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Using a variational approach, we present the full solutions of the simplifiedone-dimensional two-fluid hydrodynamic equations for a unitary Fermi gastrapped in a highly elongated harmonic potential, which is recently derived byStringari and co-workers {[}Phys. Rev. Lett. \textbf{105}, 150402 (2010){]}. Wecalculate the discretized mode frequencies of first and second sound along theweak axial trapping potential, as a function of temperature and the form ofsuperfluid density. We show that the density fluctuations in second soundmodes, due to their coupling to first sound modes, are large enough to bemeasured 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 onthe form of superfluid density, the high precision of the measured second soundfrequencies may provide us a promising way to accurately determine thesuperfluid density of a unitary Fermi gas, which so far remains elusive.

First and second sound of a unitary Fermi gas in highly elongated harmonic traps