Nice Bevo. I'm seeing the conveyor belt/airplane thread developing. I actually just submitted this to Mythbusters!
Damn it. Recoil increases by a factor of 2, not 4.
Yes, if you use the velocity of only one barrel in the equations, the total increases by two. But, you have to sum the velocities for the equation to be correct. You look like you worked the equations properly, but you left out the doubled, summed, total velocity. To use only the velocity of one barrel discounts the other.
Let's think of it another way. Use the scenario that barrel #1 shoots at 1000 fps and barrel #2 shoots at 1001 fps. Assume both shots leave the barrel under the .003 seconds required (for the average length shotgun barrel) to make the recoil considered one impulse.
What is used as the velocity for the equation? If it's averaged, then you have to average everything that comes out of the barrels; wad, shot, powder, gas, etc. But, the reason that doesn't work is because the individual shot for each barrel is traveling at the 1 fps different speed. It's also possible that the powder charge is slightly different, the wadding weighs slightly different, etc, etc. So, if you average the values of everything then you've just taken the recoil of one barrel completely out of consideration.
The impulse of recoil is the big part here, because if the shots are far enough apart then the shots are considered separate and the total recoil is the sum of the two barrels. The shooter would feel this as two distinct impulses. However, if the shots leave the barrel within the same time frame it produces a single impulse of recoil, which is what would cause the total recoil to increase by a factor of four, because the velocities have to be summed for the single impulse.
The other thing to consider is that it's possible to have charges that aren't exactly the same produce velocities that are the same, due to other external factors such as heat, barrel condition, etc.
That's why the velocity has to double, because if it doesn't you're removing the factors contributed by both barrels.