# Sound power

Sound measurements
Sound pressure p, SPL
Particle velocity v, SVL
Particle displacement ξ
Sound intensity I, SIL
Sound power Pac
Sound power level SWL
Sound energy
Sound exposure E
Sound exposure level SEL
Sound energy density E
Sound energy flux q
Acoustic impedance Z
Speed of sound
Audio frequency AF

Sound power or acoustic power Pac is a measure of sound energy E per time t unit. It is measured in watts and can be computed as sound intensity (I) times area (A):

$P_{\mathrm{acoustic}} = I \cdot A$

When the acoustic wave approaches the measurement surface at an angle, the area is taken as the area times the projection of the wave direction upon the normal of the surface.

The difference between two sound powers can be express in decibels (logarithmic measure) using this equation:

$L_\mathrm{w}=10\, \log_{10}\left(\frac{P_1}{P_0}\right)\ \mathrm{dB}$

where $P_1$, $P_0$ are the sound powers. The sound power level SWL, LW, or LPac of a source is expressed in decibels (dB) relative to a reference sound power. In air this is normally taken to be ${P_0}\,$ = 10−12 watt, that is 0 dB SWL.

Unlike sound pressure, sound power is neither room dependent nor distance dependent. Sound power belongs strictly to the sound source. Sound pressure is a measurement at a point in space near the source, while sound power is the total power produced by the source in all directions.

## Table of selected sound sources

Situation
and
sound source
sound power
Pac
watts
sound power
level Lw
dB re 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
.01 100
Large diesel vehicle
Heavy city traffic
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
10−9 30
Whisper
Wristwatch ticking
10−10 20
Human breath 10−11 10
Threshold of hearing
Reference Power Level
10−12 0

[1]

Usable music sound (trumpet) and noise sound (excavator) both have the same sound power of 0.3 watts, but will be judged psychoacoustically to be different levels.

## Sound power measurement

A frequently used method of estimating the sound power level of a source $(L_\mathrm{W})$ is to measure the sound pressure level $(L_\mathrm{p})$ at some distance $r$, and solve for $L_\mathrm{W}$:[citation needed]

$L_\mathrm{W} = L_\mathrm{p}-10\, \log_{10}\left(\frac{1}{4\pi r^2}\right)\,$ if the source radiates sound equally in all directions into free space

or

$L_\mathrm{W} = L_\mathrm{p}-10\, \log_{10}\left(\frac{2}{4\pi r^2}\right)\,$ if the source is on the floor or on a wall, such that it radiates into a half sphere.

Using these equation though requires the sound pressure level to use a corresponding reference level. From the fact that the mean square sound pressure equals characteristic acoustic impedance Z0 times power per unit area, we see that if the power is measured in watts, the distance in metres, and the pressure in pascals, then we need a correction of log10Z0 with Z0 in N∙s/m3:

$L_\mathrm{W} = L_\mathrm{p}-10\, \log_{10}\left(\frac{1}{4\pi r^2}\right)\,-\log_{10}Z_0$

For air at 25°C, the impedance is 409 N∙s/m3, so this correction is −26. However, sound pressure level is usually measured against a reference of 20 μPa (introducing a correction of −94 dB), and sound power level is (as mentioned above) usually measured against a reference of 10−12 watts (introducing a correction of 120 dB). These corrections cancel, so we may use (for the case of radiation into free space):

$\text{dBSWL} = \text{dBSPL}+10\, \log_{10}\left(4\pi r^2\right)$

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.

## Sound power with plane sound waves

Between sound power and other important acoustic values there is the following relationship:

$P_\mathrm{ac} = \xi^2 \cdot \omega^2 \cdot Z \cdot A = v^2 \cdot Z \cdot A = \frac{a^2 \cdot Z \cdot A}{\omega^2} = \frac{p^2 \cdot A}{Z} = E \cdot c \cdot A = I \cdot A\,$

where:

Symbol Units Meaning
p Pa sound pressure
f Hz frequency
ξ m particle displacement
c m/s speed of sound
v m/s particle velocity
ω = 2πf rad/s angular frequency
ρ kg/m3 density of air
Z = c · ρ N·s/m3 acoustic impedance
a m/s2 particle acceleration
I W/m2 sound intensity
E W·s/m3 sound energy density
Pac W sound power or acoustic power
A m2 area

## Sound power level

Sound power level or acoustic power level is a logarithmic measure of the sound power in comparison to a specified reference level. While sound pressure level is given in decibels SPL, or dB SPL, sound power is given in dB SWL. The dimensionless term "SWL" can be thought of as "sound watts level,"[2] the acoustic output power measured relative to 10−12 or 0.000000000001 watt (1 pW). As used by architectural acousticians to describe noise inside a building, typical noise measurements in SWL are very small, less than 1 watt of acoustic power.[2]

The sound power level of a signal with sound power W is:[3][4]

$L_\mathrm{W}=10\, \log_{10}\left(\frac{W}{W_0}\right)\ \mathrm{dB}\,$

where W0 is the 0 dB reference level:

$W_0=10^{-12}\ \mathrm{W}=1\ \mathrm{pW}\,$

The sound power level is given the symbol LW. This is not to be confused with dBW, which is a measure of electrical power, and uses 1 W as a reference level.

In the case of a free field sound source in air at ambient temperature, the sound power level is approximately related to sound pressure level (SPL) at distance r of the source by the equation

$L_\mathrm{p} = L_\mathrm{W}+10\, \log_{10}\left(\frac{S_0}{4\pi r^2}\right)\,$

where $S_0 = 1\ \mathrm{m}^2$.[2] This assumes that the acoustic impedance of the medium equals 400 Pa·s/m.

## References

1. ^ "Sound Power". The Engineering Toolbox. Retrieved 28 November 2013.
2. ^ a b c Chadderton, David V. Building services engineering, pp. 301, 306, 309, 322. Taylor & Francis, 2004. ISBN 0-415-31535-2
3. ^ Sound Power, Sound Intensity, and the difference between the two - Indiana University's High Energy Physics Department
4. ^ Georgia State University Physics Department - Tutorial on Sound Intensity