Question

A one meter 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

The speed of sound in air at STP
$v_{\text{air}}=300m/s$
The velocity of sound in gas,
$v_S=\sqrt{\frac{\gamma P}{\rho }}$
$\frac{v_{\text{gas}}}{v_{\text{air}}}=\sqrt{\frac{{\rho }_{\text{air}}}{{\rho }_{\text{gas}}}}\Rightarrow \frac{v_{\text{gas}}}{300}=\frac{1}{\sqrt{2}}$
$\Rightarrow v_{\text{gas}}=\frac{300}{\sqrt{2}}\Rightarrow v_{\text{gas}}=150\sqrt{2}$
$f_{\text{nth harmanic}}=\frac{nv}{2L}\left(\text{open organ pipe}\right)$
$\therefore f_1-f_0=\frac{v_{\text{gas}}}{2L}=\frac{\left(150\sqrt{2}\right)}{(2\times 1)}=75\sqrt{2}Hz$
$=106.06Hz$
Final answer: (106.06 𝑡𝑜 106.07 )