# Sound power

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{{short description|Rate at which sound energy is reflected or transmitted per unit time}}
{{Sound measurements}}

'''Sound power''' or '''acoustic power''' is the rate at which [sound energy](/source/sound_energy) is emitted, [reflected](/source/reflected), [transmitted](/source/Acoustic_transmission) or received, per unit time.<ref name=clinical>{{cite book|url=https://books.google.com/books?id=ElPyvaJbDiwC&q=sound+power+loudness&pg=PA94|title=Clinical Measurement of Speech and Voice|author=Ronald J. Baken, Robert F. Orlikoff|publisher=Cengage Learning|year=2000|isbn=9781565938694|page=94}}</ref> It is defined<ref>{{cite web | url = https://www.iso.org/obp/ui/#iso:std:iso:80000-8:ed-3:v1:en | title = ISO 80000-8(en) Quantities and Units - Acoustics | publisher = [ISO] }}</ref> as "through a surface, the product of the [sound pressure](/source/sound_pressure), and the component of the [particle velocity](/source/particle_velocity), at a point on the surface in the direction [normal](/source/Normal_(geometry)) to the surface, [integrated over](/source/Surface_integral) that surface."  The [SI unit](/source/International_System_of_Units) of sound power is the [watt](/source/watt) (W).<ref name=clinical/> It relates to the power of the sound force on a surface enclosing a sound source, in air. 

For a sound source, unlike sound pressure, sound power is neither room-dependent nor distance-dependent. Sound pressure is a property of the field at a point in space, while sound power is a property of a sound source, equal to the total power emitted by that source in all directions. Sound power passing through an area is sometimes called ''sound [flux](/source/flux)'' or ''acoustic flux'' through that area.

==Sound power level ''L''<sub>WA</sub>==

[[File:Atlas Copco XAHS 347-pic7-Max. sound power level.jpg|thumb|Maximum sound power level ([''L''<sub>WA</sub>](/source/A-weighting)) related to a portable [air compressor](/source/air_compressor)]]Regulations often specify a method for measurement<ref>{{cite web | url = https://www.iso.org/obp/ui/#iso:std:iso:3744:ed-3:v1:en | title = ISO 3744:2010(en) Acoustics — Determination of sound power levels and sound energy levels of noise sources using sound pressure — Engineering methods for an essentially free field over a reflecting plane | publisher = [ISO] | access-date = 22 December 2017 }}</ref> that integrates sound pressure over a surface enclosing the source. ''L''<sub>WA</sub> specifies the power delivered to that surface in decibels relative to one picowatt. Devices (e.g., a vacuum cleaner) often have labeling requirements and maximum amounts they are allowed to produce. The [A-weighting](/source/A-weighting) scale is used in the calculation as the metric is concerned with the loudness as perceived by the human ear. Measurements<ref>{{cite web | url = http://blog.nti-audio.com/measurement/eu-sound-power-regulation-vacuum-cleaners | title = EU Sound Power Regulation for Vacuum Cleaners | date = 19 December 2017 | publisher = [NTi Audio] | access-date = 22 December 2017 }}</ref> in accordance with ISO 3744 are taken at 6 to 12 defined points around the device in a hemi-anechoic space. The test environment can be located indoors or outdoors. The required environment is on hard ground in a large open space or hemi-anechoic chamber (free-field over a reflecting plane.)

===Table of selected sound sources===
Here is a table of some examples, from an on-line source.<ref>{{cite web | url = http://www.engineeringtoolbox.com/sound-power-level-d_58.html | title = Sound Power | publisher = The Engineering Toolbox | access-date = 28 November 2013 }}</ref> For omnidirectional point sources in free space, sound power in ''L''<sub>WA</sub> is equal to [sound pressure level](/source/sound_pressure_level) in dB above 20 micropascals at a distance of 0.2821 m<ref>{{cite web | url = http://www.sengpielaudio.com/calculator-soundpower.htm | title = Sound Power Level}}</ref>

