View source for Amplitude

{{Short description|Measure of change in a periodic variable}}
{{About|amplitude in classical physics}} 

The '''amplitude''' of a [[Periodic function|period]]ic [[Variable (mathematics)|variable]] is a measure of its change in a single [[Period (mathematics)|period]] (such as [[frequency|time]] or [[Wavelength|spatial period]]). The amplitude of a non-periodic signal is its [[Magnitude (mathematics)|magnitude]] compared with a reference value. There are various definitions of amplitude (see below), which are all [[function (mathematics)|function]]s of the magnitude of the differences between the variable's [[Maxima and minima|extreme values]]. In older texts, the [[Phase (waves)|phase]] of a periodic function is sometimes called the amplitude.<ref>{{Cite book | author1=Knopp, Konrad| author2= Bagemihl, Frederick | author-link1=Konrad Knopp | title=Theory of Functions Parts I and II | year=1996 | publisher=Dover Publications | isbn=978-0-486-69219-7 | page=3}}</ref>

==Definitions==
[[File:Sine voltage.svg|thumb|A [[sine wave|sinusoidal]] curve
{{ordered list|
| list_style=list-style-position:inside; margin:0;
| Peak amplitude (<math>\scriptstyle\hat u</math>),
| Peak-to-peak amplitude (<math>\scriptstyle2\hat u</math>),
| Root mean square amplitude (<math>\scriptstyle\hat u/\sqrt{2}</math>),
| [[Wave period]] (not an amplitude)
}}]]

===Peak amplitude and semi-amplitude===
For symmetric periodic waves, like [[sine wave]]s, [[square wave]]s or [[triangle wave]]s ''peak amplitude'' and ''semi amplitude'' are the same.

====Peak amplitude====
{{anchor|Peak amplitude}}In [[audio system measurements]], [[telecommunication]]s and others where the [[wikt:measurand|measurand]] is a signal that swings above and below a reference value but is not [[Sine wave|sinusoidal]], peak amplitude is often used. If the reference is zero, this is the maximum [[absolute value]] of the signal; if the reference is a mean value ([[DC component]]), the peak amplitude is the maximum absolute value of the difference from that reference.

====Semi-amplitude====
{{anchor|Semi-amplitude}}<!-- This section is the target of [[Semi-amplitude]].-->Semi-amplitude means half of the peak-to-peak amplitude.<ref name="Tatum">Tatum, J. B. ''[http://orca.phys.uvic.ca/~tatum/celmechs/celm18.pdf Physics &nbsp;– Celestial Mechanics].'' Paragraph 18.2.12. 2007. Retrieved 2008-08-22.</ref> 
The majority of scientific literature<ref>Regents of the [[University of California]]. ''[http://cse.ssl.berkeley.edu/light/measure_amp.html#measure4 Universe of Light: What is the Amplitude of a Wave?]'' 1996. Retrieved 2008-08-22.</ref> employs the term ''amplitude'' or ''peak amplitude'' to mean semi-amplitude.

It is the most widely used measure of orbital wobble in [[astronomy]] and the measurement of small [[radial velocity]] semi-amplitudes of nearby stars is important in the search for [[exoplanet]]s (see [[Doppler spectroscopy]]).<ref>Goldvais, Uriel A. [http://img2.tapuz.co.il/forums/1_109580628.pdf Exoplanets] {{Webarchive|url=https://web.archive.org/web/20210303160140/http://img2.tapuz.co.il/forums/1_109580628.pdf |date=2021-03-03 }}, pp. 2–3. Retrieved 2008-08-22.</ref>

====Ambiguity====
In general, the use of ''peak amplitude'' is simple and unambiguous only for symmetric periodic waves, like a sine wave, a square wave, or a triangle wave. For an asymmetric wave (periodic pulses in one direction, for example), the peak amplitude becomes ambiguous. This is because the value is different depending on whether the maximum positive signal is measured relative to the mean, the maximum negative signal is measured relative to the mean, or the maximum positive signal is measured relative to the maximum negative signal (the ''peak-to-peak amplitude'') and then divided by two (the ''semi-amplitude''). In electrical engineering, the usual solution to this ambiguity is to measure the amplitude from a defined reference potential (such as [[ground (electricity)|ground]] or 0&nbsp;V). Strictly speaking, this is no longer amplitude since there is the possibility that a constant ([[DC component]]) is included in the measurement.

