With moving source, power shift?

[monade]

When the source is moving with respect to the observer, the emitted wave undergoes this well-known frequency shift (Doppler shift). But isn't there also a power shift?

Let a source emit a signal (sound or light) with constant power, say P_0, and moves towards the observer with constant speed. So, the distance between source and obs decreases. Now, the emitted wave travels with a finite speed. So, the traveling duration of the signal decreases with time. So at the observer the energy arrives with an increased rate. So, similar to the frequency, the received power P_r is increased: P_r = P_0 + dP.

It is here of course assumed that there is no free-space (geometrical) loss. That is, the emitted power is entirely received by the observer, no matter what the separation distance is.

If there is indeed a power shift, why is this effect so rarely mentioned? Could it be because the power shift is generally negligible compared to the power changes caused by separation-distance changes (just like a point source gets brighter as it gets nearer)?

Thanks for your comments.

 

[Crosson]

Interesting analysis, but we are no longer talking about the power emitted by the source, but rather the power received by the observer. But this received power is equal to the product of the intensity and the area of the detector, and since the intensity is increasing geometrically this is by far the dominant effect i.e. the sound gets louder primarily because the source is getting closer, and only in by a very small amount does the velocity make the sound louder then it would otherwise be (in the absence of a more subtle mitigating effect having to do with the medium of propagation).

 

[monade]

But the intensity variations caused by the changing geometry of the power transmission may not always be "by far the dominant effect". In some cases, the relative velocity may induce significant intensity/power shifts. Let us consider a star which is far away and moves very fast with respect to the observer. I didn't do any calculation but I can imagine significant shifts.

In fact, for light, power shift may be directly related to frequency shift through the photon model: the energy of a photon is proportional to its frequency.
But here I still have a problem since, in my representation, photons would arrive to the observer at an increased rate (for a source moving towards observer). So, we would have a double power-shift effect: photons with higher energy arriving at an increased rate... But I guess this last problem belongs more to the atronomy forum or the quantum-physics forum.