Two open organ pipes of fundamental frequencies `n_(1) and n_(2)` are joined in series. The fundamental frequency of the new pipes so obtained will be

Options

A. `upsilon_(1)+upsilon_(2)`
B. `(upsilon_(1)upsilon_(2))/((upsilon_(1)+upsilon_(2)))`
C. `(upsilon_(1)upsilon_(2))/(upsilon_(1)-upsilon_(2))`
D. `sqrt((upsilon_(1)^(2)+upsilon_(2)^(2)))`

Answer

A

Explanation

Let `L_(1)andL_(2)` be the lengths of two pipes. Then `upsilon_(1)=(v)/(2L_(1))" "...(i)`
and `upsilon_(2)=(v)/(2L_(2))" "…(ii)`
When these two pipes are joined in series, the fundamental frequency of the new pipe is
`upsilon=(v)/(2(L_(1)+L_(2)))=(v)/(2L_(1)+2L_(2))=(v)/((v)/(upsilon_(1))+(v)/(upsilon_(2)))` (Using (i) and (ii))
`=(upsilon_(1)upsilon_(2))/(upsilon_(2)+upsilon_(1))`