A Classical Approach of Newtonian Mechanics

13 WAVE MOTION 13.6 The Doppler effect

that the first is emitted at time t = 0 , and the second at time t = T, where

T = 1/f is the wave period in the frame of reference of the source. At time t, the

first wave crest has traveled a distance d 1 = v t towards the observer, whereas

the second wave crest has traveled a distance d 2 = v (t − T) + vs T (measured

from the position of the source at t = 0 ). Here, we have taken into account the

fact that the source is a distance vs T closer to the observer when the second wave
crest is emitted. The effective wavelength, λJ, seen by the observer is the distance
between neighbouring wave crests. Hence,

λJ = d 1 − d 2 = (v − vs) T. (13.49)

Since v = fJ λJ, the effective frequency fJ seen by the observer is

fJ =

f

1 − vs/v

, (13.50)

where f is the wave frequency in the frame of reference of the source. We con-
clude that if the source is moving towards the observer then the wave frequency

is shifted upwards. Likewise, if the source is moving away from the observer

then the frequency is shifted downwards. This manifestation of the Doppler effect

should be familiar to everyone. When an ambulance passes us on the street, its

siren has a higher pitch (i.e., a high frequency) when it is coming towards us than

when it is moving away from us. Of course, the oscillation frequency of the siren

never changes. It is the Doppler shift induced by the motion of the siren with

respect to a stationary listener which causes the frequency change.

The general formula for the shift in a wave’s frequency induced by relative

motion of the observer and the source is

fJ =

1 ∓ vo/v

f, (13.51)

1 ± vs/v

where vo is the speed of the observer, and vs is the speed of the source. The
upper/lower signs correspond to relative motion by which the observer and the
source move apart/together.

Probably the most notorious use of the Doppler effect in everyday life is in

police speed traps. In a speed trap, a policeman fires radar waves (i.e., electro-
magnetic waves of centimeter wavelength) of fixed frequency at an oncoming