Negative experiment result:

I tried to record a reflection of a sound wave off a piece of cardboard. The goal was to see if I could measure the distance to the cardboard by measuring the time between the original squarewave coming from a loudspeaker, and the reflected echo. The first few tries, I got no noticeable echo. Finally, by reducing the period of the squarewave, I could see a weak echo in the sound recording, but the return signal was very weak relative to the driving signal. That's going to turn into lots of noise in the software when trying to extract distance info from the recording. So I'm going to abandon this approach entirely for now.

And, more generally, I'm going to abandon using time delays or frequency modulation. I'm just not going to get the resolution I need, given a sampling rate of 44.1 KHz.

Now I'll turn to using amplitude modulation. I'm hopeful this will work much better.