Question

The ratio of frequency of the 3rd mode of vibration for the closed organ pipe and the open pipe of length \( L \)?

• A. 1:2
• B. 2:1
• C. 1:1
• D. 3:2

Hint:

For pipes with different boundary conditions (open vs closed), the frequency of vibration differs. For a closed pipe, only odd harmonics exist, while for an open pipe, all harmonics are present.

Answer 1

The Correct Option is B

The frequencies of vibration for pipes depend on whether they are open or closed. For a closed organ pipe, the frequency for the \( n \)-th mode is given by: \[ f_n = \frac{n v}{4 L} \] Where: - \( n \) is the harmonic number (for the 3rd mode, \( n = 3 \)), - \( v \) is the speed of sound in air, - \( L \) is the length of the pipe. For an open organ pipe, the frequency for the \( n \)-th mode is given by: \[ f_n = \frac{n v}{2 L} \] Now, comparing the frequency for the 3rd mode in both cases: - For the closed pipe, \( f_{\text{closed}} = \frac{3 v}{4 L} \), - For the open pipe, \( f_{\text{open}} = \frac{3 v}{2 L} \). The ratio of the frequencies is: \[ \frac{f_{\text{closed}}}{f_{\text{open}}} = \frac{\frac{3 v}{4 L}}{\frac{3 v}{2 L}} = \frac{1}{2} \] Thus, the ratio of the frequencies is \( 1:2 \), meaning the open pipe's frequency is twice that of the closed pipe for the 3rd mode.