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# Frequency

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Frequency (symbol $f$), most often measured in hertz (symbol: Hz), is the number of occurrences of a repeating event per unit of time. 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 2. The period (symbol T) is the interval of time between events, so the period is the reciprocal of the frequency: .

Frequency is an important parameter used in science and engineering to specify the rate of and phenomena, such as mechanical vibrations, (), , and .

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 ).

Definitions and units
For cyclical phenomena such as , , 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. The period T is the time taken to complete one cycle of an oscillation or rotation. The frequency and the period are related by the equation $f = \frac{1}{T}.$

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 (Hz), named after the German physicist by the International Electrotechnical Commission in 1930. It was adopted by the (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 . 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. 60 rpm is equivalent to one hertz.

Period versus frequency
As a matter of convenience, longer and slower waves, such as ocean surface waves, are more typically described by wave period rather than frequency. Short and fast waves, like and radio, are usually described by their frequency. Some commonly used conversions are listed below:

1 mHz (10−3 Hz)1 ks (103 s)
1 Hz (100 Hz)1 s (100 s)
1 kHz (103 Hz)1 ms (10−3 s)
1 MHz (106 Hz)1 μs (10−6 s)
1 GHz (109 Hz)1 ns (10−9 s)
1 THz (1012 Hz)1 ps (10−12 s)

Related quantities
• Rotational frequency, usually denoted by the Greek letter ν (nu), is defined as the instantaneous rate of change of the number of rotations, N, with respect to time: it is a type of frequency applied to rotational motion.
• Angular frequency, usually denoted by the Greek letter ω (omega), is defined as the rate of change of angular displacement (during rotation), (theta), or the rate of change of the phase of a waveform (notably in oscillations and waves), or as the rate of change of the argument to the :
$y(t) = \sin \theta(t) = \sin(\omega t) = \sin(2 \mathrm{\pi} f t)$ $\frac{\mathrm{d} \theta}{\mathrm{d} t} = \omega = 2 \mathrm{\pi} f .$
The unit of angular frequency is the per second (rad/s) but, for discrete-time signals, can also be expressed as radians per sampling interval, which is a dimensionless quantity. Angular frequency is frequency multiplied by 2.
• Spatial frequency, denoted here by ξ (xi), is analogous to temporal frequency, but with a spatial measurement replacing time measurement, e.g.: $y(t) = \sin \theta(t,x) = \sin(\omega t + kx)$ $\frac{\mathrm{d} \theta}{\mathrm{d} x} = k = 2 \pi \xi.$
• or wavelength is the spatial analog to temporal period.

In wave propagation
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 , λ (). Even in dispersive media, the frequency f of a is equal to the v of the wave divided by the wavelength λ of the wave: $f = \frac{v}{\lambda}.$

In the of electromagnetic waves in , then , where c is the speed of light in vacuum, and this expression becomes $f = \frac{c}{\lambda}.$

When monochromatic waves travel from one medium to another, their frequency remains the same—only their wavelength and change.

Measurement
Measurement of frequency can be 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 period. For example, if 71 events occur within 15 seconds the frequency is: $f = \frac{71}{15 \,\text{s}} \approx 4.73 \, \text{Hz}.$ 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.
(2024). 9788184312065, Technical Publications. .
The latter method introduces a into the count of between zero and one count, so on half a count. This is called gating error and causes an average error in the calculated frequency of $\Delta f = \frac{1}{2T_\text{m}}$, or a fractional error of $\frac{\Delta f}{f} = \frac{1}{2 f T_\text{m}}$ where $T_\text\left\{m\right\}$ is the timing interval and $f$ 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.

Stroboscope
An old method of measuring the frequency of rotating or vibrating objects is to use a . This is an intense repetitively flashing 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.

Frequency counter
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 . It uses to count the number of cycles during a time interval established by a precision time base. Cyclic processes that are not electrical, such as the rotation rate of a shaft, mechanical vibrations, or , can be converted to a repetitive electronic signal by 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.

Heterodyne methods
Above the range of frequency counters, frequencies of electromagnetic signals are often measured indirectly utilizing (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 . This creates a 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
Visible light is an electromagnetic wave, consisting of oscillating and traveling through space. The frequency of the wave determines its color: 400 THz ( Hz) is red light, 800 THz () is violet light, and between these (in the range 400–800 THz) are all the other colors of the . An electromagnetic wave with a frequency less than will be invisible to the human eye; such waves are called (IR) radiation. At even lower frequency, the wave is called a , and at still lower frequencies it is called a . Likewise, an electromagnetic wave with a frequency higher than will also be invisible to the human eye; such waves are called (UV) radiation. Even higher-frequency waves are called , and higher still are .

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 vacuum at the same speed (the speed of light), giving them wavelengths inversely proportional to their frequencies. $\displaystyle c=f\lambda,$ 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
Sound propagates as mechanical vibration waves of pressure and displacement, in air or other substances. In general, frequency components of a sound determine its "color", its . When speaking about the frequency (in singular) of a sound, it means the property that most determines its pitch.
(2024). 9780470167946, For Dummies. .

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 have different hearing ranges. For example, some dog breeds can perceive vibrations up to 60,000 Hz.

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
In , , , southern , most of , and , the frequency of the alternating current in household electrical outlets is 50 Hz (close to the G), whereas in 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 above the European frequency). The frequency of the '' in an can show in which of these general regions the recording was made.

Aperiodic frequency
Aperiodic frequency is the rate of incidence or occurrence of non-cyclic phenomena, including random processes such as radioactive decay. It is expressed with the unit of reciprocal second (s−1)
(2024). 9781420009002, CRC Press.
or, in the case of radioactivity, . sub§2.3.4, Table 4.

It is defined as a rate, f = Nt, involving the number of entities counted or the number of events happened ( N) during a given t); it is a physical quantity of type .

Notes

Sources

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