Modelling noise with distance

I was wondering about the relation between noise with distance, assuming a point source, using sound as the method for communication and air as the medium of communication. Obviously as the distance from the sound source increases, noise should increase- but what is the nature of this relationship? Is it linear or non-linear?

Any ideas? Is there any other way noise can be modeled (in the context of communication between individuals)? Thanks!

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Your relation that noise should increase with distance is misleading. What really happens is that the intensity of the signal of interest is dropping as the inverse of the square of the distance, with as a consequence that environmental noise that is originating from close by will be drowning the signal. But noise intensity itself will have the same relationship to distance as any other signal. – Raskolnikov Apr 17 '11 at 9:21
What makes You think that "noise" will behave different from "sound"? – Georg Apr 17 '11 at 9:47
@Raskolnikov thanks for the input! your logic is certainly sound. "noise intensity itself will have the same relationship to distance as any other signal" can you back up your claim? any references or papers? you might also want to put that down as an answer, which i will gladly accept! – Dhruv Gairola Apr 17 '11 at 11:10
""can you back up your claim? any references or papers?"" What kind of school did You pass? This is most basic (introductory) textbook level! – Georg Apr 17 '11 at 11:20
@Dhruv: I think the reason noise seems to increase with distance is not so much because noise really gets stronger over distance, but because the signal weakens over distance, whereas there are several different noise sources along the path. But everything will depend on what exactly you want to model. I'll see if I can find some good resources about modelling of noises. Do you have a particular system in mind though? Electrical, acoustic, electromagnetic, other? – Raskolnikov Apr 17 '11 at 13:25

I assume that what you're asking is about the signal to noise ratio. Clearly the noise will not depend on distance; the problem is that the signal drops below the noise floor.

Suppose you're transmitting with frequency $\lambda$. As the distance increases the intensity drops but the relationship is not so simple as is suggested in the comments.

For the case where sound propagates in three dimensions over a distance $r$ and $\lambda << r$, the sound energy is spread over the surface of a sphere of radius $r$ and so decreases proportional to $r^2$.

Same situation, but $r << \lambda$, the sound energy does not decrease. This is a near-field effect. The place in the real-world where I've noticed this effect is when walking near a jack-hammer. As you approach the jack-hammer from a great distance, the sound steadily increases. But when you get very close to it (and I've spent my share of time "balling that jack"), the high frequencies dominate. This is because the low frequencies can't increase substantially as one gets nearer to the source.

Another way of describing this effect is to note that at long distances, the energy of the sound is transmitted in a single direction, away from the source. But very close to the source of the sound, energy is transmitted both towards the source and away from it.

An example of sound waves produced from a small source but with a very long wave length is a room with a cylinder of compressed gas that steadily leaks. This produces an increase in pressure but the pressure is essentially the same throughout the room (despite the small size of the source of the pressure).

Finally, in situations where the sound spreads in less than 3 dimensions, the energy in the sound decreases at a slower rate. In 2 dimensions it generally (for $\lambda << r$) goes down as $1/r$, and in 1 dimension the sound does not decrease (other than absorption which applies to all these cases).

A classic example of sound restricted to 2 dimensions is the sound produced by a volcanic explosion, as heard many thousands of miles away.

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The lack of dispersion in 1-d sound trasmission is the reason why the tin-can treehouse phones work, btw. – Jerry Schirmer Apr 18 '11 at 1:16
@Jerry Yes. But "dispersion" is not quite the right word. I think that "dispersion" implies a difference in speeds of different frequencies. I would begin your comment as "The lack of losses". – Carl Brannen Apr 18 '11 at 1:30
Interestingly at wattsupwithat.com there is a thread now with the phenomenon of "atmospheric audio ducting": sound flowing through a trapped atmospheric level. – anna v Apr 18 '11 at 4:38