# Proof without curves

Say, what’s that in the distance?

Could be a road full of cars or a wire full of electricity, or whatnot, but this much is for sure:
It is straight and infinitely long.

Sure, but why does the picture have 3 of those infinitely straight things?
Oh it’s just the same old thing at different distances.

Are the types of cars and their specific spacing important?
Only in their drivers’ minds, not to this puzzle.   Consider everything as being evenly distributed.

So what’s all this supposed to prove anyway?

It derives a relationship well known in physics but also experienced by many who try to sleep at what would seem like a large distance from a busy superhighway. (Care to explain?)

I think this is elucidating

the fact that if you have a line of "sources" in the plane obeying an inverse-square law then the size of its effect goes down only like 1/d rather than 1/d^2 (because as you increase the distance, the amount of stuff within a given angle increases proportionally to the distance).

So, for instance,

if the "sources" are cars emitting sound (which obeys an inverse-square law) then at 200m from the road the intensity of sound is only half (rather than 1/4) what it is at 100m from the road. (Though if the falloff is less than expected I suggest that this is also because of the nonlinearity of human perception; even if the intensity were 1/4 as much, it still wouldn't feel like a factor-of-4 decrease.)

Perhaps more startling is

the related fact that if you have a plane of such sources in space, the effect doesn't diminish with distance at all.

• Your side notes are music to my ears
– humn
Jan 19 '17 at 16:14
• but is my solution correct? Jan 19 '17 at 16:53
• And then some! Officially so in another 23 hours. You also presaged the next one like this in mind, but the graphics look to be too messy.
– humn
Jan 19 '17 at 16:55