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{{other uses}}
{{redirect|Frequencies|the film|Frequencies (film)|the album|Frequencies (album)}}
{{broader|Temporal rate}}
{{Infobox physical quantity
| name = frequency
| image = | caption =
| unit = [[Hz]]
| symbols = ''f'', ''ν''
| baseunits = [[Second|s]]−1
}}

'''Frequency''' is the number of occurrences of a repeating event per [[unit of time]].<ref>{{cite web|url=http://www.merriam-webster.com/dictionary/frequency|title=Definition of FREQUENCY|publisher=|accessdate=3 October 2016}}</ref> It is also referred to as '''temporal ''frequency''''', which emphasizes the contrast to [[spatial frequency]] and [[angular frequency]]. The '''period''' is the duration of time of one [[Turn (geometry)|cycle]] in a repeating event, so the period is the [[Multiplicative inverse|reciprocal]] of the frequency.<ref>{{cite web|url=http://www.merriam-webster.com/dictionary/period|title=Definition of PERIOD|publisher=|accessdate=3 October 2016}}</ref> For example, if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second (that is, 60 seconds divided by 120 [[heart sound|beat]]s). Frequency is an important parameter used in science and engineering to specify the rate of [[oscillation|oscillatory]] and [[vibration|vibratory]] phenomena, such as mechanical vibrations, [[audio signal]]s ([[sound]]), [[radio wave]]s, and [[light]].

== Definitions ==
[[File:FrequencyAnimation.gif|thumb|These three dots are flashing, or ''cycling'', periodically—from lowest frequency (0.5 hertz) to highest frequency (2.0 hertz), top to bottom. For each flashing dot: {{nowrap|"f" is}} the frequency in [[hertz]], (Hz)—or the number of events per second (cycles per second)—that the dot flashes; while {{nowrap|"T" is}} the ''period'', or ''time'', in seconds (s) of each cycle, (the number of seconds per cycle). Note {{nowrap|T and}} f are [[Multiplicative inverse|reciprocal values]] to each other.]]
[[File:Sine waves different frequencies.svg|thumb|As time elapses—here moving left to right on the horizontal axis—the five [[Sine wave|sinusoidal wave]]s vary, or cycle, regularly at different [[rate (mathematics)|rates]]. The red [[wave]] (top) has the lowest frequency (cycles at the slowest rate) while the purple wave (bottom) has the highest frequency (cycles at the fastest rate).]]

For [[Turn (geometry)|cyclical]] processes, such as [[rotation]], [[oscillation]]s, or [[wave]]s, frequency is defined as a number of cycles per unit time. In [[physics]] and [[engineering]] disciplines, such as [[optics]], [[acoustics]], and [[radio]], frequency is usually denoted by a Latin letter ''f'' or by the Greek letter ''<math>\nu</math>'' or [[Nu (letter)|''ν'' (nu)]] (see e.g. [[Planck constant|Planck's formula]]).

The relation between the frequency and the period <math>T</math> of a repeating event or oscillation is given by
:<math>f = \frac{1}{T}.</math>

== Units ==
The [[SI derived unit]] of frequency is the [[hertz]] (Hz), named after the German physicist [[Heinrich Hertz]]. One hertz means that an event repeats once per [[second]]. A previous name for this unit was [[Cycle per second|cycles per second]] (cps). The [[SI]] unit for period is the second.

A traditional unit of measure used with rotating mechanical devices is [[revolutions per minute]], abbreviated r/min or rpm. 60 rpm equals one hertz.<ref>{{Cite book
| last = Davies
| first = A.
| authorlink =
| publisher = Springer
| year = 1997
| location = New York
| doi =
| id =
| isbn = 978-0-412-61320-3
| url = https://books.google.com/?id=j2mN2aIs2YIC&pg=RA1-PA275
| title = Handbook of Condition Monitoring: Techniques and Methodology}}</ref>

== Period versus frequency ==
As a matter of convenience, longer and slower waves, such as [[ocean surface wave]]s, tend to be described by wave period rather than frequency. Short and fast waves, like [[sound|audio]] and [[radio]], are usually described by their frequency instead of period. These commonly used conversions are listed below:

{| class="wikitable"
|-
! Frequency
| 1 mHz (10−3 Hz)
| 1 Hz (100 Hz)
| 1 kHz (103 Hz)
| 1 MHz (106 Hz)
| 1 GHz (109 Hz)
| 1 THz (1012 Hz)
|-
! Period
| 1 ks (103 s)
| 1 s (100 s)
| 1 ms (10−3 s)
| 1 µs (10−6 s)
| 1 ns (10−9 s)
| 1 ps (10−12 s)
|}

== Related types of frequency ==
{{Other uses}}
[[File:Commutative diagram of harmonic wave properties.svg|thumb|Diagram of the relationship between the different types of frequency and other wave properties.]]

