Sound power

File:Atlas Copco XAHS 347-pic7-Max. sound power level.jpg) related to a portable air compressor]]Regulations often specify a method for measurement{{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 }} that integrates sound pressure over a surface enclosing the source. L_WA 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 scale is used in the calculation as the metric is concerned with the loudness as perceived by the human ear. Measurements{{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 }} 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.)

Here is a table of some examples, from an on-line source.{{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 }} For omnidirectional point sources in free space, sound power in L_wA is equal to sound pressure level in dB above 20 micropascals at a distance of 0.2821 m{{cite web | url = http://www.sengpielaudio.com/calculator-soundpower.htm | title = Sound Power Level}}

Sound power, denoted P, is defined byLandau & Lifshitz, "Fluid Mechanics", Course of Theoretical Physics, Vol. 6

:$P\; =\; \backslash mathbf\; f\; \backslash cdot\; \backslash mathbf\; v\; =\; Ap\backslash ,\; \backslash mathbf\; u\; \backslash cdot\; \backslash mathbf\; v\; =\; Apv$

where

• f is the sound force of unit vector u;
• v is the particle velocity of projection v along u;
• A is the area;
• p is the sound pressure.

In a medium, the sound power is given by

:$P\; =\; \backslash frac\{A\; p^2\}\{\backslash rho\; c\}\; \backslash cos\; \backslash theta,$

where

• A is the area of the surface;
• ρ is the mass density;
• c is the sound velocity;
• θ is the angle between the direction of propagation of the sound and the normal to the surface.
• p is the sound pressure.

For example, a sound at SPL = 85 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.

Sound power is related to sound intensity:

:$P\; =\; AI,$

where

• A stands for the area;
• I stands for the sound intensity.

Sound power is related sound energy density:

:$P\; =\; Acw,$

where

• c stands for the speed of sound;
• w stands for the sound energy density.

{{Other uses|Sound level (disambiguation){{!}}Sound level}}

Sound power level (SWL) or acoustic power level is a logarithmic measure of the power of a sound relative to a reference value.

Sound power level, denoted L_W and measured in dB,[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. is defined by:{{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}}

:$L\_W\; =\; \backslash frac\{1\}\{2\}\; \backslash ln\backslash !\backslash left(\backslash frac\{P\}\{P\_0\}\backslash right)\backslash !~\backslash mathrm\{Np\}\; =\; \backslash log\_\{10\}\backslash !\backslash left(\backslash frac\{P\}\{P\_0\}\backslash right)\backslash !~\backslash mathrm\{B\}\; =\; 10\; \backslash log\_\{10\}\backslash !\backslash left(\backslash frac\{P\}\{P\_0\}\backslash right)\backslash !~\backslash mathrm\{dB\},$

where

• P is the sound power;
• P₀ is the reference sound power;
• {{nowrap|1=1 Np = 1}} is the neper;
• {{nowrap|1=1 B = {{sfrac|2}} ln 10}} is the bel;
• {{nowrap|1=1 dB = {{sfrac|20}} ln 10 }} is the decibel.

The commonly used reference sound power in air isRoss Roeser, Michael Valente, Audiology: Diagnosis (Thieme 2007), p. 240.

:$P\_0\; =\; 1~\backslash mathrm\{pW\}.$

The proper notations for sound power level using this reference are {{nobreak|L_{W/(1 pW)}}} or {{nobreak|L_W (re 1 pW)}}, but the suffix notations {{nobreak|dB SWL}}, {{nobreak|dB(SWL)}}, dBSWL, or dB_SWL are very common, even if they are not accepted by the SI.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]

The reference sound power P₀ is defined as the sound power with the reference sound intensity {{nowrap|1=I₀ = 1 pW/m²}} passing through a surface of area {{nowrap|1=A₀ = 1 m²}}:

:$P\_0\; =\; A\_0\; I\_0,$

hence the reference value {{nowrap|1=P₀ = 1 pW}}.

The generic calculation of sound power from sound pressure is as follows:

:$L\_W\; =\; L\_p\; +\; 10\; \backslash log\_\{10\}\backslash !\backslash left(\backslash frac\{A\_S\}\{A\_0\}\backslash right)\backslash !~\backslash mathrm\{dB\},$

where:

$\{A\_S\}$ 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 (SPL) byChadderton, David V. Building services engineering, pp. 301, 306, 309, 322. Taylor & Francis, 2004. {{ISBN|0-415-31535-2}}

:$L\_W\; =\; L\_p\; +\; 10\; \backslash log\_\{10\}\backslash !\backslash left(\backslash frac\{2\backslash pi\; r^2\}\{A\_0\}\backslash right)\backslash !~\backslash mathrm\{dB\},$

where

• L_p is the sound pressure level;
• A₀ = 1 m²;
• $\{2\backslash pi\; r^2\},$ defines the surface area of a hemisphere; and
• r must be sufficient that the hemisphere fully encloses the source.

Derivation of this equation:

:$\backslash begin\{align\}$

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}

For a progressive spherical wave,

:$z\_0\; =\; \backslash frac\{p\}\{v\},$

:$A\; =\; 4\backslash pi\; r^2,$ (the surface area of sphere)

where z₀ is the characteristic specific acoustic impedance.

Consequently,

:$I\; =\; pv\; =\; \backslash frac\{p^2\}\{z\_0\},$

and since by definition {{nobreak|1=I₀ = p₀²/z₀}}, where {{nobreak|1=p₀ = 20 μPa}} is the reference sound pressure,

:$\backslash begin\{align\}$

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}

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.

{{Reflist}}

• [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]

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Category:Acoustics

Category:Sound

Category:Sound measurements

Category:Physical quantities

Category:Power (physics)