Recent wideband observations of Langmuir waves in the auroral ionosphere show complex structure (McAdams and LaBelle, 1999) interpreted as resulting from discretization of the normal modes in the presence of a parabolic density cavity (McAdams et al., 2000). We formalize the theory of discrete Langmuir waves trapped within a one-dimensional density cavity or enhancement in a magnetized plasma, assuming a Lorentzian density profile, and show that discrete spectrum of electric field is largely determined by the electrostatic wave equation although a very low level of electromagnetic waves is also present. In the overdense case (fpe > fce), the Langmuir waves are trapped in density depletions, and the predicted frequency spacings of the eigenmodes are comparable to those estimated by McAdams et al. (2000) using an approximate theory. We furthermore calculate the eigenfunctions associated with the first four eigenmodes in a Lorentzian density structure, showing that an Airy function approximation provides a close match to the exact solutions obtained numerically. Finally, we analyze synthesized time series simulating the signals which would be obtained by a rocket electric field receiver traversing the wave structures. This analysis reveals subtle predictions of the theory which can be measured with appropriate rocket or satellite observations. Specifically, the analysis predicts that there are nulls in the spectral lines of observed discrete emission which should be detectable in principle.