Loudness perception
45 Intensity and Distance
Intensity and distance
As you move away from a sound source, the sound gets quieter- especially when you are outdoors. This is not surprising. Much like light, sound spreads out as it travels away from the source. Unless there are surfaces for the sound to reflect from, the sound’s intensity becomes less and less as you get further from the source.
Inverse square law and the paint can analogy
If you’ve ever used a can of spray paint, you know that distance is important- the closer you are to the nozzle, the more concentrated the paint stream is. Hold the can further away and the same amount of paint gives a thinner coat that covers a larger area.
Suppose you squirt a dose of paint on a wall. You then deliver the same dose of paint from twice as far away and compare. How much larger is the new spot? How thick is the paint?
What did you answer? Twice the area? Half as thick? If you did, you got the answers wrong! Remember that area is length times width. Since the new paint spot is both twice as wide and twice as high as the original, it covers four times the area. As a result, the paint is only one quarter as thick as before. At triple the initial distance, the paint spot has nine times the original size and the paint is only one ninth as thick, and so on. In math speak, the concentration of the paint follows an inverse square law.
Sound radiating into open space works the same way. As you get farther and farther from the source, the power put out by the sound source gets spread over a larger and larger area. As a result, the sound intensity follows an inverse square law. Mathematically,
[latex]I \propto 1/d^2[/latex]
If you are uncomfortable with proportion notation, you can use the equation below instead:
[latex]\dfrac{I_2}{I_1}= \Big( \dfrac{d_1}{d_2} \Big)^2[/latex]
The picture below shows how it works. Notice that units are not given for distance. Because of the way that ratios work, any distance unit can be used. If you are two distance units from the source, the intensity is one fourth what the intensity is at one distance unit away from the source.
Stop to think 1
The sound intensity 1 foot away from a sound source is 200 μW/m2.
- What’s the intensity 2 feet away from the source?
- What’s the intensity 10 feet away from the source?
- What’s the intensity 6 inches away from the source?
Inverse square law and decibels
You can combine inverse square law and the rules of thumb for decibels. Each time distance is doubled, intensity is cut by a factor of four. Since each time intensity is cut in half the sound level decreases 3 dB, it follows that doubling distance reduces the sound level by 6 dB. Following the same logic, increasing the distance by a factor of ten reduces the sound level by 20 dB.
Stop to think 2
The sound level 1 foot away from a sound source is 70 dB.
- What’s the sound level 2 feet away from the source?
- What’s the level 10 feet away from the source?
- What’s the level 6 inches