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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.

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Such a device is actually realised in synchrotron radiation sources as the so-called figure-8 undulator; this paper is the earliest reference I was able to find, and a Google search yields many (freely-available) descriptions of the method. These devices superpose two linearly-polarized oscillations on the electron's motion, one having half the period of the other. The electron therefore traces a figure-8 when its motion is projected onto a plane perpendicular to both planes of polarization.

This is all very well for electrons, but how about your chain example? I strongly suspect that it will be impossible to realise in any non-ideal physical system. If your chain were perfectly elastic, with rigid end-points, and were vibrating in zero-gravity, then in principle I don't see why it shouldn't support such a mode. The real chain however will not have identical propagation speeds for transverse waves in the horizontal and vertical directions; curvature of the chain under gravity, and the non-ideal nature of reflection at the end-points, will stymie any attempt to support different mode numbers in each plane.

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  • $\begingroup$ Why wouldn't the same problem stymie the same mode number in each plane? $\endgroup$ – MJD Jul 6 '15 at 16:43
  • $\begingroup$ I guess the fact that the two wavelengths are different is significant because it means the phase relationship depends on the dispersive properties of the chain (in each plane). In an ideal case, the chain is non-dispersive and the first harmonic has exactly twice the frequency of the fundamental. This breaks down however when you have finite dispersion. So even if the fundamental has a (practically) identical frequency in each plane, the same need not apply to the first harmonic. $\endgroup$ – tok3rat0r Jul 6 '15 at 17:07

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