Q: The lengths of two open organ pipes are l and l + Δl(Δl << l ). If v is the speed of sound, find the frequency of beats between them.

Sol: Beat frequency $\large = \nu_1 – \nu_2$

$\large = \frac{v}{2l} – \frac{v}{2(l+\Delta l)} $

$\large = \frac{v}{2l}[1-(1+\frac{\Delta l}{l})^{-1}] $

$\large = \frac{v}{2l}[1-1+\frac{\Delta l}{l}] = \frac{v \Delta l}{2l^2}$