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]

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.

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.

Table of selected sound sources

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

Here is a table of some examples.^[3]

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 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

Relationships with other quantities

Sound power is related to sound intensity:

P=AI,

where

A is the area;
I is the sound intensity.

Sound power is related sound energy density:

P=Acw,

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]

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^[5]

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

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 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:

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:

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^[7]

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;
A₀ = 1 m²;
${2\pi r^{2}},$ 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 z₀ is the characteristic specific acoustic impedance.

Consequently,

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

and since by definition I₀ = p₀²/z₀, where p₀ = 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.

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

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