It is shown that a distribution of electrons in resonance with traveling waves, but colliding with background distributions of electrons and ions, evolves to a steady state. Previously, the existence of such solutions had been assumed, but not proved, in numerical and other calculations. Details of the steady state are given analytically in the asymptotic limit of high electron energy and are compared with numerical solutions. The asymptotic analytic solution may be useful for quickly relating emission data to likely excitations and is more reliable than conventional numerical solutions at high energy. A method of improving numerics at high energy is suggested.