Question

Two sound sources are placed along the diameter of a circle of radius $R\left(R\gg 4\lambda \right)$ . How many minima will be heard as one moves along the perimeter of the circle?

• A. $16$
• B. $12$
• C. $4$
• D. $8$

Answer 1

Answer: (A)

In this figure $S_1$ and $S_2$ are sound sources.

$S_1{S}_2=4\lambda$
$\therefore \Delta x=S_{2}P-S_{1}P=4\lambda sin\theta$
For minima $\Delta x=(2n+1)\frac{\lambda }{2}$
$\therefore 4\lambda sin\theta =(2n+1)\frac{\lambda }{2}$
$\Rightarrow sin\theta =\frac{(2n+1)}{8}$
$n=0;$ $sin\theta =\frac{1}{8}\Rightarrow \theta ={\left(sin\right)}^{-1}\left(\frac{1}{8}\right)$
$n=1;sin\theta =\frac{3}{8}\Rightarrow \theta ={\left(sin\right)}^{-1}\left(\frac{3}{8}\right)$
$n=2;sin\theta =\frac{5}{8}\Rightarrow \theta ={\left(sin\right)}^{-1}\left(\frac{5}{8}\right)$
$n=3;sin\theta =\frac{7}{8}\Rightarrow \theta ={\left(sin\right)}^{-1}\left(\frac{7}{8}\right)$
For $n=4;sin\theta >1$ it is impossible
In the quarter circle, $4$ minima are heard, in the entire circle total number of minima will be $16$ .