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 ________.

A closed and open pipe of same length. Find $$a$$ if the ratio of their 7th overtones is $$\frac{a-1}{a}$$.

A closed pipe produces only odd harmonics: 1st, 3rd, 5th, 7th, ...

The $$n$$th overtone corresponds to the $$(2n+1)$$th harmonic. The 7th overtone = 15th harmonic.

$$f_{\text{closed}} = \frac{15v}{4L}$$

An open pipe produces all harmonics. The 7th overtone = 8th harmonic.

$$f_{\text{open}} = \frac{8v}{2L} = \frac{4v}{L}$$

$$\frac{f_{\text{closed}}}{f_{\text{open}}} = \frac{15v/(4L)}{4v/L} = \frac{15v}{4L} \times \frac{L}{4v} = \frac{15}{16}$$

$$\frac{15}{16} = \frac{a-1}{a} \implies a = 16$$

The correct answer is 16.