Douglas Heckrotte wrote, "I had thought then that the volume of
air that it took to allow the pouch to rise would be made available
by a vacuum pulse traveling down the tube, rather like sound or the
resonance between cylinder and exhaust volumes used in the design of
internal combustion engines to more efficiently and quickly exhaust
gasses."

That is not far from the correct explanation.  I don't claim to know
the exact theory on this complex subject, but I do know how it relates
to church organ pneumatic actions.

The "charge" system uses pressure at the console to actuate pouches in
the organ.  There is a definite limit for the length of the tubing runs
before an unacceptable delay is encountered.

However, if the action is "exhaust" -- similar to the player pianos
vacuum action -- the console can be at the opposite end of the church
to the organ.  The notes snap "on" when the air in the tubes is
released by depressing a key, thus starting a continuous flow of
air through the tube until the key is released.  At that point the
"water-hammer" effect comes into play shutting off the action at the
organ end almost instantaneously.

John Page, UK