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

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1. Mar 16, 2019 at 11:15 PM

### Miles123K

1. The problem statement, all variables and given/known data

https://imgur.com/lGas78X

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. Mar 17, 2019 at 12:19 AM

I agree.