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Find the correct frequency so two waves are in phase

  1. Mar 7, 2013 #1
    1. The problem statement, all variables and given/known data

    A man is sitting between two speakers, 5 meters from the first and 6.5 meters away from the second.

    Speaker 1~~~~~~(6.5m) ~~~~~~ Man ~~~~ (5m) ~~~~ Speaker 2

    They both create sound waves in phase. What's the lowest frequency for which the man hears both sounds in phase?
    The velocity of sound is 340m/s.

    2. Relevant equations



    3. The attempt at a solution

    We have two speakers. Speaker 1 is positioned at x = 0, creating waves with a phase of
    ϕ1(x,t) = kx - wt
    propagating in the positive direction.
    Speaker 2 is positioned at x = 11.5, emiting waves with a phase
    ϕ2(x,t) = kx + wt
    which propagate in the negative direction.

    The man is sitting at x=6.5.
    We know that both speakers emit in phase, therefore for any given instant t we have:
    ϕ1(0,t) = ϕ2(11.5,t)

    We also want the man to receive both waves in phase, therefore:
    ϕ1(6.5,t) = ϕ2(6.5,t)

    If we solve this equations we'll get:
    ϕ1(0,t) = ϕ2(11.5,t) ⇔ 0*x - w*t = 11.5*k + wt ⇔ 2*wt = -11.5k
    ϕ1(6.5,t) = ϕ2(6.5,t) ⇔ 6.5k - wt = 6.5k + wt ⇔ wt = 0

    The result doesn't make any sense. I've probably assumed something wrongly, but I can't quite figure out what. If anyone could point my logic's holes I'd appreciate.

    Thanks.
    Daniel
     
  2. jcsd
  3. Mar 7, 2013 #2
    Jesus !!
    Such a long solution ?

    Why do you not use,

    Δψ=2π*Δx/λ

    In phase can be when there is constructive interference. In this case, what will be phase difference, Δψ ?
     
  4. Mar 7, 2013 #3
    I'm afraid I did not understand your answer. If Δψ is the phase difference then it will be zero, since both waves must arrive with the same phase. Therefore Δψ = 0 = 2π*Δx/λ , which can't be true.
     
  5. Mar 7, 2013 #4

    haruspex

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    "In phase" only requires that the phase difference is a whole number of cycles.
     
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