===Peak-to-peak amplitude{{anchor|Peak-to-peak}}=== 
'''Peak-to-peak amplitude''' (abbreviated '''p–p''') is the change between peak (highest amplitude value) and [[Crest and trough|trough]] (lowest amplitude value, which can be negative). With appropriate circuitry, peak-to-peak amplitudes of electric oscillations can be measured by meters or by viewing the waveform on an [[oscilloscope]]. Peak-to-peak is a straightforward measurement on an oscilloscope, the peaks of the waveform being easily identified and measured against the [[Oscilloscope#Graticule|graticule]]. This remains a common way of specifying amplitude, but sometimes other measures of amplitude are more appropriate.

===Root mean square amplitude===
{{Further|Root mean square#In common waveforms{{!}}RMS of common waveforms}}

[[Root mean square]] (RMS) amplitude is used especially in [[electrical engineering]]: the RMS is defined as the [[square root]] of the [[mean]] over time of the square of the vertical distance of the graph from the rest state;<ref>Department of Communicative Disorders [[University of Wisconsin–Madison]]. ''[http://www.comdis.wisc.edu/vcd202/rms.html RMS Amplitude] {{Webarchive|url=https://web.archive.org/web/20130911063155/http://www.comdis.wisc.edu/vcd202/rms.html |date=2013-09-11 }}''. Retrieved 2008-08-22.</ref>
i.e. the RMS of the [[AC waveform]] (with no [[DC component]]).

For complicated waveforms, especially non-repeating signals like noise, the RMS amplitude is usually used because it is both unambiguous and has physical significance. For example, the ''average'' [[power (physics)|power]] transmitted by an acoustic or [[electromagnetic wave]] or by an electrical signal is proportional to the square of the RMS amplitude (and not, in general, to the square of the peak amplitude).<ref>Ward, ''Electrical Engineering Science'', pp. 141–142, McGraw-Hill, 1971.</ref>

For [[alternating current]] [[electric power]], the  universal practice is to specify RMS values of a sinusoidal waveform. One property of root mean square voltages and currents is that they produce the same heating effect as a [[direct current]] in a given resistance.

The peak-to-peak value is used, for example, when choosing rectifiers for power supplies, or when estimating the maximum voltage that insulation must withstand. Some common [[voltmeter]]s are calibrated for RMS amplitude, but respond to the average value of a rectified waveform. Many digital voltmeters and all moving coil meters are in this category. The RMS calibration is only correct for a sine wave input since the ratio between peak, average and RMS values is dependent on [[waveform]]. If the wave shape being measured is greatly different from a sine wave, the relationship between RMS and average value changes. True RMS-responding meters were used in [[radio frequency]] measurements, where instruments measured the heating effect in a resistor to measure a current. The advent of [[microprocessor]]-controlled meters capable of calculating RMS by [[Sampling (signal processing)|sampling]] the waveform has made true RMS measurement commonplace.

===Pulse amplitude===
In [[telecommunication]], ''pulse amplitude'' is the [[Magnitude (mathematics)|magnitude]] of a [[pulse (signal processing)|pulse]] parameter, such as the [[voltage]] level, [[Electric current|current]] level, [[field intensity]], or [[Power (physics)|power]] level.

Pulse amplitude is measured with respect to a specified reference and therefore should be modified by qualifiers, such as ''average'', ''instantaneous'', ''peak'', or ''root-mean-square''.

Pulse amplitude also applies to the amplitude of [[frequency]]- and [[phase (waves)|phase]]-modulated [[waveform]] envelopes.<ref>{{FS1037C}}</ref>

==Formal representation==
In this simple [[wave equation]]
:<math>x = A \sin(\omega [t - K]) + b \ ,</math>

*<math>A</math> is the amplitude (or [[#Peak amplitude|peak amplitude]]),
*<math>x</math> is the oscillating variable,
*<math>\omega</math> is [[angular frequency]],
*<math>t</math> is time,
*<math>K</math> and <math>b</math> are arbitrary constants representing time and displacement offsets respectively.

==Units==
The units of the amplitude depend on the type of wave, but are always in the same units as the oscillating variable. A more general representation of the wave equation is more complex, but the role of amplitude remains analogous to this simple case.

For waves on a [[string vibration|string]], or in a medium such as [[water]], the amplitude is a [[Displacement (geometry)|displacement]].

