For a plasma with finite cross section of the sort which occurs typically in laboratory plasma waves experiments, in a constant magnetic field, the Landau damping (or growth) is obtained by a perturbation procedure in terms of the plasma velocity distribution function and potential and density profiles. The result is applied to the damping associated with the upper and lower branches of the dispersion curve for longitudinal electron plasma waves in the case of a Maxwellian velocity distribution in slab and cylindrical geometries. Application is also made to the growth rates resulting from a low‐density electron beam with radially dependent energy superposed on a Maxwellian.