Editing Resonance (section)

== 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}}