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

$P = \frac{A p^2}{\rho c} \cos \theta,$

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, from an on-line source.^[3] 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.

Situation and sound source	Sound power (W)	Sound power level (dB ref 10^−12 W)
Saturn V rocket	100,000,000	200
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
Alarm clock	0.0001	80
Noisy office Vacuum cleaner	10^−5	70
Busy restaurant Hair dryer	10^−6	60
Quiet office Average home	10^−7	50
Refrigerator low voice Quiet home	10^−8	40
Quiet conversation Broadcast studio	10^−9	30
Whisper Wristwatch ticking	10^−10	20
Human breath	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

In the case of a free field sound source 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]

$L_W = L_p + 10 \log_{10}\!\left(\frac{4\pi r^2}{A_0}\right)\!~\mathrm{dB},$

where

• L_p is the sound pressure level;
• A₀ = 1 m².

Derivation of this equation:

$\begin{align} L_W &= \frac{1}{2} \ln\!\left(\frac{P}{P_0}\right)\\ &= \frac{1}{2} \ln\!\left(\frac{AI}{A_0 I_0}\right)\\ &= \frac{1}{2} \ln\!\left(\frac{I}{I_0}\right) + \frac{1}{2} \ln\!\left(\frac{A}{A_0}\right)\!. \end{align}$

For a progressive spherical wave,

$z_0 = \frac{p}{v},$

$A = 4\pi r^2,$

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,

$\begin{align} L_W &= \frac{1}{2} \ln\!\left(\frac{p^2}{p_0^2}\right) + \frac{1}{2} \ln\!\left(\frac{4\pi r^2}{A_0}\right)\\ &= \ln\!\left(\frac{p}{p_0}\right) + \frac{1}{2} \ln\!\left(\frac{4\pi r^2}{A_0}\right)\\ &= L_p + 10 \log_{10}\!\left(\frac{4\pi r^2}{A_0}\right)\!~\mathrm{dB}. \end{align}$

The sound power estimated practically does not depend on distance, though theoretically it may diminish with distance due to viscous effects in the propagation of sound.

References

1. Ronald J. Baken, Robert F. Orlikoff (2000). Clinical Measurement of Speech and Voice. Cengage Learning. p. 94. ISBN 9781565938694.
2. Landau & Lifshitz, "Fluid Mechanics", Course of Theoretical Physics, Vol. 6
3. "Sound Power". The Engineering Toolbox. Retrieved 28 November 2013.
4. "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.
5. Ross Roeser, Michael Valente, Audiology: Diagnosis (Thieme 2007), p. 240.
6. 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
7. Chadderton, David V. Building services engineering, pp. 301, 306, 309, 322. Taylor & Francis, 2004. ISBN 0-415-31535-2

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