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A muzzle shroud is a sleeve (either circular or otherwise) that extends beyond a weapon's muzzle. It is similar to a barrel shroud with the difference that it only surrounds the far end of the barrel known as a muzzle. It can be a muzzle extension or muzzle device.
Essentially, it could be considered a flash suppressor without circumferential openings.
Sound waves are semi-directional. They consist of both Point- and Line-wave propagation. The projectile emanates mostly line-propagation properties after leaving the barrel. At the moment that the projectile is exiting the barrel it produces a mainly point-biased wave generation. (This is largely due to the shock waves traveling forward along the interior of the barrel, thus preventing any backward travel from the projectile into the barrel.) Sound waves travel in an ever-expanding bubble emanating from the muzzle when the propellant gases are traveling at supersonic speeds. This is called the "muzzle blast." Except in extremely low pressure loads, the exiting propellant gasses generate much higher sound pressure levels (dBa) than any other aspect of the weapon firing. (e.g., hammer, sear, bolt, supersonic flight signature, etc.) The supersonic "ballistic crack" is typically 136-138 dB, whereas "muzzle blast" may reach 168 dB. By encircling the muzzle, the shroud funnels and focuses the sound waves along the projectile flight axis, away from the shooter. This alone can partially protect the shooter from 12-18 dB of blast returning to the shooter's ears.
In the United States, the ATF has ruled that since the sound waves are not dampened and are metered at the same levels at two measuring points (1 m (3.3 ft) to the left and 0 m in front of the muzzle and 1 m (3.3 ft) left and 5 m (16 ft) forward of the muzzle) this is not a sound suppressing device. The only action which a muzzle shroud performs is shielding the area directly behind the muzzle from the worst blast pressures. Muzzle brakes direct the escaping gases rearward toward the shooter, thus increasing blast pressure directed toward the operator's ears. The same principle that lessens felt recoil also acts to enhance the sound levels reaching the shooter. Some of the muzzle brakes designed for heavy-caliber rifles (.338 Lapua, .408 CheyTac, .50 BMG) incorporate "blast shields" projecting laterally from the sides of the muzzle crown. Thus, reflecting sound waves traveling rearward via the muzzle brake blast chambers. These small leaves, however, do not fully enclose the muzzle diameter, and so block only the worst blast path, while allowing the majority of escaping gases to circumvent the shield by traveling around them, still reaching the shooter's ears, causing unacceptable hearing damage.