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3rd harmonic of a column of air with one end enclosed

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  1. Mar 16, 2019 at 11:15 PM #1
    1. The problem statement, all variables and given/known data
    [​IMG]
    https://imgur.com/lGas78X
    lGas78X.jpg
    The solution to this question says 450Hz. However, when I attempted to compute the frequency using the wave equation and find the normal mode solutions, I get 750Hz

    2. Relevant equations

    I suspect that the solution could be wrong, is that the case?

    3. The attempt at a solution

    ## v^2 \frac {\partial^2 \psi} {\partial x^2} = \frac {\partial^2 \psi} {\partial t^2}##
    ##\omega = v k##
    Boundary conditions:
    ##\psi (0, t) = 0## and ## \frac {\partial \psi} {\partial x} (57 \times 10^{-2}, t) = 0##
    Normal mode solution should look like:
    ##\psi (x, t) = A sin(kx)cos(\omega t - \phi)##
    ## \frac {\partial \psi} {\partial x} (57 \times 10^{-2}, t)
    = A cos(57 \times 10^{-2} k ) cos(\omega t - \phi) = 0##
    ## 57 \times 10^{-2} k = \frac {2n-1} {2} \pi ##
    ## k = \frac {2n-1} {2 \times 57 \times 10^{-2}} \pi##
    Plug in ##n=3## and using ##\omega = 2 \pi f##
    ##\omega = \frac {5} {2 \times 57 \times 10^{-2}} \pi \times 342##
    ##f = \frac {\omega} {2 \pi} = 750 Hz ##
    From my working, for the frequency to be 450, it had to be the 2nd harmonic, not the third.
     
  2. jcsd
  3. Mar 17, 2019 at 12:19 AM #2

    haruspex

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    I agree.
     
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