Question

If they are driven coherently but out of phase by ${180}^{\circ }$, the intensity at point P is $22\times {10}^{-x}W{m}^{-2}$. Find the value of x.

Answer 1

Intensity of sound at a distance $r$ , from the point source of sound of power $P$ is given by ,

$I=P/4\pi r^2$ ,

given $P_1=P_2=1MW={10}^{3}W,r_1=2m,r_2=3m$ ,

intensity of first speaker ,

$I_1=P_1/4\pi r_1^2={10}^3/(4\times 3.14\times 2^2)=19.9W/m^2$ ,

intensity of second speaker ,

$I_2=P_2/4\pi r_2^2={10}^3/(4\times 3.14\times 3^2)=8.84W/m^2$ ,

as source are coherent therefore they will produce interference at point P , therefore resultant intensity at point P ,

$I=I_1+I_2+2\sqrt{I_1{I}_2}\cos \varphi$ ,

or $I=19.9+8.84-2\sqrt{19.9\times 8.84}\cos 180$ , (given , $\varphi ={180}^o$) ,

or $I=19.9+8.84-26.5=2.2W/m^2$ ,