# Sound power

Sound measurements | |
---|---|

Characteristic |
Symbols |

Sound pressure |
p, SPL |

Particle velocity |
v, SVL |

Particle displacement |
δ |

Sound intensity |
I, SIL |

Sound power |
P, SWL |

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]} The SI unit of sound power is the watt (W).^{[1]} It is the power of the sound force on a surface of the medium of propagation of the sound wave. For a sound source, unlike sound pressure, sound power is neither room-dependent nor distance-dependent. Sound pressure is a measurement at a point in space near the source, while the sound power of a source is 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.

## Mathematical definition

Sound power, denoted *P*, is defined by^{[2]}

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

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.

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.

## Table of selected sound sources

Here is a table of some examples, from an on-line source.^{[3]} The top three or four lines seem questionable. A 150-dB source should be audible 100 times as far away as a 110-dB source, and a 200-dB source 10 000 times further than a 120-dB source. This would imply that a turbofan aircraft at take-off would be audible 100 times further than a rock concert, and a Saturn V 10 000 times further than heavy thunder. However this simple analysis does not account for the effect of atmospheric absorption effects and how they change with frequency.

Situation and

sound sourceSound power

(W)Sound power level

(dB ref 10^{−12}W)Saturn V rocket **100,000,000**200 Project Artemis Sonar **1,000,000**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

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 a loud Alarm clock **0.0001**80 a 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

## Relationships with other quantities

Sound power is related to sound intensity:

where

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

Sound power is related sound energy density:

where

*c*is the speed of sound;*w*is 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^{[4]}

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^{[5]}

The proper notations for sound power level using this reference are *L*_{W/(1 pW)} 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.^{[6]}

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}:

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:

where: 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^{[7]}

where

*L*_{p}is the sound pressure level;*A*_{0}= 1 m^{2};- defines the surface area of a hemisphere; and
*r*must be sufficient that the hemisphere fully encloses the source.

Derivation of this equation:

For a *progressive* spherical wave,

- (the surface area of sphere)

where *z*_{0} is the characteristic specific acoustic impedance.

Consequently,

and since by definition *I*_{0} = *p*_{0}^{2}/*z*_{0}, where *p*_{0} = 20 μPa is the reference sound pressure,

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 2 Ronald J. Baken, Robert F. Orlikoff (2000).
*Clinical Measurement of Speech and Voice*. Cengage Learning. p. 94. ISBN 9781565938694. - ↑ Landau & Lifshitz, "Fluid Mechanics", Course of Theoretical Physics, Vol. 6
- ↑ "Sound Power". The Engineering Toolbox. Retrieved 28 November 2013.
- ↑ "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. - ↑ Ross Roeser, Michael Valente,
*Audiology: Diagnosis*(Thieme 2007), p. 240. - ↑ 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 - ↑ Chadderton, David V.
*Building services engineering*, pp. 301, 306, 309, 322. Taylor & Francis, 2004. ISBN 0-415-31535-2

## External links

- Sound power and Sound pressure. Cause and Effect
- Ohm's Law as Acoustic Equivalent. Calculations
- Relationships of Acoustic Quantities Associated with a Plane Progressive Acoustic Sound Wave
- NIOSH Powertools Database
- Sound Power Testing