Velocity waves in spherical geometries

I am currently working on velocity waves in spherical geometries: I am considering a 1D many-particle system confined on a circle with a global drift leading to rotations, similar to this simulation of so called "phantom traffic jams": https://www.youtube.com/watch?v=Q78Kb4uLAdA

In this simulation you can clearly see velocity waves in the system, meaning that cars on one side of the ring are always faster than the those one the other side. This profile is propagating through the system leading to waves. These waves come from relatively simple assumptions about the cars, but show a very particular behaviour (including shocks which are jumps in the density and velocity) which is realistic for the particular system but not very general: I think a general velocity wave would have a sinusoidal velocity profile and arbitrary propagation speed.

My goal is to understand more generally how such a perfect sinusoidal velocity profile is emerging and how it is propagating through the system. Especially what the relative speed between global drift and wave propagation is. Also what the wavelength of such a wave would be. Very interesting would be understanding how they emerge from particle interactions, but this is probably hard. Already a "simple" differential equation for the velocity field (Like some Navier-Stokes equation) would really help.

So here is my question: Does anybody know other real system where those velocity waves appear? I thought about sound waves but they do not show drift. Or does anybody know work on differential equations which could maybe show propagating velocity waves in such a geometry?

Thanks,
Tom