0^2} V_\text{in}(s),</math>

using the same definitions for ''ω''<sub>0</sub> and ''ζ'' as in the previous example. The transfer function between ''V''<sub>in</sub>(''s'') and this new ''V''<sub>out</sub>(''s'') across the inductor is
<math display="block">H(s) = \frac{s^2}{s^2 + 2\zeta\omega_0s + \omega_0^2}.</math>

This transfer function has the same poles as the transfer function in the previous example, but it also has two zeroes in the numerator at {{Nowrap|''s'' {{=}} 0}}. Evaluating ''H''(''s'') along the imaginary axis, its gain becomes
<math display="block"> G(\omega) = \frac{\omega^2}{\sqrt{\left(2\omega\omega_0\zeta\right)^2 + (\omega_0^2 - \omega^2)^2}}.</math>

Compared to the gain in Equation ({{EquationNote|6}}) using the capacitor voltage as the output, this gain has a factor of ''ω''<sup>2</sup> in the numerator and will therefore have a different resonant frequency that maximizes the gain. That frequency is
<math display="block">\omega_r = \frac{\omega_0}{\sqrt{1 - 2\zeta^2}},</math>

So for the same RLC circuit but with the voltage across the inductor as the output, the resonant frequency is now ''larger'' than the natural frequency, though it still tends towards the natural frequency as the damping ratio goes to zero. That the same circuit can have different resonant frequencies for different choices of output is not contradictory. As shown in Equation ({{EquationNote|4}}), the voltage drop across the circuit is divided among the three circuit elements, and each element has different dynamics. The capacitor's voltage grows slowly by integrating the current over time and is therefore more sensitive to lower frequencies, whereas the inductor's voltage grows when the current changes rapidly and is therefore more sensitive to higher frequencies. While the circuit as a whole has a natural frequency where it tends to oscillate, the different dynamics of each circuit element make each element resonate at a slightly different frequency.

==== Voltage across the resistor ====
Suppose that the output voltage of interest is the voltage across the resistor. In the Laplace domain the voltage across the resistor is
<math display="block">V_\text{out}(s) = RI(s),</math>
<math display="block">V_\text{out}(s) = \frac{Rs}{L\left(s^2 + \frac{R}{L}s + \frac{1}{LC}\right)} V_\text{in}(s),</math>

and using the same natural frequency and damping ratio as in the capacitor example the transfer function is
<math display="block">H(s) = \frac{2\zeta\omega_0s}{s^2 + 2\zeta\omega_0s+\omega_0^2}.</math>

This transfer function also has the same poles as the previous RLC circuit examples, but it only has one zero in the numerator at ''s'' = 0. For this transfer function, its gain is
<math display="block"> G(\omega) = \frac{2\zeta\omega_0\omega}{\sqrt{\left(2\omega\omega_0\zeta\right)^2 + (\omega_0^2 - \omega^2)^2}}.</math>

The resonant frequency that maximizes this gain is
<math display="block">\omega_r = \omega_0,</math>
and the gain is one at this frequency, so the voltage across the resistor resonates ''at'' the circuit's natural frequency and at this frequency the amplitude of the voltage across the resistor equals the input voltage's amplitude.

==== Antiresonance ====
{{Main|Antiresonance}}

Some systems exhibit antiresonance that can be analyzed in the same way as resonance. For antiresonance, the amplitude of the response of the system at certain frequencies is disproportionately ''small'' rather than being disproportionately large. In the RLC circuit example, this phenomenon can be observed by analyzing both the inductor and the capacitor combined.

Suppose that the output voltage of interest in the RLC circuit is the voltage across the inductor ''and'' the capacitor combined in series. Equation ({{EquationNote|4}}) showed that the sum of the voltages across the three circuit elements sums to the input voltage, so measuring the output voltage as the sum of the inductor and capacitor voltages combined is the same as ''v''<sub>''in''</sub> minus the voltage drop across the resistor. The previous example showed that at the natural frequency of the system, the amplitude of the voltage drop across the resistor ''equals'' the amplitude of ''v''<sub>''in''</sub>, and therefore the voltage across the inductor and capacitor combined has zero amplitude. We can show this with the transfer function.

The sum of the inductor and capacitor voltages is
<math display="block">V_\text{out}(s) = (sL+\frac{1}{sC})I(s),</math>
<math display="block">V_\text{out}(s) = \frac{s^2+\frac{1}{LC}}{s^2 + \frac{R}{L}s + \frac{1}{LC}} V_\text{in}(s).</math>

Using the same natural frequency and damping ratios as the previous examples, the transfer function is
<math display="block">H(s) = \frac{s^2+\omega_0^2}{s^2 + 2\zeta\omega_0s + \omega_0^2}.</math>

This transfer has the same poles as the previous examples but has zeroes at
{{NumBlk||<math display="block">s = \pm i\omega_0.</math>|{{EquationRef|7}}}}

Evaluating the transfer function along the imaginary axis, its gain is
<math display="block">G(\omega) = \frac{\omega_0^2-\omega^2}{\sqrt{\left(2\omega\omega_0\zeta\right)^2 + (\omega_0^2 - \omega^2)^2}}.</math>

Rather than look for resonance, i.e., peaks of the gain, notice that the gain goes to zero at ''ω'' = ''ω''<sub>0</sub>, which complements our analysis of the resistor's voltage. This is called '''antiresonance''', which has the opposite effect of resonance. Rather than result in outputs that are disproportionately large at this frequency, this circuit with this choice of output has no response at all at this frequency. The frequency that is filtered out corresponds exactly to the zeroes of the transfer function, which were shown in Equation ({{EquationNote|7}}) and were on the imaginary axis.