{| class="wikitable"
! Situation and<br>sound source !! Sound power<br>([W](/source/Watt)) !! Sound power level<br>([dB](/source/Decibel) ref 10<sup>−12</sup> W)
|-
| [Saturn V](/source/Saturn_V) rocket<ref>{{Cite journal|title=NASA Technical Reports Server (NTRS)|url=https://ntrs.nasa.gov/citations/20120003777|access-date=2021-03-24|website=NASA|date=15 February 2012|quote=the largest sound power levels ever experienced at NASA Stennis was approximately 204dB, which corresponded to the Saturn S‐IC stage on the B‐2 test stand.|last1=Allgood|first1=Daniel C.}}</ref>|| align="right" | {{val|100000000}} || align="right" | 200
|-
| [Turbojet](/source/Turbojet) engine || align="right" | {{val|100000}} || align="right" | 170
|-
| [Turbofan](/source/Turbofan) aircraft at take-off || align="right" | {{val|1000}} || align="right" | 150
|-
| [Turboprop](/source/Turboprop) aircraft at take-off || align="right" | {{val|100}} || align="right" | 140
|-
| [Machine gun](/source/Machine_gun) <br>Large [pipe organ](/source/pipe_organ)|| align="right" | {{val|10}} || align="right" | 130
|-
| [Symphony orchestra](/source/Symphony_orchestra)<br>Heavy [thunder](/source/thunder)<br>[Sonic boom](/source/Sonic_boom) || align="right" | {{val|1}} || align="right" | 120
|-
| [Rock concert](/source/Rock_concert)<br>[Chain saw](/source/Chain_saw)<br>Accelerating [motorcycle](/source/motorcycle)|| align="right" | {{val|0.1}} || align="right" | 110
|-
| [Lawn mower](/source/Lawn_mower)<br>Car at highway speed<br>[Subway steel wheels](/source/Rapid_transit) || align="right" | {{val|0.01}} || align="right" | 100
|-
| Large [diesel vehicle](/source/Diesel_engine) || align="right" | {{val|0.001}} || align="right" | 90
|-
| Loud [alarm clock](/source/alarm_clock) || align="right" | {{val|0.0001}} || align="right" | 80
|-
| Relatively quiet [vacuum cleaner](/source/vacuum_cleaner) || align="right" | {{val|e=-5}} || align="right" | 70
|-
| [Hair dryer](/source/Hair_dryer) || align="right" | {{val|e=-6}} || align="right" | 60
|-
| Radio or TV || align="right" | {{val|e=-7}} || align="right" | 50
|-
| [Refrigerator](/source/Refrigerator)<br/>Low voice || align="right" | {{val|e=-8}} || align="right" | 40
|-
| Quiet conversation || align="right" | {{val|e=-9}} || align="right" | 30
|-
| Whisper of one person<br>Wristwatch ticking|| align="right" | {{val|e=-10}} || align="right" | 20
|-
| Human breath of one person || align="right" | {{val|e=-11}} || align="right" | 10
|-
| Reference value || align="right" | {{val|e=-12}} || align="right" | 0
|}

==Mathematical definition==
Sound power, denoted ''P'', is defined by<ref>Landau & Lifshitz, "Fluid Mechanics", Course of Theoretical Physics, Vol. 6</ref>
:<math>P = \mathbf f \cdot \mathbf v = Ap\, \mathbf u \cdot \mathbf v = Apv</math>
where
*'''f''' is the sound force of unit vector '''u''';
*'''v''' is the [particle velocity](/source/particle_velocity) of projection ''v'' along '''u''';
*''A'' is the area;
*''p'' is the [sound pressure](/source/sound_pressure).

In a [medium](/source/Transmission_medium), the sound power is given by
:<math>P = \frac{A p^2}{\rho c} \cos \theta,</math>
where
*{{mvar|A}} is the area of the surface;
*{{mvar|ρ}} is the [mass density](/source/mass_density);
*{{mvar|c}} is the [sound velocity](/source/sound_velocity);
*{{mvar|θ}} is the angle between the direction of propagation of the sound and the normal to the surface.
*{{mvar|p}} is the [sound pressure](/source/sound_pressure).
For example, a sound at SPL = 85&nbsp;dB or ''p'' = 0.356 Pa in air (''ρ'' = {{val|1.2|u=kg.m-3}} and ''c'' = {{val|343|u=m.s-1}}) through a surface of area ''A'' = {{val|1|u=m2}} normal to the direction of propagation (''θ'' = 0°) has a sound energy flux ''P'' = {{val|0.3|u=mW}}.

