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Q. If the length of an open organ pipe is $33.3\, cm$, then the frequency of fifth overtone is $(v_{sound} =333\, ms^{-1})$

MHT CETMHT CET 2005Electromagnetic Waves

Solution:

An open organ pipe is open at both ends.
Wavelength of vibration in open organ pipe is given by
$\lambda = \frac{2L}{n},$ where $n = 1, 2, 3$, ...
Let $\lambda_1$ be the wavelength of stationary waves set up in the open organ pipe corresponding to $n = 1$.
$\lambda_1 = \frac{2L}{1} $ or $L = \frac{\lambda_1}{2}$
The frequency of vibration in this mode is given by
$v_1 = \frac{v}{\lambda_1} = \frac{v}{2L}$
or $v_1 = \frac{v}{2L}$
This is the lowest frequency of vibration and is called the fundamental frequency. The note or sound of this frequency is called fundamental note or first harmonic.
Similarly for
$n = 2, v_2 =\frac{2v}{2L} = 2v_1$
For $n = 3, v_3 =\frac{3v}{2L} = 3v_1$
In general, the frequency of vibration in nth normal mode of vibration in open organ pipe would be
$v_n = nv_1 $
Hence, frequency of fifth overtone
$= 6 \times $ fundamental frequency
$ = 6 \times \frac{v}{2L}$
$= \frac{6 \times 333}{2 \times 33.3 \times 10^{-2}} = 3000\, Hz$