  We theoretically investigate first and second sound modes of a unitary Fermi
gas trapped in a highly oblate harmonic trap at finite temperatures. Following
the idea by Stringari and co-workers {[}Phys. Rev. Lett. \textbf{105}, 150402
(2010){]}, we argue that these modes can be described by the simplified
two-dimensional two-fluid hydrodynamic equations. Two possible schemes - sound
wave propagation and breathing mode excitation - are considered. We calculate
the sound wave velocities and discretized sound mode frequencies, as a function
of temperature. We find that in both schemes, the coupling between first and
second sound modes is large enough to induce significant density fluctuations,
suggesting that second sound can be directly observed by measuring
\textit{in-situ} density profiles. The frequency of the second sound breathing
mode is found to be highly sensitive to the superfluid density.
