Doors made specifically for blocking sound typically utilize 1 2 inch thick particle board sound board an interior layer of lead and integral interlocking thresholds and sweeps.
Sound insulation under door.
The thicker the door the better it reduces noise transfer.
Just be aware that replacing a 1 inch door with a thicker one will involve some carpentry modifications to the door jambs.
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As with the front door gaps under interior doors let a lot of sound through as though the door is partially open.
The key is to make sure that the fabric you use is thick and multilayered the more layers the better as this helps muffle sound waves you could even get a consider getting a specially made acoustic curtain.
Using curtains for soundproofing a door works in very much the same way as it does for insulating a door.
Though most interior doors are 1 3 8 inch thick exterior doors are typically 1 3 4 inch thick.
Fill the gaps with a draft stopper either purchased or homemade.
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You can actually go one step further than a solid core door and buy a sound blocking door.