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 $\Delta x=\left(\right.2n+1\left.\right)\frac{\lambda }{2}$
$\therefore 4\lambda sin\theta =\left(\right.2n+1\left.\right)\frac{\lambda }{2}$
$\Rightarrow \, sin\theta =\frac{\left(\right. 2 n + 1 \left.\right)}{8}$
$n=0;$ $sin\theta =\frac{1}{8}\Rightarrow \theta =\left(s i n\right)^{- 1}\left(\frac{1}{8}\right)$
$n=1; \, sin\theta =\frac{3}{8}\Rightarrow \theta =\left(s i n\right)^{- 1}\left(\frac{3}{8}\right)$
$n=2; \, sin\theta =\frac{5}{8}\Rightarrow \theta =\left(s i n\right)^{- 1}\left(\frac{5}{8}\right)$
$n=3; \, sin\theta =\frac{7}{8}\Rightarrow \theta =\left(s i n\right)^{- 1}\left(\frac{7}{8}\right)$
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 $16$ .