S : Source O : Observer V : velocity of the wave v O vovo S vsvs v vsvs v O vovo S v vsvs v vovo S v O DOPPLER EFFECT

S Velocity of the source : v s, frequency of the wave at the source : f s Velocity of the observer : v o, frequency of the wave at the observer : f o Velocity of the wave : v, medium (air) is not moving The initial distance between the source and the observer is x vsvs O vovo t = t 1 S vsvs O vovo t = t 2 S vsvs O vovo t = t 3 S vsvs O vovo t = t 4 v v x Example : An ambulance moving to the right direction with the velocity v s. The siren observed by an observer that is also moving to the right with the velocity v o as shown in the diagram below ;

The distance passed by the wave as  t = t2 t2 - t1 t1 is v (t 2 - t 1 ) = x + vo vo (t 2 - t1)t1) x S vsvs O vovo t = t 1 S vsvs O vovo t = t 2 v v (t 2 - t 1 ) x = ( v - v o ) ( t 2 - t 1 ) 1. The wave sent at t 1 is received at t 2 2. The wave sent at t 3 is received at t 4 3. The number of the wave sent between (t 1 – t 3 ) is equal to number of the wave received between (t 2 – t 4 )

SO t = t 1 S vsvs O vovo t = t 2 SO t = t 3 S vsvs O vovo t = t 4 v v (t 4 – t 3 ) The distance passed by the wave as  t = t 4 – t 3 is v (t 4 – t 3 ) = x – v s (t 3 – t 1 ) + v o (t 4 – t 1 ) x

(v - v o ) (t 4 – t 2 ) = (v - v s ) (t 3 - t 1 ) The number of waves sent by the source between (t 3 - t 1 ) is equal to the number of waves received by the observer between (t 4 – t 2 ) : f s (t 3 - t 1 ) = f o (t 4 – t 2 )   x = (v - v o ) (t 2 - t1)t1) v (t 4 – t 3 ) = (v - v o ) (t 2 - t 1 ) - vs vs (t 3 - t 1 ) + vo vo (t 4 - t1)t1) v - t 2 ) - v (t 3 - t 1 ) = vo vo (t 4 - t 1 ) - vo vo (t 2 - t 1 ) - vs vs (t 3 - t1)t1) v (t 4 – t 2 ) - v (t 3 - t 1 ) = vo vo (t 4 – t 2 ) - vs vs (t 3 - t1)t1)

When the medium (wind) is moving with its velocity vm vm : v o, v s and v m are positive when they are parallel to the v, they are negative when antiparallel to the v

Exercise : An ambulance activated its siren while moving to the north with a constant speed 10 m/s. At the same time an observer was moving to the south with a constant speed 4 m/s. At the time the wind was moving horizontally to the south-east with a constant speed 2 m/s. If the frequency of the siren produced by the ambulance is100 kHz, what is the frequency received by the observer when : 1. The observer was at the front of the ambulance 2. The observer was behind of the ambulance ambulance wind

ambulance