Up: Introduction science education project Previous: Doppler effect, made simple
Up: Introduction science education project
Previous: Doppler effect, made simple
Dated: 11 August 2024
This is a follow-up on my discussion of the Doppler effect, which might be called basic or primitive Doppler effect for further specification, because here I will deal mostly with the so-called double Doppler effect, that arises in Doppler radar experiments and can be interpreted dynamically (which is difficult for the basic effect). In the context of dynamic effects, the question of the role of energy conservation makes much more sense and therefore it will be considered here.
To some extent, this is also motivated by a recent (unpublished) paper by Quattrini,
CONSERVATION LAWS -LIMITS OF THE EQUIVALENCE OF INERTIAL FRAMES, July 2024 ,
in which
he purports to demonstrate inapplicability of the equivalence of inertial frames and claims the standard result for the double Doppler effect to be an approximation leading to a violation of energy conservation, moreover the recoil effect neglected in that standard result to be the physical basis for the Doppler effect. He also dismisses the usual reading of the (primitive) Doppler effect as a misinterpretation, arising from "superficial analysis".
In my essay, I demonstrate that the standard double Doppler formula is indeed exact even in the presence of recoil and that it does not lead to a violation of energy conservation; rather it can be derived using energy conservation (because the double Doppler effect leads to a frequency shift in a single frame of reference, not one between frames), whereas the ordinary Doppler effect is not a direct consequence of conservation laws. Quattrini's other claims turn out to be incorrect as well; in particular, it is easy to devise thought experiments that show recoil not to be at the origin of the Doppler effect.
Since the first deposition of this on my website, I have largely extended my essay
One reason is that the consideration of two special inertial frames did not seem sufficient to bring my points across. Therefore, I now have added a derivation of the Doppler radar formula, treating the collision between the photon and the mirrors in an inertial frame with arbitrary velocity (parallel to the photon wave vector). This also gives rise to a nice interpretation of the whole double Doppler shift as composed of three frequency shifts, in general, with only two of them Doppler shifts. I.e., the experiment is not suited for a discussion of the dependence of the primitive Doppler effect on energy conservation.
My second reason is that Quattrini has presented alternative formulas for the double Doppler effect in the presence of recoil (looking different from the standard result) in an attempt to demonstrate the approximative nature of my result. I give a link to version 3 of his paper, where this is done
CONSERVATION
LAWS and applicability OF THE EQUIVALENCE OF INERTIAL FRAMES: the Doppler
RADAR .
His formulas are correct (even though their derivation is not cleanly done, there is a sign error in the denominator of formula (e), corrected by taking the factor N in the numerator to equal the one in the denominator -- in formula (e), these factors would be different; also the second term containing h f0 in the denominator of the formula of the last line is there only due an error, so "neglecting" it in the "approximation" rather renders the formula exact).
But my formula is correct and exact, too. I show the equivalence with his formulas in the addendum of the essay.
Up: Introduction science education project Previous: Doppler effect, made simple
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