This is the parameter one would be interested in when converting noise back into usable energy, along with any losses in the capturing device.

==Relationships with other quantities==
Sound power is related to [sound intensity](/source/sound_intensity):
:<math>P = AI,</math>
where
*''A'' stands for the area;
*''I'' stands for the sound intensity.

Sound power is related [sound energy density](/source/sound_energy_density):
:<math>P = Acw,</math>
where
*''c'' stands for the [speed of sound](/source/speed_of_sound);
*''w'' stands for the sound energy density.

==Sound power level==
{{Other uses|Sound level (disambiguation){{!}}Sound level}}
'''Sound power level''' (SWL) or '''acoustic power level''' is a [logarithmic measure](/source/Level_(logarithmic_quantity)) of the power of a sound relative to a reference value.<br>
Sound power level, denoted ''L''<sub>''W''</sub> and measured in [dB](/source/Decibel),<ref name=IEC60027-3>[http://webstore.iec.ch/webstore/webstore.nsf/artnum/028981 "Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units"], ''IEC 60027-3 Ed. 3.0'', International Electrotechnical Commission, 19 July 2002.</ref> is defined by:<ref>{{cite book |vauthors=Attenborough K, Postema M|title=A pocket-sized introduction to acoustics|date=2008 |publisher=University of Hull|location=Kingston upon Hull|url=https://hal.archives-ouvertes.fr/hal-03188302/document|isbn=978-90-812588-2-1|doi=10.5281/zenodo.7504060}}</ref>
:<math>L_W = \frac{1}{2} \ln\!\left(\frac{P}{P_0}\right)\!~\mathrm{Np} = \log_{10}\!\left(\frac{P}{P_0}\right)\!~\mathrm{B} = 10 \log_{10}\!\left(\frac{P}{P_0}\right)\!~\mathrm{dB},</math>
where
*''P'' is the sound power;
*''P''<sub>0</sub> is the ''reference sound power'';
*{{nowrap|1=1 Np = 1}} is the [neper](/source/neper);
*{{nowrap|1=1 B = {{sfrac|2}} ln 10}} is the [bel](/source/Decibel);
*{{nowrap|1=1 dB = {{sfrac|20}} ln 10 }} is the [decibel](/source/decibel).

The commonly used reference sound power in air is<ref>Ross Roeser, Michael Valente, ''Audiology: Diagnosis'' (Thieme 2007), p. 240.</ref>
:<math>P_0 = 1~\mathrm{pW}.</math>
The proper notations for sound power level using this reference are {{nobreak|''L''<sub>''W''/(1 pW)</sub>}} or {{nobreak|''L''<sub>''W''</sub> (re 1 pW)}}, but the suffix notations {{nobreak|dB SWL}}, {{nobreak|dB(SWL)}}, dBSWL, or dB<sub>SWL</sub> are very common, even if they are not accepted by the SI.<ref name=NIST2008>Thompson, A. and Taylor, B. N. sec 8.7, "Logarithmic quantities and units: level, neper, bel", ''Guide for the Use of the International System of Units (SI) 2008 Edition'', NIST Special Publication 811, 2nd printing (November 2008), SP811 [http://physics.nist.gov/cuu/pdf/sp811.pdf PDF]</ref>

The reference sound power ''P''<sub>0</sub> is defined as the sound power with the reference sound intensity {{nowrap|1=''I''<sub>0</sub> = 1 pW/m<sup>2</sup>}} passing through a surface of area {{nowrap|1=''A''<sub>0</sub> = 1 m<sup>2</sup>}}:
:<math>P_0 = A_0 I_0,</math>
hence the reference value {{nowrap|1=''P''<sub>0</sub> = 1 pW}}.

