In this article, we will discuss the Doppler Effect. We will know how the Derivation of the Doppler Effect is important for waves and optics. After learning this we can justify the reasons behind the sound propagation mechanism in real-life situations.

The purpose of this article is to understand the Derivation of the Doppler Effect. You will also find many other important information that you might need to know. If you do not have time to read all of the information, then I would suggest reading the introduction section first.

Introduction

When there is relative motion between the source and observer the frequency of sound or light heard or seen by the observer is different from the actual frequency of the sound source. This phenomenon is called Doppler’s effect. For example, let’s take an example of a moving car. A boy is standing on the side of the road which is an observer. The car is moving with the velocity v and beeping horn of a certain frequency. Observers will hear the sound with a different frequency as compared with the frequency of the sound coming from the car’s horn. This difference is caused by the shrillness effect of the sound.

For better understanding let’s take two situations

1. Let the car is parked at point A and the observer is at d distance from the car. Now the car blows a horn which emits 100 waves per second. In this case, the observer will receive 100 waves of the horn. This means the sound’s frequency emitted from the car and received by the observer is the same.
2. Now the car is moving with velocity v towards the observer or the observer is moving towards the parked car with velocity v. In this case the waves received by the observer are more than 100. The frequency of sound received by the observer is called apparent frequency. In this case, the apparent frequency is increased which causes an increase in shrillness and pitch.
3. Let the car is moving in the opposite direction to the observer. In this case, the sound waves received by the observer will be less than the sound waves emitted by the car.

Assumptions

Following pre-assumptions will be used for the relation between true frequency and experienced frequency.

• Velocity of the source and observer should be less than the velocity of sound.
• Observers should be able to hear.

Derivation of Doppler’s effect

Case-1

Consider a source is stationary and the observer is moving with velocity V_o.

Fig- 1

Initially consider both the car and observer in rest.

Here, C is the velocity of the sound wave.

λs is the wavelength of the source

f_s is the frequency of sound waver

If the observer moves with speed Vo then the relative speed of the sound will be

Frequency of the sound will be

Here wavelength remains unchanged because the source is in rest.

Since,

So,

Now, is the observed frequency of the sound for this case. It is also called the apparent frequency of the sound.

Now, if the observer moves opposite to the source

The relative velocity between the source and observer is

And the apparent frequency should be

Case -2

Consider a source is moving with velocity Vs and the observer is stationary

Fig- 2

• In this case the source velocity is not imparted with the net velocity of the source.
• Here wavelength of the sound will change according to the position.

Where

C is wave velocity

λs is the wavelength of the source

T is the time taken by the wave

So,

Let d is the distance between the car and stationary observer.

. . . . . . . (1)

Here, Vs is velocity by which source is moving towards a stationary observer

The wavelength observed by the observer

From equation (1)

So,

Thus is the change in wavelength of the sound.

Now,

This will be the formula for observed frequency by the observer.

When the source moves opposite to the observer

Case-3

When the observer and source are both in motion.

In this case, it has the following cases.

• When observer and source both moves towards each other

Fig- 3

Here observer and source both imparted to increase the frequency of sound.

• If the observer and source both move in the same direction. The source moves away from the observer while the observer wants to catch the source.

Fig- 4

In this case observe wants to catch the source and increase the frequency. But the source is moving in the opposite direction and imparted to decrease frequency.

• Observer moving out from the source while source wants to catch observer

Fig- 5

Here observer imparted to decrease the frequency while source increases it.

• While observer and source both moves in opposite directions to each other.

Fig- 6

Here observer and source both imparted to decrease the frequency of sound.

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