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Is a resonant frequency any of the harmonics?

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  1. Nov 29, 2014 #1
    1. The problem statement,λ all variables and given/known data
    A drum skin is stretched over one end of a pipe, creating a resonant air column with one open end and one fixed end. How long must the pipe be to achieve a resonant frequency of 280.0 Hz? (Use 343 m/s for the speed of sound.)

    2. Relevant equations
    V = fλ

    3. The attempt at a solution
    My assumption is that a resonant air column means that any of the harmonics for a fixed and open end could be used in order to solve for pipe length.

    1. Solve for wavelength:
    V = fλ
    343/280 = λ
    1.225m = λ

    2. Solve for "pipe length"
    Knowing that the first harmonic is a 1/4 λ for a fixed and free end system, I divided 1.225 by 4 to get ~0.30m, which the book states is correct. But if that works for a resonant air column, the rest of the harmonics should as well. So I multiplied 1.225 by 3/4 to get ~0.92m. The book also states this is correct, but these are the only two lengths the book shows

    However, could I not keep going to other harmonics to get greater and greater lengths? For example, the 3rd harmonic, I multiply 1.225 by 5/4 to get ~1.5m, which the book does not list. But is this not also correct?

    Thanks for your help
     
  2. jcsd
  3. Nov 29, 2014 #2

    haruspex

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    I agree with you. Or you could read the question as "how long does the pipe need to be", suggesting a minimum length.
     
  4. Nov 29, 2014 #3
    Thanks for your help! :)
     
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