Doppler Effect

Introduction

Waves of all types suffer an interesting phenomenon as the speed of their source is varied. You have likely encountered this phenomenon in your own life. Recall what happens to the sound of a train’s horn as it passes by you at very high speed. This is an example of a Doppler shift in sound waves.

A similar phenomenon is observed for light waves as well. The Doppler effect plays a massive role in the calculation of the distance of stars and planets from our Earth. The same phenomenon is used by the speed radars that traffic police officials use to catch speeders. In this tutorial, we will discuss the Doppler effect in detail.

What is the Doppler Effect?

Doppler effect is also known as the Doppler shift, and it mention as the apparent change in the frequency of waves of any kind when their source is moving at a certain velocity for the observer. The phenomenon of the Doppler shift is named after Christian Doppler, an Austrian physicist who studied this effect in detail and described it in 1842.

If you cannot recall the sound of a train passing you by, then try to remember when a car or bike sped by you while sounding its horn. The horn's pitch rises and falls as the bike or car approaches you and recedes away from you.

As you can see in the image above, wavefronts emitted by a moving car are pressed closer together in the direction of its motion. That is, the wavelength appears to be smaller in the front than in the back. Since frequency and wavelength have an inverse relation, the sound heard by an observer in front of the car will have a higher frequency.

Apparent Frequency

You must have noticed how we have used the word “apparent” while discussing the Doppler shift. This is because the frequency of the waves only appears to change to the observer. In reality, the source continues to emit the sound at the same frequency as before. Thus, the Doppler effect can also be termed an illusion.

The term “apparent frequency” is used to refer to the changed frequency observed due to the Doppler effect. It is higher than the original frequency when the source is approaching the observer, and lower than the original frequency when the source is receding away.

You should remember that the value of this apparent frequency depends on the velocity of the wave in question, as well as the velocity of the source for the observer. Further, if the wave is traveling in a medium that is not stationary, the latter’s velocity will also come into the picture.

The Formula for the Doppler Effect

The most general formula for the Doppler effect is given as follows

$$\mathrm{f=\left ( \frac{c\pm v_{r}}{c\pm v_{s}} \right )f_{0}}$$

Here,f is the observed or apparent frequency, $\mathrm{f_{0}}$ is the original frequency,$\mathrm{c}$ is the velocity of the waves in the medium,$\mathrm{v_{r}}$ is the velocity of the observer or receiver, and $\mathrm{v_{s}}$ is the velocity of the source. Note that this formula is valid only when the speed of the medium isn’t significant. The velocities of the receiver and the source are measured relative to the medium.

Note − The plus-minus sign in the above equation may be explained as follows

• When the source is going towards the receiver, we take the + sign for $\mathrm{v_{r}}$ and the – sign for $\mathrm{v_{s}}$.

• When the source is going away from the receiver, take – sign for $\mathrm{v_{r}}$ and + sign for $\mathrm{v_{s}}$

The formula is greatly simplified when either the receiver or the observer is at rest.

Applications

The Doppler effect has vast applications in both scientific and consumer scenarios. Here are a few simple examples

Radars

Doppler radars fire a microwave beam and measure the change in frequency observed after the wave reflects from the target. From this change in frequency, the speed of the target in question can be easily measured. Traffic police officials use the Doppler radar on a very common basis.

Applications in Astronomy

To calculate how rapidly stars and galaxies are moving away, or to detect exoplanets, astronomers use the concept of the Doppler effect observed in electromagnetic waves. The change in frequency, in this case, is too minute to be visible to the naked eye, but spectroscopic measurements can lead to accurate estimates.

Sirens of Emergency Vehicles

The reason emergency vehicles employ sirens that constantly rise and fall in pitch is that emergency vehicles are supposed to move at urgent speeds. Thus, if the siren were to be emitted at a constant frequency, it would hit bystanders when the emergency vehicle approached them.

To avoid the adverse effects of sudden changes in frequencies, emergency sirens constantly change their frequency.

Limitations

• Upper cap on the velocity of the source or observer

  For the Doppler effect to be tangible, the velocities of the source and the receiver must not exceed the velocity of the wave. The most common example of this limitation is the sonic boom created by objects traveling at supersonic speeds.

• Limitation of range

  Doppler radars have a limited range in which they can function. If the object in question is too far away, then the radar will not be able to pick it up.

Conclusion

This effect, also known as the Doppler shift, refers to an apparent change in the frequency of waves when their source is not stationary for the observer. Doppler effect is an illusion, and the frequency of the wave in question is not altered. Rather, it only appears different from its actual value. Doppler effect is observed in waves of all types, including but not limited to sound waves, light waves, waves in liquids, etc. The general relation between observed and actual frequency is given by

$$\mathrm{f=\left ( \frac{c\pm v_{r}}{c\pm v_{s}} \right )f_{0}}$$

The Doppler effect has a vast array of applications, with the most prominent one being the Doppler radars, which use microwave beams to measure the speed of their targets. The detection of exoplanets also utilizes the Doppler effect. For the Doppler effect to be valid, the speeds of the source and receiver must not exceed the speed of the wave in question. Sonic booms observed as jet planes pass by are a common example of this limitation.

FAQs

Q1. What happens when the source is moving at an angle for the observer?

Ans. This situation is known as transverse the Doppler effect and it is observed only in relativistic scenarios, i.e. when the speed of the source/receiver is comparable to the speed of light. The formula in this case reads

$$\mathrm{f=f_{0}\left ( 1-\frac{vcos\:cos}{c} \right )^{-1}}$$

Q2. What are blueshift and redshift?

Ans. The blue color lies on the higher frequency end of the visible spectrum. The Doppler shift in frequency of light waves towards a higher frequency is known as blueshift.

Conversely, Doppler shift towards a lower frequency is known as redshift.

Q3. What happens when the source or the observer is stationary?

Ans. The formula, in this case, remains the same as before. We just need to substitute zero for the velocity of the source or the observer. Note that if both source and observer are at rest, then the Doppler effect does not occur.

Q4. What is the formula used by Doppler radars?

Ans. In Doppler radars, since the microwave beam is affected by the Doppler effect twice (while going towards the target, once while returning), the formula reads

$$\mathrm{\Delta f=\frac{2\Delta v}{c}f_{0}}$$

Q5. What happens when the source and the receiver are moving with the same velocities?

Ans. Doppler effect only occurs when there is relative motion between the source and the observer. If the source is at rest for the observer, then the Doppler effect won’t occur. On the other hand, the Doppler effect will occur even when the source and receiver are moving away from each other at the same speeds since there is still relative motion involved.