actually buck, there is no one product that will completely deaden sound. why you ask? well ill use some quotes and ellaborate where need be to explain...."Damper – by far the most common type of material. It uses mass loading to lower panel resonance and absorb structural vibrations, converting them into low level heat." they are talkin about asphalt here, thats what it does... it absorbs sound/vibration and turns it into heat (on a very minute scale), next we got "Absorber – as the name suggests, these products literally absorb airborne sound waves. The sound waves have to propagate through the absorber and the open/closed cell foam will slow it’s progress reducing the overall level in the process." this is a foam, like you were sayin. and last "Barrier – as the name suggest, these products form a barrier or wall, which blocks airborne noise from passing through. The denser the barrier, the more effective it is, which makes lead an excellent choice for a barrier" and this is the lead vinyl stuff i was talkin about. its stretchy kinda heavy rubber cloth.
so what that all means is that if you wanna truly deaden sound, and you are bent on doin a home job, then a layer of asphalt sheeting first to reduce case vibration. then a layer of the lead fabric to physicaly stop/block the sound waves, then a layer of like neoprene or some other dense foam for slowing/blocking initial sound, all very thin layers,.