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)
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