Question

Sound waves of frequency $13.6\text{kHz}$ are emitted by two small coherent sources of sound placed d=1 m apart as shown at the centre of a circular toy train track. A person riding the train observes 4 maxima/sec when the train is running at a speed of $72\text{km/hr}$. The radius of the track is $\frac{100k}{\pi }m$. Find k. (velocity of sound in air is $340\frac{m}{s})$

Answer 1

The wavelength of the sound wave in air is,
$=\frac{340}{13.6\times {10}^3}m=25\times {10}^{-3}m$

The separation between the sources, d = 1 m = $40\lambda$.
n interference patterns are produced on the circumference of the circular track,
here, $d\sin \theta =n\lambda ,\theta$ representing the position of the nth maximum as shown in the figure i.e.
$40\lambda \sin \theta =n\lambda$ or , $\sin \theta =\frac{n}{40}$,
Since $\sin \theta$ lies between 0 & 1, there are 40 maxima in $\frac{1}{4}th$ of the circumference or 160 maximum on the entire circle.
The (observer ) train travels at $72\text{km/hr}$ or, $\frac{72\times {10}^3}{3600}\text{m/s}$
i.e $20\text{m/s}$
and observes 4 beats /seconds.
160 beats are observed in $\frac{160}{4}=40\sec$. When the observer travels along the entire circle.
The length of the track $=40\times 20m=800m$
$\therefore$ The radius of the track $=\frac{800}{2\pi }m$