Question

Loudness of sound from an isotropic point source at a distance of $70\, cm$ is $20 \,dB$. What is the distance (in metre) at which it is not heard ?

Answer 1

Let at distance $r_{1}=0.7\, m$, intensity is $I_{1}$
$\therefore 20=10 \log _{10} \frac{I_{1}}{I_{0}}$...(i)
Let at distance $r_{2}$, sound level is zero.
Then, intensity at that point must be equal to $I_{0}$.
(where, $I_{0}=10^{-12} Wm ^{-2}$ )
$I \propto \frac{1}{r^{2}} $
$\Rightarrow \frac{I_{1}}{I_{0}}=\left(\frac{r^{2}}{r_{1}}\right)^{2}$
Using this in Eq. (i), $20=10 \log _{10}\left(\frac{r_{2}}{r_{1}}\right)^{2}$
Or $\left(\frac{r_{2}}{r_{1}}\right)^{2}=10^{2} $
$\Rightarrow r_{2}=\sqrt{100} r_{1}=10(0.7)=7 \,m$