Question

An open organ pipe and a closed organ pipe have the frequency of their first overtone identical. The ratio of length of open pipe to that of closed pipe is

• A. 3:4
• B. 1:2
• C. 2:1
• D. 4:3

Hint:

The first overtone of an open pipe corresponds to the second harmonic, while for a closed pipe, it corresponds to the third harmonic. The ratio of their lengths is determined by these harmonic relationships.

Answer 1

The Correct Option is D

Step 1: Understanding the overtone frequencies.
The frequency of the first overtone for an open pipe is \( f_2 = 2f_1 \), where \( f_1 \) is the fundamental frequency, and for a closed pipe, the first overtone is the third harmonic, \( f_2 = 3f_1 \).
Step 2: Frequency and length relationship.
The frequency of an organ pipe is inversely proportional to its length. So, for the same overtone frequencies, the ratio of the lengths of the pipes is given by: \[ \frac{L_{\text{open}}}{L_{\text{closed}}} = \frac{4}{3} \]
Step 3: Conclusion.
The ratio of the lengths of the open and closed pipes is \( 4:3 \), so the correct answer is (D).