The inverse square law and the sound intensity as sound energy quantity sound pressure - sengpielaudio Checker
 
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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. p2I is true for progressive plane waves.

Compare also the inverse distance law 1/r,
when using sound pressure as sound field quantity.

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 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 µW/m2 = 1 × 10−12 W/m2.
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 with 6 dB.

If you measure at distance
r1   = m = ft
a sound intensity level (SIL1)
Lp1 = dB,
then at distance
r2   = m = ft
the inverse square law 1/r2 predicts
a sound intensity level (SIL2)

Lp2 = 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.

Sound intensity level and Sound intenity

Enter a value in the left or right box, then press the TAB bar or make
a mouse click at an empty space at the side, to get the solution.
The calculator works in both directions of the
sign.

Sound intensity level LI:
dB-SIL
 ↔  Sound intensity  I:
W/m2
Formula 1   Formula 2
Standard reference sound intensity I0 = 1 pW/m2 = 10−12 W/m2 ≡ 0 dB

Inverse square law 1/r2
Inverse Square Law

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

I1 / I2 = r22 / r12

I2 = I1 (r12 / r22)

Where:
I1  =  sound intensity 1 at r1
I2  =  sound intensity 2 at r2
r1  =  distance 1 from source
r2  =  distance 2 from source

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 1/2˛ = 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; Ip2

Sound Energy Quantities
Sound intensity, sound energy density,
sound power, electric power.


Inverse Square Law 1/r²
Sound Field Quantities    AnimatedLaughingSmiley
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 wrong statements in the context of sound sizes
and the distance of the sound source

Wrong expression Correct version
Sound intensity decreases inversely as the
distance increases with 1/r
from the sound source.
Sound intensity decreases inversely as the square
of the distance increases with 1/r2
from the sound
source.
Sound intensity level decreases inversely as the
square of the distance increases with 1/r2 from the
sound source with 3 dB per doubling.
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 decreases inversely as the square
of the distance increases with 1/r2
from the
sound source.
Sound pressure decreases inversely as the
distance increases with 1/r
from the sound source.
Sound pressure level decreases inversely as the
square of the distance increases with 1/r2
from the
sound source with 3 dB per doubling.
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?

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