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Field line resonances with frequencies in the range 1 to 4 mHz have recently been observed by the JHU/APL HF Doppler radar during quiescent geomagnetic conditions. These structures are observed to have stable frequencies for durations of several hours, leading us to the conclusion that they may be standing waves (in the radial direction, as opposed to standing waves along a field line) of the magnetosphere driven by the solar wind. Using this premise, the propagation of fast mode ULF waves in the magnetosheath and the near earth magnetosphere are examined in an ideal, linearized MHD context. A model is presented in which fast waves propagate in the equatorial plane between the flanks of the bow shock and a turning point deep within the magnetosphere. Due to the magnetic field gradient near the earth, a field line resonance develops between the turning point and the plasmapause. Using a realistic set of magnetospheric parameters, it is possible to reproduce the set of observed frequencies and the respective positions of their field line resonances within the ionosphere (assuming a dipole mapping). However, because the model cavity frequencies are sensitive to magnetosheath parameters, this model does not explain the extreme stability with respect to geomagnetic conditions of the observed frequencies.