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

Correct Answer: 16

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: $16$.