An organ pipe `P _(1)` , closed at one end and containing a gas of density `rho _(1) ` is vibrating in its first harmonic . Another organ pipe `P_(2)`, open at both ends and containing a gas of density `rho _(2)` is vibrating in its third harmonic. Both the pipes are in resonance with a given tuning fork . If the compressibility of gases is equal in both pipes , the ratio of the lengths of `P_(1)` and `P_(2)` (assume the given gases to be monoatomic)

Options

A. `(1)/(3)`
B. 3
C. `(1)/(6) sqrt((rho _(1))/(rho_(2))`
D. `(1)/(6) sqrt((rho _(2))/(rho_(1))`

Answer

D