First slide

Standing waves

Question

Two organ pipes both closed at one end, have lengths $l$ and $l + Δl$ . If the velocity of sound in air is $ν$ , then the number of beats per second produced is :

Moderate

A

$\frac{υ}{4 l}$
B

$\frac{υ}{2 l}$
C

$\frac{υ}{4 l^{2}} Δ l$
D

$\frac{υ}{2 l^{2}} Δ l$

Solution

$n_{1} = \frac{V}{4 l} n_{2} = \frac{V}{4 (l + Δl)}$
$Beats n_{1} - n_{2} = \frac{V}{4 l} [\frac{1}{l} - \frac{1}{(l + Δl)}]$
$= \frac{V}{4 l} [1 - \frac{1}{(1 + \frac{Δl}{l})}]$
$n_{1} - n_{2} = \frac{V}{4 l} [1 - {(1 + \frac{Δl}{l})}^{- 1}]$
$= \frac{V}{4 l} [1 - (1 - \frac{Δl}{l})]$
$= \frac{V}{4 l} \frac{Δl}{l} = \frac{VΔl}{4 l^{2}}$
Short cut
$n = \frac{v}{4 l} \Rightarrow dn = - \frac{v}{4 l^{2}} dl$

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