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  4. A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at STP. Assuming the speed of sound in air at STP is 300 m/s, the frequency difference between the fundamental and second harmonic of this pipe is Hz.

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Q. A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at STP. Assuming the speed of sound in air at STP is $300\, m/s$, the frequency difference between the fundamental and second harmonic of this pipe is__________ Hz.

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Solution:

$v_{s}=\sqrt{\frac{\gamma P}{\rho}}$
$\frac{v_{gas}}{v_{air}}=\sqrt{\frac{\rho_{air}}{\rho_{gas}}} \,\Rightarrow \frac{v_{gas}}{300}=\frac{1}{\sqrt{2}}$
$\Rightarrow v_{gas}=\frac{300}{\sqrt{2}}$
$\therefore v_{gas}=150\sqrt{2}$
Now $n_{2}-n_{1}=\frac{v_{gas}}{2\ell}=\frac{150\sqrt{2}}{2\left(1\right)}=45\sqrt{2}$
$\Rightarrow \Delta n=106.06\,Hz$