Doppler effect, different cases of doppler effect, practice problems, FAQs

Have you seen a speed gun? It is used to check the speed of a moving vehicle. It has a transmitter and a receiver. The speed gun transmits a signal toward the moving vehicle and it receives the signal back after it bounces from the target. If a vehicle is approaching the gun, the returning frequency will be higher and if the vehicle is moving away, the frequency will be lower. With this change in frequency guns calculate the speed of the vehicle. This change in frequency can be explained by the doppler effect. Let’s see what the doppler effect is!

Table of content

• Doppler effect
• Source moving and observer stationary
• Observer Moving and source stationary
• Both source and observer moving
• Practice problem
• FAQs

Doppler effect

Doppler effect is a physical phenomenon in which change in frequency is observed when there is relative motion between the source and observer.This is also called doppler shift.

Doppler shift can be understood by taking various cases as described below.

Source moving and observer stationary

Let a source S moving with velocity and an observer is stationary in a frame in which the medium is also at rest.Taking the direction from the observer to the source as the positive direction of velocity. Let the speed of a sound of angular frequency and period , both measured by an observer at rest with respect to the medium, be . Assuming that the observer has a detector that counts every time a wave crest reaches it. 

Let a source S moving with velocity and an observer be stationary in a frame in which the medium is also stationary.Take the direction from the observer to the source as the positive direction of velocity. Let the speed of a sound of angular frequency and period , both measured by an observer who is stationary with respect to the medium, be . Assuming that the observer holds a detector that counts every time any wave crest reaches it.

Initially at time t=0 the source is at point , Which is at a distance L from the observer. If Source emits a crest that will reach the observer at time . At time the source has moved a distance and is at point ,Which is at a distance from the observer. At , the source emits a second crest. This reaches the observer at

The observer will listen crest at

Hence in time interval 

the detector will count n crests with the period given by

As we know so rewriting the equation

Where is the frequency of wave when source is stationary and is the frequency when the source is moving. 

If is small as compare to taking binomial expansion to terms in first order and neglecting higher power may be approximated, giving

If the source is moving toward the observer we replace by and observed frequency

Observer Moving and source stationary

Let us consider the source is stationary and oveserveris moving with velocity towards the

Source.

So the source and medium are approaching at speed and the speed with which the sound wave approaches is . The time interval between the arrival of the first and the the crests will be

The period of wave measured by observer,

is small as comcare to so can be neglected

As we know so rewriting the equation

Using binomial theorem

If the observer is moving away from the source we replace by and observed frequency

From the above equations we can see Doppler shift is almost same whether it is the observer or the source moving

Both source and observer moving

Now when both the source and the observer are moving. Taking the direction from the observer to the source as the positive direction. Let the source and the observer be moving with velocities and respectively . 

Let at time t = 0, the observer is at and the source is at and the distance between them is L. The source is emitting a sound wave of velocity , of frequency and period measured by an observer at rest with respect to the medium. The source emits the first crest at t=0. Because the observer is moving, the velocity of the wave relative to the observer is . Therefore, the first crest reaches the observer at time . 

At time , both the observer and the source have moved to their new positions and respectively. The new distance between the observer and the source will be . At the source will emit a second crest and time to reach the observer is

At time the source emits its crest and this will reach at

Hence in the time interval

The observer will count n crests so the period

Hence frequency observed

This is the general formula for finding the doppler shift.

Practice problem

Q1.A cell phone emitting sound of 350 Hz is dropped from a balloon rising vertically upwards with constant velocity 5 m/s. The frequency of sound after 2s as observed by man in a balloon is (Velocity of sound in air is 335 m/s and acceleration due to gravity is dropped from a balloon moving vertically upwards with constant speed 5 m/s. The frequency of sound after 2 s which is observed by man in a balloon is (Velocity of sound in air can be taken as 335 m/s and the value of acceleration due to gravity is 10 m/s).
Answer. Velocity of observer

Velocity of sound

Using doppler shift equation 

As observer is moving opposite to the direction of source so put in place of

Q2. A sound speaker is moving towards man with a speed of 20 m/s and having frequency 240 Hz and the man is moving towards a source with a velocity of 20 m/s. What is the apparent frequency heard by the man, if the velocity of sound is 340 m/s?

Answer. Velocity of observer

Velocity of sound

Using doppler shift equation 

As source is moving toward to the source so put in place of

Q3. An observer is moving towards a stationary source of sound, with a velocity one-fourth of the velocity of sound. What is the percentage increase in the apparent frequency?
Answer. Given

Using doppler shift equation 

The increase in the apparent frequency is

%

Q4. A sound source moves towards stationary listener with of the speed of sound. What is the ratio of apparent to real frequency is:

Answer. Given

Using doppler shift equation 

FAQs

Question 1. What is the limitation of the doppler effect?
Answer:  Velocity of source and observer should be less than the velocity of sound.

Question 2. For which sound source doppler effect is not applicable?
Answer: it is not applicable for supersonic sound sources because they produce shock waves.

Question 3.What is the use of the doppler effect?
Answer:  Doppler effect used in various form

1. In Flow Measurement
2. In Vibration measurement
3. In Velocity profile measurement
4. In Military

Question 4. Does wave speed change in doppler effect?
Answer: No, wave speed depends only on the medium.