Interesting thread! I'm not a mathematician, just an IT guy with barely enough math knowledge to be dangerous, but hear me out.
It always seemed to me that a waveform is something that can only be approximated. Unless it's generated by a function, that is (like a sine wave.) In music, the waveforms can only be approximated since the underlying sounds are very complex and chaotic.
Have you considered how waveforms not generated by a function are created? It isn't like they are created by some algorithm used to approximate sound.
The actual complexity of a waveform makes it a very fractal-like structure. As such, it might make more sense to assign a fractal dimensionality to waveforms, which probably lies somewhere between 1 and 2?
Last post by Porcus -
This wasn't correct about fractals either ... but I think this is getting increasingly irrelevant to the OP's issue. (That need not be a bad thing, morphing threads isn't uncommon ...) If we get down to the molecular level, and define the "sound" in a gas medium to be the position (x,y,z) of each particle at each time - i.e. two variables (t,n) as input, and three variables (x,y,z) as output - then one could certainly discuss whether the "dimension" is 2 or 5 of a number in between, but neither answer will help explaining an analogy to (motion?) pictures. At least not to me.
I kinda suspect that the OP's friend sees only one particular model of sound, and which isn't sufficiently consistent with any model of picture to get any idea on the key differences. (Are they in the physical phenomenon, or in what could possibly reach the brain through our sensory organs? I.e., is inaudible sound still sound and different shades of infrared still a picture?)