* [[Angular frequency]], usually denoted by the Greek letter [[Omega (letter)|ω (omega)]], is defined as the rate of change of [[angular displacement]], ''θ'', (during rotation), or the rate of change of the [[phase (waves)|phase]] of a [[Sine wave|sinusoid]]al waveform (notably in oscillations and waves), or as the rate of change of the [[Argument of a function|argument]] to the [[sine function]]:

::<math>y(t) = \sin\left( \theta(t) \right) = \sin(\omega t) = \sin(2 \mathrm{\pi} f t)</math>

::<math>\frac{\mathrm{d} \theta}{\mathrm{d} t} = \omega = 2 \mathrm{\pi} f</math>

: Angular frequency is commonly measured in [[radian]]s per second (rad/s) but, for [[discrete-time signal]]s, can also be expressed as radians per [[sampling interval]], which is a [[dimensionless quantity]]. Angular frequency (in radians) is larger than regular frequency (in Hz) by a factor of 2π.

* [[Spatial frequency]] is analogous to temporal frequency, but the time axis is replaced by one or more spatial displacement axes. E.g.:

::<math>y(t) = \sin\left( \theta(t,x) \right) = \sin(\omega t + kx)</math>

::<math>\frac{\mathrm{d} \theta}{\mathrm{d} x} = k</math>

: [[Wavenumber]], ''k'', is the spatial frequency analogue of angular temporal frequency and is measured in radians per [[meter (unit)|meter]]. In the case of more than one spatial dimension, wavenumber is a [[vector (mathematics and physics)|vector]] quantity.

== In wave propagation {{anchor|Frequency of waves}} ==
{{further|Wave propagation}}

For periodic waves in [[Dispersion relation|nondispersive media]] (that is, media in which the wave speed is independent of frequency), frequency has an inverse relationship to the [[wavelength]], ''λ'' ([[lambda]]). Even in dispersive media, the frequency ''f'' of a sinusoidal wave is equal to the [[phase velocity]] ''v'' of the [[wave]] [[division (mathematics)|divided]] by the wavelength ''λ'' of the wave:
:<math>
f = \frac{v}{\lambda}.
</math>

In the [[special case]] of electromagnetic waves moving through a [[vacuum]], then ''v = c'', where ''c'' is the [[speed of light]] in a vacuum, and this expression becomes:
:<math>
f = \frac{c}{\lambda}.
</math>

When [[wave]]s from a [[monochrome]] source travel from one [[medium (optics)|medium]] to another, their frequency remains the same—only their [[wavelength]] and [[phase speed|speed]] change.

== Measurement ==
{{see also|Frequency meter}}

Measurement of frequency can done in the following ways,

=== Counting ===
Calculating the frequency of a repeating event is accomplished by counting the number of times that event occurs within a specific time period, then dividing the count by the length of the time period. For example, if 71 events occur within 15 seconds the frequency is:
:<math>f = \frac{71}{15 \,\text{s}} \approx 4.73 \, \text{Hz}</math>
If the number of counts is not very large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time.<ref>{{cite book
| last = Bakshi| first = K.A. |author2=A.V. Bakshi |author3=U.A. Bakshi
| title = Electronic Measurement Systems
| publisher = Technical Publications
| year = 2008| location = US| pages = 4–14
| url = https://books.google.com/?id=jvnI3Dar3b4C&pg=PT183
| isbn = 978-81-8431-206-5}}</ref> The latter method introduces a [[random error]] into the count of between zero and one count, so on [[average]] half a count. This is called ''gating error'' and causes an average error in the calculated frequency of <math>\Delta f=\frac{1}{2T_m}</math>, or a fractional error of <math>\frac{\Delta f}{f}=\frac{1}{2fT_m}</math> where <math>T_m</math> is the timing interval and <math>f</math> is the measured frequency. This error decreases with frequency, so it is generally a problem at low frequencies where the number of counts '''''N''''' is small.

