I'm trying to figure out how I can possibly know the real equation of the intensity of the diffraction pattern of a double slit (I don't care about interference for now). Here are my thoughts:
Textbook say the intensity is $D = (\sin(\alpha)/\alpha)^2$ with $\alpha = kd\sin x$ for both single and double slit. (for double slit, you just multiply with the interference pattern, but the envelope stays the same).
Now, if we put both slits far apart, we should see two bumps in the diffraction pattern but I can't get those bumps with that equation. What I did is change the $x$ in the diffraction for $x + \Delta$ and then add the two patterns with the correct spacing. The problem with that method is if I bring the two slits back close together, I don't get the same result as before which proves me wrong.
I understand that the equation I use have some constraint and I might not fit in, but any ideas on how I can solve the problem? I have done the experiment and I can really see a shift in the pattern if I close on of the two slit, but can't figure out how the get the theory about it.
Here is an example of the 2 bumps I'm talking about: In orange and green are the two pattern. Red is the sum of both with the two bums and blue is the theory equation I gave earlier.
After reducing the distance (and dividing by two the intensity) I get: