Editing Spacetime (section)

==== Longitudinal Doppler effect ====
The classical Doppler analysis deals with waves that are propagating in a medium, such as sound waves or water ripples, and which are transmitted between sources and receivers that are moving towards or away from each other. The analysis of such waves depends on whether the source, the receiver, or both are moving relative to the medium. Given the scenario where the receiver is stationary with respect to the medium, and the source is moving directly away from the receiver at a speed of ''v<sub>s</sub>'' for a velocity parameter of ''β<sub>s</sub>'', the wavelength is increased, and the observed frequency ''f'' is given by
: <math>f = \frac{1}{1+\beta _s}f_0</math>

On the other hand, given the scenario where source is stationary, and the receiver is moving directly away from the source at a speed of ''v<sub>r</sub>'' for a velocity parameter of ''β<sub>r</sub>'', the wavelength is ''not'' changed, but the transmission velocity of the waves relative to the receiver is decreased, and the observed frequency ''f'' is given by
: <math>f = (1-\beta _r)f_0</math>

[[File:Spacetime Diagram of Relativistic Doppler Effect.svg|thumb|Figure 3–6. Spacetime diagram of relativistic Doppler effect]]
Light, unlike sound or water ripples, does not propagate through a medium, and there is no distinction between a source moving away from the receiver or a receiver moving away from the source. Fig.&nbsp;3-6 illustrates a relativistic spacetime diagram showing a source separating from the receiver with a velocity parameter <math>\beta ,</math> so that the separation between source and receiver at time <math>w</math> is <math>\beta w </math>. Because of time dilation, <math>w = \gamma w' .</math> Since the slope of the green light ray is −1, <math>T =  w + \beta w = \gamma w' (1 + \beta) .</math> Hence, the [[relativistic Doppler effect]] is given by<ref name="Bais" />{{rp|58–59}}
: <math>f = \sqrt{\frac{1 - \beta}{1 + \beta}}\,f_0.</math>