I was teaching a class about waves, working with a few examples on desmos i decided to try it out some different functions, so i did $$f(x) = a\cos(kx+b\sin(\omega x))$$
I noticed that in some specific values of $\omega$ it is possible to notice a transverse wave in the graph, that the "thickness" of this wave in some sense can be controlled by the "b" constant. I could not explain why this behavior emerges.
For some $k$ values you also get these interesting cylindrical patterns:
why?