What are the eigenfunctions and eigenvalues of a moebius strip? A Moebius strip is a simple example of a non orientable surface.  Suppose a very thin metal model of one, say of length $2\pi L$ (midline), width a, thickness negligible is perturbed ("kicked"), and then left to freely vibrate transversely.   How do its vibrations differ compared to those of a simple cylinder of the same dimensions?   What about the eigenvalues, i.e. allowable frequencies?   (This is not a homework problem, but could be.)  For the free boundary case, what about torsional waves about the center longitudinal axis of the strip?   Naively, it feels like there should be a difference, i.e. doubling of wavelengths.      Is that right?
 A: They would be some waves (obviously, so we may describe them by using complex exponentials)
$$\psi(x,y) = \exp i(l_xx + l_yy)$$
where x denotes the "periodic" coordinate and y the width of the band.
all that is left to do is to impose some boundary conditions onto the band. Let us say the the band has length $L$ (in the x direction) before it reconnects to itself. We must write
$$\psi(x,y) \sim \psi(x+L,y')$$
More specifically the image tells us that up becomes down and left becomes right after one revulsion such that the correct boundary conditions are:
$$\psi(x,y) = \psi(x+L,-y) \rightarrow \exp(2il_y y) = \exp(i l_x L) \forall y$$
At first sight I don't see any other solutions than $l_y = 0$ and $l_x = \frac{2\pi n_x}{L}$
Conclusion
If nobody finds a mistake in my reasoning I conclude that the topology of the mobius band eliminates oscillation along its y. Oscillations along the x direction do not pose any problem.
In fact this is consistent with intuition. Just grab a piece of paper and make a Mobius band from it. You will see that you cannot create any modes(= bends) in its y direction due to the boundary conditions !
message to OP: thanks for the interesting question :)

A: I believe that they look something like this

In 2006 I was curious about the normal modes of graphene -- as sheets, ribbons, tubes, etc. I folded the structure of a narrow graphene nanoribbon into a mobius strip and gave it a "kick" of random velocities. These are the resulting disturbances that propagated around the ring in a molecular dynamics simulation.
