A point source of sound emits a constant power with intensity inversely proportional to the square of the…
A point source of sound emits a constant power with intensity inversely proportional to the square of the distance from the source. By how many decibels does the sound intensity level drop when you move from point P1 to P2 . Distance of P2 from the source is two times the distance of source from P1.
Solution
Sol. Let the two points be labelled as 1 and 2. The difference in sound intensity level is given by
$\beta_2 - \beta_1 = (10\ \mathrm{dB})\left(\log\frac{i_2}{i_0} - \log\frac{i_1}{i_0}\right)$
$\quad\;= (10\ \mathrm{dB})[(\log i_2 - \log i_0) - (\log i_1 - \log i_0)]$
$\quad\;= (10\ \mathrm{dB})\log\left(\dfrac{i_2}{i_1}\right)$
Now, $i \propto \dfrac{1}{r^2}$ (Given)
$\displaystyle \frac{i_2}{i_1}=\left(\frac{r_1}{r_2}\right)^2=\frac{1}{4}\qquad(\therefore\; r_2=2r_1)$
$\therefore\; \beta_2-\beta_1=(10\ \mathrm{dB})\log\left(\frac{1}{4}\right)=-6.0\ \mathrm{dB}$
Answer: $-6.0\ \mathrm{dB}$