Как правильно пишется частота колебаний - Правописание и грамматика

ЧАСТОТА КОЛЕБАНИЙ

величина, обратная периоду колебаний T(f=1/Т), т. е. равная числу периодов колебаний (числу колебаний), совершаемых в единицу времени. Обычно Ч. к. измеряется в герцах: 1 Гц соответствует одному колебанию в секунду. Часто используется также величина w=2pf, наз. циклической или круговой частотой. В случае гармонич. волновых процессов

u(x, t)=Acos(wt± kx),

где А — амплитуда колебаний, t — время, х — координата. Круговая Ч. к. связана с волновым числом k и фазовой скоростью v соотношением w=kv.

Физический энциклопедический словарь. — М.: Советская энциклопедия.
.
1983.

Смотреть что такое «ЧАСТОТА КОЛЕБАНИЙ» в других словарях:

частота колебаний — число колебаний в 1 с. Обозначается f или ν. Если Т период колебаний, то f = 1/T; измеряется в герцах (Гц). Угловая частота колебаний ω = 2πf = 2π/T рад/с. * * * ЧАСТОТА КОЛЕБАНИЙ ЧАСТОТА КОЛЕБАНИЙ, число колебаний в 1 с. Обозначается f или n.… … Энциклопедический словарь
частота колебаний — (f[v]) Величина, обратная периоду колебаний. [ГОСТ 7601 78] частота колебаний Число колебаний в единицу времени [Терминологический словарь по строительству на 12 языках (ВНИИИС Госстроя СССР)] частота колебаний частота Количество периодов… … Справочник технического переводчика
ЧАСТОТА КОЛЕБАНИЙ — число колебаний в 1 с. Обозначается f или ?. Если T период от колебаний, то f = 1/T; измеряется в герцах (Гц). Угловая частота колебаний ??= 2?f = 2?/T рад/с … Большой Энциклопедический словарь
Частота колебаний — 14. Частота колебаний v Средняя частота спектра лазерного излучения в пределах интервала частот линии спонтанного излучения Источник: ГОСТ 24453 80: Измерения пара … Словарь-справочник терминов нормативно-технической документации
частота колебаний — svyravimų dažnis statusas T sritis automatika atitikmenys: angl. oscillation frequency vok. Schwingungsfrequenz, f rus. частота колебаний, f pranc. fréquence d oscillation, f … Automatikos terminų žodynas
частота колебаний — virpesių dažnis statusas T sritis automatika atitikmenys: angl. oscillation frequency vok. Schwingungsfrequenz, f rus. частота колебаний, f pranc. fréquence d oscillations, f … Automatikos terminų žodynas
частота колебаний — svyravimų dažnis statusas T sritis Standartizacija ir metrologija apibrėžtis Svyruojamojo judesio dažnis. atitikmenys: angl. oscillation frequency vok. Schwingungsfrequenz, f rus. частота колебаний, f pranc. fréquence d’oscillations, f ryšiai:… … Penkiakalbis aiškinamasis metrologijos terminų žodynas
частота колебаний — virpesių dažnis statusas T sritis Standartizacija ir metrologija apibrėžtis Tam tikros sistemos virpėjimo dažnis. atitikmenys: angl. oscillation frequency vok. Schwingungsfrequenz, f rus. частота колебаний, f pranc. fréquence d’oscillations, f… … Penkiakalbis aiškinamasis metrologijos terminų žodynas
частота колебаний — virpesių dažnis statusas T sritis chemija apibrėžtis Virpesių skaičius per 1 sekundę. atitikmenys: angl. oscillation frequency rus. частота колебаний … Chemijos terminų aiškinamasis žodynas
частота колебаний — svyravimų dažnis statusas T sritis fizika atitikmenys: angl. oscillation frequency vok. Schwingfrequenz, f; Schwingungsfrequenz, f rus. частота колебаний, f pranc. fréquence d’oscillations, f … Fizikos terminų žodynas

Источник

Толковый словарь русского языка. Поиск по слову, типу, синониму, антониму и описанию. Словарь ударений.

частота колебаний

ЭНЦИКЛОПЕДИЧЕСКИЙ СЛОВАРЬ

Частота́ колеба́ний — число колебаний в 1 с. Обозначается f или ν. Если Т — период колебаний, то f = 1/T; измеряется в герцах (Гц). Угловая частота колебаний ω = 2πf = 2π/T рад/с.

* * *

ЧАСТОТА КОЛЕБАНИЙ — ЧАСТОТА́ КОЛЕБА́НИЙ, число колебаний в 1 с. Обозначается f или n. Если T — период от колебаний, то f = 1/T; измеряется в герцах (Гц). Угловая частота колебаний w = 2pf = 2p/T рад/с.

