Solutions to the wave equation on non orientable surfaces like a mobius strip 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?
 A: 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.
A: The following paper by: Kousuke Yakubo,Yshai Avishai,and Doron Cohen describes in section II the solution of the Helmholtz equation on a flat rectangular surface having the topology of a Mobius strip. The solution is given in terms of the admissible wave numbers.
