  Characterising the fast variability in black-hole low-mass X-ray binaries
(BHXBs) can help us understand the geometrical and physical nature of the
innermost regions of these sources. Particularly, type-B quasi-periodic
oscillations (QPOs), observed in BHXBs during the soft-intermediate state
(SIMS) of an outburst, are believed to be connected to the ejection of a
relativistic jet. The X-ray spectrum of a source in the SIMS is characterised
by a dominant soft blackbody-like component - associated with the accretion
disc - and a hard component - associated with a Comptonizing region or corona.
Strong type-B QPOs were observed by NICER and AstroSat in GX 339$-$4 during its
2021 outburst. We find that the fractional rms spectrum of the QPO remains
constant at $\sim$1 per cent for energies below $\sim$1.8 keV and then
increases with increasing energy up to $\sim$17 per cent at 20$-$30 keV. We
also find that the lag spectrum is "U-shaped", decreasing from $\sim$1.2 rad at
0.7 keV to 0 rad at $\sim$3.5 keV, and increasing again at higher energies up
to $\sim$0.6 rad at 20$-$30 keV. Using a recently developed time-dependent
Comptonization model, we fit simultaneously the fractional rms and lag spectra
of the QPO and the time-averaged energy spectrum of GX 339$-$4 to constrain the
physical parameters of the region responsible for the variability we observe.
We suggest that the radiative properties of the type-B QPOs observed in GX
339$-$4 can be explained by two physically-connected Comptonizing regions that
interact with the accretion disc via a feedback loop of X-ray photons.
