Question

The 3rd overtone of a closed organ pipe is in unison with 3rd overtone of an open pipe. The ratio of the length of the closed pipe to the length of open pipe is

• A. \( \frac{7}{8} \)
• B. \( \frac{4}{3} \)
• C. \( \frac{6}{5} \)
• D. \( \frac{7}{9} \)

Hint:

In harmonic analysis, remember that closed pipes only have odd harmonics, while open pipes have all harmonics.

Answer 1

The Correct Option is A

Step 1: Understanding the overtone relation.
In a closed organ pipe, the harmonics are given by odd integers, and in an open organ pipe, the harmonics are given by all integers. The 3rd overtone for the closed pipe corresponds to the 6th harmonic, and the 3rd overtone for the open pipe corresponds to the 6th harmonic as well.
Step 2: Using the harmonic formula.
For the closed pipe, the wavelength of the 6th harmonic is \( \lambda_{\text{closed}} = \frac{4L}{3} \), and for the open pipe, the wavelength of the 6th harmonic is \( \lambda_{\text{open}} = \frac{2L}{3} \). Thus, the ratio of the lengths of the closed to the open pipe is \( \frac{7}{8} \).
Step 3: Conclusion.
The correct answer is (A), \( \frac{7}{8} \).