Question

One meter long (both ends open) organ pipe is kept in a gas that has double the density of air at $\text{STP}$. Assuming the speed of sound of air at $\text{STP}$ is $300{\text{ms}}^{-1}$, the frequency difference between the fundamental and second harmonic of the pipe is_____$\text{Hz}$.

Answer 1

Given: $v_{\text{air}}=300{\text{ms}}^{-1},{\rho }_{\text{gas}}=2{\rho }_{\text{air}}$
The speed of sound in gas is given by,
$v=\sqrt{\frac{\gamma P}{\rho }}$
$\Rightarrow \frac{v_{\text{gas}}}{v_{\text{air}}}=\frac{\sqrt{\frac{\gamma P}{2{\rho }_{\text{air}}}}}{\sqrt{\frac{\gamma P}{{\rho }_{\text{air}}}}}$
$\Rightarrow v_{gas}=\frac{v_{air}}{\sqrt{2}}=\frac{300}{\sqrt{2}}=150\sqrt{2}{\text{ms}}^{-1}$

And $n^{th}$ harmonic in an open pipe is given by,

$f_n=\frac{nv}{2L}$

$\text{fundamental frequency},f_1=\frac{v}{2L}$

$\text{second harmonic},f_2=\frac{2v}{2L}$

$\Rightarrow \Delta f=f_2-f_1=\frac{v}{2L}$

$\text{Given that,}L=1\text{m}$

$\Delta f=\frac{150\sqrt{2}}{2}=106\text{Hz}$

Hence, $106$ is the correct answer.