 Solution

The common Neumann boundary conditions define a $(p+1)$-dimensional timelike worldvolume: one time direction and $p$ spatial directions. The remaining Dirichlet boundary conditions fix its transverse position. An open string therefore ends on a D-brane of spatial dimension $p$, a D$p$-brane. For the critical bosonic string theory there are $25-p$ transverse directions.

If the Dirichlet constants agree at the two endpoints, both ends lie on the same planar D-brane. If they differ, the string stretches between two parallel D-branes with the same orientation but different positions. Neumann directions are tangent to the brane; Dirichlet directions are transverse to it. The brane is dynamical: tangential massless open-string polarizations supply a gauge field, while transverse ones describe its displacement, as in worldvolume fields from open-string massless states. The unprojected bosonic brane also has a tachyon; the boundary-condition interpretation does not make that ground state supersymmetrically stable.