- #1
Miles123K
- 57
- 2
Homework Statement
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. Homework 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.