Doppler  effect    When  there  is a relative motion between the source of sound and the listener, due to either the motion of the source,  or  the  motion  of the listener, the frequency of sound appears  different  to  the  listener.  This  is  called Doppler effect.  The  Doppler  effect  occurs  in  all  types  of  waves including light.

We  consider  only  a  special  case  when the motion of the source  and  the listener  are along the line joining them. The following  convention  of  signs  is followed for velocities. We take  the  positive direction as that from the position of source to  position of listener (see fig.d14). Consider first the source moving towards the listener, with velocity V_s which is positive. In  an  interval  of time t, it will produce n t crests in the medium, where n is the frequency. In the same interval of time the  source  will move a distance V_s t, towards the source. Therefore,  within  a  distance (V - V_s) t, there will be n t crests,  where V is the velocity of sound. This implies that the wavelength of sound will be shortened to

l ¢ = [(V-V_s) t]/(tn ) =(V - V_s)/ n

The corresponding frequency is

n ¢ = n ( V/(V - V_s))

will be higher than the frequency of sound emitted by the source.

Now,  consider  the  motion  of  the  listener  with  source stationary. If the listener is moving with velocity -V_L, towards the  source, it will encounter larger number of crests than if it had been stationary. In this case n ¢ is,

n ¢ =n (V+ V_L)/V

The  apparent frequency when both, the source and the listener is in motion with velocities + Vs and +V_L is

n ¢ = n [( V- V_L)/(V - V_s)]

If the medium is also in motion, its velocity being V_M, then

n ¢ = n [( V +V_M- V_L)/(V+V_M - V_s)]

For  light waves the situations `source receding from observer' or `observer receding from source' are exactly identical, since there exist no medium  relative to which the source and observer move. We must also take into account the effect of time dilation of relativity, for  relative  velocities  of  source  and observer comparable to velocity of light. The Doppler's frequency thus calculated is,

n ¢ = n [( c- V)/(c + V)]^1/2 (receding)

where  V is the relative velocity between the source and  observer.  If  the source and observer are approaching then,

n ¢ = n [( c + V)/(c - V)]^1/2 (approaching)