  The Doppler effect in Fabry-Perot cavities with suspended mirrors is
analyzed. Intrinsically small, the Doppler shift accumulates in the cavity and
becomes comparable to or greater than the line-width of the cavity if its
finesse is high or its length is large. As a result, damped oscillations of the
cavity field occur when one of the mirrors passes a resonance position. A
formula for this transient is derived. It is shown that the frequency of the
oscillations is equal to the accumulated Doppler shift and the relaxation time
of the oscillations is equal to the storage time of the cavity. Comparison of
the predicted and the measured Doppler shift is discussed, and application of
the analytical solution for measurement of the mirror velocity is described.
