The problem of the plane wave oscillation of a one‐dimensional plasma is considered. It is shown that for stable plasmas with unperturbed distribution functions in a general class which includes Maxwellians as a special case, the electric field does not damp exponentially fast, as predicted by Landau. Instead, the electric field tends to zero as some reciprocal power of time, where the exact power depends on the smoothness of the initial data. To obtain a more conventional result it is assumed that all distribution functions vanish for particle speed greater than some constant, as they must relativistically. Then for wavenumber less than a certain constant the plasma oscillations do have an undamped part corresponding to a conventional normal mode with a phase speed greater than the fastest particle speed. For wavenumbers larger than that constant the oscillations tend to zero as some reciprocal power of time.