The vibrations of 4 air columns are as shown in the figure. The ratio of the frequencies $n_{p} : n_{q} : n_{r} : n_{s}$

12 : 6 : 3 : 4

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1 : 2 : 4 : 3

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4 : 2 : 3 : 1

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6 : 2 : 3 : 4

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Solution

The correct option is B 1 : 2 : 4 : 3
In closed pipe the frequencies are given by:

$n_{c} = \frac{v (2 n - 1)}{4 L}, n = 1, 2, 3..$ .
In open pipe the frequencies are given by:
$n_{o} = \frac{v p}{2 L}, p = 1, 2, 3..$ .
In the question, the pipes shown are respectively close, open, open, close and they are vibrating in respectively in normal, 2nd harmonic, 4th harmonic, 3rd harmonic. So the frequency ratio will be:
$n_{p} : n_{q} : n_{r} : n_{s} = 1 : 2 : 4 : 3$ .

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n_{p} : n_{q} : n_{r} : n_{s}

f_{p} : f_{q} : f_{r} : f_{s}

Scores	$2$	$4$	$6$	$8$	$10$	$12$	$14$	$16$
Frequency	$4$	$2$	$3$	$4$	$5$	$6$	$5$	$4$

\frac{41 x - 12}{x^{2} - 16} = \frac{4 x + 3}{x - 4} .

\frac{2^{\frac{1}{2}} \times 3^{\frac{1}{3}} \times 4^{\frac{1}{4}}}{10^{- \frac{1}{5}} \times 5^{\frac{3}{5}}} \div \frac{3^{\frac{4}{3}} \times 5^{- \frac{7}{5}}}{4^{- \frac{3}{5}} \times 6}

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