  Low mass X-ray binaries (LMXBs) show strong variability over a broad range of
time scales. The analysis of this variability, in particular of the
quasi-periodic oscillations (QPO), is key to understanding the properties of
the innermost regions of the accretion flow in these systems. We present a
time-dependent Comptonisation model that fits the energy-dependent
rms-amplitude and phase-lag spectra of low-frequency QPOs in black-hole (BH)
LMXBs. We model the accretion disc as a multi-temperature blackbody source
emitting soft photons which are then Compton up-scattered in a spherical
corona, including feedback of Comptonised photons that return to the disc. We
compare our results with those obtained with a model in which the seed-photons
source is a spherical blackbody: at low energies the time-averaged, rms and
phase-lag spectra are smoother for the disk-blackbody than for a blackbody,
while at high energies both models give similar spectra. In general, we find
that the rms increases with energy, the slope of the phase-lag spectrum depends
strongly on the feedback, while the minimum-lag energy is correlated with the
disc temperature. We fit the model to a 4.45-Hz type-B QPO in the BH LMXB MAXI
J1438-630 and find statistically-better fits and more compatible parameters
with the steady-state spectrum than those obtained with a blackbody
seed-photons source. Furthermore, we successfully apply the model to the type-C
QPO in the BH LMXB GRS 1915+105, and thus conclude that this
variable-Comptonisation model reproduces the rms and phase-lags of both type B
and C low-frequency QPOs in BH LMXBs.
