Many important properties of ultralow frequency (UFL) magnetic pulsations (~1 to 10 minute periods) observed in the magnetosphere and by ground-based magnetometers have been interpreted successfully in terms of resonant field line models. These models describe the pulsations as transverse hydromagnetic waves standing on dipole flux tubes with fixed ends in the ionosphere. Signal structure and frequency are consistently explained, but the models fail to provide convincing exptanations of the discrete spectra often observed. In this letter we propose that the spectrum of excitations is dominated by frequencies that correspond to eigenfrequencies of the fast (compressional) mode in the outer magnetosphere. In the presence of plasma inhomogeneities, the eigenmodes of the fast mode damp, even in the absence of dissipation, through coupling to transverse waves. We discuss the properties of a damped eigenmode, pointing out the analogy to the properties of damped surface eigenmodes. The coupled waves exhibit field line resonance behavior on the magnetic shell where the transverse mode dispersion relation is satisfied, but the spectrum is dominated by fast mode eigenfrequencies.