Constructive and destructive wave interference

[kathyt.25]

EQUATIONS USED
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!

 

[Chewy0087]

The problem you're having is confusing distance away & phase difference, for example, in (1), they may be 1/2 a wavelength AWAY from each other, but the actual conditions for these equations is where the waves MEET if the interference is 1/2...etc, do you see what that means? And does that answer your question?

 

[kathyt.25]

The problem you're having is confusing distance away & phase difference, for example, in (1), they may be 1/2 a wavelength AWAY from each other, but the actual conditions for these equations is where the waves MEET if the interference is 1/2...etc, do you see what that means? And does that answer your question?

I'm not sure if I understand what you mean by where the waves meet. And what's the difference between "phase difference" and "wavelength"?

Also, how do I know what to set "n" as in the constructive/destructive interference condition equations?

 

[Chewy0087]

Hmm, phase difference is the difference between where each waves are in their cycle, for example two waves A & B, if they both have the same frequency & wavelength, and one wave is at the bottom, and the other is at the top of the cycle they're said to be 1/2 a wavelength out of phase.

Now look at your example in (1), you see that both waves go from middle - top at the same time, in their first wave, these waves are considered to have a phase difference of 0, because they are essentially exactly the same in opposite directions. They will peak together & move together. Similiarly, if you imagine the red speaker was 1 whole wavelegnth further back, the distance between them would be 1 1/2 wavelegnths, however they would still be in phase with one another. So you see the distance between them is irrelevant, it's only the phase (or path) difference that counts.