When a chain hangs between two short posts, I can use my hand to oscillate it up and down in a standing wave. It's easy to demonstrate three of the possible normal modes, at the fundamental frequency and the first two harmonics. For mechanical reasons it's hard to get the chain to oscillate in a higher mode, but I think there's no problem in principle, and I could do it if the chain were lighter.
I can also vibrate the chain horizontally obtaining a sort of horizontal polarization of the wave.
I can vibrate the chain circularly, obtaining a circular polarization of the wave; it's as if the chain was oscillating horizontally and vertically at the same time, with the vibrations out of phase. Each link of the chain travels around in a circle in a plane perpendicular to the line between the chain's endpoints.
It seems to me that I ought to be able to induce the chain to vibrate at the fundamental frequency in the vertical direction, and simultaneously at the first harmonic (double) frequency in the horizontal direction. A single point on the chain would then follow a path in space that was a parabola instead of a circle. (Depending on the phase offset, it could also be a figure-eight design.)
I haven't been able to get the chain to oscillate in this hypothetical parabolic polarization. Is it just a mechanical difficulty, like the reason I can't get the fifth harmonic, or is there some reason why it's impossible in principle, or possible but unstable?
If the parabolic polarization exists, what's it called? Google search for parabolic polarization only yields information about parabolic antennae.