{{multiple image
| align = right
| direction = vertical
| header =
| image1 = Resonant reed frequency meter.jpg
| caption1 =
| image2 = Czestosciomierz-49.9Hz.jpg
| caption2 =
| width = 300
| footer = A resonant-reed frequency meter, an obsolete device used from about 1900 to the 1940s for measuring the frequency of alternating current. It consists of a strip of metal with reeds of graduated lengths, vibrated by an [[electromagnet]]. When the unknown frequency is applied to the electromagnet, the reed which is [[resonance|resonant]] at that frequency will vibrate with large amplitude, visible next to the scale.
}}

=== Stroboscope ===
An older method of measuring the frequency of rotating or vibrating objects is to use a [[stroboscope]]. This is an intense repetitively flashing light ([[strobe light]]) whose frequency can be adjusted with a calibrated timing circuit. The strobe light is pointed at the rotating object and the frequency adjusted up and down. When the frequency of the strobe equals the frequency of the rotating or vibrating object, the object completes one cycle of oscillation and returns to its original position between the flashes of light, so when illuminated by the strobe the object appears stationary. Then the frequency can be read from the calibrated readout on the stroboscope. A downside of this method is that an object rotating at an integral multiple of the strobing frequency will also appear stationary.

=== Frequency counter ===
{{main|Frequency counter}}
[[File:Frequency counter.jpg|thumb|left|Modern frequency counter]]

Higher frequencies are usually measured with a [[frequency counter]]. This is an [[electronic instrumentation|electronic instrument]] which measures the frequency of an applied repetitive electronic [[signal (electronics)|signal]] and displays the result in hertz on a [[digital display]]. It uses [[digital logic]] to count the number of cycles during a time interval established by a precision [[quartz clock|quartz]] time base. Cyclic processes that are not electrical in nature, such as the rotation rate of a shaft, mechanical vibrations, or [[sound wave]]s, can be converted to a repetitive electronic signal by [[transducer]]s and the signal applied to a frequency counter. Frequency counters can currently cover the range up to about 100 GHz. This represents the limit of direct counting methods; frequencies above this must be measured by indirect methods.

=== Heterodyne methods ===
Above the range of frequency counters, frequencies of electromagnetic signals are often measured indirectly by means of [[heterodyning]] ([[frequency changer|frequency conversion]]). A reference signal of a known frequency near the unknown frequency is mixed with the unknown frequency in a nonlinear mixing device such as a [[diode]]. This creates a [[heterodyne]] or "beat" signal at the difference between the two frequencies. If the two signals are close together in frequency the heterodyne is low enough to be measured by a frequency counter. This process only measures the difference between the unknown frequency and the reference frequency. To reach higher frequencies, several stages of heterodyning can be used. Current research is extending this method to infrared and light frequencies ([[optical heterodyne detection]]).

== Examples ==

=== Light ===
{{main article|Light|Electromagnetic radiation}}

[[File:EM spectrum.svg|thumb|Complete spectrum of [[electromagnetic radiation]] with the visible portion highlighted]]

Visible light is an [[electromagnetic wave]], consisting of oscillating [[electric field|electric]] and [[magnetic field]]s traveling through space. The frequency of the wave determines its color: {{val|4|e=14|ul=Hz}} is red light, {{val|8|e=14|u=Hz}} is violet light, and between these (in the range 4-{{val|8|e=14|u=Hz}}) are all the other colors of the [[visible spectrum]]. An electromagnetic wave can have a frequency less than {{val|4|e=14|u=Hz}}, but it will be invisible to the human eye; such waves are called [[infrared]] (IR) radiation. At even lower frequency, the wave is called a [[microwave]], and at still lower frequencies it is called a [[radio wave]]. Likewise, an electromagnetic wave can have a frequency higher than {{val|8|e=14|u=Hz}}, but it will be invisible to the human eye; such waves are called [[ultraviolet]] (UV) radiation. Even higher-frequency waves are called [[X-ray]]s, and higher still are [[gamma ray]]s.

All of these waves, from the lowest-frequency radio waves to the highest-frequency gamma rays, are fundamentally the same, and they are all called [[electromagnetic radiation]]. They all travel through a vacuum at the same speed (the [[speed of light]]), giving them [[wavelength]]s inversely proportional to their frequencies.

:<math>\displaystyle c=f\lambda</math>

where ''c'' is the speed of light (''[[speed of light|c]]'' in a vacuum, or less in other media), ''f'' is the frequency and λ is the wavelength.

In [[Dispersion (optics)|dispersive media]], such as glass, the speed depends somewhat on frequency, so the wavelength is not quite inversely proportional to frequency.

