NOTES
There's a difference between the doubling rate in atmosphere versus in the water, because water is a lot denser than air; as a submarine descends, it gets closer to the center of the Earth, increasing gravity, but less of the water is contributing to gravity, which decreases the rate of increase somewhat. For the aircraft case, I'm neglecting the mass of the air entirely because the effect of air wouldn't change the doubling rate by very much at all. I'll go into this in more detail later on.