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Sound waves - Destructive Interference

  1. Apr 8, 2013 #1
    1. The problem statement, all variables and given/known data
    Two identical loudspeakers are located at points A & B, 2m apart. The loudspeakers are driven by the same amplifier (coherent and are in the same phase). A small detector is moved out from point B along a line perpendicular to the line connecting A & B. Taking speed of sound in air as 332 m/s, find the frequency below which there will be no position along the line BC at which destructive interference occurs.


    2. Relevant equations



    3. The attempt at a solution
    I am not sure how would I approach this problem. I started with calculating the path difference when the detector is at a distance x from its initial position,
    [tex]\Delta x=\sqrt{4+x^2}-x[/tex]
    For destructive interference,
    [tex]\sqrt{4+x^2}-x=\left(n+\frac{1}{2} \right)\lambda[/tex]
    where ##\lambda## is the wavelength of the wave.

    I don't know how should I proceed from here.

    Any help is appreciated. Thanks!
     

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  3. Apr 8, 2013 #2

    ehild

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    Think:
    What does lowest frequency mean for the wavelength?
    How does the pathlength - difference change if the detector moves along x?

    ehild
     
  4. Apr 8, 2013 #3
    Lowest frequency means highest wavelength. For that should I substitute n=0? And then I will get ##\lambda## as a function of x. So should I find maximum of this function?
     
  5. Apr 8, 2013 #4

    ehild

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    Yes, do it...

    ehild
     
  6. Apr 8, 2013 #5
    I tried that but had no luck. Substituting n=0,
    [tex]\lambda=2(\sqrt{4+x^2}-x)[/tex]
    Differentiating w..r.t x
    [tex]\frac{d\lambda}{dx}=2(\frac{x}{\sqrt{4+x^2}}-1)=0[/tex]
    This equation has no solution for x. :(

    EDIT: Looks like the function has its maximum value at x=0. And this gives me the right answer. Thanks ehild! :smile:
     
  7. Apr 8, 2013 #6

    ehild

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    There is no local maximum, but the function decreases with x, and x≥0. The function takes the maximum at the boundary.

    ehild
     
  8. Apr 8, 2013 #7
    Yes I realised it when I examined the derivative. :)
     
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