Anonymous asked: The Doppler Shift formulae for sound waves are pretty easy to visualize and derive, but why can't the same principles be used for electromagnetic waves? What are the resulting formulae?
Yes, there are many good pictures and animations of the Doppler effect for physical waves!
You know the Doppler shift formulae are based on the relative velocities of a wave source and observer. Where sound and other mechanical waves are concerned, the velocity of the medium must also be taken into account, and the velocities of the source and observer are relative to the medium. Electromagnetic waves do not need a medium, so the shift formulae only require information about the relative velocity between the source and observer, but they need to be in terms of the speed of light, c.
Actually the “classical” Doppler effect model for sound waves is approximately valid for electromagnetic waves when the speeds of the source and observer are much smaller than c. The ratio of the observed or received frequency fo to the source or emitted frequency fs is
fo/fs = 1 - vrel/vwave = 1 - v/c (≅ 1)
where vrel is the relative frequency between the source and observer, and vwave is the speed of propagation of the wave (here, the speed of light). Otherwise, the classical formulae change to follow the tenets of special relativity:
(1) the laws of physics are same in any reference frame,
(2) the speed of an e.m. wave (light) is finite and constant in the same medium regardless of the motion of either observer or source.
Basically you formulate everything in factors of the speed of light, add in the Lorentz factor γ = 1/√1-(v/c)2 for time dilation, and you are good to go:
fo/fs = (1-v/c)γ = √[(1-v/c)/(1+v/c)]
In the limit where the relative velocity between the source and observer is much smaller than the speed of light, v « c, this formula reduces to the classical one. For a light source moving away from you, the frequency appears to decrease, the wavelength lengthens and the observed light is shifted toward the red end of the visible spectrum. See the recent post on this wavelength shift in relation to astronomy!
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