First slide

Standing waves

Question

An organ pipe P₁, closed at one end vibrating in its first harmonic and another pipe P₂, open at both ends vibrating in its third harmonic are in resonance with a given tuning fork. The ratio of the length of p₁ to that of P₂, is

Moderate

A

8/3
B

3 /8
C

1 /2
D

1 /3

Solution

Let l₁and l₂be the respective lengths of the closed and open organ pipes. The frequency of vibration of a closed pipe in r ^th mode of vibration is given by
$n_{r} = (2 r â 1) \frac{V}{4 l_{1}} For firs t harmo n i c, r = 1 n_{1} = \frac{V}{4 l_{1}} For open pipe, n_{r} = \frac{r V}{2 l_{2}} For third harmonic, = \frac{3 V}{2 l_{2}} \frac{V}{4 l_{1}} = \frac{3 V}{2 l_{2}} or \frac{l_{1}}{l_{2}} = \frac{1}{3}$

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