Waves on a circle

In the last couple of posts we've imagine the different ways in which a guitar string vibrates, and we've seen that each of the different ways contains a different number of vibrating segments, and there is always a whole number of segments.
The next idea is tricky, but imagine if the guitar string was instead a hoop of steel. It turns out that even for a hoop, the rules about vibrations are the same as for a straight string. You can get a hoop of steel to vibrate in different ways, but there is always a whole number of vibrating segments on the hoop.
You might be able to visualise this by imagining that the guitar string was actually a stiff rod of springy steel. It would still vibrate like the guitar string does, so if you bend the stiff rod into a hoop it still vibrates in the same way.