Question

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 $300m/s$, the frequency difference between the fundamental and second harmonic of this pipe is__________ Hz.

Answer 1

Answer: 106.00

$v_s=\sqrt{\frac{\gamma P}{\rho }}$
$\frac{v_{gas}}{v_{air}}=\sqrt{\frac{{\rho }_{air}}{{\rho }_{gas}}}\Rightarrow \frac{v_{gas}}{300}=\frac{1}{\sqrt{2}}$
$\Rightarrow v_{gas}=\frac{300}{\sqrt{2}}$
$\therefore v_{gas}=150\sqrt{2}$
Now $n_2-n_1=\frac{v_{gas}}{2\ell }=\frac{150\sqrt{2}}{2\left(1\right)}=45\sqrt{2}$
$\Rightarrow \Delta n=106.06Hz$