Question

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

Answer 1

Hint : The Doppler Effect is observed when there is a relative motion between a source of a wave and the detector/ observer. The frequency of the wave increases when the source and observer move towards each other and decreases when they move away from each other.

Complete step by step answer
The Doppler Effect is the shift in the original frequency to the observed frequency of the wave emitted by a source when there is a relative motion between the observer and the source.
When a source moves towards the observer or the observer moves towards source, there is a relative motion between them. Same is the case when the observer and the source move away from each other. This causes the wave crests to either bunch together or to spread out. Which causes the Doppler effect.
Let $ S $ be a source of sound moving with a velocity $ {v_s} $ .
Let $ O $ be an observer moving with a velocity $ {v_o} $ .
The relative velocity of the sound with respect to the observer is given by-
 $\Rightarrow {\vec v_r} = {\vec v_s} - {\vec v_o} $
Suppose that the wave moves towards the observer, the distance covered by the source in the time period of complete cycle $ T $ is $ = {v_r}T $ .
When observer is at rest,
 $\Rightarrow {v_r} = {v_s} $
This movement of the source causes the crests formed in the wave to be closer than they normally would, thus the wavelength of the sound decreases. Due to this a wavelength will be observed to be shorter than usual by a value equal to the difference between original wavelength of the sound and the distance travelled by the source in that time period.
 $\Rightarrow \lambda ' = \lambda - {v_s}T $
The frequency of the wave can be written as-
 $\Rightarrow f = \dfrac{1}{T} $
Substituting this value,
 $\Rightarrow \lambda ' = \lambda - \dfrac{{{v_s}}}{f} $
 $\Rightarrow \lambda ' = \dfrac{{\lambda f - {v_s}}}{f} $
We know that for a wave the velocity,
 $\therefore v = \lambda f $
Therefore here,
The absolute velocity of the wave $ v = \lambda f $ ,
Substituting this value in the equation we have,
 $\Rightarrow \lambda ' = \dfrac{{v - {v_s}}}{f} $
The apparent frequency corresponding to this wavelength is given by dividing both sides with the velocity of the wave and performing a reciprocal,
 $\Rightarrow \dfrac{{\lambda '}}{v} = \dfrac{{v - {v_s}}}{{vf}} $
 $\therefore f' = \dfrac{v}{{\lambda '}} = \dfrac{{vf}}{{v - {v_s}}} $.

Note
The magnitude of the apparent frequency depends on the relative velocity of the source with respect to the observer. If the observer and the source move towards each other, the relative velocity will be positive, paired with the negative sign it becomes negative and apparent frequency becomes higher. Similarly, when they move away, the sign of the relative velocity is negative, which becomes overall positive and the frequency becomes lower.