БОЛЬШОЙ ЭНЦИКЛОПЕДИЧЕСКИЙ СЛОВАРЬ

ЧАСТОТА КОЛЕБАНИЙ — число колебаний в 1 с. Обозначается f или ?. Если T — период от колебаний, то f = 1/T; измеряется в герцах (Гц). Угловая частота колебаний ??= 2?f = 2?/T рад/с.

ПОЛЕЗНЫЕ СЕРВИСЫ

Источник

Частота в физике, теория и онлайн калькуляторы

Частота

Определение частоты

Определение

Частотой называют физическую величину, характеризующую периодический процесс.

Она равна числу повторений или реализации событий за единицу времени. Обозначают частоту $nu ,$ могут встречаться другие варианты обозначений частоты, например $f$ или $F$.

Частота (наряду со временем) — это наиболее точно измеряемая величина.

Частота колебаний

Частота служит одним из основных параметров, характеризующих колебания.

Определение

Частота — это физическая величина обратная периоду колебаний (T). Частота — это число полных колебаний, которые совершаются за
единицу времени.

[nu =frac{1}{T}left(1right).]

В Международной системе единиц (СИ) частота измеряется в герцах или обратных секундах:

[left[nu right]=с^{-1}=Гц.]

Герц — единица измерения частоты периодического процесса, при которой за время в одну секунду протекает один цикл процесса. Единица измерения частоты периодического процесса получила свое наименование в честь немецкого ученого Г. Герца.

Частота биений, которые возникают при сложении двух колебаний, происходящих по одной прямой с разными, о близкими по величине частотами (${nu }_1 и {nu }_2$) равна:

[{nu =nu }_1- {nu }_2left(2right).]

Другой характеристикой колебаний является циклическая частота, которая равна:

[{omega }_0=2pi nu left(3right).]

Циклическая частота измеряется в радианах, деленных на секунду:

[left[{omega }_0right]=frac{рад}{с}.]

Частота колебаний тела, массой$ m,$ подвешенного на пружине с жесткостью $k$ равна:

[nu =frac{1}{2pi sqrt{{m}/{k}}}left(4right).]

Выражение (4) выполняется для упругих, малых колебаний. Масса пружины должна быть мала в сравнении с массой тела.

Частота колебаний математического маятника, длина нити которого $l$:

[nu =frac{1}{2pi sqrt{{l}/{g}}}left(5right),]

где $g$ — ускорение свободного падения.

Частота колебаний физического маятника:

[nu =frac{1}{2pi sqrt{{J}/{mgd}}}left(6right),]

где $J$ — момент инерции тела, совершающего колебания относительно оси; $d$ — расстояние от центра масс маятника до оси колебаний.

Формулы (4) — (6) приближенные. Чем меньше амплитуда колебаний, тем точнее результаты дают эти формулы.

Частота дискретных событий, частота вращения

Определение

Частотой дискретных колебаний ($n$) — называют физическую величину, которая равна количеству действий (событий) в единицу времени.

Если время, которое занимает одно событие обозначить как $tau $, то частота дискретных событий равна:

[n=frac{1}{tau }left(7right).]

Единицей измерения частоты дискретных событий является обратная секунда:

[left[nright]=frac{1}{с}.]

Секунда в минус первой степени равна частоте дискретных событий, если за время, равное одной секунде происходит одно событие.

Частотой вращения ($n$) — называют величину, равную количеству полных оборотов, которое совершает тело в единицу времени. Если $tau $ — время, затрачиваемое на один полный оборот, то:

[n=frac{1}{tau }left(8right).]

Примеры задач с решением

Пример 1

Задание. Частица совершает гармонические колебания, которые описывает следующий закон: $x=6{sin left(frac{pi }{4}t+frac{pi }{3}right) }(м)$. Какова частота этих колебаний?

Решение. Рассмотрим уравнение движения частицы:

[x=6{sin left(frac{pi }{4}t+frac{pi }{3}right)left(1.1right). }]

Из этого уравнения мы видим, что амплитуда колебаний точки равна: $x_m=6 left(мright);;$ циклическая частота колебаний равна ${omega }_0=frac{pi }{4}(frac{рад}{с})$; начальная фаза колебаний: ${varphi }_0=frac{pi }{3}(рад)$. Частоту найдем, используя формулу:

[{omega }_0=2pi nu left(1.2right),]

из которой имеем:

[nu =frac{{omega }_0}{2pi }left(1.3right).]

