Sound power

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

Sound power or acoustic power P_ac is a measure of sonic energy E per time t unit.

It is measured in watts and, can be computed as sound intensity (I) times area (A):

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

The difference between two sound powers can be express in decibels using this equation:

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

where

P₁, P₀ are the sound powers.

The sound power level SWL, L_W, or L_Pac of a source is expressed in decibels (dB) and is equal to 10 times the logarithm to the base 10 of the ratio of the sound power of the source to a reference sound power. It is thus a logarithmic measure.

The reference sound power in air is normally taken to be ${P_0}\,$ = 10⁻¹² watt, that is 0 dB SWL.

Sound power is neither room dependent nor distance dependent. Sound power belongs strictly to the sound source.

1 Table of some sound sources
2 Sound power measurement
3 Sound power with plane sound waves
4 Sound power level
5 References
6 External links

Table of some sound sources

Situation and sound source	sound power P_ac watts	sound power level L_w dB re 10⁻¹² W
Rocket engine	1,000,000 W	180 dB
Turbojet engine	10,000 W	160 dB
Siren	1,000 W	150 dB
Heavy truck engine or loudspeaker rock concert	100 W	140 dB
Machine gun	10 W	130 dB
Jackhammer	1 W	120 dB
Excavator, trumpet	0.3 W	115 dB
Chain saw	0.1 W	110 dB
Helicopter	0.01 W	100 dB
Loud speech, vivid children	0.001 W	90 dB
Usual talking, Typewriter	10⁻⁵ W	70 dB
Refrigerator	10⁻⁷ W	50 dB

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 by Daniel $(L_W)$ is to measure the sound pressure level $(L_p)$ at some distance $\mathit{r}$ , and solve for $(L_W)$ :

$\mathit{L_W} = \mathit{L_p}-10\, \log_{10}\left(\frac{1}{4\pi r^2}\right)\,$ If the source is in free space

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

The sound power estimated this does not diminish or increase with distance, unless reflections are present.

Sound power with plane sound waves

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

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

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
P_ac	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,"^[1] the acoustic output power measured relative to a very low base level of watts given as 10^-12 or 0.000000000001 watts. 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.^[1]

The sound power level of a signal with sound power W is:^[2] ^[3]

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

where W₀ is the 0 dB reference level:

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

The sound power level is given the symbol L_W. 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

$L_\mathrm{p} = L_\mathrm{W}%2B10\, \log_{10}\left(\frac{S_0}{4\pi r^2}\right)\,$

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

References

^ ^a ^b ^c Chadderton, David V. Building services engineering, pp. 301, 306, 309, 322. Taylor & Francis, 2004. ISBN 0415315352
^ Sound Power, Sound Intensity, and the difference between the two - Indiana University's High Energy Physics Department
^ Georgia State University Physics Department - Tutorial on Sound Intensity