`n_(1)` is the frequency of the pipe closed at one and `n_(2)` is the frequency of the pipe open at both ends. If both are joined end to end, find the fundamental frequency of closed pipe so formed

Options

A. `(2n_(1) - n_(2))/(n_(1)n_(2)) Hz `
B. `(n_(1)n_(2))/(2n_(1)+n_(2)) Hz `
C. `(n_(1) + n_(2))/(n_(1) - n_(2)) Hz`
D. `(n_(1) n_(2))/(2n_(2)+n_(1)) Hz`

Answer

B

Explanation

Let v be velocity of sound in air, then for the pipe closed at one end ,
`n_(1) = v/(4L_(1)) :. L_(1) = v/(4n_(1))`
and for the pipe open at both the ends, ltBrgt `n_(2) v/(2L_(2)) " or " L_(2) = v/(2n_(2))`
When they are joined end to end, the new length of the pipe is `L = L_(1) + L_(2) `
` = v/(4n_(1)) + v/(2n_(2)) = v/2 [1/(2n_(1)) + 1/(n_(2))]`
`:. L_(1) + L_(2) = v/4 [(n_(2) + 2n_(1))/(n_(1) n_(2))]`
` :. ` The new fundamental frequency of the pipe closed at one end is given by
`n = v/(4(L_(1) + L_(2))) = (v xx 4n_(1)n_(2))/(4 xx v (n_(2) + 2n_(1)))`
` :. n = (n_(1)n_(2))/((2n_(1) + n_(2)))`