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My theory on the second (or more) round ringing the barrel is that the second bullet traps air between it and the first bullet. It compresses that air as it travels, until it reaches the base of the first bullet. At that point, the trapped air has compressed into zero space, and the pressure is immense. This pressure is highly concentrated at the point of the eventual ring. The barrel "gives" slightly, until the pressure drops below the yield strength of the steel, and then the diameter of the barrel rebounds until it reaches its new, larger, diameter in the area of the ring.
This is the only reason I can think of for the ring to form, since all other possibilities seem to spread the pressure over an area larger than the ring.
This is the only reason I can think of for the ring to form, since all other possibilities seem to spread the pressure over an area larger than the ring.