Burke's Barrage is based around a pattern called "cross arm tennis" To learn this, hold two balls in your right/strong hand, & one in the left. Arms crossed, right UNDER left.
Note that balls A & C stay in the right and left hands respectively - ball B is thrown continually from hand to hand over the top of the pattern.
Right - Burke's Barrage is simply cross arm tennis with where balls A & C are flourished each time they are caught and thrown. Describing the flourish is the hard bit - it is a continuation of each clawed catch, and the ball should be carried through 1 1/2 circles before being thrown again in a under-arm throw. To describe this flourish -
Imagine that you're sitting in a car, driving forwards, and that the front wheels are each turned slightly in towards each other. (left wheel points slightly right, and right wheel points slightly left, and yes, the wheels are still turning.)
Ball A should follow the path of the circumference of the left wheel, revolving in the same direction, for 1 1/2 revolutions, while Ball C should follow the right wheel.
Does this make sense?
The upshot is that the flourished balls are never still - they are always in mid-air, or being carried. I don't know whether this makes sense, but for what its worth, that's the Barrage.
According to Ben, Burke's Barrage is "not complex, and after it, Rubenstein's Revenge is a simple extension."
First of all, what "Beyond the Cascade" calls Cross-Arm Tennis, is 95% of Burke's Barrage. All you have to do is make forward circles with each wrist when that hand is crossing on top. If you don't have "BtC", read on. I'll teach you a trick I call the Swoop. Burke's Barrage is just alternating Swoops . A Swoop is basically a reverse chop, except that your hands truly cross, and the next throw is part of the Swoop.
Juggle three balls. Notice how you throw a ball from your right hand and it lands in your left? Suppose your right hand decides to race against its own throw, and zips across under the ball it threw. You can then make your left hand catch and throw from under the right arm. But the right hand doesn't get lazy. It has been making a graceful circle, and zips home in time to continue the regular cascade. That left hand under the arm throw goes basically straight up, and the right hand moves underneath it back to normal cascade position. You can go back to the cascade by letting it exchange in the right, or you can force the issue by making the left grab it. This begins the cycle on the other side, and includes the two throws in a row by the same hand which is part of Burke's Barrage.
To do two in a row, I think it helps to get in the habit of starting a Swoop with an outside (reverse cascade throw). You then have an easier time racing against that throw, and you can aggressively grab the ball you're catching. With this idea, the Swoop goes like this: reverse throw with the right (not too far, sort of aimed toward the center), grab a ball with the right, drag it down and across, make an under arm throw with the left, and circle the right back home. You finish with a ball descending just right of center. Now grab that ball with the left, by making a reverse throw with the left (not too far). (Make sure you've tried the general Swoop on the left alone.) This begins the left hand Swoop.
Alternating Swoops (starting each side with a reverse throw) is Cross-Arm Tennis. (Be sure you are not accidentally doing reverse chops, where there are no two throws in a row by the same hand. If you are, slow down and be sure you are throwing under the arm.) To complete the trick, make graceful circles and do the wrist action.