Editing Sound power (section)

==Sound power level==
{{Other uses|Sound level (disambiguation){{!}}Sound level}}
'''Sound power level''' (SWL) or '''acoustic power level''' is a [[Level (logarithmic quantity)|logarithmic measure]] of the power of a sound relative to a reference value.<br>
Sound power level, denoted ''L''<sub>''W''</sub> and measured in [[Decibel|dB]],<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]];
*{{nowrap|1=1 B = {{sfrac|2}} ln 10}} is the [[Decibel|bel]];
*{{nowrap|1=1 dB = {{sfrac|20}} ln 10 }} is the [[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]] (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 [[Acoustic impedance#Characteristic specific acoustic impedance|characteristic specific 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.