Question

What is the Doppler effect? Obtain the expression for the apparent frequency of the sound heard when the source is in motion with respect to an observer at rest.

Answer 1

Hint: In this question we will proceed with giving the definition of Doppler Effect and then according to the given condition in question we will proceed with using the formula which relates frequency, wavelength, and speed of sound.

Formula used:
$$f={\displaystyle \frac{v}{\lambda }}or\lambda ={\displaystyle \frac{v}{f}}$$
Where $$f$$ is the frequency, $$\lambda$$ is wavelength and $$v$$ is the velocity of the sound wave.

Complete step by step answer:
Given Data: -
Let us suppose a source of sound $$S$$ is emitting sound waves of frequency $$f$$. These waves are traveling in the medium with velocity $$v$$. The observer is at rest.
Doppler Effect: -
It is defined as the phenomena of apparent change in frequency of sound due to relative motion between the source of the sound and observed.
From the above given condition there may be two cases
Case 1:
When the source is moving with velocity $$v_s$$ towards the observer at rest then,
the relative velocity of sound waves reaching to the observer $$=v-v_s$$
now the apparent wavelength will be as follows:
${\lambda }^{\prime }={\displaystyle \frac{v-v_s}{f}}$
so, the apparent frequency is as follows
$\Rightarrow f^{\prime }={\displaystyle \frac{v}{{\lambda }^{\prime }}}={\displaystyle \frac{v}{v-v_s/f}}$
$\Rightarrow f^{\prime }={\displaystyle \frac{v}{v-v_s}}f$
Case 2:
When the source is moving with velocity $$v_s$$ away from the observer at rest then,
the relative velocity of sound waves reaching to the observer $$=v-(-v_s)=v+v_s$$
now the apparent wavelength will be as follows:
$${\lambda }^{\prime }={\displaystyle \frac{v+v_s}{f}}$$
so, the apparent frequency of the sound waves reaching to observer is as follows
$\Rightarrow f^{\prime }={\displaystyle \frac{v}{{\lambda }^{\prime }}}={\displaystyle \frac{v}{v+v_s/f}}$
$\Rightarrow f^{\prime }={\displaystyle \frac{v}{v+v_s}}f$

$\therefore$ The expression for the apparent frequency of the sound heard when the the source is in motion with respect to an observer at rest is $f^{\prime }={\displaystyle \frac{v}{v+v_s}}f$.

Note:
Before doing this question we should know the Doppler effect first. The apparent frequency is greater than the actual frequency as the source moves towards the listener and the apparent frequency is less than the actual frequency as the source moves away from the listener.