A soundproofed brick wall should combine rigidity and flexibility. The brick façade gives the wall a solid, rigid surface that sound waves will not penetrate. That very rigidity, however, reflects sound, so any sound waves that get through or come from the interior will bounce, which can actually increase sound. A soundproofed brick wall needs to be backed with a sound-absorbent material.
A thicker wall is more soundproof. Use 2-by-6-inch rather than 2-by-4-inch studs behind a brick façade to reduce sound penetration through the wall. The extra depth provides more space to absorb sound. Insulate the wall cavity with fiberglass batts or other loose sound-absorbing material rather than closed-cell foam, which is more reflective.
Properly backing a brick wall will increase soundproofing. Gypsum wallboard is a great absorber of sound because it is less rigid and absorbs sound. Thicker drywall behind the brick facade will absorb more sound. Use 5/8-inch rather than 1/2-inch, or add a second layer. Increase the soundproofing by installing drywall on furring strips rather than directly to the wood studs, which will create a sound-absorbing air gap.
You can improve the brick soundproofing by making a double-thick façade, with two layers of brick. This will increase construction costs, however, and may not be necessary if you have a good single-layer brick façade. Installing the façade on oriented strand board sheathing with an air gap of at least 1 inch between brick and wood will boost soundproofing.
Picking the right kind of brick for a façade will help soundproofing. You must get a brick rated for exterior use, but brick hardness varies. Choosing a somewhat softer brick will help absorb rather than reflect sound. Used brick is frequently a good choice; it is attractive as well as more soundproof than some newer bricks.