Question

The speakers are coherent but have a phase difference of ${180}^{\circ }$ ?
Give answer as multiples of individual intensity of speakers

Answer 1

Path difference $\Delta x=\frac{\lambda }{2}$
So, phase difference $\Phi =\frac{2\pi }{\lambda }\Delta x$
$\therefore$ $\varphi =\frac{2\pi }{\lambda }.\frac{\lambda }{2}=\pi$
Since the speakers are coherent and out of phase initially, thus total phase difference is $\pi +\pi =2\pi$.
This indicates that constructive interference takes place.
Resultant intensity $I_R=I_1+I_2+2\sqrt{I_1{I}_2}\cos 2\varphi$
Here, $I_1=I_2=I_o$
$\therefore$ $I_R=I_o+I_o+2\sqrt{I_o{I}_o}\left(1\right)=4I_o$
So, if the speakers are coherent and out of phase initially, sound of intensity $4I_o$ is heard.