Frequency (symbolf), most often measured inhertz (symbol: Hz), is the number of occurrences of a repeating event perunit of time.[1] It is also occasionally referred to astemporal frequency for clarity and to distinguish it fromspatial frequency. Ordinary frequency is related toangular frequency (symbolω, with SI unit radian per second) by a factor of 2π. Theperiod (symbolT) is the interval of time between events, so the period is thereciprocal of the frequency:T = 1/f.[2]
For example, if a heart beats at a frequency of 120 times per minute (2 hertz), the period—the time interval between beats—is half a second (60 seconds divided by 120).
Apendulum with a period of 2.8 s and a frequency of 0.36 Hz
For cyclical phenomena such asoscillations,waves, or for examples ofsimple harmonic motion, the termfrequency is defined as the number of cycles or repetitions per unit of time. The conventional symbol for frequency isf orν (the Greek letternu) is also used.[3] TheperiodT 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 termtemporal frequency is used to emphasise that the frequency is characterised by the number of occurrences of a repeating event per unit time.
TheSI unit of frequency is thehertz (Hz),[4] named after the German physicistHeinrich Hertz by theInternational Electrotechnical Commission in 1930. It was adopted by theCGPM (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 thesecond.[5] A traditional unit of frequency used with rotating mechanical devices, where it is termedrotational frequency, isrevolution per minute, abbreviated r/min or rpm.[6] Sixty rpm is equivalent to one hertz.[7]
As a matter of convenience, longer and slower waves, such asocean surface waves, are more typically described by wave period rather than frequency.[8] Short and fast waves, likeaudio and radio, are usually described by their frequency. Some commonly used conversions are listed below:
Diagram of the relationship between the different types of frequency and other wave properties. In this diagram,x is the input to the function represented by the arrow.
Angular frequency, usually denoted by the Greek letterω (omega), is defined as the rate of change ofangular displacement (during rotation),θ (theta), or the rate of change of thephase of asinusoidal waveform (notably in oscillations and waves), or as the rate of change of theargument to thesine function:
For periodic waves innondispersive media (that is, media in which the wave speed is independent of frequency), frequency has an inverse relationship to thewavelength,λ (lambda). Even in dispersive media, the frequencyf of asinusoidal wave is equal to thephase velocityv of the wavedivided by the wavelengthλ of the wave:
In thespecial case of electromagnetic waves invacuum, thenv =c, wherec is thespeed of light in vacuum, and this expression becomes
Whenmonochromatic waves travel from onemedium to another, their frequency remains the same—only their wavelength andspeed change.
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 arandom error into the count of between zero and one count, so onaverage half a count. This is calledgating 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 countsN 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 anelectromagnet. When the unknown frequency is applied to the electromagnet, the reed which isresonant at that frequency will vibrate with large amplitude, visible next to the scale.
An old method of measuring the frequency of rotating or vibrating objects is to use astroboscope. 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 afrequency counter. This is anelectronic instrument which measures the frequency of an applied repetitive electronicsignal and displays the result in hertz on adigital display. It usesdigital logic to count the number of cycles during a time interval established by a precisionquartz time base. Cyclic processes that are not electrical, such as the rotation rate of a shaft, mechanical vibrations, orsound waves, can be converted to a repetitive electronic signal bytransducers 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 utilizingheterodyning (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 adiode. This creates aheterodyne 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 anelectromagnetic wave, consisting of oscillatingelectric andmagnetic 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 thevisible spectrum. An electromagnetic wave with a frequency less than4×1014 Hz will be invisible to the human eye; such waves are calledinfrared (IR) radiation. At even lower frequency, the wave is called amicrowave, and at still lower frequencies it is called aradio wave. Likewise, an electromagnetic wave with a frequency higher than8×1014 Hz will also be invisible to the human eye; such waves are calledultraviolet (UV) radiation. Even higher-frequency waves are calledX-rays, and higher still aregamma 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 calledelectromagnetic radiation. They all travel through vacuum at the same speed (the speed of light), giving them wavelengths inversely proportional to their frequencies.wherec is the speed of light (c in vacuum or less in other media),f is the frequency andλ is the wavelength.
Indispersive media, such as glass, the speed depends somewhat on frequency, so the wavelength is not quite inversely proportional to frequency.
Thesound wave spectrum, with rough guide of some applications
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", itstimbre. When speaking about the frequency (in singular) of a sound, it means the property that most determines itspitch.[11]
The frequencies an ear can hear are limited to aspecific range of frequencies. Theaudible 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. Otherspecies have different hearing ranges. For example, some dog breeds can perceive vibrations up to 60,000 Hz.[12]
In many media, such as air, thespeed of sound is approximately independent of frequency, so the wavelength of the sound waves (distance between repetitions) is approximately inversely proportional to frequency.
Aperiodic frequency is therate of incidence or occurrence of non-cyclic phenomena, including random processes such asradioactive decay. It is expressed with the unitreciprocal second (s−1)[13] or, in the case of radioactivity, with the unitbecquerel.[14]
^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.ISBN9781420009002.
