In this note we comment and extend the results of a previous analysis in which the non-linear behaviour of one-dimensional electrostatic oscillations in a homogeneous, unbounded, collisionless and fully ionized plasma was considered. The evolution of a monochromatic wave of small, but finite amplitude is studied by expanding the dependent variables as well as the independent variable tin the form of asymptotic series; an ordering parameter e proportional to the initial amplitude of the electric field is introduced. The expansion of the independent variable in such a series allows us to eliminate secular terms from the part of the distribution function which does not depend on the free-streaming terms. This, in turn, allows us to determine corrections to the complex frequency a. Results of a previous note on non-linear Landau damping for an initially Maxwellian. distribution function are confirmed, but it is indicated that they apply to values of time up to a value τ1 rather than for all times. One can proceed to larger values of time in the manner of the multiple time-scale method. In particular it is found that the Landau damping is increased with respect to the linear value only initially during the first time scale.