Annus MirabilisEinstein in 1905
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special relativity

The equation explained

ν′=ν1−cos⁡φvV1−(vV)2.\nu' = \nu \frac{1 - \cos \varphi \frac{v}{V}}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} .
In wordsThe frequency the moving observer perceives is the frequency of the source times one minus the cosine of the angle between the line from the source and the observer's velocity times the speed over the velocity of light, divided by the square root of one minus the square of that speed ratio.OverviewDoppler's principle for any speed: the frequency an observer moving with velocity vv perceives from a very distant source, in terms of the angle between the line from the source and the observer's velocity.Full explanationThis is the equation for ω′\omega' read as a statement about an observer. The frequency is ν=ω2π\nu = \frac{\omega}{2 \pi}, and the factor 2π2 \pi cancels. The source is infinitely distant and at rest in KK; the observer moves with velocity vv, and φ\varphi is the angle between the line from source to observer and the observer's velocity, measured in the source's system. That line is the direction the light travels, so its cosine with the direction of motion is a=cos⁡φa = \cos \varphi. An observer moving away from the source (φ=0\varphi = 0) perceives a lower frequency, one moving towards it (φ=π\varphi = \pi) a higher one. The square root makes the formula exact at every speed below VV; to first order in vV\frac{v}{V} it is the older ν′=ν(1−cos⁡φvV)\nu' = \nu \left( 1 - \cos \varphi \frac{v}{V} \right).Every stepω′=ωβ(1−avV)\omega' = \omega \beta \left( 1 - a \frac{v}{V} \right)The angular frequency in kk, from p. 911.ω=2πν,ω′=2πν′\omega = 2 \pi \nu, \quad \omega' = 2 \pi \nu'An angular frequency is 2π2 \pi times the number of oscillations per second.a=cos⁡φa = \cos \varphiThe wave travels along the line from the source to the observer, so the cosine of its direction with the motion is the cosine of the angle φ\varphi.ν′=ν1−cos⁡φvV1−(vV)2\nu' = \nu \frac{1 - \cos \varphi \frac{v}{V}}{\sqrt{1 - \left( \frac{v}{V} \right)^2}}Cancel 2π2 \pi and write β\beta out.Also

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