Question

If the speakers are driven coherently and in phase, the intensity at point P is $553\times {10}^{-x}W{m}^{-2}$. 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 and in phase therefore they will produce constructive interference at point P , therefore resultant intensity at point P ,

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

or $I=19.9+8.84+2\sqrt{19.9\times 8.84}$ ,

or $I=19.9+8.84+26.5=55.24W/m^2$ ,