Question

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

Answer 1

Answer: 7

Let at distance $r_1=0.7m$, 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, $I_0=10^{-12}W{m}^{-2}$ )
$I\propto \frac{1}{r^2}$
$\Rightarrow \frac{I_1}{I_0}={\left(\frac{r^2}{r_1}\right)}^2$
Using this in Eq. (i), $20=10{\log }_{10}{\left(\frac{r_2}{r_1}\right)}^2$
Or ${\left(\frac{r_2}{r_1}\right)}^2=10^2$
$\Rightarrow r_2=\sqrt{100}{r}_1=10(0.7)=7m$