==== Relationships between resonance and frequency response in the RLC series circuit example ====
These RLC circuit examples illustrate how resonance is related to the frequency response of the system. Specifically, these examples illustrate:
* How resonant frequencies can be found by looking for peaks in the gain of the transfer function between the input and output of the system, for example in a Bode magnitude plot
* How the resonant frequency for a single system can be different for different choices of system output
* The connection between the system's natural frequency, the system's damping ratio, and the system's resonant frequency
* The connection between the system's natural frequency and the magnitude of the transfer function's poles, pointed out in Equation ({{EquationNote|5}}), and therefore a connection between the poles and the resonant frequency
* A connection between the transfer function's zeroes and the shape of the gain as a function of frequency, and therefore a connection between the zeroes and the resonant frequency that maximizes gain
* A connection between the transfer function's zeroes and antiresonance

The next section extends these concepts to resonance in a general linear system.

=== Generalizing resonance and antiresonance for linear systems ===

Next consider an arbitrary linear system with multiple inputs and outputs. For example, in [[state-space representation]] a third order [[linear time-invariant system]] with three inputs and two outputs might be written as
<math display="block">\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} =
 A \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} + B \begin{bmatrix} u_1(t) \\ u_2(t) \\ u_3(t) \end{bmatrix},</math>
<math display="block">\begin{bmatrix} y_1(t) \\ y_2(t) \end{bmatrix}
= C \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} + D \begin{bmatrix} u_1(t) \\ u_2(t) \\ u_3(t) \end{bmatrix},</math>
where ''u''<sub>''i''</sub>(''t'') are the inputs, ''x''<sub>''i''</sub>(t) are the state variables,  ''y''<sub>''i''</sub>(''t'') are the outputs, and ''A'', ''B'', ''C'', and ''D'' are matrices describing the dynamics between the variables.

This system has a [[transfer function matrix]] whose elements are the transfer functions between the various inputs and outputs. For example,
<math display="block">
\begin{bmatrix} Y_1(s) \\ Y_2(s) \end{bmatrix} = 
\begin{bmatrix} 
H_{11}(s) & H_{12}(s) & H_{13}(s) \\
H_{21}(s) & H_{22}(s) & H_{23}(s)
\end{bmatrix}
\begin{bmatrix} U_1(s) \\ U_2(s) \\ U_3(s) \end{bmatrix}.
</math>

Each ''H''<sub>''ij''</sub>(''s'') is a scalar transfer function linking one of the inputs to one of the outputs. The RLC circuit examples above had one input voltage and showed four possible output voltages–across the capacitor, across the inductor, across the resistor, and across the capacitor and inductor combined in series–each with its own transfer function. If the RLC circuit were set up to measure all four of these output voltages, that system would have a 4×1 transfer function matrix linking the single input to each of the four outputs.

Evaluated along the imaginary axis, each ''H''<sub>''ij''</sub>(''iω'') can be written as a gain and phase shift,
<math display="block">H_{ij}(i\omega) = G_{ij}(\omega)e^{i\Phi_{ij}(\omega)}.</math>

Peaks in the gain at certain frequencies correspond to resonances between that transfer function's input and output, assuming the system is [[exponential stability|stable]].

Each transfer function ''H''<sub>''ij''</sub>(''s'') can also be written as a fraction whose numerator and denominator are polynomials of ''s''.
<math display="block">H_{ij}(s) = \frac{N_{ij}(s)}{D_{ij}(s)}.</math>

The complex roots of the numerator are called zeroes, and the complex roots of the denominator are called poles. For a stable system, the positions of these poles and zeroes on the complex plane give some indication of whether the system can resonate or antiresonate and at which frequencies. In particular, any stable or [[marginal stability|marginally stable]], complex conjugate pair of poles with imaginary components can be written in terms of a natural frequency and a damping ratio as
<math display="block">s = -\zeta\omega_0 \pm i\omega_0\sqrt{1-\zeta^2},</math>
as in Equation ({{EquationNote|5}}). The natural frequency ''ω''<sub>0</sub> of that pole is the magnitude of the position of the pole on the complex plane and the damping ratio of that pole determines how quickly that oscillation decays. In general,{{sfn|Hardt|2004}}
* Complex conjugate pairs of ''poles'' near the imaginary axis correspond to a peak or resonance in the frequency response in the vicinity of the pole's natural frequency. If the pair of poles is ''on'' the imaginary axis, the gain is infinite at that frequency.
* Complex conjugate pairs of ''zeroes'' near the imaginary axis correspond to a notch or antiresonance in the frequency response in the vicinity of the zero's frequency, i.e., the frequency equal to the magnitude of the zero. If the pair of zeroes is ''on'' the imaginary axis, the gain is zero at that frequency.

