Question

An open pipe and a closed pipe have the same length. The ratio of frequencies of their $n^{th}$ overtone is:

• A. $\frac{n+1}{2n+1}$
• B. $\frac{2(n+1)}{2n+1}$
• C. $\frac{n}{2n+1}$
• D. $\frac{n+1}{2n}$

Answer 1

The correct option is B $\frac{2(n+1)}{2n+1}$

Let $f$ be the length of the pipe and $v$ the speed of the sound. The frequency of the open organ pipe of $n^{th}$ overtone is ,

$f_1=(n+1)\frac{v}{2l}$

And frequency of closed organ pipe of $n^{th}$ overtone is,

$f_2=(2n+1)\frac{v}{4l}$

The desired ratio is thus,

$\frac{f_1}{f_2}=\frac{2(n+1)}{2n+1}$