Radio emissions emanating from the Earth and other planets are often characterized by discrete frequency structures. For example, recent ground-based observations of auroral roar, an auroral radio emission which occurs near the second and third harmonic of the electron cyclotron frequency, show that it consists of fine frequency structure similar to that of auroral kilometric radiation and other planetary radio emissions. These auroral roar fine structures, sometimes as narrow as a few hertz, often occur in multiplets separated by the order of ≤ 1 kHz which drift up and down in frequency. Theoretical and experimental efforts to explain the generation of auroral roar suggest that in the source region near the F region peak, the quasi-electrostatic Z mode (or upper-hybrid) waves are first excited, partly converted to free-space radio waves and subsequently observed on the ground. Using WKB-type calculations of the wave mechanics of upper-hybrid modes in a cylindrical field-aligned density structure, we show that discrete frequency eigenmodes are a natural consequence of such density structures. Discrete eigenmodes can exist within density enhancements but not within depletions. Cylindrical field-aligned structures the order of 100 m to several kilometers diameter result in eigenmodes spaced by a few hundred hertz as observed for auroral roar. Since structure of this scale size often occurs in the Earth's auroral ionosphere at F region altitudes, it seems possible that the observed auroral roar fine structure results from this mechanism.