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Sound power

Sound power, denoted P, is defined by^[1]

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

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

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

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.

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

Here is a table of some examples.^[2]

Situation and sound source	Sound power (W)	Sound power level (dB ref 10−12W)
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:

${\displaystyle P=AI,}$

where

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

Sound power is related sound energy density:

${\displaystyle P=Acw,}$

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^[3]

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

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^[4]

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

The proper notations for sound power level using this reference are L_{W/(1 pW)}orL_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.^[5]

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^[6]

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

where

• L_p is the sound pressure level;
• A₀ = 1 m²;
• ${\displaystyle {2\pi r^{2}},}$ 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

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