Doppler Effect in Light

Difference between Doppler Effect in Light and Sound

There is a difference between the Doppler effect in light and that in sound because sound waves will travel in some medium (e.g. air) which will determine their speed. Light waves do not require a medium to propagate and hence can move through a vacuum. Thus, we have one variable more to consider in the case of sound. The following example best illustrates the difference between the Doppler effect in light and sound.

From the example above, we see that there are not only quantitative differences between the Doppler effect in light and sound but also qualitative differences in some circumstances. The speed of the objects with respect to the medium in the case of sound is essential, while there is no medium at all for light waves. In this sense, the Doppler effect in light is simpler than the Doppler effect in sound.

The Formula of the Doppler Effect in Light

The Doppler effect in light all has to do with the speed of light, particularly the speed between the emitter and the observer relative to the speed of light. Thus, if we call the speed of light \(c\) (as everyone does) and the speed between the observer and the emitter \(v\), then we can define the speed between the observer and the emitter as a fraction of the speed of light as

\[\beta=\frac{v}{c}.\]

This speed is positive if the observer and the emitter move away from each other (which is logical because their distance is increasing) and negative if they move toward each other.

We can now formulate the formula of the Doppler effect in light. This formula relates the frequency that the observer observes, \(f_{r}\), to the frequency that the emitter emits, \(f_{e}\), using the relative speed between the emitter and the observer of the light. The formula is

\[f_\text{r}=\sqrt{\frac{1-\beta}{1+\beta}}f_\text{e}.\]

Let’s see if this is a logical formula. Firstly, we see that the relative frequency difference is only affected by the speed between the emitter and the observer. This makes sense because there aren’t any other quantities of interest in this situation.

Secondly, we want to check the formula qualitatively. If the emitter and the observer are moving toward each other, then \(\beta\) is negative, so \(f_\text{r}>f_\text{e}\). This is correct: The observer measures a higher frequency than the emitter just like the Doppler effect in sound. We do the same check for the situation in which the emitter and the observer are moving away from each other, and we conclude that this formula is indeed qualitatively correct. One more check could be that if the observer and emitter have zero relative speed, then \(\beta =0\) and they agree on the frequency of the light signal.

It turns out that for low speeds relative to the speed of light, we get the following simple approximation:

\[\frac{\Delta f}{f_\text{e}}\simeq \frac{v}{c},\]

where we introduced the frequency difference \(\Delta f=f_\text{e}-f_\text{r}\). This is actually the exact result for the Doppler effect for sound waves (but then where \(c\) is the speed of sound). So for low relative speeds, the Doppler effect in light is very similar to the Doppler effect in sound.

Redshift and Blueshift in the Doppler Effect in Light

Remember that the frequency of light in the visible spectrum determines its color. The lower frequencies are red and the higher frequencies are blue. The difference in the observed frequency of a visible light wave between the emitter and the observer then means that the emitter and the observer of the light wave both see different colors if they have a speed relative to each other!

We can rephrase this as a blueshift being a situation in which \(\Delta f<0\) and a redshift being a situation in which \(\Delta f>0\). Only when \(\Delta f=0\) is there no Doppler shift at all.

If the emitter and observer are moving toward each other, the light is blueshifted, and if the emitter and observer are moving away from each other, the light is redshifted. We show an animation of Doppler shifts below, where a yellow ball (the emitter) appears to change color depending on its speed relative to us (the observer).

Doppler shift of the light emitted by a yellow ball, Wikimedia Commons CC BY-SA 3.0.

We see that the yellow ball undergoes a blueshift when it is moving toward us: the yellow gets shifted toward a bluer color, which in this case is green. When the yellow ball moves away from us, it undergoes a redshift: the yellow gets shifted toward a redder color, orange.

Examples of the Doppler effect in light

A Doppler shift takes place for every relative speed between an emitter and observer of light. However, from the formula above we can see that the frequency change of light due to the Doppler effect only starts to be noticeable when the speed between the emitter and the observer is quite large compared to the speed of light, i.e. when \(\beta\) is not very small.