 Neumann-Dirichlet open-string boundary condition

A transverse open string coordinate has an ND boundary condition when one endpoint obeys a Neumann boundary condition and the other a Dirichlet boundary condition. On a strip of width $\pi$, its nonconstant spatial eigenfunctions are $\cos ((n+1/2)\sigma )$ when the free endpoint is at zero. They solve the endpoint equations because their derivatives vanish at zero and their values vanish at $\pi$. Reversing the endpoints gives sine eigenfunctions with the same frequencies. The fixed endpoint removes the dynamical worldsheet zero mode.