The inverse square law and the sound intensity 1/r² - sound energy quantity not for sound pressure

Sound intensity I and the inverse square law 1/r²

● How does the sound intensity decrease with distance from the sound source? ●
How does the sound intensity level decrease with doubling the distance from the source?
Sound pressure is not sound intensity. I ~ p² is true for progressive plane waves.
Compare also the inverse distance law 1/r,
when using sound pressure as sound field quantity.

This is an approximation when the venue is not a direct sound field or an anechoic room

In the real world, the inverse square law (squared distance law) I ~ 1/r² is always an
idealization because it assumes exactly equal sound intensity or acoustic intensity I as
sound energy propagation in all directions. If there are reflective surfaces in the sound field,
then reflected sounds will add to the directed sound and you will get more sound intensity at
a field location than the inverse square law predicts. If there are barriers between the source
and the point of measurement, you may get less than the square law predicts. Nevertheless,
the inverse square law is the logical first estimate of the sound intensity you would get at a
distant point in a reasonably open area. The reference sound intensity level SIL = 0 dB is the
acoustic intensity of I₀ = 1 pW/m² = 1 × 10⁻¹² W/m².
Note: Since the sound intensity level (energy quantity) is difficult to measure, it is common
to use sound pressure level (field quantity) measured in decibels instead. Doubling the
sound pressure raises the sound pressure level by 6 dB.
Sound pressure is really not sound intensity.

If you measure at distance
r₁   = m = ft
a sound intensity level (SIL₁)
L_I1 = dB,
then at distance
r₂   = m = ft
the inverse square law 1/r² predicts
a sound intensity level (SIL₂)
L_I2 = dB.

You can explore numerically to confirm the 1/r² law that doubling the distance drops the sound
intensity I to a quarter (0.25) by a sound intensity level of about 6 dB and that 10 times the
distance drops the sound intensity I to a hundredth (0.01), that is a level drop by 20 dB.

Note: The radiated sound power (sound intensity) is the cause -
and the sound pressure is the effect.
The effect is of particular interest to the sound engineer.
The effect of temperature and sound pressure.

Acousticians and sound protectors (noise fighters) need the sound
intensity (acoustic intensity). As a sound designer you don't need
that energy quantity.
Look out more for the sound pressure that makes an effect to the
eardrums of our hearing and to the diaphagms of microphones.

Sound pressure and Sound power − Effect and Cause
Sound intensity level and Sound intensity

Enter a value in the left or right box. The calculator
works in both directions of the ↔ sign.

Sound intensity level L_I:
dB-SIL ↔ Sound intensity I:
W/m²

Standard reference sound intensity I₀ = 1 pW/m² = 10⁻¹² W/m² ≡ 0 dB

Inverse square law 1/r²

Law for Sound Energy Quantities

Distance ratio Sound Intensity I ∝ 1/ r²

1   1/1² = 1/1 = 1.0000

2   1/2² = 1/4 = 0.2500

3   1/3² = 1/9 = 0.1111

4 1/4² = 1/16 = 0.0625

5 1/5² = 1/25 = 0.0400

6 1/6² = 1/36 = 0.0278

7 1/7² = 1/49 = 0.0204

8 1/8² = 1/ 64 = 0.0156

9 1/9² = 1/81 = 0.0123

10 1/10² =1/100 = 0.0100

Calculating sound intensity with the inverse square law

Relationship of sound intensity I, sound pressure p, and the square law:
(r is the distance from the sound source)


This obviously means

Formulas for distance attenuation − Sound intensity calculation

The value of the sound intensity increases inversely squared with
increasing distance from the sound source, that means with 1/r²:

Where:

I₁ = sound intensity 1 at closer distance r₁ from the sound source

I₂ = sound intensity 2 at more far distance r₂ from the sound source

r₁ = closer distance r₁ from the sound source

r₂ = more far distance r₂from the sound source

Note! Since the sound intensity level is difficult to measure, it is common
to use sound pressure level measured in decibels instead. Doubling the
sound pressure raises the sound pressure level (SPL) by 6 dB.

The sound pressure level L_p to plot against
the distance of the sound source r

D: direct field of the spheric source
R: reflected field (diffuse field)
r_H: critical distance

If we double the distance, the value for the sound pressure falls to a half of its initial value.
If we double the distance, the value for the sound intensity falls to a quarter of its initial value.
This corresponds to a decrease in level by (−)6 dB.
For the level change in dB we get:

A doubling of distance from the sound source in the direct field will reduce the "sound level"
by (−)6 dB, no matter whether that are sound intensity levels or sound pressure levels! This will
reduce the sound intensity I (energy quantity) to 1/2² = 1/4 (25 %) and the sound pressure p
(field quantity) to 1/2 (50 %) of the the initial value.
The inverse square law 1/r² shows the distance performance of energy quantities
and the inverse distance law 1/r shows the distance performance of field quantities.
Energy quantities are propotional to squared field quantities  –  e.g. I ~ p².

