The inverse square law sound intensity rule sound energy size sound pressure - sengpielaudio
 
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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?
Point sources of gravitational force, electric field, light, or radiation obey the inverse square law.

 
 
Avoid using the psychoacoustical terms loudness perception and volume.
This subjective sound-sensation is not clearly measurable without ambiguity.

The term "loudness" or "volume" is a problem because it belongs to psycho-
acoustics and this personal feeling is not correct definable.

Loudness as a psychological correlate of physical strength (amplitude) is also
affected by parameters other than sound pressure, including frequency,
bandwidth, spectral composition, information content, time structure, and the
duration of exposure of the sound signal. The same sound will not create the
same loudness perception by all individuals (people).

 
 
As psycho-acoustic parameters to describe the "loudness" there is the
"loudness level" with the unit phon and the "loudness" with the unit sone.
 
Sound pressure p is not sound intensity I.   I ~ p2 is true for progressive plane waves.
 
Compare also the inverse distance law 1/r, when
using
sound pressure as sound field quantity.
 
Sound pressure is inversely proportional to the distance of the point of measurement
from the source, so that if we double the distance we halve the sound pressure.

 
How does the sound decrease with distance?
Damping of sound level with distance
 
This is an approximation when the venue is 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 we 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, we may get less than the square law predicts. Nevertheless,
the inverse square law is the logical first estimate of the sound intensity we 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 in Pa is really not sound intensity in W/m².

 If we measure at a distance
 r
1   = m = ft
 a sound intensity level (SIL1)
 LI1 = dB,
 and then at distance
 r2   = m = ft
 the inverse square law 1/r2 predicts 
 a sound intensity level (SIL2) of

 LI2 = dB in the free field.
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.
 
There is no noise decrease or sound drop per meter.
We get a sound level drop of 6 dB per doubling of distance.
Sound power or sound power level has nothing to do with the distance from the sound source.
 
Decrease of sound pressure level
 
 
Note: The radiated sound power (sound intensity) is the cause
and the
sound pressure is the effect.
where the sound engineer is particularly interested in the effect.
The effect of temperature and sound pressure:
Sound pressure and Sound power – Effect and Cause
.
 
 
 
Acousticians and sound protectors ("noise fighters") need the sound
intensity (acoustic intensity) – but sound engineers and sound
designers ("ear people") don't need that sound energy quantity.

 
Who is involved in audio engineering, should rather take care of the
sound field quantity, that is the sound pressure or the sound pressure
level (SPL) as an effect at the eardrums of our hearing and on the
diaphragms of the microphones, and the corresponding audio voltage
and its voltage level.

 

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 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
The graphs shown are normalized
Distance
ratio 
Sound Intensity
I proportional1/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)
 
             Intensität-Abstand
This obviously means Schalldruck-Abstand
Aha!

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:
 
Formel Pegelabnahme Schalldruck
 
Formula Distance Intensity
 
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 r2 from 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.
Doubling the sound intensity raises the sound intensity level by 3 dB.
 
The sound pressure level Lp to plot against
the distance of the sound source
r
Schallfeld
D: direct field of the spherical source
R: reflected field (diffuse field)
rH: critical distance
 
If we double the distance, the value for the sound pressure falls to
a half (50%) of its initial value.
If we double the distance, the value for the sound intensity falls to
a quarter (25%) of its initial value.
This corresponds to a decrease in level by (−)6 dB.
For the level change in dB we get:
 
Formel Pegelabnahme Schalldruck
 
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 proportional to squared field quantities  –  e.g.
I ~ p2.
 
 
 How is the sound level dependent from the distance to the sound source?
 The sound pressure level shows in the free field situation a reduction of 6 dB per
 doubling of distance; that means the sound pressure drops to a half and not a quarter.
 It is the sound intensity, that drops to a quarter of the initial value.

 

Sound Energy Quantities
Sound intensity, sound energy density,
sound energy, acoustic  power.
(electrical power).

Inverse Square Law 1/r²
         Sound Field Quantities    AnimatedLaughingSmiley
Sound pressure, sound or particle velocity,
particle displacement or displacement amplitude,
(voltage, current, electric resistance).

Inverse Distance Law 1/r

Conversions and Calculations - Sound Quantities and their Levels
Conversion of sound units (levels)
Damping of Sound Pressure Level with Distance

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

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

Sound pressure is not intensity

Neither the sound power nor the sound power level decreases in doubling
the distance. Why is this so?

The sound power level quantifies the totally radiated sound energy from an object.
Different to the sound pressure the sound power is independent of the distance to
the sound source, the surrounding area and other influences.

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

 
Differentiate: Sound pressure p is a "sound field quantity"
and sound intensity I is a "
sound energy quantity". In
teachings these terms are not often separated sharply
enough and sometimes are even set equal.
But I ~ p2.
 

Changing of sound power with distance is nonsense
 
 
Question: How does the sound power decrease with distance"? Answer: "April fool -
The sound power does not decrease (drop) with distance from the sound source."

 
Levels of sound pressure and levels of sound intensity decrease equally with the
distance from the sound source.
Sound power or sound power level has nothing
(!) to do with the distance from the sound source.
Thinking helps: A 100 watt light bulb has in 1 m and in 10 m distance really always
the same 100 watts, which is emitted from the lamp all the time.
Watts don't change with distance.

 
A frequent question: "Does the sound power depend on distance?" The clear
answer is: "No, not really."

 
We consider sound fields in air which are described by the scalar quantity p (sound
pressure) and the vector quantity v (sound velocity) as a sound field quantity.

 

Decrease of the soundfield
 
Pressure, velocity, and intensity of the sound field near to and
distant from a spherical radiator of the zeroth order
 
Sound engineers and sound designers (ear people) are mainly
interested in sound field quantities and consider more the sound pressure
drop at distance doubling (Schalldruckabfall - Entfernungsverdopplung).
Acousticians and sound protectors (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!   AnimatedLaughingSmiley   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.
I ~ p2.
Intensity is proportional to the square of the amplitude of the sound
pressure.
 
Our hearing (eardrum) is directly sensitive to the sound pressure. From the
historical perspective, the level differences for stereo listening were called
"intensity" differences. However, sound intensity is a specifically defined
quantity that 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 we have to work as a sound technician to check the sound quality by ear,
then think of the sound waves, which move the eardrums using the sound
pressure as a sound field quantity. There is also the advice: Try to avoid
the use of sound power and sound intensity as sound energy quantities.
 
How does the sound decrease with distance?
 
Some more useful links:
The human perception of loudness - Loudness factor (volume)
Damping of sound level with distance
Decrease in level of sound pressure and sound intensity with distance
Sound pressure p and the inverse distance law 1/r
How does the sound decrease (fall-off or drop) with distance?
Conversion of sound units (levels)
Subjectively perceived loudness and objectively measured sound pressure
Sound sizes, their Levels and References - Calculations, and Formulas
Relationship of acoustic sizes
Comparative representation of sound field sizes and sound energy sizes
Sound pressure and Sound power − Effect and Cause
Table of Sound Levels (dB Scale)
The Formula Wheel - Formulas of Acoustics (Audio)
Acoustic equivalent for ohm's law - ohm's law as equivalent in the acoustics
 
 
In audio, electronics and acoustics use only the word "damping" and not the wrong word "dampening".
 
 
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