Question

A closed and an open organ pipe have same lengths. If the ratio of frequencies of their seventh overtones is $\left(\frac{a-1}{a}\right)$, then the value of $a$ is ______.

Answer 1

For a closed organ pipe}, the frequency of the seventh overtone is:
$f_c = (2n + 1) \frac{v}{4\ell}, \quad n = 7.$
$f_c = 15 \frac{v}{4\ell}.$
For an open organ pipe, the frequency of the seventh overtone is:
$f_o = (n + 1) \frac{v}{2\ell}, \quad n = 7.$
$f_o = 8 \frac{v}{2\ell}.$
The ratio is:
$\frac{f_c}{f_o} = \frac{15}{16} = \frac{a - 1}{a}.$
Solving:
$a = 16.$
Final Answer: $16$.

Answer 2

For a closed organ pipe}, the frequency of the seventh overtone is:
$f_c = (2n + 1) \frac{v}{4\ell}, \quad n = 7.$
$f_c = 15 \frac{v}{4\ell}.$
For an open organ pipe, the frequency of the seventh overtone is:
$f_o = (n + 1) \frac{v}{2\ell}, \quad n = 7.$
$f_o = 8 \frac{v}{2\ell}.$
The ratio is:
$\frac{f_c}{f_o} = \frac{15}{16} = \frac{a - 1}{a}.$
Solving:
$a = 16.$
Final Answer: $16$.