It is well known that the pressure fluctuations on both sides of a cylinder and those in its oscillating near-wake region are both sound sources at low Reynolds and Mach numbers. However, assessment of the propagating capacity and quantification of the radiating versus non-radiating components of these two sound sources are not currently available for this important benchmark aeroacoustic problem. Here, we isolate the radiating acoustic sound sources from the non-radiating hydrodynamic pseudo-sounds by applying the wavelet decomposition technique of Mancinelli et al. (J. Fluid Mech., vol. 813, 2017), previously used in subsonic jet-noise experiments, to decompose the cylinder near-field pressure fluctuations obtained from our direct numerical simulations. Rigorous independence and convergence analyses of the wavelet decomposition procedure are performed. It is found that the radiating acoustic component strongly dominates over the non-radiating hydrodynamic component at near-field locations above and upstream of the cylinder. In the oscillating near-wake region, the hydrodynamic component dominates over the acoustic component at most frequencies, except at the vortex shedding frequency where they exhibit comparable strengths. Furthermore, within the oscillating near-wake region, the overall sound pressure level associated with the hydrodynamic pressure fluctuations exceeds that associated with the acoustic pressure fluctuations. Away from the oscillating near-wake region, the hydrodynamic pressure fluctuations decrease dramatically while the acoustic counterparts decay slowly, demonstrating that the hydrodynamic pressure fluctuation does not propagate, and that the acoustic pressure fluctuation is the only component to propagate to the far field.