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In attempting to find a reference for explaining sound propagation, I came across this article in Wikipedia:

Stokes law(sound attenuation)

What I learnt in high school is that sound is propagated by an inverse square law.

My question is, what (if any) is the relationship between Stoke's law and the inverse square law? Is the inverse square law simply an approximation/generalization of Stoke's law? Or do they in fact have nothing to do with each other?

Lastly, if my assumption that the inverse square law is an approximation of Stoke's law, can anyone point me to a reference?

Ali
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Anthony Truskinger
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    I believe Stoke's law is the energy loss of the sound wave due to scattering and absorption. The inverse square law however applies to the reduction in intensity of the sound due to the wave spreading through space. In reality both factors apply to reduce the intensity of sound. – Kenshin Oct 05 '13 at 12:10

2 Answers2

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As Chris and leftaroundabout said, these are really independent things.

The $1/r^2$ law comes from the fact that as the energy in the sound wave travels away from its source, it gets spread out over a larger and larger sphere. Since the total energy must remain the same (assuming no dissipation), the amount of energy in each unit of area on the sphere must decrease. The rate it must decrease is $1/r^2$. So this law is really a result of conservation of energy.

The other law says what happens if there is dissipation. Then as the energy travels through the medium (say, air), then some of the energy in the sound will be lost to heat. The way it works is that after a sound wave has traveled a distance $l_d$, the amplitude of the wave decreases by a factor of $e$.

Putting these two together, we expect intensity $I$ as a function of distance to the source $r$ to have the form $I(r) = I_0 R_0^2 \frac{e^{-2r/l_d}}{r^2}.$ Now why didn't your teachers tell you about this. Well if you calculate $l_d$ according to the formula on wikipedia, you get about six miles. So four sounds that you ordinarilly hear, it is not an important effect.

Now as leftaroundabout said, the attenuation law was derived assuming a plane wave, but I'm fairly certain that you will get the functional form I said for the spherical wave. The decay length might be slightly different, but it will just be an $O(1)$ correction.

Brian Moths
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They do in fact have nothing to do with each other. Inverse-square is a simple conservation-of-energy argument for point sources, i.e. it basically assumes that no absorption takes place anywhere – which is a pretty good approximation for dry air, provided of course there are no solid obstacles anywhere.

Stokes' law is much more specific in that it starts with a plane wave, while inverse-square amounts to a spherical-wave model. Now, any wave looks locally like a plane wave if you zoom in close enough, but Stokes' law only makes sense if you look at an area much larger than the wavelength. Which you can do with a spherical wave, but you need to move out far away from the source, where $r^2$ becomes basically constant1.

For audio applications, Stokes' law is largely irrelevant, because a) you'll hardly get to a proper plane-wave scenario and b) the viscosity of air is so low that the predicted absorption is usually much smaller than any losses at solid boundaries, and to the inverse-square spreading. Where Stokes' law is important is at high frequencies in fluids with significant viscosity, e.g. in medical ultrasonic scans.

Inverse-square itself needs to be handled with care: in lots of applications, reflection plays a significant role, so the spreading is limited and the intensity drops slower than $1/r^2$, or there are solid obstacles between source and observer so the intensity is obviously lower.

1For the purpose of its effect on magnitude only. Of course, the variations in $r$ still act on the phase, where a large offset doesn't play any role.

leftaroundabout
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