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Obviously, the only way to make sound is to move air, no matter how much or little, and both BA and dynamic drivers move air. Tinyman is correct about the tuning aspect (shaping of the housing, size of internal airspace, many other things), but fundamentally, a dynamic driver moves air differently than BA driver, mostly due to the design: it moves a pushes a diaphragm.
Many BA driver units move a smaller amount of air, which means there is often less bass impact, but there can be plenty or even more bass extension (down to 15 Hz, 10 Hz, etc).
Think about a home stereo speaker with it's round dynamic driver, then watch
this video from a Westone technician about how BA drivers work.
I need a source for this... All my sources show BAs moving as much air as dynamics. Remember, air moved = dB produced from the driver. It's definition, no questioning that one. Now, think of it this way, pressure = force / area. Dynamic drivers tend to have a larger area, so they need more force. BA's tend to have less area, but don't need to exhert as much force. So they actually cancel each other out quite nicely.
Let's take a deeper look this way and compare the Yamaha EPH-100 and the Heir Audio 3.Ai in terms of their ability to move air. The 3.Ai tops out at around 102 dB @ 70 Hz while the EPH-100 tops out at 105 dB at 40 Hz. The amount of air being moved here is actually very close and very similar.
Why is this happening? Well, let's put it this way, the dynamic drive is larger, so it puts out a larger force, but has more area to fill. So we get a larger force, but a large area to pressurize. However, the BA has a very small area to deal with, but can't create a very large force. In general, a larger force, but bigger area will have the same output as a small force with small area. There are many ways to think about this, but the pressure really equalizes itself out either way. So to simplify, let a = small area, A = large area, f = small force, and F = large force (case sensitive), we get:
F/A = f/a
Since their ratios essentially cancel out.
Let F = 3, A = 6
F/A = P
3 / 6 = 0.5
P = 0.5
Let f = .75 < F = 3, a = 1.5 < A = 6
F/A = P
.75 / 1.5 = 0.5
P = 0.5
Obviously these numbers are staged, but I'm just trying to make the point that the ratios really cancel each other out at the end of the day. Now, obviously, we can play with this model to make the dynamic win out, we can also do the opposite to make the BA win out. They thing is, playing with this model would be the equivalent of tuning a driver. Driver size decisions, power, force outputs, etc, are all tuned into the driver. It relies 100% on the tuning. However, saying that because BAs push less area and therefore don't move as much air (pressure) is not strong statement.