In the RLC circuit example, the first generalization relating poles to resonance is observed in Equation ({{EquationNote|5}}). The second generalization relating zeroes to antiresonance is observed in Equation ({{EquationNote|7}}). In the examples of the harmonic oscillator, the RLC circuit capacitor voltage, and the RLC circuit inductor voltage, "poles near the imaginary axis" corresponds to the significantly underdamped condition ζ < 1/<math>\sqrt{2}</math>.

== Standing waves ==
{{main|Standing wave}}
[[File:Animación1.gif|thumb|A mass on a spring has one [[natural frequency]], as it has a single [[degrees of freedom (engineering)|degree of freedom]]]]
A physical system can have as many natural frequencies as it has [[degrees of freedom (engineering)|degrees of freedom]] and can resonate near each of those natural frequencies. A mass on a spring, which has one degree of freedom, has one natural frequency. A [[double pendulum]], which has two degrees of freedom, can have two natural frequencies. As the number of coupled harmonic oscillators increases, the time it takes to transfer energy from one to the next becomes significant. Systems with very large numbers of degrees of freedom can be thought of as [[continuum mechanics|continuous]] rather than as having discrete oscillators.{{citation needed|date=January 2021}}

Energy transfers from one oscillator to the next in the form of waves. For example, the string of a guitar or the surface of water in a bowl can be modeled as a continuum of small coupled oscillators and waves can travel along them. In many cases these systems have the potential to resonate at certain frequencies, forming [[standing wave]]s with large-amplitude oscillations at fixed positions. Resonance in the form of standing waves underlies many familiar phenomena, such as the sound produced by musical instruments, electromagnetic cavities used in lasers and microwave ovens, and energy levels of atoms.{{citation needed|date=January 2021}}

===Standing waves on a string===
[[File:Standing wave 2.gif|thumb|alt=animation of a standing wave |A [[standing wave]] (in black), created when two waves moving from left and right meet and superimpose]]
When a string of fixed length is driven at a particular frequency, a wave propagates along the string at the same frequency. The waves [[Reflection (physics)|reflect]] off the ends of the string, and eventually a [[steady state]] is reached with waves traveling in both directions. The waveform is the [[superposition principle|superposition]] of the waves.{{sfn|Halliday|Resnick|Walker|2005|p=432}}

At certain frequencies, the steady state waveform does not appear to travel along the string. At fixed positions called [[Node (physics)|nodes]], the string is never [[Displacement (geometry)|displaced]]. Between the nodes the string oscillates and exactly halfway between the nodes–at positions called anti-nodes–the oscillations have their largest amplitude.{{sfn|Halliday|Resnick|Walker|2005|pp=431&ndash;432}}{{sfn|Serway|Faughn|1992|p=472}}<ref>{{cite AV media | date = May 21, 2014 | title = String Resonance | url = http://digitalsoundandmusic.com/video/?tutorial=oZ38Y0K8e-Y | access-date = August 22, 2020 | publisher = Digital Sound & Music | id = YouTube Video ID: oZ38Y0K8e-Y}}</ref>

[[Image:Standing waves on a string.gif|thumb|upright|Standing waves in a string&nbsp;&ndash; the [[fundamental frequency|fundamental]] mode and the first 5 [[harmonic]]s.]]

For a string of length <math>L</math> with fixed ends, the displacement <math>y(x,t)</math> of the string perpendicular to the <math>x</math>-axis at time <math>t</math> is{{sfn|Halliday|Resnick|Walker|2005|p=432}}
<math display="block"> y(x,t) = 2y_\text{max}\sin(kx) \cos(2\pi ft), </math>

where
*<math>y_\text{max}</math> is the [[amplitude]] of the left- and right-traveling waves interfering to form the standing wave,
*<math>k</math> is the [[wave number]],
*<math>f</math> is the [[frequency]].

The frequencies that resonate and form standing waves relate to the length of the string as{{sfn|Halliday|Resnick|Walker|2005|p=434}}{{sfn|Serway|Faughn|1992|p=472}}
<math display="block"> f = \frac{nv}{2L}, </math>
<math display="block"> n = 1,2,3,\dots</math>

where <math>v</math> is the speed of the wave and the integer <math>n</math> denotes different modes or [[harmonic]]s. The standing wave with {{math|1=''n'' = 1}} oscillates at the [[fundamental frequency]] and has a wavelength that is twice the length of the string. The possible modes of oscillation form a [[Harmonic series (mathematics)|harmonic series]].{{sfn|Halliday|Resnick|Walker|2005|p=434}}

==Resonance in complex networks==

A generalization to complex networks of coupled harmonic oscillators shows that such systems have a finite number of natural resonant frequencies, related to the topological structure of the network itself. In particular, such frequencies result related to the eigenvalues of the network's Laplacian matrix. Let <math>{\bf A}</math> be the [[adjacency matrix]] describing the topological structure of the network and <math>{\bf L}={\bf K}-{\bf A}</math> the corresponding [[Laplacian matrix]], where <math>{\bf K}={\rm diag}\, \{ k_{i}\}</math> is the diagonal matrix of the degrees of the network's nodes. Then, for a network of classical and identical harmonic oscillators, when a sinusoidal driving force <math>f(t)=F_{0}\sin \omega t</math> is applied to a specific node, the global resonant frequencies of the network are given by <math>\omega_{i}=\sqrt{1+\mu_{i}}</math> where <math>\mu_{i}</math> are the [[Eigenvalues and eigenvectors|eigenvalues]] of the Laplacian <math>{\bf L}</math>.<ref name="Bartesaghi2023">{{cite journal |last1=Bartesaghi |first1=Paolo |title=Notes on resonant and synchronized states in complex networks |journal=Chaos |date=2023 |volume=33 |issue=3 |page=033120 |doi=10.1063/5.0134285 |pmid=37003810 |arxiv=2207.11507 |bibcode=2023Chaos..33c3120B |s2cid=251040250 |issn=1054-1500}}</ref>

