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The pressure of what? The atmosphere? I don't think so.

Well then, imagine this. You have a room that is a total vacuum. You open a window on one side of the room. If you have pressure sensors all the way across the room, do think they would show the psi snapping to full atmospheric pressure in a wave, or do you think they would show a the lowest pressure at the farthest point, increasing as the distance to the window decreased?

Imagine you have a total vacuum in your room, but instead of opening a window, you remove an entire wall. Now, at t₀, you have a room at 0 atmospheres, with an "outside" pressure of 1 atmosphere, but that's not going to last. The atmosphere will move from high pressure to low pressure, and it's not going to slow down three inches inside the room and decide to take a leisurely pace the rest of the way - your leading edge is going to keep on trucking right on into the rest of the room, because the rest of the room, minus the first three inches, is lower pressure than everything else. And in a tube 100 miles (160,000 meters) long, there's pretty much only one direction for it to go - down the tube, and tout suite at that. ;)