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 Question:
We must calculate the frequency of the 5th overtone for two different pipe setups (one closed, one open) using the same length $L$, and find their ratio.

Step 2: Detailed Explanation:
Let the length of the air column be $L$.
Let the velocity of sound be $v$.
Case 1: Pipe closed at one end.
A closed pipe produces only odd harmonics.
Frequencies are given by: $f_k = (2k + 1) \frac{v}{4L}$, where $k$ is the overtone number ($k = 0, 1, 2, \dots$).
For the 5th overtone, $k = 5$:
Harmonic number = $(2(5) + 1) = 11$.
Frequency ($f_c$) = $11 \left( \frac{v}{4L} \right)$.
Case 2: Pipe open at both ends.
An open pipe produces all integer harmonics (both even and odd).
Frequencies are given by: $f_m = (m + 1) \frac{v}{2L}$, where $m$ is the overtone number ($m = 0, 1, 2, \dots$).
For the 5th overtone, $m = 5$:
Harmonic number = $(5 + 1) = 6$.
Frequency ($f_o$) = $6 \left( \frac{v}{2L} \right)$.
To compare $f_c$ and $f_o$ easily, convert $f_o$ to the same denominator ($4L$):
$f_o = 6 \left( \frac{v}{2L} \right) \times \frac{2}{2} = 12 \left( \frac{v}{4L} \right)$.
Calculate the Ratio:
$\text{Ratio} = \frac{f_c}{f_o}$
$\text{Ratio} = \frac{ 11 \left( \frac{v}{4L} \right) }{ 12 \left( \frac{v}{4L} \right) }$
$\text{Ratio} = \frac{11}{12}$
(Notice that the actual length $17$ cm and velocity $340$ m/s are distractor variables; they cancel out completely during the ratio step).

Step 3: Final Answer:
The ratio is $\frac{11}{12}$, matching option (c).