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As far as I've seen, E&M waves are sinusoidal. Our eyes observe these patterns and give us colors, but in a situation like such: Why do electrons emit radiation? There is still a wave, or at least a pulse, but it's not necessarily sinusoidal, and I'm not sure what 'wavelength' would be used. Nonetheless, radiation produced in this way does have a wavelength-it is used in synchrotrons. So what am I missing?

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When electromagnetic radiation comes in pulses, as in synchrotron radiation, the pulse contains many frequencies because the angle of emission defines the radiation, and the angle is not unique, so there is a spread in frequencies.

enter image description here

Frequency distribution of radiated energy

given in the plot as a ratio on the critical frequency. The calculations are not simple but the plot shows the frequency spread in a synchrotron pulse.

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But how do you pull a frequency out of some half movement of an electron? It doesn't move in sinusoidal pattern, or even complete a full cycle. How does that come to be recognized as a wavelength? –  Anthony Jul 8 '13 at 4:23
    
This is a different question. The electron radiates photons, not continuous electromagnetic waves. Photons have a unique frequency given by E=h*nu, where E is the energy lost by the electron in radiating the photon. There are billions of photons in each dx in the plot above. They act as an ensemble of particles and miraculously ( not really, see motls.blogspot.com/2011/11/… ) this ensemble adds up to the electromagnetic wave of maxwell's equations with exactly the same frequency. –  anna v Jul 8 '13 at 5:22
    
Isn't there a way to view the electron as emitting continuous waves? –  Anthony Jul 8 '13 at 7:40
    
No. The electron is an elementary particle.Light is composed of photons, and the photon is an elementary particle. The treatment in the link is about beams of electrons, an ensemble of same energy electrons. Maxwell's equations are fine. At the individual interaction level photons are radiated one by one and their ensemble builds up the observed classical electromagnetic wave. It is a many body build up, not trivial and the way it happens is described in the link given before. –  anna v Jul 8 '13 at 11:46

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