==Types==

===Mechanical===
{{Main|Mechanical resonance|String resonance}}
[[File:Resonating mass experiment.jpg|thumb|upright|School resonating mass experiment]]

[[Mechanical resonance]] is the tendency of a [[mechanics|mechanical system]] to absorb more energy when the [[frequency]] of its oscillations matches the system's natural frequency of [[vibration]] than it does at other frequencies. It may cause violent swaying motions and even catastrophic failure in improperly constructed structures including bridges, buildings, trains, and aircraft. When designing objects, [[engineers]]  must ensure the mechanical resonance frequencies of the component parts do not match driving vibrational frequencies of motors or other oscillating parts, a phenomenon known as [[mechanical resonance#Resonance disaster|resonance disaster]].

Avoiding resonance disasters is a major concern in every building, tower, and [[bridge]] [[construction]] project. As a countermeasure, [[shock mount]]s can be installed to absorb resonant frequencies and thus dissipate the absorbed energy. The [[Taipei 101]] building relies on a {{convert|660|t|ST|adj=mid|pendulum}}—a [[tuned mass damper]]—to cancel resonance. Furthermore, the structure is designed to resonate at a frequency that does not typically occur. Buildings in [[seismic]] zones are often constructed to take into account the oscillating frequencies of expected [[ground motion]]. In addition, [[engineer]]s designing objects having engines must ensure that the mechanical resonant frequencies of the component parts do not match driving vibrational frequencies of the motors or other strongly oscillating parts.

[[Clock]]s keep time by mechanical resonance in a [[balance wheel]], pendulum, or [[Quartz clock|quartz crystal]].

The cadence of runners has been hypothesized to be energetically favorable due to resonance between the elastic energy stored in the lower limb and the mass of the runner.{{sfn|Snyder|Farley|2011}}

====International Space Station====

The [[rocket engine]]s for the [[International Space Station]] (ISS) are controlled by an [[autopilot]]. Ordinarily, uploaded parameters for controlling the engine control system for the Zvezda module make the rocket engines boost the International Space Station to a higher orbit. The rocket engines are [[hinge]]-mounted, and ordinarily the crew does not notice the operation. On January 14, 2009, however, the uploaded parameters made the autopilot swing the rocket engines in larger and larger oscillations, at a frequency of 0.5&nbsp;Hz. These oscillations were captured on video, and lasted for 142 seconds.<ref>{{cite news |last1=Oberg |first1=James |title=Shaking on Space Station Rattles NASA |url=http://www.nbcnews.com/id/28998876/#story |archive-url=https://web.archive.org/web/20130815215631/http://www.nbcnews.com/id/28998876#story |url-status=dead |archive-date=August 15, 2013 |access-date=1 January 2021 |work=NBC News |date=4 February 2009}}</ref>

===Acoustic===
{{Main|Acoustic resonance}}
[[Acoustic resonance]] is a branch of mechanical resonance that is concerned with the mechanical vibrations across the frequency range of human hearing, in other words [[sound]]. For humans, hearing is normally limited to frequencies between about 20&nbsp;[[Hertz|Hz]] and 20,000&nbsp;Hz (20&nbsp;[[kHz]]),{{sfn|Olson|1967|pp=248&ndash;249}}  Many objects and materials act as resonators  with resonant frequencies within this range, and when struck vibrate mechanically, pushing on the surrounding air to create sound waves.  This is the source of many percussive sounds we hear.

Acoustic resonance is an important consideration for instrument builders, as most acoustic [[Musical instrument|instruments]] use [[resonator]]s, such as the [[string resonance|strings]] and body of a [[violin]], the length of tube in a [[flute]], and the shape of, and tension on, a drum membrane.

Like mechanical resonance, acoustic resonance can result in catastrophic failure of the object at resonance. The classic example of this is breaking a wine glass with sound at the precise resonant frequency of the glass, although this is difficult in practice.<ref name="UCLA">{{cite web |last1=UCLA Physics & Astronomy Department |title=50. Breaking Glass with Sound |url=http://demoweb.physics.ucla.edu/content/50-breaking-glass-sound |website=Lecture Demonstration Manual |publisher=[[University of California, Los Angeles]] |access-date=1 January 2021}}</ref>

===Electrical ===
{{Main|Electrical resonance}}
[[Image:Tuned circuit animation 3.gif|thumb|upright=1.1|Animation illustrating electrical resonance in a [[tuned circuit]], consisting of a [[capacitor]] (C) and an [[inductor]] (L) connected together. Charge flows back and forth between the capacitor plates through the inductor.  Energy oscillates back and forth between the capacitor's [[electric field]] ({{mvar|E}}) and the inductor's [[magnetic field]] ({{mvar|B}}). ]]

[[Electrical resonance]] occurs in an electric circuit at a particular ''resonant frequency'' when the [[Electrical impedance|impedance]] of the circuit is at a minimum in a series circuit or at maximum in a parallel circuit (usually when the transfer function peaks in absolute value). Resonance in circuits are used for both transmitting and receiving wireless communications such as television, cell phones and radio.

