# Doppler Effect | Derivation of Doppler effect formula

In this lecture, we will go to learn about what is Doppler effect and how the radar works on the principle of the Doppler effect.

## Introduction to Doppler Effect

Radar works on the principle of the Doppler effect are,

• A pulse Radar, which is transmitting the burst of pulses of EM energy and received back the reflected signal from the target.
• The time taken by the EM energy to travel up to the target and come back to the radar antenna will be considered.
• In other words, we can say that the separation of the signal of the echo and the energy transmitted by the transmitter is made on the basis of the difference in time.
• If the radar transmitter transmits energy continuously then the separation of transmitted energy and the received energy is not possible as it is done in the pulsed radar.
• We have to use a separate antenna for transmission and reception both and separating of the received signal from the transmitted signal in terms of frequency not in the time as in the pulsed radar.

You can also learn the Topics

## Doppler Effect

• Everyday life has multiple examples of the Doppler phenomenon with sound; the whistle from a moving train is a good example.
• As the train approaches a stationary listener, the pitch(frequency) of the whistle sounds higher than when a train passed by, at which time the pitch sounds the same as if the train were stationary. As the train recedes from the listener, the pitch decreased.
• so, as the listener on the train approaches the stationary horn, the pitch of the horn sounds higher, as the train recedes from the stationary horn, the pitch sounds lower.
• Electromagnetic waves radiated by radar, as well as sound waves, obey the Doppler principle, although electromagnetic waves travel at the speed of light, and audio waves travel at the speed of sound.
• The Doppler Effect is a frequency shift that results from relative motion between a frequency source and a listener.
• If both source and listener are not moving with respect to each other then no Doppler effect will be observed even if they are traveling at the same speed in the same direction.
• If the source and listener are moving closer to each other then the listener will perceive a higher frequency- the faster the source or receiver is approaching the higher the Doppler Shift.
• If the source and listener are moving away from each other then the listener will perceive a lower frequency- the faster the source or receiver is moving away the lower the Doppler Shift.
• The Doppler shift is directly proportional to the speed between the source and listener, the frequency of the source, and the speed at which the wave travels.

## Doppler Effect equation

• Suppose the range of the target is R and the wavelength \lambda.
• The energy travel from the radar antenna to the target and from the target to the radar then the total wavelength is given by \frac{2R}{\lambda} and there is 2\pi radian is the phase change.
• The unit for R and the \lambda is considered the same.
• As the energy travel two-way so than the total phase change \phi is,

• The R will change continuously as the target is in motion so the phase will also change accordingly. A change in phase with respect to the time may be called angular frequency w may be given by,

w=2\pi f_d= \frac{d\phi}{dt}…….Eq.(1)

= (\frac{4\pi }{\lambda}).(\frac{dR}{dt})…….Eq.(2)

therefore,

\boxed{w= \frac{4\pi V_r}{\lambda}}…….Eq.(3)

where,

• f_d=Doppler frequency shift
• V_r=radial velocity or rate of change of range with time
• If the angle between the target and the radar antenna is \theta, then the radial velocity may be written as vcos\theta as shown in the figure below,
• from the eq.(1) and eq.(3) we can write the modified equation s below;

2\pi f_d= \frac{4\pi V_r}{\lambda}

\boxed{f_d= \frac{2 V_r}{\lambda}}…….Eq.(4)

\boxed{f_d= \frac{2 f V_r}{c}}…….Eq.(5)

• Eq. (4) & (5) are the final equation of the Doppler Frequency.

where,

• f=transmitted frequency
• c=velocity of EM waves =3 x 108 m/sec
• f_d=Doppler Frequency Shift  Join Our Telegram Group!