First slide

Stationary waves

Question

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.

Moderate

A

1150 Hz
B

1450 Hz
C

1250 Hz
D

1650 Hz

Solution

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 $n_{H_{2}} = \frac{(2 n + 1) v_{H_{2}}}{4 l}$
Resonating frequency of stationary wave formation in O₂ is $n_{O_{2}} = \frac{(2 m + 1) v_{O_{2}}}{4 l}$
for stationary wave to be produced in both gases we use $n_{H_{2}} = n_{O_{2}}$
$\frac{(2 n + 1) v_{n_{2}}}{4 l} = \frac{(2 m + 1) v_{α_{2}}}{4 l} \Rightarrow \frac{2 m + 1}{2 n + 1} = \frac{1100}{300} = \frac{11}{3}$
for min frequency
$2 m + 1 = 11 2 n + 1 = 3$
$\Rightarrow n_{min} = \frac{3 v_{h 2}}{4 l} = \frac{3 \times 1100}{4 \times 0 .5} = 1650 Hz$

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