  We present a comprehensive observational and theoretical analysis of the
amplitudes and profiles of oscillations that occur during thermonuclear X-ray
bursts from weakly-magnetized neutron stars in low mass X-ray binaries. Our
sample contains 59 oscillations from six sources observed with the Rossi X-ray
Timing Explorer. The oscillations that we examined occurred primarily during
the decaying portions of bursts, and lasted for several seconds each. We find
that the oscillations are as large as 15% during the declines of the bursts,
and they appear and disappear due to intrinsic variations in their fractional
amplitudes. However, the maxima in the amplitudes are not related to the
underlying flux in the burst. We derive folded profiles for each oscillation
train to study the pulse morphologies. The mean rms amplitudes of the
oscillations are 5%, although the eclipsing source MXB 1659-298 routinely
produces 10% oscillations in weak bursts. We also produce combined profiles
from all of the oscillations from each source. Using these pulse profiles, we
place upper limits on the fractional amplitudes of harmonic and half-frequency
signals of 0.3% and 0.6%, respectively (95% confidence). We then compare the
pulse morphologies to theoretical profiles from models with one or two
antipodal bright regions on the surface of a rotating neutron star. We find
that if one bright region is present on the star, it must either lie near the
rotational pole or cover nearly half the neutron star in order to be consistent
with the observed lack of harmonic signals. If an antipodal pattern is present,
the hot regions must form very near the rotational equator. We discuss how
these geometric constraints challenge current models for the production of
brightness variations on the surface of a neutron star. (abridged)