===Relationship with sound pressure level===
The generic calculation of sound power from sound pressure is as follows:
:<math>L_W = L_p + 10 \log_{10}\!\left(\frac{A_S}{A_0}\right)\!~\mathrm{dB},</math>
where:
<math>{A_S}</math> defines the area of a surface that wholly encompasses the source.  This surface may be any shape, but it must fully enclose the source. 
 
In the case of a sound source located in free field positioned over a reflecting plane (i.e. the ground), in air at ambient temperature, the sound power level at distance ''r'' from the sound source is approximately related to [sound pressure level](/source/sound_pressure_level) (SPL)  by<ref name=Chadderton>Chadderton, David V. ''Building services engineering'', pp. 301, 306, 309, 322. Taylor & Francis, 2004. {{ISBN|0-415-31535-2}}</ref>
:<math>L_W = L_p + 10 \log_{10}\!\left(\frac{2\pi r^2}{A_0}\right)\!~\mathrm{dB},</math>
where
*''L''<sub>''p''</sub> is the sound pressure level;
*''A''<sub>0</sub> = 1 m<sup>2</sup>;
*<math> {2\pi r^2},</math> defines the surface area of a hemisphere; and 
*''r'' must be sufficient that the hemisphere fully encloses the source.

Derivation of this equation:
:<math>\begin{align}
L_W &= \frac{1}{2} \ln\!\left(\frac{P}{P_0}\right)\\
        &= \frac{1}{2} \ln\!\left(\frac{AI}{A_0 I_0}\right)\\
        &= \frac{1}{2} \ln\!\left(\frac{I}{I_0}\right) + \frac{1}{2} \ln\!\left(\frac{A}{A_0}\right)\!.
\end{align}</math>
For a ''progressive'' spherical wave,
:<math>z_0 = \frac{p}{v},</math>
:<math>A = 4\pi r^2,</math> (the surface area of sphere)
where ''z''<sub>0</sub> is the [characteristic specific acoustic impedance](/source/Acoustic_impedance).

Consequently,
:<math>I = pv = \frac{p^2}{z_0},</math>
and since by definition {{nobreak|1=''I''<sub>0</sub> = ''p''<sub>0</sub><sup>2</sup>/''z''<sub>0</sub>}}, where {{nobreak|1=''p''<sub>0</sub> = 20 μPa}} is the reference sound pressure,
:<math>\begin{align}
L_W &= \frac{1}{2} \ln\!\left(\frac{p^2}{p_0^2}\right) + \frac{1}{2} \ln\!\left(\frac{4\pi r^2}{A_0}\right)\\
        &= \ln\!\left(\frac{p}{p_0}\right) + \frac{1}{2} \ln\!\left(\frac{4\pi r^2}{A_0}\right)\\
        &= L_p + 10 \log_{10}\!\left(\frac{4\pi r^2}{A_0}\right)\!~\mathrm{dB}.
\end{align}</math>

The sound power estimated practically does not depend on distance. The sound pressure used in the calculation may be affected by distance due to viscous effects in the propagation of sound unless this is accounted for.

==References==
{{Reflist}}

==External links==
*[https://www.bksv.com/en/knowledge/blog/sound/sound-power-sound-pressure Sound power and Sound pressure. Cause and Effect]
*[http://www.sengpielaudio.com/calculator-ak-ohm.htm Ohm's Law as Acoustic Equivalent. Calculations]
*[http://www.sengpielaudio.com/RelationshipsOfAcousticQuantities.pdf Relationships of Acoustic Quantities Associated with a Plane Progressive Acoustic Sound Wave]
*[http://wwwn.cdc.gov/niosh-sound-vibration/default.aspx NIOSH Powertools Database] {{Webarchive|url=https://web.archive.org/web/20091112010908/http://wwwn.cdc.gov/niosh-sound-vibration/default.aspx |date=2009-11-12 }}
*[https://archive.today/20130208115754/http://www.lmsintl.com/testing/testlab/acoustics/sound-power-testing Sound Power Testing]

{{Authority control}}

Category:Acoustics
Category:Sound
Category:Sound measurements
Category:Physical quantities
Category:Power (physics)

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Adapted from the Wikipedia article [Sound power](https://en.wikipedia.org/wiki/Sound_power) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Sound_power?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.