# Adding a 3rd Source to a 2 Source Interference Pattern

1. The problem statement, all variables and given/known data
Consider the double slit pattern shown above where the distance between slits is d. Each lettered point is labeled according to ΔD, the difference in distances from the slits to that point. ΔD is expressed in terms of wavelength, λ. Suppose that a third slit were inserted between the two slits such that the distance between adjacent slits becomes d/2. Would each point X,Y,Z be a max, min, or neither?

2. Relevant equations
ΔD=dsinΘ
ΔD=mλ constructive interference
ΔD=λ/n destructive interference, where n= # of slits
ΔD=mλ/n neither constructive nor destructive, where m=0,n,2n,3n,…

3. The attempt at a solution
At first I tried playing around with those equations but I ended up confusing myself. Then I realized that for a fringe to be a principal maximum, all of the light sources must be in phase there. At each point of constructive interference for two sources, S1 and S3 will still be in phase, but each will be out of phase with the new source S2 (halfway between S1 and S3). Similarly, S1 and S3 will interfere destructively at a point of destructive interference for two sources, but both will not be out of phase with S2. So I’m thinking none of the points will be max/min. Is my reasoning/answer correct?

Let me know if my amazing drawing/description are not clear enough and I will try to elaborate.

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