An eigenmode analysis of the response of an empty ring-down cavity to an arbitrary laser excitation is presented. By explicitly taking into account both the mode structure of the ring-down cavity and the spectral content of the laser pulse, it is found that the complicated ring-down signals commonly observed in the laboratory can be interpreted in terms of cavity mode beating. Some conclusions drawn from this analysis are verified experimentally by measurements of the time and frequency response of empty cavities. These observations provide clear evidence for the existence of longitudinal and transverse mode structures in ring-down cavities.