The wheels on the bus go...

It is well known that in videos of moving cars, the wheels often appear to be rotating unnaturally slowly or even in reverse. This is explained well here, and I thought of it today, while studying signal processing.

Several years ago, I was surprised to see the same phenonemon in real life, not in a video. I was in a bus on a highway at night, and looking out the window, the wheels of the car next to me appeared to be barely rotating at all. How could this possibly happen? Was this the Matrix and was I being shown a video of a car next to me, not the car itself?

Eventually I thought of a good explanation. Since it was nighttime, the only lighting was provided by overhead streetlights. Being fed by AC power, they apparently flickered on and off 100 times per second (twice 50 Hz, the Israeli power frequency). I saw cars only when the light flickered on, and while it was off, each wheel traveled a whole revolution but I was unable to see it.

It was nice to have that explained, but some time later, an even weirder thing happened. I was in Istanbul (enjoying a flight layover on an Israel-NY trip), and was on a train traveling parallel to a freeway. Looking at the cars on the freeway, I saw their wheels turning backwards. My flickering-light theory was inapplicable, because the time was 11am!

But it wasn't long before I found an explanation for this too. Between the cars and train was a fence, made up of vertical bars with gaps between them. I didn't see it at first because we were moving fast enough that it became a blur. But apparently, this fence blocked one part of a car's wheel at any given time, and a moment later blocked another part. For each part of the wheel, there was a time gap in which I could not see its rotation. Thus, the same effect described above took place and the wheels appeared to be rotating backwards.

Some of the complications and weird features of this explanation:
1) Even though I couldn't see the fence itself due to the blurring, I could see its effects on the wheels.
2) Only part of the wheel was obscured at any one time, yet the same effect seemed applied consistently to the whole wheel.
3) As the wheel rotated, different parts of it moved to different places, some of which did not have the same fence-obscurance pattern as the original place - yet the effect of the fence remained unchanged. This rotational movement must be a "second-order effect" which is negligible when you analyze the equations. Though it's not obvious why it should be negligible, since rotation as a whole is a central part of the effect.

What can I say... the world is an interesting and confusing place.