Why is there Proportionality instead of Equality in Young's Experiment?

[Syrus]

Homework Statement

The picture below shows line 2.14, which is derived based on some of the discussion from the text above (I assume it's fairly obvious the experiment is Young's). My question is, why is there a proportionality instead of an equality? No amplitudes are mentioned for the constituent waves eminating from each of the slits, so I figure this should be an equality?

Also, some lines below, the sentence "Note that we are adding the wave amplitudes here." appears. I don't, however, see any amplitude additions. Is this a mistake for "..adding the wave expressions here"?

Homework Equations

The Attempt at a Solution

 

[jedishrfu]

It looks right. They are adding the wave intensities from each slit to determine the intensity on the screen and they show it from a top down perspective.
the sqrt expressions are the pythagorean distance from a given slit to a point on the screen (notice they take the center between the slits as x=0 and so each slit is +-s/2 from the center.

 

[Syrus]

Yes, I unserstand that much, the only thing I am unsure of is why a proportionality is used as opposed to an equality. Any ideas?

 

[haruspex]

They don't know and don't care what the actual intensities are. because they're assuming only small deviations from the midpoint, the two intensities will be almost the same.