# A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at STP.

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 $300 \mathrm{~m} / \mathrm{s}$, the frequency difference between the fundamental and second harmonic of this pipe is____ $\mathrm{Hz}$.

Solution:

(106) Given : $V_{\text {air }}=300 \mathrm{~m} / \mathrm{s}, \rho_{\text {gas }}=2 \rho$ air

$U \sin g, V=\sqrt{\frac{B}{\rho}}$

$\frac{V_{\text {gas }}}{V_{\text {air }}}=\frac{\sqrt{\frac{B}{2 \rho_{\text {air }}}}}{\sqrt{\frac{B}{\rho_{\text {air }}}}}$

$\Rightarrow V_{\text {gas }}=\frac{V_{\text {air }}}{\sqrt{2}}=\frac{300}{\sqrt{2}}=150 \sqrt{2} \mathrm{~m} / \mathrm{s}$

And $\mathrm{f}_{\text {nth }}$ harmonic $=\frac{n v}{2 L}$ (in open organ pipe)

( $\mathrm{L}=1$ metre given)

$\therefore f_{2 \text { nd }}$ harmonic $-f_{\text {fundamental }}=\frac{2 v}{2 \times 1}-\frac{v}{2 \times 1}=\frac{v}{2}$

$\therefore f_{2 \mathrm{n}}$ harmonic $-f_{\text {fundamental }}=\frac{150 \sqrt{2}}{2}=\frac{150}{\sqrt{2}} \approx 106 \mathrm{~Hz}$