  In this letter we develop a theory for the first and second sound in a
two-dimensional atomic superfluid across the superfluid transition based on the
dynamic Koterlitz-Thouless theory. We employ a set of modified two-fluid
hydrodynamic equations which incorporate the dynamics of the quantised
vortices, rather than the conventional ones for a three-dimensional superfluid.
As far as the sound dispersion equation is concerned, the modification is
essentially equivalent to replacing the static superfluid density with a
frequency dependent one, renormalised by the frequency dependent "dielectric
constant" of the vortices. This theory has two direct consequences. First,
because the renormalised superfluid density at finite frequencies does not
display discontinuity across the superfluid transition, in contrast to the
static superfluid density, the sound velocities vary smoothly across the
transition. Second, the theory includes dissipation due to free vortices, and
thus naturally describes the sound-to-diffusion crossover for the second sound
in the normal phase. With only one fitting parameter, our theory gives a
perfect agreement with the experimental measurements of sound velocities across
the transition, as well as the quality factor in the vicinity of the
transition. The predictions from this theory can be further verified by future
experiments.
