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!
 A: 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.
