# Sound power

Sound measurements
Characteristic
Symbols
Sound pressure  p, SPL
Particle velocity  v, SVL
Particle displacement  ξ
Sound intensity  I, SIL
Sound power  P, SWL, Q
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 a measure of the total sound energy radiated by a source per second.[1] Similarly, the portion of this total sound power which reaches a certain area is called the sound energy flux. In SI units, sound power is measured in Watts.[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 sound power is the total power produced by the source in all directions.

Sound power is a rough measure of the loudness of a particular sound, but the human perception of sound is affected by many other factors, such as the sound's frequency or the loss of sound intensity due to distance from the sound's source.

## Mathematical definition

Sound power, denoted P and measured in W, is given by:

$P = \mathbf f \cdot \mathbf v = A p\, \mathbf u \cdot \mathbf v$

where:

### Sound energy flux

The rate at which a specified area receives a fraction of the total sound energy from the source is measured by the sound energy flux, denoted Q and measured in W, given by:[2]

$Q = \int_A p \mathbf v \cdot \mathrm d\mathbf A$.

In a medium of density ρ for a progressive plane sound wave with speed c, the sound energy flux Q through a surface of area A with an angle θ between the direction of propagation of the sound and the normal to the surface, corresponding to an effective sound pressure p is:

$Q = \frac{p^2A} {{\rho}c}\cos{\theta}.$

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 Q = 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 (LWA) related to a portable air compressor.
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
10−9 30
Whisper
Wristwatch ticking
10−10 20
Human breath 10−11 10
Reference power level 10−12 0

[3]

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.

## Equations in terms of other measurements

Sound power can be related to other sound measurements:

$P = \frac{A p^2}{\mathfrak{R}(z)} = A \mathfrak{R}(z) v^2 = A I = c A w.$

For sine waves with angular frequency ω, the amplitude of the sound power can be related to those of the particle displacement and the particle acceleration:

$P_\mathrm{m}(\mathbf r) = A \omega^2 z_\mathrm{m}(\mathbf r) \xi_\mathrm{m}(\mathbf r)^2 = \frac{A z_\mathrm{m}(\mathbf r) a_\mathrm{m}(\mathbf r)^2}{\omega^2}.$
Symbol Unit Meaning
c m·s−1 speed of sound
v m·s−1 particle velocity
z Pa·m−1·s specific acoustic impedance
A m2 area
p Pa sound pressure
P W sound power
I W·m−2 sound intensity
w J·m−3 sound energy density
ξ m particle displacement
a m·s−2 particle acceleration

## Sound power level

For other logarithmic measures of sound-related quantities, see sound level.

Sound power level (SWL) or acoustic power level is a logarithmic measure of sound intensity compared to one picowatt.(1 picowatt is 10−12 Watts) Sound power is given in dB SWL. The dimensionless term "SWL" can be thought of as "sound watts level".[4] 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.[4]

Sound power level, denoted LW and measured in dB, is given by:[5][6]

$L_W = 10 \log_{10}\left(\frac{P}{P_0}\right)~\text{dB}$

where:

• P is the sound power, measured in W;
• P0 is the reference sound power, measured in W.

The reference sound power is P0 = 10−12 W. This is not to be confused with dB W, 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

$L_p = L_W + 10 \log_{10}\left(\frac{S_0}{4\pi r^2}\right)$

where $S_0 = 1~\text{m}^2$.[4] This assumes that the acoustic impedance of the medium equals 400 Pa·s·m−1.

### Estimating sound power level from sound pressure level

Based on the assumption that sound radiates equally in all directions without loss, the sound power level of a source $L_W$ can be estimated using the sound pressure level $L_p$ at some known distance $r$:[citation needed]

$L_W = L_p - 10 \log_{10}\left(\frac{1}{4\pi r^2}\right)$

Alternatively, if the source is on the floor or on a wall, such that it radiates into a half sphere, the sound power level can be estimated using:

$L_W = L_p - 10 \log_{10}\left(\frac{2}{4\pi r^2}\right)$.

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 10 log10 Z0 with Z0 in N∙s∙m−3:

$L_W = L_p - 10 \log_{10}\left(\frac{1}{4\pi r^2}\right) - 10 \log_{10} Z_0.$

For air at 25 °C, the impedance is 409 N∙s∙m−3, 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 three corrections cancel, so we may use (for the case of radiation into free space):

$\text{dB SWL} = \text{dB SPL} + 10 \log_{10}\left(4\pi r^2\right).$

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.

## References

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