Joe,
Rule of thumb on soundproofing; Doubling the mass OR the air gap will improve the STL of a MAM partition by 3 to 5 db in practice.
When you build the 'new' wall in front of the plastic door, it must be as massive as the exterior walls & you must seal it 360 degrees around it's perimeter. THEN you must build new massive walls INSIDE this. 2" air gap between frames is very common. The interior walls do not need to be decoupled from the floor slab since it is on the earth. The interior ceiling must either sit on the new interior walls or be hung from the structure via resilient hanger like Mason HD. The devil is in the details and this is not all of them.
Cheers,
John
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John H. Brandt
Recording Studio Design/Consulting, Acoustics, & Electronics
Jakarta, Indonesia
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