This paper presents the validity and limitation of the thin-shell approach for the analysis of elastic wave propagation in a pipe with nonviscous liquid. The phase velocities calculated by the thin-shell approach were compared with those calculated by the thick-cylinder approach. In contrast to the case of the empty pipe, where only two modes were obtained and the first mode was calculated in a limited frequency range, the results for the liquid-filled pipe exhibits a large number of modes due to the large number of branches of the apparent liquid mass. Several of the lowest modes of the waves in a liquid-filled pipe were calculated for various pipe thicknesses in a low frequency range. The thin-shell approach was valid for a reasonable range of pipe thicknesses.