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I have some conceptual questions about constructive/destructive interference based on the equations:

∆x= nλ

∆x= (n+1/2)λ

Where n is any integer

MY CONFUSION & EXAMPLE

I don’t understand why the condition to get constructive interference is ∆x= nλ, and the condition to get destructive interference is ∆x= (n+1/2)λ.

I’ll demonstrate why I’m confused using this diagram. The boxes are speakers that generate sound waves. When they’re moved farther apart, they either create destructive/constructive interference, based on how far apart they’re moved, or how many wavelengths are produced within that distance, L.

http://i4.photobucket.com/albums/y111/kathy_felldown/wavelengths.jpg

(I apologize for the messy diagram... waves are hard to draw in Microsoft Paint!)

(1)Both speakers are ½ a wavelength away from each other. They create the same wavelength, both with the same positive amplitudes, so that crests correspond with crests, and they add up -->constructive interference!

(2)Both speakers are 1 wavelength away from each other. They create waves that are opposite to each other – so a crest for one wave corresponds with a trough for another wave, so that they cancel each other out -->destructive interference!

(3)The speakers create waves that add up, not cancel out. Here, they are 3/2 of a wavelength -->constructive interference.

(4)The speakers create waves that cancel each other out. Here, there are 2 wavelengths-->destructive interference.

So you can see from the pattern that I created, that destructive interference is only created when the wavelengths are WHOLE numbers (ie. λ=1,2,3…)

And that constructive interference is only created when the wavelengths are FRACTIONS (ie. λ=1/2, 3/2, 5/2…). You only get constructive interference whenever a speaker is moved at a distance that fits a wavelength that increases by 1λ from λ=1/2, as the "initial condition".

MY MAIN QUESTION

Therefore, what I need explanation for is:

Why the equation/condition for constructive interference is ∆x= nλ, and for destructive interference is ∆x= (n+1/2)λ. Because these defined conditions are completely opposite of the patterns that I created with the speakers, above!

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# Constructive and destructive interference

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