This article on The Physics of Medieval Archery simplifies away one of the most interesting elements — but the diagram hints at it:
The bow — any bow — is basically a spring. The archer does work on this spring as he draws the bow, storing potential energy in the elastically deformed bowstave. When he releases the string, some of this potential energy is converted into kinetic energy of the arrow, through the action of the tension in the bowstring accelerating the arrow, the arrow leaves the bow at high speed and wings its way towards its target. Its orientation is stabilised by three fletchings at the rear of the arrow.If we draw a graph of the force F needed to draw the arrow back through a distance x, the area under the graph represents the work done on the system and hence the potential energy stored in the bow. If the graph is a straight line through the origin (i.e. the bow behaves like a spring that obeys Hooke’s law), this energy will be equal to Fx/2 (see diagram).
In fact, the graph of F against x is usually a curve, because of the complicated shape of the bowstave (it is thicker in the middle and thinner at the ends) and the fact that the tension in the bowstring does not always pull in the same direction relative to the ends of the bow. We deal with this by introducing an efficiency term e, and writing the total energy stored as eFx/2. While a modern bow made of composite materials can have an efficiency greater than 1, a medieval longbow would have had an efficiency of about 0.9.
So the goal is to maximize the potential energy stored in the bow, which is represented by the area under the curve, but the bow, since it behaves like a simple spring, offers next to no resistance at the beginning of the pull. A hefty longbow with a 100-pound draw does not have a constant 100-pound draw; it culminates with 100 pounds of resistance near the end of the pull.
What we want is something closer to constant resistance, so that with the same peak resistance we can store twice as much energy — because the area under the curve is now a rectangle, not a triangle.
This is what composite, recurve bows do — or almost do. By mixing materials — traditional compound bows are made of wood, horn, and sinew, not just wood — and adding additional curves, these bows provide more resistance earlier in the pull.
Modern bows take all this a step further, by using pulleys with specialized cams to not only even out the resistance but to drop it off dramatically at the very end of the pull, so they can easily be held “loaded” and ready to shoot — which is not practical with a historic longbow.