Question 465: Waves

Question

Two open pipes of different lengths and same diameter in which the air column vibrates with fundamental frequencies ' $n_1$ ', and ' $n_2$ ' respectively. When both pipes are joined to form a single pipe, its fundamental frequency will be

Accepted Answer

The fundamental frequency of an open organ pipe of length $ l $ is given by the formula:
$ n = \frac{v}{2l} $
where $ v $ is the velocity of sound in the pipe.
For an open pipe with frequency $ n_1 $ and length $ l_1 $, we have:
$ n_1 = \frac{v}{2l_1} \Rightarrow l_1 = \frac{v}{2n_1} \quad 	ext{(Equation 1)} $
Similarly, for an open pipe with frequency $ n_2 $ and length $ l_2 $:
$ n_2 = \frac{v}{2l_2} \Rightarrow l_2 = \frac{v}{2n_2} \quad 	ext{(Equation 2)} $
When both pipes are joined to form a single pipe, the total length $ L $ is:
$ L = l_1 + l_2 \quad 	ext{(Equation 3)} $
Let the fundamental frequency of the joined pipe be $ N_0 $. Therefore, we can write:
$ L = \frac{v}{2N_0} = l_1 + l_2 $
Substituting from Equations 1 and 2, we have:
$ \frac{v}{2N_0} = \frac{v}{2n_1} + \frac{v}{2n_2} $
By solving the equation above, we find:
$ N_0 = \frac{n_1 n_2}{n_1 + n_2} $

Answer

$\frac{n_1+n_2}{n_1 n_2}$

Answer

$\frac{n_1 n_2}{2 n_2+n_1}$

Answer

$\frac{2 n_2+n_1}{n_1 n_2}$

Answer

$\frac{n_1 n_2}{n_1+n_2}$

Waves

Two open pipes of different lengths and same diameter in which the air column vibrates with fundamental frequencies ' n_1 ', and ' n_2 ' respectively. When both pipes are joined to form a single pipe, its fundamental frequency will be

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