Question

An air column is of length 17 cm. The ratio of frequencies of 5th overtone if the air column is closed at one end to that open at both ends is (velocity of sound in air = 340 ms$^{-1}$) ______.

• A. $\frac{9}{11}$
• B. $\frac{5}{7}$
• C. $\frac{11}{12}$
• D. $\frac{13}{9}$

Hint:

Overtone mapping:
Closed pipe $k^{\text{th}}$ overtone = $(2k+1)^{\text{th}}$ harmonic.
Open pipe $k^{\text{th}}$ overtone = $(k+1)^{\text{th}}$ harmonic.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
For a closed pipe, frequencies are $(2p+1)n_c$. For an open pipe, frequencies are $(p+1)n_o$.

Step 2: Formula Application:
5th overtone for closed pipe ($p=5$): $f_c = (2 \times 5 + 1) \frac{v}{4L} = \frac{11v}{4L}$.
5th overtone for open pipe ($p=5$): $f_o = (5 + 1) \frac{v}{2L} = \frac{6v}{2L} = \frac{12v}{4L}$.

Step 3: Explanation:
Ratio $\frac{f_c}{f_o} = \frac{11v/4L}{12v/4L} = \frac{11}{12}$.

Step 4: Final Answer:
The ratio is 11 : 12.