Question

Second harmonic of an open pipe and third harmonic of a closed pipe of length \(75\text{ cm}\) produce \(n\) beats per second. If the fundamental frequency of the open pipe is \(167\text{ Hz}\), then the value of \(n\) is (Speed of sound in air \(=340\text{ ms}^{-1}\)):

• A. 6
• B. 9
• C. 8
• D. 4

Hint:

For a closed pipe only odd harmonics are present. The third harmonic is three times the fundamental frequency.

Answer 1

The Correct Option is A

Concept: For an open organ pipe, \[ f_n=nf_1 \] For a closed organ pipe, \[ f_n=(2n-1)f_1 \] Beat frequency is \[ n=|f_a-f_b| \]

Step 1: Find the second harmonic of the open pipe. Given fundamental frequency \[ f_1=167\text{ Hz} \] Therefore, \[ f_{\text{open}}=2f_1 \] \[ =2(167) \] \[ =334\text{ Hz} \]

Step 2: Find the third harmonic of the closed pipe. Length of closed pipe \[ L=75\text{ cm}=0.75\text{ m} \] Fundamental frequency of closed pipe: \[ f_1=\frac{v}{4L} \] \[ =\frac{340}{4\times0.75} \] \[ =\frac{340}{3} \] \[ =113.33\text{ Hz} \] Third harmonic: \[ f_{\text{closed}}=3f_1 \] \[ =3(113.33) \] \[ =340\text{ Hz} \]

Step 3: Calculate beat frequency. \[ n=|340-334| \] \[ n=6 \] \[ \boxed{n=6} \]