First slide

Standing waves

Question

If $n_{1}, n_{2} a n d n_{3}$ are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency n of the string is given by

Moderate

A

$\frac{1}{n} = \frac{1}{n_{1}} + \frac{1}{n_{2}} + \frac{1}{n_{3}}$
B

$\frac{1}{\sqrt{n}} = \frac{1}{\sqrt{n_{1}}} + \frac{1}{\sqrt{n_{2}}} + \frac{1}{\sqrt{n_{3}}}$
C

$\sqrt{n} = \sqrt{n_{1}} + \sqrt{n_{2}} + \sqrt{n_{3}}$
D

$n = n_{1} + n_{2} + n_{3}$

Solution

$n_{1} = \frac{1}{2 l_{1}} \sqrt{\frac{T}{μ}}$ . . . . (i)
$n_{2} = \frac{1}{2 l_{2}} \sqrt{\frac{T}{μ}} . . . . . (i i)$
$n_{3} = \frac{1}{2 l_{3}} \sqrt{\frac{T}{μ}}$ . . . . . (iii)
and
$n = \frac{1}{2 l} \sqrt{\frac{T}{μ}}$ . . . . . (iv)
$From eqn. (iv), we get$
$\frac{1}{n} = \frac{2 l}{\sqrt{\frac{T}{μ}}}$
$As l = l_{1} + l_{2} + l_{3}$
$∴ \frac{1}{n} = \frac{2 (l_{1} + l_{2} + l_{3})}{\sqrt{\frac{T}{μ}}} = \frac{2 l_{1}}{\sqrt{\frac{T}{μ}}} + \frac{2 l_{2}}{\sqrt{\frac{T}{μ}}} + \frac{2 l_{3}}{\sqrt{\frac{T}{μ}}}$
$= \frac{1}{n_{1}} + \frac{1}{n_{2}} + \frac{1}{n_{3}}$ (using (i), (ii) and (iii))

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