Question

A rock band gives rise to an average sound level of 102 dB at a distance of 20 m from the centre of the band. As an approximation, assume that the band radiates sound equally into a sphere. The sound power output of the band is $8\times {10}^x$ watts. Find x.

Answer 1

We have loudness of sound ,

$L=10lo{g}_{10}\left(I/I_0\right)$ ,

or $\left(I/I_0\right)={10}^L$ ,

given $L=102dB=10.2B$ , and we know , $I_0={10}^{-12}W/m^2$ ,

therefore ,

$I/I_0={10}^{10.2}$ ,

or $I={10}^{-12}\times {10}^{10.2}={10}^{-12}\times {10}^{10}\times {10}^{0.2}={10}^{-1.8}W/m^2$ ,

now $A=4\pi r^2=4\times 3.14\times {20}^2{m}^2$ , (given , $r=20m$)

power output , $P=I\times A$ ,

so $P={10}^{-1.8}\times 4\times 3.14\times {20}^2=80W$