===Optical ===
{{Main|Optical cavity}}

An [[optical cavity]], also called an ''optical resonator'', is an arrangement of [[mirror]]s that forms a standing wave [[cavity resonator]] for [[light wave]]s. Optical cavities are a major component of [[laser]]s, surrounding the [[gain medium]] and providing [[feedback]] of the laser light. They are also used in [[optical parametric oscillator]]s and some [[interferometer]]s. Light confined in the cavity reflects multiple times producing standing waves for certain resonant frequencies. The standing wave patterns produced are called "modes". [[Longitudinal mode]]s differ only in frequency while [[transverse mode]]s differ for different frequencies and have different intensity patterns across the cross-section of the beam. [[Optical ring resonators|Ring resonators]] and [[Whispering gallery|whispering galleries]] are examples of optical resonators that do not form standing waves.

Different resonator types are distinguished by the focal lengths of the two mirrors and the distance between them; flat mirrors are not often used because of the difficulty of aligning them precisely. The geometry (resonator type) must be chosen so the beam remains stable, i.e., the beam size does not continue to grow with each reflection. Resonator types are also designed to meet other criteria such as minimum beam waist or having no focal point (and therefore intense light at that point) inside the cavity.

Optical cavities are designed to have a very large [[Q factor|''Q'' factor]].<ref name="q factor">{{cite web |title=''Q'' factor, quality factor, cavity, resonator, oscillator, frequency standards |url=http://www.rp-photonics.com/q_factor.html |website=Encyclopedia of Laser Physics and Technology |access-date=1 January 2021}}</ref> A beam reflects a large number of times with little [[attenuation]]—therefore the frequency [[line width]] of the beam is small compared to the frequency of the laser.

Additional optical resonances are [[guided-mode resonance]]s and [[surface plasmon resonance]], which result in anomalous reflection and high evanescent fields at resonance. In this case, the resonant modes are guided modes of a waveguide or surface plasmon modes of a dielectric-metallic interface. These modes are usually excited by a subwavelength grating.

===Orbital ===
{{Main|Orbital resonance}}

In [[celestial mechanics]], an [[orbital resonance]] occurs when two [[orbit]]ing bodies exert a regular, periodic gravitational influence on each other, usually due to their [[orbital period]]s being related by a ratio of two small integers. Orbital resonances greatly enhance the mutual gravitational influence of the bodies. In most cases, this results in an ''unstable'' interaction, in which the bodies exchange momentum and shift orbits until the resonance no longer exists. Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of [[Jupiter]]'s moons [[Ganymede (moon)|Ganymede]], [[Europa (moon)|Europa]], and [[Io (moon)|Io]], and the 2:3 resonance between [[Pluto]] and [[Neptune]]. Unstable resonances with [[Saturn]]'s inner moons give rise to gaps in the [[rings of Saturn]]. The special case of 1:1 resonance (between bodies with similar orbital radii) causes large Solar System bodies to [[Clearing the neighbourhood|clear the neighborhood]] around their orbits by ejecting nearly everything else around them; this effect is used in the current [[definition of planet|definition of a planet]].

=== Atomic, particle, and molecular ===
{{Main|Nuclear magnetic resonance|Resonance (particle physics)}}

[[File:HWB-NMR - 900MHz - 21.2 Tesla.jpg|thumb|upright|[[Nuclear magnetic resonance|NMR]] Magnet at HWB-NMR, Birmingham, UK. In its strong 21.2-[[Tesla (unit)|tesla]] field, the proton resonance is at 900&nbsp;MHz.]]
[[Nuclear magnetic resonance]] (NMR) is the name given to a physical resonance phenomenon involving the observation of specific [[quantum mechanics|quantum mechanical]] [[magnetism|magnetic]] properties of an [[atom]]ic [[atomic nucleus|nucleus]] in the presence of an applied, external magnetic field. Many scientific techniques exploit NMR phenomena to study [[molecular physics]], [[crystallography|crystal]]s, and non-crystalline materials through [[NMR spectroscopy]]. NMR is also routinely used in advanced medical imaging techniques, such as in [[magnetic resonance imaging]] (MRI).

All nuclei containing odd numbers of [[nucleon]]s have an [[Spin (physics)|intrinsic angular momentum]] and [[magnetic moment]]. A key feature of NMR is that the resonant frequency of a particular substance is directly proportional to the strength of the applied magnetic field. It is this feature that is exploited in imaging techniques; if a sample is placed in a non-uniform magnetic field then the resonant frequencies of the sample's nuclei depend on where in the field they are located. Therefore, the particle can be located quite precisely by its resonant frequency.

[[Electron paramagnetic resonance]], otherwise known as ''electron spin resonance'' (ESR), is a spectroscopic technique similar to NMR, but uses unpaired electrons instead. Materials for which this can be applied are much more limited since the material needs to both have an unpaired spin and be [[paramagnetic]].

