What is Doppler’s effect? Derive an expression for the apparent frequency received by observer when alone source is in motion. (Assume media to be at rest).

Doppler's effect is the change in the pitch when there is a relative motion between source and observer.
Let a source which is at rest be emitting waves of frequency f and wavelength λ.
Let V be the velocity of the wave emitted by source,
                            V=fλ. 
        
Let the distance between source and observer be V.
Therefore, the wave will take one second to reach the observer.
Now let the source start moving with velocity u towards the observer.
The wave, which was just emitted by the source, starts moving will take one second to reach the observer and in a second source, will travel 'u' distance. 
The source will be at a distance V–u just after 1 second and there will be f waves in V–u distance.
Therefore wavelength of wave when source starts moving is, 
                     
The apparent frequency  is, 
                     
                                   ...(1) 
When source starts moving away from observer then replace u by –u.
Therefore, the apparent frequency is, 
                                    ...(2) 
From equation (1), the apparent frequency increases when the source moves towards the observer.
From equation (2) the apparent frequency decreases when the source moves away from the observer.