Adding a 3rd Source to a 2 Source Interference Pattern

[kosovo dave]

http://imageshack.com/a/img834/5921/7iap.jpg

Homework Statement

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?

Homework 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,...

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.

 

[kosovo dave]

Also, sorry for the image size. Is there a way to shrink it or would I have to reupload a smaller image?

 

[Redbelly98]

I agree with your reasoning for points X & Z.

For point Y, the path lengths from S1 and S3 differ by __×λ. Therefore, each of those path lengths differs from the S2 path length by ___×λ.

If you fill in the blanks above, I think you will arrive at a different conclusion for point Y.

Hope that helps!

p.s. I have replaced the image in your original post with a link to it. Unfortunately, our forum software does not provide for resizing of images. You can upload a resized -- and cropped -- drawing if you wish. If you do, I could then edit your 1st post to include the reduced drawing. (A 640-pixels-or-less image width would definitely fit okay here.)

 

[kosovo dave]

It took me awhile to see what you were saying, but Y should have constructive interference right? Since S1 and S3 are separated by 2 wavelengths, S2 is out of phase by one wavelength with each.

 

[Redbelly98]

Yes, correct.