Подставим значение циклической частоты, полученное из уравнения (1.1) в формулу (1.3), получаем:

[nu =frac{frac{pi }{4}}{2pi }=frac{1}{8} left(Гцright).]

Ответ. $nu =frac{1}{8}Гц$

Пример 2

Задание. К упругой пружине прикрепили маленький груз, при этом она растянулась на $Delta x$ (м). Какой будет частота колебаний грузика, если он будет совершать свободные колебания? Затуханием колебаний пренебречь.

Решение. Сделаем рисунок.

В нашей задаче мы имеем колебания пружинного маятника, частоту которого можно найти как:

[nu =frac{1}{2pi sqrt{{m}/{k}}}left(2.1right).]

Рассмотрим состояние равновесия тела, которое прикреплено к пружине (рис.1). Запишем второй закон Ньютона для сил, действующих на это тело в состоянии равновесия:

[moverline{g}+{overline{F}}_u=0 left(2.2right).]

Запишем проекцию уравнения (2.2) на ось Y:

[F_u=mgleft(2.3right).]

Так как колебания груза на пружине малые, то выполняется закон Гука и мы можем считать, что:

[F_u=kDelta x left(2.4right).]

Из (2.3) и (2.4) найдем отношение ${m}/{k}:$

[mg=kDelta x to {m}/{k}={Delta x}/{g}left(2.5right).]

Подставим полученный в (2.5) результат в (1.1), частота колебаний тела на пружине равна:

[nu =frac{1}{2pi sqrt{{Delta x}/{g}}}.]

Ответ. $nu =frac{1}{2pi sqrt{{Delta x}/{g}}}$

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From Wikipedia, the free encyclopedia

Frequency
A pendulum making 25 complete oscillations in 60 s, a frequency of 0.416 Hz
Common symbols	f, ν
SI unit	hertz (Hz)
Other units	cycle per second (cps) revolution per minute (rpm or r/min)
In SI base units	s⁻¹
Derivations from other quantities	f = 1 / T
Dimension	${displaystyle {mathsf {T}}^{-1}}$

Frequency 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 is distinct from angular frequency. Frequency is measured in hertz (Hz) which is equal to one event per second. The period is the interval of time between events, so the period is the reciprocal of the frequency.^[2]

For example, if a heart beats at a frequency of 120 times a minute (2 hertz), the period, T—the interval at which the beats repeat—is half a second (60 seconds divided by 120 beats). 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.

Definitions and units[edit]

A pendulum with a period of 2.8 s and a frequency of 0.36 Hz

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 vibrations per unit of time. The conventional symbol for frequency is f; the Greek letter ν (nu) is also used.^[3] The period T is the time taken to complete one cycle of an oscillation or rotation. The relation between the frequency and the period is given by the equation^[4]

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

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 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]

Period versus frequency[edit]

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 (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)

[edit]

Diagram of the relationship between the different types of frequency and other wave properties.

Angular frequency, usually denoted by the Greek letter ω (omega), is defined as the rate of change of angular displacement (during rotation), θ (theta), or the rate of change of the phase of a sinusoidal waveform (notably in oscillations and waves), or as the rate of change of the argument to the sine function:

${displaystyle {frac {mathrm {d} theta }{mathrm {d} t}}=omega =2mathrm {pi } f.}$

The unit of angular frequency is the radian 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 2π.
Spatial frequency, denoted here by ξ, is analogous to temporal frequency, but with a spatial measurement replacing time measurement,^{[note 1]} e.g.:

${displaystyle {frac {mathrm {d} theta }{mathrm {d} x}}=k=2pi xi .}$

In wave propagation [edit]

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:

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

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

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

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

Measurement[edit]

Measurement of frequency can be done in the following ways:

Counting[edit]

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:

${displaystyle f={frac {71}{15,{text{s}}}}approx 4.73,{text{Hz}}.}$

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.^[10] 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 ${textstyle Delta f={frac {1}{2T_{text{m}}}}}$ , or a fractional error of ${textstyle {frac {Delta f}{f}}={frac {1}{2fT_{text{m}}}}}$ where ${displaystyle T_{text{m}}}$ 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.

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 resonant at that frequency will vibrate with large amplitude, visible next to the scale.

Stroboscope[edit]

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.

Frequency counter[edit]

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.

Heterodyne methods[edit]

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 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[edit]

Light[edit]

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 (4×10¹⁴ Hz) is red light, 800 THz (8×10¹⁴ 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 4×10¹⁴ 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 8×10¹⁴ Hz will also be invisible to the human eye; such waves are called ultraviolet (UV) radiation. Even higher-frequency waves are called X-rays, and higher still are gamma rays.

