A distant rumble

The last post talked about how we perceive low pitches as varying more with absolute pressure levels than high pitches seem to.  It’s a big effect, a major perceptual bias.  Why does such a bias exist?  What is it for?  It’s too big an effect to be an accident, so there much be some evolutionary advantage.  Here I’ll lay out a possible explanation.

One might expect sound from a point source to decrease by 1/r^{2}, like light or any sort of other wave expanding outwards in the shape of a sphere with area r^{2}.  And this is true – as we move away from a sound the volume decreases rapidly, with sound pressure for an ideal point source outside dropping by 6 dB as distance doubles (dB is a logarithmic measure).

If sound were immune to friction creating heat, and there is nothing near the point source to reflect sound (perhaps a speaker hanging from a crane), then spherical expansion would be all we need to know.  But reality is very different – higher frequencies are attenuated with distance more.  Much more.

We have all heard boom cars and thunder from far away, so this is intuitive.  However, the physics is not.  Searching for the reasons for this brought up a ton of very bad, misinformative web pages.  (It appears that lots of people use Yahoo answers for their physics homework, and the kids answering clearly don’t know they don’t know the answer either.  It’s a virtuous cycle – as a teacher I might choose to pose questions for which answers blindly copied from the internet are obviously lame, making it easy to discern whether the student has a true understanding.)

There are many mechanisms at work.  Besides absorption in air due to friction varying with frequency, there are differences in how longer wavelengths refract around local objects, bending due to temperature gradients, foliage, ground absorption and reflection, wind, and so on.

There is a careful comprehensive ISO standard for modeling these effects.  True to form for a useful document prepared with public money, it can’t be downloaded for free, but I found a copy hosted by the FAA as part of an environment impact study.  Here’s a table showing some sample values of attenuation, in dB per kilometer.


Absorption by the air alone is a very significant effect at high frequencies.  At 8khz, 15 degrees C, 20% RH there is a 202dB drop in a single kilometer.  (Twenty orders of magnitude, is that to be believed?  I think they are expressing are dB in power, which means only ten orders of magnitude in pressure.)  This is an exponential effect that quickly swamps the loss due to mere spherical expansion of energy.  But low frequencies don’t feel it so much.

I can imagine other effects that might apply in a natural setting.  Outdoors among trees, each branch is a potential reflector of sound.  When we are very close to a point source, the reflections will be far away, relatively small in energy and therefore not significantly affect any frequency.  As we move away from the source, the reflections make up more and more of the total energy from the point source that reaches us.  Furthermore, as we continue to get farther away we will hear more and more 2nd, 3rd and higher order reflections.

Adding lots of reflections in the limit becomes convolution with a Gaussian.  Crisp impulses in the original signal turn into a smoothed-out lumps.  It’s easy to see that this attenuates higher frequencies.  I only brought up a Gaussian because it is algebraically special by being its own Fourier transform, so it makes a low pass filter, as seen here (image taken from this pdf):


Ok, I’ll admit that last part is pure handwaving.  In any event, we know low frequencies carry farther.

So, my theory is just this: in natural environments, being farther away 1. makes things softer, and 2. attenuates high frequencies; therefore, our nervous system evolved to compensate.  Lower tones and softer volume will both let me know that the mammoth is bellowing far away, but it should continue to sound like the same animal as it approaches.

The modern world has volume knobs, and these confuse our sensory apparatus because the world we evolved in didn’t have any.

I find this idea simple and elegant.  It’s therefore probably wrong – I’d love to hear what others think.

One response to “A distant rumble

  1. Anonymous

    But isn’t it said that the easiest, simplest answer is more times than not the correct answer? Nature doesn’t try to be complicated nor does it set out to confuse you. It simply just is and your explanation seems straightforward and reasonable. If it was more complicated and confusing than that, then there is a good chance that “Man” not “Nature” created it.

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