Question

$2$ loudspeakers are emitting sound waves of wavelength $\lambda $ with an initial phase difference of $\frac{\pi }{2}$ . At what minimum distance from $O$ on line $AB$ will one hear maxima?

• A. $25 \, \lambda $
• B. $\frac{100 \lambda }{\sqrt{15}}$
• C. $50 \, \lambda $
• D. $\frac{25 \, \lambda }{3}$

Answer 1

Answer: (D)

$\Delta \phi=2n\pi $
$\Rightarrow \frac{\pi }{2}+\frac{2 \pi }{\lambda }dsin \theta =2n\pi $
$\frac{2 \pi }{\lambda } \, dsin \theta =\left(2 n - \frac{1}{2}\right)\pi $
$sin\theta = \, \left(2 n - \frac{1}{2}\right)\frac{\lambda }{2 d}=\frac{1}{2}\times \frac{\lambda }{2 \times 3 \lambda }=\frac{1}{12}$
$\Rightarrow \frac{y}{\sqrt{\left(100 \lambda \right)^{2}}}=\frac{1}{12}$
$144y^{2}=\left(100 \lambda \right)^{2}$
$y\approx \frac{100 \lambda }{12}=\frac{25 \lambda }{3}$