Question:
What does the recursive bend in the bow actually accomplish?
Answer:
It sure sounds like you're talking about a "recurve bow". You are correct about the physical advantage, but you haven't thought further about why it's an advantage.
Because human strength has a limit, the amount of power available to a bow is limited by how much force the human can pull. With a straight longbow, the pull force starts off low and increases to a maximum when pulled all the way back. With a recurve bow, the pull force starts off high and remains roughly constant all the way back. In both cases, the amount of force when pulled all the way back is the same. However, with a recurve bow and it's near constant level of force, the total amount of energy stored in the bow is greater.
Imagine what happens when the bows are released in slow motion. In both cases, the string starts off pushing the arrow with a lot of force. However, with the longbow the push gets weaker and weaker. With the recurve bow, the push stays strong all the way. As a result, the recurve bow's arrow gets more of a push overall.
