Once you understand that all vibrational energy is actually circular motion, then a number of otherwise mysterious things start to become very simple.
For example, the other day a friend posed the following puzzler: We all know that when you make a short burst of sound (say, by clapping your hands together), the sound impulse radiates out into an ever expanding sphere — since the sound goes in all directions. Logically it would seem that the loudness of that sound, when heard from some particular distance, should drop off as 1 / distance2, since the surface of the expanding sphere increases as distance2, so the sound should be diffused by that much. For example, you’d think that a clap from twice as far away would sound only one fourth as loud.
But that’s not what happens. In fact, the loudness of the sound drops off only as 1 / distance. For example, a clap from twice as far away sounds half as loud.
What’s going on here? It happens this way because a sound wave is actually a kind of circular motion. I’ll explain the rest tomorrow.
Energy per unit perpendicular area as a function of distance from ac acoustic source in 3D with a homogeneous medium goes as
E=O(r^{-2})
Perceptual “loudness” goes as
L=log E=O(1) – O(log r)
which is decreasing pretty slowly with distance.
Oh, I’m sorry. My terminology was imprecise. I was referring to the actual sound pressure that impinges on the ear, leading to the perception of sound.
Subjective loudness, or sound pressure level, is indeed measured on a logarithmic value, usually measured in decibels. My fault entirely.