Question

A bird is singing on a tree and a man is hearing at a distance ‘r’ from the bird. Calculate the displacement of the man towards the bird so that the loudness heard by the man increases by $20\,{\text{dB}}$. [Assume that the motion of man is along the line joining the bird and the man]
A.$\dfrac{{9r}}{{10}}$
B.$\dfrac{r}{{10}}$
C.$\dfrac{{3r}}{5}$
D.$\dfrac{{4r}}{5}$

Answer 1

Use the formula for the loudness of the sound and intensity of the sound at a distance from the source of sound. Derive an equation for the loudness of sound in terms of distance from the source using these two equations. Take subtraction of the loudness of the sound at distance r and after movement of the man and determine the displacement of the man.

Formulae used:
The loudness $\beta$ of the sound is given by
$\beta = 10\log \dfrac{I}{{{I_0}}}$ …… (1)
Here, $I$ is the intensity of the sound and ${I_0}$ is the minimum intensity of the sound detectable by the human ear.
The intensity $I$ of the sound at a distance $r$ is given by
$I = \dfrac{P}{{4\pi {r^2}}}$ …… (2)
Here, $P$ is the power output of the source of the sound.

Complete step by step answer:
Rewrite equations (1) and (2) for the loudness ${\beta _1}$ and intensity ${I_1}$ of the sound at a distance from the bird.
${\beta _1} = 10\log \dfrac{{{I_1}}}{{{I_0}}}$ and ${I_1} = \dfrac{P}{{4\pi {r^2}}}$
Here, $P$ is the power output of the sound from the bird.
Substitute $\dfrac{P}{{4\pi {r^2}}}$ for ${I_1}$ in the equation for loudness ${\beta _1}$.
${\beta _1} = 10\log \dfrac{{\dfrac{P}{{4\pi {r^2}}}}}{{{I_0}}}$
$\Rightarrow {\beta _1} = 10\log \dfrac{P}{{4\pi {I_0}{r^2}}}$ ……. (3)
Rewrite the above equation for the loudness ${\beta _2}$ of the sound heard by the man when he moves towards the bird.
${\beta _2} = 10\log \dfrac{P}{{4\pi {I_0}r{'^2}}}$ …… (4)
Here, $r'$ is the distance of the man from the bird when he moves towards the bird.
Subtract equation (4) from equation (3).
${\beta _2} - {\beta _1} = 10\log \dfrac{P}{{4\pi {I_0}r{'^2}}} - 10\log \dfrac{P}{{4\pi {I_0}{r^2}}}$
$\Rightarrow {\beta _2} - {\beta _1} = 10\log P - 10\log 4\pi {I_0}r{'^2} - 10\log P + 10\log 4\pi {I_0}{r^2}$
$\Rightarrow {\beta _2} - {\beta _1} = 10\log 4\pi {I_0}{r^2} - 10\log 4\pi {I_0}r{'^2}$
$\Rightarrow {\beta _2} - {\beta _1} = 10\log \dfrac{{4\pi {I_0}{r^2}}}{{4\pi {I_0}r{'^2}}}$
$\Rightarrow {\beta _2} - {\beta _1} = 10\log \dfrac{{{r^2}}}{{r{'^2}}}$
$\Rightarrow {\beta _2} - {\beta _1} = 20\log \dfrac{r}{{r'}}$
The difference in the loudness heard by the man at two different positions is $20\,{\text{dB}}$.
$20\,{\text{dB}} = 20\log \dfrac{r}{{r'}}$
$\Rightarrow 1 = \log \dfrac{r}{{r'}}$
Take antilog on both sides of the above equation.
${\text{Antilog}}\left( 1 \right) = {\text{Antilog}}\left( {\log \dfrac{r}{{r'}}} \right)$
$\Rightarrow 10 = \dfrac{r}{{r'}}$
$\Rightarrow r' = \dfrac{1}{{10}}r$
The displacement of the man is the difference between the positions $r$ and $r'$ of the man.
$\Delta r = r - r'$
Substitute $\dfrac{1}{{10}}r$ for $r'$ in the above equation.
$\Delta r = r - \dfrac{1}{{10}}r$
$\therefore\Delta r = \dfrac{9}{{10}}r$

Therefore, the displacement of the man is $\dfrac{9}{{10}}r$.Hence, the correct option is A.

Note: The value of the position of the man obtained after movement is not the required displacement of the man. We have asked to determine the displacement of the man and not the final position of the man. The students should not forget to take the difference between the original position and changed position of the man to determine the displacement of the man.