Integer-mode expansion with parallel D-brane endpoints ($\cos n\sigma \text{for NN},\sin n\sigma \text{for DD}$)

On $0\le \sigma \le \pi$, a coordinate with Neumann boundary conditions at both ends has a constant position, linear time zero mode and cosine oscillators. A coordinate with Dirichlet boundary conditions at both ends has a linear spatial interpolation between endpoint positions and sine oscillators, with no momentum zero mode. Both have integer frequencies. This differs from an ND boundary condition, which applies different conditions at the two ends of the same coordinate and gives half-integer frequencies.