  We study the propagation of the two sound modes in two-dimensional Bose gases
across the Berezinksii-Kosterlitz-Thouless (BKT) transition using
classical-field dynamics, which is motivated by recent measurements of
Christodoulou et al. Nature 594, 191 (2021). Based on the dynamic structure
factor, we identify the two sound modes as the Bogoliubov (B) and the
non-Bogoliubov (NB) sound mode below the transition, and as the diffusive and
the normal sound mode above the transition. The NB sound mode velocity is
higher than the B sound mode velocity, which we refer to as the weak-coupling
regime of the sound modes. We excite the sound modes by driving the system as
in the experiment and by perturbing the density with a step-pulse perturbation,
as a secondary comparison. The driven response depends on the driving strength
and results in higher velocities for the B sound mode at high temperatures near
the transition, compared to the sound results of the dynamic structure factor
and step-pulse excitation. We show that the higher mode velocity has a weak
temperature dependence across the transition, which is consistent with the
experimental observation.
