Doppler effect, made simple

Next: Double Doppler effect and conservation laws   Up: Introduction science education project   Previous: Sagnac effect supporting relativity

Dated: 29 July 2024

Recently, I came across the following question on Research Gate, posted back in 2015 by Stefano Quattrini:

Is the doppler effect of light an actual energy shift of photons or it is only a relativistic connection of different reference frames?

The wording of the question already suggests a somewhat poor understanding of what is going on physically in the Doppler effect and this impression is reinforced by the detailed elaboration of the question, proposing that the effect is actually due to the energy exchange between the photon and an absorber, so it should be a consequence of conservation of energy rather than a matter of frequency evaluation in different frames of reference.

The author seems to be oblivious to the extremely general nature of the Doppler effect as a wave phenomenon that allows one to assess the question in quite different contexts than that of electromagnetism. Suppose I am at a beach and water ripples are rolling towards myself at a frequency of one per second. I can determine this with a stop watch by counting the crests arriving in a time interval. Then I run into the sea at the same speed at which the waves are moving towards me, or even better, there is a footbridge on which I can run against the wave train without touching the water. At what frequency will I see the crests now coming towards me? Well, I am running against the wave at a speed of one wavelength per second and the wave is moving towards me at one wavelength per second, so I will see two wavelengths of the wave pass under my feet per second, i.e., I observe a frequency of two crests per second. That is the Doppler effect! By what energy exchange have I modified the frequency of the water wave? My only interaction with it is that I see the light coming from it, hardly a strong enough influence to increase its frequency by a factor of two! Moreover, my friend has stayed at the beach with his stopwatch and finds that the frequency of the same ripples that I observe coming to me at 2 Hz is still one per second for him (as he is not moving towards the wave). By what token would the 1 Hz be a more real frequency than the 2 Hz or vice versa?

Now, there are of course modifications, when we deal with electromagnetic waves rather than with water waves, because relativistic aspects will be more important with the former than with the latter. But, as I show in the following small essay, these do not significantly change the picture. All that happens is that time dilation effects have to be added to the regular Doppler effect.

In order to bring out the physics in a transparent way, I derive the effect first for sound, using geometrical and kinematical aspects of wave propagation, and show then the minor modifications that are added by relativity. I hope, this direct kind of approach will be clearer to readers than some more elegant derivations based on the invariance of the wave phase.

Simple derivation of the Doppler effect, 25.07.2024

What I found a bit shocking is that in the nine years since posting the question the author has not changed his opinion. Instead he declares the question solved, but in the wrong sense. That is, he did not seriously try to understand the physics of the effect (for example by looking at various derivations or experimental realizations). Rather, he wrote his own paper trying to prove his view, using energy conservation arguments in the Doppler radar experiment. Of course, energy and momentum are conserved in that experiment, no doubt about that. But that does not mean that the Doppler effect is explained by these conservation laws.

In the Doppler radar case, the effect appears twice, once for the the radar signal sent out to the object to be detected/measured and a second time for the reflected ray returning to the sender. The process, to which energy and momentum conservation arguments are applied, however, is the reflection event, i.e., something between the two wave phenomena undergoing a Doppler effect... One may also describe the reflection as absorption and reemission, then energy conservation refers to the state of the electromagnetic field plus the absorber having the same energy before absorption as after reemission. But this is not the time interval, in which one can locate the two Doppler effects.

Conservation laws should be discussed in a fixed frame of reference, because the conserved quantities keep their values under the dynamics of the system, not under a change of frame (energy and momentum both normally change under a change of the frame of reference). So if we look at energy conservation in the frame of the sender (of the radar signal) first, we note that the returning signal has a different frequency from that sent out. (It is usually also much weaker, so its total energy is smaller than the total energy emitted. But that need not bother us, we may discuss just the reflected part of the signal or focus on energy densities instead of total energies.) What this means is simply that the reflection of the signal did not only lead to a momentum transfer between the electromagnetic field and the reflector but also to an energy tranfer. The reflector of course lost the energy that it transmitted to the field (if it moved towards the sender) or gained the energy that the field transmitted to it (if it moved away from the sender). However, the (double) Doppler effect is not caused by energy conservation and energy tranfer to/from the radar field. The Doppler effect is a phenomenon between frames, energy and momentum conservation hold in one frame.

There is of course also energy and momentum conservation in the frame of the reflector, and here the incoming frequency is the same as the outgoing frequency, so the energy (per photon) of the incoming and reflected rays are the same, while the momenta are of course changed during reflection. Yet, there is no Doppler effect involved at all in energy conservation at fixed frequency...

Next: Double Doppler effect and conservation laws Up: Introduction science education project    Previous: Sagnac effect supporting relativity

Datenschutzerklärung der Otto-von-Guericke-Universität Magdeburg nach DSGVO