# Doppler Effect Phenomenon

Doppler effect is a common phenomenon that we observe in our everyday lives. For instance, if you are standing on the footpath, you hear an ambulance approaching in the distance. First, the sound of the siren is faint and it gets louder as the vehicle reaches closer to you. Once the ambulance crosses you and goes ahead the sound gradually decreases again. This can be described as the Doppler effect.

Generally, the Doppler effect can be described as the change in wave frequency (whether it is light or sound) during relative motion between the source of the wave and the observer. This effect can be observed every time a source wave is moving in relation to the observer.

## What is Doppler Effect?

When the source and observer are moving relative to each other, the frequency observed by the observer (fa) is different from the actual frequency produced by the source (f0). This is basically the Doppler effect. Here, when the source of waves is moving towards the observer they will have an upward shift in frequency. As for the observers from whom the source is receding, there will be a downward shift in frequency.

However, it should be noted that the effect does not occur as a result of the actual change in the frequency of the wave source. The Doppler effect can be observed in both sound waves and light waves.

The main reason that we experience this effect is that as the wave source moves toward the observer, each new wave crest that is formed from the source is emitted from a location that is closer to the observer. Therefore, as the source moves closer and closer the waves will now take less time to reach the observer or the time between the arrivals of new wave crests is reduced. This further causes an increase in frequency. Similarly, when the source of waves is going away, the waves are emitted from a farther location thus increasing the arrival time between each new waves. This leads to a reduction in frequency.

Nonetheless, from what we have learnt above, we can summarize that the Doppler effect could result from factors such as the motion of the observer, the motion of the source, or motion of the medium. This is mainly true for sound waves.

Whereas, for waves that can travel in any medium, such as light, we need to consider only the relative difference in velocity between the source and the observer.

## Doppler Effect Formula

The general form of the Doppler Effect formula is expressed as:

$f = \left ( \frac{c\pm v_{r}}{c\pm v_{s}} \right ) f_{o}$

C = propagation speed of waves in the medium;

Vr = speed of the receiver relative to the medium, +c if the receiver is moving towards the source, -c if the receiver is moving away.

Vc = speed of the source relative to the medium, +c if the source is moving away -c if the source is moving towards the receiver.

## Doppler Effect Cases

When we say source and observer are moving relative to each other, we get different cases as follows.

Case I: An observer moving away from the stationary source.

Case II: An observer moving towards the stationary source.

Case III: A source moving away from the stationary observer.

Case IV: A source moving towards the stationary observer.

Case V: A source and an observer moving towards each other.

Case VI: A source and an observer moving away from each other.

Case VII: A source and an observer moving in the same direction along the direction of the velocity of sound (v).

Case VIII: A source and an observer moving in the same direction opposite to the direction of the velocity of sound (v)

When the source and observer are relatively at rest with respect to each other, then the frequency heard by the observer is equal to the actual frequency produced by the source.

fa = fo

In this situation, the number of cycles passing through the point of observer remains constant with time. As the observer moves towards the source (or) source moves towards the observer, the observer experiences sound waves in relatively lesser time as the relative separation decreases. So from the formula$F=\frac{1}{T}$, the observer experiences a frequency more than the actual frequency produced by the source

$\Rightarrow \,{{f}_{a}}>{{f}_{o}}$

As the observer moves away from the source (or) source moving away from the observer, the observer hears sound waves after a relatively long time as the relative separation increases, so from the formula $F=\frac{1}{T}$ the observer experiences a frequency less than the actual frequency produced by the source.

$\Rightarrow \,{{f}_{a}}<{{f}_{o}}$

Considering,

Case I: In this case, the observer is moving away from a stationary source. It implies that

${{f}_{a}}<{{f}_{o}}$ $\Rightarrow \,{{f}_{a}}=k\,{{f}_{o}}\,\,\,\,\,\,\,\,\,\,where\,(k<1)$

Let k be a constant which is less than 1

The relative velocity of sound with respect to moving observer is v – vo.

