The existence and structure of large amplitude, stationary, longitudinal plasma oscillations are studied for the case of a simple waterbag distribution of electrons and an immovable background of ions. The analysis employs the one‐dimensional Vlasov equation for a plasma of infinite spatial extent. An expression for the maximum amplitude of the oscillations is derived. This maximum amplitude decreases monotonically as the ratio of the electron thermal velocity to the wave phase velocity increases. The structure of the oscillations is expressed analytically in terms of hyperelliptic integrals.