How can we use the Doppler effect equation?

Hint: We need to understand what is the Doppler effect in order to derive the equation for the effect and understand how to use it in a given situation. We can use the Doppler’s effect in finding the frequency changes in the sound waves which are in relative motion.

Complete Solution:
The Doppler’s effect is a phenomenon which is related to a wave which is in relative motion with respect to its listener or the source itself is in motion. It is said that a sound wave of a particular frequency seems to have a change in its frequency as listened by the listener when the source and the listener are in a relative motion. This change in frequency resulting in an apparent frequency as heard by the listener is called the Doppler’s effect in sound.

Consider a source of sound S, which moves with a velocity \[{{v}_{S}}\] and a listener which moves with a velocity \[{{v}_{l}}\]. Now, let ‘v’ be the speed of sound waves in the given medium. If ‘f’ is the actual frequency of the sound waves produced, then the apparent frequency is given as –
\[{{f}_{a}}=f\dfrac{v+{{v}_{S}}}{v+{{v}_{l}}}\]

Now, let us consider the different situations.
Case I: When the source is at rest and the listener is moving towards the source.

i.e.,
\[{{f}_{a}}=f\dfrac{v}{v+{{v}_{l}}}\]

Case II: When the source is at rest and the listener is moving away from the source.

i.e.,
\[{{f}_{a}}=f\dfrac{v}{v-{{v}_{l}}}\]

Case III: When the listener is at rest and the source is moving towards the listener.

i.e.,
\[{{f}_{a}}=f\dfrac{v+{{v}_{s}}}{v}\]

Case IV: When the listener is at rest and the source moves away from the listener.

i.e.,
\[{{f}_{a}}=f\dfrac{v-{{v}_{s}}}{v}\]

These are the different situations in which the frequency of the sound can change.
This is the required solution.

Note:
The Doppler’s effect is valid for any kind of waves, be it transverse or longitudinal. We have seen the sound waves which are longitudinal in nature. Even the light waves also possess a Doppler’s effect which results in the celestial red shifts or blue shifts.