Frequency  

Common symbols  f, ν 
SI unit  hertz (Hz) 
Other units 

In SI base units  s ^{−1} 
Derivations from other quantities 

Dimension 
Frequency (symbol f), most often measured in hertz (symbol: Hz), is the number of occurrences of a repeating event per unit of time.^{ [1] } It is also occasionally referred to as temporal frequency for clarity and to distinguish it from spatial frequency . Ordinary frequency is related to angular frequency (symbol ω, with SI unit radian per second) by a factor of 2π. The period (symbol T) is the interval of time between events, so the period is the reciprocal of the frequency: T = 1/f.^{ [2] }
Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.
For example, if a heart beats at a frequency of 120 times per minute (2 hertz), the period—the interval between beats—is half a second (60 seconds divided by 120 beats).
For cyclical phenomena such as oscillations, waves, or for examples of simple harmonic motion, the term frequency is defined as the number of cycles or repetitions per unit of time. The conventional symbol for frequency is f or ν (the Greek letter nu) is also used.^{ [3] } The periodT is the time taken to complete one cycle of an oscillation or rotation. The frequency and the period are related by the equation^{ [4] }
The term temporal frequency is used to emphasise that the frequency is characterised by the number of occurrences of a repeating event per unit time.
The SI unit of frequency is the hertz (Hz),^{ [4] } named after the German physicist Heinrich Hertz by the International Electrotechnical Commission in 1930. It was adopted by the CGPM (Conférence générale des poids et mesures) in 1960, officially replacing the previous name, cycle per second (cps). The SI unit for the period, as for all measurements of time, is the second.^{ [5] } A traditional unit of frequency used with rotating mechanical devices, where it is termed rotational frequency , is revolution per minute, abbreviated r/min or rpm.^{ [6] } 60 rpm is equivalent to one hertz.^{ [7] }
As a matter of convenience, longer and slower waves, such as ocean surface waves, are more typically described by wave period rather than frequency.^{ [8] } Short and fast waves, like audio and radio, are usually described by their frequency. Some commonly used conversions are listed below:
Frequency  Period 

