Annabelle
In other words, when audio is carefully analyzed, is there a specific shape of waveform that represents a fade-in or fade-out?
Yes it looks like a less than/greater than sign, except not so linear.
Some sound waves when looked at (say a drum hit) would start from a flat horizonal plane and increase upward and downward very sharply, but in the basic form of the < less than sign. And the time it would take for that upward/downward increase would be in milliseconds. That would be the fade in.
A cymbal wash would start out on that horizonal plane and slowly increase upward and downward over a few seconds, but it would still follow that less than sign <.
And for the fade out it would be the opposite.
I hoped I have helped. If I'm off base, just ignore me! : )