The [[Mössbauer effect]] is the resonant and [[recoil]]-free emission and absorption of [[gamma ray]] photons by atoms bound in a solid form.

[[Resonance (particle physics)|Resonance in particle physics]] appears in similar circumstances to [[classical physics]] at the level of quantum mechanics and [[quantum field theory]]. Resonances can also be thought of as unstable particles, with the formula in the [[#Universal resonance curve|Universal resonance curve]] section of this article applying if ''Γ'' is the particle's [[Particle decay#Decay rate|decay rate]] and <math>\omega_0</math> is the particle's mass ''M''. In that case, the formula comes from the particle's [[Propagator (Quantum Theory)|propagator]], with its mass replaced by the complex number ''M''&nbsp;+&nbsp;''iΓ''. The formula is further related to the particle's decay rate by the [[optical theorem]].

== Disadvantages ==

A column of soldiers marching in regular step on a narrow and structurally flexible bridge can set it into dangerously large [[amplitude]] [[oscillation]]s. On April 12, 1831, the [[Broughton Suspension Bridge]] near [[Salford, England]] collapsed while a group of British soldiers were marching across.<ref name="Bishop">{{cite book |author=Bishop, R.E.D. |title=Vibration |edition=Second |publisher= Cambridge University Press, London |year=1979}}</ref> Since then, the British Army has had a standing order for soldiers to break stride when marching across bridges, to avoid resonance from their regular marching pattern affecting the bridge.<ref name="MEN">{{cite news |title=Broughton Bridge is falling down! |last=Smith |first=Alan |date=12 April 1975 |work=Manchester Evening News }}</ref><ref>{{cite book |last=Braun |first=Martin |title=Differential Equations and Their Applications: An Introduction to Applied Mathematics |year=1993 |edition=4 |page=175 |url=https://books.google.com/books?id=USWV3PP3b08C&q=Differential+Equations+and+Their+Applications:+An+Introduction+to+Applied+Mathematics |isbn=0-387-97894-1 |access-date=30 May 2009 |publisher=Springer-Verlag |location=New York}}</ref>

Vibrations of a motor or engine can induce resonant vibration in its supporting structures if their [[natural frequency]] is close to that of the vibrations of the engine. A common example is the rattling sound of a bus body when the engine is left idling.

Structural resonance of a suspension bridge induced by winds can lead to its catastrophic collapse. Several early suspension bridges in [[Europe]] and [[United States]] were destroyed  by structural resonance induced by modest winds. The collapse of the [[Tacoma Narrows Bridge (1940)|Tacoma Narrows Bridge]] on 7 November 1940 is characterized in physics as a classic example of resonance.<ref name="Forbes">{{cite news |last1=Siegel |first1=Ethan |author1-link=Ethan Siegel |title=Science Busts The Biggest Myth Ever About Why Bridges Collapse |url=https://www.forbes.com/sites/startswithabang/2017/05/24/science-busts-the-biggest-myth-ever-about-why-bridges-collapse/?sh=41b3a01a1f4c |access-date=3 January 2021 |work=[[Forbes]] |date=24 May 2017}}</ref> It has been argued by [[Robert H. Scanlan]] and others that the destruction was instead caused by [[aeroelasticity#Flutter|aeroelastic flutter]], a complicated interaction between the bridge and the winds passing through it—an example of a [[self oscillation]], or a kind of "self-sustaining vibration" as referred to in the nonlinear theory of vibrations.{{sfn|Billah|Scanlan|1991|p=}}

== Q factor==
{{Main|Q factor}}
[[File:High and low Q factor.svg|thumb|High and low Q factor]]
The ''Q'' factor or ''quality factor'' is a [[Dimensionless quantity|dimensionless]] parameter that describes how under-damped an [[oscillation|oscillator]] or resonator is, and characterizes the [[bandwidth (signal processing)|bandwidth]] of a resonator relative to its center frequency.{{sfn|Harlow|2004|p=2.216}}{{sfn|Tooley|2006|pp=77&ndash;78}}
A high value for ''Q'' indicates a lower rate of energy loss relative to the stored energy, i.e., the system is lightly damped. The parameter is defined by the equation:
<math display="block"> Q =2\pi \text{ } \frac{ \text{ maximum energy stored}}{ \text{total energy lost per cycle at resonance}} </math>.<ref name="MIT">{{cite web |title=Frequency response: Resonance, Bandwidth, Q factor |url=https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/resonance_qfactr.pdf | archive-url = https://ghostarchive.org/archive/20221009/https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/lecture-notes/resonance_qfactr.pdf |archive-date=2022-10-09 |url-status=live |publisher=[[Massachusetts Institute of Technology]] |access-date=3 January 2021}}</ref>

The higher the Q factor, the greater the amplitude at the resonant frequency, and the smaller the ''bandwidth'', or range of frequencies around resonance occurs. In electrical resonance, a high-''Q'' circuit in a radio receiver is more difficult to tune, but has greater [[selectivity (radio)|selectivity]], and so would be better at filtering out signals from other stations. High Q oscillators are more stable.<ref name="MIT" />

