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 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):

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

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, L_W, or L_Pac 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 some sound sources

Situation and sound source	sound power Pac watts	sound power level Lw 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 at 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 is in 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.

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:

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

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," 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.

The sound power level of a signal with sound power W is:

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

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 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

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

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

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 — pdf
• NIOSH Powertools Database
• Sound Power Testing