AB is a cylinder of length 1 m fitted with a thin flexible diaphragm C at the middle and two other thin flexible diaphragm at ends A and B which do not allow outside air to mix with H₂ and O₂ gas as shown. Find the minimum frequency of a tuning fork which can establish sustained stationary waves in cylinder with diaphragm C as a node. Given that velocity of sound in H₂ is 1100 m/s and in O₂, is 300 m/s.

 

 

For sustained stationary wave in both gas, independently stationary wave must setup in  H₂ & in O₂
Resonating frequency of stationary wave formation in H₂ is ${\mathrm{n}}_{{\mathrm{H}}_2}=\frac{(2\mathrm{n}+1){\mathrm{v}}_{{\mathrm{H}}_2}}{4\mathrm{l}}$
Resonating frequency of stationary wave formation in O₂ is ${\mathrm{n}}_{{\mathrm{O}}_2}=\frac{(2\mathrm{m}+1){\mathrm{v}}_{{\mathrm{O}}_2}}{4\mathrm{l}}$
for stationary wave to be produced in both gases we use ${\mathrm{n}}_{{\mathrm{H}}_2}={\mathrm{n}}_{{\mathrm{O}}_2}$
$\frac{(2\mathrm{n}+1){\mathrm{v}}_{{\mathrm{n}}_2}}{4\mathrm{l}}=\frac{(2\mathrm{m}+1){\mathrm{v}}_{{\mathrm{\alpha }}_2}}{4\mathrm{l}}\Rightarrow \frac{2\mathrm{m}+1}{2\mathrm{n}+1}=\frac{1100}{300}=\frac{11}{3}$
for min  frequency

 $$\begin{gathered} 2\mathrm{m}+1=11 \\ 2\mathrm{n}+1=3 \end{gathered}$$
$\Rightarrow {\mathrm{n}}_{min}=\frac{3{\mathrm{v}}_{\mathrm{h}2}}{4\mathrm{l}}=\frac{3\times 1100}{4\times 0.5}=1650\mathrm{Hz}$