system
Given the following post, respond as if you're commenting on Reddit:
human
When talking to someone through two cups attached by a string, why do you hear their voice louder than just talking through air at the same distance?

Any formulas to help explain this would be awesome!
gpt
Ignoring friction, the energy in soundwaves is conserved as they travel.

If you use a tin can telephone, the soundwave travels almost entirely through the string. The power received at the other end is equal to the one you input in the string.

If you don't use it, sound travels through the air in all directions. Even though total energy of the wave is constant, this energy is spread on a sphere centered on you and growing in size. The area of the sphere goes as R^(2), so at distance R the power received by any given receiver goes as R^(-2).

Bonus round:

We have considered the case of sound spreading through 1-dimensional and 3-dimensional shapes. But we can generalize: imagine a point source emits a wave (any kind of wave) of given energy which travels on a D-dimensional medium. The wavefront will be a (D-1)-dimensional sphere travelling outwards, with surface ~ R^(D-1). So the intensity at distance R goes as 1/R^(D-1).