There are two different, yet completely equivalent, ways that I look at this problem.
The first, as noted in the comments and in many other places, is to us Huygen's principle. Why does Huygen's principle work? Because the assumed incident wave function is a plane wave. Huygen's principle is one way to create a plane wave, and then select what parts of it continue to propagate. The math then gives the diffraction pattern in a straightforward way.
The second equivalent way is to again start with an incident plane wave. By Fourier transform, this means that the transverse momentum of the wave function contains all frequency components. The slits, or half edge, or whatever else is in the way then serves as a frequency filter, only letting through the frequency components that correspond to the Fourier transform of their shape. Apply that filter in momentum space, and transform back to position space, and it all works out the same.
Now, the second method is more general, since you don't have to start with an ideal plane wave, but can instead model actual sources - you just have to do the FT, apply the transverse momentum filter, and transform back.
As an aside, thinking in terms of transverse momenta also makes looking at diffraction gratings, diffraction patterns in crystals, and many other similar problems much easier. Just be careful - nobody can hear you scream in reciprocal space...
