# Sound Intensity

## Acoustic power of sound per unit area.

### Sound Intensity

The sound intensity is the sound power transmission through a surface *(W/m ^{2}) - *a vector quantity with direction through a surface.

*I = N / A (1)*

*where *

*I = sound intensity (W/m ^{2}) *

*N = sound power through surface area (W)*

*A = surface area (m ^{2})*

### Sound Intensity Level

The dynamic range of human hearing and sound intensity spans from *10 ^{-12} W/m^{2}* to

*10 - 100 W/m*. The highest sound intensity possible to hear is

^{2}*10,000,000,000,000*times as loud as the quietest!

This span makes absolute values for sound intensity impractical in normal use. A more convenient way to express sound intensity is the relative logarithmic decibel scale with reference to the lowest human hearable sound -

*10*

^{-12}W/m^{2 }

*(0 dB)*.

**Note!** In US the reference *10 ^{-13} watts/m^{2}* is commonly used.

Sound Intensity Level can be expressed as:

L_{I}= 10 log (I / I_{ref})

=(2)10 log_{10}(I) + 120

where

L_{I}= sound intensity level (dB)

I= sound intensity (W/m^{2})

I_{ref}= 10^{-12}- reference sound intensity - the threshold of hearing (W/m^{2})

The logarithmic sound intensity level scale match the human sense of hearing. Doubling the intensity increases the sound level with *3 dB* *(10 log (2))*.

### Example - Sound Intensity

The difference in dB for intensity *10 ^{-8} watts/m^{2} *and

*10*(

^{-4}watts/m^{2}*10000 units*) can be calculated as

ΔL_{I}= 10 log ((10^{-4}watts/m^{2}) / (10^{-12}watts/m^{2})) - 10 log (( 10^{-8}watts/m^{2}) / ( 10^{-12}watts/m^{2}))

= 40 dB

Increasing the sound intensity by a factor of

*10 raises its level by 10 dB**100 raises its level by 20 dB**1000 raises its level by 30 dB**10000 raises its level by 40 dB**and so on*

**Note!** Since the sound intensity level may be difficult to measure it is common to use sound pressure level measured in decibels instead. Note that doubling sound pressure raises the sound pressure level with *6 dB*.

### Loudness

Sound loudness is a subjective term describing the ear's perception of a sound.

Sound intensity and feeling of loudness:

*110 to 225 dB - Deafening**90 to 100 dB - Very Loud**70 to 80 dB - Loud**45 to 60 dB - Moderate**30 to 40 dB - Faint**0 - 20 dB - Very Faint*

### Sound Power, Intensity and Distance to Source

Sound intensity decreases with the distance to the source. Sound intensity vs. distance can be expressed:

I = N / 4 πr^{2 }(3)

where

N=sound power(W)

π= 3.14

r= radius or distance from source (m)

### Sound Intensity and Sound Pressure

The relation between Sound Intensity and Sound Pressure can be expressed as

I = p^{2}/ ρ c(4)

where

p= sound pressure (Pa)

ρ= density air (1.2 kg/m^{3}at 20^{o}C)

c= speed of sound (331 m/s)

The relation between Sound Intensity Level and Sound Pressure Level is

*L _{I} = L_{p} - 0.2 (5)*

*where *

*Lp = sound pressure level (dB)*