Pipelines are used widely to convey fluids and gases in the petrochemical, water, and energy sectors. Vibration, noise and fatigue failure of pipelines are serious problems in many engineering situations, especially in ship and ocean engineering areas. In this paper, the acoustic propagation and attenuation characteristics of axisymmetrical eigenmodes in a cylindrical, elastic, fluid-filled pipe are investigated analytically. Pipe equations for n=0 axisymmetrical wave motion are derived, and analytical expressions of wavenumber are obtained for s=1 and s=2 wave, which correspond to a fluid dominated wave and an axial shell wave, respectively. The numerical results for wavenumber of pipe and fluid are obtained and discussed. It shows that the frequency dependent phase velocity of this mode depend strongly upon the shell thickness/radius ratio and the density of the contained fluid. It seems that this analysis method is simple and effective.