1 mHz (10^{−3} Hz)  1 ks (10^{3} s) 
1 Hz (10^{0} Hz)  1 s (10^{0} s) 
1 kHz (10^{3} Hz)  1 ms (10^{−3} s) 
1 MHz (10^{6} Hz)  1 μs (10^{−6} s) 
1 GHz (10^{9} Hz)  1 ns (10^{−9} s) 
1 THz (10^{12} Hz)  1 ps (10^{−12} s) 
For periodic waves in 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 divided by the wavelength λ of the wave:
In the special case of electromagnetic waves in vacuum, then v = c, where c is the speed of light in vacuum, and this expression becomes
When monochromatic waves travel from one medium to another, their frequency remains the same—only their wavelength and speed change.
Measurement of frequency can be done in the following ways:
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 period. For example, if 71 events occur within 15 seconds the frequency is: 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.^{[ citation needed ]} 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 , or a fractional error of where is the timing interval and 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.
An old 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 integer multiple of the strobing frequency will also appear stationary.
Higher frequencies are usually measured with a frequency counter. This is an electronic instrument which measures the frequency of an applied repetitive electronic 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 time base. Cyclic processes that are not electrical, such as the rotation rate of a shaft, mechanical vibrations, or sound waves, can be converted to a repetitive electronic signal by transducers and the signal applied to a frequency counter. As of 2018, frequency counters can 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.
Above the range of frequency counters, frequencies of electromagnetic signals are often measured indirectly utilizing heterodyning (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 convert higher frequencies, several stages of heterodyning can be used. Current research is extending this method to infrared and light frequencies (optical heterodyne detection).
Visible light is an electromagnetic wave, consisting of oscillating electric and magnetic fields traveling through space. The frequency of the wave determines its color: 400 THz (4×10^{14} Hz) is red light, 800 THz (8×10^{14} Hz) is violet light, and between these (in the range 400–800 THz) are all the other colors of the visible spectrum. An electromagnetic wave with a frequency less than 4×10^{14} Hz 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 with a frequency higher than 8×10^{14} Hz will also be invisible to the human eye; such waves are called ultraviolet (UV) radiation. Even higherfrequency waves are called Xrays, and higher still are gamma rays.
All of these waves, from the lowestfrequency radio waves to the highestfrequency gamma rays, are fundamentally the same, and they are all called electromagnetic radiation. They all travel through vacuum at the same speed (the speed of light), giving them wavelengths inversely proportional to their frequencies. where c is the speed of light (c in vacuum or less in other media), f is the frequency and λ is the wavelength.
In dispersive media, such as glass, the speed depends somewhat on frequency, so the wavelength is not quite inversely proportional to frequency.
Sound propagates as mechanical vibration waves of pressure and displacement, in air or other substances.^{ [10] } In general, frequency components of a sound determine its "color", its timbre. When speaking about the frequency (in singular) of a sound, it means the property that most determines its pitch.^{ [11] }
The frequencies an ear can hear are limited to a specific range of frequencies. The audible frequency range for humans is typically given as being between about 20 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.^{ [12] }
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.
In Europe, Africa, Australia, southern South America, most of Asia, and Russia, the frequency of the alternating current in household electrical outlets is 50 Hz (close to the 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 tones B♭ and B; that is, a minor third above the European frequency). The frequency of the 'hum' in an audio recording can show in which of these general regions the recording was made.
Aperiodic frequency is the rate of incidence or occurrence of noncyclic phenomena, including random processes such as radioactive decay. It is expressed with the unit reciprocal second (s^{−1})^{ [13] } or, in the case of radioactivity, with the unit becquerel.^{ [14] }
It is defined as a rate, f = N/Δt, involving the number of entities counted or the number of events happened (N) during a given time duration (Δt);^{[ citation needed ]} it is a physical quantity of type temporal rate.
In physics, the cross section is a measure of the probability that a specific process will take place in a collision of two particles. For example, the Rutherford crosssection is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.
The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids, the fundamental frequency is the lowest frequency sinusoidal in the sum of harmonically related frequencies, or the frequency of the difference between adjacent frequencies. In some contexts, the fundamental is usually abbreviated as f_{0}, indicating the lowest frequency counting from zero. In other contexts, it is more common to abbreviate it as f_{1}, the first harmonic.
The hertz is the unit of frequency in the International System of Units (SI), equivalent to one event per second. The hertz is an SI derived unit whose expression in terms of SI base units is s^{−1}, meaning that one hertz is the reciprocal of one second. It is named after Heinrich Rudolf Hertz (1857–1894), the first person to provide conclusive proof of the existence of electromagnetic waves. Hertz are commonly expressed in multiples: kilohertz (kHz), megahertz (MHz), gigahertz (GHz), terahertz (THz).
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity. The phase velocity is given in terms of the wavelength λ (lambda) and time period T as
In optics, the refractive index of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings. Wavelength is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter lambda (λ). The term "wavelength" is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.
The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the dimensionless change in magnitude or phase per unit length. In the context of twoport networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.
In the physical sciences, the wavenumber, also known as repetency, is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance. It is analogous to temporal frequency, which is defined as the number of wave cycles per unit time or radians per unit time.
In optics and in wave propagation in general, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency; sometimes the term chromatic dispersion is used for specificity to optics in particular. A medium having this common property may be termed a dispersive medium.
Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or superposition, of plane waves. It has some parallels to the Huygens–Fresnel principle, in which the wavefront is regarded as being made up of a combination of spherical wavefronts whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium.
A sine wave, sinusoidal wave, or sinusoid is a periodic wave whose waveform (shape) is the trigonometric sine function. In mechanics, as a linear motion over time, this is simple harmonic motion; as rotation, it corresponds to uniform circular motion. Sine waves occur often in physics, including wind waves, sound waves, and light waves, such as monochromatic radiation. In engineering, signal processing, and mathematics, Fourier analysis decomposes general functions into a sum of sine waves of various frequencies, relative phases, and magnitudes.
In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiation, and to quantify emission of neutrinos and other particles. The SI unit of radiance is the watt per steradian per square metre. It is a directional quantity: the radiance of a surface depends on the direction from which it is being observed.
In radiometry, irradiance is the radiant flux received by a surface per unit area. The SI unit of irradiance is the watt per square metre (W⋅m^{−2}). The CGS unit erg per square centimetre per second (erg⋅cm^{−2}⋅s^{−1}) is often used in astronomy. Irradiance is often called intensity, but this term is avoided in radiometry where such usage leads to confusion with radiant intensity. In astrophysics, irradiance is called radiant flux.
In physics, a wave vector is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave, and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction of wave propagation.
In radiometry, radiant intensity is the radiant flux emitted, reflected, transmitted or received, per unit solid angle, and spectral intensity is the radiant intensity per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. These are directional quantities. The SI unit of radiant intensity is the watt per steradian, while that of spectral intensity in frequency is the watt per steradian per hertz and that of spectral intensity in wavelength is the watt per steradian per metre —commonly the watt per steradian per nanometre. Radiant intensity is distinct from irradiance and radiant exitance, which are often called intensity in branches of physics other than radiometry. In radiofrequency engineering, radiant intensity is sometimes called radiation intensity.
In physics and engineering, a phasor is a complex number representing a sinusoidal function whose amplitude, and initial phase are timeinvariant and whose angular frequency is fixed. It is related to a more general concept called analytic representation, which decomposes a sinusoid into the product of a complex constant and a factor depending on time and frequency. The complex constant, which depends on amplitude and phase, is known as a phasor, or complex amplitude, and sinor or even complexor.
In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.
Optical resolution describes the ability of an imaging system to resolve detail, in the object that is being imaged. An imaging system may have many individual components, including one or more lenses, and/or recording and display components. Each of these contributes to the optical resolution of the system; the environment in which the imaging is done often is a further important factor.
An isotropic radiator is a theoretical point source of waves which radiates the same intensity of radiation in all directions. It may be based on sound waves or electromagnetic waves, in which case it is also known as an isotropic antenna. It has no preferred direction of radiation, i.e., it radiates uniformly in all directions over a sphere centred on the source.
In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space. The spatial frequency is a measure of how often sinusoidal components of the structure repeat per unit of distance.