=== Sound ===
{{main article|Audio frequency}}
[[File:Ultrasound range diagram.svg|thumb|425px|right|The [[sound wave]] spectrum, with rough guide of some applications]]

[[Sound]] propagates as mechanical vibration waves of pressure and displacement, in air or other substances.<ref>{{cite web|url=http://www.merriam-webster.com/dictionary/sound|title=Definition of SOUND|publisher=|accessdate=3 October 2016}}</ref>. In general, frequency components of a sound determine its "color", its [[Timbre|timbre]]. When speaking about the frequency (in singular) of a [[sound]], it means the property that most determines [[Pitch (music)|pitch]].<ref>{{Cite book|last1= Pilhofer |first1=Michael |title=Music Theory for Dummies|url=https://books.google.com/books?id=CxcviUw4KX8C|year=2007|publisher=For Dummies|page=97|isbn= 9780470167946}}</ref>

The frequencies an ear can hear are limited to a [[threshold of hearing|specific range of frequencies]]. The [[audible frequency]] range for humans is typically given as being between about 20 [[Hertz|Hz]] and 20,000 Hz (20 kHz), though the high frequency limit usually reduces with age. Other [[species]] have different hearing ranges. For example, some dog breeds can perceive vibrations up to 60,000 Hz.<ref name="Physics Factbook">{{cite web|url=http://hypertextbook.com/facts/2003/TimCondon.shtml|title=Frequency Range of Dog Hearing|last=Elert|first=Glenn|author2=Timothy Condon|year=2003|publisher=The Physics Factbook|accessdate=2008-10-22}}</ref>

In many media, such as air, the [[speed of sound]] is approximately independent of frequency, so the wavelength of the sound waves (distance between repetitions) is approximately inversely proportional to frequency.

=== Line current ===
{{main article|Utility frequency}}
In [[Europe]], [[Africa]], [[Australia]], Southern [[South America]], most of [[Asia]], and [[Russia]], the frequency of the [[alternating current]] in [[mains electricity|household electrical outlets]] is 50 Hz (close to the [[Musical note|tone]] G), whereas in [[North America]] and Northern [[South America]], the frequency of the alternating current in household electrical outlets is 60 Hz (between the [[Musical note|tones]] B♭ and B; that is, a [[minor third]] above the European frequency). The frequency of the '[[mains hum|hum]]' in an [[audio recording]] can show where the recording was made, in countries using a European, or an American, grid frequency.

== See also ==
{{see also|Frequency (disambiguation)|Category:Units of frequency}}
{{columns-list|colwidth=30em|
*[[Audio frequency]]
*[[Bandwidth (signal processing)]]
*[[Cutoff frequency]]
*[[Downsampling]]
*[[Electronic filter]]
*[[Frequency band]]
*[[Frequency converter]]
*[[Frequency domain]]
*[[Frequency distribution]]
*[[Frequency extender]]
*[[Frequency grid]]
*[[Frequency modulation]]
*[[Frequency spectrum]]
*[[Interaction frequency]]
*[[Natural frequency]]
*[[Negative frequency]]
*[[Periodicity (disambiguation)]]
*[[Pink noise]]
*[[Preselector]]
*[[Radar signal characteristics]]
*[[Signaling (telecommunications)]]
*[[Spread spectrum]]
*[[Spectral component]]
*[[Transverter]]
*[[Upsampling]]
}}

== Notes and references ==
{{reflist|30em}}

== Further reading ==
* {{Cite book | last=Giancoli | first=D.C. | title=Physics for Scientists and Engineers | publisher=Prentice Hall | year=1988 | edition=2nd | isbn=0-13-669201-X | postscript= }}

== External links ==
{{Wiktionary|frequency|often}}
*[http://www.sengpielaudio.com/calculator-wavelength.htm Conversion: frequency to wavelength and back]
*[http://www.sengpielaudio.com/calculator-period.htm Conversion: period, cycle duration, periodic time to frequency]
*[http://www.sengpielaudio.com/calculator-notenames.htm Keyboard frequencies = naming of notes - The English and American system versus the German system]
*[http://www.acoustics.salford.ac.uk/schools/index1.htm Teaching resource for 14-16yrs on sound including frequency]
*[https://web.archive.org/web/20070218175557/http://www.ikalogic.com/freq_meter.php A simple tutorial on how to build a frequency meter]
*[http://www.diracdelta.co.uk/science/source/f/r/frequency/source.html Frequency - diracdelta.co.uk] – [[JavaScript]] calculation.
*[http://onlinetonegenerator.com A frequency generator with sound, useful for hearing tests]

{{Acoustics}}
{{Portal bar|Physics}}

[[Category:Acoustics]]
[[Category:Mechanical vibrations]]
[[Category:Physical quantities]]
[[Category:Wave mechanics]]
[[Category:Filter frequency response]]
[[Category:Temporal rates]]

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ClueBot NG: Reverting possible vandalism by 2601:192:8800:CA90:6496:6E14:EA01:2A05 to version by JackintheBox. Report False Positive? Thanks, ClueBot NG. (3513555) (Bot)