Sound power

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Sound measurements
Characteristic
Symbol
 Sound pressure  p · SPL
 Particle velocity  v · SVL
 Particle displacement  ξ
 Sound intensity  I · SIL
 Sound power  P · SWL
 Sound energy  W
 Sound energy density  w
 Sound exposure  E · SEL
 Sound energy flux  Q
 Acoustic impedance  Z
 Speed of sound  c
 Audio frequency  AF

Sound power or acoustic power is a measure of sound energy per time unit. 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.

Mathematical definition[edit]

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:

Table of selected sound sources[edit]

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
0.01 100
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
Broadcast studio
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.

Equations in terms of other measurements[edit]

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
ω rad·s−1 angular frequency
ξ m particle displacement
a m·s−2 particle acceleration

Sound power level[edit]

For other uses, see Sound level.

Sound power level (SWL) or acoustic power level is a logarithmic measure of sound intensity in comparison to a 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]

Sound power level, denoted LW and measured in dB, is given by:[3][4]

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.[2] This assumes that the acoustic impedance of the medium equals 400 Pa·s·m−1.

Sound power measurement[edit]

A frequently used method of estimating the sound power level of a source L_W is to measure the sound pressure level L_p at some distance r, and solve for L_W:[citation needed]

L_W = L_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_W = 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.

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[edit]

  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

External links[edit]