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Doppler Effect Equation

The Austrian Physicist Christian Doppler formulated the Doppler effect in 1842. According to him, Doppler effect is nothing but the change in wave frequency when an observer is moving with respect to the source. Let us consider an example, if a stationary observer on a platform listen to the sound of the whistle of a train which is coming into this platform. When it is closer to the observer the sound is increases.As the train approaches the platform, an increase in the pitch of the sound will be observed. If this train is going away, the sound will be reduced. This is due to the Doppler effect of the sound wave.


Doppler Effect

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Doppler Effect Formula

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Doppler effect is given by the change observed in the frequency of sound wave, when the observer and the source are in a relative motion. The changes in the wave frequency can be described as follows. If the source is moving towards the observer, every crest is generated from a closer position to the observer. So the wave will reach to the observer very fast. Therefore, the arrival time is comparatively less, hence the frequency increased. If the object is moving away from the observer,  wave is generated from the farther position compared to the observer. So, there is a time lag in the arrival of the wave to the observer. Hence there is a reduction in the frequency of the propagating wave.
Doppler Effect Sound Source

The equations related to the Doppler shift is given below:

1. When the source is moving towards a observer at rest

f'=($\frac{v}{v-v_{s}}$)f
2. When the source is moving away from the observer at rest

f'=($\frac{v}{v+v_{s}}$)                                 
3. When observer is moving towards the stationary source

f'=($\frac{v+v_{0}}{v}$)                                
4. When observer moving away from a stationary source

f'=($\frac{v-v_{0}}{v}$)f                                 
5. When both Source and observer moves towards each other

f'=($\frac{v+v_{0}}{v-v_{s}}$)f  
                              
6. When both Source and observer move away from each other

f'=($\frac{v-v_{0}}{v+v_{s}}$)f    
                              
7. When the Source is approaching the Stationary observer and observer moving away from observer
f'=($\frac{v-v_{0}}{v-v_{s}}$)f
                                   
8. When the Observer is approaching the Stationary source

f'=($\frac{v+v_{0}}{v+v_{s}}$)f                                   
Where $v_{s}$= Velocity of the Source,
           $v_{0}$= Velocity of the Observer,
           v = Velocity of sound or light in medium,
           f = Real frequency,
           f' = Apparent frequency.

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