Question

Ratio between the frequencies of the third harmonics in the closed organ pipe and open organ pipe of the same length is:

• A. 2:1
• B. 1:2
• C. 1:4
• D. 4:1
• E. 1:5

Hint:

Remember, the terminology in acoustics precisely defines "harmonics" and "overtones." The first overtone is the second harmonic in an open pipe and the third harmonic in a closed pipe.

Answer 1

The Correct Option is B

In acoustics, an open organ pipe can produce both odd and even harmonics, while a closed organ pipe only produces odd harmonics. The third harmonic in an open organ pipe corresponds to three times the fundamental frequency of the pipe. For a closed organ pipe, the fundamental frequency (first harmonic) is \( f \), the third harmonic (since only odd harmonics are possible) is the third multiple of the fundamental frequency, or \( 3f \). 
However, in an open organ pipe of the same length, the fundamental frequency is the same \( f \). The third harmonic for an open pipe would similarly be \( 3f \), but here, it is important to note that the open pipe will support harmonics at every integer multiple of \( f \), so the third harmonic is also \( 3f \). 
Given that both types of pipes produce a third harmonic frequency of \( 3f \), the frequencies are the same. This appears to contradict the supposed correct answer, suggesting either a misunderstanding in the phrasing of the question or a need to specify more clearly which overtone or harmonic is being referred to. If the question implies comparing the frequency of the third overtone (which would be the fifth harmonic in an open pipe), then the frequencies would differ: 
For an open organ pipe, the fifth harmonic is: \[ f_{{open}} = 5f \] Comparing this with the third harmonic of the closed pipe (\( 3f \)), the ratio would be: \[ {Ratio} = \frac{f_{{closed}}}{f_{{open}}} = \frac{3f}{5f} = \frac{3}{5} \] 
However, if the question correctly asks for the ratio of the third harmonics, both being \( 3f \), the ratio is 1:1, not 1:2 as listed. To align with the provided answer choice (B) and assuming the correct interpretation, it seems necessary to clarify the question's phrasing or reconsider the provided answer.