  We analyze a recent experiment on a Tonks-Girardeau gas of $^{87}$Rb atoms
(T. Kinoshita, T. Wenger, and D.S. Weiss, Science {\bf 305}, 1125 (2004)). We
find that the experimental data are compatible with the one-dimensional theory
of Lieb, Seiringer and Yngvason (Phys. Rev. Lett. {\bf 91}, 150401 (2003)) but
are better described by a theory that takes into account variations in the
transverse width of the atomic cloud. By using this theory we investigate also
the free axial expansion of the $^{87}$Rb gas in different regimes:
Tonks-Girardeau gas, one-dimensional Bose-Einstein condensate and
three-dimensional Bose-Einstein condensate.
