For a certain organ pipe,the first three resonance frequencies are in the ratio of $1:3:5$ respectively. If the frequency of the fifth harmonic is $405 \, Hz$ and the speed of sound in air is $324 \, ms^{-1}$,the length of the organ pipe is $.......... m$.

$A$ pipe closed at one end has a length of $83 \ cm$. The number of possible natural oscillations of the air column whose frequencies lie below $1000 \ Hz$ are (velocity of sound in air $= 332 \ m/s$).

$A$ uniform narrow $1.95 \, m$ long pipe is open at both ends. It resonates at two successive harmonics of frequency $275 \, Hz$ and $330 \, Hz$. The speed of sound in the tube is ...... $m/s$.

$A$ closed organ pipe of length $L_1$ and an open organ pipe contain diatomic gases of densities $\rho_1$ and $\rho_2$ respectively. The compressibilities of the gases are same in both pipes,which are vibrating in their first overtone with the same frequency. The length of the open organ pipe is (Neglect end correction).

In a physics lab,a student is performing an experiment with a resonance tube to find the speed of sound and its end correction. For this,he used a resonance tube of length $120 \ cm$. When the length of the air column in the tube is $16 \ cm$ and $50 \ cm$,he obtains the $I$ and $II$ resonance respectively,while a tuning fork of frequency $500 \ Hz$ is sounded just above the tube. Match the parameters in List-$I$ with their suitable values in List-$II$.

List-$I$	List-$II$
$A$. Wavelength of sound $(cm)$	$p$. $1$
$B$. Height of liquid column at $II$ resonance $(cm)$	$q$. $2$
$C$. Speed of sound $(m/s)$	$r$. $340$
$D$. End correction $(cm)$	$s$. $68$
$E$. Minimum level of liquid column at resonance $(cm)$	$t$. $70$

Explain the formation of stationary waves in a closed pipe and derive the equations for natural frequencies (normal modes).