Sound Power

Sound Power (W)

Sound power is the energy rate, or energy of sound per unit of time (J/s or W in SI-units) - emitted by a source.

When sound propagates through a medium acoustic sound power is transferred. The sound intensity is the sound power transmission through a surface (W/m²) - a vector quantity with direction through a surface. The sound power radiated from a source can therefore be calculated by integrating the acoustic intensity over a surrounding surface.

N = ∫_S  I·n dS                         (1)

where

N = acoustic power radiated from source (W)

I = sound intensity - sound power through surface (W/m²)

n = unit vector normal to surface area

S = surface area surrounding source (m²)

For a source propagating sound in all directions - through a virtual spherical surface - the acoustic power can be modified to

N = 4 π r² I                         (1b)

where

r = radius in sphere  (m)

Sound Power Level (dB)

It can be practical to express the sound power relative a reference power - 10^-12 W - in a logarithmic "decibel" scale as

L_N = 10 log₁₀(N / N_ref)        

     = 10 log₁₀(N) + 120                          (2)

where

L_N = sound power level (decibel , dB)

N = sound power (W)

N_ref = 10^-12 - reference sound power (W)

• decibel calculator

The human ear is able hear sound powers that spans from 10^-12 W to 10 - 100 W, a range of 10/10^-12 = 10¹³.

Example - Sound Power Level

The sound power from a from a tool is 0.0015 W. The sound power level can be calculated as

L_N = 10 log₁₀ ((0.0015 W) / (10^-12 W)) 

             = 91.8 dB 

The sound power level from a machine is rated to 100 dB. The sound power can be calculated by rearranging (1) to

N = 10^{((L_N - 120) / 10)} 

   = 10^{(((100 dB) - 120) / 10)}   

   = 0.01 W

Sound Intensity Level (dB)

Sound intensity can also be expressed relative to a reference intensity - the threshold of hearing - 10^-12 W/m² - in a logarithmic "decibel" scale as

L_I = 10 log₁₀(I / I_ref)        

     = 10 log₁₀(I) + 120                          (3)

where

L_I = sound intensity level (decibel , dB)

I = sound intensity (W/m²)

I_ref = 10^-12 - reference sound intensity (W/m²)

The value of sound intensity level and sound pressure level is almost the same

L_I = L_p - 0.2            (4)

where

L_p = sound pressure level (dB)

Example - Sound Intensity and Pressure Level - Distance from Source 

The sound intensity - sound power transfer per unit surrounding surface - in a distance 1 m from the 100 dB machine in the example above can be calculated by rearranging (1b) to

I =  N / (4 π r²)

   = (0.01 W) / (4 π (1 m)²)

   = 0.0008 W/m²  

The sound intensity level can be calculated with (3) as 

L_I ≈ L_p ≈ 10 log₁₀(0.0008 W/m²) + 120

    = 89 dB

The sound intensity in distance 10 m

I =  N / (4 π r²)

   = (0.01 W) / (4 π (10 m)²)

   = 0.000008 W/m²  

The sound intensity level can be calculated with (3) as 

L_I ≈ L_p ≈ 10 log₁₀(0.000008 W/m²) + 120

    = 69 dB

Typical Sound Power

Sound power in watts and sound power level in decibels from some common sources are indicated below:

Temperature ^oC
K
^oF

Length m
km
in
ft
yards
miles
naut miles

Area m²
km²
in²
ft²
miles²
acres

Volume m³
liters
in³
ft³
us gal

Weight kg_f
N
lbf

Velocity m/s
km/h
ft/min
ft/s
mph
knots

Pressure Pa
bar
mm H₂O
kg/cm²
psi
inches H₂O

Flow m³/s
m³/h
US gpm
cfm