If you were to take a board with uniform dimension throughout its length and bend by putting pressure on both tips and in the middle, the board would bend the most in the middle and hardly at all toward the tips.
Since most of the bend would be concentrated in a small area in the middle, it would be most likely to break in the middle. If you want to bend the piece of wood by the same amount without the wood breaking, you'd have to find a way to distribute the load so that each part of the wood does the same amount of work. That reduces the stress on any one section.
Obviously, the thinner the wood, the easier it bends. If a perfectly parallel piece of wood bends the most in the middle, then to get the wood to bend evenly all along its length, you'd have to have it thickest in the middle and tapered toward the tips.
That's where "roughing it out" comes in. When your bow is finished, it will be tapered from the handle to the tips. It can be tapered only on the belly; it can be tapered only on the sides; or it can be tapered on both the sides and the belly. Since we know already what the basic shape of the bow will be, we can cut out a lot of work by roughing it out. That is, we can taper it to begin with before we even start tillering it.
The question is, what dimensions should we use? Well, every piece of wood is different, and every bow is different. The best way to determine what your initial dimensions should be is to practice and see what works best. But if it is your first bow, it never hurts to have a set of preconcieved dimensions that somebody else has figured out. Keep in mind, though, that these are just guidlines. You don't have to do it like this.
For this build along, I'm going to use a piece of red oak that is 72" long, 1.5" wide, and 3/4" thick.
The first thing you should do is find the exact dead center of the bow. That means you have to find the intersection of the center of the length and the center of the width. Since it's 72" long, you know the center is going to be at 36". But just in case you ever use anything different, there's an easy way to find the center without measuring the length and dividing by two. Put the yard stick flush against one end, and mark the wood at the other. Then do the same thing on the other end. Now you will have two marks that are closer together. Hopefully, they are within 12 inches of each other. Now just put a ruler on them and center the two lines on either side of the ruler. Put a mark at the 6" spot. That's the center of the length.
The center of the width is found by centering the ruler on the width and marking the wood at the 6 inch mark. Now you should have a + sign at the dead center of the bow.
I put a "C" on there since this build along is idiot proof.
It's very important when making a bow that the tips line up perfectly through the handle. It could be that your board is just a little bit warped from side to side. If so, then you can't just center the tips on either end. Doing so will result in tips that are misaligned, the string will be misaligned, the bow will twist, and it will have a better chance of breaking. There's a better way to get the tips lined up perfectly. Just run a string along the length of the bow and hang weights off either end.
I'm using squeeze clamps for weights. Now tweek each end, trying to get the string as close to the middle of the ends as possible while making sure it runs straight through the dead center of the bow.
Now make a mark on each end right under the string. You can use these marks for reference points when you draw your tips.
Now here's the dimensions I'm going to use for this build along.
The area in red will be removed. I'll explain how later. After you remove the red, sort of smooth over that transition so there isn't that sharp turn. Know what I mean? Just sort of round it out.
Remember that these are just guidelines. For example, you don't have to taper from the 15" mark. I've done it anywhere from 12" to all the way back at the handle. Speaking of handles, that's what the 10 inches in the middle are for. I'm going to glue on a 10" riser.
Some people may go a little wider at the tips. That allows you to tweek the tips later in case you need to bring the string closer to the center of the handle. You could do that by shaving wood off one side, making the tip more narrow. Do whatever works for you.