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Homework Help: Walking away from two loudspeakers-destructive interference

  1. Jan 29, 2010 #1
    1. The problem statement, all variables and given/known data

    You are standing 2.5 m directly in front of one of the two loudspeakers shown in the figure. They are 3.0 m apart and both are playing a 686 Hz tone in phase. As you begin to walk directly away from the speaker, at what distances from the speaker do you hear a minimum sound intensity? The room temperature is 20 degrees Celsius.

    2. Relevant equations
    v=343 m/s at 20 degrees
    destructive interference: l1-l2=(n+1/2)lambda where n=0, 1, 2...
    v=f*lambda

    3. The attempt at a solution
    First I found the wavelength which is 0.5 m. Then I drew a right angle triangle, where the distance between the speakers is 3 m and the other side is 2.5 m. Then I determined the hypotenuse, which is 3.91 m. For destructive interference, the sounds from both speakers should be out of phase and this happens when the person is a distance of (n+1/2)lambda away from one speaker and a distance of lambda away from the other (I think). So then I took the value 3.91, and added lambda/2 to get 4.16 for the hypoteneuse and solved for the other side to get 2.9 m. I did this another two times by adding 3/2 lambda and then 5/2 lambda to 3.91 (since the question is asking for the third value...this isn't explicitly stated, but I tried entering my answer with only 1 or 2 distances but I got a message saying I need to enter more values. When I entered 3 of them, a message came up saying I should re-check my calculations). I was wondering if there is something wrong with my logic.
     
  2. jcsd
  3. Jan 29, 2010 #2

    Doc Al

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    OK. At that initial position, what's the path length difference between the sound coming from each speaker? That's the key. What happens to that path difference as you move?
    That's much too restrictive a criteria. To get destructive interference the sounds must be out of phase--all that's required for that is to have the path length difference be (n+1/2) lambda. (What's the first value of n that you reach as you walk away?)
     
  4. Jan 29, 2010 #3
    So at the initial position, one speaker is 3.91 m away while the other is 2.5 m away, right? So the difference would be 3.91-2.5 which is 1.41. As you move, the path length difference decreases.

    The first value of n that you reach is 3, right?
     
  5. Jan 29, 2010 #4
    I tried something but I'm not too sure whether it makes sense.
    using L1-L2=(n+1/2)lambda, I plugged in 1.41 for L1-L2 and solved for n since we know lambda is 0.5. By doing this I got n=2.32...would this confirm that the first value of n that you reach as you are walking away is 3?
     
  6. Jan 29, 2010 #5

    Doc Al

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    Sounds good.

    How did you determine that?
     
  7. Jan 29, 2010 #6

    Doc Al

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    Remember that the path difference decreases as you walk away.
     
  8. Jan 29, 2010 #7
    I think I understand what you're saying...n should be 2, right?
     
  9. Jan 29, 2010 #8

    Doc Al

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    Right. Now figure out the distance you need to be to make that happen.
     
  10. Jan 30, 2010 #9
    I see. So now I'm trying to solve for the distance...I tried doing this by first solving for the path difference by using L1-L2=(n+1/2)lambda, since we know n=2 and lambda=0.5. When I did this, I got the path difference as 1.25. I am now trying to solve for L2 (the distance from the speaker you are standing directly in front of). I tried to find a relationship between the sides using the Pythagorean theorem (since it is a right angle triangle) but nothing is working out. Is this the right approach? I can't think of any other way to find the distance without determining the path difference first.
     
  11. Jan 30, 2010 #10

    Doc Al

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    That's the right approach. Do this: Call the distance X. What's the hypotenuse in terms of X? What's the path difference? Set up and then solve the equation for the path difference.
     
  12. Jan 30, 2010 #11
    Ok, I solved it the way you said and everything worked out :) The only problem is that they're asking for three values...how would I find the other 2?
     
  13. Jan 30, 2010 #12
    I just thought of something...would I use n=1 and then n=0 to find the other 2 values?
     
  14. Jan 30, 2010 #13

    Doc Al

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    Now you're cooking. :approve: :wink:
     
  15. Jan 30, 2010 #14
    Thanks for all the help! :)
     
  16. Feb 7, 2010 #15
    I am working on the same problem, I don't understand what to do for post #10 and beyond. Can someone please help me!
     
  17. Feb 8, 2010 #16

    Doc Al

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    Start by expressing the path difference in terms of X as described in post #10. Use the Pythagorean theorem.
     
  18. Nov 26, 2010 #17
    okay for the same question how would we solve "Suppose the speakers are reconnected so that the 686 Hz sounds they emit are exactly out of phase. At what positions are the intensity maximum and minimum now?"
     
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