DISCLAIMER!
Box jumps can hurt you! Unless you want to be competitive, you should probably stick to step downs whenever permitted and consider step ups when permitted. Back in 2012, I was trying to bound and tore into my shin to the tune of 7 stitches and bone exposure.
BACKSTORY
My box jumps used to suck. A few years ago in the CF open they allowed step ups (you've always been allowed to step down) and we found out that for many folks it actually was faster than jumping up. Then there were the freaks that could bound up and down and so stepping up and down for them was much slower.
So I started to step up all the time from that point forward. Our box programmed a WOD one day and Rx was jumping not stepping. I was pissed off because my box jumping was #1 slower and #2 more tiring. I ended up switching in the middle of the WOD back to step ups and complained to anyone who would listen.
Then I had a shift.....As I wrote in the beginning, if you want to be competitive, you should probably learn how to bound.
METHOD
Around August 1st, I started to bound on the 20 inch box each day for 30 seconds as part of my warm up. Mind you, prior to this I had done double unders, pistols, etc..and some of my usual mobility so I wasn't doing these cold.
I got to where I could get 18 in the 30 seconds. At that point, I moved up to 24 inches. I was getting 13 at the start. Then the other day I did 19. I'm moving up to 30" next week.
TECHNIQUE
1. Shoulder blades (this is the KEY that made all the difference to me)
You cannot use your arms to do bounding box jumps. You CAN use them to box jump and get the initial jump, but the explosiveness is just too fast for you to get your arms forward upon impact if you are bounding. The way that you do it is through the movement in the shoulder blades.
At the top of the movement, standing on the box, set the blades down and back. At this point you drop to the floor. Then on impact as your heels kiss the ground, you initiate the explosion with shoulder blades moving up and apart. Reset them as you are about to drop to the floor again.
2. Corner jumps
I learned this from Invictus. Angle the box so that you are jumping to the corner. It will cut your forward trajectory down so that you are literally jumping more "up" and then bringing the feet together vs. jumping with feet in the same spacing but up and forward. It also cuts down on the catastrophe that can result from a miss.
3. Abdominal tension
This should go without saying, but if you can't maintain tension in the core, then you are not going to have a stable place to brace the impact from and create the rebound force needed to bound back up.