Why does rotating the Michelson-Morley apparatus cause a fringe shift?

[genxhis]

I am having difficulty understanding the rotation in the michelson-morley experiment. I tried reading from several sources, but I think I am missing something each source regards as obvious. I can understand why the travel times involved in each light path must differ by T = Lv²/c³ (approximately). But I don't understand how rotating the apparatus by 90 degrees would change anything other than which path is labeled what. In particular I don't see how this causes a fringe shift.

 

[jtbell]

When arm A is parallel to the "ether wind", the fringe shift was expected to be in one direction; when arm A is perpendicular to the "ether wind", the fringe shift was expected to be in the opposite direction. And when arm A is at some intermediate angle, the fringe shift was expected to be somewhere in between; in particular, if the arms are at a 45-degree angle, there should have been zero fringe shift. So if you rotate the apparatus gradually, and watch the fringes as you go along, it was expected that the fringes would shift back and forth according to the angle of the arms of the interferometer with respect to the "ether wind."

But they didn't, of course...

 

[genxhis]

Thanks. I understood it a bit better after reading what you said. It also gave me an idea. To make myself more confident, I made a quick simulation on Mathematica (using roughly real data) that had the fringes shifting back-and-forth by 0.37 cycles as the apparatus rotated.