Proper acoustic waves associated with the Rankine vortex of low Mach numbers are studied by solving the compressible Navier-Stokes equations and the continuity equation in cylindrical coordinate system. The vortex Mach number (VMN) is introduced as the ratio of the maximum swirling speed to the sound speed. The density modulation is expressed in terms of the cylindrical functions, which, in contrast to the plane waves, do generally not keep the mass of the fluid invariant. Axisymmetric solutions subjected to mass conservation condition (MCC) are surveyed at low VMN and are shown to exist. The consequence of the MCC is that the wave numbers are complex and discretized in accordance with the distribution of the zeros of the first derivative of the Hankel function. Consequently, the generated sound propagates as a wave packet. The wave length and the width of the wave packet increases and decreases, respectively, with the VMN.
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