Sound Energy Quantities
Sound intensity, sound energy density,
sound energy, acoustic power.
(electrical power).
Inverse Square Law 1/r²          Sound Field Quantities
Sound pressure, sound or particle velocity,
particle displacement or particle amplitude,
(voltage, current, electric resistance).
Inverse Distance Law 1/r

Conversions and Calculations - Sound Quantities and their Levels

Frequently used false statements in the context of
sound values and the distance of the sound source

Wrong expression Correct version

Sound intensity (energy) falls inversely
proportional to the distance 1/r from the sound
source.                                                               wrong Sound intensity (energy) falls inversely proportional
to the square of the distance 1/r² from the sound
source.

Sound intensity level decreases inversely as the
square of the distance increases per doubling of
sound source with (−)3 dB per doubling.       wrong Sound intensity level decreases by (−)6 dB per
doubling of distance from the source to 1/4 (25 %)
of the sound intensity initial value.

Sound pressure falls inversely proportional
to the square of the distance 1/r² from the sound
source.                                                               wrong Sound pressure falls inversely proportional
to the distance 1/r from the sound source.
That is the 1/r law or distance law.

Sound pressure level decreases as the
distance increases per doubling of distance
from the source by (−)3 dB.                             wrong Sound pressure level decreases by (−)6 dB per
doubling of distance from the source to 1/2 (50 %)
of the sound pressure initial value.

Neither the sound power, nor the sound power level decreases in
doubling the distance up to a value or up to any dB. Why is this so?

How many decibels (dB) level change is twice (double, half) or three times as loud?

Decrease of the sound field with distance from the source

Pressure, velocity, and intensity of the sound field near to and
distant from a spherical radiator of the zeroth order

"Ear people" like sound engineers and sound designers are mainly
interested in sound field quantities and consider more the sound pressure
drop at distance doubling (Schalldruckabfall - Entfernungsverdopplung).
Acousticians and noise fighters are mainly interested in sound energy
quantities and consider here the sound intensity drop at distance doubling.
They all view together the same line!      Isn't that beautiful?
Nevertheless, the drop in sound intensity goes with 1/r² and the decrease
of sound pressure is 1/r. This should be understood quite well.

Our hearing (eardrums) is directly sensitive only to the sound pressure.
From a historical perspective, the level differences at the stereo listening
were called "intensity" differences. But sound intensity is a specifically
defined quantity, which can not be picked up by a microphone, nor would
it be useful for a sound recording. So call the "intensity" stereophony
better level difference stereophony.

If you are a technician checking the sound quality by listening with your
hearing, think of the sound waves that move your eardrums only by the
soundpressure as sound field quantity. For this there is the advice:
Avoid the use of the sound power and the sound intensity as sound
energy quantities.

How does the sound decrease with distance?

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Sound intensity level L_I: dB-SIL	↔	Sound intensity I: W/m²

Standard reference sound intensity I₀ = 1 pW/m² = 10⁻¹² W/m² ≡ 0 dB

Where:
I₁	=	sound intensity 1 at closer distance r₁ from the sound source
I₂	=	sound intensity 2 at more far distance r₂ from the sound source
r₁	=	closer distance r₁ from the sound source
r₂	=	more far distance r₂from the sound source

Wrong expression	Correct version
Sound intensity (energy) falls inversely proportional to the distance 1/r from the sound source. wrong	Sound intensity (energy) falls inversely proportional to the square of the distance 1/r² from the sound source.
Sound intensity level decreases inversely as the square of the distance increases per doubling of sound source with (−)3 dB per doubling. wrong	Sound intensity level decreases by (−)6 dB per doubling of distance from the source to 1/4 (25 %) of the sound intensity initial value.
Sound pressure falls inversely proportional to the square of the distance 1/r² from the sound source. wrong	Sound pressure falls inversely proportional to the distance 1/r from the sound source. That is the 1/r law or distance law.
Sound pressure level decreases as the distance increases per doubling of distance from the source by (−)3 dB. wrong	Sound pressure level decreases by (−)6 dB per doubling of distance from the source to 1/2 (50 %) of the sound pressure initial value.