Sound power

Sound power or acoustic power is the rate at which sound energy is emitted, reflected, transmitted or received, per unit time.[1] It is defined[2] 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).[1] 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 measurements
Characteristic
Symbols
 Sound pressure p, SPL,LPA
 Particle velocity v, SVL
 Particle displacement δ
 Sound intensity I, SIL
 Sound power P, SWL, LWA
 Sound energy W
 Sound energy density w
 Sound exposure E, SEL
 Acoustic impedance Z
 Audio frequency AF
 Transmission loss TL

Sound power level LWA

Regulations often specify a method for measurement[3] that integrates sound pressure over a surface enclosing the source. LWA 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[4] 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.[5] 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. This doesn't seem to be the case, at least not when the sound has to travel along the surface of the earth. For omnidirectional sources in free space, sound power in LwA is equal to sound pressure level in dB above 20 micropascals at a distance of 0.2821 m[6]

Situation and
sound source
Sound power
(W)
Sound power level
(dB ref 10−12 W)
Saturn V rocket100,000,000200
Project Artemis Sonar1,000,000180
Turbojet engine100,000170
Turbofan aircraft at take-off1,000150
Turboprop aircraft at take-off100140
Machine gun
Large pipe organ
10130
Symphony orchestra
Heavy thunder
Sonic boom
1120
Rock concert (1970s)
Chain saw
Accelerating motorcycle
0.1110
Lawn mower
Car at highway speed
Subway steel wheels
0.01100
Large diesel vehicle0.00190
Loud alarm clock0.000180
Relatively quiet vacuum cleaner10−570
Hair dryer10−660
Radio or TV10−750
Refrigerator
Low voice
10−840
Quiet conversation10−930
Whisper of one person
Wristwatch ticking
10−1020
Human breath of one person10−1110
Reference value10−120

Mathematical definition

Sound power, denoted P, is defined by[7]

where

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.
  • p is the sound pressure.

For example, a sound at SPL = 85 dB or p = 0.356 Pa in air (ρ = 1.2 kgm−3 and c = 343 ms−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.

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

Sound power level definition

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 LW and measured in dB, is defined by[8]

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[9]

The proper notations for sound power level using this reference are LW/(1 pW) or LW (re 1 pW), but the suffix notations dB SWL, dB(SWL), dBSWL, or dBSWL are very common, even if they are not accepted by the SI.[10]

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:

hence the reference value P0 = 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[11]

where

  • Lp is the sound pressure level;
  • A0 = 1 m2;
  • 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 z0 is the characteristic specific acoustic impedance.

Consequently,

and since by definition I0 = p02/z0, where p0 = 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. Ronald J. Baken, Robert F. Orlikoff (2000). Clinical Measurement of Speech and Voice. Cengage Learning. p. 94. ISBN 9781565938694.
  2. "ISO 80000-8(en) Quantities and Units - Acoustics". [ISO].
  3. "ISO 3744:2010(en) Acoustics — Determination of sound power levels and sound energy levels of noise sources using sound pressure — Engineering methods for an essentially free field over a reflecting plane". [ISO]. Retrieved 22 December 2017.
  4. "EU Sound Power Regulation for Vacuum Cleaners". [NTi Audio]. Retrieved 22 December 2017.
  5. "Sound Power". The Engineering Toolbox. Retrieved 28 November 2013.
  6. "Sound Power Level".
  7. Landau & Lifshitz, "Fluid Mechanics", Course of Theoretical Physics, Vol. 6
  8. "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.
  9. Ross Roeser, Michael Valente, Audiology: Diagnosis (Thieme 2007), p. 240.
  10. 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
  11. Chadderton, David V. Building services engineering, pp. 301, 306, 309, 322. Taylor & Francis, 2004. ISBN 0-415-31535-2
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