The design of a bow would use both mechanical engineering and materials engineering skills. The materials engineer would be working to change the strength and stiffness and fatigue resistance (how many times you can string the bow before it might get a crack). The mechanical engineer would be figuring out the forces (loads) on the bow and changing the shape or the material to get improved bow response.
I recommend starting to read some of the many magazines about engineering that exist (the Smithsonian has one for Air and Space, and there are many others). You can get an idea from those stories if that is the kind of engineering you are interested in.
I have two favorite parts of my job. One is when we discover something new in the laboratory that no one knew before. The other is when I see students grow and mature in both their interest and expertise in their field .