Many features of the magnetospheric hydromagnetic wave spectrum are fully understood, but a remaining puzzle is why selected field line resonances appear to be excited by a broadband source. In this paper, we propose that the preferential resonances correspond to global eigenmodes of the magnetospheric cavity. In order to understand the behavior of ultralow frequency waves in the terrestrial magnetosphere (geomagnetic pulsations), we describe the structure of global eigenmodes in the hydromagnetic box model of an inhomogeneous plasma. Global modes are large-scale modes whose frequencies match field line resonance frequencies somewhere in the system. The coupling between localized field line resonance effects and the large-scale mode leads to damping of the latter. In the box model the eigenmode equation yields a spectrum of discrete eigenfrequencies. For each eigenfrequency, the equation contains singular points where the field line resonance occurs. We base much of our discussion on earlier works which have used the same equation in other contexts. We describe solutions and then use the results to outline features of the modes that may be observable.