Sound absorption function, in regards to reducing/preventing reverb/reflections, is different from sound transmission blockage. The backing(behind the absorber) is playing a critical part of changing the velocity of the wave so that it can be maximally absorbed. If you, for example, put an absorber in the middle of a room, it won't absorb nearly as low as it can, near a boundary surface.
If you wanted to physically block sound from a path; the the absorber needs it's own solid back and the absorber needs to be physically decoupled from main wall/structure to prevent vibration transmission, and the absorber needs to be significantly larger than the wavelength being absorbed(so that the wave does not simply wrap around the absorber). However, even this is not so simple, as sound is usually radiated over a very wide area and in different directions - and the lower the frequency - the more difficult it will be to deal with. To actually produce substantial isolation, the entire thing needs to be air-tight/enclosed and de-coupled from the main environmental structure, less vibrations transmit throughout and act as a breach. Proper industrial sound isolation rooms have every possible air leak sealed, and the entire room is suspended from the main structure of the building on a big shock absorber. Even in home audio applications, where full isolation/suspension is not possible, all of the walls/ceiling/etc. are hung from special shock absorber brackets. The easiest way to do some decent isolation at home is to build a full surrounding constrained layer wall/ceiling assembly - using drywall, a layer of Green Glue and then another layer of drywall. If you do this in a room with a concrete floor, it will work superbly, otherwise, you also have to deal with the floor, which further complicates things. You still have to make sure no air leaks occur. Then you have to make special accommodation for heating/cooling. Special acoustic isolation baffle systems are typically used to connect to the room to prevent most sound from breaching the HVAC to the rest of the house.
-Chris