# Sound power

Sound power or acoustic power is the rate at which sound energy is emitted, reflected, transmitted or received, per unit time. It is defined 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). 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 L$_{WA}$

Regulations often specify a method for measurement 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 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. 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
| Situation and sound source | Sound power (W) | Sound power level (dB ref 10$^{−12}$ W) |
| Saturn V rocket | 100000000 | 200 |
| Turbojet engine | 100000 | 170 |
| Turbofan aircraft at take-off | 1000 | 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
    $P=\mathbf {f} \cdot \mathbf {v} =Ap\,\mathbf {u} \cdot \mathbf {v} =Apv$
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
    $P={\frac {Ap^{2}}{\rho c}}\cos \theta ,$
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 m$^2$ 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:
    $P=AI,$
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.

## Sound power 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, is defined by:
    $L_{W}={\frac {1}{2}}\ln \!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {Np} =\log _{10}\!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {B} =10\log _{10}\!\left({\frac {P}{P_{0}}}\right)\!~\mathrm {dB} ,$
where
* P is the sound power;
* P$_0$ 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
    $P_{0}=1~\mathrm {pW} .$
The proper notations for sound power level using this reference are L$_{W/(1pW)}$ or L$_W$ (re 1 pW), but the suffix notations dB SWL, dB(SWL), dBSWL, or dB$_{SWL}$ are very common, even if they are not accepted by the SI.
The reference sound power P$_0$ is defined as the sound power with the reference sound intensity I$_0$ = 1 pW/m$^2$ passing through a surface of area A$_0$ = 1 m$^2$:
    $P_{0}=A_{0}I_{0},$
hence the reference value P$_0$ = 1 pW.

### Relationship with sound pressure level

The generic calculation of sound power from sound pressure is as follows:
    $L_{W}=L_{p}+10\log _{10}\!\left({\frac {A_{S}}{A_{0}}}\right)\!~\mathrm {dB} ,$
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) by
    $L_{W}=L_{p}+10\log _{10}\!\left({\frac {2\pi r^{2}}{A_{0}}}\right)\!~\mathrm {dB} ,$
where
* L$_p$ is the sound pressure level;
* A$_0$ = 1 m$^2$;
* ${2\pi r^{2}},$ defines the surface area of a hemisphere; and
* r must be sufficient that the hemisphere fully encloses the source.
Derivation of this equation:
    ${\begin{aligned}L_{W}&={\frac {1}{2}}\ln \!\left({\frac {P}{P_{0}}}\right)\\&={\frac {1}{2}}\ln \!\left({\frac {AI}{A_{0}I_{0}}}\right)\\&={\frac {1}{2}}\ln \!\left({\frac {I}{I_{0}}}\right)+{\frac {1}{2}}\ln \!\left({\frac {A}{A_{0}}}\right)\!.\end{aligned}}$
For a progressive spherical wave,
    $z_{0}={\frac {p}{v}},$
    $A=4\pi r^{2},$ (the surface area of sphere)
where z$_0$ is the characteristic specific acoustic impedance.
Consequently,
    $I=pv={\frac {p^{2}}{z_{0}}},$
and since by definition I$_0$ = p$_0$$^2$/z$_0$, where p$_0$ = 20 μPa is the reference sound pressure,
    ${\begin{aligned}L_{W}&={\frac {1}{2}}\ln \!\left({\frac {p^{2}}{p_{0}^{2}}}\right)+{\frac {1}{2}}\ln \!\left({\frac {4\pi r^{2}}{A_{0}}}\right)\\&=\ln \!\left({\frac {p}{p_{0}}}\right)+{\frac {1}{2}}\ln \!\left({\frac {4\pi r^{2}}{A_{0}}}\right)\\&=L_{p}+10\log _{10}\!\left({\frac {4\pi r^{2}}{A_{0}}}\right)\!~\mathrm {dB} .\end{aligned}}$
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.