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Doppler Effect - an Illustration

 Illustration of Doppler Effect

Introduction

There are multiple articles and videos available on the internet which explain the Doppler effect, mostly using the fact that the waves get compressed when an object making sound approaches you and that the waves are stretched when the object moves away from you. These explanations are good enough, but don’t provide a deeper understanding. Also derivation of the formula for frequency change is left to the reader. This post is an attempt to explain in a layman’s terms the effects on the frequency of sound we hear. 

Subsonic & Mach 1 speeds

In all the examples below, it’s assumed that the police car (Well, considering the speeds in the examples, it would be more like a jet!) running the siren is 12 seconds away initially, it approaches and subsequently passes you. You stop hearing the sound once it gets 12 seconds away from you. 

Approaching car

Case 1 - Cop car traveling at 1/4th the speed of sound

The initial sound emitted by the car will reach you in 3s (4 times the speed of the car), whereas when the car passes you after a total of 12s, you will immediately hear the sound. This means, all the sound waves created by this car in these 12s reach you in 9s (3s to start hearing and 12s when it passes). Given that all these 12s worth waves are compressed in 9s, the frequency of the sound you will hear is 

12/9 or 4/3*(Fs)


Case 2 - Cop car traveling at 1/3rd the speed of sound

The initial sound emitted by the car will reach you in 4s (3 times the speed of the car), whereas when the car passes you after a total of 12s, you will immediately hear the sound. This means, the sound waves created by this car in these 12s reach you in 8s (4s to start hearing and 12s when it passes). Given that all these 12s worth waves are compressed in 8s, the frequency of the sound you will hear is 

12/8 or 3/2*(Fs)


Case 3 - Cop car traveling at 1/2 the speed of sound

The initial sound emitted by the car will reach you in 6s (2 times the speed of the car), whereas when the car passes you after a total of 12s, you will immediately hear the sound. This means, the sound waves created by this car in these 12s reach you in 6s (6s to start hearing and 12s when it passes). Given that all these 12s worth waves are compressed in 6s, the frequency of the sound you will hear is 

12/6 or 2*(Fs)


In general, we can come up with a formula

Frequency heard (Fh)  = (speed of sound)/(speed of sound - speed of car) * Fs


So in case 1, 

Fh = (speed of sound)/(speed of sound - ¼speed of sound)*Fs = 1/(1-¼)*Fs = 4/3*Fs


You may verify the same for other cases.


Case 4 - Cop car travels at the speed of sound

The initial sound emitted by the car will reach you in 12s (same as the speed of the car), whereas when the car passes you after a total of 12s, you will immediately hear the sound. In fact, any sound emitted by the siren during these 12s will reach you at the same time when the car passes you. That means ALL these waves are compressed in that moment, theoretically making the heard frequency infinite (that’s a reason traveling at exactly the speed of sound makes the vehicle unstable). You can verify it with the formula we derived above.


Departing car

Case 1 - Cop car traveling at 1/4th the speed of sound

The initial sound emitted by the car will immediately reach you whereas the sound emitted 4s later will reach you in another 1s (total of 5s); sound emitted 8s will reach you in another 2s (total of 10s) and sound emitted 12s after it passes you will reach you in another 3s (total of 15s). So you will hear the sound generated in those 12s over 15s, which means the waves are rarefied by a factor of 1.25, decreasing the heard frequency that much making it 0.8. Now,


Fh = (speed of sound)/(speed of sound + speed of car) * Fs, which is

Fh = (speed of sound)/(speed of sound + ¼ speed of sound)*Fs = 1/(1+¼)*Fs = 0.8*Fs


Similar explanation for other cases when the car is departing at ⅓ or ½ or same speed as sound.


Supersonic speeds


It gets interesting from here - what happens when the object is traveling at supersonic speeds.


Take the same starting point - the car is 12s away from you. Since the car is traveling at supersonic speed, sound reaches you AFTER the car has passed you. For the explanation below, the car (or jet?) is assumed to travel at Mach 2. 


This means, the sound emitted by the siren 12s back will reach you in 24s (12s after the car passes you). The sound emitted when the car was 10s away will reach you in 20s; when it was 8s away, the sound will reach you in 16s. See this interesting phenomenon!


So you won’t hear anything until the car passes you and then you suddenly start hearing it in REVERSE! 


If the car comes to a stop just next to you, we are good to stop here (or are we - why the cop stopped near you!)


If the cop car departs at the same speed, we go into an even more interesting situation. The sound emitted when the approaching car was 1s away will reach in 2s (or 1s after the car passed you). This sound will be superimposed by the sound which was emitted by the departing car 1/3s after it passed you (which would have taken another ⅔s to reach you). You can extrapolate it further. This means if the car plays a song in reverse on approach at normal frequency and at three times the frequency after it departs, the subject will hear it with double the amplification, only if they can hear between the sonic boom created.

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