  The discovery of nearly coherent brightness oscillations during thermonuclear
X-ray bursts from six neutron-star low-mass X-ray binaries has opened up a new
way to study the propagation of thermonuclear burning, and may ultimately lead
to greater understanding of thermonuclear propagation in other astrophysical
contexts, such as in Type Ia supernovae. Here we report detailed analyses of
the ~580 Hz brightness oscillations during bursts from 4U 1636-536. We
investigate the bursts as a whole and, in more detail, the initial portions of
the bursts. We analyze the ~580 Hz oscillations in the initial 0.75 seconds of
the five bursts that were used in a previous search for a brightness
oscillation at the expected ~290 Hz spin frequency, and find that if the same
frequency model describes all five bursts there is insufficient data to require
more than a constant frequency or, possibly, a frequency plus a frequency
derivative. Therefore, although it is appropriate to use an arbitrarily
complicated model of the ~580 Hz oscillations to generate a candidate waveform
for the ~290 Hz oscillations, models with more than two parameters are not
required by the data. For the bursts as a whole we show that the
characteristics of the brightness oscillations vary greatly from burst to
burst. We find, however, that in at least one of the bursts, and possibly in
three of the four that have strong brightness oscillations throughout the
burst, the oscillation frequency reaches a maximum several seconds into the
burst and then decreases. This behavior has not been reported previously for
burst brightness oscillations, and it poses a challenge to the standard burning
layer expansion explanation for the frequency changes.
