Acoustic intensity, which measures sound power per unit area, decreases with the square of the distance from a point source. This geometric attenuation law explains why sound becomes rapidly quieter as you move away from its source.
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How to measure it in class
With the FizziQ app, it is possible to experimentally verify the 1/r squared decay law.
Steps:
- Open FizziQ and select the “Sound Level Meter” instrument
- Use a stable sound source (speaker with continuous sound, electronic tuning fork)
- Measure the sound level at different distances: 1 m, 2 m, 4 m, 8 m
- Plot the values on a graph (level in dB vs distance)
- Verify that the level decreases by approximately 6 dB when the distance doubles
- Discuss deviations: reflections, absorption, non-point source
Scientific activities on this topic
- Verification of the 1/r squared law: https://www.fizziq.org/en/activities/chloe-at-the-concert/
- Study of attenuation indoors vs outdoors
- Modeling of road noise propagation
Learn more
Physical origin:
A point sound source emits spherical waves. The emitted energy is distributed over a sphere surface of radius r. Since the surface area of a sphere is 4 pi r squared, the intensity (energy per unit area) is proportional to 1/r squared.
Attenuation in decibels:
The 1/r squared law translates into a simple rule in decibels:
- Distance x 2 leads to -6 dB
- Distance x 10 leads to -20 dB
Model limitations:
This law assumes:
- A point source (not a highway or a wall of speakers)
- A homogeneous medium without obstacles
- No reflections (free field)
Indoors, reflections from walls create a reverberant field that attenuates less with distance.
Other attenuation factors:
In addition to geometric divergence:
- Atmospheric absorption (especially high frequencies)
- Absorption by ground and vegetation
- Meteorological effects (wind, temperature gradients)
Formula
Intensity as a function of distance:
I(r) = P / (4 pi r squared)
Sound level as a function of distance:
L(r) = L(r0) - 20 x log10(r/r0)
where:
- I: acoustic intensity (W/m squared)
- P: source power (W)
- r: distance from the source (m)
- L: sound level (dB)
- r0: reference distance
Application examples
- A conversation at 1 m (60 dB) will be at 54 dB at 2 m and 48 dB at 4 m
- Road noise at 10 m (80 dB) will be at 74 dB at 20 m and 60 dB at 100 m
- Concerts use delay systems to compensate for attenuation
- Noise barriers are placed near sources to maximize their effectiveness
FAQ
Q: Why does the 6 dB rule not always work indoors? A: In a room, reflections create a diffuse sound field that adds to the direct sound. Attenuation is lower and depends on reverberation.
Q: Does a linear source (road, railway) follow the same law? A: No, for an infinite linear source, attenuation is 1/r (not 1/r squared), which is -3 dB per doubling of distance.
Q: At what distance is a shout no longer audible? A: A shout (100 dB at 1 m) reaches the hearing threshold (0 dB) around 100 km in theory, but atmospheric absorption and obstacles limit the actual range to a few hundred meters.
Q: Can FizziQ measure acoustic intensity? A: FizziQ measures sound pressure level (dB SPL). Intensity can be calculated if the acoustic impedance of the medium is known.
Related concepts
Sound Intensity - Sound Level - Decibel - Sound Propagation - Attenuation - Noise Pollution