The simultaneous occurrence of locally resonant toroidal pulsations over a range of L shells complicates the interpretation of observations of these waves made from elliptically orbiting spacecraft. A theoretical model is developed which shows that measurements of ULF wave frequencies from spacecraft may be distorted due to the combined effects of spacecraft motion and spatial gradients in wave frequency, phase velocity, and the oscillation onset time. Spectral broadening and frequency shifts due to sampling biases caused by spacecraft acceleration and nonlinear variations in wave frequency with position are also considered. For toroidal field line resonances observed from AMPTE/CCE, distortions due to spatial gradients in wave frequency are the most important. For commonly observed standing Alfv¿n waves, the measured frequency is lower than the true frequency on the outbound leg of the orbit while it is anomalously high on the inbound pass. The predicted frequency error for the AMPTE/CCE orbit ranges from ≈15% for L=5 to ≈4% for L=8. Comparison of fundamental mode frequency measurements made on 10 inbound and 9 outbound passes during low to moderate Kp (2- to 4+) in the morning sector (0500 to 0900 MLT) confirms the predictions of the model. The average effective ringing time is estimated to be 1000¿300 s. The agreement of this result with theoretical calculations of damping rates due to ionospheric Joule dissipation (Newton et al., 1978) indicates that phase mixing between adjacent toroidal oscillations is not the dominant damping mechanism. The variation of pulsation frequency with L is also used to examine the dependence of plasma mass density on r. The form &rgr;∝r-q is found to fit the data reasonably well in the outer magnetosphere with q ranging from 3.0 to 5.5 for individual passes and q=4.0 determined from the combination of all measurements. ¿ American Geophysical Union 1989