  Using a variational approach, we present the full solutions of the simplified
one-dimensional two-fluid hydrodynamic equations for a unitary Fermi gas
trapped in a highly elongated harmonic potential, which is recently derived by
Stringari and co-workers {[}Phys. Rev. Lett. \textbf{105}, 150402 (2010){]}. We
calculate the discretized mode frequencies of first and second sound along the
weak axial trapping potential, as a function of temperature and the form of
superfluid density. We show that the density fluctuations in second sound
modes, due to their coupling to first sound modes, are large enough to be
measured in current experimental setups such as that exploited by Tey
\textit{et al}. at the University of Innsbruck {[}Phys. Rev. Lett.
\textbf{110}, 055303 (2013){]}. Owing to the sensitivity of second sounds on
the form of superfluid density, the high precision of the measured second sound
frequencies may provide us a promising way to accurately determine the
superfluid density of a unitary Fermi gas, which so far remains elusive.