Examples that normally have a low Q factor include [[door closer]]s (Q=0.5). Systems with high Q factors include  [[tuning fork]]s (Q=1000), [[atomic clock]]s and lasers (Q≈10<sup>11</sup>).<ref name="NIST">{{cite journal |last1=Physical Measurement Laboratory |title=Time and Frequency from A to Z, Q to Ra |url=https://www.nist.gov/pml/time-and-frequency-division/popular-links/time-frequency-z/time-and-frequency-z-q-ra |journal=NIST |date=12 May 2010 |publisher=[[National Institute of Standards and Technology]] (NIST) |access-date=1 January 2021}}</ref>

==Universal resonance curve==
[[File:Universal Resonance Curve.svg|thumb|upright=1.3|"Universal Resonance Curve", a symmetric approximation to the normalized response of a resonant circuit; [[abscissa]] values are deviation from center frequency, in units of center frequency divided by 2Q; [[ordinate]] is relative amplitude, and phase in cycles; dashed curves compare the range of responses of real two-pole circuits for a ''Q'' value of 5; for higher ''Q'' values, there is less deviation from the universal curve. Crosses mark the edges of the 3&nbsp;dB bandwidth (gain 0.707, phase shift 45° or 0.125 cycle).]]

The exact response of a resonance, especially for frequencies far from the resonant frequency, depends on the details of the physical system, and is usually not exactly symmetric about the resonant frequency, as illustrated for the [[simple harmonic oscillator]] above.

For a lightly damped linear oscillator with a resonance frequency <math>\omega_0</math>, the ''intensity'' of oscillations <math>I</math> when the system is driven with a driving frequency ''<math>\omega</math>'' is typically approximated by the following formula that is symmetric about the resonance frequency:{{sfn|Siegman|1986|pp=105&ndash;108}}<math display="block">I(\omega) \equiv |\chi|^2  \propto \frac{1}{(\omega - \omega_0 )^2 + \left( \frac{\Gamma}{2} \right)^2 }.</math>

Where the susceptibility <math>\chi(\omega) </math> links the amplitude of the oscillator to the driving force in frequency space:{{sfn|Aspelmeyer|Kippenberg|Marquardt|2014}}<math display="block"> x(\omega) = \chi(\omega) F(\omega) </math>

The intensity is defined as the square of the amplitude of the oscillations. This is a [[Lorentzian function]], or [[Cauchy distribution]], and this response is found in many physical situations involving resonant systems. {{math|Γ}} is a parameter dependent on the damping of the oscillator, and is known as the ''linewidth'' of the resonance. Heavily damped oscillators tend to have broad linewidths, and respond to a wider range of driving frequencies around the resonant frequency. The linewidth is [[Proportionality (mathematics)|inversely proportional]] to the ''Q'' factor, which is a measure of the sharpness of the resonance.

In [[radio engineering]] and [[electronics engineering]], this approximate symmetric response is known as the ''universal resonance curve'', a concept introduced by [[Frederick E. Terman]] in 1932 to simplify the approximate analysis of radio circuits with a range of center frequencies and ''Q'' values.{{sfn|Terman|1932|p=}}{{sfn|Siebert|1986|p=113}}

==See also==
{{Portal|Electronics|Physics}}
{{cols|colwidth=26em}}
* [[Cymatics]]
* [[Driven harmonic motion]]
* [[Earthquake engineering]]
* [[Electric dipole spin resonance]] 
* [[Formant]]
* [[Limbic resonance]]
* [[Nonlinear resonance]]
* [[Normal mode]]
* [[Positive feedback]]
* [[Schumann resonance]]
* [[Simple harmonic motion]]
* [[Stochastic resonance]]
* [[Sympathetic string]]
* [[Resonance (chemistry)]]
* [[Fermi resonance]]
* [[Resonance (particle physics)]]
{{colend}}

