Sound Power

Sound power or acoustic power is the rate at which sound energy is emitted, reflected, transmitted or received, per unit time. It is defined 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). 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

Regulations often specify a method for measurement 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 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. 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

Mathematical definition

Sound power, denoted ''P'', is defined by
:$P = \mathbf f \cdot \mathbf v = Ap\, \mathbf u \cdot \mathbf v = Apv$
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
:$P = \frac \cos \theta,$
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.
*''p'' is the sound pressure.
For example, a sound at SPL = 85 dB or ''p'' = 0.356 Pa in air (''ρ'' = and ''c'' = ) through a surface of area ''A'' = normal to the direction of propagation (''θ'' = 0°) has a sound energy flux ''P'' = .

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:
:$P = AI,$
where
*''A'' stands for the area;
*''I'' stands for the sound intensity.

Sound power is related sound energy density:
:$P = Acw,$
where
*''c'' stands for the speed of sound;
*''w'' stands for 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,"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. is defined by:
:$L_W = \frac \ln\!\left(\frac\right)\!~\mathrm = \log_\!\left(\frac\right)\!~\mathrm = 10 \log_\!\left(\frac\right)\!~\mathrm,$
where
*''P'' is the sound power;
*''P''₀ is the ''reference sound power'';
* is the neper;
* is the bel;
* is the decibel.

The commonly used reference sound power in air isRoss Roeser, Michael Valente, ''Audiology: Diagnosis'' (Thieme 2007), p. 240.
:$P_0 = 1~\mathrm.$
The proper notations for sound power level using this reference are or , but the suffix notations , , dBSWL, or dB_SWL are very common, even if they are not accepted by the SI.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), SP81
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The reference sound power ''P''₀ is defined as the sound power with the reference sound intensity passing through a surface of area :
:$P_0 = A_0 I_0,$
hence the reference value .

Relationship with sound pressure level

The generic calculation of sound power from sound pressure is as follows:
:$L_W = L_p + 10 \log_\!\left(\frac\right)\!~\mathrm,$
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) byChadderton, David V. ''Building services engineering'', pp. 301, 306, 309, 322. Taylor & Francis, 2004.
:$L_W = L_p + 10 \log_\!\left(\frac\right)\!~\mathrm,$
where
*''L''_''p'' is the sound pressure level;
*''A''₀ = 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:
:$\begin L_W &= \frac \ln\!\left(\frac\right)\\ &= \frac \ln\!\left(\frac\right)\\ &= \frac \ln\!\left(\frac\right) + \frac \ln\!\left(\frac\right)\!. \end$
For a ''progressive'' spherical wave,
:$z_0 = \frac,$
:$A = 4\pi r^2,$ (the surface area of sphere)
where ''z''₀ is the characteristic specific acoustic impedance.

Consequently,
:$I = pv = \frac,$
and since by definition , where is the reference sound pressure,
:$\begin L_W &= \frac \ln\!\left(\frac\right) + \frac \ln\!\left(\frac\right)\\ &= \ln\!\left(\frac\right) + \frac \ln\!\left(\frac\right)\\ &= L_p + 10 \log_\!\left(\frac\right)\!~\mathrm. \end$

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 EffectRelationships of Acoustic Quantities Associated with a Plane Progressive Acoustic Sound WaveNIOSH Powertools Database

Sound Power Testing

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Acoustics
Sound
Sound measurements
Physical quantities
Power (physics)