$\Rightarrow \,\,\,\,V-{{V}_{o}}\,<\,v$

Therefore,

$\,\,\,\,\,\,\,\,\,\frac{v-{{v}_{o}}}{v}<1$

Comparing this with k < 1

We get,

${{f}_{a}}=\left( \frac{V-{{V}_{O}}}{V} \right){{f}_{o}}$

Case II: In this case, the observer is moving towards the stationary source, it implies that

Fa > Fo

$\Rightarrow \,\,\,{{F}_{a}}=k{{F}_{o}}\,\,where\,\,(k>1)$.

Let k is a constant which is greater than 1.

The relative velocity of sound with respect to a moving observer is V + Vo.

$\Rightarrow \,v+{{v}_{o}}>v$

Therefore,

$\,\,\,\,\frac{V+Vo}{V}>1$

Comparing this with (k > 1)

We get,

${{f}_{a}}=\left( \frac{V+{{V}_{O}}}{V} \right){{f}_{o}}$

Case III: In this case, source moving away from the stationary observer. It implies that,

Fa < fo

$\Rightarrow \,\,fa=k\,fo\,\,\,\,\,\,\,\,\,\,where\,(k<1)$

Let k be a constant which is less than 1.

The relative velocity of sound with respect to moving source is V + Vs

$\Rightarrow \,\,V+Vs>V$

Therefore,

$\,\,\,\,\frac{V}{V+Vs}<1$

Comparing this with k < 1

We get,

${{f}_{a}}=\left( \frac{V}{V+Vs} \right){{f}_{o}}$

Case IV: In this case, the source is moving towards the stationary observer. It implies that

Fa > f

$\Rightarrow \,\,fa=k\,fo\,\,\,\,\,\,\,\,\,\,where\,(k>1)$

Let k be a constant which is greater than 1.

The relative velocity of sound with respect to moving source is V – Vs

$\Rightarrow \,\,\,V-Vs

Therefore,

$\,\,\,\,\,\frac{V}{V-Vs}>1$

Comparing this with (k > 1)

We get,

${{f}_{a}}=\left( \frac{V}{V-Vs} \right)fo$

Case V: In this case, source and observer are moving towards each other that implies Fa > fo

(or) fa = kfo where (k > 1)

Let k be a constant which is greater than 1

The relative velocity of sound with respect to the moving source is V – Vs.

The relative velocity of sound with respect to the moving observer is V + Vo.

$\Rightarrow \,\,\,V-VsV$ $\,\,\,\frac{V}{V-Vs}>1\,\,\,\,\,\,\,\,and\,\,\frac{V+Vo}{V}>1$

From above we can say that

$\frac{V}{V-Vs}\times \frac{V+Vo}{V}>1$ $\Rightarrow \,\,\frac{V+Vo}{V-Vs}>1$

Comparing this with k > 1

We get, ${{f}_{a}}=\left( \frac{V+Vo}{V-Vs} \right)fo$

Case VI: In this case, source and observer are moving away from each other which implies that, Fa < Fo

(or) Fa = KFo where (K < 1)

Let k be a constant which is less than 1

The relative velocity of sound with respect to the moving source is V + Vs.

The relative velocity of sound with respect to moving observer is V – Vo.

$\Rightarrow \,\,\,\,V+Vs>V\,\,\,\,\,\,\,and\,\,\,\,V-Vo $\,\,\,\,\,\frac{V+Vs}{V}>1\,\,\,\,\,\,\,\,\,and\,\,\,\,\frac{V-Vo}{V}<1$

From above we can say that,

${}^{\frac{V-Vo}{{V}}}/{}_{\frac{V+Vs}{{V}}}\,\,\,<1$ $\frac{V-Vo}{V+Vs}<1$

Comparing this with k < 1

We get, $Fa=\left( \frac{V-Vo}{V+Vs} \right)Fo$

Case VII: in this case, both source and observer are moving in the same direction along the direction of the velocity of sound (v).

