Neumann-Dirichlet open-string boundary condition (${\partial }_{\sigma }X(0)=0,X(\pi )=y$)

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.

For one free boson, the mixed ND open-string sector has strip vacuum energy $E_0=1/48$. The strip-to-plane transformation relates this energy to the plane Virasoro algebra zero mode by $E_0=h-c/24$, with central charge $c=1$. Hence its ground conformal weight is $h=1/48+1/24=1/16$. For $d$ independent ND coordinates the weights add to $d/16$. At $d=24$, the plane matter ground weight is $3/2$, while the strip oscillator vacuum energy is $1/2$. The plane physical condition $L_0^{\mathrm{plane}}-1=0$ is therefore equivalent to ${\alpha }^{\prime }{p}_{\parallel }^2+N+1/2=0$. The apparent different constants are coordinate/constraint conventions, not different mass spectra.