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Constructive and destructive wave interference

  1. Jun 15, 2009 #1
    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

    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.
    (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".

    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!
  2. jcsd
  3. Jun 15, 2009 #2
    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?
  4. Jun 15, 2009 #3
    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?
  5. Jun 15, 2009 #4
    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.
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