We will use the following 

speed of sound, v 

speed of source, v_{s} 

speed of observer, v_{o }(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 O_{1} and O_{2}
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 O_{1} 

for waves near observer O_{2} 


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 O_{1} 

for waves near observer O_{2} 




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. 
