Free-end reflection of a transverse string wave (${\partial }_x\psi =0,r_{\psi }=+1$)

At a free termination of a taut string, zero transverse force imposes a Neumann boundary condition. Incident and reflected displacement amplitudes add with equal sign so their slopes cancel. The endpoint is a displacement antinode, all incident power is reflected, and the flux $-T{\psi }_x{\psi }_t$ through the boundary vanishes. This is a transverse-wave boundary condition, distinct from the quantum open-string usage of similar notation.