Editing Relativistic Doppler effect (section)

== Relativistic Doppler effect for sound and light ==
[[File:Doppler spacetime diagram for sound.svg|thumb|Figure 10. The relativistic Doppler shift formula is applicable to both sound and light.]]
First-year physics textbooks almost invariably analyze Doppler shift for sound in terms of Newtonian kinematics, while analyzing Doppler shift for light and electromagnetic phenomena in terms of relativistic kinematics. This gives the false impression that acoustic phenomena require a different analysis than light and radio waves.

The traditional analysis of the Doppler effect for sound represents a low speed approximation to the exact, relativistic analysis. The fully relativistic analysis for sound is, in fact, equally applicable to both sound and electromagnetic phenomena.

Consider the spacetime diagram in Fig.&nbsp;10. Worldlines for a tuning fork (the source) and a receiver are both illustrated on this diagram. The tuning fork and receiver start at O, at which point the tuning fork starts to vibrate, emitting waves and moving along the negative x-axis while the receiver starts to move along the positive x-axis. The tuning fork continues until it reaches A, at which point it stops emitting waves: a wavepacket has therefore been generated, and all the waves in the wavepacket are received by the receiver with the last wave reaching it at B. The proper time for the duration of the packet in the tuning fork's frame of reference is the length of OA while the proper time for the duration of the wavepacket in the receiver's frame of reference is the length of OB. If <math>n</math> waves were emitted, then {{nowrap|<math>f_s = \frac{n}{|OA|}</math>,}} while {{nowrap|<math>f_r = \frac{n}{|OB|}</math>;}} the inverse slope of ''AB'' represents the speed of signal propagation (i.e. the speed of sound) to event ''B''. We can therefore write the speed of sound as<ref name=Brown_a/>

<math display="block">c_s = \frac{x_B - x_A}{t_B - t_A},</math>
and the speeds of the source and receiver as
<math display="block">\begin{align}
v_s &= -\frac{x_A}{t_A}, &
v_r &= \frac{x_B}{t_B},
\end{align} </math>
and the lengths
<math display="block">\begin{align}
|OA| &= \sqrt{ t_A^2 - (x_A/c)^2 }, \\
|OB| &= \sqrt{ t_B^2 - (x_B/c)^2 }.
\end{align}</math>

<math>v_s</math> and <math>v_r</math> are assumed to be less than <math>c_s,</math> since otherwise their passage through the medium will set up shock waves, invalidating the calculation. Some routine algebra gives the ratio of frequencies:

{{NumBlk||<math display="block"> \frac{f_r}{f_s} = \frac{|OA|}{|OB|} = \frac{1 - v_r/c_s}{1+v_s/c_s} \sqrt{ \frac{1 - (v_s/c)^2 }{ 1-(v_r/c)^2 }} </math>|{{EquationRef|9|Eq. 9}}}}

If <math>v_r</math> and <math>v_s</math> are small compared with <math>c</math>, the above equation reduces to the classical Doppler formula for sound.

If the speed of signal propagation <math>c_s</math> approaches <math>c</math>, it can be shown that the absolute speeds <math>v_s</math> and <math>v_r</math> of the source and receiver merge into a single relative speed independent of any reference to a fixed medium. Indeed, we obtain {{EquationNote|1|Equation&nbsp;1}}, the formula for relativistic longitudinal Doppler shift.<ref name=Brown_a/>

Analysis of the spacetime diagram in Fig.&nbsp;10 gave a general formula for source and receiver moving directly along their line of sight, i.e. in collinear motion.

[[File:Doppler shift for sound with moving source and receiver.svg|thumb|Figure 11. A source and receiver are moving in different directions and speeds in a frame where the speed of sound is independent of direction.]]
Fig.&nbsp;11 illustrates a scenario in two dimensions. The source moves with velocity <math> \mathbf{v_{s}}</math> (at the time of emission). It emits a signal which travels at velocity <math>\mathbf{C}</math> towards the receiver, which is traveling at velocity <math>\mathbf{v_r}</math> at the time of reception. The analysis is performed in a coordinate system in which the signal's speed <math>|\mathbf{C}|</math> is independent of direction.<ref name=Brown_b/>

The ratio between the proper frequencies for the source and receiver is

{{NumBlk||<math display="block"> \frac{f_r}{f_s} 
= \frac{1 - \frac{|\mathbf{v_r}|}{\mathbf{|C|}} \cos ( \theta_\mathbf{C,v_r}) } 
       {1 - \frac{|\mathbf{v_s}|}{\mathbf{|C|}} \cos ( \theta_\mathbf{C,v_s})  
       }  \sqrt{ \frac{1-(v_s/c)^2}{ 1-(v_r/c)^2 } } </math>|{{EquationRef|10|Eq. 10}}}}

The leading ratio has the form of the classical Doppler effect, while the square root term represents the relativistic correction. If we consider the angles relative to the frame of the source, then <math>v_s = 0</math> and the equation reduces to {{EquationNote|7|Equation&nbsp;7}}, Einstein's 1905 formula for the Doppler effect. If we consider the angles relative to the frame of the receiver, then <math>v_r = 0</math> and the equation reduces to {{EquationNote|6|Equation&nbsp;6}}, the alternative form of the Doppler shift equation discussed previously.<ref name=Brown_b/>