Due to differences in solar illumination, a geomagnetic field line may have one footpoint in a dark ionosphere while the other ionosphere is in daylight. This may happen near the terminator under solstice conditions. In this situation, a resonant wave mode may appear which has a node in the electric field in the sunlit (high conductance) ionosphere and an antinode in the dark (low conductance) ionosphere. Thus, the length of the field line is one quarter of the wavelength of the wave, in contrast with half-wave field line resonances in which both ionospheres are nodes in the electric field. These quarter waves have resonant frequencies that are roughly a factor of 2 lower than the half-wave frequency on the field line. We have simulated these resonances using a fully three-dimensional model of ULF waves in a dipolar magnetosphere. The ionospheric conductance is modeled as a function of the solar zenith angle, and so this model can describe the change in the wave resonance frequency as the ground magnetometer station varies in local time. The results show that the quarter-wave resonances can be excited by a shock-like impulse at the dayside magnetosphere and exhibit many of the properties of the observed waves. In particular, the simulations support the notion that a conductance ratio between day and night footpoints of the field line must be greater than about 5 for the quarter waves to exist.