When both are moving in the same direction, if suddenly observer comes to rest then the source will be moving towards the observer.

$\Rightarrow Fa=\left( \frac{V}{V-Vs} \right)Fo\,\,\,\,\,\to \,\,(a)$

When both are moving in the same direction, if suddenly source comes to rest, then the observer will be moving away from the source.

$\Rightarrow \,\,Fa=\left( \frac{V-Vo}{V} \right)Fo\to (b)$

Combining (a) and (b) we get

${{F}_{a}}=\left( \frac{V-Vo}{V=Vs} \right)Fo$

Case VIII: in this case, both source and observer are moving in the same direction but opposite to the direction of the velocity of sound (v).

When both are moving in the same direction, if suddenly observer comes to rest, then the source will be moving away from the observer.

$\Rightarrow Fa=\left( \frac{V}{V+Vs} \right)Fo\,\,\,\,\,\to \,\,(a)$

When both are moving in the same direction, if suddenly source comes to rest, then the observer will be moving towards the source.

$\Rightarrow \,\,Fa=\left( \frac{V+Vo}{V} \right)Fo\to (b)$

Also Read: Longitudinal and Transverse Waves

## Effect Of Motion Of Medium On Apparent Frequency

Till now we assumed medium at rest, if the medium is in motion, then two cases arise

(i) When medium moves from source to observer:

If the wind is blowing with speed Vw from the source to the observer that is in the direction of the velocity of sound, then the velocity of sound v is taken as V + Vw in all the above formulae which are derived taking medium at rest.

(ii) When medium moves from observer to source:

If the wind blows with a velocity V – Vw in all the above formulae derived taking medium at rest.

### Special cases:

Case I: Doppler’s effect in reflected sound

Consider a source of sound moving towards a stationary wall with a speed Vs emitting a sound of frequency fo. To a stationary observer o, the apparent frequency fa of sound heard by the observer directly from the source is

$Fa=\left( \frac{V}{V+Vs} \right)Fo$

To find the frequency of the reflected sound heard by the observer, assuming the image of the source s1 is the wall, then the apparent frequency is,

$F{{a}^{1}}=\left( \frac{V}{V-Vs} \right)Fo$

Beat frequency heard by the observer is,

$F=F{{a}^{1}}-Fa=\left( \frac{V}{V-Vs}-\frac{V}{V+Vs} \right)Fo$ Ɵ

$F=\left( \frac{2VVs}{{{V}^{2}}-V_{s}^{2}} \right)Fo$

Case II: Sound source and observer are moving in a direction making an angle with the line joining them.

Till now we discussed only the cases in which source and observer are along the same line and joining them. If the motion is along some other direction, join source and observer by a straight line then the components of velocities along the line joining them must be considered.

For example, from the figure,

$Fa=\left( \frac{V+Vo\cos \theta }{V+Vo\sin \theta } \right)Fo$

Case III: Source (or) observer rotating:

1. When the source is rotating:

When the source moves towards the observer, then the frequency heard is maximum

${{F}_{\max }}=\left( \frac{V}{V-Vs} \right)Fo$

When the source is moving away from the observer, the frequency heard is minimum. ${{F}_{\min }}=\left( \frac{V}{V+Vs} \right)Fo$

If the observer is situated at the centre of rotation then there will be no change in frequency of the sound heard.

2. When the observer is rotating:

When the observer moves towards source then the frequency heard is maximum and is given by

${{F}_{\max }}=\left( \frac{V+Vo}{V} \right)Fo$

When the observer moves away from the source, then the frequency heard is minimum and is given by

${{F}_{\min }}=\left( \frac{V-Vo}{V} \right)Fo$

When the source is at the centre of rotation then there will be no change in frequency of the sound heard.

## Application of Doppler Effect

Some of the common applications of Doppler effect include;

• Astronomy