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where we have the source S moving with speed v_s  to the right. It sound at  frequency f

with one wavefront at time 0 and the next one at time  T  (the period). The viewed distance between the fronts, ?  will change because of the Doppler shift.

 
                                      

If we follow the wave fronts for a time t  we note that the first front moves  a distance  v_sat
while the second moves a distance  v_sat(t?T) (since it was emitted time T later), and the source moves a distance v_st . From the diagram  we see that we have the relation

                                      
which gives
                                        
                                               T = 1/f_s

                                             

The wavelength  

                               , is measured by someone at rest in the medium.


 
                        Now relate this new wavelength to the frequency heard by the  detector. But the speed of sound is the same for the detector, so we still have  the relation 

                                        , and

                                           
                               This is the doppler formula for a moving source.