Given a mobius strip, what do the solutions of the wave equation look like qualitatively? How do they differ from solutions on the equivalent strip glued together as a cylinder? Any refs, particularly to symmetry?



Solutions to the wave equation on non orientable surfaces like a mobius strip
Any solution to the wave equation on a Möbius strip lifts to a solution on its orientation double cover, which is a cylinder of equal width but twice the circumference. In order for a solution on the cylinder to descend to the Möbius strip, it is necessary and sufficient that it be invariant under a certain order two symmetry. If the cylinder has coordinates given by $[0,\pi a]$ in the free direction and $[0, 2\pi L]$ in the loop direction, then the solutions are linear combinations of products $AB$, where $A$ has the form $\cos (kn\_a t)\cos(\frac{n\_a}a x)$ or $\sin (kn\_a t)\cos(\frac{n\_a}a x)$, and $B$ has the form $\cos(kn\_Lt)\cos(\frac{n\_L}{L}y)$, $\sin(kn\_Lt)\cos(\frac{n\_L}{L}y)$, $\cos(kn\_Lt)\sin(\frac{n\_L}{L}y)$, or $\sin(kn\_Lt)\sin(\frac{n\_L}{L}y)$. Here, $n\_a$ and $n\_L$ are nonnegative integers, and $k$ is a constant. Invariance under the symmetry is equivalent to $n\_a + n\_L$ being an even number. In contrast, solutions on the cylinder of the same dimensions correspond to solutions on the double cover such that only $n\_L$ is even.


