# Sound power

Sound measurements
Characteristic
Symbols
Sound pressure p, SPL,LPA
Particle velocity v, SVL
Particle displacement δ
Sound intensity I, SIL
Sound power P, SWL, LWA
Sound energy W
Sound energy density w
Sound exposure E, SEL
Acoustic impedance Z
Speed of sound c
Audio frequency AF
Transmission loss TL

Sound power or acoustic power is the rate at which sound energy is emitted, reflected, transmitted or received, per unit time.[1] It is defined[2] as "through a surface, the product of the sound pressure, and the component of the particle velocity, at a point on the surface in the direction normal to the surface, integrated over that surface." The SI unit of sound power is the watt (W).[1] 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 or acoustic flux through that area.

## Sound power level LWA

Maximum sound power level (LWA) related to a portable air compressor.

Regulations often specify a method for measurement[3] that integrates sound pressure over a surface enclosing the source. LWA 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[4] 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.[5] For omnidirectional sources in free space, sound power in LwA is equal to sound pressure level in dB above 20 micropascals at a distance of 0.2821 m[6]

Situation and
sound source
Sound power
(W)
Sound power level
(dB ref 10−12 W)
Saturn V rocket 100,000,000 200
Project Artemis Sonar 1,000,000 180
Turbojet engine 100,000 170
Turbofan aircraft at take-off 1,000 150
Turboprop aircraft at take-off 100 140
Machine gun
Large pipe organ
10 130
Symphony orchestra
Heavy thunder
Sonic boom
1 120
Rock concert (1970s)
Chain saw
Accelerating motorcycle
0.1 110
Lawn mower
Car at highway speed
Subway steel wheels
0.01 100
Large diesel vehicle 0.001 90
Loud alarm clock 0.0001 80
Relatively quiet vacuum cleaner 10−5 70
Hair dryer 10−6 60
Radio or TV 10−7 50
Refrigerator
Low voice
10−8 40
Quiet conversation 10−9 30
Whisper of one person
Wristwatch ticking
10−10 20
Human breath of one person 10−11 10
Reference value 10−12 0

## Mathematical definition

Sound power, denoted P, is defined by[7]

${\displaystyle P=\mathbf {f} \cdot \mathbf {v} =Ap\,\mathbf {u} \cdot \mathbf {v} =Apv}$

where

In a medium, the sound power is given by

${\displaystyle P={\frac {Ap^{2}}{\rho c}}\cos \theta ,}$

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 = 1.2 kg⋅m−3 and c = 343 m⋅s−1) through a surface of area A = 1 m2 normal to the direction of propagation (θ = 0°) has a sound energy flux P = 0.3 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:

${\displaystyle P=AI,}$

where

• A is the area;
• I is the sound intensity.

Sound power is related sound energy density:

${\displaystyle P=Acw,}$

where

## Sound power level definition

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 LW and measured in dB, is defined by[8]

${\displaystyle L_{W}={\frac {1}{2}}\ln \!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {Np} =\log _{10}\!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {B} =10\log _{10}\!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {dB} ,}$

where

• P is the sound power;
• P0 is the reference sound power;
• 1 Np = 1 is the neper;
• 1 B = 1/2 ln 10 is the bel;
• 1 dB = 1/20 ln 10 is the decibel.

The commonly used reference sound power in air is[9]

${\displaystyle P_{0}=1~\mathrm {pW} .}$

The proper notations for sound power level using this reference are LW/(1 pW) or LW (re 1 pW), but the suffix notations dB SWL, dB(SWL), dBSWL, or dBSWL are very common, even if they are not accepted by the SI.[10]

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

${\displaystyle P_{0}=A_{0}I_{0},}$

hence the reference value P0 = 1 pW.

### Relationship with sound pressure level

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

${\displaystyle L_{W}=L_{p}+10\log _{10}\!\left({\frac {A_{S}}{A_{0}}}\right)\!~\mathrm {dB} ,}$

where: ${\displaystyle {A_{S}}}$ 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) by[11]

${\displaystyle L_{W}=L_{p}+10\log _{10}\!\left({\frac {2\pi r^{2}}{A_{0}}}\right)\!~\mathrm {dB} ,}$

where

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

Derivation of this equation:

{\displaystyle {\begin{aligned}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{aligned}}}

For a progressive spherical wave,

${\displaystyle z_{0}={\frac {p}{v}},}$
${\displaystyle A=4\pi r^{2},}$ (the surface area of sphere)

where z0 is the characteristic specific acoustic impedance.

Consequently,

${\displaystyle I=pv={\frac {p^{2}}{z_{0}}},}$

and since by definition I0 = p02/z0, where p0 = 20 μPa is the reference sound pressure,

{\displaystyle {\begin{aligned}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{aligned}}}

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

1. ^ a b Ronald J. Baken, Robert F. Orlikoff (2000). Clinical Measurement of Speech and Voice. Cengage Learning. p. 94. ISBN 9781565938694.
2. ^
3. ^
4. ^ "EU Sound Power Regulation for Vacuum Cleaners". [NTi Audio]. Retrieved 22 December 2017.
5. ^ "Sound Power". The Engineering Toolbox. Retrieved 28 November 2013.
6. ^
7. ^ Landau & Lifshitz, "Fluid Mechanics", Course of Theoretical Physics, Vol. 6
8. ^ "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.
9. ^ Ross Roeser, Michael Valente, Audiology: Diagnosis (Thieme 2007), p. 240.
10. ^ 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 PDF
11. ^ Chadderton, David V. Building services engineering, pp. 301, 306, 309, 322. Taylor & Francis, 2004. ISBN 0-415-31535-2