Frequency

Sinusoidal waves of various frequencies; the bottom waves have higher frequencies than those above. The horizontal axis represents time.

Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency. The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency.

1 Definitions and units
2 Measurement
- 2.1 By timing
- 2.2 By stroboscope effect, or frequency beats
3 Frequency of waves
4 Examples
5 Period versus frequency
6 Other types of frequency
7 See also
8 References
9 External links

Definitions and units

For cyclical processes, such as rotation, oscillations, or waves, frequency is defined as a number of cycles, or periods, 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 a Greek letter ν (nu).

In SI units, the unit of frequency is hertz (Hz), named after the German physicist Heinrich Hertz. For example, 1 Hz means that an event repeats once per second, 2 Hz is twice per second, and so on ^[1]. This unit was originally called a cycle per second (cps), which is still sometimes used. Heart rate and musical tempo are measured in beats per minute (BPM). Frequency of rotation is often expressed as a number of revolutions per minute (rpm). BPM and rpm values must be divided by 60 to obtain the corresponding value in Hz: thus, 60 BPM translates into 1 Hz.

The period is usually denoted as T, and is the reciprocal of the frequency f:

$T = \frac{1}{f}.$

The SI (as well as English) unit for period is the second (s).

Measurement

By timing

To calculate the frequency of an event, the number of occurrences of the event within a fixed time interval are counted, and then divided by the length of the time interval.

In experimental work (for example, calculating the frequency of an oscillating pendulum) it is generally more accurate to measure the time taken for a fixed number of occurrences, rather than the number of occurrences within a fixed time. The latter method introduces — if N is the number of counted occurrences — a random error between zero and one count, so on average half a count, causing an biased underestimation of f by ½ f / (N + ½) in its expected value. In the first method, which does not suffer this particular error, frequency is still calculated by dividing the number of occurrences by the time interval; however it is the number of occurrences that is fixed, not the time interval.

By stroboscope effect, or frequency beats

In case when the frequency is so high that counting is difficult or impossible with the available means, another method is used, based on a source (such as a laser, a tuning fork, or a waveform generator) of a known reference frequency f₀, that must be tunable or very close to the measured frequency f. Both the observed frequency and the reference frequency are simultaneously produced, and frequency beats are observed at a much lower frequency Δf, which can be measured by counting. This is sometimes referred to as a stroboscope effect. The unknown frequency is then found from $f=f_0\pm \Delta f$ .

Frequency of waves

Frequency has an inverse relationship to the concept of wavelength, simply, frequency is inversely proportional to wavelength λ (lambda). The frequency f is equal to the phase speed v of the wave divided by the wavelength λ of the wave:

$f = \frac{v}{\lambda}.$

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:

$f = \frac{c}{\lambda}.$

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

Examples

In music and acoustics, the frequency of the standard pitch A above middle C on a piano is usually defined as 440 Hz, that is, 440 cycles per second (Listen) and known as concert pitch, to which an orchestra tunes.
A baby can hear tones with oscillations up to approximately 20,000 Hz, but these frequencies become more difficult to hear as people age.
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 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 where the recording was made — in countries utilizing the European, or the American grid frequency.
Visible light from deep red to violet has frequencies of 430 to 750 THz.

Period versus frequency

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

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

Other types of frequency

Angular frequency ω is defined as the rate of change in the orientation angle (during rotation), or in the phase of a sinusoidal waveform (e.g. in oscillations and waves):

$\omega=2\pi f\,$ .

Angular frequency is measured in radians per second (rad/s).

Spatial frequency is analogous to temporal frequency, but the time axis is replaced by one or more spatial displacement axes.
Wavenumber is the spatial analogue of angular frequency. In case of more than one spacial dimension, wavenumber is a vector quantity.

References

↑ Accidentally, 1 hertz is the approximate frequency of a human heart (Herz in German language)