Once you've selected a location for the partition wall, marked and measured for the wall, and framed the partition wall by using 2x4-inch pieces of lumber in a staggered configuration, you will need to seal the edges of the partition wall with an acoustic ceiling and weave soundproof blanket insulation between the 2x4-inch studs to help dampen or lessen the noise that is able to pass through the partition wall.
Step 1 - Caulk the Lumber Along the Adjacent Support Wall
Use the caulk gun and acoustic sealant to create a seal that will prevent sound from coming in between the cracks of the adjacent support wall and your new soundproof partition.
When shopping for acoustic sealant, try to find a type that is as close a match as possible to the color of your existing walls. When using the caulk gun, try to create as small a caulk bead as possible while filling the gaps between the existing adjacent support wall and the support brace on the new partition wall frame. Apply acoustic sealant to both sides of the piece of 2x6 inch lumber.
Step 2 - Apply Acoustic Sealant Along the Top Plate and Ceiling Line
You should also apply acoustic sealant with the caulk gun along the 2x6 inch piece of lumber that is used for your first top plate. Again, don't apply too much acoustic sealant, just apply enough to fill any gaps that may be caused by warps or bowing in the lumber.
Step 3 - Weave the Soundproof Blanket Insulation between the Studs
When weaving the soundproof blanket insulation in between the 2x4 inch studs, it is important to take your time and consider where the drywall will be installed later. Before you begin weaving the soundproof blanket insulation, take a few minutes to visualize how the drywall will be applied. For example, when you later apply drywall to one side of the partition wall, you will only nail or screw into the 2x4-inch studs that are even with the face of the 2x6-inch frames that you are facing.
This means that you will need to pass the soundproof blanket insulation behind the studs that are edged on the face of the frame that you're facing, and in front of the studs where the drywall will be installed on the opposite side of the partition.
When weaving the soundproof blanket insulation between the studs, don't pack it too tightly. This type of installation works best when it is loose-fitting and slightly crumpled. You should only use nails or staples to attach the blanket insulation to the studs if the insulation can not stand on its own. Otherwise, allow the soundproof blanket insulation to simply stand edge to edge in between the spaces in the studs.