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The Doppler Effect: Calculations
We will use the following  
speed of sound, v  
speed of source, vs  
speed of observer, vo (all speeds relative to the air)  
transmitted frequency, f  
apparent or perceived frequency, f'  
   
Moving Source, Stationary Observer  
To simplify things, we will consider observers positioned along the direction of motion of the source, as shown above.  
   
 
   
Note that the two observers O1 and O2 receive sounds of different wavelength, as a result of the motion of the source.  
The magnitude of the velocity of the waves relative to the source is given by (see relative speed and velocity)  
 
for waves near observer O1
for waves near observer O2
 
In general (see here)  
 
so, in this case, we have  
   
The apparent frequency will be given by     
   
which means that  
   
 
Moving Observer, Stationary Source  
Again, for simplicity, we will consider observers moving along the line joining observer and source.  
   
   
   
In this case there is, of course, no change in the wavelength.   
However, the speed of the waves relative to the observer is now given by  
   
  for waves near observer O1
  for waves near observer O2
 
   
In general, the frequency of a wave can be found by  
   
   
therefore, the apparent frequency, f' is given by   
 
and since   
   
we have   
   
   
Moving Source and Moving Observer  
In this case, we can imagine applying both the above equations to the situation.   
Find an apparent frequency due to the motion of the source and then put that into the second equation (in place of f) to find the "final" apparent frequency.  
Thus we have   
   
which of course gives   
   
Here we see that if the speeds of observer and source are the same (and in the same sense) then f = f.  
In other words, the Doppler effect depends on the relative speed (or velocity) of source and observer.   
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