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.

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[edit]

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.^[11] 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.^[12]

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.^[13]

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[edit]

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 where the recording was made, in countries using a European, or an American, grid frequency.

Aperiodic frequency[edit]

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)^[14] or, in the case of radioactivity, becquerels.^[15]

It is defined as a ratio, f = N/T, involving the number of times an event happened (N) during a given time duration (T); it is a physical quantity of type temporal rate.

Notes[edit]

^ The term spatial period, sometimes used in place of wavelength, analogously corresponds to the (temporal) period.^[9]

References[edit]

^ «Definition of FREQUENCY». Retrieved 3 October 2016.
^ «Definition of PERIOD». Retrieved 3 October 2016.
^ Serway & Faughn 1989, p. 346.
^ ^a ^b Serway & Faughn 1989, p. 354.
^ «Resolution 12 of the 11th CGPM (1960)». BIPM (International Bureau of Weights and Measures). Archived from the original on 8 April 2020. Retrieved 21 January 2021.
^ «Special Publication 811: NIST Guide to the SI, Chapter 8». Nist. 28 January 2016. Retrieved 2022-11-08.
^ Davies 1997, p. 275.
^ Young 1999, p. 7.
^ Boreman, Glenn D. «Spatial Frequency». SPIE. Retrieved 22 January 2021.
^ Bakshi, K.A.; A.V. Bakshi; U.A. Bakshi (2008). Electronic Measurement Systems. US: Technical Publications. pp. 4–14. ISBN 978-81-8431-206-5.^{[permanent dead link]}
^ «Definition of SOUND». Retrieved 3 October 2016.
^ Pilhofer, Michael (2007). Music Theory for Dummies. For Dummies. p. 97. ISBN 978-0-470-16794-6.
^ Condon, Tim (2003). Elert, Glenn (ed.). «Frequency range of dog hearing». The Physics Factbook. Retrieved 2008-10-22.
^ Lombardi, Michael A. (2007). «Fundamentals of Time and Frequency». In Bishop, Robert H. (ed.). Mechatronic Systems, Sensors, and Actuators: Fundamentals and Modeling. Austin: CRC Press. ISBN 9781420009002.
^ Newell, David B; Tiesinga, Eite (2019). The international system of units (SI): (PDF) (Report). Gaithersburg, MD: National Institute of Standards and Technology. doi:10.6028/nist.sp.330-2019. sub§2.3.4, Table 4.

Sources[edit]

Davies, A. (1997). Handbook of Condition Monitoring: Techniques and Methodology. New York: Springer. ISBN 978-0-412-61320-3.
Serway, Raymond A.; Faughn, Jerry S. (1989). College Physics. London: Thomson/Brooks-Cole. ISBN 978-05344-0-814-5.
Young, Ian R. (1999). Wind Generated Ocean Waves. Elsevere Ocean Engineering. Vol. 2. Oxford: Elsevier. ISBN 978-0-08-043317-2.

External links[edit]

Keyboard frequencies = naming of notes – The English and American system versus the German system
A frequency generator with sound, useful for hearing tests

From Wikipedia, the free encyclopedia

Frequency
A pendulum making 25 complete oscillations in 60 s, a frequency of 0.416 Hz
Common symbols	f, ν
SI unit	hertz (Hz)
Other units	cycle per second (cps) revolution per minute (rpm or r/min)
In SI base units	s⁻¹
Derivations from other quantities	f = 1 / T
Dimension	${displaystyle {mathsf {T}}^{-1}}$

Definitions and units[edit]

A pendulum with a period of 2.8 s and a frequency of 0.36 Hz

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

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.

Period versus frequency[edit]

Frequency	Period
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)

[edit]

Diagram of the relationship between the different types of frequency and other wave properties.

Angular frequency, usually denoted by the Greek letter ω (omega), is defined as the rate of change of angular displacement (during rotation), θ (theta), or the rate of change of the phase of a sinusoidal waveform (notably in oscillations and waves), or as the rate of change of the argument to the sine function:

${displaystyle {frac {mathrm {d} theta }{mathrm {d} t}}=omega =2mathrm {pi } f.}$

The unit of angular frequency is the radian 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 2π.
Spatial frequency, denoted here by ξ, is analogous to temporal frequency, but with a spatial measurement replacing time measurement,^{[note 1]} e.g.:

${displaystyle {frac {mathrm {d} theta }{mathrm {d} x}}=k=2pi xi .}$

In wave propagation [edit]

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

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

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

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