Jack, bows with limbs of different cross sections do not produce different energy. Energy is not produced by a bow. The energy is produced by the archer when he draws (applies force to) the bow and gives the limbs potential energy. What limb cross-section changes is the efficiency by which the limb's potential energy is transferred to the arrow as kinetic energy.
And Tom (assuming this is your real name), I think I understand your post after reading it a few times and doing something else for awhile. First you assert that there is a flaw in the equation I used because it doesn't account for the bow being braced. I disagree. I didn't make up this equation, I got it off a physics website that listed different potential energy equations. And I think comparing two braced bows is also perfectly valid. They're both likes springs at rest, even when braced. The equation I gave was developed for springs, and bows behave like springs in the way they generate elastic potential energy. Bows have resting lengths, even when braced.
In my second post I said specifically that you could adapt the equation for an unstrung bow, but to compare bows with the same resting lengths you'd need to have both with tips in the same place (in front of or behind the handle) when the bows are unstrung. So to compare two similar bows so that the only thing that differs is the fact that one is deflexed, you'd have to reflex the tips of the deflexed bow so that the tips are in the same place as those of a non-deflexed bow. I thought I made this clear, but maybe not. If you applied this equation to two unstrung bows, one deflexed (without reflexed tips) and one that is reflexed, of course they'll generate different amounts of potential energy when strung and drawn, because their resting lengths will be different. You aren't holding everything save the presence of a deflex in the handle constant.
Tom, you say, "half of the energy imparted to an arrow occurs in the last 2/3rds of the time it's on the string". This is true enough, but look at an F/D curve. Assuming you use D (draw length) on the horizontal axis and the 'curve' is actually linear starting at 6 inches on the horizontal axis and heading to higher force and draw weights at a diagonal, the incremental area under each inch of the triangle is small at lower draw lengths and increases at higher draw lengths. This means the bow is easier to draw early on (the archer needs to apply less force), and also stores less energy. If you were to release an arrow from an underdrawn bow, it would be given a very small amount of energy relative to if it had been fully drawn. Most of us should agree on this.
On the other hand, when you fully draw the bow, that first inch of travel that the string moves through after release imparts a huge amount of energy into the arrow, and the incremental amount of energy given to the arrow for each further inch of movement diminishes as the string inches back towards brace height. If, for the sake of easy math, we assume that the F/D curve is linear with a positive slope, then 3/4 of the total energy stored in the limbs has already been released into the arrow by the time it passes the halfway point back to brace height. So, Tom and Gordan, as I said, "most of the arrow's energy will be given to it very soon after release." This doesn't break any physical laws, and the arrow doesn't have to accelerate by itself. If you look at an F/D curve and read it from right to left to interpret how energy flows from limbs to arrow through the release process, this shouldn't be a controversial statement.
So while the arrow is being given energy from the time you first release the string to the point at which the arrow leaves the string, the incremental energy has fallen off so much as the string approaches brace height that I don't think it does much to the momentum and velocity of the arrow. The early stages of release for the most part define the ending velocity, while that last half just fine-tunes it.
And Badger, see my above clarification about resting lengths for two bows you want to compare.
Take care folks,
-Eric
