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)

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