Physics - Sound Wave Question with Solution | TestHub

Let the length of both pipes be $L$ and the velocity of sound be $v$.

For a closed pipe, the fundamental frequency is $f_c=\frac{v}{4L}$.
The frequency of the $n^{\text{th}}$ overtone for a closed pipe is given by the $(2n+1{)}^{\text{th}}$ harmonic:
$f_{c,n}=(2n+1)f_c=\frac{(2n+1)v}{4L}$

For an open pipe, the fundamental frequency is $f_o=\frac{v}{2L}$.
The frequency of the $n^{\text{th}}$ overtone for an open pipe is given by the $(n+1{)}^{\text{th}}$ harmonic:
$f_{o,n}=(n+1)f_o=\frac{(n+1)v}{2L}$

Taking the ratio of the frequency of the $n^{\text{th}}$ overtone for the closed pipe to that of the open pipe:
$\text{Ratio}=\frac{f_{c,n}}{f_{o,n}}=\frac{\frac{(2n+1)v}{4L}}{\frac{(n+1)v}{2L}}$

$\text{Ratio}=\frac{2n+1}{4}\times \frac{2}{n+1}=\frac{2n+1}{2(n+1)}$

Answer: $\frac{(2n+1)}{2(n+1)}$