The eigenfrequencies of all modes of oscillation of a water-bag distribution of electrons confined in an external parabolic electrostatic potential are determined by analytical methods for all values of a parameter η, which measures the ratio of the electron density to the strength of the external electric field. Two approximations are studied, one for η 1 (low electron space charge) and the other for (1 - η) 1 (high electron space charge). All eigenfrequencies turn out to be real, but `damping' results from the fact that for η ≠ 0 and η ≠ 1, the frequencies ωn(η) of the various modes n are not simply related. If the wavelength of the perturbation is much less than a Debye length {n (2/π) log [12(2 - √3)/(1 - η)]}, [ωn(η)/n] is not strongly dependent on n, while if the wavelength is much greater than a Debye length {n (2/π) log [12(2 - √3)/(1 - η)]}, ωn(η) is near the plasma frequency ωp, as expected for a cold plasma, or for the long wavelength limit of an infinite plasma at finite temperature. The focusing effect of the parabolic electrostatic potential in an electron rest frame is similar to that of a periodic alternating magnetic field, such as is used in the NRL Sozotron, in a laboratory frame. It is also similar to the sum of external and ion-produced potentials in which the trapped electrons in an ion-rich plasma diode find themselves. The eigenfunctions of the perturbation current are also obtained, so that the development in time of a given initial perturbation current can be determined.