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

Answer

a

Explanation

Frequency of closed pipe `n_(1)=v/(4l_(1))rArr l_(1)=v/(4n_(1))`
Frequency of open pipe, `n_(2)=v/(2l_(1))rArr l_(2)=v/(2n_(2))`
When both pipe are joined, then length of closed pipe
`l=l_(1)=l_(2) rArr v/(4n)=v/(4n_(1))+v/(2n_(2))`
`rArr 1/(2n)= 1/(2n_(1)) + 1/(n_(2)) or 1/(2n) = (n_(2)+2n_(1))/(2n_(2)n_(2))`
`rArr n=(n_(1)n_(2))/(n_(2)+2n_(1))`