  Recent studies of scalar and gravitational waveforms from high-eccentricity,
extreme mass-ratio black hole binaries show the presence of quasi-normal bursts
(QNB) -- lingering high frequency decaying oscillations (also known as
``wiggles'') -- soon after each periapsis passage. One puzzle associated with
these QNB is that in the case of a nearly-extreme rotating central black hole
the frequency of the QNB has been found to be in a range which is lower than
the corresponding range of relevant quasi-normal modes. We reproduce these
results using a different approach and perform a detailed analysis to find
evidence for the resolution of the puzzle and for the origin of the QNB. We
find that the QNB frequency as measured at future null infinity evolves in
(retarded) time and approaches the dominant quasi-normal frequency
exponentially in time. We also show that the QNB amplitude decays inversely in
(retarded) time. We discuss the time dependence of both the QNB waveform
frequency and its amplitude and argue that this behavior arises as a result of
the excitation of many quasi-normal overtones and the summation thereof.
