Sound power

Sound power or acoustic power is the rate at which sound energy is emitted, reflected, transmitted or received, per unit time.<ref name=clinical>Template:Cite book</ref> It is defined<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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).<ref name=clinical/> 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

File:Atlas Copco XAHS 347-pic7-Max. sound power level.jpg

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

Regulations often specify a method for measurement<ref>{{#invoke:citation/CS1|citation

|CitationClass=web }}</ref> 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<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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

Mathematical definition

Sound power, denoted P, is defined by<ref>Landau & Lifshitz, "Fluid Mechanics", Course of Theoretical Physics, Vol. 6</ref>

<math>P = \mathbf f \cdot \mathbf v = Ap\, \mathbf u \cdot \mathbf v = Apv</math>

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

<math>P = \frac{A p^2}{\rho c} \cos \theta,</math>

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 = Template:Val and c = Template:Val) through a surface of area A = Template:Val normal to the direction of propagation (θ = 0°) has a sound energy flux P = Template:Val.

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:

<math>P = AI,</math>

where

• A stands for the area;
• I stands for the sound intensity.

Sound power is related sound energy density:

<math>P = Acw,</math>

where

• c stands for the speed of sound;
• w stands for the sound energy density.

Sound power level

{{#invoke:other uses|otheruses}} 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,<ref name=IEC60027-3>"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.</ref> is defined by:<ref>Template:Cite book</ref>

<math>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},</math>

where

• P is the sound power;
• P₀ is the reference sound power;
• Template:Nowrap is the neper;
• Template:Nowrap is the bel;
• Template:Nowrap is the decibel.

The commonly used reference sound power in air is<ref>Ross Roeser, Michael Valente, Audiology: Diagnosis (Thieme 2007), p. 240.</ref>

<math>P_0 = 1~\mathrm{pW}.</math>

The proper notations for sound power level using this reference are Template:Nobreak or Template:Nobreak, but the suffix notations Template:Nobreak, Template:Nobreak, dBSWL, or dB_SWL are very common, even if they are not accepted by the SI.<ref name=NIST2008>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</ref>

The reference sound power P₀ is defined as the sound power with the reference sound intensity Template:Nowrap passing through a surface of area Template:Nowrap:

<math>P_0 = A_0 I_0,</math>

hence the reference value Template:Nowrap.

Relationship with sound pressure level

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

<math>L_W = L_p + 10 \log_{10}\!\left(\frac{A_S}{A_0}\right)\!~\mathrm{dB},</math>

where: <math>{A_S}</math> 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<ref name=Chadderton>Chadderton, David V. Building services engineering, pp. 301, 306, 309, 322. Taylor & Francis, 2004. Template:ISBN</ref>

<math>L_W = L_p + 10 \log_{10}\!\left(\frac{2\pi r^2}{A_0}\right)\!~\mathrm{dB},</math>

where

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

Derivation of this equation:

<math>\begin{align}

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{align}</math> For a progressive spherical wave,

<math>z_0 = \frac{p}{v},</math>

<math>A = 4\pi r^2,</math> (the surface area of sphere)

where z₀ is the characteristic specific acoustic impedance.

Consequently,

<math>I = pv = \frac{p^2}{z_0},</math>

and since by definition Template:Nobreak, where Template:Nobreak is the reference sound pressure,

<math>\begin{align}

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{align}</math>

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.

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 Template:Webarchive
• Sound Power Testing