==Notes==
{{Reflist}}

==References==
* {{cite journal |last1=Aspelmeyer |first1=M |last2=Kippenberg |first2=Tobias J. |last3=Marquardt |first3=Florian |author1-link=Markus Aspelmeyer |title=Cavity optomechanics |journal=Reviews of Modern Physics |date=30 December 2014 |volume=86 |issue=4 |page=1391 |doi=10.1103/RevModPhys.86.1391 |arxiv=1303.0733 |bibcode=2014RvMP...86.1391A |hdl=11858/00-001M-0000-002D-6464-3 |s2cid=119252645 |url=https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.86.1391  |url-access = subscription}}
* {{cite journal |last1=Billah |first1=K. Yusuf |last2=Scanlan |first2=Robert H |author2-link=Robert H. Scanlan |title=Resonance, Tacoma Narrows Bridge Failure, and Undergraduate Physics Textbooks |journal=American Journal of Physics |date=1991 |volume=59 |issue=2 |pages=118&ndash;124 |doi=10.1119/1.16590 |url=http://www.ketchum.org/billah/Billah-Scanlan.pdf |archive-url=https://web.archive.org/web/20000919163924/http://www.ketchum.org/billah/Billah-Scanlan.pdf |archive-date=2000-09-19 |url-status=live |access-date=1 January 2021 |bibcode = 1991AmJPh..59..118B}}
* {{cite book |last1=Ghatak |first1=Ajoy |author1-link=Ajoy Ghatak |title=Optics |date=2005 |publisher=Tata McGraw-Hill |location=New Delhi |isbn=978-0-07-058583-6 |edition=3rd |url=https://books.google.com/books?id=jStDc2LmU5IC}}
* {{cite book |last1=Halliday |first1=David |last2=Resnick |first2=Robert |last3=Walker |first3=Jearl |author1-link=David Halliday (physicist) |author2-link=Robert Resnick |author3-link=Jearl Walker |title=Fundamentals of Physics |date=2005 |publisher=John Wiley & Sons Ltd |isbn=978-0-471-71716-4 |volume=part 2 |edition=7th |url=https://archive.org/details/fundamentalsofph0007ed_part2hall_o7t2/page/n1/mode/2up |url-access = registration}} 
* {{cite web |last1=Hardt |first1=David |title=Understanding Poles and Zeros |url=https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf |archive-date=2022-10-09 |url-status=live |website=2.14 Analysis and Design of Feedback Control Systems |publisher=[[Massachusetts Institute of Technology]] |access-date=18 April 2020 |date=2004}}
* {{cite book |editor1-last=Harlow |editor1-first=James H. |title=Electric Power Transformer Engineering |date=2004 |publisher=CRC Press |location=London |isbn=978-0-8493-1704-0 |url=https://books.google.com/books?id=DANXjaoaucYC}}
* {{cite book |last1=Ogata |first1=Katsuhiko |title=System Dynamics |date=2005 |publisher=Pearson |location=Harlow |isbn=978-1-292-02608-4 |edition=4th }}
* {{cite book |last1=Olson |first1=Harry F. |author1-link=Harry F. Olson |title=Music, Physics and Engineering |volume=2 |date=1967 |publisher=Dover Publications |location=New York |url=https://archive.org/details/musicphysicsengi0000olso |isbn=978-0-486-21769-7 |url-access = registration}} 
*{{cite book| title = College Physics| last1 = Serway| first1 = Raymond A.| last2= Faughn| first2 = Jerry S.| edition = 3rd| publisher = Saunders College Publishing| date = 1992| isbn = 0-03-076377-0}}
* {{cite book |last1=Siebert |first1=William McC. |title=Circuits, Signals, and Systems |date=1986 |publisher=MIT Press' McGraw Hill Book Company |location=London; New York |isbn=978-0-262-19229-3 |url=https://books.google.com/books?id=zBTUiIrb2WIC}}
* {{cite book |last1=Siegman |first1=A. E. |author1-link=Anthony E. Siegman |title=Lasers |date=1986 |publisher=University Science Books |isbn=978-0-935702-11-8 |url=https://archive.org/details/lasers0000sieg/page/n5/mode/2up |url-access = registration}}
* {{cite journal |last1=Snyder |first1=Kristine L. |last2=Farley |first2=Claire T. |title=Energetically optimal stride frequency in running: the effects of incline and decline |journal=[[The Journal of Experimental Biology]] |date=2011 |volume=214 |issue=12 |pages=2089&ndash;2095 |doi=10.1242/jeb.053157 |url=https://jeb.biologists.org/content/214/12/2089 |pmid=21613526|doi-access=free|bibcode=2011JExpB.214.2089S }}
* {{cite book |last1=Terman |first1=Frederick Emmons |author1-link=Frederick Terman |title=Radio Engineering |date=1932 |publisher=McGraw-Hill Book Company |location=New York |oclc=1036819790 |edition=1st |url=https://archive.org/details/radioengineering00term/page/n7/mode/2up |url-access = registration}}
* {{cite book |last1=Tooley |first1=Michael H. |title=Electronic Circuits: Fundamentals and Applications |date=2006 |publisher=Taylor & Francis |location=Oxford |isbn=978-0-7506-6923-8}}

==External links==
{{Commons category}}
*[https://feynmanlectures.caltech.edu/I_23.html The Feynman Lectures on Physics Vol. I Ch. 23: Resonance]
*[http://www.lightandmatter.com/html_books/lm/ch18/ch18.html Resonance] {{Webarchive|url=https://web.archive.org/web/20170103120102/http://www.lightandmatter.com/html_books/lm/ch18/ch18.html |date=2017-01-03 }} - a chapter from an online textbook
*[[Brian Greene|Greene, Brian]], "''[https://www.pbs.org/wgbh/nova/elegant/resonance.html Resonance in strings]''". [[The Elegant Universe]], [[Nova (American TV program)|NOVA]] ([[Public Broadcasting Service|PBS]])
*[http://hyperphysics.phy-astr.gsu.edu/hbase/sound/rescon.html#c1 Hyperphysics section on resonance concepts]
*[http://users.ece.gatech.edu/~mleach/misc/resonance.html Resonance versus resonant] (usage of terms)
*[http://www.johnsankey.ca/bottom.html Wood and Air Resonance in a Harpsichord]
*[http://www.acoustics.salford.ac.uk/acoustics_info/glass Breaking glass with sound] {{Webarchive|url=https://web.archive.org/web/20081202121519/http://www.acoustics.salford.ac.uk/acoustics_info/glass |date=2008-12-02 }}, including high-speed footage of glass breaking

{{Authority control}}

[[Category:Resonance| ]]
[[Category:Antennas (radio)]]
[[Category:Oscillation]]