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The Doppler Effect
The diagram below represents waves emitted by a source of sound, S, which is stationary relative to the air.  
We will assume that S behaves like a point source of waves.  
This means it produces spherical wavefronts, of which small cross-sectional segments are shown.  
The speed of propagation of the waves, relative to the air, is v.  
 
The next diagram represents waves emitted by a source which is moving (with velocity vs) relative to the air.  
Motion of the source produces a change in the wavelength, of the sound.  
The source is "catching up" with the disturbances which it sends to the right and "running away from" those sent to the left (if you will forgive these rather unscientific sounding terms!).  
This results in shorter wavelength in front of source, longer wavelength behind.  
  
We now return to a situation where the source is stationary.  
However, we will consider the person listening to the sound (the observer) be moving relative to the air.  
 
In the case shown, we have chosen to have the observer move towards the source.  
As S is stationary (relative to the air) the wavelength of the sound is the same everywhere.  
Even so, the observer will be intercepting the waves with greater frequency than if he/she was not moving (relative to the air).  
 
Both these effects (change of wavelength due to motion of source and change of frequency due to motion of observer), result in a change in the apparent frequency (and hence the pitch) of the sound perceived by the observer.  
 
This perceived change in pitch is called the Doppler effect, after the Austrian physicist, Christian Doppler).  
   
You can observe this change in pitch every time a police car or ambulance moves rapidly past you with its siren sounding.  
As the vehicle moves past you, you hear a decrease in pitch, assuming it's moving fast enough.  
You might also be able to notice it by running past a stationary siren... if you happen to be world champion runner!  
   
To summarize:
The Doppler Effect is the apparent change in the frequency of a sound wave due to the relative motion of source and observer.  
The Doppler shift, Δf, is defined as the difference between the transmitted (f) and perceived (f') frequencies  
 
NB  
The magnitude of this shift depends on  
1. the transmitted frequency  
2. the relative speed of source and observer.  
   
The relative Doppler shift, defined as follows  
 
is a useful quantity as it can be used to measure the relative speed of two bodies.  
   
However, it should be noted that actual Doppler speed measurement usually uses electro-magnetic waves which exhibit a similar effect but for which the mathematical analysis is a little different:
 
Doppler effect in sound waves  
Doppler effect in electro-magnetic waves  
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