## JEE main and advanced : The Doppler Effect

In this blog I will be talking about the Doppler Effect for JEE Main and Advanced.

The Doppler effect, also known as Doppler displacement, is a phenomenon that occurs in any kind of wave, being it electromagnetic waves, like light, or mechanical waves, like sound, when a relative motion exists between the emission source of the waves and the observer or receptor.

The phenomenon was first described in 1842 by the Austrian physicist Christian Doppler, and consists in an apparent change in frequency and wave length provoked by the relative movement between the transmitter and the receptor. If the transmitter goes near the receptor, the length of the wave appears to decrease and the frequency appears to increase. If both the transmitter and the receptor stray away from each other, the wave length appears to increase and the frequency appears to decrease. All of this happens without the emitted wave length and frequency actually changing during the displacement of the emission source.

## Doppler effect in sound

In daily life, and according to the human perception capacity, the Doppler effect can be easily experienced with some kinds of mechanical waves. For example, the mechanical waves that make up sound. One of the typical examples are the sirens in an ambulance or a police car.

Let us suppose that we are standing in the side walk and an ambulance comes down the street and drives next to us, at 100 km/h, which is 8% of the speed of sound (1235 km/h).

When the ambulance is approaching, we hear the sound getting more acute, due to the shortening of the sound wave length. But this is an apparent shortening. Every wave emitted in front of the ambulance is closer to the previous wave by the movement of the ambulance in that same direction.

If the ambulance moves forward at a speed equivalent to 8% of the speed of sound, the wave length will be reduced by 8%, if compared with the real length wave emitted by the siren. On the frequency it has the opposite effect, it will increase by 8%, and that is why we hear the siren with acuter tones.

Once the ambulance passes in front of us and moves away, we start to hear that the sounds turns out to be more like a bass, due to the relative increase in the wave length and the decrease of the frequency. Behind the ambulance, every emitted wave would be 8% further away from the previous one.

If the ambulance managed to move exactly at the speed of sound, then there wouldn’t be waves of sound in front of it. The same ambulance will move just in front of the sound waves, and the wave length will be compensated by the ride of the ambulance. We would not be able to listen to the ambulance until it was right next to us.

If the ambulance could travel at a faster speed than sound, the front of the sound waves will be behind the ambulance, since the waves would go slower than the ambulance itself, and they would be left behind as soon as they were emitted.

In the next animations we can see the Doppler Effect in sound waves emitted by a stationary object, and a moving object in subsonic, sonic and supersonic speeds.

## Doppler effect in light

To experience the Doppler effect in light, and other electromagnetic waves, it’s necessary to have relative travel speeds between the transmitter and the observer that are far greater than the ones we studied in the previous paragraphs, since the speed of light is faster than the speed of sound.

The changes in the wave length of light are very small at ordinary speeds, like the ones we as human experience. Even at the speeds at which stars and galaxies move it’s necessary to use high sensitivity devices to detect the Doppler Effect in the light they emit.

Precisely in astronomy, the Doppler effect is responsible for the displacement towards the red of the light of the stars and galaxies that are far away. This stars and galaxies are moving away from us at high speeds and their light reaches us displaced towards longer length waves, meaning, towards the zone of the electromagnetic spectrum that corresponds to the red.

The displacement or shift to the red color in the light of far away stars and galaxies is one of the first observational tests that the actual Universe is in expansion. And not only that, this also indicates that the farther away a galaxy is, the fastest is moving away from us, which corresponds with the Inflationary Universe theory.

In the case of the displacement of electromagnetic waves in vacuum, the Doppler effect only is consequence of the relative displacement between source and observer. In the case of waves that obligatorily need a medium to move, such is the case of the mechanical waves that constitute sound per example, the characteristics and movement of the medium can also affect the Doppler effect.

## Formulas

Let us suppose an observer at rest and a wave transmitter source with a determined frequency f0. The speed of the wave propagation is independent from the displacement of the source, and the speed of the observer is zero. Then, the observed frequency in a given moment is described by the following formula:

Where:

• f is the observed frequency and f0 the real wave frequency emitted by the source
• c is the propagation speed of the wave
• vs is the displacement speed of the transmitter

If the observer is in motion and the source is at rest, the previous formula would be like:

Where vr would be the displacement speed of the observer or receptor.

Both cases would respond to the general formula: