= Integer-mode expansion with parallel D-brane endpoints
{title2=$\cos n\sigma\ \text{for NN},\quad\sin n\sigma\ \text{for DD}$}
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.
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