Sound Intensity

Sound Intensity

The sound intensity is the sound power transmission through a surface (W/m²) - a vector quantity with direction through a surface.

I = N / A                                             (1)

where

I = sound intensity (W/m²)

N = sound power through surface area (W)

A = surface area (m²)

Sound Intensity Level

The dynamic range of human hearing and sound intensity spans from 10^-12 W/m² to 10 - 100 W/m². The highest sound intensity possible to hear is 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/m² (0 dB).

Note! In US the reference 10^-13 watts/m² is commonly used.

Sound Intensity Level can be expressed as:

L_I = 10 log (I / I_ref)    

   = 10 log₁₀(I) + 120                              (2)

where

L_I = sound intensity level (dB)

I = sound intensity (W/m²)

I_ref = 10^-12 - reference sound intensity - the threshold of hearing (W/m²)

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)).

The Decibel Scale

Example - Sound Intensity

The difference in dB for intensity 10^-8 watts/m² and 10^-4 watts/m² (10000 units) can be calculated  as

ΔL_I = 10 log ((10^-4 watts/m²) / (10^-12 watts/m²)) - 10 log (( 10^-8 watts/m²) / ( 10^-12 watts/m²))

= 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 πr²                                   (3)

where

N = sound power (W)

π = 3.14

r = radius or distance from source (m)

• inverse square law

Sound Intensity and Sound Pressure

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

I = p² / ρ c                                (4)

where

p = sound pressure (Pa)

ρ = density air (1.2 kg/m³ at 20^oC)

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)