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 is 300 m/s, the frequency difference between the fundamental and second harmonic pipe is ____

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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 is 300 m/s, the frequency difference between the fundamental and second harmonic pipe is _____ Hz.

(Round-off to the nearest integer)

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answer is 106.

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Detailed Solution

Given: $V_{a i r} = 300 m / s, ρ_{g a s} = 2 ρ a i r$
Using, $V = \sqrt{\frac{B}{ρ}}$
$\frac{V_{g a s}}{V_{a i r}} = \frac{\sqrt{\frac{B}{2 ρ_{a i r}}}}{\sqrt{\frac{B}{ρ_{a i r}}}}$

$\Rightarrow V_{g a s} = \frac{V_{a i r}}{\sqrt{2}} = \frac{300}{\sqrt{2}} = 150 \sqrt{2} m / s$

And $f_{nth} harmonic= \frac{nv}{2L} (in open organ pipe)$
(L = 1 metre given)
$∴ f_{2 n d} harmonic - f_{f u n d a m e n t a l} = \frac{2 v}{2 \times 1} - \frac{v}{2 \times 1} = \frac{v}{2} ∴ f_{2 n} harmonic - f_{f u n d a m e n t a l} = \frac{150 \sqrt{2}}{2} = \frac{150}{\sqrt{2}} \approx 106 H z$