  We present a comprehensive study of the properties of nearly coherent
brightness oscillations (NCOs) in a large sample of type-I X-ray bursts
observed from low mass X-ray binary 4U 1728-34. We have analyzed ~545ks of data
of this source obtained with the Rossi X-ray Timing Explorer over a 3 year span
from 1996-1999. The data contain 38 bursts, 16 of which show NCOs. We find no
NCOs present when the inferred mass accretion rate of the system is lowest.
Furthermore, we define a measure of the strength of the oscillations and find
that this strength increases with increasing mass accretion rate. This
correlation is particularly evident within bursts detected only a few weeks
apart, and becomes less clear for bursts separated in time by several months to
years. The correlation we find for 4U 1728-34 between the NCOs and the inferred
accretion rate of the system is similar to that found for KS 1731-260 by Muno
et al. (2000), where the NCOs are only present at relatively high mass
accretion rates. However, unlike the case for KS 1731-260, we find an
anti-correlation in the bursts of 4U 1728-34 between the existence of episodes
of photospheric radius expansion and the inferred mass accretion rate of the
system. Moreover, we distinguish between NCOs present in the rise and in the
decay phase of the burst and find that the bursts with NCOs only in the decay
phase occur at intermediate accretion rates, while those with NCOs in both the
rise and the decay phases are concentrated at high inferred mass accretion
rates. We discuss these results in the context of the theory of thermonuclear
bursts and propose intrinsic differences in the neutron stars in KS 1731-260
and 4U 1728-34 as the origin of the different behaviors.