The amplitude of sound waves and audio signals (which relates to the volume) conventionally refers to the amplitude of the [[Sound#Sound Pressure Level|air pressure]] in the wave, but sometimes the amplitude of the [[Particle displacement|displacement]] (movements of the air or the diaphragm of a [[loudspeaker|speaker]]) is described.<ref name="VEDANTU 2020">{{cite web | title=Amplitude Frequency Period Sound | website=VEDANTU | date=2020-06-23 | url=https://www.vedantu.com/physics/amplitude-frequency-period-sound | access-date=2022-06-30 | quote=Amplitude is the maximum displacement of the particles of a sound wave. Frequency is the number of vibrations made by a sound wave per second.}}</ref><ref name="BYJUS 2016">{{cite web | website=BYJUS| title=amplitude-frequency-period-sound | date=2016-08-05 | url=https://byjus.com/physics/amplitude-frequency-period-sound/ | access-date=2022-06-30|quote=The amplitude of a sound wave is the measure of the height of the wave. The amplitude of a sound wave can be defined as the loudness or the amount of maximum displacement of vibrating particles of the medium from their mean position when the sound is produced.}}</ref> The [[logarithm]] of the amplitude squared is usually quoted in [[decibel|dB]], so a null amplitude corresponds to −[[infinity|∞]]&nbsp;dB. [[Loudness]] is related to amplitude and [[Sound intensity|intensity]] and is one of the most salient qualities of a sound, although in general sounds it can be recognized [[Neuroscience of music|independently of amplitude]]. The square of the amplitude is proportional to the intensity of the wave.

For [[electromagnetic radiation]], the amplitude of a [[photon]] corresponds to the changes in the [[electric field]] of the wave. However, radio signals may be carried by electromagnetic radiation; the intensity of the radiation ([[amplitude modulation]]) or the frequency of the radiation ([[frequency modulation]]) is oscillated and then the individual oscillations are varied (modulated) to produce the signal.

==Amplitude envelopes==
Amplitude envelope refers to the changes in the amplitude of a sound over time, and is an influential property as it affects perception of timbre. A flat tone has a steady state amplitude that remains constant during time, which is represented by a scalar.  Other sounds can have percussive amplitude envelopes featuring an abrupt onset followed by an immediate exponential decay. <ref>{{Cite web |title=amplitude envelope |url=https://maplelab.net/overview/amplitude-envelope/ |access-date=2023-10-30 |website=MAPLE Lab |language=en-US}}</ref> 

Percussive amplitude envelopes are characteristic of various impact sounds: two wine glasses clinking together, hitting a drum, slamming a door, etc. where the amplitude is transient and must be represented as either a continuous function or a discrete vector. Percussive amplitude envelopes model many common sounds that have a transient loudness attack, decay, sustain, and release. <ref>{{cite journal |last1=Schutz |first1=Michael |last2=Gillard |first2=Jessica |title=On the generalization of tones: A detailed exploration of non-speech auditory perception stimuli |journal=Scientific Reports |date=June 2020 |volume=10 |url=https://www.nature.com/articles/s41598-020-63132-2}}</ref>

==Amplitude normalization==
With waveforms containing many overtones, complex transient timbres can be achieved by assigning each overtone to its own distinct transient amplitude envelope. Unfortunately, this has the effect of modulating the loudness of the sound as well. It makes more sense to separate loudness and harmonic quality to be parameters controlled independently of each other.

To do so, harmonic amplitude envelopes are frame-by-frame normalized to become amplitude ''proportion'' envelopes, where at each time frame all the harmonic amplitudes will add to 100% (or 1). This way, the main loudness-controlling envelope can be cleanly controlled.<ref name="pitt.edu">{{cite web|url=http://www.pitt.edu/~rdb37/synthparentwebpage_main.html|title=Additive Sound Synthesizer Project with CODE!|website=www.pitt.edu}}{{Dead link|date=August 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

In Sound Recognition, max amplitude normalization can be used to help align the key harmonic features of 2 alike sounds, allowing similar timbres to be recognized independent of loudness.<ref>{{cite web|url=http://www.pitt.edu/~rdb37/ssar.html|title=Sound Sampling, Analysis, and Recognition|website=www.pitt.edu}}{{Dead link|date=August 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref>{{cite web|url=https://www.youtube.com/watch?v=ZtWGXyYcs-A| archive-url=https://ghostarchive.org/varchive/youtube/20211108/ZtWGXyYcs-A| archive-date=2021-11-08 | url-status=live|title=I wrote a Sound Recognition Application|last=rblack37|date=2 January 2018|via=YouTube}}{{cbignore}}</ref>

==See also==
{{Wiktionary|amplitude}}
*[[Complex amplitude]]
*[[Wave]]s and their properties:
**[[Envelope (waves)|Envelope]]
**[[Frequency]]
**[[Wavelength]]
**[[Crest factor]]
*[[Amplitude modulation]]
*Body [[Thermal amplitude (medical)|thermal amplitude]]
*[[Atmospheric temperature#Temperature range|Atmospheric thermal amplitude]]
*[[Wave height]]

==Notes==
{{Reflist}}

{{Authority control}}

[[Category:Physical quantities]]
[[Category:Sound]]
[[Category:Wave mechanics]]