Question

Sound waves from a tuning fork $A$ reach a point $P$ by two separate paths $A B P$ and $A C P$. When $A C P$ is greater than $A B P$ by $11.5 \,cm$, there is silence at $P$. When the difference is increased to $23 \,cm$ the sound becomes loudest at $P$ and again when it increases to $34.5 \,cm$ there is silence again and so on. Calculate the minimum frequency (in $Hz$ ) of the fork if the velocity of sound if taken to be $331.2\, m / s$.

Answer 1

Given that $\Delta_{1}=11.5\, cm =(2 n+1) \frac{\lambda}{2} $ (Destructive)
and $ \Delta_{2}=23 \,cm =(2 n+1) \lambda $ (Constructive)
$\Delta_{3}=34.5\, cm =(2 n+1) \frac{3 \lambda}{2}$ (Destructive)
If there is no maxima or minima between $\Delta_{1}, \Delta_{2}$ and $\Delta_{3}$ that means $n=0$ for maximum wavelength or minimum frequency.
$\Rightarrow \frac{\lambda}{2}=11.5 \,cm $
$\Rightarrow \lambda=23 \,cm$
$\Rightarrow n=\frac{v}{\lambda}=\frac{331.2}{0.23}=1440 \,Hz$