Doppler Effect

DOPPLER EFFECT

The pitch or frequency of the whistle of a moving train appears to increase as the sound source approaches a stationary observer. The pitch or frequency decreases as the source of sound recedes away from the stationary observer. This apparent change in frequency was first observed by Doppler in 1845.

Definition

The phenomenon of the apparent change in the frequency of the sound due to relative motion between the source of sound and the observer is called Doppler effect.

1. Both source and observer at rest

Suppose S and O are the positions of the source and the observer respectively. Let n be the frequency of the sound and v be the velocity of sound.

In one second, n waves produced by the source travel a distance SO = v as shown in fig. 3.21

The original wavelength is λ = v/n      ………(1)

The original frequency n = v/λ          ………..(2)

2. When the source moves towards the stationary observer

If the source moves with a velocity vs towards the stationary observer then after one second the source will reach Sʹ such that SSʹ = vs.

Now n waves emitted by the source will occupy a distance (v - vs ) only as shown in fig. 3.22

Therefore, the apparent wavelength of the sound,

The apparent frequency

Comparing equations (2) and (4) we can conclude that n' > n, the pitch or frequency of the sound appears to increase.

3. When the sources moves away from the stationary observer

If the source moves away from the stationary observer with velocity vs

The apparent frequency

Comparing equations (2) and (5) we can conclude that n' < n, the pitch or frequency of the sound appears to decrease.

For a moving source, the change in frequency occurs because although the source emits waves at constant rate, the waves emitted in a specified time occupy a shorter or longer length than when the source is stationary. Consequently, there is apparent change in wavelength and hence the frequency changes.

4. Source is at rest and observer in motion

S and O represent the position of source and observer respectively. The source S emits n waves per second having a wavelength

Consider a point A such that OA contains n waves which crosses the ear of the observer in one second as shown in figure 3.23 (a).

That is, when the first wave is at the point A, the nth wave will be at O where the observer is situated.

5. When the observer moves towards the stationary source

Suppose the observer is moving towards the stationary source with velocity v0. After one second the observer will reach the point Oʹ such that OOʹ = v0.

The number of waves crossing the observer will be n waves in the distance OA in addition to the number of waves in the distance OOʹ which is equal to as shown in figure 3.23(b).

Therefore, the apparent frequency of sound

The apparent frequency of sound

Comparing equations (2) and (6) we can conclude that n' > n, the pitch of the sound appears to increase.

6. When the observer moves away from the stationary source

Suppose the observer is moving towards the stationary source with velocity v0.

Therefore, the apparent frequency of sound

The apparent frequency of sound

Comparing equations (2) and (7) we can conclude that n' < n, the pitch of the sound appears to decrease.

Similarly for a moving observer the change in frequency occurs because although the source emits at constant rate, the waves emitted in a specified time is heard by the observer or listener receives longer or smaller number of waves than the observer is stationary relative to the source.

Consequently there is apparent change in wavelength and hence the frequency changes.

Suppose the wind is moving with a velocity ω in the direction of propagation of sound, the equation for the apparent frequency becomes

Applications of Doppler effect 

(i) To measure the speed of an automobile

An electromagnetic wave is emitted by a source attached to a police car. The wave is reflected by a moving vehicle, which acts as a moving source. There is a shift in the frequency of the reflected wave. From the frequency shift using beats, the speeding vehicles are trapped by the police. 

(ii) RADAR (Radio detection and ranging).

A RADAR sends high frequency radiowaves towards an aeroplane. The reflected waves are detected by the receiver of the radar station. The difference in frequency is used to determine the speed of an aeroplane.

(iii) SONAR (Sound navigation and ranging)

Sound waves generated from a ship fitted with SONAR are transmitted in water towards an approaching submarine. The frequency of the reflected waves is measured and hence the speed of the submarine is calculated.

(iv) Blood flow meter

Ultrasonic sounds are transmitted towards organs the frequency change in reflected waves used to measure blood flow rate.

(v) Tracking a Satellite

The frequency of radio waves emitted by a satellite decreases as the satellite passes away from the Earth. The frequency received by the Earth station combined with a constant frequency generated in the station gives the beat frequency. Using this, a satellite is tracked.

(vi) Stars moving towards the earth or away from the earth

There is an apparent change in wavelength of spectral lines emitted by a moving star. If the star is moving away from the Earth, there is a shift towards the red end and if it is approaching towards the Earth, a shift towards the violet end of the spectrum is indicated. This spectral shift enables the velocity of star to be computed along the line of sight.