(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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

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