The potential and charge particle distributions are calculated for a collisionless sheath bounding a plasma adjacent to a wall which emits secondary electrons under both floating and current-carrying conditions. The corresponding conditions at the sheath-plasma boundary are determined in terms of the parameters secondary electron beam density/plasma density and beam velocity/plasma electron thermal speed. This 'steady state initial condition' is then examined for instability to excitation of the beam-plasma instability using particle-in-cell computer simulations. Results are given and summarized in the appropriate parameter space showing when the instability can be classified as weak, strong, or violent. The work is related to other work on the beam-plasma instability-theoretical, computational and experimental- and on large amplitude electron plasma waves.