# 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 $P_2$ is (assume the given gases to be monoatomic):

$\begin {array} {1 1} (1)\;\frac{1}{3} & \quad (2)\;3 \\ (3)\;\frac{1}{6} \sqrt {\frac{\rho_1}{\rho_2}} & \quad (4)\;\frac{1}{6}\sqrt {\frac{\rho_2}{\rho_1}}\end {array}$

$(4)\;\frac{1}{6}\sqrt {\frac{\rho_2}{\rho_1}}$